NASA Conference Publication 3270 

Third International Workshop 

on Squeezed States and 

Uncertainty Relations 

Edited by 

D. Han 

Goddard Space Flight Center 

Greenbelt, Maryland 

Y. S. Kim 

University of Maryland 

College Park, Maryland 

N. H. Rubin 

Y. Shih 

University of Maryland Baltimore County 

Baltimore, Maryland 

W. W. Zachary 

Howard University 

Washington, D. C. 

Proceedings of a workshop held at the 

University of Maryland Baltimore County 

Baltimore, Maryland 

August 10 - 13,1993 

National Aeronautics and 

Space Administration 

Scientific and Technical 

Information Branch 
Principal Organizers: 

Morton H. Rubin (University of Maryland Baltimore County) 

Yanhua Shih (University of Maryland Baltimore County) 

Organizing Committee: 

C. Alley 

V. Dodonov 

D. Han 

v. Isakov 

Y. S. Kim 

C. S. Lee 

M .A. Man' ko 

V. I. Man'ko 

P. McGrath 

H. S. Pilloff 

A. V. Vinogradov 

W. W. Zachary 

(Univ. of Maryland, College Park) 

(Moscow Physical Institute of Technology) 

(NASA Goddard Space Flight Center) 

(Lebedev Physical Institute) 

(Univ. of Maryland, College Park) 

(Univ. of Maryland, College Park) 

(Lebedev Physical Institute) 

(Lebedev Physical Institute) 

(Laboratory for physical Science) 

(Office of Naval Research) 

(Lebedev Physical Institute) 

(Howard University) 

This workshop was supported by the American Physical Society, the Goddard Space Flight 

Center, National Aeronautics and Space Administration, The International Science Foundation, 

the Laboratory of Physical Sciences, the National Science Foundation, and the Office of Naval 

Research. Spectral Physics Corporation provided funds for refreshments. 
PREFACE 

The Third International workshop on Squeezed States and Uncertainty Relations was held at the 

University of Maryland Baltimore County on August 10-13, 1993. This workshop was jointly 

organized by the University of Maryland and the Lebedev Physical Institute of the Russian 

Republic. These workshops were initiated by Y. S. Kim of the University of Maryland, College 

Park. The first of these workshops was held in 1991 at the University of Maryland, College Park 

and the second in Moscow in 1992. 

The purpose of these workshops is to bring ational selection of scientists to 

discuss the latest developments in branches of physics, and in the 

understanding of the foundations third Workshop special attention 

was given to the influence that quantum optics is having on our understanding of quantum 

measurement theory. 

The fourth meeting in this series will be held in the People's Republic of China. The principal 

organizer will be Q. C. Peng of the Shanxi University at Taiyun, P.R.C. 
REGISTERED PARTICIPANTS 

Alley, Carroll O., Dept. of Physics, University of Maryland, College Park, MD 20742 

Amoldus, Henk, Dept. of Physics, Villanova University, Villanova, PA 19085 

Artoni, Maurizo, Photonics Branch, NASA Goddard Space Flight Center, Greenbelt, MD 20771 

Bandilla, Arno, Arbeitsgruppe Nichtklassische Strahlung der Humboldt Universitat, Berlin, 

Germany 

Bly, Darren, Dept. of Physics, University of Maryland Baltimore County, Baltimore, MD 21228 

Boivin, Luc, Massachusetts Institute of Technology, Room 36-497, Cambridge, MA 02139 

Brandt, Howard, U.S. Army Research Laboratory, 2800 Powder Mill Road, Adelphi, MD 20783 

Braunstein, Samuel, Dept. of Physics, Technion-Israel Institute of Technology, Haifa 32000, 

Israel 

Bulatov, Alexi, Dept. of Physics, CCNY of CUNY, New York, NY 1003 1 

Cabrera, Guillermo, Instituto de Fisica, Universidade Estadual de Campinas, Campinas, SPY 

Brazil, 13083-970 

Cappetta, Luisa, Dept. Ing. Info. and Math. Appl., University of Salerno, Via Ponte Don 

Melillo, I84084 Fisciano (SA), Italy 

Castaiios, Octavio, Instituto de Ciencias Nucleares, UNAM, A.P. 70-543 Mexico D.F., Mexico 

Ceperley, Peter, Electrical and Computer Engineering, George Mason University, Fairfax, VA 

22030 

Chizhov, Alexei, Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Head 

Post Office P.O. Box 79, Moscow 10I000, Russia 

D'Ariano, Giacomo, Dipartimento di Fisica "A. Volta", Universita di Pavia, Via Bassi 6, Pavia 

1-27100, Italy 

de Muynck, Willem, Theoretical Physics, Eindhoven Institute of Technology, Eindhoven, The 

Netherlands 
DeFacio, Brian, Department of Physics and Astronomy, University of Missouri, Columbia, MO 

6521 1 

De Martini, Francesco, Dept. of Physics, Universita di Roma, Piazza A. Moro, 2 Roma, Italy 

Dowling, Jonathon, Research, Development, and Engineering Center, U.S. hy Missile 

Command, Redstone Arsenal, AL 35898 

Enrico, Montanari, Dept. of Physics, Universita Degli Studi di Ferrara, Via Paradiso 12, Ferrara 

44 1 00, Italy 

Fedele, Renato, Dept. of Physical Sciences, University of Naples, Naples 1-80125, Italy 

Cabrera, David Jose Fernandez, Dept. of Physics, Centro de Investigacion y Estudios, 

Avanzados del IPN, A.P. 14-740 07000, Mexico D.F. 

Fivel, Daniel, Dept. of Physics, University of Maryland, College Park, MD 20742 

Fortini, ~ierluik, ~eit. of Physics, University of Ferrara, Via Paradiso 12, Ferrara 44100, Italy 

Fortunato, Mauro, Dept. of Physics, University of Rome "La Sapienza", Rome 2-00185, Italy 

Franson, James, Applied Physics Laboratory, Johns Hopkins University, Laurel, MD 20723 

Fry, Edward, Dept. of Physics, Texas A&M University, College Station, TX 77843-4242 . 

Gallis, Michael, Dept. of Physics, Pennsylvania State University Schuylkill, Schuylkill Haven, 

PA 17972 

Garza, Octavio, Instituto de Ciencias Nucleares, UNAM, A.P. 70-543 Mexico D.F., Mexico 

Giovannini, Massimo, Instituto Nazionale di Fisica Nucleare Torino, Italy ' 

Greenberger, Daniel, Dept. of Physics, CCNY, New York, NY 1003 1 

Haus, Herman, Electrical Engineering and Computer Science, Massachusetts Institute of 

Technology, Cambridge, MA 021 39 

Hegerfeldt, Gerhard, Institut fir Theoretische Physik, Universittit Giittingen, Germany 

Hillery, Mark, Dept. of Physics and Astronomy, Hunter College of CUNY, New York, NY 

10021 

Horne, Michael, Dept. of Physics, Stonehill College, North Easton, MA 02357 

Illuminati, Fabrizo, Dipartimento di Fisica, Universita di Salerno, Salerno, Italy 
Isakov, V., Lebedev Physical Institute, Leninsky Prospect 53, Moscow 117924, Russia 

Joneckis, Lance, Laboratory for Physical Sciences, University of Maryland, College Park, MD 

20742 

Joobeur, Adel, Electrical and Computer Engineering, University of Wisconsin, Madison, WI 

53705 

Jorgensen, Thomas, Chemical Physics, The Technical University of Denmark, Lyngby, Denmark 

Kaertner, Franz, Research Laboratory of Electronics, Massachusetts Institute of Technology, 

Cambridge, MA 02 139 

Kakazu, Kiyotaka, Dept. of Physics, University of Maryland, College Park, MD 20742 

Karassiov, V. P., Lebedev Physical Institute, Leninsky Prospect 53, Moscow 117924, Russia 

Kauderer, Mark, Rome Laboratory, Grifith Air Force Base, Rome, NY 13441 

Ketov, Sergei, Institute of Theoretical Physics, University of Hannover, Hannover, Germany 

Kiess, Thomas, Dept. of Physics, University of Maryland, College Park, MD 20742 

Kim, Abjung, Dept. of Physics, Northwestern University, Evanston, IL 60201 

Kim, Chonghoon, 2 145 Sheridan Road, Evanston, IL 60208-3 1 18 

Kim, Y. S., Dept. of Physics, University of Maryland, College Park, MD 20742 

King, Sun-Kun, Dept. of Physics, National Tsing hua University, Hsinchu 30043, Taiwan 

Klimov, Andrei, Fisico-Mathematicas, Universidad de Guadalajara, 44 100 Guadalajara, Jalisco, 

Mexico 

Klyshko, David, Dept. of Physics, Moscow State University, Moscow 119899, Russia 

Kostelecky, Alan, Dept. of Physics, Indiana University, Bloomington, IN 47405 

Kouznetsov, Dmitri, Centro de Instrumentos, Universidad Nacional Autonoma de Mexico, AP 

70-1 86 hlexico DF, Mexico 

Krenn, Gunther, Atominstitut der Osterreichischen Universitaten, Vienna, Austria 

Kryuchhyan, G., Institute for Physical Research, Armenian Academy of Sciences, Ashtarak 2, 

Armenia 
Kulagin, V., Steinberg Astronomical Institute, Moscow State University, Moscow 119899, 

Russia 

Kumar, Prem, Electrical Engineering and Computer Science, Northwestern University, Evanston, 

IL 60208-3 1 18 

Kwiat, Paul, Dept. of Physics, University of California, Berkeley, CA 94720 

Larchuk, Todd, Electrical Engineering, Columbia University, New York, NY 10027 

Lee, C. T., Dept. of Physics, Alabama A&M University, Normal, AL 35762 

Lee, Kang-Soo, Dept. of Physics, Sogang University, Seoul, Korea 

Lerner, Peter, MS K723, Los Alamos National Laboratory, Los Alamos, NM 87545 

Li, Hui, Dept. of Physics, Tsinghua University, 100084 Beijing, China 

Ling, Hong Yuan, Dept. of Chemistry and Physics, Rowan College, Glassboro, NJ 08023 

Lvovsky, Alexander, c/o. Dr. A. Rabinovich, 34-40 79th Street, #5, Jackson Heights, NY 11372 

Man'ko, Margarita, Lebedev Physical Institute, Leninsky Prospect 53, Moscow 117924, Russia 

Man'ko, Vladimir, Lebedev Physical Institute, Leninsky Prospect 53, Moscow 117924, Russia 

Nancini, Stevano, Dept. of Physics, Universita di Camerino, Camerino 62032, Italy 

Mandel, Leonard, Dept. of Physics and Astronomy, University of Rochester, Rochester, NY 

14627 

Mann, Ady, Dept. of Physics, Technion-Israel Institute of Technology, Haifa, Israel 

Martens, Hans, Dept. of Physics, Eindhoven University of Technology, Eindhoven, The 

Netherlands 

Martin, Manuel, Instituto de Fisica, Universidad Autonoma de Puebla, Puebla, Mexico 

Matacz, Andrew, Dept. of Physics, University of Maryland, College Park, MD 20742 

McDermott, Roger, Applied Mathematics, The Open University, Milton Keynes MK7 7BH 

Bucks, United Kingdom 

McGrath, Paul, Laboratory for Physical Sciences, University of Maryland, College Park, MD 

20742 
Mertens, Christopher, Dept. of Physics, Georgia Institute of ~echnol&gy, Atlanta, GA 3 03 32- 

0430 

Miaemm, Mmin, Dept. of Physics, The City College ofNew York, New York, W 1003 1 

Mdi, Solomon, Dep ento de Fisica, Universidade Federal de Sao Carlos, 235 Sao Carlos, 

SP Br&l13565-905 

Morris, Randall, MW System Design, Martin Marietta, Moorestown, NJ 08057 

Mulhem, John, Dept. of Physics, University of New Hampshire, Durham, NH 03824 

Nikonov, Dmitri, Dept. of Physics, Texas A&M University, College Station, TX 778434242 

Orszak, Jeffrey, Degt. of Physics, University of Maryland Baltimore County, Baltimore, 

2 1228-5398 

Ortolan, Antonello, Laboratori Nazionali di Eegnaro, Legnaro, Italy 35020 

Ou, Jeffrey, Dept. of Physics, Indiana University - Purdue University at Indianapolis, 

Indianapolis, IN 46202 

Ozawa, Masanao, Dept. of Mathematics, Nagoya University, Chikosako, Nagoya 464-01, Japan 

Panunto, Michael, Degt. of Physics, University of Maryland Baltimore County, Baltimore, 

21228-5398 

PilloE, Herschel, Office of Naval Research, 800 North Quincy Street, Arlington, VA 22217-5660 

Pittman, Todd, Degt. of Physics, University of Maryland Baltimore County, Baltimore, MI3 

2 1228-5398 

Popescu, Sandu, Theoretical Physics, Universite Libre de Bruxelles, Brussels, Belgium 

Puri, Wavinder, Dept. of Physics and Astronomy, University of Rochester, Rochester, W 14627 

Reifler, Frank, lhAW System Design, Martin Marietta, Moorestown, NU 08057 

Ritus, V., Lebedev Physical Institute, Leninsky Prospect 53, Moscow 117924, Russia 

Rous, Phillip, Dept. of Physics, University of Maryland Baltimore County, Baltimore, RID 

21228-5398 

Roger, htoine, Genie Physique, Ecole Polytechnique, Montreal, Qukbec, Canada 
Rubin, Morton H., Dept. of Physics, University of Maryland Baltimore County, Baltimore, MD 

21228-5398 

Ryff, Luiz Carlos, Instituto de Fisica, Universidade Federal do Rio de Janeiro, Rio de Janeiro, 

Brazil 

Saleh, Bahaa, Dept. of Engineering and Computer Science, University of Wisconsin, Madison, 

WI 53706-1691 

Sandoval, Lourdes, Instituto de Fisica, Universidad Autonoma de Puebla, Puebla, Mexico 

Santos, Emilio, Fisica Moderna, Universidad de Cantabria, Santander, Spain 

Schraner, Daniel, Dept. of Physics, University of Illinois, Urbana, IL 61801 

Schrade, Guenter, Abteilung fur Quantenphysik, Universitat Ulm, Ulm, Germany 

Scully, Marlan, Dept. of Physics, Texas A&M University, College Station, TX 77843 

Sergienko, A., Dept. of Physics, University of Maryland Baltimore County, Baltimore, MD 

2 1228-5398 

Shapiro, Jeffrey, Dept. of Electrical Engineering and Computer Science, Massachusetts Institute 

of Technology, Cambridge, MA 021 39-4307 

Shepard, Scott, Dept. of Physics, Baylor University, Waco, TX 76798-73 16 

Shih, Yanhua, Dept. of Physics, University of Maryland Baltimore County Baltimore, MD 

21228-5398 

Shimony, Abner, Dept. of Physics and Philosophy, Boston University, Boston, MA 02215 

Spiridonov, Vyacheslav, Centre des Recherches Mathe matique, Universite de Montrkal, 

Montreal, Quebec H3C 307, Canada 

Stroucken, Tineke, Dept. of Theoretical Physics, Technical University of Eindhoven, Eindhoven, 

The Netherlands 

Sudarshan, E. C. G., Dept. of Physics, University of Texas, Austin, TX 78712 

Summers, Geoffrey, Dept. of Physics, University of Maryland Baltimore County, Baltimore, 

MD 21228-5398 

Summhammer, Johann, Atominstitut der d sterreichischen Universitaten, Vienna. Austria 

Tanatar, Bilal, Dept. of Physics, Bilkent University, Ankara, Turkey 
Tapster, Paul, Defense Research Agency, St. Andrews Road, Malvern Worcs, United Kingdom 

Teich, Malvin, Dept. of Electrical Engineering, Columbia University, New York, NY 10027 

Torre, Amalia, ENEA, Dip. INN, Settore Elettroottica e Laser, Frascati, Rome, Italy 

Torres - Vega, Gabino, Departamento de Fisica, Centro de Investigaci6n y de Estudios 

Avanzados del IPN,07000 Mkxico D.F 

Tsue, Yasuhiko, Dept. of Physics, Kochi University, Kochi 780, Japan 

Uffink, Jos, Dept. of History and Foundations of Mathematics and Science, University of 

Utrecht, The Netherlands 

Vaidman, Lev, Dept. of Physics, Tel-Aviv University, Tel Aviv 69978, Israel 

Vasilyev, Michael, Lebedev Physical Institute, Leninsky Prospect 53, Moscow 1 17924, Russia 

Vitiello, Guseppe, Dept. of Physics, University of Salerno, Salerno, Italy 

Vogel, Karl, Abteilung fiir Quantenphysik, Universitiit Ulm, Ulm, Germany 

Vourdas, Apostolos, Dept. of Electrical Engineering, University of Liverpool, Liverpool, United 

Kingdom. 

Wahiddin, Mohamed, Dept. of Mathematics, University of Malaya, Kuala Lumpur, Malaysia 

Wang, Lijun, Dept. of Physics, Duke University, Durham, NC 27708-0305 

Wu, Ling-An, Institute of Physics, Chinese Academy of Sciences, Bejing 100080, China 

Xiao, Min, Dept. of Physics, University of Arkansas, Fayetteville, AR 72701 

Yeon, Kyu Hwang, Dept. of Physics, Chungbuk National University, Cheongju, Korea 

Yilmaz, Huseyin, Hamamatsu Photonics, 105 Church Street, Winchester, MA 01890 

Youn, Sun-Hyun, Dept. of Physics, Seoul National University, Seoul, Korea 

Yu, Li-Hua, Brookhaven National Laboratory, NSLS, Upton, NY 1 1973 

Yuen, Horace, Dept. of Electrical Engineering and Computer Science, Northwestern University, 

Evanston, IL 60208 

Zachary, Woodford W., Computational Science and Engineering Research Center, Howard 

University, Washington, D.C. 20059 
Zeilinger, Anton, Institute for Experimental Physics, University of Innsbmck, Innsbmck, Austria 

Zhang, Yuhong, Center for Biologies and Evaluation Research, Food and Drug Administration, 

8800 Rockville Pike, Bethesda, MD 20892 
TABLE OF CONTENTS 

SECTION 1 SQUEEZED STATES 

Properties of Two-Mode Squeezed Number States. ............................ A. V. Chizhov and B. K. Murzakhmetov 

Supersymmetry and Radial Squeezed States for 

Rydberg Wave Packets. .............................................. R. Bluhm and V. A. Kostelecky 

Effect of Dispersion Forces on Squeezing With 

RydbergAtom ..................................................... S. K. Ng, M. R. Muhamad, and 

M. R. B. Wahiddin 

Higher-Order Squeezing of the Quantum Electromagnetic Field 

and the Generalized Uncert,ainty Relations in Two-Mode 

Squeezed States. ................................................... Li Xizeng and Su Baoxia 

Q-Derivatives, Coherent States and Squeezing ............................... E. Celeghini, S. DeMartino, S. De Siena, 

M. Rasetti and G. Vitiello 

Sum-Frequency Generation From Photon Number 

Squeezed Light. ................................................... L. A. Wu, C. Du, M. Wu, and S. Li 

Spectral, Noise, and Correlation Properties of Intense Squeezed 

Light Generated by a Coupling in Two Laser Fields. ........................... G. Yu. Kryuchkyan and K. V. Kheruntsyan 

Multimode Squeezing, Biphotons anduncertainty Relations 

in Polarization Quantum Optics. ......................................... V. P. Karassiov 

A New Approach to the Photon Localization Problem .......................... D. Han, Y. S. Kim, and M. E. Noz 

Production of Squeezed States for Macroscopic Mechanical 

Oscillator ........................................................ V. V. Kulagin 

Squeezing in a 2-D Generalized Oscillator .................................. 0. Castafios, R. L6pez-Pefia, and V. I. Man'ko 

Simultaneous Two-Component Squeezing in Generalized 

q-Coherent States. ................................................. R. J. McDermott and A. I. Solomon 
Photon Statistics of a Two-Mode Squeezed Vacuum. .......................... 99 

G. Schrade, V. M. Akulin, 

W. P. Schleich, and V. I. Man'ko 

SECTION 2 QUANTUM MEASUREMENT THEORY 

On the Theory of Quantum Measurement .................................. 107 

H. A. Haus and F. X. KSirtner 

Uncertainty Relations as Hilbert Space Geometry. ............................ 117 

S. L. Braunstein 

Time-Dependent Distinguishability: Choosing To Be 

a Wave or a Particle ................................................ 123 

T. P. Grayson, X. Y. Zou, D. Branning, 

J. R. Torgenson, and L. Mandel 

Homodyning and Heterodyning the Quantum Phase. .......................... 129 

G. M. D'Ariano, C. Macchiavello, 

and M. G. A. Paris 

Joint and Angle-Covariant Spin Measurements 

With a Quadrupole Magnetic Field. ...................................... 149 

H. Martens and W. M. de Muynck 

Two New Kinds of Uncertainty Relations. ................................. 155 

J. Uffink 

Measurement of Very Small Phase Fluctuations 

by Means of the Operational Approach. ................................... 16 1 

A. Bandilla 

The Quantum Measurement of Time. ..................................... 167 

S. R. Shepard 

Manipulation of Quantum Evolution. ..................................... 173 

D. J. Ferndndez C. and B. Mielnik 

SECTION 3 AMPLIFLIER AND CAVITY 

ELECTRODYNAMICS 

Quantum State Engineering. ........................................... 18 1 

K. Vogel, V. M. Akulin, and W. P. Schleich 

4 

Squeezed Light from Multi-Level Closed-Cycling 

Atomic Systems. .................................................. 189 

M. Xiao and Y. F. Zhu 
A Method of Measuring the Amplitude-Modulated Vacuum Field 

Near a Conducting Mirror ............................................ 195 

S. H. Youn, J. H. Lee, and J. S. Chang 

How to Characterize the Nonlinear Amplifier?. .............................. 203 

D. K. Kallistratova and C. F. Cotera 

Phase Properties of Multicomponent Superposition States 

in Various Amplifiers ............................................... 207 

K. S. Lee and M. S. Kim 

Sub-Poissonian Light and Photocurrent Shot-Noise Suppression 

in Closed Opto-Electronic Loop ........................................ 2 1 3 

A. V. Masalov, A. A. Putilin, and V.V . Vasilyev 

Modes in Light Wave Propagating in Semiconductor Laser. ...................... 219 

M. A. Man'ko 

SECTION 4 DISTRIBUTIONS IN PHASE SPACE 

Squeezing Generated by a Nonlinear Master Equation and by 

Amplifying-Dissipative Hamiltonians. .................................... 227 

V. V. Dodonov, M. A. Marchiolli, 

S. S. Mizrahi, and M. H. Y. Moussa 

Phase-Space Analysis of Charged and Optical Beam Transport: 

Wigner Rotation Angle .............................................. 239 

G. Dattoli and A; Torre 

Phase Space Flow of Particles in Squeezed States. ............................ 25 1 

P. H. Ceperley 

Multivariable Hermite Polynomials and 

Phase-Space Dynamics .............................................. 257 

G. Dattoli, A. Torre, S. Lorenzutta, 

G. Maino, and C. Chiccoli 

Understanding Squeezing of Quantum States With the 

Wigner Function .................................................. 269 

A. Royer 

On the Possibility of a Quantum Mechanical Phase Space 

Representation.. .................................................. 275 

G. Torres-Vega 

o Particle Localization, Spinor Two-Valuedness, and Ferrni 

Quantization of Tensor Systems ........................................ 28 1 

F. Reifler and R. Morris 
SECTION 5 UNCERTAINTY RELATIONS 

Quantum Phase Uncertainties in the Classical Limit ........................... 289 

J. D. Franson 

How to Detect an Excited Atom Without Disturbing It 

or How to Locate a Super-Mine Without Exploding It ......................... 299 

L. Vaidrnan 

The Stokes Line Width and Uncertainty Relations. ........................... 307 

A. I. Nikishov and V. I. Ritus 

Maximum Predictive Power and the Superposition 

Principle ........................................................ 315 

J. Summhammer 

Correlated Quadratures of Resonance Fluoresence and the 

Generalized Uncertainty Relation. ....................................... 32 1 

H. F. Arnoldus, T. F. George, 

and R. W. F. Gross 

Minimum Uncertainty and Squeezing in Diffusion Processes 

and Stochastic Quantization ........................................... 33 1 

S. De Martino, S. De Siena, 

F. Illuminati, and G. Vitiello 

SECTION 6 FIELD THEORY AND GENERAL INTEREST 

Applications of Squeezed States: Bogoliubov Transformations 

and Wavelets to the Statistical Mechanics of Water 

and Its Bubbles. ................................................... 339 

B. DeFacio, S.-H. Kim, and A. Van Nevel 

Coherent States on the m-Sheeted Complex Plane as 

m-Photon States. .................................................. 353 

A. Vourdas 

Some Rules for Polydimensional Squeezing ................................ 365 

V. I. Man'ko 

Schriidinger Operators With the q-Ladder Symmetry 

Algebr .......................................................... 373 

S. Skorik and V. Spiridonov 

The Wave Function and Minimum Uncertainty Function P 

of the Bound Quadratic Hamiltonian System. ............................... 38 1 

K. H. Yeon, C. I. Um, and T. F. George 

The Thermal-Wave Model: A Schrijdinger-Like Equation 

for Charged Particle Beam Dynamics. .................................... 387 

R. Fedele and G. Miele 
Causality Problems for Fermi's Two-Atom System ........................... 393 

G. C. Hegerfeldt 

Pseudomaster Equation for the No-Count Processin a 

Continuous Photodetection. ........................................... 401 

C. T. Lee 

Squeezed States and Graviton-Entropy Production 

in the Early Universe. ............................................... 407 

M Giovannini 

Time-Dependent Variational Approach in Terms of Squeezed 

Coherent States--Implication to Semi-Classical Approximation--. ................... 413 

Y. Tsue 

Decoherence and Dissipation for a Quantum System 

Coupled to a Local Environment. ....................................... 4 19 

M. R. Gallis 

General Properties of Quantum Optical Systems in a 

Strong Field Limit ................................................. 425 

S. M. Chumakov and A. B. Klimov 

Probing the Antiferromagnetic Long-Range Order With 

Glauber Spin States ................................................ 430 

G. G. Cabrera 

Berry Phase in Heisenberg Representation. ................................ 437 

V. A. Andreev, A. B. Klimov, and P. B. Lerner 

SECTION 7 INTERACTION OF LIGHT 

AND MATTER 

Quantum Propagation in Single Mode Fiber. ................................ 45 1 

L. G. Joneckis and J. H. Shapiro 

Quantum Amplification and Quantum Optical Tapping With 

Squeezed States and Correlated Quantum States. ............................. 465 

Z. Y. Ou, S. F. Pereira, and H. J. Kimble 

Quantum Noise Limits to Matter-Wave Interferometry 

M. 0. Scully and J. P. Dowling .................................... 475 

Blwh Vector Projection Noise ......................................... 485 

L. J. Wang, A. M. Bacon, H. Z. Zhao, 

and J. E. Thomas 

New Quantum Properties of Photons and 

Their Detection. ................................................... 495 

M. Artoni and J. L. Birman 
Quantum Statistics of Raman Scattering Model 

With Stokes Mode Generation. ......................................... 501 

B. Tanatar and A. S. Shumovsky 

Evolution of Wave Function in a Dissipative 

System ......................................................... 507 

L. H. Yu and C-P. Sun 

Nonresonant Interaction of Ultrashort 

Electromagnetic Pulses With Multilevel 

QuantumSystems .................................................. 515 

E. Belenov, V. Isakov, and A. Nazarkin 

SECTION 8 EPR PROBLEM, BELL'S 

INEQUALITIES AND 

MULTIPHOTON INTERFEROMETRY 

Complementarity and Path Distinguishability: 

Some Recent Results Concerning Photon Pairs .............................. 523 

A. Shimony and G. Jaeger 

Further Evidence for the EPNT Assumption ................................ 535 

D. M. Greenberger, H. J. Bernstein, 

M. Home, and A. Zeilinger 

The Photon: Experimental Emphasis on Its 

Wave-Particle Duality ............................................... 545 

Y. H. Shih, A. V. Sergienko, M. H. Rubin, 

T. E. Kiess, and C. 0. Alley 

The Generation of Entangled States From 

Independent Particle Sources. .......................................... 563 

M. H. Rubin and Y. Shih 

Nonlocal Effects on the Polarization State of a Photon, 

Induced by Distant Absorbers. ......................................... 569 

L. C. B. Ryff 

The Strong Bell Inequalities - A Proposed 

Experimental Test. ................................................. 575 

E. S. Fry 

On the Nonlocal Predictions of Quantum Optics. ............................. 581 

T. W. Marshall, E. Santos, 

and A. Vidiella-Barranco 

A Loophole-Free Bell's Inequality Experiment. .............................. 591 

P. G. Kwiat, A. M. Steinberg, 

R. Y. Chiao, and P. H. Eberhard 
The Probabilistic Origin of Bell's Inequality. ............................... 603 

G. Krenn 

Einstein-Podolsky-Rosen-Bohrn Experiment 

and Bell Inequality Violation Using Type 11 

Parametric Down Conversion .......................................... 609 

T. E. Kiess, Y. H. Shih, A. V. Sergienko, 

and C. 0. Alley 

BANQUET TALK 

The Relation between Physics and Philosophy. .............................. 617 

A. Shimony 
SECTION 1 

SQUEEZED STATES 
PROPERTIES OF TWO-MODE SQUEEZED NUMBER STATES 

A. V. Chizhov and B. I<. Murzakhmetov 

Boyolubov Tlzeoret ical Laboratory, Joint Institute for ~Vucleai. Research 

Head Post Ofice, P.O. Box 79. 101 000 Aoscow, R.rr.ssian Federafion 

Abstract 

Photon statistics and phase properties of two-mode squeezed number states are studied. It is shown that photon number distribution and Pegg-Barnett phase distributioil for 

such states have similar (AT + 1)-peak st.ructure fo~ nonzero value of the &fference in the 

number of photons between modes. Exact analytical formulas for phase distributions based 

on different phase approaches are derived. The Pegg-Barnett phase distribution and the 

phase quasiprobability distribution associated with the Wigner function are close to each 

other, while the phase quasiprobability distribution associated with the Q function carries 

less phase information. 

1 Introduction 

Recent developments in quantum optics have led to new proposals t.o generate nunlher states 

of the electromagnetic field using conditioned measurements teclll~iques [l] or the properties of 

atom-field int.era.ctions in micro~vave cavities in the micronlaser [2]. The precisely defined twolnode photon number state I:lr + q, N) can he used as an input field in a squeezing device, such 

as a parametric amplifier. The model involves a signal and an idler modes driven by a. classical 

pump. The Hamiltonian for the two coupled nlodes is taken to be [3, 11 (we set tl = 1) 

fi = wa&t& -t-t .+wait& - i{g6&exp(iwt) - g'b a exp(-id)}, 

where w is the pump frequency and g is the effective interrnode coupling constant. If we consider 

exact resonance w = w, +wb then the Hamiltonian may be transformed into the interaction picture 

In this picture the time-evolution opera.tor is 

exp(-iHrt) = esp{-gtbi + g=t9bt}. 

'and is immediately identifiable as a time-dependent two-mode squeezed opera.t.or: 

with squeezing parameter < = gt. The output state at time t will be the two-nlo(le squeezed 

number state 
The properties of this state are phase dependent and it should be interesting to study them. 

The problem of the quantum description of the optical field phase has been the subject of 

considerable study for many years [5]. This is connected with the difficulty in constructing a 

Hermitian phase operator. Within the past few years the notion of phase variables in quantum 

systems has been greatly clarified. Pegg and Barnett [GI-[S] have shown how such an operator can 

be defined for quantized electromagnetic fields. This new formalism makes it possible to describe 

the quantum properties of optical phase in a direct way within quantum mechanics on the basis 

of the Hermitian phase operator and its eigenstates. 

A quite different approach to the concepts of the phase variable has also been widely used 

in quantum optics [9]-[11] and which involves quantum quasiprobability distributions such as 

the Q function and the IVigner function rather than Hermitian operators and their eigenstates. 

These quasiprobability distributions depend upon the complex eigenvalue a of the non-Hermitian 

annihilation operator, which can be expressed in terms of a radial variable lcvl and a "phase" 0 

both of which are real. If we integrate over the radius, the resulting distributions are periodic 

in the phase angle and, for the most of states they satisfy all properties required by a proper 

phase distribution. In recent papers, the Pegg-Barnett phase distribution have been compared 

with those distributions obtained from the \Yigner and Q functions by integrating them over the 

radius for the multi-photon down-conversion [12], displaced number states and displaced thermal 

states [13], squeezed number states and squeezed thermal states [li]. In this paper we extend 

such comparison onto two-mode case. 

The purpose of this paper is to study photon statistics and phase properties of the twomode squeezed number states which can be considered as a natural generalizat ion of a two-mode 

squeezed vacuum state. 

2 Photon number statistics 

Consider two modes of the electromagnetic field, which have annihilation operators 6 and 

9. A two-mode squeezed number state (TMSNS) is defined by acting with the squeeze operator 

S(r, 9) on the two-mode number state IN + q, M), that is 

where q is the difference in the number of photons between two modes and 

In problems in which photons are either created in pairs or destroyed in pairs the \:slue of q remains 

constant. N0t.e that in many applica.tions where pair creation occurs starting from vacuum, the 

parameter q will be zero. The number state decon~position of TMSNS can be writ.t,en as 
where 

(tanh r)N+n b, = (N!(N + q)!n!(n + q)!)'I2 (cosh r)l+q 

and 

with y being a phase of squeezing. The above amplitude is obtained by using the factored form 

of the two-mode squeeze operator [15] 

$(r, y) = (cosh r)" exp[-ltite2" tanh r] 

x exp[-(ltb + it&) ln(cosh r)] e~~[hk-~'" tan11 r]. (6) 

' The mean number of photons in the TMSNS is 

The joint probability to find n, photons in mode a and nb photons in mode b is given by 

Using (3) and (4), we get 

where 

As we can see in Fig. 1, photon number distribution Pq(n) has an oscillatory behaviour. Such a 

behaviour is a consequence of interference in four-dimensional phase space [lti]. \Ve \vould like t.o 

emphasize a presence of (N + 1) peaks in the photon number distribution. The sinlilar behaviour 

of the photon number distribution was observed for the displaced number states [IT]. It should 

be stressed that such a peak structure for TMSNS can be revealed only for those values of the 

parameter q greater than a certain number. This number depends on the value of .rV and for large 

i'V we ought to choose large values for such a number. Otherwise, some adjacent peaks in the 

photon number dist.ribution might overlap and thus (N + 1)-peak structure cannot be certainly 

discerned. 
FIG. 1. Photon number distribution for the two-mode squeezed number state with r = 0.5, q = 50 

and (a) N = 0, (b) N = 1, (c) N = 2. 
3 Quasiprobability distributions 

In this section we exanline the representation of ThlSNS by quasiprobability phase-space 

distributions. For convenience we choose the squeezing parameter to be real, [ = r. The two-mode 

quasiprobability distributions are formed by a natural generalization of those for the single-mode 

fields [IS]. The Glauber-Sudarshan ?' function, the \I1igner function and the (2 function are 

obtained by evaluating t.he Fourier transforms 

from the characteristic functions 

where s = 1 if V = P, s = 0 if k' = 11- and s = -1 if I/' = Q. \\e \vould like t.o llotice that 

there no exists well-defined Glauber-Sudarshan 'P function for states under consideration owing 

. t.o t.heir nonclassical nat.ure [IS, 191. 

According to ref. PO], the Q function can alternatively be defined as 

From this definition we see that the Q function is always non-negative. Using the definition of 

t.he density ma.trix of TMSNS 

and the fact.ored form of the squeeze operator (6), we obtain 

10 129 

Q(Q,@) = K2(Cosh r)4.V+29+2 esp[-(a3 + ~'3') tanh r] exp[-(la 1' + 1312)] 

.?I .v (i sinh r)n+k.~!(~ + q)!(ax,.j=)"-n(a J)-\'-' 

n!k!(N - n)!(N - k)!(N + - n)!(:\' + - k)!. n=O k=O 

As to t.he \Viper funct.ion. it can also be represellted as [20] 

khere D,(y) and Db(y) are the displacement operators for modes a and b respectively. It is 

straightfor\vard to evaluate the \Viper function using eq. (14) and the operator tralisforniations (151 
and their Hermitian conjugates. We find a quite simple analytical form for the Wigner function 

4 w(%/3) = -(-l)q exp[-2 cosh 2r(laI2 + 1/312) - 2 sinh 2r(cra + a*Pa)] r, 

x LN ((2 sinh rl~l)~ + (I cosh ~181)~ + 2 sinh 2r(o3 + oap')) 

XE.~+~ ((2cosh rial)* + (2sinh ~1~1)~ + 2 sinh 2r(aB + asp)) . (19) 

where L,(x) is the Laguerre polynomial of order n. From eqs. (15) and (19) one can see that the 

Q function and the Wigner function depend on the sum of the phases 8, + Bb only. This fact clearly 

exhibits the correlated nature of the two-mode squeezed number states. In the next section we 

will employ the quasiprobability functions in consideration of phase properties of these states. 

4 Phase distributions 

Now we employ the two-mode Pegg-Barnett phase formalism ['21], [2'2] to find the phase 

distribution function for such states. This formalism is based on the observation that the Hermitian 

phase operator can be defined in a finite-dimensional state space, spanned by the number states. 

The main idea of the Pegg-Barnett formalism is to evaluate all necessary expectation values on 

this finite-dimensional state space, and only after that the dimension of the space is allowed to 

tend to infinity. Having the number state decomposition (3) of T3ISSS we can determine the 

continuous joint phase probability distribution for the continuous phase variables 8, and Ob, which 

is given by 

where b, are given by Eq. (4). The distribution (20) is normalized such that 

One important phase property of TMSNS is seen directly from the form of formula (20). It is 

clear that the joint probabi1it.y distribution depends on the sum of the two phases only 

This means the strong correlations of the two modes. Integrating P(8,: Ob) over one of the phases 

gives a marginal phase distribution P(8,) or P(Ob) for the phases 8, or Ob. which are uniformly 

distributed 
Thus the phases 8, or eb of the individual modes are unifor~nly distributed, and the only nonuniformly distributed phase quantity is the phase sum O+ = 8,+Ob. In Fig. 3 n7e plot. the Pegg-Barnet to 

phase distribution for TMSNS in polar coordinates for different values of parameter q. For nonzero 

values of q the phase distribution shows (N + 1)-lobe structure, and the greater q the nlore distinct lobes become. However, when q = 0 the phase distributioll has only onc lobe for all .IT. It 

is important to notice a remarkable resemblance in a behaviour of the phase distribution and the 

photon number distribution for TMSNS: they both display the (N + 1)-peak structure. Another 

significant feature of the joint phase distribution is a property of the phase locking - the phase 

sum is locked to the argument of the squeezing parameter in the limit of large squeezing [21, 231. 

Now consider phase quasiprobability distributions which can be obtained by integrating 

quasiprobability distribution functions (11) over the radial variables [9], [lo]. As we have noticed 

above, P function is not well-defined function for the states under corlsideration and therefore 

it is impossible to determine corresponding phase quasiprobability distribution. As a result of 

integration of Q(cr, p) and IV(a, ,d) over la1 and !PI, we arrive to the following formula: 

where the coefficients G(")(n, k) distinguish between two distributions. a.nd they are: 

(i) for the Q function 

(ii) for the Wigner function 

where X = min(n, k), v = max(n, k). All the coefficients G(")(n, k) are symmetrical, G(")(n, k) = 

G(")(k, n), and G(")(n, 12) = 1. Note, that such expressions (20), (25) for the phase distributions 

are valid for all two-mode states with the number state decomposition like in (3). In Fig. 3, 

we show the plots of the three phase distributions in polar coordinates for Th4SNS calculated 

according to formulas (20) and (25) with the coefficients (26) and (27) for different values of AT 

and nonzero q. It is seen that the Pegg-Barnett phase distribution and P("')(O+) are similar 

and have the N + 1 lobes, while P(Q)(O+) is much broader and has only one lobe. In the case 

q = 0 all three distributions have the same form of one lobe. So, as in the case of displaced 

number states [13], there is an essential difference in the phase information carried by P(Q)(O+) and 

P(lV)(O+). Because of the averaging procedure with the "prol~al~ilities" C;(Q)(n, k)G'(Q)(~j +(I, k+q) 
FIG. 2. Phase distribution P('~)(B+) for the two-mode squeezed number state with r = 0.5, 

N = 2 and q = 0 (solid line), q = 3 (long-dashed line) and q = 6 (short-dashed line). 
FIG. 3. Phase distribution P('~)(B+) (solid line) and phase quasiprobability distributions 

P("')(O+) (short-dashed line) and P(Q)(B+) (long-dashed line) for the two-mode squeezed number 

state with r = 0.5, q = 6 and (a) N = 0, (b) .\r = 1, (c) N = 2. 
some phase information is lost in p(Q)(B+). The Pegg-Barnett phase distribution is very close 

to the distribution P(~)(B+), although it is not identical to it. The phase peaks of P("')(B+) are 

slightly na.rro\x7er than those of P('~)(B+). The grea.ter the difference in iiulnbes of photons q the 

closer these two distributions. Basically they carry the same phase information. This similarity 

is in agreement with the area of overlap in phase space arguments, which are that the Mfigner 

function represents quantum states in the phase space [lo]. However, the Wigner function can 

take on negative values and the positive definiteness of P(~)(O+) isnot automatically guaranteed, 

while there are no such problems with the Pegg-Barnett phase distribution. 

We have discussed photon statistics and phase properties of the two-mode squeezed number states showing that the photon number distribution and the Pegg-Barnett phase distribution 

for such states exhibit the similar N + 1-peak structure for nonzero values of the difference in 

the number of photons q between modes. We have compared the Pegg-Barnett phase distribution with the phase quasiprobability distributions P(Q)(B+) and P(")(B+ ) obtained by integrating 

the Q function and the Wigner function over the radial coordinates. \Are have shown that the 

Pegg-Barnett phase distribution and the distribution P(W)(B+) carry basically the same phase 

information, while the distribution P(Q)(B+) loses an essential part of the phase information. 

References 

[l] C. I<. Hong and L. Mandel, Phys. Rev. Lett. 56, 58 (1986). 

[2] P. Filipowicz, J. Javanainen and P. Meystre, J. Opt. Soc. Am. B 3. 906 (1986). 

[3] S. M. Barnett and P. L. Knight, J. Opt. Soc. Am. B 2, 467 (1955). 

[4] W. H. Louisell, A. Yariv and A. E. Siegman, Phys. Rev. 124, 1646 (1961). 

[5] P. Carruthers and M. M. Nieto, Rev. Mod. Phys. 40, 411 (1968); 

S. M. Barnett and D. T. Pegg, J. Phys. A 19, 3849 (1986). 

[6] D. T. Pegg a.nd S. M. Barnett, Europhys. Lett. 6, 4813 (1988). 

[7] S. M. Barnett and D. T. Pegg, J. Mod. Opt. 36, 7 (1989). 

[S] D. T. Pegg and S. M. Barnett, Phys. Rev. A 39, 1665 (1989). 

[9] S. L. Braunstein and C. 14. Caves, Phys. Rev. A 42, 4115 (1990). 

[lo] W. Schleich, R. J. Horowicz and S. Varro, Phys. Rev. .A 40, 740.5 (1YS9). 

[ll] IY. Schleich, A. Bandilla and H. Pa.ul, Phys. Rev. A 45, 6652 (1992). 

[12] R. TanaS and Ts. Gantsog, Phys. Rev. X 45, 5031 (1992). 
[13] R. Tanai, B. I<. Murzakhmetov, Ts. Gantsog and A. V. Chizhov, Quantum Optics 4, 1 (1992); 

A. V. Chizhov, Ts. Gantsog, B. I<. hiurza.khmetov and R. Tana.6, J. Quant. Nonliilear Phenomena (in press). 

[14] A. V. Chizhov, Ts. Gantsog and B. Ii. f\lurzakhmetov, Qua.ntum Opt. 5, 55 (1993). 

[15] C. M. Caves and B. L. Schumaker, Phys. Rev. A 31, 3068 (1985); 

B. L. Schumaker and C. M. Caves, Phys. Rev. A 31, 3093 (1985). 

[16] C. M. Caves, Chang Zhu, G. J. Milburn and IV. Schleich, Phys. Rev. A 43, 3854 (1991). 

[17] F. A.M. de Oliveira, M. S. Kim, P. L. Knight and V. Buiek Phys. Rev. A 41, 2645 (1990). 

[18] S. M. Barnett and P. L. Knight, J. Mod. Opt. 34, 841 (1987). 

[19] B. R. Mollow and R. J. Glauber, Phys. Rev. 160, 1076 (1967); 

160, 1097 (1967). 

[20] I<. E. Cahill and R. J. Glauber, Phys. Rev. 177, 1857 (1969); 177, 1883 (1969). 

[21] S. M. Barnett and D. T. Pegg, Phys. Rev. A 42, 6713 (1990). 

[22] Ts. Gantsog and R. Tanai, Opt. Commun. 82, 145 (1991). 

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SUPERSYMMETRY AND RADIAL SQUEEZED STATES 

FOR RYDBERG WAVE PACKETS 

Robert Bluhma and V. Alan Kosteleckfbl 

"Physics Department, Colby College 

Waterville, ME 04901, U.S.A. 

Physics Department, Indiana University 

Bloomington, IN 47405, U.S.A. 

Atomic supersymmetry provides an analytical effective-potential model useful for 

describing certain aspects of Rydberg atoms. Experiments have recently demonstrated the existence of Rydberg wave packets localized in the radial coordinates 

with p-state angular distribution. This talk shows how atomic supersymmetry can 

be used to treat radial Rydberg wave packets via a particular analytical type of 

squeezed state, called a radial squeezed state. 

1. Introduction 

Irradiation of a Rydberg atom with a short laser pulse can produce a radially 

localized wave packet with a p-state angular distribution. The time evolution of such 

a state initially exhibits some attributes of the classical radial motion, including the 

Kepler period [I, 2, 3, 41. After several radial oscillations, the packet disperses. At 

various later times, the quantum wave recombines into single- or multiple-coniponent 

packets called revivals [5, 6, 7, 8, 91. 

The localization of the initial packet suggests a theoretical description via a coherent or squeezed state [lo]. However, a direct approacli meets technical obstacles 

[ll] or generates quantum packets that do not describe the characteristics of radial 

p-state excitations of Rydberg atoms in the absence of external fields. 

This talk provides a summary of our recently developed framework for the analytical study of Rydberg wave packets [12, 131. Our approach incorporates nonhydrogenic aspects of the packets using atomic supersymmetry [14,15]. This provides 

an effective central potential along with analytical wavefunctions ktIt and exact Rydberg eigenenergies E,2t for a Rydberg electron, expressed in terms of shifted quantum 

numbers n* and I*. Recent summaries of the methods and results of atomic supersymmetry and references to the literature can be found in Ref. [16]. 

l Speaker 

PAGE @&AM mMB 
We use the analytical wavefunctions of atomic supersymmetry to construct a family of analytical squeezed states, called radial squeezed states, that form representations of radial Rydberg wave packets. The procedure begins by mapping the classical 

physics associated with the effective potential into the form of a harmonic oscillator. 

After the conversion to quantum physics, squeezed states can be derived for the resulting uncertainty relation. The method is based on an extension of the approach of 

Ref. [17] for cirular states of hydrogen. Tbe procedure is outlined in sections 2 and 3 

below. The time evolution of the resulting radial squeezed states is briefly discussed 

in section 4. In what follows, we use atomic units with tL = e = me = 1. 

The reader is referred to Refs. [12,13] for more details about the subjects presented 

in this talk and for more references to the background material. 

2. Classical Physics 

The classical theory corresponding to atomic supersymmetry uses a central potential that leads to the effective one-particle radial hamiltonian 

where p, = 7: is the radial momentum. The classical continuous variable I* is a 

shifted value of the classical angular momentum I, arising from the incorporation of 

the effects of the central potential. At the quantum level, Z*2 becomes the quantized 

quantity 1*(1" + 1) of atomic supersymmetry. The energy E* has for convenience 

been expressed in terms of a continuous classical variable n*, which at the level of 

quantum physics converts to the quantized, shifted principal quantum number of 

atomic supersymmetry. 

When E* is negative, the particle is bound and oscillates between outer and inner 

apsidal points, 1.1 and 1.2, say. The classical orbital period T:, which is the time taken 

to move from rl to r2 and back, is given via Eq. (1) as 

The classical orbit can be shown from Eq. (1) to be a precessing ellipse, obeying 

the equation 

Here, e = J-, do is a constant of the integration, and f = 1*/1. For the 

radial Rydberg wave packets we discuss below, l*/n* is small and so the corresponding 

classical orbits are highly elliptical. For simplicity in what follows, we impose fOo = $. 
Direct efforts to obtain coherent or squeezed states based on this model face 

technical obstacles. Instead, we map the theory into a harmonic-oscillator formalism. 

We convert to a new set of classical variables, R and P, chosen to have sinusoidal 

variation with the angle 0 and given by 

11 e r2 el R----= -sinf0 , p --R = -- 

f 1*2 

cos f 0 . r 1*2 

In terms of these variables, the classical equations of motion become 

Equation (1) then takes the form of the energy equation for a simple harmonic oscillator of frequency I and energy e2/2 f 2: 

3. Quantum Physics 

At the quantum level, the new classical variables R, P become quantum operators 

obeying the comrnut ation relation 

The uncertainty product A RAP is 

11 ARAP 2 - - zf(r2) . 

At a given time, any minimum-uncertainty wavefunctions must therefore obey the 

equation 

(R - (R))1C, = iA(P - (P))1C, , (10) 

where the real constant A is given by 
For convenience in what follows, we introduce the quantities 

In terms of these quantities, the normalized minimum-uncert ainty solutions to Eq. 

These wavefunctions form a three-parameter family of squeezed states, called radial 

squeezed states. Their construction shows they are suitable candidates for radial 

Rydberg wave packets in any atom correctly modeled by atomic supersymmetry. 

The relatively simple analytical form of the radial squeezed states permits the 

derivation of a number of useful results. Among these are the following expectation 

values: 

2a + 3 (r) = - 

270 (PT) = -71 , and 

702 (2702~*(1* + 1) + a + 1) - - 70 712 

(H) = 2(a + 1)(2a + 1) +- a+l 2 (15) 

Requiring finiteness of these expressions and the separate normalizability of the kinetic and potential energies of the radial squeezed states restrict the parameters to 

1 a,> -5 and yo >O. 

The uncertainty relation for the usual variables r and pT can be calculated for the 

radial squeezed states. It is 

As expected, this is not a minimum-uncertainty product. However, a relatively large 

value of a is needed to reproduce the experimental situation as the wave packet makes 

its first pass through the outer apsidal point, so the uncertainty product is close to 5 

at that time. 

4. Time Evolution 

The analytical nature of the radial squeezed states has made it possible for us 

to study their time evolution both analytically and numerically. Here, we restrict 

ourselves to describing briefly some results and providing a few representative figures 

to illustrate the key features. 

Specifying a ,particular radial squeezed state means selecting values for the three 

quantities a, yo, and 71 at some initialization time. Since the uncertainty product 
ArAp, is expected to be minimized at the first pass through the outer turning point [I] 

we choose this as the initialization time and define it as t = 0. The three parameters 

a, 70, and 71 are determined by imposing conditions on the three physical quantities 

(p,), (r), and (H) at t = 0: 

Here, r:ut is the outer apsidal point of the quantum orbit, and Eii+ = -1/2ii*2 is the 

energy expectation of the central state ii excited by the short laser pulse. 

In the context of radial squeezed states for atomic supersymmetry, our analysis 

shows that the radial squeezed state initially oscillates with the classical orbital period 

Tz and gradually disperses. For a radial squeezed state dominated by shifted principal 

quantum number ii* and including a range of shifted principal quantum numbers 6n*, 

the packet is effectively dispersed at an interference time tLt E fi*T;l/36n*. Much 

later, near the revival time tt$, E fi*T;/3, the radial squeezed state recombines into a 

packet called a full revival that is similar to the initial one and that orbits with a period 

Ti. At times t: = t;rk,/r between t:nt and t:e,, the radial squeezed state combines 

into r spatially separated packets called fractional revivals, with wavefunction period 

T; = ST;. 

Some of these behaviors are illustrated in Figure 1, which plots the unnormalized 

radial probability density f (r) = r2 l$ii,I,l (r) l2 at different times, for a hydrogenic 

radial squeezed state with ii* = ii = 85 and I* = 1 = 1. In this case, the initialization 

parameters are a E 168.225, yo E 0.01 17465, and yl = 0. This corresponds to a 

classical orbit of high eccentricity, e ^I 1, and periodicity TA E 93.3 psec. 

Figure la displays the radial squeezed state at t = 0. It is situated near the 

outer quantum apsidal point r:ut 14450 a.u. The initial uncertainty product is 

ArAp, E 0.501, which is close to the minimum of 0.5, as expected for a smooth packet. 

Initially, the packet moves towards the origin. Figure lb shows its shape at t = T;/4, 

while Figure lc shows it at t = T,*1/2. As the packet approaches the origin it spreads 

and develops oscillations in the radial probability distribution, thereby indicating the 

transition from a classical to a quantum object. The uncertainty product halfway 

through the first classical period is ArAp, E 59.5. 

After reflection off the inner apsidal point, the packet recombines into a more 

classical object as it moves towards the outer apsidal point. Figures Id and le show 

the packet at t = 3T34 and t = T;, respectively. Some decoherence is visible, but the 

packet is still well defined. The amount of dispersion increases with successive orbits, 

until a time of order ti,, E 4T:. Subsequently, the probability distribution no longer 

exhibits a unique and well-defined peak at any point in the orbit. For illustration, 

the situation at t = 5T; is displayed in Figure If. 
(b) 

Figure 1 
(dl 

Figure 1 
(f) 

Figure 1 
(h) 

Figure 1 
At later times, fractional and full revivals appear. The fractional revival consisting 

of two separate peaks, which appears at t,* = t:e,/2 = 1.3 nsec., is shown in Figure 

lg, while the full revival appearing at t:e, 21 2.6 nsec. is shown in Figure lh. These 

revivals exhibit all the features expected from the classical analysis, including, for 

example, the expected values for the wavefunction periodicity. 

Along with the additional analysis contained in Refs. [12,13], these results indicate 

that radial squeezed states are useful models for radial Rydberg wave packets. 

5. References 

1. J. Parker and C.R. Stroud, Phys. Rev. Lett. 56, 716 (1986); Phys. Scr. T12, 

70 (1986). 

2. G. Alber, H. Ritsch, and P. Zoller, Phys. Rev. A 34, 1058 (1986); G. Alber 

and P. Zoller, Phys. Rep. 199, 231 (1991). 

3. A. ten Wolde, L.D. Noordam, A. Lagendijk, and H.B. van Linden van den 

Heuvell, Phys. Rev. Lett. 61, 2099 (1988). 

4. J.A. Yeazell, M. Mallalieu, J. Parker, and C.R. Stroud, Phys. Rev. A 40, 5040 

(1989). 

5. J.A. Yeazell, M. Mallalieu, and C.R. Stroud, Phys. Rev. Lett. 64, 2007 (1990); 

J.A. Yeazell and C.R. Stroud, Phys. Rev. A 43, 5153 (1991). 

6. D;R. Meacller, P.E. Meyler, I.G. Hughes, and P. Ewart, J. Phys. B 24, L63 

(1991). 

7. A. ten Wolde, L.D. Noordam, H.G. Muller, and H.B. van Linden van den 

Heuvell, in F. Ehlotzky, ed., Fundamentals of Laser Interactions II (SpringerVerlag, Berlin, 1989). 

8. I.Sh. Averbukh and N.F. Perelman, Phys. Lett. 139A, 449 (1989). 

9. M. Nauenberg, J. Phys. B 23, L385 (1990). 

10. See, for example, J.R. Klauder and B.-S. Skagerstam, eds., Coherent States 

(World Scientific, Singapore, 1985); D. Han, Y.S. Kim, and W.W. Zachary, eds., 

Squeezed States and Uncertainty Relations (NASA, Washington, D.C., 1992). 

11. E. Schrodinger, Naturwissenschaften 14, 664 (1 926). 

12. R. Bluhm and V.A. I<ostelecki, Radial Squeezed States and Rydberg Wave Packets, Phys. Rev. A, Rapid Communications, in press. 

13. R. Bluhm and V.A. I<osteleck~, Atomic Supersymmetry, Rydberg Wave Packets, 

and Radial Squeezed States, Indiana Uiliversity preprint IUHET 256 (August 

1993). 
14. V. A. Kosteleckf and M. M. Nieto, Phys. Rev. Lett. 53, 2285 (1984); 

Phys. Rev. A 32, 1293 (1985). 

15. V. A. Kostelecki and M. M. Nieto, Phys. Rev. A 32, 3243 (1985). 

16. V.A. Kosteleckf, in I). Han, Y.S. Kim, and W.W. Zachary, eds., Proceedings 

of the Workshop on Harmonic Oscillators (NASA, Washington, D.C., 1993), p. 

443; Supersymmetry, Coherent States, and Squeezed States for Rydberg Atoms, 

Indiana University preprint IUHET 254 (July 1993), to appear in the Proceedings of the International Symposium on Coherent States, Oak Ridge, TN, June 

1993. 

17. M.M. Nieto, Phys. Rev. D 22, 391 (1980); V.P. Gutschick and M.M. Nieto, 

Phys. Rev. D 22, 403 (1980). 
EFFECT OF DISPERSION FORCES ON SQUEEZING 

WITH RYDBERG ATOMS 

S.K. Ng 

M.R. Muha~nad 

Department of Physics, IJniversity of Malaya, 59100 Ihala Lumpur, A4alaysia 

M.R.B. Walliddin 

Department of h!athematics, University of Afalaya, 59100 I<uala Lumpur, Malaysia 

Abstract 

We report exact results concerni~lg the effect of dipole-dipole interaction (dispersion 

forces) on dynamic and steady-state characteristics of squeezing in the emitted fluorescent 

field from two identical coherently <lriven two-level atoms. The atomic system is subjected to 

three different damping baths in particl~lar the nor~nal vacuum, a broad band thermal field 

and a broad band squeezed vacuum. The atomic model is the Dicke model, hence possible 

experiments are most likely to agree with theory when performed on systems of Rydberg 

atoms making microwave transitions. The presence of dipole-dipole interaction can enhance 

squeezing for realisable values of the various para~neters involved. 

The fundamental importance of squeezecl light aild its interaction with matter as well as potential 

applications have become the focus of considerable attention in the last few years both theoretically 

and experimentally. Indeed, the coupling of a two-level atomic system to squeezed light for example 

can lead to interesting new phenonlena as reported by a numnber of authors [1,2,3]. In this paper we 

consider the possibility of generating squeezing in the enlitted fluorescent field from two identical 

coherently driven two-level atoms with dipole-dipole interaction (dispersion forces). The atomic 

. system is subjected to three different datnping baths in particular, the normal vacuum, a broad 

band squeezed vacuuln and a broad band thermal bath. We report exact results concerning the 

time evolution as well as steady-state emitted field quadratures which may lead to squeezing. It 

is shown that squeezing n~ay be enhanced for certain realisable values of the system parameters. 

The dipole-dipole coupling is expected to play a significant role especially in the high atomic 

density limit [4]. It is especially i~nporta~lt because it generally breaks the pernlutation symmetry 

of the atom-field coupling [5] whicll is the basic assumption of the collective spontaneous emission 

process (superradiant) involvix~g NA-atom system. As a consequence, the S2 conservation (where 

S is the total 'spin' of NA two-level atoms) \\fllic.h appear inherently in the point like Dicke model 

[6] is affected significantly. Since the atomic llloclel is the coherintly driven two-atom Dicke model, 

possible experiments ?re most likely to agsee ivit.11 theory \\?lien performed on systeins of Rydberg 

atoms making micro\vave transitions - see e.g. [7] and referc.l~c.es therein. 
2 Description of the model 

Our starting point is the inaster equation for the reduced atoinic density operator in a frame 

rotating at WL, the laser frequency - see equation (5) of ref. [8]: 

where Sf, Sz are the usual collective atoinic operators; 2R is the Rabi frequency characterizing 

the strength of the driving field, y is the single at0111 Einstein A coefficient, a+ is the static 

dipole-dipole interaction potential such. that 

and A = w~ - WL - a+ (wA is the atoinic trailsitioil frequency). 

Note that A = -a+ represeilts the on-resonant case. AT and A4 are pa.rameters of the squeezed 

vacuum describiilg squeezing with IA!l" lV(M + 1) holding for llliniilluill uncertainty squeezed 

state. @ is the rela.tive pl~ase between the squeezed vacuum and tlle laser. In what follows we 

set IMI2 = N(N + 1). Note that if N = 0, then the atoills are dainped by the normal vacuum 

whilst the case N # 0, A4 = 0 corresl>onds to the-atoms dainped by a broad band thermal field 

such that N -, ii which is the mean occul>ation iiulllber of the resonant field mode. The master 

equation is solved using the fourth order Runge-I<utta method. 

3 Squeezing in resonance fluorescence 

We now discuss the possibility of gellera.ti~lg squeezing in the elllitted fluorescent radiation field 

and the effect of a+ on it. The "in l~hase" and "phase quadrature" components of the emitted 

fluorescence field denoted by F' and F, are respectively 

with 
Dimensionless Time 

Fig. 1 Fx and Fy against -r for atoms damped b a 

n a 2 normal vacuum ; - =0.5 , + = 1.0 and - =0.0 

Y Y Y 

Dispersion Forces 

a + Fig.2 Steady- state Fx versus for atoms damped by a n A - a+ normal vacuum ; - =0.5 and - -- - 

Y Y Y 
0.1 - 

0.08 - 

0.06 - 

0.04 - 

0.02 - Thermal Field ( ii = 0.05 ) 

E 

-0.08 - 

-0.14 0 2 4 6 8 ~1'0~~12i~14i~ 

Dispersion Forces 

a+ Fig3 Steady-state Fx versus - for atoms damped either 

by a normal vacuum, a brdd bang thermal field gr a 

broad band squeezed vacuum ; - = 10 and - = 15 

Y Y 

Dimensionless Time 

Fig.4 Components of Fx and Fy as functions of -r for 

atoms damped by a broad band squeezed vacuum ; 

52 A - =0.1 , 7 =0.5 , - =O , N=O.OS and m =O 

Y Y 
as there is a linear functional relation between the atolnic operators and the field operators [9]. 

Squeezing occurs whenever F, or F, < 0. 

Fig. 1 shows the time evolution of F' and F, for the atonlic system danlped by the normal 

vacuum (N = 0) and driven by an off-resonant weak laser with weak dipole-dipole interaction: " = 0.5; 7 = 1.0. In transient, both F, and F, display squeezing but do not occur at the same 

7 

time due to Heisenberg Uncertainty relation, with nlaxilnuln squeezing occurring in F,. In this 

case steady-state squeezing is only predicted in F,. Increasing a+ further suppresses the squeezing 

in F, while F, is positive in the steady-state. In Fig. 2 all the parameters are the same as in 

Fig. 1 but now the atoins are driven by a resonant laser (A = -a+). There the steady-state 

F, is plotted against 7. Clearly, squeezing is obtained when 7 > 1.7 and the squeezing keep 

increasing with a+ and reaches its maxi~~~um value at 7 z 2.5. But it is reduced when a+ is 

increased further. 

The effects of an off-resonant strong laser (: = 10) on steady-state squeezing in Fz for the three 

damping baths are sho~vn in Fig. 3. The role of the dispersion forces (dipole-dipole interaction) 

to produce optilnu~n squeezing is best seen for the nor~nal vacuum, thermal field and squeezed 

vacuum (Q = n/2, N = 0.05) cases. In Fig. 4 we plot the the components of F, and F, as 

functions of dimensionless time (r = ~t) for a detuoed weak laser (: = 0.1, = 0.5) driving 

the atomic system and dainped by a broad band squeezed vacuum (N = 0.05, Q = 0). Here, 

squeezing is evident in F, in the transient as well as in the steady-state since it satisfies the 

squeezing condition < $ I (S') I. The steady-state squeezing achieved is estimated about 

12%. 

In conclusion, the dipole-dipole interaction proves to be important in the analysis of squeezing 

in the emitted fluoresce~lt field from a cooperative systenl coherently driven and damped either 

by a normal vacuum, broad band thermal fielcl or a broad band squeezed vacuum. 

4 Acknowledgement 

This research is supported by the Malaysia IRPA RhD 04-07-04-169. MRBW is grateful to 

University of Malaya for the support to attend the present conference. 

References 

[I] C.W. Gardiner, Phys. Rev. Lett. 56, 1917 (1986) 

[2] G.h4. Palma and P.L. I<night, Phys. Rev. A 39, 1962 (1989) 

[3] G.S. Xgarwal and R.R.. Puri, Phys. Rev. .A 41, 3782 (1990) 
[4] G.S. Agarwal, L.M. Narducci and E. Apostolidis, Optics Co~ml~un. 36, 285 (1981) 

[5] B. Coffey and R. Friedberg, Phys. Rev. A 17, 1033 (1978) 

[6] R.H. Dicke, Pllys. Rev. 93, 99 (1954) 

[7] G.P. Hildred, R.R. Puri, S.S. Hassan and R.IC. Bullougll, J. Phys. B 17, L535 (1984) 

[a] W.A.M. Othmari and M.R.B. Wahiddi11, J. Mod. Opt. 40, 7 (1992) 

[9] M.R.B. Wahiddin, B.M. Garraway and R.I<. Bullough J. Mod. Opt. 34, 1007 (1987) 
Higher-order squee~ing of the quantum electromagnetic Geld and 

the generalized weertainty relations in two-mode squeezed states 

La' Xizeng Sor BBOZ~Q 

Departme nf of Phyeiea, Tianjin University, Tianfin 80007R, P.B. China 

It is found that the two-mode output quantum electromagnetic ficld in two-mode squeezed 

states exhibits higher-order squeezing to all even orders. And the generalized uncertdnv 

relations are also presented for the first time. 

The concept of higher-order squeezing of the single-mode quantum electromagnetic field 

was first introduced and applied to severd processes by Hong and Mandel in 1985113. Lately 

. Li Xizeng and Shan Ying have calculated the higher-order squeezing in the process of degenerate four-wave mixing8 and presented the higher-order uncertainty relations of the fields in 

single-mode squeezed states4. In this paper we generalize the above work to the higher-order 

squeezing in two-mode squeezed states. The generalized uncertainty relations are also presented 

for the first time. 

1 Definition of higher -order squeezing in two-mode 

squeezed states 

The definition of two-mode squeezed states was given by Caves and Schurnakers: 

Where ~($1 is the two-mode squeezed operator 

(CY+,CU- > is the two-mode coherent state, ii* are two-mode annihilation operators, o* are 

eigenvalues of ii* in ]a+, a- >. 

Define the two-mode squeezed annihilation operators by &, 

where 

p = coehr, v = eibeinhr, 

= re" is the squeeze parameter. 
Then the two-mode squeezed states are the eigenstates of dr, 

and a* are the eigemalues of A+. 

The red two-mode output field k can be decomposed into two quadrature components 

and &, which are canonical conjugates. The ontput field k! exhibits higher-order squeezing to 

any higher-order (Nth order) in E~, if there exists such a phase angle 4 that the higher-order 

moment < (A E~)~ > in a two-mode squeezed state is smaller than its value in a completely 

tow-mode coherent state, viz., 

This is the definition of higher-ordet squeezing in two-mode squeezed states. 

2 The quadrature components of the two-mode output 

field I!? 

The electric field operator for the two-mode output field has the form of 

E (z, 1) = E(+)(z, t) + D'-'(2, t). (6) 

Where 

We now introduce two Hermitian quadrature components and E$ of the electric field 

defined by 

j?~~ (x, t) = ~(i) eilWt-=)-4 + E(-I~-~Iw:-~-~I (9) 

Then, E(X, t) can be decomposed into two quadrature components E] and E$, which are 

canonical comjugates 

Where fl is the carrier frequency 

w+ + w- n = 2' 

and q4 is an arbitrary phase angle that may be chosen at will. 
The units are chosen so that h = c = 1. 

Substituting Eqs. (7) and (8) into (9), we obtain 

where 

and 

€=W+-iI=il-w- (14) 

is the modulation frequency. 

From Eq.(3), we get 

B* = p*B* - vA$. 

Substituting (15) to (12), we obtain bl in terms of 

Where 

Define 

Then 

3 Higher-order noise moment < > and 

Higher -order squeezing 

By using the Campbell-Baker-Hausdorff formula, we get 

+ . . + (IV - l)!!c,N/'. (Pi is even) (20) 

where N(~) = N(N - 1) . . . (N - r + 1), 4 = &[E,, E~] = [B, B+], ' :: " denotes normal ordering 

with respect to B and B+. 

Now we take the two-mode squeezed states, then 
€3 a = IS, B+I = lhli2 + ~~1' = (ls+la + ls-ia)(lrla + IYI') - - 3(~*fi-aid + ai4 + pv e ). (22) 

horn (20), (4) and (13), we get 

If 4 is chosen to satisfy cos(8 - 24) = 1, then Eq(23) leads to the result 

when coshr < !, the right hand side is smaller than (N - l)!!flN/', which is the corresponding 

Nth order moment for two-mode coherent states. 

It follows that the two-inode output field exhibits higher-order squeezing to all even orders. 

4 The generalized uncertainty relations 

[A]. Higher-order noise moment < (A&)" > 

E* can be regarded as a special case of E,, in which if 4 is replaced by 9 + x/2, then from 

(23) it follows that 

If 4 is chosen to satisfy co8(8 - 24) = 1, then (I 

When corhr < 0, the right hand side is greater than (X - l)!!flN/'. 

[B). Generalized uncertainty relations 

Rorn (24) and (261, we obtain 

6' 

< > < (A&)" >= {(Ar - 1)!!]'nN[1 + --st'nh'(2r)lN. n = 
Equation (27) shows that < > and < > in two-mode squeezed states 

can not be made arbitraily small simultaneously. We call Eq.(27) the generalized uncertainty 

relations in two-mode squeezed states, and the right hand side (constant) is dependent, on 

N, r, 52, and s. 

Since 

r" 1 + -crinh"2r) > 1 0' 

SO 

< (~4)~ > < (AE~~N > > [(N - l)!!J2n" 

If r = 0, the two-mode squeezed states become two-mode coherent states, then 

< (AEI)" >,,. . < >c,8= [(N - I)!!]' . fly. 

This is the generalized uncertainty relations in two-mode coherent states. 

If c=O?N = 2, we obtain 

This is just the usual Heisenberg uncertainty relations in relevant reference^^^^^^^^. 

5 Application 

As an application of the above result, we calculate the generation of higher-order squeezing 

by non-degenerate four-wave mixing (NDFWM). It can be shown that the field of the combined mode of the probe wave and the phase-conjugate wave exhibits higher-order squeezing 

to all even orders, and the generalzied uncertainty relations still hold in NDFWM process. 

6 Acknowledgments 

This research was supported by the National Natural Science Foundation of China. 

References 

[I] C.K.Hong and L.Mande1, Phys. Rev. Lett. 54, 323 (1985) 

[2] C.K.Hong and L.hlande1, Phys. Rev. A32, 974 (1985) 

131 Li Xizeng and Shan Ying, Phys. Rev. A40, 7384 (1989) 

[4] Li Xizeng and Shan Ying, Technical Digest of the XVI International Conference on Quantum Electronics, Tokyo, Japan (July 1988) p.180 

[5] C.M.Caves and B.L.Schumaker, Phys. Rev. AS1, 3086(1985) 
Q-DERIVATIVES, COHERENT STATES 

AND SQUEEZING 

E.Celeghini l, S.De Martino 2, S.De Siena 2, M.Rasetti and G.Vitiello 

Dipartimento di Fisica - Uniuersitci di Firenze and INFN-Firenze, I50125 Firenze, Italy 

Dipartirnento di Fisica - Uniuersitd di Salerno and INFN-Napoli, I841 00 Salerno, Italy 

Diprtimento di Fisica and Unitci INFM - Politecnico di Torino, I10129 Torino, Italy 

Abstract 

We show that the q-deformation of the Weyl-Heisenberg (q-WH) algebra naturally arises 

in discretized systems, coherent states, squeezed states and systems with periodic potential 

on the lattice. We incorporate the q-WH algebra into the theory of (entire) analytical 

functions, with applications to theta and Bloch functions. 

1 Introduction 

The general properties of q-algebras [I] [2] have been widely studied, in particular in connection 

with specific physical models. In this paper we will show [3] that the q-deformation of the WeylHeisenberg (q-WH) algebra naturally arises in discretized quantum systems, coherent states, 

squeezed coherent states and systems with periodic potential on the lattice. 

q-algebras are deformations of enveloping algebras of Lie algebras and, like the latter, they 

have Hopf algebras properties. The q-deformation of the Weyl-Heisenberg algebra Cq-WH), 

as well as the WH algebra, is not even a Hopf algebra; it has only the properties of a Hopf 

superalgebra 141. 

In our study of q-deformations we want to preserve the analytic structure of the corresponding Lie algebras and therefore we need to operate in a frame where analyticity is ensured: this 

is guaranteed by working in the Fock-Bargmann representation (FBR). In this representation 

it is immediate to show that finite difference operators possess the algebraic structure of q-WH 

algebra: As a result we recognize that a q-deformation of the algebra occurs whenever a finite 

length is involved in a physical system, the q-parameter being related with the finite spacing. 

The q-deformation is also expected in the presence of periodic conditions, since periodicity is a 

special form of invariance under finite difference operators. 

We use the well known mapping of the q-algebra into the universal enveloping algebra of 

a corresponding Lie structure; to be specific, the mapping of finite difference operators into 

functions of differential operators, which can be indeed achieved only by operating on COO 

functions, namely by working in the FBR. 

We would like to stress that we succeed into incorporating q-deformation of the WH algebra 

into the theory of (entire) analytical functions, with specific applications to theta functions and 

to Bloch functions, a result which may deserve by itself much attention. 
In this paper we will use dimensionless units for all physical quantities. 

2 Finite difference operators 

The FBR operators, solution of the WH commutation relations [a, at] = 1, [N, a] = -a, [N, at] = 

at , are [5]: 

d 

a++., 

d 

N+z-, a+ -. dz dz (2.1) 

The Hilbert space 3' is identified with the space of the entire analytical functions. Wavefunctions are expressed as +(z) = Zz=o c~u~(z), Ez=o 1~~1~ = 1, u~(z) - - " (n~ Z+) . The 

set {un(z)) provides an orthonormal basis in 7. The finite difference operator D, 

with q = et, E C, may be written on F as D, = ((q- l)~)-'(~'& - 1). D, is the well known [B] 

[7] [8] [9] q-derivative operator and, for q -, 1 (i.e. < -t 0) , it reduces to the standard derivative. 

We have the algebra 

and observe that it is nothing but the q-deformation of the WH algebra. In fact, this can be 

seen by introducing the following operators in the space 3 

where clearly &, = = at and liq,l a, = a. The quantum version of the WeylHeisenberg algebra is thus realized in terms of these operators {a,, i?,, N; q E C) with relations 

111 [21: N [N,a,]=-a,, [N,it,]=ii, , [a,,&,]=aqitq-&,a,=q . (2-5) 

Equivalently, by introducing a, - iid-*12 , the q-WH algebra eq. (2.5) is rewritten in the more 

familiarformas [N,a,] = -a, , [N,n,] =% , a,a, -q-:~,a, =qtN. 

The finite difference operator algebra (2.3) in the FBR thus provides a realization of the 

q-WH algebra (2.5). 

The notion of hermiticity associated with (2.5) has been studied in ref. [lo] in connection 

with the discussion of the squeezing of the generalized coherent states (GCS),, defined in the 

usual Fock space 3. 

We note that the commutator [a,, iiq] acts in 3 as follows 

In conclusion, the strict relation of the q-WH algebra with the finite difference operator 

Dq (q # 1) suggests. that whenever one deals with some lattice or discrete structure, then a 

deformation of the operator algebra acting on 3 should arise. 
3 Coherent states, theta functions and squeezing 

We summarize now the relation of q- WH algebra with the customary coherent states (CS) )z > 

[5], with theta functions and with squeezing. Eq.(2.6) is the key relation to establish our results. 

For sake of shortness we only report the relevant relations [3]: 

where D (z) denotes the usual CS generator. 

We observe that [a,,&,] acts as mapping operator from lz > to Iqz > up to a phase 

factor. On the other hand, it acts the z-dilatation operator (z -, qz) in the space of entire 

analytic functions. When q = ef , with I pure imaginary, = id, then [a,, d,] : z -, eiez , generates the U(1) group of phase transformations in the z-plane. We also observe that eqs. 

(3.2) and (3.3) provide a non linear realization of the quantum algebra (2.5) in terms of a and 

at. Vice-versa, the nonlinear operator D (2) is represented by the linear form [a,, a,]. 

In the framework of the formalism of CS on the von Neumann lattice L the defining functional 

equation for the theta function is 151 

0, (z + z,) = exp (irF, (-m)) exp () exp(zmz) 6, (2) , 

with z, = mlwl + mzwz an arbitrary lattice vector and F,(m) = mlm2 + m~el + mzez . A 

solution of (3.5) can be expressed as 

0, (r) = e-'r~c(m) exp (- y) exp (- zm r) f (r) , m 

where f (z) is an arbitrary entire function such that the series (3.6) is converging. 

To establish the relation between q-WH algebra and theta functions, we write q = qm = e*vlL , with cm a vector on the lattice L and, by setting zm = (qm - l)z, we have 131 
Eqs. (3.7-9) show that theta functions span indeed a space of representations for the qalgebra (2.5). 

Finally, we study the relation of q-WH algebra with squeezing. Let fi, = -i& and [i,fi.] = i, 

over a Hilbert space of states $(z) identified with the space of entire analytic functions 7. 

I Introduce the operators a = ?(i + ifi.) , a = (2 - i) [a, at] = I. It is immediate to 

observe that 

with S (c) denoting the squeezing generator [ll], < = log q the squeezing parameter and $s (z) 

the squeezed state. We therefore conclude that the operator [a,, a,] is the squeezing generator 

for CS in the FBR, thus confirming the conjecture previously [lo] formulated whereby q-groups 

are the natural candidates to study the squeezed CS. 

4 Quantum mechanics on the lattice 

Our purpose is now to show that q-WH algebra is underlying the physics of lattice quantum 

systems. Lattice Quantum Mechanics (LQM) is characterized by the E(2) commutator algebra, 

which in the momentum space is written as [3] 1121 

d [2,, cos(kc)] = [i- , cos(kc)] = -ic sin(kc) . dk (4.1) 

[fie, cos(kc)] = [c-' sin(kc), cos(kc)] = 0 . 

where 2, and fi,, denote the (one-dimensional) lattice position operator and the lattice momentum operator, respectively (extension to higher dimensions is straightforward). The corresponding uncertainty relations are 

where A2(d) = (A2) - with = (A) = J dkq*(k)A~(k). We observe that these relations 

go, in the continuum limit c -+ 0, to the usual ones. In this connection we observe that the 

continuum limit is, in fact, an isometric and conformal mapping of the torus on the plane. 

Following the usual procedure [13], the states minimizing the uncertainties (4.2) and (4.3) 

are found to be, in momentum space 

Q(k) = G-i exp [7cos(tk) - i~rk] . (4.4) 
The normalization constant G is given by G = $4(27) , lo denoting the modified Bessel function 

of the first kind of order 0. We adopted the notation: X = A€-', 7 = X = (0,) + i7(ii,) , 

and 7 is connected with the mean square roots of position and momentum. 

In the continuum limit, i.e. for small E, one recovers in the space of configurations, 1 $(x) = (7~)-2 exp {- [(z7)-l(x - (2))l+ i(@)(x - ($))I } , which is the minimum uncertainty 

wave-function given by Schrodinger [14]. The \E(k)'s are the lattice coherent states. 

In order to see the relation with the q-algebra we consider the conformal image # of the 

Hilbert space obtained upon introducing the variable z = eid (9 = kc , -gr 5 9 5 A), such that 

-%a . d d = -215s = sz$ . The functions in # are assumed to be entire squareintegrable analytic 

functions. We have 

f(9 + r) = ei4"(9) = qNf(z) = j(q~) , (4-6) 

with q = e". The realization (2.4) has been adopted in the FBR, with z restricted t6 the unit 

. circle. The E(2) algebra (4.1) is realized by 

with f E #, and the identifications 

z+2 z-f L1 = - , L2=- d 

2 , L3=z-, L+=z, L-=X. 23 dz 

One can see that [aq,$] is nothing but the group element eitL3 of E(2). The algebraic 

structure of LQM is thus intimately related with the q-WH algebra, the deformation parameter 

q being determined by the discrete lattice length E = -i log q. 

We finally note that zn = eind, n integer, is the eigenfunction of L3 associated with the 

eigenvalue n of the number operator in the FBR: L3zn = Nzn = nzn . The functions z = ei4 play also a r61e in the Bloch functions theory. Suppose we have a 

periodic potential V(x,) = V(xn + E) on the lattice. Bloch theorem ensures the existence of 

solutions of the related Schriidinger equation of the form $(xn) = efikZnvk(xn), with vk(xn) = 

vk(xn + E). $(xn) is the Bloch function. Let us limit ourself to consider for simplicity the plus 

sign in the exponentials. +(xn) has the property 

The choice of the variable z = eik"urns out .to be natural in the case of periodic potentials: 

Since zn = (zn) and qN (%)k = (qZ,)k = @(n+l) = c+l 9 

which shows that the Bloch functions provide indeed a representation for the q-WH algebra. 
References 

[I] L.C Biedenharn, J. Phys. A22, 117 (1989). 

[2] A.J. Macfarlane, J. Phys. A22, 4581 (1989). 

[3] E.Celeghini, S.De Martino, S.De Siena, M. Rasetti, and G.Vitiello, Mod. Phys. Lett B, 

(1993), to be published; Florence University Preprint DFF188/9/93. 

[4] E.Celeghini, T.D.Palev and M.Tarlini, Mod. Phys. Lett B5, 187 (1991). 

[5] A. Perelemov, Generalized Coherent States and Their Applications, (Springer-Verlag, Berlin 

Heidelberg, 1986). 

[6] L.C. Biedenharn, and M.A. Lohe , Comm. Math. Phys. 146, 483 (1992). 

[7] F.H. Jackson, Messenger Math. 38,57 (1909). 

[8] T.H. Koornwinder, Nederl. Acad.. Wetensch. Proc. Ser. A92, 97 (1989). 

[9] D. I. Fivel, J. Phys. A24, 3575 (1991). 

[10] E. Celeghini, M. Rasetti, and G. Vitiello, Phys. Rev. Lett. 66, 2056 (1991). 

[ll] H.P. Yuen, Phys. Rev. A13, 2226 (1976). 

[12] F.Guerra, Statistical Mechanics Methods in Quantum Field Theory, Proceedings of the 

International School of Mathematical Physics, G. Gallavotti ed., Universita' di Camerino 

(1974). 

[13] P.Carruthers, and M.N. Nieto, Rev. Mod. Phys. 40, 411 (1968). 

[14] E.Schriidinger, Naturwissenschaften 14, 664 (1926). 
SUM-FREQUENCY GENERATION FROM PHOTON NUMBER 

SQUEEZED LIGHT 

Ling-An Wu 

Institute of Physics, Chinese Academy of Sciences, 

P. 0. Boz 603, Beijing 100080, China 

Congshi Du Meijuan Wu Slliqun Li 

Department of Modern Applied Physics, Tsinghua University, 

Beijing 100084, China 

Abstract 

We investigate the quantum fluctuations of the fields produced in sum-frequency (SF) 

generation from light initially in the photon number squeezed state. It is found that, to the 

fourth power term, the output SF light is sub-Poissonian whereas the quantum fluctuations 

of the input beams increase. Quantum anticorrelation also exists in SF generation. 

1 Introduction 

In recent years squeezed states of light have been successfully produced in several laboratories 

and certain applications also demonstrated.' In future applications it may be desirable to change 

the frequency of a light beam already in the squeezed state, for example by frequency doubling, 

parametric down conversion or four wave mixing. Three wave processes are preferable since the 

second order nonlinear susceptibility is much larger than the third order susceptibility. Second 

harmonic generation from a pump beam initially in a squeezed state has been diicussed2n3 as 

well as SF/DF generation and degenerate parametric down conversion of quadrature squeezed 

light?** In this paper we shall consider SF generation from input fields initially in the photon 

number squeezed or sub-Poissonian state. 

2 Sum-frequency generation 

As shown in Fig. 1 we take both input fields, of frequency wl and w2, and the sum-frequency 

ws field (w3 = wz + wl) to be in a collinear geometry. The input beams are incident on a 

nonlinear medium such as a crystal with an effective second order nonlinear coefficient x'. The 

field Hamiltonian for ideal SF generation ignoring losses is 

where al, a2 and a3 are the annihilation operators for the two input beams and the resultant 

SF beam, respectively. 
FIG. 1 Sum-frequency generation in a nonlinear crystal. 

We introduce the slowly varying operators 

where the subscript j = 1,2 or 3 refers to the two input or SF waves, respectively, as throughout 

our paper. For propagation along the z-axis at a velocity u in a lossless medium we make the 

conversion z = ut. Since w3 = wl + w2 and substituting x = X'/U we deduce the equations of 

motion for the slowly varying operators from Eq. (1) to be 

In the short path approximation for xz c;:= 1 we assume the solution of Eq. (3) to take the form 

where cj(0) denotes the incident fundamental field at t = 0. It may be seen that cj(z) still 

satisfies the commutation relation 

[c~(z),c:(z)] = 4,. (5) 

Substituting into Eq. (3) we obtain the solutions for cl (z), c2 (z) and c3(z). 

To discuss the photon number fluctuations of the quantized fields we consider the variance 

(An: (2)) or the Fano factor 

aj = (An; (2)) 

(ni (2)) 

where nj(z) = ci(z)cj(z), and (An:(z)) E (n: (2)) - (nj(z))l. To obtain the above expressions 

we t3 3 need to solve for the factors tick, ef2c: and ck ck, where ck = ck(0) and k = 1 or 2, then find 

their expectation values. If the input fields are in a photon number squeezed state we adopt the 

formalism given by Kitagawa and Yamamoto5, that is, 

Here nk = alak is the number operator and ~k, r)k are constants chosen to maximise the squeezing 

for a given input field intensity proportional to IakJ2. Using Eqs. (7) and after some tedious 

calculations we may obtain the expectation values (c:ck) and higher terms with respect to the 
eigenstates of ak, noting that cs(0) is a vacuum field. The Fano factors may thus be computed 

numerically for explicit values of ak, 7k and qk. It is evident that the expressions for the Fano 

factors of the two input beams are identical if the subscripts k = 1 and 2 are interchanged. 

Another quantity that reflects the quantum fluctuations of the light fields is the quantum 

correlation between the sum of the input photon numbers, nl(z) + n2(z), and the SF photon 

number ns (z) . That is, if we define (n(z)) = (nl (2) + n2 (z) + ns (z)) , then 

and we define the Fano factor for the total sum of the photon numbers as 

The Fano factors for different pairs of the two input beam intensities are plotted against the 

effective interaction length xz in Figs. 2 to 4. It can be seen that as becomes less than 1 in 

all three cases, that is, the SF produced becomes sub-Poissonian. Moeover, as the interaction 

length increases the photon number fluctuations are reduced even further but at the expense of 

the input fields. 

FIG. 2 Fano factors of the input and SF beams versus effective interaction length 

for sub-Poissonian inputs of equal intensity. Subscripts 1,2 and 3 denote the two 

inputs and the SF beam, respectively. a is the Fano factor for the total photon 

number of the three beams. 

As shown in Fig. 2, when the two input beams are of the same intensity, that is, lal l2 = la2 l2 

and assuming that their initial statistical properties are identical, then a1 and a* are always 

the same, as may be expected. However, when one frequency e.g. w2 has a greater number of 

photo& than the other, its Fano factor is less affected (see Fig. 3) since there is a greater number 

of unconverted photons left. The Fano factor a for the sum of all three intensities also increases 

with interaction length, but it is always better i.e. less than that for the individual beams. This 

can be understood from the correlation between the three beams, since the creation of one SF 

photon is always associated with the annihilation of two input photons, one from each input 

beam. 
FIG. 3 Fano factors of the input and SF beams versus effective interaction length 

for sub-Poissonian inputs of different intensities. Subscripts 1,2 and 3 denote the 

two inputs and the SF beam, respectively. a is the Fano factor for the total photon 

number of the three beams. 

In Fig. 4 we have taken one input beam w2 to be sub-Poissonian whilst the other wl is in a 

coherent state, with both having the same intensity. It can be seen'that the curves of 02 and 03 

behave similarly to those in the previous figure, but al remains constant at 1. This implies that 

the coherent input beam remains coherent throughout the interaction, with only the originally 

squeezed beam suffering increased quantum fluctuations. The physical reason for this is yet to 

be fully understood. 

- 2=1.~~6 coherent I E: 1 2-l. 0E6 squeezed 

- /' 

i 

- 

- 

I 

a1 :=i.0~6 coherent - =1.OE6 squeezed O1 

-- 

- 

- 

I I I I t I 

FIG. 4 Fano factors of the input and SF beams versus effective interaction length 

for inputs of equal intensities but different statistical nature. Subscripts 1,2 and 3 

denote the two inputs and the DF beam, with wl in a coherent state and wz in a 

sub-Poissonian state. a is the Fano factor for the total photon number of the three 

beams. 
We have investigated the photon number fluctuations in SF generation with sub-Poissonian light 

as input and small effective interaction length. We find that the generated SF beam becomes subPoissonian and the quantum fluctuations decrease as the effective interaction length increases, 

but at the expense of the two input beams which become noisier. The Fano factor of the sum 

of the intensities of the three beams increases also with interaction length, but it will always 

be less than that of the other beams. This can be understood on the grounds of the quantum 

correlation between the input and SF fields. When one of the input beams is coherent and the 

other sub-Poissonian, the coherent beam remains coherent throughout the interaction, whilst 

the other becomes noisier. The generated SF beam, however, still becomes sub-Poissonian. 

4 Acknowledgments 

This work was supported by the National Natural Science Foundation of China. 

References 

[I] See the feature issues J. Opt. Soc. Am. B4 (lo), (1987); Appl. Phys. B55 (3), (1992). 

[2] A.V. Belinskii and A.S.Chirkin, Opt. Spectrosc. (USSR) 88, 695, (1989). 

[3] Wu Ling-An, Zhang Weidong, Wu Meijuan and Li Shiqun, Proc. Shanghai Int. Symp. Quantum Optics, SPIE 1726, p.151 (Shanghai 1992); Wu Meijuan, Li Shiqun and Wu Ling-An, 

ibid p.92. 

[4] Li Shiqun and Wu Ling-An, Chinese Phys. Lett. 8,222, (1991). 

[5] M. Kitagawa and Y. Uamarnoto, Phys. Rev. A34, 3974, (1986). 
SPECTRAL, NOISE ANWCORKtSLATIOEJ FIKZPERT lR:i OF j.N'ITXNSH SqEEZKi:i 

LIGHT GENEMTED B'f A COUPLING Ifi3 LASER Ff%i,DS 

Institute fur Pf2ys.ic-aZ Resear-cf2, Nt.~t.io>~t~Z ..~lctrdemy of Sc.iet?c:.i-~i-~s 

Ashtarak- 2, 37i3Li.IO9 Repub1 it: t> f Ar.}>?enitr 

Two scheme= af foa.r-wave mixing olcci.f.l~.t~or~: witkt ~tc~i-i<.J.~~gc.rirr:i.-utc$ 

Fmmps are proposed for above-t,Prr.e~Pio id geri~:rist5.urt of ~qut::ezciQ .t i phi, 

with nu-nzero mneiwi-f-ield acpl-it,~xdf:~- Hoi.~e arid co-r-~r-,j.atiof~. yrycjy;~t,i~%s 

arnd opt-ical spect~a of squf=ezed-light; ems cjcnerated .L:ri " t, TLG ----. b~.: 

schenies are diseuesed - 

1, Introduction 

The squeezed li&t generated to date has been in the main either 

squeezed vacuum or squeezed light with an extremely mall mean field 

amplitude, Therefore there is much current intierest; in searct.rirtg cjf 

new schemes to generate squeezed light with a large coherent compo 

nent of the field, 

In this report we would like to present nome nonlinear optical 

schemes for generation of intense squeezed fight with nanzeru mean 

amplitude- 3%; t type of coherent squeezed--state i is called 

sometimes bright squeezed lightOne of the important schemes for the generation a squeezed 

light, realized experimentally, is based the process of nondegene 

rate four-w,i;?e mixing (FWF3) in a cavity- In this process an intense 

pap field wiLh frequency (*\D {for certainty) interacts via a noalinear. medium with two .rriu&es af radiated field with re w 1 ? 

w such that 2Lt>*-k"* +dl2 

2' 

In contrast with this stm~dapd scl~eme af WM, we propose f,u eonsider the process of FWF3 under the pumping by kwo laser fields of 

different frequencies cai, (2.2 - AS a result of ccriuplirrg in rronlinear 
medium, i% pai~ uf ptiot~XiE3 of ~WU pus~i~1 f LefeZs I-(: .tri%~i~f c.,Z%i %c> ;i IT 2 

pair of ptrotc>rks of spon.tar~eous1y ge~iieratcd signs L f i.e I c.3 with dt>gz~zic: 

rate freq-ilericy ia such tPiak, ct: -ti9 -k-%b:, _ 0' i 2 0 

We st;udg twa dif f ere-rrt I'rju~-w;~ve mi-x-ing cot-if' igi.xrilr.;it; icir-i ~5 . 'f'hi-t f i. r-<-• . i * 'l 

of them ( aee Pig - 1 ) corisis-bs csf a ncir-il inear medi iem irt a ?:in$< i:ijv i t, Y , 

which cou-zslef; two ~10rioet-~.ro1yiat ic capruyagat, ing yi'iinJT:+ beams u f iA .i f ! '' . P;. --. -' 

pen% colors ifivi~x3encies LC, 1.0 1 with c+n ini;x%ai:avii;y cX@~n! nladc nt. 1- 2 

the tialf -=urn freque.rk~y LI;D= (~i) .j+3 f/Z - We ~~r~~i.dfjy fj- , .ie . i.: - S-LJF; t---- 0 i- f:; p:3 f.i l, /i 12 

news excitation of a sing1 e c;.avi.i;y.--r.esonant mode - So we i~3~<+ 

dealing with a nearly cnl linear wave-vectar matching crmiii.t.ie>n 

Fig-$- Scheme of the double-cnXur-pumped FWM crerri lfaktrr wi tt~ 

single cavity-mode excitaliion, 

In the first part ojF our report (nee Seczlon 2) we sJ1;111 pr'esm~t 

the results, related to the configuration af Fig-1- At; the begirlning 

of this part we wuuld like to describe brief f y same af the Be lawthreshold ~esults [1,23- 

2-1, Squeezing af the central line of the resonance f lucir>escunce 

in a bichromatic field in a cavity 

We consider an ensr?mbIe of two-level atanln S.ntcrni:t;ing Sn at1 up-- 

tical cavity with a biclrromatir: pump field and with a cavity mode of 

radiation field- This pump field is treated classically and chosexi 
We have calculated the cavity-output, intensity ia the process of 

resonance fl~xorescence f 13 - At this we do not require the execution 

of any phase-matching condikions, TIfe result for the inelastic part 

of the intensity is showx~ in Fig-2 as a function of the detuning of 

the rsavit:y Pesrsnarscrs f reqlienc y o:; f ram %he atomic trmsi t ion f re-- 

~)k~er~ey 0.3 . The curve is plotted for particula~ values of the pwp 5% 

i.n tt.:nsit y parametex- % ='W Jh ard yara~tete~e. Tf6, dh f br ie. ttt~: mat~tx 

e lcrrien t of dim 1.e %ra:nsituri , c? .'E the at;ornlc ab~%u~yt iwfs cuef -- 

cw, - 0,)/r 

Fi.n.2. Cavity output intensity vergus (wc-wo )/I. : < =5, b/y=6, 

We see that the intensity consist of a series &f peaks with a 

eaxif;i;ant spacing 6, They are symmetrically located about a central 

ye&, coFric:idi_ng with the atfrwic trmaitfori frequency wO - intensity spectruai was experimentally studied by Y-Zhu et al, f31 for 
two-level- like Ba atow driven g7y two strong, equa.3-~inp.ti.t;ude fields 

with frequency separation 2E.- Note that the results of our calculations are 2x1 agreement with the f:xperi~teri%ajt r:uUFvesLet us tm riow to the results , re lateb to %tie qua-r~-t,urci f lust~~ations in -the process uf FWH- We consider a ~ei-~.era'tir~ri of the ~i-ocif: 

with frequency equal to the resonarice ffuuresceni~e ia line 

w =((.I:$ w2 )/2 The calculation of the quadrature-phase f l~xctuatioiis 0 1 

show that this wO-msde is excited in a squeezed state, The opt;j.ma'l 

value of the squeezing spectfum S f LC' 1 is realized a% zero f~equerfcy 

and the maximal sqaeezirm may reach 35%- The deperide~l~ of t,he q-uwi--- 

tity S Iu=O) on the pump -Lnt,epit;y ya~m-tier P -zWfij is pfot,ted 3.n mLr7 

Fig-3 t23, 

Pig-3- Peak squeezing Srnlr,(0) verms 5 fur rpL"=O,f. - (a), 

rl&=O -0% - fb) - The squeezing -i;~ absent far: val~es of f . for which the yrubabilitles af one and twin ccao -&sotun 

radiation prmesses cancell each utherr, 

In order to interpret this resulk, note that tkir: exaitatiwn WE 

the ~c,~-mode in a cavity is caused by a nonlinear spcmt;anecl.l~s process 

of two-photon radiation by an atom in a bichrmatic field, f n a :Low 

order of interaction it may be represented by the fallowing grxpkis 

(see Fig-43, f t is essential thae there is a stmng pair corpelatiun 

between tiie photons 0% freqaeiscy cck, This eorrekatiori has a super-- 

bunching behavior a-nd manifestis itself iri tkr: reduction uf quad~;lture fluctuations belaw the shut-noise ie-vel- 
f?ig _ 4, 1l~u.stsatiosrr of %Pie 'procese 0% two-phmtorr ab~orwtion at 

frequencies G) -13 56 of the pump field^ with the 1,2 0 

emiosion of two photons at f requer~ciy wo= (w,u2 )/?. 

2-2. Above thr.e~tiu.ld rc~ults iri a parmetric model of EWf 

irr t'tre presence of pkrase mucfulation 

Now we consider a simple parametric model of four-wave mixing 

-under tichroma-t.i.c, pumping irr order to obtain the ~queezkrrg re~ultc 

above -the generation thre~kiold, Irr our. analys3.s 'we irrclucle -the ef - 

f ects of self -phac~, ~~:mdula.tiun and cross-yt.l..izae mudulatiunWe d.essrihe the nonlinear medium y~re~~~urnemlogically by the third 

(3) order susceptibility ,y - So we start from the following Hamiltonial 

where a+,u are creation ond annihilatiorl operatorrr of the intracavity mode, c+ is the cravi2;y resonant fmq3rency- e second term in C 

Eq. (2 ) describes; 'tkte f a~x~..-vave interaction with cuuplli-m! cufi~tar~t; x , 

p~oyortionnl to xt3' - 'The third and forth term deg~~ibe the selfphase modulatiorl arid cruris-phase modulation, The fifth term accounts 

far the coupling of the cavity mode with reservoir, which will give 

rise to -the cavity dan~pkrig ~on~~axl; y - The q'uaritities E, , E2 a~s the 
cortiplex amplitudes of the pmiy fields st, f.r=ti,qxis.fici.~<:< c+ tcr; - At, iic: 9 ' ;? 

arly coliirrear phase matching cunditicrjri tf-iesc .r;r3~~ fi~f.ds gentirate 

an in.t+.ac:avi.ty signal mde with f requf:rtc:y 1t!, -- ti.+ 4.i~~ 'j/2 - 1-51 th if; c:iiri-. 

0 i 

f ikwrat ion the p~~~ip fie l& travel ti-iru-acl,')ri -Lfte niedj:mi or.i 1.y c~ai':.e . 3 J 

i-t, is possible tu neglect %Ye pmip depsletiiuri ard treat 21%, E 2 ~3.z 

f i xed constants - 

With use of standard methods, we obtain the foI.lowing differc?r~ti - 

af. equations for the stuctta~tij-c a~ipll.T.xibe of the slgcie i awdc ;.L, Is :: 

where -43 is ttxe csxv-ity betwriir~g paraniet.er - "iltie noit:= t;~-r+rn K 0 c 0 

has the f o l. luwirig cox-re latur 

Nate that witfiout the phase-.-mdu la% .icm terat~ eq-uat i orrs 13f P~LC'~ ti i ~.i FI 

would be the same as for- the degerierste paraate-dp-Lc osci .f.l;i-txir aid 

degenerate f our-wave ~ii.xing be lm threshul.d - 'The nuve ii f ~+I~,IZP~!:E mid 

results iri our ~y stem are co-naected. .with the i.ncr_lr.pcjrxL -j un -t $;tic: 

self -phase mdultiticm term- ?%is temi rm2.%~- 3.n %ire atsxw----t. , , k sect .. -.; ticj t t! 

generatiori of %he signal. f ieid- kt, t~s curiaider the s-L~ib1.e sLeady.--. 

state solution fur the out.put, intensiky , It is eq-uaf to 

In Fig, 5 we plot the nomnaliaed output interisitg an a fact l~r* r~f' 

the parameter s2=S2~/p for Lhe case of equal ampli.tuber; of t;krc_t jXiEip 

fields iE51=lf2/=E. 

T'he zero intensity solution is stable below tbe gencsat.isn thrc-- 

2 2 2 2 shld at E <s andt well above.-tkme~Pioid at, c k-, , whex-c A 

The r.~urizeru-ir~tencity sohxbiun given by Eq, f 5 ) -is stat lr. 5x3. the x-e 

2 2 giun 3<s <& and ha~e a mari2-n~ fur B less *,,han yl/z .. We see .t;hn.t; 5 

B 

2 2 bistable khaviour of the output if; realized in %tie rr-.,g<.r~~~ 3.e -5:~ A 

Now let us turn out to the pr~blern of sqa.eezl.rrg. The S;F~:C%~WI 0f 

tfie uutpat field akve tbesf-ruld is ol%aiwied in a ~tar~dr'~.rd line-.- 

arized treatment a£ q-uantmi f luct-aa%iuris <=i;ix.we:.; fox-- sqxreezi~r~ spec - 

tra S[~c).f V~TE;US aid the spectrami value S(c3 ) a-f; t.f.1~ ~Tnts of c-pi. 
opti~lal frequency i~~=i.i~~~ YRPSUS the prmp field irltsrlri i ty pi\~'a~i-i<?i;i?~. 

cZ~"xf;~, (E=lE I =[E ) ) are plotted in Pig.5--(a),(tl). 1 2 

Fig. 5, Cutput intensiey uf khc EWM usci l latex. VC?T':;~Z:S tali<? i~it.~~l 

5;it.y of the yui.xi fieldti:: Af'j rS, 

L 

Fig - 6 - f a j -- Squeezrirg apJct~-wn -JeyEilfi c,~2h/ f cjp A J;. - --r- . > .+: :: 1. . 8 

2 (dotked line? ar14 -E =4.8 Csililid line); (b) dependeslcc isf 

2 -the quaat ity S f~.i ) on. E -far A ,Q, ::s - 

opt 

It slxx.~ld be noted that %he expex*iruental mcnourc?mczrt. of i.1u.n rril i:;c 

on tlie qudrature component in a similar €%@I rx>rrfigisratibr~ has 2~ccttt 

carpied out by D-Grandclemeni; et a1 [45 - They did rlczL efjnd kha sqic 

ezed noise reduction- This is not so surpr-icing because, as fo I l own 

from our analysis, squeezing is realizi?d for properly cl~osr?n ~alut?:i 

2 of parameters AJy and K - 
3 - Intracavity gmrarcetr-ic EWM with noricfegerre-pa LC: yLiEiyG. 

Abave--t.iireshuld results on bright ~qtreezirig of .tkii? tfiree 

intracavity mc~rks 

This part of our r*ewr.t is devoted tcj ttir: resuttrs oft. i:~r:i.gtit, sq:ii:- 

ezing in the configur-ation uf WM. wfie-re the two Lasex.- d.r.i.vj.rrg 

f ielde pro-pagate in tfie direction of the cavity axis {see Fig-7 3 - 

These driving fields feed two intracwity puaip ~f~t3de:r; aii the frcq-i~c5:l.- 

cies o:~ to , The pump inodes generate a signal mode at. their ha:! f.--:sum 

iZ 2 

f x>equsr~~y LO -(w ftzt )/2 and the wave-vccf;ctr matching ci>t~d i ti un i-~.r~? 0 iz 

exscutc?d exactly (%*-.+k2=-2%). So the configuration of I33JM with tvei: 

intracav'rt;y resunant ~i~de,c of f r.eq-uei.cle~ CLi it? 3rd I: :i?t::.t i L~zc,cL.~. 2 I:> 

In this cunfigurat.3.on the effects of ~ixxtual inf Lac:-ice ~:t: --Lire p-t.;rrrlp 

and signal modes are esse:.ti-al, So we take 2rrt.o zic:eutll'ri.t, the pcimp 

depletion- The cunsideratiun is simpli f iecS tiowever. irk theit we ignore 

the phase mdmlati.on termsNote that tfre advantages of Chis scheme af BVM, as c~ru;~rcd I.o 

the standard nondegenerate FlM with a sir-tg~le pxmp mc>de, are caused 

by the following, As shown below the effect uf phase di.f.f-uni.o:n is 

absent here- As a result, the output field for each of t.he flki.t+ee 
modes have nonzero mean lltudes with a definite phases, 

Thus we start from the folluwing Hamiltonian 

2 

-% H = ), ii~.a.a. + it) 4-2 

JJ3 120 120 

As compared to the previous model, nuw we take into account the 

quantization of the zmm~ modes and incornrate: (i ) the coupling of 

the pump modes with two external coherent driving fields of amplitudes El, E2 and ($5 ) the decay of the three cavity modes due to the 

coupling with reservoirs, 

With use of standard arethods, a FokkerPlanck equation in positive P-representation for. the system is found, from which stochastic 

differential equations for the complex field amplitudes are obtained 

Here yo, y=yl-=u, are the dampi- constants for the modes w, and %, 

o respectively, E is the amplitude of the driving fields 2 

E exi2 # ape aPbitrary phases of the drivine 

1,2 

fields, R. are Gaussian noise terms wlth the fof lowing nonzero corJ 

relations 

&,(t )Ro(ig )> = ka a 6(t-t' ). CRl(t)R2(tg )> = - $ <6ft-t* 3- 

12 (93 

In order to analyze the puantum fluctuations of the modes we apply a linearized treatment of fluctuations about the stable steadystate solutions- It is wurth noting that, as pppoeed to the standard 

scheme of PWM, for the present system there exist three types of 

stable steady-state mlutione, They correspond to the three possible 

regimes of oscillation: one belcrw the generation threshold E<E, and 

two different above-threshold reg-s at Ei <E <ZEt and E .<ZEt, The 

threshold value of the amplitude E is EL= y (ydx lLR - 
The n-esults for the cavity-output int~xisit,ies &""' girt phrjf,cisi 

J 

number units per unit t%~ie) for the anodes ~3) in %tie abuve- t,fir.e::f ii: J d 

regime are f u l lowi rrg - Fxi the regiun ]t-:.5 1;2 {z rE/E j we iii*~t< i,: t: , 6 i : L 

The skeady-state phases y) of all the Lhree modes a.= i.fitf irleri 

J 

above threshold ancf equal tu 

Y;=+* 7 ~'~=4~ ' "*=t@i*2)/2 {:ll: 

So, they a.re determined by the phases of tPte dri.-gi.-r~g fields 4,. I +*. We calculate the qua.c;frat;ur-e Eluctwt iuwt va~j-avrce for n.11 i,Re 

three irntraca.vity mucks arid currespunctirr6 ecyuaezing spectra for t,he 

cavity-outpt fields, We would like to pointti out at mice *.,hat r7.n 

effective squeezing occurs for. each of kiie tPtr.ee ~fiodes iltmve thr-e 

shold E5,61, 'l'his is an ext~emely intereeing f ea-ture of t;he buublr-,- 

color-pumped IWM oscillatorThe maximal squeezing is realized for the phase of the locsn.i 

oscillakor equal to -3 =ay and is determined by the f luet.uation:; of 

3 32 

phase variables 

s (w) = 1 r- 8yn 6~) (--t.~~)6~1 (~3)> , (32) 3 J J J J 

where n is the intracavity steady-state phatun number cxf the 

J 

w . -mode , 3 

Examples of the tswves of the squeezfng spectrun~ for the cigriaJ 

mode are plotted in Fig-8 -~WP dlf f erent values c~f ya,rxie%en.e .:' arid 

r- yo/y - 

Our analysis show that a noise bucki ion 'below tke shot;--noise I.uvel may reach approximately 100% in the whole abcwe--threshrs3_d rcgicxr 

e>f. and for r-2 10.. The higher yufi the better the squcea+rig- ifuw~-i?er, 

the intensity of this field 5s limited by Ckie velue Jv =* ~,/.u. G:fi~2 

0 O 

ration of more inten- light in the squeezed state ucc-ura ;l't LFje 
p~lmp f i t3.I €1. f req-acnc-:ies - Tfre curx-csparidix-ig a-utgu% irite=--=: L~D A. + td 3-e~i ' WFO-+Z 

.w i.Lh i ncrr:a se uf the i.m: ide.nt fie lds , Himever the ma;*=%~gr l squesz i'rrg 

rtrzry r,a;icrlk approxjxii;it,f: iy 5G% f OF certain vaiue~ sf the fati3 2 .o/?, . 

-, . I! 1-g. 8 - Sc~uee~ixw. S~~C~PUTT~ S o (~3 ) V~Y.SUS ~3fp o : E= 1-1, I-- --2 -- ( sul id 

line); --4, rL=lf) - (dot;Cecll. line), 

We resC~icl; ouu~ie2ves by representation of the squeezing spclrum 

S (LL- ) a% the pui.n%~ of miriirria ~tl=c3 Tkie depzridence of thP~ 

L,2 crpi c.pi 

quiwrtity on the yzrame.tr,~ r.. kf; plotted TFrr Plg.9- 

'l%txs we find tha% the duuble-color-pumped FWH osciXlatars is 

extreme 1 y pi-om1 sing for %he above- thx-eshuld generation of onc-mode 

briglrt squeezed light- 
Now we sfml.X present the results or1 %,he sub G~ITS~--XIQP.F;E c0e.r-aLa%i --. 

ons in the above-threstiufd ~egime- We coriei&e.r the COFTC~E~-~,~C~TLE tie--

tween both the intensities arid the phases c~f the i:r~.te.pr.ac t, kt~g m~fitider;;. 

In the well-known processes of riundegenerate L;WM arid -paraie2tx j.c 

dm-conversion, the pfrotoris of two gerierated ziade8 ape CZ-eated Iri 

pairs arid a positive correlation <Sn 6t-r >Xi betweeri. %he pkicjtwi .riuni 

12 

ber f fuctuations of the~e ~~~des occurs- 2-L result,s in the x*=d.u~ti~zl 

of qzlantwn f luctuatio'ns below the shat-riu5.se level. ixi ttic: inLeasiby I 

diff erence of the modes, This yhencmena has heen ~Lf~erved far. tii~ 

first the in 2-ntraczavity -riundf;?generate parmett.i.t. osci 1.. 1.n Lc-r. t, y 

A-Heidmann et al C71- 

1x1 our nonfinea~ system we have f'uund another manifest;ai;jurr c>f-' 

such an effect, It consists of the raeductiun of qt.xant;um n~i~i~ in i;-lic-t 

inkensity swn of the pump modes E87 - The cxp2anaticx1 ui this phi?^^<> 

mena is following, The photoz-rs of two pump mtlldes are axu;~i.hilat;ed in 

pairs and the pump modes acquire correlated skatisticaf yro~c?rtriez, 

which are characteristic for two-phu%on &sox-ptiarr- A3 a result;, tik~t? 

correlation between initially uncorrelatcd coherent pump f ri.i:lltiu be --. 

comes negative da.r16n2><0, And this fact rcesults .in .the sub s53ot-- 

noise fluctuations in the intensity sum of the cavity tx&.p~rt beamsWe shall not dwell 013 the particular quantitative resu3ts- Note (3111.3r 

that the maximal noise -duetion may reach appraximat;ely iUO% irli Lhe 

limit E-42, &en the pump depletion is maximal, 

A more interesting featu~ of our FWH configuralitm is ~antrei.:~ed 

with the correlations between the phases uf t,hc pump maden- T%iay are 

studied in terms of the q.uadratum phase operatars, as applied to 

the twin hamsdying experimentaJ measurementsIn general the variance of the stxu or difference 0.f the q~i+dr;lit;~r-- 

re component operator 

contains the coritribut ions froxi bath the ixitensiky and pkiase var i :J.- 

ble fluctuations of the mudes, 

In our system we can select; properly the phases a£ &he locia1 on.- 
: ; 1 tat~jrs to get %he vapixzice as expressed "in temm of the phase 

f-l~rc.:t;ust-ionf; ortLy, The T-esutt, cwi witteri as fulfows 

Ni-jt,~?, ti~aC slick a prsssibility do no% exist; in parrnetpic praeesses 

w i t,Et phase di ffusicrn ef feet. The nonclassical correlations between 

Ltii: phase f Itrct-r~atit~ris ape ztm~ifezted iri t;Pe va~imce t' (2 ) 

12 

ctnc3 so w2 obtain a reduction sf quantum f fuctuations below the c11otnc~i r;e t eve l 'I_n the dlf rererice <c,f the quad~a.t.ure ykiaaes: I/',, <Z , 

A simpLr, axial-ytica2. r.r,fn;tf% is obta-i+ed aim a tihe ~iie~~-iz.retf 

. cavf ty - oLtZ;put fields, Tkie cu-~respuwiding spectrum of the quad~ature 

phase dif f e~eriee f luctuat-ions in the regicsri I&- <2 is f ulhwi~ig 

We see Lho.t. the rioise reduetion up to 100% i~ yu~sitfe Ln the lZmit 

r-+2 at zeru frequency, In the region s32 the 8fiape uf the s~ctr.iwr 

is ccrrqplicated arid the noise level is i~icreaeedFinal Ly .we preserit come rewlts concernirjg the op%ical spectra of 

t,he cavri.tiy-ox~tyut squeezed light, We consider the intensity spectra 

of each of tb: tt~ree nuwlasslcal light Plrt;tms aromd ttie f~eg-~ericies 

to (.i-3, I,Z), IB~ese spectra contain a delta-function peak correswnJ 

ding t,o cc3heren.f; part of radiat;ion and a bruadened noxicukerent part- 

'f%e broadened parts of spectra are caused by the quant;~ fluctuations af &he field, In the lowes% order in quantm fluctuations they 

contain kwo conkrib~itions, arising from the tempuraf correlations of 

~rstgl f ttr 16~~ j t f 3. Depending o~ %he contribution strength, a two- or 

3 J 

four--paked s%r~uctux+e uf noncohex-err% part of spectra arise in the 

case nf oseill at ins c11arac.i;er of these correlations - 

At1 excm$ple of f our-peaked spectrum for the pump TiePd is represented in Fig-IO- Here the one pair of %he peaks is caused by the 
phase flur:tuat,,ions arid the otk~er one - kg t,hc Lrrt,c:rieit-y ~~ii:~:Li.j~iLions- Ttie fact, that is interesting here, 3-R t,#~~f s(rjp;ir;it;iiiti irt f'r*i:qtt 

eney of 4,heee two corktriht,icsris- Su it sf:esis Lo P3e pa~sibi~+ :,I) iiiierthe information about the pkiase f ltrct~xatioti:~ f rsom t,i.i+: ucr~~i i CJF~, j 4 ::A i 

spctm - 

Fig - f 0, 122x3 r-iorrcoheront part aT tkte kr?iLenr.; i ty spec tnxm c;i tJre 

pmp made .5-=2,2, t-=O.Q!j. 

1 - G, Yu KTycachkyan, ISuar~tm f3ptics 3, 209 ( 199 Z ) 

2, G, Yu K3cy-uchkyan axad K, V, Kf~erwfitsyari, CUyT, _ G~rirm- 87, 3-81 ( Li3fr32 f 

3, Y-Zhu, B-M, A,lr;tzan~~, D-J-Gauthier, ad T-W,Hossterg, Pf-iyc-Xit=vA4 1 , 6574 ( X990 ) 

4- Lr,Grandrslement, H-Pinard, and f;-Grynberg, XEEE J ..&ziarrt-E: 1uct.r.orl. 

25, 580 ( 1989 ) , 

5, G-Yu Xrgruclikyarz and #,V,Kl~eruntsyan, Q~lar~tuai Optics -2. 28$1 1 191f2) 

6, G-Yu Kryrchkyan and K-V-Eheruntayar), JETP 76, $3 (39933 

7, A-Heihanxl, H,J,Hamwics, S-Heynaud, E-Giacc>binc7, C-It'nbrke, aiiil 

G-Camy, Phys-Rev-Lett- 59, 2555 (1987'1 

8, G-Yu Kryxclikyan and fS-V,~eruntsyan~ Op%,Cumm, 93, 328 fTEI92I 
MULTIMODE SQUEEZING, BPPHBTONS AND UNCERTAINTY 

RELATIONS IN POLARIZATION QUANTUM OPTICS 

V. P. Karassiov 

Lebedev Physical Institute, 

Leninsky Prospect 53, Moscow 11 7924, Russia 

Abstract 

The concept of squeezing and uncertainty relations are discussed for multimode quantum 

light with the consideration of polarization. Using the polarization gauge SU(2) invariance 

of free electromagnetic fields, we separate the polarization and biphoton degrees of freedom 

from other ones, and consider uncertainty relations characterizing polarization and biphoton 

observables. As a consequence, we obtain a new classification of states of unpolarized (and 

partially polarized) light within quantum optics. We also discuss briefly some interrelations of 

our analysis with experiments connected with solving some fundamental problems of physics. 

'J1 Introduction 

Polarization properties of light were widely investigated long ago when examining some fundamental problems of quantum mechanics including "hidden" variable theories, Bell's inequalities 

and Einstein-Podolsky-Rosen (EPR) paradox, different topological phases etc. (see, e.g., [I-81 and 

references therein). Herewith, as a rule, the polarization structure of light has been described 

in terms of the field correlation functions, associated Stokes parameters and the Poincare sphere 

which are well adapted to classical optics experiments [7,9] but are not quite adequate to specific quantum ones (photon counting)[3]. Such a description also ignores a polarization SU(2) 

symmetry[lO-121 of light fields though it has been widely used implicitly - through the Stokes 

2 1/2 parameters sd which determine, in particular, the polarization degre degP = [s: + si + s,] /so 

of monochromatic plane wave light beams[1,3,9,313]. 

But recently a new formalism[lO-121 was proposed for a description of polarization structure 

of multimode quantum light fields using the polarization SU(2) symmetry and a related concept 

of the P-quasispin which generalizes the Stokes vector notion at the quantum level and is closely 

related to the Stokes operators defined in [13]. This approach enabled us to gain a new insight 

into the polarization structure of light and quantum mechanisms of its depolarization[l2,13]. 

At the same time, so-called squeezed states of light are intensively examined now within quantum optics (see, e.g.,[10,14-171 and references therein) since these states have attractive properties 

of the "noise reduction" in measurements of some quantum mechanical observables. However, we 

note that squeezed states have been studied sufficiently well only for' the single-mode fields[15,14] 

whereas for multimode fields it is not the case since even the definition of the concept of multimode 

squeezing is not unique that is due to a lot of the choices of measurable quantities[l6,17]. 
The aim of this report is to give an analysis of the concept of squeezing of the multimode light 

related to polarization degrees of freedom by using the above mentioned formalism of P-quasispin. 

Specifically, we will show that there exist new quantum states of light beams exhibiting , in a sense, 

an absolute squeezing in polarization degrees of freedom. Such states are generated by specific 

unpolarized biphoton clusters and have all characteristics of usual unpolarized light, but unlike the 

latter, new quantum states of unpolarized light are "polarizationally noiseless" [lo-12,181. Besides 

we discuss briefly some generalizations and applications of new non-classical states of light to 

setting up new optical experiments related to some fundamental problems of physics. 

2 Polarization P-quasispin of electromagnetic fields and 

unpolarized biphotons 

In quantum optics the free transverse electromagnetic(em) field with "m" spatiotemporal modes 

is described by the vector potential[l,3,12,13] 

where a,(j)/a;f;(j) are destruction/creation operators for j-th spatiotemporal and a-th polarization modes of the field, &(j) are the polarization unit vectors adapted to the helicity basis, 

Z3(j) = &/wj, V is a quantization volume, etc. With the help of Eq. (2.1) one determines correlation tensors [3] 

G!::~!~;~~..~~~({F~, ta}; {G, t:}) = TT[~E~L)(F~ , tl)-..Ej8-)(Fs7 ts) E:;)(c9 t~)...~:~)(~, ti)], 

which correspond to different physical quantities, measurable in optical experiments, and are 

expressed in terms of quantum expectations of ordered polynomials in operators aa(j) and a:(j)[6]. 

We note that quantum expectations of any physical quantities are calculated by averaging on the 

space Lphys = LF(m) spanned by basis vectors 

which are generated by the creation operators a;f;(i) of photons with transverse (a = +, -) polarizations (helicities) only (that corresponds to a standard form of the gauge condition for transverse 

radiation fields in quantum electrodynamics[l2,13]). 

The most important of such measurable quantities is the field Hamiltonian 
which determines the time-evolution of other field observables[3]. But in polarization quantum 

optics there are specific observables which characterize proper polarization properties of light 

beams and correspond to to the group U(2) of a specific polarization gauge invariance of the 

Narniltonian (2.4)[10-121. This continious polarization group U(2) is closely related to discrete 4 4 

symmetries of em fields(rnirror reflection B : af (j) a$(j), 6 = -kj and the spatial inversion 

fi) since all these (chiral) symmetries act in a natural manner on a 2-dimensional "polarization 

spinor" {e',(i), a! = f) spaces[ll,l3]. 

The generators of the polarization group U(2) are of the form 

Pi = x a: ((i)ai (i) = x P* (i), N = x x a:(i)a,(i) = x N(i) 

where N is the total photon number operator and operators Pa are generators of the SU(2) 

subgroup defining the polarization (P) (quasi)spin [lo-121. The operators Pp and N satisfy com- 

' mutation relations 

[N,Pa]=O, [Po,P*]=fP*, [P+,P-]=~Po (2.6) 

and in the case m = 1 coincide up to the factor 1/2 with Stokes operators C, : C1 = 2P2, C2 = 

-2Po, C3 = -2 PI [13]. As is clear from Eqs (2.5) the total P-quasispin of the em field is obtained 

by adding of the appropriate quasispin quantities for single spatiotemporal modes. However, 

from the experimental viewpoint the total P-quasispin of the em field enable us to examine new 

interesting physical phenomena connected with correlations of different modes, in particular, with 

so-called "entangled states" which are widely discussed in multiparticle interferometry [2,5,19]. 

Note that the operators Pa do not commute with + components S, of the gauge non-invariant 

(and hence locally non-observable) ordinary spin S = (S*, S2, S3) of the em field wich define the 

field transformations with respect to the SO(3) c SL(2C) group of rotations in the usual 3- 

dimensional space and are expressed in terms of the i(i(i t) Fourier components Ai*)(j) as follows 

where cabc is the fully antisymmetric tensor ( €123 = 1). Specifically, from Eqs.(2.1),(2.7) one easily 

finds relations specifying "rotation" properties of different physical operators[l2]. For example, 

in the case of plane wave beams, when in (2.1) esa(j) = S3a, a = 1,2,3, e*3(j) = 0 (e,,(i) is the 

projection (directing cosine) of &(i) on the "a"-th axis of a fixed spatial frame of reference with 

the axe 0k3 being parallel to all zj) and S3 = 2Po, one finds a relation 

defining transformations of P-spin components under rotations around the light beam axis. 

From Eqs (2.5), (2.7) it follows that the 9-quasispin formalism has evident advantages in 

comparison with the ordinary spin for describing properly polarization properties of light since 

its components have a clear physical meaning and are measurable in quantum optics polarization 

experiments related to counting photons with definite polarizations[l2]. In particular, the total 
Figure 1: Scheme of the measurement of P-quasispin components 

helicity 2 Po of the field is the difference (N+ - N-) of the right- and left- handed photon numbers 

and Hermitian operators 2P1 = (P+ + P-) and 2P2 = i(P+ - P-) determine (cf.[9,6]) differences 

of photon numbers for two pairs of orthogonal linear polarizations which are connected with the 

helicity basis by the linear transformations[l2] 

1 1 

6) if (j) = -{a$ (j) + a$(j)), &t(j) = -{-a? (j) + at(j)) & 4 (2.9b) 

implemented, for example, with the help of phase plates and polarization rotators [9,6,7]. (From 

the formal viewpoint components PI and P2 correspond to the choice of different subgroups 

SO(2) c SU(2) unlike the helicity subgroup U(1) for Po. Moreover, basis wave functions with 

linear polarization defined by Eqs (2.9) are eigenstates of operators describing the abovementioned 

discrete symmetries 8, f' of light fields.) 

A typical principal scheme[l8] of the measurement of components P, of P-quasispin is presented on Fig. 1, where we use the following notations: PP denotes phase plates, PLS stands 

for polarization light beam splitters, PA, and PD, are, respectively, polarization analyzers and 

photodetectors for polarization modes "aN. We note that this scheme can be realized in both 

singlemode (m = 1) and multimode (m > 1) regimes. However, as it will be seen later, the use 

of multimode regimes enables us to reveal new interesting physical phenomena, in particular, an 

absolutely unpolarized quant um light [lo- 121. 

Since in the case of the monochromatic plane waves quantum expectations < P, > are proportional to the Stokes parameters s, =< C, >, a = 1,2,3; so =< N > [13], then in general cases 

one can consider that quantities < P, >, < N > determine the polarization degree degP of light 

beams with arbitrary wave fronts and frequencies by the relation 

2 1/2 degP = 2[ (< P, >) ] / < N > (2.10) 

a=0,1,2 

generalizing the appropriate definition for one-mode light beams[3]. At the same time the quantum 

averages < lP21 >= P(P + 1) of the SU(2),,[ Casimir operator P2 = (1/2)(P+ P- + P- P+) + Po2 
are connected by the relation 

with the variances a, =< I Pa > -(< lP, 1 >)2 determining "polarization noises" [3,12,18] and 

different uncertainty measures for operators P, (cf. [20-221). 

Therefore, one may use P-spin (Pa) as an adequate tool for studying proper polarization 

properties of quantum light fields in parallel to the usual apparatus of the correlation functions 

and Stokes vector s' = (sl/so,s2/so,s3/so) running on the Poincare sphere[3]. But unlike the 

latter, the use of the P-spin formalism allows us to gain a more deep insight into the inner nature 

of the polarization struture of light beams with arbitrary wave fronts. 

Indeed,as it was shown in [lo-121, one can decompose the Fock space LF(m) spanned by the 

vectors (2.3) into the direct sum 

LF(~) = C L(Pn) (2.12) 

P,r 

of infinite-dimensional subspaces L(Pn) which are specified by eigenvalues P, ?r of the P-spin and 

Po respectively and spanned by basis vectors IP?r; n, X > of the form 

where C a; = 2l?rl, C /Iij = (P - I?rl), -yij = n/2 - P. For example, in the cases rn = 1 and 

m = 2 we have the following expressions[23] 

a) ~PT >= [(P - ?r)!(P + ~)!]-'l~(a:(l))I~l+?(af(l))ld-~(~)~-~~II~ >, (2.14~) 

b)(P?r = f P; n,t >= [(n + l)!(n - 2P)!(P - t)!(P + t)!/(2P + I)!]-'I2 

(~f(1))~" (af (2))P-t(~&)n12-P10 >, 2t = n(1) - n(2) (2.14b) 

for some of such vectors. In general, the coefficients C( ...) in (2.13) are determined from the 

defining equations 

and some equations for fixing an extra (vector) label X (see [12,11] and references therein). Operators 

1 Y+ = -(a:(i)af(j) + af(i)a:(j)),x,$ = a:(i)af(j) - af(i)a:(j) 

t3 2 (2.16) 

in (2.13), (2.14) are the solutions of the operator equations 

and may be interpreted as creation operators of Po-scalar and P-scalar biphoton kinematic clusters, 

respectively. 
From Eqs(2.16), (2.17) one easily obtains that < Pa >= O,CY = 0,1,2, in states generated by 

actions on the vacuum vectors 10 > of operators (x$)'(~T)' only ( and spanned by vectors (2.13) 

with n = 0); these states are examples of entangled states of multiparticle interferometry[5,19]. In 

general, tie states (2.13) describe light beams representing a mixture of both usual (uncoupled) 

photons a:(j) and unpolarized P- and Po-scalar biphoton clusters Xi:, k;: 110-12). As it follows 

from (2.13), the total number operators Nph, NX , NY , respectively, of uncoupled photons and 

X-and Y-type biphotons are given as follows, 

We, however, note that biphotons 5; exist for any number "m" of spatiotemporalal modes whereas 

X.$ # 0 only for "mN 2 2. We also emphasize that in contrast to the usual photon operators 

a$(j), aa(j) the operators X, = (Xi:)+, Xi:, Yjj = (l$)+,xr satisfy not the canonical commutation relations but trilinear commutation relations for quanta of generalized parastatistical fields 

(these operators, however, can be transformed in some "particle-like" quanta ones)[ll]. 

Further, the decomposition (2.12) is invariant with respect to the Lie algebra so*(2m) generated 

by biphoton operators XG,X$ and commuting with the polarization invariance algebra su(2) = 

Span{P,}[12,11]. Therefore, states > belonging to a subspace L(Pn) with given P, n at 

initial time will be in it for the time evolution governed by the interaction Hamiltonians Hint = 

H: ({xi,, X$ }) what is similar to the situationin the theories with spontaneosly broken symmetry; 

examples of such Harniltonians are given by those of some parametric processes [lo-121. Extending 

the algerbra so* (2m) by adding operators Yjj, YY,: we get the algebra u(m, m) commuting with the 

polariztion subalgebra u(1) = Span{Po) c 42) and associated with interaction Hamiltonians 

Hi,,* = H,Lt({xj9 Y,;; Xij, X$)) (describing, for example, light propagation in Kerr media) which 

keep invadant for time evolution subspaces ~'(r ) = Cp>llKl L(PR) (with fixed r) [12]. If we restict 

ourselves by biphoton operators xj, k;: only we obtain7he subalgebra sp(2m9 R) c u(m, m). So, 

algebras so*(2m), sp(2m, R) and u(m, m) describe specific P-and Po-scalar degrees of freedom of 

light fields which are complementary, in a sense, to.polarization ones. 

3 Squeezing in polarization quantum optics. A new classificat ion of unpolarized light 

The decomposition (2.12) implies a new classification of the polarization states of quantum light 

fields from the physical viewpoint [11,12]. This classification is closely related to a specific sort of 

squeezing of multimode light beams with consideration of polarization. 

In fact, a definition of squeezing in quantum mechanics is based on an analysis of different 

uncertainty relations for expectations < I (A;)" I > of a set {A;, i = 1, . .. , T > 1) of non-commuting 

Hermitian operators Ai representing some quantum observables[l,l4-17,20-251. These relations 

are connected with specific measures of admissible quantum fluctuations ("noises") for joint measurements of all observables A; in a state I > which characterize differences betveen quantum 

observables and their classical analogs(< [Ail >) and are displayed with the help of different 

quasiprobability functions[3,14,25] and generalized coherent states[3,10,20-221. Specifically, the 

most widespread uncertainty relation (of the Heisenberg type) has the form[1,14,20-221 
where (AA)2 G BA =< I(A)21 > -(< JAl >)2 is a standard quadratic measure (variance) of a 

deviation of the quantum quantity A from its classical analog. Then the problem of squeezing 

consists in finding quantum states minimizing both the product AAiAAj of two "individual" 

uncertainty measures (the condition of a joint quasiclassical behaviour of A; and Aj) and one (say, 

84) of them (the condition of properly squeezing). 

If the right side of inequality (3.1) is a c-number this problem is easily solved and leads to 

a definition of the usual concept of squeezing related to generalized coherent states of the group 

SU(1,l) [14-171. For example, it is the case for single-mode em field when we use as observables Ai 

two quadrature components A1 r X1 = (a;(j) + a,(j))/& A2 I X2 = i(az(j) - a.(j))/fi(a, j 

are fixed) [14,15]. However, for multimode em fields the situation becomes more complicated 

since in this case we have a more vast set of observables which obey non-trivial commutation 

relations[l0,16,17]. Therefore, there arc: many possibilities of definition of squeezing related to 

a choice (from physical considerations) of some subsets of observables (and adequate joint uncertainty meausures for them) for which a solution of this problem is comparatively simple. As 

we established above, in polarization quantum optics it is natural to take as such subsets components P, of the P-quasispin obeying the commutation relations (2.6) of the su(2) algebra as well as 

. subsets of unpolarized biphoton operators (2.16) of X- and Y- types (related to the "biphoton algebras" SO*(~~Z), sp(2nz) R) and u(m, m)). That enables us to define a specific polarization squeezing 

which is closely related to a new physical phenomenon of biphoton unpolarized light(UL)[12]. 

Since operators Pa are similar to angular momentum operators J,, a = 1,2,3 obeying the s42) 

commutation relations, one can apply analysis[20,21,24] of uncertainty relations and an appropriate concept of squeezing for operators J, to analysis of those for operators Pa, a! = 1,2, 3(P3 = Po). 

As is known[20-221, the Heisenberg uncertainty relations (3.1) for A; = Ji (i = 1,2,3) are minimized 

on the SU(2) generalized coherent states[20] I(; fj >= exp(CJ+ - C* J-)I j; fj >, C = -f exp(+) 

where Ij; fj > is the highest (or lowest) vector of the SU(2) irreducible representation Dj. The 

states I<; f j > are maximally close to clas~ica~l ones[20] and minimize a SU(2)-invariant (cf. 

(2.11)) "radial" uncertainty nleausure Ci OJ, (xi a~, = nziiz = j on the states 15; f j >) which is 

an adequate characteristic of q~asicla~ssical behaviour of a whole set {J;)[20,21]. Besides, these 

states are used for a definition of polarization analogs Q(8,4; p)ltp = I < 5; f P(plC; f P > I2(p 

is a density matrix of a light bea,m) [23] of Q-functions of quasiprobability[3,14] which are well 

adapted for displaying squeezing properties of oscillator systems. Evidently, for physical systems 

with a fixed value of j (e.g., for usual spin systems) we obtain an "absolute" squeezing for {J;), 

characterized by relations 

C oJ, = 0, AJ, = o =< I J;I > Vi, a 

only for the unique vector li; 0 >= 10; 0 >. But for en1 fields the situation is quite different because 

of the decomposition (2.12) for LF (172). 

Specifically, as is seen from Eqs (2.13), (2.17), the states I >E L(O0) = Spaiz(100; 12, X >) satisfy 

Eqs (3.2a) and provide an "absolute" nlininlunl of both the aforementioned "radial" uncertainty 

measure xi ap, as well as uncertaint.y relations of the (3.1) type for operators Pi; besides these 

states form the infinite-dimensional space on which three non-commuting operators Pa behave 

themselves as c-numbers exhibiting an "absolute squeezing" and totally classical behaviour in 

polarization degrees of freedom (that it is of interest for designing different experiments related to 

the EPR-paradox and "hidden va.riable" t4heories[l ,2,5,1 I]). We note that in LF(nz) there exists 
another class of quantum states displaying a similar (though more weak, than (3.2a)) property of 

polarization squeezing. Namely, for states I >E L1(n = 0) we find from (2.12)-(2.14) 

As it follows from Eqs (3.2), states [PO; ... >E L(P0) C ~'(n = 0) and (00; ... >E L(O0) possess 

the characteristic property (< JP,j >= 0) of UL (cf.[3,9]). Besides, the calculations [ll] showed 

that < IS,I >= 0 for all cr and correlation tensors $")(r',~F',t) have for these states a form 

corresponding to UL beams with, in general, arbitrary wave fronts. But unlike classical (chaotic) 

UL, for the states (00; ... > and [PO; ... > we have additional characteristics of light depolarization 

which follow from Eqs (2.12)-(2.14), (2.17) and are expressed in terms of higher moments for Pa: 

showing the absence of appropriate polarization "noisesn (< 1 (Pa)" I > -(< I(.P,) I >)$. a = 0919 2 

for I >E L(O0) and cr = 0 for I >E Lf(n = 0)) of any order measured by appropriate noises of 

difference photocurrents in schemes of Fig. 1; herewith, as it follows from Eq. (2.8), for axial 

(plane wave) light beams results of measurements do not depend on rotations of analyzers around 

beam axis. 

So, for states ( >E E(O0) all proper polarization properties are identical with those for vacuum 

state 10 >, but unlike the latter the light intensity < lHf 1 > in these states(with the Hamiltonian 

Hf from Eq. (2.3)) is not equal to zero. Consequently, they may be recognized as states describing 

absolutely unpolarized light while the states I >E L'(O) have a hidden polarization structure 

revealed in measurements of linear polarization noises. Therefore, states 14 > E ~'(0) generated 

by biphotons yf, X: and >>E L(O0) c ~'(0) generated only by biphotons XG describe new 

types of UL due to strong quantum phase correlations rather than random mixing Pighe beam 

as it is the case for the classical UL [3,9]. Examples of such states are yielded by generalized 

coherent states of the above biphoton algebras (and appropriate groups) related with interaction 

Hamiltonians Hint = H:,, + H;, where Hk, = CiCj(gijXjj + g:,X:), HL, = Cij(fijKj + f~%:) 

describing some specific parametric processes[ll]. In particular, generalized coherent states of the 

SO"(2m) group orbit type 

discussed together with some related models in [lo-111 are generated by If:;',, whereas H:,, produces 

generalized coherent states of the group Sp(2m, R) c U(m, m) 

coinciding in the case m = 1 with two-mode squeezed states introduced in [15] and related to the 

SU(1,l) group [11,12]. In general cases states (3.4), (3.5) display some properties of specific multimode squeezing associated with biphoton algebras so*(2m), sp(2m, R), u(m, m) (cf.[lO-12,161). 

Therfore, operators Sx , Sy can be called as biphoton squeezing operators. Without dwelling here 
Figure 2: Scheme of production of biphoton unpolarized light 

on analysis of all their properties we note that operators Sx commute with the "proper9' polarization squeezing operators Sp(() = exp((P+ - (*P-) while it is not the case for the operators 

SY. 

Physical realizations of such states, connected with actions of Po- and P-scalar biphoton 

squeezed operators Sy({/?ij}) and Sx({xj}) on the vacuum vectors 10 >, are represented schematically on Fig. 2 where POG stands for parametric oscillator generators corresponding to the 

operators Sy, Sx and other notations are the same as on Fig 1. We note that, in practice, it is 

easier to realize such schemes corresponding to Eq. (3.5) rather than Eq. (3.4) because the latter 

require parametric oscillator crystals with highly anisotropic properties. Therefore, for production 

of P-scalar light it is preferable to combine more simple schemes of production of Po-scalar light 

together with some interferometric schemes[5,18,19]. 

Thus, our analysis displays inner mechanisms of the light depolarization at the quantum level 

by contrast to the generally accepted viewpoint [9] that randomization is the only way of obtaining 

Uk. Besides, the P-spin formalism yields (see (2.17) and (2.18)) some new natural measurable 

quantitative characteristics of light depolarization, namely, degrees depp = (1 - 2Plfi) and 

depp,, = (1 - 12iillR) of the content of P-scalar and of Po-scalar biphotons where p,?i = &, fi 

denote expectation values of appropriate operators; herewith F = -112 + [1/4+ < (P21 >>I1/"s 

determined from Eqs (2.10), (2.11) as a function of degP, fi and variances a,. Evidently, depp, 

is connected with thewell-known degree of circular polarization 1 < N+ > - < N- > I/ < M > 

whereas depp provides a new quantitative characteristic of polarization structure of light related 

to measurements of polarization noises. 

We also note that analysis above can be extended by considering modifications of the decomposition (2.12) where any other Hermitian operator Pii = Sp(((n'))Po(Sp(((n')))+ is diagonalized instead of Po (Sp(((5)) = exp(C(n')P+ - (*(n')P-), ((G) = -!exp(-iq5) and vector 

n' = (sin 8 cos 4, sin 0 sin 4, cos 6) corresponds to a position of the Stokes vector s" on the Poincare 

sphere). Specifically, one can diagonalize Hermitian operators Pa, a = 1,2 corresponding to 

a linear polarization basis of light beams[l2]. Such extensions lead to new states of quantum UL generated by Pc(e.g., PI- or P2) - scalar biphotons xt(n') = (Sp(((n')))+Y;:Sp(((I)) 

of the (2.16) type and having characteristics similar to those described by Eqs (3.2)-(3.3) but 

with some peculiarities concerning their "rotation" properties determined by Eqs (2.8); for example, the condition AP1(2) = 0 is valid only for quite definite angle positions of polarization analyzers. We also note that usual multimode Glauber coherent states i{at, a;} >= ni + + exp(ai a+(i) + a,a'(i) - ot'a+(i) - ar*a- (i)) 10 >, af # 0, which are in general cases states 
of partially polarized light, contain ( for special values of parameters a?) a subclass of states 

corresponding to UL. In particular, all such states display properties(< 1 Pa I >= 0, ap, # Ova) of 

usual UL, when the condition la: 1 = laf I is fulfilled[8,18]. 

All this leads to a new classification of states of UL within quantum optics which can be 

represented by a chain of embedded subsets 

with the following typical density matrices for each subset: 

a)ULO -+ pth, 

b)ULc 4 pc = I{%} >< {adI, b+I = 10-1, (3.7b) 

C)UL" 4 pbp = IX >< XI, IX >= exp(XP+ - XP-)I >, I >E L1(a = 0), (3.7~) 

d)uLP0 -t pp, = I >< 1, I >= S~({/3i~))lO >E L'(* = 0), (3.7d) 

e)u~' -t PP = I >< 1, I >= Sx({7ij})10 >E L(OO) (3.7e) 

where pth is a density matrix for the thermal radiation[3], p, describes coherent UL whereas 

pbp, ppo, pp correspond to different kinds of biphoton UL[12]. We note that all these classes of UL 

are distinguished by values of depp = (1 - 2~1~) and deppo = (1 - 12?il/N). 

4 Generalizations and conclusion 

Thus, in the previous sections we have shown that in the Fock space LF(m) of multimode light 

with consideration of polarization one can pick out with the help of Eq. (2.12) subspaces (L(P = 

On = 0), L'(T = 0) and someones related to them) of quantum states describing different new types 

of UL light and, simulbaneously, manifesting specific forms of squeezing in polarization optics. All 

other subspaces L(Pa), ~'(a), T > 0, in the decomposition (2.12) describe, generally speaking, 

states of partially depolarized quantum light (see [10,11] where we also examined various types of 

polarization generalized coherent states of light). 

However, in real physical experimental situations states of light beams do not belong to a single 

subspace L(Pn) but are superpositions of states from different subspaces L(Pa). Therefore, it is of 

interest to study polarization squeezing properties (with using measurement devices of schemes on 

Fig. 1) of partially polarized light beams obtained by actions of the biphoton squeezing operators 

Sy, SX together with the "proper" polarization squeezing operators Sp(() on states ]in >,hY, of 

some physical input light beams 

that is presented schematically on Fig 3. As a result we can obtain new (non-classical) sets of 

states of partially polarized light which can be called as partially polarized squeezed light(PPSL). 

Specifically, taking as lin >,hYs usual multimode Glauber coherent states I{at, a;} >, a' # 0, 

we get in such a manner states of PPSL which contain (at the condition la;! = [ail) a subclass 

of states corresponding to ULC in the classification above. In general, transmitting different input 
Figure 3: Scheme of production of partially polarized squeezed light 

beams through physical devices corresponding to different combination of the above squeezing 

operators in (4.1), one can obtain new classes of partially polarized light distinguished by values 

of depp = (1 - 2PIN) and depp, = (1 - 12?il/~) by analogy with UL. 

In conclusion we emphasize that the above results give a more deep insight into polarization 

structure of light beams enabling to determine new nonusual states in quantum optics. In a sense, 

the results of section 2,3 and those of papers [lo- 121 yield all necessary prerequisites for developing 

a quantum description of unpolarized light waves whose existence has not yet an adequate solution 

within the classical optics[26]. All this opens some possibilities in setting new optical experiments 

related, in particular, to "hidden" variables, "entangled states" and EPR paradox [1,2,5,6,19], 

polarization chaos, spontaneous symmetry breaking and bistability [8,11,12], "optical atoms" and 

reduction of quantum noises [4,6,11,12,19] etc. From other lines of possible applications of the 

results above we point out precise measurements in spectroscopy of anisotropic media[l8] and 

studies of interaction of light in different new polarization states with optically active biological 

macromolecules (using the interrelations between the above chiral symmetries su(2), b, a of em 

fields and chiral properties of such molecules)[27]. 

5 Acknowledgements 

The author thanks V.P. Bykov and A.V. Masalov for stimulating discussions and M.H. Rubin and 

Y. Shih for interest in the work. A partial financial support of the American Physical Society 

is acknowledged. The gratitude is also expressed to the staff of Facultad de Ciencias FisicoMatematicas de Universidad de Guadalajara, where the final version of the paper was prepared, 

for hospitality. 

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1590 (1988) 

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[9] A. Gerrard and J.M. Burch, Introduction to matrix methods in optics, (Wiley, London, 1995) 

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P.N. Lebedev Phys. Inst.), 211, 161 (1991) 

[ll] V.P. Karassiov, J. Sov. Laser Research, 12, 147 (1991); ibidem, 431 (1991) 

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1959) 

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[17] C.K. Hong and L. Mandel, Phys. Rev., A32,974 (1985) 

[18] V.P. Karassiov and A.V. Masalov, Optika and Spectr., 94, N 4 (1993)[Russian] 

[19] D. Greenberger, M. Horne and A. Zeilinger, Phys. Today,46, N8, Pt.1, 22 (1993) 

[20] A.M. Perelomov, Generalized coherent states and their applications,(Nauka, Moscow, 1987) 

[21] R. Delbourgo, J. Phys.,AlO, 1837 (1977) 

[22] V.V. Dodonov and V.I. Man'ko, Trudy FIAN, 183,5 (1987) 

[23] V.P. Karassiov, Bull. Lebedev Phys. Inst., N 516, 00 (1993) 

[24] C. Aragone, E. Chalbaud and S. Salamo, J. Math Phys., 17, 1963 (1976) 

[25] G.S. Agarwal, D. Home and W. Schleich, Phys. Lett., A 170, 359 (1992) 

[26] H. Hepp and H. Jensen, Sitzber. Heidelberg. Acad. Wiss., Mat.-Naturw. K1.,4, 89 (1971) 

[27] S.F. Mason, Molecular optical activity and the chiral discrimination, (University Press, Carnbridge, 1982) 
A NEW APPROACH TO THE PHOTON 

LOCALIZATION PROBLEM 

D. Han 

National Aeronautics and Space Administration, Goddard Space Flight Center, 

Code 902.3, Greenbelt, Maryland 20771 

Y. S. Kim 

Department of Physics, University of Maryland, College Park, Maryland 20742 

Marilyn E. Noz 

Department of Radiology, New York University, New York, New York 10016 

Abstract 

Since wavelets form a representation of the Poincare group, it is possible to construct 

a localized superposition of light waves with different frequencies in a Lorentz-covariant 

manner. This localized wavelet satisfies a Lorentz-invariant uncertainty relation, and also 

the Lorentz-invariant Parseval's relation. A quantitative analysis is given for the difference 

between photons and localized waves. It is then shown that this localized entity corresponds 

to a relativistic photon with a sharply defined momentum in the non-localization limit. 

Waves are not particles. It is confirmed that the wave-particle duality is subject to the 

uncertainty principle. 

1 Hntroduc$ion 

We propose a quantitative approach to the photon localization problem. Photons are relativistic 

particles requiring a covariant theoretical description. Classical optics based on the conventional 

Fourier superposition is not covariant under Lorentz transformations [I]. On the other hand, 

wavelets can be regarded as representations of the Lorentz group [2, 3, 41. In Ref. [I], we 

discussed the difference between waves and wavelets without mentioning the word "wavelet." We 

have seen there that the lack of covariance of light waves is due to the lack of Lorentz invariance 

of the integral measure. We concluded there that an extra multiplicative factor is needed to make 

the Fourier optics covariant. We would like to point out that this procedure corresponds to the 

wavelet formalism of wave optics. 

In spite of the covariance of wavelets, photons are not wavelets. Instead, we shall make a 

quantitative analysis of the difference between these two clearly defined physical concepts. The 

advantage of this quantitative approach is that we can see how close they are to each other. In 

this way, we can assert. that photons are waves with a proper qualification. 

Another convenient feature of the localized wavelet representation is that it is possible to 

introduce an cut-off procedure in a covariant manner, so as to preserve the information given in 
the distribution. By introducing the concept and word "window" [5,6,7,8], it is possible to define 

the region in which the frequency distribution is non-zero. We can then compare the "windowed" 

wavelet to the photon operators in quantum field theory to pinpoint the difference between the 

photons and wavelets. 

2 Localized Light Wavelets 

For light waves, we start with the usual expression 

Unlike the case of the Schrodinger wave, w is equal to k, and there is no spread of wave packet. 

The velocity of propagation is always that of light. We might therefore be led to think that the 

problem for light waves is simpler than that for nonrelativistic Schrodinger waves. This is not the 

case. In Ref.[l], we have considered the following questions. 

(1). We would like to have a wave function for light waves. However, it is not clear which 

component of the Maxwell wave should be identified with the quanta1 wave whose absolute square 

gives a probability distribution. Should this be the electric or magnetic field, or should it be the 

four-potential? 

(2). The expression given in Eq.(l) is valid in a given Lorentz frame. What form does this equation 

take for an observer in a different frame? 

(3). Even if we are able to construct localized light waves, does this solve the photon localization 

problem? 

(4). The photon has spin 1 either parallel or antiparallel to its momentum. The photon also has 

gauge degrees of freedom. How are these related to the above-mentioned problems? 

Even though light waves do not satisfy the Schrodinger equation, the very concept of the superposition principle was derived from the behavior of light waves. Furthermore, it was reconfirmed 

recently that light waves satisfy the superposition principle [9]. It is not difficult to carry out a 

spectral analysis on Eq.(l) and give a probability interpretation. The question then is whether 

this probability interpretation is covariant. We addressed this question in Ref. [I]. We concluded 

in effect that the localized light wave is covariant if we use the wavelet form. We regret however 

that we were not aware of the word "wavelet" at that time. In this section, we shall translate 

what we did in Ref. [I] into the language of wavelets. 

The expression given in Eq.(l) is not covariant if g(k) is a scalar function, because the measure 

dk is not invariant. If g(k) is not a scalar function, what is its transformation property? We shall 

approach this problem using the light-cone coordinate system. We define the light-cone variables 

as 

s = (2 + t)/2, u = (2 - t) . (2) 

The Fourier-conjugate momentum variables are 

If we boost the light wave (or move against the wave with velocity parameter P), the new coordinate 

variables become I I sl=a+s, ul=a-U, k,=a-k, ku=a+IC,, (4) 
where 

a* = [(I P)l(l r P)I'/~ 

If we construct a phase space consisting of s and k, or u and k,, the effect of the Lorentz boost will 

simply be the elongation and contraction of the coordinate axes. If the coordinate s is elongated 

by a+, then k, is contracted by a- with a+a- = 1. 

In the case of light waves, k, vanishes, and ku becomes k or w. In terms of the light-cone 

variables, the expression of Eq. (1) becomes 

We are interested in a unitary transformation of the above expression into another Lorentz frame. 

In order that the norm / Ig(k)12dk (7) 

be Lorentz-invariant, F(u) and g(k) should be transformed like 

Then Parseval's relation: 1 lF(u)12du = j lg(k)12dk 

will remain Lorent z-invariant . It is not difficult to understand why u in Eq.(2) and k = ku in Eq.(3) are multiplied by a+ 

and a- respectively. However, we still have to give a physical reason for the existence of the 

multipliers (a*)'i2 in front of F(u) and g(k). In Ref. [lo], Kim and Wigner pointed out that 

the multipliers in Eq.(8) come from the requirement that the Wigner phase-space distribution 

function be covariant under Lorentz transformations [ll, 71. 

Let us illustrate this point using a Gaussian form. We can consider the g(k) function of the 

form 

which leads to the F(u) of the form 

F(u) = (:) 'I4 exp (h2/2) exp (ipu) , 

where b is a constant and specifies the width of the distribution, and p is the average momentum: 

Under the Lorentz boost according to Eq.(8), g(k) becomes 
Thus, according to the transformation law given in Eq.(8), the transformed F(u) becomes 

We note here that the average momentum p is now increased to a+p. The average momentum 

therefore is a covariant quantity, and a- can therefore be written as 

Q- = alp , (15) 

where a is the average momentum in the Lorentz Game in which a* = 1, and P = 0. 

As a consequence, in order to maintain the covariance, we can replace F(u) and g(k) by F1(u) 

and gl(k) respectively, where 

These functions will satisfy Parseval's equation: 

in every Lorentz frame without the burden of carrying the multipliers 6 and &. We can 

simplify the above cumbersome procedure by introducing the form 

where the procedure for the Lorentz boost is to replace p by a+p, and k in g(k) by a-k. This is 

precisely the wavelet form for the localized light wave, and this definition is consistent with the 

form given in earlier papers on wavelets [2, 31. 

3 Windows 

There are in physics many distributions, and we have a tendency to choose "smooth" or analytic 

functions to describe them. These functional forms usually extend from minus infinity to plus 

infinity. However, the distribution function of physical interest is usually concentrated within a 

finite interval. On the other hand, it is not uncommon in physics that mathematical difficulties 

in theory come from the region in which the distribution function is almost zero and is physically 

insignificant. Thus, we are tempted to ignore contributions from outside of the specified region. 

This is called the "cut-off" procedure. 

One of the difficulties of this procedure is that a good cut-off approximation in one Lorentz 

frame may not remain good in different frames. The translational symmetry of wavelets allows us 

to define the cut-off procedure which will remain valid in all Lorentz frames. 

We can allow the function to be nonzero within the interval 

while demanding that the function vanish everywhere else. The parameter w determines the size 

of the window. The window can be translated or expanded/contracted according to the operation 

of the affine group [8]. Indeed, it is possible to define the window in such a way that the boundary 

condition be covariant, and the information contained in the window be preserved [8]. 
4 Photon Locallization Problem 

The wavelet formalism allows the description of a localized wave function for light waves in a 

Lorentz covariant manner with Lorentz-invariant normalization. We shall now examine how close 

the wavelet can be to photons. First of all, it should be noted that there are no physical laws 

which dictate the functional form for a(k) or g(k). This depends on initial conditions. If we 

choose an analytic function, this is purely for mathematical convenience. If we choose functions 

which vanish near k = 0 and k greater than a certain value, this also satisfies our criterion for 

mathematical convenience. Indeed, the concept of window plays a decisive role in this form of 

localization. 

In quantum electrodynamics, we start with the form 

This is a covariant expression in the sense that the norm 

is invariant under Lorentz transformations, because the integral measure (1 /w )dk is Lorentzinvariant. 

We are quite familiar with the expression of Eq.(l) for wave optics, and with that of Eq.(20) for 

quantum electrodynamics. The wavelet form of Eq.(18) satisfies the same superposition principle 

as Eq.(l), and has the same covariance property as Eq.(20). It is quite similar to both Eq.(l) and 

Eq.(20), but they are not the same. The difference between F(u) of Eq.(l) and the wavelet G(u) 

is insignificant. Other than the factor Ja where a has the dimension of the energy, the wavelet 

G(u) has the same property as F(u) in every Lorentz frame [I]. However, the difference between 

G(u) and A(u) is still significant. 

It is possible to give a particle interpretation to Eq.(20) after second quantization. However, 

A(z,t) cannot be used for the localization of photons. On the other hand, it is possible to give 

a localized probability interpretation to F(z,t) of Eq.(l), while it does not accept the particle 

interpretation of quantum field theory. 

A(u) of Eq.(20) and G(u) of Eq.(18) are numerically equal if 

where the window is defined over a finite interval of k which does not include the point k = 0. It 

is thus possible to jump from the wavelet G(u) to the photon field A(u) using the above equation. 

However, the above equality does not say that a(k) is equal to g(k). The photon intensity distribution is not directly translated into the photon-number distribution. This is the quantitative 

difference between wavelets and photons. 

Of course, this difference becomes insignificant when the window becomes narrow. The narrower window in k means a wider distribution of the wave in the u coordinate system. We 

conclude therefore that photons become waves in non-localization limit. Particles are not waves. 

The wave-particle duality is subject to the uncertainty principle. The relation given in Eq. (22), 

together with the appropriate window, is a statement of this uncertainty relation. 
Acknowledgments 

We would like to thank B. DeFacio and G. Hegerfeldt for very helpful comments and criticisms. 

We thank also 6. Kaiser for sending us reprints of his articles. 

References 

[I] D. Han, Y. S. Kim, and M. E. Noz, Phys. Rev. A 35, 1682 (1987). 

[2] E. W. Aslaksen and J. R. Klauder, J. Math. Phys. 9, 206 (1968) and 10, 2267 (1969). 

[3] I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, 

Philadelphia, PA, 1992). 

[4] G. Kaiser, Phys. Lett. A 168, 28 (1992). 

[5] G. Kaiser, in Wavelets and Applications (Toulouse, 1994), Proceedings (1992), and the references contained in this paper. 

[6] H. H. Szu and H. J. Caulfield, Optical Engineering 31, 1823 (1992). 

[7] For the phase-space picture of wavlets, see B. DeFacio, C. R. Thompson, and 6. V. Welland, 

in Digital Image Synthesis and Inverse Optics, Proc. of SPIE - The International Society for 

Optical Engineerg (Society of Photo-Optical Instrument ation Engineers, Bellingham, WA 

1992). See also B. DeFacio, in Workshop on Squeezed States and Uncertainty Relations, Proceedings, D. Han, Y. S. Kim, and W. W. Zachary, eds. (NASA Conference Publications No. 

3135, 1992). 

[8] D. Han, Y. S. Kim, and M. E. Noz, in Second International Workshop on Squeezed States 

and Uncertainty Relations, edited by D. Han, Y. S. Kim, V. I. Man'ko (NASA Conference 

Publications No. 3219, 1993). 

[9] A. Aspect, P. Grangier, and G. Roger, J. Optics (Paris) 20, 119 (1989). 

[lo] Y. S. Kim and E. P. Wigner, Phys. Rev. A 36, 1293 (1987). 

[ll] Y. S. Kim and M. E. Noz, Phase Space Picture of Quantum Mechanics (World Scientific, 

Singapore, 1991). 
PRODUCTION OF SQUEEZED STATES FOR MACROSCOPIC MECHANICAL 

OSCILLATOR 

V.V. Kulagin 

Sfernberg Astronomical Institute, Moscow State University, 

Universitetsky prospect 19, I1 9899, Moscow, Russia. 

Abstract 

The poeeibility of squeezed states generation for macroecopic mechanical oscillator is discussed. It 

is ahown that one can obtain mechanical dator in squd state via coupling it to electromagnetic 

oscillator (Fabry-Perot resonator) and pumping thie Fhbry-Perot resonator with a field in squeezed 

state. The degradation of squeeaing due to mechanical and optical loeaes is also analysed. 

Realization of quantum states such aa squeezed, amplitude squeezed and others in real physical 

systems is of great importance for confirmation of predictions of quantum mechanics and its future 

development. There are a lot of papers concerning generation of squeezed states of electromagnetic 

fields [I, 2,3]. However realbation of such states in other systems, for example, mechanical is also 

of great importance. Tbis problem arises in different high precirrion measurements, especially in 

gravitational wave experiment [4]. 

Let's consider a system of two coupled oscillato~s [5,6]: electromagnetic (represented by FabryPerot resonator with a laser pump beam amplitude EL and frequency wp near one of the reeonant 

frequencies of resonator wb) and mechanical (represented by a moving mirror of Fabry-Perot 

resonator connected to a spring). Pump beam enters the resonator through fixed mirror with 

reflectance approaching 1 (this assumption is not critical for results and used only for simplicity, 

we also suppose that I tj I--, 0 but EL. I tj I=const., where tj is amplitude transmittance 

coacient of fixed mirror). Usually one treats such a system in Hamiltonian formalism framework 

[7] when equations of motion are conservative and the system is in free evolution. However in 

our case it is more convenient to use Langevin approach with evolution of the system under the 

action of input fielde: EL (classical laser field) and Eb, (quantum field from field controller - device 

allowing to generate electromagnetic field in appropriate state). In our analysis Ea, is squeezed 

noise with two point of squeezing and it enters the resonator through moving minor with nonzero 

amplitude transmittance coefficient t,. Then bearised equations of motion for such system have 

the following form: 

where the following parameters are introduced 
& asrB + axe the mpEtude abnd the phase of the Geld opticd resonator due to the action of 

"Baser forcen E5 ~th &romfixed, S is the beam cross section, il is the length of opeicd resonator, 

6, is opGcs9 dmphg of remmtor due to the bhe of idde 5dd tlarou& mosijlng 

refilecfi%rity less thm 1), rn ad 61, are mws and hquency of mecb&cal adator arid f i~ its 

coor&ate operator. we =me that one can onnit the dmeg 

ten We edso adls could be represent& in term of their qu 

operators wifh mud c tation rehtions [8, 91: 

Index " ba" tits a field entered the syetem from field controller and hdex "out" me= that the 

field outputs from Fabry-Berot resonator though moving mirror. 

htroducing quadratme components for d ator though eq~~afion 

(frequency A CI. up is not exactly equal to the frequency w, of &g]le me&~cd mdator ad 

match- the presence of co between two oaciuators) one can eablily obth the 

equations of motion (we s the ayetern is in steady stah therefore d&,,/& = 0) 

It is ob.riow horn (5) that fluctuati0118 of 3, and 2, depend on fluctuations of controller 6eBd 

quadrature components only near frequencim w FZ. 0 and w m 2 1 A I that means that controller 

field must consist of two modes with different frequencies wo ajnd wo + 2A. Then one can btroduce 

where &, &, kc; are quadrature component operators of two modes in narrow non overlapping 

bandwidths (6w <I A 1) near left andl right sidebands which are detuned from the pump hequency 

up by A : wo = wp - A and wo + 2A = up + A. Then introducing the coefficients 

one could obtain the following system of equations 

where phase C = a~ctan(2A/6~) represents the delay of the pump field inside the cavity with 

respect to the laser field. 
Let's rsssme that ccontroUer field Eb, is in squeezed state with < >= 0 and dispersions of 

quardrature components If), 51 

where go >. I is the squeezing coefficient of action field for mode with frequency %, No is 

vaculm level of dispersioo and we use the fact that 2 1 A la 2w,, < w. The same expressions me 

did for $., & with otnious snbatitution go -+ g2 and correlations between &, &. and $., & 

ane aero. Factors go and g, depends on the structure of back action field contrder. 

The andanid would be in squeesed etate only 8 the following special conditions 

are dd. The first concerns the det g h of the eptem: it mwt satiafy the equation 

In this case contribution of "noiay wenponeat'' kc to i, %anishe*. For the contribution of mothex 

'n&y component" & would be dm noimportant one must &me the phase 4 according to the 

following equation: 

x-<-2+= kr/2 k=0,&1,f2 ... (11) 

that means that special phase correspondence must take place. In physicd language thie meam 

that one xnuat compensate the delay of electromagnetic field inaide Fabry-Perot resonator witfa 

regwd to the pamp field. This can be done by appropriate phase correspondence between the 

pump beam and the 6EeI.d Eb &om the squeezed state controller. Then the system equations of 

motion occur (k = 1) 

Therefore we obtain specid quantum nondemolition coupling between optical and mechanical 

oscillator: one quadtature component of mechanical oecillator couples only with one quadrature 

component of opticai fidd (on frequencies and + 28) and the larger the squeezing of the 

field of two electruxna.gnetic modes (with frequency wo and wo + 2A) the greater the squeexe factor 

of mechanical osciIiabr. It is worth mentioning that the squeezing g2 of the mode with frequency 

wo + 2A conld be sder than the squeeeing go of the mode with frequency q of dectromagnetic 

fieid provided 6, < A (because of the filtration of fluctuations by narrow bandwidth opticd 

resonator) otherwise the squeezing ef two rrdes must be equal. 

Let's &ws the the of operatifig regime achievement and the iniiuzncc of mechecal SP and 

optical coherent (due to difraction and mirror absorbtion) 6, losses. In practice optical damping 6, 

due trtursdt tzcnce through moving minor is much greder than mechanical damping 6,. Then the 

field inside resonator becomes squeesed through time 6;'. After that through time (2~~/w,)-'I" 

(that is iuverse value of coupling constant) the state of meckcal oscillator becomes also squeezed 

and the hiitid state is forgotten. It is obvious that mechanical losses must not be very large: 

6, < (2~ y/wp)112/g (g - required squeesing &tor for mechanical oscillator) otherwise the rate 

of coherent pumping though Iorraes would be larger than the rate of squeeeing through action of 

contrder fieid. Simiiarly 6, mt be ernaller than 6,/ max(go, g3). 
In conclusion let's discuss the structure of back &ion field controller. In accordance with the 

ideas of papers [lo, 11, 121 it rust contain two circulators for uncoupling the fields Ewt and Eh, 

the load (absorber black body with sea0 temperature) and tw~ squeezers with pump frequencies 

2uo and 2(~ + 2A) (for example, degenerate parametric mplifiere or four wave mixers). Then the 

field Ed comes through moving resonator mirror and two circulators to the load and diseipates in 

it. Zero fluctuations of the load entera through fitit circulator the squeeser with pump frequency 

2wo, then gets through second circulator to the squeezer with pump frequency 2(~ + 2A) and 

then enters the system through moving mirror. 

References 

[I] D. F. Walla, Nature (L), 306, 141 (1983). 

[Z] B, Yurke, Pnp. Rev. A, 32, 300 (1985). 

[3] H. A. Haua, Y. Y~mmoto, Php. Rev. A, 34, 270 (1986). 

[4] 0. M. Caves, K. S. Thane, R. W. P. Drever, V. D. Sandberg, M. Zimmermann, Rev. Mod. 

Phye., 62,341 (1980). 

[q V. V. Kulagin, V. N. Rudenko, Nuovo Cimento C, 10,601 (1987). 

[6j V. V. Kd*, V. K. hdenko, Sov. Phys. JETP, 67,617 (1908). 

G. J. Milburn,A. S. Lane, D. F. Walls, Phya.Rev. A, 27, 2804(1983). 

[8] C. M. Caves, Php. Rev. D, 26, 1817 (1982). 

[9] C. M. Caves, 8. L. Schlm&~, Bhys. Rev. A, 31,3008 (1985). 

[lo] B. Yurke, J. Opt. SOC. Am. B, 2, 732 (1985), 

[ll] B. Yurke, J. Denker, J. Phpica B, 108, 1359 (1981). 

[12] B. Yurke, L. R. Corruccini, Phye. Rev. A, 30, 895 (1984). 
SQUEEZING IN A 2-D GENERALIZED OSCILLATOR 

Octavio Castaiios and Ram6n L6pez-Peiia 

Instituto de Ciencias Nucleares, UNAM 

Circuito Exterior, C.U., Apdo. Postal 70-543 

0451 0 Mkxico, D. F., Mdxico 

Vladimir I. Man'ko 

Physical Institute of the Russian Republic 

Leninsky Prospect 53. Moscow, Russia 

Abstract 

A two-dimensional generalized oscillator with time-dependent parameters is considered 

to study the two-mode squeezing phenomena. Specific choices of the parameters are used 

to determine the dispersion matrix and analytic expressions, in terms of standard hermite 

polynomials, of the wavefunctions and photon distributions. 

1 Introduction 

In the miadle of the sixties and beginning of the seventies a set of quantum states of the electromagnetic field were observed which have less uncertainty in one quadrature than a coherent 

state [I-31. These one-mode squeezed states have generated big expectations in optical communication systems 141. In some quantized fields, the interaction hamiltonians occur only between 

pairs of modes and then to understand the illail1 features of the system, one restricts to study 

one and two normal modes . In the last decade two-mode squeezing phenomena have attracted 

attention to study properties of noise and correlations [5-81. Recently the accidental degeneracy 

of a two-dimensional (2-D) harmonic oscillator with frequency wo plus an interaction proportional 

to the z-th projection of the angular momentum was studied [9]. This system was called the 

generalized 2-D harmonic oscillator because presents a bigger accidental degeneracy depending on 

the strength X of the angular momentum interaction. This model was generalized [lo] to include 

time-dependent parameters, n-L = n-LOf(t) and A = wOXO(t). If we take f(0) = XO(0) = 1; the 

hamiltonian, for t = 0, represents a charged particle moving in a constant magnetic field . 

The aim of this work is to study two-mode squeezing phenomena with this model because it 

demonstrates the change of dispersions due to variation of the mass and coupling constant during 

the evolution. In the framework of quantuill optics the hainiltonian is built by: the operator 

(11 f + f)$ . a', that causes a time-dependent exchange of kinetic and potential energies within 

each mode; the interaction 1/2(l/ f - f)($ Zf + ii. Z), which describes a degenerate two-photon 

interaction; and the potential iXo(t)(aial - aia2), that is a mode mixing operator. 

The solution of the corresponding time depe~ldent Schroedinger equation is obtained through 

the theory of integrals of motion [ll]. By means of Noether's theorem, using a special variation 
[lo], we construct the linear time dependent integrals of the motion. The resulting quantum 

invariants are given in terms of the positions and momenta operators [10,11] by 

with the initial conditions P(0) = $and Q(0) = cj', so that the 2 x 2 matrices previously introduced 

satisfy XI (0) = X4(0) = I2 and A3(0) = A2(0) = 0. The operators x(t) = 1/fi [Q(t)/l+ il/fiP(t)] 

and its hermitean conjugate, can be constructed with the matrices 

with 1 = d& defining the oscillator length. These integrals of motion also are given in terms 

of the creation and annihilation photon operators 

With the initial conditions x(0) = ii and xt(0) = $, the matrices defined in (3) comply with 

Ml(0) = M4(0) = I2 and M3(0) = M2(0) = 0. The Ak's, Mk's, A, and A, are entries of symplectic 

matrices in four dimensions because the invariants (1) and (3) satisfy the commutation relations 

of Heisenberg-Weyl algebras. 

In the present work, we study the behavior of the model for Ao(t) an arbitrary function of time 

and considering two kinds of varying masses, i. e., two choices for the function f (t), namely: 

1 7 tso 

f (t) = Icosh2 not , 06tiT . {%(t - T) sinh OOT + cosh O0TI2 , T 5 t 

(5) 

For these two cases the XI, matrices take the general form 

where the definition 0 = J; woXo(r)dr was used. The analytic expressions for the pr's functions 

are given in Ref. [lo]. In the next sections we determine the coherent and Fock-like states, the 

photon distributions and the dispersion matrices in terms of these pk's. 

2 Squeezed Coherent and Fock States 

The coherent-like states are obtained by solving the differential equation ~(t)~~(if,t) = 0 with 

/i(t) given in Eq. (3). This solution yields the vacuum state of the physical system, and its phase 

is chosen to guarantee that satisfies the time dependent Schroedinger equation. The expression 

for the ground state wavefunction is 
To get the last expression the relation (2) was used and the functions /+ = -& (gpl + f W) and 

pq = 5 (gp2 + f p4) were defined. To obtain the general expression for the eigenstates in the 

coordinate representation one needs to apply the unitary operator ~(a) = exp{B. zt - ti!* . A), 

which is an invariant, to the vacuum wavefunction (7), i.e., 

These are expressed in terms of multi-dimensional Hermite polynomials [12] through the relation 

Substituting the last expression into (8) and using the form of the coherent-like states in the 

Fock-like representation, we get the Fock-like eigenstates in the coordinate representation: 

These multi-dimensional Hermite polynomials are rewritten as a product of two standard onedimensional Hermite polynomials [12] as follows: 

1 

[cos 0 ql + sin 8 q2] 1 

1 [- sin 8 ql + cos 8 q21 

where we use the explicit expression of matrix R. These Fock (10) and coherent (8) -like states 

represent squeezed and correlated eigenstates of the system as it will be shown further. 

3 Propagator 

The propagator in the coherent state representation is given by the matrix elements of the evolution 

operator U(t), which will be obtained by means of the theory of time dependent integrals of 

motion [ll]. If f(t) is an integral of motion then satisfies f(t)~(t) = ~(t).f(~). Taking its matrix 

elements with respect to the coherent states, we get a linear system of differential equations, which 

can be solved. Thus the propagator takes the form 

For the cases (4) and (5), the following relations are satisfied 
Substituting these relations into the Eq.(12) we get the propagator, which through Eq. (9) can 

be expressed in terms of multi-dimensional Hermite polynomials. If we compare with the power 

series expansion of the propagator we get the probability amplitude for having nl and n2 photons 

in the coherent-like state 17, t), i.e., 

By means of the Eq.(12) this amplitude can be rewritten in terms of standard Hermite polynomials 1121. The squared absolute value of this amplitude yields the photon distribution function 

of the system, WnIn2(7, t) = I(nln21y', t)I2. This will let us calculate, at least formally, the mean, 

(Nk), and the mean squared fluctuation of the number of photons, AN^)^, in direction k, which 

are present in the coherent state 17, t). The expectation values of Nk and N; are evaluated directly 

using the expressions of the creation and annihilation photon operators in terms of the integrals 

of the motion (3), and the commutation properties for these invariants. For the vacuum state one 

has 

With these expressions, we evaluate the ratio of the mean squared fluctuation (ANk)2 and the 

mean number of photons (Nk), which determines the nature of the distribution function of the 

system: 

For the cases (4) and (5) the ratio is greater than one when t > 0, which implies that we have 

a super-Poissonian photon distribution function. For t = 0, there is a discontinuity in the ratio, 

which is obtained by comparing the following limiting procedures: malting t -, 0 and then G -+ 0, 

and conversely. 

4 Dispersion Matrices 

The dispersion matrix can be written in terms of 2 x 2 matrices characterizing the dispersions 

in the positions and momenta operators and the correlation, between them. Besides for the cases 

under study, due to (6), they talte the form 
The corresponding correlation matrices for the creation and avihilation operators are obtained 

immediately from the last expressions; they are given by 

time time 

time time 

time time 

Fig. 1. Dispersion and correlation matrices behavior in positions and momenta 

space for tile studied cases in this paper: (a) corresponds to Eq.(4), and (b), to Eq.(5). 
The behavior of the dispersion matrices is illustrated in Fig. 1. For the case (4), we choose the 

parameters y = 0.1 and mo = wo = 1. It is seen that there is squeezing for the coordinates and 

stretching for the momenta. Also one notes that a, is a negative function and therefore there 

are one-mode correlations between the coordinates and the momenta. If we reverse the sign of y, 

the roles between the dispersion for coordinates and momenta are interchanged, and a, becomes 

positive. In the case (5), we use the parameters Ro = 0.15, T = 10, and mo = wo = 1. In spite 

of the mass is different that in the previous example, the general trends are similar. For example, 

the app is an increasing function of time starting from its minimum value at t 5 0, and there is 

squeezing for the a,,. The main difference appears in the correlation apq: in this case, it can be 

positive for large times, while in the previous one is negative or zero for any time. 

5 Acknowledgments 

Work supported in part by projects UNAM-DGAPA IN-103091 and CONACYT 1570-E9208. 

References 

[I] D. R. Robinson, Commun. Math. Phys. 1, 159 (1965). 

[2] D. Stoler, Phys. Rev. D 4, 1925 (1971). 

[3] E. Y. C. Lu, Lett. Nuovo Cimento 2, 1458 (1971); 4, 585 (1972). 

[4] H. P. Yuen and J. H. Shapiro, IEEE Trans. Inform. Theory IT 24, 657 (1978); 26, 78 (1980). 

[5] C. M. Caves and B. L. Schumaker, Phys. Rev. A 31, 3068 (1985); B. L. Schumaker and C. 

M. Caves. Phys. Rev. A 31, 3093 (1985). 

[6] B. Yurke, S. L. McCall and J. R. Klauder. Phys. Rev. A 33, 4033 (1986). 

[7] Y. S. Kim and V. I. Man'ko, Phys. Lett. A 159 (1991), 226. 

[8] R. Glauber, in Proc. of Quantum Optics Conference, Hyderabad, January 5-10, 1991, Eds. 

G. S. Agarwal and R. Ingua, Plenum Presss, 1993. 

[9] 0. Castaiios and R. L6pez-Peiia, J. Phys. A: Math. Gen. 25, 6685 (1992). 

[lo] 0. Castaiios, R. L6pez-Peiia, and V. I. Man'ko: Noether's Theorem and Time-Dependent 

Quantum Invariants. Submitted for publication. 

[ll] V. V. Dodonov and V. I. Man'ko, in: Proc. P. N. Lebedev Physical Institute, Vol. 183. Invariants and evolution of non-stationary quantum systems, Ed. M. A. Markov (Nova Science, 

Commack, N. Y., 1987). 

[12] V. V. Dodonov, V. I. Man'ko, and V. V. Semjonov, Nuovo Cimento B 83, 145 (1984). 
SIMULTANEOUS TWO COMPONENT SQUEEZING IN 

GENERALIZED *COHERENT STATES 

Roger J. McDermot t and Allan 1.Solomon 

Faculty of Mathematics, Open University 

Walton Hall, Milton Keynes, MK7 6AA, U. K. 

Abstract 

Using a generalization of the q-commutation relations, we develop a formalism in which 

it is possible to define generalized q-bosonic operators. This formalism includes both types 

of the usual q-deformed bosons as special cases. The coherent states of these operators show 

interesting and novel noise reduction properties including simultaneous squeezing in both 

field components, unlike the conventional case in which squeezing is permitted in only one 

component. This also contrasts with the usual quantum group deformation which also only 

permits one component squeezing. 

1 Deformed Commutation Relations 

Consider the single particle deformed commutation relation [I] 

where at and a are generalized creation and annihilation operators, N is the number operator such 

that Nln) = nln), f is a real function, and the vacuum 10) is defined by a10) = 0. We define a 

normalized one-particle state by at 10) = 11). This formalism incorporates the deformation schemes 

previously encountered in the literature as special cases. 

Examples: 

1. f(N) = 1. 

This is the usual commutation relation of the Heisenberg-Weyl algebra and describes ordinary quantum mechanical bosonic systems such as the the harmonic oscillator. 

2. f (N) = q. 

The so-called q-oscillator, first suggested by Arik and Coon [2]. It has since been studied 

in detail by several authors e.g. Jannussis et a1 [3], Kuryshkin [4], Kulish and Damaskinsky 

PI. 

q*+2+1 

3. f (N) = q(qN+l) - 

This gives a deformed commutation relation equivalent to that of the q-boson first discovered 

by Macfarlane ~[6] and Biedenharn in connectioa with the representation theory of quantum 

groups. 
4. This form of deformed commutation relation can also be related to the extensive work 

of Bonatsos, Daskaloyannis and others, [7] and refs. therein, on the generalized oscillator 

formalism as well as the recent work of Jannussis [8]. 

Building up normalized eigenstates of the number operator N by repeated application of the 

generalized creation operators in (I), we obtain 

where the function [n] is defined recursively by 

[n + I] = 1 + f (n)[n] 

with initial condition [0] = 0. 

Explicitly, we see 

[n] = 1 + f(n-1) + f(n-l)f(n-2) + f(n-l)f(n-2)f(n-3) 

+..- +f(n-l)f(n-2).-.f(2)f(l) (4) 

The functions [n] can be thought of as generalizations of the basic numbers of q-analysis [9]. They 

obey a highly non-linear arithmetic but for appropriate choice of the function f, they tend in 

some limit to the ordinary integers. 

2 Coherent States 

Conventional coherent states of the oscillator obeying the undeformed commutation relation 

( f ( N ) = 1) may be defined by 

alX) = XlX) (6) 

or equivalently 

where the exponentional function, by definition, has the property 

These definitions of coherent states have been used to generalize the concept to the cases where 

the commutation relations have been deformed. 
Given the q-commutation relation aat - ata = 1, we may define coherent states I A) by 

To achieve the alternative definition given by (7), it is necessary to introduce a q-derivative 

operator [9], ,D, such that 

D E (Ax) = AE,(Ax) 9x9 (10) 

where E,(x) is the Jackson q-exponential. When this is done, we see that 

The same procedure can also be used to define q-coherent states for the Macfarlane-Biedenharn 

oscillator (although in this case the generalization of the exponential function is different from 

that of Jackson). 

, For [n] (defined by (5)), an analytic function of the variable n, it is possible to extend the above 

analysis to the case of bosonic creation and annihilation operators obeying the general commutation relations (1). 

We define an operator D, such that 

1 D, = ; [.$I . 

This acts as a generalized differential operator 

The eigenfunctions of D, given by 00 -n 

are well-defined provided the function f satisfies the appropriate convergence criteria. If f (n) > 1 

as n + oo then E (x) converges for all real values of x. If f (n) < 1 as n + oo then convergence is 

ensured for a certain range of x dependent on the precise nature of the function f. 

Since a E(Xat) 10) = AE(Xat) lo), we can use E(x) to define analogues of coherent states as normalized eigenstates of the generalized annihilation operator. 

3 Noise Reduction Properties 

We consider conventional (undeformed) bosons. 
The electromagnetic field components x and p are given by 

1 I 

x = -(a + at) and p = -(a -at). & ~1/2 

As usual, we define the variances (Ax) and (Ap) by 

(AX)~=(X~)-(X)~ and (A~)'=(p~)-(p)~. (17) 

In the vacuum state 1 1 1 1 

(A 40 = and (Ap)o = - a* 

and so 

1 

(A x)o(A P)" 5. (19) 

The commutation relation for a and at leads to the following uncertainty principle 

Thus the vacuum state attains the lower bound for the uncertainty, as do the coherent states. 

While it impossible to lower the product (A x)(Apj below the vacuum uncertainty value, it is 

nevertheless possible to define squeezed states Ill] for which (at most) one quadrature lies below 

it's vacuum value, i.e. 

If we now consider the generalized bosonic operators given by (I), using the same definitions for 

the the field quadratures, x and p, as in (16) we find that, just as in the conventional case, the 

vacuum uncertainty product (A X)~(A~)~ = is alower bound for all number states. 

However, unlike the conventional case, it is not a global lower bound. 

Consider the quadrature values in eigenstates of the generalized annihilation operator. 

Then 

and 

where 
If we choose 0 < f (n) < 1, then it can be shown that E~,~\A[~ E (0,l) for X within the radius of 

convergence of the generalized exponential. 

Hence 1 

(A 2):. = Z{l - €fillXI2) 

Evaluating the variance for the other component, we find that (Ap): = (A $1: SO 

However, it can also be shown that 

Thus we see that these generaliged q-coherent states satisfy a restricted form bf the Minimum 

Uncertainty Property (M.U.P.) of the conventional coherent states. Additionally we see that there 

is a general noise reduction in boih'quadratures compared to their vacuum value. In conventional 

coherent states there is no noise ieduction relative to the vacuum value. In conventional squeezed 

states, there is noise reduction in only one component. 

4 Special Cases 

We can apply the preceding analysis to the q-deformed bosons recently studied in connection with 

quantum groups (e.g. [5]). 

a). 'Physics' q-bosons 

First consider the q-bosons described by Macfarlane and Biedenharn [6]. 

These use the definition of the generalised number, in], recently discussed in the Physics literature 

and so will be termed 'physics' q-bosons. They are characterised by the deformed commutation 

relation 

t aa - qata = q-N. 

This can be rewritten fl] as 

aat - f (N) ata = 1 

N+2+1 where f (N) = :(*N+l) 

In this case, for normalizable eigenstates, the function ef,l is negative and so noise reduction does 

not take place. This is in agreement with the findings of Katriel and Solomon [12]. 
b). 'Maths' q-bosons 

We now consider the q-boson described by Arik and Coon [2]. This uses the generalized number function found in classical q-analysis and will therefore be termed a 'maths' q-boson. It is 

characterised by the deformed commutation relation 

For q E (0, I), the Jackson q-exponential Eq(1AI2) converges, provided cqlAI2 = (1 - q)lAI2 < 1. 

Given this condition on A, we have normalizable q-analogue coherent states satisfying (6) in which 

Hence, for this type of q-boson, we do obtain noise reduction in both quadratures with respect to 

the vacuum value. 

5 Acknowledgements 

One of us (R. J.M.) would like to thank the organizers of the workshop for their generous financial 

assistance. 

References 

[I] A.I. Solomon and J.L. Birman Symmetries in Science VII ed. B. Gruber, 

(Plenum Press, N.Y., 1993). 

[2] M. Arik and D.D. Coon J. Math. Phys. 17 524 (1976). 

[3] A. Jannussis, L.C. Papaloucas and P.D. Siafarikas Had. J. 3 1622 (1980) 

[4] V.V. Kuryshkin Ann. Fond. L. de Broglie 5 111 (1980). 

[5] P.P. Kulish and E.V. Damaskinsky J. Phys. A. 23 415 (1990). 

[6] A.J. Macfarlane J. Phys. A. 22 4581 (1989). 

L.C. Biedenharn J. Phys. A. 22 L873 (1989). 

[7] C. Daskaloyannis J. Phys. A. 25 2261 (1992). 

D. Bonatsos and C. Daskaloyannis J. Phys. A. 26 1589 (1993). 

[8] A. Jannussis J. Phys. A. 26 L233 (1993) 

[9] F.H. Jackson Mess. Math. 38 62 (1909). 

H. Exton q-Hypergeometric Functions and Applications, (Ellis Horwood, Chichester, 1983) 

[lo] R.J. Glauber Phys. Rev. 131 27 66 (1963). 

[ll] H.P. Yuen Phys. Rev. A. 13 2226 (1976). 

[12] J. Katriel and A.I. Solomon J. Phys. A. 24 2093 (1991). 
PHOTON STATISTICS 

OF A TWO-MODE SQUEEZED VACUUM 

- 

G. Schrade, V. M. Xkulin ': W. P. Scl~leich ' 

Abt. fur Quantenphysik, Universitut Ulm 

D-89069 C-lin, Ge~ntaizy 

V. I. Mau'ko 

Dipa~*tl:mento dl: Scienze Fisiche 

Unl:versita di ,Yapoli Federico II 

801 25 Na.poli, Italy 

Abstract 

\ire investigate the general case of the photo11 distribution of a two-mode squeezed vacuuin 

and show that the distribution of photons among the two modes depends on four parameters: two squeezing parameters, the relative phase between the two oscillators and their 

spatial orientation. The distribution of the total nunlber of photons depends only on the 

two squeezing parameters. We derive analytical expressions and present pictures for both 

distributiolls. 

1 Introduction 

Squeezing t.he quantuxn fluctuations of the radiation field has been delllollstrated experilllentally 

using various optical systems [I]. Most of them rely on two-mode squeezing [2, 3, 41. Therefore 

the properties of a two-mode squeezed state have been studied extensively [5, 61. However it 

a.ppears, t11a.t the plloton statistics of such a state has not been investiga.ted in a.11 details. Only 

some particular cases have been considered (see for example Refs. [$I and [8]). In the present 

paper we therefore extend these considerat.ions to an arbitrary two-mode squeezed vacuum and 

address the following questions: (i) \l'hat is the no st general case of the squeezed vacuuin of a 

two-mode oscillat.or, a.nd how lnany illdependent para.meters a.re needed to describe this state? 

(ii) 1Vha.t is the photon statistics in this state? This is of interest in t.he context of the degenerahe 

parametric a.~nplifier [9] since the lnatheinatical structure of the two-mode distribution function 

coincides wit11 the transition probability functioll of a two-dimensiona.1 para~netric oscillator [lo]. 

'also hloscow Institute of Pliysics and Technology Dolgoprudny, Moscow, Russia 

?also Mas-Planck-Isistifut fiir Quant.enopt.ik, D-8.5748 C;arching, Ciennany 

"pertrla~~ent. address: Lebedev inst.it.ut,e of the Academy of Sciences, hlosco\v, Russia 
2 general case of the squeezed vacuum of a two-mode 

oscillator 

\\'e start our co~lsiderations with the question (i). The linear callo~lical transfor~llation 

of the creation operators at, bt and the annihilation operators a, b to their new counterparts At, 

Bt and A, B corresponding to the two modes suggests a set of 10 parameters. Indeed, the generic 

transformation (1) of the four-dimensional vector (at, a, bt, b) into (At, -4: Bt, B) via a 4 x 4 ~natrix 

brings in 16 co~nplex values, that is 32 parameters. Since a and at, b and bt ,A and At, B and Bt 

are herinitia~l conjugate of each other, only half of these parameters are independent. Therefore 

the nu~nber of parameters reduces to 16. The condition to preserve the co~lunutation relation 

{A, B] = 0, [A, Bt] = 0, [A, At] = 1, and [B, Bt] = 1 provides additional constraints. Since the 

co~n~nutator of an operator with its hermitian conjugate is always real, the last two conditions 

provide only two constraints, whereas the first two conditions must hold for the real and the 

i~naginary part separately. This decreases the numnber of para~neters by six leaving us indeed with 

10 parameters. 

This finding is in accordance with the results of group theory. Reference [ll] shows, that 

the group of rotations and squeezing of the four-dimensional phase space, that preserves the 

pl~ase'space volunle of the two degrees of freedom, consists of 10 generators. Hence there are 10 

paranleters determining the elements of the real symplectic group Sp(4,R). 

Rut what is the physical ~neaning of these parameters? They are associated with rotation and 

squeezing transformation of phase space of two oscillators. Generic rotations of four-dimensio11a.l 

phase space are described by 6 pa.rameters but the syrnplectic rotations in four-dimensional space, 

that is the transformation preserving the co~ll~nutation relations, are described by 4 para~neters. 

They can be represented as a sequence of four rotations. The first rotation given by the transformation e'Mml, where M = i(atb - bta) is the angular monlentu~n operator, corresponds to the 

rotation in the coordinate space by the angle dl. The second and the third rotations are given by 

the operator e'""' and e'xbtb, and correspond to rotations by the angles $ and x in the respective phase-spaces of the two oscillators. The last rotation again can be taken in the forin eiM+2. 

Thus the angles $2, $ and x are chosen to be the paranleters of the sy~nplectic rotation. 

Tlne general sy~nplectic transformation also includes squeezing. We can represent the generic real 

sy~nplectic transformation (1) as consisting of three co~~secutive transformations: syml>lectic rotation (4 parameters), followed by independent squeezing of the two modes given by the squeezing 

operators 

S1 = e .,(a2-at2)/2, s2 = q(b2- bt2)/2 7 @I 

t(2 squeezing para~neters r1 and r2) and followed by another real sy~llplectic rotation (4 paralneters 

more). But how lnany of these parameters govern the two-dimensional squeezed vacuuln sta.he? 

The first four parainetric rotati011 acting on the completely synlllletric vacuum state 1eavt.s this 

state unchanged. Hence 2 squeezing paranleters and 4 paralneters associated with the second 

rotations defii~e the two-climensiond squeezed vacilum state. In atldition wc* call il~c.lutlr all overall 

cl~iautiirn plmse factor tip of this state. An explicit calculation of the gt.uc.ral case of a two-mode 

sclueezrd vacuum wave filnction is giw~ inRef. [12]. 
3 Photon statistics for the total number of photons 

We now address the question: how Inally of these parameters govern the photon statistics of 

such a two-dimensional squeezed vacuum state'? In particu1a.r) how many of them determine the 

probability of (i) cou~lti~lg a total ~lu~nber 71 of photons in the two modes, and (ii) cou~lti~~g the 

liu111ber 121 and 122 of photons in the individual modes'! 

In this section we c.0nside.r the c.a.se of (i) and ask for the tot.a.1 number n of phot.ons in both 

modes, whicl~ corresponds to the surface of a four-dimensional sphere, $(pi + x: + + x:) = n, 

centered at the origin of the four-dimensional phase space. The rotation of the phase space does 

not alter this sphere and hence only two squeezing parameters are essential. The probability of 

cou~lting the t,ot.a.l number n of photons for the case of two independently squeezed oscillators 

reads n 

W~(SI,SZ) = C wn,(s~)W,,-nl(s~), n1 =O 

(:3) 

where W,,, (sj) is the one-dimensional photon statistics [13] 

for nj odd rvn3(sj) = J=S~'/~~-~J ( 79 "1 12 ) for aj eve11 

of the squeezed vacuum nrave function 

where 

sj = tanh2(rj). 

Equation (4) reduces Eq. (3) to 

where the ~lorlllalization factor N reads 

The odd ternls of lVn vanish because of Eq. (4) . The sulll in Eq. (7) has been calculated in [I%] 

and the probability of count,ing rz = 2k photons then reads 

where describes the hypergeometric function. For the special case of identical squeezing in 

t.lle two modes, that. is, .sl = .s2 = .s, Eq. (9) yields 
I11 Fig. 1 we show the photon statistics (9) for various magnitudes of the squeezing parameters 

sl and s2. The solid and dashed curves correspond to weak and strong symmetric squeezing 

(sl = .s2), respectively. I11 accordance with Eq. (10) the photo11 distribution then displays an 

ex~onentia.1 dependence. Stronger squeezing results in a higher amount of quanta involved. The 

dotted curve shows the photon statistic for an asym~netric squeezing. 

Fig. 1. Probability of counting n = 2k photons in a two-mode squeezed vacuum 

as given by Eq. (9). For sy~lunetric squeezing (dashed line for sl = s2 = 0.5 and solid 

line for sl = sz = 0.99) the curve is a straight line and hence exponential whereas for 

asy~ll~lletric squeezing (dotted line for sl = 0.5 and s2 = 0.99 ) the photon statistics 

is non-exponential. Here we have not specified the distribution of the iz = 2k photo~ls 

among the two modes. Note that 1V,,=2k+l = 0 which we have omitted for simplicity. 

4 Photon statistics in the individual modes 

\Ve now turn to the second case and calculate the photon statistics W(nl, 1z2) = I < n1, 1z21\Ils, > I2 

in the individual modes. For this purpose we start from the generic expression of a squeezed 

va.cuum state I*,, >, whicll we produce in the following way: 

with the va.cuum wave function of the two-dime11siona.l oscillator \IIO,O(:~l, x2) = \IlO(:rl) \Il0(:c2), the 

angula,r momentum operator fi = i (atb - bta) and the operator f (y) = eap(iy(btb - ntn)) which 

clescribes a mutua.l phase shift 27 between the oscillators. The wave funct.ion of a two-diinensiona.1 

squeezed vacuuill state is a Gaussian described by the three co~nplex numbers A, B, a.nd C, which 

are fiinctioils of the four real pammeters 1-1, 1-2, 4, and r. Their explicit dependence is given in 

[I?]. One also finds there the generic two-mode squeezed va.cuum \x7ave function depeuding on two 

inore para~neters that do not affect the 1)hoton distriBution. 
\\re co~~tract Eq. (1 1) with the probability amplitude of the photon energy states h,, (XI) and 

t,!4L2(;~2) and arrive after calculati~~g the resulting double integral [12] at 

Here Pf denotes t11e associated Legendre Polynomial. As in Eq. (7) the total uunlber of photons 

12 = 11.1 $7~~ IIIUS~ be an even umber 11euce 721 and n2 are either both even or both odd. Otherwjse 

the p11oto11 distributio~~ fu~lctio~~ vaaishes. 

In Fig. 2 we display the probability to find n.1 photons in ll~ode 1 and n2 photons in Inode 2. 

This is the generic case of the photon distribution of a two-mode squeezed t7acuum. It depe~~ds 

on four pa,ra.meters: two squeezing paranlet.ers, the orientation of the distribution fu~~ction with 

respect to our 1a.bora.tory system, and the correlation betweeten the two modes. Reside the evenodd oscillatio~~s the maxima lie on curves 1vhic11 are sylnmetric with respect to the lllai~l diago~lal 

121 = 722. This behavior is sinlilar to the distribution fu~~ctio~~ of a displaced two-mode squeezed 

state tliscussed in Ref. [i, S]. 

Fig. 2. Probal>ility of cou~~til~g 121 and 712 p11otons in the two-mocle squeezed -, - . - vacllllrn calculated via Erl. (12). \Ve 11ave chosen 7.1 = 3. 1.2 = :I. o = ?. and -j = L. , - 9 

I Ile wavy struct,llrr of this distl.ih~~tion is confined to an angle in the photon nilmher 

plane. 
5 Conclusion 

We conclude by su~lm~arizing our main results. The wave function of the two-mode squeezed 

vacuum depends on 6 parameters (besides the phase factor). However only 4 of them, - the 

squeezing parameters rl and 1-2 of the two modes, the phase difference 7 between the two oscillators, 

and the rotation of the reference systenl by the angle 4 - manifest thelllselves in the generic case in 

the distribution of the photons among the two modes. This distribution function can be expressed 

explicitly in terms of Legendre polynomials. Only two parameters rl and r2 govern the distribution 

of the total nuinber of photons, which we express explicitly in terms of a hypergeollletric function. 

In conclusion we want to make the remark that similar consideration for N-mode squeezed vacuuln 

state shows that the photon distribution IV(nl, n2, ..., nN) depends on N2 parameters. 

6 Acknowledgments 

One of the authors (V.I.M.) thanks the University of Ulm and in particular the Hans-Kupczyk 

. Foundation for the support which made this research possible. 

References 

[I] For a review on squeezed states see, for example, special issues on squeezed states in J. Opt. 

Soc. Am. B 4, No. 10 (1987); and Appl. Phys. B 55, No. 3 (1992). 

[2] C. M. Caves and B.L. Schumaker, Pl~ys. Rev. A 31,3068 (1985). 

[:3] S. M. Barnett and P. L. Iinight, J. Opt. Soc. Am. B 2, 467 (1985). 

[4] R. Yurke, S. McCall, and J. R. Iilauder, Phys. Rev. A 33, 4033 (1986). 

[5] B. L. Schumaker, Phys. Rep. 135, 317 (1986). 

[6] A. K. Ekert and P. L. Knight, Am. J. Phys. 57, 692 (1989). 

[7] C. M. Caves, C. Zhu, G. J. Milburn, and IV. Schleich, Phys. Rev. A 43, 3854 (1991). 

[S] M. Artoni, U. P. Ortiz, and J.L. Birman, Phys. Rev. A 43, 3954 (1991). 

[9] For the photon statistics of a parametric oscillator see for example R. Vyas and S. Singh, 

Phys. Rev. A 40, 5147 (1989); and Opt. Lett. 14, No. 20, 1110 (1989); 

[lo] I. A. Malkin, V.I. Man'ko, and D. A. Trifonov, J. Math. Phys. 14, 576 (1973); V. V. Dodonov 

and V. I. Man'ko in Invariaiats and the ecolutioiz of iaonstatioizary quaiatum systrm,~, ed. by 

14. A. Alarkov, (No\:a Science Publ., New E'ork, 1989). 

[I I] (:. J. 12lilbur11, 3. Phys. A 17, 737 (1983). 

[I:'] G. Schrade, \:. 31. Xkulin, V. I. Ylan'ko. ant1 \V. P. Scllleicl~, Phys. Rev. A 48, 2398 (1993). 

[I:{] \V. Schlcicl~ and J. A. \l:'heeIer, Sature 326, 573 (1987). 
SECTION 2 

QUANTUM MEASUREMENT TaEORY 
ON THE THEORY OF QUANTUM MEASUREMENT 

Hermann A. Haus 

Franz X. KZrtner 

Department of Electrical Engineering and Computer Science 

Research Laboratory of Electronics 

Massachusetts Institute of Technology 

Cambridge, MA 02159 

Abstract 

Many so called paradoxes of quantum mechanics are clarified when the measurement 

equipment is treated as a quantized system. Every measurement involves nonlinear processes. 

Selfconsistent formulations of nonlinear quantum optics are relatively simple. Hence optical 

measurements, such as the quantum nondemolition (QND) measurement of photon number, 

are particularly well suited for such a treatment. It shows that the so called 'collapse of the 

wave functionn is not needed for the interpretation of the measurement process. Coherence 

of the density matrix of the signal is progressively reduced with increasing accuracy of the 

photon number determination. If the QND measurement is incorporated into the double slit 

experiment, the contrast ratio of the fringes is found to decrease with increasing information 

on the photon number in one of the two paths. 

1 Introduction 

The Theory of Quantum Measurement has a long and venerable history. Many of the original 

discussions of the founders of quantum mechanics are contained in the reprint volume of Wheeler 

and ~urek[lI. Yet, inspite of its long history, the issues raised in these well known discussions 

have not been fully settled. 

In this paper we attempt to make a modest contribution to this weighty problem. In doing so 

we are guided by a quote of Niels Bohr which reads: "... one sometimes speaks of "disturbance 

of phenomena by observation" or "creation of physical attributes to atomic objects by measurement." Such phrases, however, are apt to cause confusion, since words like phenomena and observation, just as attributes and measurements, are here used in a way incompatible with common 

language and practical definition. On the lines of objective description, [I advocate using] the 

word phenomenon to refer only to observations obtained under circumstances whose description 

includes an account of the whole experimental arrangement. In such terminology, the observational problem in quantum physics is deprived of any special intricacy and we are, moreover, directly reminded that every atomic phenomenon is closed in the sense that its observation is based 

on registrations obtained by means of suitable amplification devices with irreversible functioning 

such as, for example, permanent marks on a photographic plate, caused by the penetration of 
electrons into the emulsion" [Ref. I, p. 31. We have underlined the words that we consider 

particularly worthy of note. Bohr requires a description of the whole experimental arrangement. 

Further, if one is to state the outcome of the experiment in classical language, large amplification 

is required. 

At the risk of making statements that may be considered even more controversial by the adherents of the Einsteinian school, we should like to strengthen Bohr's quote by saying: "Physical 

reality cannot be formulated until the measurement equipment used to determine the observables 

is specified and treated as a quantum system. The large gain of the measurement equipment 

provides the classical interface at the output of the measurement apparatus." 

Much of the controversy involving quantum measurements is the consequence of the fact that 

it is very difficult to describe well the measuring equipment, according to our interpretation, to 

describe it quantum mechanically. 

In quantum optics we have made great progress in describing optical components quantum 

mechanically. The theory has been well tested experimentally. The squeezing by a parametric 

amplifier is well understood theoretically and amply confirmed experiment Less extensively explored, yet also tested, is the self-phase modulation and squeezing in optical fibers via 

the optical Kerr effect['-']. Hence it appears natural to use the well tested quantum description 

of optical devices to construct a measurement apparatus and test some of the predictions of 

quantum .mechanics using such a measurement apparatus. This is the main objective of this 

paper. We start by describing a Quantum Nondemolition Measurement of the photon number 

of a signal via a nonlinear Mach-Zehnder interferometer. We follow the development of the 

composite wave function of the signal and measurement apparatus to the output. We shall see 

that the photon number in the signal can be determined with a negligible probability of error 

if the gain of the measurement apparatus is large enough. Further, when this is the case, the 

density matrix of the signal, obtained by tracing over the (Hilbert) coordinates of the measurement equipment, is diagonalized. Finally, since the probability of error of measuring a particular 

photon number approaches zero, each measurement, and not the whole ensemble, can be interpreted as yielding an interpretable result. This corresponds to the von Neumann projection 

operator interprbtation. However, when the gain is not very large, the signal density matrix 

does not decohere, it is not diagonalized. This is consistent with Bohr's dictum that we can put 

the measurement results into classical language only if the gain of the measurement equipment 

is very large. 

When no measurement is performed, and the signal and "measurement" beams are passed 

on into a second nonlinear Mach-Zehnder interferometer with a Kerr coefficient of opposite 

sign, the entire action of the first interferometer can be undone; the wave functions emerge 

disentangled! This confirms the reversibility of quantum mechanics. 

We conclude with the double slit experiment. We put a nonlinear Mach-Zehnder measurement apparatus in each of the two light beams. As the accuracy of the photon number 

determination is systematically increased, the contrast of the interference fringes decreases accordingly. 
2 The Quantum Nondemolition Measurement 

Figure 1 shows a nonlinear Mach-Zehnder interferometer. The signal beam 8: at one frequency 

and the probe beam Pn at another frequency enter a Kerr medium through a dichroic mirror. At 

the end of the Kerr medium they are again separated by another dichroic mirror. A portion of the 

probe beam has been passed on directly for interference. Classically, the Kerr medium produces 

a phase shift on the probe beam that can be measured giving an indication of the intensity of 

the signal beam. Quantum mechanically, the process is described by the Hamiltonian of the 

Kerr 

I? = fiK&!ii8htb 0) 

where K is a factor proportional to the Kerr coefficient; 4 is the annihiation operator of the 

signal photons, & that of the probe photons. They obey the usual commutation relations: 

[&,@I = 1 (3) 

It should be noted that the Hamiltonian (1) does not account for a self-phase shift. This has 

been left out for convenience. A medium resonant at the sum frequency of signal and probe 

would be described by such a simplified Hamiltonian. 

The two portions of the probe beam are combined by a beam splitter with the Hamiltonian: 

As usual, one may consider the wave packets to evolve in time as they propagate along the 

system. If the beam splitter is 50150, the parameter M must be chosen 

where l is the length of the medium and v, is the group velocity, l/v, is the travel time. 

From the known Hamiltonian one may determine the evolution of the wave function 1 $), Jp) 10) 

of the three input ports. They are products at the input, and become entangled at the output. 

We denote the output annihiation operators by f* and 6. The balanced photodetector measures 

the expectation values of the difference current operator 4 = ftf* - iti and its moments[l11. 

(i) = l@12 sin(~ii!ii) = ~vl'ii!$ (6) 

The expectation value traced over the Hilbert space of the probe yields the sine of the signal 

photon operator. If the sine function can be expanded to first order, it becomes the photon 

operator. The mean square fluctuations follow from the second moment and are[''] 

if the signal is in a photon number state. This is shot noise since IPI2 is the photon number in 

the probe beam. 
/ Dichroic Mirrors , 5 Ken. Medium '4 a, out - Signal 

Subsystem 

Measurement P' 

Subsystem 

d. 

I 

FIG. 1. Schematic of nonlinear Mach-Zehnder interferometer and balanced detector. 

FIG. 2. Two nonlinear Mach-Zehnder interferometers with media of equal and 

opposite Kerr coefficients. 
The probability of error follows from the mean square fluctuations (7) that approach gaussians in the large photon number limit[ll]: 

If lnp12 >> 1, the probability of error can be made arbitrarily small. The physical meaning of 

this quantitity can be fathomed as follows. nlPI2 is the phase shift due to the probe photons, 

n itself is the phase shift due to one photon. The geometric mean of these two products has to 

be made very large. If we used fiber interferometers, these operating parameters are not easily 

achieved. Here, however, we are not concerned with the practical realization of the measurement 

apparatus, but only with the theoretical conclusions that can be drawn from it. In particular, 

we find that the probability of error can be made arbitrarily small, for IrcPl = 10, it is This 

means that each measurement has vanishing error probability. Hence one may interpret every 

measurement, and not only the ensemble, as yielding a definite result. This is analogous to the 

von Neumann projection postulate which interprets a measurement as projecting the state into 

' 

an eigenstate. Pursuing this interpretation further, we can say that a measurement with the 

Mach-Zehnder interferometer at large gain projects the signal into a photon state. 

3 The Density Matrix 

The trace of the density matrix over the measurement system part at the output of the signalmeasurement system of Fig. 1 can be evaluated for a signal wave function[ll]: 

m,kO error probability 

In the limit of large gain, the density matrix traced over the measurement equipment becomes 

diagonal at the same rate as the probability of error approaches zero (note the exponential 

factor!). Hence, again, we see that the signal acquires a classical (decohered) appearance when 

the gain of the measurement system ([&PI) is made very large. 

4 Reversibility 

If one does not perform a measurement on the probe beam, but reintroduces it in the second 

Mach-Zehnder as shown in Fig. 2, which has a Kerr coefficient of opposite sign, one can 

disentangle entirely the wave functions. This shows, of course, the reversibility of quantum 
mechanics if no measurement intervenes in the process. Of course, no measurement could have 

been undertaken, because the probe beam was completely recycled. This brings us back to 

the act of measurement. A measurement is an irreversible process that prevents recycling. 

Indeed, in the present example the probe beam is passed into a balanced detector in which it 

is absorbed. Only then can one apply the homodyne photon detection formula to evaluate the 

current operator statistics. 

Tracing, Decoherence and the Act of Measurement 

The density matrix of the signal system becomes diagonal in the signal Hilbert space when traced 

over the probe space. Tracing is a mathematical operation which, according to the postulates of 

quantum mechanics, evaluates expectation values. In the context of the derivation of the signal 

density matrix, the reduced density matrix can be interpreted as a "Gedankenexperimentn on 

the density matrix of the signal after passage through the Mach-Zehnder. Accompanied by the 

statement that the signal and probe systems would never be combined again, the entanglement 

that in fact exists between the two systems could never be reversed. In this sense, the reversibility 

of quantum mechanics is broken. In an actual measurement, of course, the apparatus works on 

the probe subspace, causes partial or total decoherence in that space, and leads "de facto" to 

an irreversible action. 

Two Slit Experiment 

Finally, 1et.u~ look at the "two-slit" interference experiment of Fig. 3. The two slits are here 

replaced by the two arms of an interferometer. A phase shifter in one of the arms changes the 

phase of the superimposed beams. If the two beams were perfectly coherent, the intensity at 

the detector would have to show perfect extinction. However, we mount two QND apparati in 

each of the arms to ascertain the number of photons passing through them individually. The 

gain of the apparati can be adjusted, thus changing the accuracy of the measurement of the 

photon number passing through each arm. One can then compute the expectation value of the 

contrast and finds it to be[l1] (see Fig. 4) 

Thus, a similar exponential factor as the one that appears in the error probability determines 

the extinction of the contrast. The factor is squared, because two measurements are being 

performed. Here again we find that the transition between the behavior of the photon as a wave 

and that of a particle is a continuous one. The accuracy of the determination of the photon 

number determines how much the photon behaves as a particle. 
FIG. 3. An interferometer representing two-slit interference and attached QND 

measurement apparati. 

FIG. 4. Expectation value of detector current versus phase and error probability 

of photon number determination. 
We started with the postulate that a proper formulation of a quantum measurement has to 

quantize the measuring apparatus as well. The quantum formalisms developed for optical components enable one to do a full quantum analysis of an optical measurement apparatus. The 

measurement apparatus of photon number with infinite gain yields results that can be described 

in classical language: photons behave as particles (since we chose a particle measurement apparatus). When the gain is not infinite, the behavior is more duplicitous, it is not what one would 

call the behavior of a classical particle. This confirms Bohr's statement that it is necessary 

to have large gain to obtain measurement results that can be put into classical language. We 

also found that a measurement with infinite gain is equivalent to a projection operation on the 

signal. 

If no measurement is undertaken, the entanglement of the signal and probe states can be 

fully undone by an inverse apparatus. 

Finally, the "double-slit" experiment can also be described in terms of partial knowledge of 

the photon number in each of the paths. If the knowledge is only partial, there can still be 

interference of the two beams. 

8 Acknowledgments 

This work was supported by the Office of Naval Research, Grant N00014-92-5-1302. 

9 References 

[I] Quantum Theory and Measurement, J. A. Wheeler and W. H. Zurek, Eds., Princeton 

University, Princton, NJ, 1983. 

[2] R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 

55, 2409 (1985). 

[3] L. W. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986). 

[4] M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, and H. J. Kimble, Phys. Rev. Lett. 59, 

2520 (1987). 

[5] M. W. Maeda, P. Kumar, and J. H. Shapiro, Opt. Lett. 12, 161 (1987); J. Opt. Soc. Am. 

B 4, 1501 (1987). 

[6] B. Yurke, P. Grangier, R. E. Slusher, and M. J. Potasek, Phys. Rev. A 35, 3586 (1987). 

[7] R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, Phys. Rev. 

Lett. 57, 691 (1986). 

[8] M. Shirasaki and H. A. Haus, J. Opt. Soc. Am. B 7, 30 (1990). 
[9] K. Bergman and H. A. Haus, Opt. Lett. 16, 663 (1991). 

[lo] N. Imoto, H. A. Hans, and Y. Yarnamoto, Phys. Rev. A 32, 2287 (1985). 

[ll] F. X. Kartner and H. A. Haus, Phys. Rev. A 47,4585 (1993). 
UNCERTAINTY RELATIONS AS WILBElRT SPACE GEOMETRY 

Samuel L. Braunstein 

Department of Physics, Technion-lsrael Institute of Technology, 

32 000 Haifa, Israel 

Abstract 

Precision measurements involve the accurate determination of parameters through repeated measurements of identically prepared experimental setups. For many parameters 

there is a "natural" choice for the quantum observable which is expected to give optimal 

information; and from this observable one can construct an Heisenberg uncertainty principle 

(HUP) bound on the precision attainable for the parameter. However, the classical statistics 

of multiple sampling directly gives 

us tools to construct bounds for the precision available for the parameters of interest (even 

when no obvious natural quantum observable exists, such as for phase, or time); it is found 

that these direct bounds are more restrictive than those of the HUP. The implication is that 

the natural quantum observables typically do not encode the optimal information (even for 

observables such as position, and momentum); we show how this can be understood simply 

in terms of the Hilbert space geometry. 

Another striking feature of these bounds to parameter uncertainty is that for a large 

enough number of repetitions of the measurements allvquantum states are "minimum uncertainty" states - not just Gaussian wave-packets. Thus, these bounds tell us what precision 

is achievable as well as merely what is allowed. 

1 Introduction 

I want to start by pointing out that the very term ('precision measurement" implicitly refers to an 

idealization of our use of quantum theory. It assumes that there are some c-number parameters 

(ordinary commuting numbers instead of operators) "out there" in the laboratory which have 

no, or at least negligible, quantum uncertainty for their values. For instance, in the case of 

the precision interferometric measurement of some path difference it is thus assumed that the 

end-mirror masses are so heavy that the initial uncertainty in their location and the quantum 

"diffusion" of their position throughout the entire measurement procedure are negligible. In this 

talk I will discuss the fundamental limits quantum theory places on our ability to determine cnumber parameters through measurement. We might, for instance, be interested in the precision 

determination of fundamental constants such as the gravitational coupling G, the speed of light 

c, etc., or of laboratory parameters such as a time difference T, some path difference D, a phase 

difference @, etc. These latter parameters are just settings on some machine in our laboratory. 

The type of machine we envision is one which has an inexhaustible supply of raw material 

which it uses to make a "train" of identical quantum states. These states will have encoded in 

them in some way the value of the parameter we seek. Thus, if our machine has some setting 

PAGE WAM mT mMm 

tRITE&TiOIVALLY 'EM& 117 
which controls the value of a parameter X, then it will generate an inexhaustible supply of states 

alternatively, if the setting is X' then it will generate a supply of new states Our job 

is to use some observable [ to determine the parameter's value. 

Let us consider a simple example where our machine sets some "location" parameter X which 

determines the location of the center of the peak of the wavefunctions generated: 

where i is a sequential label for replicas of our state generated by the laboratory machine at some 

fixed setting X. Because we are seeking the "location" of the state we might expect that the 

natural observable of position [ = i is optimal, so this is the observable we shall use. In seeking 

fundamental limits to measurement in quantum theory it is worth seeing what the Heisenberg 

uncertainty principle (HUP) says. For a single measurement we would simply take the value of 

position measured and use that as our estimate of the location parameter X of our state. In this 

way, the uncertainty for the parameter X would just be the uncertainty in our observable for this 

state, so 

by the HUP, for a single measurement. Alternatively, for N measurements (each on a new replica 

of our state) we might take the average location we observe i = ($1 + . . + SN)/N as an estimate 

for our parameter, which is an operator with conjugate momentum j = & + . . + jN so the N-shot 

HUP will read 

1 ti AX=AX~--, fi 2Ap1 

in terms of the uncertainty Am for a single state. This result shows a simple 110 improvement 

over the single-measurement HUP. 

To summarize, we have learned that precision measurements deal with the language of cnumber parameters - this being an idealization. We have learned that we must make some 

choice of observable to determine our parameter of interest, and where it exists we might choose 

the natural observable. Finally, we saw that the process of extraction of an estimate for our 

parameter from observation requires some form of data analysis - even if it be only the completely 

transparent procedure of using the value observed. 

2 Bounds to data analysis 

The previous section considered a crude multiple measurement uncertainty principle. It is surprising that much more powerful versions may be obtained from well known classical bounds to the 

efficiency of performing data analysis. The simplest of such bounds is called the Cram&-Rao lower 

bound [I], which we shall derive here. (Helstrom [2] was the first to place this bound in quantum 

language and recognize its significance; in this paper we broaden our physical understanding of 

this bound.) 

Consider an arbitrary kind of data analysis based upon the N observations xl, . . ., x~ which 

yields an estimate Xat for the parameter X via a function 

Xest = Xest (21 , . ., XN) 
which is independent of the value of the parameter sought, and for which each of the observations 

x; is "drawn" independently from the probability distribution 

To ensure that this data analysis has some connection with our parameter we impose the condition 

that on average it yields the correct value for X, i.e., 

By taking the derivative of this with respect to X we obtain 

where AXest E XWt - X. Then applying the Schwartz inequality yields the Cramhr-Rao bound 

with F the Fisher information given by 

Before we look at the consequences for this theorem from classical data analysis, we shall state 

one more theorem which applies to the specific kind of data analysis called maximum likelihood 

(ML) analysis. Fisher's theorem [3] states that as N + oo ML analysis is asymptotically efficient 

(since it asymptotes towards the Cramhr-Rao bound), i.e., 

thus, at least one method of data analysis may reach the classical bound when a large enough 

sample of data is used. What is not given by this theorem is just how large a data sample is 

required to achieves this. 

The simplest implication of the Cram&-Rao bound for a quantum system involves measurements of multiple copies of the pure state 

where X is a so-called translation parameter. The result may be directly obtained from the 

uncertainty in the momentum for this state 
yielding from the Cram&-Rao bound 

This result is rather startling, since it shows that the classical bound is always more restrictive 

than the quantum bound; even for N = 1 when we would expect to have no qualms about the 

usual HUP. Further, it directly yields a parameter uncertainty principle for an arbitrary kind 

of data analysis, and hence is more general than the N-shot HUP we discussed in the previous 

section. To see why the classical bound can be more restrictive than the HUP we turn now to the 

metric on Hilbert space. 

3 Quantum and classical metrics 

Consider two ensembles I$x) and lt,bxt) which differ only in their values of the parameter X. For 

pure states there is a natural metric, measuring the distance in Hilbert space between a pair of 

, states; since pure states are just rays of unit length, any suitable function of the angle between 

these rays will determine a Hilbert space metric. In particular, we shall choose the Study-Fubini 

metric [4] 

d (l$x), l$xt)) = 1 - I($xl$xt)12 ; 

as the states overlap more they become less distinguishable which is reflected in a decrease in the 

metric. 

For the observable i the two ensembles we are studying can be completely characterized by 

the probability distributions p(t(X) and p([lX1) (p and p' respectively). We can thus construct 

a classical metric to behave like the quantum one with respect to the distinguishability of states. 

Classical distinguishability, however, is limited by the Cram&-Rao bound, so if the nearby ensembles differ in 

the parameter X by SX then they are permitted to be distinguishable so long as NF(p) (SX)2 2 

1. This gives a natural classical metric for nearby ensembles as 

where the scale factor is chosen for convenience. This result is equivalent to the metric introduced 

by Wootters [5] to lowest order. 

To compare these two metrics, classical and quantum, we shall write the states in a descrete 

basis (labeled by j) 

Then to second order we may calculate the quantum metric to be 
where is the "momentum" operator generating infinitesimal translations in the parameter 

X. The first term in square brackets is just the classical metric. The second and third terms 

correspond to the dispersion in the rate of change of the wavefunction's phase; this is similar to 

Eq. (I), and reduces to it exactly when X is a translation parameter. 

This relationship shows that classical distances between nearby ensembles are shorter than the 

corresponding quantum distances. Because the ensembles are harder to distinguish classically, the 

Cram&-Rao bound is more restrictive than the HUP. The classical distances are shorter than the 

quantum ones. We might ask whether the greater distances inherent in the quantum metric are 

accessible at all for the purposes of precision determination of parameters? We shall see that the 

answer is typically yes. 

4 Seeing a quantum distance 

The quantum metric 

d (I$)) I$')) = d(p) P') (AV')~(&X)~ 9 

exceeds the classical one only by the variance of the rate of change of the phase of the wavefunction. 

Thus, by a suitable choice of basis the phase can be made trivial so that the classical part of the 

metric makes up the entire contribution. This may seem a little strange since we typically will 

start with what we thought was the "natural" observable for a particular parameter. Yet since 

in general the phase will give some non-zero contribution this argument tells us that natural 

observables are rarely optimal. 

We can see this surprising effect quite simply for a squeezed state. For the purposes of this 

example we will scale the units of position and momentum so that the ordinary HUP reads 

AxAp 2 1. Suppose we wish to find the location parameter X for the squeezed state 

On a phase-space diagram this corresponds to the squeezed vacuum (squeezed by e-' in along the 

x-axis) rotated counter-clockwise by 6' radians, and then displaced along the positive x-axis by 

the distance X. The natural observable for the position of the state would be = 2. So our mean 

signal would be 

S(?)=X, 

i.e., a measurement of 2 directly yields an estimate for the parameter X. Further, the uncertainty 

(or "noise") in this value after a single measurement will be 

which includes the reduced noise from the squeezed quadrature, plus an admixture (for 6' # 0) 

from the amplified quadrature. 

By contrast, suppose we use the observable i ZE it = R(B)PR~(B). The new signal will be on 

average 

sf = (2') =xcose, 
which is reduced by a trigonometric factor, and the uncertainty in this value after a single measurement will be simply 

Ax1 = e-' 

A comparison shows that the signal-to-noise ratio has been improved by using a non-natural 

observable with 

for e2' sin(28) > 1 (an improvement for all 8 # 0 is possible using a more complicated rotation 

angle for the new observable). 

In general then, so long as we can choose a basis where the phases are trivial we can "see" the 

full quantum distance for the new observable. In this ideal observable we will have a bound of 

ti 

Axest, opt 2 

~~APX ' 

which if X is a translation parameter will be just our original N- shot HUP. Now, however, Fisher's 

theorem tell's us that this lower bound can always be achieved so long as we take a large enough 

. sample size (i.e., N -, oo) and that we use the maximum likelihood method for our data analysis. 

This means that for the proper choice of observable (typically not the natural one) allV quantum 

states are minimum uncertainty states! This removal of so-called natural observables from their 

central place in quantum theory has practical implications for the precision measurement of time, 

phase; etc., where natural observables for these measurements are somewhat problematic. 

5 Conclusion 

Precision measurements naturally assume a language involving the existence of c-number parameters. The bounds to classical data analysis form a natural extension of the Heisenberg uncertainty 

principle from operator dispersions to parameter uncertainties; and generalize it to allow arbitrary 

data analysis on the results of multiple measurements of identically prepared systems. In general, 

it is found that the "natural" observables do not yield maximal information about the parameters 

sought; however, with a suitable observable and using maximum likelihood data analysis all quantum states (not just gaussian wavepackets) are minimum uncertainty states for a large enough 

number of measurements. 

References 

[I] H. Cram&, Mathematical Methods of Statistics (Princeton Univ. Press, Princeton, NJ,1946) 

pp. 500-504. 

[2] C. W. Helstrom, Quantum Detection and Estimation Theory (Academic, New York,1976). 

[3] R. A. Fisher, Proc. Camb. Soc. 22, 700 (1925) . 

[4] J. Samuel and R. Bhandari, Phys. Rev. Lett. 60, 2339 (1988). 

[5] W. K. Wootters, Phys. Rev. D 23, 357 (1981). 
TIME-DEPENDENT DISTINGUISHABILTY: 

CHOOSING TO BE A WAVE OR A PARTICLE 

T.P. Grayson, X.Y. Zou, D. Branning, J.R. Torgerson and L. Mandel 

Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627 

Abstract 

Interference experiments with connected parametric down-converters have demonstrated 

that the possibility, in principle, of identifying the photon path through the interferometer 

is sufficient to wipe out all interference, irrespective of whether the identification is actually 

made. The distinguishability of the photon path can be controlled by a time-dependent 

shutter, which leaves the choice whether the photon behaves as a wave or as a particle in the 

experimenter's hands. By contrast, in some more recent experiments involving the addition 

of a 1ow-Q cavity, each idler photon makes the choice whether the associated signal photon 

behaves like a wave and exhibits interference, or like a particle. 

1 Introduction 

In this paper we briefly review some recent interference experiments in which several non-classical 

and non-local effects show up. The basis for all the experiments is the process in non-linear 

optics known as parametric down-conversion that generates pairs of signal and idler photons 

simultaneously in an entangled quantum state [I, 21. The experiments outlined below all make 

use of two down-converters, and the prototype of the arrangement is illustrated in Fig.1 [3, 41. 

Fig. 1. Outline of the experimental set-up underlying all the experiments [Reproduced from Zou et a1 (1991)l. 

Here NL1 and NL2 are two similar crystals with a X(2) non-linear susceptibility functioning 

as down-converters. Both crystals are optically pumped by mutually coherent light beams of 
amplitudes &, & derived from the same laser beam. As a result down-conversion can occur at 

NL1 with the emission of a signal sl and an idler il photon simultaneously, or down-conversion 

can occur at NL2 with the emission of a pair of s2, i2 photons. The crystals are so oriented that 

il passses through NL2 and then is colinear with i2. At the same time the sl and s2 beams are 

brought together at the output beam splitter BS, where they are mixed, and the mixed signal 

beams fall on signal detector D,. We are interested to know whether sl and s2 exhibit mutual 

coherence and interfere. If they do, then the photon counting rate of Ds will oscillate as BS, is 

slowly translated in a direction perpendicular to its face. 

The results of the experiment are illustrated in Fig 2. [3]. It appear that sl and s2 do indeed 

Displacement of BS in pm 

oa 0.4 Q(I OO.0 1 

0 1 2 3 4 5 8 

Phase in multiples of ~r 

Fig. 2. Experimental Results of the Interference Experiment [Reproduced from 

Zou et a1 (1991)l. 

interfere (curve A ) when il and i2 are aligned, but that all interference disappears when il is 

blocked by a beam stop (curve B). The average rate of photon emission is, however, the same 

in both cases. If we argue that il induces coherence between sl and s2 in some sense, then 

this induced coherence is unusual because it is not accompanied by any induced emission. If, 

instead of blocking il, we insert a filter NDF in the path of ill as shown in fig 1, then the 

resulting interference pattern of sl, s2 is found to have a visibility proportional to the absolute 

transmissivity 7 of the filter. It can be shown that under somewhat idealized conditions, and 

with equal signal sl and sz intensities, the degree of coherence 1 y,,,, 1 between sl and s2 or the 

visibility is given by [4] 

where 'yoc is the normalized second order auto-correlation function of the down-converted field. 

70, 72, 71 are propagation times from NL1 to NL2, from NL2 to D,, and from NL1 to D,, 

respectively and 70 + 72 - 7-1 = 0 when the interferometer is balanced, in which case cyDc(0) = 1. 

It is possible to understan,d the absence of mutual coherence between sl and s2 when il is blocked 

in terms of the potential distinguishability of the photon sources. With the help of an auxiliary 

measurement with detector Di shown in Fig. 1, that does not disturb the interference between sl 

and s2, one can determine the source of the detected signal photon [3]. 
Effect of a Differential Time Delay 

As is well known, the insertion of a differential time delay ID in one interferometer arm generally 

lowers the visibility, and if ID exceeds the coherence time T, of the light the visibility drops close 

to zero. But according to Eq. (I), the effect of incrementing 72 by TD should be exactly the same 

as the effect of leaving 72 unchanged and incrementing 70 instead, even though 7-0 relates to the il 

path which is not really part of the interferometer. The reason is the quantum entanglement of 

signal and idler photons, which makes the effect of a delay on the signal virtually indistinguishable 

from the effect of the same delay on the idler. As the experimental results shown in Fig. 3 indicate, 

the observed visibility falls with increasing TD in accordance with Eq. (I), although it is the idler 

1 which is being delayed [5]. 

5 a 

delay of idler1 (ps) 

Fig. 3. Experiment Results showing the effect of a time delay imposed on the il 

idler photons [Reproduced from Zou et a1 (1993)l. 

3 Effect of a Time-Dependent Filter 

So far we have dealt only with steady state situations. An interesting variant of the foregoing 

arises if the filter of transmissivity 7(t) shown in Fig. 1 is allowed to vary in time. Indeed we may 

think of the filter as a time-dependent shutter that opens and closes at certain times. We now 

ask how the transmissivity affects the visibility of the interference contributed by a signal photon 

which is detected by D, at time t. 

This problem has recently been examined theoretically [6]. With the help of a spectral analysis 

of the fields and the filter response function it was shown that the visibility of a signal photon 

detected by D, at time t is completely determined by the filter transmissivity 7 at the earlier 

time t - 72 - 7:. Here T: is the propagation time of photons from the filter to NL2. The time 

t - r2 - T: is therefore the time at which a photon from NL1 on the way to NL2 and then on 

to D, would have passed the filter. Of course there is no such photon. But the time t - 72 - T{ is 

the time when il would pass the filter if the photons originate in NLl. Provided the filter is then 

open (7 = 1) this photon is indistinguishable from an i2 photon from NL2. It does not matter 

at all what the transmissivity is at any other time. 
4 Effect of a Resonant Cavity around the Idler Beams 

Consider the experimental arrangement shown in Fig. 4 [7]. Here a beam splitter BS; has been 

inserted in the il beam between NLI and NL2, and light reflected from BSi is detected by an 

additional detector Di such that the propagation time from NL1 to Di is also 71. It follows that an 

sl, il photon pair emitted by NL1, of which the il photon is reflected by BSi and passed to D;, will 

result in coincident detections by both D, and Di. Needless to say, these photons do not contribute 

to the signal interference because their source is known. Mirrors M3, M4 are introduced so as 

to form an optical cavity resonant with the idlers. An il photon which is transmitted through 

BSi may then propagate to M4, where it may be reflected and again pass through BSi ; it may 

then be reflected from M3 and return to BSi, where it is reflected and passed to D;. Similarly, 

an iz photon emitted from NL2 may traverse the cavity and end up being reflected by BS; and 

detected by D;. Needless to say, an il or i2 photon which has made one or more trips around 

the cavity in this way will be detected by Di later that the conjugate signal photon is detected 

by D,. Moreover, if the interferometer is balanced, the time delay between detections by D; and 

D, will be the same whether the photons originate in NL1 or NL2. As a result the sources of 

these photons are indistinguishable and interference is to be expected, whereas the sources are 

distinguishable for coincident signal-idler detections. 

IF, 

Fig. 4. Outline of the experiment with a resonant idler cavity [Reproduced from 

Grayson et a1 (1993)l. 

In order to measure the delay time intervals TD between D, and Di detections the photoelectric 

pulses from D, and D; are passed to the 'start' and the 'stop' inputs of a time-to-digital converter 

(TDC) that measures and digitizes the intervals and accumulates the data in channels determined 

by TD. The number of events accumulated in delay channel TD is then a measure of how many 

photon pairs have a time separation 70. In addition by varying the optical path difference through 

displacement of BS, we can extract the visibility of the interference pattern formed by the two 

signal beams. 

The results of the experiment together with theoretically expected values are shown in Fig. 

5. [7]. Fig. 5a gives the accumulation of photon pairs as a function of the delay TD. The peak 

centered at TD = 0 corresponds to idler il photons that emerge from the optical cavity without 
making any round trip. The observed visibility is shown in Fig. 5b. Those photons that emerge 

without extra delay are counted but exhibit no interference, because they originate in NL1. Those 

photons that emerge after one cavity roundtrip behave like waves and interfere with about 50% 

visibility. 

0 

-4 -2 0 2 4 6 8 10 12 

Relative Time Delay T,, (ns) 

Fig. 5. Experimental results showing how the photon pair rate and the visibility 

vary with the delay TD. The cavity round trip time is 6 nsec. [Reproduced from 

Grayson et a1 (1993)l 

Because the idlers are registered by the TDC after the signal photons, the experiment has 

some of the character of a delayed choice experiment. But the 'choice' is here made not by the 

experimenter but by the idler photons. Those idlers that are reflected by BSi at the first encounter 

cause the conjugate signal photons to behave like particles; those idlers that make one round trip 

before emerging from the cavity cause the conjugate signal photons to behave like waves and to 

interfere. We note that both aspects of nature are here exhibited by different photons in the same 

apparatus. 
5 Acknowledgments 

This research was supported by the National Science Foundation and by the U.S. Office of Naval 

Research. 

References 

[l] D.N. Klyshdo, Zh. Eksp. Teor. Fiz. 55,1006 (1968) [Sov. Phys. JETP 28, 522 (1969)l. 

[2] D.C. Burnham and D.L. Weinberg, Phys. Rev. Lett. 25, 84 (1970). 

[3] X.Y. Zou, L. J. Wang and L. Mandel, Phys. Rev. Lett. 69, 318 (1991). 

[4] L.J. Wang, X.Y. Zou and L. Mandel, Phys. Rev. A 44,4614 (1991). 

[5] X.Y. Zou, T.P. Grayson, G.A. Barbosa and L. Mandel, Phys. Rev A 47, 2293 (1993). 

[6] L.J. Wang, X.Y. Zou and L. Mandel, J. Opt. Soc. Am B 9, 605 (1992). 

[7] T.P. Grayson, X.Y. Zou, D. Branning, J.R. Torgerson and L. Mandel, to be published (1993). 
HOMODYNING AND HETERODYNING 

THE QUANTUM PHASE 

G. M. D'Ariano, C. Macchiavello and M. G. A. Paris 

Dipartimento di Fisica 'Alessandro Volta' 

Via Bassi 6,1427100 Pavia, Italy 

ABSTRACT: The double-homodyne and the heterodyne detection schemes for phase shifts between 

two synchronous modes of the electromagnetic field are analyzed in the framework of quantum 

estimation theory. The probability operator-valued measures (POM's) of the detectors are evaluated 

and compared with the ideal one in the limit of strong local reference oscillator. The present 

operational approach leads to a reasonable definition of phase measurement, whose sensitivity is 

actually related to the output r.m.s. noise of the photodetector. We emphasize that the simplehomodyne scheme does not correspond to a proper phase-shift measurements as it is just a zero-point 

detector. The sensitivity of all detection schemes are optimized at fixed energy with respect to the 

input state of radiation. It is shown that the optimal sensitivity can be actually achieved using suited 

squeezed states. 

1 Introduction 

Weak forces on macroscopic bodies in interferometric arrangements or, more generally, minute variations 

of environmental parameters in optical fibers are detected through the induced changes in the optical 

paths of the light beams. The detection of the induced phase-shift represents one of the most sensitive 

measurement on radiation in order to monitor such small perturbations. The back-action effect on the 

measured parameter due to the radiation pressure imposes limitations on the radiation intensity, and 

improvements of the sensitivity can only be achieved by suited preparation of the input quantum state. 

In this paper a narrowband analysis of some relevant phase detection schemes is presented (a multimode 

wideband analysis can be found in [I]). Classically a phase-shift measurement in an interference experiment 

can be directly related to the polar angle between two quadratures of one field-mode, which in turn are 

given by two output photocurrents. Quantum mechanically the quadratures of the field are noncommuting 

observables and their relative polar angle cannot be interpreted as a selfadjoint operator, as in the early 

Dirac's heuristic approach [2]. The os~illator phase is not an observable in usual sense, and the problem 

of identifying its quantum dynamical counterpart has provoked many discussions in the literature (see, for 

example, [3, 4, 51 and references therein). Among the numerous attempts the limiting procedure of Pegg 

and Barnett [3] has become the most popular technique, because it allows the evaluation of expected values 

with very simple and reliable rules. However, despite its simplicity and effectiveness as a mathematical 

tool, this approach has no obvious physical interpretation, and leaves most of the conceptual problems 

on phase detection still open. The actual problem of a phase measurement description does not concerns 

with a definition of a selfadjoint operator, but with an operative recipe to evaluate the phase statistics in 

a real measurement, starting from the knowledge of the density matrix of the input radiation. On these 

lines the most appropriate approach to the phase detection of the field is the quantum estimation theory 

of [4]. Even though it easy to show that this method is equivalent in the end to the Pegg and Barnett 

procedure (see [8] for more details), nevertheless it provides a physically meaningful scheme for the phase 

measurement where all conceptual problems disappear. Despite it has long been recognised as the most 

natural framework for analyzing any kind of quantum detection, the quantum estimation theory has not 
gained the necessary popularity yet, perhaps due to the fact that its main ingfedient-the probability 

operator mmme (P0M)-is generally a nonorthogonal spectral decomposition, and thus appears to be 

in conflict with the conventional dictum of quantum mechanics that only "observables "-i.e. orthogod 

POM9s-w be mwured. This point has been well clarified in some papers (see for example [8]), where 

it is shorn tht nonorthogod POM9s correspond to actual observables on a larger PIilbert space wEch 

includes &o modes pert the measuring apparatus (all together referred to as Qproben). It is dea~ 

that this assertion provides the proper una setting for the operational point of view of [Ill, whese the 

dependence of the measwed operator detection scheme just corresponds to the involvement of the 

probe variables in the measuring process, involvement which becomes unavoidable when the phase of the 

field is detected. 

Any quantum measurement needs a classical final stage and for measurement on radiation this is essentidy 

a photodetection. Moreover a proper measurement of the quantum phase has to be related to the detection 

of a quantity which itself is a phase, i.e. is defined on the unit circle. In this sense we distinguish between 

two main different classes: the genuine phase detection schemes and the measurements of a single phedependent observable. In the former class, the phase-shift of the field is related to the polar angle between 

two measured pho\ocurrents which, in turn, correspond to a couple of two conjugated quadratures of 

the field. Such scheme is the only viable one for phase detection, and corresponds equivalently to either 

heterod or double-homodyning the field. This also clarifies the subtle nature of the phase itself 

which, despite being a single real parameter, nonetheless requires a joint measures of two conjugated 

operators. In contrast, in the second class of measurements, only a single observable is detected-typically, 

when homodyning a single-quadrature of the field. Here we want emphasize that a single-quadrature 

measurement cannot be used to infer the value of the phase, because the knowledge of a quadrature would 

require au additional measurement on the field-essentially its intensity-which unavoidably would destroy 

the information on phase. Thus, the single-homodyning scheme can be used only as a zero-phase monitoring 

technique, which, however, is the essential of a typical interferometric measurement. In order to stress the 

operational nature of POM approach here we also present, as an example, a numerical simulation of a real 

experiment, reproducing the classical output photocurrents due to a particular quantum state, and then 

evaluating the phase statistics as the polar angle distribution. 

After selecting a measurement procedure-either ideal or feasible-the statistics of the detected phase can 

be further improved at fixed total energy by looking for optimal states of the field. We show that the r.m.s. 

phase sensitivity versus the average photon number fi is bounded by the ideal limit At$ N A-l, whereas 

for the feasible schemes the bound is A4 N fi-2/3, in between the shot noise level 84 - fi-'l%d the 

ideal bound. The latter can actually be achieved by single-homodyning suitable squeezed states, but only 

in the neighborhood of a fixed zero-phase working point. The requirement of detecting the whole phase 

probability distribution makes the proper phase measurement less sensitive than in the case of a zero-point 

detection and the state achieving the two bounds are dramatically different: they are weakly squeezed 

(about 2% of squeezing photons) for the double-quadrature measurement, whereas they become strongly 

squeezed (50%) for one-quadrature detection. 

Sect.2 is devoted to the theory of phase measurements, with a detailed analysis of the various schemes. 

Subsect. 2.1 is a brief review of the quantum estimation approach. Subsect. 2.2 presents some remarks 

and criticisms about the different definitions of sensitivity. Detection of the phase through simultaneous 

measurement of two quadratures of the field are discussed in Subsects. 2.3 and 2.4, which are dedicated 

to double-homodyne and heterodyne detection (it is shown that the two schemes are fully equivalent). 

Subsect. 2.5 examines the measurement of the quantum sine and cosine of the phase, with a comparison 

between the present quantum estimation approach and earlier treatments [7].In Subsect. 2.6 we analyze the 

homodyne detection scheme. In Sect .3 the optimal states of the field are given, which maximize sensitivity 

for all the schemes of Sect.2, also indicating how to actually achieve such states. 
2 Quaraturn measurement of the phase 

At presatly it is fairly universally accepted that the quantum description of a phase measurement on 

a single mode of the e. m. field be approached by means of the usual concept of observaible. 

As a matter of fact, even though int operators can be defined on the Fo 

carn appropriatdy describe the actd statistics of phase measurements. The quant 

of Helstrom [4] provides the most general-nonetheless operational-framework to 

measurement in the quantum context, and, in particular, the detection of a phase differences between 

synchronous oscillators. The main ingredient of such theory is represented by the mathematical concept 

of probability-operator-valued measure (POM) on the Hilbert space 'lis of the system, which extends the 

conventional description by selfadjoint operators. Using a notation which is fdar to physicists--even 

though not strictly legitimate from the mathematical point of view [8]- given a set of (generally complex) 

parameters s to be measured, a POM di(z) is a self-adjoint measure with the following properties 

where Z denotes the space of the parameters z. Eqs.(2.1) assure that using a POM one has an operational 

recipe which univocally relates the density-matrix state 1 of the system to a probability distributions of 

the parameters z corresponding to a particular experimental setup. In formulas one has 

Also a set of selfadjoint operators A can be defined 

h 

and, more generally, operator functions f (A) as 

When the POM di(z) = Is) (zldz is given in terms of orthogonal Iz)'s, then it corresponds to the customary 

measure of the commuting observables A, whose corresponding selfadjoint operators obey the function 

calculus 

On the contrary, the relation (2.5) no longer holds true for a generic nonorthogonal POM. As a consequence, 

the selfadjoint operators A only provide the expected values of the parameters s, whereas the higher 

moments of the probability distributions differ from the corresponding moments of the operators A 

themselves, and can be evaluated only through Eq.(2.2). One should notice that, despite the POM's 

generally describe measurements that do not correspond to observables in the usual sense, nonetheless 

there is no conflict with the basic assertion of quantum mechanics that only observables can be measured. 

In fact, the Naimark theorem assures that every POM can be obtained as a partial trace of a custommy 

projection-valued measure on a larger Hilbert space [$I which itselfrepresents the original system interacting 

with an appropriate measuring apparatus. Upon denoting by I+(z)) E 'lis @ 7ip a complete orthonormal 

set of eigenvectors of commuting selfadjoint operators acting on the enlarged space-including also the 

apparatus (probe) space 'lip-the probability distribution 
comesponds to the BOM on the mbert space 7ds of the system only 

di(s) = trp (bpl$(~))($(s)l) ($(e)lbpI$(&)) e (2.7) 

Notice that the above atensions of the system HPilbert space-hence, physically, the experinnental 

seallization of the POM-is not necessaaily unique. 

Hnn Subsect. 2.1 we review the qumtum. esthation theory of the phase which leads to the opt 

representing the most accurate measurement. This, however, is only an ideal knit as no viable schmes 

hplmenting such POM has been envisaged yet. Therefore, in Subsects.2.3 and 2.4 experimentdy 

achievable detection schemes which correspond to a sub-optimal POM (double-homodyne and heterodyne) 

are dyzed in detail, wheseas in Subsect.2.6 the customary homodyne detection is considered. The latter 

exits &om the present quantum estimation approach, however it is in order, due to the relevance of this 

scheme in any intdmometric setup. 

The quantum estimation theory analyzes the possible strategies for estimating a parameter on the basis 

of an error-cost-hction: the optimum POM is the one which minimizes the total average cost. For a 

maximum-likelihood criterion, the optimum POM for the phase is 

lei@) being the Susskind-Glogower phase states [7] 

It is worth noticing that in the present case the maximum likelihood criterion is equivalent to the Gaussiancost-function one for the bounded case (sin2-cost) [4]. Such generality of the optimal POM db(4) justifies 

the term Canonical Measurement here adopted for this POM approach. Also notice that the Pegg and 

Bmett [3] approach is totally equivalent to the present one as regards evaluations of statistics. However, 

there is no physical interpretation for the mathematical tricks on which their method relies. Some examples 

of commuting pairs of self-adjoint operators achieving the optimum P OM (2.8) on a system-probe Hilbert 

space have been proposed in [9] and in [5]: however, 'no viable method for experimentally implementing a 

corresponding setup has been devised yet, and hence the POM (2.8) only represents an ideal limit. 

Corresponding to the optimal POM (2.8) one defines the selfadjoint phase operator 

1 4 dP(4) = -i c (-)"-'- 

n-m In> (ml 

n#m 

and the squared operator 

Notice that, as announced, 

PzP, 

and more generally f3) # f(4). Such fail of the operator function calculus also holds true for the 

selfadjoint operators defined through the experimentally feasible non-optimal POM's. The fact that there 

is no orthogonal optimum POM for the phase, physically corresponds to the impossibility of defining the 

measurement of the phase independently on the apparatus. This assertion clarifies and formalizes the 

operational nature of the phase detection which has been pointed out by Mandel et al.in [ll]. 
2 Phase Sensitivity 

Usudy the sessitivity of a measujrement of a parameter say z E IEe is assumed equident to the r.m.s. of 

the eqmhentd probabst y distribution dP(a), namely 

On the other had, the phase miable 4 is defined in the bounded domain [-u,u) with 2x-periodicity: 

tMs pecdar propwty of the phase has lead many authors to the conclusion that the r.m.s. of the phwe is 

not the appropriate quantity to be considered as an evaluation of the phase sensitivity, 

not inmiat undw phase miable translation 4 - q4 + X. Thus, different definitions for the sessitivity 

have beem adopted, whjich would be equident for an unbounded Gaussian-distributed miable. Here, 

Qbftw aiticd review- such quantities, we show that an operational definition of a phase measupment 

proacedue leads uneqiuivocdy to adopt the r.m.s. itself as the correct sensitivity parameter. 

a. %"Bacrse dispepso'ooz 0[12, 131. 

Dispersion D is defined as follows 

where cn are the coefficient of the number representation of the state and the sine and cosine operators are 

defined according to Eq.(2.4) as follows 

and coincide with the sine and cosine operators of Susskind and Glogower [7]. The definition (2.14) 

follows from elementary error-propagation calculus, the phase 4 being regarded as a function of the two 

%dependent variables" sin 4 and cos 4 as follows 

In Eq.(2.17) the correct logarithm branch is selected in order to obtain the desired domain for 4. A part 

- the minor point that Eq.(2.14) would lead to dispersion D = 1 for constant distributions-instead of 

Atp2 = u2/3-the main criticism is that sin 4 and cos 4 cannot be considered as independent variables, 

because they correspond to a noncommuting pair of operators which are jointly measured when detecting 

4. 

b. Reciprocal peak likelihood 64[5]. 

The peak likelihood P(#) is the maximum height of the probability distribution. Its inverse, namely 

has been introduced in [5] as a measure of the width of the distribution, coherently with the maximumlikelihood strategy used in the quantum estimation theory. Here, the following criticisms are in order: i) 64 

is a locd criterion, namely it checks only one point of the distribution, whereas there is no control on the 

global behaviour as, for example, on the eventual occurrence of high tails. The most degenerate situation 

occurs when the tails are so high that the distribution itself converges to the P(4) = 1/2r apart from 
one point with infinite probability density and zero integral [14], thus leading to vanishing 64 instead of 

64 = r2/3; ii) the coherence of this sensitivity definition with the maximum-likelihood strategy [5] cannot 

be considered as a did argument, in view of the aforementioned equivalence between the likelihood strategy 

and the (quasi)-Gaussian one; iii) recent numerical results [S] have shown that the simulated sensitivity 

does actually not correspond to 64. 

c POM p.m.#. (A?$) 

Given a physical apparatus (or an ideal detector) one has a corresponding POM and, in turn, a probability 

distribution dP(4) according to Eq.(2.2). Such probability has a r.m.s. error (2.13) given by 

Here (. . .) denotes the ensemble quantum average on the system space Xs, and the operators 4 and 

depend on the considered POM (for the optimum POM they are given in Eqs.(2.10,2.11)). Notice that 

there is no ambiguity in choosing between the two operators @ # p, because Eq.(2.19) directly follows 

from the probability (2.2). For a random-phase state-namely a constant probability distribution--one 

correctly has (A?$) = u2 13. 

As regards the problem of invariance under phase shifts, here we stress that this actually is not a problem. 

In fact, the only concern is the correspondence between experimental and theoretical quantities, and the 

circular topology of the phase arises at both experimental and theoretical levels in the same way. Whatever 

procedure is considered for measuring the phase, the information on it has always to be inferred from a 

joint sine-cosine measurement, and hence the experimental equipment itself has to be tuned on a selected 

2%-window. Once the domain is fixed, the experimental noise is, by definition, the r.m.s. noise on such 

domain. Therefore, different choices of the 2?r-window actually lead to different experimented amounts of 

noise, and also theoretically the r.m.s. noise has to be evaluated on the chosen domain (hereafter we will 

always use the [-u, u) window). 

2.3 Double-homodyne Detection 

The double-balanced-homodyne [11] (DBH) detection provides a way for simultaneously measuring a couple 

of field-quadratures for one mode of e. m. field. The schematic diagram of the experimental set-up is 

reported in Fig. 1. There are four 50-50 beam splitters and four identical photocounters, and a. 1p/2 phase 

shifter is inserted in one arm. The mode supporting the phase is a, whereas a; stable reference for the 

phase is provided by a local oscillator (LO) which is synchronous with a and is prepared in a highly excited 

coherent state Iz). 

The DBH scheme can also perform a phase measurement, however with a probability distribution which 

does not correspond to the ideal case due to unavoidable addition of "instruzmental" noise. The DBIP 

phase distribution is obtained through the following procedure. Each experimental event consists of a 

simultaneous detection of the two difference photocurrents il = h6 - A5 and i2 = .ii4 - hg which "trace" 

two field-quadratures. Each event thus corresponds to a point plotted in the complex plane of the field 

amplitude. The phase relative to the event is nothing but the polar angle of the point itself. An experimental 

histogram of the phase distributions is thus obtained upon dividing the plane into small ("infinitesimaln) 

angular bins of equal width 64, from -u to u, then counting the number of points which fall into each bin. 

In formulas, one has the statistical frequency Pn for the n- th bin 8, [-?r + 7264, -u + (n + 1)64) 

1 P - -(# ofevents withIl =pcos$,12 =psin#,#E 6,) , *-N 

where p = 41; + I; and N is the total number of experimental points. 

In Fig. 2, as an example, a computer simulation of the above experimental procedure is illustrated for a 

squeezed state with equal number (n) = 10 of signal and squeezing photons. The experimental histogram 
Figure 1: Outline of scheme of a double-homodyne detectors 

(lo4 events) is compared with the theoretical results from the POM for the DBH detection. This can 

be obtained as follows. The difference photocurrents il and i2 are commuting operators with factorized 

probability P(I1, 4) = P(II)P(12). Introducing the reduced current f = f/lzl for each homodyne detector, 

one has the probability distribution in terms of the Fourier-transform of the generating function for the 

moments (eiG) 

The phase distribution is the marginal probability integrated over the modulus p 

Using Eq.(2.21) one has 

6s being the density matrix of the mode a (the system) and 

the density matrix of the probe. From Eqs.(2.6,2.7) one can see that the "experimental" POM is obtained 

upon tracing over the probe Hilbert space Xp, thus obtaining the operator which acts solely on the system 

space Xs 

rlzl dp dfio(+) = d4 j0 / - J*"' trp{is B PP e ip(.ii -P cos d)+iu(fz -p sin 4) 

o -+I 2~ -rlz1 2~ 1. 
Figure 2: Computer simulation of the DBH experimental procedure for a squeezed state with equal number 

(n) = 10 of signal and squeezing photons. The experimental histogram (lo4 events) is compared with the 

theoretical results from the POM in Eq.(2.26) 

Using the coherent state resolution of the identity, the following closed formula is obtained (the detailed 

derivation is reported in the appendix) 

where F(x) is the Eulero9s gamma function. 

The POM in Eq.(2.26) for the DBH detector corresponds to an effective measured phase operator which 

is given by 

and the squared one 

h 

needed for evaluation of the instrumental sensitivity (A&). For any state of the mode a one can simply 

verify that 

(A~D) 2 (69 9 (2.29) 

namely the DBH scheme adds extrinsic instrumental noise, as it does not implement the optimal canonical 

measurement of the phase. However, we stress again that the DBH detection is the best available method 

for detecting the phase. In Fig.3 a comparison between the canonical (ideal) and DBH (feasible) phase 

probability distributions is given for the same state of the computer simulation in Fig.2, showing that the 

former is sharper and higher than the latter. 
Figure 3: Comparison between the ideal and double-homodyne phase probability distributions for the same 

squeezed state of Fig.2 

2.4 Heterodyne Detection 

The fist proposed method to perform simultaneous measurements of two field-quadratures was the 

heterodyne detection. Here we synthetically analyze this scheme, only in ordef to make a connection 

with the double-homodyne detector and show that that the two apparatus are completely equivalent from 

the point of view of the measured physical quantities. The input field EzN impinges into a beam splitter 

and has nonzero photon number only at the frequency wo + WZF. The local oscillator works at the Merent 

frequency wo, and the output the photocurrent hUT is measured at the intermediate frequency WZF. The 

measured photocurrent is given by 

where Ef denote the usual positive and negative frequency components of the field. The component at 

frequency WZF is given by 

For a nearly transparent beam splitter, and in the limit of strong LO in the coherent state lz) one can 

define the reduced complex current Y 

jL lim ~-'~ouT(wzF), 

v-9 l.lzl+- 

7 = cost. 
where 7 = Iz(dm. In this limit the expression of 3 is given by 

y = Izl-'(a!b, + aibi) + vanishing terms , (2.33) 

where the subscript 41 and i refer to the signal, LO and image component of the field respectively, ar are 

signal modes, b the LO modes, and the vanishing terms denote operators which do not give contfibutions 

in the strong LO limit. In the double-homodyne in the same limit the role of the complex 

is played by 

where subscript 1 refers to the input signal and subscript 2 to the local oscillator, whereas bo is the vacuum 

mode at the unused port of the beam splitter which contains the input signal. The fully equivalence 

between heterodyne and double-homodyne is apparent when comparing Eq.(2.33) and Eq.(2.34). Aa in 

the double-homodyne case, now the real and imaginary parts of the current trace the two conjugated 

quadratures ad and of the signal mode. In [16] the POM of the heterodyne detector has been 

derived in a different context, leading to the same result obtained for the double-homodyne in Subsect.2.3. 

We notice that the actual sources of extrinsic added noise are the vacuum modes ai for the heterodyne 

detector and bo for the double-homodyne: the other vacuum modes are totally irrelevant in the limit of 

strong LO. 

2.5 Measurement of the phase quadratures 

The PDM approach naturally leads to well defined operator functions of the phase which obey the 

trigonometric calculus at the operator level, and, hence, also at the level of expectation values. In parti&, 

the sine and cosine operators are defined as in (2.16). Such definitions coincide, in the case of optimum 

POM, with the sine and cosine operators i and 2 introduced by Susskind and Glogower [7] 

where i+ denote the raising and lowering operators (+In) = In + I), 4- = (i+)t. Notice, however, that 

this equivalence between operators fails for higher powers, namely 

Here some remarks are in order, regarding relevant differences between a conventional measurement of a 

single phase-quadrature-say the cosine &and a joint measurement of both sine-cosine quadratures which 

have been analyzed in previous Subsections. A single phase-quadrature measurement leads to violation of 

the trigonometric calculus for expectation values. In fact, for a general density matrix state b one has that 

whereas for a joint measurement one obtains 

We stress again that, however, the linear operators coincide in the two cases, and thus one gets the 

same average values. However, the probability distribution of the out comes from single phase-quadrature 

measurement exhibits unphysical features for nonclassical states, whereas the probability distribution from 

the joint measurement does not. In the single-quadrature measurement one has 
where the dgenstates of C are given by [7,15] 

Ic) = {:(I - e2)-'1' C sin[(n + 1) arccos c] In) 

n=O 

On the other hand, the Radon-Nikodym derivative of the joint measurement POM's leads to 

- c2)-? C(m[filn) erp [i(n - m) arccos c] , 

n,m 

for the optimum POM case, whereas for double-homodyning one obtains 

P(c) = [bdfi~(#)!?] = i(l - e2)-! ~(mlfil~) r(9 + 1) exp [i(n - m) arccos e] . (2.42) d+ dc ~r n,m Jaz 

The differences between the single-quadrature and double-quadrature probabilities become striking for 

isotropic states, as, for example, the vacuum or a general number state. In this case the above distribution 

should be compared with the Radon-Nikodym derivative of the constant distribution 

which is a concave function and has poles at the c = f 1 stationary points of the cosine. The probabilities 

(2.41) and (2.42) coincide with (2.43) for number states, whereas the probability (2.39) has the opposite 

curvature for the vacuum state, and oscillates fastly around the function (2.43) for nonvacuum number 

states. These undesired physical features disappear for highly excited coherent states, where, however, the 

main quantum features are lost. 

2.6 Homodyne Detection 

This Subsection is devoted to the customary homodyne detector, which, despite it exits from the present 

phase estimation treatment, however it is the most relevant device in any interferometric setup. Actually, 

the homodyne detector belongs to the class of the zero-point measurement schemes, and thus is not cr 

measurement of phase. The balanced homodyne scheme measures one quadrature of a field mode, which 

in turn is related to its phase difference with respect to the synchronous LO. Generally one is interested 

in the measure of the phase shift x of the signal state 

where, without loss of generality, the input state is assumed of the form 

The expectation value of the quadrature is given by 

The quadrature 4 is proportional to the cosine of the phase with a proportionality "constant " 

which can be evaluated from the knowledge of the fixed input state. Notice that, however, when the 

present scheme is regarded as a measure of the phase of the state I$), itself the state-depending "constant 
t be preventively measured without destroying the Infomation on phase. 

ch does not need any knowledge of the "constant" is the 4 - # = ~/2 

t the maxim=-derivative zero-current working point. Thus, it is not possible to detect 

Wwenee, the sensitivity depending on the particular input signal, whereas, a suitable 

=&dm is needed to follow the working point. A convenient description of the homodpe 

detector in view sf the above condderations is given in [IT], where the zero-point (zero-field) probabaty 

&atributicpa. ir revisited sas a sort of a phase probabillity distribution. 

The phme msitivity d the homodpe detector can be straightforwardly obtained from mor propagation 

dcdus us- the r&tion 

h the Biteratme on interferometry, One can see that the q5 - XI = ?r/2 worhg point 

3 Optimd states for phase measurements 

The design a phase measurement needs optimization of both the detection scheme and of the quantum 

. state which es the phase idomation. The former is the main task of quantum estimation theory, 

which leads to an ideal scheme to be compared with the feasible ones. The latter9 which is the main 

concern of this Section, depends on the detection scheme itself, and should account for the actual physical 

constraints, mainly the total power impinged into the state. Therefore, the problem is that of opt~zhg 

the r.m.s sensitivity A) Y JT (A42) for fixed average photon number, and depending on the particular 

detection scheme. 

In the following we consider, without loss of generality, a zero average phase state, with real coefficients on 

the number basis, namely 

The state optimization problem is to minimize a quantity of the form 

with the constraints 

The (red symmetric) matrix A = {A,,,} depends on the detection scheme. In particular, for n # m one 

has (A,, = 0) 

(-)"-" 

An,m = (canonical) (n - m)2 ' 

(-ln-rn r(=p + 1) An," = (DBH) . (n-m)2 &x! 

The method of Lagrange multipliers reduces the problem to that of minimizing the following expression 

7r2 

F({cn}; &PIfi) = 7 + 2 An,m cncm + A (5 C: - 1) + (2 nc; - ii) 

n#m n=O n=O 
with respect to (c,), X and /3 being the Lag~ange multipliers. The variational problem (3.6) is that of a 

ric form and is equivalent to the eigenvalue problem 

tric mtrk M = (Mnm) given by 

Eq.(3.7) for the matrix (3.8) can be numerically solved upon suitable truncation of the Rilbert space 

3.1,. The absolute minimum corresponds to eigenvalue X = ?r2/3, and A E [0, dim3t8/2] turns out to be 

a decreasing function of the running parameter /3 E [O, 11. Notice that one should consider only average 

values A < dim%,/2, such that the nurnber distribution has vanishing tail at n = dim%,, in order to avoid 

undesired nmericd boundary effects. 

Figure 4: Phase and number probability distributions of an optimal states of fi = 20 for both ideal and 

DBH detection 

3.1 Canonical Measurement 

For ideal measurement of the phase, the best phase states obtained through the above optimization 

procedure, lead to the simple power-law 

in agreement with results of [18]. The proportionality constant actually increases very slowly as a function 

of fi, and one has a variation of few percent for two decades of Ad. Eq.(3.9) can be compared with the 

result of [I$], and with the theoretical bound Ad N ll(e6) [19] obtained by means of information-theory 

arguments. One should notice that essentially the same result can be obtained for large fi (A > 10) using 
squeezed states whwe the squeezhg photon number is optimized as a hction d the avepage totd nmbm. 

out that the opt states have only N 3.7% of squeezing photons (see Rg. 5). Tis res& ir 

qdte differat from the customary 50% opthd squeezing number (which dso holds tme for the homodpe 

smitivity of the Md-Zhder htderometw [20]). 

m 

A? 

el" 

V 

\? 

A* 

d 

v 02 @=? 

11~(v~1~[1111(8111 

1Ided 

.'. 

Double-Woeass 

- /.-.-1, - 

-1. '8. a. 

r -.--... -.-.-. 

- .- .- 

squeezing photon number as a function of the average total number for both ided md 

DIBH detection. 

As eqected, an actual measmement of the phase does not achieve the ided semitivity (3.9). h the case 

of doublehomodpe (o~ eqddently heterodyne) phase detection, the resulting power-larw is 

which is obtained by ically solving Eq.(3.7) for matrix M given in Eqs.(3.8) and (3.5). In Fig. 4 the 

optimized states for both canonical and DBH detection are compared for an equal fixed average photon 

number A = 20. Om can see that the number and phase probability distributions are qualitatively susr, 

however the DBH optimum states are slightly sharper in the number distribution and larger in the phase 

one. Also the best DBH states are essentially indistinguishable from squeezed states which sre opt~zed 

in the squeezing photon number as a function of the average total number (see Fig. 5). In this case, onlJr 

less than N 2% of squeezing photons turns out to be optimal. 
Figure 6: Uncertainty product versus the average photon number for optimal states. 

3.3 Heisenberg Uncertainty Product for phase quadrature 

The customary Heisenberg uncertainty relation 

AAAB 2 f l~r {jj[~, B]) I (3.11) 

refers to the situation in which the quantum system is prepared in a state with fixed uncertainty say AA 

and the other observable is measured. For the case of a joint 3-B measurement, however, a generalized 

uncertainty relation hold, where the 112 factor on the right side of Eq.(3.11) is dropped, corresponding to 

an added noise of 3 dB [21]. For the phase-quadratures one has the commutation relation 

corresponding to the joint-measurement uncertainty relation 

In Eq~(3.11) and (3.13) the uncertainties are defined in the usual way, namely A02 = (d2) - (6)2. On the 

other hand, in the POM approach the correct uncertainty (namely the measured quantity) is defined as 

A02 = (5) - (o)~, where 5 # o2 is defined as in Eq.(2.4). In general, by means of Schwartz inequality, 

one obtains 
I Ideal detection I Double-homodyne I Homodyne I Ideal Sine detection 

I I I 

Optimired (1.38 f O.O1)ii-l.OOf ." (1.00 f 0.01)fi-0.6bi.01 $%-I 

Squeered 

Stat- %.,/A - 4% A.,/ii 2% fi,,/fi = 50% 

'Table 1: Aspptotic smsitivities versus the average photon number (results from numerical calculations 

are given with error esthation) 

ThesePore, one should not expect that the optimum states for phase detection achieve the rninhunr 

~lneertainty product (3.13), even though phase detection corresponds to a joint sine-cosine measurement. 

Pn Fig. 6, the ' ' uncertainty product in Eq.(3.13) is compared with the actual uncertainty product 

of the optimal states for canonical measurement. One can see that the minimum uncertainties are 

never achieved, despite canonical detection is ideal. In the same figure the uncertainty product of the 

optimal states for DBN detection is also reported for comparison, showing additional noise due to nonideal 

measurement . 

3.4 Homodyne detection 

FOE homodp detection the well known sensitivity A# = iii-I is achieved only near the zero-current 

working point. As a consequence, it happens that it is better than the sensitivity achieved by the true 

phase measurement, either in the double-homodyne or the ideal case it self. This is due to the fact that the 

measuremmt of the fidd-quadrature near the zero-current working point partially underestimates the tails 

of the phase distribution at qb = f T. The latter are enhanced by large squeezing, and thus one also finds 

that the optirnal number of squeezing photons is only a few percent of ii for the true phase measurements, 

whereas it is 50% for single homodyning. It is interesting to notice that the measurement of a single 

phasequadpatwe, say sinqb, also exhibits a small optimal squeezing fraction (N 5%), as, in some sense, it 

is more faithful obsefarable than the field quadrature. In Table1 the above results are reported along with 

the pbe sensitivity for other quantum state and different detection schemes in the limit of large average 

photon numbers 6. 

4 Conclusion 

We have analyzed ideal and actual detection schemes for the quantum phase, We have also considered the 

homodyne detectors as it can be used as zero-phase measurement apparatus. The working conditions for a 

phase detectors have been discussed showing that in an actual measurement the phase shift corresponds to 

the polax angle between two real output photocurrents. We have analyzed in detail the double-homodyne 

scheme of [Ill, giving the POM of the apparatus. The equivalence of this schemes with heterodyne one in 

detecting the phase it has also been shown. 

A. critical revision of various adopted definitions of sensitivity has been reported, we have concluded that 

the usual r.m.s. noise is the right quantity to be considered. 
The sensitivity of the detection schemes Bas been optimized at fixed energy with respect to input 

qumtum stater of ra&ation. We have shown that the sensitivity versus the average photon nmber 6 is 

bounded by the ideal limit A4 N A'l, whereas for double-homodyne detection the bound is &bD A-~/~, 

in between the shot noise level A4 N md the ideal bound. The optimal states achieving the best 

sensitivity for hed energy have been numerically obtained, and we have shown that they are very close 

to coherent states weakly squeeged either for ideal detection or for double-homodyne. This resdt, which 

is in contraet with the 50% of opthal squeezing photons for single-homodyning, is due to the sensitivity 

of the double-homodyne detection to the whole phase distribution including tails, which are 

increasing squeeeing. 

Appendix: Eduation of Double-Homodyne POM 

The phase distribution is the margind probability integrated over the modulus p , namely 

Using Eq. (2.21) one has 

Bs being the density matrix of the mode a (the system) and 

the density matrix of the probe. From Eqs. (2.6) and (2.7) one can see that the POM is obtained upon 

tracing over the probe Hilbert space 7ip, thus obtaining the operator which acts on the system space Hs 

only 

Using the coherent state resolution of identity one has 

where R is the matrk element 

R = (~,~,O,OIU+~~(~~+~'~)~~O,O,Z,W). 

In Eq. (A.6) 0 denotes the unitary evolution operator of the detector, which acts on the state (A.6) as 

follows 

The explicit expression of the matrix element R is given by 
g the strong 10 limit 111 -r m and introducing the complex variable a = f (v + ip)eids) one gets 

Substituting Eq.(A.9) into Eq.(A.5) leads to 

0°11 

= @CaG&pt - @a 1p exp(p&eib)la) (ale""" e~p(p&te-~b) 

P=o 

Using: the cohermt resolution of the identity and integrating over p one obtains 

where r(z) is Eulero's Gamma function. The normal ordered representation of the vacuum state 

leads to 

From Eq.(A.13) one obtains the POM of the detector in form of a double series 

Alternatively, using the I?-function integral representation one can write 

e-d e~*at 10)(01e~-i4a = 2 ~Wpdp ~pe+)(~e'dl. 

a 
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[21] E. Arthurs, M. S. Goodman Phys. Rev. Lett. 68, 244'7 (1988) 
Joint and Angle-eavariant Spin Measurements 

with a Quadrupole Magne.$ic Field 

Bans Martens and Willem M. de Muynck 

Theoretical Physics Group, Department of Physics, Eindhoven University of Technology, 

PO Boz 513, 5600 MB Eindhoven, The Netherlands. 

Abstract 

We study a Stern-Gerlach type setup, with a quadrupole magnetic field, for neutral 

particles of arbitrary spin. The Namiltonian is of a form proposed for joint measurements of 

incompatible observables. The measurement results are discussed, showing the limitation of 

such Harniltonians. Some remarks are made on the relevance of covariance as a criterion for 

measurement schemes. 

1 Introduction 

The canonical form for the Hamiltonian of a measurement interaction is (operators are careted) 

where the subscripts o and p denote object and pointer, respectively. The operator A, is measured, whereas iLb is conjugate to the read-out observable 6,. One of the simplest measurement 

arrangements, often used as an example [l, 2, 31, is the Stern-Gerlach, where an inhomogeneous 

magnetic field effects an interaction amounting to [2, 41 

Here the coupling constant. K is proportional to the strength of the inhomogeneity. The spin 

degrees of freedom represent the object, and the spatial degrees of freedom are the "measuring 

device". Thus we measure the spin in the x-direction, by reading out the x-momentum component. 

If we assume both fix and 8, to be conserved in the absence of interactions, the Harniltionian (2) 

causes in the Heisenberg picture (ti = 1) 

fix (t ) = fix (0) + K&,t . (3) 

Thus, as soon as t is larger than the Sx-width, the spatial part is separated into s packets, according 

to 6, eigenvalue. Conversely, a read-out of fix means an accurate measurement of 6,. 

In order to be able to measure two incompatible observables jointly, Arthurs & Kelly [5] 

extended the basic scheme (1) to 
In this interaction Hamiltonian, 8; and 4 are two compatible pointer system operators, with 

read-outs $ and hP, respectively. The operators A, and &, not necessarily compatible, are to be 

measured. Arthurs & Kelly applied (4) to joint position-momentum measurements. They found 

that the probabilities of finding a result (a, b) are given by 

where la) is a coherent or squeezed state, depending on the interaction balance rcA/rcB. Therefore 

the a marginal is related to the x probability distribution by a convolution 

with a Gaussian f, and analogously for b and p. This relation, providing the basis for the interpretation of b in terms of x, we have termed "non-ideality" elsewhere [6]. Sets of operators such 

as !la) (a1 , which generate probability distributions but are not orthogonal, are called positive 

operator-valued measures (POVMs) [7]. 

If we combine (4) with (2), we get a Hamiltonian like 

Such a Hamiltonian may be realized using a quadrupole magnetic field, which around the origin 

in the x, y-plane satisfies 

Am (0,0,zy) +- ii o; (x,-y,~). (8) 

In the present paper we will investigate the properties of a measurement scheme based on the 

quadrupole Stern-Gerlach (7), where K = K, = K,. Neutral particles of arbitrary spin s will be 

used. We shall focus especially on its relation to (6) and (5). 

2 General description 

If we consider the quadrupole Hamiltonian (7), we see that its rotational symmetry immediately 

implies that 4, = 2, - C?* is conserved. It is therefore profitable to change into a polar momentum 

representation (p, p). Denote the eigenvalues of 8, by m,. Now we can eliminate cp by writing the 

m, component of the state IQ) as 

The Harniltonian, seen as an operator on the (2s + 1)-dimensional spin Hilbert space, then 

becomes 

We have taken K = 2m = 1, without loss of generality [B]. Note that r?y3z is not Hermitian, but 

then neither is i&; the overall expression (10) is Hermitian. 
FIG. 1. Two views on the measurement results. (The Gaussian spatial part initially 

had variance Ap2 = 0.01 and L, = 0.) 

a. Typical output distribution. The density of markers indicates the momentum 

3 probability density at t = 1. The input state was a s = t state, with mx = - - 2' 

b. Matrix element (ilk(pX)li) for s = 4 as a function of time. The quality of the 

measurement can be characterized by the integral of the matrix element over px > 0. 

Note that this value does not approach 1. 

We integrated eq. (10) numerically. In view of the fact that the process is intended as a spin 

measurement, we took the initial state to be a product of spatial and spin parts (denoted by 

subscripts T and a when necessary), 

The spatial part Id), was taken to be a axially symmetric Gaussian. The final state then turned 

out to be structured into a number of expanding rings, one for each Iml; only rn = 0 (for integral 

spin particles) leads to a hump remaining around the origin. Fig. 1.a is a typical example, where 

s = i, and we have 3 rings. In the figure the spin part was initially directed in the +-direction, 

with m, = -2 2, and thus we see that the distribution is peaked in the left part of the middle ring. 

More generally, states with spin initially directed in the xy-plane lead to correspondingly oriented 

distributions; only spins initially in the z-direction evolve into complete ring distributions. In the 

rings the spins are directed roughly outward, with appropriate magnitude. 

The scheme is meant to be seen as a spin measurement. Accordingly, we trace out the spatial 

variables and generate the outcome probabilities from the spin density operator i, by means of a 

POVM ~(p, p) on spin Hilbert space, 

probge (P, p) = Tr[iju M(P, $41 ; 

MP, p) = I (P TP I 8 0) 0) I ; 0) = P- (12) 

In the next two sections we will discuss some properties of the POVM M@, p). 
The output variables of the measurement are (p,,p,). Naively, one would suspect on the basis 

of (3) and (7) that a read-out of 6, can be seen as a measurement of b,. Accordingly, we are 

interested in the p, marginal of a, 

Consider first the s = case. Here the spin operators anti-commute. Thus the interaction 

Hamiltonian (7) satisfies 

$&St = & ; 8 = iy ~~(23,) . (14) 

Hence, if the initial spatial state is i,-symmetric, the p, marginal of the final (p, ,py ) distribution 

must depend only on 6,. Accordingly we may write (13) as 

where the fm are positive functions and hm(0) denote the projectors onto the & eigenstates. Thus 

we get an analog of the convolution (6), reproducing the basic result (3) with additional noise terms 

that do not depend on the spin degrees of freedom [4]. As is easily verified, an analogous relation 

holds between PY and dy. In fig. 1.b we plotted the probability that a particle with m, = +; will 

give the measurement result p,. We see that the noise terms assure that the measurement quality 

is limited, in contrast to (3). This is necessary on account of the uncertainty principle [9, 101, as 

we are jointly measuring the two incompatible observables 6, and 6,. 

Nevertheless, (15) means that an unambiguous relation between the li, and b, distributions 

exists. It allows for the m, estimation from p, that we aimed at [ll], albeit an imperfect one. In the 

spin-$ case, the spin observables are Fourier-pairs [12] and therefore close analogs of the positionmomentum pair studied by Arthurs & Kelly [5]. Accordingly, the above conclusion matches theirs. 

We might think that for s > this result may be generalized. In fig. 2.a the diagonal elements 

of the POVM (13) in 5, representation are plotted vs. p,. We see that indeed the various m, 

values are roughly correlated to different p, regions, as expected. Thus, it appears that from p, 

an estimate of m, can be made, just as for s = f. But there is a catch: neglecting the p2-term 

(strong or impulsive interaction approximation), it follows after some calculation that 

sin rt 

pix (t) = pi, (0) + 5,(0) [t COS~ 8+ - r sin"] 

1 sinrt cos rt - 1 + Gy(0) [1 - t] sin 28 + b,(0) r 

sin 8 ; 

where we used the polar position representation (x, y) = (r cos 8, r sin 8). Clearly 6, contains extra 

spin terms that do not commute with 8,. Indeed the non-diagonal elements of k(px) are not zero, 

as is evidenced by fig. 2.b. Consider again (16). The desired effect, t6,, is contained in the second 

term , whereas the last two terms are the problematic ones. As the rings expand more and more, 

the ! terms will decay roughly as l/t2. Still, even as t 4 oo, a significant term containing 6, 

remains: k and 8, are incompatible. 
FIG. 2. The POVM ~(p,) in bX representation. (Data as in fig. 1.) 

a. Diagonal matrix elements (rnl&(p,) jm) = (-rnj&f(-px) j - m>- 

b. Independent non-zero non-diagonal matrix elements (mlh(-p,) lm'). 

But all the problematic terms in (16) are seen to contain a factor sin28 or sin8. Denote 

reflection in the sr-plane by i,. The sine terms in (j,) vanish if we choose the spatial part of 

the initial state i,-symmetric, like we do in our calculations. But if we consider (s), terms with 

factors like 8: sin2 28 emerge. Such terms do not vanish, so that higher moments do not commute 

with &. Only if s = f , we have 6: K i so that (fiz) commutes with ex after dl, in accordance 

with our earlier result (15). 

But in what sense can we consider this scheme to be a measurement of & if s > i? Clearly 

no analog of (15) holds; only the expectation value ($,) is free of incompatible spin terms, so that 

we can use the measurement only to estimate the 5, expectation value. Higher moments, or even 

m, probabilities, cannot be established. 'Thus it is a 6, measurement only in the weak sense of 

expectation value estimation [l 11 . 

The measurement is therefore not generally a useful joint measurement. But it may have another 

use. Remember, the Hamiltonian (7) is rotationally symmetric. If we now consider the outcomes 

in polar coordinates, it is easily derived that the POVM k(p, rp) is angle covariant: 

Covariance is a criterion that is often used to characterize classes of rneasurelfients e.g. time 

or photon phase measurements. Here we therefore speculate that &f@, cp) realizes some kind of 

spin-angle measurement. Define the projector onto the eigenstate of the operator cos 8&, $ sin 86, 

with eigenvalue m as &($). Then we choose as spin-angle observable 1131 
where the summation runs over all relevant lml values, and c, = & for m = 0 and otherwise. 

Accordingly, c,am(0) defines a POVM that may be seen as a "spin-anglen observable [7]. We can 

now attempt to link (18) to (17). As in the previous section, this is possible for s = f. There a 

convolution-type relation between the realized angle measurement and the ideal (18) holds [4]. But, 

as in the previous section, this cannot be generalized to higher spins: k and 18 are incompatible, 

although the incompatibility is generally smaller than that of fig. 2.b. Thus, although the POVM is 

angle-covariant, it is not clear in what sense the "angle of spin" is measured, if at all. Analogously 

we may therefore conclude that photon-phase covariance and time covariance must also give many 

POVMs without unambiguous interpretation. Like spin-angle covariance, they are weak criterions. 

5 Acknowledgments 

This work was sponsored by the Foundation for Philosophical Research (SWON), which is financially supported by the Netherlands Organization for Scientific Research (NWO). 

. References 

[I] R. Feynman, R. Leighton & M. Sands, The Feynman Lectures on Physics, vol. 3 (AddisonWesley, Reading, Mass., 1965). 

[2] G. Ludwig, EinfQhrung in die Grundlagen der Theoretischen Physik, vol. 3 (Vieweg, Braunschweig, 1976). 

[3] M. Scully, B.-G. Englert & J. Schwinger, Phys. Rev. A 40, 1775 (1989); R. Levine & R. Tucci, 

Found. Phys. 19, 175 (1990). 

[4] H. Martens & W. de Muynck, J. Phys. A 25, 2001 (1993). 

[5] E. Arthurs & J. Kelly, Bell Syst. Techn. J. 44, 725 (1965). 

[6] H. Martens & W. de Muynck, Found. Phys. 20, 255 (1991). 

[7] C. W. Helstrom, Quantum Detection and Estimation Theory, (Academic, NY, 1976). 

[8] A suitable choice of time and position scales removes the factors in front of k, $ and p. 

[9] H. Martens & W. de Muynck, Found. Phys. 20, 355 (1991). 

[lo] P. Busch, Phys. Rev. D 33, 2253 (1986). 

[ll] H. Martens & W. de Muynck, Phys. Lett. A 157, 441 (1991). 

[12] J. Schwinger, Proc. Nut. Acad. Sc. 46, 570 (1960). 

[13] The Fourier transform of &, is another candidate, but it fares even worse than (18) for the 

present (low) spin values: the incompatibility with M is much larger. 
TWO NEW KINDS OF UNCERTAINTY RELATIONS 

Jos Uffink 

Department of History and Foundations of Mathematics and Science 

University of Utrecht, P.O. Box 80.000, 3508 TA Utrecht, the Netherlands 

Abstract 

We review a statistical-geometrical and a generalized entropic approach to the uncertainty principle. Both approaches provide a strengthening and generalization of the standard 

Heisenberg uncertainty relations, but in different directions. 

The purpose of this note is to introduce two approaches to the uncertainty principle which have 

been developed recently, a statistical-geometrical approach and a generalized entropic approach. 

But before we go into this, let us consider why one would need a new approach at all. In other 

words, what is unsatisfactory with the traditional approach to the uncertainty principle? In the 

standard textbook approach the uncertainty principle for position and momentum is expressed by 

the inequality 

tt V$ : A+PA+Q 2 - 

2 (1) 

or more generally: 

for arbitrary observables A and B. Here, A*A etc. is defined as: 

There are three problems. First, uncertainty relations as (1) or (2) presuppose that all observables for which one wants to write down an uncertainty relation can be represented as self-adjoint 

(or at least normal [I]) operators. Unfortunately this is not always the case. Notorious examples 

are time and energy, and phase and photon number. Further, in relativistic quantum theory, even 

the status of the position observable becomes dubious. There is no self-adjoint position (vector) 

operator for photons [2]. 

Secondly, note that the right-hand side of (2) still depends on $. It may become zero, even if 

A and B do not commute. In fact, this always happens in an eigenstate of A or B. Then, taken 

as a general statement about A+AA+B, the inequality only says that this product is greater than 

zero for some states and equal to zero for others. That, however, is true also in classical physics. 

To read off more from (2), one needs to know the state. But then, when $ is given, one can also 

calculate A*A and A*B directly, without using the inequality at all. 
Even in the case where the right-hand side is always strictly greater than zero, as in relation 

(I), there are further problems, due to the properties of the standard deviations. In the definition 

(3) of the standard deviation, the probability density is integrated with a quadratic weight factor 

that puts most emphasis on the tails of the distribution. As a result, the standard deviation can 

become very high, even if 99% or more of the probability distribution is concentrated in a very 

small interval, and the remainder is located in long tails, as e.g. in a Breit-Wigner lineshape. 

Thus a large standard deviation does not necessarily prevent a probability distribution from being 

very sharply concentrated, and a bound on the product of standard deviations by itself does not 

prevent both observables from being as precisely determined as we please. In the next sections we 

ask whether there are more stringent inequalities that improve on the above aspects. 

2 Statistical-geometrical approach 

It is usual to assume in quantum theory that the state of the system is given. But in this section 

we consider an inverse problem. Suppose we don't know the state of the system. Our problem is 

to make a statistical inference about this state from given measurement results. 

For definiteness, let us assume that some partial information about the state is given: it belongs 

to a given set of (pure) states labeled by an index parameter 8. To be more specific, it is asumed 

that these states are generated by some unitary group: 

where A is a self-adjoint operator. We can think of this set of states as describing a curve in state 

space. The problem of statistical inference is now equivalent to that of estimating the value of 8. 

It is clear that a detailed discussion of this estimation problem should involve the kind of measurements performed, the results obtained, and criteria distinguishing 'good' from 'bad' estimates. 

However, even without going into details of statistical theory [3,4], it can be made plausible that a 

fundamental bound for the estimation accuracy is obtained by considering the overlap I($el$e+se) I. 

If this overlap is high the states resemble each other much and a typical measurement result which 

would be probable or improbable in one state would likewise be probable or improbable in the 

other. Then one cannot expect to discriminate the states by any measurement procedure. It is 

only when the overlap begins to fall off that there are observables whose probability distributions 

for the states $@ and differ enough to allow for accurate discrimination. 

This suggests the following definition. Choose some fixed value P < 1 and define the estimation 

inaccuracy S*8 as the smallest value of 68 for which 

Due to the particular choice (4), this overlap does not depend on the value of 8. One can then 

show: [5, 6, 7, 81 

V : S*8A*A > 275 arccos ,B (5) 

This then represents a useful uncertainty relation. It says that an unlimited increase in the 

estimation accuracy of 8 is only possible at the expense of an increased spread in A. 

Several remarks are called for. First, these relations are applicable to any one-parameter unitary group. Obvious examples are the translations in time, represented by the evolution operators 
U(t) = exp(-itH/h) or the translations in space, exp(ixP/fi), where P is the momentum operator; or we can consider an angle of rotation and angular momentum, or phaseshifts and photon 

number. In short, we find statistical uncertainty relations of the type (5) in every case where A 

is the generator of a unitary group and 0 the group parameter. This approach is in fact ideally 

suited for a relativistical treatment of quantum theory in which one starts from the construction 

of unitary groups from the symmetries of the system. 

Secondly, relation (5) is asymmetrical; S$O is an inaccuracy of estimation of a parameter (i.e. 

a c-number). A4A on the other hand is the r.m.s. spread of a quantum mechanical observable 

(self-adjoint operator). Since one does not need a pair of operators to obtain relation (5) there 

are no problems when such a pair does not exist. 

However if one does exist, e.g. in the case of non-relativistic position and momentum, it is 

possible to take advantage of that fact. Then there is a second, independent, uncertainty relation 

for the spread in position and the estimation accuracy of the parameter in the group of kicks 

{U(p) = exp(-ipQ/ti) ) , i.e. shifts in momentum. This restores symmetry between position and 

momentum. More importantly, we note that the position operator mimicks the parameter x of 

the translation group in the sense that 

($zlQl$z) = x (6) 

(assuming ($IQI$I) = 0) i.e. the position operator acts as an unbiassed estimator of the location 

parameter. From this it follows: [9] 

combined with (5), where A is interpreted as momentum, this result implies the standard uncertainty relation (1). In fact, as a bonus, we obtain (1) not only for the position operator proper, 

but for any other operator acting as unbiassed estimator of x as well. 

There is only one problem of those nlentioned in the previous section that is not solved by the 

relations (5): they still rely on one standard deviation, and thus become useless for states in which 

this diverges. To fix this problem, the standard deviation can be replaced by an interquantile range, 

i.e. the smallest size W4(A) of an interval W on which a fraction a < 1 of the total probability 

distribution for A is concentrated: Jw I($la)12da = a. A variation of the proof of (5) gives [6, 81 

S40W+(A) > h arccos 1sp-a 

a 

Finally we note that this concept of estinlation inaccuracy fits into a general geometrical 

approach to statistical inference on the basis of the Fisher information metric [4]. Let it suffice 

here to note that this metric equips Hilbert space with a statistical distance between states which 

equals 

d($, $7 = arccos I ($ I $') I 

and that the geometrical background of (5) is the simple fact that the distance between $e and 

$# (i.e. the right hand side of (5)) is less than the length of the curve (4) connecting these points. 

This also point the way to how the relations are to be generalized in cases where the curve is not 

generated by a unitary group. 
3 Generalized entropic approach 

For a discrete probability distribution p = (pl , . . . p,), with p; 2 0, xi p; = 1, the Shannon entropy 

represents, roughly speaking, a measure of whether the distribution is 'spread out' or 'peaked'. If 

A and B are quantum observables for which an uncertainty principle holds, it is natural to ask 

for a lower bound of the sum of the Shannon entropies for the probabilities 1 (+la;) l2 and (+lbj) l2 

[lo, 111. Here we assume a discrete spectrum and la;) and Ibj) denote the eigenstates of A and B. 

It turns out that 

This entropic uncertainty relation limits the concentration of both probability distributions by a 

bound which is independent of +. 

There is a class of expressions that share many properties with the Shannon entropy, and also 

represent useful measures of 'peakedness' or concentration: 

Their properties have been studied extensively by Hardy, Littlewood and Pcilya and by Renyi [12]. 

( - log MT is known as Renyi entropy.) Special cases are: 

Where # counts the nulllber of elements in a set. The generalized entropic uncertainty relation 

then reads: [ll, 131 

which contains the relations (9) as the special ca.se with I. = s = 0. 

Remarks: The above inequalities apply to any pair of discrete observables and yield a nontrivial bound iff these observables do not share an eigenstate. (A condition which is slightly 

stronger than mere non-commutativity. ) I11 the case of a two-dimensional Hilbert space, the most 

restrictive bounds are obtained by the choice 9- = -112, s = m or V.V. 

Secondly, in the proof of (12) it is not necessa.ry to assume that observables are represented by 

self-adjoint operators. It is sufficient to demand the existence of the sets of "eigenstates" {la;)) 

and {I bj) ) , possibly non-orthogonal, such that 

Thus, the approach of this section is also applicable if one accepts unitary operators [14] or even 

more generally, positive-operator-valued measures ( POVM's) [15] as bona fide representations of 

observables. As an example, we mention a phase observable below. 
In the case of continuous observables, it appears necessary to replace (8) and (10) by a relative 

notion of entropy: 

where apldp is the Radon-Nikodym derivative of a probability measure p with respect to a 'background measure' p. These expressions reflect whether the probability distribution p is concentrated 

in comparison with p. In the discrete case, the absolute entropies are recovered by choosing p to 

be the counting measure #. In the case of continuous observables it is natural to take for p the 

Lebesgue measure, and apldp becomes an ordinary probability density. 

For continuous nondegenerate operators A and B a theorem of Haussdorf and Young analogous 

to (12) leads to [17, 131 

MT($, AIp)Ms(($, Q, IP) := (/ I($I~)I~(~+T)~~)~~~ (1 l(+lb)~~(l+~)db) l" 5 sup l(aIb)l2 

(Still assuming r = -s/(2s + I).) For position and momentum a slightly stronger inequality 

MT($, P lp)Ms($, Q la) 2 2(1 + r)(l + 2r)-(1+2T)1(2T)(2nh)-1 is obtained by a theorem of Beckner. 

These inequalities are all sharp for Gaussian 'minimum uncertainty' states and strictly imply the 

standard uncertainty relations (1). 

Let N be the photon number operator of a single electromagnetic field mode, with eigenstates In) and C In) (nl = I. A description of a phase observable by means of a POVM @ w$s 

constructed by Lkvy-Leblond and by Susskind and Glogower [14, 161 from the non-orthogonal 

improper "eigenst ates" 

This yield a resolution of identity S,2" ddl4)(41 = I in analogy with (13). Here one finds an 

analogous uncertainty relation 1 

where 

Let us finally compare the two different approaches. Both improve on the standard approach in 

the sense that they yield bounds which are state independent and strictly imply A+PA+Q > h/2. 

The statistical/geometrical approach is restricted to conjugate pairs of quantities: timelenergy, 

location/momentum, anglelangular momentum etc. All such pairs are treated, on the same 

footing, as consisting of a parameter and an observable. It is essential that the observable is 

represented as a self-adjoint operator because of its role in (4). The approach is relativistically 

invariant. The entropic approach, on the other hand, is applicable to pairs of arbitrary observables, 

not necessarily conjugate pairs or even self-adjoint operators. It is sufficient that they do not share 

eigenst ates. However, these uncertainty relations do not seem to have a relativistic generalization. 

Also, the last problem mentioned in the introduction is not completely overcome in this approach. 

IN the example of the Breit-Wigner distribution one still has the result MT(plp) = 0 for r 5 0 

and the inequality is not very restrictive. 
References 

[l] M. Bersohn, Am. J. Phys. 34, 62 (1966). 

[2] A.S. Wightman, Rev. Mod. Phys. 34, 845 (1962); K. Kraus, in Uncertainty Principle and 

Foundations of Quantum Mechanics, W.C. Price and S.S. Chissick (eds), (Wiley, London, 

1977). 

[3] A. Bhattacharyya, Bull. Calcutta Math. Soc. 35, 99 (1943); C.R. Rao, Bull. Calcutta Math. 

SOC. 37, (1945). 

[4] W.K. Wootters, Phys. Rev. D 23, 357 (1981). J. Hilgevoord and J. Uffink, Found. Phys. 

29, 323 (1991). 

[5] L. Mandelstam and I. Tamm, J. Phys. (USSR) 9, 249 (1945); G.N. Fleming, 

Nuovo Cimento A 16, 263 (1973); K. Bhattacharyya, J. Phys. A 16, 2993 (1983). 

[6] J. Uffink and J. Hilgevoord, Found. Phys. 15, 925 (1985); J. Hilgevoord and J. Uffink 

in Microphysical Reality and Quantum Formalism, A. van der Merwe et al. (eds) (Kluwer, 

Dordrecht, 1988) and in Sixty-two years of Uncertainty, A. Miller (ed), (Plenum, N.Y., 1990). 

[7] J. Anandan and Y. Aharonov, Phys. Rev. Lett. 65, 1697 (1991); L. Vaidman, Am. J. Phys. 

60, 182 (1992); A. Uhlmann, Phys. Lett. A 161, 329 (1992). 

[8] J. Uffink, Am. J. Phys. (to appear). 

[9] J-M. L6vy-Leblond, Phys. Lett. A 111, 353 (1985). 

[lo] D. Deutsch, Phys. Rev. Lett. 50, 631 (1983); 1.Bialynicki-Birula, inQuantum Probability and 

Applications 11, L. Accardi and W. von Waldenfels (eds.), (Springer, Berlin, 1984); K. Kraus, 

Phys. Rev. D 35, 3070 (1987). 

[ll] H. Maassen and J. Uffink, Phys. Rev. Lett. 60 1103 (1988). 

1121 G.H. Hardy, J.E. Littlewood and G. Pblya, Inequalities, (Cambridge U.P., Cambridge, 1934); 

A.Renyi, Wahrscheinlichkeitsrechnung, (Deutscher Verlag der Wissenschaften, 1962). 

[13] J. Uffink, Measures of Uncertainty and the Uncertainty Principle, Ph. D. thesis, University 

of Utrecht (1990). 

1141 J-M. Lkvy-Leblond, Ann. Phys. 101, 319 (1976). 

[15] E.B. Davies, Quantum Theory of open Systems (Academic Press, London, 1976); A.S. Holevo 

Probabilistic and Statistical Aspects of Quantum Theory (North-Holland, Amsterdam, 1982). 

[16] L. Susskind and J. Glogower, Physics 1, 49 (1964). 

[17] 1.1. Hirschman, Am. J. Math. 79, 152 (19.57); I. Bialynicki-Birula and J. Micielski, Comm. 

Math. Phys. 44, 129 (1975); W. Beckner, ,41111. Math. 102, 159 (1975). 
MEASUREMENT OF VERY SMALL PHASE FLUCTUATIONS BY MEANS OF 

THE OPERATIONAL APPROACH 

A. Bandilla 

A,G "Nichtklassisclie Stralilung" der MPG, Humboldt-Univeristat Berlin, 

Rudower Cliaussee 5, 0-1 199 Berlin, Germany 

Recexitly Noh, Foughres and Mandel (NFM) [I] have improved tlie operational approacli to the qua~ltum pliase problem substantially and measured the phase dispersion 

of coherent liglit dowli to very small mean photon numbers of tlie order of This has 

prompted many other investigatiolis and clarified some important questions in relation 

to what is. actually measured. Although their treatment is rather general, we confine 

ourselves here to the case of a strong local oscillator (LO) axid reproduce their measurement scheme in Fig. l. Surprisingly enough, this simultaneous measurement of the 

sine and the cosine of the pliase difference is completely equivalent to an old proposal 

to measure the pliase after strong linear amplification [2] realized experimentally by the 

Welling group [3]. The reason for this rests on the fact, that in both cases the results are 

determined by tlie Q fu~ictioii of the signal. This was shown for amplification in [4] and 

for tlie measure~nent after beam splitting by Lai and Haus [5] and aIso in [6, 7, 81. The 

, measured pliase dispersion is given by 

where the double bracltets mean a classical average over the radius integrated Q functio~l 

of the signal aiid the b,', are defined by 

These coefficielits are all s~naller than one and broaden therefore tlie phase distribution 

of the pure state 1?1, > 

In s1iowil1.g that the b,i result from tlie calculation of tlie dispersion with the help of the 

NFM operators we found tlie expa~ision [8] 

that evidently proves the above ~neiitio~ied property b: < 1 and lim,,,, bi = 1. Tllis 

exl>a~ision converges excellently ancl is very useful because eq. (1) recluces to the PeggBarnett (PB) dispersioli [9] by putting all b,', equal to one, the zeroth approximation of 

eel. (4). 

Now, very small phase fluctuatio~isilppose great photon number fluctuations and the 

last can cliange the iilterference signal. Of sucll kind is the situation for states near to the 

so-called pliase optimized states (POS) [10,11] which are characterized by the relation 
Vacuum 

I 

Input 2 

h/4 

plate 

Input 1 

Vacuum 

Fig. 1 Outlitze of thc cxperii~reiitc~l schet~rc used By Noh, Fougires aizd Marzdcl where the sirze 

aitd tlre cosiize of the phasc difleretzce are nreasured sit~wltaizeously. BS; are identical. 

50/50 Beain sp1iltct.s uiid Dj arc photodctectors. Iitput 2 is the local oscillator. 

where N is the illeall photon number. Note that coherent states lead to 

For clarity the subscripts PB indicate that eqs. (5) and (6) are Pegg-Barnett dispersioils 

i. e., they are calculated by rel>lacing the classical average in eq. (1) by the qua~ltum 

ense~nble average < ei@ > wit11 the Hermitiail phase operator @ [9]. 

The question is now how to deteriniiie such sinall dispersioils from ~lleasuremeiits 

in the operatio~lal appro:tcll, i. e., by illeasuriiig via the radius integrated Q fuilctioii 

(eqs. (1)) (2) and (4)). The answer is that wit11 the help of the expailsion (4), some 

liinitatio~ls and an additional illeasureille~lt of tlle photo11 distributioil in the scheme of 

Fig. 1 it is possible to infer the PB dispersioil of states with a (69)%oml~arable to 

the value of eq. (5). Note that tlle illeasureinent of eq. (1) aloile cailiiot give adequate 

iilforinatio~l about pliase dispersioils near to POS. This is illustrated in Fig. 2 for twophoton coherellt states (TCS) that call be optiillized to come close to POS for certain 

degrees of,squeezing s at a fixed illeall photon ilu~llber N [12]. 

The followi~lg invc:stigations are rather ailalogous to calculatio~ls made by Ritze [13] 

in his differe~lt prol~osal to il~easure extreillely sillall phase fluctuations. First, one has to 

find a suitable reference 1)liase in orcler to inake the c,, in eq. (1) real. This correspo~lds 

to < sill@ >= 0, where wc? suppress the phase of the LO. Seconcl, oilly such input fields 

call be adillitted that allow a, truncation of the expa~lsio~l (4): 
G .WiT 

coherent state 

Fig. 2 Plzase dispersions for TCS at a mean photon rzurnber of N = 1500 and increasing 

squeezing paranteter s. Note that s = 1 describes coherent ligltt. The Q function based 

dispersion (by)$ starts at 1/2N, coines close to the coherent state value 1/4N and increases again. The Pegg-Barnctt dispersion (bv)gB begins at 1/4N for s = 1, decreases 

sharply and reaclzes its tnitzimunz near to tlte POS level at strongly di#erent s t~alues than 

t 6v)& 
Third, we assuine a smooth c,, distribution and approximate the c,,+l by 

where ci, denotes the derivative of c,, with respect to n. With eq. (8) we call deterillille 

the second suill in eq. (7): 

where .iz. is the photoil ilu~nber operator and < . . . > the ~lor~nal qualltu~ll average. Due 

to the Schwarz inequality we find in addition [14,2] 

and therefore (bv)bB > c'i. It turns out that POS fulfil (6~);~ >> c'i and our truilcatioil 

assumption requires the same. Therefore 4 can be neglected in eq. (9) and we obtain 

evelltually 

It is evident that for the deterilliilatioil of the averages over the ~luinber operator expressions on the right-hand side of eq. (11) we need the kilowledge of the photo11 number distribution. This is not surprising because very small pl-lase fluctuatiolls require 

enhailced photon ~lulllber fluctuatiolls that affect the iilterference signal. During the 

lneasure~lleilt of such phase dispersioils we nlust co~lseque~ltly also illo~litor the photon 

iluinber fluctuations. 

The situatioil cliallges remarkably, if we onlit the first bean1 splitter in Fig. 1 and put 

the signal into each chaililel as illustrated in Fig. 3. This makes sense if we co~lfiile ourselves to two- yhoton coherellt states (TCS) because than the radius integrated Wigiler 

function is illeasured as was shown for coherent states by Freyberger ancl Schleich [6] and 

generalized to T(;S by Leonhardt and Paul [Is]. The lneasured dispersion is now 

where the subscript W poiilts to the Wigiler fuilctio~l and the A!L call be expaildecl illto 

the series 

(-1)l1 1 - (-1)" -5 A;~ = 1 + -+... . 4(n + 1) + 32(n + 12S(n + 

Eq. (13) shows clearly the oscillatio~ls about one what anloullts to the fact that eqs. (12) 

and (6) give exactly corresponding results in this order (l/N). However, for TCS's near to 

POS (eq. (Ci)), as introduced in [12], the next order, 1/N< is dominating. Here, the term 

1/32(n+ 1)hf the A,', plays an importmt role. The result is that the measurecl disl>ersion 

(bP)& call be smalle~ thail the corresponding PB result. Thus the measurement following 

the scheme of Fig. 3 yields for C-oherent states with N >> 1 the Pegg-Barnett result while 

second-0;-der effects can change the measured disgersion for ogtiillized TCS drastically 

for moderate. M(S 50). For very large N (6~):~ and (Sp)k coiilcide for TCS [lG]. 
vacuum 

Fig. 3 Modified l~otrzodyne dctcction scherrzc with two irzput ports whcrc two identically prepared 

sig7zals are itzcidcrzt. Tlic X/4-platc inakcs, as itz Fig. I, a n/2 pilase sltift in ordcr to 

measure simulta~~cously the siile and the cosine of the phase difleretzcc. Tlte sigttal is here 

rzot corztarrtinated by thc vacuunt frorrl tlic urzused port. Tllus, therc is iro physical reason 

for aizy broadetzit~g as it1 Fig. I. 

[I] J. W. Noh, A. Fougires and L. Mai~clel, Phys. Rev. Lett. 67, 1426 (1991); 

Pllys. Rev. A 45,424 (1992) and Pllys. Rev. A 46, 2840 (1992). 

[2] A. Bandilla and I-I. Paul, Ann. Physik 23, 32:3 (1969) 

[3] H. Gerhardt, H. Mfelliug a~ld D. Fri5licl1, Appl. Phys. 2, 91 (1973) and H. 

Gerharclt, U. Biichler and G. Litfin, Phys. Lett. 49 A, 199 (1974) 

[4] W. Schleicl~, A. Bandilla and H. Paul, Phys. Rev. A 45, 6652 

[5] Y. Lai and H. A. I-Iaus, Quailturn Opt. 1, 99 (1989) 

[6] M. Reyberger and IV. Schleich, Phys. Rev. A47, R 30 (1993) 

[7] U. Leoilhardt and H. Paul, to be published in Phys. Rev. A, 1992 

[8] A. Bandilla, to be published in the Sllecial Issue of Physica Scripta, 1993 

[9] D. T. Pegg and S. M. Barnett, Europhys. Lett. 6, 483 (1988) and S. M. 

Barnett and D. T. Pegg, J. mod. Opt. 36, 7 (1989) 

[lo] G. S. Summy and D. T. Pegg, Opt. Commun. 77, 75 (1990) 

[ll] A. Bandilla, H. Paul and H.-H. Ritze, Qua~ltulll Opt. 3, 267 (1991) 

[12] A. Banclilla, Opt. Commun. 94, 273 (1992) 

[13] H.-H. Ritze, Opt. Commun. 92, 127 (1992) 

[14] H. Paul, W. Brull~~er u. G. Richter, Ann. Physik 12, 325 (1963) 

[I51 U. Leonhardt and I-I. Paul, to be published in Phys. Rev. Lett., 1993 

1161 A. Balldilla a. H.-H. Ritze, sub~llitted for publication 
TH%E QUANTUM MEASUREMENT OF TIME 

Scott R. Shepard 

Department of Physics, 

Baylor University, Waco, Texas 76798-731 6 

Abstract 

Traditionally, in non-relativistic Quantum Mechanics, time is considered to be a 

parameter, rather than an observable quantity like space. In relativistic Quantum Field 

Theory, space and time are treated equally by reducing space to also be a parameter. 

Herein, after a brief review of other measurements, we describe a third possibility, which 

is to treat time as a directly observable quantity. 

1 The Measurements of Space and Momentum 

Here we postulate the existence of position eigenstates, such that their corresponding eigenkets (defined by 2lx) = xlx), where 2 is the position operator) resolve the identity operator 

-OD 

for the Hilbert space that they span, 5 tt {lx) : x c [-=,+=I}. Thus, any state I y) E Y 

can be expressed as a weighted sum.of position eigenstates: 

+- 

I Y) = jhlx) Y(X) (2) -00 

where ~(x) (XI w) is Schrodinger's time independent wavefunction, the magnitudesquare of which gives the probability of obtaining the value x in what we call the 

measurement of 2. 

An infinitesimal translation in physical space, denoted by ~(SX), is defined by its 

mapping of the position eigenstates: 

f(6x)lx) = lx + Sx), (3) 

and linear canonical momentum, denoted as j, is defined to be the generator of these 

translations: 

lim f(6x) = fx - i$Sx / h . 6x+O 

It is important to note that in writing equation (3) we are assuming that space is unbounded, i.e. x c [--,+-I. From equations (3) and (4) we can easily prove [I] that there 

is a Fourier transform relation between y(x) and ~(p) = (PI y), where $1~) = plp). 

Rayleigh's theorem [2] then guarantees that 1 is a norrnalizable distribution and 
hence (since it must also be positive semi-definite) it corresponds to a probability distribution function (pdf) that describes the "measurement of j." This can also be seen by 

virtue of the fact that a normalized IV(~) 1 implies that the momentum eigenkets must 

also resolve the identity operator: 

+- 

In the case of the measurements of space and momentum our descriptions are complete in 

the sense that both operators are Hermitian thereby ensuring that each has an orthogonal 

set of eigenkets so that the corresponding wavefunctions can collapse. 

2 "Time-like99 Measurements 

Time evolution, by an infinitesimal amount, st, denoted as fi(~t), isdefined by 

fi(sr)l y,t) = I y,t + st), (6) 

where I y,t) is the ket I y) when the time parameter takes on the value t. This is very different than the interpretation of equation (3), i.e. we are NOT postulating that I y,t) is an 

eigenket of a time operator. Thus, Schrodinger's time-dependent wavefunction, 

y(x, t) = (x 1 y, t) , still describes the measurement of space (not time) as is clear from the 

fact that it is still the completeness of the position eigenkets that allows us to interpret 

2 Iy(x,t) 1 as a pdf for x so that at each instant in time we have 

+- 

2 

dxly(x,t)l =1. -00 

(7) 

We can obtain "time-like" information by performing this measurement of space (on 

identically prepared systems) at different values of the time parameter, but we are only 

inferring time information from the spatial measurement. 

Similarly, we can perform measurements of other quantities, such as a spin component (e.g. jZ) at different times, but here again we would be inferring rather than directly measuring the temporal information. For example, consider a particle of spin s. We 

have an identity operator 

is= E lm)(ml 

m=-s 

(8) 

for the spin part of our state space, 3C, tt {lm) : m = -s,-s + 1, ..., s - 1,s) , where 

izlm) = mfilrn). Thus the measurement of iz (performed at time t) is still described by a 
S 

probability mass function for m, so that for every instant in time we have .Z P, (m) = 1, 

2 where P, (m) = l(m 1 yl, t)l . 

3 The Direct Measurement of Time 

We can parallel the discussion of the spatial measurement given in section 1 under 

the substitutions of t for x, the Hamiltonian fj for j, and fi(6t) for ~(sx), thereby obtaining a temporal wavefinction [3], yl(t), that is complementary to the energy represen7 

tation: yl(t) u yl(E) , i.e. 

where yl(E) = (E I yl) and fil E) = EIE). As it stands however, this rather obvious 

approach does not give a complete description of the measurement of time [4] due to the 

existence of a lower bound on the energy eigenspectra (i.e. a "ground state"). 

For the sake of definiteness, consider a single harmonic oscillator of ("rigged") 

Hilbert space HI i, {Inl) : nl = 0,1,2, ...'.), where 211nl) = nl(nl), fjl = fio ($1 + kfl), 00 

and fl = Z I nl)(nl 1. The temporal wavefunction in this case is 

n,=O 

dt) = eui@12 y19 e -in#= e-ig/2 

nl =o 

(10) 

where @ = o t , yln, = (n, 1 yl), and the gauge-induced topological phase, e-'#I2, is not 

2 observable without performing interference with another system since I yl(r) 1 ' = I yl(@) I . 

Clearly the lower bound on photon number (nl 2 0) prevents yl(t) (or y(@)) from collapsing to a delta-function. This implies that the underlying phase kets {I@)} are not 

ofiogonal, (~1142) * a(& - $2) (where w(@)= (91 Y)) and yet 

2 

so that &Iv(@) 1 is a perfectly valid pdf (as can also be seen fiom Parseval's theorem 

[2]). This pdf must somehow correspond to a realizable quantum measurement (as can 

also be seen from the formalism of probability operator measures or POMs 151) and yet 

our description of the measurement is "incomplete" in the sense that we do not have 

wavefunction collapse (the {I$)} is not an orthogonal set) and likewise, the nonHermitian operator associated with this measurement does not commute with its adjoint. 
4 Tke Complete Description of The Direct Measurement of Time 

In order to get a complete description of the direct measurement of time we must 

deal with a filbert space that is larger than 311. For example, we can use the product 

space for two oscillators, E 311 @ H2. If we are willing to restrict our states to a 

subset of Hlg2 where each value of the energy difference (m = nl - n2) occurs with a 

unique value of the energy sum (j = nl + n2) then there are an infinite number of 

Hermitian time operators since there is an infinite number of such subsets (one such 

example, where we restrict to 31' c {311B2 : nl% = 0). is discussed in [6], [7], and [8]). If 

we do not wish to restrict our states, then there are two physically reasonable alternatives. 

We perform a relative time measurement that treats the different j states as either: (1) 

distinguishable; or (2) indistinguishable. These two procedures also correspond to 

performing: (1) a "marginal measurement" in which we average over all values of absolute time; and (2) a "conditional measurement" in which we determine the relative time 

distribution at an instant in absolute time, as we now demonstrate. 

Complementarity suggests that for an arbitrary two-mode excitation, with number 

representation y,+ I 1 ,(% 11 y),,, we take a two-dimensional Fourier transform 

J=2 

Rewriting this in terms of @= - + @,)I2 and I (k - ~~)/2, we have 

where 

00 +j 

I@A,~)E Z Z li,m)eimheUh and 1j,m)=ln~)~ln~)~1 j=O m=- j=n,+n2 , m=n,-n2 (14) 

j 

We see that the ez part of v(@~,@=) cannot collapse due to the boundedness of its 

complement, the energy sum ( j 2 0). 

We can eliminate @= to obtain a complete description of the measurement of @A 

on Hlg2 by applying an "absolute time average" to I @A, @z)(@A, 1 resulting in 
Note that since both of the inner sums use the same value of j, interference among the 

different j states is excluded and we have (for pure states) the pdf 

where: Tr() denotes trace; fi is the .density matrix, and 

This procedure treats the j states as distinguishable (adding the non-interfering 

probabilities that each contributes to the measurement of @A). 

We can also eliminate @= by conditioning 1 @A, @z)(@Ay @= 1 to = 0 resulting in 

where the renormalization constant is 

Herein we are taking a "snapshot" in absolute time so that the inner sums use different 

values of j thereby permitting interference among the different j states so that (for pure 

states) we have the pdf 

This measurement treats the j states as indistinguishable (adding the interfering 

amplitudes that each contributes). 

5 Discussion 

It may be of interest to note that in the "marginal measurement" defined by 

equation (15) (which reduces to equation (16) for pure states) we are directly measuring 

the relative phase angle between our two "clock arms." Thus, two uniformly (randomly) 

distributed clocks result in a uniform (random) distribution in @A. This is different than 

what one would obtain from the marginal pdf calculated from the joint distribution of our 

two clock arms. Rather than directly measuring a phase difference, the marginal pdf 
would describe the procedure of first measuring G1 and G2, and then subtracting the 

results of these two measurements (resulting in a non-uniform distribution for the case of 

two random clocks, due to the mod 2z range of G1 and G2). 

It may also be of interest to note that physical intuition regarding the connection 

between the issue of distinguishability and "absolute time average" versus "snapshot" can 

be reinforced by contrasting electromagnetic field moments with the angular 

measurement (which is equivalent to the measurement of when the two oscillators are 

the right and left circularly polarized modes of an electromagnetic wave [3]). 

Furthermore, in the case of the "snapshot" (equations (18) and (19)) we determine the 

angular distribution of the field vectors (and their quantum fluctuations) at a point in 

absolute time, whereas the "absolute time average" (equations (15) and (16)) traces out 

the quantum version of the polarization ellipse. 

6 References 

[I] J. J. Sakurai, Modern Quantum Mechanics (Benjamin/Cumrnings, Menlo Park, CA, 

1985). 

[2] A. B. Carlson, Communication Systems (McGraw-Hill, New York, NY, 1975). 

[3] S. R. Shepard, Workshop on Harmonic Oscillators (NASA CP-3197, Washington, D. 

C. 1992). 

[4] Actually, the measurement of space suffers the same limitation since any experiment 

must be performed in a room of finite extent. The mathematical procedures and physical 

interpretations presented herein are therefore applicable to x as well as t. 

[5] C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New 

York, NY, 1976). 

[6] S. R. Shepard, Ph.D. Thesis Proposal (Massachusetts Institute of Technology, 

Cambridge, MA, 1990). 

[7] J. N. Shapiro, S. R. Shepard, and N. C. Wong, Coherence and Quantum Optics VI, L. 

Mandel, E. Wolf, and J. N. Eberly, eds. (Plenum, New York, NY, 1990). 

[8] S. R. Shepard, Workshop on Squeezed States and Uncertainty Relations (NASA CP3135, Washington, D. C., 1991). 
MANIPUkATPON OF QUANTUM EVOLUTION 

David J. Fern Andez 6. 

Departamento de Fisica, CINVESTAV 

A. P. 14-740, 07000 MLxico D.F., MEXICO 

Bogdan Mielnik 

Institute of Theoretical Physics, Warsaw University 

Warszawa, ul. Hoia 69, POLAND 

and 

Departamento de Fisica, CINVESTAV 

A.P. 14-7.0, 07000 Mexico D.F., MEXICO 

Abstract 

The free evolution of a non-relativistic charged particle is manipulated using timedependent magnetic fields. It is shown that the application of a programmed sequence 

of magnetic pulses can invert the free evolution process, forcing an arbitrary wave packet to 

"go back in time" to recover its past shape. The possibility of more general operations upon 

the Schrodinger wave packet is discussed. 

Introduction 

We expect that in a near future the problem of particle trapping will be replaced by a wider 

manipulation problem concerning the purposeful operations on quantum states. This involves 

the inverse evolution problem: given a unitary operator acting in the Hilbert space of states 

of a quantum system, one asks if there exists a realistic (possibly time-dependent) Namiltonian 

inducing this operator as a result of a dynamical evolution process. The importance of the subject: 

the unitary operations which can be dynamically induced, can also be used to control the wavelike behaviour of quantum objects, e.g. during the preparation of a measurement. The so defined 

subject was put forward by Lamb [I] and it has been subsequently developed by his followers 

(Lubkin [2], Mielnik [3, 41, Royer [5], Brown 161, Ferndndez [7], and other authors.) A key to 

the manipulation of a quantum state lies in the possibility of trapping the particle in a circular 

dynamical process called an evolution loop [4] (EL). In an EL the evolution operator U(t) becomes 

the identity for a finite time interval. The subsequent perturbation of the EL can induce arbitrary 

unitary operations on the wave packet as the result of the cummulative process involving the small 

precessions of the distorted loop [3]. 

The unitary transformations that will be discussed in this paper are: 

evolution loop U(T) = 1, T > 0, 

rigid displacement of the wave packet U(T) = eia*pfh, 

the quantum time machine U(T) = e -iT'p2/2mA , -oo < T' < CQ. 

173 
2 The evolution loops 

Consider a non-relativistic spinless particle of charge e evolving under the action of a homogeneous 

time-dependent magnetic field B(t). The Hamiltonian of the system can be expressed as: 

where L is the angular momentum operator and rl = r- (r a) 8 is the part of r orthogonal 

to B(t). Here, B(t) will be taken as the sequence of identically shaped orthogonal pulses in 

the three directions XI, $2, 23 defined by the right-handed orthonormal set of basis vectors 

{el, e2, es): 

B(t)el for t E 10, T), 

B(t - T)e2 for t E [T, 2T), 

B(t - 2T)es for t E [2T, 3T), 

with B(t) = B@(t/T), S,' @(tt)dt' = 0. The generic evolution can be determined through the operator U(3T9 0) = U(3T, 2T)U(2T, T)U(T, 0). In dimenssionless units (t' = t/T, tj = J* q, @ = 

p) it takes the form: 

Above, Q(q,p) = 7 {exp (-i fi ~(t')dt')) is the evolution operator induced by the one-dimensional harmonic oscillator of variable frequency a(t1) = (E) /3(t1) = a@(t1) and Hamiltonian B(t1) = 

p2/2 + c~(t')~~~/2, q and p are two canonically conjugated operators such that [q, p] = i and 7 is 

the time ordering operator. 

Now, because W; depends on Hamiltonians which are quadratic in the canonical variables 

(&,fi;) but it doesn't involve (sj,fij) j # i, it is possible to represent it by a 2 x 2 matrix w; (the 

"Heisenberg picture" ): 

where i = 1,2,3. The kind of dynamical process (3) depends on the algebraic type of the matrices 

w;, and due to the form of the operators W; it is determined just by one c-number invariant called 

the discriminant: 

A(,@) = Tr(wl) = Tr(w2) = Tr(w3). (5) 

Whenever this invariant accepts one of the distinguished values: 

27rI A(@) = 2 cos -, I, n = fl, f2,. . . , n 
the matrices wi fulfill the algebraic identity wq = 1 [4] + kV'; = 1. Hence: 

It can be shown [8] that for any piece-wise continuous bounded real function @(ti), 0 5 t' 5 N 

with N finite, there exist pulse amplitudes a for which the discriminant A(a@) accepts any of the 

special values (6). Whenever this happens, the n repetitions of our magnetic pulse pattern (2) 

generate the evolution loops in the space of states L2(R3) at the loop period T = 3nT. 

As an illustration we restrict the discussion to the case of rectangular pulses: 

@(t') = 0(1/2 - ti)O(t') - O(t' - 1/2)0(1 - t'), (8) 

where 8(x) is the step function. In this case, the discriminant A(ap) can be analytically determined taking the form A(a@) = 2 cos 2a - a sin 2a. The simplest EL is achieved making 

n = 4, 1 = 1 in (6); the solution for a becomes: 

The loop period is T = 12T, and the orders of magnitude of the field strenght and T must satisfy 

the relation B = 2amcleT with the value of a in (9). 

3 Rigid displacement of the wave packet 

The evolution loops provide a convenient method to generate arbitrary unitary transformations 

of quantum states. Suppose, e.g., we want to produce a rigid displacement of the wave packet. 

To this end, we take the loop induced by the rectangular pulses (8) with the a-value (9) as the 

unperturbed system. The loop is then perturbed by a homogeneous time-dependent external force 

F(t) = eE(t). The total Hamiltonian becomes H(t) = Ho(t) - er . E(t), where Ho(t) is the loop 

part and -es , E(t) is the perturbation. The evolution operator within one loop period T = 12T 

can be evaluated in the interaction picture. Hence: 

7 

U(T) = exp (f F(t) . ro(t)dt) = exp [i (a . p + b - r)] , 

where ro(t) is the triplet of canonical operators xl(t), x2(t), x3(t) in the Heisenberg frame of 

Ho(t). By taking F(t) = F sin 2rtl.r it is possible to obtain explicitly a and b in (10) [S]. With 

the aim of produce the pure rigid displacement, with b = 0, we selected F(t) as a sequence of 

pulses of rectangular force. It was found that a single pulse of rectangular force F = eE in the 

xl-direction acting within the interval [9T, lOT] displaces the packet against the applied force by 

a, = -1.658693FT2/m (the boomerang eflect). Other possibilities are discussed elsewhere [8]. 

4 The quantum time machine 

As the next example we shall discuss the quantum time machine. In this scheme, the acceleration, 

slowing or inversion of the free evolution of the charged particle is possible. Once again, the 
technology is based on a sequence of pulses of homogeneous magnetic field of the form: 

B(t)el for t E [O, 2T), 

B(t - 2T)ez for t E [2T,4T), 

B(t-4T)q fort E [4T,6T), ~;~B(t)dt=O, 

with B(t) given by: 

Bl fort E [O,tl), (0 < t, < T) 

B2 fort E [tl,T), 

-B2 fort E [T,T+t2), (t2 = T -tl), 

-Bl fort E [T+t2,2T). 

The key evolution operator U(T = 6T, 0) in dimensionless coordinates and time t1 = t/T takes a 

similar form as in (3): 

U(6,O) = 0'6 8 OGgZ 8 GflZ9 (13) 

where gZ = R(2) and G = G(2) describe the evolution of the oscillator of variable frequency and 

the free evolution in [O, 21 respectively. A11 the calculations are made in the matrix representation, 

working with the angular parameters 71 = alt; and 72 = a2t',, where a1 = eBlT/2mc, a2 = 

eB2T/2mc9 t', = tl/T, t', = t2/T. It turns out that when the amplitudes and times of the pulses 

of magnetic field satisfy the relations [$I: 

a1 = 71 tan 71 - 72 tan 72 

tan 71 3 

a2 = 71 tan 71 - 72 tan 72 

- tan y2 7 

t; = 71 tan 71 

71 tan 71 - 72til1l-y~' 

t; = 

-72 tan 72 

71 tan 71 - 72 tan y2 ' 

the free evolution operator is produced at the end time of the sequence (1 1): 

i p2 i p2 

~(r) = exp (-%%TI) = exp (---rX) ti 2m , 

where the "effective" time TI = TX = 6Tx depends on the distorsion coeficient: 

As from definition ti and ti must be positive, then yl and -y2 must lie in intervals of different 

parity, i.e. nn < 71 < (n + 1/2)n and (m - 1/2)n < 7' < mn, m,n E Z+ or vice versa. The 

distorsion coefficient x as function of rl and y2 is plotted in the Fig.1. As can be seen, one can 

generate three different situations: 

free evolution acceleration 

O < x 5 1 free evolution slowing 

free evolution regression 
FIG. 1. The "chessboard of distorted time" for the free evolution of a charged 

particle manipulated by the magnetic field (1 1-12). The level curves for the distorsion 

coefficient x are ploted as functions of the angles 71 and and mark the zones for 

which at T = 6T we obtain the acceleration (X > I), slowing (0 < x 5 1) or regression 

(X 5 0) of the free evolution. 

As a final remark we would like to point out that this kind of manipulations is not restricted 

to rectangular pulses of magnetic field. The same possibilities can be found if smooth fields are 

used, although we won't have analytic expressions for the discriminant anymore. 

Acknowledgments 

The authors acknowledge to CONACYT, Mkxico for financial support. 
References 

[I] W.E. Lamb Jr., Phys. Today 22(4), 23 (1969). 

[2] E. Lubkin, J. Math. Phys. 16, 663 (1974); ibid 15, 673 (1974). 

[3] B. Mielnik, Rep. Math. Phys. 12, 331 (1977). 

[4] B. Mielnik, J. Math. Phys. 27, 2290 (1986). 

[5] A. byer, Phys. Rev. A 36, 2460 (1987). 

[6] L.S. Brown, Phys. Rev. A 36,2463 (1987). 

[7] D.J. Ferniindez C., Nuovo Cim. 10733,885 (1992). 

[8] D.J. Ferniindez C. and B. Mielnik, Controling Quantum Motion, (preprint CINVESTAV, 

1993). 

QUANTUM STATE ENGHNEEMNG 

K. Vogel,.V. M. Akulinl, and W. P. Schleich2 

Abteilung fur Quantenphysik) Universitut Ulm 

Albert-Einstein-Allee 1 1, D-89069 Urn, Germany 

Abstract 

We show how to create an arbitrary field state in a cavity by sending appropriately 

prepared two-level atoms through the cavity and subsequently detecting them in their ground 

state. 

Two typical questions in quantum mechanics are (i) how does a given initial quantum state evolve 

in time? (ii) what are the properties of a given quantum state? Usually, the question of how 

to prepare these states is not answered. One notable exception is the operational approach by 

W. E. Lamb [I]. He prepares an arbitrary quantum state of a particle by "catching" it in an 

appropriate potential constructed out of the corresponding wave function. In quantum optics, 

however, quantum states of the radiation field are of central interest. Here this scheme does not 

work, since there is only a limited variety of Hamiltonians describing the interaction between 

matter and the radiation field. During the last few years, the preparation of nonclassical states 

of the radiation field has attracted a lot of interest. However, the investigations were limited to 

certain classes of quantum states. In particular, the generation of squeezed states [2], number 

states[3, 41 and Schrodinger cat states [4, 51 was discussed. 

So far two approaches have been used: (i) Find an appropriate Hamiltonian which transforms 

via unitary time evolution a given (simple) initial state to the desired final state. (ii) Make a 

measurement on one of two entangled quantum systems and obtain the state of the other system 

by the corresponding state reduction. Although in principle we can always construct the necessary 

Hamiltonians and entanglements, the variety of states which we can obtain in this way is limited 

since the Hamiltonians and entanglements based on physical interactions are limited by nature. 

Nevertheless, we give a recipe how to construct an arbitrary quantum state of the radiation field 

starting from the vacuum state by repeatedly using a simple Hamiltonian and subsequent state 

reduction [6]. 

2 The Ingredients for Quantum State Engineering 

The ingredients for our method of creating an arbitrary quantum state are two-level atoms and 

a cavity for the electromagnetic field. The atoms interact with a resonant mode of the cavity via 

lalso Moscow Institute of Physics and Technology, Dolgoprudny, Moscow, Russia. 

2also Max-Planck-Institut fiir Quantenoptik, D-85748 Garching, Germany. 
the Jaynes-Cummings Hamiltonian. We start with a cavity field which is initially in the vacuum 

state and consecutively inject atoms in such a way that there is at most one atom in the cavity at 

a time. Before an atom enters the cavity we prepare it in a specific superposition of the excited 

state la) and the ground state Ib). This superposition has to be chosen appropriately [7] in order 

to drive the state of the cavity field towards the desired state. A measurement of the internal 

state of each atom after it has passed through the cavity leaves the quantum field in a pure state. 

Let us consider one step of this process, that is, the interaction of the k-th atom with the 

cavity field. Before we inject the k-th atom the cavity field is in a state 

The k-th atom enters the cavity in the superposition state la) + ickl b) controlled by the complex 

number ~k [8]. During its flight through the cavity the atom interacts with the cavity field 

according to the Jaynes-Cummings Hamiltonian. After it has left the cavity, the state of the 

combined atom-field system reads 

Here CAk) = cos(g~kdn+l) and sik) = sin(g~kJX), where .rr, is the interaction time of the 

k-th atom with the field and g is the atom-field coupling constant. 

Obviously the state (2) is an entangled state. In order to obtain a pure field state we make 

a measurement on the k-th atom and detect it either in the excited state or in the ground state. 

If we detect the atom in the excited state our method cannot create the desired field state. We 

therefore have to go back to the vacuum state and start the procedure again. However, if we find 

the k-th atom in the ground state we continue the process. In this case the new field state [8] 

reads 

The coefficients for the new state, (PL~), and the coefficients for the old state, (P;~-'), are related 

via the recurrence relation 

(k) (k-1) (k) = SF~(P~:') - ckCn-i(Pn (Pn 

which follows from Eq. (2). 

3 The Art of Quantum State Engineering 

Each atom which has passed through the cavity and has been detected in the ground state increases 

the number of Fock states building up the cavity field state by one. Therefore, after N atoms 

have passed through the cavity, the field state which initially was the vacuum state Icp(O)), is a 

superposition of the N+ 1 number states lo), [I), . . ., IN). But how do we get a desired combination 
The key idea of our method is to find a combination 

of N number states lo), Il), . . ., IN) which yields the field state after the N-th atom prepared 

in an appropriate internal state la) + ieNlb) has passed through the cavity and has been detected 

in the ground state. From Eq. (4) we find for the N + 1 unknowns, that is, the N coefficients 

cpLN-') and the parameter EN, the following set of N + 1 equations: 

We express the unknown values cpLN-l) in terms of the known values dn, starting with the first 

equation of the set (7), and obtain 

In addition we have to satisfy the last equation of the set (7). We therefore substitute the 

coefficients rpiN~l) into the last equation of the set (7) and obtain 

as the characteristic equation for EN. 

We solve the characteristic equation numerically and choose one value EN from the N roots 

of Eq. (9). Equation (8) immediately gives us the corresponding coefficients cpLN-') of the state 

(c#~-')). We take (y~(~-')) as a new desired state which we have to obtain by sending N - 1 

atoms through the cavity. For the state Icp(N-l)) we do the same calculations as for the state l$d) 

and obtain the parameter EN-1 and state with N - 1 coefficients SoLN-2). We repeat the 

calculations until we end up with the vacuum state. A string of complex numbers ~1, 62,. . . , EN 

defines the internal states of a sequence of N atoms we have to inject into .the cavity in order to 

obtain the desired state from the vacuum state. 

We illustrate this method by creating a superposition of the number states 12) and 17), 

In Table I we give the values el, EZ, . . ., EV calculated for identical interaction parameters grr, = x/5. 

In order to give an impression about the individual steps of the evolution of the field state from the 
TABLE I. Internal state la) + ;lak[ evklb) of the k-th atom needed to obtain the 

state (12) + 17))/4, Eq. (lo), for a fixed interaction parameter gr = 1715. The right 

column gives the probability P:'), Eq. (14), to find the k-th atom in state Ib) after its 

interaction with the cavity field provided all earlier atoms have been detected in the 

state I b). In this case the probability P7, Eq. (16), to find all atoms in the ground 

state is P7 = P:') . pb(l) . pj7) = 0.00801. 

vacuum state to the desired state, Eq. (lo), we plot in Figure 1 the Q-function for the field state 

Icp(k)) after the k-th atom has passed through the cavity and has been detected in the ground 

state. 

4 Probabilities 

But what is the probability to create the state, that is, what is the probability Plv to find all atoms 

in the ground state after they have passed through the cavity? So far we have used unnormalized 

states for the atoms and the field because it was convenient for calculating &I, and cpLk). However, 

when we need probabilities we have to use normalized field states 

and atomic field states (la) + ieklb)) / J-. 

For the coefficients $Ak) we obtain equations similar to Eqs. (7) which read 
FIG. 1. Q-function Q(a) = l(a19(k))]2/7r for the field state Icp(k)) after the k-th 

atom has interacted with the field and has been detected in the ground state. The 

parameters for the internal states of the incoming atoms are given in Table I. 
Here the normalization constant 

1 &= Jm 

consists of two parts: The factor 114- which takes into account the normalization of the 

internal state of the k-th atom, and the factor 1/m which is due to the normalization of the 

field state after the state reduction. Here 

is the probability to find the k-th atom in the ground state. From the first equation of (12) follows 

Since we start from the vacuum state we have = 1. Moreover, we note that for a normalized 

desired state we have $AN) = dn. We substitute Nk from Eq. (13) into Eq.(15) and obtain for the 

probability PN to find all N atoms in the ground state 

The probability PN depends on the choice of roots of the characteristic equation, Eq. (9), 

and the interaction times ~k. Can we use these "degrees of freedom" to optimize the probability 

PN? To get an idea of the possibilities of this optimization let us consider the simplest case 

of identical interaction times ~r, = T for the example of a superposition of the number states 

12) and 17), Eq. (10). The dependence of the probability P7 on the interaction parameter g.r is 

shown in Figure 2. For this curve we have chosen for each atom the ek with the smallest absolute 

value. We note that P7 increases for increasing interaction parameter g~ and reaches its maximum 

P7 x 0.00944 at g~ x 0.2219~ and then decreases. Moreover, trapping states, that is, interaction 

parameters gr with sin (976) = 0 (n = 1,2,. . . ,7), manifest themselves in vanishing probabilities 

P7 as apparent from Eq. (16). As a general rule the maximum value for the probability occurs for 

interaction parameters smaller than those corresponding to trapping states. 

In the next step of the optimization we allow each atom to have its individual interaction time 

rk with the cavity field. In Table I1 we have chosen 7-k such that the probability P7 to find all 

seven atoms in the ground state has a maximum. Using this strategy we increase P7 up to the 

value P7 x 0.02630. 
FIG. 2. Probability P7 to find all seven atoms in the ground state as a function 

of the interaction parameter gr for the superposition state, Eq. (10). Here we have 

chosen ck with the smallest absolute value. Note the occurrence of trapping states 

where, according to Eq. (16), the probability P7 vanishes. 

TABLE 11. Internal state la) + il~~l eiPklb) of the k-th atom needed to obtain the 

state (12) + 17)) 14, Eq. (10). Here we have optimized the interaction parameters gm 

as to maximize the probability P7, Eq. (16), to find all atoms in the ground state. The 

right column gives the probability pik), Eq. (14), to find the k-th atom in state 1 b) 

after its interaction with the cavity field provided all earlier atoms have been detected 

in the state Ib). In this case we have P7 = 0.02630. 
In conclusion we emphasize that we can construct any superposition of the first M + 1 number 

states from the vacuum state by injecting IV appropriately prepared atoms into a cavity and 

detecting all of them in the ground state after they have interacted with the field. Furthermore, 

we note that the Jaynes-Cummings Hamiltonian is not crucial for this method. Similar interactions 

between field and atom can also be used provided that they allow for energy exchange between 

field and atoms. 

6 Acknowledgments 

One of us (K. V.) wants to thank the Deutsche Forschungsgemeinschaft for travelling support. 

The work of W. P. S. was partially supported by &ED Associates. 

References 

[I] W. E. Lamb, Physics Today 22 (4), 23 (1969). 

[2] For a review on squeezed states, see the special issues J. Mod. Opt. 34, (1987), 

J. Opt. Soc. Amer. B 4 (10) (1987), and Appl. Phys. B 55 (3) (1992). 

[3] J. Krause, M. 0. Scully, and N. Walther, Phys. Rev. A 36, 4547 (1987); J. Krause, 

M. 0. Scully, T. Walther, and H. Walther, Phys. Rev. A 39, 1915 (1989); P. Meystre, 

Opt. Letters 12, 669 (1987); P. Meystre, in: Squeezed and Nonclassical Light, edited by 

P. Tombesi and E. R. Pike (Plenum, New York, 1988); H. Paul, J. Mod. Opt. 38, 515 (1989); 

M. Brune, S. Haroche, V. Lefevre, J. M. Raimond, and N. Zagury, Phys. Rev. Lett. 65, 976 

(1990); M. J. Holland, D. F. Walls, and P. Zoller, Phys. Rev. Lett. 67, 1716 (1991). 

[4] P. Meystre, in: Progress in Optics XXX, 261 (North-Holland, Amsterdam, 1992). 

[5] M. Brune, S. Haroche, J. M. Raimond, L. Davidovich, and N. Zagury, Phys. Rev. A 45, 5193 

(1992); P. Meystre, J. J. Slosser, and M. Wilkens, Opt. Commun. '39, 300 (1990). 

[6] These results have been published in: K. Vogel, V. M. Akulin, and W. P. Schleich, 

Phys. Rev. Lett. 71, 1816 (1993); another method to create an arbitrary quantum state 

in a cavity uses the adiabatic passage of atoms to transfer the coherence between atomic Zeeman sublevels to the cavity field, see: A. S. Parkins, P. Marte, P. Zoller, and H. J. Kimble, 

Phys. Rev. Lett. 71, 3095 (1993). 

[7] We can obtain any superposition of the excited state la) and the ground state Ib) from the 

state la) or Ib) by applying an appropriate classical field. 

[8] In order to keep the mathematics simple we use here unnormalized atomic states and field 

states. 
SQUEEZED LIGHT FROM MULTI-LEVEE 

CLOSED-CYCLING ATOMIC SYSTEMS 

Min Xiao 

Department of Physics, University of Arkansas 

Fayetteville, Arkansas 72701 

Yi-fu Zhu 

Schlumberger-Doll Research 

Ridgefield, CT 06877 

Abstract 

Amplitude squeezing is calculated for multi-level closed-cycling atomic systems. 

These systems can lase without atomic population inversion in any atomic bases. Maximum squeezing is obtained for the parameters in the region of lasing without inversion. 

A practical four-level system and an ideal three-level system are presented. The latter 

system is analysed in. some detail and the mechanism of generating amplitude squeezing 

is discussed. 

1 INTRODUCTION 

Generation of squeezed states of light has attracted lots of attention in the past decade 

due to the posssible applications in various fields of physics. Squeezed light was generated 

in several different systems (atomic multi-wave mixing, degenerate parametric oscillator, 

optical fibers, optical diodes, second harmonic generation, and et al) and in several different 

forms (c.w. and pulsed) in laboratories around the world.[l] Although, this kind of quantum 

state is generated routinely in the laboratories, to make a compact, efficient, reliable, and 

cw squeezed coherent light source is still a challenge. Diode laser is a very good device to 

generate amplitude squeezed light with high efficiency[2], but the wavelength selection is 

very limited for applications in atomic spectroscopy. 

Another area of recent interest in physics is to achieve lasing without atomic population 

inversion in multi-level atomic systems.[3-71 The possible applications of these new lasers 

include reaching new wavelengths and getting " quiter" laser output intensity. Several theoretical works were published in studying quantum statistical properties of A-type three-level 

atamic system and multi-level Raman systems. [8-111 In this paper, we discuss quantum statistical properties of multi-level closed-cycling atomic systems, which exhibits lasing without 

atomic population inversion. 

We have studied two particular atomic systems. The first one is a practical four-level 

closed-cycling atomic system.[ll] A brief discussion of the steady-state behaviors and lin-. 

earized fluctuations in this system are given. Conditions achieving good squeezing are described. 
To understand the mechanism of generating squeezing in this relatively complicated system, we constructed a simple, idealized three-level model which eliminates the intermediate 

level 14> and neglects decay from the upper lasing level /2> to the ground state 11 >. After 

adiabatic elimination of extra atomic variables (under condition y31>>tc, with tc as cavity 

decay rate), we obtain a set of equations which are similar to an effective two-level atomic 

system. Then steady-state correlation functions of the amplitude fluctuations are calculated, analytically, to find best conditions for maximum squeezing. Some interesting effects 

are discussed and compared to the results of previous four-level model 

Four-Level Cycling System 

Our model consists of an essemble of N closed four-level atoms confined in a single-mode 

cavity with photon loss rate 2tc. The transition I I> t, 13> of frequency ~31 is driven by a 

laser of frequency wl with Raby frequency 20. 2yij (i, j=1-4) are the spontaneous decay 

rates from state li> to state Ij>. Using standard procedure, a set of stochastic differential 

equations are derived. The steady state behaviour and conditions to achieve lasing without 

inversion were discussed in our earlier publication.[ll] When y21/%4<1, lasing will start from 

population inversion. As the laser intensity building up, the population of the upper lasing 

level 12> will be depleted and the lasing will be sustained by the coherence induced between 

levels 12> and 13>. This transition from lasing with inversion to lasing without inversion in 

the same system is an interesting phenomena to study. At the opposite limit, i.e. 'y21/y34>1, 

the laser will always operate with no population inversion. 

We calculated, numerically, the amplitude fluctuations of the system by linearization 

around steady-state solutions and found that large degree of squeezing (about 80%) at the 

laser output can be obtained with relative low pumping power and very small decay rate 

from the upper lasing state 12> to the lower lasing state Il>. 

3 Three-Level Idealized Cycling System 

Due to the complication in the four-level system, numerical calculation has to be used in 

calculating the laser intensity fluctuations. In order to understand the mechanism and the 

limiting conditions for achieving optimal squeezing, we simplify the four-level model to be 

an ideal three-level closed-cycling system. 

Since we are only interested in the optimal conditions for generating squeezing, the decay 

from upper lasing level 12> to the lower lasing level [I> is neglected. To simplify our 

calculation, we also eliminate the intermediate state 14> and neglect decay from level 13> 

to level Il>. Since the effective coupling between the atomic transition and the intracavity 

field is fig instead of g, we can increase the coupling by putting large number of atoms in 

the cavity mode, which is usually true for a laser system. This will justify the approximation 

of neglecting decay rates but of keeping finite dipole coupling between level 13> and level 

I I> and between level 12> and level I I>. Careful choice of the atomic element as the gain 

medium can also help to satisfy this approximation. The decay rate y from level 13> to level 

12> is the only one to keep. 
Using the same standard procedure, we derive a set of stochastic differential equations 

for this system from Harniltonian, as following: 

JI~= inJJ3 + iag(J2p - Jll) + r3(t), . t 

J12= -inJ23 - iatg(J22 - Jii) + rio(t), 

Ji3= -7513 + ifl(J33 - Jii) + iqJ23 + r4(t), . t 

J13= -7Jf9 - iR(J33 - Jll) - iatgJJ3 + Ts(t), 

ti= -ma.-- igJ12 + rl(t), .t 

a = -nat + igJf, + r2(t), 

where < ri(t)rj(tl) >r DijG(t - t') describing the correlation of the fluctuations. There are 

36 nonzero Dij terms for this .particular system. To save space,. these nonzero diffusion terms 

are not given explicitly here. For a closed system, we have condition 

These ten differential equations are still too complicated to calculate correlations for 

the fluctuations analytically. However, in good cavity limit, i.e. 7 >> K (which is a good 

approximation in a realistic laser system), some of the atomic variables (J13, J?, 523, ~f~, J22, 

and J33) can be adiabatically eliiminated from equation (1). We can do it by letting time 

derivatives over these atomic variables go to zero, because they decay much faster to their 

t steady-state values comparing to J12, ~f ,, and the field variables. We can then solve J13, Jig, 

J23, J?~, JZ2, and J33 together with their corresponding fluctuation terms from equation (1) 

and substitute them into equations for J12, ~f~, a, and d. After some algebra, we arrive at 
Where the new normalized variables and parameters are defined as 

The new fluctuation terms are still quite complicated. 

The steady-state equation can be easily solved to give 

where G=Ng2/rcy is the normalized coupling strength between the field and atoms. In this 

calculation, we have assumed the resonance condition between the cavity mode and the 

transition frequency from level 12> to level Il>. The normalized steady-state population 

distributions are 

To calculate correlation fuctions of the fluctuations, we need to linearize equation (3) 

around their steady-state solutions. From the numerical calculation of the four-level system 

and the steady-state solutions of this system, we can determine the area in the parameter 

space where best squeezing occurs. For Xo>>l, and small, but finite, Y, equations in (3) 

can be linearized to give 

where 
The intracavity correlation functions and, therefore the amplitude squeezing is easily calculated by standard method. The intracavity squeezing is given by 

The steady-state intensity Xo is related to the pumping intensity Y and coupling strength 

G through the steady-state equation (5). 

It is easy to show that S+ is limited to the minimum value of -0.5, which corresponding ' 

to a 50% squeezing inside cavity. The output spectrum of squeezing has also been calculated numerically. The maximum squeezing is also limited to a 50% level at around carrier 

frequency. 

4 Discussion 

We have calculated two atomic models for generating squeezed states of light and conditions 

for lasing without inversion. The four-level system can start to lase with or without population inversion depending on the relative decay rates. Far above threshold the lasing is onky 

sustained by the coherence induced between level 13> and level 12>. For the three level ideal 

model, the laser will always operate without the population inversion. 

The four-level system can generate 80% squeezing at the output for relatively low pumping power. As for the simplified and idealized three-level cycling system, even without decay 

from the upper lasing state to the lower lasing state, the maximum output squeezing and the 

total intracavity squeezing are both limited to the 50% level. we realize that 50% squeezing 

comes from the suppression of spontaneous emission of the upper lasing level. The extra 30% 

squeezing is due to the pumping regulation through the incoherent decay processes between 

level 13> & level 14> and between level 14> & level 12>. So, the extra level 14> actually 

enhances the available squeezing for the laser output. 

References 

[I] See, e.g., the feature issue on Squeezed States of the Electromagnetic Field, J. Opt. 

Soc. Am. B4, 1450-1741, (1987); and Quantum Fluctuations in Optical Systems, Progress 

in Optics, XXX, 1-85 (1992). 

[2] Y. Yamamoto, S. Machida, and Y. Itaya, Phys. Rev. Lett. 58, 1000 (1987). 

[3] S.E. Harris, Phys. Rev. Lett. 62, 1033 (1989); A. Imanoglu, Phys. Rev. A40, 4135 

(1989). 
[4] M.O. Scully, S.Y. Zhu, and A. Gavrielides, Phys. Rev. Lett. 62, 2813 (1989); E.F. 

Fill, M.O. Scully, and S.Y. Zhu, Opt. Commun. 77,36 (1990). 

[5] A. Nottelmann, C. Peters, and W. Lange, Phys. Rev. Lett. 70, 1783 (1993). 

[6] E.S. Fry, el al, Phys. Rev. Lett. 70, 3235 (1993). 

[7] W.E. van der Veer, el al, Phys.Rev.Lett.70,3243 (1993). 

[8] G.S. Agarwal, Phys. Rev. Lett. 67, 980 (1991). 

[9] K.M. Gheri and D.F. Walls, Phys. Rev. Lett. 68, 3428 (1992); Phys. Rev. A45,6675 

(1 992). 

[lo] H. Ritsch, M.A.M. Marte, and P. Zoller, Europhys. Lett. 19, 7 (1992); H. Ritsch, 

M.A. Marte, Phys. Rev. A47, 2354 (1993). 

[ll] Yifu Zhu and Min Xiao, Phys. Rev. A48, 3895 (1993). 
A METHOD OF MEASURING THE AMPLITUDE-MODULATED 

VACUUM FIELD NEAR A CONDUCTING MIRROR 

Sun-Hyun Youn, Jai-Hyung Lee, Joon-Sung Chang 

Department of Physics 

Seoul National University, Seoul 151- 742, Korea 

Abstract 

Electromagnetic fields of the vacuum mode near a conducting mirror are modified with 

respect to those in free space, with their amplitudes having a sinusoidal spatial dependence 

from the mirror. Therefore if we combine this spatially amplitude-modulated vacuum field 

mode and intense corehent light with a beam splitter, we may detect this fluctuation of the 

vacuum mode in a homodyne detection scheme. It will give new method to produce squeezed 

states of light with a single mirror placed close to an unused port of a beam splitter. We 

show that the amplitude fluctuation of the combined light can be reduced by a factor of 2 

below that of the coherent light. We also discuss the limitations due to the finite line width 

of the laser and the effective absorption length of the photodiodes. 

1 Squeezed light generation with a conducting mirror 

The characteristics of vacuum fluctuations in a confined space have been studied in several cases, 

for example near an infinite plane conducting mirror [I], between two parallel mirrors and in a 

spherical cavity [3]. Here we consider a tangential vacuum-field mode with its wave vector k normal 

to the conducting surface ( z-direction) [2], and we choose the polarization and the propagation 

of the electric field along the x and z-axis, respectively. Then the electric field operator for the 

vacuum field entering the beam splitter (see Fig. 1) becomes 

where uk is the frequency for the mode k(wk = ck), h and eo have the usual meanings, and V 

is the normalization volume [I]. Note that the sinusoidal spatial-dependence sin(kz) for the field 

amplitude comes from the boundary condition that the tangential components of the electric field 

modes on the conducting surface should vanish. 

We now consider a homodyne detection scheme to measure the quantum mechanical noise 

of the signal (i.e. amplitude-modulated vacuum field ) as shown in Fig. 1. Let us assume, for 

simplicity, that the electric field for the local oscillator has a single mode(we will consider the 

multimode effects in the next section). Then the local oscillator field can be written as 
with 

where w is the laser frequency ( w = ck,). Here we have decomposed the coherent light El into 

ECl and $ [4]. ECl is the classical analogy of the coherent light which has a definite amplitude 

and phase, whereas $ is the quantum fluctuation in the electric field of the coherent light which 

is simply equivalent to the vacuum fluctuation. 

Considering the vacuum mode relation in the Fig. 2, we can get the electric field in fluctuating 

vacuum modes at the detector 1 

where zl (Z1) represents the length for the beam path between the single mirror (laser) and the detector 1. We have separated the electric field into positive- E(+) (-- eiWt) and a negative-frequency 

components E(-) (-- e-'wt), and note the propagation-direction of traveling wave. We also add the 

factor 2 for the normalization of the vacuum fluctuation. Jz 

Using photodetection theory 151, we obtain the photocurrent i as 

t 

j=L, dtth(t - t1)k(+)(z, t') E(-)(z, t'), 

where h(t - t') is the photodetector response function [6]. Assuming instantaneous response of 

the detector, we can approximate h(t - t') as hS(t - t'). Then the photocurrent induced on the 

detector 1 is 

Since the expectation value of a*a is much greater than that of &t& and &ti, we keep only the 

terms which contain a or a*. If the reflectivity (R) of the beam splitter approaches to 1, the Eq. 

(5) surely represents the positive frequency part of the Eq. (1). In that case, hl and 62 are totally 

decoupled and we can find the standing vacuum mode in the port 1. 

Using the condition in Eq. (5), il is. obtained as 

where j," is the time-averaged photocurrent which is normalized with respect to h, and we have 

replaced a with I a I eim. Note that in Eq. (8) we have included only one vacuum field mode 
identical to the laser light with its wavevector k and frequency w = clc, since only that mode 

survives as a result of the time average effect on the detector. We have neglected the a*a term 

in 1; , since it does not include any fluctuation and corresponds to a constant dc current which 

can be filtered out by ac coupling. And the average values of j,~ are expected to be zero, since the 

operators and ii2 both represent the fluctuating vacuum field modes. 

The sum and difference of these pliotocurrents I:, 1: are obtained from the Eq. (8). It becomes 

We now evaluate the square of each quantity to find its fluctuations. Squaring the Eq. (8), we 

gather the terms (8'6:) which contains the nonzero vacuum expectation value. Then the results 

become 

and the modulation effect due to the cos(Z; + z;) term will vanish , since the origins of the Zl and 

Z2 are not an absolute one for the traveling local oscillator mode. However, the origin of the zl 

and z2 is absolutely fixed to the mirror position so we will keep only this modulation effect due 

to cos(2zl12) term. Together with the relation R + T = 1, Eq. (10) finally become 

These results , Eq. (ll), clearly show that the intensity fluctuations measured at each photodetector exhibit a sinusoidal spatial dependence. The photocurrent fluctuations for each detector 

comes partly from the vacuum field in the local oscillator itself, and partly from the vacuum 

field modified by the perfect mirror. Note that the sinusoidal modulation in Eq. (ll), which is 

responsible for amplitude squeezing, is totally due to the vacuum field mode that has the same 

frequency as the laser oscillator but is altered by the perfect mirror. 

For the balanced homodyne detection, assuming T = R = 1/2 , the resulting quantum fluctuation of the modulated light may fall below that of the coherent state i.e. become squeezed 

as shown in Fig. 2. The fluctuation of the coherent state without the perfect mirror can be 

calculated by replacing c~s(k,z~,~) by its average value zero so that Eq. (11) simply becomes a 

constant value as the result of the usual beam splitter . On the other hand, if we consider 

the situation such that the distance zi(i = 1,2) between the perfect mirror and each detector is 

2 

well resolved and satisfies ic,z; = n*(n: positive integer), then Eq. (11) reduces to y. In other 

words, the measured quantum fluctuations may fall below that of the coherent vacuum state by as 

much as 50%. This squeezing limit comes from the intrinsic fluctuations of the coherent state of 

the laser itself, which is combined with the modified vacuum field with its amplitude suppressed 

at the detector position such that kozi = nr(i = 1,2). 
The quantum fluctuations of the square of the sum and difference of the photocurrents If and 

I," are similarly obtained from Eq. (9). Squaring this Eq. (9) and keeping the non vanishing 

terms, we can obtain 

< (I," f I;)~ >EE (Il f 12)2 

where we have used the relation Zl - zl = Z2 - 22. Using the relation T + R = 1 , we simplify 

this equation (12) like this, 

We find that fluctuations of the sum of I," and 1; in Eq. (13) is not dependent on zl,zz, since this 

fluctuation has come from the local oscillator. On the other hand, as in the case of and E, the 

fluctuation of (Il - 12)2 which comes from the modulated vacuum field contains the important 

spatial modulation with a period of n/ k,. The spatially-averaged fluctuation of (Il - 12)2 is I CY 12, 

which is just the quantum fluctuation of the free-space vacuum field without the mirror. However, 

the fluctuation of the difference at the detector position such that kozi = nr(i = 1,2) becomes 

zero. The resulting fluctuations of (Il - &)2 in Eq. (13) are also plotted in Fig. 2. In this Fig. 

2, we have replaced the cos(koz2) with zero, which represents the average value, and have plotted 

(Il - 12)2 as a function of zl. 

2 Practical limits: finite linewidt h and absorption length 

So far, we have considered a monochromatic coherent light for the local oscillator. The laser 

light, however, always has a finite linewidth, so that the modulation depth in Eqs.(ll) and (13) 

may be decreased as the linewidth increases. In this section, we will consider the practical limits 

to squeezing due to the finite active layer depth of the photodetector as well as the finite laser 

linewidth. 

The effects of the line broadening can be calculated from the Gaussian probability density 

function [7] 

Another practical limit comes from the finite thickness of effective absorption-layer of the photodetectors. The period of the spatial modulation of the quantum fluctuations is of the order of 

the optical wavelength, whereas the effective depth of the detectors is typically much larger than 

the wavelength. The probability that a photon is converted into an electron-hole pair at a distance 

< from the surface of the detector's active region can be written as [8] 

where K the absorption coefficient of the detector material. 
We can evaluate the expectation values of the fluctuations in Eqs. (11) and (13), with respect 

to this probability function and obtain 

and 

where = arctan(%) and D is the thickness of the depletion layer of the photo detector. 

Note that we also have used the fact that the variation of e- (z+t)2"k2 in the region [O,D] is so 

small that we can extract this term as a constant out of the integrand in the above equations. 

We can easily see from Eqs. (16) ind (17) that all the modulation terms which have 4,22 

dependence are reduced by a scale factor JGZ2. In Figure 3, we plot the fluctuations 

of and (Il - 12)2 in Eqs.(l6) and (17) with respect to the absorption coefficient K and Azl. To 

observe deep modulations more than anything else, we need a large absorption coefficient K for the 

detector. For an Ar+ laser light (A = 514.5nm or k, = 122 122cm-I ), the absorption coefficients 

of the Si and Ge are about 104cm-I and 4 x lO'~rn-~, respectively. The scale factors are thus 0.04 

and 0.85 for Si and Ge detectors. Therefore it would be possible to observe the squeezing effects 

with a Ge type detector. However, it might be hard to measure a modulation of the quantum 

fluctuations for the Ar+ ion laser with a silicon detector . If we make a very thin depletion layer 

for the silicon detector in order not to wash out the spatial modulation, the quantum efficiency 

7 will decrease, which implies the fact that all the photons are not converted into electron-hole 

pairs. Some extra noise will then add up with a relative amplitude (1 - q), and this extra noise 

will dso degrade the modulation of the quantum fluctuations. 

The linewidth of a single-mode frequency-stabilized Ar+ laser is better than 1 MHz , which 

is equivalent Ak = 0.0002~na-~. The scale factor for this linewidth e-Ak2z2 is 0.9996 for z R 

100cm. Therefore the practical limit to the modulation or squeezing effects is mainly due to the 

characteristics of the photodetectors in use. 

3 Conclusion 

We have proposed the possibility to produce squeezed state of light simply with a single conducting 

mirror without elaborate experimentation to produce squeezed vacuum. The electromagnetic field 

modes near a perfect mirror are modified with respect to those in free space due to the cavity 

QED effects: the modes of the vacuum fluctuations have sinusoidal spatial dependence from the 
mirror. These modified vacuum-field modes, when combined with a coherent light, may produce 

spatially modulated squeezing effects for the coherent light, which can be measured in a balanced 

homodyne detection scheme. If we divide a coherent local oscillator with a 50:50 beam splitter 

and combine with the modulated vacuum field, we estimate that the quantum fluctuations of the 

combined light are reduced by as much as 50% below the intrinsic fluctuations of the coherent 

light at distances z = nX (n : positive integer) between the photodetectors and the perfect mirror. 

Moreover, at those position, we reduced the fluctuation of the difference of two beams which are 

come from the unused beam splitter. In other words, we can obtain the highly sub-Poissonian 

fluctuation of the difference which may come from the totally correlated beams such as photon 

twins generated by the parametric amplification [9]. 

The finite laser linewidth degrades the spatial modulation by a factor e-Z2Ak2, but this factor 

can be neglected if we use a narrow linewidth laser. Decreasing the distance between the perfect 

mirror and the detector will also help. The imperfect reflectivity of the mirror may slightly decrease 

the modulation of the vacuum field. But, since the reflectivity of a metal-coated mirror at the 

optical frequency is about 97% , this also gives negligible effects. The most important practical 

limit comes from the characteristics of the photodetector. We need a detector whose quantum 

efficiency is close to unity and whose absorption coefficient is large enough that the depth of the 

effective absorption region is smaller than the wavelength (e.g., Ge type photo detector). Including 

all these limits we have calculated the quantum fluctuation of the light intensity in the last section 

and'shown in Fig. 3. The results indicate that we may increase the ratio of the average intensity 

to the intensity fluctuation using this homodyne detector with a good conducting mirror. 

References 

[I] D. Meschede, W. Jhe and E.A. Hinds, Phys. Rev. A. 41,1587 (1990) 

-. 

[2] We consider only the tangential components with the wave vector k normal to the mirror, 

which can couple with the local oscillator at the detector in our experimental configuration 

of Fig. 1. 

[3] W. Jhe, D. Meschede, and E.A. Hinds, to be published. 

141 B. Yurke,Squeezed Light , (AT&T Bell Laboratories Murray Hill, New jersey,1989) 

[5] P.D. Drummond, Phys. Rev. A 35, 4253 (1987) 

[6] B. Yurke, Phys. Rev. A 32, 311 (1985) 

[7] A.E. Siegman, Laser (Oxford University Press, Oxford, 1986) 

[8] S.M. Sze , Semiconductor Devices Physics and Technology (AT&T Bell Laboratories Murray 

Hill, New Jersey, 1985) 

[9] A. Heidmann, R.J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre and G. Carny, Phys. Rev. 

A 31, 2049 (1987) 
Fig. 1 Homodyne detector. Dl, D2 : photo detector, B.S. : bean) splitter. E, and EL represent 

the signal and local oscillatbr, respectively. 

Fig. 2 The vacuum mode relations in the beam aplit tt.1. with a conrlact.ing mirror. 
Fig. 3 The amplitude fluctuation of (a)z and (b) (I, - 12)2, ,normalized to / a 12, as a 

function of log K ( K : absorption coefficient in units of crn-' and A:, is displacemen.) Note that 

the intrinsic quantum fluctuation without the mirror correspond to 0.5 ill (a) and 1.0 in (b) on 

the vertical axis. The arrow indicates for the Ge detector whicll has K = 4 x 105cm-' (N.F. : 

Normalized Fluctuation ) 
HOW TO CHARACTERIZE THE NONLINEAR AMPLIFIER? 

Drnitri Kouznetsov Kallistratova 

Centro de Instrmmentos, UNAM,Ap.70-186 Cd. Universitaria 04510 DF,Mexico 

and Lebedev Physical Institute, Leninsky 54, 117924 Moscow 

Carlos Flores Cotera 

Centm de Instrumentos, UNAM,Ap.70-186 Cd. Universitaria 0451 0 DF, Mexico 

Abstract 

The conception of the amplification of the coherent field is formulated. The definition 

of the coefficient of the amplification as the relation between the mean value of the field at 

the output to the value at the input and the definition of the noise as the difference between 

the number of photons in the output mode and square of the modulus of the mean value 

of the output amplitude are considered. On the simple example it is shown that by these 

definitions the noise of the nonlinear amplifier may be less than the noise of the ideal linear 

amplifier of the same amplification coefficient. Proposals to search the other definition of 

basic parameters of the monlinear amplifiers are discussed. This definition should enable us 

to formulate the universal fundamental lower limit of the noise which should be valid as for 

linear quantum amplifiers as for nonlinear ones. 

1 Introduction 

In the development of the modern communication systems the tendencies to reduce the energy of 

the signal and to increase their frequency take place. The question about the minimal energy of 

the signal which can carry the information without the significant bit error rate is important. If 

the amplitude coding, the phase of the field carries no information, and there is no fundamental 

limit on the noise of amplifiers: in principle it is possible to count the number n of photons and 

to construct the state of G2n photons, where G2 is the intensity amplification coefficient. The 

uncertainty of the number of photons at the output in the ideal case should be exactly G2 times 

greatly than at the input. But the face of the field sometimes is important. For example, if the 

3-dimensional picture should be transferred, the "exact" amplification of number of photons in 

each mode causes the complete loss of the phase information, and the output picture is plane. So, 

the question of the phase properties of quantum states and their amplification is very important. 

For the case of the linear amplification the inicially coherent state the minimal fundamental noise 

is determined [l] by the amplification coefficient: 

N,;, = GG* - 1, 

where N is the intensity of the output noise 
while G is the amplification coefficient: 

Here subscripts "in" and "out" denote the initial state (before the amplification) and the output 

state (after the amplification). We assume that these states are related by the unitary transformat ion 

I UI >in; U+U = 1. (4) 

In the simplest case of the single-mode linear amplifier the transformation of the field operator is 

linear: 

U+~U = Ga + F, (5) 

where G is a c-number amplification coefficient and F is an operator which transforms the state of 

the amplifier only and does not touch the mode of the field. So, all commutators and correlators 

of all degrees of F and F+ with all degrees of a and a+ are zero. 

The aim of this work is to apply the definitions (1-4) to the example of the nonlinear amplifier. 

(For the nonlinear amplifier the relation (5) is not valid). Here we don't interested by the squeezing 

properties discussed in [4],[1], the fundamental limit of the noise is the problem which should be 

investigated. 

2 The parametric amplification the saturation 

Consider the model example of the quantum amplifier with small nonlinearity. The parametric 

amplifier with the saturation of the pump may be considered as the nonlinear amplifier. In the 

two-mode approximation the Harniltonian can be written in the form 

where a is the operator of the mode of the field, b is the operator of the idler mode, and the symbol 

:. ..: denotes the normal ordering. All correlators and all commutators of all degrees of a, a+ with 

all degrees of b, b+ are zero because the amplifier knows nothing about the phase of the field which 

should be amplified. The c-number parameter c describes the depletion of the classical pump. In 

the ordered form the hamiltonian (6) can be rewritten as 

In the following consideration the parameter c is assumed to be small. Consider the transformation 

of the field defined by the operator 

U = exp(-iHt), (8) 

where t is the c-number parameter. It may be considered as dimensionless time of the interaction. 

The Heisenberg equations have the form 
Consider the perturbation theory by the parameter e. Let 

In the 0-th order of the perturbation theory the introduction of (1 1) into (9),(10) gives: 

ao(t) = a(O)cosh(t) + b(o)+sknh(t), (12) 

bo(t) = b(Q)cosh(t) + a(O)+sinh(t); (13) 

Use notations a = a(O), b = b(0) , s = sinh(t), c = cosh(t); for the first order: 

Let the input we have the coherent state with amplitude a; the mean value of the number of 

photons x = aa* . The calculation of the amplification coefficient (3) and of the mean value of 

photons at the output gives: 

< a+a >,&= c2x + s2 - 2e(x(2 + x)c(c3/3 - c + 213) + 2(x + l)s4/3) + 0(e2). (16) 

By the formula (1) the output noise of the amplifier is 

while the noise of the ideal linear amplifier with same amplification coefficient G by formula (15) 

The difference of two last formulas gives 

This difference is negative at positive E: and all values of x. It proves that the noise of the nonlinear 

amplifier may be less than the noise of the ideal linear one with the same amplification coefficient. 

3 Discussion 

The-example of the "better than linear" quantum amplifier is constructed in the previous section. 

During the discussion at the Workshop it was suggested to consider another example of the nonlinear amplifier - the system of identical resonant 2-level atoms interacting with the single mode of 

the boson field. This system has the "exact" solution [3] and the dependence of the noise intensity 

on the amplification coefficient and the input intensity may be presented also in the extremely 

nonlinear case. It indicates the same possibility to realize the "better than linear" amplifier. Infortunately, the consideration is not yet finished at the time of the deadline of the submission. 
The paper on the noise properties of the "Tavis-Cummings amplifier" [3] should be published 

elsewhere. Formulas of the previous section based on the simplest generalization of definition. of 

the amplification coefficient and the noise intensity to the case of the nohear amplifier. 

(19) shows that in the sense of these definitions the nonlinear amplifier may be better then the 

ideal linear one. Either the real system with "better then linear9' information capabilities can be 

constructed or the another, more appropriate definition of the amplification coefficient and (or) 

the noise of the nonlinear amplifier shouldbe investigated. One of the alternative possibilities 

suggested at the Workshop - to define the amplification coefficient by 

instead of formula (3). It is under investigation now. The possibility of the other definition of 

the noise (instead of (2)) may be considered too. Of course, no "better then linear" amplidier 

should break information limits of quantum states ground in [4]. So it is important to work out 

appropriate parameters to characterize the nonlinear quantum amplifier of general kind. 

We are grateful to Roberto Ortega, Javier Mondragon, Marvin Mit telman, Alexei Bulatov and 

other participants of the Workshop for the discussions. This work is partially supported by the 

Comitet Nacional de Ciencia y Technologia, Mexico. 

References 

[I] C. Caves, Phys. Rev. D 26, 1817 (1982). 

[2] H. Haus, 6. Mullen, Phys. Rev. 128, 2407 (1962). 

[3] M. Tavis, F. W. Cummings, Phys. Rev. 170, 379 (1968). 

[4] Y. Yamamoto, H. Haus, Rev. of Modern Physics, 58, 1001 (1986). 
PHASE PROPERTIES OF MULTHCOMPONENT 

SUPERPOSITION STATES IN VARIOUS AMPLIFIERS 

K.S.Lee and M.S.Kim 

Department of Physics, Sogang University, CPO BOX 1142, Seoul, 121-742, Korea 

Abstract 

There have been theoretical studies for generation of optical coherent superposition states. 

Once the superposition state is generated it is natural to ask if it is possible to amplify it 

without losing the nonclassical properties of the field state. We consider amplification of the 

superposition state in various amplifiers such as a sub-Poissonian amplifier, a phase-sensitive 

amplifier and a classical amplifier. We show the evolution of phase probability distribution 

functions in the amplifier. 

. 1 INTRODUCTION 

The superposition principle lies at the heart of quantum mechanics according to Dirac [I]. In this 

paper we consider the amplification of optical superposition states. As a result of interaction of 

a single mode coherent state with a nonlinear Kerr medium, the coherent state input becomes a 

generalized coherent state [2 - 31. The dynamics of a single mode field propagating in the Kerr 

medium is governed by the effective Hamiltonian [2] H = wA + XA2, where X is the non-linear 

factor and A is the photon number operator. Under the influence of the nonlinear interaction the 

initial coherent state la) of the amplitude a evolves at time t into the state 

where Ik) is a Fock state. At the interaction time t = TINA, we can rewrite Eqn(l) as a form 

which is a superposition of N coherent component states located on a circle with the centre at 

the origin of phase space. If we decompose the state (2) into the Fock basis and compare it with 

Eqn(1) we obtain an equation for the arguments, Cn, of the coefficients of coherent components 

for an arbitrary value of N. For example when N = 4 we have 

which we call the Yurke-Stoler state throughout the paper. 

Such superpositions of multicomponent coherent states may be generated not only in the 

amplitude dispersive medium but also in the micromaser type experiment. Recently Garraway et 

al. have proposed a method to prepare quantum superposition of multicomponent states [4, 53 
for the eventual goal of preparation of a Fock state. A stream of three-level atoms are injected 

into a high-Q micromaser cavity. It is assumed that there is just one atom at a time in the 

cavity and the initial coherent cavity field is tuned to the two-photon resonance with the atomic 

transition. A superposition state of two coherent component states is generated by a conditional 

measurement of the atomic excitation after an interaction time that determines the relative phase 

of the component states. By a sequence of the conditional measurements the superposition of 

multicomponent coherent states may be generated. As a special case, the initial interaction time 

is chosen to create a superposition state of two component states separated in phase space by ?r 

and the interaction times are reduced by one half after each interaction. The second conditional 

measurement creates a superposition of four component states 

where N is a normalization constant. Eventually, the Nth measurement creates the superposition 

of 2N components separated by 2~12~ in phase space. We call the state (4) as the Garraway 

state. 

2 PHASE PROBABILITY DISTRIBUTION FUNCTION 

There are quasiprobabili t y distributions according to ordering of system operators. One is the 

Glauber P representation which is quasi probability distribution function for the normal ordering 

of the system operators and another is the Q function &(reie) for antinormal ordering of system 

operators [6]. The Wigner function W(reie)for the symmetrical ordering can be negative. Since 

the P function cannot be defined in the nonclassical regime, the P function is not dealt with in 

this paper. 

We can study the phase probability of the system with help of the quasiprobabilities. We 

derive two phase probability distributions from the Q and Wigner quasiprobability distributions 

[TI : 

Buzek et al. [7] compare these two probability distributions with phase probability defined by Pegg 

and Barnett [8]. As for the Wigner function, the Wigner phase probability distribution can have 

negative values, which indicate nonclassical nature of the system. For the quantum superposition 

state, the quantum interference between component states are best illustrated by the Wigner 

phase distribution as it can become negative for the quantum interference of component states. 

As shown in Fig.la the quantum interference for the Yurke-Stoler state is reflected by the negative 

values of the Wigner phase probability function. However the Wigner function for the Garraway 

state is always positive as in Fig.lb. We can therefore conclude that the quantum interference 

is not necessarily represented by negative values in the Wigner phase distribution. The Q phase 

distribution function is always positive differently from the Wigner phase distribution function as 

shown in Fig.1. 
Fig.1. Wigner and Q phase distribution of the Yurke-Stoler (a) and Garraway (b) 

state. Solid line shows the Wigner phase distribution and dashed line shows the Q 

phase distribution. (a = 3) 

AMPLIFIED SUIIPBERPOSITION STATES 

The simplest way to amplify the state of the single-mode field is to displace it by the displacement 

operator D(o) [6]. This operator shifts a field state by a given amplitude in phase space. The 

displacement of a field state can be implemented by driving the field by a classical current. The 

displaced superposition state is expressed by 

D = exp (odt - 86) , 

where 140) is the initial superposition state. From definitions of the quasiprobability distributions 

it follows that their shapes are invariant with respect to the action of the displacement operator. The only difference consists in the shift of the Wigner function of the state in phase space 

along the action of the displacement operator. One of consequences of this invariance is that the 

displacement of the state does not change the quadrature-squeezing properties associated with 

the original state. As the whole picture is displaced along one direction in phase space the phase 

distribution will be differed by the action of displacement as shown in Fig.2. The Wigner phase 

distribution becomes to have negative values by displacement. 

Recently the correlated (phase-sensitive) amplifier has been realized experimentally [9]. In 

this section we study the evolution of the phase probability distribution of the superposition 

states amplified by the phase-sensitive amplifier. The dynamics of the field mode coupled to the 

phase-sensitive amplifier is in the Born and Markov approximation governed by the Fokker-Planck 

equation for the Q function, which in the interaction picture can be written as [lo] 
Fig.2. Wigner phase distribution for the displaced Garraway superposition state. 

Solid line is for the initial state and dashed line is for the displaced state (a = 1, a = 3). 

where y is proportional to the coupling between the field mode and the environment, and No and 

Mo are the parameters determined by nature of the amplifier. If the phase-sensitive parameter 

Mo is equal to zero then the Fokker-Planck equation (8) reduces into the equation describing the 

phase-insensitive amplification of the single mode field. The gain G of the amplifier is defined as 

G = exp(yt). 

Fig.3 clearly shows that the choice of the M value determines into which quadrature noise 

should be added. When M is positive the phase information at d = 0 and 7r is kept while the 

phase information at 6 = 7r/2 and 3~/2 is lost very rapidly. However, when M is negative the 

information at 6 = n/2,3~/2 is kept at the expense of the rapid loss of the information at the 

other quadrature. 

We consider a stream of atoms injected into a micromaser cavity with an infinite &. We assume 

that there is just one atom at a time in the cavity and the atom makes the two-photon transition 

of frequency wo between the nondegenerate ground and excited states via a single intermediate 

level. The cavity is assumed to be tuned to the two-photon resonance with the excited and ground 

levels and the intermediate level is so detuned not to be excited, so that one photon transitions 

can be neglected which means that the transition between the ground and excited levels can be 

considered as a two-photon process. 

Let us assume that the field mode is initially prepared in a pure state 

and the atom is prepared in the excited state. During the time evolution the atom and the field 

become strongly entangled [4]. Nevertheless, at some particular moments they become dynamically 

disentangled and are again in their pure states. One of those moments is identical to the revival 
Fig.3. Wigner phase distribution for the Yurke-Stoler state in the phase sensitive 

amplifier characterized by No = 3, Mo = (a) and by No = 3, Mo = -m (b). 

Dashed line is for the initial state and solid line is for the state at G = 1.22 (a = 3) 

Fig.4. Part of the Wigner phase distribution of the Yurke-Stoler state in the subPoissonian amplifier. Line a is for the initial state, line b is for the state with two extra 

photons and line c is for the state with four extra photons. (a = 3) 
time tR (see ref. [4] for details) when the atom is (approximately) in its ground state and the field 

can be described by the state vector 

At this moment exactly two photons are transferred from the atom to the field, so the mean 

photon number of the field is fi(1) = fi(0) + 2, where ~(i) is the mean photon number of the 

cavity field after i atoms pass the cavity. Analogously, after a sequence of M atoms each of which 

interacts for time tR with the cavity field, the state vector of the field can be written as: 

We call the process during which the exact number of photons are transferred to the field as the 

sub-Poissonian (amplitude-squeezed) amplification. 

In Fig.4 we can see that the Wigner phase distribution function is smoothened by the amplification. This means that the phase uncertainty is enlarged as the number uncertainty (so that 

the energy uncertainty) is reduced by the sub-Poissonian amplification. 

This paper was supported by NON DIRECTED RESEARCH FUND, Korea Research Foundation, 1993. 

References 

[I] P.A.'M. Dirac, Principles of Quantum Mechanics, 4th Ed., (Oxford University Press, London, 

1954). 

[2] B. Yurke and D. Stoler, Phys. Rev. Lett., 57, 13(1986). 

[3] Ts. Gansog and R. Tanas, Phys. Rev. A 44, 2086(1991). 

[4] K.S. Lee, M.S. Kim, S.-D. Lee and V. Buzek, J. Korean Phys. Soc., 26, 197(1993); K.S. Lee, 

M.S. Kim and V. Buzek, submitted to J.Opt.Soc.Am. 

[5] B.M. Garraway, B. Sherman, H. Moya-Cessa, P.L. Knight and G. Kurizki, submitted to 

Phys.Rev.A. 

[6] R.J. Glauber, Phys.Rev., 131, 2766(1963). 

[7] V. Buzek, Ts. Gantsog, M.S. Kim, submitted to Phys.Scrip. 

[8] D. T. Pegg and S. M. Barnett, Europhys.Lett. 6, 483 (1988) 

[9] A.Y. Ou, S.F. Pereira and H.J. Kimble, Phys.Rev.Lett. 70 3239 (1993). 

[lo] M. S. Kim, K. S. Lee and V. Buzek, Phys. Rev. A 47, 4302(1993). 
SUB-POISSONHAN LIGHT AND PHOTOCURRENT SHOT-NOISE 

SUPPRESSION IN CLOSED OPTO-ELECTRONIC LOOP 

A.V.Masalov, A.A.Putilin, M.V.Vasilyev 

Lebedev Plzysical Institute, Leninsky pr.53, 1 17924 Aloscow, Russia 

Abstract 

We examine experimentally photocurrc~lt noise rcductio~i in the opto-clcctronic closed 

loop. Photocurrent noise density 12.5 dl? bclow the shot-noise level was observed. So large 

suppression was not reached in previous experilncnts [I, 2, 3, 41 and can not be explained in 

terms of an ordinary sub-Poissonian light in the loop. We propose the concept of anticorrelation state for the description of liglit in tlie loop. 

We study closed opto-electronic loop consisting of light source (single-mode, single-frequency 

argon laser with 1W output power at the wavelengtl~ 514 nm), clectro-optical modulator with halfwavelength voltage 170V, p-i-n pliotodiocle having quantum eficiency 0.68 and electronic feedback 

circuit including two amplifiers with total 62-63 dI3 amplificatior~ in the 1-100MI-Iz frequency 

range and noise tcinpcrature less tlian 3001< (Fig.1). 'l'lie modulator operates ill the regime of the 

amplitude modulation, where feedback strengtli A is proportior~al to the light power and can be 

easily changed. The feedback strength has been controlled by the value of average photocurrent: 

A/ (i) = 0.6-0.7 mA. 

Experimental noise spectra of the feedbacl; pliotodiode current (Fig.2) are in a good agreement 

with formula 

where is the standard shot-noise level, T is loop round-trip time. Eliminating the noise 

amplification areas by a narrow-bandpass filter we have obtained the photocurrent noise reduction 

up to 12.5 dB (=factor 17.8), i.e. 94.4% of standard shot noise has been suppressed (Fig.3). This 

noise reduction is beyond the limit imposed by non-unity quantum eaciency of the feedback 

photodiode: (hi2) / (SiiNL) should be larger tllan 1 - 71. Pliotocurrerit noise reduction does not 

depend on tlie quanturn eficiency of tlie fecdbacl< pliotodiode and can be ~nade arbitrary large 

(Fig.4). Our data can not be explained in tcrx~is of an ordinary sub-Poissonian light in the loop 

and require an i~ltroduction of tlie ncw coi~cel>t ofliglit. 

The light bcarn extracted from tlic loop csl~ibits a. iloisc above the sta~idard quantum limit 

(Fig.5). This noise is shown in Fig.6 versus tlie fecdbaclc strc~igtll for two values of the feedback 

photodiode quantum efficiency. The data arc in a good agrccrnent with the theory [5], shown by 

solid line. 

To describe the state of liglit in tlic loop we offer the concept of a.nticorrclation state: the 

temporal variations of average liglit iritcrlsity l>roduccd Ly electrically driven nlodulator are anticorrelated to its clunntum fluctuations. 1)uc to this n~~ticorrcl;lt,io~t tllc total fluctuations of in-loop 
light intensity can be suppressed: 

where (61ioL) is the standard quantum lcvcl. I11 addition tllc part of tlie intensity modulation 

component is a posteriori anticorrelated to photocurrent fluctuations arising in the diode due to 

non-unity quantum efficiency. Thus the pliotocurrent noise is suppressed, and this suppression 

is more effective than in the light because the closcd opto-electronic loop is a device stabilizing 

the fluctuations of the current at the modulator. The light here serves only as a feedback signal 

carrier. 

When a beam splittcr is inserted into the loop to extract the liglit, the intensity modulation 

component contains a term anticorrelated to tlie quantum fluctuations brought by the beam 

splitter. However in the extracted beam the corresponding quantum fluctuations are correlated 

rather than anticorrelated to tliis ~liodulation term. Tliercfore cxtractcd beam noise is always 

super-Poissonian: 

The principal difference between tlle anticorrelation light and ordinary sub-Poissonian light is 

seen from the following example: the ordinary sub-Poissonian light decreases its sub-Poissonian 

quality when passes through a beam splitter; the anticorrelation light with sub-Poissonian noise 

passes through a beam splitter and improves its sub-Poissonian factor while the reflected beam 

gains super-Poissonian statistics. In principle, this paradoxical situation may be observed by QND 

measurements. 

References 

[l] S.hllasliida, Y.Yamamoto. Optics Com~nun. 57, 290 (1956). 

[2] Y.Yamamoto, N.Imoto, S.hlashida. Pllys. Rev. A 33, 3243 (1986). 

[3] P.R.Tapster, J.G.Rarity, J.S.Satchel1. Pliys. Rev. A 37, 2963 (1988). 

[4] Ya.A.I;bfanov. I<vantovaya Elektronilia (Russian) 16, 2593 (1989). 
Spectrum I Analyzer dc 

Fig. 1. Experimental setup: a - liicasurelilc~lts of pllotocurrellt rroisc ill tlie closcd 

bpto-electronic loop, 13 - the sa.mc for tlic bealn extracted from the loop. 13D - pllotodiodes, EOhl - electro-optical modulator, US - beam splitter, A - aml~lifiers, Att 

attenuator. 
Frequency, MHz 

Fig. 2. Noisc spcctra of thc fccdltack pliotodiodc current at different values of the 

feedback strength; a - amplifier iloisc, loop rduxltl-tril> ti~lle T = 4711s. 
19.0 19.5 

Frequency, MHz 

Fig. 3, a - photocurrent noise spectral density in tlie loop with a narrow-band 

filter in the feedback circuit, b - standard shot-noise level, c - amplifier noise. 

' 0 1 2 3 

Photocurrent (i), mA 

Fig. 4. Shot-noise suppression factor at tl~e freque~icy 19.2 A4112 versus feedback 

strength in the units of average photocurrent. Dashed line is the limit due to the 

quantum efficiency of tlie feedback photodiode 71 = 0.68. 
0 2 0 40 60 80 I00 

Frequency, MHz 

Fig. 5. Pllotocurrent noise spectrunl for tlie ~)l~ot,odiode in the extracted beam 

(a) and the relevant standard shot-iioisc levcl (b); c and d are tllosc for feedback 

photodiode. 

0 1 2 3 

Photocurrent (i ), mA 

Fig. 6. Noise spectral density of the extracted beam photocurrent, normalized to 

the standard sliot-noise level, vcrsus tlic fecdba.ck stre~igtli (in tlie units of tlie average 

fcedbacli pliotocuirrcnt). 
MODES IN LIGHT WAVE PROPAGATING IN 

SEMICONDUCTOR LASER 

M. A. Man'ko 

Ee bedev Physics Institute, Leninsky pr., 53, Moscow, 11 7333 Russia 

The study of semiconductor laser based on an analogy of the Schradinger equation and 

an equation describing light wave propagation in nonhomogeneous medium is developed.The 

active region of semicondnctor laser is considered as optical waveguide confining the electromagnetic field in the cross-section (x, y) and allowing waveguide propagation along the 

laser resonator (%).The Inode structure is investigated taking into account the transversal 

and what is the important part of the suggested consideration longitudinal nonhomogeneity 

of the optical waveguide& is shown that the Gaussian modes in this case correspond to 

spatial squeezing and co~selation.Spatially squeezed two-mode structure of nonhomogeneous 

optical waveguide is given explicitly.Distribution of light among the laser discrete modes is 

presented.Properties of the spatially squeezed two-mode field are described. The analog of 

Franclr-Condon principle for finding the maxima of the distribution function and the analog 

of Ramsauer effect for control of spatial distribution of laser emission are discussed. 

Introduction 

The aim of the talk is to study the possible modes of the electromagnetic field propagating in a 

semiconductor laser taking into account nonhomogeneous longitudinal structure of the m&m 

in the optical waveguide of the 1aser.The Gaussian modes in such structures may demonstr& a 

squeezing phenomenon in the electromagnetic field distribution in transversal cross-section of the 

semiconductor laser.We will consider the light propagation along the active layer of the semiconductor laser understanding that the refractive index of the media of this layer has such dependace 

on the transversal coordinates z and y that provides the waveguiding conditions confining the electromagnetic field in the transversal section of the laser waveguide.We also assume that the value of 

the longitudinal coordinate z which varies along the laser resonator axis influences the refractive 

index.Our goal is to show that the refractive index dependence on the longitudinal coordinate z 

produces the change of the widths of the Gaussian electromagnetic field beam that may influence 

the far field distribution.The equation for propagating field of the fixed frequency w in paraxial 

approximation is derived from the Helmholz equation 

for which the structure of the field looks as the amplitude A which has slow dependence on the 

longitudinal coordinate z and the fast oscillating exponential.Due to this the equation for the 
amplitude A(x, y, z) is reduced to the Schrijdinger-like equation (Fock-Leontovich approximation 

[Illor parabolic approximation) 

in which formally the coordinate z plays the role of time and the refractive index plays the role of 

the potential energy.If there is no dependence of refrective index on the coordinate z the equation 

(2) is equivalent to the SchrXnger equation for the stationary states which in case of waveguide 

describe the possible modes of electromagnetic field propagating along the laser resonator. If the 

z-dependence of refractive index takes place the structure of the modes changes essentially.For 

optical fibers this phenomenon has been discussed,for example,in [2]. Our aim is to consider 

this phenomenon for the semiconductor laser waveguide since the physical picture of the light 

propagation in semiconductor laser has its own specifics and applications. 

2 Spatial Squeezing of Semiconductor Laser Beam 

As a model we will suppose that the refractive index has parabolic profile in transversal coordinates with z-dependences of the formal "frequency" -coefficients in parabolic terms wi(z)x2/2 and 

wi(n)y2/2.We will apply the result known in the theory of nonstationary quantum oscillator to the 

case of optical waveguide of semiconductor laser with the parabolic refractive index of the active 

1ayer.It was shown [3] that the width of the Gaussian packet describing at the initial moment the 

ground state of quantum oscillator due to the time dependence of the frequency has the following 

value at t-moment 

where the function e satisfies the equation 

Dots mean the-derivatives.13'e can use the complete analogy of our equation for the fight propagation with Schrijdinger equation for the nonstationary oscillator.Due to this malogy the foll 

conclusion may be made for the influence of z-dependence of the refractive index on the behaviour 

of the fundamental mode.Its width in transversal coordinate x has the form 

and in transversal coordinate y it has the form 

where the functions e,(z) and e,(z) satisfy the equations 

<(z) + u:(Z)ex(~) = 0, 

e,(z) + wi (z)ey (z) = 0 
and the initial conditions 

where z = 0 is the coordinate of the left mirror of the laser resonator. The formulae (9) and (8) 

are given in dimensionless variables. It is known that the functions e, and e, may become less 

than 1 for appropriate z-dependence.In our case it means that for appropriate refractive index 

we may obtain squeezing of the light spot on the right mirror of laser resonator in coqasison 

with the width of the field distribution on the left mirror in both transversal directions.0gpmite 

phenomenon also may be present.It means that changing somehow the media properties dong 

the resonator axis we could form the far field distributi0n.h particular there may exist spatid 

squeezing in the light beam propagating in semiconductor laser. 

3 Spatially Squeezed Two-Mode Structure in the Optical 

Waveguide of Semiconductor Laser 

We have discussed above the spatial squeezing for Gaussian modes,because the squeezing for 

modes which differ from fundamental one are described by the same formulae (5) and ($).Xow 

we will consider the spatial squeezing in discrete modes of semiconductor laser using the same 

analogy with time-dependent oscillator.h generic case these discrete modes $,, are described by 

Hermite polynonnials of two %xxiables.'In our case it may be shown that the widths of light spots 

on the right mirror of the laser resonator may be given by formulae 

We see that the width ia both direction may be essentially reduced by choosing an approp~ate 

z-dependence of refractive index.This spatial squeezing may be achieved both for Gaussia mode 

and discrete modes simltusl.he appropriate z-dependence of the refractive index comesponds to such behaviour of the function ex and e, for which their moduli become much ]less than 

1 on the right anirror.In these cases we have the spectral squeezing of the light patterns related to 

the discrete mode structure of the semiconductor laser beam. 

4 Analogs of Fr ck- Condon Principle and msauer Ec 

feet for Semiconductor Laser Nonhomogeneous dong 

the Optical Axis 

Since we have established that for the field in the semiconductor laser the formal role of potentid 

in Schrijdinger equation is played by the dielectric constant,while the role of time is played by 

the coordinate dong the cat3ty axis [4J, [5] ,[6] we have a complete analogy of the character of a 
stepwise change in the properties of the active region along the laser optical axis to the character of 

change in vibrational motion of nuclei after the electron transition in a polyatomic mo1ecule.h order to use this analogy,we will exploit existing results in quantum mechanics to explain the various 

operating conditions of a laser emphasizing that the role of the field is played by the wave function @.In quantum mechanics, bound states exist when the probability density l2 diminishes 

exponentially away from the potential energy minimum.For a semiconductor laser,these bomd 

states with quantum energy levels correspond to mode solutions of the wave equation.Hence,my 

conclusion regarding the transitions between energy levels due to changes in a potential over time 

may be reformulated analogously for the case of light redistribution among the modes when two 

(or more) different parts of the active region are joined corresponding to inhomogendty of the 

dielectric constant along the cavity axis.A continuous junction (slow change of refractive index 

along the cavity) is possible together with stepwise junction (a step in refractive index profile in the 

vicinity of the junction) .In quantum mechanics of nonstationary potential the SchrGdinger quation is solved in some cases for varying gotentia1,but such cases are few and they include a specific 

dependence of the potential on the coordinates(see [?I). For time irregular changes in potentidyin 

quantum mechanics the behavior of a system for the case of polyatomic molecules is predicted 

. by the Franck-Condon principle for potentials arbitrary depending on the coor&nates.Hence9tEs 

formulation of the analog of the Franck-Condon principle for semiconductor laser operation is of 

interest, since the qualitative predictions obtained on the basis of this principle are independent 

of models describing the dielectric constant of the laser active region [$I. 

For taking into account the nonhomogeneities of heterojunction laser along the resonator we 

proposed in [4], [ti] the modd in which the active region was considered as a set of several opticd 

waveguides with different refractive indices connected to each other in such a way that the couphg 

between the separate sections of the active region was achieved with the help of mode couphg 

coefficients Cnm.%et us consider the laser consisting of two end-joined active regions (each one 

homogeneous along the optical axis).So,we consider in fact two connected lasers each having their 

own refractive index described by their own "potential curven. Each of these connected lasers 

is described by its own gpotential welln with the "energy levels" corraponding to the descrete 

laser modes. If both potential wells are plotted in the same figure the analog of the Frm&- 

Condon principle can predict which mode in the second laser (the second part of the laser) GU 

be excited with the maximd probability,when the field energy was contain4 in a given mode of 

the first laser (the first part of the laser).This analog of the Franck-Condon psinciple produces a 

qualitative prediction for finding the maximum of the square of the modulus of the mode coughg 

coefficient of two lasers (two parts of the laser) 

i.e.,an analog of the Franck-Condon factor 

for diatomic molecules (the overlap integral of the vibration wave functions before and after the 

electron transition).This factor describes the portion of energy contained in n-mode of the second 

laser (the second part of the laser) if all energy of the first laser (the first part of the laser) was 
concentrated in its m- mode.Let us note that this rule is valid in a semiconductor laser when the 

imaginary part of the dielectric constant of its optical waveguide does not significantly change the 

spectrum of mode "levelsn (in language of quantum mechanics,the imaginary part of the complex 

potential does not change the energy level spectrum, but rather is responsible only for broadening 

of the levels and their lifetime; however, the widths of the levels are small compared to the distance 

between levels). 

Another analogy that would allow using results from quantum mechanics in the physics of 

nonhomogeneous semiconductor lasers is the analog of the Ramsauer effect [9].Let us discuss the 

practical important case when distribution of dielectric constant in one transverse direction (along 

the x-axis) has the step-like form (in the case of heterojunction laser,the distribution of ~(x) in 

the direction of current flow is determined by the difference between the refractive indices of the 

active region and the wide-band layers of the heterostructure) and in the other direction (along 

the y-axis) it is approximated by parabola (symmetric waveguide). The dielectric constant in the 

resonator is described by the formula: 

and the equation for electromagnetic field in the laser formally coincides with the Schriidinger 

equation for the harmonic oscillator with variable frequency.The analog of the Franck-Condon 

factor (15),(16) for our model of longitudinal nonhomogeneous laser consisting of two homogeneous 

parts may be presented in the form,calculated in [7]: 

where PL(x) are the Legendre polynomials,and the parameter R can be interpreted formall:- as 

the reflection coefficient from a potential whose form is determined by the time dependence of 

the oscillator ftequency.1~ our case the role of time dependent frequency is formally played by 

the longitudinal dependence a(%) in formula (17).It is possible to find such dependence of the 

"potential" a(%) to satisfy the conditions for which the equality R = 0 takes place (analogue of 

the Ramsauer effect in quantum mechanics) and Ty,, = 6nm (there is no redistribution of the 

energy of the electromagnetic field over the modes).In this case the structure of nonhomogeneity of such longitudinal nonhomogeneous laser is such that there is "transparencyn in the point 

of connection of two lasers (two parts of the laser).With a specific dependence of ~(y, z) at the 

connection between two part of the laser one can regulate the transparency of the conriection.It 

is possible to use this effect for control of the output characteristics of the semiconductor 1aser.h 

the presence of dielectric insert in such a laser one can control the output characteristics of the 

laser by varying the characteristics of the dielectric insert [lo]. 

The formula (18) may be considered as the mode distribution function of the laser light energy 

in the second laser (the second part of the laser) between its modes with index m, (m = 0,1,2. ...) 

if the light in the first laser (the first part of the laser) is exactly in the mode state labeled by 

the index n.Such distribution function emerges if the connection of two lasers is described by the 

change of the discussed aboi-e "frequency1' related to z-dependence of the refractive index along 

the laser optical waveguide. More general mode distribution function may be expressed in terms 
of Mermite polinomials of two variables in complete analogy with the parametric oscillator theory 

given in (1 1],[12]. 

Acknowledgments 

The author would like to acknowledge the University of Maryland at Baltimore County for %indl 

hospitality and the American Physical Society for travel support. 

References 

[I] A. M. Leontovich and V. A. Fock, JETP 16, 557 (1946). 

[2] V. I. Man'ko, Lie Methods in Optics, Lectures Notes in Physics, 250,p.193,eds.J.S.Mondragon 

and K.-B. Wolf, (Springer Verlag,1986). 

[3] V. V. Dodonov and V. 1- Man'ko, Invariants and Evolution of Nonstationary Quantum Systems, Proceedings of Lebedev Physics Institute,l83,ed. M. A. Markov, (Nova Science Publishers, Cornmack, N.Y..1989). 

[4] M. A. Man'ko and G. T. Mikaelyan, Sov. J. Quantum Electron., 16, 985 (1986). 

[5] M. A. Man'ko and 6. T. Mikaelyan, Nonlinear Optics of Semiconductor Lasers, Proceedings of Lebedev Physics Institute, 166,p.l26,ed. N. G. Basov,(Nova Science Publishers,Comrnack,N.U., 1987). 

[6] M. A. Man'ko, ECOOS.4-90 Quantum Optics, p.247, ed. M. Bertolotti, (IOP Publishers Etd., 

1991). 

[7] I. A. Malkin and V. I. Man'ko, Dynamical Symmetries and Coherent States of Quanfum 

Systems (Nauka, Moscow, 1979, in Russian). 

[8] M. A. Man'ko and G. 'I. Mikaelyan, Sov. J. Quantum Electron., 16, 201, (1986). 

[9] M. A. Man'ko,Group Theoretical Methods in Fundamental and Applied Physics, p.247, 

U.H.Kopvillem, ed. (Nauka ,Moscow,l988, in Russian). 

[lo] M. A. Man'ko, Soviet Physics-Lebedev Physics Institute Reports, N12, p.38, (Allerton 

Press,N.Y.,1985). 

[ll] V. V. Dodonov, V. I. h.Ianlko, and V. V. Semjonov, Nuovo Cimento, B83, 145 (1984). 

[12] V. V. Dodonov, 0. V. Man'ko, and V. I. Man'ko, Phys. Rev.A (1993 submitted). 

SQUEEZING GENERATED BY A NONLINEAR MASTER 

EQUATION AND BY AMPLIFYING-DISSIPATIVE 

HAMILTONIANS 

V.V. Dodonov 

Moscow Institute of Physics and Technology 

16, Gagarin street, 1401 60 Zhukovskiy, Moscow region, Russia 

M.A. Marchiolli, S.S. Mizrahi and M.H.Y. Moussa 

Bepartamento de Flsica, Universidade Federal de Siio Carlos 

Via Washington Luiz km 235, 13565-905, ,560 Paulo, SB, Brslsil 

Abstract 

In the first part of this contribution we show that the master equation derived from the 

generalized version of the nonlinear Doebner-Goldin equation leads to the squeezing of one 

of the quadratures. In the second part we consider two familiar Hamiltonians, the Bateman - Caldirola - Kanai and the optical parametric oscillator; going back to their classical 

Lagrangian form we introduce a stochastic force and a dissipative factor. From this new 

Lagrangian we obtain a modified Wamiltonian that treats adequately the simultaneous amplification and dissipation phenomena, presenting squeezing, too. 

1 The nonlinear master equation 

Our search for a new class of equations was inspired by the model proposed recently by Doebner 

and Goldin [I]. They looked for the most general Schrijdinger equation compatible with the 

Fokker-Planck equation for the probability density p(x, t) = I $(x, t) 12, 

(where D is a constant positive diffusion coefficient), and derived the nonlinear equation 

for a particle with mass m moving in a scalar potential V(x). The advantage of eq. (2), comparatively to other ones [2], is its group theoretical origin, the nonlinear term was not simply 

added to the usual Schrodinger equation ad hoc, but its structure was derived from the analysis 

of representations of the Di$(R3) group, proposed as a "universal quantum kinematical group" 

[I]. The only drawback of eq. (2) is its limited range of applicability, since it can be used only in 
the coordinate representation. It is desirable to have a more general equation valid in arbitrary 

representations. To obtain such a generalization (heuristically), first, we remove the Cartesian 

coordinates dependence, introducing an arbitrary set of states I z), which form a complete system 

with respect to some measure dp(z), i.e., 

Secondly, we replace the two V - operators in the additional term of eq. (2) by two arbitrary 

(linear) operators and 8. By this way we arrive at the equation, whose nonlinear part, in 

general, is neither Hermitian, nor anti-Hermitian, 

Multiplying both sides of this equation by ($ I z) and integrating over dp(z) with account of (3), 

one can check that the normalization of the wave function is preserved in time since 

JW I I ~P(z) = 0, (5) 

where (z I fl{$)) is the non-Hamiltonian term in the right-hand side of eq. (4). Evidently, the new 

equation satisfies the homogeneity condition. Moreover, the "separability property" also holds: 

the wave function of noninteracting particles is factorized for all times, if it was factorized at the 

initial moment. Nonetheless, the form (4) is not the most general, in [3] we showed that the more 

general form for Q{$} is 

where X and r may be arbitrary complex numbers satisfying the restriction X + r = 1. In papers 

[4, 51 we investigated eq. (4) with B = A+ , A , B being lowering and rising operators for the 

two-level system and for the harmonic oscillator. It was demonstrated that this equation, written 

in the discrete energy (Fock) basis I n), describes the relaxation to "pseudothermal" stationary 

states, possessing the Planck distribution for the populations of energy levels, but nonzero offdiagonal elements of the density matrix. This is not surprising, since eq. (4) relates to pure 

quantum states. However, it is more natural to describe relaxation processes in terms of density 

matrices, in order to investigate the evolution of mixed quantum states. Therefore, our next goal 

is to obtain nonlinear master equations originating from eq. (6) and its special cases. 

Starting from equation (6) and considering the evolution of the pure state density matrix 

(Z I p+ I z') = (z I $)($ I 2') governed by this equation and replacing I $)($ I by P we obtain 

- DX(z1 BAP~z') - DX*(zIjA+B+Iz1) - ~(zI [I?, P] Iz'), (7) h 

which preserves the trace, normalization and hermiticity for an arbitrary initial density matrix. 
2 Application: The harmonic oscillator in the coordinate 

representation 

This section is devoted to investigating a special case of the general equation derived in Sect. 1. 

We shall limit ourselves to one dimension, identifying the ket-vector I z) x) and apply the new 

equation to the harmonic oscillator Hamiltonian 

we choose operators A and B as A = B+ = a ax + sx, where s, D,r,A, are assumed to be real 

constants. Now, designating the elements of the density matrix in the coordinate representation 

as p(x, y ) = p*(y , x) and introducing the notation 

we can write eq. (7) as 

All derivatives in this equation must be calculated for the independent variables x and y, and only 

after that one should consider x = y in the functions px, p,, and p,,. 

For the probability density P(x) r p(x,x) eq. (11) results in the following Fokker-Planck 

equation 

dP aJ d 

D dt dx -(xP)) dx , 

i A where J(x) = - ?;;I [p,(x, x) - py(x,x)] is the current density, 2sDx is the drift velocity, and 

D is the diffusion coefficient. Eq. (12) clearly shows that the total probability is conserved in 

time. Eq. (11) admits a special class of solutions in the form of Gaussian exponentials, 

with c and @ being real functions of time, while a and b may be, in general, complex. Putting 

expression (13) into eq. (1 1) we get the equations for the coefficients a( t) and c(t), which do not 
contain neither the force f (t), nor the function b(t). Therefore, let us assume, for simplicity, that 

f = b = 0. Then, instead of eq. (11) we have the following set of ordinary differential equations 

for the real functions a = Rea, /? = Ima, c, and @, 

Here the parameter ( = r - A is introduced . The trace of the Gaussian density matrix equals 

consequently, the condition of its conservation is the equation 

and it is easy to check that this equation is fulfilled. The difficulty of treating the equations for 

the coefficients a, p, c is connected with their nonlinearity even in the absence of nonlinear terms 

in the master equations (i.e., when D = 0). It is well known, however, that in the latter case the 

equations of motion for average values and the second order moments are linear for any quantum 

system with quadratic Hamiltonian. Therefore we replace the equations for coefficients a, P, c, by 

the equivalent equations for the variances 

these quantities are related to the coefficients of the Gaussian density matrix (13) by 

tL2 2 

Ox, tL2 a2 + p2 - c2 

a,, = -(a + c) + - = - 2 axx 2 a-c 

Another quantity characterizing the quantum state is 

4 a+c A = - tL2 (oxxupp - a:,) = - 

a-C 
Its importance is explained by two factors. First, any quantum state must satisfy the generalized 

Robertson - Schrodinger uncertainty relation [6] A 2 1. Second, any positively definite density 

operator must satisfy the inequality 

but for any Gaussian state [7] 

Tr(P2) - A-112 

[Tr(i9I2 

Consequently, the parameter A characterizes the "degree of coherencen of the Gaussian state: 

A = 1 for pure states, and A > 1 for mixed states. 

The relations inverse to eqs. (21) - (23) read 

so one can check that eqs. (14) - (16) result in the following equations for the variances, 

(j3Cp = 

1 

- a, - w2 m ox, - 4Ds ox, + D 9 [2 + E(1 + A)] , m 0x1 

2 4, = - 2w m a,, + (62 - 44,) Uxx 

As a consequence, the equation for the parameter A is 

We perceive that A is conserved in time if D = 0. Alternatively, if D # 0, then A tends to 

the unit value at t -+ oo, provided both coefficients, D and s, are positive. This means that any 

initial state exhibits the relaxation to a pure state! 

The advantage of the equations for the variances is clear: The nonlinear terms are multiplied 

by the diffusion coefficients, which are small in all reasonable situations. Therefore we may use 

the solutions corresponding to D = 0 as the zero approximation, and develop some perturbative 

scheme. 

If D = 0, getting rid of a,, and a,,, one arrives at a single third-order equation for the 

coordinate variance. 

whose solution reads 

- A + B e2iwt + B* e-2iwt axx(t) - 7 
where A and B are constant coefficient. It is natural to suppose that for D 4 0 the evolution of 

aXs is given by the same expression, but with the slowly varying in time coefficients: 

where 

B*(t) = B(t) e"' f B'(t) e-2'wt 

Now inserting (33) into eq. (28) and neglecting terms of the order of 0(D2) (guessing that the 

n-th derivatives of A(t) and B(t) are proportional to Dn), we can express the covariance a,, as 

follows, 

m (A + Be2iwt + i)* e-2iwt 

a,, = iu m B- + - ) + 2mDs (A + B+) - mD , (36) 2 

and with the same accuracy we find the momentum variance for eq. (30): 

Since in this expression the function A is multiplied by the small parameter Dl we must calculate 

it in the zeroth approximation. Thus we get 

being both, R and A, slowly varying functions of time. 

Now, we make the crucial step first proposed in [8, 91: we put the expression for a*, in terms 

of A and B into eq. (30) and average both sides with respect to the fast oscillations of frequency 

w . Then we arrive at the equation, which contains functions A and R only 

and where a. = h/ (2wm) is the coordinate variance in the oscillator ground state, while I? = mD/h 

is the dimensionless diffusion coefficient. 

Now, averaging eq. (31) we arrive at a closed equation for R(t), 

Consequently, eq. (39) actually is the first order linear nonuniform equation with respect to 

function A(t), which can be easily solved provided the solution to eq. (40) is known. 

Thus, averaging the equations for the variances over the fast oscillations leads to the effective 

linearization of the equations governing the evolution of the coefficients of the Gaussian density 

matrix. To get the equation for B(t), one should first multiply both sides of the equation for tr,,, 
by exp(-2ibcrt), and then average over the fast oscillations. We confine ourselves, however, to the 

evolution of functions A(t) and A(t) (the absolute value of B(t) can be extracted from eq. (38)). 

It is worth to mention that A(t) is nothing but the energy of quantum fluctuations (up to the 

constant factor and small corrections of the order of O(D)): 

Now if we consider a pure state, R = 1, then the solution for eq. (39) is 

A(t) = 00 + [A(O) - go] exp [- 4rw(2~ s - ()t] . (42) 

We see that the energy of fluctuations can increase or decrease, depending on the sign of the term 

in the exponential: 

1)If 200 s - ( > 0, we verify that A(w) = oo , SE(oo) = hw/2 and B(m) = 0, therefore 

asymptoticaly o,, and 5, do not oscillate. 

2)If 200s - ( < 0 and since reasonably rw << 1, A(t) becomes a slowly increasing function with 

time (the same for SE(t)). From eq. (38), asymtoticaly we have A(t) z 2B(t) >> 0:. If B(t) is real 

at t = 0 then it is real for any t > 8, so from eqs. (34) and (35) o,,(t) = A(t) + 2B(t) cos(2wt) 

or A(t) - 2B(t) 5 a,, 5 A(t) + 2B(t). But from eq. (38) A(t) - 2B(t) = oi/(A(t) + 2B(t)) N 

o:/(2A(t)). Therefore asymtoticaly a,, oscillates between two values, 

thence, as time goes on the solution of the nonlinear Schrijdinger equation becomes highly squeezed. Note that a similar behavior of the variances (i.e., an exponential increase of the squeezing 

coefficient) is observed in the case of the usual Schrodinger equation for the parametrically excited 

oscillator, when its frequency changes in time [7]. However, in the present case all the coefficients 

in the generalized Schrodinger equation with the functional (6) (or its master equation counterpart 

eq.(7)) do not depend on time, and the increase of fluctuations is caused by the nonlinear terms. 

3 Amplifying - Dissipative Hamiltonians 

Here we shall consider the theory developed many years ago by P. Havas [lo], which is quite 

suited to construct Hamiltonians that take into account dissipation and apply it to two examples, 

confining ourselves to the one-dimensional case. 

3.1 The Bateman-Caldirola-Kanai (BCK) Hamiltonian 

The harmonic oscilator with an exponential time-dependent mass m(t) = mo exp((t), is known 

as the BCK Hamiltonian [11] and we shall introduce the phenomenon of friction in it. According 

to [lo] when friction is present the Lagrangian that describes the motion, ( that reproduces the 

classical equation of motion) is 
Here V(p; t) = jm(t)w: p2 and F(t) is a time-dependent external force. The exponential factor 

introduces the friction effects (also external) on the system and for the specific Lagrangian one 

has 

'Yo 'Y(t) = 7 (1 - exp(-Ct)) , (45) 

as limt-.m ~(t) = T~/( and limC,or(t) = yet, so, recovering previous results [12]. We shall 

consider F(t) as being a stochastic force whose mean in an ensemble is null, (F(t)) = 0, and that 

its correlation is Markovian, (F(t)F(t1)) = 2dS(t - t'). 

According to the quantization procedure of [12] one obtains the system Harniltonian 

where i), Q are canonicaly conjugated operators, [Q, i)] = ih. The physical position and momentum operators are related to those through 

&hys = &ex~(-'Y(~)/z) 7 jphys = P~XP(-'Y(~)/~) (47) 

In order to obtain the equations of motion for the operators P and Q in the Heisenberg picture we first do an unitary transformation in Schrodinger equation with operator S(CtI2) = 

exp(-%{Q, i)}), which leads to the new Hamiltonian 

,ir2 (,t) = - 

1 1 + - mowip + - (yoexp(-(t) + I){?,+} 2mo 2 4 

1 

+ f F(t) exp [+y(t) - Ct)] (48) 

with 

- 1 

&hys - f exp [- Z('Y(t) + ~t)] 

1 

jphys = *e~[-~('Y(t)-~t)] . (49) 

The equations of motion for f (t) and ,ir(t) are solved assuming f = u(t)fo + v(t)iio + w(t), 

leading to the set of linear differential equations 

where R; = $ - W: and with the following initial conditions 

1 

u(0) = 1 and u(0) = 2(w + () 

1 

1 v(0) = 0 and C(0) = - 

mo 
The exact solutions of the above differential equations are : 

where I,(.) and Ku(.) are the modified Bessel functions of the first and third kind, respectively 

[13]. The parameter v is 

Similarly the momentum operator in the Heisenberg picture is h(t) = p(t)& + @(t)+o + E(t), 

where 

Considering that the energy increasing prevails over energy dissipation, yo/C < 1, one has the 

approximative solutions, 
and the variances for Qphys and fiphys become 

5 5 

am (AA?,,) - exp [- (5 - 2Ro)t] {: (% - 1 t+oo 1600 

eW(-2) 5. (i 2~o + 1) (1 + P)} + o (A > 1) 

verifying squeezing even in occurence of a weak dissipation. 

3.2 The optical parametric oscillator (OPO) 

' Using the same method we can treat the OPO introducing in the Hamiltonian the dissipation of 

the cavity, hence, 

where the mathematical operators A+ and A are related to the physical creation and destruction 

operators by a = A exp (-Xt/2) and a+ = A+ exp (-Xt/2). Moreover, fl (t) = wo is the 

mode frequency in the cavity, f2 = K exp(-2iwt) + iX/4, where K and w are, respectively, 

the intensity and the frequency of the pumping field, while X is the damping constant of the 

cavity. f3(t) = F(t) eXt/(2wo)1/2. The force F(t) is assumed to be a Markovian stochastic force: 

< F(t) > = 0, < F(t)F*(t1) > = < F(t)*F(t1) > = 2dS(t - t') and < F(t)F(tt) > = 0, and the 

parameter d is related to the temperature of the cavity, 

Considering the pumping at resonance, wo = w, and for d/wo << 1, the solution of Heisenberg 

equations for operator AH(t) is 

where 

u(t) = e-'wOt cosh(2rct) + [~-'wo' (sinh2 y cosh(2nt) - z sinh y cosh y sinh(2nt)) - c.c.] (68) 

v(t) = z sinh(2nt) (cosh2 y e-'WOt - sinh2 y eawt) - 1 sinh(2y) sinh(2nt) cos(2wot) (69) 

and tanh(2y) = Xl(2wo). 
With the above solution we present the asymptotic (t -+ w) mean values of several quantities, 

after taking the average over the high frequency oscillations of the field (wo) and considering 

4K/X < 1: 

d lim (8) = u~~-"(A$A~), N - 1 

t-+m 1 - (4~1~)~ 

2)Variances of the two quadratures, 

d lim (~21,~)~ N - 1 

t--roo 2Xwo 1 - (4~1~)~ 

Although the asymptotic values of the variances of the quadratures are both above the value 

112, at initial times squeeze is seen in the fast oscillations (before averaging over the mode frequency). In eqs. (71) and (72) one verifies the effects of the dissipation-amplification process 

through the quantity ~K/A: 

i) If ~K/X << 1 (very weak pumping compared with the cavity dissipation), the thermalization, 

represented by the parameter d, dominates in the physiqal expressions at equilibrium. 

ii) On the other side, when ~K/X is close to 1 (strong pumping) the factor (1 - (~K/X)~)-' dominates the strepghts of the asymptotic values, increasing dramatically the energy and fluctuations, 

as is expected to occur at resonance. 

4 Summary 

We have presented two different formulations of quantum dynamical equations that show squeezing 

in the variances of the conjugate canonical operators. In the first one we considered a generalization of the Doebner-Goldin nonlinear extension of the Schrodinger equation and we verified 

that although the parameters that enter the nonlinear part of the equation are constant in time, 

squeezing occurs, essentially due to the nonlinearity. Moreover, the master equation shows the 

surprising feature that any initial mixed state relax to a pure state! 

In the other approach we introduced the dissipation phenomenon into the Harniltonian formalism by starting with a conveniently defined Lagrangian, as proposed by P. Havas [lo]. We 

considered two familiar time-dependent Harniltonians, the BCK and the OPO. The BCK Hamiltonian has a time-dependent mass and it displays amplification of energy and squeezing of variance 

of momentum or of position, although uncertainty is preserved. The dissipation was introduced 

and the effects are seen in eqs. (63)-(64). 

The second Hamiltonian is the OPO, describing a single mode in an electromagnetic cavity 

with pumping at resonance. Dissipation is introduced to take into account the loss in the cavity 

walls. As an expressive result we verify that the asymtotic physical expressions depend, essentially, 

on the factor ~K/A, K representing the pumping and A, the dissipation. 

5 Acknowledgments 

V.V.D. and M.A.M. are grateful to the Funda~Zio de Amparo i Pesquisa do Estado de SZo Paulo 

(FAPESP) for full financial support to elaborate the present work, and S.S.M. acknowledge travel 
financial support. S.S.M. also acknowledge partial financial support from the Conselho Nacional 

de Desenvolvimento Cientifico e Tecnolbgico (CNPq), Brasil. 

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equation to appear in. J. Phys. A. 

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new nonlinear Schroedinger equation preprint UFSCar. 

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E. Schriidinger, Sitzungsber. Preuss. Akad. Wiss, Phys. Math. 24, 296 (1930). 

[7] V.V. Dodonov and V.I. Man'ko, Invariants and the evolution of nonstationary quantum systems, (Proc. Lebedev Phys. Inst., vol. 183, Nova Science, Commack, 1989). 

[8] V.V. Dodonov, Comm. Theor. Phys. (Allahabad, India) 2, 61 (1993). 

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a damped harmonic oscillator in a magnetic field to appear in Phys. Lett. A. 

[lo] P. Havas, Nuovo Cimento Suppl. 5, 363 (1957). 

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PHASE-SPACE ANALYSIS OF CHARGED AND 

OPTICAL BEAM TRANSPORT: 

WIGNER ROTATION ANGLE 

G. Dattoli and A. Torre 

ENEA, Dip. INN., Settore Elettroottica e Laser, 

CRE Jhzscati, C.P. 65 - 00044 hscati, Rome, Italy 

Abstract 

The possibility of using the phase space formalism to establish a correspondence between the dynamical behaviwr of squeezed states and optical or charged beams, propagating 

through linear systems, has received a great deal of attention during the last years. In this 

comection, it has been indicated how optical experiments may be conceived to measure the 

Wigner rotation angle. In this paper we address the topic within the context of the paraxial 

propagation of optical or charged beams and suggest a possible experiment for measuring 

the Wigner angle using an electron beam passing through quad~poles and drift sections. 

The analogous optical system is also discussed. 

1 Introduction 

Lorentz group is the basic language of special relativity [I]. It has been recognized as a powerful 

tool in many other fields of modern physics as well. Many dynamical symmetry groups, underlying 

specific branches of physics, as quantum optics, classical and quantum mechanics, Hamiltonian 

optics, are locally isomorphic to the group SO(2,l) of Lorentz transformations in twospace and 

one-time dimensions. 

Actually, the (3 + 1)-dimensional Lorentz group SO(3,l) is the full space-time symmetry 

group. However, we seldom discuss Lorentz transformations in the three-dimensional coordinate 

system, since computing for instance velocity additions and successive Lorentz boosts is quite 

complicated, 4 x 4 matrices being involved. Restricting thereby the problem from SO(3,l) to 

SO(2,l) may simplify significantly calculations. 

Furthermore SO(2,l) has both a physical and mathematical interest. It is indeed the little group, which leaves a space-like four-momentum invariant. Accordingly, studying SO(2,l) 

amounts to studying free particles travelling faster than light (the so-called tachyons), which are 

intrinsically interesting from a theoretical point of view. 

Above all, the group SO (2,l) gave rise to a great amount of interest, since, as already remarked, 

it is locally isomorphic to other groups, as Sp(2), SU(1, I), SL(2, C), and has therefore a very 

rich mathematical and physical content. The isomorphism with the above quoted groups offers 

many advantages from both analytical and theoretical point of view. In fact, it allows further 

simplification in the calculations involving Lorentz transformations, the groups Sp(2), SU(1, I), 

SL(2, C) consisting of 2 x 2 matrices. In particular, the algebraic analogy with the symplectic 
group Sp(2) offers a further advantage, as Sp(2) consists of real matrices. Hence, it is possible 

to visualize Lorentz transformations in terms of transformations in a two-dimensional geometry, 

making a correspondence between Lorentz transformations in real space and linear canonical 

transformations in phase-space or, equivalently, between volume-preserving transformations and 

area-preserving transformations. 

In addition, as firstly recognized by Han, Kim and Noz [2], the quoted isomorphism allows 

to conceive a kind of analog computer for testing the Lorentz group properties. In particular, as 

is well known, instead of rotations, pure Lorentz transformations do not form a subgroup. As 

a consequence, the product of two boosts along different directions is not a boost but a boost 

preceded or followed by a rotation. The angle of rotation is known as the Wigner angle and 

provides the kinematic basis for Thomas precession in atomic physics [I, 21. 

Many suggestions have indeed proposed in order to perform optical experiments to observe the 

optical analog of the Wigner angle [3, 41. 

In this connection, the paper is devoted at suggesting a possible experiment for measuring the 

Wigner angle within the context of electron beam transport [5]. The paraxial propagation of a 

charged particle along a magnetic channel is indeed governed by the symplectic symmetry. It can 

. be therefore conceived an experiment involving electron beams for measuring the Wigner angle. 

The formal analogy between the propagation of charged beams and that of light beams through 

optical systems in the Gaussian approximation suggests to discuss the topic in full generality. 

However, the specific language used through the paper is that usually adopted in accelerator 

physics, whilst the symbology is just that of ray-optics. 

The paper is organized as follows. Sec. 2 is devoted to a preliminary analysis of the problem, in 

order to introduce the formalism relevant to symplectic symmetry. In Sec. 3 the analogy between 

linear canonical transformations and optical systems is developed, thus leading to a specific design 

of the experiment for measuring the Wigner angle within the electron-beam optics, as illustrated 

in Sec. 4. Concluding remarks are given in Sec. 5. 

2 SO (2,l) and Sp(2) 

As remarked in ref. [6], the symplectic group, originally introduced by Weyl in 1938, plays a central 

role in many branches of physics, as a consequence of that symplectic transformations preserve the 

skew symmetric products, which frequently appear in physics. In particular, symplectic geometry 

is the mathematical theory underlying Hamiltonian mechanics. It emerges especially in phasespace picture. The phase-space of a mechanical system is indeed recognized as a symplectic 

manifold and the time evolution of a conservative dynarnical system is a one-parameter family of 

symplectic diffeomorphisms, or, linear canonical transformations. 

Phase space formalism is becoming the unifying language for both classical and quantum 

mechanics.It is basic to the Hamiltonian formulation of classical mechanics. Within this context, 

indeed, the evolution of a dynamical system is described by a number n of independent coordinate 

variables and on the same number of canonically conjugate momenta. The cartesian space of these 

2n coordinates is just .the phase-space. 

Correspondingly, phase space picture of quantum mechanics is becoming increasingly popular. 

Although, the concept of phase-space is not compatible with quantum mechanics, ij and f? being 

noncommuting operators, the Wigner phase-space representation allows to overcome this prob- 
lem, since in this representation both the coordinate and momentum variables are enumbers. 

Accordingly, it is possible to perform phase-space canonical transformations as in the case of 

classical mechanics, which correspond to unitary transformations in the Schrijdinger picture of 

quantum mechanics. 

Phase space concept appears therefore as the unifying context, where classid as well as 

quantum mechanics can be naturally framed, thus suggesting the possibility to transfer concepts 

and methods from quantum to classical mechanics and viceversa. 

Furthermore, as discussed in ref. [2], phase-space picture provides the natural language for 

quantum optics as well, offering a geometrical view to coherent and squeezed states as circles and 

ellipses respectively. In this respect, taking advantages from the symmetry of the relevant Wigner 

phase space distribution function it is possible to calculate expectation values and transition 

probabilities for the above quoted states [2]. 

In the present paper, we are interested in the paraxial propagation of optical or charged-particle 

beams through optical systems [7, 81. We are thereby led to consider the harmonic oscillator-type 

Hamiltonian .l 

with q and p being canonically conjugate variables. As noticed, the Hamiltonian (1) models the 

paraxial propagation of electron beams as well as light beams through optical systems. The p2- 

term describes the free propagation for both electron beams and light rays, the variable p being 

understood as the particle-momentum and the ray reduced slope, respectively. On the other 

hand, the q2-term accounts for the propagation through optical systems as quadrupoles or lenslike medium, the corresponding strength being measured by the coefficient k(s). The coordinate 

s is measured along the symmetry axis of the system, also assumed as the direction of the beam 

propagation. 

The basic tools of classical mechanics are the Poisson brackets and the canonical transformations. The latter can be derived from the former. 

The properties of Poisson brackets, indeed, as the antisymmetry, the derivation property and 

the Jacobi's identity, assure that Poisson brackets make any commutative ring of functions defined 

on a domain X c R2n into a Lie algebra. It is thereby possible to associate with any Hamiltonian 

H the operator iH = {H, . . .}, which for the one-degree of freedom writes as 

Accordingly, the dynamics ruled by (1) is naturally framed within the group structure generated 

by the operators associated with the quadratic polynomials: 
Embedding the above operators into the following 

it is easy to recognize the syrnplectic structure of the corresponding group, as displayed by the 

commutation relations obeyed by the above introduced operators: 

[KllK2]= ~3 [KI,K~]=-~~ , [~2,~3]=-Kl. (5) 

As elements of Sp(2), the operators K1, k2, K3 are amenable of the matrix representation 

Finally, let us say that the operators (4) are the generators of the canonical transformations 

in phase-space, which will be analysed with some details in the next section, within the context 

of the specific problem of charged beam motion in magnetic fields. 

3 Linear canonical transformations and optical systems 

Before analysing the specific role of the operators K~, i = 1,2,3 and discussing the optical analogs 

of the associated canonical transformations, let us make some preliminary considerations in order 

to visualize the phase-space canonical transformations within the specific context of electron-beam 

transport physics. 

As is well known, an invariant quadratic form 

can be associated to any dynamics described by quadratic Harniltonians in canonical coordinates 

and momenta. In passing, it is worth stressing that within a quantum context the quadratic form Z 

is reported as the Ermakov-Lewis invariant [9], the vector a: containing obviously the position and 

momentum operators, whilst in classical mechanics it is known as the Courant-Snyder invariant 

1101, firstly introduced in the analysis of electron beam motion through magnetic channels. In the - - 

above expression, the two component vector 11: r (:) is acted by the real 2 x 2 matrix T, - which 

furthermore is required to be symmetric and unimodular: det T = 1. Just to share the language 

of accelerator physics, we refer to T - as the Twiss matrix1 andwrite it in the form 

'The quadratic form (7) can also be regarded as the transcription in phase space of the quantum invariant, the 

vector a: being formed by the expectation values of the position and momentum operators and the matrix T - being 

linked to the quantum covariance matrix. 
a+O 2a tan (24) = -- B-Y 

Figure 1: Phase-space ellipses for different values of the Twiss parameters a, ,6, y. 

The entries a, By 7 usually named as Twiss parameters, play an important role in dksigning transport channels. 

The quadratic form (7) can be depicted in the phase space as an ellipse, whose size and 

orientation are determined by the Twiss coefficients. The area of the ellipse, which is just the 

value of the invariant Z, is usually denoted in accelerator physics as Z = TE, E being referred to as 

the beam emittance. It plays a crucial role in characterizing the quality and the dynamics of the 

e-beam. In a single particle analysis, a, p and 7 define the particle trajectory, as in an ensemble 

analysis they define the second order momentum of the phase space distribution function, thus 

providing information on its extent and maximum localization. Explicitly, 

the averages being understood on the distribution function. Accordingly, the emittance can be 

given the further meaning: 

2 2 2 2 E = aqqo, - a, . 

According to the above considerati~ns~the dynamics of charged beams passing through transport 

channels naturally leads to a visualization of the problem in terms of circles and ellipses in the 

phase-space, which on the other hand have been recognized as useful pictures for the coherent 

and squeezed states of quantum optics as well (Fig. 1). 

Acting on the vector : amounts to acting on the phase-space ellipse and correspondingly on 

the Twiss parameters. To be more precise, let us say that a linear canonical transformation U, 

represented by the matrix U - = ( E), change the Twiss parameters as [8] 

After these introductory remarks, let us consider the specific effect of the transformations generated by the operators k, i = 1,2,3 and recognize the corresponding optical systems. 

Using the matrix representation (6), we immediately get 
Figure 2: Phase-space canonical transformation (a) and optical system (b) corresponding to the 

operator en%. 

which represents an elongation in the q-direction and correspondingly a stretching in the pdirection, turning for instance a circle into an ellipse, as sketched in Fig. 2a. 

It is needless to say that the transformation (12) preserves phase-space area, as a consequence 

of that Sp(2) matrices leave invariant the cross products, naturally associated with areas. 

In the optics of electron or light beams the transformation described by S(v, 0) is realized by 

means of telescopic systems, consisting of two thin lenses appropriately combined, according to 

the scheme shown in Fig. 2b. In this regard, let us recall that the symbology for optics of electron 

beams and light beams is the same. Hence, thin lenses in ray optics correspond to quadrupoles in 

electron-beam optics. 

The parameter M, which is just equal to minus the ratio of the foci of the two lenses, is 

reported in optics as the magnification of the system. It produces indeed an image magnification 

and a ray-angle demagnification. 

As to the operator K~, it is easy to obtain the associated transformation as 

cos 412 - sin 9512) 

e-'l R' '(4) = ( sin 412 cos 412 7 
f = -cot $12 

Telescopic system 

Figure 3: Phase-space canonical transformation (a) and optical system (b) corresponding to the 

operator . 

immediately recognized to describe the rotation around the origin in the phase space by the angle 

412, as further confirmed by the transformation law (11) for the Twiss parameters according to 

which the associated ellipse rotates in the counterclockwise direction by 4/2. 

Since any group elements can be appropriately factorized, the canonical transformation corresponding to the matrix R(4) can be realized by means of an appropriate sequence of the basic 

optical elements, that is thin lenses, drift sections and telescopic systems, which however are just 

a combination of the first two. 

Accordingly, the optical system corresponding to a rotation in phase-space can be realized by 

means of the following sequence (Fig. 3): 

1. telescopic system with magnification M = cos $12 

2. drift section of length d = - sin 4 

3. thin lens of focus f = -cot 412 

In this connection, it is worth stressing that the above scheme is only one of the possible ones, 

which can be obtained changing the ordering in the operator factorization, thus allowing to satisfy 

specific requests on the parameters of the optical components. 

Finally, the operator enK2, which can be represented by the matrix: 

enK2 = S(q, 90') = cosh q/2 sinh 712 

sinh q/2 cosh q/2 
Telescopic system 

---------- 

Direction q3 Entity q Direction q3 

of the squeeze of the squeeze of the squeeze 

Figure 4: Optical analog of the squeeze S(q,t$). 

is easily identified according to the transformation law (11) as producing a squeeze in the direction 

making an angle of 45" with the q-axis. 

The optical analog can be realized for instance by the same sequence as before, the relevant 

parameters being now M = coshq12, d = sinh 7112, f = tanh-' 7112. 

Finally, let us discuss the squeeze S(q, 4) in the direction making an angle 4/2 with respect to 

the q-axis. Since it can be obtained combining rotations and squeeze along the q-axis, as formally 

expressed by the composition 

it is easy to get the well known matrix representation [2] 

cosh 712 + cos 4 sinh 712 sin 4 sinh 712 

'(" = ( sin 4 sinh 712 cosh 712 - cos 4 sinh q/2 (16) 

According to the above discussion, a possible optical configuration can be realized by a telescopic 

system, preceded and followed by the same sequence of thin lens-drift section, symmetrically 

disposed. It is worth stressing that the magnification of the telescopic system is determined by 

the entity q of the squeeze as the parameters of the optical system drift section-thin lens are 

determined by the squeeze direction 4 (Fig. 4). The quantities q, q5 are usually combined into the 

squeeze parameter < - 7ei4, so that q and 4 can be regarded as the modulus and the phase of the 

squeeze parameter C. 

In csnclusion, we have stated a correspondence between the linear canonical transformations in 

phasespace, as squeezes and rotations, and optical systems, which can be conceptually conceived 

as realizing such transformations, acting effectively on electron or light beams. 
4 Electron-beam transport channels and Wigner angle 

The correspondence between linear canonical transformations and Lorentz transformations has 

been already recognized [2]. Boosts correspond indeed to squeezes in phase-space and rotations 

in real space to rotations in phase-space and rotations in real space to rotations in phase-space. 

Similarly, as the product of two boosts along different directions is not a boost, but a boost 

preceded or followed by a rotation, so the product of two squeeze along different directions does 

not result into a single squeeze, but into a squeeze and a rotation. It can be verified that 

S(X, 4)S(rl, 0) = S(J¶ W(w) . (17) 

The parameters J, 8, specifying the entity and the direction of the resulting squeeze, and the 

angle w, referred to as the Wigner angle, are determined by A, 4, q according to the well-known 

formulae: 

cosh J = cosh q cosh X + cos 4 sinh X sinh q , sin +[sinh X + cos + tanh q(cosh X - I)] tane = cos 4 sinh A + tanh q[l + cos2 d(cosh X - I)] ' 

w sin 8 sinh % sinh 4 

tan- = 2 cosh 4 cosh f + cos 4 sinh 4 sinh f ' 

Within the context of the optical analogy, developed in the previous section, the above relations 

can be recast in terms of the parameters of the optical systems corresponding to S(X,4) and 

S(q, 0) , according to 

coshJ = 1 

4M;Mf { 1) 1) + (1 -) (M- 1) 1)) , 

(Mi - l)(Mi + 1) - (1 - y) (M; - l)(Mi - 1I2 

tan26 = 2d 

(Mj- l)(Mt+ 1) (1 - $1 + (M; - 1) [2~f + (1 - 7) (Mf - 1l2] 9 (19) 

w (Mi - l)(Mf - 1) tan- = 2d 2 (M; + l)(Mf + 1) + (1 - $) (M; - l)(Mi - 1) ' 

where M, specifies the magnification of the optical system S(q, 0) , whilst MA, d and f denote the 

parameters of the system S(X, 4) according to the scheme of Fig. 4. 

Accordingly, one can design an appropriate sequence of quadrupoles and drift sections to 

perform the transformation represented by the product of three squeeze: 

S(-6, e)s(X, 4)S(v, 0) , (20) 

with J, 8 given according to (19). 

As stated in (17), such a transformation does not leave unchanged the Twiss parameters of 

the beam, which indeed change as in a rotation of the angle w/2. Explicitly, 

& 2w = a1 sinw + ,& cos - + 71 sin2 5 2 2 ' 

2w = -a1 sin w + Dl sin - + cos2 . 2 
It is evident that if ol = 0 and 71 = = 1, that is depicted in phase-space by a circle of 

unitary radius, the rotation does not have any effect, since a2 = 0 and 72 = ,02 = 1 as well. As a 

consequence, a beam having different variance in the q and p directions should be used; in other 

words, sharing the language of quantum optics, a squeezed beam should be used to produce a 

rotation of the beam ellipse. 

In that case, indeed, the initial ellipse will rotate in the phase-space just by the angle w/2. 

Measuring then the Twiss parameters of the electron beam before and after being acted by the 

optical system, performing the transformation (20), it is possible to infer the Wigner angle, w, for 

which the following link with the Twiss parameters can be deduced: 

tanw =. 2 ~2(& - n) - ~(p2 - r2) 

(P2 -72)(P1-71) +20102 ' 

5 Concluding remarks 

The consideration developed in the previous sections are basically grounded on the algebraic 

analogy between the SO(2,l) Lorentz group and the symplectic group Sp(2), which is basic to 

the Hamiltonian dynamics. Exploiting this analogy, it is possible to conceive and design an optical 

system for electron beams, which allows to get a measure of the Wigner angle by detecting the 

variations occurred in the electron-beam Twiss parameters as a result of the motion through the 

magnetic channel. In this connection, the two-slits method [ll] may offer an appropriate tool to 

visualize the rotation of the beam ellipse and thus to measure the Wigner angle. However, the 

measure is strongly limited by space-charge effects and transverse coupling, induced by sextupolar 

contributions to the quadrupole magnetic field. 

The discussion has been put forward in full generality comprehending also light beams, whose 

paraxial propagation through optical systems is governed by the symplectic symmetry as well. An 

experiment using light beams can be conceptually conceived, but its realization is rather difficult, 

since the measure of the corresponding Twiss parameters, which in the specifically optical context, 

can be understood as linked to the beam spot-size and divergence, is limited by diffraction effects. 

In conclusion, let us stress the relevance of the above results, according to which an analog 

computer for the Lorentz group can be recognized within the purely classical context of electron 

beam transport or opti$l ray propagation. 
References 

[I] Y.S. Kim and M.E. Noz, Theory and Applications of the Poincart Group, (Reidel, Dordrecht, 

1986); F.R. Halpern, Special Relativity and Quantum Mechanics, (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1968). 

[2] D. Han, Y.S. Kim and M.E. Noz, Phys. Rev. A37, 807 (1988); D. Han, Y.S. Kim and M.E. 

Noz, Phys. Rev. A40, 902 (1989); D. Han, Y.S. Kim and M.E. Noz, Phys. Rev. A41, 6233 

(1990); Y.S. Kim and M.E. Noz, Phase Space Picture of Quantum Mechanics, (World Publ., 

Go., Singapore 1991). 

[3] D. Han, E.E. Hardekopf and Y.S. Kim, Phys. Rev. A39, 1269 (1989). 

[4] P.K. Aravind, Phys. Rev. A42, 4077 (1990). 

[5] F. Ciocci, G. Dattoli, C. Mari and A. Torre, Phys. Rev. A46, 5149 (1992). 

[6] Y.S. Kim and M.E. Noz, Am. J. Phys. 51, 368 (1983). 

[7] V. Guillemin and S. Sternberg, Symplectic Techniques in Physics (Cambridge University 

Press, Cambridge, 1984). 

[8] G. Dattoli and A. Torre, "A general view to Lie Algebraic Methods in Applied Mathematics, 

Optics and Transport Systems for Charged Beam Accelerators", in Dynamical Symmetries 

and Chaotic Behaviour in Physical Systems (World Publ. Co., Singapore, 1991). 

[9] H.R. Lewis Jr., Phys. Rev. Lett. 18, 510 (1967); H.R. Lewis Jr., J. Math. Phys. 9, 1976 

(1968); H.R. Lewis Jr. and W.B. Reisenfeld, J. Math. Phys. 10, 1458 (1969). 

[lo] E.D. Courant and H.S. Snyder, Ann. Phys. 3, 1 (1958). 

[ll] F. Bordoni, T. Letardi and M. Placidi, Nucl. Instr. and Meth. 65, 72 (1968). 
PHASE SPACE FLOW OF PARTICLES IN SQUEEZED STATES 

Peter H. Ceperley 

Physics Department 

George Mason University, Fairfax, VA 22030 

Abstract 

The manipulation of noise and uncertainty in squeezed states is governed by the wave 

nature of the quantum mechanical particles in these states. This paper uses a deterministic 

model of quantum mechanics in which real guiding waves control the flow of localized particles. This model will be used to examine the phase space flow of particles in typical squeezed 

states. 

1 Introduction 

The study of squeezed states and uncertainties is exciting. It is exciting because of its potential 

applications to low-noise instrumentation and communication. It is exciting because it represents 

a new frontier of physics, giving us new understanding of quantum mechanics. However I hear 

again and again that the end of many new experiments is to verify quantum mechanics. Should it 

really be so exciting to re-verify quantum mechanics for the ten thousandth and first time if we are 

so sure that the present ideas of mechanics are indeed correct. Or perhaps we are still 

haunted by the ghosts of de Broglie and Einstein and their insistence that quantum mechanics be 

deterministically based. The non-verbalized and forbidden question seems to be "Is it possible to 

construct a localizable, deterministic model that is consistent with the observations that quantum 

mechanics explains so well?" And in spite of the repeated claims by physicists that they are ardent 

non-determinists, their pursuits seems to be strongly focused at finding a weakness in quantum 

mechanics or at least some inkling of a microscopic, deterministic world that has been previously 

hidden to physics. Certainly modern quantum optics experiments have this potential. 

It is not that there is anything incorrect with quantum mechanics. It is just that it is a 

statistical theory and there are many situations and many applications in which it would be very 

useful to have a deterministic, single particle theory also. At any rate, it would seem that this 

workshop would be incomplete without at least one person reiterating the challenge of de Broglie 

and Einstein. In this vein, I will present one possible deterministic model for quantum mechanics, 

and then go on to relate this model to squeezed states and uncertainties. Phase diagrams will be 

included. 

2 The Model 

To begin with, this model assumes that particles have real, and localizable, continuous, existence. 

This is consistent with the feelings of many, if not most, physicists. However, the model also 
assumes, contrary to popular belief, that the waves of quantum mechanics are also real. After 

a century of discussing wave-like phenomena, perhaps the physics community might allow the 

possible existence of some real waves. I assume that a source that emits particles, also emits 

waves in rough proportion to the wave intensity, and anything that absorbs particles also absorbs 

the waves in proportion. The model also assumes that the waves entrap the ensemble (or phase 

space of) emitted particles and force them to flow along with the wave energy, such that a setup 

that splits the waves will also split the particles in rough (or statistical) proportion to the split of 

wave energy, thereby insuring that the particle density stays proportional to @. In a sense, the 

particles are nearly massless specks, carried at will by the flow of waves. All large-scale dynamics 

are controlled by the waves. What is there left for the particles to do? Without them, the 

wave fields would pass through each other. The particles represent small local, non-linear mixing 

sites that allow wave fields to interact with each other for the processes of scattering, transitions, 

absorption, emission, and detection. Thus in this model[l] propagation is determined by the 

waves while interactions are determined by the particles. This model is similar to models by de 

Broglie[2], Bohm[3], Einstein[4], and others, although it is generally more specific and physical 

than the previous models. 

What about a wave equation? I find the easiest wave equation to work with is the Klein-Gordon 

Equation: 

which can be changed into a plasma equation: 

with the substitution of: 

An interpretation of this is: for each species of particle, such as electrons, space supports an 

associative guiding wave governed by (2) with the 'plasma' frequency given by (3). There need not 

be any real plasma, only a space resonance at this frequency which may be fundamentally linked 

to the existence of the particles themselves. Furthermore, I am not hypothesizing agreement of 

two independent, physical quantities in (3), the mass and the plasma frequency. In quantum 

mechanics there is no physical, classical mass and the mass is used only as a number in various 

equations. In this model also there is no physical mass associated with the classical mass numbers, 

but instead the physics is tied up in the 'plasma frequencies'. 

One can hypothesize an interaction potential between the waves and particles, and generate 

the phase space orbits shown in Fig. 1. In order that the orbits stay bound to the waves, even 

as the waves spread out and decline in amplitude, it is necessary to assume that the microscopic 

mass of the particles be in proportion to the waves. This can be rationalized[l] as a mass due to 

the energy content of a local resonance centered about the particle. Fig. la shows phase space 

orbits for particles trapped in a standing wave field. Fig. lb shows similar orbits for traveling 

waves hitting a finite height barrier. Evanescent waves partially penetrate the barrier, causing 

some particle orbits to be carried into the barrier, while a fraction of these waves and particles 

tunnel completely through the barrier. 
FIG. 1. Phase space orbits for particles captured in a) standing wave fields, and 

b) a tunneling situation. On the vertical axis is plotted the microscopic momentum of 

the particles which is very different from the macroscopic, observable momentum. 

Just to mention a few more details of this model from Ref.1: mixing equations show that 

during interactions, frequency sums and vectorial wave number sums are conserved and form the 

basis ,for macroscopic energy and momentum conservation. Microscopic energy and momentum, 

are not so useful since the waves and the associated microscopic energy tend to spread out while 

propagating. The spread out remnant wave energy creates the vacuum fields, as well as being 

redeposited around particles, i.e. in populated states. 

3 Uncertainties 

The Heisenberg uncertainty principle states that; 

However, since in this model, waves control particle motion, in order to localize a particle or group 

of particles, we need to first localize the entrapping wave. The correct uncertainty relation for 

localizing waves is given by: 

AxAk>2~, (5) 

where 6k is the spread in wave number. Just as quantum mechanics operationally does, this model 

replaces macroscopic momentum with wave number. Similarly, the related uncertainty: 

can be visualized as in Fig. 2, where a traveling wave pulse is entrapping particle orbits to itself. 

Suppose there is a detector at some point to the right that the pulse will impinge on and that 

we are discussing the 6t in the time spread of the detected particles. In order to narrow the time 

spread of particles, we must narrow the time duration the wave pulse will be at the detector and 

this requires use of the relationship: 

AtAw22~. (7) 
This is again like conventional quantum mechanics (at least operationally): in this case replacing 

energy with a frequency. So we see that the Heisenberg uncertainty relations fall out very naturally 

from this model. 

FIG. 2. Sketch of phase space of particles trapped in a traveling wave pulse. 

4 Squeezed States: Amplitude and Phase 

One of the most common squeezing operations involves reducing the amplitude fluctuations in a 

beam of particles. Fig. 3 shows a hypothetical micro-phase space of a beam. Here we assume a 

perfectly level traveling wave field (of completely well defined amplitude and phase) and illustrate 

the statistical distribution of particles that might be filling this wave field. The particles are 

randomly placed with respect to position and microscopic phase, and move horizontally to the 

right (on unplotted) orbits on the plot. The irregular line plots the fluctuating density of these 

particles as a function of position, which 'squeezing' is often called on to smooth out. If we 

selectively amplify or attenuate certain parts of the beam to level the density of particles, two 

undesirable things will also occur. First, we will not (in most cases we cannot) also homogenize 

the beam in the microscopic momentum coordinate. This will adversely effect certain other 

measurements, such as phase as we shall shortly see. Secondly, any attenuation or amplification 

process that changes particle density necessarily also attenuates or amplifies the associated wave. 

If the wave started out perfect, it will be degraded by this process. 

X 

FIG. 3. Hypothetical micro-phase space of a beam of particle entrapped by a 

uniform traveling wave field. 
FIG. 4. Elemental phase detector showing mapping of various regions of phase 

space of the incoming beams into different parts of the interference pattern. 

Figure 4 shows an elemental or simplistic phase detector. It involves injecting two signals 

into opposite ends of an attenuating medium and looking for spacial interference in the middle. 

A moving detector would be used to map out the particle density as a function of position to 

determine the relative phase of the two beams. Changing the relative phase of the two signals 

will move the interference maxima. This scheme was chosen because, unlike most other schemes, 

all particle paths are along a single line, i.e. in one dimension. This frees the other dimension for 

plotting microscopic momentum and greatly simplifies our phase space plotting. The interesting 

feature of the plot is that the phase space orbits map different microscopic momentum regions of 

the incoming signals into different interference maxima. In the figure here, only the uppermost 

momentum particles make it into the central, most useful maximum. This illustrates that inhomogeneities in the particle density in the momentum direction will adversely effect the interference 

pattern and thus the phase. Irregularities in the waves themselves will also have a similar adverse 

effect. 

5 Postscript: Modern Non-local Experiments 

Dr. Shimony raised the question about recent non-locality experiments being in conflict with the 

model presented here. My answer was that the experiments I had examined carefully[5] are in 

fact not inconsistent with this model. I wish to expand on that here. For example, the Franson 

experiment can be explained deterministically[6][7]. Even with delayed choice experiments where 

beams are changed after they become separated from each other, a change in the beam will affect 

both the particles in that beam and the waves around the particles. Since the waves determine the 

dynamics, it is not surprising that we get wave-like behavior in going through subsequent filters, 

beam splitters, and polarizers. 

Surprisingly, the hardest experiments to reconcile with this model are photon cancellation 

experiments[8]. However even for this there is a possible classical explanation [5]: the IF filters, 
which most quantum optics experiments use, are resonators and as such selectively accept and 

reject photons. The math of the time correlation of this classical acceptance and rejection process 

, has an uncanny resemblance to the math of entangled states. In a nutshell, when a Mach-Zehnder 

interferometer is balanced so as to prevent coincident detection through its outputs, the wave 

fields (of this model, as well as those of normal quantum mechanics) are phased so as to load 

I up only one IF filter at a time, thereby preventing the passage photons through the IF filters of 

I both detectors simultaneously. The conclusion is that these experiments are not conclusive proof 

~ of non-locality, particularly if one has a local model, such as the one presented here, where the 

I particles are entrapped by and guided by waves. 

These effects of IF filters also should be considered with reference to the Franson and the deI layed choice type experiments discussed above to give mathematically correct agreement between 

this model and experiments. We shall all look forward to the day when the quantum optics experiments are done with no IF filters after the down conversion process. Also, detection of practically 

all photons entering the experimental apparatus is essential if one wishes to analyze the system 

one-dimensionally (with branches). This requires large (no lost beam due to collimation or finite 

detectors, etc), high-efficiency detectors with large angular acceptance. Like good accountants, 

we need to see where everything is going. 

References 

[I] P. H. Ceperley, Annales de la Foundation Louis de Broglie 18, 197 (1993). 

[2] Louis de Broglie, Non-Linear Wave Mechanics, A Causal Interpretation, (Elsevier Publishing 

Co., Amsterdam, 1960). 

[3] D. Bohm, Phys. Rev. 85 166 (1952). 

[4] Einstein, Podolsky, and Rosen, Phys. Rev. 49, 777 (1933). 

[5] P. H. Ceperley, Bull. Am. Phys. Soc., 38 No. 2, 1004 (1993). 

[6] J. D. Franson, "Quantum Phase Uncertainties and the Classical Limit", Third International 

Workshop on Squeezed States and Uncertainity Relations, 1993. 

[7] P. G. Kwiat, W. A. Vareka, C. K. Hong, H. Nathel, and R. Y. Chiao, Phys. Rev. A 41,2910 

(1990). 

[8] C. K. Hong, Z. Y. Ou, and L. Mandel, Phys. Rev. Lett. 59, 2044 (1987). 
MULTIVARIABLE HERMITE POLYNOMIALS 

AND PHASE-SPACE DYNAMICS 

G. Dattoli, A. Torre 

ENEA, Dip. INN., Settore Elettroottica e Laser, 

CRE hscati, C. P. 65 - 00044 hscati, Rome, Italy 

S. Lorenzutta, 6. Maino 

ENEA, INNSVIL Divisione Calcolo, Bologna, Italy 

C. Chiccoli 

INFN, Sezione di Bologna and CNAF, Bologna, Italy 

Abstract 

The phase-space approach to classical and quantum systems demands for advanced analytical tools. Such an approach characterizes the evolution of a physical system through a 

set of variables, reducing to the canonically conjugate variables in the classical limit. It often 

happens that phase-space distributiops can be written in terms of quadratic forms involving 

the above quoted variables. A significant analytical tool to treat these problems may come 

from the generalized many-variables Hermite polynomials, defined on quadratic forms in Rn. 

They form an orthonormal system in many dimensions and seem the natural tool to treat the 

harmonic oscillator dynamics in phase-space. In this contribution we discuss the properties 

of these polynomials and present some applications to physical problems. 

1 Introduction 

Classical special functions play a central role in both pure and applied mathematics. Usually 

conceived for the solution of very specific problems, they gave rise to a far-reaching theory, being 

part of, and frequently motivation for, important general theories. 

The trigonometric functions, for instance, originally introduced to deal with specific problems 

of astronomy and navigation, are the basis for the theory of Fourier series and Fourier integral, 

which have applications to many parts of physics. 

Similarly, Bessel functions firstly appeared in mathematical physics in the 1738 Bernoulli's 

memoir, containing enunciations of theorems on the oscillations of heavy chains. Then, they 

reappeared in mechanical problems as the vibration of a stretched membrane or the symmetrical 

and unsymmetrical propagation of heat in solid cylinders and spheres, as well as in astronomical 

problems, related for instance to the elliptic motion of a planet about the sun. Presently, Bessel 

functions have a very wide field of applications, from abstract number theory and theoretical 

astronomy to concrete problems of physics and engineering [I]. 
Correspondingly, classical orthogonal polynomials (Jacobi, Legendre, Hermite), also introduced 

in connection with astronomical problems, are now of great importance in mathematical physics, 

approximation theory as well as in the theory of mechanical quadrature. In addition, they find 

significant applications in quantum mechanics on the determination of discrete energy spectra and 

the corresponding wave functions in fundamental problems [2]. Orthogonal polynomials imse indeed 

essential tool to deal with the problems of the harmonic oscillator and the motion of particles in 

a central field. Furthermore, the classical orthogonal polynomials of a discrete variable are of 

interest in the theory of difference methods [2]. 

Theory of classical special functions is rather well settled [3]. Recursion relations, addition 

theorems, integral representations, generating functions, asymptotic formulae, differential equations are collected in an organic body, which is however continuously refined and enriched by new 

investigations and new theoretical approaches [4]. 

Let us recall for instance the method illustrated in ref. [I], which suggests a generalization of 

the Rodriguez formula for the classical orthogonal polynomials, thus allowing to obtain explicit 

integral representations of all the special functions and to derive their basic properties. Similarly, 

the possibility of framing special functions within the context of group theory [4, 51 revealed a 

powerful tool permitting derivation of new results and a rational classification of old results, as 

well as suggesting to introduce new classes of functions, related to the recently discovered algebraic 

systems, such as the supergroups and the quantum groups [6]. 

Furthermore, we note the description of orthogonal polynomials by their recursion relations, 

which once regarded as eigenvalue equations allow to look at orthogonal polynomials from the 

viewpoint of scattering theory [4, 71. 

Recently, interest in special functions has greatly increased in connection with the possibility 

of generalizing the well known functions of mathematical physics to more than one variable and/or 

more than one index. In this regard, the generalization amounts to introducing functions with 

properties analogous to those of the one-variable counterpart. Generating functions are usually 

the key-note for many-variable generalizations of special functions. 

The multivariable Bessel functions, for instance, originally introduced by Appell [8] in connection with the problem of the elliptic motion of planets [9], have revealed a wealth of possible 

applications to physical and/or purely mathematical problems, as the scattering of laser radiation by free or weakly bounded electrons, the emission of e.m. radiation by relativistic electrons 

passing through magnetic undulators [lo] as well as problems related to the queuing theory [Ill. 

Also, they proved their relevance in multiphoton emission and absorption processes by quantum 

systems, which are of interest for the investigation of squeezed states, the relevant Hamiltonian 

operator containing indeed powers of the annihilation and creation operators [12]. 

Correspondingly, the multivariab1.e generalization of orthogonal polynomials has attracted a 

great amount of interest. In particular, as to the Hermite polynomials, let us recall that in ref. 

[13] a procedure has been developed, which generalizing that proposed by Gould and ~o~~ei [14] 

allows to define multivariable generalized Hermite polynomials, providing a complete orthonormal 

set in L2(Rn) space of square sommable functions with n variables. 

The present paper concerns with the classical many-variable functions introduced by Herrnite 

[15], whose application within the context of the phase space approach to physical problems is 

suggested. Accordingly, in Sec. 2 a general view on the many-variable Hermite polynomials is 

presented. The possibility of exploiting the developed formalism within the context of the phase 
space picture, which is becoming the context where many physical problems are naturally framed, 

is investigated in Sec. 3. In particular, the basic model of the harmonic oscillator is considered. 

General considerations and extensions are presented in Sec. 4. 

2 Many-variable Hermit e polynomials 

One-variable Hermite polynomials 7in(x)l can be defined by means of the relation 

where they appear as the coefficients of the series expansion of the exponential of a quadratic form 

defined on the red domain. 

The above expression can also be recast in the more usual form 

from which, exploiting the series development of the exponential function and appropriately rearranging the summation, the well known expression of the Hermite polynomials in form of a finite 

sum can be easily drawn: 

where [v] denotes the largest integer 5 v. 

Also, according to the formula for the Taylor expansion, (2) provides the Rodriguez formula: 

Taking the derivatives of both sides of (1) or (2) with respect to x and t, it is easy to infer the 

recursion relations 

linking the polynomial of order n to the contiguous ones. The prime denotes derivative with 

respect to x. 

It is immediate to get from the above relations the differential equation obeyed by ?in's: 

which allows to understood 3-1, as eigenfunction of the operator 

A d2 'D=-- d 

&.Z fXz 2 (7) 

'The use of the script 'H, to denote the Herrnite polynomial is in order to avoid confusion with the more common 

polynomial Hn : Hn(x) = 2n/27fn(fix). 
n being the corresponding eigenvalue. 

The opergtor 5 is not self adjoint; the adjoint operator 

admits the eigenfunction 3-1:) explicidy given as 

H,C (x) = an (~)e-~'/~ (9) 

belonging to the same eigenvalue as a,. Accordingly, the functions (7in, H:) form a biorthogonal 

set in the usual sense that they satisfy the orthogonality relation 

Generalizing the relation (1) to involve a bilinear form defined in Rn, Hermite introduced manyvariable functions [15]. Adopting a matrix notation, we can write down 

-- - = e -1xTMx 2- =- e- - 

mi, 

where 3 and h are elements of the vector space Rn: 

the superscript T meaning transpose. 

Accordingly, M - is a real n x n matrix: - = (aij), i, j = 1,. . . , n, which is required to be 

symmetric: aij = aji, not degenerate, i.e. det M 2 0 and positive definite: ~i > 0, i = 1,. . . , n. 

The expression (11) can be rewritten in thFalternative form 

which is the n-variable analog of (2). 

In passing, it is worth noticing that the above expression suggests considering the more general 

bilinear form 

with the matrices =) A = B being symmetric but in general different from each other. In this respect, 

let us recall that the Grassman Hermite functions have been introduced using the above quadratic 

form, with A and & being antisymmetric matrices and z, h anticommuting variables. A further 

extension of(14) hGe been considered in ref. [6], with the intent of obtaining a class of functions, 

related to quantum groups. 
In full analogy with the one-variable case, taking the derivative of (11) or (13) with respect to 

the components of both vectors g and h, we get the recursion relations satisfied by Rml,...,mn (z): 

witki i ranging from 1 to n. 

Since corresponding to any quadratic form cj(2) = gTM: we can associate the adjoint form 

defined in terms of the inverse matrix, i.e. $(c) = -- EM-^, - it is possible to associate with the 

Rml,...,mn'~ the adjoint polynomials Gml,...,mn (2). The definition resembles the relation (1 I), but 

involves the transformed vectors: 

Explicitly, we have indeed 

,-+(g-~~g-'(g-~ = e-+Tg-lg ky . . . 7Gml,...,mn(~) k? . C - 

m,.. 1 mn. 

- If M is the identity matrix, the two polynomials coincide: Nml,..,,mn - Gml,,.,,mn. Furthermore, 

theyturn into the product of n one-variable Hermite polynomials, i.e. 

Taking the derivative of (17) with respect to : and &, the following set of recursion relations can 

be obtained: 

with A r det M and Aij the minor relevant to the element aij. 

The relevance of the polynomials Gql,..., is clarified by the orthogonality relation, which can 

be proved in the form 

For the explicit derivation of the above relation the reader is addressed to ref. [8]. 

The orthogonality relation (20) can be conveniently exploited to express a given n-variable 

function p(g) in form'of a series involving 7imll...lmn and Gmll...,mn. Accordingly, let us put 
with the coefficients Am ,.., , Bml mn being specialized according to (20) into 

The explicit values of the entries of the matrix M - should be suggested by the specific problem 

under study. 

It is needless to say that the theory of many-variable Hermite polynomials is very rich. However, the above considerations represent all the machinery we need for testing the possibility of 

using these functions as basis for the phase-space analysis of physical problems. For a more 

detailed discussion the interested reader is addressed to refs. [8, 151. 

3 Two-variable Hermite polynomials and phase space 

picture of dynamical problems 

Phase space picture is becoming the unifying language for both classical and quantum mechanics. 

Phase space formalism is indeed basic to the Hamiltonian formulation of classical mechanics. In 

this connection, the evolution of a dynamical system is described by a number n of independent 

coordinate variables and on the same number of canonically conjugate momenta. The cartesian 

space of these 2n coordinates is just the phase-space. 

Correspondingly, phase space picture of quantum mechanics is becoming increasingly popular. 

Although, the concept of phase-space is not compatible with quantum mechanics, q^ and fi being 

noncommuting operators, the Wigner phase-space representation allows to overcome this problem, since in this representation both the coordinate and momentum variables are c-numbers. 

Accordingly, it is possible to perform phase-space canonical transformations as in the case of 

classical mechanics, which correspond to unitary transformations in the Schrodinger picture of 

quantum mechanics. 

Phase space concept appears therefore as the unifying context, where classical as well as 

quantum mechanics can be naturally framed, thus suggesting the possibility to transfer concepts 

and methods from quantum to classical mechanics and viceversa. 

Furthermore, as discussed in ref. [16], phase-space picture provides the natural language for 

quantum optics as well, offering a geometrical view to coherent and squeezed states as circles 

and ellipses respectively. In this connection, taking advantages from the symmetry of the relevant Wigner phase space distribution function it is possible to calculate expectation values and 

transition probabilities for the above quoted states [16]. 

Finally, let us recall that phase space is the context where the dynamics of electron beams 

moving through magnetic channels is studied and the Harniltonian optics can be conveniently 

reformulated. 
As already remarked, the paper is aimed at investigating the possibility of using the manyvariable Hermite polynomials, briefly discussed in the previous section, as analytid tool in the 

phase-space approach to dynamical problems. 

In the quantum framework, one-variable Hermite polynomials are intimately related to the 

harmonic oscillator dynamics. It is therefore natural to analyse, as first step for our investigation, 

the harmonic oscillator dynamics in the phase-space representation. 

Let us consider therefore the quadratic Hamiltonian 

the variable s playing the role of time. It is needless to stress the relevance of the above Hamiltonian as basic model for many physical problems as well as approximation in many of the existing 

theories. 

As already stressed, in quantum mechanics the Hamiltonian (24) rules the evolution of a 

harmonic oscillator of unit mass and time-dependent frequency k(s), 4 and fi being the position 

and momentum operators. In classical mechanics, it describes for instance the betatron motion of 

a charged particle through a magnetic quadrupole, as in the ray optics it governs the propagation 

of an optical ray through a nonhomogeneous medium with a quadratic profile of the refractive 

index. 

It is interesting to notice that in the configuration space picture the evolution of the system 

describid by the Hamiltonian (24) is analysed within the context of conceptually and formally 

different approaches: the Schrodinger equation for the quantum wave function and the Hamilton 

equation of motion for the canonically conjugate variables q and p. Conversely, the Von Neumann 

equation for the Wigner distribution function and the classical Liouville equation for the phasespace distribution are of the same form, so long as the Hamiltonian of the quantum system is 

quadratic. Hence, time evolution of the Wigner function can be obtained directly from the solution 

of the equation of motion of the corresponding classical system. Harmonic oscillator provides a 

unique example in which classical and quantum phenomenology overlap to a large extent. 

Let us approach the problem within the context of classical mechanics. Accordingly, the 

Liouville equation for the phase space distribution function p(q, p; s) is immediately written down 

with an assigned initial condition: p(q,p; s) = po(q,p). 

As introduction to the forthcoming discussion, let us recall that an invariant quadratic form 

can be associated to any dynamics described by quadratic Hamiltonians-in canonical coordinates 

and momenta. In passing, it is worth stressing that within a quantum context the quadratic 

form (26) is reported as the Ermakov-Lewis invariant [17], the vector g containing obviously the 

position and momentum operators, whilst in classical mechanics it is reported as the CourantSnyder invariant [18], firstly introduced in the analysis of electron beam motion through magnetic 

channels. In the above expression, the two component vector : E (z) is acted by the real 2 x 2 
matrix T, which furthermore is required to be symmetric and unimodular: detz = 1. Just to 

share th; language of accelerator physics, we refer to 21 - as the Twiss matrix2 andfeport it in the 

form 

The entries a,P,r, usually named as Twiss parameters, play an important role in designing 

transport channels. 

The quadratic form (26) can be depicted in the phase space as an ellipse, whose size and 

orientation are determined by the Twiss coefficients. The area of the ellipse, which is just the 

value of the invariant I, is usually denoted in accelerator physics as I = re, e being named as 

the beam emittance. It plays a crucial role in characterizing the quality and the dynamics of the 

e-beam. In a single particle analysis, a,P,y defines the contour of the particle trajectory, as 

in an ensemble analysis they define the second order momentum of the phase space distribution 

function, thus providing information on its extent and maximum localization. Explicitly, 

the averages being understood on the distribution function. Accordingly, the emittance r: can be 

given the further meaning: 

E2 = a2 a2 - , 2 

99 PP r w • (29) 

Let us stress now that the Liouville equation admits as particular solution the distribution function 

shaped indeed in form of a Gaussian and therefore with a maximum localization within the ellipse 

of area E. 

zTT x The above considerations suggest to use the functions 3-l,,,e-z- =- as basis in the phaseplane, the entries of the matrix 2 being chosen as the second order momentum of the distribution. 

In other words we can use the agve quoted functions to approximate a generic function defined in 

the phase space by means of two-variable Gaussians, further modelled by the Hermite polynomials 

Nm,n 9 which give a different maximum localization. The process is perfectly similar to that used 

for one-variable functions using the Hermite functions. 

Let us consider therefore generic distribution function p(q,p; s). According to the results of 

Sec. 2, we can express p(q, p; s) in form of a series: 

2~he quadratic form (26) can also been regarded as the transcription in phase space of the quantum invariant, 

the vector : being formed by the expectation values of the position and momentum operators and the matrix 2: 

- 

being linked to the covariance matrix. 
with am,n(s) and ~:,~(q, p)being explicitly given as 

The superscript W specifies that the polynomials 3C,,, have been used in the series expansion. 

The alternative expansion in terms of the adjoint polynomials Gm,n(q, p) can be also used, thus 

leading to 

P(P, pi s) = C bm,n~E,n(q, P) m 9% 

(33) 

with 

and the superscript G specifying that the adjoint polynomials Gm, have been used. 

The entries of the Twiss matrix appearing in the above expression can be conveniently chosen, 

according to the previous discussion, as the second-order momenta of the distribution function 

po(q,p) at the initial time. Then, the emittance E is just obtained according to (29). 

Inserting the expression (31) into the Liouville equation (25), we get the set of equations for 

the coefficients a,,, (s) , namely 

where the relations 

and 

have been used, obtained from the recursive relations obeyed by ?fm, (see Appendix A). 

Let us consider for instance the particular case where the quadrupole strength does not change 

along the direction of motion: k(s) = k; and the initial -distribution function po (q, p) has the form 

The coefficients a,,n(0) are then given as 
In addition, assuming that the electron beam is matched to the quadrupole, namely 

it is easy to prove that the distribution function does not change during the motion: 

More general situations require the numerical handling of eq. (35), from which the evolution of 

p(q,p; s) can be inferred according to the expansion (31). 

4 Concluding remarks 

The analysis developed in the previous sections has been aimed at testing the possibility of using 

many-variable Hermite polynomials as analytical tool, within the context of phase-space picture 

to dynamical problems. 

The discussion has been limited, for illustrative purposes, to the harmonic oscillator dynamics. 

However, it can be easily realized that Harniltonians containing higher order terms, accounting 

for non linear forces, can be treated by means of the same formalism as well. 

Also, within a quantum mechanical context, the formalism developed might be used in connection with the Von Neumann equation, which, as already noticed, rules the evolution of the 

Wigner distribution function W (q, p; s) according to [I91 

Furthermore, let us say that, although in Sec. 3 we have considered 1-dimensional dynamics, 

problems with more than one degree of freedom can be analysed, as, for instance, the motion of 

electron-beams along magnetic channels with transverse coupling. In that case, the radial and 

vertical motions cannot be separated and the dynamics should be analysed in a 4-dimensional 

phase-space, the relevant elements consisting of the conjugate variables x, p,, y , p,. 
APPENDIX A 

Two-variable Hermite polynomials 

This Appendix is devoted to discuss with some details the properties of the two-variable 

Hermite polynomials, which are used in Sec. 3 as illustrative of the usefulness of such functions 

within the context of the phase-space formalism. 

Firstly, let us write down the explicit form of the recursion relations, reported in (15) for the 

general case of many-variable polynomials: 

From the above relations after some algebra the following partial differential equation can be 

deduced a2 a2 a2 

q-+p- ( 8q )] xm,n = ~(m + n)xmln , (A-2) 

in some sense reminiscent of eq. (6), which can be recovered in correspondence with M - = 1. - 

Consequently, the polynomials am, can be understood as Bgenfunctions of the operator 

with eigenvalue A (m + n) . Let us consider now the adjoint polynomials Gm,n(q,p), for which the general relations (19) 

specialize into 

which provide the differential equation 

the same as for 

Finally, let us note that the orthogonality relation (20) in the case we are considering specialize 
REFERENCES 

[I] G.N. Watson, A Treatise on the Theory of Bessel Functions, (Cambridge University Press, 

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Special Functions and the Theory of Group Representation, (Academic Mat hematical Society, 

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[6] R.J. Finkelstein, J. Math. Phys. 33, 4259 (1992). 

[7] K.M. Case, J. Math. Phys. 15, 2166 (1974). 

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A. Torre, S. Lorenzutta, G. Maino and C. Chiccoli, Nuovo Cimento B106, 21 (1991). 

[ll] W. Miller Jr., Comm. Pure Appl. Math. 18, 493 (1965). 

[12] G. Dattoli, C. Chiccoli, S. Lorenzutta, G. Maino, M. Richetta and A. Torre, J. Sc. Comp. 8, 

69 (1993). 

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Polynomials, submitted for publication. 

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Noz, Phys. Rev. A40, 902 (1989); D. Han, Y.S. Kim and M.E. Noz, Phys. Rev. A41, 6233 

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Antoine Royer 

Dkpartement de Genie Physique, Ecole PoIyfechnique 

Ag,,&Ca19 Qukbec H3C 3A 7, Canada 

The Wigner function is argued to be the only natural phase space function evolving 

classically under quadratic Hamiltonians with time-dependent bilinear part. This is used to 

understand graphically how certain quadratic time-dependent ltonians induce squeezing 

of quantum states. The Wigner representation is also used to generalize Ehrenfest's theorem 

to the quantum uncertainties. This makes it possible to deduce features of the quantum 

evolution, such as squeezing, from the classical evolution, whatever the Hamiltonian. 

Be Wigner function [1] can be used to get a visual understanding of why certain timedependent Hamiiltonians induce squeezing of quantum states. We first address the question: My the 

Wigner function, rather than, say, the Husimi function [2]? It will be showed that although other phase 

space functions may evolve classically under certain specific quadratic Hamiltonians, the Wigner 

funcriola is the only one to do so under quadratic Hamiltonians having a he-dependent bilinear part. 

We consider, as an exmple, squeezing by a peridically modulated harmonic oscillator. We then 

discuss a generalization of Ehrenfest's theorem applying to the quantum uncertainties. This allows to 

deduce aspects of the quantum evolution, such as squeezing, from the classical evolution even in the 

Phase space variables (position and momentum) are denoted by q and p. A caret is used to 

identify operators. Thus, [q, pf = i, where we take h = 1. The time variable is denoted by t. 

2 Qoadratic Hamiltonims and phase space functions 

Ehrenfest's theorem expresses the time derivatives of the expectations (q) and (p) in a way 

formally similar to Hamilton's equations. In the case of a Hamiltonian = $*2/m + V($, it reads 
where F(q) = -aV/aq is the force. If the potential V(q) is quadratic, hence F(q) linear (or, if the 

particle is in a state sufficiently localized on the scale of non-harmonic variation of V), then (F(i)) = 

I?((¶)), and (1) become identical to Hamilton's equations, implying that (¶) and (p) follow classical 

trajectories in phase space. This stays me in the case of a general quadratic Hamiltonian 

H(t) = iil(t) + &(t) (2) 

where gl(t) is linear, and &(t) is bilinear in i and j, (a, P, a, b, c scalar functions of time): 

Can one push this further, and associate with state vectors Iv), or state operators 6, a phase space 

function f(q,p,t) whose quantum evolution is classical for such Mamiltonians? That is, such that 

where (qbp3 is the point which classically evolves into (q,p) in the time interval (O,t), under the Hamiltonim H(qq,t). It is natural to ask that f(q,p) be linear in eitker I~J) or 6, so that it has the fom 

where I$*) and ihp are kets and operators pmmeaized by phase space. 

lnwduce the unitary time-evolution operator 6(t,tt), defined by 

and similarly c1 and c2 corresponding to fil and g2. Let us first assure classical evolution under 

linear time-independent Hamiltonians HI = a¶+Pp: Here, Cl(t,0) = e-it@Q+p~), and one gets classical 

evolution, f(q,p,t) = f(q-Pt, p+at, O), up to a phase, iff 

'Ow) = r),l$), ;hp = Ijqp~,-l 

where n 

D, = eipq-ic@ 

are phase space displacement operators, and I$) and A are some fiducial ket and operator. We now 

note the relations ([I+ denotes anticommutators, { )pB Poisson brackets): 
where H1,2 = Hl;~(q,p). Using these results, and refering to (3, one finds that 

where c2 is the time evolutor for fi2, and ~(q,p,t) is a phase determined by the equation 

One verifies (11) by verifying that both sides of C(0,t)6&2(~0) = 6qtPtei~ satisfy the same 

differential equation. We then get, from (6) and (1 1): 

* 

One sees that the function f$ evolves classically iff I$$ is stationary, i.e., if H2 is time-independent, 

and I+) = !En) is an eigenket of it. For instance, if 62 is a harmonic oscillator, and I$) = I&) is the 

ground state, then I@@) are coherent states, and 

is the Husimi function [2], which is well known to evolve classically under that flz of which I) is n 

an eigenket. Clearly, no function f+ linear in state vectors can possibly evolve classically if H2 is 

time-dependent. Consider now (14): Again, f~ evolves classically iff At is stationary: For instance, if 

H2 is time-independent, and = IEn)(En*I or A = g&2), then the evolution is classical - but only for 

thatspecific time-independent H2. IS there an operator A (apart from the unit operator) such that 

(cp a real phase) for tim-dependent H2(t)? Yes, the parity operator if, since in = -ni9 = -& 

imply k2, n]=0. Setting A = fi in (14) yields the Wignerfunction [1,3] 

The Wigner function is the only phase space function which evolves classically under (any) timedependent H2(t). One can see this as follows: Represent any A by its Weyl symbol Aw(q,p), where 
the Weyl symbol of an operator A is defined as [3,4] 

We want 4, hence its Weyl symbol, to be stationary for any $(t). Now, just as the Weyl symbol 

(17) of fi evolves classically under f12(t), so does that of it. Ihe only possible way for (hdW(q,p) 

to be stsationiury under classical evolution with &fferent El2 's (e.g., with orbits which are ellipses or 

hygerbolas of different eccentricities) is that it be be concenarated at the origin: We must thus have 

A,(q,p) = (mst)Q)6(p); this implies [5b] that A = (const)lii[. 

%he classical evolution of the Wigner function under quadsattic M(t) is very useful for understanding the quantum in terns of the classical evolution. As an example, consider ah 

If a=l (in suitable units), the classical orbits M(q,p) = constant are circles in the phase plane (q,p). If 

a # I, the orbits me ellipses (Fig.l), the ratio of the q semi-axis to the p semi-axis being a. We will 

now let a alternate between two values, y and y -1 (where y > I), at every quarter of a period 2x10, 

while keeping o fixed (i.e., only m changes). Let m = y -1 6 1 for the fmt quarter period: D~ng that 

time intenval, a point initially on the positive q axis (beginning of arajectory 1 on Fig.2) moves to the 

negabve p axis, while receding away from the origin by a factor y. ;Then let a = y > 1 dusing the next 

quarter period: %he point moves to the negative q axis, receding away from the origin by another factor 

y. And so on. One here has parametric mplification. On the other hand, points initially on the p axis 

(arajectory 2) close in on the origin. Thus, classically, the phase plane gets squeezed into the rotating q 

axis, by a factory at each quarter period. Whence a corresponding squeezing of quantum states [5a,6]. 

Fig. 2 t 
The above concerned quadratic Hamiltonians, It will now be indicated that the Wigner-Weyl 

) ~presenation allows to reinterpret Ehrenfest's theorem, and to extend it to quadratic obseavables, hence to the quaturn uncentainbes. This makes it possible to deduce features of the quantum 

evolution from the classical evolution even in the case of arbitrary Hamiltonians. Let us first recall that 

repnsentation, expectation values of operators A have the "classical" form [1,7] 

In particular, the quantum expectations of 4 and f, are 

and the uncemhty matrix is 

A qumrum state (wave packet) may be roughly represented in phase space by an uncertainty ellipse 

(u - (u)) C-l (U - (u)) = 1 where u = 

We also need the result [7] 

[A, i]w(q,p) = i (Aw, BW)pB(q,p) if A or 6 is quadratic 

that is: The Weyl synnbol of the commutator of two operators, one of which is quadratic, is equal to the 

Poisson bracket of the individual Weyl symbols. Let now ;\ be an observable quadratic in ¶ and 6. 

We then have, by (20) and (24), for any fI(t): 

= Jd¶d~ Aw(q,~)(Hw, ~W)PB (A quadratic, any H) (25) 

where in the last line we performed an integration by parts. Eq.(25) says that the rate of change of the 

expectation of a quadratic observable is classical, in the sense that it is the same as if each point in 
Wigner phase space instantaneously followed a classical trajectory. This does not mean that ($ 

evolves classically over finite time intervals, because fw(q,p,t) in (25) is the exact qumtaldy evolved - 

Wigner function at time t, not one evolved classically during some finite time (unless H is quadratic); 

more specifically, the first time derivative of (i) is classical, but not the higher order derivatives. By 

letting A stands for 4 or i, in (25), one gets Ehrenfest's theorem: 

The last expntssions in (26a,b) are the usual statement of Ehrenfest's theorem [equivalent to (1) if H = 

$%n + ~(^ql], giving a formal quantum-classical analogy. The fit expressions in (26a,b) tell us 

much more: That the rates of change of (q) and (b) are classical, but relative to a phase space 

distribution function. Eq.(25) generalizes Ehrenfest's theorem to quadratic observables, and thus to the 

uncefiainties: Indeed, according to (21), (22) and (25), the uncertainty ellipse (23) evolves exactly as if 

each point in Wigner phase space instantaneously followed a classical trajectory. For instance, if the 

ellipse gets squeezed classically, during some small (infinitesimal) time internal, then so does it 

quantdly. In general, one may expect that if the classical motion during a finite time interval squeezes 

the unceaainty ellipse, then so does the quantum evolution. The latter statement is of course rigornusly 

true if k(t) is quadratic, in view of (17); it is also approximately me, in the case of arbi 

fw(q,p,t) is sufficiently localized on the scale of non-harmonic variation of H(q,p,t), for the evolution 

of fw(q,p,t) is then approximately classical [5b]. 

Let US mention, finally, that Ehrenfest's theorem for quadratic observables can also be written h 

a fom comsponding to the last expressions in (26a,b), namely [5c] 

a- x@) = ((A,, HwIm(h9i)w) (A quadratic, any H) (27) 

where the subscript w on the function {Aw, Elw) ,(i,p) of the non-commuting operators 4 and p 

signifies that they are ordered according to Weyl's ordering rule [4]. 

1. E.P. Wigner, Phys. Rev. 40,749 (1932) 

2. K. Husimi, Broc. Phys. Math. Soc. Jpn. 22,264 (1940) 

3. A. Grossmann, Commun. Math. Phys. 48, 191 (1976); A. Royer, Phys. Rev. A 15,449 (1977) 

4. N. Weyl, me Thwry of Groups and Quantum Mechanics (Dover, New York, 1950), pp. 272-276 

5. A. Royer, (a) Phys. Rev. A 42,560 (1990);(b) ibid 43,44 (1991);(c) Found. Phys. 22,727 (1992) 

6. J. Janszky, P. Adam and I. Foldesi, Phys. Lett. A 174,368 (1993) 

7. J.E. Moyal, Proc. Cambr. Phil. Soc. 45,99 (1949) 
ON THE PedSSPBICLITY OF A QUANTUM MECHANICAL 

PHASE SPACE REPRESENTATION 

Go. Torres-Vega 

Departamento de Fisica, Centro de Investigacidn y de Estudios Avanzados del IBM9 

Apartado Postal 14-740, 07000 M&im D.F. 

Probability densities in phase space containing quantum dynamics information are very 

useful. However, is there a true phase space representation of Quantum Mechanics? 

. 1 A Phase Space Representation 

As in other representations, let us postulate that there exist a set of basis vectors (TI = (p, gI and 

its adjoints (P) = Ip, -q), where F = (p, q) denotes a point in phase space. This set of vectors is 

oAhonorma1, (I" IT) = 6(P - I' ), and there is a closure relation, i = J dTlT) (TI, for them. The 

imer product between a bra ($1 and a ket 14) is defined as (44) = J Sdr V(F)+(T), falEnag 

the expected properties for an inner product. Hence, the projection of an abstract ket I$), onto 

the basis vector (TI, results in a wave function dependent on both p and p, $(k) = (TI$). 

The momentum operator P is defined as (p/2) - ih(8/8q), and the coordinate operator 0 is 

given by (q/2) + &(d/dp). Momentm and coordinate operators comply with the commutator 

relationship, [Q, $1 = &f. These operators provide the foundation for the development of a 

quantum representation in phase space. In what follows, we present the consequences of thk 

choice of operators. 

These operators are the generators of change of phase and translation, exp(iqp/h)lp, p) 

= exp (i~l2h) IP, g - r)) and exp (Y Qlh) l P, q) = exp (iCd2h) IP + F, g) 

With the definitions given above, we can write a Schrodinger equation in phase space, 

8 (t) = [;m(; - - - ih- sg)2+v(f+~& 

This is the equation on which the quantum theory in phase space is based. 

The physical interpretation of the wave function in Eq. (1) is that its square magitude 

times the volume element, (+(r)l2dr, gives the probability of finding the quantum system with 

moment= between p and p + dp and coordinate between q and g + dq before it is pertwbed 

in any way. Even though the wave function depends on both coordinate and momentuuaa, it 

is not possible to determine (to measure) them simultanwusly (see below). Fig. 1 shows the 

interpretation of the Heisenberg's uncertainty principle in phase space. The diagonal matrix 
elements of the quantw Eiouville equation, when the potential function can be written as a 

power series in its argwraent, V(x) = Czo V,xn, is given by 

where Ft is the density operator at time t. 

FIG. 1. Meisenberg's uncertainty relation, ApAq 2 ti, in phase space. 

2 Time Reversal Properties 

Time reversal is achieved by the replacements t - t, p + -p, q 4 q and by t&hg the 

complex conjugation of wave hnct ions. Under this tr ansfor mat ion, the momentum operator 

(p/2) - ifi(dl8q) transforms to its negative, the coordinate operator (q/2) + %(8/8p) remaim 

with no change, and the Schrodinger equation (1) becomes 

Thus, if +(p, q; t) is a solution to the Schrodinger equation, (-p, q; - t) is also a solution. 

Similarly, if (p, qqlFt(p, q) is a solution to the quantum probability conservation equation, then 

(-p, qlp^-tl - p, g) is also a solution to it. 

3 Eigenfunctions of Momentum and Coordinate 

Operators 

Let us consider the eigenfunctions of coordinate and momentwm operators in phase space. A 

solution to the Eigenvalue equation (p/2 - ifitLa/aq)(rl%) = p'(rl%) is given by (TI .pl) = 

(2rti)-'I2 exp [ig($ - p/2)/h]. On the other hand, a solution to the eigenvalue equation (g/2 + 

&a/ap)(I'lud) = q-'(rlq) is given by (I'(ud) = (27~h)-l/~ em[-ip(d - g/2)/h]. As expected, 

there is no simultaneous eigenfunctions of coordinate and momentum operators. 
4 Simultaneous Measurement of Momentum 

and Coordinate 

Following the standard convention regarding measuring process, if one proceeds to deter 

the configuration of a given quantum system, the wave function, (r) $) reduces to the eigemtate 

(rlq) of i) with probability I(% I$) ,)I2 of obtaining the value q'. If we then want to measure the 

momentm, we need to express the new state (rlq) as an expansion in terms of the eigenstates 

According to this, once one determines the configuration of a quantum system, one can find it 

with any value for the momentum. Therefore, a wave function dependent on p and q need not 

violate the uncertainty principle. 

5 Reduction $0 Momentum and Coordinate 

Representations 

Coordinate or mnnenkum representations can be recovered from the phase space representation 

proposed in this work. By taking the Fourier transform of the phase-space Schrodinger equation 

(I), over g, with kernel (4~ti)-l/~ eq(-ipg/2ti), we obtain 

where we have mspmned that the potential function can be mitten as a power series in its argument, V(z) = En Kxn. Mow, by taking the Piburier transform over p, with kernel 

(4n~)-'I2 exg(ipq/2h), we obtain 

6 Classical Correspondence 

En order to find what the classical analogues to quantum wave functions are, let us compare 

the quantum probability conservation equation in phase space (2) with the classical Liouville 

equation of motion, 

8 P a 

- p(r; t) = - -- + 

at [ma, 

We observe that the right hand side terms of the phase space quantum probability conservation 

equation (2) are just the symmetrizations of the classical products p p(r; t) and F(,)p(r; t) 

which appear in the classical Liouville equation (7). Therefore, the classical analogue to the 

quaraturn probability density I $(r; t)I2 is the corresponding classical probability density p(r; t). 
The comparison between the quantum probability conservation equation in moment= rep 

resentation, 

and the coordinate average of the classical Liouville equation, 

leads to the conclusion that the classical analogue to the quantum momentum probability density, I $(p; t) 12, is the coordinate average of the classical probability density; J dq p(r; t). Similarly, 

the classical analogue to the quantum coordinate probability density I$(q; t)I2 is the momentuna 

average of the classical probability density, J dp p(r; t). 

7 Step Potential 

As a simple illustration, let us examine the dynamics of the scattering of a Gaussian wave packet, 

+(r; 0) = (2W2 exp[-(9 - q0)~/4 - (p - Jd2/4 + i(qn - m)/2], with (Po, qo) = (JZ, -3.5), 

from a potential step. In this calculation, f he energy of the wave packet is 1 and the barrier 

height is 1. This model system illustrates the development of nodal structure due to quantu 

interference between the advancing and reflected parts of the wave packet when it encounters 

the step potential. 

-12-8 -4 0 4 8 12 -12 -8 -4 0 4 8 12 -12 -8 -4 0 4 8 12 

(J 

FIG. 2. Snapshots of the quantum evolution of the Gaussian density which is 

being scattered from a step potential at q = 0, in a dimensionless phase space. 

Figure 2 shows density plots in phase space of the square magnitude, I+(r; t = 0)12, of the 

time evolution of the Gaussian density. We can see that the wave packet breaks into three 

parts as it is evolving in time. First, there is a part whose momentum in near zero and which, 

consequently, moves very slowly. A second part of the wave packet smoothly changes its direction 

as it is reflected from the step, and combines with the part that stayed behind. Finally, a third 
part of the wave packet has enough energy to overcome the barrier and continues moving past 

the step with smaller momentum. 

FIG. 3. Same calculation as the one in Fig. 2, but made in the coordinate 

representation. 

Figure 3 shows the same type of calculation as in Figure 2 but made in coordinate representation. The analysis in phase space allows us a better understanding of the dynamics. Each 

peak appearing in the coordinate density corresponds to the different parts of the wave packet 

in phase space. These pictures suggest that, in this case, "quantum interference" effects may be 

partly explainable in classical terms. Each piece of the reflecting and transmitting wave packet 

resides in a different region of phase space and it is only when the wave function is projected 

onto coordinate or momentum space that one loses this perspective. 

-12 -8 -4 0 4 8 12 16 -12 -8 -4 0 4 8 12 16 -12 -8 -4 0 4 8 12 16 

q 

FIG. 4. Snapshots of the classical evolution of the classical analogue of the initial 

density in Fig. 2. 

4 

2 

Po- 

-2 

-4 

In a classical calculation, we have taken the Gaussian probability density p(r; 0) = (291=)-l/~ 

exp[- (q - qol2/2 - (p - ~)~/2], with (po, qo) = (fi, -3.5), which is the classical analogue to 

the one in Fig. 2a, as an initial density to be propagated according to the classical Liouville 

equation 7. In Fig. 4, snapshots of the evolution of this density are shown at the same times as 

11111111 

- - 

- p(p,q;t) 

- 

- - 

t=Q - - 11111111 

ltllllll 

- - 

- - 

- - 

- - 

t=l5Q 

- - 11111111 
in Fig. 2, and, in Fig. 5, the momentw average of the density in Fig. 4 (the classical analogue 

to the density in Fig. 3) is shown. By comparing figures 2, 3, 4 and 5 we conclude that the 

classical and quantum behavior are qualitatively the same with small quaditative differences. 

Since the classical averaging process is over real-valued functions, only constructive addition 

of the pieces of the distribution are permitted, thus, the interference effect wont appear in the 

classical analogue to the coordinate quant urn density. 

FIG. 5. Momentum averages of the densities in Fig. 4. 

8 Acknowledgements 

We would like to acknowledge financial support from CONACyT and SN, M6xico. 

References 

[I] Go. Torres-Vega and J.M. Frederick, J. Chem. Phys. 93, $862 (1990). 

[2] Go. Torres-Vega, J. Phys. Soc. Jpn. 60, 714 (1991). 

[3] Go. Torres-Vega and John M. Frederick, Phys. hv. Lett. 6'9, 2601 (1991). 

[4] Go. Torres-Vega and John W. Frederick, J. Chem. Phys. 98, 3103 (1993). 

[5] Go. Torres-Vega, J. Chem. Phys. 98, 7040 (1993). 

[6] Go. Torres-Vega, J. Chem. Phys. 99, 1824 (1993). 
PARTICLE LOCALIZATION, SPINOR PNO-VALUEDNESS, 

AND %;EREb/lI QUANTIZATION OF TENSOR SYSTEMS 

Frank Reifler 

Randall Morris 

Martin Marietta 

Government Electronic Systems 

Borton Landing Road 

Moorestown, NJ 08057 

Recent studies of particle localization show that square-integrable positive energy bispinor fields in a 

Minkowski space-time cannot be physically distinguished from constrained tensor fields. In this paper 

we generalize this result by characterizing all classical tensor systems, which admit Fermi quantization, as those having unitary Lie-Poisson brackets. Examples include Euler's tensor equation for a 

rigid body and Dirac's equation in tensor form. 

It is a common misconception that fermions can only be represented in quantum field theory by bispinor fields. Recent studies of particle localization [I], [2], [3], [4] have shown that particle wave 

functions cannot vanish in regions of positive measure for any set of times of positive measure. This 

demonstrates that gedanken experiments, designed to observe the two-valuedness of bispinors, are not 

physically realizable since such experiments require absolute isolation of particle wave functions [4], 

[51, 161. 

Moreover, in previous work we showed that square-integrable positive energy bispinor fields in a 

Minkowski space-time cannot be physically distinguished from constrained tensor fields. That is, the 

non-localization of particle wave functions implies that the two-valuedness of bispinors is unobservable in a Minkowski space-time. Thus, beam splitting experiments designed to observe the rotation 

properties of bispinors, in fact, describe the rotation properties of constrained tensor fields [4]. 

Furthennore, it was shown that Dirac's bispinor equation can be expressed, in an equivalent tensor 

form, as a constrained Yang-Mills equation in the limit of an infinitely large coupling constant. It was 

also shown [4], [7] that the free tensor Dirac equation is a completely integrable classical Hamiltonian 

system with (non-canonical) unitary Lie algebra type Poisson brackets, from which Fermi quantization can be derived directly without using bispinors. 

In this paper we generalize this result by characterizing all classical tensor systems which admit Fermi 

quantization. As shown in Section 2, these tensor systems have Lie algebra type Poisson brackets 
associated with a unitary symmetry group acting on the classical phase space. Two examples of classical tensor systems which admit Fermi quantization are Euler's equations for a rigid body and the 

tensor form of Dirac's equation. It is well known that Euler's equation can be Fermi (or Bose) quantized [8]. It is less well known that Dirac's equation can be written in a classical tensor form which 

can be directly Fermi quantized in the same manner as Euler's tensor equation. 

2. FERMI QUANTIZATION OF CLASSICAL TENSOR SYSTEMS 

In this section, Fermi quantization is derived for the Euler and Dirac tensor equations by representing 

their classical Lie-Poisson brackets as commutators of Heisenberg operators on a Fock space of Fermi 

occupation states. 

We define a Fock space, that is, a Hilbert space H of occupation states of a single field, which is suitable for both fermions and bosons, as follows. We suppose that there exists a denumerable set of 

operators A,, where p - 1, 2, 3, . . ., such that all A, and their adjoints 4 are defined on an invariant dense subspace 2, C H. For the ferrnion case, the operators A, and A; will be bounded, in which 

case ?) - H. 

For each pair of indices p and q we define: 

'which is an operator defined on 9. The following can be taken as a set of axioms which are satisfied 

by fermions and bosons alike, when p = 1, 2, 3, . . . is interpreted as an index laklling the degrees of 

freedom (modes) of a single field. Np - Npp as the occupation number operator, and $ and A, as 

creation and annihilation operators for the mode p. 

a) There is a zero-occupancy state in 9, denoted by I*, such that for all modes p: 

Furthermore, there are at least two distinct modes p and q such that A, f A,, and none of the 

operators A,: I> -+ ID are the zero operator. 

b) There are no states in H (except 0) which are orthogonal to all the occupation states: 

In,, n,, . . .> = (A;)"' (A;)"' . . . lo> (3) 

where np = 0, 1, 2, . . . is the occupation number for the mode p, and all but a finite number of 

np are zero. 

C) For all modes p, q, and r, the operators N, and A, satisfy the following commutation relation 

on 9: 

[N,, A,] - -A, 8, 
and its adjoint on ID 

where 6, equals one if p = q and equais zero otherwise. 

Note that axiom (c) is required for a quantum field theory in which field operators (Fermi or 

Bose) satisfy Heisenberg's equation [9]. Note also that axiom (c) does not involve any anticommutation relations, and hence is applicable to tensor systems. We can prove that any Fock 

space satisfying (a), (b), and (c) is either Fermi or Bose, and for a given set of modes, the Fermi 

and Bose Fock spaces satisfying axioms (a), (b), and (c) are unique up to isomorphism [lo]. 

Axioms (a) and (c), or equivalent axioms, are assumed in more general Fock spaces which have 

been used to derive parastatistics [lo]. However, the parastatistical Fock spaces do not assume 

axiom (b). That is, in a parastatistical Fmk space, there is more than one state for each set of 

occupation numbers. Since from the current state of knowledge, one can assume that the occupation numbers n,, n,, . . . determine a unique state In,, n,, . . .>, we do not consider Fock spaces 

with alternatives to axiom (b). 

For a Femi Fock space we add: 

d) For each mode p, the occupation nurnber n, = 0, 1. 

As a consequence of axiom (dl, 4 and 4 are bounded opersators defined on H (i.e., 21 -H). 

Fermi quantization of tensor fields is derived from the following thmrem: 

Given any denumerable set of m&s p, q, a, s, . . ., there exists a Fenni Fock space H (unique up to 

isornophism) satisfying (a), (b), (c), and (d). Moreover, let be a self-&joint operator whose domain 

is a dense slabspace of H. Then ahere exist operators &(t) defined on H, indexed by each pair of 

modes p an q, and dependkg on time t E R satisfying: 

i) The adjoint relation: 

ii) The (equal time) commutation relation: 

iii) The Heisenberg equation: 
PROOF: 

The (unique) Fermi Fock space exists by explicit construction [lo]. Since is self-adjoint, we may 

define the following operators on H: 

iht N e-iht b(t) = e pq 

Formulas (6) and (7) follow from formulas (I), (4), and (5). Formula (8) follows by differentiating 

formula (9) with respect to time t. Q.E.D. 

Note that by formulas (6) and (7), the operators &(t) generate a unitary Lie algebra [I 11. For example, in the case of two modes, linear combinations of &(t) for p, q = 1, 2 satisfy the commutation 

relations of angular momentum operators [12], which allows Euler's tensor equation to be Fermi 

quantized [8]. The following corollary of Theorem 1 characterizes all classical systems that can be 

Fermi quantized. 

COROLLARY: 

Classical systems that can be Fermi quantized are described by bimodal complex amplitudes1 %(t) 

satisfying: 

(where the bar denotes ordinary complex conjugation), and a Hamiltonian function PI - PI(%) which 

depends on the amplitudes &(t), such that the Wamiltonian equation is given by: 

where the Lie-Poisson brackets { , ) are defined by: 

Furthermore, Fermi quantization of such classical systems is unique up to isomorphism. 

The chief application of Theorem 1 is the Fermi quantization of Dirac's equation in its tensor form. 

As previously shown [4], there is a double covering map which takes a bispinor field @ to a constrained set of SL(2,C) X U(1) gauge potentials At and a complex scalar field p, where Lorentz indices are denoted by a, P, y = 0, 1,2,3 and gauge indices by J, K, L = 0, 1,2,3. Repeated indices will 

be contracted using Minkowski metrics gap and g,,. Since the Lie algebra of SL(2,C) is regarded as 

the complexification of the Lie algebra of SU(2), the gauge potentials A; for j = 1, 2, 3 are complex, 

while the U(l) gauge potential A: is real. A: and p satisfy the following constraint: 

1 Note that the set of observables (%) defines a complex phase space PC for the classical system, 

which is a Lie algebra under the Lie-Poisson brackets (12). In general, the "physical" phase space 

P need not coincide with PC. All that is needed is that P have Lie-Poisson brackets, and there is a 

homomorphism of PC onto P. All observables of P thus become observables of PC. For a more complete discussion of Lie-Poisson brackets see reference [13]. 

(and similarly for saPY and J") where the sum is over all pairs of fermion modes p and q, and the 

amplitudes %(t) are complex functions of time satisfying the complex conjugate relation (10). The + +' coefficients of the amplitudes a,,(t), denoted by Tg (x ), are fixed complex functions of x E K. Thus, 

at any time t, the amplitudes %(t) suffice to specify TP (and similarly, saPY and Ja), and hence can 

be considered as classical phase space variables. Substituting (19) into formula (18), we get: 

where o, is the frequency of the mode p. The classical Hamiltonian equations (which are equivalent 

to the Euler-Lagrange equation for the tensor Dirac Lagrangian (16)) are given by formulas (lo), (1 l), 

(12), and (20) which, by the corollary to Theorem 1, can be (uniquely) Fermi quantized. 

This then reproduces the existing second quantized theory for fermions. This also shows that bispinors 

are not more fundamental than the tensor Hamiltonian equations (lo), (1 I), (12), and (20), which we 

derived from the tensor Dirac Lagrangian (16). 

REFERENCES 

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wave packets," Phys. Rev. D, Vol. 22, No. 2 (15 July 1980), pp. 377-384. 

2. G.C. Hegerfeldt, "Difficulties with causality in particle localization," Nuclear Physics B (Proc. 

Suppl.) 6 (1989), pp. 23 1-237. 

3. A. Vogt and F. Reifler, "Unique continuation of some dispersive waves," Communications in 

Partial Differential Equations, (to appear). 

4. F. Reifler and R. Morris, "Unobservability of Bispinor Two-Valuedness in Minkowski Spacetime," Ann. Physics (N.Y.), Vol. 215, No. 2 (1992), pp. 264-276. 

5. R. Geroch, "Spinor structure of space-times in general relativity I," J. of Math. Phys., Vol. 9, No. 

1 1 (1968), pp. 1739-1744. 

6. R. Penrose and W. Rindler, Spinors and Space-Time, Volume 1, Cambridge University Press, 

Cambridge (1986), pp. 46-47. 

7. F. Reifler and R. Morris, "The Hamiltonian structure of Dirac's equation in tensor form and its 

Fermi quantization," Workshop on Squeezed States and Uncertainty Relations, NASA 3135 

(1992), pp. 381-383. 

8. D. Park, Classical Dynamics and its Quantum Analogues, Springer-Verlag, Berlin (1990), pp. 

274-278. 

9. F. Mandl and G. Shaw, Quantum Field Theory, John Wiley and Sons, New York (1984), pp. 

62-64. 

10. O.W. Greenberg and A. Messiah, "Selection rules for parafields and the absence of para-particles 

in nature," Physical Review, Vol. 138, No. 5B (1 965.), pp. 1 155-1 167. 

11. B.G. Wybourne, Classical Groups for Physicists, John Wiley and Sons, New York (1974), pp. 

270. 

12. G. Baym, Lectures on Quantum Mechanics, Benjamin, Reading (1978), pp. 380-386. 

13. J.E. Marsden, Lectures on Mechanics, Cambridge University Press, Cambridge (1992), pp. 4-1 1. 
SECTION 5 

UN RELATIONS 
3.D. Franson 

Applied Physics Laboratory, The Johns Hopkins University, 

Laurel, Maryland 20723 

Several sources of phase noise, including spontaneous emission noise and the loss of coherence due to 

which-path information, are examined in the classical 

limit of high field intensities. Although the origin of 

these effects may appear to be quantum-mechanical in 

nature, it is found that classical analogies for these 

effects exist in the form of chaos. 

There are several sources of phase noise that may appear to be 

inherently quantum-mechanical in nature. One example is spontaneous emission noise, which is often attributed to vacuum f luctuations. Another example is the loss of coherence in which-path 

experiments, which can be shown to be due to the entanglement of 

one particle with another. 

This paper addresses 'the question of whether or not these 

effects continue to exist in the macroscopic limit of highintensity fields. If so, do they agree with the predictions of 

classical physics in that limit? 

One motivation for considering these questions is to gain 

further insight into the origin of these effects. In addition, any 

disagreement with classical physics in the macroscopic limit would 

suggest an interesting experimental test of quantum mechanics in a 

new and untested situation. 

It will be found that a classical analysis of these systems 

does give analogous effects due to classical chaos. This suggests 

that there is at least a loose connection between quantum noise and 

classical chaos. 

On the other hand, classical physics cannot provide any 

analogy for nonlocal effects such as violations of Bell's inequality. The generalization of two-photon interferometry to highintensity fields will be briefly discussed as an example of a 

situation in which no classical description exists even in the 

PAGE &AM[ WT FKMa 

289 
macroscopic limit. 

2. Which-Path Experimeats 

Wave-particle duality suggests that we cannot determine the 

path that a particle has taken through an interferometer without 

destroying the interference pattern. In most cases, it can be 

shown that the loss of coherence is actually due to the entanglement of the particle's wave function with a second particle or 

system located in one path or the other. No actual observation of 

the path taken is necessary in order to eliminate the interference 

pattern. An interesting feature of these which-path experiments is 

that it is often possible to restore the interference pattern using 

a "quantum eraserw1. 

An excellent example of a which-path experiment is shown in 

Figure 1. As suggested by scully2 et al. , a single atom is incident 

upon a beam splitter that divides its wave function along two 

separated paths. A microwave cavity is located in each path and is 

coupled to the atom in such a way that a low-energy microwave 

photon will be emitted into whichever cavity the atom passes 

through. 

Fig. 1. A which-path experiment suggested by Scully 

et al. (Ref. 2) in which a microwave cavity is located in 

each arm of an atomic interferometer. 

The interference pattern must be destroyed, since the path of 

the atom can be determined by detecting the location of the photon. 
It can be shown that the change in the center-of-mass wave function 

of the atom has no significant effect and that the coherence is 

destroyed by the entanglement of the atom with the photon. 

It is obvious that there can be no classical analogy for this 

kind of which-path experiment because an atom cannot be described 

by a wave in classical physics. But this begs the question of what 

is really responsible for the loss of coherence. 

In order to allow a comparison with classical physics, 

consider instead the situation shown in Figure 2 in which the roles 

of the atom and photon have been interchanged3. Now a single photon 

is incident upon a beam splitter and its wave function propagates 

along two separated paths, A thin chamber containing gas atoms is 

located in each path and it is assumed that the photon is inelastically scattered, producing a secondary photon of low energy. The 

initial photon propagates with somewhat reduced energy toward a 

beam splitter and a single-photon detector. Once again, it can be 

shown that the change in the photon" wave function is irrelevant 

as long as 6k6x <e R, where 6k is the change in wave number and 6x 

is the thickness of the two chambers. The advantage of this 

which-path experiment is that it does allow a classical analysis if 

a large number of photons are incident, which corresponds to a 

classical light wave, 

~hin Chamber I 

Fig, 2. A modified wmch-path experiment in which 

the roles of the atoms and photons have been interchanged 

to allow a comparison with classical physics. 

The quantum-mechanical calculation is straightforward and has 

been described in more detail elsewhere3. Consider an operator pt 
that creates a single photon in a short gaussian wave packet: 

Here 0, are complex coefficients and af creates a photon of 

frequency wi. In order to achieve the macroscopic limit af high 

intensities, the quantum state will be taken to be a coherent state 

of the form 

where a! is a complex number sufficiently large that the pulse 

contains a large number of photons. The interaction Hamiltsnian is 

given by 

Here the operators b and c annihilate photons in the two paths 

through the interferometer and eij, is a coefficient of no interest. 

The intensity at the detector can then be shown to be 

The last two terms are the only ones that depend on the relative 

phase, as reflected by the coefficients a and a/, and are proportional to the inner product of two states containing a photon in 

two different paths, which is zero. Thus the entanglement of the 

original photon with a secondary photon in one path or the other is 

responsible for destroying the interference pattern, as expected, 

It is interesting to note, however, that it is not possible, 

even in principle, to determine which path a photon has taken, 

since the quantum uncertainty in the energy and number of photons 

in the coherent state of eq. (2) makes it impossible to associate 

the detected photons with individual secondary photons, 

Any classical description of this experiment must be based on 

a nonlinear model, since a linear system cannot produce any change 

in the frequency of the light* With this in mind, consider a 

simple model consisting of three nonlinearly-coupled harmonic 

oscillators: 
*aauaa 

~4ea~sso-p #Tapour bur~oqs aqq ~ea~sse~a 5o~eue 30 -salon13 

*Z laaod ~erqaads 6q~suap (n)@ pau;eqqo ucoxj ayq 

oq ayq umquenb 'quauqeaxq 

sazpuopuer ayq aspqd so aqq ppTj UJ e xauueuc qeqq s~ osp xexyucys 

uea aq uaas qeqq aqq uo-cqour sy ayqo~ya pue =a-~qeqaypardun s~y~, Kzoqaacexq 30 ayq bu~ob-qno ppyj fx sy paqqo~d u~ arnb~d araqm 

oq "auaasa~on~3 se u~ ayq am-quenb *uoy~d~xasap aya aaeds-aseqd 

quapTauy Aauanbaxg se TTam se e qeqmaucos rapeoxq yead bu~puodsanoa 

quapyauy pTa-rg ST ufioqs u~ aznbg E pue seq e dreys yead qe ayq 

**m ay& buyq~nsax xamod pxqaads Aq~suap roj e A~quayaJ~jns asuaqu: 

,K~~e3~xantnu 103 ayq aspa uy qajqm aqq quapyouy pTaJj sey Karaanbax3 

u~ yaea qqed go axnbyd z pue aqq ahoqe qas 30 suo~qenba sem panTos 

31 seivl paucnsse qeyq e rzauyxuou ucaqsAs jo syyq pug sy paqeaol: 

san~b asrr oq reauyIuou sqaaj3a 30 s~yq *pu~y 

reauTTuou uorqaunj 30 aqq uo~q~redas 30 omq sa~a~qr~d pue A~~exnqara 

Keus aq qqxo~ 'bu~qou xo3 'a~duexa qeqq ayq. qucoTno3 aa30j SJ 

aqe3q aqq purx jo roTneqaq qeqq uea xnmo UJ Tla?oTssep esuraasAs 

asuodsax 30 ue moqe oq ue quapTauT maq 50 qq6-r~ qnq saop -snlTT 

Tapom ST qou papuaqu? 07 ap-fnozd a a-rqsTTear uog.d~xasap js ayq 

%qu~qsuoa pu~ (q)p squasa~dar ue Teunaqxa 6u~ny;rp -p~a?3 STY& 

6s aqq aa;eyq qq5y~ sueaq u? aamb-rd 'Z s pue It axe arqeqsncpe 

ayg; aa;sqq sa~auanbaxg l!rn '[rn pue % puoCgsa~z03 oq. ayq sa~ousnbaxj 
Figs 4, Chaotic phase-space tra jectory from the classical model, which randomizes the phase and destroys the 

interference pattern. 

In the limit of low drive intensities the classical model 

produces a coherent response with no fluorescence, At sufficiently 

high intensities, chaos produces fluorescence at two frequencies 

that are analogous to the secondary photon and forward-propagating 

photons in Figure 2, both of which have random phase, Intermediate 

intensities produce more complicated behavior, including partial 

coherence at rational fractions of the drive frequency. 

Thus the classical model gives loss of the interference 

pattern due to chaos in the macroscopic limit of high intensities, 

This suggests that there is at least a loose connection between 

quantum noise and classical chaos. It is important to note, 

however, that the classical model produces a random phase only for 

sufficiently high intensities, whereas a proper quantum-mechanical 

treatment eliminates the coherence for arbitrarily low intensities. 

In many systems of this kind it is possible to implement a 

B@quantllam erasergs to restore the interference pattern1. This can be 

accomplished by letting the entangled secondary systems propagate 
in time, measuring their state at some subsequent time, and 

selecting only those events for which the secondary systems were 

found to be in the same final state. For example, a quantum eraser 

can be implemented for the micro-maser cavity experiment shown in 

Figure 1 by connecting the two cavities with a small hole containing an atom and then selecting only those events for which the 

photon in one cavity or the other was absorbed by this atom. 

Surprisingly enough, it may be possible to perform a similar 

procedure in the classical model discussed above. Suppose we 

consider a subset sf the phase-space trajectories for which the 

other (non-detected) variables are the same in the two paths, i.e. 

where the primed and unprimed variables refer to the two different 

paths. In that case, it seems likely that 

If so, the out-going fields would be the same in the two paths and 

the coherence would be restored. 

3. Spontaneous Emission Noise 

The random phase associated with spontaneous emission of a 

photon by an atom is often attributed to vacuum fluctuations. Once 

again, this may seem to be inherently quantum-mechanical in nature. 

But returning to the example shown in Figure 2, it can be seen that 

both photons emitted by an atom in one path or the other of that 

interferometer are emitted by spontaneous emission. The classical 

model discussed above gave a random phase for both of these fields 

due to classical chaos in the limit of high field intensities, 

which is in qualitative agreement with the quantum-mechanical 

result. 

This further suggests that there may be some connection 

between quantum noise and classical chaos. It must be kept in 

mind, however, that the classical model cannot produce these kinds 

of results in the limit of low intensities. 

4. Nonlocal Effects 

The preceding discussion suggests that certain kinds of 

quantum phase noise may have a classical analogy in the form of 

chaos. This analogy can only be'taken so far, however, since the 

models used do not provide a realistic description of an atom and 

are qualitatively similar to the quantum-mechanical treatment only 

in the limit of high intensities. 

In addition, quantum systems can exhibit nonlocal effects that 

violate Bell's inequality and obviously have no classical analog. 

Such effects are not limited to low-intensity fields, as can be 
illustrated b considering the generalization of two-photon 

interferometr # to high-intensity fields as illustrated in Figure 

5. A somewhat similar situation involving photon polarizations has 

also been discussed by Reid and kgunro5, 

Fig. 5, Nonlocal interferometer consisting of two 

identical interferometers with a short path and a long 

path, capable of operation with high-intensity fields. 

Nonloeal interferometry with high-intensity fields has been 

discussed in detail elsewhere6 and only the main results will be 

reviewed here. Consider a quantum state of the electromagnetic 

field given by 

where 

Here ct creates a pair of entangled photons in two paths via photon 

creation operators at and bt, y is a normalization constant, cu is a 
large complex number, and the coefficients f, describe the effects 

of filters inserted into the two beams, 

Although eq. (8) resembles a coherent state, its properties 

are quite different. The probability P, of detecting a pair of 

coincident photons in the corresponding output ports of the two 

interferometers of Figure 5 can be shown to be given by 

This is the same result obtained previously for the weak-field case 

but here the field can be extremely intense and contain a large 

number of photons. 

The probability PN of detecting N pairs of coincident photons 

in the corresponding output ports of the two interferometers is 

Eq. (11) also violates Bell" inequality. The factor of N! is due 

to the different ways in which photons can pair with each other and 

greatly enhances the probability of detecting a large number of 

pairs. No single pnoton detectors are required to observe such 

events, which correspond to large bursts of energy in the corresponding interferometer ports and which could be observed, at least 

in principle, with a bolometer. These effects are truly macroscopic in nature in that sense. 

It is also possible to consider an EPR paradox involving 

quantum phase measurements performed on high-intensity fields with 

initially uncertain phases, Both classical and non-classical 

effects are obtained, as described elsewhere7. 

Several sources of quantum phase uncertainty have been 

considered in the limit of high field intensities where a comparison with a classical treatment is possible. It was found that 

classical analogies exist for the loss of coherence due to whichpath information as well as the quantum noise associated with 

spontaneous emission. In both of these cases classical chaos 

randomizes the phase in a manner that is at least qualitatively the 

same as in the quantum description. 

This suggests that there may be a loose connection between 

quantum noise and classical chaos. The classical treatment is only 

valid in the limit of high intensities, however, which is not too 

surprising in that classical physics would not be expected to 

provide an adequate description at the quantum level. In addition, 

violations of Bell's inequalities can also occur for high-intensity 

fields. Nevertheless, there does appear to be an analogy between 

quantum noise and classical chaos. 
This work was supported by the Office of Naval Research. 

1. M.O. Scully and K. Druhl, Phys. Rev. A 25, 2208 (1982). 

2, M.O. Scully, B.G. Englert, and H. Walther, Nature 351, 111 

(1991) ; M.0, Scully, Be6, Englert, and J. Schwinger, Phys. 

Rev. A 48, 1775 (1989). 

3, J.D. Franson, Phys. Rev. A 15, 8074 (1992). 

4* J.D, Franson, Phys. Rev. Lett. 62, 2205 (1989); Phys. Rev. 

Lett. 67, 290 (1991). 

5. M.D. Reid and W.J. Munro, Phys. Rev. Lett. 69, 997 (1992). 

6. J..D. Franson, to appear in Phys. Rev. A. 

7. J.D. Franson, submitted to Phys. Rev. A. 
HOW TO DETECT AN EXCITED ATOM 

without disturbing it 

OT 

HOW TO LOCATE A S'EPPE%i&-MINE 

without expbding it 

Lev Vaihm 

School of Physics and Asfronomy 

TeI-Aviv University, Tel-Aviv 69978 IS&! AEL 

Possible realistic implementations of a method for interaction-free measurements, due to a 

Elitzur and Vaidman, are.proposed and discussed. It is argued that the effect can be easily 

demonstrated in an optical laboratory. 

1 How to locate a Super-Mine 

A "Super-mine" is a device which explodes if anything touches it. Proton, electron, photon, 

neutrino,.. . anything that reaches the mine, with any energy, triggers an explosion. Our task is 

to locate the mine. We have to find out where it is, not where it was. We have to cheek that 

the mine is in a certain location without exploding it. We have no additional infomation, 

except that there is nothing else except mines in the region. We are allowed to fail in ow 

procedure, i.e. to explode a mine. In that case we can try again, in another region. But our 

measurement has to be reliable: we must not be mistaken when we say that there is a mine, 

The task seems to be impossible: the mine interacts with the external world only by explosion 

when it is "touched," so how it can be found without an explosion? In classical physics this task 

is certainly impossible, since its solution leads to a paradox: to find the mine you have to touch 

it, but if you touch it, it explodes. Nevertheless, quantum mechanics allows a simple solution, 

which was suggested recently by Elitzur and myself [I]. 

Our method is based on a particle interferometer analogous to the Mach-Zehnder interferometer used in classical optics. In principle, it can work with any type of particle. The 

particle reaches the first beam splitter which has transmission coefficient 4. The transmi$te$ 

and reflected parts of the particle's wave are then reflected by the mirrors and finally reunite 

at another, similar beam splitter (Fig. 1 a$ Two detectors collect the particles after they pass 

through the second beam splitter. We can arrange the positions of the beam splitters and the 

mirrors such that because of destructive interference, no particles are detected by one of the 

detectors, say D2, and all are detected by Dl. We position the interferometer in such a way that 

one of the routes of the particle passes through the region of space where we w&to detect the 

existence of a mine (Fig. 2b). We send a single particle through the system. There are three 
possible outcomes of this measurement: 

i) no detector clicks, ii) detector Dl clicks, iii) detector D2 clicks. 

;En. the first case, the particle explodes the The probability for this outcome is i. In the 

second case (for which the probability is $), the measurement does not succeed either. The 

paxticle could have reached Dl in both cases: when the mine is, and when the 

located in one of the anns of the interferometer. In this case there has been no interaction with 

the object, so we can try again. Finally, in the third case, when the detector D2 clicks (the 

probability for which is f ), we have achieved our goal: we know that there is a mine inside the 

interferometer without exploding it. If we wish to specify by the interaction-free procedure the 

exact position of the mine inside the interferometer, we can test (locally) that except for that 

region, the interferometer is empty. 

I 

PARTICLE 

i / 

------------------------- SOURCE 

/ / 

I 

PARnCLE 

i / 

SOURCE 

0 / 

FIG. 1. (a) If there is no any object inside the interferometer, D2 never clicks. 

(b). When DZ clicks after sending just one particle we know that the mine is inside 

the interferometer and it is still intact. 
What is the Probability for a Successful Experiment? 

Even the ideal experiment does not always succeed. We have seen that the probability for 

success here is only 1/4. But we also have the probability 1/4 not to destroy the mine wjlthout 

finding it. Trying again and again, until success or an explosion, leads to the probability 1/3 of 

locating the mine without an explosion. We have shown [I] that by modifying the traasdssion 

coefficients of the beam splitters in the interferometer we can obtain (host) the probability 

1/2 for success. 

This is, however, a gedanken experiment. A super-mine which is sensitive to evemhing, a 

Mach-Zehnder interferometer with complete destructive interference in one detect or, a single 

particle source - these are not devices that can be found in a standard laboratory. ket us now 

discuss a few points which are relevant for a realization of the idea in a real laboratory. 

We do not need a super-mine. The mine can be replaced by a fragile object which is 

nevertheless stable in the environment of the laboratory. It "explodes" (or clicks) if a photon 

with a certain energy hits it. Our task now is to detect the object without exploding it, using 

only such photons. 

We do not need a source of single photons. We assume that the explosion (the click) is loud 

enough for us to hear, so dl we need is a weak source of photons and fast switch that stops the 

beam when detector D2 clicks. (The single-photon source is necessary if we want to locate an 

object and he sure that it was not disturbed in any way whatsoever, even if the object does not 

click.) 

We do not need 100% efficiency in detector D2. Of course, low efficiency will reduce the 

probability of detecting the object, but if the detector clicks, we are still 100% sure that the 

object is there. 

If we have an ideal interferometer, then the photon which hits the object does not have to 

invariably explode it. It is enough for the photon to have just finite probability to be absorbed 

(to be scattered, to change its phase). Even the probability for the success remains unchmged, 

we only need to put more photons through the interferometer. 

If we have an ideal equal-path interferometer, we do not need a monochromatic source: 

complete destructive interference occurs for all wavelengths. 

Unfortunately, we do not have an ideal Mach-Zehnder interferometer. There is no possibility 

of obtaining 100% destructive interference in the detector D2. Sometimes we will get clicks even 

if there is no object inside the interferometer. Also, we will get wrong clicks due to noise in the 

detector D2. 

What I have understood from my interaction with experimentalists [2] is that in a modem 

laboratory one can obtain the ratio of 1: 100 for the number of clicks in the detectors D2 and Dl 

when there is no object inside the interferometer (instead of the theoretical 0 counts for detector 

Dz). This is the most import ant limitation on the proposed "interaction-free" measurement. The 

most optimistic estimate I heard [3] was the ratio of 1: 1000. Therefore, we have to complete (one 

run of) our experiment while much fewer than a 1000 photons pass through the interferometer. 
Consrder a boy trying to catch a girl in a dark room. Tn order to catch her, he has to how where 

she is. But if she knows that she has been located, she has enough time to move to mother 

location. She constantly looks in 4 direction, and any time she sees a photon, she moves. The 

girl can detect any photons the boy uses. Then, classically, the task of the boy is hopelless. 

However, quantum mechanics allows the boy to locate the girl without her being aware of, ad 

thus, to catch her. 

Let US now discuss a proposal for demonstration of such game in a laboratory. Instead of the 

girl we wiM take a high efficiency photo-detector. As I understand [4] there are detectom of up 

to 7'0% eEciency with noise of about one count per second. A student either puts or does not 

put this detector inside the interferometer. (Or, more realistically, she blocks or does not block 

the arm of the interferometer with a small mirror such that the reflected photons are absorbed 

by the photo-detector.) The output of the detector is connected to a bell. Ow task is to hd 

out if there is a detector without ringing the bell. We are allowed sometimes to fail, i.e. to ring 

the bell. Then we call the student to start everything fiom the beginning. But, when we claim 

that the detector is there, we must be correct with high probability. 

In order to achieve this goal we tune the Mach-Zehnder interferometer for maximal destructive interference in detector D2, see Fig. 2. We prepare a weak source of light (low intensity 

laser) such that it sends about lo3 photons per second. We add a fast electronic switch which 

stops the beam when the detector Dz absorbs a photon. It is easy to get a switch with time of 

operation r = lOW6sec. (Detector Dl does not play any role in the experiment, so we can omit 

it.) 

FIG. 2. A fast switch detector stops the beam when it detects a photon. We can 

learn about the existence of an object inside the interferometer by measuring the 

time it takes for the detector to click starting from the beginning of the run. 
The experiment runs as follows: we switch on the laser and measure after what time the 

detector switch stops the laser. If it happens after about a second we can safely claim that the 

detector is not there: the probability for the mistake is about 2-lm. El however, the time is 

about 10-3sec we can claim that the detector is there: the probability for a mistake now is ah 

not large, about If our device works properly, the only other probable outcome is that we 

will hear the bell first. If tse detector is there, the probability for this is about 2/3. IB this case 

we have to call the student to start again. But the other third is, roughly, the probability for 

a successful interaction-free measurement. It seems that all kind of noises which we have not 

taken into account cannot deny us a significant chance to perform this experiment successfully. 

If we are satisfied by a less reliable measurement (even 10% error is a sound experiment) we even 

do not need an extraordinarily precise Mach-Zehnder interferometer or ultrasensitive detectors. 

How to Detect an Excited Atom 

I believe that the proposed interaction-free measurement is more than just a demonstration 

of peculiarities of quantum mechanics. This is a measurement which can be performed on an 

infinitely fragile object without disturbing it in any way whatsoever. I believe that it can have 

practical applications. Now let us discuss one of the possible application: detecting an excited 

.atom without changing its state. 

Suppose we are going to investigate an exotic excited state which can be characterized by 

the ability to absorb a photon of certain energy. We want to know when an atom in such a state 

has appeared, but we do not want to change its state while detecting it. As far as we know, 

our method is the only one available. This is in zontrast with the toy experiment of previous 

section where we always could locate the detector using photons which it cannot detect. 

In order to detect an atom we can use exactly the same system, with laser and fast electronic 

switch on the detector (Fig. 2). But we encounter a serious problem: the cross section of 

absorption of a photon by an atom is much smaller than the cross section of the laser b 

Thus, many photons will come through the interferometer before one of them will be absorbed 

by the atom. Even more photons will pass before the click of detector D2 signaling the existence 

of the excited atom. When more than 1000 photons pass the interferometer, we, most probably, 

will get a click just from noise, and therefore we will not be able to detect the atom. I am not 

familiar enough with the experimental possibilities, but there is hope of £inding some focusing 

(squeezing ?) procedure to improve the ratio between the cross section for absorption and the 

cross section of the beam. 

I have more hope in finding some other experimental implementations of interaction-free 

measurements. First, the Mach-Zehnder can be replaced by a Michelson-Morley interferometer, 

or any other two- (or several-) arm interferometer. But it can also be implemented in a singlebeam interferometer with filters of polarization or some other degrees of freedom. Let me now 

state a general scheme for interaction-free measurements. 
5 Generalizatiora of Interaction-nee Measurement 

Our task is to detect a system in a certain state, say I*) . This state might cause some kind 

of explosion or destruction; destruction of a system, of a measuring device, or at least of the 

state I@) itself. The states orthogonal to I@) do not cause the destruction. Although the only 

physical effect of I @) is an explosion which destroys the state, we have to detect it without any 

distortion. If we succeed in this task, we cd the experiment an interaction-free measurement. 

Let us assume that if the system is in a state I@) and the measuring device is in a state la1), 

we have an explosion. For simplicity, we will assume that if the state of the system is ohhogond 

to I@) or the measuring device is in a state la2) (which is orthogonal to )a1)) than neither the 

system nor the measuring device changes their state: 

Now, let us start with an initial state of the measuring device 

If the initial state of the system is I*), then the measurement interaction is: 

where 1x1) = -/3*IG1) + alG2). If, instead, the initial state of the system is orthogonal to I@) , then the measurement interaction is: 

To complete our measuring procedure we perform a measurement of the measuring device wW 

distinguishes between Ix) and 1~~). Since there is no component with IxL) in the 

it can be obtained only if the initial state of the system was 19). This is also the 

the system: we do not obtain IxL) in the case of the explosion. The probabigty to obt& 1~~) 

(if the system was initially in the state I *) ) is laPI2. It is less than the probability for expII~on, 

which is laI2, but it is finite, and the ratio 1/312 can be made as close as we want to 1. h this 

case, the measurements will detect the state I*) with probability 1/2 (and with probability 112 

will be the explosion). 

A Mach-Zehnder interferometer (Fig. 3) is a particular implementation of this scheme. 

Indeed, the photon entering the interferometer can be considered a measuring device prepared 

by the first beam splitter in a state Ix) = -&(la1) + at time tl, where la1) designates a 

photon moving in the lower arm of the interferometer, and la2) designates a photon moving 

in the upper arm. Detector D2 together with the second beam splitter tests for the state 

IXL) = - 102)) at time t2. Indeed, if the state JxL) were measured at time t2, it must 

be found with certainty. 
FIG. 3. The photon passing through the interferometer and detected by D2 

can be considered as a measuring device prepared at time tl in the state Ix) = 

&(~QI) + I&)), and found at time t2 in the state 1~~) = -&(lQ1) - lQ2)). 

The difficulties of splitting and reuniting beams in the Mach-Zehnder interferometer can be 

avoided if our system is in a state 19) which is sensitive, say, to a left circular polarization of 

light: it causes some kind of explosion, while right polarization causes no change. Then we can 

start with an x-polarized photon which interacts with the system and look for a y-poldzed 

photon. If we do find such photon, we know that the system is in the state 19). 

Our method has remarkable property of not destroying i tely fragile states and it is 

applicable to a wide class of physical systems. Therefore, although now we do not know where 

it can have practical applications, we are optimistic about finding such applications in the futurce. 

6 Acknowledgements 

It is a pleasure to thank Daniel Rohrlich and Avshalom Elitzur for very helpful suggestions. 

The research was supported in part by grant 425/92-1 of the the Basic Research Fowdation 

(administered by the Israel Academy of Sciences and Humanities). 

References 

[I] A. Elitzur and L. Vaidman, Found. Phys. 23, 987 (1993). 

[2] L. Mandel, A. Zeilinger, private communication. 

[3] F. De Mwtini, private communication. 

[4] P. Kwiat , private communication. 
THE STOKES LINE WIDTH AND UNCERTAINTY RELATIONS 

A. 38. Nikishoh V. 38. fitus 

Eebedev Physical Institute, Moscow 11 7924 

For a function given by contour integral the two types (conventions) of asymptotic representations me considered: the usual representation by asymptotic series in inverse powers of 

large parameter and the special division of contou~' integral in contributions of high and low 

saddle points. It is shown that the width of the recessive term formation zone (Stokes strip) 

in the second convention is determined by uncertainty relation and is much less than the zone 

width in the first convention. The reasons of such a difference is clarified, The results of 

the work are useiul for understanding of formation region of the exponentially small process 

arising on the background of the strong one. 

Many physical quantities are represented by contour integrals depending on two (or more) real 

parmeters: 

C 

(1) 

z is one complex or a couple of two real parameters, v and a. The asymptotic representation of 

these integrals are considered when one of the parameters, v, tends to infinity where the integral 

has essential singularity, while another parameter9 a, is near the Stokes line [I] where a = 0. 

We restrict ourselves to the case when f (t, z) in (1) has only two saddle points tl, t2 where 

f1(t1,2, Z) = 0, and denote 

Then the asymptotic representation consists of two terms: 

The main (dominant) one - ef2 and the exponentially small relatively to it (recessive) one - igef17 

Re(f2 - fl) > 1 when v >> 1. 

Qualitative distinction of the two terms lies in the different rates of change of their phases 

Im fi,a(a) with a: 
Then from physical point of view D and R are the dominant and recessive waves with different 

frequencies w2 and wl . Another qualitative distinction is the appearance (or disappearance) of R 

when a crosses the Stokes line a = 0. This appearance takes place in a certain interval Acr which 

may be called the Stokes line width [2,3,4]. According to these authors the switching function of 

recessive term coincides with error-function 

where w = w(cr) is a certain odd fupction of a, depending on convention [4] about dominant 

Figure 1: 

term. The interval Aa defined by the condition 

Iw(CY)I 5 1 -+ acr 

may be called the Stokes line width. 

2 First convention about dominant and recessive terms 

The 1-st convention [2,4] is based on asymptotic series expansion. Consider it on the example of 

standard Airy function expansion [5] 

fie-( O0 2n 47r -- fie+' 2 ekC-*, - < argz < -, 22114 C o(-C)-* + I-- 

k=O 2z114 k=O 3 3 
Figure 2: 

Near the upper Stokes line r= 1 r 1 ei(2z/3+"), -C = f 1 z 13/2ei3a/2. According to the 1-st convention 

the dominant wave Sm is formed by asymptotic series trancated near its least tern, while the 

recessive wave R, is represented by the remainder: 

fie-< m-1 

=- C 4(-<)-k + 

2z114 k=, dt fm (t) [Ai(z)Ai(te-y)- 

fm(t) = (-l)m&mcm<-m-1/2e-<, m = m(z) - number of the least term. For z > 1 the 

m = m(z) > 1 and it is possible to find the asymptotic expression for &(z). The investigation 

shows that for asymptotic series whose terms behave with number k as 

r(ak + b)(~z)-~, 

the ratio 

where 
So the recessive wave is switched on when 151 2 1, or when the phase difference of dominant 

and recessive waves becomes large: 

3 Second convention and uncertainty relation 

The 2-nd convention [4] based on contour integral representation and dividing contour integral at 

the height of recessive saddle. The dominant and recessive terms of F(z) are nothing else than 

contributions of high and low saddles of the integrand. If t2 and tl are the high and low saddle 

points and z is near the Stokes line then the steepest decent lines going over the t2 and tl (§DL2 

and SDL1) on the complex t-plane are represented on the fig.3 together with the level line (LLI) 

of low saddle point tl. 

Figure 3: 

The point t,=t,(z) on the intersection of §DL2 and LL1 is a root of eqs.: 

Imf(t,z) = Imf2, 

Ref(t,z) = Refl. (13) 

The 2-nd convention defines the dominant D and recessive R terms of F = D + R as integrals 
Using in R the Taylor expansion for f (t, z) near saddle point tl 

we obtain 

where 

is a complex function of a. Hence, the switching function g(a) is complex and is given by a Fresnel 

type integral. The recessive wave switches on or off when (w( 2 1 or 

It is very natural condition: the phase difference of dominant and recessive waves is of the order 

of 1 or greater. 

Near the Stokes line the phase is linear function of a: 

Imfl,.~(a) = bf1,2(O) - wl,2(0) . a + . . . , 

and 

Im(fi - fi) = Au.a + .;. , Aw = ~~(0) - ul(0), (20) 

as on the Stokes line 1m f1(0)=1m fi(0). Two waves may be distinguished only outside of the 

Stokes line (Stokes strip) 

IIm(f1 -fi)l 2 1 or Aw-Aaz 1, (21) 

when uncertainty relation is fulfilled. That is why the Stokes line width 

may be called natural. 

In the first convention, where 

due to condition(l2) or 

the Stokes line width is much greater than natural 
aa N \/2~e(fi - fl) 1 

aw 

>> -. Aw 

The slower formation of wave & in comparison with R is caused by the fact that the last 

fi - -fi) 1- terms in Sm are coherent to Q, and the disentanglement occurs v 

slowly. This coherence disappears when cr goes out of the more wider interval Acr - @/Aw 

than natural om and then all recessive properties are concentrated in Q,. 

4 T-parity and asymptotic expansions 

T-transformation consists of the change a 4 -a and the complex conjugation. We consider here 

only the important case when functions fls in (2) satisfy the conditions 

. and dominant wave D goes into itself, D 4 D + 6, up to unimportant phase factor and exponentially small additional term 6. One can say conditionally that D has positive T-parity. At the 

same time 

and does not have definite T-parity inside the Stokes strip because 

g(-w) = 1 - g(w). 

Yet outside the Stokes strip, when Iwl >> 1, 

ig(w)efl w f1w2/(2w), Rew ) 1, 

iefi + ie.fl-w2 I), Rew< -1. 

Then before Stokes strip the R is 2J.lrlwl times less than its value Rs on the Stokes line, does 

not change at T-inversion and has the phase of dominant wave shifted by the arg(iw-I), as 

fi - w2=Refl+il[mf2. After the Stokes strip the wave R w 2Rs, has the proper phase Imfl + n/2, 

changes its sign at T-inversion and is accompanied by small additional term of the same type as 

R itself was before the Stokes strip. 

Therefore the Stokes strip is the forming region for recessive wave with frequency wl # wz and 

negative T-parity. 

As to behaviour of dominant and recessive waves at T-transformation in the representation 

F=Sm + & , then for the examples considered in [4] the Sm transforms into itself up to the same 

factor as for D but without any additional term 6,i.e. Sm + S,, while the R, behaves according 

to (29) with the change of w by real 5, see (11). Therefore the phase of forming wave R, equals 

to Imfl + 7r/2 ad its T-parity changes from positive to negative value being indefinite inside the 

Stokes strip. As D=Sm + Rm - R, the additional term 6 having indefinite T-parity inside the wide 

Stokes strip vanishes outside it as 6 ie-t2+fi/2&5. 
b Stokes line width and the method of osculating parameters 

It is instructive to see the appearance of Stokes width in the method of osculating pwmeters. 

According to this method the particular solution y(8) of the differential equation of tke second 

order with large parameter v is sought as a superposition of quasiclassical solutions * f(8) with 

the correcting coeEcient functions ~~(0) defined by the relations 

and boundary condition 

The latter means that the solution in question is +y(8). As a*($) are not differentiated in eq.(31) 

. the differential equation of the second order is reduced to the system of two differentid equations 

of the first order. This is sometimes useful for seeking out the appropriate approximation. 

In physical literature there is a tendency to treat the two terms on the r.h.s. of (30) as two 

waves with f frequencies for arbitrary 8 and not only for 8 + +m (see [6] and references therein). 

This is done on the ground that quasiclassical solutions * f conserve the sign of frequency and 

the factors a*($) should only correct the solutions. Yet this is true only in the case when + f(8), 

describing the strong wave, is taken with the accuracy up to the amplitude a- (8) of the weak wave 

a_(@) - f (8), which under considered condition is erponentially small, for example a,(0) - e-*V. 

To see this we note preliminarily that as follows from (31,32) 

We use now as an example the parabolic cylinder function y(~)=~~i,-l12(-e-'T~42fi8). The 

constant C is fixed by the condition a+(-m) = 1. The first terms of the asymptotic expansion of 

y(8) in power series in v-I can be obtained by Darwin method [7]. 

fir the n-th approximation we have 

Here Q are the real functions of 8, bounded for k 2 1 together with their derivatives and satisfying 

the relation 

ck(-8) = (-l)k~k(8). 

It follows from (3334) that a-(0) - fn(8) consists of positive- and negative-frequency terms which 

have the form 
As seen from here a- - f becomes approximately the negative-frequency wave only when the 

first term on the r.h.s. is much smaller than the second one: 

In notation of [8] 

1 cfn(8) = -2(-iv),+5 h3fl~-3n-2. (37) 

One can show that for 0 >> 1 the function ctn(8) % an2-2n8-2n-1 a, % 2"-'F(n+l), n >> 1. Then 

the condition (36) takes the form 

So for n N v >> 1 we have B2 >> 1. It is seen that with each successive step in approximation 

for * fn the width, in which positive- and negative-frequencies are not separated, shrinks quickly, 

but only at the step n N v the width approaches the barrier one - a physically reasonable result. 

References 

[I] G. 6. Stokes, Trans. Cambridge Philos. Soc., No 10, 106 (1864). 

[2] M.V. Berry, Proc. R. Soc. London, Ser. A, 422, 7 (1989), 427, 265 (1990), 434,465 (1991). 

[3] A. I. Nikishov, V.I. Ritus, in Sakharov Memorial Lectures in Physics. Proc. of the First Int. 

Sakharov Conf. on Physics, May21-31, 1991, Moscow, v.1, Nova Sci. Pub., New York. 

[4] A. I. Nikishov, V.I. Ritus, Teor. Mathem. Fiz., 92, 24 (1992). 

[5] R. B. Dingle, Asymptotic expansions: Their Derivation and Interpretation, Academic Press, 

New York (1973). 

(61 A. A. Grib, S. G. Mamaev, V. M. Mostepanenko, Vacuum Quantum Eflects in Strong Fields, 

Energoatomizdat, Moscow (1988). 

[7] M. Abramovitz, I. A. Stegun, Handbook of Mathematical Functions with Formules, Grafs, 

and Mathematical Tables, Dover (1964). 

[8] J. C. P. Miller,. in Tables of Weber Parabolic Cylinder Functions, Her Majesty's Stationary 

Office, London (1955). 
MAXIMUM PREDICTWE POWER AND THE 

SUPERPOSITION PRINCIPLE 

Johann Summharnrner 

Atominstitut der ~sterreichischen Universitiiten 

Schuttelstr. 115, A-1 020 Vienna, AUSTRIA 

Abstract 

In quantum physics the direct observables are probabilities of events. We ask, how observed probabilities must be combined to achieve what we call maximum predictive power. 

According to this concept the accuracy of a prediction must only depend on the number of 

runs whose data serve as input for the prediction. We transform each probability to an associated variable whose uncertainty interval depends only on the amount of data and strictly 

decreases with it. We find that for a probability which is a function of two other probabilities 

maximum predictive power is achieved when linearly summing their associated variables and 

transforming back to a probability. This recovers the quantum mechanical superposition 

principle. 

E Introduction 

Quantum theory is not yet understood as well as e.g. classical mechanics or special relativity. 

Classical mechanics coincides well with our intuition and so is rarely questioned. Special relativity 

runs counter to our immediate insight, but can. easily be derived by assuming constancy of the 

speed of light for every observer. And that assumption may be made plausible by epistemological 

arguments [I]. Quantum theory on the other hand demands two premises. First, it wants us to 

give up determinism for the sake of a probabilistic view. In fact, this seems unavoidable in a 

fundamental theory of prediction, because any communicable observation can be decomposed into 

a finite number of bits. So predictions therefrom always have limited accuracy, and probability 

enters naturally. More disturbing is the second premise: Quantum theory wants us to give up 

the sum rule of probabilities by requiring interference instead. However, the sum rule is deeply 

ingrained in our thought, because of its roots in counting and the definition of sets: Define sets 

with no common elements, then define the set which joins them all. The number of elements in 

this latter set is just the sum of the elements of the individual sets. When deriving the notion of 

probability from the relative frequency of events we are thus immediately led to the sum rule, such 

that any other rule appears inconceivable. And this may be the reason why we have difficulties 

accepting the quantum theoretical rule, where probabilities are summed by calculating the square 

of the sum of the complex square roots of the probabilities. In this situation two views are 

possible. We may either consider the quantum theoretical rule as a peculiarity of nature. Or, 

we may conjecture that the quantum theoretical rule has something to do with how we organize 

data from observations into quantities that are physically meaningful to us. We want to adopt the 
latter position. Therefore we seek to establish a grasp of the quantum theoretical rule with the 

general idea in mind that, given the probabilistic paradigm, there may exist an optimal strategy 

of prediction, quite independent of traditional physical concepts, but resting on what one can 

deduce from a given amount of information. We will formulate elements of such a strategy with 

the aim of achieving maximum predictive power. 

2 Representing Knowledge from Probabilistic Data 

Any investigative endeavour rests upon one natural assumption: More data from observations 

will lead to better knowledge of the situation at hand. Let us see whether this holds in quantum 

experiments. The data are relative frequencies of events. From these we deduce probabilities 

from which in turn we derive the magnitudes of physical quantities. As an example take an 

experiment with two detectors, where a click is registered in either the one or the other. (We 

exclude simultaneous clicks for the moment.) Were, only one probability is measurable, e.g. the 

probablity pl of a click in detector 1. After N runs we have nl counts in detector 1 and n2 counts 

in detector 2, with nl + n2 = N. The probability pl can thus be estimated as 

with the uncertainty interval [2] 

From pl the physical quantity x(pl) is derived. Tts uncertainty interval is 

The accuracy of x is given by the inverse of AX. With the above assumption we expect it to 

increase with each additional run, because we get additional data. Therefore, for any N, we 

expect 

Ax(N + 1) < Ax(N). (4) 

However, this inequality cannot be true for an arbitrary function x(pl). In general AX will 

fluctuate and only decrease on the average with increasing N. To see this take a theory A which 

relates physical quantity and probability by XA = p1. In an experiment of N = 100 runs and 

nl = 90 we get: AxA(lOO) = .030. By taking into account the data from one additional run, where 

detector 2 happened to click, we have AxA(lO1) = .031. The differences may appear marginal, 

but nevertheless the accuracy of our estimate for XA has decreased although we incorporated 

additional data. So our original assumption does not hold. This is worrisome as it implies that 

a prediction based on a measurement of XA may be more accurate if the data of the last run are 

not included. Let us contrast this to theory B, which connects physical quantity and probability 

by XB = p16. With N and nl as before we have ~~(100) = .106. Incorporation of the data from 

the additional run leads to ~~(101) = .104. Now we obviously don't question the vallie of the last 

run, as the accuracy of our estimate has increased. 
The lesson to be learnt from the two examples is that the specific functional dependence of 

a physical quantity on the probability (or several probabilities if it is derived from a variety of 

experiments) determines whether our knowledge about the physical quantity will increase with 

additional experimental data, and that this also applies to the accuracy of our predictions. This 

raises the question what quantities we should be interested in to make sure that we get to know 

them more accurately by doing more experiments. From a statistical point of view the answer is 

straightforward: choose variables whose uncertainty interval strictly decreases, and simply define 

them as physical. And from a physical point of view? Coming from classical physics we may 

have a problem, as concepts like mass, distance, angular momentum, energy, etc. are suggested 

as candidates for physical quantities. But when coming from the phenomenology of quantum 

physics, where all we ever get from nature is random clicks and count rates, a definition of physical 

quantities according to statistical criteria may seem more reasonable, simply because there is no 

other guideline as to which random variables should be considered physical. 

Pursuing this line of thought we want to express experimental results by random variables 

whose uncertainty interval strictly decreases with more data. When using them in predictions, 

which are also expressed by variables with this property, predictions should automatically become 

. more accurate with more data input. Now a few trials will show that there are many functions 

x(pl) whose uncertainty interval decreases with increasing N (eq.(3)). We want to choose the one 

with maximum predictive power. The meaning of this term becomes clear when realizing that in 

general AX depends on N and on a1 (via pl). These two numbers have a very different status. 

The ;umber of runs, N, is controlled by the experimenter, while the number of clicks, nl, is solely 

due to nature. Maximum predictive power then means to eliminate nature's influence on AX. For 

then we can know AX even before having done any experimental runs, simply upon deciding how 

many we will do. From eq.(3) we thus get 

which results in 

x = C arcsin(2pl - 1) + D 

where C and D are real constants. The inverse is 

showing that the probability is periodic in X. Aside from the linear transformations provided by 

C and D any other smooth function a(x) in real or complex spaces will also fulfill requirement 

(5) when equally sized intervals in x correspond to equal line lengths along the curve a(x). One 

particular curve is 

X .x a(x) = sin(-)e12, 2 (8) . which is a circle in the complex plane with center at i/2. It exhibits the property = pl known 

from quantum theory. But note, that for instance the function p = sin(x/2) does not fulfill the 

requirement that the accuracy only depend on N. Therefore the complex phase factor in eq.(8) 

is necessary [3] [4]. 
Distinguishability 

We have now found a unique transformation from a probability to another class of variables 

exemplified by x in eq.(6). These unique variables always become better known with additional 

data. But can they be considered physical3 We should first clarify what a physical variable is. A 

physical variable can assume different numerical values, where each value should not only imply 

a different physical situation, but should most of all lead to a different measurement result in a 

properly designed experiment. Within the probabilistic paradigm two measurement results are 

different when their uncertainty intervals don't overlap. This can be used to define a variable which 

counts the principally distinguishable results of the measurement of a probability. Comparison 

of that variable to our quantity x should tell us how much x must change from a given value 

before this can be noticed in an experiment. Following Wootters and Wheeler [5][6] the variable 

0 counting the statistically distinguishable results at detector 1 in N runs of our above example 

is given by 

~l(~l) dp n' 

e(nl) = - = Jij [arcsin(2p1 - 

AP(P> ') + 21 

where Ap is defined as in eq.(2). When dividing 0 by IV1I2 it becomes identical to x when in 

eq.(6) we set C = 1 and D = $. This illuminates the meaning of X: It is a continuous variable 

associated with a probability, with the particular property -that anywhere in its domain an interval 

of fixed width corresponds to an equal number of measurement results distinguishable in a given 

number of runs. With Occam's dictum of not introducing more entities than are necessary for 

the description of the subject matter under investigation, x would be the choice for representing 

physical situations and can rightly be called physical. 

4 A Simple Prediction: The Superposition Principle 

Now we return to our aim of finding a strategy for maximum predictive power. We want to see 

whether the unique class of variables represented by x indicates a way beyond representing data 

and perhaps affords special predictions. For the sake of concreteness we think of the double slit 

experiment. A particle can reach the detector by two different routes. We measure the probabilty 

that it hits the detector via the left route, p~, by blocking the right slit. In L runs we get nL 

counts. In the measurement of the probability with only the right path available, p~, we get n~ 

counts in R runs. From these data we want to make a prediction about the probability ptot, when 

both paths are open. Therefore we make the hypotheses that ptot is a function of p~ and p~. What 

can we say about the function ptot(pL,pR) when we demand maximum predictive power from it? 

This question is answered by reformulating the problem in terms of the associated variables XL, 

XR and xt,t, which we derive according to eq.(6) by setting C = 1 and D = ;. The function 

xtot(xL, xR) must be such that a prediction for xtOt has an uncertainty interval SxtOt, which only 

depends on the number of runs, L and R, and decreases with both of them. (We use the symbol 

Sxtot to indicate that it is not derived from a measurement of ptot, but from other measurements 

from which we want to predict ptot.) In this way we can predict the accuracy of xtOt by only 

deciding the number df runs, L and R. No actual measurements need to have been done. Because 
maximum predictive power is achieved when 

We want to have a real function xtot(xL, xR), and therefore we get 

where a, b and c are real constants. Furthermore we must have c = 0 and the magnitude of both 

a and b equal to 1, when we wish to have xt0t equivalent to XR or to XL when either the one or 

the other path is blocked. So there is an ambiguity of sign with a and b. When rewriting this in 

terms of the probability we get - 

2 XL~ XR ptot = sin ( 2 ). 

This does not look like the sum rule of probability theory. Only for p~ + p~ = 1 does it coincide 

with it. We may therefore conclude that the sum rule of probability theory does not afford maximum predictive power. But neither does eq. (1 3) look like the quantum mechanical superposition 

principle. However, this should not be surprising because our input were just two real valued 

numbers, XL and XR, from which we demanded to derive another real valued number. A general 

phase as is provided in quantum theory could thus not be incorporated. But let us see what we 

get with complex representatives of the associatea variables of probabilities. We take a(x) from 

eq. (8). Again we define in an equivalent manner a~, a~ and atot. From p~ we have for instance 

(from (8) and (7) with C = 1 and D = f) 

and 

If we postulate a relationship atot(aR, aL) according to maximum predictive power we expect 

the predicted uncertainty interval Satot to be independent of a~ and CYR and to decrease with 

increasing number of runs, L and R. Analogous to (11) we must have 

yielding 

Qtot = SQL + ta~ + U, 

where s, t, and u are complex constants. Now u must vanish and s and t must both be unimodular 

when ptot is to be equivalent to either p~ or p~ when the one or the other route is blocked. We 

then obtain 

ptot = latd12 = 1s~~ + ta~1~ = PL + PR + 2.\/= COS(~), (I8) 
where 4 is an arbitrary phase factor containing the phases of s and t. This is exactly the quantun 

mechanical superposition principle. What is striking is that with a theory of maximum predictive 

power we can obtain the general form of this principle, but cannot at all predict prOt even when 

we have measured p~ and p~, because of the unknown phase 4. So we are lead to postulate 4 as 

a new measurable quantity in this experiment. 

We have tried to obtain insight into the quantum mechanical superposition principle and set 

out with the idea that it might follow from a most natural assumption of experimental science: 

more data should provide a more accurate representation of the matter under investigation and 

afford more accurate predictions. From this we defined the concept of maximum predictive power 

which demands laws to be such that the uncertainty of a prediction is solely dependent on the 

number of experiments on which the prediction is based, and not on the specific outcomes of these 

experiments. Applying this to the observation of two probabilities and to possible predictions 

about a third probability therefrom, we arrived at the quantum mechanical superposition principle. 

Our result suggests nature's law to be such that from more observations more accurate predictions 

must be derivable. 

6 Acknowledgments 

1 thank the Austrian Science Foundation (FFW) for financial support of ion double slit experiments 

(Project P8781-PHV) whose analysis led to this paper. 

References 

[I] 6. J. Whitrow, The Natural Philosophy of Time, 2nd Ed., (Clarendon Press, Oxford, 1984). 

[2] The uncertainty interval is derived from Chebyshev's inequality, e.g. William Feller, An Introduction to Probability Theory and its Applications, 3rd Ed., (Wiley and Sons, New York, 

1968), p.233. For reasons of simplicity we are using only the approximate form valid for large 

N. 

[3] More details and variables that can be used to represent several probabilities can be found 

in: J. Summhammer, Found. Phys. Lett. 1, 113 (1988). 

[4] The statistical properties of x are analyzed in: J. Summhammer, Phys. Lett. A136,'183 

(1989). 

[5] W. K. Wootters, Phys. Rev. D23, 357 (1981). 

[6] J. A. Wheeler, Int. J. Theor. Phys. 21, 557 (1982). 
CORRELATED QUADRATURES OF RESONANCE FLUORESCENCE 

AND THE GENEMLIZED 'OUNCERTAmTY RELATION 

Henk F.holdus 

Deparment of Physics, Mendel Hall, Villanova University 

Villanova, Pennsylvania 19085 

Thomas F. George 

Deparments of Chemistry and Physics, Washington State University 

Pullman, Washington 991964-1046 

Rolf W. F. Gross 

Aerospace Corporation, P.O.'Box 92957, ',s Angeles, California 90009-2957 

Resonance fluorescence from a two-state atom has been predicted to exhibit quadrature 

squeezing below the Heisenberg uncertainty limit, provided that the optical parameters (Rabi frequency, 

detuning, laser linewidth, etc.) are chosen carefully. When the correlation between two quadratures of 

the radiation field does not vanish, however, the Heisenberg limit for quantum fluctuations might be an 

unrealistic lower bound. A generalized uncertainty relation, due to SchrMiger, takes into account the 

possible correlation between the quadrature components of the radiation, and it suggests a modified 

definition of squeezing. We show that the coherence between the two levels of a laser-driven atom is 

responsible for the correlation between the quadrature components of the emitted fluorescence, and that 

nger uncertainty limit increases monotonically with the coherence. On the other hand, the 

fluctuations in the quadrature field diminish with an increasing coherence, and can disappear completely 

when the coherence reaches 112, provided that certain phase relations hold. 

We consider a two-state atom, with excited state le>, ground state Ig>, and level 

separation oo, illuminated by an intense laser. The electric field at the position of the 

atom is assumed to have the form 

gL (t) = E. Re EL exp[-i(aLt + ~(t)] , (1.1) 

where $(t) is a stochastic phase which takes into account the laser linewidth. We shall 

assume that this is a diffusion process with independent increments. The driven atom 
will emit resonance fluorescence, and the positive-frequency part of the electric field of 

this radiation is 

with dt(t) = lg >< el the atomic lowering operator, and y an overall constant. The 

fluctuations in the amplitude (or phase) are measured by a homodyne detector with the 

~hg laser as local oscillator. The slowly-varying operator under measurement is 

therefore1 

with 8 the adjustable mixing angle of the detector. Essentially, the detector measures the 

variance of this quadrature operator, denoted by var(Ee). 

The fluctuations in Ee(t) will be expressed in terms of the parameter re, defined 

as 

Clearly, this parameter lies in the range 0 I re I 1. For re = 0 we have var(EO) = 0, 

which corresponds to no fluctuations at all, and for re = 1 we have < E8 > = 0. This case 

corresponds to a pure random phase of the field (like in a number state). 

For Heisenberg's uncertainty relation we compare the quadrature field Ee with the 

field Eel, which is the same field except with a different mixing angle. Then the 

uncertainty relation can be written as 

with LH the Heisenberg lower limit, given by 
If we ksea the exp~ssion for the quadrature field in LH, it follows that the dependence 

on the ' g angles 8 and e' has the form 

and therefore we take 8' = 8 + nx with n as an integer. Then Eo is said to be squeezed if 

in view of (2.1). 

The fluctuation parameter re can be expressed in tems of matrix elements of the 

ato~c density operator B (either in the rotating frame or not). We find 

wi& the mgk 6 equal to 

The Heisenberg lower limit can be whitten as 

in tems of the populations ne and ng of the excited state and ground state, respectively. 

From (2.5) it then follows that the fluctuations in Ee are minimum if the Hlixing angle 8 

is chosen such that 6 becomes an integer multiple of x. This corresponds to maximum 

squeezing, given o, because Lw dms not depend on 8. For 6 = n.n the condition for 

squeezing becomes 

Wether squeezing occurs or not depends therefore on the populations of the atomic 

levels and the coherence between the levels. 
men the atom has spent a sufficient mount of he in the laser field, its density 

operator wiU reach a steady state. The Rabi frequency of the transition is defined as 

dth fi the &pole moment operator of the atom, and the laser linewidth h will be 

parme~zed (hrough the combination 

where A is the Einstein coefficient for spontaneous decay. The detuning between the 

laser and the atomic resonance is 

In terms of these parameters, the absolute value of the coherence becomes2 

and the difference in level population is 

which equals the Heisenberg lower limit. The condition for squeezing, for 6 = 0, then 

becomes 

as in Ref. (2). It is easy to see that for 4 > 1, which is h > A/2, squeezing never occurs. 

The fluctuation parameter r is, for 6 = 0, 
For a given deru~ng A and laser'linewidth A,, the fluctuations are ~nimum if we t&e the 

Rabi frequency, which is proportional to the laser power, equal to 

,'- 

Then r becomes 

which is independent of A. This minimizes for h = 0, which gives r = 1/2 and 

a2 = 2(~~ + f12 14). 

A convenient parametrization follows by introducing the new variable 

dimensionless and proportional to the laser power. Then r and can be written as 

and the condition for squeezing becomes 

The dependence of r and LH on 5 is shown in Fig. 1 for 8 = h = 0. The difference s = r - 

LH is also shown, and a negative value of s corresponds to squeezing in the quadrature 

field below the Heisenberg uncertainty limit. Squeezing occurs for 6 e 2, and the 

maximum squeezing appears for 6 = 213, which gives s = -118. Notice that the minimum 

in r is located at 6 = 2, corresponding to s = 0. Hence, the best relative squeezing 

(minimum s) does not coincide with the smallest relative fluctuations in the field 

(minimum r). 
Fig. I. Plot ofparmeters r, LH, ad s as rawtion of 6. 

Heisenberg's uncertainty relation (2.1) sets a lower bound on the product of two 

variances. It is well known, however, that this is not the sharpest lower bound. A 

different uncertainty relation, due to SchrMinger, is3 

The Schriidinger lower bound is related to the Heisenberg lower bound by 

where the correlation coefficient c is defined as 
f <E&~+E,E~>-cE~>~E,> 

c = 

Jvw(~8) vw(~,) 

It can be v&id that c lies in the range 

which then gives 

This shows that the possible higher bound in Schriidinger's relation is due to the 

correlation between quadratures of the field with different values of 8. We shall always 

take '8' = 8 + n / 2, as before. 

For resonance fluorescence the correlation coefficient can be expressed in terms 

of the matrix elements of the atomic density operator. We obtain 

showing that c is determined by the coherence between the levels only, and not by the 

populations of the atomic states. The relation between the Schriidinger limit and the 

Heisenberg limit then becomes 

When 6 is an integer multiple of x12 or when the coherence is zero, we have LS = LH. 

When 6 is not an integer multiple of n/2, the Schriidinger limit can become arbitrarily 

large when the coherence approaches 112 (any coherence, in absolute value, is smaller 

than 112 in a two-level system). Figure 2 shows the ratio LS/LH for 6 = x14, and as a 

function of the atomic coherence. 

In terms of the parameters 6 and q the relation (4.7) becomes 
The laser power that minimizes s for q = 112 and 6 = 0 is 6 = 213. Then the minimum 

value of s is -118 and the mininnw of r is 5/8. Figuse 3 illustrates the behavior of r, LH, 

and LS as a fuwajIoern of d for hs vdues of d q. 

Fig. 2. Ratio LJLH for 6 = Id4 as ahmion of the coherence. 

Fig. 3. Plot of the Heisenberg- and Schrodinger limits and the fluctuation parameter r a,~ 

a function of 6. 
We have studid the possibilities for squeezing in the qu&ame components of 

=son- fluorescence from a two level atm. It was shown that the coherence betwwn 

the two levels gives rise to a correlation ktwwn quadrame fields with a different 

ing angle (in homodyne detection). This implies that the uncertainty limit on 

quantum fluctuations which is set by Schrijdinger's relation can be considerably higher 

than the comsponding lipnit in Heisenberg's relation. It appears that in the steady state 

both Mts are very close, as illusbrat& in Fig. 3. P;or pulsed-laser excitation, however, 

the coherence can approach its limiting value of 112, and this would increase the 

Sc ger limit &matically. Then the Heisenberg lower bound is an unrealistic lower 

limit, and squeezing should be defined with respect to the SchrMnger uncertainty 

relation. 

supported in part by the National Science Foundation under Grant No. 

1. M. J. Collett, D. IF. Walls and P. Zoller, Opt. Commun. 52, 145 (1984). 

2. H. F. Arnoldus and G. Nienhuis, Optica Acta 30, 15'73 (1983). 

3. V. V. Dodonov, E. V. Kurmyshev and V. I. Man'ko, Phys. Lett. 79A, 150 (1980). 
MINIMUM UNCERTAINTY AND SQUEEZING HN DIFFUSION 

PROCESSES AND STOCHASTIC QBTANTHZATION 

S. De Martino, S. De Siena, F. Illuminati, and G. Vitiello 

Diprtirnento di Fisica, Universitd di Salerno, and 

INFN Sezione di Napoli, 84 081 Baronissi (§a1 erno), It alia 

We show that uncertainty relations, as well as minimum uncertainty coherent and squeezed 

states, are structural properties for diffusion processes. Through Nelson stochastic quantization we derive the stochastic image of the quantum mechanical coherent and squeezed 

states. 

It is well known that the theory of stochastic processes is a powerful tool in the study of the 

interplay between probabilistic and deterministic evolution [I]. In quantum mechanics, and in 

particular in quantum optics, such interplay is expressed by the states of minimum uncertainty? 

the coherent [2] and squeezed states [3], which are viewed as the "most classical" states. 

In this paper we report on a recent derivation [4] of uncertainty relations for classical stochmtic processes of the diffusion type, and we determine the diffusion processes of minimum uncertainty (MUDPs). We find that a special class among them is associated to Gaussian probability 

distributions with time-conserved covariance and mean value with classical time evolution: we 

refer to them as strictly coherent MUDPs. We will also identify Gaussian MUDPs with timedependent covariance and conserved expectation value: we refer to them as broadly coherent 

MUDPs. By exploiting Nelson's stochastic quantization scheme [5], we will show that the 

strictly coherent MUDPs provide the stochastic image of the standard quantum mechanical 

coherent and squeezed coherent states, while the broadly coherent MUDPs are associated with 

the phenomenon of t ime-dependent squeezing. 

Our study is motivated by the possibility that the formalism of stochastic processes offers to 

treat on the same footing, in a unified mathematical language, the interplay between fluctuations 

of different nature, for instance quantum and thermal [6]. 

Beyond the case of diffusion processes, it is interesting to note that coherence and squeezing 

have recently emerged in other contexts wider than quantum mechanics ([a], [8]). 
2 Uncertainty and Coherence in Diffusion Processes 

In what follows9 without lack of generality, we will consider a one-dimensional random vmiablle 

go The associated diffusion process q(t) obeys 

where v(+) (q(t), t), is the fomard drift, v(q(t), t) is the diffusion coefficient, and dw(t) is a 

Gaussian white noise, superimposed on the otherwise deterministic evolution, with expectation 

E(dw(t)) = O and covariance E(dw2(t)) = 2dta The forward and the backward drifts v(+)(x,t) 

and v(-) (x, t) are defined as 

v(+)(z,t) = lim E 

At-+O+ At 

The definitions of v(+) and pr(-) are not independent, but related by [5] 

It is now convenient to define the osmotic velocity u(x, t) and the current velocity v(x, t) 

From the former definitions it is clear that u(x, t) "measures" the non-differentiability of the 

random trajectories, controlling the degree of stochasticity. In the deterministic limit u vanishes, 

and v(x, t) goes to the classical velocity v(t). 

Finally, we have the continuity equation 

It is straightforward to check that E(v(+)) = E(v(-1) = E(v), and E(u) = 0. Further, 

d E(v)=&E(q) Vt. 

For the product qu, we have IE(qu(q, t)) 1 = E(u(q, t)). By Schwartz's inequality, the r.m.s. 

deviations Aq and Au satisfy 

AqAu L E(v(q, t)). (9) 
Inequality (7) is the uncertainty relation for any diffusion process. Equality in (7) defines 

the MUDPs. Saturation of Schwartz's inequality yields u(x, t) = C(t)(x - E(q)), where C(t) is 

an arbitrary function of time. Considering constant v and time-dependent u, in both cases we 

obtain a Gaussian minimum uncertainty density: 

where 2(Aq)2 = -v(t)/C(t). 

From eq. (5) we can determine the current velocity: 

where 

,Eqs. (8)-(10) lead to the stochastic differential equation obeyed by any MUDP: 

It is interesting to observe that (11) defines the so-called linear processes in narrow sense. 

When A(t) = 0 they are the time-dependent Ornstein-Uhlenbeck processes. These last ones 

play a natural role in the theory of low noise systems [I], which are thus found to be related 

with MUDPs. 

The possible choices of E(q) and Aq in (8) are not independent: taking the expectation 

value of in (9)-(lo), and reminding (6) one has that either 

where j(t) and k(t) are arbitrary functions of time, and we have chosen for simplicity q(t = 

0) = 0. Consider first case (12): Aq does not spread; also, it is immediate to verify that the 

expectation value-of the process E(q) follows a classical trajectory: 

As a consequence, MUDPs of the form (8) obeying (12) and (14) are coherent in a sense 

precisely analogous to that of quantum mechanical coherent states: we will refer to them as 

strictly coherent MUDPs and to processes (8) obeying (13) as broadly coherent MUDP~. 
It is possible to discriminate on physical grounds the strictly coherent MUDPs from the 

broadly coherent ones by observing that (12) and (14) come as immediate consequence on 

imposing the Ehrenfest condition 

so that the strictly coherent MUDPs can be viewed as the most deterministic semi-classical 

processes. 

Consider the scale transformation z + e-8x which authomatically implies u -, esu, where s is 

the scale parameter. The Gaussian distribution (8) is form-invariant under this transformation, 

while the uncertainty product (7) is strictly invariant with Aq + e'"Aq and Au -+ esAu. We 

will show next that in the framework of Nelson stochastic quantization this transformation is 

just the squeezing transformation of quantum mechanics. In this context, broadly coherent 

MUDPs are of special interest when considering time-dependent squeezing. 

. 3 Nelson Diffusions 

A very important class of diffusion processes (Nelson diffusions) in physics has been introduced 

by Nelson in his stochastic formulation of quantum mechanics 151. 

Togeach singkparticle quantum state P = exp [R + is], Nelson stochastic quantization 

associates the diffusion process q(t) with 

where m is the mass of the particle. At the dynamical level, the Schroedinger equation with 

potential V (x, t) is equivalent to the Hamilton-Jacobi-Madelung equation 

It is well known [9] that for Nelson diffusions the uncertainties Aq and Au are related to the 

quantum mechanical uncertainties A8 and Afi of the position and momentum operators 8 and 

a by 

A8 = Aq , (A$)' = ~[(Au)' + (AV)~] , 

Minimum uncertainty Nelson diffusions (MUNDs) are MUDPs. Correspondingly, we will 

speak of strictly and broadly coherent MUNDs. By solving (17) for MUNDs we obtain V(x, t) 

and the classical equations of motion for E (9). For strictly coherent MUNDs (12) we have 
When the arbitrary constants f (t) and Vo(t) vanish, eqs. (19) are those of the classical 

harmonic oscillator and the associated quantum states are the standard Glauber coherent states; 

when f (t) = const'. we have the Klauder-Sudarshan displaced oscillator coherent states; finally, 

when f (t) is truly time-dependent, we obtain the Klauder-Sudarshan driven oscillator coherent 

states [lo]. 

For broadly coherent MUNDs we have instead 

where g(t) = (Aq)-ldAq/dt must be such that w2(t) is positive. Eq. (20) describes the parametric oscillator potential, associated to the feature of time-dependent squeezing. 

Furthermore, we can identify among MUNDs those corresponding either to Heisenberg or to 

Schroedinger minimum uncertainty. The key relation, easy to prove, is 

where < (a, .} >+ denotes the expectation of the anti-commutator in the state Q, i.e. the 

Schroedinger part of the quantum mechanical uncertainty. 

Eqs. (18) and (21) show that the strictly coherent MUNDs (19) exhaust the Heisenberg 

minimum uncertainty states, while the broadly coherent MUNDs (20) form a subset of the 

Schroedinger minimum uncertainty states. 

Finally, we investigate the possibility of letting v be time-dependent in the context of quantum mechanics. From the first of equations (16) this means letting either m or ti be functions 

of time. 

This latter case seems a bit speculative at this stage. We thus fix our attention on the case 

of time-dependent mass m(t) and constant h. 

For such systems it can be immediately verified that the Nelson scheme (16)-(17) still holds 

with m(t) replacing m. Considering the most interesting case of strictly coherent MUNDs, 

which means choosing C (t) oc u(t), and solving (17) we obtain 
where f(t), Vo(t) are arbitrary functions of time. Eqs. (22) supplemented with m(t) = rnoer(') 

define the dynamics of the damped parametric oscillator. The stochastic approach thus sheds 

new light in a unified treatment on the study of quantum dissipative oscillators, for it allows to 

derive for the expectation value the dynamical equation (22) that was so far unknown. 

In conclusion, we have shown that the quantum mechanical concepts of uncertainty, coherence, and squeezing can be imported in the probabilistic arena of diffusion processes. This 

appears to be possible because of a subtle interplay between fluctuations, control, and optimization. Conversely, we may also say that these features of quantum mechanics can be traced back 

and related to general properties of diffusion processes. 

Work on this subject is in progress, and includes application of our scheme to polymer 

dynamics and chemical reactions, uncertainty relations in field theory and dynamical systems 

on lattices and manifolds. 

References 

[I] C. W. Gardiner, Handbook of Stochastic Methods, (Springer, Berlin, 1985). 

[2] J. R. Klauder and B. S. Skagerstam, Coherent States, (World Scientific, Singapore, 1985). 

131 D. Stoler, Phys. Rev. Dl, 3217 (1970); H. P. Yuen, Phys. Rev. A13, 2226 (1976). 

[4] S. De Martino, S. De Siena, F. Illuminati, and G. Vitiello, Digusion Processes and Coherent 

States, Preprint, University of Salerno, June 1993. 

[5] E. Nelson Dynamical Theories of Brownian Motion (Princeton University Press, Princeton, 

1967); Quantum Fluctuations, (Princeton University Press, Princeton, 1985); F. Guerra, 

Phys. Rep. 77,263 (1981); G. Parisi, Statistical Field Theory, (Addison-Wesley, New York, 

1988). 

[6] P. Ruggiero and M. Zannetti, Phys. Rev. Lett. 48, 963 (1982). 

[7] E. C. G. Sudarshan, in these Proceedings. 

[8] E. Celeghini, M. Rasetti, and G. Vitiello, Phys. Rev. Lett. 66, 2056 (1991); E. Celeghini, 

S. De Martino, S. De Siena, M. Rasetti, and G. Vitiello, in these Proceedings. 

[9] D. De Falco, S. De Martino, and S. De Siena, Phys. Rev. Lett. 49, 181 (1982); S. De 

Martino and S. De Siena, Nuovo Cimento B79, 175 (1984). 

[lo] J. R. Klauder and E. C. G. Sudarshan, Fundamentals of Quantum Optics (Benjamin, New 

York, 1970). 
SECTION 6 

FIELD TaEORY GENERAL INTEREST 
APPLICATIONS 0l? SQUEEZED STATES: BOGOLIUBOV 

TRANSFORMATIONS AND WAVELETS TO THE 

STATISTIGAG MECHANICS OF WATER AND ITS BUBBLES 

B. DeFacio, S.-H. Kim and A. Van Nevel 

Department of Physics & Astronomy 

Missouri University 

Columbia, MO 65211 

ABSTRACT 

The squeezed states or Bogoliubov transformations and wavelets are applied to two problems in non-relativistic statistical mechanics: the dielectric response of liquid water, e(?, w), 

and the bubble formation in water during insonnification. The wavelets are special phasespace windows which cover the domain and range of L1 n L2 of classical causal, finite energy 

solutions. The multi-resolution of discrete wavelets in phase space gives a decompostion 

into regions of time and scales of frequency thereby allowing the renormalization group 

to be applied to new systems in addition to the tired 'usual suspects' of the lsing models 

' and lattice gasses. The Bogoliubov transformation: squeeze transformation is applied to 

the dipolaron collective mode in water and to the gas produced by the explosive cavitation 

process in bubble formation. 

1. INTRODUCTION 

Water is extremely important in chemistry and biology both as a solvent and as a neutral medium which is rich in pos~ibilities.~-'~ The dielectric response of a medium, e(q', w) 

is related to the index of refraction as, n(q', w) = JETi, whose real part controls wave propagation and whose imaginary part gives the attenuation. An August 1993 dissertation at 

Missouri University by one of us, S.-H. Kim19, formulates a new model which is rather 

successful in accounting for the complex index of refraction n(w) + ia(w), over 14 to 15 

decades of frequency. This model is a number-conserving relaxation-time,7 Boltzmann 

transport equation model in phase space596t9 based on Kubo's fluctuation-dissipation theorem l2 and a Kerr approximation.11 The free rotation gives an IR peak at the correct 

frequency which is much larger than the one found experimentally by Simpson et al.15 A 

phase-space analysis of our model based on wavelets will be presented here. 

The problem of bubble formation in liquids is both old and diffi~ult.~O-~~ It isnow 

known that two different mechanisms of bubble formation exist: cavitation, where the 

growth is explosive and nucleation where diffusion of molecules yields a slow growth. 

This work will be restricted to cavitation. All bubbles are interesting for electromagnetic 

wave propagation in water because they give additional spatial dispersion $dependence to i the index of refraction of the medium through their Mie scattering. 

The approach taken here will be to form a mean-field theory of a a4 theory in 4 spaceI time dimensions. It is a variant of an idea due to Kau~,~~ Wilhelm~son~~ andG1inu-11~~ t that bubbles are non-linear, coherent effects in fluids. Kaup's clever title is, "Cavitons 

are solitons" for a plasma driven by both sound and electromagnetism. Our equations are 

I completely different and have solitary wave solutions rather than solitons. These solitary 

waves will have finite lifetimes in contrast to the infinite lifetimes of Kaup's cavitons, which 
he first showed were an integrable system. However, since both models are subject to other 

perturbations, this difference is not actually as serious as it first appears.29 

There is also a practical reason for trying to understand bubble formation. Apfe130j3', 

Nyborg32, Rose and G01dberg~~ have shown that ultrasound at the frequencies used for 

medical diagnostics forms bubbles in water and tissue. They have raised questions about 

the safety of ultrasound but have not yet shown any evidence that these bubbles (or any 

other effect of ultrasound) are harmful. Tre~ena~~ has suggested that bubble collapse may 

occur through shock waves, which could damage cells. 

Both of these problems will be analyzed using wavelet^^^-^' which are a special class of 

windows 

(i) which cover the domain R1 or R2 in refs. (36-49) and the Lorentzian manifold of 

special relativity in ref (50,51); and 

(ii) are dense in the range of L2 n L1. 

The first space L2 is required classically for finite energy signals and for a probability interpret ation of Hilbert space states in non-relativistic quantum mechanics, and the 

space L1 is required classically for causality and is a major technical requirement in nonrelativistic quantum mechanics which provides absolute continuity. Clearly, (i) assures the 

existence of measurability of the underlying space and (ii) places additional restrictions on 

the solutions for the correct dynamics and symmetries. In particular, the symmetry group 

is the set of dilations. In the continuous one-dimensional case dilations are generated by 

the transformation D 

on the coordinates x and solutions $ E L2 n L1, where a E R!,. and b E R1 . In the discrete 

case (k, j) E (the integers) with a = 2 (one choice, but not the only one) 

In the discrete case, L2 -+ C2 and L1 + C1 and qkj "live in" C2 n C1. The choice of a = 2 

gives the Lebeque covering of R1 and extends to Rn and thus any other integer with this 

property will suffice. To satisfy (2) the conditions (2a) completeness 

(2b) mean zero for $, mean one for 4 
where the multiresolution decompostion of Mallet satisfies Vj = Vj-lGWj-l, i.e. U 

Wj-l = tj. and q-1 n Wj-l = (0). For fixed j, eqs. (3,4) reduce to a translation on 12. 

t2 ne' 

If k is considered as a time (space) translation, kto, and 23 as a frequency (wavenumber) 

scale, 2jwo, with towo 2 1 then each fixed jo corresponds to a discrete translation group 

at frequency scale 2j0wo and changing jo changes the frequency scale. Even if ?,b or $ are 

time (space) signals, the dilation operators Dab or Djk map them into the time (space)- 

frequency (wave number) phase space. Thus 

implies that 

The interpret at ion of the 's spanned by the 4)'s is as a low band-pass window at frequency 

scale 2jwo whereas the higher frequencies are in the Wj's spanned by t,bj.'s at that frequency 

scale. 

From this short discussion two things should be obvious. One is that although a wavelet 

is a window, it is a great deal more as it must cover each space-time point in order 

not to miss any point particles, or sources of charge or mass. The other is that all of the 

solutions allowed by the symmetries and dynamics of the system must be expressed in 

the wavelet basis, i.e. they must be a complete set. Kaiser50 has formulated relativistic 

electrodynamics in terms of a wavelet based on analyicity and coherent states. D'Aranio 

and DeFa~io~~ have formulated a class of squeezed states of quantum optics and provided 

a general inversion structure for a general density operator in terms of a suitable window, 

which was expressed in terms of a general basis of observables. Han, Kim and Noz51 have 

presented a study of the relativistic phase space of light using compactly supported wavelet 

windows to clarify the relation between photons and light waves. Since there are many 

wavelet windows for any given problem, it is important to know which ones are natural 

and, if possible, if a "best one" exists. 

In this paper, the application of wavelets and state squeezing, Bogoliubov transformations to the two statistical mechanics problems: E (q',w) for water in Section 2 and 

bubble formation will be described in Section 3, and our conclusions will be presented in 

Section 4. 
2. STATISTICAL MECHANICS OF WATER, THE DIELECTRIC RESPONSE 

In collaboration with Professor 6. Vignale, we have published studies of a classical 

water-like polar fluid whose properties are chosen to mimic water. The starting point is the 

number-conserving relaxation-time approximation to the Boltzmann transport equation. 

Up to the first order deviation from equilibrium it has the form 

where BT BT overdots are time rate of changes, q, = -, pa = -z, fO = e-mI.2 is the equilibri- a~~ 

urn Boltzmann distribution function. T is the k~netic energy of the molecule and includes 

translations, rotations and small amplitude vibrations, and Z is the partition function. 

fi is the first order correction term to fo by a perturbed applied potential. The static 

response of the electric susceptibility ~(w = 0) is a 3 x 3 real matrix-valued quantity given 

by -Pfo(qa)- 

Upon determining the dynamical self-response function x,(w) self-consistently from ff'~ 

where the Kubo fluctuation dissipation theorem for the single particle self-electric susceptibility 

relates the electric susceptability to the van Hove correlation function G(w). The total 

response from the Kubo theorem is 

where Q is the local field factor which describes the coherent many particle interactions. It 

is the analog of Lorentz's local field factor, in modern many-body theory. We have explored 

several choices and found Wei and Patey17 to provide the best agreement with experiment. 

Indeed, various computer intensive molecular dynamics and Monte Carlo calculations gave 

eT(6,0) as 25-71 instead of the experimental value of 90.8 f 3.2! The large experimental 

uncertainty is dominated by the spontaneous dissassociation H20 + H+ + OH- , which 

at 20°C yields an ion concentration of 8.33 x moles/liter. The high mobility of the 

H+ species allows it to form a positively charged surface layer at the negative electrode 

which can distort the measured value of the electric permittivity, ~(w). By taking a limit 

as the moments of inertia approach zero, the Stockmayer fluid calculations were found 

comparable to the best computer studies. The causal dielectric permittivity values are 

and 
where (T, L) denote the transverse and longitudinal components of the dielectric tensor e 

and the electric susceptability tensor X. The directions (T, L) are taken from the direction 

of the applied (vacuum) electric field 3. Remark: A wave propagation or scattering 

experiment is required to measure the q" dependence of n and a, in contrast to the ac 

capacitance bridge which measures only the angular frequency, w, dependence. A graph 

from ref. (18) of the real part of the index of refraction will be shown later. 

There is a collective mode in EL which Lobo, Robinson and Rodrigues9 first suggested 

and Pollock and Alderlo found in their simulation of a Stockmayer fluid model of water. 

It occurs in our model both in the symmetric rotor "waterlike" case1' and in the realistic 

asymmetric rotor case. l9 

Considering the water molecule as a two-level system for simplicity, the composite Boson 

creation and destruction operators which create and destroy water molecules (recall that 

this analysis is at frequencies that are far below the breakup threshold energies) are written 

as (Z,, ZJ) where their directions are those of the electric dipole moment, 9, of the molecule 

and a, p = 1,2 label which level. The collective dipolaron boson is created and destroyed 

by the Bogoliubov transformed operators (b,, b;). The frequency of the dipolaron is 

complex-valued 

and lies in the lower half w-plane because the response function l/eL(;, w) is causal. The 

frequency in eq. (15) is determined from 

and the width from 

provided that a,, >> r,,. The peak and its width are shown in Fig. 1, where one can 

see that Qq, >> I?,, is satisfied. The mechanism of this collective mode is high frequency 

coherent oscillations of the electric dipoles in self-consistent fields in the liquid. In order 

for the mode to be longitudinal, they oscillate 180" out of phase. Both a,, and r,, were 

shown by Kim, et a1.18, to be almost dispersion-free (they only change by 1% over the 

entire range of validity). This mode depends on q = (4 being small but non-zero so it 

cannot be measured from any ac capacitance experiment such as those in ref. (3) and must 

await a light or neutron scattering experiment. Such an experiment would verify or falsify 

the model formulated in refs. (18,19). 
Fig. I. The dipolaron collective mode in water. 

Next, the-reader is reminded that eq. (10) includes translations, rotations and vibrations 

in the j& (p) terms. It is well known52~53 that the renormalization group is based upon a Pa 

scaling invariance. By expanding the single particle distribution function fl(.) in a wavelet 

series 

where (.) can stand for either the translation I, rotation (f3,4) or vibrations (ql, q2, q3) in 

the real index of refraction n(w). The three normal modes for vibration are given by group 

theory and are used in our calculation of n(w). In Fig 2 the frequencies up to 10lOHz 

are dominated by the translations, from 1010 N 1012 Hz at (A, A') the collisions dominated 

Debye relaxation time dominates and at 1013Hz the free rotor peak occurs at (B1, B'). 

The free rotor peak at (B') in our theory is much too large, so work is underway to try 

to bring this into better agreement with the experiment. Between 1013 and 1014Hz the 

collective mode (C') occurs with coherent oscillations of the dipoles as its mechanism. The 

small optical peaks at (Dl, D2) fail to resolve the two higher frequency modes in (02) but 

we think that future experiments will. Thus, the wavelet phase space allows us to analyze 

the Boltzmann transport equation with a non-trival interaction. 
Fig. 2. The real index of refraction of water. 

3. BUBBLE FORMATION IN WATER BEING INSONIFIED BY ULTRASOUND 

The idea behind this calculation is to adapt the idea of Kaup to a solitary wave model 

by generalizing the work of Hammer, Shrauner and DeFacio where 4 is a 18 1 fluid density 

field 

with ui = AIB where A, B are real and positive coupling constants and A is the strength 

of the negative mass, B is the strength of the positive nonlinear quartic interaction to three 

space dimensions with a radial time independent potential. In 3 @ 1 space-time dimensions 

where LF is the Lagrangian of the fluid, LA is the acoustic Lagrangian for the sound wave 

and LI is the interaction Lagrangian. If ,u z 0 and the sound waves are small enough in 

amplitude to be linear (see eq. (30a)) the field equations become 

The key role played by the nonlinear "potential" which acts as a convection term is our 

justification for calling this approach a strong convection model. The mechanism is that 
the constant sound wave in the fluid produces gas under pressure as a new phase in the 

fluid. This is a non-equilibrium, open system and the gas density can be described by one 

Bogoliubov transformation and the non-zero temperature, O°C to 100°C, can be obtained 

from another Bogoliubov transformation. The entropy of this situation was discussed by 

DF, VN and Professor Brander63 of the Institute for Theoretical Physics at Chalmers in 

the Gothenberg. The l = 0, radial solutions to eqs. (21a,b) are Jacobi elliptic functions 

with modulus kl, 0 5 kl 5 1, Sol is a density defect where the local mass density is a 

minimum so that it is a site for cavitation and (u, uol) are the amplitudes for the density 

and pressure wave, respectively. 

Next a time-independent mean field theory for the density # is formed by linearizing 

about the Jacobi elliptic functions according to 

and 

B#~+B<<~>#+BU~[S~(K(~+~)I~)]~ . 

This should be valid, provided it is not applied too close to a caviton 'explosion', and leads 

to the linearized field eqaution for the density # 

where 

Separation of variables using 

and 

u(r) R(r) = - 

r (25b) 

reduce eq. (25) to 

This ODE satisfies Dirchlet zero boundary conditions on the dense domain H2(R:) where 

R: is the positive half-line and H2 is the Sobolev space of L2 functions whose first two 

derivatives exist and are L2. Thus,.eq. (26) can be solved numerically using the Numerov 

algorithm. Then the density is constructed from the relation 

From eq. (27) the sound wave has modulus kl 0. One thing about this strong convection 

model which seems new is the inclusion of l 2 1 multipoles which give deviations from 

spherical symmetry. The calculus of variation "proof" that bubbles are spherically shaped 
only holds for thermal and fluid equilibrium, whereas the continous quasi-monochromatic 

sound wave makes this an open, non-equilibrium system. In Fig. 3 VMFT is plotted and 

Fig. 3. An example of a mean field theory potential, T/MFT. 

Fig. 4. The water density in a 3mm cell calculated from the mean field theory 

potential in the previous figure. 

in Fig. 4 a bubble is shown which was found in the density, using the convection potential 

of Fig. 3. The parameters are consistent with those of water with a sound wave amplitude 

which is physically realizable. The bubble is centered at S = 0.75mm; and has radius 

roe 3001.1 (p = 10-~m). The detection of bubble interfaces using wavelets is under study. 

Since some authors have questioned the safety of ultrasound in tiss~e,3~>~* it is important 

to understand bubble formation. In addition, the Mie scattering from bubbles adds spatial 

dispersion to the effective index of refraction. 
4. CONCLUSIONS 

Two different problems in non-equilibrium statistical mechanics were discussed: 

(i) the dielectric response of water, e (q', w), and 

(ii) bubble formation in water under insonnification using Bogoliubov transformations: 

state squeezing maps and wavelets. 

The multi-resolution structure of phase space permits the sort of analysis which the 

renormalization group has provided for lsing models and lattice gas models and little else. 

Thus, "squeezed states" are useful for more than quantum optics. 

ACKNOWLEDGEMENTS 

This work was supported in pqt by the Missouri University Department of Physics 

and AFOSR grants 90-307 and 91-203. Professors Giovanni Vignale and Peter Pfeifer 

are thanked for many useful conversations. The figures were produced by the NeXT and 

Silicon Graphics Personal Iris work stations purchased by the two AFOSR grants. The 

organizers of the Third International Workshop on Squeezed States, Professors M.H. Rubin 

and Y.-H. Shih of UMBC and Dr. D. Han of NASA, Greenbelt are thanked for organizing 

an excellent conference which continues the traditions initiated by Professor Y .S. Kim and 

Professor W. W. Zachary. 

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N95- 13940 

Coherent States on the m-sheeted Comolex Plane as m- hoto on stateg 

Department of Electrical Engineering and Electronics 

University of Liverpool, P.O. Box 147,Liverpool, L69 3BX 

Abstract 

Coherent states on the m-sheeted complex plane are introduced and 

properties like overcompleteness and resolution of the identity are studied. 

+ They are eigenstates of the operators a m tam which create and annihilate 

clusters of m-particles. Applications of this formalism in the study of 

Hamiltontans that describe m-particle clustering are also considered. 

1. introduction 

Apart from the original (Glauber) coherent states which are associated 

with the Weyl group, other types of coherent states associated with other 

groups (e.g. SU(2), SU(1,l) etc.) have also been studied. In a recent 

publication [l] we extended these ideas in a different direction and 

introduced coherent states on the m-sheeted covering group of SU(1,l). From 

a physical point of view it can be used for the description of m-particle 

clustering. Here we extend the Glauher coherent states into coherent states 
in the m-sheeted complex plane. The properties of these states 

(overcompleteness, resolution of the identity etc.) are explicitly 

considered. Using these states we extend the Bargmann [2] analytic 

representation into a new formalism that we call Bargmann analytic 

representation in the m-sheeta complex plane. Using this representation we 

+ introduce new creation and annihilation operators a 

m lam which create and 

annihilate clusters of m particles and show that the properties of our 

coherent states with respect to them, are similar to the properties of the 

ordinalcy (Glauber) coherent states with respect to the usual creation and 

+ annihilation operators a ,a. 

The above ideas are used in the description of m-particle clustering. 

which is a generalisation of the concept of pairing. They could be used to 

generalise two-photon states into m-photon states with even better 

properties. Some work in this direction but from a different point of view 

has already been presented [3, 41 

2. Goherent states on the m-sheeted com~lex ~lang 

We consider the Riemann surface 

* 

where C is the punctured complex plane I * 

354 
and Z is the discrete group of the integers modulo m. The punctured complex m 

* plane C is the m-sheeted covering surface of the Riemann surface Rm. 

* The sheet number s(z) of a complex number z in C is defined as 

where IP stands for the integer part of the number. s(z) takes integer 

values from 0 to 81 - 1 (modulo m). 

We also consider the harmonic oscillator Nilbert space N and express it 

as 

H is an infinite-dimensional subspace spanned by the number eigenstates a 

H1- ( I Nm+1> ; N-0, 1. 2, ..... ) (5) 

We call n the projection operators on HI f 
We now introduce the states 

where s(z) is the sheet number of z defined in equ.(3). It is clear that if 

z belongs to the zero sheet the states lm~n should be used; if z belongs to 

the first sheet the states fmN+l> should be used; etc. The state iz> 

belongs to the Hilbert space H 

s(z) ' 

We refer to the states (7) as coherent 

states on the m-sheeted complex plane. We can prove 

where S is tbe kronecker delta. 

In order to give a resolution of the identity for these states we first 

grove a resolution of the identity within the Wilbert space H I : 

Sl is the l-sheet and rl is the projection operator (6). S 

gives the resolution of the identity: 
The states (r;& with z in the sheet SI, form an overcomplete set within 

the Hilbert space Hl. 

3. Extended Barnman re~resentation on the m-sheeted com~lex lane 

Bargmann (2) introduced analytic representations in the complex plane 

which are based on ordinary coherent states. In refs. [5] analytic 

representations in the unit disc which are based on the SU(1,l) Perelomov 

coherent states, have been studied. This formalism has been extended in 

ref.[I] into analytic representations in the m-sheeted unit disc. In this 

section we study an analogous extension from the Bargmann representation in 

the complex plane into an analytic representation in the m-sheeted complex 

plane. 

We generalise the Bargmann representation by representing the arbitrary 

(normalised) state 10 with the function 

Ih-zfN I N> 

* 

where z takes values in C and s(z) is the sheet number of z (equ.(3)). The 

f(z;m) is analytic in the interior of each sheet and has discontinuities 

across the cuts C I ' 
As an example we consider the number eigenstates [M - d+I> where N, I 

are the integer part and remainder of H divided by m, correspondinglly. They 

are represented by the function 

f(z:m) - (,s()) z" (NI).' (14) 

The kronecker B(R,s(z)) ensures that this function is non-zero only in 

the I sheet. 

As a second example we consider the states lz ;m> of equ. (7) which are 

0 

represented by. -ehe. function 

* 

f(z;m;zo) - exp 6z ;mlz ;& - 

0 

We next consider the operators 

It is seen that they act as creation and annihilation operator within 
each one of the Hilbert spaces Hi. 

The operator 

can be considered as a number operator within the Hilbert space H R ' 

We also consider the operator 

which we call "remainder operatorn or "number modulo m operator". Its 

eigenstates are the number states and its eigenvalues the remainder%-l of the 

division of the number of the state over m. The operator R commutes with the 

operators a a +: m' m 

[R, am] - [R, am+] - o 

+ 

Note that the operators a a commute with the projection operators r of m' m P 

equ. (7) : 

+ 

A consequence of that is that an "arbitrary" function of a ,a leaves each m m 

of the Hilbert spaces HI invariant in the sense that when it acts on a state 

which belongs in HI it produces another state which also belongs in the same 
space. 

We next consider the displacement operators with respect to the a 

9 

a lam 

+ * 

Dm(zo) exp [zo am - z o am1 (26) 

+ They displace the operators am,a by a constant and they commute with 

m 

the operator R or equ.(22) 

[Dm (zo). Rl - 0 (29) 

We now act with the operators Dm(zO) on the number eigenstates Il> (0 d 

1 5 m-1) and get the coherent states on the m-sheeted complex plane (7) : 

Iz - (z~)~"~ ; m>- D~ (zo) I l > (30) 

The subscript i indicates that among the m roots, the one which belongs 

to the sheet Sl should be chosen. Equ.(30) can also be writen as 

lz;~ - Dm (zm) Is(.)> (31) 

where Is(z)> is number eigenstate and s(z) the sheet number of z (equ. (3)). 

Using equs. (27), (31) we prove that the I Z;U are eigenstates of am 

a IZ;U - zmlz;m> m (32) 

+ It should be pointed out that operators similar to a ,a have been 

m m 

considered in [6] and used in refs. [4j where the following states have been 
studied 

They are a subset of our coherent states associated with the Hilbert space H I 

with I - 0; or equivalently with the zero sheet of our m-sheeted complex 

plane. It has been shown in [4] that the states (33) have very interesting 

quantum statistical properties. 

4. m- hoto on states 

We consider the Hamiltonian 

and express H as: 

We easily see that the eigenvectors and eigemralues of H are: 

The physical significance of thrs Hamiltonian lies in the fact that the 
4- 

operators am lam create and annihilate clusters of m particles. Pairing of 

particles plays a very important role in various contexts in Physics. 

Generalisation from pairing into m-particle clustering can be described with 

various Hamiltonians. The obvious choice is 

+ * m 

H-na a+r (a')m+r a 

There are certain difficulties associated with this Hamiltonian [3] and in 

any case it is useful to explore alternative models, especially if they are 

based on some symmetry which can be exploited to handle these highly 

non-linear terms. In ref. [I] a Hamiltonian associated with the SU(1,l) 

group, which describes m-particle clustering has been studied. In this 

section the Hamiltonian (34) which is associated with the Weyl group and 

which also describes m-particle clustering, has been studied. 

5. Discussion 

Coherent states on the m-sheeted complex plane have been introduced in 

equ,(7). The Hilbert space has been split into m subspaces (equ.(4)) and 

+ 

operators a a which play the role of creation and annihilation operators m' m 

+ in each subspace, have been introduced. The am , a are different from the 

m 

+ usual creation and annihilation operators a ,a. It has been shown that our 

coherent states have the usual properties of coherent states with respect to 
+ the a 

m lam. They are eigenstates of a (equ.(32)); and they can be 

m 

expressed as the product of the displacement operator times the lowest state 

(equ. (31)) 

All these ideas can be used for the description of m-particle 

clustering. This is a generalisation of the concept of pairing which plays 

an important role in areas like squeezing in quantum optics, 

superconductivity, superfluidity, phase transistions etc. Consequently this 

formalism might be used for generalisations in all these areas. A 

Hamiltonian that describes m-particle clustering has been considered in 

equ.(34) and its eigenvalues and eigenfunctions have been calculated. 

1. A. Vourdas, J.Math.Phys. 34, March (1993). 

2. V. Barpann, Commun.Pure Appl.Math. 14, 187, (1961). 

3. R.A. Fisher, M.M. Nieto, V.D. Sandberg, Phys.Rev. D29, 1107, (1984). 

S.L. Braunstein, R.I. McLachlan, Phys.Rev., A35, 1659, (1987). 

M. Hillery, Phys.Rev., A42, 498 (1990). 

4. J. Katriel, A.I. Solomon, G. D'Ariano, M. Rasetti, Phys.Rev. D34, 2332 

(1986). 

J. Katriel, M. Rasetti, A. Solomon, Phys.Rev. D34, 1248, (1987). 
G. D'Ariano, H. Rasetti, Phys.Rev. D35, 1239, (1987). 

G. D'Ariano, S. Morosi, H. Rasetti, J. Katriel, A.I. Solomon, Phys.Rev. 

D36, 2399, (1987). 

5. T. Paul, J.Math.Phys. 25, 3252, (1984). 

J.R. Klauder, Ann.Phys. 188, 120, (1988). 

A. Vourdas, Phys.Rev. A41, 1653, (1990); Phys.Rev. A45, 1943, (1992). 

J.P. Gazeau, V. Hussin, J.Phys. A25, 1549, (1992). 

6. R. Brandt, O.W. Greenberg, J.Math.Phys. 10, 1168, (1969). 
SONIE RULES FOR POLYDIMENSIONAL 

SQUEEZING 

V. I. Man'ko 

Dipartimento di Scienze Fisiche Uniuersita di Napoli "Federico 1I"and I.N.F.hT.,Sez.di Napoli 

and 

Le bedev Physics Institute, Leninsky pr.,53, Moscow, 11 7333 Russia 

Abstract 

The review of the following results of the Refs. [l] - [5] is presented: For mixed state 

light of N-mode electromagnetic field described by Wigner function which has generic Gaussian form the photon distribution function is obtained and expressed expliciltly in terms of 

Hermite polynomials of 2N-variables.The momenta of this distribution are calculated and expressed as functions of matrix invariants of the dispersion matrix.The role of new uncertainty 

relation depending on photon state mixing parameter is elucidated.New sum rules for Hermite polynomials of several variables are found.The photon statistics of polymode even and 

odd coherent light and squeezed polymode Schrodinger cat light is qiven explicitly.Photon 

distribution for polymode squeezed number states expressed in terms of multivariable Hermite polynomials is discussed. 

1 Introduction 

In the Ref.[l] it was shown that the matrix elements of density matrix in number state basis for 

polymode oscillator are expressed in terms of Hermite polynomials of several variables for the 

density operator in the canonically transformed t hermd st ate of the oscillator.In the recent works 

[2], (31 the photon distribution function for the generic Gaussian light described by the Wiper 

function which is the most generic Gaussian in quadrature phase space was found and expressed in 

terms of Nermite polynomials of 2 variables for one-mode case [2] and 2N variables for polymode 

case [3].The physical meaning of mixed Gaussian state of the light may be understood if one takes 

into account that the pure rnultimode Gaussian state corresponds to the generalised correlated 

st ate introduced by Sudarshan [6] who related those states to the symplectic dynarnical group-The 

mixed Gaussian states studied above may be considered as the mixture of generalised correlated 

states plus thermal noise acting on each mode which has its own temperature. The photon 

distribution function for even and odd coherent states [7] or Schrijdinger cat state. [8] subject 

to squeezing both in one-mode and polymode cases has been found in Ref.[4].The polymode 

Schriidinger cat states and photon distributions for light in these states were introduced in [5&The 

aim of this work is to give a review of the photon distribution functions and related sum rules 

for one-mode and polymode Gaussian light and for the even and odd coherent states light using 

the results of [2] - [5]. It should be emphasized that the squeezed light is worth to be used in 

interferometric gravitational antennas [9] and the even and odd coherent light may play alternative 

role in gravitational wave experiment [lo] ,El 11. 
2 Photon Distribution for Polymode Gaussian Light 

The mixed squeezed state of the N-mode light with a Gaussian density operator ,ij is described 

by the Wigner function (see,for example,[l2]) 

where 2N-dimensional vector Q = (p, q) consists of N components pl , ..., p~ and N components 

q19 ..*, QN. 2N parameters < p; > and < q; >, i = 1,2,. . . , N, combined into vector < Q>, are the 

average values of the quadratures.h real symmetric quadrature dispersion matrix M has 2N2 +N 

parameters 

They obey certain constraints, which are nothing but the generalized uncertainty relations [12]. 

. The photon distribution function in this state has the form [3],[1] 

where vector n consists of N nonnegative integers: n = (nl, 712,. . . ,nN). The function Hgl(y) 

is the Hermite polynomial of 2N variables. We introduced also notations 

The symmetric 2N-dimensional matrix R and the 2N-dimensional vector y are given by the 

relations 

R = U' - 2M) (IZN + 2~)-' Ua, 

where the 2N-dimensional dtary matrix 

is introduced.The matrices IN and lZN are identity matrices of corresponding dimensions. 

The probability to have no photons has the form 

1 Po = [dd (M + ZI1~)]-i exp [- < Cj > (2M + 12N)-l< 4 >] (7) 

It may be shown [3],[4] that the multivariable Hermite polynomial is the even function if the 

sum of its indices is the even number and the polynomial with the odd sum of indices is the odd 

function. Due to this pairity property of the polydimensional Hennite polynomials the "diagonal" 

multivariable Hermite polynomial is the even function since the sum of its indices is always even 
number.@onsequently the above photon distribution function is the even function. Thus the 

photon distribution function for generic mixed Gaussian light found in[3] is expressed in t 

multivariable Hermite polynomials and it depends on the quadrature means and dispersions.a'ho 

photon number means and dipersion matrix corresponding to the found distribution (3) are of the 

form 

where 3 and dj are the trace and the determinant of the 2x2-matrix Mi,describing only j-th 

mode, and the 2-vector Qj has the components (pj, qj). 

3 Pure Polymode States 

The photon distribution for polymode squeezed correlated state may be expressed in terms of 

s ymplectic transform parameters relating boson operators as follows 

where R is a symplectic 2Nx2lV-matrix consisting of four N-dimensional complex square bbdks, 

and d is a complex N-vector.Then we have for photon distribution in squeezed correlated polyrnode 

state IP > labeled by the complex number vectaz with N-components 

where 1 1 1 

Fo(,B) = (det <)-; exp [58t)*~-1/3 + P(d* - t)*C-ld) + ;idt)*C-ld - 5ldl (10) 

The photon doistribution function of the squeezed number state Jm > is descibed by the fonnula 

Pn = I det ~1-l exp [~e(d~*<-'d) - ld12] !HE}(L))~ 

n!m! ' 

Here m is the label of the state, whereas n is a discrete vector variable. 2Nx2N-matrix R and 

2N-vector L are expressed now in terms of blocks of matrix R and vector d as follows, 

4 Even and Odd Coherent States 

The one-mode even and odd coherent states have been introduced in Ref.[7].The polymode even 

and odd coherent states have been introduced in Ref.[S].The squeezed and correlated even and odd 
coherent states have been introduced and studied in Ref.[4]. We will discuss the photon statistics of 

the light in these states which are also called Schriidinger cat states [8] .The multimode Schriidinger 

cat states are defined by the relation [5] 

where the multimode coherent state I A > is 

and D(A) is the multimode displacement operator creating coherent state fiom the vacuum.The 

normalization constants are 

where complex number A has the form 

The photon distribution function has the form 151 

and the photon means corresponding to these distributions are 

< A+ 1 n; 1 A+ > = 1 a; l2 tanh 1 A 12, 

< A- I n; I A- > = 1 a; la coth 1 A l2 . 

The photon number dispersion matrix has the matrix elements 

a2 = I a; 121 ~k I2 sech2 I A I2 + I ai l2 tanh I A I2 Q, 

0,; = - I ai 121 Pk I2 cosech2 I A l2 + I a; l2 ~0th I A l2 6ik. 
5 Squeezed Schradinger Cat States 

Let us find out the photon statistics of squeezed polymode Schriidinger cat state labeled by the 

complex N -vector P.To do that let us difine transition amplitude from the polymode squ 

and correlated state I@ > to the polymode photon number state In > 

where 

Then the photon distribution function for polymode light in the squeezed Schriidinger cat state 

(even and odd) is given by the formula [4] 

If the shift parameter d = 0 the formula is simplified 

if we have equality 

N 

Cni = 2k, 

i=l (24) 

and for even states 

P,+k+l(p> = 07 (25) 

if one has 

N Eni =2k+ 1, 

i=l 

(26) 

where Pn(p) is given by the formula (3).For the light in the odd squeezed Schfidinger cat state 

the photon distribution is 

7'G+l(P) = 4NYn(P), (27) 

if the indices satisfy the equality (26) and 

if the indices satisfy relation (24).Thus the squeezed Schr6dinger cat states if the shift parameter 

is equal to zero have highly oscillating distribution function.The influence of shift parameter 

decreases the oscillations of the distribution function. For Hermite polynomials the following sum 

rule may be found [3] 

1 

= [det (dXzR + IZN)]-~ exp [ZZ (ACzR + &A%] . 
Were a = (zl, z2, ... z2~), the 2Nx2N matrix C, is the 2N-dimensional andog of the Pauli matrix 

oz9 and the diagonal 2Nx2N - matrix A has the matrix elements Xi in j-th and (N+j)-th rows. 

Let us consider the one mode case.Then the formula for the photon distribution function in terms 

of Hermite polynomials of two variables may be expressed in terms of usual Hermite polynolPials 

(T + 1)z + [a, - a,, - 2iup,] z* 

Hn-k 

((2T + 4d + 1) [B, - a,, - 2iap,]) a 

Here the parameters app, B,,, (~p, are matrix elements of the quadrature dispersion matrix M, d is 

determinant of this matrix and T is the trace of the matrix. The complex number z is determined 

by the relation 

1 z=-(<q>+i<p>). JZ (31) 

Formula (30) can be used also to illustrate the generalized uncertainty relation (for the Gaussian 

states). Indeed, it is obvious that the probability to find n photons must be nonnegative. On the 

other hand, all but one terms in the right-hand side of eq. (30) are positive independently on the 

concrete values of the parameters determining the quantum state. The only exception is the term 

Consequently, to guarantee the positiveness of the photon distribution function for all conceivable 

combinations of the parameters one should impose the restriction d 2 $.This inequality is the 

Schrijdinger uncertainty relation (see, 161, [12] ). 

Acknowledgments 

The author would like to achowledge the University of Maryland at Baltimore County for kind 

hospitality and I.N.F.N.,Sez. di Napoli for support. 

References 

[l] V. V. Dodonov, V. I. Man'ko, and V. V. Semjonov, Nuovo Cimento B 83,145(1984); see also: 

V. I. Man'ko, Proceedings of the International Workshop on Harmonic Oscillator, University 

of Maryland at College Park,25-28 March,1992, D.Han and Y.S.Kim, eds. (Nasa Conference 

Publications,l993) 

[2] V. V. Dodonov, 0. V. hlan'ko, and V. I. Man'ko, "Photon distribution for one-mode mixed 

light with generic Gaussian Wigner function", University of Naples Preprint INFN-NA-IV93/35? DSF-T-93/35(1993) (submitted to Phys. Rev. A ). 
[3] V. V. Dodonov, 0. V. Man'ko, and V. I. Man'ko, "Multidimensiond Hermite polynomials 

and photon distribution for polymode mixed light", University of Naples Preprint INFN-NAIV-93/36, DSF-T-93/36(1993) (submitted to Phys. Rev. A ). 

[4] V. V. Dodonov, V. I. Manlko,and D.E.Nikonov,"Even and odd Coherent States (Schr6dinges: 

cat states) for Mult irnode Parametric Systems" (in preparation). 

[5] Nadeem A. Ansari and V. I. Man'ko,"Photon statistics of multimode even and odd coherent 

light", University of Naples Preprint INFN-MA-IV-93134,DSF-T-93/34(1993); (submitted to 

Phys. Rev. A). 

[6] E. 6. G. Sudarshan, Pmceedings of Second International Ml'orkshop on Squeezed States and 

Uncertainty Relations, Moscow, 25-29 May, 1992,D.Han, Y. S. Kim, and V. I. Man'ko, eds., 

(NASA Conference Publication,3219,p.2419 1993). 

[7] V. V. Dodonov, I. A. Malkin, and V.I.Man'ko,Physica 72,597 (1974). 

[S] M. Brune, S. Haroche, J. M. Raimond, L. Davidovich, and N. Zagury, Phys. Rev. A 45,5193 

(1992); C. C. Gerry and E. E. Hach 111, Phys.Lett A 74,185(1993). 

[9] C. M. Caves, Phys.Rev. D 23, 1693 (1981). 

[lo] N. A. Ansari, L. Di Fiore, M. Man'ko, V.Man'ko, R. Romano, S. Solimeno, F. Zaccaria, 

Technical Digests of EQEC793-EQUAP793,2, Firenze, 10-13 Septernber,l993, P.De Natale 

and R.Meicci, eds. 

and S. Pelli, p.688 (1993); see also: N. A. Ansari, L.Di Fiore,M.Man'ko,V.Man'ko,R.Fbmano, 

S.Solimeno,and F.Zaccaria," Optimal laser intensity for interferometric detection of gravitational wavesn, University of Naples Preprint INFN-NA-IV-93116,DSF-T-93/16(1993). 

[ll] N. A. Ansari, L. Di Fiore, M. Man9ko, V. Man'ko, R. Romano, S. Solimeno, F.Zaccaria, 

"Quantum limits in interferometric GW antennas in the presence of even and odd coherent 

states", University of Naples Preprint INFN-NA-IV-93/20,DSF-T-93/20(1993); Phys. Rev. 

A (in press). 

[I21 V. V. Dodonov and V. I. Man'ko, Invariants and Evolution of Nonstationary Quantum Systems, Proceedings of Lebedev Physics Institute 183, M. A.Markov, ed. (Nova Science Publishers, Commack, N.Y., 1989). 
SCHR~DINGER OPERATORS WITH THE 

¶-LADDER SYMMETRY ALGEBRAS' 

Sergei Skorik 

Department of Physics, University of Southern California, 

Los Angeles, CA 90089-0484, U.S. A. 

Vyacheslav Spiridonov2 

Centre de Recherches Mathhatiques, Universite' de Montreal, 

C.P. 6128-A, Montrkal, Qukbec, H3C 3J7, Canada 

Abstract 

A class of the one-dimensional SchrZjdinger operators L with the symmetry algebra LB* = 

qf2~f L, [B+, B-] = PN(L), is described. Here Bf are the 'q-ladder' operators and PN(L) 

is a polynomial of the order N. Peculiarities of the coherent states of this algebra are briefly 

discussed. 

1 Introduction 

Exactly solvable spectral problems are of great importance. They have numerous applications in 

classical and quantum mechanics. In the last decades the theory of solitons once again exhibited 

their universal character. However, the definition of the notion of solvability (or integrability) itself 

is quite delicate. In particular, it involves the definition of functions which are allowed to enter the 

solution of the ~roblem (in a sense these are two complementary things - it is common to define 

functions as solutions of some equations). Let us take for example the standard one-dimensional 

Schriidinger equation: 

L$(x) (-d2/dx2 + u(x))+(x) = A+(x), (1) 

endowed with some boundary conditions. The widely used tacit definition of solvability of this 

spectral problem consists in the requirement for $(x) to be a finite sum of the hypergeometric 

functions 2Fl(a, b; c; x) [I], or of their descendants. On the one hand, the 2F1 indeed occupies a 

distinguished place among the classical special functions, but on the other hand, there are more 

complicated objects whose global structure has been well understood. 

In the theory of nonlinear evolution equations the smooth bounded potential u(x) is said to be 

solvable if it has a spectrum consisting of the N + 1 permitted bands, N of the finite width and 

one infinitely large (these are the regions of A for which the wave functions $(x) are bounded). 

In some cases this condition leads to the Lam6 equation, which is a simplest generalization of the 

lTalk presented by V.S. at the Third International Workshop on Squeezed States and Uncertainty Relations 

(Baltimore County, August 10- 13, 1993) 

20n leave of absence from the Institute for Nuclear Research, Russian Academy of Sciences, Moscow, Russia 
second order differential equation for hypergeometric function. Another type of functions with 

known global analytical properties can be defined with the help of the nonlinear second order 

differential equations: the Painlevk transcendents PI-PVI are the simplest examples. Different 

and more rich classes of functions are defined by the second order (linear or non-linear) finitedifference equations, an example is given by the basic, or q-hypergeometric function. In the latter 

two cases, the corresponding special functions may be taken for definition of potentials and then 

the condition of solvability of (1) can be thought as the requirement for wave functions +(x) to 

be related to u(x) by some simple formulae which do not lead to the essentially new objects. 

This short discussion and the systems to be described below demonstrate that the problem of 

classification of all exactly solvable problems is far from completion even for a simple equation 

(1). 

The integrability of a problem is related to its symmetry properties. Unfortunately, most of 

the symmetries are "hidden" and, as a result, the group-theoretical treatment of special functions 

often emerges as a secondary problem. Rarely new function was introduced primarily from the 

symmetry principles. The situation however changes when the classification problem is dealt with. 

The so-called characterization theorems are targeted to the enumeration of specific properties 

which define a taken system uniquely within the given class of equations. Using them one can 

see what properties should be abandoned in order to get more general systems. E.g., one may 

ask what are the most general potentials u(x) for which there exist a differential operator A of 

the order N such that [L,A] = 0. For odd N this happens to lead to the finite-gap potentials 

mentioned above (even N cases contain generically a functional non-uniqueness). 

Recently, the characterization of all potentials leading to the ordinary ladder algebra: 

where A is the N-th order differential operator, has been given in [2]. For N = 1 it is easy to find 

that u(x) cc x2, i.e. a harmonic oscillator potential. For N = 2 one gets the singular oscillator 

potential u(x) cx ax2 + /?/x2. The N = 3 case corresponds to the quite complicated situation 

when u(x) involves the Painlevk IV transcendental function [2]. 

Already from (2) one expects that the spectrum of Hamiltonian L is purely discrete and 

equidistant. This, however, depends on the structure of zero modes of A. If all of them are 

physical eigenstates of L with different eigenvalues, and the conjugated operator A+ does not 

break normalizability, then one has a spectrum composed from N independent arithmetic series. 

The wave functions are explicitly expressed through f (x), f'(x) and J f (x)dx, where f (3) is defined 

by a system of N first order nonlinear differential equations, i.e. the problem is solved exactly in 

the above sense. Although it is much harder to calculate physical observables of these systems than 

for the potentials related to the hypergeometric function, the principally important characteristics 

- the spectrum - is very simple and elegant. 

The aim of this note is, however, to present even more complicated potentials than just mentioned ones [3-71. The first class of them is connected with the algebra (2) but the operator A is 

now a differential-difference operator [3]. The resulting potentials are more general than those of 

[2] since the corresponding characterization theorem does not apply. The second class is based on 

the q-deformed ladder relation: 
which simply can not be realized with the help of differential operators of the finite order. Note 

that the limit q + 1 does not necessarily mean that the operator A will be an integral of motion 

- there may be a diverging constant entering additively into L such that for q = 1 one gets (2) 

rather than [L, A] = 0. 

Let us briefly describe the definition of potentials leading to (3). The basic tool is the factorization, 

or dressing method based on the Darboux transformations. One takes a set of Harniltonians, 

Lj = -@/d;C2 + u~(x), and represents them as products of the first-order differential operators, 

Up to Some constants Aj: 

Lj = A:A; + Aj, (5) 

i.e. uj(x) = ff(x) - fj(x) + 4. Then one imposes the following intertwining relations: 

. which constrain the difference in spectral properties of Lj and Lj+1 and are equivalent to the 

equations: 

Aj+l + A- j+l + = A;A;+ Aj. (7) 

Substitution of (4) into (7) yields the chain of differential equations [8]: 

which is called the dressing chain. 

The potentials we are interested in are defined by the following self-similarity constraints 

imposed upon the dressing chain [3, 61: 

The simplest example, defined by the reduction fj (x) = q3 f (qjz), Aj = q2jA, has been found in 

[41. 

At the operator level, the relations (9) lead to the Schrodinger operators with non-trivial 

q-deformed symmetry algebras. Let us consider the products: 

which generate the interwinings 

The structure relations complimentary to (11) look as follows: 
These identities show that if the operators Lj and Lj+~ are related to each other through some 

simple transformation, e.g. 

Lj+N = q2ULjU-1 + W, (I3) 

where U is an invertable operator, then the combinations B: = M:U, By E U-'M,', map 

eigenfunctions of Lj onto themselves, i.e. they are symmetry operators for Lj. The form of U is 

restricted by the requirement that the Ljls be of the Schrodinger form. Taking U to be the &ne 

transformation generator, U f (x)U-I = f (qx + r), fixing the indices (e.g., j E 1) and removing 

them, we get the symmetry algebra [6]: 

For N = 1 this is a q-analog of the Heisenberg-Weyl algebra which for special values of the 

parameters serves as the spectrum generating algebra [5]. For N = 2 this is a q-deformation of 

the su(1,l) algebra, and for N > 2 we get polynomial relations describing symmetries of the 

self-similar potentials. Note that the limit q + 1 is not trivial. If the parameter r in (9) is not 

zero then we get the realizations of the algebra (2) which generalize the ones described in [2]. 

For q # 1 the parameter r may be set equal to zero. If the operators Bf are well defined and 

have N normalizable zero modes, then the self-similar potentials have spectra consisting of N 

independent geometric series. Moreover, u(x)'s are reflectionless and represent initial conditions 

for the infinite-soliton solutions of the KdV equation. 

We conclude that the N = 1 case describes the deformation of harmonic oscillator potential, 

the N = 2 case corresponds to the q-deformed conformal quantum mechanics [3], and the M 2 3 

cases correspond to the q-deformation of the PainlevC type equations. 

An interesting situation takes place when the parameter q is a root of unity, i.e. qn = 1. 

Generically these cases are related to the hyperelliptic potentials, the N = 1 system has been 

analyzed in detail in [?I. Depending on whether q is a primitive root of unity of odd or even 

degree, the solution may be unique or non-unique. The q = -1 system exists only when the initial 

condition f(0) = 0 is imposed and it provides a non-standard realization of the Heisenberg-Weyl 

algebra. Indeed, the equation arising from (8), (9) at N = 1, q = - 1: 

has the general solution f (x) = ~212. This corresponds to the operators Bf satisfying [B-, B+] = 

p with the explicit form: 

where P is the parity operator P$(x) = +(-x). 

The general q3 = 1 solution exists for arbitrary initial condition and is given by the equianharmonic Weierstrass function: u~(x) = 2p(x + Qj), (p'l2 = 4(p3 - I), where Qj+3 = Qj, and 

qjf2j = w2 - the real semiperiod of the doubly periodic function p(x). The analytical solutions at 

q4 = 1 exist for special initial condition but they contain functional non-uniqueness. The particular 
subcase of the q4 = 1 system is defined by the (pseudc+)lemniscatic Weierstrass function satisfying 

the equation (P')~ = 4p3 f p. SO, the group-theoretical treatment of the Schriidinger equation 

with these specific elliptic function potentials naturally leads to the q-oscillator algebra at roots 

of unity. Note that the algebra of symmetry operators in these cases does not have the spectrum 

generating meaning. 

3 Coherent States of the q-ladder Algebras 

Coherent states are interesting objects of quantum mechanics [9]. Originally proposed for the 

harmonic oscillator potential, eventually they were generalized in many directions. Let us discuss 

briefly coherent states of the algebra (14), (15) which we define as eigenstates of the "annihilationn 

operator B-: 

B-$a(x) = a$a(x). (18) 

Looking at the definition of B- one can realize that this is quite complicated functional equation. 

The simplest case (N = 1) has the following explicit form: 

where f (x) is a smooth solution of the differential equation with the deviating argument: 

The fi factor appeared because we took U to be unitary operator so that (B-)t = El+. We 

also assume that O < q2 5 1. (The q2 > 1 choice is equivalent to analytical continuation of the 

q2 < 1 potentials to the imaginary axis. This brings pole singularities into the potential and, as 

a result, operators B* are not well defined.) Coherent states of the q-oscillators have been widely 

discussed (see, e.g., [lo]), but the realization (19) is a principally new one since it deals with the 

ordinary Schrodinger equation. Unfortunately, the structure of functions $,(x), their minimal 

complete subsets, and many other things are not known at present. 

As it was noticed in [7] there is a particular coherent state among $,(x) which happens to be 

an eigenstate of the Harniltonian! (Such situation is characteristic for the whole algebra (14), (Is), 

i.e. for any self-similar potential at q2 < 1.) Tt corresponds to the zero energy state, L$d(x) = 0, 

the formal existence of which follows from the boundedness of the potential Jzm lu(x)ldx < oo. 

So, we have the following representation of the q-oscillator algebra: 

i.e. B* are pure complex conjugated numbers. Possible existence of such Uclassicaln states of the 

q-oscillator algebra has been noticed also (in the different context) in [ll]. Since for $d(x) we 

have two equations: (19) with a = re-ie and 
we can remove the derivative part and get pure q-difference equation: 

~1q1-1'2[e-" lql$,l(qx) + eieq-l$cl(q-lx)] = (f (x) + Q-' f (~-l X))&~(X), (21) 

which, however, again is not easy to solve. 

Finally, let us consider the case q = -1, i.e. the coherent states associated with the realization 

(17). We put for convenience p = 2 and renormalize B* + JZB*. The +,(x) states form a 

subset of solutions of (B-)2$,(x) = a2+,(x). Because (B*)' are purely differential operators, 

one can easily solve this equation. Picking out the proper linear combination of the corresponding 

two independent solutions, one can find: 

where la) are the canonical coherent states of a harmonic oscillator: 

x la) = 7r-ll4 exp ( " - la!' - (- - a)'). 2 a 

The states (22) are not minimal uncertainty states for the variables x and p G -id/dx for a # 0: 

((Ax)') = (I - (a - a*)2 - (a + a*)2e-41aP)/2, 

((Ap)') = (1 + (a + a*)' + (a - ~r*)~e-'I~1~)/2, 

((Ax)?((Ap)') > 1/45 

where Ax = x - (x), (3) = xI+,(x)12dx, etc. However, it is easy to construct other canonical 

variables 

where P is the parity operator, for which +,(x) minimize their uncertainties, ((A$)')((An)') = 

114. A detailed consideration of the properties of these coherent states will be given elsewhere 

[12Is 

Finally, we would like to point out that the procedure of determination of potentials with fixed 

symmetry properties presented here may be generalized to other spectral problems. In [13] it was 

applied to the second order finite-difference equation. In that case many features of the continuous 

considerations maintain but there are also few new ones. E.g., the algebras at q > 1 may now 

have spectrum generating meaning, i.e. one can find the systems with the exponentially growing 

spectra. Another advantage consists in the possibility to realize q-analogs of the compact unitary 

algebras like 42). 

4 Acknowledgments 

The authors benefitted from the stimulating discussions of the outlined results (during this 

workshop and the last years) with D.Five1, A.Its, V.Kac, V.P.Karassiov, A.Mann, F.W.Nijhoff, 

A.Shabat, E.C.G.Sudarshan, S.Suslov, L.Vinet, and A.Zhedanov. The work of V.S. is supported 

by the NSERC of Canada and by the Fonds FCAR of QuCbec. 
References 

[I] Erdelyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F.G., Bateman Manuscript Project. 

Higher Transcendental Functions. Vol. 2. (McGraw-Hill, 1953). 

[2] Veselov, A.P. and Shabat, A.B., Funkt. Anal. i ego Pril. 29, n. 2, 1-21 (1993). 

[3] Spiridonov, V., in: Proc. of the Workshop on Harmonic Oscillators (College Park, March 

25-28, 1992). Eds. D.Han, Y.S.Kim, and W.W.Zachary (NASA Conf. Publ. 3197, 1993) pp. 

93-108; Mod. Phys. Lett. A'i', 1241-1251 (1992). 

[4] Shabat, A., Inverse Prob. 8, 303-308 (1992). 

[5] Spiridonov, V., Phys. Rev. Lett. 69, 398-401 (1992). 

[6] Spiridonov, V., in: Proc. of the XIXth ICGTMP (Salamanca, 29 June - 4 July 1992). Eds. 

M.A. del Olmo, M.Santander, and J.M.Guilarte (CIEMAT/RSEF, Madrid 1993) Vol. I, pp. 

198-201; Proc. of the CAP-NSERC Workshop on Quantum Groups, Integrable Models and 

Statistical Systems (Kingston, July 13-18, 1992). Eds. J.LeTourneux and L.Vinet (World 

Scientific, 1993). 

[7] Skorik, S. and Spiridonov, V., Lett. Math. Phys. 28, 59-74 (1993). 

[8] Infeld, L. and Hull, T.E., Rev. Mod. Phys., 23, 21-68 (1951). 

[9] Klauder, J.R. and Skagerstam, B.-S. (eds), Coherent States - Applications in Physics and 

Mathematical Physics (World Scientific, 1985). 

[lo] Arik, M. and Coon, D., J. Math. Phys. 17, 524-527 (1976); Biedenharn, L.C., J. Phys. A22, 

L873-L878 (1989); Kulish, P.P. and Damaski~sky, E.V., J. Phys. A23, L415-L419 (1990); 

Gray, R.W. and Nelson, C.A., J. Phys. A23, L945-L950 (1990); Floreanini, R. and Vinet, 

L., Lett. Math. Phys. 22, 45-54 (1991); Phys. Lett. A180, 393-401 (1993); Bracken, A.J., 

McAnally, D.S., Zhang, R.B., and Gould, M.D.., J. Phys. A24, 1379-1391 (1991); Fivel, D.I., 

J. Phys. A24, 3575-3586 (1991); Quesne, C., Phys. Lett. A153, 303-307 (1991); Zhedanov, 

A., J. Phys. A24, L1129-L1133 (1991)) Teor. Mat. Fiz. 94, 307-315 (1993); Askey, R. and 

Suslov, S.K., J. Phys. A26, L693-L698 (1993). 

[ll] Granovskii, Ya.1. and Zhedanov, A.S., Mod. Phys. Lett. A8, 1029-1035 (1993). 

[12] Spiridonov, V., in preparation. 

[13] Spiridonov, V., Vinet, L., and Zhedanov, A., Lett. Math. Phys. 29, (1993). 
THE WAVE FUNCTION AND MINIMUM UNCERTAINTY 

FUNCTION 

OF THE BOUND QUADRATIC HAMILTONIAN SYSTEM 

Kyu Hwang Yeon 

Department of Physics, Chungbuk National University 

Cheongju, Chungbuk, 360- 763, Korea 

Chung In Um 

Department of Physics, College of Science, Korea University 

Seoul, 136- 701, Korea 

T.F. George 

Department of Chemistry and Physics, Washington State University 

Pullman, Washington 991 64-1046 

Abstract 

The bound quadratic Hamiltonian system is analyzed explicitly on the basis of quantum 

mechanics. We have derived the invariant quantity with an auxiliary equation as the classical 

equation of motion. With the use of this invariant it can be determined whether or not the 

system is bound. In bound system we have evaluated the exact eigenfunction and minimum 

uncertainty function through unitary transformation. 

1 Introduction 

In recent years an extensive effort has been devoted to obtaining an exact solution for the oscillator 

systems with time-dependent Hamiltonian and especially dissipative system, i. e ., damped free 

particle 2), damped 3, or damped driven harmonic oscillator 4, and driven time-dependent harmonic 

oscillator5). After Lewis and R,iesenfeld6) first derived the relation between the eigenstates of the 

dynamical invariant and the solution of the Schrodinger equation, many authors have applied the 

dynamical invariant method to investigate the time-dependent oscillator system. The dynamical 

invariant is related directly to an auxiliary equation as the classical equation of motion for the 

Hamiltonian system, which is given as nonlinear second-order differential equation. Therefore the 

dynamical invariant can be determined by the particular solution to the auxiliary equation. 

In this paper, employing the operator method, we derive the wave function and the minimum 

uncertainty function for the quadratic Hamiltonian system which includes canonical variables with 

time-dependent coefficients. Recently, we have investigated this system to obtain the wave function 

and propagator through path integral method 7). In Sec. 2, we derive the dynamical invariant 

from the equation of motion. We classify whether or not the system is bound in the consideration 

of our system as classical system and then find the conditions for bound and unbound. In Sec. 3, 

using the quantum invariant operator, we define the creation and annihilation operators and then 

evaluate the wave function and propagator of our system. In Sec. 4, we introduce new creation 

and annihilation operators from the old ones in Sec. 3, and evaluate eigenfunction of the unitary 

transformed system and minimum uncertainty function. Finally, we give the summary in Sec. 5. 
2 The bound quadratic Hamiltonian system 

The quadratic Hamiltonian of the system is given as 

1 

H = -IA(t)p2 2 + B(t)(pq + qp) + C(t)q2] (1) 

where p and q are canonical variables. A(t), B(t) and C(t) is continuously differentiable function 

but A(t) is nonzero. The classical equation of motion can be obtained from the Hamilton's 

equation of motion: 

+ C(t)q + t(t)q = 0 (2) 

with 

The general solution for Eq. (2) can not be found, but we may take the solution in the following 

form: 

q = p(t)eiy(t) (5) 

here, ~(t) and y(t) are the functions to be determined from Eq. (2). The functions are real and 

depend only on time. Substitution of Eq. (5) in Eq. (2) offers the real and imaginary parts of 

this equation as 

p - pi2 + C(t)P + t(t)p = 0. (6) 

and 

+ 2p;u + C(t);up = 0. 

The time invariant quantity can be found from Eq. (7) in the form: 

with auxiliary condition as the classical equation of motion, Eq. (2). With the use of Eq. (8), the 

nonlinear differential equation [Eq. (6)] can be writ ten as 

We may find another classical time invariant quantity with an auxiliary equation as classical 

equation of motion. We assume that this invariant quantity depends on p, q and t. Then, from 

Hamilton's equation of motion, the time derivative of I(p, q, t) becomes 
Combining Eq. (10) with Eq. (1) we may obtain the time invariant quantity as 

I(q,p, t) is an invariant quantity and thus we can express it in phase space. For !2 = 0, Eq. (11) 

becomes a linear line in phase space and canonical variables q and p can occupy every region 

in phase space. Therefore the motion of the system is unbound. On the other and, for 0 # 0, 

Eq. (11) becomes ellipse in phase space because the coefficient matrix of it has positive real 

eigenvalues. The canonical variables q and p can occupy some finite region in phase space, and 

thus the motion of the system is bound. 

3 The Wave Function, Propagator and Uncertainty Values 

For quantum mechanical treatment of our system, we may replace the canonical variables with the 

corresponding quantum operators in Eq. (7) and then we may also obtain the quantum invariant 

operator of the system as the same form of Eq. (1 1). 

In order to obtain the eigenfunctions and eigenvalues of the invariant operator, we define the 

creation and annihilation operator, a and a t with auxiliary equations, Eq. (8) and Eq. (9) as 

A 1 

at = F{- 2ti+ A [+i (B- :)] -ip}. 

The invariant operator [Eq. (ll)] can be expressed in terms of a and a t as 

Since the a and at satisfy the commutation relation, the normalized eigenstates and eigenvalues 

of Eq. (14) are given by 

The ground state must satisfy the condition that 

auo = 0. 

Solving Eq. (17) for u~ we obtain 
Operating a t continuously to Eq. (18) we may obtain the nth excited states: 

where Hn is a nth order Hermite polynomial. 

Eq. (19) is an eigenfunction of the invariant operator [Eq. (14)] with an auxiliary equation as 

classical equation of motion, but is not the solution of the Schrodinger equation; 

ih- - 

a a 

Comparision of Eq. (19) with (20) offers the exact wave function of the system: 

Making use of the Mehler's formula together with Eq. (21), we can easily evaluate the propagator of the system given by 

where p' = p(t'), 7' = +y(tl), A' = A(tf), and B' = B(tl). Eq. (22) is the result that we obtained 

previously7). 

The uncertainty relation is defined by 

With the help of Eq. (21), we can express Eq. (23) as 
4 The Minimum Uncertainty Function 

The minimum value for n = 0 in Eq. (24) is larger than K/2 and thus the coherent state of our 

system is not a minimum uncertainty state. To obtain minimum uncertainty state, we introduce 

the new creation and annihilation operators defined by 

for a pair of c numbers p, Y obeying 

(p12 - IuI2 = 1. 

The canonical transformation [Eq. (25)] which keeps the commutator invariant, is a unitary 

transformation. The properties of the b and bt are the same as those of a and at '1. 

Performing the same procedures in Sec. 3, we can obtain the wave function for nth excited 

states: 

x {- 2KAJp " - vI2 [++ -i [(B - f) Ip - v12 + i(pu* - u~*) +]]} (27) 

Substituting Eq. (26) into Eq. (24) and evaluating the diagonal element, the uncertainty relation 

for (n,n) states can be obtained: 

From Eq. (27) we can also find the condition of p and u for the minimum uncertainty 

where 

0 = tan" 

and 

Here, k is real and positive, and o must have the same sign of f (B - i). We can confirm that 

the minimum uncertainty is a function of one continuous parameter in the finite region. 
5 Summary 

Introducing the quadratic Hamiltonian system given .in Eq. (I), we have derived the classical 

invariant quantity with an auxiliary equation as the classical equation of motion. With the use 

of this invariant, we can distinguish whether or not the system is bound. We transformed the 

invariant into an operator in the replacement of the creation arid annihilation operator [Eq. (14)] 

and then evaluated the corresponding eigenfunction and eigenvalues. However, this eigenfunction 

is not the Schrodinger solution of the system. Though we obtain the exact wave function of the 

system [Eq. (21)] and propagator [Eq. (22)] the minimum uncertainty constructed by this wave 

function is larger than h/2 and thus the coherent states of the system is not minimum uncertainty 

st ate. To obtain the minimum uncertainty we introduce the canonical transformation, which keeps 

the commutator invariant. Through this unitary transformation we obtained the eigenfunction 

and minimum uncertainty state of the system. 

Acknowledgments 

This work was supported by the Center for Thermal and Statistical Physics, KOSEF under Contact 

No. 93-08-00-05 and by the BSRI Program, Ministry of Education, Republic of Korea, and by the 

U.S. National Science Foundation under Grant No. CHE-9 196214. 

References 

(1) J. G. Hartley and J. R. Ray, Phys. Rev. D 25, 382(1982); D. C. Khandekar and S. S. 

Lawande, J. Math. Phys. 20, 1870(1979): P. Camiz, A. Gerardi, C. Marehioso, E. Presutti and E. Scaeciatelli, J. Math. Phys. 12, 2040(1971); P. 6. L. Leach, J. Math. Phys. 

21,300(1980). 

(2) K. H. Yeon and C. I. Um, J. Korean Phys. Soc. 23, 82(1990). 

(3) K. H. Yeon, C. I. Um and T. F. George, Phys. Rev. A 36,5287(1987); V. V. Dodonov, 

T. F. George, 0. V. Man'ko, C. I. Um and K. H. Yeon, J. Soviet Laser Research 12(5), 

385(1991); L. F. Landovitz, A. M. Lavine. E. Ozizmis and W. M. Schreiber, J. Chem. Phys. 

78, 291(1983); 78, 6133(1983). 

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W. H. Kahng, J. Phys. A: Math. Gen. 20, 611(1987). 

(5) C. F. Lo, Phys. Rev. A 47, 115(1993). 

(6) H. R. Lewis, Jr. and W. B. Riesenfeld, J. Math. Phys. 10, 1458(1969). 

(7) K. H. Yeon. K. K. Lee, C. I. Um, T. F. George and L. N. Pandey, Phys. Rev. A in print. 

(8) H. P. Yuen, Phys. Rev. A 13,2226(1976); J. von Neumann, Math. Ann. 104, 570(1931) 
THE THERMAL-WAVE MODEL: 

A SCHRODINGERLIKE EQUATION 

FOR CHARGED PARTICLE BEAM DYNAMICS 

R. Fedele and G. Miele 

Dipartimento di Scienze Fisiche, Universitci di Napoli "Federico II", 

and INFN Sezione di Napoli, Mostra d'oltremare Pad. 20, 80125, Napoli, Italy 

Abstract 

In this paper we review some results on longitudinal beam dynamics obtained in the 

framework of the Thermal Wave Mode2 (TWM). In this model, which has recently shown the 

capability to describe both longitudinal and transverse dynamics of charged particle beams, 

the beam dynamics is ruled by Schrodinger-like equations for the beam-wave-functions, whose 

squared modulus is proportional to the beam density profile. Remarkably, the role of the 

Planck constant is played by a diffractive constant e, the emittance, which has a thermal 

nature. 

1 Introduction 

Recently, on pure basis of analogy with other similar subjects of physics, a new technique to derive 

the equation of motion for a thermal system, like a charged particle beam at finite temperature, 

which is able to take into account the collective behaviour of the ensemble has been obtained 

PI- [51. 

The starting point of this technique, the Thermal Wave Quantization (TWQ), are the equations 

of motion of the considered system, in the so called single-particle approximation, from which is 

possible to obtain the single-particle hamiltonian of the system. At this point, the formal analogy showed in the case of the transverse dynamics for relativistic charged particle beams, with 

the electromagnetic optics in paraxial approximation and with the two dimensional nonrelativistic quantum mechanics [I], suggests to replace the single-particle hamiltonian, with a differential 

operator, and the hamilton-equations with a Schrijdinger-like equation, in which coordinate and 

particle momentum are replaced by a beam-wave-function. 

This technique, applied to the longitudinal and transverse beam dynamics, has led to the formulation of the Thermal Wave Model (TWM) for relativistic charged particle beam propagation, 

which represents a useful quantum-like description of the total beam optics [l]-[5]. This model has 

already been successfully applied for estimating the effects of the aberrations in linear colliders 

[3], [5], as well as for describing nonlinear beam-plasma interaction [Z], and nonlinear longitudinal 

dynamics in circular accelerating machines [4]. 

'presented by R. Fedele 
2 TWM for longitudinal dynamics 

Let us consider a single relativistic particle of electric charge q, within a stationary bunch, travelling 

with longitudinal velocity pc (P = 1) in a circular accelerating machine of radius & = cTo/2T (To 

being the revolution period). Its longitudinal motion is described, neglecting radiation damping 

and quantum excitations, by a pair of equations which, defining s G d (t being the time), can be 

put in the following dimensionless form [6] 

where x is the longitudinal particle coordinate and P = AE/Eo is the dimensionless longitudinal 

energy variation, after a turn in the ring. Note that x (-xRo 5 x 5 TRO) and P are both evaluated 

with respect to the synchronous particle (AE = O), and Eo is the synchronous particle energy. 

The quantity AV represents the total voltage variation after a turn and it takes into account 

the interactions of the particles with the surrounding medium (RF-cavities, pipe, kickers, etc.). 

Consequently, the equations (1) and (2) describe the longitudinal bunch dynamics on time scale 

t >> To. Furthermore, in (1) the parameter q is defined as r] r (Aw/wo)/(AE/Eo) (wo = c/& 

and Ao being the frequency shift with respect to wo). By defining the momentum compaction 

a G (AR/Ro)/(AE/Eo), where AR is the orbit radius variation with respect to &, it can be 

easily proved that q = l/y2 - a. From (1) and (2) we can easily write the following dimensionless 

single-particle harniltonian 1 H=~P*+U , (3) 

where 

Ur- /. qAV dx' . cToE0 o 

In order to find an equation which describes the longitudinal evolution of the beam, taking into 

account its thermal spreading (longitudinal emit t ance) while it interacts with the surrounding 

medium (potential well and wake fields), we follow the quantum-analogy, which suggests to use in 

(3) the following correspondence rules 

6 a 6 

P 4 P E -ieL- a and H 4 H E ieLax as ) 

where EL is the longitudinal beam emittance. Thus, by considering (3) and (5), the following 

Schrijdinger-like equation for the beam wave function (bwf) 8 can be assumed 

where H = 8p2 + U. Consequently, (6) becomes 
Note that (7) describes the longitudinal beam dynamics in terms of the bwf 9, which we assume to be related to the longitudinal density number X(x, s) through the following relation: 

X(x, s) = XoJiP(x, s)I2, where A. = iV/Ro (N being the total number of particle in a bunch). 

According to the previous definitions, ]@I2 gives the longitudinal beam density profile. Furthermore, the circular topology of the ring should requires periodic solutions for Q, with respect to 

x (9(nRo, s) = 9(-nRo, s) and az9(nRo, s) = a,@(-a&, s)). In these conditions, from (7) the 

norm squared (A@) of bwf, defined as 

is conserved (U is assumed a real function), and it has been fixed for simplicity equal to &. This 

result is compatible with the physical requirement that JT:~ X(x, s) dx = N. However, in the 

following we restrict our analysis to consider bunched beam whose effective length is much smaller 

than &. Under this assumption, the above conditions of periodicity of Q do not have a relevant 

role, because in this limit, for the bunch, the ring looks like an infinite linear accelerator. Thus 

we can define the effective bunch length (TL, and the expectation value of the momentum P as 

and in complete analogy to quantum mechanics the uncertainty principle 

holds. Furthermore, note that (7) 117 plays the role of an eflective mass. 

2.1 The interaction potential and synchrotron oscillations 

As it has been shown in Ref. [4], the potential U can be split in two parts (RF + Self-interaction) 

and (4) becomes: U = URF + US. Note that whereas in general URF is a known function of x 

and s, the explicit expression for Us, depending on the bunch density (collective effects), needs 

particular assumptions about the beam interaction with the surrounding medium to be specified. 

This interaction can be parametrized in terms of the longitudinal coupling impedance 171. In the 

1K 2 special case of a linear approximation for the RF-potential, URF = lnx , where K is the cavity 

strength (m is the synchrotron wave number), and for a purely reactive longitudinal coupling 

impedance X, the equation for bwf becomes 

where I is the beam current, and n is the so-called harmonic number [8]. In the simplest case of 

(X = 0) and for Ro >> a~, the Eq. (11) can be exactly solved, and the normalized solutions for 

bwf are the well known Hermite-Gauss modes as it occurs in complete analogy for electromagnetic 

optics in paraxial approximation [9] 
In (12) the functions aL(s), pL(s) and ~L(s) are solutions of the following system of differential 

eauat ions 

and &(x) are the Hermite-polynomials with m a non-negative integer. Note that (@,I2 for rn .= 0 

(fundamental mode) gives a Gaussian particle distribution. Remarkably, it can be easily proved 

that (13) is completely equivalent to 

- d2a; +4~ai = 4E with E = 

ds2 = const. 

In this form it is easily to recognize that the equation for aL(s) (16), i.e. (13), describes the 

synchrotron oscillations. The equilibrium condition d2aL/ds2 = 0 gives 

where a: and or are the equilibrium value of o~ and a:, respectively, and v8 is the synchrotron 

number given by the ratio between the synciirotron frequency 0. = cm and the revolution 

frequency wo [lo]. Equation 17) recovers the well known relationship between the bunch length 6 a: and the energy spread aL [lo]. Since for the resent case the bwf is Gaussian, in obtaining 

(17) we have introduced the minimum value o:o$= eL/2 of the product aLo; consistently with 

disequality reported in section 2. 

3 Coherent stability criterion and soliton solution 

In this Section, we develop, within the framework of the thermal wave model, an analysis of some 

collective effects occurring when the bunch interacts with the surrounding medium. To this end, 

we consider the special case of RF cavity 08 and take into account both the space charge effect 

and a purely inductive coupling impedance. Consequently, (1 1) becomes 

Note that (18) is formally identical to the cubic NLS equation which describes the propagation of 

an e.m. pulse through a nonlinear medium in paraxial approximation [11],[12],[13]. In this analogy, 

the factor EL^ plays the role of the diffraction parameter (the inverse of the wave number), s 

corresponds to the time, and -[qqI(X/n)/(2~E~)]13)~ corresponds to a nonlinear refractive index. 

Thus, an analysis of the bunch coherent instability (stability) can be made in complete analogy 

to the electromagnetic case I11],[12]. To this aim, applying the well known method developed in 
nonlinear e.m. optics to search for the sufficient conditions of modulational instability, we show 

that coherent instability for particle bunches is fully equivalent to modulational instability for e.m. 

bunches. Moreover, a soliton-envelope solution, very interesting for accelerator physics, is found. 

As an example, we analyze the instability of a plane-wave (!Po($, s) = po exp[i(kox - Qos)], 

where po is a positive constant) solution of Eq. (18), when a small perturbation around it is 

introduced. Let 

*(x, S) = [Po + Pl(X, s)] exp[i(kox - 520s) + iel(x, s)] (19) 

be the perturbed solution, being pl and 8.1 real functions, and 

EL77 2 - no = Tko 2IrfLE0 " ((f) P: a 

In order to obtain the dispersion relation of the system we can assume 

pl(x, s) = py cos (kx - Rs + 40) el (x, S) = 0: sin (kx - Rs + 40) , (21) 

where p?, 8:, k, 52 and are real constants. By imposing that (19) is a solution of the linearized 

(18) we obtain the following dispersion relation 

Reminding that unstable modes occur for S(R) # 0, namely 

we get stability for qX < 0 and instability for fl > 0. This result recovers the well-known 

condition for coherent stability (instability) for monochromatic charged particle beams [14], in 

addition we remark that the above condition is fully similar to the Lighthill criterion, valid for 

modulational instability of an e.m. plane-wave travelling in a nonlinear medium [11],[12],[13]. 

For a bunched beam (oL << &), a solitary solution of (18) is found by looking for a solution 

of a relativistic envelope form: 

with XO, wo, and Po real numbers. Thus, according to the general theory of NLS equations [ll], 

the following soliton-like solution for the beam density (A = XoG2), which satisfies (8), is possible 

under the condition qX > 0: 

N~~~R - 1 (;) x sech2 [ Nq2R (f) (x-as)] , ') = ~E~T~E~ 77 ~E~TOEO 77

where 

1 N~~R 2 

EL77 2 

wo= ,xo--[ 277~~~LToEo (f)] . 
4 Conclusions 

In this paper, we have reviewed some results and applications of the Thermal Wave Model, showing 

in particular, how it is possible to give a novel approach to the study of the nonlinear longitudinal 

dynamics of a relativistic particle bunch in circular accelerating machines. Neglecting radiation 

damping and quantum excitation, we have shown that the nonlinear interaction between the bunch 

and the surroundings (potential well and wake fields) is governed by an appropriate NLS equation 

(equation (ll)), fully similar to the one that holds for the propagation of an e.m. bunch in a 

nonlinear medium in paraxial approximation [l 11 , [12], [13]. Much remains to be done - like, for 

instance, the extension to 2- or even to 3-D, or the development of an iterable formulation - 

to make this model really interesting for the study of the typical, still unsolved, beam-dynamics 

problems. Nevertheless, its very innovative feature of allowing the treatment of the whole beam 

at the same time, makes it look extremely promising for a new, and more complete approach to 

the subject. 

References 

[I] R. Fedele and G. Miele, Nuovo Cimento D 13, 1527 (1991). 

[2] R. Fedele and P.K. Shukla, Phys.Rev. A 44, 4045 (1992). 

[3] R. Fedele and G. Miele, Phys.Rev. A 46, 6634 (1992). 

[4] R. Fedele, G. Miele, L. Palumbo and V.G. Vaccaro, Phys. Lett. A 179, 407 (1993). 

[5] R. Fedele, G. Miele, and F. Galluccio, INFN/TC-93/12, to be published in Proc. 1993 Particle 

Accelerator Conference. Washington D .C. May 17-20, 1993. 

[6] B.W. Montague, CAS Proc., CERN 77-13, 63 (1977). 

[7] L. Palumbo and V.G. Vaccaro, LNF-89/035(P), May 1989. 

[8] S. Hansen et al. , CERNIISR-RF-DI-TH-OP/75-15, March 1975. 

[9] S. Solimeno, B. Crossignani, and P. Di Porto Guiding, Difiaction, and Confinement of Optical 

Radiation, (Academy Press, London, 1986). 

[lo] M. Sands, SLAC Report 121 (1971). 

[ll] G.B. Whitham, Linear and Nonlinear Waves, (J. Wiley, New York, 1974). 

[12] Y.R. Shen, The Principles of Nonlinear Optics, (Wiley-Interscience Publication, New York, 

1984). 

[13] S.A. Akhmanov, A.P. Sukhuorukov, and R.V. Khokhlov, Sov.Phys.Usp. 93, 609 (1968). 

[14] A. Hofmann, CAS Proc., CERN 77-13, 139 (1977). 
CAUSALITY PROBLEMS FOR FERNHISS TWO-ATOM SYSTEM 

Gerhard C. Hegerfeldt 

Institut fur Theoretische Physik 

University of Gottingen, D37073 Gottingen, Germany 

Abstract 

Let A and B be two atoms or, more generally, a 'source' and a 'detector' separated by 

some distance R. At t = 0 A is in an excited state, B in its ground state, and no photons 

are present. A theorem is proved that in contrast to Einstein causality and finite signal 

velocity the excitation probability of B is nonzero immediately after t = 0. Implications are 

discussed. 

To study and check finite signal velocity, Fermi [I] considered two atoms A and B separated by a 

distance R. At time t = 0 atom A is assumed to be in an excited state leA) and B in its ground 

state IgB), with no photons present. Atom A will decay to its ground state under the emission of 

a photon which may then be absorbed by atom B. Fermi asked when atom B will notice A and 

start to move out of its ground state. In accordance with Einstein causality, i.e. no propagation 

faster than the speed of light, he expected this to occur after a time t = R/c. This was indeed 

what Fermi found by his calculation. 

More than thirty years later Shirokov [2] pointed out that Fermi's 'causal' result was the 

artefact of an approximation. Indeed, Fermi had replaced an integral over positive frequencies by 

an integral ranging from -00 to m. Without this approximation his calculation would not have 

given the expected result. 

Moreover, Fermi had calculated the probability for a transition to A nonexcited, B excited and 

no photons, i.e. the transition probability from the state leA) lgB) loph) to the state IgA) leB) loph), 

which requires measurements on A, B and photons simultaneously. Hence this 'exchange' probability does not refer to finite signal velocity or Einstein causality but to 'local' or 'nonlocal' 

correlations. What is really needed for finite signal velocity is the probability of finding B excited, 

irrespective of the state of A and of possible photons.This will be called the excitation probability 

of B. 

Fermi's problem was investigated by many authors in this or in a related form, e.g. by Heitler 

and Ma [3], Hamilton [4], Fierz [5], Ferretti [6], Milonni and Knight [7], Shirokov [2] and his 

review 181, Rubin [9], Biswas et al. [lo], and Valentini [ll]. The older papers confirmed Fermi's 

conclusion, while the results of the later papers depend on the model and the approximations 

used. At present there seems to be agreement that Fermi's 'local' result is not correct, but that 

this nonlocality cannot be used for superluminal signal transmission since measurements on A and 

B as well as on photons are involved. 
Usually previous authors have used 'bare' states and a Harniltonian of the form 

where HAF and HBF represent the coupling of atoms A and B to the quantized radiation field. 

The Hilbert space is simply a tensor product, 

The initial state is then 

Idob") = I~A) ISB) 10ph) . 

The probability of finding B in some excited state, irrespective of the state of A and photons, is a 

sum over all excited states leB) of B, over all states liA) of A and over all photon states I{n)), i.e. 

where the completeness relation for orthonormal bases has been used. The operator 

represents the observable "B is in a bare excited state", and it is a projection operator. The 

expectation value of 6:~ gives the excitation probability of B. 

For bare states, however, there is a serious difficulty. Even with atom A absent and no photons 

present atom B will be immediately excited under simultaneous emission of photons! This wellknown unphysical behavior is a consequence of the interaction term HBF because then the bare 

ground state IgB)lOph) is no longer an eigenstate of the bare Hamiltonian. Therefore, all results 

for bare states have to be considered with caution. 

Valentini [ll] and also Biswas et al. [lo] have found the following interesting result for bare 

states by using perturbation theory and cutoffs. They calculated that for t 5 Rlc the bare ground 

state of B behaves as if the excited atom A were not present. This result seems to indicate a causal 

behavior and suggests a similar result for a properly renormalized theory. This, however, will be 

shown not to be the case. 

Fermi's problem of finite signal velocity will now be treated under very plausible assumptions 

without bare states. Although a renormalized theory has yet to be constructed only two basic 

properties of such a supposedly existing theory are needed. The first is that the states of such 

a theory form a Hilbert space, denoted by 'FIren. The other property needed is a renormalized 

selfadjoint Hamiltonian H, which is bounded from below, e.g. by 0. The assumption of positive 

energy is standard and physically well-motivated. 

In general Hren is no longer a tensor product, 
and the initial state, denoted by I&), will not be a simple product state, 

Similarly, if the observable "B is in an excited state" makes sense and is represented by an 

operator OeB then in general 6, # 0:;. However, QeB will still be a projection operator since 

its eigenvalues are 1 for 'yes' and 0 for 'no'. The excitation probability of B at time t is then given 

by the expectation value 

($t I Oeg I $t) . Alternatively one may assume that the excitation probability of B is an expectation value of some 

positive operator, or one may measure the excitation through a positive observable which vanishes 

for the ground state, e.g. some operator related to the square of the dipole moment [12]. In all 

these cases one will run into difficulties with Einstein causality. 

No point-like localization of A and B is required. As a generalization of Fermi's set-up A and B 

may be systems initially localized in two regions separated by a distance R with no (real) photons 

present. The ground state of B may be degenerate. 

We note that measurements of the excitation probability of B involves measurements on B 

only and that Pi(t = 0) = 0. One would expect, as Fermi, that 

Pi(t) = 0 for 0 5 t 5 Rlc . (7) 

However, in a slightly different context a theorem of the author [13] as well as prior [14] and 

later results [15, 16, 17, 181 showed difficulties with causality in particle localization [19]. Although 

the theorem is not applicable here - it applies to free particles or to the center-of-mass of systems 

- it makes one wary. Indeed, as a complement to this first theorem I will now show a second 

theorem which includes interactions. 

Theorem. Let the Hamiltonian be positive or bounded from below and let the initial state 

at time t = 0 be 

A in an excited state, I+o) = B in a ground state, no photons. 

Let Pg(t) be the probability of finding B excited, 

where OeB is a projection operator or, more generally, a positive operator. 

Then either 

(i) The excitation probability of B is nonzero for almost all t, and the set of such t's is dense 

and open. 

(ii) The excitation probability of B is identically zero for all t. 

Remarks. Alternative (i) means that B starts to move out of the ground state immediately 

and is thus influenced by A instantaneously, in contrast to Einstein causality. Alternative (ii) is 

clearly unphysical since in this case B is never excited so that B is never influenced by A. 
The proof is basically very simple and uses only the positivity of H,,,, or rather its boundedness 

from below, and the fact that one deals with the expectation value of a positive selfadjoint operator. 

Proof of theorem. Since I$t) is continuous in t, so is Pi(t). Hence, if for some tl one has 

P;(tl) > 0 then this also holds in a small interval around tl, and therefore the set is open. Now 

let us assume that the set of t's with Pi(t) > 0 is not dense. Then there is a small but finite 

interval I such that 

P;(t) = O for t E I. (9) 

It will now be shown that this implies that alternative (ii) holds. Eq. (9) can be written as 

($tl OeB l$t) = O for t E I . (10) 

If OeB is a projection operator then (Oe,)2 = OeB. Therefore Eq. (10) can be written as 

($tI ('eB)2 I$t) = 11 OeB l$t) 112 

= 0 for ~EI. 

This means that 

OeB j$t) = 0 for t E I. 

For O,, a positive operator the argument is similar [20]. Now let 4 be any fixed vector and define 

the auxiliary function Fd(t) by 

Then, by Eq. (12), 

Fd(t) = 0 for t E I. 

Since Ifmn 2 - const, one has that the operator 

is well-defined for y < 0. Putting z = t + iy one sees that F4(z) can be defined as a continuous 

function for Im z < 0, and, moreover, Fd(z) is analytic for Im z < 0. However, such an analytic 

function cannot have boundary values vanishing on a real interval unless 

for Im z # 0 [21]. But then, by continuity, one also has F4(t) = 0 for all real t. Hence the right 

side of Eq. (13) vanishes for all t. Since 4 was arbitrary one has 

OeBl$t) - 0 for all t 

and this gives Pi(t) r 0, i.e. case (ii). 

This proves that Pi@) is either nonzero on a dense open set or that it vanishes identically. 

In a slightly more sophisticated way it will now be shown directly that Pg(t) is either nonzero 

for almost all t or vanishes identically. Let the set of zeros of Pi(t) be denoted by No. The 
same argument as before shows that F'(t) vanishes there too. As a boundary value of a bounded 

analytic function F4(t) satisfies, unless it vanishes identically, the inequality [22] 

00 

dt log IF+(t)l/(l + t2) > -oo . 

If & had positive measure the integral would be -oo and thus Fd(t) would vanish identically in 

t, for each 4. This would again imply case (ii). Hence if case (ii) does not hold Pi(t) can only 

vanish on a null set [23]. This completes the proof of the theorem. 

A typical behavior of the excitation probability of B according to (i) is shown in Fig. 3. No 

estimate of the actual magnitude of Pi(t) is provided by the above argument, except that is 

nonzero for almost all t. It follows trivially for alternative (i) that the set of zeros of Pi(t) is not 

only of measure 0 but also nowhere dense. 

It should be noted that the above proof makes no use of any spatial separation of the two 

subsystems nor of its photon content. In fact, the theorem is a mathematically rigorous result 

which holds for any initial state I+o), any positive Harniltonian and expectation value of any 

positive operator [24]. Physics comes in only when one thinks of as representing two spatially 

separated subsystems with no photons. Of course, if the systems are not spatially separated part 

(i) of the theorem comes as no surprise. 

Extensions. The derivation does not need that A and B are atoms. The result clearly extends 

to more general situations: 

a) Larger systems: A may be some "source" of photons and B a "detector". 

b) A and B may move. 

c) Other particles and other interactions may be included. 

Other positive observables can be considered. E.g., for an excited localized atom (or system) 

with no real photons initially one obtains an acausal result for photons and electromagnetic energy 

in regions not containing the atom. This is in contrast to a result by Kikuchi 1251 who, at the 

suggestion of Heisenberg, had studied this problem using the same approximation as Ferrni [I]. 

The general case of a decaying particle or system can also be treated by the above approach. 

Discussion. If the effect implied by the theorem were real it could in principle be used for 

superluminal signals, with all the well-known consequences. However, the result may also be 

viewed as a difficulty for the formulation of the underlying theory. The theorem is of the 'if-then' 

type. To avoid its physical consequences one has to check whether its conditions or any additional 

physical assumptions are fulfilled in a given situation. There are several possible ways out. 

a) Systems localized in disjoint regions might not exist as a matter of principle, so that strictly 

speaking they always 'overlap'. Then an immediate excitation may evidently occur. 

b) Renormalization will introduce a sort of photon cloud around each system. This essentially 

implies an overlap of the systems, leading back to case a). 

c) The notion of 'ground state of B' in the presence of A may not make sense. Without A 

present one will expect a lowest energy state to exist for the system B plus radiation field, with 

no real photons. However, with A present, the lowest state of the complete system may change. 

Thus the 'ground state of B' may not be preparable independently of A. Effectively this also leads 

back to case a). 
These possible ways out suggest implicitly that the problem is not well-posed, i.e. an experimental set-up to check the theorem might not be feasible. But without disjointly localizable 

sources and detectors how to check finite signal velocity at all? 

One may argue that any violation of Einstein causality would be so rare or so small as to be 

unobservable in practice. But then a good theory should contain this from the beginning. Should 

quantum mechanics with its Hilbert space structure and its idealized measurements at sharp times 

therefore be modified? The above result is based on the use of Hilbert space and a selfadjoint 

time-development operator. This might not be appropriate any longer for systems with infinitely 

many degrees of freedom. 

Conclusion. Fermi's original question on finite signal velocity has been generalized and malyzed in a model-independent way, without the use of any 'bare' theory or any approximations. 

Only positivity of the energy has been used. It has been shown that this leads to violation of 

Einstein causality if one assumes that two subsystems, 'source' and 'detector', can be localized in 

disjoint regions at some initial time. The view has been taken that this difficulty is of a theoretical 

nature, and possible ways out have been discussed. 

References 

[I] E. Fermi, Rev. Mod. Phys. 4, 87 (1932) 

[2] M.I. Shirokov, Yad. Fiz. 4, 1077 (1966). [Sov. J. Nucl. Phys. 4, 774 (1967)l 

[3] W. Heitler and S.T. Ma, Proc. R. Ir. Acad. 52, 123 (1949) 

[4] J. Hamilton, Proc. Phys. Soc. A 62, 12 (1949) 

[5] M. Fierz, Helv. Phys. Acta 23, 731 (1950) 

[6] B. Ferretti, in: Old and New Problems in Elementary Particles, edited by G. Puppi (Academic 

Press, New York, 1968), p. 108 

[7] P.W. Milonni and P.L. Knight, Phys. Rev. A 10, 1096 (1974) 

[8] M.I. Shirokov, Sov. Phys. Usp. 21, 345 (1978) 

[9] M.H. Rubin, Phys. Rev. D 35, 3836 (1987) 

[lo] A.K. Biswas, G. Compagno, G.M. Palma, R. Passante, and R. Persico, Phys. Rev. A 42, 

4291 (1990) 

[ll] A. Valentini, Phys. Lett. A 153, 321 (1991) 

[12] An operator is called positive if all its expectation values are nonnegative. It is then automatically selfadjoint and bounded if defined everywhere. For the expectation values to give 

probabilites the operator has to be bounded by 1, but this will not be used in the following. 

[13] G.C. Hegerfeldt, Phys. Rev. D 10, 3320 (1974) 
1141 A.S. Wightman and S.S. Schweber, Phys. Rev. 98, 812 (1955); B. Gerlach, D. Gromes and 

J. Petzold, Z.Phys. 221, 141 (1969) 

[15] B. Skagerstam, Int. J. Theor. Phys. 15, 213 (1976); J.F. Perez and I.F. Wilde, Phys. Rev. D 

16, 315 (1977) 

[16] G.C. Hegerfeldt and S.N.M. Ruijsenaars, Phys. Rev. D 22, 377 (1980) 

[17] G.C. Hegerfeldt, Phys. Rev. Lett. 54, 2395 (1985); Nucl. Phys. B 6, 231 (1989) 

[18] B. Rosenstein and M. Usher, Phys. Rev. D 36, 2595 (1987) 

[19] Sometimes it is argued more directly that only the noncausally propagating positive-frequency 

field components are actually measured, as in the photon counting probabilities of Glauber. 

This argument is, however, not quite conclusive since positive-frequency components are 

a free-field concept and since, moreover, these counting probabilities hold only in an approximate way; cf. J.R. Klauder and E. C.G. Sudarshan, Fundamentals of Quantum Optics 

(Benjamin, New York, 1968). 

[20] The positive root of a positive selfadjoint operator is uniquely defined and selfadjoint. Eq. 

(11) is then replaced by 

ll(0eB)1'21$o)112 = O 

which in turn implies Eq. (12). 

[21] To see this directly one defines an extension of F4 to the upper half plane by F4(z) = F+(z*)*, 

for Im z > 0. Since F4(t) is real for t E I it follows that the extension is continuous on I, 

and from this one can show that it is analytic for z f lR\I. Hence I is contained in the 

analyticity domain, and since Fd(z) = 0 for z E I it vanishes identically. This is a special 

case of the Schwarz reflexion principle, cf. e.g. N. Levinson, and R.M. Redheffer, Complex 

Variables (Holden-Day, San Francisco, 1970) 

[22] G. Barnet, Bounded Analytic Functions (Academic, New York, 1981), p. 64 

[23] No being a null set this implies also that its complement, the set of t's with Pi(t) positive, 

is dense. However, the proof of this fact given before is more transparent and has therefore 

been included. 

[24] A frequently used model in quantum optics uses an electric dipole interaction, the rotatingwave approximation and cutoffs. Then the Hamiltonian is bounded from below, and thus the 

theorem applies. Hence without any further calculations one knows that in this model there 

is a violation of Einstein causality. 

[25] S. Kikuchi, Z. Phys. 66, 558 (1930) 
PSEUDOMASTER EQUATION FOR THE NO-COUNT 

PROCESS IN A CONTINUOUS PHOTODETECTION* 

Ching %sung Lee 

Department of Physics, Alabama A & M University, Normal, Aliabama 35762 

The detection of cavity radiation with the detector placed outside the 

cavity is studied. Each leaked photon has a certain probability of propagating away without being detected. It is viewed as a continuous quantum 

measurement in which the density matrix is continuously revised according 

to the readout of the detector. The concept of pseudomaster equation for 

the no-count process is introduced; its solution leads to the discovery of the 

superoperator for the same process. It has the potential to become the key 

equation for continuous measurement process. 

Quantum theory of measurement has been a highly controversial topic ever since the pioneer 

work of von Neumann. The process discussed in this original work is now classified as the 

first-kind (nondestructive) measurement, which is described by an instantaneous projection with 

projection operators as its principal mathematical tools. On the other hand, photodetection is 

a second-kind (destructive) measurement. To deal with photodetection the concept of quantum 

measurement must be extended to include continuous measurement process in which the quantum 

state of the subject system must be continuously revised to reflect our most up-to-date knowledge 

about the system resulting from the ongoing measmment process. Superoperators have been 

the principal mathematical tools for such study. It is the purpose of this paper to introduce a 

simple equation which has the potential to become the basic equation to describe the continuous 

measurement process. 

There have been two different approaches to the theory of photoelectric detection. The fim 

approach was initiated by -Wandel [I] and followed by Kelley and Kleiner [2] and by Glauber 

[3]. The second approach was initiated by Mollow [4] and followed by Scully and Lamb [q, 

by She pherd [6], and by Srinivas and Davies [7]. In the first approach, light propagates in 

the open space, it encounters the photodetector, and any unabsorbed photons propagate away. 

In the second approach, both the radiation and the photodetector are enclosed in a cavity, andl 

any photons not absorbed by the detector at one time are available for detection at later times. 

For obvious reason these two different approaches are classified by Mandel [8] as open-system 

model for the former and closed-system model for the latter. 

The formula for the photoelectron distribution obtained in the open-system model was 

criticized by Srinivas and Davies [7] as being a short-time approximation only, it may lead 

to unphysical and meaningless results when time is large. In his response, Mandel [8] pointed 

out that, for the open-system model, it should be understood that the normalization volume must 

be large enough to satisfy the condition L >> ct at all times. He also pointed out that the 

circumstances modelled by the closed system are much less commonly encountered in practice. 

In this paper we introduce a new model that is the hybrid of the -two previous approaches. 

The radiation source is kept inside a cavity but the photodetector is taken outside of the cavity 

to detect only those photons leaked out of one end of the cavity, and any unabsorbed photons 

propagate away. We believe that this hybrid model not only keeps the merits of both previous 

models but also is a closer simulation of the circumstances more frequently encountered in 

practice. For example, if we want to study the statistical properties of photons in the output of 

a laser system, this hybrid model will describe the situation most closely. 
In 1981 Srinivas and Davies [7] first considered the closed-system photodetection as a 

continuous quantum measurement. The information provided by the readout of the detector is 

used to adjust the density matrix continuously so that it always represents our most current 

knowledge about the cavity radiation. The counting process consists of no-count and one-count 

processes, the former lasts a finite duration while the latter occurs instantaneously. The effects 

of the two processes on the density matrix can be represented by two superoperators postulated 

to be as follows: (i) The one-count process is described by the superoperator 3 such that 

where b(t) and @(t+) are the density matrix for the radiation field immediately before and after 

the detection of a photoelectron while and at are the annihilation and creation operators, 

respectively, of a photon. (ii) The no-count process lasting for a duration T is described by the 

superoperator ST such that 

A-t. x -t- exp (-Za a~)j(t) exp (-Za a~) 6(t + T) = S~;(t) ~-J~[B(~) exp (-~ata~)1 9 

where X is the coupling constant. It is obvious from Eqs. (1) and (2) that the evolution of the 

quantum state for the cavity radiation under continuous photon counting is nonunitary. These 

two postulates were verified very recently by Imoto et al. [9] by using some specific microscopic 

model. The system consists of a cavity radiation and a stream of two-level atoms described by 

the well-known Jaynes-Cumming Harniltonian. More detailed theory has been further developed 

by Ueda et al. [lo]. 

These two superoperators are valid for the closed-system model. To find the corresponding 

superoperators for the hybrid model is tEe initial motivation of the present study. We have 

discovered a general method to solve this type of problems, and we shall call it the method of 

pseudomaster equation. 

For simplicity, we assume that the radiation inside the cavity is kept inactive except for 

possible leakage through one side of the cavity due to the less than perfect reflectivity of the 

mirror. Let ,LL be the rate of photon leakage or the inverse of the decay time. It can be expressed 

in terms of the speed of light c, the reflectivity of one end mirror R and the distance between 

the two end mirrors d as [ll] 

Let p(n, t) be the probability that n photons remain inside the cavity at time t. Except for the 

difference in the physical meaning of the constant, the master equation for p(n, t) in the hybrid 

model is exactly the same as the one first derived by Scully and Lamb [5] for the closed-system 

model, i.e., 

which describes the so-called non-referring measurement process, namely, the "free" evolution 

of the photon-num ber distribution without detection or without any adjustment of the quantum 

state according to the readout of the detector. 

Assuming each leaked photon has the probability 5 of being detected, then we believe the 

no-count process can be described by the following difference-differential equation 
where (1 - 5) is the probability that a leaked photon propagates away without been detected. 

Equation (5) is a modification of Eq. (4). One important characteristic of Eq. (5) is that it does 

not conserve the total probability. Because of this characteristic we call it pseudomaster equation. 

The physical meaning of q(n, t) will become clear in later development. 

By the way, the pseudomaster equation for the closed-system model is 

which is quite trivial, so we shall focus our attention on Eq. (5) for the hybrid model. 

The solution to Eq. (5) can be easily obtained through Laplace transformation. Let the 

no-count period begin at t = tl and let us make the identification 

as the initial condition, then the solution to Eq. (5) at a later time t = t2 can be written as 

The first physical meaning of q(n, t) is that it is the unnonnalized form of p(n, t) in nocount process, namely, 

Let Pm(tl, t2) be the probability that m photoelectron^ are detected during the period (tl, t2). We 

are particularly interested in the special case m = 0. We can obtain from Eq. (8) the probability 

that no photoelectron is detected during the period (t l, t 2) as 

This is the second physical meaning of q(n, t). 

To be consistent, the no-count probability must satisfy the condition 

Po(t1, t2)P0(~21 t3) = P0(tl,t3)- 

Using Eqs. (8) and (10) in Eq. (9) we obtain 

Replacing tl and t2 in Eq. (10) by t2 and t3, respectively, and using Eq. (12), we obtain 
so the condition of Eq. (11) is satisfied. 

There are two kinds of waiting times, conditional and unconditional. We pick an arbitrary 

instant and ask: "How long do we have to wait until we detect a photoelectron?"is is 

the unconditional waiting time. If the waiting begins immediately after the detection of a 

photoelectron, then it is called conditional waiting time. From the expression for Po (tl , t?) we 

can also derive the unnormalized distributions for the two different waiting times accordmg to 

[I21 

and 

where (n), is the average number of photons remaining at time t. 

The difference between unconditional and condition waiting times is that the latter begins 

immediately after the detection of a photons. Therefore we can see the effect on the quantum 

state of the radiation due to the detection of a photon by comparing Eqs. (14) and (15). If we 

replace p(n, tl) in Eq. (14) by 

we obtain Eq. (15). On the other hand, if we try to use Eq. (I), we also have 

which is identical to Eq. (16). So we conclude that the effect on the quantum state due to the 

one-count process in the hybrid model is the same as that in the closed-system model, and the 

superoperator for the one-count process in hybrid model remains the same as given in Eq. (1). 

We now try to find the superoperator for the no-count process. Equation (12) provides the 

clue for this search. It turns out to be quite sophisticated because it involves the exponential of 

a superoperator. First let us define a superoperator R(r, p, 5) such that 

Then we can define the exponential of this superoperator as 

which must be applied to both sides of the operand in lock step and cannot be split up into 

two separate parts, with one being the Hermitian conjugate of the other. Let us define another 

superoperator S(r, p) such that 

P At- S(T, p)~(t) = exp (-:iitiir) ~(t) exp ar) , (20) 
which is slightly different fkom that defined in Eq. (2) in the physical meaning of the constant p 

and in that it does not include the renormalization. We are now ready to present the superoperator 

Q(T, 5) for the no-count process such that 

Then the density operator at the end of the no-count process can be written as 

where the difference between the two superoperators Q(T, p, 5) and Q,q(r, p, 5) is that the later 

includes the renormalization. 

It should be pointed out that this superoperator must satisfy the following conditions: 

POP, t + 7) = Tr(Q(7, P, 5)i(t)l, (2) 

and 

Q(r1 + 72, P, 5) = Q(s, P, [)&(TI, P, 5). (25) 

The last semigroup condition is very critical because we have found at least two less sophisticated 

expressions for &(T, p, C), which satisfy the first two conditions, but must be abandoned because 

they do not satisfy the last condition. It can be easily verify that the expression given in Eq. (21) 

satisfy the first two condition. That it also satisfies the last condition can be shown as follows: 

where we used the operator identity [13] ex6tv((8, dt)e-xati = f (de-X, dtex) to change the 00 Ca co k 

orders of some operators and used the summation identity x x A(k, 4) = x x A(k - &, &) 

E=O e=o k=o e=o 

to carry out one summation. 

For a perfect detector with 5 = 1, we have QN(T, A, 1) = ST as given in Eq. (2) for the 

closed-system model. This means the present model with perfect detector is identical to the 

closed-system model as far as mathematics is concerned. On the other hand, when 5 = 0 we 

obtain the superoperator for the non-referring measurement process, which always preserves the 

total probability, i.e., Tr[Q(r, p, O)$(t)] = 1; SO no renormalization is necessary. 

We can also consider a more general "closed-system" model with the detector inside the 

cavity but the cavity has some leakage. The pseudomaster equation for the no-count process 

of such a system can be written as 
where A is the coupling constant between the radiation field and the cavity and ,.L denotes the rate 

of photon leakage. Except for a slight change of constants, the calculations of this generalized 

closed-system model are almost exactly the same as the hybrid model. We just list the most 

significant results as follows: 

and 

Q(T, A, P) = S(T, -I- P)~XP [W, + P, A/(A + ,.L))I. (30) 

In conclusion, we have introduced a hybrid model for photodetection which is the mixture of 

the previous open-system and closed-system models. It is without the defects of the former and 

more realistic than the latter. We have also introduced the concept of pseudomaster equation, the 

solution of which provides the clue to discovering the superoperator for the no-count process. It 

is obvious that, comparing with the superoperator approach, the method of pseudomaster equation 

is simpler, easier to see how to write it down and easier to handle. So we believe it has the 

potential to become the fundamental equation for analyzing various models of photodetections 

as continuous quantum measurement process. 

*This work was supported by the U. S. Navy, Office of Naval Research, under Grant #N00014- 

89-J-1050. 

[I] E. Mandel, Roc. Roy. Soc. London "1, 1037 (1958); ibid. 74, 233 (1959). 

[2] P. C. Kelley and W. H. Kleiner, Phys. Rev. AP36, 316 (1964). 

131 R. J. Glauber, in: Quantum Optics ~nd Electronics, ed. by C. deWitt, A. Blanch, and 

C. Cohen-Tannoudji, Gordon and Breach, New York, 1965, p. 63. 

[4] B. R. Mollow, Phys. Rev. 168, 1896 (1968). 

[5] M. 0. Scully and W. E. Lamb, Phys. Rev. 179, 368 (1969). 

[6] T. J. Shepherd, Optica Acta 28, 567 (1981); ibid. 31, 1399 (1984). 

[7] M. D. Srinivas and E. B. Davies, Optica Acta 28, 981 (1981); ibid. 29, 235 (1982). 

[8] L. Mandel, Optica Acta 28, 1447 (1981). 

[9] N. Irnoto, M. Ueda and T. Ogawa, Phys. Rev. A 41, 4127 (1990). 

[lo] M. Ueda, Quantum Optics 1,131 (1981); Phys. Rev. A 41,3891 (1990); M. Ueda, N. Imoto 

and T. Ogawa, ibid. 41, 3891 (1990). 

[ll] For a rigorous derivation of this expression see, for example, C. T. Lee, Opt. Communi. 27, 

277 (1978). The expression for the decay time is given in Eq. (33) as T = -2t/c In R, but it 

should be pointed out that this formula is for leakage through both ends and the separation 

between the two ends is 21. 

[12] C. T. Lee, Phys. Rev. A 46, 6100 (1992). 

[13] See, for example, W. H. Louisell, Quantum Statistical Properties of Radiation, John Wiley 

& Sons, New York, 1973, p. 154. 
SQUEEZED STATES AND GRAV'bCTON-ENTROPY PRODUCTION 

IN THE EARLY UNIWRSE 

Massirno Giovannini 

Dipedimento di Fisica Teorica , Universita' di Ton'no 

and IMFM, Sezione di Ton'no, Turin, Italy 

Squeezed states are a very useful framework for the quantum treatment of tensor perturbations (i.e. gravitons production) in the early univem. In pafticcular, the non equilibaium 

entropy growth in a c~mological proccess of pair production is completely determind by the 

iatd squeezing parameter and is insensitive to the numkr of particles in the initid 

. The total produced entropy may reprwnt a significant frxtion of the entropy stord 

today in the casmic blackmy rdiation, provided pair prductiora oridnate from a change 

in the bxkground metric at a curmture scale of the Planck order. Within the formalism 

of qumd thermd state it is ah po~~ible to dhuas the stimulaM emission of gavitoas 

from an initial thermal bath,undea the action of the carnie gravitationd bwucksound field, 

We find that at low enerw the grwiton psodluction is enhanced, if compare$ with spow 

tanmus creation from the muurn; as a conwuence , the inflation scale must be lowerd, 

in order not to excd the ohm& @agB qudmpoie anbtropy. This edlpect k important, 

in particular, for model on a symmet~y-breaking tranition which quire, as initial 

condition, a state of thermal quilibdaam a& kmlperataara higher than the inflation wde and 

in which inflation has a minimd dur*bra. 

In order to discuss the gavitofa prduction indud by a cbsmological background transition, the 

stating point is the Enwid wave quation for a t perturbation. For the sake of generdity 

we will t&e i h the wiation of atmt [I], w&ch co gotads, in a string 

cwmologicd to a time-dependence of the &adkin-TwEn dilaton field Ht), The 

linearized wave aqludion is then, in the B Dicke (Strinw) frame [I] 

where hi is the graviton field describing a t eurbation on a given curved background, 

represent& by a homogeneous diagonal metric in which d dimensions escpmd with the scde 

factor a(t) and n dimension contract with the scale factor b(t): 
(conventions: p, v == 1, ..., D = d + n + 1; ij=l, ..., d; a,b=l, ..., n; t is the cosmic time coordinate, 

ad y;j, xb are the metric tensors of two maximally symmetric euclidean manifolds, parametrized 

respectively by "internal" and "external" coordinates xi and y4). 

In terms of the conformal time coordinate 77, defined by dt/dq = a9 equation (1) becomes [I] 

. d-1 where +! = hia~bte-* and V is 

(d-l)aN nb" 1 4- -- + -(d - l)(d - 3) I= 26 4 (f)l +L(n 4 - 2) 

1 1 a'b' 1 a' n b' + -f2 + 5n(d - 1)-- - -(d - 1)-4 - --+' ab 2 a 2 b 

This effective potential takes into account the contribution of the expanding dimensions (a' # O), 

of the contacting dimensions (b' # 0) and of the possible variation in time of the gravitational 

coupling constant(+' # 0). In the case of four expanding dimensions (without dilaton field and 

without contracting dimensions) we recover the standard result, a minimally coupled scalar field 

equation. 

The quantum description of the amplification of scalar or tensor fluctuations, as disc 

(for other references see e.g. [ll]), is based on the separation of the field into background 

solution and first order perturbations, and on the expansion of the solution to the perturbed wave 

equation into 1 in) and 1 out) modes. The complex coefficients of this expansion are interpreted in 

second quantization formalism as annihilatioq and creation operators for a particle (b, bt) and the 

corresponding antiparticle (6,6$). The relation between I in) and I out) mode solution can t hua be 

expressed for each mode k as a Bogoliubov transformation between the I in) operators (b, bt, 6, @) 

and the out ones (a, at, iit, 6) [3) 

a* = c+(k)bk + c:(k)6Lk , i!, = c-(k)bk -I- c;(k)&Ik (5) 

where Ic+12 - Ic-I3 = 1. As noted by Grishchuk and Sidorov (31, by parametrizing the Bogoliubov 

coefficients 4 in terms of the two real numbers r 2 0 and 8, 

c+(k) = coshr(k) 9 cz (k) = ezi@h sinh r(k) (6) 

the relations (I) can be re-written as unitary transformations generated by the (momentumconserving) bmode squeezing operator Ck, 

(r is the so-called squeezing parameter) as 

(and related expressions for B+, 5, ii) 
The mean number of ~roduced gravitons is then, according to equation (8) 

From equation (5) is possible to compute the spectral energy density 

w is the proper frequency related to the comoving one k by w = k/a(t) where a(t) is the scale of 

the expanding background metric. 

We then insert the known expression of c,(w) in eq.(9) and measure p(w), as usual, in units 

of critical energy density p,, defining Q(w) = p(w)/p,. We have, in four dimensions (D = 4) 

expanding with scale factor a(q) "- 7-" in conformal time [4] 

R,(to) zz 10"' is the fraction of the critical energy density present today in the form of radiation; 

a 3 1 is a coefficient parametrizing (in conformal time) the power-law behaviour of the scale 

factor; HI E H(tl) is the curvature scale at the time tl marking the end of inflation and the 

beginning of the radiation-dominated era; wo 2 10'l8 Hz is the minimum amplified frequency 

crossing today the Hubble radius Hg'; wa 2 lQ2wo is the frequency corresponding to the matter 

radiation transition; wl, finally, is the maximum amplified frequency, related to the inflation scale 

by wl ~b) 1011(Hl/Mp)112 Hz (Mp is the Planck maw). 

The computed spectra are constrained by the CMB anisotropy, by the pulsar timing data and 

by the closure density. The bounds on the variation of the spectral energy density becomes bounds 

on the variation of the squeezing parameter, which is given by [1],[3] 

(161 is a model-dependent number of order of unity). 

2 Ther modification of the graviton spectrum 

In this picture the crucial assumption is that the initial state of the gravitons is precisely the 

vacuum. The vacuum, however, is not the moat general initial state for a gravity wave or for a 

generic scalar perturbation[5]. We can mimic a generic initial state with a squeezed number state, 

or, better with a statistical mixture of two mode squeezed number states [6]. In particular, any 

inflationary model based on a temperature dependent phase transition require as initial condition 

a homogeneous thermal state. So, a particularly relevant case is that of a thennal mixture of 

number states. Such initial condition will modify the mean number of particles and the spectral 

energy density . 
For the spmtral enerm density we obtain 16) 

Here Po"" PP'"to) is the proper temperature of the initial thermal state, diabatically rescald 

down to the present obwrvaeion t ime to [po is defined in terms of the comoving temperature as 

@(to) = &(to)]. The effect of the initial finite temperature is to enhance graviton production at 

low fquency with respt to the high fretquency sector of the spectrum. This effect dewnds on 

the due of the initid temperature which, in the context of inflationw models b 

symmetry breding, is greater thw the inflation scale. However, the modification of the sptmm 

is seleviant only if the inflationwy perid is not too long (see [6] for a detailed discussion). 

3 Entropy production from the cosmological amplification 

of vacuum fluctuations 

Unlike the particle spetmm, whi& depends on the initial state, the non eq~libriusrp entmpy 

peh, associatd with the prww of paastick production [7], is not dwted by the mimla 

choice of the initial conditions. 

It is pmsibb indd to inatduce a rasn quilibrim entraps did 

ed states, in which the loss to the reducd demsity mine~x 

disgemioxs in the suductuabnt operators I, 5 whaxse vaarriaarnce! is 

initial due [$I, [9]. In terns of thw oprabeors a and G have the 

i 

ak = -eide [(ah rk - sinh rk)(t - is) + (cmh vk + sinh rk)(& - d&)] 2 (16) 

(the wlative phe hm with respt t n sad$ a way to identifq. the a: aand 5 

operatom with the suductu& ones) arnd the squ number mvehetiona (in the bmis of 

whe~ L, and H, Laguem and the W te polynomials, ly @k zz @-3Pb. It 

sbdd be noted that lations, this wavehction t be simply fmtorized 
in terms of two dwoupled squeezed wcillators in an excited state, which are known to provide 

the usud representation for the on@ mode squ-d nmbr waveforection. 

It is intemting to point out, in pasing, that the wave functions of a two mode squ-4 state, 

as weU as the trmsition pmbability betwen a generic two mde mmkr state andl a &m mode 

squ-4 nugglbrtr state, in the superfluctua~rt wiabla representation (2 , Z) are the same as the 

co~aponding qumtities obt&nd in the conte~ of the "sq " Ladau levels pdbm for the 

elmtron in a uniform mwetic field [12]. Hn the Lmdau 1 lem the two quaturn numb= 

lakllisg the one partick wave functions are the enerm of the electron and the component of the 

angulas momentum pepndiculaas to the plane of the clwid motion of the elmtron. Here the 

two qumtum numkm in the many pwticle wavefunction are rapectively the nuder of gravitom 

with fow-aaaomenturna k (nk) asld with four-momentw -k (n-k). Also to be mentioned is the fact 

that it is pasible, within this fomdism, to comider more generd wavefunctions with nk # n-k 

This is physicdly quident to consider, as initid condition for the gavitons, a state with 

non-zero amber of paticla and non zero four-momentuna. 

The entropy growth for a generic squ naixture of number st&= is the same as for the 

squwd muum [10), 

where p,k is the rduc4 density matrix for the mode k, and the integrated entropy over dl the 

graviton spmtmm is [$I, [9] (fm q. (191, (13)): 

whem SV .v [a(t)mV(t)j3 -- msf is the usud bld-body entropy of the CMB radiation (in te 

able to eqlin the ob94:p"vd cosrmoled 

idation rdiation trmsition is of the order of 

m de is bunadd hraa the CMB 

okp"vatiofa would seem to mle out 

the enkropy of the ur;livem. On the 

(or the fadiation dominated phase) 

ure, like in the *pmbig- models [2] which mi= 

natmdy in duality- string camology, and in which the ure scale can app 

dues (HI 5 MPL) [I] ,[2). 

I am grata to M.Gasperini and G.Veneziano for a fruitful collaboration which led to the many, 

results reviewed in this paper, and for many helphl discuasiom. I whish also to thank the lstituto 

Nazionale di Fisica Nucleare, Sezione di Torino, for the financial mpport. 
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G.Veneziano, Phys.Lett.B 265, 287(1991); 

K.A.Meissner and G.Veneziano, M0d.Phys.Lett.A 6, 3397 (1991); 

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(1970) 
TIME-DEPENDENT VARIATIONAL APPROACH IN TERMS OF 

SQUEEZED COHERENT STATES 

-IMPLICATION TO SEMI-CLASSICAL APPROXIMATIONYasuhiko Tsue 

Department of Physics, Kochi University, Kochi 780, Japan 

Abstract 

A general framework for time-dependent variational approach in terms of squeezed coherent states is constructed with the aim of describing quantal systems by means of classical 

mechanics including higher order quantal effects with the aid of canonicity conditions developed in the time-dependent Hartree-Fock theory. The Maslov phase occurring in a semiclassical quantization rule is investigated in this framework. In the limit of a semi-classical 

approximation in this approach, it is definitely shown that the Maslov phase has a geometric 

nature analogous to the Berry phase. It is also indicated that this squeezed coherent state 

approach is a possible way to go beyond the usual WKB approximation. 

1 Introduction 

In many-body problems, a great interest is paid to describe quantal systems in terms of a few 

classical variables because we are especially interested in some particular characteristic motions 

in quantal systems, for example, nuclear collective motions in nucleus and the dynamics of soliton 

models of baryons as the low energy effective theory of QCD. As is well known, in various quantal 

systems, if one takes the limit of "large N", the quantum theories are well described as the classical 

ones [I]. However, since, for example, we are interested in the nuclei as finite quantum manyparticle systems, it should be noticed that the deviations from classical dynamics can never be 

neglected. 

With the aim of establishing a possible framework for the classical description of quantal 

systems, we give a rather general framework to describe quantal systems by means of classical 

mechanics including the higher order quantal effects. Our basic idea is formulated with the use of 

the time-dependent variational principle utilizing the squeezed coherent states [2], paying strong 

attention to canonicity conditions developed in the TDHF theory [3][4]. 

In this paper, first, we briefly review our time-dependent variational approach with squeezed 

coherent states developed in Refs.[2] and [5]. Secondly, we show that, when we take a semiclassical limit in our framework, it is clearly realized that the Maslov correction occurring in the 

semi-classic$ quantization procedure in the usual WKB method can directly be interpreted as 

the Berry phase [6]. Although it has originally been pointed out that the Maslov correction is a 

kind of the Berry phase [7], it is possible to take account of the higher order quantum effects than 

that of the semi-classical approximation in our framework. Furthermore, it is understood that our 

approach is a possible way to go beyond the usual WKB approximation. 
2 Formulation 

In this section, we give the framework of the time-dependent variational approach in terms of 

squeezed coherent states [5][6]. We start with the general squeezed coherent state as 

Here, 10) is a vacuum state with respect to boson operators 6k, and ak and Bkkl are the timedependent c-number variables. The state I@(P)) is called the squeezed vacuum. In the following 

consideration, we are restricted ourselves to deal with boson systems composed of one kind of 

boson. If we want to consider the systems described by su(2)-algebra such as the Lipkin model, it 

is enough to express the algebra by the use of two-kinds of boson operators, the representation of 

which is well known as Schwinger boson representation. Then, Bkki is taken as Bkki = Bkbkk1(lC = 

42) PI. 

With the aid of definitions of coordinate-momentum operators Q = m(h + 81) and a = 

(-i)@(6 - at), the above squeezed coherent state can be rewritten as the following Gaussiantype state : 

where we. define the following variables as 

q - &(a + a*) , p = (-i)/$ - a*) , 

Here, c-number variable 52 is divided into real and imaginary parts, and for later convenience, 

the part "1" which represents the width of the wave packet of the original vacuum is extracted 

from real part. In the following, we will start with this expression of the squeezed coherent state 

in Eq.(3). Thus, we treat the variables q,p, G and IT as dynamical ones. Here, note that the 

variable G is positive definite and never takes zero. This fact is important in order to present an 

interpretation of the usual WKB approximation within our framework. 

In general, we can calculate the expectation values for arbitrary operators in terms of the 

Wigner transform : 

(@(t)ldl@(t)) = exp{h~}~w(q, P) (8) 
Here, the derivative operator 5 and the Wigner transform Ow(q,p) are defined as 

where the relation Q lq) = qlq) is satisfied. The Wigner transform Ow only depends on q and p 

and the variables G and JI are' introduced by the operation of $. 

We need to determine the time-development of the variables q(t), p(t), G(t) and II(t), so that 

the time-development of the state I@(t)) is determined. We can carry this out with th; aiddof the 

time-dependent variational principle similar to the TDHF theory : 

Furthermore, we impose the canonicity conditions developed in the TDHF theory [4] in order to 

extract canonical variables. Taking the freedom of canonical transformations into account, we can 

express the canonicity conditions in the following form : 

where dF = d/dF is defined and s(X, Y) which represents the freedom of the canonical transformation is an arbitrary function of canonical variables X and Y. We can take possible solutions of the 

above canonicity conditions as (X, Y) = (q,p) and (hG, JI). Therefore, the resultant equations 

of motion derived from the time-dependent variational principle are nothing but the canonical 

equations of motion due to the canonicity conditiom : 

Here, the dot denotes the time-derivative and the c-number Hamiltonian function H is defined 

by H = (@(t)l~l@(t)) = e"~~(g,~). Thus, our main task is reduced to solving the classical 

equations of motion under appropriate initial conditions in the canonical form. As is seen from 

Eqs.(l3) and (14), roughly speaking, the variables q and p represent the classical motion and G 

and H may be regarded as the classical images of quantum fluctuations, 

3 Maslov Phase as Berry Phase 

In this section, we give a relation between the usual WKB approximation and our framework of 

the time-dependent variational approach with squeezed coherent states. Then, it is clearly shown 
that the Maslov correction occurring in the semi-classical quantization procedure in the usual 

WKB method can directly be interpreted as the Berry or geometric phase. 

In our framework, it is necessary to choose the initial conditions for newly-introduced variables 

as the classical image of quantum fluctuations, that is G and II. We adopt two criteria developed in 

our papers [2][6], namely the requirements of "Least Quantal Effects" and "Minimal Uncertainty" 

at initial time. As for the "classical parts" q(t) and p(t), we may select the initial conditions in a 

similar way to the usual TDHF theory [9]. 

Now, if the limit of h -, 0 is taken in Eq.(13), then these equations are reduced to the usual 

classical Hamilton's equations of motion. Thus, it is expected that the variables G and II represent 

the quantum fluctuations around the above-mentioned classical motions. Therefore, it is realized 

that the semi-classical limit in our framework is to take the limit of h + 0 in the equations of 

motion in Eqs.(13) and (14). In this limit, we can solve the equations of motion for G and 11 

in Eq.(14) and express these solutions in terms of the classical orbit (q(t),p(t)). The results are 

obtained as follows : 

where A E dq/dqo, B I dq/dpo, C dp/dqO and D dp/dpo are defined and the variables 

with subscript 0 represent the initial values. Since the variables G and 11 are always accompanied 

with h, the expectation value of Hamiltonian should also be taken into account up to the order 

of h in this semi-classical limit. Namely, as the approximate energy expectation value, we adopt 

H - Hd(q, P) + hHql(q, P, G, 

In the usual WKB considerations, the energy is kept in the classical form which does not 

include h. Therefore, in our framework of the time-dependent variational approach with the 

squeezed coherent states, h J dtHql in the action integral should be combined with the requantized 

phase factor J dt(@(t)lihdJ&l@(t)) in order to compare our treatment with the usual WKB one 

properly. Thus, action function is written as 

LC {Pi + h AB 4G - AB] -Hcl} (+total time-derivative term) , 

where it is assumed that the classical orbit is a periodic one, the period of which is written by 

Td. According to the requantization procedure similar to the TDHF theory, we set the modified 

action integral except for the part of "energy" to integer n times 27rh : 

We rewrite the above relation as 

where I' is defined and is explicitly calculated with the relation G r IzI2 J2 : 
v : integer 

The above expression is nothing but a requantization condition in the semi-classical approximation. 

Here, z (G) never passes through the point of migin z = 0 (G = 0) as is previously mentioned. 

Then, G or z undergoes the time-evolution accompanied with the classical motion q(t) through 

the variables A and B. The integer v, which corresponds to the Maslov correction occurring in the 

usual semi-classical quantization procedure in the WKB method, appears as the winding number 

around the origin z = 0 associated with the classical motion. These situations are analogous to 

the case encountered for the Berry phase [10][11]. Namely, it is understood that, in our squeezed 

coherent state approach, the Maslov correction or the Maslov "phase" corresponds to the Berry 

phase and the classical orbit plays a role of an "external parameter." The coefficient ./r in the 

Maslov phase I' may be interpreted as a half of the solid angle that subtends at the "singular 

point" G = 0 (z = 0). Furthermore, the parameter governing the approximation is h7 so that h 

plays a role of an "adiabatic parameter" in the consideration of the Berry phase. Therefore, it 

is clearly realized in our approach that the Maslov correction has the similar geometric aspect to 

the Berry phase. It is thus shown that the quantum effects are automatically contained in the 

semi-classical limit in our squeezed coherent states approach. 

4 Beyond the WKB Approximation 

In the usual WKB method, the energy of the system is kept in the classical form which does 

not include ti and the quantum effects are taken into account only through the requantization 

condition. On the other hand, in our time-dependent variational approach with the squeezed coherent states, the energy is' the expectation value of the Hamiltonian with respect to the squeezed 

coherent state itself, that is H = (~(t)l~l~(t)), so that the higher order quantum effects of h 

are already included. Thus, under the conception of our squeezed coherent states approach, the 

E ECKART 

EXACT 

I This Case 1 I 

I I 

FIG. 1. The energies are shown in the case of Eckart V(Q) = 

-Uo/cosh2 aQ, in which we set the parameters (Io = 1 and ru = 0.1 for simplicity. 

"This Case" represents the energy calculated numerically in our squeezed coherent 

state approach. "WKB7' and "Exact" represent the energies obtained by the usual 

W K B approximation and exact eigenvalue of the ground st ate, respectively. 
energy is calculated as follows : First, we analytically or numerically solve the self-consistent 

equations of motion in Eqs.(13) and (14). Secondly, we calculate the energy expectation value 

of the Hamiltonian with respect to the squeezed coherent state which includes the higher order 

effects of ti than the quantum effects in the WKB approximation. 

For example, in the case of Eckart potential, V(Q) = -u0/ cosh2 a~, the energy expectation 

value calculated numerically in our framework is compared with the exact energy eigenvalue 

and the usual WKB energy in Fig.1. Here, the initial conditions in our approach are taken as 

q0 = po = 0. Therefore, the energy thus obtained corresponds to the ground state one. It can be 

seen from Fig.1 that our treatment gives a fairly good result owing to the incorporation of the 

higher order effects of ti. 

In summary, we have given the framework of the time-dependent variational approach in terms 

of the squeezed coherent states with the aim of describing quantal systems by means of the classical 

dynamics. In our squeezed coherent states approach, the Maslov correction that appears in the 

usual semi-classical quantization procedure is clearly realized as the Berry or geometric phase. 

Furthermore, our approach is a possible way to go beyond the WKB approximation. 

5 Acknowledgments 

The author is greatly indebted to Dr. Y. Fujiwara, Professor A. Kuriyama and Professor 

M. Yamarnura for the collaboration at the early stage of this work. He also would like to thank 

Professor H. Horiuchi, Dr. T. Fukui and Dr. H. Fujii for fruitful discussions and comments. 

References 

[I] L. G. Yaffe, Rev. Mod. Phys. 54, 407 (1982). 

[2] Y. Tsue, Y. Fujiwara, A. Kuriyama and M. Yamamura, Prog. Theor. Phys.'85, 693 (1991). 

[3] T. Marumori, T. Maskawa, F. Sakata and A. Kuriyama, Prog. Theor. Phys. 64, 1294 (1980). 

[4] M. Yamamura and A. Kuriyama, Prog. Theor. Phys. Suppl. 93 (1987), and references therein. 

[5] Y. Tsue and Y. Fujiwara, Prog. Theor. Phys. 86, 443, 469 (1991). 

[6] Y. Tsue, Prog. Theor. Phys. 88, 911 (1992). 

[7] R. G. Littlejohn, Phys. Rev. Lett. 61, 2159 (1988). 

[8] M. Yamamura, A. Kuriyama and Y. Tsue, Prog. Theor. Phys. 88, 719 (1992). 

[9] K.-K. Kan, J. J. Griffin, P. C. Lichtner and M. Dworzecka, Nucl. Phys. A332, 109 (1979). 

[lo] M. V. Berry, Proc. Roy. Soc. London A392, 45 (1984). 

[ll] Geometric Phases in Physics, edited by A. Shapere and F. Wilczek, (World Scientific, Singapore, 1989). 
DECOHEmNCE AND DISSIPPATION FOR A QUANTUM 

SYSTEM COUPLED TO A LOCAL ENVIRONMENT 

Michael R. Gallis 

Department of Physics, Penn State University/Schuylkill Campus 

Schuylkill Haven, Pennsylvania 17972 

Internet:mrg3@psuvm.psu. edu 

Abstract 

Decoherence and dissipation in quantum systems has been studied extensively in the 

context of Quantum Brownian Motion. Effective decoherence in coarse grained quantum 

systems has been a central issue in recent efforts by Zurek and by Hartle and Gell-Mann 

to address the Quantum Measurement Problem. Although these models can yield very 

general classical phenomenology, they are incapable of reproducing relevant characteristics 

expected of a local environment on a quantum system, such as the characteristic dependence 

of decoherence on environment spatial correlations. I discuss the characteristics of Quantum 

Brownian Motion in a local environment by examining aspects of first principle calculations 

and by the construction of phenomenological models. Effective quantum Langevin equations 

and master equations are presented in a variety of representations. Comparisons are made 

with standard results such as the. Caldeira-Leggett master equation. 

1 Introduction and Motivation 

Decoherence via coarse graining has been studied in the context of quantum measurement theory 

by Zurek[l] and by Hartle and Gell-Mann[2] as a mechanism which leads to the emergence of classical properties. Recent efforts have focused on the decoherence effects of a heat bath, which has 

also been examined in detail in the study of quantum brownian motion. Decoherence is identified 

as the (effective) suppression of interference terms in the density operator (p(x , x') , x # 2). It has 

been pointed out that most of the models which have been considered are somewhat simplistic and 

cannot reproduce the phenomenological features expected of a system which interacts locally with 

a homogeneous and isotropic environment [3]. In this paper I describe the perceived shortcomings 

of existing models and illustrate the construction of a phenomenological quantum master equation 

which contains many features expected from local coupling to a homogeneous environment 141. 

Although decoherence is the most interesting feature of the effects of a heat bath, dissipation 

(and other effects) also generally appear in the dynamics of the density operator: 

ap(x7 t, = Hamiltonian terms + Dissipation terms + . - - g(x, x1)p(x, x'; t). at 

The decoherence term appears as a (spatially dependent) decay term in the evolution equation, 

and can be understood in terms of effective fluctuating forces, or potentials: [5, 61 
Typical models have a quadratic form, g(x, y) oc (x - y)2 for the decoherence term, corresponding 

to a fluctuating force which is independent of position. However, for a local bath one expects 

the correlation function to die off at some characteristic length scale (the correlation length of 

the environment), which has some important ramifications for decoherence. For a quadratic form 

of decoherence, the decay rate of the interference terms in the density matrix increases without 

bound, while for a local model the decay rate saturates at separations (between x and x') much 

larger than the correlation length of the environment, reflecting the independence of environment 

fluctuations at large separations. As it turns out, the quadratic form can be considered a short 

length scale approximation of a more detailed model. 

To consider the decoherence effects of an environment, simultaneous treatment of dissipation 

is necessary since decoherence and dissipative effects both generally arise from the same source 

(the interaction with a heat bath). For simplicity, I consider only linear dissipation, that is 

As an example of quantum dissipative evolution, Dekker[7] has constructed a phenomenological 

master equation which includes ohmic dissipation and quadratic decoherence: 

The Caldeira-Leggett [8] master equation is obtained from a first-principle calculation for the effects 

of a simple thermal bath. With an appropriate choice of parameters for the Dekker model, the 

Calderia-Leggett master equation can be reproduced. 

Many open system models can reduce to the same classical phenomenology, particularly in the 

Markov regime, and yet have significant differences for a quantum sytem in that same regime. To 

illustrate this "richness" of quantum dissipative models, consider a rather generic oscillator bath 

model (following Zwanzig[9]) : 

The classical calculations (the results of which are presumably reproduced in at least some limit of 

the quantum model) are relatively straightforward. The classical fluctuation-dissipation relation 

between the fluctuating forces and the nonlinear dissipation kernal emerges naturally, and in the 

usual Markov limit becomes: 

and a simple langevin equation can (at least in principle) be obtained: 

For a homogeneous environment, the dissipation constant would be independent of position. 

Some observations about the Markov limit are in order. For the classical picture, the spatial 

correlations of the fluctuating forces are irrelevant. After all, the particle can only be in one place 
at one time. For a quantum system one must consider superpositions between the particle at 

different locations, i .e. superpositions between different trajectories for the particle. My point is 

that different models may produce the same classical phenomenology, but have some important 

differences for the quantum case, in particular for the effective decoherence due to the environment. 

In order to help motivate some choices which will be required for the construction of the new 

model, consider a particle locally coupled to a scalar field. This particular model is a natural 

extension of one considered by Unruh and Zurek[lO]. The action for this model is given by: 

This model produces approximately ohmic dissipation in one dimension[8, 111. In addition, one can 

extract from the influence functional the effective correlation function of the fluctuating forces[5, 

111: 

< F(x, T) . F(y, s) > 0 = - hE2 / ddkk2 {cothLF) cos(wt) cos(k (x - y)) 2(2r)d 

This correlation function results from independent contributions from each mode of oscillation of 

the field. With some of the chasacteristics suggested by this local environment in mind, I now 

turn to the actual construction of the model. 

2 The Phenomenological Model 

The initial form of the evolution of the density orerator is taken to be in the Lindblad[l2] form 

(Schrodinger picture) : 

for which there is a corresponding form for the Heisenberg picture L*[O] which can readily be 

obtained from the cyclic properties of the trace. Fbr a finite dimensional Hilbert space, this form 

is the most general for a completely positive dynamical semigroup. For infinite dimensional Hilbert 

spaces, it is a reasonable starting point. I will be focusing on the nonunitary part of the evolution, 

AL. 

The construction of the model is essentially the determination of the operators V, subject to 

the constraint that the dissipation is ohmic (expressed as an operator condition). This constraint 

produces the "correct" classical phenomenology, but does not completely determine the model. 

However, linear dissipation almost forces the V, to be at most linear in momentum, that is 

Homogeneity and isotropy also serve to constrain the model. Assuming some sort of mode by 

mode interaction with a field, a reasonable choice is given by: 
The discrete index p has been replaced by the continuous index k. The model is then completely 

specified by the complex functions a and P. 

The resulting nonunitary contribution to the Schrodinger equation is given by the expression: 

The position representation of the new model is given by: 

"""; t, = Hamiltonian terms - (1 dkY(1- cos k(x - 2'))) p(x, XI; 1) at 

The first nonhamiltonian term is responsible for decoherence. The corresponding noise spatial 

correlation is determined by a(k). The characteristic length should be on the order of the inverse 

of the "width" of la(k)I2 in k space. The second nonhamiltonian term is responsible for the 

dissipation. Clearly the dissipation and other terms are more complicated in this new model. 

However, that would also be expected from a more elaborate first principle calculation. 

By examining the Eherenfest relations of physical quantities using L* , some interesting physical 

features of the new model emerge. By construction, the average position and momentum obey 

relations corresponding to ohmic dissipation: 

where 

With only limited constraints on cr and P (y must be positive), the kinetic energy is seen to be 

t hermalized: 

where 

The effective temperature is determined by cr and P. 
A low length scale approximation of the new model can be obtained by expanding the exponential terms in powers of k . x: 

p . x, (k . p, p}] + J&k 'm(a(k)"(k)) x, [k . p, p]]. (19) 

ti 

The lowest nonvanishing terms are second order, which exactly reproduces the Dekker master 

equation for 1 dimension. As a result, we can think of the Dekker or Caldeira-Leggett equations 

as a low length scale approximation for more general models. 

On the other hand, the Caldeira-Leggett master equation, 

can be considered a special case of the Dekker master equation, with the Dxp terms equal to zero 

(which Dekker has argued should be the case) and an additional low momentum approximation 

which ignores the Dxx term. With this type of special case in mind, we can construct a low 

momentum approximation for the new model which includes only the decoherence and dissipation 

terms: 

This would seem to be a likely starting point for applications of this model. However, this 

approximation is not a positive form for the dynamics. 

Finally, I would like to look at the Wigner representation of the new model, which has some 

interesting features. If we expand the terms of the evolution equation in powers of ti (in the same 

manner as is typically done with the potential), 

the lowest order terms correspond exactly to the Wigner representation of the Dekker equation. 

The Wigner representation of the Dekker equation is a standard classical type of diffusion equation. This illustrates the idea that the "classical" nature of the system emerges when coherent 

superpositions are not important in the dynamics. In this case, the relevant superpositions are 
between different locations separated by distances on the order of the environment correlation 

length. 

The convolution theorem can also be used to write down the Wigner representation of the 

evolution: 

W(q, p) = (Hamiltonian terms) - - /~#(P-P')w(~,P-P') P'Reb* (G) P (%)I 

ti ti2 

la12(k) I. (9 l2 

- (J~~~)W(~,PI+ J~P~w(P,P-PI) ti2 

\ I 

Il.Im[a* (f) a (%)I 

ti2 

1 a2 p218(f) l2 + 2Wq,pp(23) + DXX(~~-$)W~~P)+/~P~ n2 (48 

One apparent effect in the new model is a spreading induced by these convolution terms. 

In summary, a new phenomenological master equation for ohmic dissipation and decoherence 

has been constructed which has completely positive dynamics. The new model has the features 

expected from local coupling to a homogeneous environment: specifically, the evolution is istropic 

and translationally invariant. Spatial correlations of the environment appear explicitly in the 

models. The new model also includes previous results as low length scale approximations. 

References 

[I] W. H. Zurek, S. Habib and J. P. Paz, Phys. Rev. Lett. 70, 1187 (1993). W. H. Zurek, Prog. 

Theor. Phys. 89, 281 (1993). 

121 M Gell-Mann and J. B. Hartle, Phys. Rev. D 47, 3345 (1993). 

[3] M. R. Gallis and G. N. Fleming, Phys. Rev. A 42, 38 1990. 

[4] Some of these results have appeared in: M. R. Gallis, Phys. Rev. A 48, 1028 (1993). 

[5] M. R. Gallis, Phys. Rev. A 45, 47 (1992). 

[S] L. Dicisi, Phys. Lett. A 112, 288 (1985). 

[7] H Dekker, Phys. Rep. 80, 1 (1981). 

[8] A. 0. Caldeira and A. J. Leggett, Physica (Utrecht) 121 A, 587 (1983). 

[9] R. Zwanzig, J. Stat. Phys 9, 215 (1973). 

[lo] W. G. Unruh and W. H. Zurek, Phys. Rev. D40, 1071 (1989). 

[ll] M, R. Gallis, unpublished. 

[12] G. Lindblad, Commun. Math. Phys. 48, 119 (1976). 
GENERAL PROPERTIES OF QUANTUM OPTICAL SYSTEMS 

IN A STRONG FIELD LIMIT 

S. M. Chumakov 

I.I.M.A.S. -Cuernavaca , Universidad Nacional Autdnoma de Me'scico , Apartado Postal 139- B, 621 91, Cuernavaca, Mor, Mkxico 

A.B.Klimov 

Fac. de Ciencias Fisico-Matemciticas 

Universidad de Guadalajara, 

Corregidora 500, 44100 Guadalajara, Jal, M6xico 

Abstract 

We investigate the dynamics of an arbitrary atomic system (n -level atoms or many n-level 

atoms) interacting with a resonant quantized mode of an em field. If the initial field state 

is a coherent state with a large photon number then the system dynamics possesses some 

general features, independently of the particular structure of the atomic system. Namely, 

trapping states, factorization of the wave function, collapses and revivals of the atomic energy 

oscillations are discussed. 

1 Introduction 

The Jaynes - Cummings Model is of principle import-ance in quantum optics. It consists of a single 

atom interacting with a single mode of quantized em field in a lossless cavity. The properties of 

JCM in the region of a strong coherent quantum field are: 

1. Collapses and revivals of atomic inversion oscillations [I]. 

2. The existance of trapping states [2]. (These are initial atomic states which lead to the 

constant mean value of atomic energy in the course of the interaction with the field). 

3. Wave Function (WF) factorization in the trapping states [3]. (This very unusual property 

of the JCM trapping states means that the field and atomic subsystems remain to a high 

accuracy in pure states, in spite of the presence of interaction). 

4. WF factorization for an arbitrary initial atomic state at a half revival time. For this time 

moment the field state is a coherent superposition of macroscopically different states (so 

called Schrodinger cat) [4, 31. 

Here, we address to ourselves the following question: 

Which phenomena from this list will survive for an arbitrary atomic system interacting with a 

strong quantum field ? 
2 What is an arbitrary atomic system ? 

We start with some examples: 

1. many two-level atoms in a cavity (the Dicke Model, [5]); 

2. many three-level atoms of arbitrary level configurations in a cavity; 

3. many n-level atoms of arbitrary degeneracy of the levels. 

All these systems are described by Hamiltonians of the form: 

Here, ht , d are photon operators; is the bare Hamiltonian of the atomic system; 2+, k- are 

atomic transition operators. We suppose, that the following commutation relations are valid: 

Ah [A, k+] = R+ [h, X-] = -x- (2) 

The model formulated in such a way implies the Rotating Wave Approximation and, therefore, 

the excitation number conservation: 

[H, A] = 0, N = hth + A. (3) 

It is usually the case in Quantum Optics. We do not impose any conditions on the commutator 

which leads to a large freedom in the specification of the atomic subsystem. 

We adopt also the exact resonance condition. It means that transition frequencies between 

neighboring levels are equal to the field frequency w. This condition is imposed *ith a sake of 

simplicity (the arbitrary detunings can be also involved in our approach). Stress, that the cavity 

is supposed to be a perfect one. We do not discuss the dissipation processes here. 

3 The classical field limit 

For to approach the classical field limit one has to take a coherent field states (CS) with large 

photon numbers. Then one may substitute 

where a fie" is a CS parameter. Then the quantum interaction Hamiltonian becomes proportional to the operator 

H~~~ + HCl = eidk+ + ewi4R-. 

Hcl has the sense of the atomic Hamiltonian in a constant ,classical field. We will call the eigenvectors 

kl I p_ >at = A; I p_ >at (4) 

the semiclassical eigenvectors. 
4 Factorization of the Wave Function 

We now formulate our principle result [6]. Let initial atomic state is semiclassical eigenstate and 

initial field state is CS with a large photon number: 

Then the WF will be approximately factorized for the time up to gt N K. The factorized WF has 

a form: 

Here, pn - initial CS amplitudes, C is the energy ground level of the bare atomic system, e-ict 

is a phase factor which we do not write down explicitly here. 

We do not give here the proof of eqs.(6) (see [6]). Instead, let us qualitatively describe the 

system behavior. Eqs.(6) stand that the field and the atomic subsystem remain approximately in 

pure states in the course of the evolution (not in the mixed ones!). However, the two subsystems 

essentially interact. The field evolution depends on the semiclassical atomic eigenfrequency Xi. 

The field state rotates along the circle of radius &? in the phase space slowly loosing the shape 

of the initial CS (spreading in phase). 

The atomic subsystem is rotated by the free atomic hamiltonian ir with the angular velocity 

dependent on the initial photon number E. 

We have proved eqs.(6) by means of the perturbation theory with the operator-valued small 

parameter l/(k + 112) . We neglected the terms of order O(ii-'I2) the amplitudes. However, we 

kept higher accuracy in the frequencies (neglecting the terms of order 0 (11~) ) . It means that 

our approximation holds for times up to gt K. It is a usual situation, that the system dynamics 

is more sensitive to the small corrections in frequencies, than in amplitudes. 

5 Trapping states 

It follows immediately from eqs. (6), that all the semiclassical eigenstates of an arbitrary atomic 

system are trapping states, i.e. the mean value of the atomic inversion does not evolve with time: 

6 

at < 2 I h I 2 >at = const. 

This result holds with the accuracy 0(ii-'I2), due to our accuracy for the transition amplitudes. 
6 Collapses and Revivals 

This phenomenon, discovered initially for the JCM [I] appears for an arbitrary atomic system in 

a strong coherent field. Indeed, an arbitrary initial state of the atomic system can be expanded 

in the basis of the semiclassical eigenstates (4): 

Then, for the atomic inversion we get: 

The atomic inversion is determined by the scalar products of the field states (6): 

This is a direct generalization of the well-known unharmonic series for the JCM, which also 

contains collapsing and reviving Rabi oscillatiorrs. The collapse, revival times and envelopes can 

be easily found. 

The atomic matrix elements entering in eq.(6) are slowly varying functions oscillating with the 

frequency eq.(7). They may modulate the revival envelopes. However, for the Dicke model case 

these matrix elements to be equal to zero if p # q f 1 and the correspondent revivals disappear. 

If p = q f 1 , the atomic matrix elements equ~ls to 1 just for the time moments of revivals. 

7 Schrodinger Cat. 

For the JCM case there are only two factorized states. An arbitrary initial state leads to the 

superposition of them. The two JCM atomic states I 4(t) >, p = 0,l are nothing but two 

different spin-1 /2 coherent states and they can be transformed one to another by rotation. It just 

happens at a half revival time tR/2 [3]: 

and then the system WF is factorized for an arbitrary initial atomic state: 

Therefore, at this time moment the field WF is the quantum superposition of macroscopically 

different states I Bp(t) >, p = 0,l. 

It is clear, that for larger atomic system (say for spin-1) not all different atomic semiclassical 

eigenstates can be transformed one into another by rotation, and the Schrodinger cat is absent 

for arbitrary initial atomic state (as it has been recently noticed in the work [7] 
8 Conclusions 

We have solved the problem of the interaction of an arbitrary atomic system with the strong quantized em field in a lossless cavity. The key point of our solution is the wave function factorization 

(6) for the specially chosen initial states (4). 

Since these states form a complete basis, this gives possibilities for an exhaustive description 

of the system dynamics. 

Being reduced to the JCM, our results reproduce the treatment of J. Gea-Banacloche 131. 

For the case of the Dicke model, they correspond to the first two orders of the perturbation 

theory proposed in the work [8] and can be treated as a direct generalization of that scheme for 

the arbitrary atomic system. From the mathematical point of view, our treatment is connected 

with the concept of dynamical symmetry group for the quantum optical systems [9]. (Note, that 

this dynamical symmetry is approximate rather than exact one). 

Stress, that our method allows to make explicit analytical calculations of any physical quantities 

for the systems under study. 

9 Acknowledgments 

S.Ch. and A.K. acknowledge the support from CONACYT. 

References 

[I] N.B. Narozhny, J.J. Sanchez-Mondragon and J.H. Eberly, Phys.Rev.A 23, 236 (1981). 

[2] K. Zaheer, M.S. Zubairy, Phys.Rev.A 39, 2000 (1989); 

[3] J. Gea-Banacloche, Phys.Rev.A 44,5913 (1991); ibid. 46, 7307 (1992); ibid. 47, 2221 (1993). 

[4] S.J.D. Phoenix, P.L. Knight, Phys.Rev.A 44, 6023 (1991). 

153 R.H. Dicke, Phys.Rev. 93, 99 (1954). 

[6] S.M.Chumakov, A.B.Klimov and J.J.Sanchez-Mondragon, submitted to Phys.Rev. A (1993). 

[8] M. Kozierowski, A.A. Mamedov and S.M. Chumakov, Phys.Rev. A 42, 1762 (1990); M. 

Kozierowski, S.M. Chumakov, J. ~wiatlowski and A.A. Marnedov, Phys.Rev. A 46, 7220 

(1992). 

[9] I.A. Malkin, V.I. Man'ko, Dynamical symmetries and coherent states of quantum systems 

(Nauka, Moscow, 1979, in russian ). 
PROBING THE ANTIFERROMAGNETIC LONG-RANGE 

ORDER WITH GEAUBER SPIN STATES 

G. G. Cabrera 

Institute de Fisica "Gleb Wataghinn, 

Universidade Estadual de Campinas (UNICAMP), C. P. 61 65, 

Campinas 13083-970, SP, Brazil 

Abstract 

It is well known that the ground state of low-dimensional antiferromagnets deviates from 

NBel states due to strong quantum fluctuations. Even in the presence of long-range order, 

those fluctuations produce a substantial reduction of the magnetic moment from its saturation value. Numerical simulations in anisotropic antiferromagnetic chains suggest that 

quantum fluctuations over N6el order appear in the form of localized reversal of pairs of 

neighboring spins. In this paper, we propose a coherent state representation for the ground 

state to describe the above situation. In the one-dimensional case, our wave function corresponds to a two-mode Glauber state, when the NCel state is used as a reference, while 

the boson fields are associated to coherent flip of spin pairs. The coherence hfests itself 

through the antiferromagnetic long-range order that survives the action of quantum fluctuations. The present representation is different from the standard zero-point spin wave state, 

and is asymptotically exact in the limit of strong anisotropy. The fermionic version of the 

theory, obtained through the Jordan-Wigner transformation, is also investigated. 

1 Introduction 

The Heisenberg model has been extensively studied for many years as a non trivial many-body 

problem in quantum magnetism. For low dimensional systems, the ground state deviates from NCel 

ordering due to strong quantum fluctuations [I]. The determination of this ground state represents 

a fascinating problem, that in one dimension originated a whole branch of Mathematical Physics 

based in the so called "Bethe-Ansatz" technique [2]. However, exact solutions are extremely 

intricate, very often not susceptible of a direct physical intuition, and in the case of the Heisenberg 

model, they are restricted to one dimension. 

In this contribution, we would like to present a novel approach based in a localized description 

of quantum fluctuations. If one takes as a reference the Ising limit, with a ground state of 

N&l type, switching the transverse part of the Heisenberg Hamiltonian may be visualized as 

a disordering process, where pairs of neighboring spins are simultaneously flipped, the ground 

state being a quantum superposition of admixtures contained in the manifold of total S, = 0. 

This effect has been systematically observed in numerical simulations for anisotropic Heisenberg 

chains [3], and was used by Lagos and the author as the heuristic base for the construction of a 

trial solution [I]. To fix ideas, we will concentrate in the case of spin 112, and most of the examples 

will refer to a one-dimensional system. The theory can be extended to arbitrary dimension [4], 
and to arbitrary value of the spin [53. The antiferromagnetic Heisenberg Hamiltonian with axial 

anisotropic exchange can be written as: 

where i, j are site indexes for nearest neighbors, J > 0 is the antiferromagnetic exchange, the S's 

are spin 1/2 operators, and a is the axial anisotropy parameter. Special cases of Hamiltonian (1) 

are: i) cr = 0, the Ising case; ii) cr = 1, the isotropic Heisenberg model; iii) a -, m, the so called 

'XY-model'. 

It is important to stress that the approach that will be proposed here is not perturbative, in 

spite that the Ising limit is considered as a departure point for its formulation. The method works 

successfully in the axial anisotropic regime, and is asymptotically exact for high anisotropy. Near 

the isotropic point, however, a delocalization transition occurs, and the linear spin wave theory 

becomes a better approximation when compared with exact results or numerical simulations [6]. 

However, the treatment can be extended in a variational way to account for the isotropic case, 

or the Heisenberg-XY regime [7]. In particular, for two dimensions, isotropic exchange, and spin 

S = 1/2, the ground energy deviates less than 0.5% [7] from results obtained by elaborate Monte 

Carlo calculations [8]. 

The Wave Function 

The Hamiltonian (1) is said to represent the so called XXZ model, with the axial-anisotropy 

region confined to the interval 0 5 a < 1. In the Ising limit (a = O), the ground state is of N6el 

type. For the infinite chain, there is a broken symmetry, and one of the two possible N4el states 

has to be chosen as a reference state. They both are connected by time inversion, the ground 

state of the infinite system being a doublet in the anisotropic region. The phase transition, with 

the presence of long-range order and a symmetry broken ground state, requires degeneracy. In 

contrast, in finite chains, the spectrum is not degenerate and the ground state is an eigenstate 

of the time-inversion operator (symmetric or antisymmetric, depending on the number of spins), 

with equal admixtures of both Nbl kets. 

For our developments here, we will choose the Nkel state In/), where the eigenvalues of 

the Sz(m) operators for the linear chain are (-1)". With the usual definition of spin ladder 

operators S* (m) , we define boson-like operators by: 

4 = &{:&N+ C s+(m)s-(mil) m odd 

where N is the total number of sites in the chain. Operators defined by (2) and (3) flip pairs of 

neighboring spins when applied to the reference Nkel state In/). Two sequences with translational 

symmetry are possible, which we label by even and odd. It is apparent that a similar construction 
can be realized with the other NCel state IN)' with S, -+ $ (-l)m+l, interchanging the roles of 

operators (2) and (3). 

In the quasi-Ising limit, the ground state is close to IN, and under this assumption we obtain 

the following algebra for the 4's: 

which are boson-like commutation relations. Within the same approximation, and restricting 

ourselves to the manifold S, = 0, the Heisenberg Hamiltonian (1) can be written as a two-mode 

harmonic oscillator Hamiltonian [I]: 

where E, (a) is the ground state energy. 

The Nkel state satisfies the relations: 

showing that the NCel state, or quasi-classical state, can be represented as a Glauber state [9], in 

terms of the 4's operators. The eigenvalue in (6), that also enters in the definitions (2,3), has 

been chosen so as to cancel the linear terms in Hamiltonian (5). Using a standard notation, we 

define translation operators by: 

~(z) = exp {z$ - ~4+}, (7) 

where 4 may be the even or odd operator. A coherent state is thus obtained as: 

Is, 22) = ~XP (21 h - a;$!) exp (~24~ - z;d!) lo), (8) 

with z1,22 two arbitrary complex numbers. If we write 

we get a closed expression for the N6el state as a minimum uncertainty wave packet of the @s 

operators. Of course, an equivalent representation can be constructed with the Nkl ket as the 

vacuum, just by shifting the definitions of the 4's in a constant, and thus introducing linear terms 

in Hamiltonian (5). 

Since the N6el state is a well defined state, we would like to represent the ground (0 >, in 

terms of fluctuations over the N6el state IN). This can be accomplished in closed analytic form, 

by inverting expression (8): 

The structure displayed by (9) is quite interesting. Quantum fluctuations over the quasi-classical 

state IN) are induced by the 4's operators. The distribution of fluctuations is Poissonian, the 
anisotropy parameter a being related to the width of the wave packet. For ar sufficiently small, 

the state (9) displays long-range order in spite of quantum fluctuations, but the effective magnetic 

moment is reduced from its saturation value. A vanishing magnetic moment signals at a phase 

transition as a function of the parameter a. 

In spite that the algebra given by (4) is obtained in the Ising limit, the trial state (9) results 

to be extremely accurate in describing the energy and correlation functions in the whole interval 

0 5 a < 1. The one dimensional case represents the most stringent test for the wave function, 

since, as we will sketch below, the accuracy of the method improves with the dimension [4]. The 

energy per spin and the staggered magnetic moment corresponding to our trial state (9) can be 

put in closed analytical form in terms of Bessel functions of integer order [5]: 

The generalization to higher dimensions is rather straightforward [4]. If one assumes that the 

lattice is bipartite, i.e. not frustratedfor N&l ordering, the corresponding boson-like operators are 

defined as: - - 

where R labels sites in a sublattice, (6) the set of nearest neighbors, and z is the coordination 

number. The reference N6el state IN >, in this case, assigns up spins to the R sublattice, and 

down spins to R + 6. The interesting formula is the analogue of (9) for the ground state: 

Due to the factor in the exponential of (13), one realizes that the importance of quantum 3-1' fluctuations iminishes with the coordination number z, and correspondingly with the dimensionality. However, closed form expression for arbitrary dimension and spin have not been obtained 

yet, even for the simplest lattice structure (like the square lattice) [lo]. In spite of the above 

fact, using the Hellmann-Feynman theorem [5], one can show that the error in the energy, as 

calculated with the state (13), is proportional to a4. To go towards the isotropic point and to the 

Heisenberg-XY region, one has to generalize the theory in a variational way [7]. 

3 The Jordan- Wigner transformation 

The one-dimensional Heisenberg model for spin 112 can be mapped onto a spinless fermion model 

through the so called Jordan-Wigner transformation [ll]. The two spin states are mapped onto 

fermion states with occupation numbers 0 and 1. Two fermions can not occupy a single site (one 

may think in a very large on-site Coulomb repulsion), and they hop in a lattice with nearest 

neighbors interactions. The transformation is accomplished by: 
where the C, Ct are fermion operators. We readily obtain the relations: 

that substituted in (1) yield the following fermion Hamiltonian: 

where t s Jal2, V = J. If J > 0, the fermion interaction V is repulsive. For positive a, 

Harniltonian (15) describes hole-like fermions. An electron-like system is obtained by changing 

a -+ -a. We already know that the properties of the Heisenberg Hamiltonian (1) are invariant 

under such a change 1121. Antiferromagnetic ordering implies that the total number of fermions 

is f , and the fermion ket that corresponds to the N6el state is the one with 

We will call this state (N > to keep continuity with the previous sections. If one translates our 

ground state (9) into fermion language, one gets: 

with 

and mean occupation number given by 

where CY = 2tlV. As in the spin case, most physical interesting quantities can be calculated in 

closed analytical form using the trial ket (16). For the fermion-fermion correlation functions, this 

has been done in Ref.[l3]. A quantity which is important to investigate possible Fermi liquid 

behavior, is the one-particle momentum distribution 1141. In our case, this can be obtained 

analytically with (16), yielding: 

1 

(Nk)/goles = - [l - sin cos k)] , 2 (17) 

for hole-like fermions, and 

1 (Nk),, = 5 [1+ sin (2~ cos k)] , (18) 

for a particle-like system. The above distributions display a soft variation with the wave number 

k, and therefore no Fermi liquid behavior. The highly correlated limit CY -+ 0, yields the constant 

value 1 /2, with the complete destruction of the Fermi surface of non-interacting particles. 
4 Acknowledgments 

The author acknowledges support from Fundask de Amparo & Pesquisa do Estado de Sib Paulo 

(FAPESP, Brazil), and Fundo de Apoio ao Ensino e Pesquisa (FAEP, UNICAMP, Brazil). 

References 

[I] M. Lagos and G. G. Cabrera, Phys. Rev. B 38, 659 (1988). 

[2] H. A. Bethe, Z. Phys. 71, 205 (1931); R. Orbach, Phys. Rev. 112, 309 (1958); M. Gaudin, 

La fonction d'onde de Bethe, (Masson, Paris, 1983). 

131 D. Medeiros and G. G. Cabrera, Phys. Rev. B 43, 3703 (1991); ibid. 44, 848 (1991). 

[4] G. G. Cabrera, M. Lagos, and M. Kiwi, Solid State Commun. 68, 743 (1988). 

151 D. Gottlieb and M. Lagos, Solid State Commun. 79, 551 (1991). 

[6] G. G. Cabrera, Phys. Rev. B 43, 13476 (1991). 

[7] D. Gottlieb, M. Lagos, and M. Montenegro, Solid State Commun. 81, 729 (1992). 

[8] T. Barnes, D. Konchan, and E. S. Swanson, Phys. Rev. B 39,4357 (1989). 

[9] R. Glauber, Phys. Rev. 131, 2766 (1963). 

[lo] It appears that this problem is related to the formulation of the Wigner- Jordan transformation 

in two dimensions. See for example, Y. R. Wang, Phys. Rev. 46, 151 (1992). 

[ll] P. Jordan and E. Wigner, 2. Phys. 47, 631 (1928); E. Lieb, T. Schulta and D. Mattis, Ann. 

Phys. (N. Y.) 16, 407 (1961). 

[12] J. des Cloiseaux and M. Gaudin, J. Math. Phys. 7, 1384 (1966). 

[13] M. Lagos, M. Kiwi, E. R. Gagliano, and G. G. Cabrera, Solid State Commun. 67,225 (1988). 

1141 G. G. Cabrera, to be published. 
BERRY PHASE IN HEISENBERG REPRESENTATION 

V.A. Andreev, A.B. Klimov 

Optics Division 

Levedev Institute of Physical Sciences 

Moscow 1 1 7924 

and 

P.B. Lerner 

T-4, MS K723 

Los Alamos National Laboratory 

Los Alarnos, NM 87545 

Abstract 

We define the Berry phase for the Heisenberg operators. This definition is motivated by 

the calculation of the phase shifts by different techniques. These techniques are: the solution 

of the Heisenberg equations of motion, the solution of the Schrodinger equation in coherentstate representation, and the direct computation of the evolution operator. Our definition 

of the Berry phase in the Heisenberg representation is consistent with the underlying supersymmetry of the model in the following sense. T'he structural blocks of the Kamiltonians of 

supersymmetrical quantum mechanics ("superpairs") [1,2] are connected by transforniations 

which conserve the similarity in structure of the energy levels of superpairs. These transformations include transformation of phase of the creation-annihilation operators, which are 

generated by adiabatic cyclic evolution of the parameters of the system. 

1 INTRODUCTION 

The equivalence of the Schrodinger and Heisenberg pictures in quantum mechanics is something 

that is taken for granted. The specific choice of the Schrodinger, Heisenberg or interaction picture 

is usually regarded as a matter of convenience. Berry phase was defined initially and mostly for the 

Schrodinger picture as nonintegrable phase factor appearing in the wave-function after (adiabatic) 

evolution of the system's parameters. Here we investigate how one should define the analog of the 

Berry phase in the Heisenberg representation. 

The traditional introduction to Berry phase includes the following construction. The Hamiltonian for the particular system H(X) is introduced, where the set of parameters, X, is considered 

to be changing adiabatically with time. Until now, most of the applications had in mind the 

discrete, though maybe degenerate, spectrum of the Hamiltonian, for all values of the pararnters. 

Then, after the adiabatic cyclic evolution of the parameters, each eigenstate Gn of a corresponding 

stationary problem 

~(x)Gn = ~n(X)+n 
acquires a nonintegrable phase: q5, 4 ei4n$,. This phase usually admits geometric interpretation 

in terms of some contour in parameter space. 

In the Heisenberg representation, the operators of observables, and not the eigenstates, do 

evolve in time. One can assume, that for particular observable, a(t), the cyclic evolution of 

parameters results in the following relation after the period, T: 

where S is a unitary operator. 

One supposes, that the operator, St, in a basis of the eigenfunctions of the Eq. (I) has the 

It represents what can be naturally called matrix of a Berry phase in Heisenberg picture, 

although now it is a unitary operator, not a number. The rest of this paper is dedicated to 

demonstrating the usefulness of this notion. 

We shall demonstrate for a simple exactly solvable model that this operator can be defined 

consistent with the (super)symmetry of a model. We cannot prove it in a general way, though we 

believe that the following proposition is true for any Hamiltonian H, for which Berry phase could 

be defined. Namely, if there exists the Hamiltonian, ~~(x(t)), after adiabatic cyclic transformation 

of its parameters it will be transformed into new Hamiltonian, H2, which is a superpair to the 

initial Hamiltonian, HI. The adiabatic theorem reappears in this approach as the identity of the 

eigenvalues of HI, and H2 (however, the degeneracy of zero eigenvalue, in general, may change). 

The wavefunctions, may, however undergo some unitary transformation. 

This is in accord with general ideology of supersymmetric quantum mechanics, which usually 

includes the compound Hamiltonian, M , formed by two simpler Hamiltonians, HI and H2 [1,2,3,4], 

The properties of the Hamiltonians, HI and Hz, are closely related. Normally, these Hamiltonians 

have identical spectral structure, but, perhaps, only a finite number of energy levels. Consequently, 

almost all of the levels of the Hamiltonian W are doubly degenerate. 

Previously [5], in this context, we studied the supersymmetric structure of the Jaynes-Cummings 

Model (JCM). The formal introduction of the supersymmetry in the JCM includes the replication 

of the JCM with a trivial phase transformation performed on the creation-annihilation operators 

of the boson: 

a -+ ei4a, at -+ ate-'6 (5) 

When q5 is a real function of time, the Hamiltonians, Hl and Hz, have identical spectra, and all 

the energy levels of the Hamiltonian H are doubly degenerate. 

The physical meaning of the phase shift, q5, is not clear. It would be desirable to interpret it, 

analogous to Berry's interpretation of the Aharonov-Bohm effect [6], as a manifestation of a Berry 
phase. Indeed, the JCM has a nontrivial Berry phase with potentially observable ramifications 

[7,8]. The interpretation of the phase shift (5) as Berry phase for some cyclic evolution in the 

JCM is not yet proven, however we demonstrate that this is true for somewhat simpler model of 

Section 3. 

Thus, for our purposes we seek the transformations of phases of the Hermitian operators. 

rather than wavefunctions. Therefore, we calculate Berry phase for our model using both the 

Schrodinger formalism for the wavefunctions and the Heisenberg formalism for the operators. Also, 

we calculate the phase shifts by an explicit expression of the evolution operator through SU(1,l) 

group operators. Both approaches are shown to be equivalent as the result of our analysis. 

Because performing the adiabatic cyclic evolution of the system can be, under certain conditions, a valid quantum measurement, the result of the paper can be put in other form. Namely, the 

distinguishability of the systems described by the Hamiltonians forming a superpair is equivalent 

to the nontriviality of the Berry phase obtained during a cyclic evolution of the system's parameters. The separate measurements of the Berry phase of those two systems during the evolution 

distinguishes these systems. 

The structure of paper is as follows. In Section 11, we make a general definition of the cyclic 

. evolution in the Heisenberg representation. In Section 111, the supersymmetric family of Hamiltonians, similar to the Hamiltonian for degenerate parametric two-photon optical interaction, is 

considered. The calculation of the evolution of the Heisenberg operators shows that besides the 

dynamic phase shift (called self-modulation of phase in nonlinear optics) and amplitude change 

(amplification and de-amplification), there is an additional term. Unlike the previous ones this 

term has a nonzero value in the case of adiabatic cyclic evolution. However, the explicit proof 

of the identity of this phase to the phase, which is obtained by the wavefunction in course of 

adiabatic cyclic evolution is required. This is doqe by obtaining the explicit WKB solution of 

the Schrodinger equation in the coherent state formalism and comparing the results (Section IV). 

Following the recent tradition supported by the papers of Aharonov and Anandan, and Samuel 

and Bhandari [9,10] we identify the "slow" physical time of adiabatic evolution as a parameter of a 

closed contour. This definition is supported by the calculation of the Berry phase from Heisenberg 

equation of motion and through the explicit expression of the evolution operator in Section V. 

2 BERRY PHASE IN THE HEISENBERG PRESENTATION 

We consider the Hamiltonian, H, with time-dependent parameters, i(t): 

We assume that periodicity and adiabaticity of the evolution of the Hamiltonian can be represented 

in two ways. First, one can follow the evolution of H(t) as a function of the time-dependent 

parameters. In this case, the Hamiltonian will undergo trivial transformation after the evolution. 

The second way is to consider the change of the Hamiltonian as an observable with physical 

meaning of energy under the action of the evolution operator U(t) which is engendered by the 
Hamiltonian H(t) 

In this case the transformed Hamiltonian &(T) can be different, unitarily equivalent to the 

initial Hamiltonian. 

The spectra of these Hamiltonians are therefore identical. However, other features can be 

different. This distinction can be extremely important if there are lines of singularity or other 

topological complications in the space of parameters. 

In the case of adiabatic cyclic evolution, this operator doesn't depend on particular law of 

evolution of the system's parameters and we shall define this unitary operator as Berry phase in 

Heisenberg representation. It implies, that the difference from H(T) and H(T) can be expressed 

in terms of certain phase factors (eigenvalues of the operator U(1")). As a rule, they appear for 

a certain components of the Hamiltonian H. These phases are closely connected with the Berry 

phases in Schrodinger representation, indeed, in the example of the Section 111 they are identical. 

The fact that the Hamiltonians H(t) and H(T) are forming a superpair is of primary importan&. One of the simplest ways of representation follows from Eq. (7), which connects H(T) and 

fi(~) 

One can then demonstrate using the formalism of that relationship implies superstructure [2,3]. 

This relation holds only in the case of nonnegative spectra of the Hamiltonians. Thus, we restrict 

ourselves by the Hamiltonians bounded from below, in which case we shift the zero level of energy 

to avoid negative eigenvalues. 

The superstructure could be introduced in more sophisticated ways. A particular example 

would be demonstrated as well. The paper is dedicated to the connection between Berry phase 

and supersymmetry on the example of the simple model. More complicated examples such as the 

Jaynes-Cummings model (JCM) will be studied elsewhere. 

In the paper [ll], the authors proposed the generalization of Berry's concept, interpreting 

their phase, a factor similar to our symbol V (Eqs. (20)-(21)), as a gauge transformation of the 

wavefunction. It is induced by the reparametrization of the Hamiltonian. The main result of 

this paper can be formulated in our language in the following way. The gauge transformation of 

Wilzcek and Zee while being applied to the Hamiltonian, regarded as observable, can result in 

other Hamiltonian, even if the evolution is adiabatic and cyclic. However, both Hamiltonians are 

unitary equivalent and the observable quantities are identical for both of them. The adiabatic 

theorem in the general quantum mechanics formalism is represented by supersymmetry in our 

paper and by the gauge invariance in their formalism. 
3 BOSONIC FIELDS WITH SELF-ACTION: THE SUPERSYMMETRY OF THE SQUEEZING HAMILTONIANS 

The problem is formulated for the simple model of two identical noninteracting bosonic fields 

described by the Hamiltonian 

The superpair of this Hamiltonian is 

The constant c is defined from the equation 

The system described by the Hamiltonian H in the form (4), where HI and H2 have the form 

(12) and (13), respectively, are supersymmetric. This supersymmetry is generated by the following 

supercharges [5]: 

Hi = q1q2 Hz = 4241 

The second supercharge has the form 
This finishes the formal description of supersymmetry. The procedure of assigning a supersymmetric structure for an initial model remind our prescription for the JCM [5]. One should 

make a constant phase shift of the creation and annihilation operators and consider a replica of 

the initial system with transformed operators as a superpair for the initial system. 

The described procedure has an obscure physical meaning. To clarify it we explicitly describe 

the unitary operator, which connects the components of the Hamiltonian in subsequent sections. 

4 THE OPERATOR BERRY PHASE 

Now we are prepared to study the Berry phase of the Heisenberg operators. We suppose the three 

parameters w, ReX, ImX* to undergo adiabatic periodic change in time. First, we calculate the 

nonintegrable phase attained by the wavefunction of the model, described in previous Sections, in 

coherent-state representation. Below, we shall demonstrate, that all the components of the Hamiltonian H, defined by Eqs. (12) and (13) can be obtained from one component of this Hamiltonian 

by applying the adiabatic cyclic evolution of the parameters of the system. Thus, the superpairs 

are connected by a unitary operator, which we call the Berry phase in Heisenberg representation. 

A single component of the Hamiltonian N, corresponds to the Schrodinger equation 

The contribution of the scalar term ho(t) is removed by the transformation 

The wavefunction V is represented in the coherent state formalism by 

where T is time ordering and Icy), I/?) are coherent states: 

ala) = ale) alp) = (slat = (cylcy* (22) 

For the quadratic Hamiltonian (5) the calculations can be done explicitly. The solution 

V(cy*, p, t) of the Eq. (19) in the coherent-state formalism has the form 

where a,b are the parameters of the initial state. As usual for coherent-state representations 

[12], the solution is expressed in terms of auxiliary functions c(t), ~(t) which satisfy the following 

system of the equations [13]: 
with the initial conditions ((0) = 1,q(0) = 0. 

The following system of equations is valid for the functions (, q 

X* i* ii - -9 A* + [ih - iw- A* + ~2 - 41~l21~ = 0. 

i i ( - -i + [ih + iw- + 3 - 41~I~]( = 0. X X (25) 

Below we shall demonstrate that WKB-solution of these equations will provide a multiplier 

with a nonintegrable phase. 

The Hamiltonians HI, H2 include the creation-annihilation operators distinguished by some 

deterministic phase shift. For a description of the evolution of these operators we use one of 

the four components of the Hamiltonian H, because all four Hamiltonians are independent and 

unitary-equivalent to each other, 

The Heisenberg equations have the form [14], 

It is again assumed that the parameters w, A, A* are slowly varying periodic functions of time. 

The operators a, at are satisfying the following equations: 

i* i* ii - -it + [iw - iw- + w2 - 41X12]a = 0, A* A* 

One observes that equations (25), for the functions [,q are identical to the system of equationi 

(28) for the operators at, a. By substitution 

one obtains for operator b and its Hermitian conjugate, if the terms 

(i*/~*)~, $();*/A*) in the equation for b are neglected 
A* 8 + [iw - iw- + w2 - 41X12]b = 0, A* 

X it + [-G + iw- + w2 - 41X12}bt = 0. X 

The WKB approximation for the solution of Eqs. (28) yields 

with 

For cyclical evolution of the parameters the contribution of the function f (t) is zero, because 

it is an exact differential. However, the integral of the function $(t) yields the Berry phase. 

and JB is given by the same expression with k -+ $ = 6 + 8. The initial conditions can be 

chosen in such a way, that the phase transformations of the operators take place as a result of the 

evolution 

a -+ aeU~, at --, .t,-i+~ 

then 

The calculation of the Berry phase for the model Hamiltonian H is now complete. 

One observes that the superpairs HI and H2, composing the supersymmetric Hamiltonian H, 

are distinguished by the Berry phase. As the Hamiltonian Hz transforms as a result of cyclic 

evolution, the phases () + 8) and (4 + 8) evolve in a different way. Berry phase is independent of 

the particular law of evolution, nevertheless it depends on the choice of the contour in parameter 

space. The contours can be chosen in such a way that the following relation is valid [15]. 

Now we have found the geometric phase for the Hamiltonian H using the Beisenberg representation. 
5 THE CALCULATION OF THE BERRY PHASE USING EVOLUTION OPERATOR 

We have shown that creation-annihilation operators obtain phase shift according to Eqs. (34)- 

(35) as a result of adiabatic cyclic evolution which is identical with the Berry phase gained by the 

coherent phase. The substitution of the transformed operators in the Hamiltonian h gives 

This result can be obtained directly from computation of the action of evolution operator on 

Hamiltonian H. 

One can rewrite the Hamiltonian H in the terms of SU(1,l) operators [16,17]: 

where 

KO = 1/2(ata + aat), K- = a2, K+ = (at)2 

[KO, K+] = +K+, [KO, K-] = -K-, [K+, K-] = 2Ko. 

To get rid of the T-exponent in the evolution operator: 

one can rewrite it in the form [12]: 

This decomposition expresses the evolution operator in terms of three time-dependent functions 

a(t), P(t)andy (t). Using the commutation relations one obtains the following equations for these 

functions: 

The quantity a(t) satisfies the equation 

Cr + iwa + ipa2 = -ip*, 

which can be rewritten after the substitution a = (l/ip)($/+) in the form 

ti K = $exp[: Jf dr(-- + iw)] 

P 

ti 2 + k[w2 - 41pI2 + iw- - iw] = 0. 

c1 
The equations (43) coincide in the form with the equations (25), (28) which were our starting 

point of obtaining expression of the Berry phase. The phase given by the equation (43) is identical 

to the Berry phase as defined above. Now one can study the behavior of the functions a, P, 7 under 

cyclic adiabatic evolution of the parameters. The function n is the solution of the equation with 

the periodic potential. This implies that this function is quasiperiodic: 

where q5~ is independent of t. Thus the ration n/n is a periodic function as well. 

The function P = -i J dw - ln+ is not periodic. According to Eq. (32) after the cyclic evolution 

,f? changes by a constant q5~ 

The function 7 is not periodic as well, however, it is not very important to us here. 

One can now investigate the transformation of the primary Hamiltonian h in course of evolution. Since we decomposed h in the sum of three operators KO, K+ , K-, we can consider the action 

of the evolution operator on the single terms of the sum. It is quite necessary for Eqs. (41) not 

to be symmetrical with respect to the functions a, p, 7. We can use the decomposition from Eq. 

.(40) only for the study of the transformation of the operator K- : 

K-(T) = U-I (T)K-(O)U(T). (46) 

The initial conditions of the evolution were chosen as a(0) = P(0) = 0 as a consequence of 

periodicity it implies a(T) = 0. This means that eQK+ and eyK- do not contribute to the periodic 

evolution of group operators. The only term with which changes is ePKO. Taking into account 

their relation P(T) = -q5T one has 

Similarly, choosing a different order of multiplication in the decomposition (40) one gets the 

transformation property for the other operators. 

Consequently, the Hamiltonian H is transformed under the evolution as 

This expression is the same one we obtained from the WKB solution of the operator equations 

(28). The phase shift is identical to the phase shift gained by coherent-state representation of 

the wavefunction. The reconciliation of all methods proved the correctness of the definition of 

Berry phase in Heisenberg picture. Because all of the components of the Hamiltonian H can be 
obtained through the constant phase shift of the creation-annihilation operators or sign reversal 

of the phase factors O,r$ we finish up with the proposition that the trivial phase shifts in the 

supersymmetric Hamiltonian can be regarded as the Berry phases. 

CONCLUSION 

The present paper proposes the following definition of Berry phase in Heisenberg representation. 

Berry phase is the unitary operator connecting the Hamiltonian of the system with the initial 

Hamiltonian after adiabatic cyclic evolution of the system's parameters. We study the simplest ' 

model in which this unitary operator is independent of the particular quantum numbers of the 

system and reduces to a certain c-number phase factor. This definition is motivated by the identification of this number to the adiabatic phase factor gained by the wavefunction of the system 

in coherent-state representation. Since the model is exactly soluble, the direct computation of the 

evolution operator is possible and it confirms the calculations made by two other different methods: the WKB solution of the Heisenberg equations and the WKB-solution of the Schrodinger 

equation in coherent-state representation. This definition allows us to interpret the supersyrnmet- ' ric compound Hamiltonian as if it has been formed by images of the initial Hamiltonian, obtained 

as a result of an adiabatic cyclic evolution over a different contours in a parameter space. 

... 

7 ACKNOWLEDGEMENTS 

The early stages of this work were sponsored in part by the University of Michigan Distinguished 

Visitors' fund and the Office of Naval Research Initiative for Accelerated Research. 
References 

[I] E.Witten, Nucl. Phys. B 185 (1981) 513. 

[2] A. Lesniewski, M. Lewenstein, A. Jaffe, Ann. Physics 78 (1989) 

[3] I.E. Gendenstein, I.V. Krive Sov. Phys.-Uspechi 28 (1985) 645. 

[4] R.W. Haymaker, A.R.P. Rau Am. Journ. Phys. 54 (1986) 928. 

[5] Andreev V.A., Lerner P.B., Phys. Lett. A. 134 (1989) 507. 

[6] Berry M.V. Proc. by. Soc. A 392 (1984) 45. 

[7] Andreev V.A. Klimov A.B. Lerner P.B. JETP Lett, 50 (1989) 

[8] ibid., Europhysics Lett. 12 (1990) 101. 

[9] Y. Aharonov. J. Anandan. Phys. Rev. Lett. 58 (2986) 1574. 

[lo] J. Samuel, R. Bhandari, Phys. Rev. Lett. 60 (1988) 2339. 

[ll] F. Wilczek A. Zee, Phys. Rev. Lett. 52 (1984) 2111. 

[12] A.M. Perelomov. Coherent states and their applications. (Springer-Verlag, Berlin, 1986.) 

[13] Dodonov V. V., Man'ko V.I. Sov. Phys. Proc. FIAN, 183, (1988) 71. 

[14] Abdalla M.S., Ahmed M.M.A., Obada A.-S.F. Physics A 162 (1990) 215. 

[15] Andreev V. A., private communication 

[16] K. Wodkiewicz, J.H. Eberly, JOSA B2 (1985) 458. 

[17] Chris C. Gerry, P.K. Ma, E.R. Vrscay Phys. Rev. Lett. A 39 (1989) 668. 
SECTION 7 

WCITQN OF P3Gm 
QUANTUM PROPAGATION IN SINGLE MODE FIBER 

Lance G. Joneckis 

Laboratory for Physical Sciences 

University of Maryland 

College Park, MD 20740 

Jeffrey H. Shapiro 

Department of Electrical Engineering and Computer Science 

Massachusetts Institute of Technology 

Cambridge, MA 021 39-4307 

Abstract 

This paper presents a theory for quantum light propagation in a single-mode fiber which 

includes the effects of the Kerr nonlinearity, groupvelocity dispersion, and linear loss. The 

theory reproduces the results of classical self-phase modulation, four-wave mixing, 

and classical soliton physics, within their respective regions of validity. It demonstrates the 

crucial role played by the Kerr-effect material time constant, in limiting the quantum phase 

shifts caused by the broadband zero-point fluctuations that accompany any quantized input 

field. Operator moment equations-approximated, numerically, via a terminated cumulant 

expansion-are used to obtain results for homociyne-measurement noise spectra when dispersion is negligible. More complicated forms of these equations can be used to incorporate 

dispersion into the noise calculations. 

1 Introduction 

Optical fibers have long been considered for the generation of squeezed-state light, starting with the 

pioneering work of Levenson and coworkers in the mid 19807s, who observed 0.58 dB of continuouswave (cw) squeezing [I], to recent measurements exhibiting over 5 dB of short-pulse squeezing 121. 

In this paper we present a theory for quantum light propagation in single-mode optical fiber. Our 

development, which includes the effects of the Kerr nonlinearity, group-velocity dispersion, and 

linear loss, is guided by two overarching principles: the theory must include all relevant prior results, both classical and quantum mechanical, and, within reasonable limits, it must accommodate 

arbitrary input states. This theory [3] is an extension of our prior work on quantum propagation 

in a dispersionless, lossless, Kerr medium 141. In that earlier study it was shown that a material 

time constant is crucial to a correct description of quantum nonlinear phase shifts beyond the 

four-wave mixing regime, a conclusion similar to that reached by Blow and coworkers [5]. The use 

of a finite Kerr-effect time' constant is retained in the current treatment of lossy, dispersive fiber. 

This paper is organized as follows. In Section 2 we review quantum propagation in a Kerr 

medium, concentrating on the structure of the theory and the necessity of a finite Kerr-effect time 

constant. We recount the principal result of 141, namely the limits on squeezed-state generation 
in lossless, dispersionless fiber. Section 3 expands this theory to incorporate dispersion and linear 

loss via a split-step approach. Section 4 introduces the terminated cumulant expansion (TCE) 

as a technique for closing the infinite chain of coupled moment equations generated by the full 

theory. Using the TCE, the limits on quadrature-noise squeezing in lossy, dispersionless fiber 

are quantified. Finally, in Section 5, we discuss the relationship of our approach to other quantum propagation theories for single-mode fiber, focusing on the necessity of the Kerr-effect time 

const ant. 

2 Quantum SelfiPhase Modulation 

Our attention, in this section, is restricted to a linearly polarized field propagating in a lossless, 

dispersionless, single-mode fiber that exhibits the Kerr nonlinearity. Classically, the refractive 

index in this fiber can be written as follows, 

where z is the axial coordinate along the fiber, t is time, no is the linear refractive index, n2 is the 

Kerr coefficient, A is the fiberls effective cross-sectional core area, and E(z, t) is the normalized 

complex envelope of the single-mode field within the fiber. The field normalization we employ is 

such that E(z, t) has units 4-. Note that the introduction of the photon energy, hw, 

is strictly a convenience at this classical stage; it has no quantum significance as yet. 

Input Plane Output Plane 

(z=O) (z=L) 

Fig. 1. Schematic configuration for Kerr-effect 

propagation in lossless, dispersionless fiber. 

The classical propagation problem in lossless, dispersionless, Kerr-effect fiber is sketched in 

Fig. 1. The fiber is excited, at z = 0, by an input field EIN(t) that launches a +z-going wave 

E(z, t) satisfying E(0, t) = EIN(t). In a reference frame moving at the group velocity, v,, the 

complex field envelope within the fiber satisfies the following differential equation [6], 

dE(z, t') =i~E*(z,t')E(z,t')E(z,t'), forz20, dz 

where t' = t - zlv, is the retarded time, 
is the nonlinear, phase shift per unit length per unit photon flux, and X = 2nc/w is the center 

wavelength of the light. The iiitensity is a constant of motion for Eq. 2, so it is easily shown that 

Using the initial excitation condition bhat specifies E(0, t'), and directing our attention to EouT(t) = 

E(L, t), the field coupled out of the fiber at x = L, we obtain the classical input-output relation 

for a length L lossless, dispersionless fiber exhibiting the Kerr nonlinearity, namely, 

where 

r = KL, 

is the nonlinear phase shift per unit photon flux and, for convenience, we have dropped the Llv, 

group delay. 

Two well known results follow directly from Eq. 5: spectral broadening through self-phase 

modulation (SPM), and four-wave mixing (FWM). As an immediate consequence of Eq. 5, we 

see that an optical pulse propagating through the fiber acquires a time-varying phase shift, which 

is proportional to the pulse's intensity. The derivative of this time-varying phase constitutes an 

intensity-dependent instantaneous-frequency variation, implying, for sufficiently intense pulses or 

long fibers, significant spectral broadening. On the other hand, when the input field comprises a 

strong monochromatic pump, ETN, at frequency w plus weak sidebands at frequencies wf 0, FWM 

couples the sidebands. The input-output relation for classical FWM can be obtained, from Eq. 5, 

by replacing the exponential term with its two-term Taylor series approximation, and assuming 

that all nonlinear phase shifts other than the pump x pump term are small: 

where ~(t) = E(t) - EP for the input and output fields. 

In the quantum theory of the Kerr interaction the photon-units complex field envelope, E(z, t), 

becomes a photon-units field operator, E(Z, t). The input field ~~erator-~~~(t) = ~(0, t)-is a 

single-spatial mode, multitemporal-mode free field, and hence must satisfy the following 6-function 

commutator rule [7], 

[EIN (t) 9 EiN (u)] = 6(t - 21) , (8) 

for photon-units field operators. The output field ~~erator-~~~~(t) E E( L, t)-is also a singlemode free field, whose commutator must therefore mimic Eq. 8. This requirement is automatically 

met by quantizing Eq. 2 as follows, 

~E(z, t')

dz =i~~t(~,t')~(~,t')~(z,t'), for z>o. 

Quantum FWM emerges from Eq. 9, in a manner that ensures commutator preservation, by 

decomposing the input field operator, EIN(t), into a onumber pump, EL, plus a 6-function A 

commutator field operator, eIN (t) . Then, assuming that all nonlinear phase shifts other than the 
pumpxpump term are small, Eq. 9 can be linearized and solved to yield the quantum version of 

Eq. 7, namely, 

Equation 10 is commutator preserving, does not require a Kerr-effect time constant, is consistent with the classical FWM theory, and agrees with experimental results in fiber squeezed-state 

generation [I]. 

We find that it is not possible to treat quantum SPM from Eq. 9, in a way that recovers the 

classical limit, without a modification of the theory. To highlight the failing of instantaneousinteraction quantum SPM, let us calculate the mean output-field for a coherent-state input field 

[4]. By direct substitution, 

can be shown to be the solution to Eq. 9, leading to 

as the quantum version of Eq. 5 under an instantaneous Kerr-interaction model. The output mean 

field, 

for a coherent-state input field reduces to 

because a coherent-state input field obeys the eigenfunction relation 

For the rest of the calculation we employ a limiting argument, 

where we have exploited the characteristic function for coherent-state photon counting [8] to obtain 

Eq. 17. 

Equation 18 predicts that the instantaneous Kerr nonlinearity has no effect whatsoever on the 

mean field of a coherent-state input, regardless of how large the classical, peak nonlinear phase shift 

becomes! This result contradicts the classical theory, Eq. 5, and, more importantly, it contradicts 
experiments which have confirmed the classical spectral-broadening predictions. In retrospect, 

the failure of Eq. 9 to reproduce SPM in the appropriate limit should not be too surprising as it 

arises from applying an instantaneous, spectral-broadening nonlinearity to a quantized field whose 

vacuum-st ate fluctuations extend to infinite bandwidth. 

The classical SPM result can be recovered from our quantum mean-field calculation by introducing a phenomenological Kerr-effect time constant, TK, as a lower-bound for T in Eq. 16. In 

particular, with T = TK > 0 and r/rK << 1, Eq. 17 becomes 

- t 

EOUT(t) * exp (~T/TK) d7 IEIN(T) 12] EIN(~) [ LTK 

whenever TK is smaller than any classical time scale present in the field. 

To incorporate TK into our quantum propagation theory for the lossless, dispersionless, Kerr 

effect in single-mode fiber, we employ the following partial mode expansion for the input-field 

operator: 

where { 6iN. : -oo < n < oo ) is a set annihilation operators, and 

( 0, otherwise, 

is a c-number, TK-secOnd duration, rectangular pulse. Suppressing the L/v,-second group delay, 

the same expansion applies to the output field operator, '@*uT(t), with the input annihilation 

operators replaced by the output annihilation operators, {&,OUT : -00 < n < oo). These output 

annihilation operators are related to their corresponding inputs by 

for -oo < n < oo, 

which should be compared with the instantaneous quantum Kerr model, Eq. 12. 

Equations 20-22 comprise a coarse-grained time model for quantum Kerr-effect propagation 

in lossless, dispersionless, single-mode fiber. Introducing the Kerr time-constant as a simple phenomenological parameter, TK, is arguably a distasteful ad hoc procedure, but the known existence 

of finite Kerr response times plus the inconsistency of an instantaneous quantum Kerr model mitigate against this deficiency. A more important consideration with respect to Eqs. 20-22 is that 

they are fundamentally incomplete; { J(t - nrK) : -oo < n < oo ) is an orthonormal temporal 

mode set, but not a complete orthonormal mode set. Hence, in the coarse-grained time model, 

the input field commutator obeys 

[&N (t) '@iN (u)] = srK (t, U) 

if(n-l)~~St,u<n~~,fornaninteger, 

0, otherwise. 
This expression is not the Sfunction commutator that a photon-units field operator should possess. 

However, as will be made clear below, it can be a reasonable approximation thereto. Moreover, 

from Eqs. 22 and 23, we see that the co.arse-grained time quantum theory is consistent in that it 

preserves S,,-function commut ators-we have that 

Given the approximate nature of the coarse-grained model's field commutator [4], what conditions suffice to ensure the accuracy of this model's moment predictions? Clearly, this model is 

a poor approximation for field dynamics occurring on time scales comparable to TK, or, equivalently, over bandwidths of the order of rgl. We expect TK to fad in the interval 1-100 fsto 

match the known time scales of n2-so our quantum input field, EIN(t), must not have excited 

(non-vacuum state) modes at these time or bandwidth scales. Furthermore, because SPM causes 

spectral broadening, we must check that this same timelbandwidth scale condition is satisfied 

by the output field operator, EOuT (t) . Finally, we must limit any photodetection measurements 

that we make on the output field to be insensitive to ~~u~(t)-behavior at TK time scales or T;' 

bandwidths. When all three of these conditions apply, we believe the coarse-grained time quantum Kerr model-with a proper value for rK-should capture the full behavior of quantum Kerr 

propagation in lossless, dispersionless, single-mode fiber. 

The output mean-field derivation-with a Kerr time-constant rK-is virtually identical to the 

analysis of the inst ant aneous-interaction case. We find that 

where we have defined the complex number, h, according to 

iR= [exp(i+,) -I]TK, for 4, =r/rK. 

Equation 25 bears a striking similarity to the classical formula, Eq. 5. Indeed, the two would be 

identical were R = r to prevail. 

We know that r is the classical nonlinear phase shift per unit photon flux. Because TK is the 

time duration of a single, rectangular-pulse mode in our coarse-grained time model, we have that 

4, is numerically the classical phase shift produced by one photon in such a mode. Equation 25 

applies to a coherent-state input field-which has vacuum-fluctuation noise in all its modes-so 

it is fair to regard $, physically, as the quantum nonlinear phase shift that is due to the vacuum 

fluctuations of a single mode. 

The predictions of the quantum and classical theories are nearly coincident when we have 

r = Re(R) and IIm(R)I << 1. These conditions hold for 4, << 1, as can be seen from Fig. 2, 

where we have plotted Re(R)/r and Im(R)/r vs. 4,. Assuming TK = 1 fs, a 1 km fused silica 

fiber will have 4, w 10-3rad; from Fig. 2 we conclude that Eq. 25, the quantum coherent-state 

mean field, will then be in excellent agreement with Eq. 5, the classical mean field result. For 

4, > 10-'rad, there is a pronounced divergence between Re(R) and r, caused by the intrinsic 

periodicity of R. Viewed as a function of $,, Eq. 26 shows that R is periodic with period 27r. 

Physically, this periodicity constitutes a quantum state-recurrence for Kerr-effect propagation in 

lossless, dispersionless fiber and occurs when L = L, - 27rTK/~; for TK = 1 fs and fused-silica 
fiber this implies L, lo4 km, an experimentally inaccessible value for our assumptions of zero 

loss and zero dispersion. 

, Fig. 2. Logarithmic plots of &(R ) /rand I~(R ) / r Fig. 3. Minimum and maximum low-frequency, 

vs. ,. The coherent-state mean field of coarse homodyne-measurement noise spectra for a cw, 

grained time SPM reduces to the classical SPM coherent-state input vs. fiber length. The solid 

result when R x r, , is the quantum nonlinear curves are the coarse-grained time SPM the- 

&ase shift, i.e., the Kerr-induced phase shift of ory; the dashed curves are the instantaneousone photon per -second mode. interaction FWM theory. The parameter values 

employed are: = 1.06 pm; A = 3 56 x 10- l1 m2; 

n;! = 3 2 x 10- 20 m2/W; pIN = 1 W, and = 

1 fs. 

Expressions for the quantum output moments, up to the second order, have been derived for 

Gaussian-state inputs [4]. We note that these moments yield the correct results in the appropriate limits. Specifically, when 4, << 1 they are in agreement with the quantum FWM results. 

Furthermore, if we let R --+ r and S,, (t, u) -+ 0 and use classical covariances in lieu of quantum covariances throughout, e.g., (EiN (~)EIN(u)) 3 ETN (t) EIN(u), etc., the resulting equations 

agree with classical stochastic SPM results. We note that none of these classical formulas depend 

on the Kerr-effect time constant, r~. This is fully consistent with our assumption that the spectrum of the classical input excitation and the implied output-field spectrum are both narrower 

than r~l. Under these circumstances we expect-in a classical t heory-t hat the Kerr interaction 

is effectively inst ant aneous. 

The implications of the full theory are readily ascertained by examining the output field's 

homodyne-detection statistics. An ideal homodyne measurement on EOUT (t) yields a photocurrent. 

whose statistics are proportional to those of the following abstract quantum measurement: 

where 4 is the local-oscillator (LO) phase. As is conventionally done in cw squeezing experiments, 

we shall focus on the minimum and maximum values of the homodyne-noise spectra as the LO 

phase is varied. 
For a coherent-state input, both the instantaneous-interaction FWM and the coarse-grained 

time SPM theories imply frequency-independent homodyne spectra out to frequencies comparable 

to ril. In Fig. 3 we have plotted Smc (0) and Smm (0) , vs. fiber length L, for both theories. These 

curves assume n2 = 3.2 x 10-~'m~/W, A = 3.56 x 10-11m2, X = 1.06pm, PIN - ~JIE~

= 

1 W, and, for the coarse-grained time case, r~ = 1 fs. Quantum FWM predicts complete noise 

suppression for one quadrature component, S,in(0) --+ 0, as the fiber length becomes infinite. 

Moreover, this transpires at minimum uncertainty product, Smin (0)Smm (0) = 1. These FWM 

features are clearly evident is Fig. 3; for long enough fibers, however, they are at odds with our 

coarse-grained time SPM theory. According to Fig. 3, the coarse-grained time SPM theory has 

an Smin(0) which reaches a nonzero minimum at L x 500m, and increases, for longer fibers, 

substantially above the coherent-state' value of unity. 

The precise location of the minimum in the low noise quadrature depends on r~ with higher 

values of r~ extending the region of agreement between the instantaneous interaction FWM and 

the coarse-grained time SPM theories. For various technical reasons, e.g., guided acoustic-wave 

Brillouin scatter, it is not feasible to perform such an experiment using a cw field. Nevertheless, 

the physical conditions corresponding to the minimum in Fig. 3, namely 1 Watt input power and 

roughly a 500-meter-long fiber, seem well within the realm of possibility for pulsed experiments. 

In fact, modeling Shelby's soliton-squeezing experiments [9] by using his peak intensity as the 

intensity in our cw theory, we find that conditions are right for a rK-dependent deviation from 

quantum FWM. However, accurate analysis of a short-pulse experiment-especially one based 

on solitons-must surely account for groupvelocity dispersion. In addition, a realistic quantum 

propagation theory for long fibers should address linear loss. 

3 Inclusion of Dispersion and Loss 

A split-step configuration for incorporating group-velocity dispersion (GVD) and linear loss (LL) 

into quantum propagation analysis for a Kerr-nonlinear fiber is shown schematically in Fig. 4. An 

infinitesimal length of fiber at z is divided into two sub-segments. The first sub-segment exhibits 

only the Kerr nonlinearity and linear loss; the second has neither Kerr effect nor loss, but suffers 

from dispersion. 

Let E (2, t) be the coarse-grained time, +z-going, photon-units field operator within the fiber, 

i.e., as before we have the mode expansion 

Over an infinitesimal, Sz-meter-long fiber segment, the split-step procedure leads to the z-to- 

(z + 62) annihilation operator transformation, 

Bn(z+6z)= 5 h[n-m;Sz]&m(z), for -m<n<m, 

m=-w 

(29) 

where 

dn(z) E exp [i(~Sz/rK)&k(z)B~(z) + (a/2)Sz] &(z) + JcllSz&(z), for -m < n < m, (30) 
characterizes SPM and LL alone over the dispersionless sub-segment, and 

-%~z, for n = -I., 1, 

27, 

1 + 962, for n = 0, 

T~ 

I 0, otherwise, 

is a discrete-time impulse response accounting for the dispersive sub-segment. In Eqs. 28-31, we 

have suppressed the groupvelocity delay, and we have introduced the fiber's dispersion coefficient, 

P2, and its power-attenuation coefficient, a. More importantly, with the inclusion of a Langevin 

noiseoperator in(z) required by the presence of the LL these equations preserve the 6,(t, u)- 

function commutator of a coarse-grained time input field operator, GIN (t). SO, if we drive the fiber 

at z = 0 with such an input-forcing E (0, t) = EIN (t)-and then iteratively apply Eqs. 28-31, we 

will arrive at an output field operator, EOuT(t) - E(L, t) at z = L with a proper coarsegrained 

time commutator. Furthermore, because the unitary operators that transform the (6, (z)} into 

the {in (a)} are known, as are the operators for changing the {in (z)} into the {d, (z + Sz)}, we 

can-in principle-calculate all the measurement statistics for EOUT (t), given any state of EIN (t). 

Input Plane Output Plane 

(z=0) f (z=L) 

K = Kerr Nonlinearity 

L = Linear Loss 

D = Group Velocity Dispersion 

Fig. 4. Schematic split-step configuration for inclusion of groupvelocity dispersion and linear loss 

into the coarse-grained time SPM theory. 

We note that the only aspect missed in this split-step approach is a term proportional to the 

commutator of the Hamiltonians governing each step. It can be shown that this commutator is 

finite and its contribution goes to zero in Sz2; hence, in the limit of Sz -4 0 this is an exact 

theory. Taking the limit of Sz - 0, the differential equation for the mode operator is 

+ n() for 0 5 z 5 L, -co < n < co. (32) 

The coarse-grained time cannot be suppressed-it is essential to preventing the mean-field contradiction exhibited in Sect. 3.1-but it can be hidden. Returning to field operator notation, Eq. 32 
can be recast as 

with the obvious, implied definitions for the rK-approximation to the second partial derivative 

with respect to time and for the Langevin noise field-operator. 

Equation 33 is extraordinarily appealing. Converting it, naively, to continuous-time classical 

form by merely dispensing with the operator carets, dropping the Langevin noise source, and the 

TK subscripts, we obtain 

a P2 d2 a -E(z, t) = i~E*(z, t)E(z, t)E(z, t) - i--E(z, t) - -E(z, t), dz 2 dt2 2 (34) 

the well-known starting point for the classical theory of solitons in fiber [6]. 

4 Moment Propagation and the Terminated Cumulant 

Expansion 

Although it is possible to calculate the exact state transformation for the preceding quantum 

theory, it is a daunting numerical task in almost all cases of interest. Thus, we have elected to 

follow a much simpler and restrictive course--moment propagation. Taking the expectation value 

of Eq. 33 we find that the mean field develops according to 

d .P2 a:K Q -(E(z, t)) = i6(et(z, t)~(z,t)~(?, t)) - 2--(~(z, t)) - ?(b(z, t)), dz 2 a,,t2 

which illustrates the fundamental problem of moment propagation-the Kerr nonlinearity couples 

each moment's differential equation to those of higher order. For example, in the single time case, 

the differential equation for the moment (Etk (t, z) (t, z)) includes terms containing the moment (etk+' (t, z) E'+' (t, 2)) , leading to an infinite progression of coupled differential equations. 

This infinite linkage of moment equations can be broken, in an approximate way, through 

a terminated cumulant expansion (TCE). (A brief review of cumulants-for classical random 

variables-is found in the Appendix. Here, we rely on normally-ordered quantum cumulants.) In 

the TCEK expansion, all normally-ordered quantum-field cumulants beyond the K-th order are 

set to zero: 

((n[et(z, t)lmk JJ[E(z, t)lnk)) = 0, when x(mk + nx) > K. 

k k k 

(36) 

The TCEK assumption provides low-order moment expressions for all field-operator moments 

beyond K-th order. For example, the third-order cumulant relation, 

((E t) t))) = ($I(,, t)E2(z, t)) - 2(~~(t, t)~(z, t))(~(z, t)) - (et(z, t)) (~~(2, t)) + ,2(Et(z, t)) (~(z, t))2, (37) 

affords the following explicit expression for the TCE-2 expansion: 

(~~(z,t)~~(z, t)) = (Et(z, t))(E2(z, t)) +2(Et(z, t)~(z, t))(~(z, t)) -2(Et(z, t))(~t(z, t))2. (38) 
Substitution of this expression into the mean-field equation, Eq. 35, eliminates the third-order 

moment and leads, ultimately, to a closed system of differential equations for 

The accuracy of the TCE approximation depends upon both the initial state and its subsequent 

propagation. For a Gaussian state, such as a coherent state, all cumulants of order three or higher 

vanish. Moreover, a Gaussian state remains Gaussian under linear propagation-even if it is lossy 

and/or dispersive-so, for example, TCE-2 is exact for Gaussian-state inputs in the four-wave 

mixing limit. Higher-order TCE approximations track deviations from a Gaussian state, hence 

they should prove useful for Gaussian-state inputs even beyond the four-wave mixing regime. This 

is demonstrated, quantitatively, in Fig. 5, where we compare coherent-state input, homodynenoise output spectra for a lossless, dispersionless Kerr-effect fiber computed via TCEK, with 

the exact solution presented in the previous section. The coarse-grained time SPM curveP(ex,act 

solution) represents a state that is very nearly Gaussian state up to the point of its minimum-noise 

curve departs from the instantaneous-interaction FWM curve. (As noted above, coherent-state 

FWM is always a Gaussian-state case.) The TCE-2 approximation misses the mark, as it always 

. represents a minimum-uncertainty Gaussian state, which is plainly a bad approximation to the 

exact solution. However, the TCE3 approximation captures the essential nature of the exact 

solution, viz. it correctly predicts the minimum noise level and its subsequent rise to the shotnoise-level. As expected, the TCE-4 and TCE5 approximations show slightly better performance, 

but the meager improvement they provide hardly justifies their added computational burden. 

Fig. 5. Minimum and maximum low-frequency, 

homodyne-measurement noise spectra vs. ,/2 

( , = 2 for a fiber of length 325 x lo5 km, 

k = lfs) at 1.55vm in a dispersionless, lossless 

fiber. QPM denotes the exact calculation, FWM 

the four-wave mixing approximation, and TCE 

the terminated cumulant expansion (K = 2,3,4,5 

shown). The input field is a 1-Watt, coherent 

state; 0 dB is the coherent-state noise level. 

Fig. 6. Minimum and maximum low-frequency, 

homodyne-measurement noise spectra vs. ,/2 

( , = 2 for a fiber of length 325 x lo5 km, 

k = lfs) at 1.55 vm in a dispersionless fiber. SPM 

denotes the exact, lossless calculation, and TCE 

the terminated cumulant expansion (K = 3,4,5 

shown) with 0.2dB/km linear loss. The input field 

is a 1-Watt, coherent state; OdB is the coherentstate noise level. 

Armed with the TCE approximation, we can obtain homodyne-noise spectra for situations in 

which the exact calculations we thwarted by moment linkage. Consider propagation in a lossy, 
dispersionless Kerr-effect fiber. In Fig. 6 we have plotted the exact SPM result for the lossless 

case (solid curve) and the TCE solutions (dashed curves) for a fiber with a power loss coefficient 

of 0.2dB/km. The regions of overlap follow the same trends seen in Fig. 4, and predict a 4dB 

increase in the noise level over the lossless case. 

Inclusion of dispersion couples time slots and greatly increases the complexity of TCE moment 

calculations, even with a coherent-state input. If we address N time samples of the field, there 

are (19N + 15N2 + 2N3)/3 complex moments, or (35N + 30N2 + 4N3)/3 real quantities, to track. 

Each real quantity obeys one of 23 types of differential equation, which contain anywhere from 

5 to 87 terms. For N = 100, there are thus 717,300 moments, or 1,434,500 real quantities, to 

be computed. We are working on these calculations at present, and expect to be reporting our 

results in the near future. 

5 Conclusions 

We have presented a general theory for quantum propagation of an optical field in a lossy, dispersive 

Kerr-effect fiber. Our approach leads to equations that are continuous in space, but discrete in 

time. The time granularity is set by a phenomenological Kerr-effect time constant needed to 

properly recover the known results of classical self-phase modulation. Other theories have been 

developed that describe propagation in such a fiber [11],[16], but ours is the first for which a 

material time constant has been specifically employed to temper the instantaneous interaction. 

It has been argued that the presence of dispersion provides a much more constrictive bandwidth 

limitation than ril, thereby eliminating the need for this Kerr-effect time constant [17]. We 

disagree, but in the interests of brevity, we shall confine our remarks to a few brief points. First, 

it has been shown that there is a four-wave mixing region in which dispersion enhances squeezing 

[la]; here we may expect that dispersion exacerbates the need for a finite TK to correctly determine 

the validity limit of FWM. On the other hand, if there are propagation regimes-such as soliton 

propagation-wherein dispersion renders a finite Kerr-effect time constant unnecessary, then that 

impotence should appear in our calculations, i.e., our noise results should be insensitive to the value 

we assign to TK. Note that, even with loss and dispersion, the value of TK is irrelevant to linearized 

noise analysis, and this includes the linearized noise theory of quantum solitons. Finally, the 

theory we have presented handles the case of an arbitrary field, in either the normal or anomalous 

dispersion regimes, and is more encompassing than those restricted to soliton propagation. 

Appendix 

For a real-valued classical random vector 2 - (XI, X2, . . . , X,) whose joint characteristic function 

is qh(3 = (exp(i8. a)), the cumulants are defined by: 

where the cumulant generating function is a($ - ln(qh(3). Higher order cumulants contain 

information of decreasing significance [lo]. 
References 

[I] R.M. Shelby, M.D. Levenson, R.G. DeVoe, S.H. Perlmutter, and D.F. Walls, Phys. Rev. Lett. 

57, 691 (1986). 

[2] K. Bergman and H.A. Haus, Opt. Lett. 16, 663 (1991). 

[3] L.G. Joneckis, and J.H. Shapiro, Quantum Electronics and Laser Science Conference, Vol. 

12 of 1993 Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 

282. 

[4] L.J. Joneckis, J.H. Shapiro, J. Opt. Soc. B 10, 1102 (1993). 

[5] K. J. Blow, R. Loudon, and S. J.D. Phoenix, J. Opt. Soc. B 8, 1750 (1991). 

[6] G.P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989). 

[7] H.P. Yuen and J.H. Shapiro, IEEE Trans. Inform. Theory IT-24, 657 (1978). 

[8] H.P. Yuen and J.H. Shapiro, IEEE Trans. Inform. Theory IT-26, 78 (1980). 

[9] R.M. Shelby, (personal communication). 

[lo] C.W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and Natural Sciences 

(Springer-Verlag, New York, 1985). 

[I 11 S. J. Carter, P.D. Drummond, M.D. Reid, and F .M. Shelby, Phys. Rev. Lett. 58, 1841 (1987). 

[12] S.J. Carter, and P.D. Drummond, J. Opt. Soc. B 4, 1565 (1987). 

[13] Y. Lai, and H.A. Haus, Phys. Rev. A 40, 844 (1989). 

[14] Y. Lai, and H.A. Haus, Phys. Rev. A 40,854 (1989). 

[15] P.D. Drummond, Phys. Rev. A 42, 6845 (1990). 

[16] S.J. Carter, P.D. Drummond, Phys. Rev. Lett. 67, 3757 (1991). 

[17] H.A. Haus, and F.X. Kaner, Phys. Rev. A 46, R1175 (1992). 

[18] B. Yurke, Phys. Rev. A 35, 3974 (1987). 
QUANTUM AMPLIFICATION AND QUANTUM 

OPTICAL TAPPING WITH SQUEEZED STATES 

AND CORRELATED QUANTUM STATES 

2. Y. Ou 

Department of Physics, Indiana University-Purdzle University at Indianapolis (IUPUI) 

Indianapolis, IN 4 6202 

S. F. Pereira and H. J. Kimble 

Norman Bridge Laboratory of Physics 12-33) California Institute of Technology 

Pasadena, CA 91125 

Abstract 

Quantum fluctuations in a nondegenerate optical parametric amplifier (NOPA) are investigated experimentally with a squeezed state coupled into the internal idler mode of the 

NOPA. Reductions of the inherent quantum noise of the amplifier are observed with a minimum noise level 0.7 dB below the usual noise level of the amplifier with its idler mode in 

a vacuum state. With two correlated quantum fields as the amplifier's inputs and proper 

adjustment of the gain of the amplifier, it is shown that the amplifier's intrinsic quantum 

noise can be completely suppressed so that noise-free amplification is achieved. It is also 

shown that the NOPA, when coupled to either a squeezed state or a nonclassically correlated 

state, can realize quantum tapping of optical information. 

1 Introduction 

It has been known since the date when optical amplification was first realized that fundamental 

principles of quantum mechanics play an important role in the noise performance of a linear 

amplifier [I, 23. For example, it was found [3, 41 that even in an ideal case when all the classical 

noise is eliminated, "extra" quantum noise from an amplifier's internal modes will add to the 

amplifier's output thus preventing noise-free amplification and degrading the output signal-tonoise ratio (SNR) relative to that of the input. Such "extra" quantum noise would destroy any 

coherent quantum superpositions that are often encountered in the microscopic world, should one 

try to amplify the microscopic quantum superposition to a macroscopic scale so as to produce a 

paradox such as Schrodinger's Cat [5]. 

However, Caves pointed out in a systematic analysis [6] of quantum noise in a linear amplifier 

that noiseless amplification is possible with a phase-sensitive amplifier (for which the gain depends 

on the phase of the input signal). On the other hand, for a phase-insensitive amplifier, although 

extra noise cannot be avoided as stated above, it may be rearranged, according to Caves' analysis. 

More specifically, a phase-insensitive amplifier is described by a general quantum model [6, 73: 
where t~"+'"~ is the annihilation operator for the input and output signal, G is the power gain 

of the amplifier and k is an operator related to the internal modes of the amplifier and satisfies 

[k, kt] = 1 - G. The quantum fluctuations in k will give rise to the "extra noise" added to the 

output signal. From Eq.(l), one can derive Caves' uncertainty relation [6] 

where Ai E ((A~;)~)/G is the input equivalent noise added to the quadrature-phase amplitudes 

Xi(i = 1,2), with XI E (h+ht)/2,X2 E (h-dt)/2i,kl E (E'+kt)/2, and k2 E (k-Pt)/2i. Thus 

the noise in amplification in one quadrature-phase amplitude where the signal is encoded can be 

suppressed while the extra noise demanded by Eq.(2) is mostly coupled into the unused conjugate 

quadrature, with their noise product satisfying Eq.(2). By following this line of reasoning, it was 

suggested 18, 9, 10, 111 that by coupling the amplifier's internal modes to a squeezed vacuum 

instead of the usual vacuum state, the suppression of added noise for one quadrature can be 

achieved as stated above. 

In the analysis of Caves, it was assumed that the input field hi" is independent of the internal 

modes of the amplifier described by E'. On the other hand, the situation will be totally different 

if ~in and k are correlated. Notice that the quantities in Eq.(l) are amplitudes of the relevant 

fields. Thus interference between the amplitudes of 2" and F may give rise to cancellation of 

their quantum fluctuations and lead to noise reduction in the amplifier's output. Quantum noise 

subtraction has been realized with various kinds of correlated quantum fields [12, 13, 141. 

The distribution of information in the modern age requires division of incoming information 

into identical pieces for sharing by many users. An optical tap is a kind of information divider 

by optical means, with which one can extract the needed information while at the same time 

leaving the information readable by other users down the line [15]. The challenge is of course to 

tap the information without degradation of the signal-to-noise ratio (SNR) for both tapped and 

transmitted information. An optical divider or tap is usually a four-port device with two inputs 

and two outputs (the law of quantum mechanics requires there to be an extra input). A typical 

divider is simply a beamsplitter: information comes in one input and is divided into two outputs. 

However, the uncorrelated quantum noise from the other unused port will add to the outputs and 

degrade their SNRs. On the other hand, it is known that quantum noise can be suppressed with 

a squeezed state. Shapiro thus suggested [15] to couple the unused port to a squeezed vacuum to 

reduce its quantum noise. Another technique is to use two correlated quantum fields as the two 

inputs. Quantum correlation between the two inputs will subtract out the quantum noise in the 

outputs. Such techniques can be used in any four-port system for information division. 

In the following sections, we will mainly discuss quantum fluctuations in a nondegenerate 

optical parametric amplifier (NOPA) which has only one internal mode called "idler". In section 

2, we first describe an experiment in which we couple a squeezed light field into the internal idler 

mode of the NOPA and demonstrate quantum noise reduction by the scheme of rearranging the 

quantum noise between two conjugate quadrature-phase amplitudes. In section 3, we will discuss 

quantum noise cancellation in amplification with a correlated quantum statk where noise-free 

amplification can be achieved with moderate correlation. In section 4, we will consider the NOPA 

as a four-port system (2 inputs and 2 outputs) and show that it can be used as a quantum optical 

information tap when the inputs are coupled either to a squeezed state or to a correlated quantum 

state. 
2 Quantum Noise Reduction in Optical Amplification 

with a Squeezed State 

A nondegenerate optical parametric amplifier (NOPA) is an optical amplifier that utilizes nonlinear 

coupling to convert energy in a pump beam(s) to a signal beam. It can be realized in either threewave mixing or four-wave mixing processes. Besides the pump beam(s) and the input-output 

signal beams, another beam called "idler" is coupled to the pump and signal beams at the same 

time. This idler beam labled as gn corresponds to the so-called internal mode of the amplifier 

discussed earlier. In terms of the quantities in Eq.(l), fi = dm f@t and G is related to the 

pump beam. Fig.1 shows a NOPA with a coherent signal input and its idler mode coupled to 

a squeezed vacuum generated by a squeezer. Detailed descriptions of each device usedin the 

diagram can be found in Ref.[l4b]. In the linear operating regime (small input signal), the pump 

beam is undepleted and does not contribute any extra quantum noise to the output [16]. his 

Eq.(l) becomes 

&out = & ;in + ,/= jpt. (3) 

We can rewrite Eq.(3) with the quadrature-phase amplitude &(8) e e-" + 2teie (c = a, b) as 

If the fields d and % are independent of each other, the output noise of the amplifier is then given 

by 

N~((B = G ~:(e) + (G - i)~p(-e), (5) 

where Nj(0) ((a - (A))2) (i = a, b). 

si@ in sipd out 

SQUEEZER I 

FIG. 1. Diagram for the experiment of quantum noise reduction with a NOPA. 

The shaded part of the noise circle for the amplified signal corresponds to amplified 

input signal noise and the rest of the noise comes from the extra noise contributed by 

the amplifier's internal modes 
Usually, the idler mode 8 is coupled to empty vacuum and NP(-81 = 1, resulting in extra 

noise G - 1 in the output. On the other hand, with the idler mode b coupled to a squeezed 

vacuum, for the squeezed quadrature of 8 = 8-, we have Nb- = Nf'(-O-) < 1, thus the extra 

noise at the output due to the idler can be reduced. For the other quadrature, however, the extra 

noise will be enhanced. Therefore, as we change the phase and look at different quadratures, 

we will obtain a phase-sensitive noise level for the output with noise reduction at some phases 

and noise enhancement at other phases, as shown in Fig.2, where we plot the signal output noise 

level as a function of local oscillator phase. It is found that the minimum noise level in the 

phase-sensitive curve ii drops below the phase-insensitive curve i, which is the output noise level 

when the idler mode is coupled to the vacuum, thus demonstrating quantum noise reduction in 

the amplification process. The phase-insensitive curve i (QG) also gives a measure of quantum 

noise gain as compared to the vacuum noise level go (G, E !DG/QO) [14b]. The dashed trace iv 

corresponds to the output noise level expected for a lossless system with perfectly squeezed idler 

at the same operating gain of the amplifier. To better quantify the noise reduction, we tune the 

phase to 8 = 8- and block and unblock the injected squeezed light field. When the squeezed light 

is blocked, it corresponds to a vacuum st ate coupled to the idler mode. In Fig.3, we plot the output 

noise level as we turn "ON" and "OFF" the squeezed light. By performing the same measurement 

at different gains of the amplifier, we can plot the amount of noise redution A- Q(8,)/QG 

against the quantum noise gain G, as in Fig.4. The best noise reduction of -0.7 dB is observed 

at G, = 2.6 (4.2 dB). The solid curve in Fig.4 is a theoretical prediction for our system with 

0.3% internal round-trip loss for the NOPA and with 30% external loss (mainly propagation and 

detection losses), as determined by independent measurements [14b]. The amount of squeezing 

-2 8- 8,+n LO Phase, 8 

FIG. 2. Spectral density of photocurrent fEuctuations for i- generated by NOPA's 

signal output Gout as a function of the local oscillator phase 8. Trace i is the amplified 

noise level QG when the idler mode is in a vacuum state, while trace ii corresponds to 

the case when the idler is in a squeezed vacuum state. Trace iii is the vacuum noise 

level lPO and the dashed trace iv corresponds to the output noise level expected for a 

lossless system with perfectly squeezed idler. 
Time (s) 

FIG. 3. Amplified noise level of the signal output for NOPA. "OFF" correponds to 

the output noise level QG for a vacuum state coupling to the idler mode. "ONn gives 

the output noise level a($-) for a squeezed state input to the idler. The noise levels 

are referenced to the vacuum noise level !Po. 

FIG. 4. Quantum noise reduction A- for the amplified output signal as a function 

of the detected quantum noise gain G, for a squeezed idler input of N- = 0.52. The 

solid curve is the theoretical prediction for our system and the dashed curve ii is for a 

t lossless system with perfect squeezing for the idler. 
that is coupled into the idler mode is also directly measured to be Np(8,) = 0.52. Thus all the 

relevant parameters in the theory for our experiment are measured independently. It is seen that 

the experimental data fit the theoretical prediction quite well. The dashed trace ii corresponds 

to the maximum possible noise reduction with a coherent signal input in a lossless system with 

perfect squeezing coupled to the idler mode (Np(8,) = 0), in which no extra noise is added to 

the amplified output. 

3 Cancellation of Quantum Fluctuations in Optical Amplification with Correlated Quantum Fields 

In the discussion of last section, we assumed that the quantum fluctuations in the signal input 

and the amplifier's internal idler mode are uncorrelated. When their quantum fluctuations are - 

correlated, however, we cannot write the output noise as in Eq.(5) because the correlation between 

2F-8) and .,@(-8) may result in cancellation (or enhancement) of their fluctuations through 

destructive (or constructive) interference. 

The quantity to describe the degree of correlation between fields 8 and 6 is the correlation 

function defined as 

Assume that aka and aZb are positively correlated, that is, Cab > 0 and that the fluctuations 

of field h are smaller than or equal to that of field &, that is, iVa = (612xa) 5 (A2&,). We will 

encode the signal only into the field h ((K) A # 0 and (%) = 0) because it has less noise. 

The signal-to-noise ratio (SNR) for the field b is then Ra = A2/iVa. On the other hand, because 

the two fields are correlated, we have for the noise in the difference of the two fields: 

where we have minimized Nd by choosing the optimized coefficient Am = (AA~AJ~~)/(A~~

A, = Cab when (~22.) = (a2%). Thus the noise in the difference of two fields is smaller than the 

noise in the single field b. So the optimized signal-to-noise ratio (SNR) is & = (*a-~mkb)2/~d = 

A2/Na(l - Czb) if both B and 6 fields are employed. Obviously, > &. 

Next let us consider the situation when the fields b and & are injected into the signal and idler 

ports of the amplifier, respectively. We adjust the phase of the pump beam so that Eq.(4) becomes 

Thus the amplified signal becomes 

The noise of the amplified signal output is calculated as 
FIG. 5. Diagram for a beamsplitter. 

which reaches minimum value of 

Naz\ = G((A~~ - X,A%)~) = G(A~~~)(I - Czb) = GNd, (10) 

when d(~ - 1)/G = A, = Cab. Note that Eqs.(8) and (9) are for the signal beam alone; mixing of 

the field 8 with the field & as required for Nd has taken place within the amplifier itself. Combining 

Eqs.(8,10), we obtain for the output SNR 

Therefore the output SNR Rpt is equal to the input SNR Rd (and > R,) with the signal amplified 

by the gain G. No extra noise is added in the amplification process. However, noise-free amplication can only be achieved at some specific gain G = 1/(1 - A:) determined by the correlation 

function Cab between the two fields 8 and &. When the two fields are close to perfect correlation 

with A, = Cab + 1, the gain G can be arbitrarily large. 

4 Quant urn Optical Information Tapping with Squeezed 

States and Correlated Quant urn Fields 

The concept of quantum optical information tapping was first discussed by Shapiro [15] for a 

beamsplitter with squeezed state coupled to one of the input ports. Consider a. beamsplitter 

shown in Fig.5, where the input port ii2 is in a squeezed state with the degree of squeezing 

denoted by S. A coherent signal of size A is injected into the other port labled as 2cl with input 

SNR RF = A2. It can be easily calculated [15] that for a beamsplitter with transrnissivity T and 

reflectivity R, the output SNR at both output ports are given as 

The efficiency of this information tapping scheme can be quantified [17] as the ratio of the output 

SNRs to the input SNRs: 
2 which has maximum value of - 

1+S 

when T = R = 112. It is seen that 

for squeezed state input at port 2 (S < 1). On the other hand, for classical state (S L: I), we 

always have q < 1 [lg]. Thus Eq.(13) is the criterion for realization of a quantum tap for optical 

informat ion. 

Quantum information tapping can also be achieved for a beamsplitter with correlated quantum 

fields &,I as the two inputs. For this case, let us assume the two fields have the same noise level, 

that is, (~~2~) = (A2Xb). For a beamsplitter, we have for the quadrature of the fields: 

where we only write down the X-quadratures, in which information is encoded. 

It is easy to show that Kt = RF when R/T = A: = C,2b as before in Section 3. For output 

port 2, we find 

Thus the information tapping efficiency 

Therefore quantum optical information tapping is achieved with correlated fields. Notice here we 

choose R, T so that RiUt = RF. Of course, we could choose R, T to maximize q. However, q will 

never be close to the perfect value of 2 even for perfect correlation. This is because of the plus 

sign in Eq.(l4b) required by unitarity for any beamsplitter; and it can not be changed to a minus 

sign no matter what you do with the relative phase of the two fields. In the following, we will see 

a different situation for the NOPA. 

For the NOPA, there are also two inputs (signal and idler) and two corresponding outputs, as 

shown in Fig.6. With proper phase adjustment of the pump, the input-output relations for NOPA 

are given as 

&out = fi &in - m Pnt, 

signal in signal out 

FIG. 6. Diagram of NOPA as a four-port device. 

Aam 1 

F 

A - bi" fl 

NOPA 

idler in idler out 

(GI 

1 gut F 

1 

/ bw' 

A 
or in terms of the quadrature-phase amplitudes 

First, let us assume that a coherent signal of size A is injected into the signal port for amplification 

and squeezed state of squeezing S is injected into the idler port. The input SNR is then Rin = A2. 

From the input-output relations in Eqs.(lG), we can calculate the output SNRs as > 

Hence the information tapping efficiency '1 has the form of 

G + (G - 1) 2G(G - 1)(1+ S) + S 2 

'1 = - N- for G2 >> S. (18) G+(G-1)s GS+G-1 G(G-1)(1+S)2+S 1+S 

, Thus quantum optical information tapping is possible (9 > 1) as long as S < 1. When G2 >> S, 

'1 approaches 2/(1 + S), which is the same as the result of Shapiro [15] for a beamsplitter. Of 

course, in this process, the signal is amplified. 

As for the situation with correlated quantum fields as the inputs, for the parameters discussed 

in section 3, we know that R,"ut = Rd. From Eqs.(lG), we can easily find out Riut for the idler 

output. For the parameters given in section 3, we have 

and 

(~~2r~) = (I - x;)[(A~~:) - (A~~E)] + GN~ = GN~ for (~'2:) = (A~R~). 

Therefore, Riut = Rd(G - 1)/G and 

where the second equality follows since G is chosen as in section 3, namely, G = 1/(1 - C:,,). 

Eq.(19) shows that we can always realize quantum optical information tapping in NOPA with 

correlated quantum fields. 

In fact, for any linear four-port device with two inputs and two outputs, we can realize quantum 

optical information tapping with input of either a squeezed state or a correlated quantum state: 

5 Acknowledgments 

The work was supported by the Office of Naval Research and by the National Science Foundatio~. 
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(1 992). 
QUANTUM NOISE LIlNIITS TO 

MATTER-WAVE INTERFEROMETRY 

Department of Physics, Texus A&M University 

College Station, Texas 77843 

and 

Max-Planck-Institut fiir Quantenoptik, W-8046, Garching, Germany 

Jonathan P. Dowling 

Research, Development, and Engineering Center, AMSMI-RD-WS-ST 

U. S. Army Missile Comnand, Redstone Arsenal, Alabama 35898 

We derive the quantum limits for an atomic interferometer in which the atoms 

obey either Bose-Einstein or Fermi-Dirac statistics. It is found that the limiting 

quantum noise is due to the uncertainty associated with the particle sorting between 

the two branches of the interferometer. As an example, the quantum-limited 

sensitivity of a matter-wave gyroscope is calculated and compared with that of 

laser gyroscopes. 

Matter-wave intederometq dates from the inception of quantum mecharmics, i.e., 

the early electron diffraction experiments [I 1. More recent neutron interferometny experiments have yielded insight into many hndmental aspects of quantm mecharmics 

[2]. Presently, atom interferometq has been demonstrated and holds promise as a 

new field of opties - matter-wave optics [3]. This field is particulasly interesting since 

the potential sensitivity of matter-wave interferometers 143 fm exceeds that of their 

Eght-wave or "photon" antecedents [5]. 

However, as was emphasized at the recent Solvay conference on quanatm optics, 

there is at present no paradigm available for calculating or esthating the quantlun 

noise limits to matter-wave interferometers, and therefore we have no basis for estimating the potential sensitivity of devices based on matter-wave interferometrgr (e.g., 

gyroscopes) [6]. 

In order to motivate the analysis and derive the quantum lhits, we proceed as follows: First, we "set the stage" by considering a simple gyroscope and deriving the rotation-induced signal in matter-wave optics. Next, we proceed to develop the theory for 

atomic interferometers, cast in an operator formalism that is well suited to a quantum 

noise analysis, and then we obtain the quantum noise limits for matter-wave interferometry. Finally, we compare current laser gyroscope sensitivity to that of near-term, 

matter-wave devices. 

We begin by considering an idealized atom interferometer used as a rotation detector 

or gyroscope, as shown in Fig. 1. From this diagram it is easy to see that the atomic 

path difference between the upper branch a and the lower branch P is given by Sd = 

ZrSZt, where G? is the angular velocity of the interferometer, r is the radius of the circle, 
FIG. 1. A schematic illustration of an interferometer with semicircular arms to be used as 

a rotation sensor or gyroscope. The loop rotates with an angular frequency l2 about an axis 

through its center and normal to the loop plane. The path difference between counter- and 

z r copropagating beams can be easily seen to be 6-4 = ri2 -, where v is the atomic velocity. v 

From these considerations the phase shift of Eq. (21) follows immediately. We may then 

use this result to estimate the minimum detectable rotation rate pin, Eq. (23). 

v the particle velocity, and t = nr lv is the particle transit time through the interferometer. This readily translates into a Sagnac phase difference of Sqap = k (A, - Ap) 

= 2dRl% v = 2ARIR v, where 2 = 7i I mu is the atomic de Broglie wavelength [7] and 

A the area enclosed by the arms. The phase signal is then given by q signal , - 

2A mRIA; independent of the interferometer shape as long as A is the total area 

enclosed. This expression holds for both atom and light interferometers, if, in the - 

photon case, we define an effective photon mass my implicitly by my d2 = ti o. Now, 

since the "mass" of a photon is governed by optical energies of a few electron volts - 

and atomic masses are of order lo3 MeV - we see that matter-wave gyroscopes 

potentially have a signal that is enhanced by many orders of magnitude, compared to 

light (laser) gyroscopes. Thus motivated, we next consider a detailed analysis of phase 

sensitivity in matter-wave interferometry. 

2. A Simple Model 

In accordance with current experiments [3], let us consider the model illustrated 

in Fig. 2. There, we see a stream of N atoms passing one-at-a-time through a beam 

splitter into a simple interferometer with upper and lower branches labelled a and P, 

respectively. Upon recombining the two beams, we inspect the resultant interference 

pattern for phase shifts induced, say, by a gravitational potential between the two 

branches or a net rotation of the system. As in the optical dual [5], one might expect 

that the overall sensitivity of the device will be limited by the quantum limits imposed 

by particle number fluctuations AN on the phase noise Aq in the interferometer. Pt is 
defector a 

;> 

*I 

detector b 

FIG. 2. We illustrate a scheme whereby a stream of N atoms is sent through a simple 

interferometer during a measurement time t,. The atoms are split at beam splitter 1, 

follow paths a or P, are reflected off the mirrors, and are then recombined at beam 

splitter 2. The recombined atoms are detected at upper detector a or lower detector b 

where interference fringes are recorded. 

often stated that dN is to be associated with the fluctuations in the arrival time of 

atoms in the input beam, i.e., AN - 6 where A is the mean number of particles. 

However, we shall show that the particle ber noise arises not from fluctuations in 

the input beam intensity but rather from beam splitter uncertainties pertaining to the 

lack of knowledge of which path, a or P , the atom has taken through the interferometer. 

3. The Quantum Signal 

Let us continue developing our simple model depicted in Fig. 2. We assume that, 

upon reflection from a beam splitter surface, the particles undergo an unimportant 

phase shift that we take to be ~12, but that in reality depends upon the structure of the 

beam splitter. Upon passage through a beam splitter, however, the atom undergoes a 

phase shift of qi, i = 1,2, for the first and second beam splitter, respectively. The 

cumulative effect in the interferometer of these various processes on the atomic wave 

function y is depicted in Fig. 3, and leads to a wave function va corresponding to the 

upper detector and yb for the lower detector, namely 

where 8, - z12 + kRa + q2 , and Bb - KRa + cpl + +'Pz, and where, without loss of generality, we let cpl = q2 = z . Here, k is the atomic wave number and la and Rp are the 
FIG. 3. Chasing phases through the interferometer accounts for ac 

in the upper or lower detectors. The phase shift upon reflection is apbihary, but we choose 

it here to be z/2 for simplicity. Upon transmission, a phase shift of 91 or 92 k assumed 

for beam splitter one or two, respectively, and without loss of generality we take 91 = 

92 = z. 

path lengths through the upper and lower branches, respectively. We imagine now 

that the beam is recombined by the second beam splitter and then the detectors a and b 

shop in Fig. 2 count the number of atoms as they arrive in the recombined upper 

beam or lower beam, respectively. If we label N atoms with the index i = 1, .. ., N, as 

those sent through the interferometer during a measurement time t,, then the appropriate state vector @,$ for the ith atom in the interferometer, after recombination, 

is given by 

where here (pap - k(da - +. We see that this state is an appropriate superposition of 

the number states Jla, Ob ) and 1 0, , lb ) corresponding to an atom incident on the 

upper or lower detectors, respectively. The state vector for the N-atom state is 

then constructed via a direct product of the individual atomic states, namely 

" f A 

Let c ,i and c , where 0 = a, b, be the creation and annihilation operators, respectively, for the number states ina, nb ) i, where, corresponding to number op- A ^f " erators nqi - %i cqi , the eigenvalues na and nb are O or 1. Then the number operaA 

tor No for the number of. upper or lower atoms is determined by 
A 

and the operators c obey the commutation relations 

where the plus or minus sign indicates Bose or Fermi statistics, respectively. %he 

statistical nature of the atoms. will be impodant in circumstances where the density 

of particles in the interferometer is so large that there is more than one atom. at a 

time within a single coherence length, or if the atoms are injected in a correlated 

A 

manner into the input ports [91. The expectation values (NJN of these number operators, Eq. (4), are given by 

This yields the expression for the mean number of atoms in the a and P branches as 

These expectations constitute the signal; we proceed to calculate the noise. 

4. A Calculation of Poisson Noise 

As noted earlier, it is frequently stated that number fluctuations dN in the in- 

- 

terferometer should be just noise of the form 4% due to fluctuations in the input 

beam. Let us briefly investigate this hypothesis. A reasonable assumption is that the 

distribution of the N atoms in the input beam is Poissonian with a distribution function P, , given by 
where H is the mean number of particles in the beam. The signal or expectation 

A 

value of the operators No is then given by 

where H = n Pn, and the upper and lower terms in braces are associated with 

n 

detectors o = a and o = b, respectively - a convention we shall use throughout. 

Hence, the fluctuations accompanying this signal are determined by the root 

A 

variance (dNd, which is found to be 

Now, to get a determination of minimum detectable phase shift, one usually 

equates the signal, Eq. (9), to the noise, given by Eq. (10). Regardless of the choice of 

qap, we see from Eqs. (9) and (10) that upon equating signal to error, the phase dependence cancels out and we have no determination of the minimum detectable 

phase. The point is that Nis not a random number, since we have the constraint N, 

+ Nb = N = constant, and fluctuation3 in the incoming atomic beam do not determine sensitivity. In other words, precise knowledge of the value of N obviates the 

need for a Poisson analysis, since N is clearly not a random variable. What, then, is 

the limiting noise mechanism in the interferometer? 

5. The Quantum Noise 

We compute the quantum noise fluctuations using the second quantized formalA 

ism developed earlier. Recalling the definitions for the number operator N, Eq. (4), 7 

and the state vector I@)N, Eq. (3), and using the commutation relations, Eq (5), we 

may write, 
where, as before, the upper and lower terms in braces correspond to cr = a or b, respectively, and the f sign refers to the statistics of the particles: a plus sign for bosons 

and a minus sign for fermions. We note that the last, statistics-dependent term of Eq. 

(11) is the sum of non-negative matrix elements and so itself is non-negative or non- , positive, according to the plus sign or negative sign, respectively. A quantitative 

analysis of the contribution of this statistics-dependent term requires a specific model of 

the coherences between at0.1~~ in a dense beam. However, one can qualitatively state 

that for sufficiently high densities the use of fermionic atoms will tend to lower the 

quantum noise limit. This is because the last term will be negative. Bosons will have the 

opposite effect. Detailed analysis of the statistics-dependent contribution is beyond the 

scope of this letter, and will be left to a later work. Hence, since, in current experiments, the beam intensity is so low that there is only one atom at a time within a single 

coherence length. In this case, the statistics-dependent second term in the last line of 

Eq. (11) is zero, and we are left with the result 

We notice that this result depends on the total number of atoms N and not the mean 

number TE as in the Poisson-distribution argument given before. Now, the signal in 

either branch Nu is given by Eq. (7). 

The quantum fluctuations in phase Aqap in the measured phase difference qap 

may be determined by [B] 
a result that is independent of p This independence might appem swf-jising at OEP ' 

first, but it is a direct result of the fact that the quantum nunnber state noise (BNJ is 

proportional to the slope of the signal QVd for the upperAower n er states cornidered here. (See, in pdcular, reference 8.) Again, we stress th is not the eqectation value but rather the total number of atoms detected in the measwement he 

t,. This is not then the expression' one would expect from application of the uneertainty principle, for in that case N would have to be replaced by m). We reemphasize that it has not been clear what form of the uncertainty p~nciple one 

should even use in an atom interferometer [GI. For light, the so called 

uncerlainty principle, BpMT 2 1, yields for a coherent state Ap g 1 / (2V) - where the 

expectation (AT) and not the total number N is used. For atoms it is not obvious at d 

what the relationship should be, and we have shown that the resdt is lunezrpected in 

that Eq. (13) depends on the total number N , that is precisely known for the atom 

interferometer, and where (iV) has no meaning. In contradistinction, in a laser 

interferometer, it is impossible to know the total number of photons and only the 

mean can be specified. Hence, the atom result, Eq. (13), is quantitatively, 

qualitatively, and philosophically different from the optical result. Hence, Eq. (13) is 

indeed a novel.resultt We note that B. Yurke obtained a similar result for F 

using spin algebra techniques [9]. 

6. Compagping Laser and Matter Gpos 

We conclude by applying this result tn the gyroscope problem. Let us note that the 

atom number N is given by j t,, where j is the atomic flux (in atoms per second) hitting the detector. We have fi-om Eq. (14) the minimum detectable phase shift, pmin = 

signal 2/1/j%and equating this to the signal derived earlier, p = 2AmR/'i)i, we find 

the minimum detectable rotation rate amin is given by 

$pin h 1 __ - (matter). 

This should be compared to the same result obtained from using an optical interferometer in which the flux j is given by the power P divided by the photon energy fiw 

[5,71, in other words 

where my is the effectme photon mass, defined by my-= tiolc2. In Table I we compare and contrast properties of the matter-wave and laser light interferometers in 

order to gauge their effetiveness in measuring R min. As mentioned before, we note 
that the typical photon effective mass gives an increase in sensitivity of 10". This 

mass factor, however, is offset by the low particle flux available for atoms. 'FBais fact 

increases the laser moscope sensitivity over that of matter-wave devices by a factor 

of around lo2. In addition, the atoms make about one &round trip" through an 

interferometer, whereas in a ring laser gyroscope the photons make many (= lo4) 

circuits around the ring and yield an additional sensitivity fador of lo4 in favor of the 

laser system. 'This still leaves the matter-wave device 104 times more sensitive. 

mary then, we conclude that the phase uncertainty arising in an atomic 

intefierometer arises fkom atomic nunaber fluctuations associated with the sorting of 

the particles between the two s of the interferometer. Applying our results to an 

interferometer used as a gyroscope, we find that a matter-wave moscope can be expected to be more sensitive to rotation by some four orders of magnitude than present 

laser devices. 

TABLE I. Compared and contrasted are different properties of matter-wave and optical 

gyroscopes in terms of their sensitivity to phase differences - or equivalently - rotation 

rates. We see that the high mass of atoms initially contributes an increase of sensitivity 

of 10l0, but that the low atomic beam intensity, compared to photon beams, removes some 

of this advantage, as does the reduced number of round trips possible in an atom 

4 interferometer. Nevertheless - a typical factor of a 10 increase in rotation sensitivity 

can still be expected using atoms rather than photons. 
The authors would like to thank J. Clauser, J. Fransen, J. Kmble, P. Knar, 

J. Mlynek, A. Overhauser, D. Pritchard, M. Reinsch, and S. Werner for many interesting discussions relating to this work. M. 0. S. thanks the Office of Naval Research and J. P. %). acknowledges the National Research Council for financial support. 

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[9] B. Yurke, Phys. Rev. Lett. 56, 1515 (1986). 
BLOCH VECTOR PROJECTHON NOISE 

E. J. Wang 

Department of Physics, Duke University, Durham, North Carolina, 27708-0905 

A. M. Bacon, H. Z. Zhm, and J. E. Thomas 

Department of Physics, Duke University, Durham, North Carolina, 67708-0305 

Abstract 

In the optical measurement of the Bloch vector components describing a system of N twolevel atoms, the quantum fluctuations in these components are coupled into the measuring 

optical field. This paper develops the quantum theory of optical measurement of Bloch vector 

projection noise. The preparation and probing of coherence in an effective two-level system 

consisting of the two ground states in an atomic three-level A-scheme are analyzed. 

1 Introduction 

The properties and generation of an opticd squeezed state have been interesting subjects of study 

for a number of years. The Bloch vector model of a two-level atomic system interacting with 

a laser field, and the use of angular momentum components Jj to represent the N two-level 

atoms [I], have also been widely investigated. It is known that quantum systems with dynarnical 

variables in the form of nonlinear products of the position and momentum operators are different 

from those involving only the position and momentum operators. For instance, in a system 

of N twolevel atoms, described by a Bloch vector spin angular momentum 3, the uncertainty 

relation, AJ1 . A J2 2 $1 (33) 1, depends on the quantum state of the system, as opposed to that 

of Ax Ap 2 $ for the two quadrature phase components of an optical field [2]. The quantum 

fluctuations in atoms hence provide an interesting system for the study of uncertainty relations, 

and are of practical importance. For example, the fluctuations in an atomic system contribute to 

noise that can in principle, limit the accuracy of atomic frequency standards 131. 

As a simple example, let us consider the spin model for a single two-level atom with a ground 

state 11) and an excited state 12). The Bloch vector operators are 

where 

hi i), i, j = 1,2. 
These operators obey the usual commutation relation for angular momentum operators, 

where rijk is the Levi-Civita symbol. It is easy to show that i describes a spin-f system. 

For a superposition state 

e 0 Irp) = cos - 11) + ei4 sin -12), 2 2 

where 8, and 4 are some angles, we have 

Clearly, the vector r = ((il), (i2), (83)) falls onto the surface of a sphere of radius i. The fluctuations in the components of s are, 

1 

(A) = (i) - () = - - (4) j = l,2,3, 4 (4) 

with the fluctuation in the total Bloch vector 

It is also easy to show that 

1 (A;:) ' (AS:) 2 al~ijk(&)12. 

Now let us consider the situation for N two-level atoms. If there is no mutual interaction 

between the atoms, the system can be described by the total spin angular momentum operator 

We assume that all atoms are in the ground state 11) initially or, equivalently, the system is in 

the angular momentum eigenstate IS = $, S3 = - $) (Fig. la). Now we have 

and 

We hence obtain 
FIG. 1. Bloch vector for an N-atom system. In a), all atoms are in state II), and 

the mean total Bloch vector (So) points down. b), (So) is rotated about an axis n to 

(S1). The cones represent the fluctuations in Bloch vectors. 

As is shown in Fig.la, the uncertainty in S forms a cone centered on (So), pointing inversely 

along axis r3, with a conic angle be r: @. When a resonant laser field is applied, the Bloch 

vector S is rotated from (So) to (S1), by angle 8 about an axis n in the 0rlr2 plane (Fig.lb). 

Now all atoms are in a superposition state as given in Eq.(2), and one can show that the mean 

square fluctuations in the components of the Bloch vector S are N times of that given by Eq.(4). 

Now let us take a closer look at ~i~.lb: When the Bloch vector S is rotated from (So) to (St), 

the cone representing the fluctuations in S is also rotated. The projection of the base of the cone, 

which represents the fluctuation, onto an axis, say rl, is merely 

One can obtain similar results for the fluctuations along other axes. The fluctuations in the 

Bloch vector components are hence the projections of the Bloch vector uncertainty onto the 

corresponding axes. It has been pointed out that the shape of the cone (Fig.lb) can be altered, 

and turned to an ellipse, by introducing a non-uniform interaction between the external field with 

the atoms [4], or by mutual interaction between the atoms. We see from Eq.(?) that the noise 

in the Bloch vector component along rl reaches a minimum point at 0 = 5 and 4 = 0 when the 

component is maximized. Yet the shot noise in the radiation field from the atomic medium is j proportional to fi and also maximized. It is predicted that the total noise consisting of the shot 

noise and the Bloch vector projection noise would reach its minimum value at 8 = % [2]. 
2 Theory 

In this section, we develop the quantum theory of optical measurement of atomic Bloch vector 

projection noise. We will consider the experimental situation of A-three-level atoms in a beam 

interacting with spatially separated laser fields. In a A-scheme three-level atom (Fig.2), we assume that dipole transitions between states 11) and lo), and 12) and 10) are both allowed, with 

resonant frequencies, ~1, wo2, and transition dipole moments dol, do2, respectively. We assume for 

simplicity that dol and do2 are orthogonal. First, a resonant optical pumping field is applied to 

pump all atoms into state 11). Two co-propagating off-resonant laser fields of frequencies ~1, u2, 

polarizations el, e2, respectively, are applied downstream to prepare the atoms into a superposition state of 11) and 12). Level 10) adiabatically follows the ground state amplitudes and can be 

eliminated. Hence, we are left with an effective two-level system consisting only of ground states, 

which do not spontaneously decay. At a later point in the atomic beam, a probe field of frequency 

w, and polarization e2 is applied. This induces an el-polarized radiation field, oscillating between 

states 10) and 11) which is homodyned with an external local oscillator field of frequency wl. We 

will show that the homodyne output is proportional to the atomic Bloch vector components and 

carries its noise characteristics. 

FIG. 2. Level diagram of a A-scheme three level atom. 

We first treat the preparation process of the atomic ground state superposition. As illustrated 

in Fig.3.a, in the lab frame, atoms in the beam moving along axis z with speed v, enter the 

coherence preparation region I, between x = 0 and x = xo. It is more convenient, however, to 

calculate the atomic state in the atomic center-of-mass (CM) frame. Let us consider an atom that 

appears at an arbitrary position x in the probe region .?I, at time t'. Referring to Fig.S.b, we see 

that the atom entered region I at a previous time t'- x/v and exits region at time t' - (x - xO)/v. 

We start from the effective Hamiltonian 

and the interaction 

P=-- hQole-'~ltlo)(ll - - h002 e-iyll 

2 2 10)(2) + H.c., (9) 
where we take 

(a) Lab frame 

- X 

0 Xo x, X,+L 

(b) CM frame 

FIG.3, Schematic illustration of the experimental situation in a) the Lab frame, and 

b), in the atomic center-of-mass frame. Regions I and 11 are the coherence preparation 

and the probe region, respectively. 

as the Rabi frequencies for the applied laser field given by 

elEl e-i',,lt e2E2 e-i*t E(t) = - 2 +- 2 + C.C. . 

Taking the energy of state lo), $ = 0, the atomic state takes the form 

I$(t)) = ao(t) 10) + al(t) eiwlt 11) + a~(t) eiw2t 12). 

We obtain from the Schriidinger equation, 

where 

Aj E wj - Woj, j=1,2 

are the detunings. When SZ1, R2 << Al, Az, we may adiabatically eliminate level 10) by defining 
where 

Eq.(ll) then becomes 

The initial condition of the atomic state as given in Eq.(lO) is 

or that the atom is in state 11) when entering region I at time t' - xlv. 

When the atomic ground splitting w21 << A1, A2, and A1 k: A2, (Qoll w lQ021, we may define 

the light shift frequency 

the Raman Rabi frequency 

and the net detuning 

A E Al -A2 = (w1 -w2) -wzl. 

Eq.(13) can be readily solved by changing variables 

which yields when A << I@,(. 

Hence we obtain the atomic state for an atom which leaves region I at time t' - (x - xo)/v and 

will arrive at position x at time tl, as 

X - 20 2-20 ei$(tl-T) eiwl(t~-~) 11) 1$(t1 - 7)) = Al(tl - -) V 

- e 0 ,-i(bl -w)(tl- T) - cos - 11) + ei%in - 2 2 12) 7 
where 

is the Raman Rabi area. All common time dependent phase factors in Eq.(17) are left out, as 

these will not affect the atomic coherence (eI2). 

Now let us closely examine the probe region. As illustrated in Fig.4, a probe field 

EP e-iwt &p(t) = e2 y + C.C. 

is incident onto the atomic beam at region 11, and introduces an interaction Hamiltonian 

for the j-th atom at position x at time t. Here Q2 = (doz. e2)Ep/ti is again the Rabi frequency for 

the probe field, and = 10) (21 is the Schrcdinger picture operator for the j-th atom. 

FIG. 4. Schematic diagram of the homodyne detection of the probe-field-induced 

Rarnan transition field. 

It is convenient to use Heisenberg equations in the atomic CM frame for operators &)(z, t), 

etc.. which are 

and 

where 7 is the spol~taneous decay rate of state 10). 7 is small in the experiment so that noise 

terms in Eq.(20) will be neglected. Eq.(19) can be readily solved for the evolution between times 
t' - (x - xo)/v when the atom leaves the preparation region I, and t' in the atom frame: 

(j) 2 - $0 

#(t) = (t' - -) e iwl [t-tt+ F]. 

v (21) 

@(t) is now determined from @(t' - (z - xo)/v) whose expectation value will be evaluated for 

the atomic state I$(t' - (x - xO)/v)) in Eq.(17). Using Eq.(21), Eq.(20) is solved 

-a; eiwlt' eiw)l 1 x - xo W -2- 

2 6g) (t' - + iA2 

where we assumed that 7 (x - xo)/v >> 1. 

Note that in the lab frame, the j-th atom is at position x at time t'. Hence we obtain from 

Eq. (22) the atomic polarization 

where 

~(x, t') cx dlo bg) (2, t3 + H.c., 

j 

under the assumption that the detuning A2 >> 7, the spontaneous decay rate. 

Now if the atomic dipole moment dlo and hence P are orthogonal to the polarization of the 

probe field E,, the optical radiation field due to the atomic polarization P 

can be separated using a polarizer from E,. Were kl is the optical thickness of the atomic beam. 

Adding the vacuum field & of the same polarization as E*, we obtain for the positive frequency 

part of the total output field from the atomic medium 

&lY(t) = 4+' (1) + hi+) (t) (26) 

where 

e-iwl A (j) x - 20 a+) (t) = iT&,(t) C 612 (x, t - -)- 

j v 

7cx aklp (do2 ' e2) (do1 . el) 

hA2 

is a dimensionless scattering coefficient, where p is the number density of the atomic beam, and 

el, ez are the polarization unit vectors of the probe field E, and the scattered field Eat, respectively. 

E~~~ is mixed with a local oscillator field Em(t) .v Em e-'wlt in the homodyne detection 

scheme illustrated in Fig.4. The output signal n can be written in the operator form as the 

difference of photo-currents Pl and 12, 

6 (t) = Qt) - Qt) 

a ELo (t) . EL:! (t) + ELO (t) E! (t). 
Using Eq.(27), and (17) for the atomic state, we obtain 

where (bg) (t - y)) is evaluated for the atomic state i$(t - T)) given in Eq.(17). The photocurrent difference (6 (t)) can be written in terms of the Bloch vector components (sl), and (s2) of 

Eq.(l). Then Eq.(29) can be further simplified 

(fi (t)) = MlTEEoEp 1 sin 9 cos[(wl - w2 - wzl) - (x - 20) 

v 

+ 401, 

where M is the number of atoms in the probe region, and 4o some reference phase. Eq.(30) gives 

the usual Ramsey fringe pattern [5]. 

Now let us evaluate the fluctuations in the atomic Bloch vector components. We calculate the 

power spectrum of the homodyne output signal n (t) 

1 S(w) = - / dr (fi (t) ti (t + 7)) e-iwT. 27r 

After some algebra, we obtain 

under the assumption that the net detuning A is small so that A . zl/v << 1, where xl is the 

distance between the preparation region I and the probe region II. IEL~~~ is the intensity of 

the local oscillator field, and the intensity of the probe field in units of photon number per 

second. A(w) is a spectral function that centers at w = 0, with a spectral width of order of the 

transit bandwidth of the probe region. - 522),+., = cos 9, and (512 + &21)cp = sin 9 cos are 

components of the Bloch vector expectation values evaluated for the atomic state 19) given in 

Eq.(2). 

Now let us closely examine the four terms in Eq.(32). The first term is clearly due to the shot 

noise in the homodyne process. The second term is the reduction of vacuum shot noise level due 

to Raman absorption. It can also be viewed as the shot noise associated with the spontaneous 

Raman transition. The third term is proportional to M2, and centers at w = 0, represents the 

power spectrum of the stationary Ramsey fringe signal of Eq.(30). The last term is the phase #- 

dependent Bloch vector projection noise given in Eqs.(4) and (7). 

3 Summary 

In this paper, we developed the quantum theory for the experimental study of the Bloch vector 

projection noise. Eqs.(30) and (32) are the primary results. In a A-three-level atomic system, when 

two off-resonant Raman fields are applied, the upper state adiabatically follows the ground state 
amplitudes and the A-scheme is hence reduced to an effective two-level system. Decayless ground 

state coherence is prepared. By probing with one optical transition, and detecting the induced 

transition between the other ground state and the upper level with a homodyne technique, we can 

measure the Bloch vector components as given in Eq.(30). The photo-current difference in the 

homodyne scheme also yields the noise characteristics of the Bloch vector components as given in 

Eq. (32). 

An experimental study is currently being conducted using a wide-angled supersonic ytterbium 

(Yb) atomic beam. The 556 nrn transition of 17'Yb 'S -+3 P transition is used. In a 2.6 kG 

magnetic field, the '"Yb ground states of nuclear spin I = i are split by 2 MHz and form a 

A-system with the upper state IF = f, F3 = i). Doppler shifts in the corresponding o+ and 

?r transition in the wide angle atomic beam are Zeeman compensated [6] simultaneously by a 

quadrupole magnetic field. With this technique, the transition linewidth is narrowed to a few 

MHz for an interaction path length 1 of 2.5 cm. The Ramsey fringe pattern of Eq.(30) has been 

observed. 

4 Acknowledgments 

This work is supported by NSF, RADC, and AFOSR. 

References 

[I] R. H. Dicke, Phys. Rev. 93, 99 (1954). 

[2] K. Wodkiewicz and J. H. Eberly, J.O.S.A. B 2, 458 (1984). 

[3] W. M. Itano et al., Phys. Rev. A 47, 3554 (1993). 

[4] D. J. Wineland et al., Phys. Rev. A 46, R6797 (1992). 

[5] N. F. Ramsey, Molecular Beams , (Oxford University Press, London, 1956). 

[6] K. D. Stokes et al. Optics Lett. 14, 1324 (1989). 
properties of phonons and their detection 

M. Artoni 

Photonics, NASAIGoddard Space Flight Center, Greenbelt, MD 20772, U.S.A. 

Joseph. L. Birman 

Physics Department, The City College of New York, Convent Avenue at 138th Street 

New York, N.Y. 10031, U.S.A. 

Abstract 

We present a theoretical investigation on new and interesting properties of the phonon 

polarization field in solids. In particular, non-classical aspects of the phonon population 

and an experimental scheme that would enable one to detect them will be discussed. 

1. Introduction 

In recent years much interest has been devoted to the investigation of quantum effects that have 

no classical analogs, of which optical squeezing is the most ubiquitous one 111. In view of the successful 

generation and detection of squeezed states of the electromagnetic (e.m.1 field it is natural to ask 

whether analogous states exist and can be observed for other boson fields. Condensed matter exhibits 

a variety of bosons that, via the interaction with an external field, can be excited into a squeezed 

state in much the same way as done with photons. 

It was previously pointed out 121 that in a phonon-polariton [3,41, a mixed mode in which an 

optical phonon is coupled to a photon, the photon component exhibits non-poissonian quantum 

statistics and optical squeezing [2,51. Owing to the quadratic nature of the transformation that takes 

one from coupled bare phonons plus bare photons to a polariton, where phonons and photons appear as 

exactly dual particles, we demonstrate that the phonon component of a polariton exhibits analogous 

properties. We further analyse an experiment that would enable one to detect such delicate features 

in solids with an appreciable effect. 

Many topics in solids combine wave and particle aspects. Exactly as the photon describes the 

particle nature of the e.m. field, the phonon describes the particle nature of a lattice vibration [6]. 

Under certain conditions these two excitations may interact: at resonance, transverse optical phonons 

and photons couple and the character of the propagation inside the crystal is entirely changed. The 

pioneering work of Pekar [n, Fano [41, and Hopfield [81 has shown that eigenstates of the coupled 

system of a lattice vibration and radiation are composite particles made of photons and phonons, i.e. 

phonon-polaritons (polaritons hereafter). This represents the quantomechanical equivalent of the 

classical work of Huang [91 who first derived the dispersion for infrared active optical lattice 

vibrations of a cubic ionic crystal, showing that the actual modes propagating in the crystal are 

radiation-lattice coupled waves. A typical dispersion curve (no spatial dispersion) for a polariton is 

shown in Fig. 1 below. 

3. Squeezed states 

Squeezed states are quantum states in which the fluctuations in one of the phase quadratures of 

the field are reduced below the vacuum noise limit [I]. Single and two-modes squeezed states have 
been extensively studied [I], and experiments [I] have already 

demonstrated their realizability in the case of an e.m. field. A 

If 6l and bl denote two independent bosons, one can 

introduce [2] a two-mode squeezed state for the mixed boson 

fl = a14 +p1il as a displaced state of the squeezed vacuum 

B(y12) and $&,(r)are two-mode displacement and squeeze 

operators 1101, respectively, whereas the other symbols have 

their usual meaning. r, the squeeze factor, mediates the 

k coupling between the two modes 1 and 2. In the states I yI2) 

Fig.2 Schematic polariton energy these modes become so tightly correlated that they no longer 

spectrum. up ~p denote upper fluctuate independently by even the small amount allowed in a 

coherent or vacuum state [I]. and lower dispersion branches. wT 

and wL are transverse and logitudinal 4. Nan-cln~sicnl ~honons 

phonon resonant frequencies. 

The objective of this section is to show that a polariton 

state is a non-classical state and that the phonon counterpart 

associated with it exhibits non-classical features. We will 

restrict ourselves to two-mode polariton coherent states. In particular, these states are most suitable 

to describe the actual experimental scheme which we will discuss below. They can be constructed 

[2,11,12] from the polariton vacuum defined as ~klO)pD, = 0, fi,, being the polariton 

transformation [8] 

Iqtk12 =Iykk~, + e2i) y;ksr12 yields the average number of polaritons in the state (2), where the 

Ykkds are the eigenvalues of the bose operator fkk = kk +eid &,. The 6's and 6's are 

respectively photon and phonon bare field annihilation operators. The real parameters 

(a: ,P:,x:, Q, rk) are mode and material dependent [21. 

One should focus at this point on the structure of the polariton vacuum state. It has been the 

object of an exhaustive study not only by us [2,131, but also other workers I141 though within different 

contexts. In this rather interesting work a crucial common result emerged: the polariton vacuum lo),, 

is unitarily related to the bare particles vacuum 10) by a transformation of squeezing {1,101, that is 
rk gives the amount of squeezing in the mode k. It is then clear from Eq.s (2) and (3) that a polariton 

coherent state is an instance of the two mode squeezed state defined in Eq. (11, 1 and 2 referring to 

counterpropagating wavevector modes. 

It also follows from Eq.s (2) and (3) that for those kls for which S n 1 the state 117,,),, reduces 

to a bare particle coherent state, obtained by displacing the bare vacuum 10); but for those kts for 

which S # 1, owing to purely quadratic terms in the photon and phonon creation and annihilation 

operators, I q*t)pol acquires a significantly more complicated structure. The non-classical character of 

a polariton state is clearly related to the parameter 5. Owing to the wide breadth of values that r, 

can take on [2,5,151 one presume to create a polariton state with strong enough non-classical character 

so as to produce a sensitive result in the detection process. To this extent we recall from the results 

reported in [I51 that, for a GaSb crystal, rk may vary across the polariton spectrum between values 

bigger than 1 to values that are even two orders of magnitude smaller. 

Non-classical phonons in a polariton would be commonly characterised by a non-classical 

probability density distribution of the number of phonons in the polariton state [1,16]. In a polariton 

coherent state, unlike a polariton number state, no definite number of polaritons exists, but a well 

' defined probability corresponds to each polariton number with a distribution of probabilities known 

to be classical [1,2,161. Nonetheless, the distribution of its phonon component is in general not 

classical, with the most striking effects occurring where the squeezed structure of a polariton is mist 

enhanced. 

Fig.2 Probability distributions for observing n phonons in a Gdb-polmiion [IS] for the two 

modes k, = 10 cm-' [r, = 2.03 (left) und kk, = 5.5. lo3 cm-' [r, = 5 . lo-'] (onset) 

The phonon number distribution in a polariton coherent state consists indeed of two contributions 1151, 
a Poisson one (l":F) whose structure remains as such through the whole spectrum, and an oscillating 

one (~1) whose size depends on the polariton dispersion. For those modes k's for which rk = 0 (Fig. 

Poirs. 2 onset) Pcoh reduces to the Poisson distribution PCoh (classical limit), whereas for those k's for 

which rk # 0 (Fig. 2) the component @:will contribute with strong oscillations in the large n side 

of the distribution (non-classical limit). In this limit, quasi-periodic oscillations give rise to a 

remarkable effect of "quantization" in the phonon population which appears to be a distinctive nonclassical feature of the phonon field in the polariton state (2). The two limits are illustrated in Fig. 2 

through a numerical evaluation of Eq. (4) for GaSb polaritons [15]. 

5. Detection 

The objective of this section is to address the question of how to probe the nonclassical 

characteristics of the phonon field of a polariton discussed above. Probing the phonon number 

distribution associated with a specific structure of the polariton state is problematic if one decides to 

use particle-counting techniques analogous to those generally employed in the optical domain. 

Conversely, it would be possible to probe directly the non-classical structure of the polariton state 

that yields a nonclassical phonon density distribution. 

The idea consists in establishing coherence between two scattering processes that involve the 

gbsorption of two different wavevector phonon modes. Coherence would then produce constructive or 

destructive interference depending on whether the polariton is in a non-classical or classical state. 

Thus a measurement of the rate with which the probe scatters off of the phonon field of a polariton 

would provide a signature of the non-classical character of a polariton state. 

This idea can be implemented as follorvs. Let a two-mode polariton be excited in a crystal, the 

modes referring to counterpropagating wavevectors of magnitude Ikol and frequency O,. Then let a 

two-components probe beam, having high coupling efficiency to the phonon part of the polariton 

(neutrons e.g.1, impinge on the crystal: both probe components have incident energy Oh,, but different 

wavevectors kd and k, (Fig. 3). The kinematics of the scattering process is described simply by the 

general conservation of energy and momentum (wavevector). Taking advantage of these laws, the 

input probe beam can be arranged so that the incoming probe is scattered into a given output state 

lout), only when absorption off of the modes ko and -ko occurs as schematically illustrated in Fig.3. 

Fig.3 Scattering processes 

involving the absorption of Wine= Wf -Wo Win== Wf-Wo two counterpropagating 

A A-L a a -L --L A 

k,= kf - ko k,= , kf+ko , w phonons with energy a,,, 

and wavevectors ko and 

4 + -k,. With the probe in the 

- k0' wo i - 

state lin),(~~. 5), the two 

processes can coherently 

interfere when the phonon 

field is squeezed. 
Now let the detector be arranged so that only a probe in a final state of momentum kf and energy Wf 

be detected and let the incoming probe be in a coherent superposition of states with wavevectors kd 

and ks, 

A,B and (Ppr are respectively the real amplitudes and relative phase of the two components probe. 

These parameters can be all made to be controllable. 

The relevant scattering rate (lowest order), when a polariton is initially excited into a coherent 

state, is 

denoting by f O the scattering amplitude for the process. The rate consists of two parts: one 

Poiss independent of ro = rko arising from the classical part of the distribution and one ro-dependent 

coming from the oscillating counterpart @z (cf. Eq. 4). The relative size of these two contributions 

does play a significant role in determining the magnitude of the scattering rate. Namely, when 

polariton modes I ko l are populated for which is ro = 0 Eq. (6) is approximated by 

In this case we can show 1151 that for suitable values of the amplitude A there exists a phase (Pp, for 

which oP"bS' + 0 SO that the lowest-order scattering can be completely inhibited. On the contrary, 

when polariton modes I kol are populated for which is ro # 0 the second contribution (- s: ) in Eq. (6) 

is not negligible and can be shown to be always positive defined [151. For these modes the rate turns 

out to be always greater or equal to &sc', but never 0. 

Hence rate measurements would permit one to discern the non-classical and classical character of 

a polariton coherent state. Destructive interference, able to suppress the rate with which a probe is 

scattered, is a signature of a classical polariton state, conversely constructive interference, yielding in 

principle a nonvanishing rate, is exhibited when the polariton state is non-classical (squeezed). 

6. Discussion 

Squeezed states, a familiy of pure quantum states having no classical analogue, have appeared 

in the literature since 1960's. In particular, extensive theoretical investigations for the realization of 

squeezed states of the electromagnetic field are also of long lasting. Only recently was the 

experimental realization of squeezed light with fewer quantum fluctuations than the vacuum 

achieved. To date there have been no reports, however, of the existence of non-classical states in 

condensed media; but the situation appears to be rather favorable for polaritons. This crystal mixed 
quasiparticle appears to be indeed a promising place where to look for non-classical states of light 

and other bosons, such as e.g. phonons, especially if extremely low loss crystals can be obtained. 

A further extension of this work would include considerations on the physical origin of the nonclassical effects discussed above. Intermode correlation resulting from the quadratic transformation 

that takes one from coupled bare photon plus bare phonon to polaritons is a plausible origin for the 

squeezed structure of a polariton state and ultimately for the non-classical features of the phonon 

component associated with it. 

In a real material elementary excitations and quasi-particles normally experience dissipation 

and phase destroying processes that may degrade the structure of the non-classical state inside the 

medium. For a complete treatment and in view of the possible experimental realizability of our 

findings the model presented here should further be extended to include various dissipative processes 

that randomize the phase of coherent superposition states on very short time scales. Such an 

investigation would afford the inclusion of irreversible couplings on the basis of the master equation. 

Nonclassical states have great fundamental significance and are extremely appealing in their 

own right as a test of basic quantum theories as well as perhaps for practical applications. The idea 

of searching for non-classical states of phonons in solids is certainly expected to add a new dimension 

to the search for nonclassical behaviour. 

[I] "Squeezed and non-classical light", Ed.s P. Tombesi & R.P. Pike, NATO ASI-B, 190, (Plenum, 

1989); J. Opt. Soc. Am. B4, No 10, (1987) and J. Mod. Opt. 34, No 6, (1987) (special issues); 

[21 M. Artoni, J.L. Birman, Phys. Rev. B, 43,4221, (1991); 

[31 "Polaritons", Eds. E. Burstein & F. De Martini, (Pergamon Press, 1972); 

[41 U. Fano, Phys. Rev. 103,1202, (1956); 

[51 M. Artoni, J. Birman, Optics Comm. 89,324 (1992); 

[61 C. Kittel, in "Quantum Theory of Solids" a. Wiley, New York, 1964) 

[7l S.I. Pekar, "Crystal Optics and Additional Light Waves", (Benjamin/Cummings, 1983); 

[81 J.J. Hopfield, Phys. Rev., 112,1555, (1958); Here we adopt the Hopfield model for a 

phonon-polariton as the starting point of our work. A considerable success [31 has been 

achieved by this rather simple theoretical model which nevertheless encompasses the 

major relevant physics; 

[91 K. Huang, Proc. Roy. Soc. (London) A 208,352, (1951); 

[lo] C. Caves and B.L. Schumaker, Phys. Rev. A, 32,3068-3111, (1985); 

[Ill J. L. Birman, M. Artoni, B.S. Wang, Physics Report 194, p. 367, (1990); 

[I21 A. Quattropani, L.C. Andreani, G.F. Bassani, Nuovo Cimento, Vol. 7D, p. 55, (1986); 

[13] M.Artoni, J.L. Birman Quantum Optics 1 (1989) 91; 

[14] P. Schwendimann, A. Quattropani, Z. Hradil (preprint) 

[15] M. Artoni, J. Birman, Optics Cornm. (submitted) (1993); 

[I61 W. Schleich, J. Wheeler, Nature, 326,574, (1987), J. of Opt. Soc. Am. B, 4, no 10 p. 1715, (1987); 
QUANTUM STATISTICS OF RAMAN SCATTERING 

MODEL WITH STOKES MODE GENERATION 

B. Tanatar and Alexander S. Shumovskyl 

Bilkent University, Physics Department, 06533 Ankara, Turkey 

Abstract 

The model describing three coupled quantum oscillators with decay of Rayleigh mode 

into the Stokes and vibration (phonon) modes is examined. Due to the Manley-Rowe relations the problem of exact eigenvalues and eigenstates is reduced to the calculation of new 

orthogonal polynomials defined both by the difference and differential equations. The quantum statistical properties are examined in the case when initially: the Stokes mode is in 

the vacuum state; the Rayleigh mode is in the number state; and the vibration mode is in 

the number or squeezed states. The collapses and revids are obtained for different initial 

conditions as well as the change in time the sub-Poisson distribution by the super-Poisson 

distribution and vice versa. 

Introduction 

Recently quantum st atistical properties of scattered light in the Raman process have attracted 

considerable interest.[l, 21 In particular, the anticorrelation between the Stokes and Rayleigh lines 

in the resonance scattering have been examined[3, 41, and the generation of squeezed light have 

been considered.[5, 6, 71 At the same time, the strong quantum fluctuations of energy have been 

observed[8, 91 experimentally. It is known that the Raman scattering is an example of optical 

parametric process in which one of the interacting waves is a medium vibration mode of boson 

type[lO] (phonons). In the case of condensed matter, such a mode is usually in thermal equilibrium 

with a given temperature. The state of that mode is determined by different mechanisms of microscopic interactions in the medium and in some cases can lead to a strong number fluctuation.[ll] 

An example is provided by a polariton-type system in which the equilibrium state is a squeezed 

one. [l2] Evidently, the statistical properties of vibration mode must have influence on the statistics 

of scattered light. 

In this paper we consider the quantum properties of scattered light, and its dependence on the 

type of statistical distribution function of the vibration mode in the Raman scattering. For 

simplicity, we suppose the resonance steady state process with generation of inelastic Stokes 

component only. The initial state of Rayleigh mode is assumed to be a number state, while 

vibration mode can be initially in a number state or in a squeezed vacuum state. The Stokes 

field is initially in the vacuum state. The simplest model of three bounded oscillators is used for 

description of process under consideration.[l3] Using the representation of Schrodinger equation 

in terms of new orthogonal polynomials[14, 151, we examine the dynamics of the Mandel's factor 

'On leave of absence from Bogolubov Laboratory of Theoretical Physics, JINR, Dubna, Moscow Region, Russia 
of scattered light, and show the qualitative difference between two choices of initial states of the 

vibration mode. 

The rest of this paper is organized as follows. We first introduce the model Hamiltonian for 

which we calculate the dynamical properties. Evaluation of the eigenvalues and eigenfunctions, and 

a discussion of how to construct the time dependent Mandel's factor is given in the next section. 

We then present our results, and conclude with an emphasis on the experimental implications. 

2 Raman scattering model 

To discuss some of the interesting statistical properties of light we study the Raman scattering 

model described by the Hamiltonian 

where a+ and a$ are the creation operators for the Rayleigh and Stokes modes, respectively, with 

the corresponding frequences w and w,, b+ is the creation operator for the vibration mode with 

frequency wb, and y is the coupling constant. We consider here only the Stokes process because we 

will examine the case of low intensity initial field for which anti-Stokes component is negligible. 

As a consequence of the Manley-Rowe relations, an exact eigenstate of the above Hamiltonian 

can be-chosen as n 

where I... >, is the number state of g-th mode. The coefficients ATym are determined by the 

recursion relation[l5] 

[(n - j) (j + 1) (m+ j + l)]'I2 = xn"A;" - A::: [(n - j + 1)j (m + j)]*12 

Here xntm = (Enym - wTn - w,m)/y, and Envm is an eigenvalue of the Hamiltonian corresponding 

to the eigenstate given above. The above relation [eq. (3)] can be represented in the form 

Pz (x) = x P;lm (x) - q;ym P;l;l (x) ; with qj = (n - j + l)j(m + j), 

defining some orthogonal polynomials P(x), which can be expressed in terms of the Bernoulli 

polynomials B as 

where the coefficients @ satisfy 

j - ) = jjj- , and @;,j+i(n - j)i = jPi-i,j . 

In terms of the polynomials P(x), the equation for eigenvalues E has now the form 

P;~:(X) = 0, 
while the coefficients of the eigenstate given by eq. (2) are determined by the expression 

The coefficient Xo is determined from the normalization condition. 

Having constructed the solution to the eigenvalue and eigenstate problem, it is not hard to 

examine the dynamics of the system. In order to discuss our results, we shall use the timedependent Mandel's Q-factor defined as 

Here af and a, are the Bose operators for the Rayleigh mode, %(afa,) is the time-dependent 

number variance, and < ... >t denotes a time-dependent expectation value. Q(t) is positive in 

the case of super-Poisson statistics and negative for sub-Poisson number distribution. Zero value 

corresponds to the coherent state (Poisson distribution). Q-factors for the other modes may 

be defined similarly. The time dependent terms in the Mandel's factor must be calculated as 

corresponding expect at ion values with time dependent wave function. 

Fig. 1. Q-factor for Rayleigh mode initially in a number state with n = 2 (solid) 

and n = 10 (dashed), while the vibration mode is in a number state with m = 2. 

3 Results and discussion 

We have obtained qualitative differences in the quantum statistical properties of the Rayleigh 

mode depending on the initial state of the vibration mode. Using a number state for the Rayleigh 

mode, we consider uncorrelated (number state) and correlated (squeezed state) phonons in the 

vibration mode, and calculate the time evolution of the Mandel's Q-factor for the Rayleigh mode. 
In Fig. 1 we show Qr(t) when the Rayleigh mode is initially in a number state with n = 2 and 

n = 10, indicated by solid and dashed lines, respectively. The vibration mode is also in the number 

state with m = 2. We observe that for a small value of n Q,(t) periodically fluctuates between 

sub-Poisson and super-Poisson statistics. As n is increased, quantum statistical distribution of the 

Rayleigh mode becomes more of super-Poissonian. The situation is qualitatively different, when 

the vibration mode is a correlated one described by a squeezed state as shown in Fig. 2. Here 

the squeezed state of the vibration mode is prepared such that the mean number of phonons is 

2, viz., lvI2 = 2. Solid, and dashed lines refer to the Rayleigh mode in the number state with 

n = 2 and n = 10, respectively. Here, we note that the distribution of photons remain mostly 

super-Poissonian, for large enough n. Similar results are obtained when the Rayleigh mode is 

initially in a coherent state. 

To render the description of the vibration mode more realistic, we also use the Bose distribution 

at a given temperature. Here the mean number of phonons m may be regarded as a parameter. 

In Fig. 3 we show Q,(t) when the Rayleigh mode is initially in a number state with n = 2. Solid, 

dashed, and dotted lines refer to vibration mode parameter m = 0.1, 1, and 10, respectively. 

We observe that for small values of m (low temperature) Q,(t) periodically fluctuates between 

sub-Poisson and super-Poisson statistics. As m is increased, quantum statistical distribution of 

the Rayleigh mode becomes entirely super-Poissonian. 

We have also carried out calculations with large numbers of n, and observed the collapse-revival 

phenomenon occurs in the system for large enough n. Since an increase in n implies an increase 

in the number of terms in the various sums, it is not surprising to observe the collapse-revival 

patterns as in the case of the Jaynes-Cummings model.[l7, 181 It should be noted that similar 

behavior was obtained in the numerical calculations of Drobny and Jex[19], for the case of initial 

coherent state of the Rayleigh mode. In this connection, we emphasize that the collapse-revival 

phenomenon is a general property of the model described by the Hamiltonian irrespective of the 

initial state of the Rayleigh mode. 

Fig. 2. Same as Fig. 1 when the vibration mode is initially in a squeezed state with 

parameter lvI2 = 2. 
We have obtained a qualitative difference in quantum statistical properties of scattered light 

depending on the statistics of the vibration mode. Our choice of initial state of the vibration 

mode can be considered as simulating correlated and uncorrelated phonons, and also phonons at 

finite temperature. We conclude stating that the experimental investigation of quantum statistical 

properties of scattered light in the Raman correlation spectroscopy with different types of incident 

light (e.g., in the number state or strongly sub-Poissonian) may yield important information about 

the correlations in the medium as well as in the molecules. 

Fig. 3. &-factor for Rayleigh mode initially in a number state (n = 2). Solid, 

dashed, and dotted lines indicate vibration mode parameter fi = 0.1, 1, and 10, 

respectively. 

Acknowledgments 

B. T. thanks the Bilkent University for travel funds. A. S. acknowledges fruitful discussions with 

Professors C. Bowden, R. BuUough, S. Carusotto, 0. Keller, F. Persico, and V. Rupasov. 

References 

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1990) p. 103. 

[2] J. Mostowski and M. G. Raymer, in: Contemporary Nonlinear Optics, eds. G. P. Agarval and 

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131 G. Agarwal and S. Jha, Z. Phys. B 25 (1979) 391. 

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[lo] Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York (1984)). 

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Evolution of Wave Function in a Dissipative System 

Li Hua Yu 

National Synchrotron Light Source, 

Brookhaven National Laboratory, N. Y.11973 

Chang-Pu Sun 

Institute of heo ore tical Physics, State University of New York 

Stony Brook, N. Y. 11 794t 

Abstract 

For a dissipative system with Ohmic friction, we obtain a simple and exact solution for 

the wave function of the system plus the bath. It is described by the direct product in two 

independent Hilbert space. One of them is described by an effective Harniltonian, the other 

represents the effect of the bath, i.e., the Brownian motion, thus clarifying the structure of the 

wave function of the system whose energy is dissipated by its interaction with the bath. No 

path integral technology is needed in this treatment. The derivation of the Weisskopf-Wigner 

line width theory follows easily. 

The simplest example of a dissipative system, an harmonic oscillator coupled to the environment, which is a bath of harmonic oscillators, has been the subject of extensive studies[l-151. 

We shall show in the present paper that in a special case, the Ohmic case (to be defined later), 

the dissipative system can be exactly treated both classically and quantum mechanically, thereby 

clarifying the sense in which the wave function is describable by an effective Hamiltonian. In this 

treatment path integral technology is not needed, and our presentation is self-contained. 

We consider the problem discussed by Caldeira and Leggett (CL)[l], an harmonic oscillator 

system (the dissipative system) with coordinate q, mass M, and frequency (wi + Aw2)lI2, interacting with a bath of N harmonic oscillators of coordinates xj, mass mj, and frequency wj., where 

Aw2 is a shift induced by the coupling already discussed by CL. The Hamiltonian of the system 

and the bath is: 

The dynamic equation for operators in the Heisenberg representation leads to the following set 

of equations of motions: 

+permanent Address: Department of Physics, Notheast Normal University, Changchun 130024, P.R.China 

507 
Now, applying the Laplace transform[2] (the bars are our notations for the Laplace transform, s 

is the Laplace transform of time t), equations (2), (3) can be used to eliminate bath variables Ij 

to obtain the equation for q 

where qo, qo, xjo, xjo are the initial values of the respective operators in the Heisenberg representation. Assuming the number of bath oscillators is large enough so that we can replace the sum 

over j by an integration over wj, the coefficient of the last term can then be separated into two 

terms: 

where p(wj) is the bath oscillator density. Following an argument similar to the one pointed 

out by CL[l] , the requirement that the system becomes a damped oscillator with frequency wo 

and damping rate 9 in the classical limit, known as the "Ohmic friction" condition, leads to the 

following constraint : 

By observing eq.(4) and eq.(5), it can be shown that the second term of eq.(3) leads to damping 

with the damping constant 9, while the first term of eq. (5) represents a frequency shift. If the 

frequency renormalization constant Aw2 is chosen to satisfy: 

the frequency is shifted to wo. Then equation (4) is simplified, and its inverse Laplace transform 

gives the quantum Langevin equation, valid at time t >= 0+: 

with the Brownian motion driving force: 

sin wjt f(t) = -C~j(xj~ COSW~~ f xjo-) . j Wi 

During the derivation, in order to carry out the integral in eq.(5), we used the requirement of the 

inverse Laplace transform that s must pass all the singular points from right of the complex plane, 

and hence real(s) >0. 

Equation (8) and (9) are the equations of a driven damped harmonic oscillator, the solution 

of which is well known as a linear combination of the initial values at qo, qo, xjo, and xjo : 
and here w = (w: -r,1~/4)l/~ isthe frequency shifted by damping. (All formulae are correct whether 

w is real or imaginary. To avoid a minor detail of the initial value problem, we have redefined the 

initial time as t=O+.) The explicit expression for bjll bj2, ail, Qi2, Pijl, and Pij2 are well known 

in freshmen physics. 

We emphasize that the use of the Laplace transform instead of the Fourier transform allows 

us to express q(t) and xj(t) explicitly in terms of the initial values, as in eq.(lO) and eq.(ll). 

The equations (10) and (11) serve as the starting point of subsequent discussions. We will 

proceed to find the ~reen's function of the full system,and hence the solution of the wave function 

in Schoedinger representation. The result tells us in what sense the damped oscillator is described 

by an efective Hamiltonian without the bath variables and gives its specific form, it also shows 

that under .this condition, the wave function can be factorized, and the main factor relevant to 

the damped oscillator is a solution of the Schoedinger equation with an effective Hamiltonian. 

Equations (10) and (11) are correct both in classical mechanics and in quantum mechanics in 

the Heisenberg representation. We notice that q(t) and xj(t) are both linear superpositions of 

qo, qo, Xjo, kjo with c-number coefficients. The commutation rules between q(t), q(t), xj(t), ij(t) 

are k(t), Q(t)] = , [xj(t), (t = $, and operators q(t) and ((t) commute with xj(t) and 

xj(t). One can prove these commutation rules by two ways: (a). By direct computation, using 

the fact -that at t=O, they are correct. (b). By the general principle that q(t), b(t), xj(t), kj(t) 

are related by a unitary transformation to qo, qo, x;o, and ijo. 

Equation (10) and (11) show that the operators q(t) and xj(t) can each be written as a sum 

of two terms: 

where Q(t) and ci(t) are linear in qo and qo and independent of xjo and xjo, and tj and Xij(t) 

are linear in xjo and kjo, and independent of qo and go. Thus Q(t) and C;(t) are operators in 

one Hilbert space SQ, while tj(t) and Xij(t) are in an independent Hilbert space Sx, and the full 

Hilbert space is a direct product SQ @ Sx. 

We shall first analyze the structure of SQ. We write that: 

To explicitly show that we are discussing the SQ space, we define Qo 1 qo, Qo e Po. The eigenfunction of Q(t) with an eigenvalue denoted by Q1, in'the Qo representation, is easily calculated 

to be: ( M ) [ iM 

UQ~ (Qol t) = 2Th sin wt exp -- 2ha2 (ai~: - 2QoQl+ d(Qi1 t))] 1 
with 4 as an arbitrary phase, i.e., a real number. This eigenfunction is related to the Green's function G(Q1, Qo;t,o) E< QilU~(t)lQo >, by uQl(Qol t) =< QolU;'(t)IQi > =< QOJU&(~)~QI >= 

G*(Ql, Qo; t, 0), where we denote the evolution operator by U(t), which is unitary when we choose 

the eigenvectors of Q(t) to be orthonormal. Thus we have: 

i Mwe sqt iM 

G(Q1, Qo; t1 O) = ( ) P [ (I - 2QoQl + $(QI.~))] 

Next, we shall determine the arbitrary phase q5(Q1,t), which is the phase of the eigenvectors 

of Q(t) Using eq.(16), we find the commutation rule for Q and Q: 

Thus we define the canonical momentum P(t) as: 

and get the commutation rule: [&(t), P(t)] = ih. The eigenfunction of P(t) can be calculated in 

two ways: (a). We can calculate the eigenvector of P(t) in the Qo representation using eq. (20) 

and then use the Green's function eq.(18) to transform it into the Q(t) representation; (b). The 

commutation rule [Q(t), P(t)] = ih requires that ~(t)=-ih&, so the eigenfunction of P(t) with 

eigenvalue PI is exp[ia~]. By comparing these two solutions, the arbitrary phase q5(Ql, t) in the 

Green's function is determined to within a phase q5(t), which is independent of Q1. $(t) is an 

arbitrary real function of time, except that $(O) = 0 so that it satisfies the condition that at t=O, 

the Green's function becomes S(Q1 - Qo). Thus we obtain the Green's function in the SQ space: 

Mwebt 

G(Q1, Qo; exp - + b2e"Qi - 24001) - ] . (21) t.0) = ( 2wih sin wt ) [ iM 2ha2 

It is then straight forward to derive the Hamiltonian HQ using the following relation: 

and remembering that the matrix elements of UQ and U;' are the Green's function and its 

conjugate. The result is: 

p2 1 2 7; 2 HQ = e-qt- + -Mwoe Q + $(t) . 2M 2 (23) 

Since q5 is arbitrary except that q5(O) = 0, we can take $(t) = 0. Therefore we have derived the 

well known effective Hamiltonian for the dissipative system. We emphasize that the expression for 

HQ is here derived, while in usual literature it is introduced by more or less heuristic arguments. 

Next, we shall analyze the effect of the bath. Similar to eq. (17) we obtain the eigenfunctions 

B~,l(~jol t) for tj. Using Dirac's notation we have: Q~ug, >= Q1luQ1 >, &lO,, >= tjllOtjl >. 
Thus luQ > IT @ (Btj > is an eigenvector of q(t), with eigenvalue of Q + 6. In other words, 

j j 

the eigenvector of q(t) with eigenvalue q is (q, {tj} >= (u > ~@16t, >. 4-xt, j 

j 

If initially the wave function is 1 Qo >= > > @ lxjo >, to calculate the wave function at 

j 

time t, we should expand Qo in terms of the eigenvectors Jq, itj) >, i.e., we should calculate 

Then the wave function in the Schoedinger representation at time t is 

Notice that $(Q, t) of eq.(24) is the wave function in the Schoedinger representation with the 

effective Hamiltonian eq.(23). Hence we have connected the effective Hamiltonian approach to the 

dissipative system problem with the other approaches that take both the system and the bath into 

account. We also notice that even though our Q(q, itj), t) is in a different representation from that 

of Q(q, {xj), t), the usual probability interpretation is still valid: J J ... J IQ(~, {b},t)12 >dtj is 

j 

the probability density of finding the particle at q. Since this solution is very simple, it provides a 

simple way to analyze other complicated problems, e.g., study the influence of Brownian motion 

on interference, which we shall not elaborate because of space. 

Under certain conditions, eg., at low temperature and when the system q is in highly excited 

states so the range of q is large enough that for all the values of tj which do not have vanishingly 

small probability, q B I x tjI, we can approximately write: Q(q, {GI, t) = $(q, t) xj(&,t). 

j j 

That is, the wave function is factorized, the dissipative system q can be described by the wave 

function $(q, t) only, and the Brownian motion can be ignored. Therefore it is interesting to 

examine the width of the argument of the wave function $, due to the Brownian motion, i.e., 

the mean value of (xtj)2 at time t. It can be calculated using its expression in the Heisenberg 

j 

representation as introduced by (10) and (18). At temperature T, this width is : 

This width is zero initially, then approaches its final equilibrium value in a time interval of the 

order of l/q. At low temperature limit, the equilibrium width is simplified to: 

If the damping rate q is much smaller than the frequency of the oscillator wo, this width happens 

to become the same as the width of the ground state of the system 6/(2rnw). 
Finally, it is interesting to see the distribution of the dissipated energy of the harmonic oscillator 

in the bath, and check if it agrees with the Weisskopf-Wigner line width theory[l6] For simplicity, 

we assume zero temperature. To calculate the energy dissipated by the system into the j'th bath 

oscillator, we use eq.(ll) and its derivative to obtain the expression of xj(t) and pj = mjij(t), 

which are then substituted into the expression of the energy of the j'th oscillator: hj = & + 

$mjw~~~. We then calculate the expectation value of hj, assuming initially the system is in the 

n'th excited state, and the bath oscillators in the ground state. We calculate the contribution to 

the expectation value of hj from the system, by keeping only terms which depend on qo, and qo. 

The result is then the energy disspation by the system into the j'th oscillator. When multiplied 

by the density of states p(wj) eq.(6), it gives the dissipated energy spectrum. It is a function of 

wj, with a narrow peak near the resonance wj = w, if the damping is small, i.e., if 7 << w. It is 

oscillatory with a frequency of 2wj. Its time average over a period is : 

where A(wj) varies slowly near the resonance, i.e., over the width 712 of the peak it changes very 

little. Therefore, we can replace wj by w near the peak, and the result is simplified to: A(w) z 1. 

Thus eq.(29) shows that the dissipated energy has a Lorenzian distribution near the resonance, in 

agreement with the Weisskopf-Wigner line width theory. 

Acknowledgements 

We thank Pr0f.C.N. Yang for drawing our attention to the problem of dissipative systems, for 

spending his valuable time in many sessions of stimulating discussions on this subject, and for 

many suggestions which are critically important for the ideas of this paper. The part of work by 

Li Hua Yu is performed under the auspices of the U.S. Department of Energy under Contract No. 

DE-AC02-76CH00016. The part of work by Chang-Pu Sun is supported in part by a Cha Chi 

Ming fellowship through the CEEC program at the State University of New York at Stony Brook, 

and in part by the NSF of China through the Northeast Normal University. 

References 

[I] A.O. Caldeira, A.J. Leggett, Ann. Phys. 149, 374 (1983), and Physica 121A, 587 (1983). 

[2] A.O. Caldeira, Helvetica Phisica Acta 61, 611 (1988). 

[3] E. Kanai, Prog. Theor. Phys. 3, 440 (1948). 

[4] S. Nakajima, Prog. Theor. Phys. 20, 948 (1958). 

[5] R.Zwanzig, J. Chem. Phys. 33, 1338 (1960). 

[6] I.R. Senitzky, Phys. Rev. 119, 670 (1960). 

[7] R.P. Feynman, F.L. Vernan, Ann. Phys. 24, 118 (1963). 
[8] G.W. Ford, M. Kac, P. Mazur, Jour. Math. Phys. 6, 504 (1965). 

[9] M. D. Kostin, J. Chem. Phys. 57, 3589 (1972). 

[lo] W.H. Louisell, Quantum Statistical Properties of Radiation, (John Wiley & Sons, 1973). 

[I 11 M. Sargent 111, M.O. Scully, W .E. Lamb, Laser Physics, (Addison-Wesley, 1974). 

[12] K. Yasue, Ann. Phys. 114, 479 (1978). 

[13] R.H. Koch, D.J. Van Harlingen, J. Clarke, Phys. Rev. Lett. 45, 2132 (1980). 

[14] H. Dekker, Phys. Report 80, 1 (1981). 

[15] R. Benguria, M. Kac, Phys. Rev. Lett, 46, 1 (1981). 

[16] V.F. Weisskopf, F.P. Wigner, Z. Physik 63, 54 (1930); 65, 18 (1930). 
NONRESONANT INTERACTION OF ULTRASHORT 

ELECTROMAGNETIC PULSES WITH MULTILEVEL QUANTUM 

SYSTEMS 

E.Belenov, V.Isakov and A.Nazarkin 

Lebedev Physics Institute, Russian Academy of Sciences, Leninsky pr. 53j 11 7924 Moscow, 

Russia 

Abstract 

Some features of the excitation of multilevel quantum systems under the action of electromagnetic pulses which are shorter than the inverse frequency of interlevel transitions are 

considered. It is shown that the interaction is characterized by a specific type of selectivity 

which is not connected with the resonant absorption of radiation. The simplest three-level 

model displays the inverse population of upper levels. The effect of an ultrashort laser pulse 

on a multilevel molecule was regarded as an instant reception of the oscjllation velocity 

by the oscillator and this approach showed an effective excitation and dissosiation of the 

molecule. The estimations testify to the fact that these effects can be observed using modern 

femtosecond lasers. 

1 Introduction 

Progress in ultrashort pulse technique allows production of laser pulses of 5-10 femtosecond duration, i.e., of a few periods of elecl~romagnetic wave [I]. The interaction of so short light pulses with 

matter must differ from the case of quasi-monochromatic resonant radiation because the pulse duration becomes shorter than the inverse frequancy of the transitions between vibrational levels of 

molecules wil or optical phonons in crystals, and under such conditions the result of interaction is 

practically independent of the real structure of matter. In particular, a Raman-active medium can 

be effectively excited by a single femtosecond pulse because its wide spectrum initially contains 

Stokes and anti-Stokes components of the field [2, 31. 

We shall discuss here two effects which may occur if the light pulse is shorter than the time 

of vibrational transitions between molecule levels, i.e. under the condition ~~wd << 1. The first 

effect displays a specific selectivity of vibrational excitation of molecules by ultrashort light pulses; 

which is not connected with resonant properties of matter. The second effect is a possibility of 

effective excitation of high vibrational levels and even dissosiation of molecules in the field of an 

ultrashort electromagnetic pulse. 
2 Selectivity of Molecule Excitation by Nonresonant Pulse 

Radiation 

The selectivity of molecule excitation is connected with the fact that the probability of transitions 

between molecule levels cannot be presented already as a function of slowly varying amplitude 

E(t) of the light field similar to the case of quasimonochromatic resonant radiation but depends 

on its real temporal structure &(t). The equations for ultrashort pulse propagation, in their turn, 

impose restrictions on such integral values as J_', E(t)dt. As shown in Ref [4], the "area" of 

electromagnetic wave of limited aperture in a free space goes to zero, Jr, E(t)dt = 0. It results 

from the Gauss' integral theorem in the case of electromagnetic wave propagation in a space 

without electric charges. If prim is a dipole moment of the transition between levels m and n, the 

amplitude of the transition n -+ m in the dipole approximation is of the form 

and tends to zero for ultrashort pulses. 

It means that excitation of vibrational levels of molecules by femtosecond pulses is mainly due 

to two-quanta processes over distant virtual levels 

1 

where rnm = - pnkpkm ( + ) is a composite matrix element responsible for 

2r3 k Wkn - W Wkm +W 

Raman effect at the transition n + m, and the probability of excitation in the second order 

perturbation theory is proportional to the pulse energy J:, I E(t)12 dt # 0. Thus, an ultrashort 

pulse provides "inverse" selectivity of molecule excitation - the transitions forbidden in the dipole 

approximation are stimulated much more effectively than the allowed ones. 

As an illustration let us consider three-level quantum system, i.e. three lowest levels of the 

deformational vibrations of the C02 molecule (00'0,01'0 and 02'0). The model assumes that the 

condition of the ultrashort interaction Tpwij << 1 holds for i,j=1,2,3 and the other levels are distant 

in the energy scale, i.e. lwik - wl >> ql, k# 1,2,3. Levels (1) and (3) are of the same parity and 

the transition between them is forbidden in the dipole approximation. 

The condition rpwij << 1 implies of course that the interaction of an ultrashort pulse with 

matter is coherent (rp << TI, T2, pulse duration is shorter than times of longitudinal and transverse 

relaxcation) and can be described by equations for the probability amplitudes of the molecule to 

be at the level j(=1,2,3): 

Let the system be at the ground level (1) at the initial instant of time, i.e. al(t = -w) = 1 

and a2(t = -00) = a3(t = -m) = 0. Under the action of ultrashort pulse, the transitions onto 
level (2) occur due to the fast oscillating field E(t) which do not have a constant component, but 

transitions onto level (3) occur due to the interaction quadratic in field and E2(t) # 0. We can 

assume that during the interaction the redistribution of population will be provided mainly by 

the transitions over level (3). Indeed, by integrating the second equation and saving here the 

amplitudes 61 and 63 whose time dependence is slow with respect to E(t), one obtains 

In other words, the population of level (2) is small (a2 - pijE/Kw) because of a nonresonant 

interaction of this level with the field of the ultrashort pulse. After substitution of a2(t) into the 

first and third equations one can obtain equations for slow components of the amplitudes iil and 

?i3, which do not contain parameters of dipole transitions 1 4 2 and 2 -+ 3: 

It is interesting to note that the solution of these equations 

iil(t) = cos @(t.), 

ii3(t) = i sin @(t), 

coincides formally with the solution for resonant coherent excitation of two-quanta transition, 

although in our case the interaction is nonresonant in principle. 

The population of molecule levels after the action of ultrashort pulse is determined by the 

following expressions: 

These equations display an evident selectivity of the interaction of an ultrashort light pulse 

with the three-level quantum system which produces the population inversion on the transition 

3 + 2. If the energy of an ultrashort pulse corresponds to the value of the rotation phase 

!l!, = 3 J:, ~EI~ dt = f (2n + I), n = 0,1,2,. . . the quantum system will be moved onto the 

upper level (3) and all the energy stored can be picked up, for example, by two sequential T-pulses 

which are resonant to transitions 3 + 2 and 2 -, 1. 

Let us make some estimations for the molecule C02. It can be easily shown that if the initial 

population n2o of the level (2) is not equal to zero, the inversion on the transition 3 + 2 appears 

if Q, > (n20/nlo)1/2. For a COz molecule hw32 N fiwZl N 700cm-I and under normal conditions 

n20/nlo z 10%. Using data on Raman cross section (r13 21 cm3, X = 488 nm 151) one can 

estimate that a 10-femtosecond pulse provides inversion between vibrational levels 02'0 and 01'0 if 

its power exeeds 1012 W/cm2. This value can be easily satisfied by modern femtosecond technique. 
3 Dissociation of Molecules by Femtosecond Pulses 

The possibility of effective excitation of high vibrational levels makes topical the problem of 

transformation, for example, dissociation of molecules under the action of ultrashort laser pulse. 

Usually, collisionless excitation and dissociation of molecules by resonant radiation are investigated 

on the basis of the chain of equations for amplitudes of population probabilities of vibrational 

levels. This analisis is very complicated because of a large amount of levels which should be 

under consideration. In our case the condition T~W,,,,, << 1 means that the result of interaction 

is not sensitive to the real vibrational structure of molecule spectrum and makes it convinient to 

investigate directly the equations for waive functions of vibrational Harniltonian regarding it as a 

function of the distance between nuclei. The statement of the problem is quite similar to that 

solved by A. Migdal on excitation and ionization of atom after its nucleus received an instant 

impact [6, page 1781. For simplicity we consider a two-atom molecule and assume the pulse to 

be so short that during its action the nuclei acquire some velocity but their positions relative to 

electrons have no time to change. If at the initial instant of time the molecule is at the ground 

state then just after the pulse action it remains at the same vibrational level and its wave function 

is related to the initial one as 

where P = -a (s) !rm_ I ~(t) J2 dt is the impulse transmitted to molecule under the action of 

ultrashort 6-pulse; KO is polarizability of the molecule, Q is vibrational coordinate. 

The quantum state described by the wave function yo(&, oo) is not already a stationary state of 

the molecule after receiving an instant impulse P. The transition of the system to the stationary 

state is accompanied by excitation of the upper vibrational levels and the probability to find 

molecule at the level v is determined by the coefficient of expansion of cpo(Q, oo) into series in 

terms of the stationary wave functions of the vibrational Hamiltonian: 

It is evident that the higher the pulse energy and consequently the transmitted momentum P, 

the more is the difference between the quantum state of the molecule just after the pulse action 

and the stationary state and the higher are the levels to be involved into transition of the molecule 

to its stationary state. 

Let us display it in framework of the model of anharmonic Morse oscillator, i.e. a particle 

of mass M which is moving in a field wth potential energy U(R) = D (e-2ffR - 2e-"Rd) [6]. The 

spectrum of positive energy of this particle is continuous and corresponds to a dissociate molecule. 

In the negative energy region there is a finite series of vibrational levels which is convergent to the 

dissociation energy D: Eu+l - Ev = hwo - v - hawo, where wo = (El - Eo)/h = a(JZii;lZT- atr)/~ 

is the fundamental frequancy of the oscillator and Awo = cr2h/M is the anharmonism of the 

molecule. The wave functions of Morse oscillator ire of the form cp, (Q) .v e-el2ts F (-v, 2s + 1, t), 

2 2MD where 6 = ce-o~, ,ti s + v = SO = a + 1 N and N is a total number of the oscillator levels. Awo 2 - The problem of excitation of anharmonic oscillator levels by an ultrashort radiation 6-pulse 

has an exact solution. The probability of finding the molecule at the level v is 
or, neglecting the difference between so and N, - has the form 

where 1 2m /3 = P(ah)-I ln- [ ah ] N J?? 1 E 1' dt. 

For weak ultrashort pulses (P -, 0) the probability of exciting vibrational levels is small and 

proportional to the square of the pulse energy: 1 - woo = p2$/(2~). 

With elevation of the pulse energy the population of high vibrational levels increases and, 

therefore, the dissociation of the molecule becomes more probable. In the limit of superpower 

ultrashort pulses (,8 >> 2N) the probability of dissociation is exponentially close to unity: 

Thus, under the action of femtosecond laser pulse molecules can be effectively excited on high 

vibrational levels, and moreover, when the pulse energy exceeds the threshold 

the inversion takes place between levels v and v-1. 

In conclusion let us present some estimations. ?or the CO molecule which is active in Raman 

scattering we shall estimate the energy of the laser pulse which provides inversion between the 

lowest levels (v = 1) 4 (v = 0). Using data on Raman cross section (ano/aQ 3 10-16m2, X = 

488 nm[5]) and spectroscopic parameters (D = 83777 em-', a = 2.3904 A-') [7] we can estimate 

the pulse energy required for the inversion to be of the order of 0.1 J/cm2. 

The last estimation is connected with the energy of the ultrashort pulse which is necessary 

for an effective (wD N 1) dissociation of a two-atom molecule. As an example we consider the 

molecule of J2 whose dissociation energy is relatively small (D z 1.5eV) and is of the same order 

of magnitude as the spectral width of a femtosecond pulse. From the condition ,8 2 2N we can 

obtain that the pulse energy required for dissociation exceeds 1 J/m2. This requirment is much 

stronger than the one for the inversion on the lowest levels but it can be satisfied by the modern 

femtosecond technique. 

4 Acknowledgments 

This research is supported in part by the Russian Basic Research Foundation under Grant No. 93-' 

02-14271 and in part by Wissenschaftskolleg in Berlin, Schweizerische National-versicherungsGesellschaft, Alusuisse-Lonza Holding AG and Mr. Helmut Hartmann (Augsburg). We would like 

to thank Professor M.H. Rubin and Professor Y.H. Shih for their kind invitation to participate in 

the Workshop. One of the authors (V.I.) is grateful to the American Physical Society for travel 

support. 
References 

[I] S.A. Akhmanov, V.A. Vysloukh and A.S. Chirkin, Optics of Femtosecond Laser Pulses. 

(Nauka, Moscow, 1988, in Russian). 

[2] Y .X. Yan, E.B. Gamble and I<.A. Nelson, J. Chem. Phys.83, 5391 (1985). 

[3] E.M. Belenov, A.V. Nazarkin and I.P.Prokopovich, JETP Lett. (Russia) 55, 223 (1992). 

[4] E.M. Belenov and A.V. Nazarkin, JETP Lett. (Russia) 53, 188 (1991). 

[5] W.R. Fenner, H.A. Hyatt et ak, J. Opt. Soc. Am. 63, 73 (1973). 

[6] L.D. Landau and E.M. Lifshitz, Quantum Mechanics, (Nauka, Moscow, 1963, in Russian). 

[7] 1.1. Tugov, Proceedings of Lebedev Physics Institute.146, 17 (Nauka, Moscow, 1984, in Russian). 
EPR PROBLEM, BELL'S INEQUALITIES 

AND 

MULTIPHOTON INTERF'EROMETRY 
Complementarity and Path Distinguishability: some recent results 

concerning photon pairs. 

A bner Shimony 

Physics and Philosophy Departments, Boston University, 

Boston, Massachussetts 0221 5 

Gregg Jaeger 

Physics Department, Boston University, 

Boston, Massachussetts 0221 5 

Abstract 

Two results concerning photon pairs, one previously reported and one new, are summarized. It was previously shown that if the two photons are prepared in a quantum state 

formed from [A) and IA') for photon 1 and IB) and IB') for photon 2, then both one- and 

two-particle interferometry can be studied. If v; is the visibility of one-photon interference 

fringes (i = 1,2) and 012 is the visibility of two-photon fringes (a concept which we explicitly 

define), then 

v; + 5 1. 

The second result concerns the distinguishabilit:. of the paths of photon 2, using the known 

2-photon state. A proposed measure E for path distinguishability is based upon finding 

an optimum strategy for betting on the outcome of a path measurement. hfandel has also 

proposed a measure of distinguishability PD, defined in terms of the density operator p of 

photon 2. We show that E is greater than or equal to PD and that v2 = (1 - E~)'/~. 

1 Introduction. 

The idea of an entangled quantum state of a composite system - i.e., a state not factorizable 

into a product of one-particle states - was discovered by Schrtidinger in 1926, and has been 

intensively studied as a result of analyses by Einstein-Podolsky-Rosen and Bell. A very convenient 

method for preparing entangled photon pairs by parametric down-conversion in laser-pumped 

nonlinear crystals was discovered by Burnham and Weinberg in 1970. Their discovery permitted 

the development of two-photon interferometry by Mandel and his school, Alley and Shih, Franson, 

Rarity and Tapster, Chiao and his school, and others.' 

For subsequent discussion. it will be useful to refer to a schematic two-photon apparatus (Fig. 

I), in which a pair of photons emerges from a source S, one of which propagates in beams A and/or 
A', and the other in beams B and/or B', where the locution ''and/orW is a brief way of referring to 

quantum mechanical superposition. For the work on path distinguishability that we shall report, 

this partial description of Fig. 1 provides the essence. For the work on the complementarity of 

one-photon and two-photon interference, some further elements are indispensable. There is an 

ideal symmetric beam splitter HI upon which each of the beams A and A' impinge, from which 

emerge beams U2 and L2. We can speak equivalently of a photon "emerging" in beams U1, L1, 

U2, L2 or of its LLdetection by an ideal photo-detector" in the respective beams. Finally, there are 

vasiable phase shifters 41 and $2 inserted in beams A and B. 

FIG.l. Schematic two-particle four-beam iilteferoineter 

2 Complementarity. 

It was noticed in the past, for instance by Harne and Zeilinge~.~ t11a.t when the photon pair is 

prepared in the entangled stake I*), 

then probabilities of single detections in the va.rious elllerging 11ea.m~ a.re independent of phase 

shifts q51 and q2? specifically. 

1 P(&-1) = P(L,) = P(L-2) = P(L,) = , .- (2) 

whereas the probabilities of joint detection depend on ol and 02. specifically, 

1 

P(ullT2) = P(LlL2) = -[1 - COS($~ + 4 (34 

Since the probabilities in Eqs.(Sa. b) vary from a minimum of zero to a non-zero maximum value, 

while those of Eq.(2) do not vary a.t all. it is reasonable to extend standard optical terminology 

and say that the visibility of one-photon "fringes" is zero and the visibility of two-photon fringes 
is unity, where "fringe" is a generic way of referring to the dependence of detection probabilities 

upon variable phase shifts. When the quantum state of the two photons has the product form 

then 

1 P(Ui) = 5(1 - sin&), i = 1,2, 

1 P(Li) = Z(l + sink), i = 1,2, 

and the probabilities of joint detection are the products of respective single detections: 

1 

p(UlU2) = P(U1)P(U2) = $1 - sin&)(l - sin&), etc. 

It is reasonable to say in this case that the visibility of one-photon fringes is unity, but the visibility 

of two-photon fringes is zero (the latter statement in spite of the fact that P(UlU2) does vary with 

$1 and $2, because of the consideration that this variation is not a genuine two-photon effect but 

is derived from the one-photon variation). 

The two extreme cases of I@) and I@) suggest that there is a complementarity of one-photon and 

two-photon interference visibility. Jaeger, Horne, and Shimony3 raised the question of a general 

complementarity relation, holding for any two-photon state expressible in terms of IA), IA'), 1 B), 

I B') . A necessary condition for investigating this question was to define explicitly the "one-photon 

visibility" v; (i = 1,2) and the "two-photon visibility" y2. The former is straightforward, simply 

adapting the standard optical concept introduced by Rayleigh. We state it here only for the beams 

Ul and U2, but parallels hold for L1 and L2 

For vl2 Jaeger et al. suggested 

The "corrected" joint probability P(u~u~) is defined as 

where the second term on the right hand side removes the variability that is derived from the 

single probabilities P(Ul), P(U2) and the third term is a correction against excessive subtraction 

in order to agree with intuition in the extreme cases of I*) and I@). 
In order to exhibit the desired complementarity relation, it is essential to calculate vi and vll 

in the most general two-photon state that can be prepared with IA), IA') as basis states for photon 

1 and IB), I B') as basis states for photon 2, namely, 

Note that only three phase angles A, p, v are used, because an overall multiplication by a phase 

factor does not change the quantum state, and this fact can be used to choose the coefficient of 

1A)IB) to be real. In Ref. 3 it was fallaciously argued that a basis change of 

ip' - IA) = erld) , IA') = e IA') , 

1 B) = e'" 18) , I B') = eid 1 B') , 

can be used to express 10) in terms of IA), (A'), I B), I B') with real coefficients. But Prof. Sheldon 

Goldstein pointed out to us (private communication) that in general only two of the three phase 

angles in Eq.(lO) can be eliminated by a basis change, and therefore the greatest simplification 

that can be achieved in full generality retains one explicit phase angle, for instance, 

So far, we have not demonstrated a complementarity relation for the general case of Eq.(12). We 

therefore report the result in the restricted case of T = 0, which we have investigated. As stated 

in Ref. 3, Eqs.(29-32), we obtain 

whence 

or equivalently, 

Inequalities (15a,b) are our expressions of the complementarity of one-photon and two-photon 

visibilities. Although we have derived them only for the special case of T = 0, we are confident 

that they hold for any T and hence for the most general 10). Work is in progress on this important 

question. 
3 Path Distinguishability. 

We return now to Fig. 1 and ask a new question. Suppose that we are allowed to make any 

observation on photon 1, which is the left-going photon that propagates in A and/or A'; what is 

the best procedure for predicting which detector will be triggered by photon 2, if ideal detectors 

are inserted in beams B and B'? This question is related to a question recently raised by Mande14 

concerning the distinguishability of the path of a photon that propagates in beams B and/or B'. 

There is, however, an important difference between Mandel's question and ours. He assumes only 

that one knows the density operator p characterizing an ensemble of photons in the beams B 

and/or B', and he asks for a measure of distinguishability expressed in terms of p. By contrast, we 

ask for a measure of distinguishability based upon the quantum state 10) of the pair of photons 

1 and 2, together with the outcome of an arbitrary measurement upon photon 1. It is possible to 

compare our result with Mandel's, because when 10) is given a density operator for photon 2 can be 

calculated5 by tracing out the appropriate variables of photon 1. But, of course, if only p is given, 

there are many possible preparations of an ensemble of photons propagating in beams B and/or B' 

that would yield the same p. In other wcrds, the preparation of the ensemble provides additional 

information that is not included in p. Consequently, we anticipate a discrepancy between Mandel's 

measure of path distinguishability and ours. 

As a preliminary to our proposed measure of path distinguishability we suppose that an observable O is measured on photon 1. Since the space of states that we have allowed for photon 1 

is two-dimensional, there is no loss of generality if we restrict the observable 0 to the form 

where 141) and (42) are orthonormal kets in the space spanned by (A) and IA'). (We are grateful 

to Prof. Lev Vaidmm for suggesting that we consider any 0, rather than just (A)(AI - IA')(AII 

as in our original preprint.) The eigenvalues of 0 are +1 and -1. Now formulate a strategy for 

betting on whether the detector in team B or in beam B' is triggered, letting the strategy depend 

upon the quantum state (0) of the photon pair and the outcome +1 or -1 of measuring 8. If in 

a single case the correct detector is predicted, the observer wins one unit of utility; if the wrong 

detector is predicted, the observer loses one unit of utility. Once the strategy is specified, it is 

straightforward to calculate from (0) the average gain per bet. Let Eo be the largest average gain 

thus calculated as the strategy is varied but O is fixed. Finally, our measure of distinguishability 

of paths, which we shall label E, is defined as 

E = rnaz Eo (over the set of allowed obserz~ables) . (17) 

E is thus the qua.nt.um mechanical estimate of the gain per bet when the optimum allowable 

strategy is followed, the bets being made concerning paths B and B'. 

To ca,lcula.te Eo we first rewrite I@), assumed to be normalized, as 

I@) = lx1)lB) + 1x2)1B1> , (18) 

where, as before, IB) and (B') are orthonormal, but Ill) and 1x2) need not be; however, 
With no loss of generality we can assume that 

which can be achieved, if necessary, by interchanging the labels B and B' of the two paths of 

photon 2. Then we can write 

1x2) = .\/XI) + 1x3) , (214 

where 

and 

If we define 

then the I%;), defined by . . 

are orthonormal. Furthermore, 

Ni(l+ !XI2) + N3 = 1 . 

Any basis in the space of allowable states of photon 1 can be expressed as 

where 

Ip12 + +vI2 = 1 . 

This basis defines the observable O of Eq.(16). It will also be useful to write 

an observable in the allowable space of states of photon 2; clearly D is observed to have values +1 

and -1 according as photon 2 is detected in path B or B'. 

If O is the observable chosen to be measured, then there are four pure strategies for bets on 

the path of photon 2: 

(1) If 0 = +1, predict B = +l; if O = -1, predict B = -1. 

(2) If 0 = +I, predict B = -1; if O = -1, predict B = +l. 

(3) Predict B = +1 regardless of the vaIue of 0. 

(4) Predict B = -1 regardless of the value of 0. 
In addition to these pure strategies there are mixed strategies, consisting of following (I), (2), 

(3), (4) with arbitrary probabilities summing to unity. But since the game is not being played 

against a rational o ponent, the average gain in a mixed strategy cannot exceed the maximum of 

the average gain Ei of the pure strategies: i = 1,2,3,4. These are calculated as follows: 

E:) = P(O = 1 and B = 1)+ P(O = -1 and B = -1) 

where 

Et) = (3) P(B = -1) - P(B = +1) = -Eo . (30) 

Note that Eg) and Et) are independent of 0. Then 

Eo = mas{l~(~p~~ - Iv12) - TIpIIvI COS(~A + 0" - Op)l , IS - 2~31). (31) 

In view of Eqs.(l7) and (31) one finds the measure E of path distinguishability by investigating 

Eo as p and v are varied, subject to Eq.(24c). We first note that for any (0) there is an 0 such 

that 

1~8'1 r IE~~)I , 

so that the second option in Eq.(31) can be neglected when we maximize over all possible 0. To 

prove these statements, it suffices in Eqs.(24a,b) to let p = 1 and v = 0, determining an 0' such 

that Eqs.(26), (27), (28) yield 
and 

IE~!I = INl(l - IA12) - N31 . 

Since Nl and N3 are non-negative, and (1 - IAI2) is non-negative by Eq.(2lc), we obtain 

the rhs being the same as IE$)~ for all 0. E is therefore obtained by maximizing the first option 

of Eq.(31) for allowable p and v, and the result is 

By Eqs. (27a), (27b), and (23) E can be rewritten as 

We can now make a comparison with Mandel's4 measure of path distinguishability Po. Mandel 

notes that in a two-dimensional Nilbert space, any density operator p can be expressed uniquely 

in the form 

p = PID PID + PD PD , (38) 

where p~ is diagonal in the I B), I B') basis, i. e. 

(after adaptation to our notation), 

and 

PID 2 0, PD 2 0. 

Since p~ is a diagonal density operator in the specified basis, one can prepare an ensemble with 

a definite proportion c~l in the state I B) and a definite proportion c22 in the state I B') such that 

this ensemble is represented by p~. It is this consideration that leads Mandel to identify PD as 

the degree of path distinguishability when p is given. Mandel also shows that 

where p;j is the i jth matrix element of p in the I B) , I B') basis. 
Now let us consider the 10) of Eq.(18), which we can rewrite as 

By the standard procedure for writing the density matrix of particle 2 of a two-particle system: 

we obtain (with the help of Eq.(23)), 

p11 = Nl , 

Hence, Eq.(37) can be rewritten as 

which can be shown as follows to be greater than or equal to PD of Eq.(42). 

Proof : First note that if x and y are real numbers in the interval [O,1] which sum to unity, 

then 1 

XYS~, (46) 

from which it follows that 

Furthermore, since, by Eq.(23) 

we have 1 

Ip12l = NllXl 5 5 . 

From Eqs.(47) and (48) we obtain 

where the lhs of this inequality is E2 and the rhs is Pi. Since both E and PD are non-negative, 

it follows that 

E>PD. (50) 

We note that when E is unity, so is Po: that is, perfect distinguishability (in our sense) on 

the basis of the two-photon state 10) implies perfect distinguishability (in Mandel's sense) on the 

basis of the density operator. There is an intuitive reason for this agreement: E = 1 implies that 

there is perfect correlation between the behavior of photon 1 and the entrance of photon 2 into 

IB) or IB'), but perfect correlation requires the orthogonality of 1x1) and 1x2) in Eq.(18). This 

orthogonality, in turn, guarantees that the density operator of photon 2 is diagonal in the I B), I B') 

basis. 
If we look at the other extreme, however, we find that Po = 0 does not imply that E = 0. 

Again there is an intuitive reason. When PD = 0, then p is a pure case, derived from a quantum 

state of the form 

I$) = clB) + cllB1) , (51) 

so that 

Then 

and this vanishes if and only if lc12 = !dl2 = f . But when the amplitudes of JB) and JB1) in the 

pure state I$) are equal, there is no strategy for betting on the path that will yield a net gain on 

the average. On the other hand, when lcI2 and (dl2 are unequal, the strategy of betting on the 

path associated with the larger coefficient will yield a net gain on the average. The advantage 

of our E over PD is the ability of the former to take advantage of inequalities in the amplitudes 

associated with the two paths. 

Mandel also relates path distinguishability to the visibility v2 of the interference pattern, where 

He obtains the inequality 

212 < PID = 1 - PD , 

with equality holding only when pll = p22. We obtain from the expressions for E and v2 in 

Eqs. (45) and (54) the equation 

which holds for any preparation of an ensemble of photons in states IB) and IB1) derived from 

a two-photon state of the form 10). Hence, for the preparation of photon 2 that we have been 

studying, the visibility vz is a natural measure of path indistinguishibility. 

4 Acknowledgments. 

We are grateful to Profs. L. Vaidman, M. Horne, S. Goldstein, and L. Mandel for suggestio~s and 

encouragement. This research was supported in part by the National Science Foundation under 

grant no. PHY-90-22345. 
References 

[I] References to the authors mentioned in this paragraph and to others are given in ""Down 

Conversion Photon Pairs: A New Chapter in the History of Quantum Mechanical Entanglement," by M. Horne, A. Shimony, and A. Zeilinger, in Quantum Coherence, edited by J. 

Anandan (World Scientific, Singapore, 1990). 

[2] M. A. Horne and A. Zeilinger, in Microphysical Reality and Quantum Formalism, Proceedings 

of the conference at Urbino, Italy, Sept .25-0ct. 3,1985, edited by A. van der Merwe, F. Selleri, 

and G. Tarozzi (Kluwer Academic, Dordrecht, 1988). 

[3] 6. Jaeger, M. A. Horne, and A. Shimony, "Complementarity of one-particle and two-particle 

interference," Physical Review A 48, 1023 (1993). 

[4] L. Mandel, "Coherence and Indistinguishability," in Optics Letters 16, 1882 (1991). 

[5] E. Beltrametti and G. Cassinelli, The Logic of Quantum Mechanics, Addison-Wesley, Reading, MA (1981), p.56. 

[6] J. von Neumann and 0. Morgenstern, Theory of Games and Economic Behavior, Princeton 

University Press, Princeton (1953), pp. 84-85, 145-146. 
Further Evidence For the EPNT Assnmgtisarm 

Bzunie1 M. rger 

City College of the City Univemity of New York 

New York, New York 10031 

Herbefi J. Bernstein 

Hampshire College, Aderst, Massachuse~s 0Im2 

Wchael Home 

Stonehill College, North Emton, iaaassachusetts 02357 

We recently proved a theorem extending the Greenberger-Home-Zeilinger (Gm) 

meorern from multi-particle systems to two-particle systems. This proof depended upon 

an auxiliav assumption, the EPW assumption (Emptiness of Paths Not Taken). 

According to this mumption, if there exists an Ehstein-Rom-Podo1sb (EPR) element of 

reality that dekmines that a path is empty, then there can be no entity associated with the 

wave that travels this path (pdot-waves, empty waves, etc.) and repom somation to the 

mpE@de, when the paths mombine. We produce some fu&er evidence in support of 

this assumption, wfich is c true in quan~um theory. The alternative is that such a 

port-wave theoy would have to vblate EPPI localitgr. 

g to extend the GMZ (Greenberger-Norne-Zeilinger) Theoremly2 

down to two-particle sys&ms3, we produced a proof that be realized depended on a 

assumption, wKch went beyond the lEPR (Einstein-Podolsb-Wosen) assumptions4. 

assumption was the EPW assumption-- the Emptiness of the Paths Not Taken. 

asswption ruled out the possibility of any kind of information-bearing entity traveling 

down a path, provided one could produce an EPR element of reality connected with the 

path being empty. The EPR criterion depends on one being able to perform an experiment 

far away, without in any way affecting any particle that could possibly be travelling down 

this path. Then if this experiment shows that the path is empty, the path must be truly 

empty, according to EPR, since one has not interfered in any way with anything along the 

fact of emptiness is then an "element of reality", because it is true independently 

an experimenter might later do that might interfere with the path or particles 

along it. 

P PAGE RAM MT mMW 

535 
One might be tempted to the EPm assumption rules out my kind of 

hkdemce at all, as when a pd bough a ham splitter, and the two paths ane 

laer raombined and ink~e~. But for a single particle, one cannot pmduce an element of 

~&ty connected with the path, because any measurement on the particle to deternine 

which path it takes will necessdy disturb it. So the EPm assluanptlon does not apply to 

one-particle systems. But for a two-particle system, one may make a masurement on one 

paureicle that determines which path the second particle takes, and here the EPNTF 

assumption does apply, and it gives results that accord with quantP4m theory. Further 

westions conce g the applicabiliv and plausibility of the EPEdT assumption are deal& 

with in Ref. (3). The reason that the assumption is worth exploring in detail is that &ere 

are other theories, such as pilot wave theories, that compek d& qua- theov and that 

do depnd on idomtion-be g empty waves for their effects. It should also be poin~ted 

out that an alternative, complementary approach to this not along the lines of 

GHZ, has been taken by L. Hardy, who can prove that for percentage of phcles 

in the beam, the GHZ theorem must be true for two particless. 

SITUAmON IN QU ORU 

One can easily show that within quantum thmry if a path is empty, it is truly ernpv, 

wGch means that if the amplitude for a particle to be travelling along a path is zero, then no 

infomation can be itted along that path. For example, in Fig. (1) we depict a em 

device which takes an incoming wave function that can be along either of the paths 1 or 

The particle enters from the left and exits to the right. 

There is a phase shifter of angle a located on path 1, 

2, with amplitudes a or b respectively, and converts it into a wave function travelling along 

the paths 3 and 4, with amplitudes c and d respectively. 

Since the device is unitary, its most general form is 

eiD .sin 

y)[; e"D+" 'sin 

If now a phase-shif&g device, that shifts the by a, is placed into beam 1, the 

incoming beam will change from amplitude a to aeia . An infinitesimal change in a will 

produce the result in beams 3 and 4, 

& = eiA cosy aeiai &, 

= ae'"i &, 

One sees that both of these terms will be zero if a is zero. This result shows that if the 

beam 1 is empty (i.e., a*), there is no way to transmit any change in a to any amplitude 
dowrpsbem of beam 1, even if fie= is a u~tsary comecdon ktwmn the bms. gfi% 

res\nla is easily generdized to any number of phcbs md mpllmdes. It shows that 

according to qumtm mechanics, no infomation can be &ansmitBd &rough an empw 

hm. This of coum is elze ewndd content of fie EPm mmpdon. 

We are ssmhg that quanm theory gives 4;omt ssdts and that it is the bden 

of my dtemative theoay to reproduce these ~sd@, The =arson &at he &aussion cwot 

stop here is that one might choose wot to believe qumm mwhdcs and say tha &e~ are 

hdmd alternative ways to produce the md@ of the thmergr withoa hg the Isni&q 

and &eariq of the theov. We shall show that md~-pdcle suppeqosi~om place a hav 

burden on any such theory. 

ON PASSED &ONG E 

We will now show that if one msumes that a hem is empv, as an EPW elernem of 

rea~ty, but one still insists that it can cq infoma~on, then if this infomation cm be 

operationally transmitted to another beam, this infoma~on can violate the unceaaintgr 

principle. 

We shall work out a particular example, but it is obvious that the thrust of the 

ugument is very general. Consider a particle at rest that can decay into tws phcles, as the . one at point 0 in Fig. (2). The two 

h intederence pattern is produced by measuring coincidences 

between detectors A, B, and C. 

particles come off in opposite directions, and are restricted by slits to the two sets of 

directions, a-a', and b-6'. So the state of the system after decay is 

w =*(I..') +lbb')). 

The primed particle paths are directed through a beam-splitter at F toward the detectors B 

and C. (We assume, for unitarity, that the reflected ray picks up a 900 phase). The paths 

of the unprimed particle are directed toward a screen with two slits, such that path a leads 

to one slit, El while path b leads to the other slit, E2. Each of the slits is of width d, and 

they are separated by a distance D, such that D >> d. The diffraction pattern formed at a 

distance L from the slits is picked up by the detector A. We also assume that L >> D, and 

that the position of A is given by x, as measured at L, perpendicularly from the center of 

the slits (see Fig. (2)). 

An important feature of multi-particle beams that applies here should be noted. If 

the detector A is moved as a function of the distance x, there will be no diffraction pattern 
obseaared, as there will be no single particle interference from this setup. Only if A is 

monitored in coincidence with the detector B or @ will a pattern apperu= We shdl co~m 

this below. The reason we have used this two-particle setup to produce diffraction is 

that we can remove the beano-spgter at F. Then if detector B fires, we know that the 

partick had aen the path a', and so its pmer must have taken the path a. Thus &is 

particle must have entered the slit El and there will be only a one-p~icle interference 

panern at A, of mgdar width 8 = il / d, where we assume for convenience that hd. 

Similaly, if detector @ fires, we know the particle must have entered the slit E2 and will 

also produce a one-particle interference pattern at A. Since we can deermine wkch path and slit the unphed particle takes, without in any way htederirng with the particle, ~s 

howledge is an EPR element of redity. h other words, according to EPR, it is an 

objecai.ve fact. So even if we do not bother to remove the beam-splitter at F, EH"W would 

conclude that the element of ~ality exists, beGause we could have removed the split&r 

~thout ~weing the particle, and so the particle actually takes one slit or the other. They 

would conclude that because the phcle takes one path or the other, but quantum theoy is 

powerless to describe this fact, that quantum theory is therefore an incomplete theoq. 

But of course, according to quantaun theory, whether we remove the km-spfimr 

or not is a crucial fact, one that completely changes the context of the exphent. E we 

remove it, then indeed the particle is in one path or the other. But if we do not remove it, 

the particle cannot be described as being in either path. 

If we perform the experiment with the beam-splitter in place at F then the wave 

function ty becomes (we are also including a n/2 phase shifter at at, purely for 

computaliond convenience) 

+ (p(kx)(ieH1 IAa') + eW2 IA~')) 

= +(p(kx ]a)(i(eW1 + eU2 )( 4 + (eH2 - e" )(c)). 

The pmbab%Q for cobciderat counts at A and B, or at A and C is 

2 kAf 2% P, = l(p(kx)(2 cos 7, PAC = l(p(kX)r sin 2 

Aa = e, - e2. 

In this equa~on (p(kx) represents the Fourier Trmsform of the single slit patkm produced 

by each of the slits, Qr wsch 8 = A/ d, and it is much wider than one of the ~o-partjcle 

a, whose width is of order 8 = A/ D. So in fact, in the region of the cenwd 

the term kilt/ 2 = nDx/ LA contributes, g>(k,) can be considered to be a 

constant. (In this region, the minimum xm of the cosine term occurs when the argunment 

equals d2, or xm = LA l2D5 LO, and 8 = A1 2D.) One can expe~entally isolate this 

central maximum from the others, and since both slits con~bute to it, one has Ax = D, 

while Lipx = p0 = pill D = fit D. Thus this central maximum is of the order of a minhm 

uncertainty packet. Note also that if one only triggers the detector A, ignoring the dewton 

B and C, one will get a number of counts independent of the path difference At beween 

the beams, P, = P, + PAC = const., which proves our original assertion that one must 

count coincidences to see the interference pattern in this experiment. 

For the case when one removes the beam-splitter, one has only the top equation 

above for y , 
y -+ C(kx)(iew l~a') + ey~b')) 

If then counter B fires in coincidence with A, one knows that the unprimed particle is in the 

beam a' and similarly if counter G fm it is in beam b', and so the coincidence counting 

rates will be P, = i(~(k]~, PA, = iIp(k)(l, which will be a constant on the scale of the 

two-particle pattern, but will fall off as 8 = ill d. This is consistent with the uncertainty 

principle, since in this case, the particle is going only through one slit, so that 

AX-d,Ap-p6- Ald. 

Now we come to the point of the argument. What if one were to believe that in the 

case of an ordinary single-particle two-slit experiment, when one can detect into which slit 

the particle enters one will obtain single slit patterns, as quantum theory predicts? But 

when one does not know which slit the particle enters there is some kind of informationbearing pilot wave that carries the information about the second slit, so that even though the 

particle travels through one slit only, nonetheless it is aware of the existence of the second 

slit through the intermediary of the pilot wave, and so one gets a two-slit diffraction 

pattern. It is for this case that we have devised our experiment above. For we can produce 

the EPR element of reality needed to prove the particle takes only one beam, by removing 

the beam-splitter. However if one believe., that the element of ~eality persists when one 

does not remove the beam splitter, and also that a pilot wave of some sort carries 

information about the second slit, so that the diffraction pattern can occur, we believe that 

this leads to a contradiction in our experiment. 

In our experiment, if one accepts another principle of EPR, that of locality, then one 

must accept the fact that the unprimed particle receives no information that can tell it 

whether in fact the beam splitter at F has been removed or not. So there is no way for the 

particle to know how to evaluate the information obtained from the pilot wave. Does it lead 

to a diffraction pattern or not? In an ordinary one-particle experiment, there is no way to 

observe that the particle has actually taken one slit or the other. But in our experiment, we 

can provide that information without disturbing the particle approaching the slit. According 

to the EgNT assumption, when one of the paths is truly empty in the EPR sense, there can 

be no information transmitted along the other path. The negation of this assumption 

implies that some information can be carried along this other path. If this is so, are there 

any experimental implications of this? If not, it is merely an idle statement. 

The way to exploit these facts is to assume that the beam-splitter is present, but that 

there is another detector present, in beam b'. This detector will fire if the particle takes 

beam b: If it does not fire, then we know that the particle is in beam a-a'. But there will 

be an empty wave in beam b-b'. Since the detector has not fired, the empty wave will 

presumably pass along to the beam splitter at F. Does it share any of its information with 

the other beam? To decide this, we assume that there is some parameter P that determines 

how much information the pilot wave carries along one beam when it is known that the 

particle takes the other beam, through the other slit. In that case, the particle will produce a 

coincidence count probability for the counters A and B to fire, 

P * f lp(kx]21e"1 +/3e"212, 

where p varies between 0 and 1. 

This would lead to a diffraction pattern with a contrast of C = 28/(1+ p2). Not 

only would this disagree with quantum theory, but also with the uncertainty principle 

directly, since then (Ap)2 - [p2 (RI D)~ + (1 - p2)(hl d)'] , while (Ax)2 - d2, since one 

knows which slit the particle actually took. For finite P, this violates the uncertainty 
principle, giving in the limit of j9 + 1 the mdt hdg + AdlD<<ti. The alternative is that 

the theov must violate EPR locality. 

The argunnent suppo*g the EPNT ssu article syskm is even 

stronger thm for two particles. Agh we shall atnple, but the resuls 

are generalizable. Consider a particle that decays If the particles are of 

the same mass, and when they are counted, it is has the same energy, 

they will come off at 1200 apart. They are now restricted by slits to three sets of 

tions, a-ar-a", b-br-b", and c-cv-c". particle 1 is unprimed, particle 2 is primed, and 

paPticle 3 is double-phed. If particle 1 takes path a, then 2 must take path a', and 3 takes 

path a", etc.) (See fig. (3)). This is a simple generahation of the two-particle process in 

DEF 

F" D' 

ter With Three TriA particle at 0 decays into three particles which take the possible paths a-ar-a", bbr-br', or c-cr-c". The three paths for each particle converge at a tritter, and then 

pass to one of three detectors. Each particle has one phase shifter in one path, a$, 

or y . In the second part of the experiment, a detector is placed at X to determine 

whether the particles have taken the paths c-c1-c". 
the previous example. Each particle is now refocussed into a unitary 

(We have previously called such devices "multi-ports", or "cri#ers", 

way device is a "tritter"6- Such devices can emulate an unitary transfornation of 

the system.) For simplicity, our particular tritter is taken to hasre a specific unitary 

transformation (see Fig. (4)). If the beams 

out 

We call this device a "trider". The three input beams a, 

6, c, are transformed unitarily into the output beams d, e, I 

a, b, c, are the incident beams, and d, e, f, are the outgoing beams, they will be related by 

rhe relations 

I4 -9 *(Id)+le)+lf)). 

21d13 whe~ 1, A, = e , and p = e4"'I3 axe the cube roots of unity, and 

p2 =a.a2 =p,ap=i, 

p*=il,A*=p,l+A+p=O. 

The actual setup is as shown in Fig. (3). mere is a tritter in the path of ewh 

particle. There is also a phase shifter, of phase a in beam a,one of phase /3 in beam b', 

and one of phase y in c". The initial wave func of the three phcles Is 

eial ao'n") + eipl bb'b") 

Each of the tritter outputs goes to a detector, labelle r particle 1, and similarly, 

with primes, for the other particles. The amplitude after passing through the' tritters and 

I E) + (F))((D') + 1 E') + (F))((D") + (E") + IF")) 

+eiraD) + dE) + YF))(~D')+ pl E')+ A1 F'))(1DM) + PIE")+ YF"))]. 

From this one can calculate the output to any set of detectors. 

Rather than write down all possible outputs explicitly, we shall merely as an 

example write down all those that involve the counters D and D', namely DD'D", DD'E", 

and DD'F". They are 
+etc. 

So for exmple, the probability of wmhg a coincidence in the detectors DD'D" is 

PDwDu = & (3 + 2 cos(a - p) + 2 COS(B - y) 9 2 cos(y - a)). 

The significant point here is that the counting rate associated with this set of coinsidence 

counts depends symmetrically on the phase shifts a,P,y. 

As with the two-particle case, coincidences ounters will lead 

to a diffraction pattern of counting. If one looks at only two one), one will 

find a flat rate. For exmple, if one adds the rates (mplitu&s r the thnee comts 

given above, one will get for the probability of a count in EID', 

PDD, = PDDtDn + Pm,p + PDD, = CQW~. 

NOW assume that a dewtor is placed into the beam c at point X, so that if it fires, 

one knows that the this path, and therefore that particle 2 took path 

c', and particle 3 to hes the path as an EPR element of reality, as one 

can determine the les by intercepting the third one. Thus in this 

experiment, the EPNT assumption applies and says that if a particdar path is empty, then 

there is no entity associated with this path that can carry infomation through the system if 

the paths later happen to rejoin. 

We shall be interested in the case where the detector is installed at X, but it does not 

fire. h this case, one knows that the particle is not located along the paths c-cl-c". In 

other words, this set of paths is empty. Because the phase shifter y is located in the path 

c", the EPEdT theorem would predict that when the counter X does not fire, the counting 

rate for coincidences cannot depend upon the angle y . The counter X will fire 113 of he 

time. If we keep the same nomalimtion as before, so that the total probability for events in 

which X does not fire is 2/3, then the wave function reaching the set of trims is 

iyx = &(eialaa'a") + eg 1 bb'b")). 

Thus we see already that, quantum mmhanically, it cannot depend on y. The wave 

funstion reaching the counters will be 

eia(p) + IE) + IF))(IIY) +IF) +IF'))~D")+IE")+ IF)) 

+eifl(lq + nlE) + @)(la) + ap3 + pl~))(Io") + a1 E") + F") 

Now the probability for a coincide ce in DD'D" will be 

eia + eg )I WD") + etc 

P,, ,,. = & (1 + cos(a - B)). 

Not only is this independent of y but by a suitable choice of angles, one can make the 

result either greater or less than the result when the detector X was absent. For example, 

when a = P, then Px,DvDn = XI, while if y = a = B, then PDD,Dw = but if 

a=p= yi-;lc,then PDD,Du =XI. 

So, because there exists an element of reality connected with the fact that the path 

containing y is empty, this probability no longer depends on y . However, if one looks at 

Fig. (4), one sees that the phase shifter y lies in the beam c", while the detector X lies in 

the beam c. So according to EPR locality, there is no way in which particle 3, on one of 

the double-primed paths, could be made aware of whether the counter X has fired or not, 

or even whether it is present or not. Thus if there existed a pilot wave that sampled the 
double-phed patlhs, there is no way in which it can have been notified to change the 

n the infomaeion it trier. In fact, the 

o renm &tween the thae now there is 

a label 8Mached to the particle expressing the EPR element that the path of the particle 

exclu~ c". hd the existence of this element &so must be respo for the fact that the 

resulting count rate no longer depends upon y . Similarly, the unprimed particle, one of 

whose paths contains the detector X, has no knowledge of y at all, and neither does the 

primed particle. So connected with this experiment can be explained by the 

EPNT assumption, but to be extremely implausible if one accepts EPR locality, and ~Ees ona pilot wave type of explanation. Of course, if one drops EPR locality, one 

can use the Bohm-Hiley theory7 to explain these events, as it is equivalent to quantum theosy, and non-local. 

We believe that if one accepts that pilot waves can exist in a local theory, then one 

mecessasily will produce effects that violate quantum theory. The alternative EPNT 

assumption rules out such effects, even in two-particle systems, and is consistent with 

quantum theory. 

REFERENCES 

1. D. M. Greenberger, M. A. Horne, and A. Zeilinger, Bell's Theorem, Quantum 

Zheory, and Conceptions of the Universe M. Katafos, ed., Kluwer Academic, 

Dordrecht, 1989. 

2. D. M. Greenberger, M. A. Horne, A. Shimony, and A. Zeilinger, Amer. J. 

Physics 58, (1990) 1131. 

3. H. Bernstein, D. M. Greenberger, M. A. Horne, and A. Zeilinger, Phys. Rev. 

A47, (1993), 78 . 4. A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, (1935), 777-780. We 

shall refer to this paper as EPR. It is reprinted in J. A. Wheeler and W. H. Zurek, 

Quantum measurement Theory Princeton U. P., Princeton, 1983. 

5. L. Hardy, Phys. ktt. A161, (1992) 326. 

6. A. Zeilinger, H. J. Bernstein, D. M. Greenberger, M. A. Horne, and M. 

Zukowski, in Proc. of Fourth International Symposium of Foundations of 

Quantum Mechanics in the Light of New Technology, H. Ezawa, Y. Murayarna, 

et al, eds., World Scientific, Singapore, 1993. 

7. This theory is discussed in great detail in D. Bohm and B. J. Hiley, The Undivided 

Universe, Routledge, London, 1993. 
THE PHOTON: EXPERIMENTAL EMPHASIS ON ITS 

WAVE-PARTICLE DUALITY 

Y. H. Shih, A. V. Sergienko and M. H. Rubin 

Department of Physics, University of Maryland 

Baltimore County, Baltimore, MD 21228 

T. E. Kiess, C. 0. Alley 

Department of Physics, University of Maryland, 

College Park, MD 20742 

Abstract 

Two types of Einstein-Podolsky-Rosen experiments were demonstrated recently in our 

laboratory. It is interesting to see that in an interference experiment (wave-like experiment) 

the photon exhibits its particle property, and in a beam-splitting experiment (particle-like 

experiment) the photon exhibits its wave property. The two-photon states are produced from 

Type I and Type 11 optical spontaneous parametric down conversion, respectively. 

We wish to report two EPR [I] type experiments. The first one is a two-photon interference 

experiment in a standard Mach-Zehnder interferometer. Another one is a two-photon beam 

splitting type experiment for the measurement of polarization correlation. It is interesting to 

see that in the interference experiment (wave-like experiment) the photon exhibits its particle 

property, and in the beam-splitting experiment (particle-like experiment) the photon exhibits its 

wave property. 

I Two-photon interference in a standard Mach-Zehnder 

interferometer. 

A pair of photons with different colors ( XI = 632.8 nm, X2 = 788.7 nm, 155.9nm difference in 

center wavelength) is directed to one input port of a Mach-Zehnder interferometer. Coincidence 
measurement is de between the two output ports of the interfaometer with the help of a 

300 psec coincidence time window. The interference behavior was studied in a wide range of 

the optical path dliRerence of the interferometer from white light condition, AE 0, to about 

AE 127m (2 2 - 103 times the coherence length of the down converted beams). Whea the 

optical delay of the intederometer is greater than the coincidence time window, the arslpEtuds i;2i 

which one photon follows the longer arm and the other follows the shorter arm of the intefperometa 

are 'cut off? by the coincidence time winaow. The particle property of the photon is demonstactd 

by means of more than 50% interference visibility. 

Ar laser 

Figure I- 1 : Schematic diagram of the experiment. 

The experimental arrangement is shown in fig. 1-1. A lOcm long Type I phase matching KDP 

crystal pumped by a single mode 351. lnm CW Argon ion laser line is used to generate collinear 

photon pairs at wavelengths 632.8nm and 788.7nm. The coherence length of the pump beans 

was measured to be longer than 5m. The 351.lnm pump beam and the down converted b& 

were polarized in the extraordinary and ordinary ray directions of the crystal, respectively. A 

Glan-Thompson prism was used to separate the collinear down converted photon beams from the 

orthogonal polarized 351.lnm pump beam. Before the 351.lnm laser line was sent to pump the 

parametric down conversion, a quartz dispersion prism was used to separate out the radiation 
lines of the laser plasma tube which are close to the 632.8nm and 788.7nm wavelengths. 

The collinear 632.8nm and 788.7nm photon pair was then injected into a standard MachZehnder interferometer. The optical path differences of the interferometer AL = L - S can 

be arranged to be shorter or longer than the coherence length, Icoh, of each beam of the down 

conversion field and the coincidence time window, c . AT&%. The collinear photon pairs were 

injected onto the beamsplit ter with an incident angle of about ten to twelve degrees (near no 

for which the reflected and transmitted intensities of the 632.8nm and 988.9nm beams were 

measured to be equal (50% - 50%) within 5%. 

Geiger mode avalanche photodiode detectors, operated at dry ice temperature, were used to 

record coincidences in the two output ports of the Mach-Zehnder interferometer. Each of the 

detector has a narrow band interference spectral filters. The central wavelengths of the filters are 

632.8nm and 788.8nm with bandwidths of I.4nm and 1.7nm9 respectively. The output pulses 

from detector A and detector B were then sent to Nl, N2 counter and a coincidence circuit to 

record coincidences. The coincidence time window ATmin was about 300psec. 

We collected data for three regions of interest. In the first region, AL < lcoh, i.e., the optid 

paths difference of the interferometer are equal to within the first order coherence length of the 

signal and idler. In the second region, Zcoh < AL < c . ATcoin. In the third region, AL > 

c. ATcoin. The following reported data are all direct measured values without any noise reductions 

or theoretical corrections. 

(1). The first region, AL < lcoh. 

Fig. 1-2 shows the normalized counting rate of Nc when the optical path difference changed 

from white light condition to about 4pm. In this region, Nl and N2 both showed clear single 

wavelength, 632.8nm and 788.7nm, respectively, first order interference pattern. However, Plr, 

shows a complecated interference pattern with 632.8nm9 788.7nm, and the beating and the sum 

frequencies. The interference visibility is close to 100% with the 300psec coincidence window. The 

solid curve in fig. 1-2 is a theoretical fitting of equation (1-10). Fig. 1-3 shows a typical first order 

interference pattern of iV2 for detector B. The interference visibility is about 90%, with a period 

corresponding to wavelength 788.7nm. 

Fig. 1-4 shows the typical interference patterns of Nc at AL E ll5pm. The 1V, patta in 

fig. 1-4 is different than that in fig. 1-2 in two ways, (A) the interference visibility is reduced and, 

(B) the beating component and the 632.8nm and 788.7nm components of the moddations ase 

reduced and the sum-frequency modulation becomes predominant. The solid line in fig. E4 is a 

theoretical curve resulting from Gaussian spectral filter functions in equation (1-10). The singIe 

detector counting rate is reported in fig. 1-5. The interference visibility is reduced to about 42%. 

(2). The second region, Iah < A L < c . AT,;,. 

In this region both Nl and N2 become constant, however, Nc shows clear interference with the 

sum frequency. Fig. 1-6 shows the interference pat tern of N, for AL 0.5cm. Compare to the 

300 psec coincidence time window and the coherent length of the down converted beams, which 

satisfying lcoh < AL < c ATcOin. The interference visibility is 44% f 3% with modulation at a 

wavelength of 351.lnm. In this region, all the measured interference patterns have modulation 

visibilities close to but less then 50%. 

(3). The third region,AL > c . ATcoin. 

The interference patterns of N, in the final region of interest, AL > c . ATmm, is presented in 

fig. 1-7. An interference visibility of 75% f 3% was measured at AL S 43cm with an interference 
BS position relative to = 0 (pm) 

Figure 1-2: Normalized coincidence counting rate of Neat near white light condition (AE S 0). 

The beting frequency,with 3.2pmperiod, and sum frequency, with 35l.lnmperiod, are evide3t 

from the graph. Signal and idler frequencies, at periods 632.8nm and 788.7nm also contribute 

The solid line is a theoretical curve of Eq. (1-10). 

BS position relative to A= 0 (~m) 

Figure 1-3: Single detector counting rate N2 at near white light condition (AL Z 0). 
0.0 I I I I I I I J 

3 .O 1.5 2.0 2.5 3.0 3.5 

BS position relative to AL= 11 5 pm (pm) 

Figure 1-4: Normalized coincidence counting rate of N, at AL 2 115 pm. Compared wirh 

fig. 1-2, the beating component and the wl and w2 components of the modulation are reduced 

and the sum-frequency modulation becomes predominant. The solid line is a theoretical cum-e 

resulting from Gaussian spectral distributions in Eq. (I- 10). 

200000 

180000 

a 160008 -- 

a, 140000 a 

5 120000 C 

S 

100000 0 

rC 

0 80000 

;u" 60000 E 

7 

Z 40000 

20000 

0 ' I I I I I I I 1 

0.0 0.5 1 -0 1.5 2.0 2.5 3.0 3.5 

BS position relative to AL- 1 1 5 pm (pn) 

Figure 1-5: Single detector counting rate N2 at AL Z 115 pm. 
period of 351.1nrn. When AL increased to about 127cm, the interference visibility was me 

to be 56% Jt 3%. In this region, no interference modulations were found for Nl and N2. 

In our earlier paper [2J a general theory for a two photon interference everiment in two 

interferometers was developed. The experiment was suggested by Franson [3]. Eqerimental study 

for two independent interferometers have demonstrated more than 50% interference visibility by 

using short time coincidence time windows [4, 51. The theory for this experiment is sidar. The 

coincidence counting rate is calculated from the field fourth order correlation function: 

G(rlt2, r2t2; r2t2, rltl) =< E{-)E~-)E~+)E{+) > (I - 1) 

where E!') is the positive frequency part of the electric field in the Heisenberg picture evaluated 

at the position rj and the time tj. E!-) is the hermitian conjugate of E;+), 

aj is the destruction operator of the photons in the jth beam and fj is the pass band of the filter in 

the beam peaked at Qj. We take nl + 522 = up, the pump frequency. In this experiment the filters 

are chosen so that each detector only detects one of the down converted beam, i.e., $21 - n2 >> a 

the band width of the filters. 

The average coincidence counting rate is given by 

h!, = - lc dtidtz~(rit1, r2t~7 ~2~2, rltl)S(tl - t2, AT-) 

(1 - 3) 

= ) SI,T dtldt2 I (0 I E!' )(tl)> ~p)(t2) I 8) I2 S(tl- t2, AT'in) 

where S(t9 ATcG,) is a coincidence detection fuilction, ATmin is the coincidence time window, and 

the integrals are over the detection time T. A two photon amplitude, which is dso called effective 

two-photon wavefunction, is defined in (1-3) by, 

The two photon part of the state that emerging from the down conversion crystal may be 

taken to be [6], 

&~s(wI+ w2 - wp)ai(wl)af (Y) 1 o > (I - 51 

where the 5 function indicate a perfect frequency phase matching condition. The wave nmba 

phase matching condition is implicit in the choice of the location of the pinholes and the detedors. 

Substitute (1-4) and (1-5) into (I-3), it is straight forward to show that, 

'@(tl, t2) = A(tl9 t2) + A(t1 - AT, t2 - AT) + A(tl, t2 - AT) + A(tl - AT, $2) (1 - 6) 

where A(tl,t2) is calculated in (I-8), AL = c - AT is the optical path difference in the two 

of the interferometer. The fist (second) term is the amplitude for which both photons follow the 

short (long) path through the interferometer, and the third (fourth) term is the amplitude for one 

photon follows the short (long) path and another photon follows the long (short) path. A simple 

calculation using Gaussian filters 
where ~j is the bandwidth of the jth filter, gives 

where IC is a constant. 

If we now substitute equations (1-6) and (I-?) into (1-3) and take 

s(t, ATcoin) = exp(- I t I /2ATcoin) 

the average counting rate may be written in the form 

where 

where A = 1/(4CATdn), C is a constant that need not concern us. We remind the reader that the 

error function erf c(x) +O as 3: + 00 and er f c(x) + 2 as s + -00. The key point to understand 

the behavior of the coincidence counting modulation is the variation of the J's with the increase 

of AT = AL/c 

(1). For AL < kwh, JO = Jl = 2 J+ = 25-. From (I-lo), the coincidence counting rate R, has 

oscillations at wl,~, and their sum and difference frequencies. The visibility is 100% in this case 

As is seen in fig. 1-2. As AL increases J1 and J+ rapidly decrease becoming negligible when AL is 

approaching kwh, the coherence length of the down converted beam. This can be seen in fig. 1-4, 

when AL = 115pm which is about one half of the coherence length lmh, the beating component 

and the 632.8nm and 788.7nm components of the modulations are reduced and the sum-hue~cy 

modulation becomes predominant. 

(2). l,h < AL < c - AT-in, as AL increases to be greater than lWh both J1 and J+ are zero 

and we left with 
B) 

o a -2 

C 

B) 

a .- 1.0 

0 

0- 

c 0.8 

0 

B 0.6 

Q) 

.- N 

"194 

E 

0.2 

Z 

0.0 

0.0 0.1 0.2 0.3 0.4 0.5 

BS position relative to &.. = 0.5 crn (p m) 

Figure 1-6: Normalized coincidence counting rate of Nc at AL S 0.5 cm for a AT& = 

300psec time window. The beating modulation and the wland w2 modulations have completel~ 

disappeared. A visibility of (44 f 3)% was measured. 

8s position relative to A i = 43 cm @m) 

Figure 1-7: Normalized coincidence counting rate of Ncat AL Z 43cm. The observed (752 31% 

interference visibility marks the quantum interference effect. The modulation at X = 0.351pm 

the sum frequency of the signal and idler light quanta. 
which indicates that the modulation is only at the sum frequency. The modulation visibility can 

only approach to a maximum value of 50%, this is because the contribution of the last two t 

in Jo which arise from the state amplitudes in which one photon follow the longer and the otha 

the shorter path of the interferometer. Fig. 1-6 clearly shows this modulation. 

(3). AL > c . ATcoin. In this region, the interference pattern looks the same as in case (2)- 

however, the interference visibility increases to more than 50%. This is because of the vanishg of 

the last two terms in Jo, the interference visibility is predicted to be 100% in idealized experimentd 

conditions. This interference behavior is clearly demonstrated in fig. 1-7. 

The above simple theory of the quantum mechanical model provides a good quantitative pmderstanding of what is happening in this experiment without the introduction of any artificial 

parameters. In the region of AL < Zcoh, all J'S contribute to the interference pattern, which is not 

distinguishable from a classical model. In this region the first order interference pattern appears in 

both Nl and N2 counting. The coincidence modulation may explained as the result of the product 

of Nl and N2 modulations. When AL increases, J1 and J+ approach zero due to the vanishing of 

the factor exp(-C2AT2). This effect may be considered also to be a classical wave behavior. h 

the second region, lcoh < AL < c . ATc0in, the coincidence interference behavior shown in (1-12) is 

expected. Since the wl and W* beams never meet at the same detector because of the filters, and 

each beam does not interfere with itself when A L > Zcoh, the coincidence modulation is a non-lod 

two photon interference effect. In the third region, it is by now well known that under condition 

AL.> e . ATcoin, the interference is a purely quantum effect. It is impossible to have a classid 

model to explain the coincidence counting rate modulation of more than 50%. Mathematically the 

increase of the visibility is due to the vanishing of the factor exp(-AT12 AiF,oin) in Jo. Physid~ 

this is due to the cut off by the coincidence time window of the state amplitudes in which one 

photon follows the longer path and other the shorter arm of the interferometer. This is equivalent 

to the projection of a quantum entangled EPR stbte [S, 81 

~EPR = A(t1, t2) + A(t1 - AT, t2 - AT) (I - 13) 

from the initial state. For AL > c ATcoin, the entangled two-photon EPR state (1-13) is realized 

by the measurement, which takes advantage of the particle nature of the light quanta in a wavelike 

experiment. 

II Einstein-Podolsky-Rosen-Bohm Experiment By 

Splitting A Pair of Orthogonally Polarized Photon. 

Type I parametric down conversion has drawn a great deal of attention since the first application 

[9] of it in an Einstein-Podolsky-Rosen-Bohm experiment [lo]. The experimental study of Type If 

photon pairs was performed before Type I in our laboratory. However, the experimental results 

seemed to suggest that the orthogonally polarized signal and idler photon pair do not have the 

expected quantum entanglement. This phenomenon has troubled us and many other physicists 

with whom we have communicated in the past [l 11. The entanglement of the Type I1 photon 

pair was demonstrated recently in our laboratory under two experimental conditions: (1) using 

a thin nonlinear crystal and (2) detecting coincidences in narrow spectral bandwidth [12, 131. In 
this section, we wish first to report the experimental study of this crystal length and detection 

bandwidth dependent entanglement of Type I1 down conversion. Then we report an experimental 

study of entangled two-photon EPR-Bohm states in Type I1 down conversion with linear, cirdar 

and elliptical polarizations. 

BBO 

Type II 

At- laser 

Figure 11-1: Schematic experimental set up. 

The experimental set up to study the effect of crystal length and detection bandwid& 

dent entanglement is illustrated in fig. 11-1. A single mode CW Argon ion laser line of 351.14am 

was used to pump a BBO (/3 - BaB204) nonlinear crystal. The BBO was cut for a TyBe 

phase matching condition to generate a pair of orthogonally polarized signal and idler photocollinearly and degenerately in 702.2nm wavelength. Two BBO crystals with lengths 

and 0.5mm, respectively, were used in the experiments. The 702.2nm pairs were sep 

the pumping beam by a UV grade fused silica dispersion prism, then directed collinearly at a OW 

normal incident angle to a polarization independent beam splitter which has 50% - 50% dection 

and transmission coefficients. In each transmission and reflection output port of the beamqIitter 

a Glan Thompson linear polarization analyzer followed by a narrow bandwidth interference spectral filter were placed in front of a single photon detector. The photon detectors are dry ice cooled 

avalanche photodiodes operated in Geiger mode. The output pulses of the detectors were then sent 
to a coincidence circuit with a 3nsec coincidence time window. The two detectors are separaeed 

by about 2m, so that compared to the 3nsec coincidence window, the detections are spacelike 

separated events. The coincidence counting rates were studied as functions of angles and d2, 

where 8; is the angle between the axis of the ith polarization analyzer and the direction, which 

is defined by the *ray polarization plane of the BBO crystal. Keep in mind that a right-hand& 

natural coordinate system with respe-ct to the ki vector as the positive direction is employed for 

the discussions in this paper. The following form of coincidence rate as a function of B1 and B2 

was observed in the experiments, 

R, = &(cos 281 sin 282 + sin 281 cos 2B2 - p sin el cos B2 sin 82 cos B1) (11 - 1) 

where p is a parameter which depends on the crystal length, the detection bandwidth, and the 

group velocities of the Q - e beams inside the crystal. If p = 2, eq. (11-1) reduces to, 

R, = & sin2(& - 82) (n - 2) 

which is the expected quantum correlation for the entangled two-photon EPR-Bohm state 

19) quantum mechanically indicates a two-photon polarization state which is a superposition of 

the quantum probability amplitudes: 

(1) / o - ray transmitted)@ 1 e - ray reflected) 

(11 - 4) 

(2) 1 e - ray transmitted)@ I o - ray ref leded) 

when the orthogonally polarized photon pair meets the beamsplitter. On the other Band, if p = 0 

the interference cross term does not contribute. State (11-3) can not be concluded and no sip of 

the entanglement of the pair can be seen from the measurement. 

Fig. 11-2 reports the measured values of p for BBO crystals with lengths of 5.65mm and O.5mm 

for different bandwidths of the filters. Note that for the 5.65mm BBO crystal p was always substantially less than 2 for the filters that used in the measurements. For the 0.5mnz BBO, p = 1-98 

was achieved with a lnm bandwidth spectrum filter. The solid curves are the fits to a theore%id 

model which will be presented below. The values of p were obtained from the measurements of 

coincidence rate as functions of O1 and 82. Fig. 11-3, fig. 11-4 are typical mmuremerats wBich 

reflect the different coincidence behavior for 5.65mm and 0.5mm BBO crystals. In fig. XI-3, 81 

was set to 4S0 and the coincidence rate was mapped out as a function of $2. In fig. 11-4, both 81 

and 82 were changed, keeping the sum of el and 82 equal to 90°. In both fig. 11-3 and fig. 114 the 

filters were Inm bandwidth. By fitting many similar curves, p = 0.72 f 0.07 and p = 1.98 -+ 0.04 

were determined for 5.65mm and 0.5nzm crystals, respectively. 

For Type I1 down conversion the two photon part of the state that exits the down conversion 

crystal may be calculated from the standard theory for parametric down conversion to be [6], 

where w and k represents the frequency and the wave vector for signal (I), idler (2), and pump 

(p). The frequency phase matching condition is explicitly displayed by the delta function, and the 
0.50 mm BBO 

Bandwidth (nm) 

Figure 11-2: Crystal Length and Detection Bandwidth Dependent Entanglement. The 

curve is a fitting of the theoretical mode. 

I I 1 1 1 1 1 

02 (Degree) 

Figure 11-3: Coincidence Measurements for Linear Polarization States when elwas set equal to 

45". 
wave number phase matching condition is implicit in the choice of the location of the deteckors, in 

this experiment we consider collinear down conversion. The function $(w) is determined from the 

standard theory of down conversion. It depends on the length of the crystal, and D = c/u, - c/h. 

We shall refer to D as the two-photon dispersion. The subscript indies o and e for the aation 

operators indicate the ordinary and extraordinary rays of the down conversion, traveling &ng 

the same direction as the pump, the i-direction. The coordinate axes z and y are chosen dong 

the the polarization direction of the o-ray and the e-ray, respectively. 

The fields at the detectors 1 and 2 are given by 

where aj is the destruction operator of the photons, j = o,e, i is in the direction of the ith 

linear polarization analyzer axis, i = 1,2, at and a, are the complex transmission and 

coefficients of the beamsplit ter. The function f;(w ) , i = 1,2, is the spectral transmission &cimt 

function of the filter in front of the ith dctector. 

The counting rate cabn be written in terms of the square of the effective two-photon wave 

function (I-3), which has been used in the calculation of the first experiment. It is straight 

forward to show from (11-5),(II-6), 

\k(tl, t2) = at~[Bl go B2 - &A(tl - 71, t2 - 72) + 2e Z2 - doA(ta - 72, tl - TI)] (11 - 7) 

where 

where we have assumed that the filter fi and f2 are peaked around ill and R2, respectively, where 

621 + i12 = ~p. For simplicity we take them to have the same shape so that f'(w) = fi(w + fix) = 

f2(w + 622). $ can be computed from the standard theory of optical parametric down conversionTaking the origin of the coordinates at the output side of the crystal, and letting $'(w) = $(w+fl~), 

we find 

The average coincidence counting rate is given by (1-2). In the following calculation we asme 

S(t, AT,;,) = 1 for a 3nsec time window (tl - t2 << AT,,;,). Taking the filters to be Gaussian 

it is not difficult to show that the coincidence counting rate becomes 

R, = Rd[cos 2B1 sin 282 + sin 281 cos 202 - p sin 81 cos O1 sin O2 cos 02] (n - 11) 

where 
with 

(II - 13) 

C- = I< dt[2er f c(t) - er f c(s + t) + er f c(t - s)I2 

where erfc is the error function, I< is a constant, and parameter 

s = UDL.. (I1 - 14) 

The only parameter that p depends is s, which shows the dependence on cr, the bandwidth ofthe 

filters, D, the two-photon dispersion, and L, the length of the crystal. By sketching the integrals, 

it is not difficult to show that for a very long crystal p S 0, because C+ - C E 0, and for a short 

crystal p Z 2, because C r 0. 

The functions in (11-13) are eas3.y evaluated numerically and fit the data accurately with no 

free parameters. The solid lines in fig. 11-2 are the theory curves for 5.65mm and 0.5mm BBO 

crystals. The curves agree with the measured values of p within reasonable experimental emrOne can achieve p 2 2 with bandwidth filters less than Inm for a 0.5mm BBO thin crystal. 

Using a 0.5mm crystal and a lnm bandwidth filter to achieve p = 2, measurements for twc~ 

photon polarization entangled EPR states were made. The use of a quarter wave plate asd a 

beamsplitter easily can demonstrate the quantum mechanical entanglement of arbitraq aplid 

polarization states in Type 11 down conversion, The experimental set up is the same as in fig. El, except a qumter wave plate is placed after the 0.5rnm BBO crystal. If the fast 

wave plate is oriented at angle @ with respect to the direction, the orthogon 

tion states 1 X) and I Y) are transformed to orthogonal elliptical polarization stasAfter the beamplitter a two-photon entangled state with elliptical polarizations is produced, 

I e) = I/& -i sin (n - 115) 

where state 19) is a superposition of the quantum probability amplitudes: 

(1). (cos @ I X') - i sin @ I Yf))transmitted 8 (sin @ 1 X') + icosa [, Yt))re f lected 

(2). (sin @ I X') + i cos I Yf)transmitted €3 (cos @ I X') - i sin @ I Y'))reflected 

when the orthogonal elliptical polarized photon pair meets the beamsplit ter. 

The coincidence counting rate for linear polarization analyzers is then, 

R, = &[sin 2(2@) cos 2(6; + 6;) + cos 2(2@) sin '(6; - &)] (n - 16) 

where 6: is the angle between the axis of the ith polarization analyzer and the IX,') direction. Care 

has to be taken to follow the rules of natural coordinate system, especially for the reflected 

Note that the direction of IX;) is opposite to that of [Xi). 
02 (Degree) 

Figure 11-4: Coincidence Measurements for Linear Polarization States when B1 + O2 = 90.w~ 

preserved. 

02 (Degree) 

Figure 11-5: Coincidence Measurements for Circular Polarization EPR-Bohm State. 
Wave02 (~egree) 

Figure 11-7: Coincidence Measurement for Elliptical Polarization State with Quarter Waveplate 

oriented at 71.5'. 
If @ = 0°, state (11-15) becomes state (11-3) which is a two-photon linear polarization entagled 

state. Quantum correlations given by eq. (11-2) were observed experimentally, see fig. 11-3 and 

fig. 11-4, with modulations about(98 f 2)%. 

For 9P = 45". State (11-15) becomes the circular polarization EPR-Bohm state, 

The expected quantum correlations 

were measured experimentally. Fig. 11-5 reports the measured results. The modulation is about 

(98 f 2)%. 

When the quarter wave plate was set to @ = 26.5O and 71.5", fig. 11-6 and fig. 11-7 report four 

typical measurements which were taken under the conditions: 6; f 6; = 90°. The solid lines in 

these figures are the theory curves of (11-18). Note, here, we use 8' system to define the angles for 

the analyzers. 

Contrary to the coincidence counting rate, the single detector counting rate remains consat 

for all the above measurement. Fig. 11-8 reports a typical counting rate for detector 2 in a 

measurement. 

A pair of orthogonally polarized light quanta enters a single port of a beamsplitter, if one of the 

photons, for example the transmitted one, is detected to be linearly polarized in a certain direction, 

4, the other one can be predicted with certainty to be linearly polarized in the direction 02. This 

makes the experiment EPR type argument. Addition to this argument, it is also'interesting to see 

that O2 is not necessarily perpendicular to 01, the value of 82 depends on the EPR state prepared 

by the observer. 

5000 

0 

0 60 120 180 240 300 360 

@ 2 (Deg) 

Figure 11-8: Single detector counting rate. 
This simple beam-splitting experiment is a particle-like experiment. Eq. (11-4) is b 

argument that the photon can be either transmitted or reflected by a beamsplitter. On the other 

hand this simple beam-splitting type experiment demonstrated the wave property of the photonThe 100% modulation of the coincidence counting rate is essentially an interference superposition 

of the two-photon amplitudes in (11-7). The overlap and non-overlap of the amplitudes A(il - 

q, t2 - T2) and A(t2 - 7-2, tl - q) is a good measure of the wave packet picture of the photon, which 

results the crystal length and detection b'andwidth dependent of the two-photon entanglement. 

We wish to qhank D.N. Klyshko for many useful discussions. This work was supported partially 

by the Office of Naval Research Grant No. NO0014 - 91 - J - 1430. 

References 

[I] A. Einstein, B. Podolsby and N. Rosen, Phys. Rev. 47, 777 (1935). 

[2] M.H. Rubin and Y.H. Shih, Phys. Rev. A 45, 8138 (1992). 

131 J.D. Franson, Phys. Rev. Lett. 62, 2205 (1989). 

[4] Y.H. Shih, A.V. Sergienko, and M.H. Rubin, Phys. Rev. A 47,1288 (1993). 

[5] P.G.Kwiat, A.M. Steinberg, and R.Y. Chiao, Phys. Rev. A 47, 2472 (1993). 

[6] D.N. Klyshko, Photons and Nonlinear Optics, Gorden and Breach Science Publishers, N.Y., 

(1988). 

[7] E.Schriidinger, Naturwissenschaften, 23, 807, 823, 844 (1935). A translation of these papers 

appears in J.A. Wheeler and W.H. Zurek, ed., Quantum Theory and Measurement, Princeton 

University Press, (1983). 

[8] M.A. Horne, A. Shimony, and A. Zeilinger, Phys. Rev. Lett. 62, 2209 (1989). 

[9] C.O. Alley and Y.H. Shih, Foundations of Quantum Mechanics in the Light of New Technology, ed. M.Namiki et. aL, p. 47 (1986). Y.H. Shih and C.O. Alley, Phys. Rev. Lett. 61,2921 

(1 988). 

[lo] D. Bohm, Quantum Theory, Prentice Hall, Englewood Cliffs, (1951). 

[ll] Private communications with J.A. Wheeler, E.P. Wigner, A. Shimony, J.D. Franson and. 

others. 

[12] T.E. Kiess, Y.H. Shih, X.V. Sergienko, and C.O. Alley, Phys. Rev. Lett., 71,3893 (1993). 

[13] Y .H. Shih, A.V. Sergienko, M.H. Rubin, T.E. Kiess and C.O. Alley, Submitted to Phys. Rev. 

A, (1993). 
THE GENERATION OF ENTANGLED STATES FROM 

INDEPENDENT PARTICLE SOURCES 

Morton H. Rubin and Yanhua Shih 

Physics Department 

University of Maryland Baltimore County, Baltimore, MD 21228-5398 

Abstract 

The generation of entangled states of two systems from product states is discussed for the 

case in which the paths of the two systems do not overlap. A particular method of measuring 

allows one to project out the nonlocal entangled state. An application to the production of 

four photon entangled states is outlined. 

1 Introdact ion 

The importance of non-local entangled states in the study of the Einstein-Podolsky-Rosen paradox 

(EPR) has been stressed by Horne, Shimony, and Zeilenger (GHZ) [I]. These states are entangled 

states of two or more particles in which at least one pair of particles has space-like separation. Such 

states may be produced from independent sources by allowing the particles to scatter from one 

another' and then separate, or by allowing their paths to overlap spatially such as by passing pairs 

of photons through a single beam splitter. The generation of entangled states from independent 

particle sources in which the entangled particles overlap has been discussed by Yurke and Stoler 

121, and by Tan, Walls and Collett [3]. We consider the case in which the entangled particles are 

produced from independent sources and never overlap. 

2 Independent Sources With Path Overlap 

We briefly review the work of Yurke and Stoler. Two independent sources each emit a particle 

which is passed through a beam splitter and phase shifter. The detection system consists of a pair 

of detectors, labeled 1 and 2. Each detector is composed of a beam splitter which directs each 

particle to one of two particle counters denoted by R and G. The initial input state is a product 

state, 

(!J >= (a >I Ip >2. (1) 

After passing through the system this state may be written as 
where the i9s come from the reflection off the beam-splitters. If we consider the case in which only 

the R detectors register, the only part of the state that we look at is 

which is an entangled state. By varying the phase shifters, interference between the two terms can 

be detected. This interference occurs because there are two distinct paths from the input state to 

the detection state that can not be distinguished by the detectors. From this point of view [4] the 

fact that "vacuum" enters the system through the beam splitters plays no role since the detectors 

do not see the vacuum states. 

3 Independent Sources Without Path Overlap 

We now turn to a different method of generating entangled states in which the independent 

sources are independently detected. The particle paths never cross. The entanglement is obtained 

by selective detection and the interference is caused by varying the phase shift in each path in a 

specific way. We shall see below that this allows us to consider the generation of an EPR state of 

the type envisioned by GHZ. We illustrate this technique for the simplest case, namely, two spin112 particles. The input to the system is two independent particles polarized along the positive 

z-axis, 

I!P>=I+z>1I+z>2. (4) 

Each particle is passed through a separate Stern-Gerlach apparatus which splits each state vector 

into a superposition of states polarized along the x-axis. After passing through the system we 

have 

where the phase shifter effect has been included. Multiplying (5) out gives 

Then we project this onto the detector states 

1 lDk >= I + zk >= -(I + x >I, +I - x >2), k=1,2. JZ 

The amplitude for detecting particle 1 at detector 1 and particle 2 at detector 2 is: 

which is still a product for independent choice of phases. Now if we choose O2 = + n, then the 

second pair of terms in (8) cancel and the first shows interference as el is varied, 
This is equivalent to producing the entangled state given by the first two terms of (6). 

In this case the two independent particles never meet. Were the interference is between the 

pair of particles that went through the phase shifter and the pair that did not. From the point of 

view of indistiguishable paths, it is the paths of these two-particle states that are indistiguishable. 

Another way to look at this particular example is to note that the choice of the angles makes the 

first two term in (6) total spin 1 states and the second two form a total spin 0 state, the detector 

only detects the spin 1 part of the state. 

This argument can be generalized to more than two particles and can be used to form the 

basis of many-particle interference experiments to test the theorem of Greenberger, Worne and 

Zeilenger [5]. The use of projection operators to pick off parts of state vectors, which is redly 

what measurements do, is nothing new. It is fundamental to quantum theory. What is different 

is the means of picking off nonlocal entangled states from independent particles. 

4 The Theory of the Four-Photon Experiment 

We now apply this to a possible four- hoto on interference experiment. We wish to show how 

' one can produce the highly correlated state necessary to carry out the experiment envisioned by 

Greenberger, Horne and Zeilenger. In fig. 1 we show the experiment envisioned. It is a double 

Franson interferometer [6]. The coincidence counting rate is proportional to 

where the integrals are over the duration of the measurement. The function S expresses the fact 

that the coincidence time window is one when t2 - tl , t3 - tl and t4 - tl are all less than some 

small coincidence time t,,;, and rapidly goes to zero when this is not true. 

where j=I and I1 . The middle two terms can be discriminated against for L - S = A L if 

A L/c >> t,,;,. Then we can confine ourselves to considering 

The amplitude in (12) is still a product, 

1 

@z(tl, t~)@zz(t3, t4) = -&{Qz(Lz, Lz)Qzz(Lzz, LIZ) + Qz(Sz, Sz)@~z(Szz, Szz)) + 

{Qz(Lz, Lz)*zz(~zI, SII) + *z(~I, SZ)QZZ(LIZ, LII)]). (13) 

This is in a form similar to that of (5). We now want. to show how the last term in curly brackets 

can be made to vanish. 

We have shown elsewhere [7] that the two-photon wave function is of the form 
The function u(t) describes the correlation of the photon pair in space and time. Its width is 

determined by the single photon bandwidth. This is usually fixed by filters in the output beams 

of the down-conversion crystal. The function v(t) is of the form 

It expresses the fact that the pair is created somewhere within the overlap of the crystal and the 

pump beam. In general, if the pump beam is nearly monochromatic, A will be a slowly varying 

function of time. 

Using these facts, the second term in curly brackets of (13) can be written as 

The key point is that we can choose the path lengths so that 

making the relative phase between the two terms in the bracket of (16) negative. Since A(t) is 

constant over the measurement time this term will vanish. The first term in curly brackets has 

two terms with a relative phase of Rp(ALz + ALzz)/c which, under the condition stated, becomes 

n + Stp2Uz/c. Therefore varying the path length while keeping their difference fixed allows one 

to see interference with oscillations at half the pump wavelength. 

The above experiment is for space-time variables. There is a more convenient way to proceed 

using the polarization of the photons. 

5 Acknowledgments 

After this paper was completed a related paper on the generation of entangled states from independent sources [8] was called to our attention. We thank Professor Zeilinger for giving us a 

preprint of this paper. 

This work was supported by Office of Naval Research Grant No. N00014-91-J-1430. 

References 

[I] M. A. Horne, A. Shimony, and A. Zeilenger, Phys. Rev. Lett. 62, 2209 (1989). 

M. A. Horne, A. Shimony, and A. Zeilenger,Quantum Coherence, edited by J. S. 

Anandan, (World Scientific Co. Pte. Ltd, Singapore,l990). 

[2] B. Yurke and D. Stoler, Phys. Rev. Lett. 68, 1251 (1992) and Phys. Rev. A 46,2229 

(1992). 

[3] S. M. Tan, D. F. Walls and M. J. Collett, Phys. Rev. Lett. 66, 252 (1991). 

[4] This point of view is the natural one from the point of view of Feynman's approach 

to quantum mechanics, R. P. Feynman, R. B. Leighton, and M Sands, The Feynman 

Lectures on Physics, Quantum Mechanics, (Addison-Wesley, Reading,MA, 1965). 
[5] D. M. Greenberger, M. Norne and A. Zeilenger, in Bells Theorem, &uantum Theory 

and Conceptions of the Universe, edited by M. Kaftos (Kluwer Academis, Dordrect, 

1989). 

[6] J. D. Franson, Phys. Rev. Lett.62, 2205 (1989). 

[7] Y. H. Shih and Morton H. Rubin, to be published 

[8] M. Zukowski, A. Zeilinger, M. A. Horne and A. K. Ekert, private communication. 
coincidence 

counter 

Figure 1: A pump beam enters two crystals which each produce a pair of photons by optical 

parametric downconversion. Each photon passes through an interferometer and is detected by 

one of the four detectors (Dk). A coincidence counter (C) gives a count only when all four 

photons arrive within a fixed time window. 
NONLOCAL EFFECTS ON THE POLARIZATION STATE 

OF A PHOTON, INDUCED BY DISTANT ABSORBERS 

L.C.B. Ryff 

Unaverso'dade Federal do Rio de Janeiro, Instituto de Fisaca 

Cazxa Postal 68528, 21945-970 Rzo de Janezro, RJ, Brazzl 

Abstract 

A variant of a Franson's two-photon correlation experiment is discussed, in which the 

linear polarization state of one of the photons depends on the path followed in the interferometer. It is shown that although the path difference is greater than the coherence length, 

the photon can be found in a polarization state represented by the superposition of the 

polarization states associated to the paths when there is coincident detection. Since the 

photons, produced via parametric down-conversion, are fairly well localized in space and 

time, the situation in which one of the photons is detected before the other can reach the interferometer raises an intriguing point: it seems that in some cases the second photon would 

have to be described by two wave packets simultaneously. Unlike previous experiments, in 

which nonlocal effects were induced by means of polarizers or phase shifters, in the proposed 

experiment nonlocal effects can be induced by means of variable absorbers. 

Ever since Bell's theorem [I] and the important paper by Clauser, Horne, Shimony, and Holt 

[2], different experiments have been performed related to quantum mechanical nonlocality. In 

these experiments the photons of a correlated pair either are made to impinge on polarizers [3] 

or are made to pass through phase shifters [4,5]. Here I would like to discuss an experiment 

(represented in Fig. 1) in which nonlocal effects can be induced by means of variable absorbers (or 

variable beam-splitters). It is a modified Franson's experiment f5]. in which a half-wave plate 

(X/2), a two-channel polarizer, and two variable absorbers (As and AL) have been included. 

Photons yl ancl yz, produced via parametric down-conversion, are in the same polarization state 

[6], which I will assume as being parallel to x. The orientation of the half-wave plate is chosen 

so that after passing it y2 is in a different polarization state, perpendicular to the state it was 

in. As has been shown [7], when there is coincident detection, the packets following the long 

(L2) and the short (S2) paths interfere. In the present proposal the relative amplitude of these 

packets is varied at a distance by means of As and -AL. For our purposes, we only need.to 

consider a field with one polarization component. When the beam-splitters HI, H2. Hi and Hi; 

the absorber As; and the half-wave plate are removed, yl and 72 can only follow the short paths. 

and the coincidence ra.te between the detectors at sites 1 and 2 is given by [8] 

where It) is the state of the field at time t; .@z)(t) is the annihilation part of the electric field 

operator at the detector at site 2; and so on, and k is a proportionality constant. If, instead 
of removing the beam-splitters, they are replaced by mirrors and -4~ is removed, yl and 72 can 

only follow the long paths, and the coincidence rate between the detectors at sites 1 and 2 is 

given by -(-I -(+I -(+I Ro = k(tlE2, (t) B!-,'(t) El, (t) E,, (t)lt) . -(+I 2p)(t) and Es (t) are related by the expression 

-(+I . -(+) EjL (t) = e'+~ Ejs (t - AT) , j = 1,2, (3) 

where AT = (L - S)/c, with L = L2 = L1 (S = S2 = Sl) representing the length of the long 

(short) path, and 4j is a phase shift. Since L - S is much greater than the coherence lengths of 

the wave packets associated with yl and 72, 

and 

Since AwAT << 1, where Aw is the uncertainty in the sum of the frequencies of yl and 72, it 

can be shown that [5] 

where wl + w2 = wlo + wzo f Aw, and wlo(w20) is the central frequency of yl(y2). 

FIG. 1. Franson's experiment in which a half-wave plate, a polarizer, and two 

absorbers have been included. 

To represent the action of the absorbers, I will introduce a parameter 8, such that: 

@i:'(t) % -(+I sin 6 Els (t) (7) 

and 

-(+I -(+I EIL (t) % COS~ EIL (t) . (8) 

Therefore, the transmissivities of the absorbers are varied in a correlated way. The field operator 

at site 1 in the experiment represented in Fig. 1. will then be given by 

*(+) 1 -(+I Ei (t) = 5 - [sine El, (t)+cosO ~il)(t)] . (9) 
If b is the orientation of the polarizer and L(b,x) = y, then when the half-wave plate is in 

place, the field operators at sites 2" a.nd 2'" are given by 

-(+I a -(+I -(+I E2,, (t) = - [sinv E2$ (t) + cosy EZL (t)] 2 (10) 

and 

P -(+I --(+I g!$!(t) = - [cosv EZs (t) + sin9 EZL (t)] , 2 

where a and ,G' are phase factors. 

It is then easy to show, using (I), (3), (4), (5), (6), (9), and (10) and choosing + +2 = 

-(WID + wzo) AT + 2nn (n is an integer), that the coincidence rate between the detectors at 

sites 1 and 2" is given by 

-(-) -(-I --(+I --(+I Ro RglI1 = k(t lEzl1 (t) El (t) El (t) E2,, (t) It) = - cos2(8 - cp) . 16 (12) 

Similarly, 

--(-I -(+I -(+I Ro R;,,, = k(tlg$,)(t) El (t) El (t) E2,,, (t)lt) = - sin2(@ - 9) . 16 (13 

We see from (12) and (13) that, for coincident detection, whenever we have detection at site 

1, if 72 follows the direction to 2, it impinges on the polarizer in a polarization state parallel to 

c, such that L(c, x) = 8. (12) and (13) then follow from Malus' law. We can also easily verify 

that whenever we have detection at site l', if y2 follows the direction to 2', it also impinges on 

the detector in a state parallel to c. By a similar procedure we also easily verify that whenever 

we have detection at site 1(l1), if 72 follows the direction to 2'(2), it impinges on the detector 

(polarizer) in a polarization state parallel to d, such that L(d, x) = -8. Since yl and 72 are 

fairly well localized in space and time [9], these results are totally counterintuitive. 

The nonlocal aspects of the experiment I am discussing can be made more evident by comparing the following two situations. In the first, the beam-splitter HI is removed. To simplify 

the argument, we can consider the ideal situation in which all photons are detected. We then 

easily see that 

12 2 Rilll = - sin 6' sin 9 8 (14) 

and 

1 

~,,,, = - cos2 6' cos2 y , 8 

which correspond, respectively. to the possibilities "yl and 72 following the short paths" and LL 71 and 72 following the long paths". In the second situation Hi is in place. Then, whenever ;l 

is detected, we know that, if Hi were not in place, yl would have been detected either at site 1 

or at site 1'. Thus, 72 must either follow the long or the short path. as when Hi is not in place. 

according to the locality assumption, since, it is irrelevant whether Hi is in place or not. As a 

consequence, one must have 

in strong disagreement with (12). I11 particular, when p = 7r/4. (16) leads to R;,,, = 1/32. 

whilst (12) leads to %trl = (1/16)cos2(4 - 7r/4). where 0 5 6' 5 n/3. 
As I have emphazised elsewhere [7], the situation in which yl is detected before 72 can reach 

the interferometer raises an intriguing point: it seems that y2 would have to be described by 

two wave packets simultai~eously. Iihen y is detected before 72 can reach the interferon~etes. 

the possibilities corresponding to "both photons following the long paths" and to "both photons 

following the short paths" remain indistinguishable, and results (12) and (13) are still obtained. 

(If this were not so, special relativity might be in trouble, since the detections of and -12 

are events separated by a space-like interval. Therefore, the order in which they occur depends 

on the Lorentzian frame in which the experiment is being described. On the other hand, the 

detection rates must be Lorentz invariant quantities). The packets associated with 72 correspond 

to the following two possibilities: (Ll)yl follows path L1; and (S1) yl follows path S1. These 

packets are split at H2, producing four packets. The packet following L2 in possibility (L1) 

- packet (L2, L1) - interferes with the packet following path S2 in possibility (S1 ) - packet 

(Sz, Sl) - producing a packet (I) in a polarization state different from those that would have 

occurred, had 72 followed either path L2 or path Sz. It is in this pacliet (I) that 72 is to be found 

when there is coincident detection. In the experiment that I an1 discussing, the anlplitudes of 

the packets (L2, L1) and (S2, S1) depend on the parameter 8. 111 the previous esperiments, one 

acted either on the polarization [3] or on the phase [4,5] of the correlated photons to induce 

nonlocal effects. In the present proposal. the action is on the amplitudes. If we were to act also 

on the phases, polarization states different from those discussed here could be produced. 

This experiment could be performed using the recently improved time resolution techniques 

[lo]. 

References 

[I] J.S. Bell, Physics (Long Island City, NY) 1, 195 (1964); J.F. Clauser and -4. Shimony. Rep. 

Prog. Phys. 41, 1881 (1978). 

[2] J.F. Clauser, M.A. Horne, A. Shimony, and R.A. Holt, Phys. Rev. Lett. 23. 880 (1969). 

[3] S.J. Freedman and J.F. Clauser. Phys. Rev. Lett. 28, 938 (1972): A. Aspect. P. Grangies. 

and G. Roger, ibid. 49, 91 (1982); IS'. Perrie, ,4.J. Dunca.n, H.J. Beyer? and H. I<leinpoppen. 

ibid. 54, 1790 (1985). 

[4] M.A. Horne, A. Shimony, and -4. Zeilinger, Phys. R.ev. Lett,. 62, 2209 (1989); J.G. Rarity 

and P.R. Tapster, ibid. 64, 2495 (1990). 

[5] J.D. Franson, Phys. Rev. Lett. 62, 2205 (1989); Z.Y. Ou. S.Y. Zou, L.J. Wang, and L. 

Mandel, ibid. 65, 321 (1990); P.G. Icwiat, W.A. Varekq C.I<. Hong, H. Sathel. and R.Y. 

Chiao, Phys. Rev. A 41, 2910 (1990). 

[6] Z.Y. Ou and L. Mandel, Phys. Rev. Lett. 61, 50 (1988). 

[7] L.C.B. Ryff, Phys. Rev. A 48, 1020 (1993). 

[8] R.J. Glauber, in Quantum Optics and Electronics, edited by C. De IVitt. -4. Bladin. alld 

C. Cohen-Tannoudji (Gordon and Breach, New York, 1965). 11.62. 
[9] D.C. Burnham and D.L. Weinberg, Phys. Rev. Lett. 25, 84 (1970): S. Friberg, C.Ii;. Hong. 

and L. Mandel, ibid. 54, 2011 (1985); R. Ghosh, C.I<. Hong, Z.Y. Ou, and L. biandel. 

Phys. Rev. A 34, 3962 (1986); Z.Y. Ou and L. Mandel, Phys. Rev. Lett. 61, 54 (1988). 

[lo] J. Brendel, E. Mohler, and 147. kiartienssen, Europhys. Lett. 20, 575 (1992); P.G. Iiwiat, 

A.M. Steinberg, and R.Y. Chiao, Phys. Rev. A 47, 2472 (1993). 
THE STRONG BELL INEQUALITIES 

A PROPOSED EXPERMENTAL TEST 

Edward S. Fry 

Physics Department, Texas A&M University 

College Station, TX 77843 

All previous experimental tests of Bell inequalities have required additional assumptions. 

The strong Bell inequalities (i.e. those requiring no additional assumptions) have never been 

tested. An experiment has been designed that can, for the first time, provide a definitive test 

of the strong Bell inequalities. Not ady will the detector efficiency loophole be closed, but, 

the locality condition will also be rigorously enforced. The experiment involves producing 

two 19%g atoms by a resonant Raman dissociation of a mercury dimer ('%Ig2) that is in an 

electronic and nuclear spin singlet state. Bell inequalities can be tested by measuring anelm 

momentum correlations between the spin one-half nuclei of the two lg9Hg atoms. The 

method used to make these latter measurements will be described 

1 Introduction 

Due to low detector efficiencies, previous experimental tests of Bell inequalities have required an 

auxiliary assumption in order to derive an experimentally testable A new experimental 

concept has ken developed that makes it possible to test the strong Bell inequalities (Bell inequalities 

without any auxiliary assumptions). It also leads directly to an extension that rigorously enforces 

locality. 

This approach is an ex erimental realization of Bohm's version of the Einstein-Podolsky-Rosen 

Gedankenexperiment. m7 6P Instead of photon pairs, it involves measurements of the correlations 

between angular momentum components of two atoms (actually spin one-half nuclei) of the isotope 

'"H~. A brief description of the roposed experiment was recently published,r71 and a very detailed 

discussion has also been provided. 81 

In general, the measurement of components of spin suggests the use of Stern-Gerlach magnets. For 

spin one-half particles, there are two possible projections, + ("spin-up") and - ("spin-down"), of the 

spin on the axes of the magnets. As indicated in Figure 1, the magnetic field gradient deflects each spin 

one-half particle into one of two detectors depending on the projection of its spin on the magnet axis. To 

measure a spin component of a particle in any specified direction, one simply rotates the axis of the 

corresponding magnet to that direction. However, such an approach is unsuitable in the present work 
for two reasons. Arst, to enforce Einstein locality, it would be necessary to rotate a magnet to an 

direction in a time short compared to the time it takes light to travel from one magnet to the other 

- a few nanoseconds. This appears to be physically impossible. Second, based on available magnetic 

fields and the magnitude of the nuclear magnetic moment, the deflection produced by 19%g nuclear 

spins is so small as to preclude this approach. 

Stern-Gerlach Stern-Gerlach 

FIG. 1. Schematic of a Stem-Gerlach apparatus for measuring spin correlations. The 

diagram assumes spin one-half particles and the magnets are oriented for measurements of 

the spin components of both particles in the Z direction. 

Other approaches to the measurement of spin components are possible. lFor example, a Bell 

inequality involving roton spin correlations has been tested using proton-proton scattering to determine 

the spin components!91 However, this type of approach requires several auxiliary assumpfons. 

Efficient detection of atoms is combined with spin analysis in the approach to be described here. 

This approach takes advantage of the hyperfine structure interaction in 199~g atoms in order to select 

nuclear spin components via electric dipole transitions. Detection efficiency is sufficiently high to cbse 

that loophole. And finally, this approach permits a very rapid change in the direction of the masmment 

of spin components; hence Einstein locality can be enforced. 

2 Overview of the Proposed Experiment 

The 19%g atoms are produced by a stimulated Rarnan excitation to a dissociating state ofthe X'Z~ 

ground state of a 19?Hg dimer. The total electron and the total nuclear spin angular momenta are both 

zero in the initial rotational state of the Hg dimers, and are not changed in the dissociation process. The 

two mercury atoms resulting from the dissociation are both in ground states. Since the electronic 
angular momentum is J=O and the nuclear spin of 199~g is I=1R, each ground state atom therefon has a 

torisl mplw momenmm F=1/2. The component of total angular momentum (which is therefore the 

component of nuclear spin) in any given direction is measured by orienting excitation laser beams at 

253.7 nm in that &=tion and using polarization selective excitation of one of the &man sublevels. 

lFoP left chulady polarized excitation, only atoms whose component of angular momentum, Mp=+l/%, 

in the direction of laser beam propagation are excited. Similarly, right circularly polarized light only 

excites atoms with Mp=- 112. Excited atoms are photoionized via an auto-ionizing state using a laser at 

197.3 nm. The resulting photoelectrons and ions are detected using Channeltron electron multipliers. 

Assuming a mercury atom has been detected if EITHER an electron OR an ion is observed, then the 

overall detection efficiency for the Hg atoms is greater than 96%. Figure 2 shows a schematic of the 

expfiment h&cabg the relative directions of the dimer, atom, and laser beams. 

Dimer Excitation 

Laser Bearm: 266 

FIG. 2. Schematic of the experiment showing the direction of the mercury dimer beam 

together with a pair of the dissociated atoms and their respective detection planes. Relative 

directions of the various laser beams are also shown. 

Rates for simultaneous detection (coincidence rates) of the two atoms are measured for components 

of their angular momenta in the directions and e2, respectively (i. e. in the directions of the 

excitation laser beams). A set of four angles can be chosen that give a maximum violation of the strong 

Bell inequality.[7* This experiment also lays the foundation for an experiment that enforces locality 
since a combination of a Pockels cell and a beamsplitting polarizer can be used to stochastically change 

the ktion of the exciting laser beams on the nanosecond time scale. 

3 Measurement of the Nuclear Spin Components 

Figure 3 shows the relevant energy levels of the mercury atom,[101 and the comsponding transitions 

at 253.7 nm for spin analysis and at 197.3 nm for detection via photoionization. The f~st transition is 

fiom the (6s2)6's0 (F = x) ground state (level 1) to the (6~6~)6~~~ (I? = x) state (level 2); the 

second transition is from level 2 to the (6p2) 3~o autoionizing state (level 3). Atom detection is via both 

the resulting ion and the photoelectron. 

Assuming a Fano profile for the autoionizing transition, a calculated value for its oscillator strength, 

and a measured value for its width, a theoretical value of -2.3~10-l4 cm2 is obtained for the peak crosssection.[79 *] This is an unusually large cross-section, but is consistent with measured cross-sections for 

analogous transitions in ~d.["] Since both transitions are so strong, laser pulse energies of only 60 pJ 

and 250 pJ, respectively, are sufficient to ionize all the atoms in a few nanoseconds. 

FIG. 3. Relevant energy levels of the mercury atom,[lol and the corresponding transitions 

for detection. 

As shown in Figure 2, the laser beams for spin analysis have a wavelength of 253.7 nm and lie in 

planes perpendicular to the direction to the source. They are at angles el, 82 to the +Z-axis. The angles 

el, €I2 of these laser beams define the directions in which each atom's angular momentum components 

is observed. 

If we choose some arbitrary direction as the quantum axis for our system, then the quantum numbers 

for the two components of angular momentum in that directiop are M~&1/2 (for an F=1/2 ground state 
199~g atom). If the 253.7 nm laser beam is propagating in the direction of the quantum axis and has 

right circular polarization (o+), then MF must increase by one in the transition. Thus, only ground state 

atoms for which the projection of the angular momentum in the direction of laser beam propagation is 

MplR can be excited to the 63~i (F = j/Z) state and ionized; see Figure 4. Similarly, for left 

circular polarization, only atoms with M~+1/2 are excited and ionized. In summary, by choosing an 

arbitrary direction and using a circularly polarized excitation laser, a Hg atom with either component of 

angular momentum in that direction can be selectively detected. 

- - -- 

FIG. 4. Analysis scheme with right circularly polarized light. Since M ~~st increase by 1 8 in the transition, only ground state atoms with MF-1R can reach the 6 Pl (F = x) state. 

4 Summary 

The use of electric dipole transitions to measure the component of nuclear spin of 199~g inan 

arbitrary direction has been described. Its application to a definitive experimental test of the previously 

untested strong Bell inequalities is discussed. This novel approach makes it feasible to rigorously 

enforce Einstein locality. 

5 Acknowledgement 

This research was supported by the Robert A. Welch Foundation grant No. A-1218 and by the 

National Science Foundation grant PHY-9221038. The author also thanks Dr. Shifang Li for many 

helpful discussions and suggestions. 
6 References 

[I] C. 0. Alley and Y. H. Shih, in Proc. 2nd Int. Symp. Foundations of Quantum Mechanics, 47 

(Tokyo, 1987). 

[2] A. Aspect, J. Dalibard, and 6. Roger, Phys. Rev. Lett. 49, 1804 (1982). 

[3] J. F. Clauser and A. Shimony, Rep. Prog. Phys. 41, 1881 (1978). 

[4] Z. Y. Ou and L. Mandel, Phys. Rev. Lett. 61,50 (1988). 

[5] J. G. Rarity and P. R. Tapster, Phys. Rev. Lett. 64, 2495 (1990). 

[a D. Bohm, Quantum Theory, (Constable, London, England, 1954). 

[7] E. S. Fry, in Proceedings of the International Conference on Lasers '92, 621 (Houston, TX, 

1993). 

[8] E. S. Fry, submitted to Phys. Rev. A (1994). 

[9] M. Lamehi-Rachti and W. Mittig, Phys. Rev. D 14,2543 (1976). 

[lo] C. E. Moore, Atomic Energy Levels, Volume 111, Circular 467 (National Bureau of Standards, 

Washington, DC, 1958). 

[ll] K. Miyazaki, T. Watanabe, and K. Fukuda, J. Phys. Soc. Japan 40,233 (1976). 
ON THE NONLOCAL PREDlCTlONS OF QUANTUM OPTICS 

Trevor. W.Marshall 

Depa~ment of Mathematics, University of Manchester, Manchester, U.K. 

Emilio Santos and Antonio Vidiella-Barranco 

Departamento de Fisica Moderna, Universidad de Cantabria, Santander, Spain 

We give a definition of locality in uantum optics based upon Bell's work, 

and show that locality has been vio 7 ated in no experiment performed u 

now. We argue that the interpretation of the Wigner function as a proba density gives a very attractive local realistic picture of uantum optics provided that this function is nonnegative. We conjecture t % at this is the 

case for all states which can be realized in the laboratory. In particular, we 

believe that the usual representation of "single photon states1# by a Fock 

state of the Hilbert space is not correct and that a more physical, although less simple mathematically, repi'esentation involves density matrices. We 

study in some detail the experiment showing anticorrelation after a beam 

splitter and prove that it naturally involves a positive Wigner function. Our 

(quantum) predictions for this experiment disagree with the ones reported in the literature. 

1. What is Ilocallty 3 

The purpose of this paper is to investi ate the conditions for the violation of 

locality in quantum optics. We shall s 71 ow that these conditions are rather 

stringent and have not been fulfilled in those experiments where violations 

of locality have been claimed. 

The first problem is that several, quite different, meanings have been given to the word "localit I' (or "nonlocality"). In fact, there are people claiming that quantum mec K anlcs never redicts non locality because it forbids 

sending signals at superluminal ve P oeity. On the other hand, some authors 

include auxiliary hypotheses, related to the Clauser et al. [I] 

"no-enhancement" assumption, as a part of the concept of locality. With such 

a definition there are a lot of locality violations in the predictions of 

quantum o tics. Here we shall use somethin intermediate between these 

work. 

extremes. e b shall define locality in the fol 9 owing form based on Bell's 

We should consider an EPW (Einstein-Podolsky-Rosen) experiment where 

some correlation is measured between properties, like spin, of two 

separated particles. Locality is satisfied ~f single probabilities ( and coincidence probabilities (p 4 ) can be obtained from a locaP%i$$eet)l variables (LHV) model, i. e., i? there are hidden variables, collectively represented by h, which determine the above probabilities by means of 
integrals of the form 

the functions PCJ, FP4 and W fulfilling the conditions : 

Normalization: Iw(~) dh=l (3) 

Positivity: W(h) 2 8, P3(h9e1 )2 0, P4(h,82) 2 0 (4 

Boundedness: P3(h981) I "$ P~(h,82) I 1. (5) 

2. How to test lacality 3 

A test of localit involves performing an experiment where quantum optics predicts the vio tion of some genuine Bell inequality. Genuine means that 

the inequality can be derived from the conditions (1) to (5) alone, without 

adding auxiliary assumptions like "no-enhancement" : 

P3(h,01) I PA (1) , P4(h,82) I P2(h) not assumed. (6) 

Here PA (h) and P2(h) mean detection probabilities when no selector (e.g. polarimer) is inserted between the source and the detector. 

Then, we stress that it is impossible to test locali in experiments measuring coincidences alone. In fact, it has been poss le to construct a 

general LHV model iving the same coincidence probabilities as quantum optics for every EP 1 -type experiment in which only coincidences are 

measured [3]. Genuine Bell inequalities, therefore, should necessarily involve both singles and coincidences. In this respect, we do not agree with the usual 

statement that there are loopholes in the experiments to disprove LHV 

theories, because the word "loophole" sug ests that the experiments have 

only practical difficulties. The fact is that t ese experiments have not been 

designed to test genuine Bell inequaltities, but inequalities involving addit~onal assumptions, like (6). Therefore they can only refute restricted 

families of LHV models, namely those fulfilling those assumptions. 

We also point out that a necessary condition [4] for the violation of locality is the existence of an entangled quantum state, i.e., a nonfactorable 

wavefunction. There are some experiments, where violations of locality have 

been claimed, which do not even fulfil this condition (e.g. the state vector 

(2) of Ref. 5 is factorable). 

3. Single photon interferometry 

The simplest "entangled state" in quantum o tics appears in experiments of 

interference of a single photon (see Fig. I) [ 8 1. In an experiment of this class, 

a photon v2 is sent to a beam splitter represented by a dashed line in Fig.1). The state of the radiation field after t e I, beam splitter is 
which exhibits entangiement between a "single photon statew and the 

"vacuum state". In the anticorrelation ex eriment, represented in Fi 

coincidences are predicted between the 8 etecton PMr and PMt, whi 

to prove that the photon goes undivided into one channel. 

FIG. 1. Triggered experiment [GI. Atoms in the source S produce pairs of 

photons. The detection of the first hoton of the cascade produces a gate during which the photomultipliers, P L t and PM,, are active. The probabil~ties of the detection during the gate are pt P: Mt/N and pr = N,/Ni for singles and 

pc = NCINl for coincidences, Nj being the de 1 ection rates. 

In the recombination experiment the two beams, produced at the beam 

splitter, are recombined at another beam splitter where they arrive with a 

drfferent phase ( which may be chan ed by means of the phase shifter) and 

the detection probability at one of t 71 e outgoins channels depends on that 

phase difference. The standard way to "explain' this phenomenon is to say that there are two possible routes for the photon and that, according to 

quantum theory, "possibilities intepfere". However, this is not a scientific 

explanation at most, it may be considered as a poetic sentence or a 

practical rule \ . Later on we shall see how the phenomenon may be really explained uslng the Wigner representation of quantum optics. 

4. The Wigner representation as a Bscal hidden variables model 

In order to understand the meaning of entanglement in quantum optics, we 

calculate the Wigner function of (7) and we obtain 

W = N ( 2 1 at + a, l2 -1 ) exp( - 2 1 at l2 - 2 1 a, l2 ), N = normalization (8) 

This function does not factorize, but this presents no problem; we may 

interpret W as a 'oint probability distribution for the ampl~tudes at and a,. We see that, in t b e W~gner representation, entanglement is just correlation, 

an obviously classical concept. Of course, there is another very well known 

roblem, namely that (8) is not positive definite, as a probability should be. h e shall return to this difficulty in detail later on, but for the moment let 

us ignore it proceed as if W were a genuine (non-negative definite) 
probability. 

In the Wigner representation fields propagate like in classical optics. In 

fact, the equations involving creation or annihilation operators become, in 

the Wigner representation, similar equations between field amplitudes: 

The picture that emerges is that of light as pure (Maxwellian) waves, but 

with a real zeropoint (background) radiation. Then, in the single photon interference experiment there is always "somethingf@in both channels, 

because both at z 8 and ar z 0. 

Anticorrelation can be understood as a result of the interference between 

signal and zeropoint at the beam splitter. In fact, as shown by Eq. (9), the 

amplitudes of the fields in the outgoing channels contain a part coming from the signal (channel 1 and a part coming from the zero oint (channel 8, 

arrow from below in iig. I). The superposition of amplitu 8 es at channels r 

and t should produce interference, which will depend on the relative phases of the incoming channels. However, in any case the interference will be 

constructive in one channel and destructive in the other one, b conservation 

of energy. Now, quantum optics predicts that detectors are on Y y sensitive to 

the intensit above the zero oint level, which explains why there is 

detection on Y y in one channel. E etection sensitive only to the intensity above 

the zeropoint level follows from the the normal ordering prescription of 

quantum optics. In the Wigner representation normal ordering becomes a 

subtraction of the zeropoint, as shown by the equality 

In this way we have a transparent picture of the anticorrelation, without any need of "photons". The explanation of the interference in the recombination 

experiment is rather easy, because we are dealing with a 

of radiation. Interference is produced between (corre zeropoint at the second beam splitter. What we want to 

the only problem of the Wigner function is the lack 

entanglement. If the Wigner function of an entangled is not because it contains correlation, but because it is not positive definite. 

5. The meaning of enhancement and other nonclassical effects 

Now it is eas to understand wh "no-enhancement" (see Eq. (6) ) is violated. 

In fact from E q. (7) it follows t at K 

and this intensity may be greater than the intensity Il of the incoming signal, if the relative phase of El and Eo is zero. A similar phenomenon happens at a polarizer. It is enough to assume that the detection probability increases monotonically with the ~ncoming intensity to explain the origin of 
The existence of a real zeropoint electromagnetic radiation is, therefore, 

crucial for the explanation of enhancement. On the other hand, all LHV 

theories in which "no-enhancement" holds true have been refuted b the 

performed experimental tests of Bell's inequalities [7-93. Consequent Y y we 

may conlcude that LWV theories not involving zeropoint field are the ones 

actually refuted by these experiements. If one is fond of LHV theories, one 

should therefore look for theories involving a real zeropoint. It is very 

interpretation above suggested, belongs to this class. 

that the Wigner represention of quantum optics, with the probabi Bood istic 

i In the wave interpretation of quantum optics that we are suggesting (taking 

a positive Wigner function as a probability distribution the interpretation of the nonclassical states of light is also transparent. d ccording the usual 

definition, "nonclassical states of light" are those not havin a positive Glauber-Sudarshan (P) representation. That is, for nonclassica B states, the 

P-representation either does not exist or it is not non-negative definite. In 

contrast, any classical state of light has a positive P-representation, P({a.}), which may be interpreted as a probability distribution of the 

amplifudes of the normal modes of the radiation. If we assume that there is 

a zeropoint radiation having a probability distribution Wo({P.)), in addition 

to the classical radiation, the question arises: What is #he probability distribution of the full radiation present?. The answer is obvious, in every normal mode the total am litude, y-, will be the sum of both amplitudes, i.e. 

yj = aj + Pj. Then the progability distribution of the full radiation will be 

This is just the Wigner function, which is known to be related to the 

P-function by Eq.(12). Therefore, classical states of light are those where 

the zeropoint is not modified; some additional radiation is added on top of 

the zeropoint. Consequently, "nonclassical states" are those where the 

zeropoint is modified. 

6. Solution of the positivity problem 

A very simple solution of the positivity problem, the problem that the 

Wigner function is not positive definite for all quantum states of light, is to 

sssume that only states with a positive Wigner function may be 

manufactured in the laboratory. That is, we assume that quantum states with a negative Wigner function are just mathematical constructions useful 

as intermediate steps in some calculat~ons. Two main objections may be put to this assumption, namely that some of tqese forbidden states have been 

actually produced, e.g. single photon states, and that there are other 

representations in quantum optics, e. . the Q -or positive P- representation, which are always positive and there r ore better candidates than the Wigner function. We shall devote the remainder of the aper to answering the first 

objection. The answer to the second is that the d' igner function has a number 

of properties that make it the only good candidate. To quote just one, it is 

the only phase-space distribution which evolves according to a classical 

Liouville equation for any Hamiltonian quadratic in the creation and 
annihilation o s. The Biouville 

Hamiltonian o from the uantum one 

symmetrical o ng 9, t e standard 

replacing the operat aj' ) by the classical 

the amplitude fi aj* e the canonical 

coordinate aj. 

Now for the first objection. In the first place we point out that we do not 

ose to inter ret the Wigner function throughout quantum mechanics as a 

ability distri ution in phase space. For instance, we do no% apply it to 

lectrons in an atom. We make the proposal just for the electromagnetic field. In contrast with what happens tn particle uantum mechanics, in 

quantum optics most of the states of the radiation R ave a positive Wigner function. For instance, this is the case for the vacuum, the coherent states, 

the chaotic state (thermal light and even the squeezed states. Amongst the usual states of light, practlca I ly only Fock states (number states) have a 

non-positive Wigner function. Then the uestion arises: can pure Fock states really be produced in the laboratory?. W at 9, we conjecture is that Fock states 

are never produced as pure states, always as mixtures having a positive Wigner function. For instance, if we have a beam of "single-photon signalsn such that within a time window w the probability of a signal is p << 1, then 

the state corresponding to the window is not the single photon state I I>, but 

the mixture represented by the density matrix p = (1 - p) I 0> e 0 + p I 1> < 1 1. 

The associated Wigner function is positive provided p e 112. 1 I p > 112 the 

probability of having more than one photon within the window becomes 

relevant and again the Wigner function is positive. It may be ar ued that, 

with some effort, it is possible to monitor the single photon signas P in such 

a way that the probability of having one in a time window is close to one 

whilst the probability of having more than one is negligible. We shall return 

to this later on. 

7. Positivity sf the Wigner function in parametric down coasverslan 

There is a general argument showing that the Wigner function may be taken 

as positive in all experiments involving parametric down conversion. These 

experiments involve one or several nonlinear crystals where, in quantum language, the rocess takes place of converting a single photon of frequency 00 into two p otons R of frequencies o and 02 = coo -w 1 . The quantum Hamiltonian contains terms with an anni h ilation operator of the first t pe of 

photons and two creation operators. However, in all practical calcu r ations 

the incoming beam (the pumpin ) is taken as classical, and the annihilation 

operator is replaced by a c P assical amplitude. Consequently, the full 

Hamiltonian becomes quadratic in the operators of creation and destruction 

of photons. 

Now, it is well known that the Wigner function evolves accordin to a 

classical Liouville equation whenever the Hamiltonian is quadratic. 8n the 

other hand, the Liouville equation preserves positivity, in the sense that if 

the Wigner function is positive at a time then it remains positive at any later time. As the initial state (before switching on the umping) is the 

vacuum, whose Wigner function is positive, the Wigner P unction remains 

positive forever. We should point out that the action of devices like lenses, 
mirrors, beam splitters, etc. is linear and, consequently, ail of them 

preserve the positivity of the Wigner function. 

It is possible to argue that, strictly speaking, the Hamiltonian associated 

with the nonlinear crystal is cubic rather than uadratic, and a cubic 

Hamiltonian does not guarantee the positivity of the % igner function. This is 

true, but then the "nonclassicalw effects due to negative values of the Wigner function should be relevant in those experiments where a clear disagreement is obtained with %he (approximate) quantum predictions obtained using a 

cDassical pumping. No experiment of this type has been pedormed to our 

knowledge. 

I 

8. State of the beam produced in an atomic source 

In the following we investigate the positivit of the Wiqnea functim in 

experiments involving photon beams produce 2 by an atom~c source. As a 

typical example we consider the experiment by Grangier el al. [6], 

represented in Fig. 1 where the authors claimed to have produced single photon signals, which seems to imply negative Wigner functions. 

. For simplicity we consider atoms with Wo states: 1 g2 (ground) and I e2 

(excited). If at time t=O we have I e> I O> (excited atom plus radiation 

vacuum), then the evolution gives (to first order pe~urbation theory) 

where N is a normalization constant, j labels the radiation mode and A+(t) is 

the creation operator of a (localized, multimode) photon . 

If we consider many atoms, which arrive at the source, are excited there (by the action of a laser) and decay at times tl , t2, ... ts, ..., we should represent the atomic beam by the state vector 

If the state of the outgoing atoms are not controlled, (14) is not the correct 

representation of the physical situation. In fact, we must take the partial trace of p =Iy> eyI over the atomic states, which leads to 

Furthermore, if the emission times are not controlled, we must average over the times tl , t2,. . . 

After some algebra [I 11, we obtain that the final density matrix is 

P chaotic = nj ( Znj njnj (t + iij)-"j-' lnj><njl } 

which represents chaotic light, no matter how weak is the beam. The. average photon number in mode J, nj, is related to the coefficient cj .Essential for 
the result (16) is to take into account the interference between "photons" 

coming from different atoms. 

It is interesting that the Wigner function of (116) is positive definite, a well 

known property of chaotic light. 

The quantum predictions for the correlation and recombination experiments 

can be easily obtained. The recombination experiment shows interference 

with 100% visibility, which is not strange because the state (16) corresponds to "classicalw light that can be treated by standard wave optics. The correlation experiment gives for the ratio of the coincidence probability to the product of singles: 

a = Pcoinc. (pr pt)-l = 2 if detection window << lifetime of excited atom 

a = Pcoincs (pr pt)-l = 1 if detection window >> lifetime of excited atom 

For intermediate situations we get values of a between 0 and 1. This result 

can be also ex lained classical optics, as is well known since the early work of Brown- f wiss [I who showed experimentally the photon bunching properties (i.e., a > 1 ) chaotic light. 

9. "Single photon signals" in an atomic beam 

The procedure used by Grangier et al. [6] in order to manufacture single photon signals was to monitor the photons by detecting them in coincidence 

with partner photon emitted by the atom in a cascade. Of course, our two 

state model for the atom is no longer adequate because a cascade implies at 

least three atomic states. However, it is still ap ropriate to represent the 

state of the beam by Eq.(15) provided that, in a 8 dition to taking the partial 

trace of the density matrix over the atomic states, we average over all 

emission times except one, say to. 

We get a "sin le phdon signal" superimposed to the chaotic light, which may be represente d by the density matrix 

Psingle photon = N { P chaotic + A+(t'tO) P chaotic A(t-tO) 1 (1 9) 

The interesting result is that the Wigner function of (17) is positive. 

A straightforward but lengthy catcuiation[.i 1 gives the quantum prediction for the commented experiments. With t e parameters of the actual 

experiment 161 ( that is a coincidence window about twice the atomic 

lifetime) we predict that, when the beam is very intense, the "single photon" effect is lost and we get the asymptotic value a + 1.57 (pure chaotic light). In the performed experiment[6] values a > 1 are not observed because, in the 

actual ex erimentai conditions, the spacial coherence in the detector is 

lost[l2]. \owever, we think that rather modest improvements of the 

a paratus will allow observing our prediction of interference between 

p I: otons emitted by different atoms[ll J. 
We have shown that the interpretation of the Wigner function as a 

probability distribution provides a very attractive local realistic view of 

uantum optics. The main difficulty for this interpretation is the fact that 

t 'F, e Wigner function is not positive definite for some quantum states. We 

conjecture that all states actually realizable in the laboratory have a 

pos~tive Wigner function. In particular we have shown that this is the case 

for two typical situations where it is claimed that '@single photon states" 

are produced, namely parametric down conversion and light beams produced by a (weak) atomic source. In the second case we argue that the correct 

representation of the light beam is by means of a density matrix, rather than 

a pure quantum state. We do not accept the so-called ignorance interpretation of the density matrix, that is as a probability distribution on the set of pure quantum states. On the contrary, we assume that most of the 

pure quantum states are not physical states. 

Even if the above conjecture is correct, and the Wigner function of all 

physical states of light is nonnegative, some problems remain. We do not 

understand yet the processes of emission and detection of light or, more 

generally, the interaction of light with atoms. That is, we do not have a local 

realistic theory of atoms. In particular, we do claim to interpret the 

Wigner function of the electrons in the atom as a probability distribution. 

We acknowledge financial support of DGICYT Project No. PB-92-0507 (Spain) 

References 

1. J. F. Clauser and M. A. Horne, Ph s. Rev. D 10, 526 (4974) 2. J. S. Bell, Physics (N.Y.) 1, 195 [I 964) 3. E. Santos, Ph s. Rev. A 46, 3646 (1992) 4. V. Ca asso, J. Fortunato and F. Selleri, Int. J. Mod. Phys. 7, 319 (1973) 

5. Z. V. 8 u and L. Mandel, Phys. Rev. Lett. 61,50 (1988) 

6. P. Grangier, G. Roger and A. Aspect, Euro hys. Lett. 1, 173 1986) 7. J. F. Clauser and A. Shimony, Rep. Prog. Bhys. 41, 1881 (19 $ 8) 8. A. Aspect, P. Grangier and G. Ro er, Phys. Rev. Lett. 47, 460 (1981); 49, 91 

(1982); A. Aspect, J. Dalibard and 8. Roger, Phys. Rev. Lett. 49, 1804 (1981) 9. J. 6. Rarity and P. R. Ta ster, Phys. Rev. Lett. 64, 2495 

10. R. Hanbury-Brown and . Q. Twiss, Proc. Roy. Soc. A 0) 

300 (1 957) ; 

243, 291 (1957 11. T. W. Marshall, I! . Santos and A. Vidiella-Barranco, to be published. 12. P. Grangier. Private communication. 
A Loophole-free Bell's Inequality Experiment 

Paul 6. Kwiat, Aephraim M. Steinberg, and Raymond Y. Chiao 

Department of Physics, U. C. Berkeley 

Berkeley, CA 94720 

Phifippe H. Eberhard 

Lawrence Berkeley Laboratory, U. C. Berkele y 

Berkeley, CA 94720 

Abstract 

The proof of Nature's nonlocality through Bell-type experiments is a topic of longstanding interest. Nevertheless, no experiments performed thus far have avoided the so-called 

"detection loophole," arising from low detector efficiencies and angular-correlation difficulties. 

In fact, most, if not all, of the systems employed to date can never close this loophole, even 

with perfect detectors. In addition, another loophole involving the non-rapid, non-random 

switching of various parameter settings exists in all past experiments. We discuss a proposal 

for a potentially loophole-free Bell's inequality experiment. The source of the EPR-correlated 

pairs consists of two simultaneously-pumped type-I1 phase-matched nonlinear crystals and a 

polarizing beam splitter. The feasibility of such a scheme with current detector technology 

seems high, and will be discussed. We also present a single-crystal version, motivated by 

other work presented at this conference. 

In a separate experiment, we have measured the absolute detection efficiency and time 

response of four single-photon detectors. The highest observed efficiencies were 70.7 f 1.9% 

(at 633 nm, with a device from Rockwell International) and 76.4 f 2.3% (at 702 nm, with 

an EG&G counting module). Possible efficiencies as high as 90% were implied. The EG&G 

devices displayed sub-nanosecond time resolution. 

1 Introduction 

It is now well known that quantum mechanics (QM) yields predictions which are inconsistent with 

the seemingly innocuous concepts of locality and reality. This was first shown by Bell in 1964 [l,2] 

fpr the case of two quantum-mechanically entangled particles, e-g., particles in a singlet-like state; 

which do not possess definite polarizations even though they are always orthogonally polarized. 

As implied by Einstein, Podolsky, and Rosen (EPR) [3], it is straightforward to construct local 

realistic models that explain certain features predicted by QM (e.g., the total anti-correlation 

between detectors measuring the same polarization component of the two particles). The quantum mechanical contradiction with local realism becomes apparent only by considering situations 

of non-perfect correlations (i.e., measuring the polarization components at intermediate, nonorthogonal angles). More recently, Greenberger, Horne, and Zeilinger (GHZ) [4] and Mermin [5] 

have shown that QM and local realism are incompatible even at the level of perfect correlations, 
for certain states of three or more particles. Mmdy has also presented a clever gedanlcen experiment using electron-positron annihilation to achieve a contradiction with local realism without the 

need for inequalities [6], and has recently proposed adll optical analog which may allow a feasible 

experimental implementation [7]. Unfortunately, none of these ingenious extensions and generalizations of the work of Bell reduces the experimental requirements for a completely unambiguous 

test. In fact, ail of them seem to mandate even stronger constraints on any real experiment than 

do the original two-particle inequalities. One exception is the recent discovery that the detection efficiency requirement can be reduced by employing a state of two particles that are not 

maximally-entangled, i.e., with an unequal superposition of the two terms [S]. 

Experimentd tests of Bell's inequalities have been extended to new systems, some relying 

on energy-time or phase-momentum entanglement [9-121; nevertheless, regardless of the type of 

entanglement employed, no test of Bell's inequalities to date has been incontrovertible, due to 

several loopholes which severely reduce the true impact such an experiment might yield. One of 

these, the angular correlation problem, has essentially been solved by turning to down-conversion 

light sources over cascade sources, leaving the fast-switching loophole and the detection loophole. 

The former concerns the space-like separation of the different parts of the experiment. Clearly, 

no claims about nonlocality can be made if the pre-detector analyzers are varied so slowly that 

a signal traveling at the speed of light could carry the analyzer-setting information back to the 

source or to the other analyzer before a pair was produced or detected. To close this loophole, the 

analyzers' settings should be rapidly and randomly changed. Only one Bell-type experiment, that 

of Aspect et al. [13], has made any attempt at all to address this locality condition, but even in that 

experiment the loophole remains. Although the experiment used rapidly-varying analyzers, the 

variation was not random, and it has been argued that the rapidity of the polarization switching 

was not sufficient to disprove a causal connection between the analyzer and the source [14, 151. 

The detection loophole arises from the non-unity detection efficiency in any real experiment 

(efficiencies in past tests were at best lo%), so that only a fraction of the emitted correlated pairs 

is detected. If the efficiency is sufficiently low, then it is possible for the subensemble of detected 

pairs to give results in agreement with quantum mechanics, even though the entire ensemble 

satisfies Bell's inequalities. Due to the non-existence of adequate detectors, experiments have so far 

employed an additional assumption, equivalent to the fair-sampling assumption that the fraction 

of detected pairs is representative of the entire ensemble [16, 171. In order to experimentally close 

this loophole, one must have detectors with sufficiently high single-photon detection efficiencies. 

Formerly, it was believed that 1183% (= 24 - 2) was the lower efficiency limit. However, one 

of us (P. H. E.) has shown that by using a non-maximally entangled state (i.e., one where the 

magnitudes of the probability amplitudes of the contributing terms are not equal), one may reduce 

the detector requirement to ~67%~ in the limit of no background [8]. 

High-efficiency single-phot on detectors 

The highest single-photon detection efficiencies to date have been observed using avalanche photodiodes in the Geiger mode; until recently these have been limited to about 40%. We have measured 

efficiencies as high as 76%, and there are indications that these may be improved to 80% or even 

90% 118, 191. The technique used to measure the absolute efficiency of a single-photon detector 

is now fairly well known. It was proposed by Iclyshko 1201, and first used by Rarity et al. [21] 
lOnm Lens SPCM 

Beam 

dump 

FIG. 1. A simplified schematic of the setup used to measure absolute quantum 

efficiencies. The 10-cm-long KDP crystal is pumped by the 351-nm line from an argonion laser. The smaller iris and interference filter on the path of the trigger photon serve 

(through phase-matching and energy conservation constraints) to define the path of 

the conjugate photons, which are all collected by the bottom detector (modulo losses 

en route). The outputs of the detectors are amplified and fed into the START and 

STOP channels of a Time to Amplitude Converter/Single Channel Analyzer. The 

coincidence rate output, as well as the two singles rates, are measured with a counter 

and stored on a computer. By comparing the coincidence and trigger singles rates, the 

efficiency of the bottom detector may be determined. 

to characterize a silicon avalanche photodiode. The pairs of photons produced in spontaneous 

parametric down-conversion are highly correlated in time, and reasonably well collimated (i.e., 

constraining the direction of one photon of a pair determines within a few milliradians the direction of the other). One photon of each pair is directed to a "trigger" detector, and the collection 

optics are arranged to catch all of the "conjugate" photons with the detector whose efficiency is 

to be measured (see Fig. 1). The singles count rates at each detector (Rt and I&) are measured, 

as well as the rate of coincidence counts (Rt,,) between the detectors. In the ideal limit of no 

accidental counts (arising from photons from diflerent pairs "accidentally" arriving within the coincidence timing window) and no background events (from unwanted external light, dark counts 

within the detector, or electronic noise), the efficiency of the "conjugate" detector is simply the 

ratio of coincidence rate to the "trigger" detector singles rate: qc = Rt,,/Rt . In the presence of 

accidentals, A, and trigger detector background, BG, the formula is modified slightly: 

In practice our correlated photon pairs resulted from pumping a potassium di-hydrogen phosphate (KDP) crystal, cut for type-I phase-matching. The down-converted photons typically exited 
the crystal at a few degrees with respect to the axis of the pump beam. Using irises and filters to select the trigger photons, we were able to measure the efficiency at the wavelength pairs 702-702nm 

and 632nm-988nm (the energies of the down-ionverted photons must sum to the energy of the 

parent ultraviolet photon at 351nm). We ned four single-photon detectors: two Single Photon Counting Modules (SPCM-200-PQ, EG&G), and two Solid State PhotoMultipliers (SSPM, 

Rockwell Internationd Corp.). The former devices use Geiger-mode silicon avalance photodiodes 

specially manufactured to have a very low "k", the ratio of hole- to electron-ionization coefficients 

[22]; our devices were also custom modified to employ a high overbias voltage of 30V. The SSPMs 

are also silicon devices, but operate using impurity-band-to-conduction-band impact-ionization 

avalanches, yielding a very sensitive response in the infrared. The avalanches are localized within 

areas several microns in size, and do not in general lead to device breakdown, so that these devices 

are capable of distinguishing between single-, doubb-, etc. photon detections [23]. 

The highest observed efficiencies were 70.9f 1.9% (with an SSPM, at 632 nm), and 76.4f 2.3% 

(with an SPCM, at 702 nm). We believe these to be the highest reported single-photon detection 

efficiencies in the visible spectrum; they are important for quantum cryptography and loopholefree tests of Bell's inequdties, as well as more prosaic applications such as photon correlation 

spectroscopy and velocimetry. It is im.portant to note that associated with each of the detectors 

there are other sources of loss, which may yet be improved. Notably, the SPCM detector is 

housed in a can with uncoated glass windows, and the detector surface itself was broadband 

anti-reflection coated. Using multi-layer, wavelength-specific coatings, it should be possible to 

essentially eliminate losses at these interfaces, implying a detection efficiency of > 82%. Moreover, 

a higher overbias is expected to increase the efficiency even further. The plastic optical fibers used 

to couple the light into the SSPMs were measured (after dismantling the apparatus) to possess 

unexpected losses-correcting for all of these losses would suggest SSPM efficiencies as high as 

93 f 7%. (Since then, relative measurements of SSPM efficiencies of 90 f 5% have been observed.) 

Work is currently underway to improve the fiber coupling scheme. In addition to high efficiency, 

a useful single-photon detector must have a low level of background, or noise. The SPCMs are 

internally cooled to about -30°C, and have rather small active areas (only (0.1mm)2); their dark 

count rates are correspondingly low, typically 65s-l. The SSPMs are cryogenically cooled to 6 K, 

but have much larger active areas ((~mrn)~); typical dark count rates are 7,000s-l. 

Previous measurements of the time correlation of the photon pairs have shown that they are 

emitted within $0 fs of each other [24]. Therefore, they can be used to accurately measure the 

intrinsic time resolution of single-photon detectors, by mapping out the coincidence rate as a 

function of electronic delay time. We found that the time profile for coincidences between the 

SSPM and the SPCM consisted of' a main 3.5ns-FWHM peak preceded by a smaller peak by 

llns. A sidar time profile between two SPCM's displayed only one peak, with 300ps FWHM. 

Afterpulses were detected in the SPCM9s at a level of less than of the counting rate, with an 

expontentid falloff time-constant of 4.5~~. 

Proposed EPR-source 

Even with unisefficiency detectors, the down-conversion schemes used until now are inadequate for 

a completely unambiguous test of Bell's inequalities, because they must perforce discard counts. 

In the simplest of the down-conversion Bell-inequality experiments [25, 263, non-collinear corre- 

FIG. 2. Schematic of a novel arrangement in which a loophole-free test of BeU's 

inequdties is feasible. a) An dtraviolet pump photon may be spontaneously downconverted in either of two nofinear cryst&, producing a pair of orthogondy-pol&zed 

photons at half the frequency. One photon from each pair is directed to each output 

port of a polasizing beam splitter. When the outputs of both crystals are combined 

with an appropsiate relative phase 6, a true singlet- or triplet-Eke state may be produced. By using a half waveplate to effectively exchange the polarizations of photons 

originating in crystd 2, one overcomes several problems arising from non-ideal phasematcEng. An additiond mirror is used to direct the photons oppositely to separated 

analyzers. b) A typical analyzer, including a ha waveplate (HWP) to rotate the polarization component selected by the analyzing beam splitter, and precision spatid filters 

to select only conjugate pairs of photons. In an advanced version of the experiment, 

the HWP could be replaced by an ultrafast polarization rotator (such as a Pockels or 

Kerr cell) to close the space-like-separation loophole. 

lated photons were directed through equal path lengths to opposite sides of a 50-50 beam splitter, 

aligned so that the transmitted mode of one photon coincided with the reflected mode of the conjugate photon, and vice versa. A half waveplate prior to the beam splitter was used to rotate the 

polarization of one of the photons (which were initially horizontally polarized) by 9Q0. Coincidence 

rates between detectors looking at the two output ports were recorded, as a function of the orientation of polasizers at the detectors. In only measuring coincidence rates, the experimenters were 

able to effectively create a singlet-like state by discarding cases where both photon exited the same 

port of the beam splitter. However, it should be stressed that because of these discarded terms, 

the detection efficiency is inherently limited to 50% (unless the detector can reliably distinguish 

one photon from two), and no indisputable test of Bell's inequalities is possible. A similar problem 

arises in the single-crystal Bell-inequality experiment presented at this conference [27], as well as 

in experiments based on energy-time entanglement [Ill and phase-momentum entanglement [12]. 

We propose here a setup which should permit for the first time closure of the loopholes. .A very 

different experiment by Edward Fry is just underway [28], using for the first time atoms from dissociated mercury dimers as the correlated particles. The advantage is that detection efficiencies 

of 95% are possible by photoionizing the atoms and detecting the photoelectrons, 

A schematic of our proposed source is shown in Fig. 2a [29]. Two nonlinear crystals are simultaneously punnped by a coherent pump beam to induce spontaneous parametric down-cornersion; 

the pumping intensity can be independently varied at each crystal. The crystals are cut for type-I1 

collinear degenerate phase matching (i.e., the down-converted photons are collinear and orthogonally polarized, with spectra at roughly twice the pump wavelength). For example, we envisage 

using 1-cm long crystals of beta-barium borate (BBO), pumped by the 325-nm line from a HeCd 

laser incident at 54O to the optic axis. For clarity, we first assume a monochromatic pump beam 

(at frequency 2wo), and a single-mode treatment of the down-converted photons. Then the state 

after the crystals is 
where we have omitted higher order terms (for the very unlikely case in which more than one 

pump photon down-converts; by reducing the pump intensity, the contribution of these terms 

can be made as small as desired). A includes the down-conversion efficiency into the modes 

we are considering, and also the pump field strength; f represents a possible attenuation of the 

pump be- incident on crystal 2. The state (2) describes a photon pair [one photon polarized 

horizontally (H), the other vertically (V)] originating with probability amplitude A/ in 

crystal 1 and with probability A~/JW in crystal 2. We now combine the modes from the two 

crystals at a polarizing beam splitter. For an ideal polarizing beam splitter, incident p-polarized 

light (horizontal in Fig. 2) is completely transmitted, while incident s-polarized light [vertical 

(out of the plane of the paper) in Fig. 21 is completely reflected; therefore, one photon of each 

pair will be directed to output port 3, while the conjugate photon is directed to output port 4. 

Including a phase shift 6 = 2woAx/c (where Ax, the difference in path lengths, may be varied by 

moving one of the mirrors slightly) between the two non-vacuum terms of (2), we then have 

. where we have omitted the (predominant, but uninteresting) vacuum term and the prefactor 

A/ Jm. For the balanced case (f = I), and for 6 = 180°, (3) reduces to the familiar singletlike'state. (In practice, one should probably use a triplet-like state (6 = 01, as this is far less 

sensitive to cross-talk effects in the polarizing beam splitter-see below.) Note that this contains 

no non-coincidence terms that must be intentionally discarded to prepare a singlet-like state. The 

source may thus find application in quantum cryptography [30,31], as it doubles the signal-to-noise 

ratio of most previous down-conversion EPR schemes. 

With the above source of correlated particles, one can now perform a polarization test of Bell's 

inequalities. Polarization analysis is performed using an additional polarizing beam splitter after 

each output port of the interferometer, and examining one or both channels of each analyzer 

with high efficiency detectors (see Fig. 2b). "Rotation" of these analyzers can be effectively 

accomplished by using a half waveplate before each one to rotate the polarization of the light. If 

the detectors are far separated from each other and from the source, and one uses some rapid, 

random means to rotate the light before the analyzers, (such as a Pockels or Kerr cell, whose 

voltage is controlled by a random signal), then one can close the space-like separation loophole. 

The signal could be derived, for instance, from the decay of a radioactive substance, or even from 

the arrival of starlight. Note that since the down-converted photons are emitted within tens of 

femtoseconds of one another [24] (unlike the photons in an atomic cascade), the limiting time 

factors will be the detector resolution (expected to be less than 10 ns) and the switching time 

(which can also be on the order of nanoseconds). 

4 Other considerations 

It has been shown that by using a non-maximally entangled state (i.e., one where the magnitudes of the probability amplitudes of the contributing terms are not equal), one may reduce 

the detector efficiency requirement [8]. The basic idea is that by making one term of (3) have 

a greater amplitude than the other, one effectively polarizes the source. For example, if f is a 
red number 5 1, then a photon travelling to port 4 possesses a net horizontal polarization, while 

a photon travang to port 3 appears somewhat vertically-polarized. By appropriately choosing 

the polarization-analyzer angles, one may reduce the contributions of the singles rates, while still 

violating a Bell's inequality, even for q as low as 67%. It should be noted, however, that the magnitude of the violation is reduced as f is reduced, increasing the relative importance of undesirable 

background counts. A background level (e.g., from any stray light or dark counts) of less than 1% 

is desired. 

One problem that arises in the above scheme is the effect of walkoff of the down-converted 

photons in the birefringent parent crystal. While the birefringence of the nonlinear crystal is 

essential for achieving phase-matching, it also results in a relative displacement of the two downconverted photons: they propagate in the same direction after exiting the crystal, but are separated 

by a distance d = L tan p, where L is the propagation distance inside the crystal, and p is 

the htra-crystd angle between the ordinary and extraordinary beams. Consequently, after the 

polarizing beam splitter, the position of a detected photon partially labels its origin, degrading 

coherence. Remmhabiy, insertion of an extra half waveplate after one of the crystals to rotate the 

pol~zation by 90" avoids this problem. Photons from either crystal exiting port 3 of the beam 

splitter would be i~tially extrmr&nary-polarized; photons exiting post 4 of the beam splitter 

would be initidy ordinary-polarized. A similar situation occurs due to longitudinal w&off: After 

propagation through some length of the birefringent down-conversion crystal, one of the downconverted photons wil% "pd ahead" of its conjugate. In the absense of the h& waveplate, one 

could in principle determine from which crystal a given pair of photons originated by examining the 

timing of the coincident detection. This dlisting~shability of contributing paths removes quantum 

interference, just as in the transverse w&off situation considered above. (This same effect was 

discussed by Sergienko et al. at this conference [32].) Fortunately, the extra h& waveplate also 

removes this longitudinal w&oE effect. With the waveplate, an infidtely-fast detector boking 

at the originally extraordlinay-polarized photons would always trigger before the detector looking 

at the originally ordinary-polarized photons; hence, interference remains. The waveplate also 

removes difficulties arising from finite bandwidth and vector phase-matching considerations [29]. 

For a plme-wave pump, the phase-matching constraints for down-conversion imply that with 

careful spatial filtering, one can in principle collect only conjugate pairs of photons from the 

crystals (i.e., essentially no unpaired photons, aside from stray light). Once we allow a more 

realistic, gaussian-mode pump, then this is no longer possible. For identical, finite-sized collection 

irises, there will always exist situations where one photon is detected while the other is not (even 

aside from the problem of inefficient detectors). This effect is mitigated by collecting over a larger 

solid angle. In particular, in order to keep the losses less than 2%, we must employ irises which 

accept light out to 30 times the pump divergence angle. For example, if we employed a 325-nrn 

pump with a beam waist radius (l/e2) of 3.5 mm, then we would need to accept all half-angles 

up to 1 milliradian. (In practice, this could be accomplished by use of a precision spatial filter 

system in each output port; see Fig. 2b.) 

We have also investigated the effects of using various non-ideal optical elements. One of these 

investigations bas led us to a novel interference effect. In considering an imperfect recombining 

polarizing beam splitter, we discovered that the singlet-like state is much more sensitive thar, the 

triplet-like state to beam-splitter "cross talk", when f = 1. By "cross talk" we mean the situation 

where photons exit the wrong port of the beam splitter (i.e., some fraction of the incident p 
crystal 1 

=O for 6 =O 

FIG. 3. The two processes that can lead to the final state of both photons going 

toward the same detector are indistinguishable; thus, we add their probability amplitudes. We assume that f = 1, and for simplicity that the transmission and reflection 

amplitudes are 114 and i/a, respectively. When 6 = 0 (triplet-state), there is 

destructive interference. 

polarized light is reflected, while some fraction of the incident s-polarized light is transmitted). 

Whenever this happens, we once again have a situation in which both photons can leave the same 

output port of the beam splitter - just as in the experiment discussed at the beginning of Sect. 3, 

this reduces the effective efficiency, making it more difficult to violate Bell's inequalities. However, 

in the special case for which the percentage of cross-talk for the two polarizations is equal (one 

example is the case of a non-polarizing 50-50 beam splitter), there is quantum interference which 

prevents the two photons from exiting the same port of the beam splitter; the interference only 

exists for the triplet arrangement (6 = 0°), not the singlet case (6 = 180") [see Fig. 31. This is, in 

some sense, the complementary effect to one in some one-crystal experiments (see, for instance, 

[24, 33]), in which the two-photons from a down-conversion crystal always take the same port of 

a 50-50 beam splitter. It would be interesting to demonstrate this phenomenon experimentally. 

Using a scheme that was motivated by two of those presented at this conference [34, 351, it is 

possible to produce this EPR-source using only one down-conversion crystal. The key is to reflect 

the pump beam back through the crystal (see Fig. 4). Although this scheme has the advantage 

that it reduces the number of optical elements (most importantly the number of crystals) it does 

have a few difficulties not present in the 2-crystal approach. First, one must use an isolator to 

prevent the pump from coupling back into the laser. Next, the return path must be kept short 

compared to the pump coherence length (typicdy 10 cm for a HeCd laser), to maintain coherence 

between the processes in which the down-converted pair originated from a left-going or a rightgoing pump photon. Findy, one must find very good dichroic mirrors to separate the pump beam 
f Dichroic Mirrors 7 

UV pump 

-a@- 

I 

Attenuator (0 \\Matching/' 

FIG. 4. Proposed one-crystal scheme to produce EPR-state. 

from the down-converted photons. One solution is to use high-quality prisms (not shown in Fig. 

4)- 

In conclusion, with the setup described herein it should be possible to produce an indisputable 

violation of a Bell's inequality. We have examined the effects of background, imperfect polarizing 

beam splitters, and phase-distorting optics. For X/2O-flatness optics, a background level of 1%, and 

custom-selected polarizing beam splitters (with an extinction ratio of at least 500: I), numerical 

calculation predicts that a violation should be possible as long as the net detection efficiency is 

greater than 82.6%. Naturally, all optics would be anti-reflection coated to minimize reflection 

losses; including a 0.25% loss for each interface, and the 2% loss from the gaussian nature of 

the beam, this means that the bare detector efficiency needs to be at least 88.6%, which may be 

achievable in light of our recent measurements [18, 191. Of course, for a safety margin, one would 

like it to be even higher. 

5 Acknowledgments 

We would like to acknowledge the invaluable assistance of Joshua Holden. One of us (P.H.E.) was 

supported by the U.S. Department of Energy, Contract No. DE-AC03-76SF00098. The rest were 

supported by the Office of Naval Research under Grant No. NO0014-90-J-1259. 
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quii~ lay uuar;y 
simple settheoretical arguments. Recently another attempt has been made by Lucien Hardy [6] 

who showed that the probability for a contradiction of the GMZ-kind can be greater than zero for 

a two-particle system. 

Nevertheless none of these approaches has provided a general argument based on the concepts 

of locality and reality which explicitly demonstrates the origin of the discrepancy between local 

realism and quantum mechanics. Thus our aim is to show the essential restrictions of local realism 

by discussing the results of a general two-particle experiment using the assumptions of locality and 

reality. As evident consequences the conditions for the fulfilment of these restrictions are equivalent 

to Bell's inequality. 

2 Predictions based on the knowledge of correlations 

We consider the following experimental setup (cf. Fig. 1): A source emits the two parts of a 

system in opposite directions. Measurements with the possible results +l and -1 are performed 

on each part by two observers A and B. Each of them may select one of two possible values of a 

measurement parameter a and P, respectively. As a consequence four different experiments can 

be made, corresponding to the four different combinations of the measurement parameters al, a2 

and &, P2. 

FIG. 1. The experimental setup consists of a source which emits the two parts of 

a system in opposite dia-ections. Measurements with the possible results +1 and -1 

are performed on each part by two observers A and B. Both A and B have a knob 

which selects one of two possible values of a measurement parameter. In such a way 4 

different experiments can be made (cf. table I). 

We assume that all four experiments have been made. The results are listed in table I. 

TABLE I. The correlations of the results of the 4 experiments are listed. They mi ht f have been observed in actual experiments or calculated by quantum mechanics. Pi is 

the probability for different results in experiment i. In consequence the probability for 

equal results Pp is 1 - Pf. 

Experiment I A I B I Correlation 
In experiment 1 with the parameters adjusted to a1 and PI the observers A and B got different 

results with probability pf . In experiments 2, 3 and 4 the probabilities for different results are 

## P, , P, and Pf, respectively. 

Knowing the correlations which have to be expected either from previous experiments or 

quantum mechanical calculations, A is able to predict the possible results of B and vice versa. 

This means that after A has for example performed a series of n measurements with the setting 

a1 he can infer all possible results of B on the basis of his experimental data and the knowledge 

of the correlations in experiments 1 and 2 by the following reasoning. 

Predicted results of experiment 4 I 

FIG. 2. A procedure is shown by which observer A (B) after having performed 

a series of n measurements is able to predict the results of observer B (A) for the 

two alternative settings of the measurement parameter /3 (a). The actually measured 

results of observers A and B are listed in the two boxes. By changing a corresponding 

number of signs (ni = pf n for experiment i) of the measured results the predictions 

are in agreement with the calculated or previously observed correlations listed in table 

I. Nevertheless it turns out immediately that the predicted results of experiment 4 

are consistent with the actual correlations of experiment 4 (cf. table I) only if the 

inequality n, + n, + n, 2 n, (equivalent to Bell's inequality) is fulfilled. 
If B selects the parameter h (experiment 1) the probability for different results has to be Pf (cf. 

table I). For n m this means that nl = Pf n results have to be different. By reversing nl 

signs of his results A arrives at a series of possible results of B which correspond to the known 

correlations. Because there are different ways of changing nl signs of n numbers A ends 

up with a list of nl!(tLnl)! different predictions for the possible outcomes of B's measurements. In 

the same way A is able to infer the possible results B could get, if B selects P2 (experiment 2) by 

changing nz = ~f . n signs of his results. In Figure 2 the results of A are listed in the box in row 

A,, . The predictions he derives from these data are symbolized by vertical lines in rows BPI and 

BAS Each line corresponds to one way of changing nl and n2 signs, respectively. By this means 

the predicted results correspond to the known correlations. 

Now let's assume that observer B actually selects the parameter PI and performs a series of n 

measurements. In figure 2 his results are listed in the box in row Bpi. Of course they correspond 

to one of the predictions by A. 

Not knowing what A has done observer B himself predicts all possible results A could get if 

he selects the parameter a1 (experiment 1) or a2 (experiment 3) (cf. table I) by considering all 

possible ways of changing nl or n3 signs of his results. Again the actual results of A correspond 

to one of the predictions by B as it is shown in figure 2. 

Bell's inequality 

In the previous section we have shown how it is possible for A to predict all results B could 

obtain and vice versa. In the following we are going to apply the locality assumption that "no real 

change can take place in the second system in Consequence of anything that may be done to the 

first system" [I]. Moreover we assume in agreement with realistic approaches that "unperformed 

experiments have results" [7] or in other words that predicted results have the status of potential 

reality. 

If we now ask what A could have measured if he had selected the parameter a2 (experiment 3) 

instead of a1 (experiment I), we just have to take into consideration the predictions by observer 

B to find the answer. Based on his actual results and the known correlation in experiment 3 (cf. 

table I) observer B has predicted all results A could have got if he had chosen a2 (cf. figure 2). As 

a consequence of the locality assumption the results of B, which are the basis of his predictions, 

are independent of anything that may be done by A. Because of this independence all of B's 

predictions have the status of potential reality, which means that if A had selected a2 he actually 

would have got one of the results predicted by B. 

In the same way we find the answer to the question what B could have measured if he had 

selected the parameter P2 (experiment 2) instead of P1 (experiment 1) by considering the predictions by observer A (cf. figure 2). It is important to notice that because of the locality assumption 

we can make independent use of the predictions by B and A to answer the question what A and 

B could have measured if they had selected a2 and P2, respectively. 

Since we know all possible results A and B could have got if they had chosen a2 and P2, 

respectively (experiment 4), we may now try to find out if these results are consistent with the 

known correlation of experiment 4 Pf (cf. table I). For this purpose we take one of the results B 

could have got if he had selected P2 (row Bp, in figure 2), change nz signs to get the actual results 
of A (box in row A,,), change nl signs to get the actual results of B (box in Bp,) and change n3 

signs to end up with one of the results A could have got if he had selected 02 (row A,, in figure 

2). Of course we could also do the same thing the other way round but anyway the results A 

could have got are connected to the results B could have got by the following transformation rule 

(cf. figure 2): Reverse n2 signs in the first, nl signs in the second and n3 signs in the last step 

or the other way round. Doing this the maximum number of signs one can change is simply 

m, + n, + n3. This result of local realistic reasoning is consistent with the observed correlation 

in experiment 4 if and only if n, + n, + n, >_ n, = ~f n . If this condition is violated, then 

not a single pair of the predicted results of experiment 4 (cf. figure 2) is correlated in agreement 

with experience because there is no pair with more than nl + n2 + n3 different signs. 

It follows immediately that this condition is equivalent to Bell's inequality: 

We get (3) by adding inequalities (1) and (2) and using the definition of the expectation value of 

the product of the results Ei in experiment i (i = 1,2,3,4). 

4 Discussion 

We have shown that just by discussing the possible results of a general two-particle experiment 

in a local realistic way one is directly led to a condition for the consistency between quantum 

mechanics and the concept of local realism. 

The crucial point in the argumentation is on the one hand the assumption that A's and B's 

data are determined locally, which means that A's (B's) results are completely independent of 

the measurement parameter selected by B (A). On the other hand by assuming that unperformed 

experiments have results A's and B's predictions can be combined in order to get a prediction of 

experiment 4 (unperformed). It turns out that this kind of counterfactual reasoning is inconsistent 

with the results one obtains by actually making experiment 4. 
5 Acknowledgments 

I would like to thank Prof.Anton Zeilinger, Dr. Johann Sumrnhammer, Dr.Marek ~ukowski and 

Clemens Ulrich for useful discussions on this and related topics. 

This work has been supported by the Austrian Fonds zur Forderung der wissenschaftlichen 

Forschung, grant No. P8781-PHY. 

References 

[I] A.Einst ein,B.Podolsky, and N. Rosen," Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?" Phys.Rev.47,777-780(1935). 

[2] John S.Bel1," On the Einstein Podolsky Rosen Paradoxn Physics 1,195-200(1964) 

[3] D.M. Greenberger,M.Home,and A.Zeilinger," Going beyond Bell's theorem" ,in Bell's Theorem,Quantum Theory,and Concepts of the Universe,edited by M.Kafatos (Kluwer Academic,Dordrecht ,The Netherlands,l989) ,pp.73-76 

[4] D.M.Greenberger,M.Horne,A.Shimony,and A-Zeilinger, "Bell's theorem without inequalitiesn,- 

Am.J.Phys.58,1131-1143(1990) 

[53 Eugen P.Wigner,"On Hidden Variables and Quantum Mechanical Probabilitiesn Am.J.Phys. 

38,1005-1009(1970) 

[6] Lucien Hardy,"A New Way to Obtain Bell Inequalities" Phys.Lett. A 161, 21-25 (1991) 

[7] Asher Peres, "Unperformed experiments have no results* Am.J.Phys. 46, 745-747 (1978) 
EINSTEIN-PODOLSKY-ROSEN-BOHM EXPERIMENT AND 

BELL INEQUALITY VIOLATION USING TYPE II 

PARAMETFUC DOWN CONVERSION 

T. E. Kiess, 

Department of Physics, University of Maryland, 

College Park, MD 20742 

Y. H. Shih, A. V. Sergienko, 

Department of Physics, University of Maryland, 

Baltimore County, MD 22228 

and C. 0. Alley 

Department of Physics, University of Maryland, 

College Park, MD 20742 

Abstract 

We report a new two-photon polarization correlation experiment for realizing the EinsteinPodolsky-Rosen-Bohm (EPRB) state and for testing BeIl-type inequalities. We use the pair of 

orthogonally-polarized lighr quanta generated in Type 11 parametric down conversion. Using 

lnm interference filters in front of our detectors, we observe from the output of a 05mm 

B - BaB204 (BBO) crystil the EPRB correlations in coincidence counts, and measare an 

associated Bell inequality violation of 22 standard deviations. The quantum state of the 

photon pair is a polarization andoe cf ,,he spin-l/2 singlet state. 

1 Introduction 

The Einstein-Podolsky-Rosen-Bohm (EPRB) gedanken[l, 21 for two quantum particles has played 

an important conceptual roIe for viewing quantum-mechanical correlations that provide intrigue 

and challenge to classical intui~ion. In brief, the EPRB correlation considered here is contained 

in a tweparticle, two detector setup in which a measurement is first made on one particle at 

one detector of a parameter thzt is indeterminate prior to the measurement. This outcome then 

implies with certainty the outcome of the measurement on the second particle. Demonstrations 

for spin-112 quanta[3], for photon polarization states [3,4, 5, 6, 7, 81, and more recently for other 

variables [9, 10, 11, 121 have all shown these EPRB correlations in the coincidence registrations 

of two detectors. 

The same quantum-mechanical state of two particles that generates an EPRB experiment also 

provides an enhancement of observed coincidence rates beyond a maximum bound set by Bell's 

two postuIates[l3, 31. Violatiozs of Bell-type inequalities are one application of EPRB states. 
We have fomd a new and convenient way to generate quantum states that exhibit these EPRB 

wmelations[I, 23, with an associated violation of two-particle Bell-type inequalities[l3, 31. Our 

mwe 51 the pair of light quanta generated in parametric down conversion with Type 11 phase 

&&img. The pair is incident on one port of a nonpolarizing beamsplitter with output ports 

mntdiieing two linear analyzer-detector packages. The pair is orthogonally polarized, without the 

nged for an interferometer or retardation plates. Our Bell Inequality violation in polarization 

'ables is as large as 22 standard deviations. 

2 Experimental Method 

Our experimental configuration is shown in Fig. 1. A 351.lnm unfocussed Argon ion laser line is 

incident on a Q.5mm long BBO crystal oriented to achieve Type 11 phase matching in parametric 

down conversion, with the 702.2nm wavelength collinear with the pump. Pairs degenerate in 

propagation direction and frequency emerge from the BBO crystal and separate from the pump at 

a quartz prism. They are split at a nonpolarizing beamsplitter and sent to two Glan-Thompson 

analyzer-detector packages. Coincidence counts are collected in a 6nsec coincidence time window 

from two avalanche photodiode detectors operating in the Geiger mode. 

Coincidence counts collected in a large time window are 

for a constant No and parameter X that is not a priori equal to two. Indeed, for filters of bandpass 

greater than lnm, or for a BBO crystal of greater length (5,65mm), X was found to be less than 

two. We discuss this bandwidth and' crystaI length dependence elsewhere in greater detail. For the 

present conditions (filters with FWHM of lnm, crystal length 0.5mm), the large visibility (> 99%) 

in Figs. 2 and 3 is predicted from (1) only for X nearly two. We obtained X = 1.98 f 0.04. 

This value of X is within la of 2, for which (1) reduces to sin2(B1 - B2), a function of only the 

difference angle, - B2 This dependence on only one variable specifies an invariance with respect 

to the other, often referred to as a rotational invariance[3, 71. 

This rotational invariance is a key property of our quantum state. Either the o-ray or the e-ray 

Gould trigger either detector, making detection of single counts ideally independent of analyzer 

angle. Once one detector has fired, the conditional probability of registering an event in the second 

detector is given by the coincidence probability, which for X = 2 can be rewritten as oc ws2(B1 - 

O2 + 90 deg). This is Malus' law for detection of a linear polarization at angle el - B2 + 90 deg. 

The polarization at the second detector is known with certainty to be linear at this angle after 

the first detector has fired. This is a correlation of the EPRB type. 

The A-dependent mixing term in (1) foils any interpretation specifying the o-ray to be localized at detector 1 and the e-ray at detector 2, or vice-versa. Both of these quantum-mechanical 

amplitudes are present, and their interference generates the A- dependent term of (1) that represents an overlap of "0-ray resolved at Dl, e-ray resolved at D2" (- cosBlsinB2) with "e-ray at 

Dl, o-ray at D2" (- sinBlcosB2). 
3 Results 

We use the (coincidence cod expression (1) to exhibit Bell inequality violations. This application 

of (1) proceeds without any necessary regard to the underlying mechanism procfucing the Xdegendent interference term. 

For the X = 2 singlet state andog, we measured the Be11 inequality expression as derived by 

Redmm[l4,3) 

I Nl2(4) - fi2('4) 15 0.25, 

N12(-, -) (2) 

for 4 = el - e2 = 22.5 deg (see Table I). 

It is possible to avoid exhibiting the aforementioned symmetry properties used to derive (2) 

by testing a more general Bell inequality form. We proceed fkom the inequality[l5,3,8] 

in which the Glauser-Horne no-enhancement assumption[l5] has already been imposed, and in 

which probabilities have been converted to coincidence counts N12 accumulated in some time 

. interval. 

Although we have generated violations of (3), we advocate a stronger version in which the 

transmission losses of the analyzers are recognized and removed. The basis for this is a generalized 

version of the no-enhancement hypothesis, in which the passive, polarization-independent analyzer 

losses are assumed not to affect the behavior of the source whose coincidence properties are under 

study. We note that these analyzer losses must be controlled[l5,3,16] in a rigorous Bell inequality 

test. For our purpose here of exhibiting the coincidence behavior of the source, we use this 

generalization to alter (3) to the form 

We deduced a choice of four angles (4 = 22.5 deg, 6$ = 135 deg, 8; = 67.5 deg, 8; = 90 deg) that 

would violate (3) or (4) maximally. Coincidence counts (Table 11) in eight minutes are collected 

for these angles. Table 111 shows the deduced violation in these counts of both (3) and (4). 

As has been noted[S], violations of (4) occur for any greater-than-zero value of the left hand 

side. For a relevant figure of merit in judging Bell inequality violations, we advocate the quantity 

Q-1, for Q the ratio of the term of (4) in square brackets to the term in parentheses. For Q-1>0, 

the form (4) is violated. We compare in Table I11 the measured Q-l to the prediction generated 

fiom A. As another consistency check, we list the ratio q2N12(01, -) : q1N12(-,02), which should 

be 1:1, within experimental error. 

In conclusion, we have identified a new source useful in realizing the EPRB gedanken and in 

testing Bell Inequalities. We achieve the EPRB correlations by virtue of a coincidence probability 

or sin2+ for q5 = $ - 02. We obtain a violation of the form (2) for polarization variables that 

is 22 standard deviations and limited here only by accumulation time. The quantum state is 

entangled[l?] in polarization variables, leading to an interference term in coincidence counts that, 

because of EPRB correlations and Bell-type inequality violations, is manifestly quantum in nature. 
4 Acknowledgements 

This work was supported in part by the Office of Naval Research Grant No. N00014-19- J-1430. 

References 

fl] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935). 

[2] D. J. Bohm, Quantum Theo y, (Prentice-Hall, Englewood Cliffs, N.J., 1951). 

(31 For a review, see J. F. Clauser and A. Shimony, Rep. Prog. Phys. 41,1881 (1978). 

[4] A. Aspect, f. Grangier, and G. Roger, Phys. Rev. Lett. 49,91 (1982), and 47, 460 (1981). 

[5] A. Aspect, J. Dalibard, and G. Roger, Phys. Rev. Lett. 49, 1804 (1982), and references 

therein. 

[6] W. Perrie, A. J. Duncan, H. J. Beyer, and H. Kleinpoppen, Phys. Rev. Lett. 54,1790 (1985). 

[7] C. 0. Alley and Y. H. Shih, in Proceedings of the Second Internationa Symposium on Foundations of Quantum Mechanics in the Light of New Technology, Tokyo, 1986, ed. M. Namiki, 

et al (Physical Society of Japan, Tokyo, 1987), p. 47; Y. H. Shih and C. 0. Alley, Phys. Rev. 

Lett. 61, 2921 (1988). 

[$I 2. Y. Ou and L. Mandel, Phys. Rev. Lett. 61, 50 (1988). 

[9] 3. D. Franson, Phys. Rev. Lett. 62,2205 (1989). 

[lo] J. Brendel, E. Mohler, and W. Martienssen, Europhys. Lett. 20, 575 (1992). 

ill] Y. H. Shih, A. V. Sergienko, and M. H. Rubin, Phys. Rev. A47, 1288 (1993). 

[12] P. G. Kwait, A. M. Steinberg, and R. Y. Chaio, Phys. Rev. A47, R2472 (1993). 

I131 J. S. Bell, Physics 1, 195 (1965); Rev. Mod. Phys. 38,447 (1966). 

[14] S. J. Freedman and J. F. Clauser, Phys. Rev. Lett. 28, 938 (1972). 

[15] J- F. Clauser and M. A. Horne, Phys. Rev. D 10, 526 (1974). 

[16] For a recent discussion, see V. L. Lepore and F. Selleri, Found. Phys. Lett. 3, 203 (1990). 

[17] M. A. Horne, A. Shimony, and A. Zeilinger, Phys. Rev. Lett. 62, 2209 (1989). 
FIG. 1. Experimental Setup. Pairs from collinear Type I1 down conversion in a 

BBO crystal are separated from the pump at a prism and directed to a 50:50 beamsplitter. The coincidence registrations in detectors 1 and 2 are recorded as a function 

of the angles O1 and O2 of the Glan Thompson analyzers, for each bandwidth filter 

installed in front of the detectors. 

613 
Table L BeU hequakity Measurements for 4 = 22.5 deg. 

Tabb II. BeU Ineguallity Measurements. 

Table HII: Bell Inequality Violations Using Counts of Table I%. 
P 

A. Shimony 
The Relation Between Physics and Philosophy 

Abner Shimony 

Physics and Philosophy Departments, Boston University, 

Boston, Massachussetts 0221 5 

It is an honor to be invited to give the banquet talk at this Workshop. The invitation by Profs. 

Shih and Rubin provides an opportunity to present to a new audience some theses to which I am 

devoted. I hope that you will hear some things that you have more or less believed for a long time 

but have never heard articulated explicitly. 

Working physicists, I believe, almost inevitably have strong philosophical interests, regardless 

of whether they have taken courses labeled "Philosophy" and whether they have liked what they 

have sampled. Their interest is implicit in the discipline of physics itself. Peter Bergmann, in the 

. introduction to his Basic Theories of Physics I attributed to Einstein the view that "a theoretical 

physicist is . . . a philosopher in workingman's clothes." I would omit the adjective "theoreticaln 

and apply the characterization to experimentalists as well. This claim depends, of course, on a 

conce$ion of what philosophy is. I propose the following: philosophy is the systematic search 

for perspective, for connections among aspects of the world, and for depth of explanation. Some 

physicists are drawn into their profession from the beginning because they have been convinced by 

a few revealing examples that the procedures of physical investigation help to achieve perspective, 

connections, and depth of explanation. These are philosophers from the start. Others are drawn 

in because of their fascination with specific phenomena. In my case, the onset of curiosity about 

the physical world, so far as I can recall, came at the age of four, when I saw my father siphoning 

wine out of a barrel, and I was amazed that the wine went up in the siphon before it descended. 

But even if the route to professional physics is via wonder at specific phenomena, an approach 

to philosophy, in the sense mentioned, is unavoidable, because the physicist's understanding of 

phenomena goes beyond the phenomena themselves to underlying causes and to connections. 

There are various ways to subdivide the discipline of philosophy, but the following will be 

convenient for our purposes: 

1. epistemology, which assesses claims to knowledge; 

2. metaphysics, which considers what kinds of things exist and what are the basic principles 

governing them; 

3. theory of value. 

I'll put greatest stress upon the relations between physics and metaphysics, because they are richer 

and more surprising than its relations to epistemology and value theory. 

Epistemology : 

Among the many problems of epistemology I shall focus on these: (i) what is the proper 

formulation of scientific method, and to what extent is that method rationally justified? (ii) To 

what extent can we disentangle subjective from objective contributions in our experience of the 

world? There were serious and highly intelligent figures in the history of thought, like Descartes, 
who maintained that these and related questions in epistemology had to be answered before the 

substantive work of the sciences could begin, because a foundation is necessary before constructing 

the edifice of science. We can confidently say, I believe, that this architectural metaphor was 

mistaken, because all attempts to establish the foundations of knowledge without any substantive 

assertions about the world seem to have failed. Instead, work on the theory of knowledge and 

work on substantive science had to proceed in tandem, and - judging by the spectacular successes 

of the natural sciences - we can say that somehow human beings are capable of this remarkable 

and somewhat paradoxical intertwining of investigations. 

As to problem (i), concerning scientific method, the great breakthrough occurred in the sixteenth and seventeenth centuries, and was inseparable from the development of the new physics 

and astronomy of Copernicus, Kepler, Galileo, Huygens, and Newton. It had been known from 

the time of the great Greek philosophers, and probably in a vague way many millennia earlier, 

that somehow human beings can learn by sense experience and also by reasoning. The division 

between Plato and his followers, on the one hand, and Aristotle and his followers on the other, concerned the primacy of sense experience or reason. In the work of the new physicists, culminating 

in Newton's Principia, the interplay of the contributions of experience and reason was clarified. 

These physicists recognized that human beings are not so constituted that they can have direct 

sensory experience of the fundamental structure of matter or of the fundamental forces among 

material things. However, if tentative propositions about fundamental physics are formulated in 

mathematical terms, then logical and mathematical reasoning leads to conclusions that can be 

checked by sense experience, notably the trajectories of terrestrial bodies and the apparent positions of planets in the celestial sphere. Newton's successful prediction that the orbit of Saturn 

would be distorted when that planet is close tc Jupiter, and Halley's successful prediction of the 

recurrence of a comet, and other spectacular verifications of predictions, indirectly confirmed the 

tentative propositions about fundament a1 physics which were incapable of direct check. Thus, 

in tandem with the substantive advance in physics and astronomy was the demonstration of the 

power of the scientific method summarized above, called the "hypothetico-deductive method." 

This method was not discredited by the significant refinements of scientific methodology in the 

centuries that followed, notably enrichment with probability theory. Reflection upon scientific 

practice was largely responsible for these refinements, since it was noticed by Gauss, Bessel, and 

others that experimental data are subject to random errors, and they realized that in addition to 

the dominant causes of a natural phenomenon there are always innumerable small perturbations 

that cannot be completely catalogued and yet can be treated rationally in terms of probability 

distributions. Further refinements of scientific method have been made in the twentieth century, 

and there are still controversial questions concerning inductive inference. Nevertheless, I would 

maintain that the great revolutions of physics of the twentieth century were accomplished essentially with a classical methodology. The propositions of relativity theory, quantum mechanics, 

elementary particle theory, etc. were established by applications of the hypothetico-deductive 

method, refined somewhat by probability theory, without a methodological revolution. 

As to problem (ii), concerning the disentanglement of subjective from objective factors in experience, there was not - so far as I can see - a single critical breakthrough, but rather a long, 

intricate, progressive, and still incomplete development. As the human observer was recognized 

to be a natural system, the experiences of that observer could be studied as the termini of causal 
chains, some of which are initiated within the observer, while others are initiated externally. Of 

course, all external stimuli eventually impinge upon the neural and psychic machinery of the 

observer. The naturalistic point of view regarding the human observer has made it possible for 

psychologists and neurophysiologists to understand such matters as optical illusions, the limited 

range of the visual and auditory spectra, and the distortion of perceptual judgments by emotional 

and conceptual bias. In turn, such understanding of the subjective element in experience permits 

scientists to take counter-measures, by instruments and procedures of observation, in order to enhance the revelation of objective factors in experience. Of course, the ultimate subjective elements 

of experience, such as sensory qualities, are entirely beyond present science, and will continue to be 

mysterious until we have a scientific world view that can accommodate consciousness. There has 

been, however, a capital negative achievement concerning the subjectivity of experience: namely, 

an accumulation of evidence that the structure of space and time are not imposed upon experience by the operation of the human mind, as Kant maintained in his doctrine of transcendental 

idealism. Kant's primary argument proceeds from the apparent necessity and universality of geometrical propositions, understood not as mere statements of a formal calculus but as assertions 

about physical space. The invention of non-Euclidean geometry by Lobatchevsky and Bolyai, the 

demonstration of its consistency relative to Euclidean geometry by Klein, and the success of its 

incorporation in general relativity theory are devastating to Kant 's argument. Furthermore, when 

his exemplary geometric instances of synthetic a priori judgments are undercut, then his further 

claim, that the principles of causality, substance, etc. are necessary because they are imposed 

by the mind, is seriously weakened; and it is weakened even further by the triumph of a nondeterministic physics. In other words, the epistemological explanation of the basis of science - 

which was a major part of Kant's program -is undermined, and we are driven to recognize that in 

so far as science is valid, even as an approximation, it is so because it is a quite good description of 

the world as it is. The foregoing anti-Kantian argument is regrettably condensed, but it suffices to 

point to my next thesis: that there are important connections between physics and metaphysics. 

Metaphysics: 

This thesis may be disconcerting to many people, because the word "metaphysics" is commonly 

used as a pejorative by physicists, even by those who are sympathetic to epistemology. Until the 

beginning of the eighteenth century there were great scientists who were also metaphysicians, like 

Aristotle and Leibniz, and a case can be made for adding Newton to the list. The historical 

consequence of Kant's work at the end of the eighteenth century and Hegel's at the beginning 

of the nineteenth was a rift between science and philosophy, regardless of their intentions; and 

Hegel's grandiose and obscurely reasoned attempt at a "System" of thought engendered a negative 

reaction among many careful and critical scientists. The positivism of Mach and the Vienna 

Circle was a consequence of the quest for clarity of expression and rigor of demonstration, and 

to some extent it was a reaction against the excesses of Hegel. The enterprise of systematic 

philosophy - aiming at perspective, connections, and depth, as I proposed at the beginning of 

my talk - became generically suspect, because of the weaknesses of exposition, the excessive 

claims, and the remoteness from scientific practice of the most influential systematic philosopher 

of the nineteenth century. The Vienna Circle, which originally called itself the "Mach verein," was 

nearly unanimous in its condemnation of metaphysics as nonsense and its desire to purge scientific 

discourse of any residue of metaphysics. At the time of the great scientific revolutions of the early 
twentieth century, positivism was a pervasive influence among scientifically oriented philosophers 

and philosophically oriented scientists. It is not surprising, then, that many of the revolutionaries 

gave an anti-met aphysical interpretation to their discoveries. Einstein for a long time expressed 

great sympathy with Mach's ideas, and some of his expositions of special relativity centered 

around the operationalist analysis of space-time concepts like "simultaneity." Heisenberg's initial 

formulation of quantum mechanics dispensed with position and momentum, which he regarded as 

unobservable on a microscopic scale, and tried to express his theory in terms of observables like 

frequency and intensity. Both Einstein and Heisenberg eventually deviated from positivism and 

both espoused some form of physical realism - Einstein in his argument for "elements of physical 

reality" and Heisenberg in a statement that the wave function of an atom can be regarded as a 

description of the atom as a "thing-in-itself." But Bohr to the end of his life gave interpretations 

of quantum mechanics with a positivistic flavor, e.g., saying of the wave function that "we are 

dealing here with a purely symbolic procedure, the unambiguous physical interpretation of which 

in the last resort requires a reference to a complete experimental arrangement." Bohr believed that 

the philosophical implications of quantum mechanics are epistemological, concerning limitations of 

human knowledge and renunciations of unitary pictures of the kind offered by classical physics. His 

consistent playing down of metaphysical interpret ations of quant um mechanics reflect a pervasive 

suspicion of metaphysical speculation in the scientific community, and it is therefore not surprising 

that he maintained a strong grip upon the attitude of physicists towards quantum mechanics 

until fairly recently. I have to confess that in spite of my thesis about the relation of physics 

to metaphysics, I have respect and sympathy for the positivists' pursuit of clarity of concepts 

and rigor of demonstrations. I am skeptical, however, that criteria for the meaningfulness of 

sentences, such as "verifiability," "confirmability," or "falsifiability." can be formulated a priori 

without drastically damaging scientific investigation. Mach was right that there was an obscurity 

in Newton's concept of "absolute motion," but the reason is Galilean invariance, which precludes 

an absolute distinction between rest and uniform rectilinear motion; however, Galilean lnvariance 

is entirely consistent with a clear concept of. absolute acceleration. In sum, the clarification of 

scientific language must proceed in tandem with the progress of scientific knowledge, analogously 

to the linkage of scientific methodology and substantive science. If physics has progressed to the 

point where some of the traditional metaphysical problems can be formulated with precision, and 

even subjected to experimental treatment, then we are presented with a wonderful opportunity, 

that ought not be neglected because of a suspicion of the sterility of metaphysics. And I contend 

that modern physics has reached such a stage of development. We are fortunate enough to be 

invited to a great feast of ideas, and it would be a self-defeating austerity to decline the invitation. 

Here is a partial list of results of modern physics having metaphysical implications. Of course, 

all of these results are subject to modification as science progresses, but it is most unlikely that 

any of them will cease to be good approximations (in the sense of the correspondence principle) 

to their successors. For this reasons, the implications that we can draw from them are likely to 

have a quasi-permanent status. 

I. Modern physics (broadly understood to include chemistry) has established the granular 

character of matter, the present candidates for elementarity being quarks, leptons,. and various 

bosons. That the immense variety of kinds in the natural world can be understood in terms of 

combinations of these elementary units is a great vindication of the atomic vision of Democritus. 
But the modern qualifications of the Democritean vision are as important metaphysically as 

the vindication, especially that none of the elementary units is immortal, but all are subject to 

creation and annihilation in allowable processes; and that certain quantities, notably energy and 

momentum, are conserved in all of these transformation processes, and others are approximately 

conserved. 

2. The overwhelming evidence that the quantum state of a physical system is a complete 

description of it, without any need or place for supplementary "hidden variables," implies that 

various features of the quantum formalism must be attributed to individual physical systems and 

not just to ensembles. There is objective indefiniteness, because in any quantum state some 

physical variables have no definite value; there is objective chance, because the different behaviors of various systems with the same quantum state cannot be attributed to differences among 

hypothetical hidden variables; there is objective probability, because the chance behavior of an ensemble of systems in the same quantum state conforms to definite probability distributions. These 

three properties conjoined are called"potentiality," in Heisenberg's terminology. That the state 

of a physical system involves potentialities and cannot be fully understood in terms of actualities 

is one of the profound metaphysical implications of quantum mechanics. This interpretation of 

the quantum formalism, it may be noted, is entirely different from Bohr's, which is epistemological in character. For instance, the unceertainty principle is not interpreted in Bohr's fashion 

as a limitation upon human ability to measure both position and momentum, but rabher as an 

acknowledgment of the objective fact that position and momentum cannot both be actual. 

3. Quantum mechanics also has a remarkable implication for the relation between parts and 

wholes. There are states of composite systems, called "entangled" by Schroedinger, that cannot 

be expressed as the product of states of the individual components. Entanglement is manifested 

by the occurrence of correlations that cannot be accounted for by product states. By contrast, 

classical physics is pervasively analytic and characterizes the state of composite systems in terms 

of the states of components. Thus quantum mechanics has brought an unprecedented kind of 

holism into our view of the natural world. 

4. When the parts of a composite system are spatially separated, the entanglement of its 

state implies a kind of non-locality: there is correlated behavior that cannot be explained by any 

propagation of causal influences that is not superluminal. Nevertheless, the probabilistic nature 

of quantum mechanics ensures that this kind of non-locality cannot be used to send a signal faster 

than light. For this reason, quantum non-locality can be characterized facetiously as "passion at 

a distance" rather than "action at a distance." In ways that are not fully understood, however, 

we can tentatively say that quantum non-locality requires some modification of classical ideas of 

causality. 

5. The combination of quantum mechanics with the intrinsic indistinguishability of elementary 

particles of a given type leads to the conclusion that that a system of n identical integral spin 

particles are in an entangled fully symmetric state (except in the special case that all are in 

the same one-particle state), whereas n identical half-integral spin particles are in an entangled 

anti-symmetric state. There are important differences between these two types of entanglement, 

especially that the Pauli exclusion principle holds in the latter case and not in the former. In either 

case, however, the status of the individual particle is problematic. The individuality or thisness 

(Latin "haecceitas") does not seem to be manifested except in interactions with macroscopic 
systems, as emphasized in Frisch's famous article, "Take a photon." (I suspect, incidentally, that 

the word "haecceitas" has seldom if ever occurred at a physics conference since the seventeenth 

century, but I use it to emphasize again the thesis that modern physics throws some light upon 

traditional metaphysical problems.) 

5. Gauge field theories provide strong constraints upon the interactions permitted among 

elementary particles. This is an advance of great importance, since a perennial obscurity of the 

Democritean point of view has been the nature of interactions. 

6. Both the special and the general theories of relativity transform our conception of space 

and time. The special theory asserts an inseparability of space and time in a much stronger 

sense than in classical kinematics. Classically, space-time is a four-dimensional affine space, with 

metric properties restricted to space separately and time separately, whereas the space-time of 

special relativity is a four-dimensional metric space (with a three-one metric). General relativity 

envisages a dynamical interaction of space-time with matter, instead of regarding the former as a 

fixed arena in which material dynamics occurs. 

7. The interface of physics and biology has a metaphysical implication of great significance: 

. that all biological processes not involving mentality can in principle be understood in terms of 

physical interactions. Even the program of pre-biotic evolution - of understanding the emergence 

of mutually catalytic molecules and eventually explaining the evolution of the genetic code in 

terms of ordinary physical processes - is very promising. I should note that the qualification 

"not involving mentalityn was not inserted for the purpose of hedging but for the positive reason 

that, in my opinion, a further scientific revolution is needed in order to understand the place of 

mentality in the natural world. 

8. Elementary particle theory and cosmology already have, or are on the verge of having, 

profound metaphysical implications. But since the assessment of these implications is beyond my 

expertise, I encourage others to make them explicit. 

Finally, I wish to warn against construing this list of metaphysical implications as indicating 

that all is settled or nearly settled in fundamental physics, and that we need only read off the 

corollaries in order to do philosophy properly. Rather, there seem to me to be dark clouds showing 

the need for radical changes in physical theory. One dark cloud is the problem of the reduction 

of the wave packet, which I prefer to call the problem of the actualization of potentialities. Of 

course, the advocates of many-worldgand decoherence theories say that there is no such problem, 

but I am skeptical of their solution (which obliterates the distinction between potentiality and 

actuality), not for reasons of sophisticated physics, but because of a straightforward philosophical 

analysis of the relevant concepts. The outcome of this philosophical analysis is to provide a 

motivation for a modification of quantum dynamics, such as the stochastic modification advocated 

by Pearle, Ghirardi-Rimini-Weber, Piron, Karolyhazy-Frenkel, Gisin, Percival, Diosi, Penrose, 

and Bell. They all hope to provide a physical, rather than an epistemological, explanation of the 

actualization of potentialities. Here is an instance in which philosophical considerations may turn 

out to provide valuable heuristics for physical investigations. Another dark cloud is the problem of 

mentality. Before one takes too literally the phrase "theory of everything," applied to the hopedfor theory that explains all forces of nature, the spectrum of elementary particles, the values of 

all parameters, and the structure of space-time, one should inquire whether such a theory could 

even in principle account for the immediacy of consciousness. 
Theory of value: 

I shall say little about the relation between physics and theory of value, partly because of my 

lack of expertise. However, I do have a few strong opinions. There is, I believe, a real danger 

that our emotional and moral senses will be dulled by the advance of technology. The case is 

not hopeless, and the humanities and social sciences have serious suggestions for confronting this 

danger. But I think that the natural sciences can also make a contribution. A sense of wonder is 

one of the great motivations of the investigations of science, and the discoveries of science should 

provide much new nourishment of the sense of wonder. If physics is to offer an antidote against 

emotional flattening, however, it is essential that its discoveries be understood as revelations of 

the real structure of the world, to a good approximation, and not just as recipes for making 

laboratory predictions. My personal experience is that penetrating even a little into the secrets of 

the universe provides an emotional satisfaction that is like the satisfaction traditionally provided 

by religion. I suspect that others share this experience. 
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Third International Workshop on Squeezed States and Uncehty Relations 

C-NAS5-30376 

Code 910.1 D. Han, Y. S. Kim, N. H. Rubin, Y. Shih, and W. W. Zachary, Editors 

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13. ABSTRACT (Maximum 200 words) 

The Third International Workshop on Squeezed States and Uncertainty Relations was held at the eTnivenity of 

Maryland Baltimore County on August 10 - 13,1993. This workshop was jointly organized by the Univenity of 

Maryland and the Lebedev Physical Institute of the Russian Republic. These workshops were initiated by Y. S. 

Kim, of the Univenity of Maryland, College Park. The first of these workshops was held in 1991 at the University 

of Maryland, College Park, and the second, in MOSCOW in 1992. 

The purpose of these workshops is to bring together an international selection of scientists to discuss the latest 

developments in Squeezed States in various branches of physics, and in the undentanding of the foundations of 

quantum mechanics. At the third workshop, special attention was given to the influence that quantum optics is 

having on ow undentanding of quantum measmment theory. 

The fourth meeting in this series will be held in the People's Republic of China. The principal organizer will be 

Q. C. Peng, of the Shanxi Univenity at Taiyun, P. R C. 

Squeezed States, Quantum Optics, Uncertainty Relations, Croup, Bell's Inequality 

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