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Workshop on

Squ .t.zed Sl:ates

and Uncertainty

Relation,"

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NASA Conference Publication 3135

Workshop on

Squeezed States

and Uncertainty

Relations

Edited by

D. Han

NASA Goddard Space Flight Center

Greenbelt, Maryland

Y. S. Kim

University of Maryland

College Park, Maryland

W. W. Zachary

COMSERC

Howard University

Washington, D.C.

Proceedings of a workshop held at the

University of Maryland

College Park, Maryland

March 28-30, 1991

N/kSA

National Aeronautics and

Space Administration

Office of Management

Scientific and Technical

Information Program

1992
Principal Organizers:

D. Han (NASA - Goddard Space Flight Center)

Y. S. Kim (University of Maryland at College Park)

W. W. Zachary (Howard University)

Local Organizing Committee:

D. R. Brill (Univ. of Maryland at College Park)

J. D. Franson (Johns Hopkins University Applied Physics Lab.)

M. Kafatos (George Mason University)

P. McGrath (Laboratory for Physical Sciences)

H. S. Pilloff (Office of Naval Research)

A. K. Rajagopal (Naval Research Laboratory)

Y. H. Shih (Univ. of Maryland at Baltimore County)

This workshop was supported in part by the Office of Naval Research, and the Goddard Space

Flight Center, National Aeronautics and Space Administration. The Maryland Academy of

Science now has the policy of cosponsoring high quality scientific meetings. This Workshop is

the first meeting cosponsored by the Academy. The administration of this Workshop was

provided in part by Westover Consultants, Inc., 6303 Ivy Lane, Suite 416, Greenbelt, MD

20770.

ii
PREFACE

TheWorkshoponSqueezedStatesandUncertaintyRelationswasheldattheUniversity

of Marylandat CollegeParkonMarch28- 30, 1991.ThisWorkshopwaslargely supportedby

theGoddardSpaceFlight Centerof theNationalAeronauticsandSpaceAdministration.The

ideato holdtheWorkshopof thisnaturewasinitiatedbytheOfficeof NavalResearch.

Thepurposeof thisWorkshopwasto studypossibleapplicationsof squeezedstatesof

light. Specifically,theWorkshopwill beconcernedwith thefollowing questions.

(1) Whatphysicscanwedowith squeezed-statelasers?

(2) Whatimpactdoesthesqueezedstategive tootherbranchesof physics?

TheWorkshopbroughttogetheractiveresearchersin squeezedstatesof light andthosewhomay

find theconceptof squeezedstatesusefulin theirresearchefforts,particularlyin thefoundations

of quantummechanics.

Theeffortmadefor thismeetingwill becontinuedby theSecondInternationalWorkshop

onSqueezedStatesandUncertaintyRelationsto beheldin MoscowonMay 25- 29, 1992.The

principalorganizersfor thisMoscowmeetingareV. I. Man'ko(LebedevPhysicalInsitute)and

Y. S.Kim (Univ.of MarylandatCollegePark). WeexpectthattheMoscowmeetingwill attract

manyresearchersfrom Europeaswell asthosefrom theUnitedStates.

iii
Registered Participants

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

Alsing, Paul M., Dept. of Physics, Univ. of Oregon, Eugene, OR 97403

Aragone, Carlos, Dept. of Physics, Universidad Simon Bolivar, Caracas 1080A, Venezuela

Aravind, P. K., Dept. of Physics, Worcester Polytechnic Institute, Worcester, MA 01609

Artoni, Maurizio, Dept. of Physics, City College of CUNY, New York, NY 10031

Ashrafi, Solyman, Computer Science Co, 10110 Aerospace Building, Lanham, MD 20706

Bergrman, Keren, Dept. of Electrical Engineering, Massachusetts Inst. of Tech., Cambridge,

MA 02139

Bernstein, Herbert J., School of Natural Science, Hampshire College, Amherst, MA 01002

Birman, Joseph L., Dept. of Physics, City College of CUNY, New York, NY 10031

Boyd, Derek, Dept. of Physics, Univ. of Maryland, College Park, MD 20742

Brandt, Howard, Harry Diamond Laboratories, Adelphi, MD 20783

Brill, Dieter R., Dept. of Physics, Univ. of Maryland, College Park, MD 20742

Brooke, James A., Dept. of Mathematics, Univ. of Saskatchewan, Saskatoon, SAS, S7N 0W0,

Canada

Buzek, Vladimir, Optics Section, Blackett Lab., Imperial College, London SW7 2BZ, United

Kingdom

Campos, Richard A., Dept. of Applied Physics, Columbia University, New York, NY 10027

Carmichael, Howard J., Dept. of Physics, Univ. of Oregon, Eugene, OR 97403

Caves, Carlton M., Center for Laser Studies, Univ. of Southern California, Los Angeles, CA

90089

Cha, Boyoung, Dept. of Physics, Univ. of Maryland, College Park, MD 20742

Chang, Chung Yun, Dept. of Physics, Univ. of Maryland, College Park, MD 20742

Chiao, Raymond Y., Dept. of Physics, Univ. of California, Berkeley, CA 94720

Choi, Hyung Sup, Dept. of Physics, City College of CUNY, New York, NY 10031

Cruze, Stephen P., TASC, 12107 Maple Forest Ct., Apt. L, Fairfax, VA 22030

PRECEDING PAGE BLANK NOT FILMED

V
D'Ariano, Giacomo Mauro, Dipto. di Fisica, Univ. di Pavia, 1-27100 Pavia, Italy

Dattoli, Giuseppe, INN.SVIL, ENEA, Box 65, 1-0044 Frascati, Italy

DeFacio, Brian, Dept. of Physics, Univ. of Missouri, Columbia, MO 65211

Debergh, Natalie, Dept. of Theoretical Physics, Univ. of Liege, B-4000 Liege, Belgium

Deutch, Ivan, Dept. of Physics, Univ. of California, Berkeley, CA 94720

Dodonov, V. V., Dept. of Physics, Moscow Physical Technical Institute, 140160 Zhukovsky,

Moscow Region, U.S.S.R.

Dowling, Jonathan P., AMSMI-WS, U.S. Army Missile Command, Weapons Science

Directorate, Redstone Arsenal, AL 35898

Ford, Lawrence, Dept. of Physics, Tufts University, Medford, MA 02155

Fortini, Pierluigi, Dept. of Physics, Univ. of Ferrara, 1-44100 Ferrara, Italy

Franson, J. D., Applied Physics Lab., Johns Hopkins University, Laurel, MD 20723

Fuchs, Christopher, Dept. of Physics, Univ. of North Carolina, Chapel Hill, NC 27599

Gallis, Michael, Dept. of Physics, Pennsylvania State University, Schuylkill Haven, PA 17972

Garrison, John C., Dept. of Physics, L-297, Lawrence Livermore, Lab., Livermore, CA

94550

Greenberg, O. W., Dept. of Physics, Univ. of Maryland, College Park, MD 20742

Greenberger, Daniel, Dept. of Physics, City College of CUNY, New York, NY 10031

Griffin, James J., Dept. of Physics, Univ. of Maryland, College Park, MD 20742

Grishchuck, Leonid, Steinberg Astronomical Inst., Moscow State University, 119899 Moscow,

U.S.S.R.

Han, Daesoo, Code 936, Goddard Space Flight Center, National Aeronautics and Space

Administration, Greenbelt, MD 20771

Hartle, James B., Dept. of Physics, Univ. of California, Santa Barbara, CA 93108

Haus, Hermann A., Dept. of Electrical Engineering, Massachusetts Inst. of Tech., Cambridge,

MA 02139

Henry, Richard W., Dept. of Physics, Bucknell University, Lewisville, PA 17837

Hillery, Mark, Dept. of Physics, Hunter College of CUNY, New York, NY 10021

vi
Hochberg,David, Dept.of Physics,VanderbiltUniversity,Nashville,TN 37235

Hope,Deborah,Dept.of Physics,AustralianNationalUniversity,Canberra, ACT 2601,

Australia

Horne,Michael,Dept.of Physics,StonehillCollege,NorthEaston,MA 02357

Hu, Bei-Lak, Dept. of Physics,Univ. of Maryland,CollegePark,MD 20742

Hwang,Euyheon,Dept. of Physics,HamptonUniversity, Hampton,VA 23668

Jacobs,Verne L., Naval ResearchLab., Code4694,Washington,DC 20375

Jin,Insik, Dept.of Physics,Univ. of Maryland,CollegePark,MD 20742

Joneckis,Lance,ResearchLab.of Electronics,MassachusettsInst.of Technology,

Cambridge,MA 02139

Kafatos,Minas,Dept. of Physics,GeorgeMasonUniversity, Fairfax,VA 22030

Kennedy,T. A. Brian, Schoolof Physics,GeorgiaInst. of Tech.,Atlanta,GA 30332

Ketov,SergayV., Dept.of Physics,Univ. of Maryland,CollegePark,MD 20742

Kiess,Thomas,E., Dept.of Physics,Univ. of Maryland,CollegePark,MD 20742

Kim, Myung Shik, Dept.of Physics,SogangUniversity,Box 1142,Seoul 100-611,S.Korea

Kim, ShoonK., Dept.of Chemistry,TempleUniversity,Philadelphia,PA 19122

Kim, Y. S.,Dept. of Physics,Univ. of Maryland,CollegePark,MD 20742

Klauder, JohnR., Dept. of Physics,Univ. of Florida, Gainesville,FL 32612

Kostelecky,Alan, Dept.of Physics,IndianaUniversity,Bloomington,IN 47405

Kuo, Chung-I, Dept. of Physics, Tufts University, Medford, MA 02155

Larchuk, Todd, 218 Wellington Place, Matawan, NJ 07747

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

Lee, Seung-Hee, Dept. of Electrical Engineering, Northwestern University, Evanston, IL

60208

Ma, Xin, Dept. of Chemistry, Univ. of Houston, Houston, TX 77204

Mandel, Leonard, Dept. of Physics, Univ. of Rochester, Rochester, NY 14627

vii
Man'ko,MargaritaA., LebedevPhysicalInstitute,53LeninskyProspect,117924Moscow,

U.S.S.R.

Man'ko,Vladimir I., LebedevPhysicalInstitute,53LeninskyProspect,117924Moscow,

U.S.S.R.

Maguire,GeraldQ., Dept.of ComputerScience,ColumbiaUniversity,New York, NY 10027

Mari, Carlo,INN.SVIL, ENEA, Box 65,1-00044Frascati,Italy

McGrath,Paul,Lab.for PhysicalSciences,4928CollegeAve.,CollegePark, MD 20740

Misner, Charles,Dept. of Physics,Univ. of Maryland,CollegePark,MD 20742

Morris, Randall,Gov.ElectronicsDiv., GeneralElectricCo., Moorestown,NJ 08057

Movshovich,Roman,NIX& T Bell Labs.,1C-247,600MountainAve.,MurrayHill, NJ

07974

Mufti, Asif Ali, Dept.of Physics,Univ. of Arizona,Tucson,AZ 85721

Nikitin, S., LebedevPhysicalInstitute,53LeninskyProspect,117924 Moscow,U.S.S.R.

Noz, Marilyn E., Dept.of Radiology,New York University, New York, NY 10016

Oneda,Sadao,Dept.of Physics,Univ. of Maryland,CollegePark,MD 20742

Ortolan,Antonello,Dept.of Physics,Univ. of Ferrara,1-44100Ferrara,Italy

Park,ShimC.,Dept.of Physics,Baylor University,Waco,TX 76798

Paz,JuanPablo,Dept.of Physics,Univ. of Maryland,CollegePark,MD 20742

Pedrotti,Leno,Dept.of Physics,Univ. of Dayton,Dayton,OH 45469

Pilloff, Herschell,Div. of Physics,Office of NavalResearch,800NorthQuincy Street,

Arlington,VA 22217

Rajagopal,A. K., Naval ResearchLaboratory,Code6830R,Washington,DC 20375

Rajalakshmi,R.,511HamptonCourt,Blacksburg,VA 24060

Ralph,Timothy,Dept.ofPhysics,AustralianNationalUniv., Canberra,ACT 2601,Australia

Raymer,MichaelG., Dept.of Physics,Univ. of Oregon,Eugene,OR 97403

Reifler, Frank,Gov.ElectronicsDiv., GeneralElectricCo., Moorestown,NJ 08057

Reiss,Howard,Dept.of Physics,AmericanUniversity,Washington,DC 20016

viii
Renaud, Rocio Jauregui, Inst. de Fisica, Univ. Nacional Autonoma de Mexico, Mexico 01000,

Mexico

Rice, Perry, Dept. of Physics, Miami University, Oxford, OH 45056

Richard, Jean-Paul, Dept. of Physics, Univ. of Maryland, College Park, MD 20742

Rubin, Morton H., Dept. of Physics, Univ. of Maryland at Baltimore County, Baltimore, MD

21228

Saif, Babak V., NASA - Goddard Space Flight Center, Code 726, Greenbelt, MD 20771

Sarty, Gordon E., Dept. of Mathematics, Univ. of Saskatchewan, Saskatoon, SAS, S7N 0W0,

Canada

Schleich, Wolfgang, Abteilung f. Theoretische Physik III, Uniw_rsitat Ulm, W-7900 Ulm,

Germany

Schmitt, Harry A., Naval Weapons Station, Code 063, Seal Beach, CA 90740

Scully, Marian O., Dept. of Physics, Univ. of New Mexico, Albuquerque, NM 87131

Sergienko, Alexandr V., Dept. of Physics, Moscow State University, 119899 Moscow,

U.S.S.R.

Shapiro, Jeffrey, Dept. of Electrical Engineering, Massachusetts Inst. of Tech., Cambrdige, MA

02139

Shepard, Scott, Dept. of Electrical Engineering, Massachusetts Inst. of Tech., Cambrdige, MA

02139

Shevy, Yaakov, Dept. of Applied Physics, California Inst. of Tech., Pasadena, CA 91125

Shih, Yan-Hua, Dept. of Physics, Univ. of Maryland at Baltimore Campus, Baltimore, MD

21228

Sica, Louis, Applied Optics Branch, Naval Research Lab., Code 6530, Washington, DC

20375

Singer, Faige, Dept. of Electrical Engineering, Columbia University, New York, NY 10027

Stoler, David, Research Center, Kearfott Guidance and Navigation Corp., Little Falls, NJ

07424

Svensmark, Henrik, NORDITA, Blegdemsvej 17, DK-2100 Copenhagen 0, Denmark

Suzuki, Norikazu, Max-Plank-Institut f. Kernphysik, W-6900 Heidelberg, Germany

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

ix
Toll, John S., Dept. of Physics, Univ. of Maryland, College Park, MD 20742

Um, Chung-In, Dept. of Physics, Korea University, Seoul 131-701, S. Korea

Van Hine, Christopher, Naval Surface Weapons Center, Silver Spring, MD 20903

Villarreal, Carlos, Inst. de Fisica, Univ. Nacional Autonoma de Mexico, Mexico 01000,

Mexico

Vinogradov, A. V., Lebedev Physical Institute, 53 Leninsky Prospect, 117924 Moscow,

U.S.S.R.

Vyas, Reeta, Dept. of Physics, Univ. of Arkansas, Fayetteville, AR 72701

Weiner, Brian, Dept. of Physics, Pennsylvania State University, DuBois, PA 15801

Wheeler, John A., Dept. of Physics, Princeton University, Princeton, NJ 08544

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

Yeon, Kyu Hwang, Dept. of Physics, Chungbuk National University, Cheongju 360-763, S.

Korea

Yi, Pang, Dept. of Physics, Univ. of Maryland, College Park, MD 20742

Yilmaz, H., Hammamatzu Photonics K.K., 105 Church Street, Winchester, MA 01890

Yin, Xiaosong, Dept. of Physics, Miami University, Oxford, OH 45056

Yuen, Horace P., Dept. of Electrical Engineering, Northwestern University, Evanston, IL

60208

Zachary, W. W., COMSERC, Howard University, Washington, DC 20059

Zaidi, H. R., Dept. of Physics, Univ. of New Brunswick, Fredericton, NBK, E3B 5A3,

Canada

Zeilinger, Anton, Dept. of Experimental Physics, Univ. of Innsbruck, A-6020 Innsbruck,

Austria

Zhu, Chengjun, Dept. of Physics, Univ. of Florida, Gainesville, FL 32612

Zorn, Gus T., Dept. of Physics, Univ. of Maryland, College Park, MD 20742
TABLE OF CONTENTS

INTRODUCTION

PAGE

............. °°°°'°'°°°'°°°'°°6"°°°°°°°°°°°°°°°*°°° .......... O,O,OOOOOOOOOOOOOOOOOOOOOOOOOO1

SECTION I MEASUREMENT PROBLEMS AND THE

EINSTEIN-PODOLS KY-ROSEN PARADOX ...................................... 3

Nonclassical and Nonlocal Effects

in the Interference of Light

L. Mandel ............................................................................................. 5

The Ultimate Quantum Limits on

the Accuracy of Measurements

Horace P. Yuen ................................................................................... 13

Violations of a New Inequality

for Classical Fields

J. D. Franson ...................................................................................... 23

Use of Entanglement in Quantum Optics

Michael A. Home, Herbert J. Bernstein,

Daniel M. Greenberger, Anton Zeilinger ......................................................... 33

EPR Experiment and Two-Photon

Interferometry -- Report of A

Two-Photon Interference Experiment

Y. H. Shih, M. H. Rubin, A. V. Sergienko .................................................... 47

The Energy-Time Uncertainty Principle

and the EPR Paradox: Experiments

Involving Correlated Two-Photon

Emission in Parametric Down-Conversion

Raymond Y. Chiao, Paul G. Kwiat,

Aephraim M. Steinberg .......................................................................... 61

Higher-Order Quantum Entanglement

Anton Zeilinger, Michael A. Home,

Daniel M. Greenberger ........................................................................... 73

Algorithmic Information Theory and

the Hidden Variable Question

Christopher Fuchs ................................................................................ 83

Photon Polarization Version of the

GHZ-Mermin Gedanken

Thomas E. Kiess ................................................................................ 87

xi
Proposalfor aQND Which-Path

MeasurementUsingPhotons

M. G. Raymer,S.Yang .... oo.°° .... °°°o .... °ooo°ooooo°o°°°°oooo°o°o .... oo° .... oooooo°°°o°oo°°°o 91

Localization of One-Photon State

in Space and Einstein-Podolsky-Rosen

Paradox in Spontaneous Parametric

Down Conversion

A. N. Penin, T. A. Reutova, A. V. Sergienko ............................................... 95

Generalized Parametric Down

Conversion, Many Particle

Interferometry, and Bell's Theorem

Hyung Sup Choi .................................................................................. 97

SECTION II SQUEEZED STATES OF LIGHT .................................................... 105

Going Through A Quantum Phase

Jeffrey H. Shapiro ............................................................................... 107

Quadratic Squeezing: An Overview

M. Hillery, D. Yu, J. Bergou .................................................................. 125

Quantum Spatial Propagation of

Squeezed Light In a Degenerate

Parametric Amplifier

Ivan H. Deutsch, John C. Garrison ........................................................... 137

Squeezed Pulsed Light from a

Fiber Ring Interferometer

K. Bergman, H. A. Haus ...................................................................... 147

Equilibrium Temperature of Laser

Cooled Atoms in Squeezed Vacuum

Y. Shevy .......................................................................................... 163

On the Measurement of Time for the

Quantum Harmonic Oscillator

Scott R. Shepard ................................................................................. 177

The Origin of Non-Classical Effects In

a One-Dimensional Superposition of

Coherent States

V. Buzek, P. L. Knight, A. Vidiella Barranco ............................................... 181

xii
Limitationson Squeezing and Formation of

the Superposition of Two Macroscopically

Distinguishable States at Fundamental

Frequency in the Process of Second

Harmonic Generation

S. P. Nikitin, A. V. Massalov ................................................................. 195

Field Quantization and Squeezed

States Generation in Resonators With

Time-Dependent Parameters

V. V. Dodonov, A. B. Klimov, D. E. Nikonov ,,.,,,o°°°ooo°°°,°.o,o°°,ooo,ooo**ooo°,oooooo* 199

Radiation Force on a Single

Atom in a Cavity

M. S. Kim ........................................................................................ 217

Squeezed Light from Conventionally

Pumped Multi-Level Lasers

T. C. Ralph, C. M. Savage .................................................................... 225

Statistical Properties of Squeezed Beams

of Light Generated in Parametric Interactions

Reeta Vyas ........................................................................................ 229

Nondestructive Measurement of Intensity

of Optical Fields Using Spontaenous

Parametric Down Conversion

A. N. Penin, G. Kh. Kitaeva, A. V. Sergienko ............................................. 233

Short-Cavity Squeezing in Barium

D. M. Hope, H.-A. Bachor,

P. J. Manson, D. E. McClelland .............................................................. 237

Amplitude Squeezed Light From a Laser

D. L. Hart, T. A. B. Kennedy ................................................................. 241

SECTION III THEORETICAL DEVELOPMENTS ................................................ 245

Squeezed States and Path Integrals

Ingrid Daubechies, John R. Klauder .......................................................... 247

Supercoherent States and Physical Systems

B. W. Fatyga, V. Alan Kostelecky,

Michael Martin Nieto, D. Rodney Truax .................................................... 261

Squeezed States, Time-Energy Uncertainty

Relation, and Feynman's Rest of the Universe

D. Han, Y. S. Kim, Marilyn E. Noz oo°°°°°ooo°°°°***°°°°°°°°°.°°°°*°°°°°°** 6°°oo°**o°°o'Q°°°°269

°°°

XUl
Wavelets and the Squeezed States of Quantum Optics*

B. DeFacio .......................................................................................... 283

Coherent States and Parasupersymmetric

Quantum Mechanics,

Nathalie Debergh ................................................................................ 297

A Gaussian Measure of Quantum Phase Noise

Wolfgang P. Schleich, Jonathan P. Dowling ................................................ 299

Ideal Photon Number Amplifier and Duplicator

G. M. D'Ariano .................................................................................. 311

Decoherence in Quantum Mechanics

and Quantum Cosmology

James B. Hartle .................................................................................. 327

Squeezed States in the Theory of

Primordial Gravitational Waves

Leonid P. Grishchuk ............................................................................ 329

Squeezed States: A Geometric Framework

S. T. Ali, J. A. Brooke, J.-P. Gazeau ........................................................ 339

Wormholes and Negative Energy From

the Gravitationally Squeezed Vacuum

David Hochberg ................................................................................. 343

Exact Solution of a Quantum Forced

Time-Dependent Harmonic Oscillator

Kyu Hwang Yeon, Thomas F. George,

Chung In Um .................................................................................... 347

Nonclassical Depth of a Quantum State

Ching Tsung Lee ................................................................................ 365

Towards A Wave Theory of Charged

Beam Transport: A Collection of Thoughts

G. Dattoli, C. Mari, A. Torre .................................................................. 369

Entropic Uncertainty Relation

at Finite Temperature

Sumiyoshi Abe, Norikazu Suzuki °.°° .... °oooooooo°°°°°°° .... o°°°°°°°°°°°°°°°°°°°° .... ° ....... 377

The Hamiltonian Structure of Dirac's Equation

In Tensor Form and Its Fermi Quantization

Frank Reifler, Randall Morris .................................................................. 381

xiv
INTRODUCTION

Squeezedstateswerepredictedtheoreticallyin the1970's.Theybecameaphysical

realityduringtheperiod1985-1988.Effortsarebeingmadetoproducemoreefficient squeezedstatelasers.Morerefinedtheoreticaltoolsarebeingdevelopedforthisnewphysical

phenomenon.Oneof thepressingquestionsduringthe 1990'swill be:What shouldwe dowith

squeezedstates?This is themainquestionwewantedto addressin thisWorkshop.

Therearemanywho saythatthepotentialfor industrialapplicationsis enormous,asthe

historyof theconventionalasersuggests.Therearealsothosewho saythatthesqueezedstate

is only afad andwill disappearwithin twoyears.In orderto find a moreaccurateanswerto the

question,letusmakethefollowing observations.

(1) All thosewhoworkedsohardto producesqueezedstatesof light arecontinuing

theireffortsto constructmoreefficientsqueezed-statelasers.Quitenaturally,theyarelooking

for newexperimentsusingtheselasers.Newexperimentsoftenrequirenewideasfrom

branchesof physicssomewhatremovedfrom theirown. For instance,theconceptof squeezed

statesarosein partfrom thedesireto detectgravitationalwaves.

(2) Thephysicalbasisof squeezedstatesis theuncertaintyrelationin Fockspace,

whichis alsothebasisfor thecreationandannihilationof particlesin quantumfield theory.

Indeed,squeezedstatesprovideauniqueopportunityforfield theoreticiansto developa

measurementtheoryfor quantumfield theory.Thepreconditionfor formulatingfield theoretic

measurementtheoryis acorrectunderstandingof theconventionalmeasurementtheoryand

relatedexperiments.

(3) Thetheoryof squeezedstatessharesacommonmathematicallanguagewith many

otherbranchesof physics.Thebasiclanguageis theLorentzgroup,whichplaysimportantroles

in quantumfield theory,thephase-spacepictureof quantummechanics,relativity,elementary

particlephysics,condensedmatterphysics,canonicaltransformationsin classicalmechanics,and

crystalandpolarizationoptics. It is possiblefor thephysicistsin thesefields to learnnew

lessonsfrom thephysicsof squeezedstates.

TheWorkshopwasattendedby manyof theoriginatorsof squeezedstatesof light as

well asthosewhospentmanyyearsstudyingthefoundationsof quantummechanicsandrelated

problems.Therewerealsomanystudents.TheEditorsareveryhappyto presentthepapersby

thoseactiveresearcherson squeezedstatesandrelatedsubjects.
I. MEASUREMENT PROBLEMS AND THE EINSTEIN-PODOLSKY-ROSEN PARADOX

3 PRECEDING PAGE BLANK NOT FILMED
N92-22 046

NONCLASSICAL AND NONLOCAL EFFECTS IN THE INTERFERENCE OF LIGHT

L. Mandel

Department of Physics and Astronomy

University of Rochester

Rochester, New York 14627

INTRODUCTION

Although we tend to think of

optical interference as a classical

wave phenomenon, recent experiments

have revealed a number of effects that

are not describable in classical

terms. This is particularly true of

interference effects involving the

detection of a photon pair. We shall

refer to them as fourth order interference, on the grounds that the joint

probability density for the detection

of one photon at _ at time t and

another at _2 at time t is proportional to the fourth order correlation

function of the field (Ref. I)

F(2'2){rlt,r2 t) = <E(i-)(rlt)EJ-)(r2t )

X ÷) ^

E_ (r2t)E_-) (rlt)> (1)

This probability is readily measured

when two photodetectors are positioned

at r_ and r 2 and the signals from the

two detectors are fed to a coincidence

counter that registers 'simultaneous'

detections by the two detectors in

coincidence.

4'th ORDER INTERFERENCE MEASUREMENTS

In the special case in which the

two points x_,x 2 lie on a llne, and

This research was supported by the

National Science Foundation and by the

U.S. Office of Naval Research.

the light is produced by two sources

A,B on a parallel line such that A

emits one photon and B emits one

photon, it can be shown that (Refs.

2,3)

r(2,2) 2 (Xl-X2) [1÷cos L ] , (2)

where L = _/0. 0 is the small angle

subtended by the two points A,B at x_

or x 2 and h is the wavelength. L is

the same fringe spacing that is

encountered in the more usual second

order interference. According to Eq.

(2) the visibility of the fourth order

interference effect can be 100%,

despite the absence of phase correlation between the two sources. By

contrast a classical field that exhibits 4'th order interference cannot

achieve a visibility higher than 50%.

(Refs. 2-4)

2

IkO I.Ii O.O u

Fig. I Experimental results showing

4'th order interference. [Reproduced

from Ref. 6.]

5

PRECEDING PAGE BLANK NOT FILMED
We have observed greater than 50%

visibility in several recent interference experiments,(Refs. 5,6) in

which the two photons were generated

together in the process of spontaneous

parametric down-conversion in a nonlinear crystal. (Ref. 7) It is

convenient to produce the interference

pattern by mixing the two incoming

photons with the help of a 50%:50%

beam splitter with a photodetector at

each output port. Figure I shows the

experimental results when the rate of

coincidence counting, after some

corrections are applied, is plotted

against the position of one detector,

while the other detector remains

fixed. The interference pattern has

the expected periodicity L, and the

observed 75% visibility shows that we

are dealing with a quantum phenomenon,

because there is no classical field

that can give rise to more than 50%

visibility.

The same mixing technique has

been applied to the measurement of the

time separation between the two

photons on a femtosecond time scale,

and to study violations of locality.

In order to understand the principle

of the method, let us consider the

symmetric beam splitter with intensity

transmissivity T and reflecticlty R

(R+T : I), shown in Fig. 2. Let a,B

label the two input ports and _,v the

two output ports. Suppose that the two

photons enter in the state I1a,IB>, in

which each photon is in the form of a

\

T

Flg. 2 The beam splitter.

short wave packet and the two wave

packets are identical and overlap

completely in time. In order to arrive

at the output state I_out> we first

note that there are three

possibilities: (a) one photon appears

at each output (11 ,I >); (b) both

photons, appear at output port

(12 ,0 >J; (c) both photons appear at

output port v (I0 2 >) It can be _,

shown (Ref. 8) that l_out > is given by

the linear superposltion

l_out > = (T-R)II ,1,_ >+i 2_

" (1%,o,,>÷1%,%>), (3)

from which it follows that when T :

I/2 : R, both photons always appear

together at one or the other output.

If there is a photodetector at each

output, there will be no coincidence

detections (other than accidentals),

because the corresponding two-photon

probability amplitude vanishes by

destructive interference. But if one

photon is delayed slightly relative to

the other one by some amount T, the

destructive interference is no longer

complete, and the coincidence probability P(_) rises from zero with

increasing T. When _ exceeds the

duration of the wave packet and the

two wave packets no longer overlap,

P(_) be comes constant and independent

of "c.

._ 800-

m

B

o 6o0.

i.oo200-

'5

:9 o

- ;_L,-C, 0 I_'_._/.:_;_

l , I

Position of beam splitter (/_rn)

Fig. 3 Measured coincidence rate as a

function of time delay in fsec between

the two photons. [Reproduced from Ref.

9.]
Figure 3 shows the result of such

a coincidence counting experiment

(Ref. 9) in which each photon wave

packet had a length of about I00 fsec.

It can be seen that the observed

probability P(_) is close to zero for

= 0, and rises to become constant

when +_ equals or exceeds about 100

fsec. We therefore have a technique

for measuring the time separation

between two pulses of light and the

length of the pulse, when each pulse

consists of a single photon. The time

resolution achieved in this experiment

was about 3 fsec, which is about a

million times shorter than the resolving time of the detectors and the

associated electronics. In some later

experiments (Ref. I0) the resolution

was further measured to about I fsec,

which is less than half an optical

period.

THE FRANSON EXPERIMENT

A number of experiments have also

been performed for which there is no

adequate classical model to explain

the 4'th order interference.(Refs. 11-

14) Let us consider the experimental

situation illustrated in Fig. 4, which

Fig. 4 The principle of the Franson

4'th order interference experiment.

[Reproduced from Ref. 13.]

was first proposed and discussed by

Franson.(Ref. 15) A source emits two

photons A and B simultaneously, each

with some center frequency _A,mB and

bandwidth Am. The photons emerge in

two different directions and fall on

two photodetectors D A and DB without

ever coming together. Some beam splitters and mirrors forming two similar

interferometers are introduced, so as

to provide two alternative paths for

each photon, as shown: a direct path

and a longer indirect path. Let the

propagation time difference between

the two paths be T+T A in channel A and

T+_ B in channel B, with T >> I/Am,

_A,_B << I/Am.

Because the path difference in

each of the two interferometers

greatly exceeds the coherence length

c/Am of the light, no second order

interference is expected. The probability tlhat a photon is detected by

D A does not change significantly when

TA is changed slightly, and similarly

for DB. However, if we calculate the

joint probability PAB that a photon is

detected by D A and by D B in

coincidence, which can be measured

with a coincidence counter, we find

that it exhibits interference of the

form

PAB _ [1+rl cos(mA_A+mB_B+const.)]. (4)

n can be 1 00% if the coincidence

resolving time T R is sufficiently

short, and it is about 50% when T R >>

T >> I/Am.

This result is best understood as

an interference of a photon pair.

There are several different ways in

which a coincidence can occur: (a)both

photons follow the short interferometer paths and arrive

simultaneously at the two detectors;

(b)both photons follow the long interferomet er paths and arrive

simultaneously at the detectors;

(c)one photon follows the long path

and one follows the short path but the
time difference T+_A (say) lies within

the coincidence resolving time TR, so

that the photons are deemedto arrive

'simultaneously'. As these probabilities are intrinsically

indistinguishable, we have to add the

corresponding probability amplitudes

and then square in order to arrive at

the probability PAB"This leads to the

result in Eq. (4). The interference

exhibits non-local features, because

the outcome of a measurement

registered by DA depends not only on

_A but also on _B' even though the

interferometer in channel B cannot

influence what happens in channel A.

This interference effect has

recently been observed (Refs. 13,14)

in experiments in which the two

photons were produced by downconversion in a non-linear crystal.

Figure 5 shows the results of such an

experiment in which one mirror was

movedpiezoelectrically and the twophoton coincidence rate was measured.

Evidently there is interference

despite the fact that the two photons

never mix and the path difference

exceeds the coherence length of the

light more than 100-fold.

-_ 0 _

i_ - _-_-'_._-_-_._._

D_plae_ent _ _i (rim)

Fig. 5 Results of the Franson-type

interference experiment. [Reproduced

from Ref. 13.]

The question whether a classical

field can give rise to this kind of

behavior has been discussed.(Refs. 16-

18) Let us attempt to describe the

experimental situation in Fig. 4 in

terms of classical waves. Let

VA(t),VB(t) be the complex analytic

signals representing the stationary

light field leaving the source. Then

the fields at the two detectors DA,D B

can be expressed in the form

WA(t) = aVA(t) + BVA(t+T+_A)

WB(t) = aVB(t) + BVB(t+T+_B)

(5)

where a,B are constants characteristic

of the beam splitters and mirrors. The

joint probability that a photoemission

occurs at DA at time t and at D B at

time t+_ is proportional to the twotime correlation

PAB(_) = <IWA(t)I21WB(t+_)I2> (6)

The integral of PAB(T) with

respect to T over the resolving time

TR of the coincidence counter yields

the coincidence counting rate, which

is proportional to

ITR/2 d_<IWA(t)I21WB(t+_)I2> (7)

C =--TR/2

With the help of Eqs. (5) it may be

shown (Ref. 16) that _c contains an

interference term of the form

J

"2B2 ITR/2 interfer. = _ dT

_-TR/2

× <VA(t)VA(t+T)VB(t+_)VB(t+T+T)>

-i(_A_A+mB_B )

x e + c.c., (8)

together with a somewhat similar

interference term involving

exp[i(cOB_B-_OA_A) ]. But _C also contains a non-oscillatory or background

contribution
background + ITR/2d_(la 14+ IBI 4)

"-TR/2

x <IA(t)IB(t+T)>+ Ia121812

x (<IA(t)IB(t+T+T)>

+ <IA(t)IB(t+T-T)>) , (9)

which represents light background for

the interference. Here IA(t) =

IV A(t) 12, etc. The presence of the

interference terms suggests that

certain classical fields can exhibit

the observed interference effect.

Let us now examine the

magnitudes. Whereas the integrand in

Eq. (8) tends to zero with increasing

T, that in Eq. (9) does not. We recall

that for any ergodic process correlations must eventually die out. It

follows that for sufficiently long

the terms in T are no longer correlated with those without _, and

therefore for a stationary field,

<VA(t)VA(t+T)VB(t+_)VB(t+_+T)>

÷ <VA(t)VA(t+T)><VB(t)VB(t+T)>

o , (lo)

because T >> I/A_. The integrand in

Eq. (9), on the other hand, tends to

the constant value (IeI_1812) 2<IA><IB >

as _ ÷ ®. Therefore if we integrate

with respect to _ over a sufficiently

long resolving time TR, the background

term will greatly exceed the interference terms, and the visibility of

the interference will become negligibly small. In ref. 16 it was argued

that the integrand in Eq. (8) has a

range in T of order I/Am. But even if

it has a longer range, so long as TR

is much longer than this range, the

visibility of the interference given

by Eqs. (8) and (9) would be very

small.

Actually, a classical model of

the light from a parametric downconverter fails for other, more

compelling reasons. It can be shown

(Ref. 19) that for any classical field

whose correlation time is much shorter

than TR,

_AB-_AB accid. < r_AA -_A accid.' (11)

where GAB is the coincidence counting

rate when signal light falls on one

detector and idler light on the other,

and _AA is the self-colncidence rate

for the signal. Accidental coincidence

contributions are subtracted on both

sides. In practice, classical inequality (11) is, however, found to be

violatedby down-converted light by

several hundred standard

deviations. (Ref. 19)

EXPERIMENTAL TEST OF THE DE BROGLIE

GUIDED WAVE THEORY

Finally, we describe a recent

experiment to test a prediction of the

de Broglle guided wave theory relating

to interference. (Refs. 20,21 )

According to this theory, which is a

hybrid of classical and quantum concepts, there exist both photons and

electromagnetic waves, with the latter

acting as guides for the former. But,

in addition to yielding the probability for detecting a photon, the

electromagnetic wave is supposed to

have a physical reality that extends

beyond being a probability wave.

Figure 6 shows the essential

features of the experlment.(Ref. 22)

Three 50%:50% beam splitters

BS_,BS2,BS , form a Michelson type of

interferometer, and BS2 can be adjusted piezoelectrically to move

through one or two microns. Any light

that penetrates BS, and BS 2 falls on

detector D, and D 2, respectively. The

counting rates R_,R_ of the two detec-
cs__._ _m,D_

D,,.._f,

Fig. 6 Outline of the interference

experiment to test the de Broglie

guided wave theory. [Reproduced from

Ref. 22. ]

tors are measured as a function of

beam splitter BS 2 displacement Ax,

together with the coincidence counting

rate R_2. The interferometer is fed

with the signal (s) and idler (i)

light produced by down-conversion in

the non-llnear crystal NLC, as shown,

and it is balanced so that the paths

of i from NLC to BS 3 and of s from NLC

to BS_ to BS 3 are equal.

Reference to Fig. 6 shows that

the idler can only reach detector D_.

On the other hand, the signal can

reach both detector D 2 and detector

D I, and moreover it can reach D_ via

the two different paths NLC to BSI to

BS 3 to BS_ to D_ and NLC to BS_ to BS 2

to BS_ to D_. If the distances BS_ to

BS 3 and BS_ to BS 2 are nearly equal,

these two paths interfere, so that

counting rate R_ of DI, which is given

by the expectation of the square of

the wave function _ at D_, depends on

Ax. On the other hand, the counting

rate R 2 of Da, which is given by

< I_2 I 2> is independent of gx.

According to the guided wave theory,

(Ref. 21) the counting rate R_2 of D_

and D2 in coincidence is proportional

to the expectation <I_i121_212>, and

since I_212 is constant and independent of Ax, whereas I_i I2 shows

interference, this would be expected

to exhibit much the same interference

as R_.

Let us compare that prediction

with the quantum mechanical one. As

there is only one signal and one idler

photon emitted at one time, and because the idler can only reach D_, it

follows that whenever a coincidence is

registered, D_ must have detected the

idler photon and D2 the signal photon.

But reference to Fig. 6 shows that, in

that case, there is no ambiguity in

the photon paths, because the wave

function _ collapses along the two

paths s to BS_ to BS 3 to BS_ and s to

BS_ to BS 2 to BS_ that interfere.

Therefore R_2 should exhibit no interference or dependence on gx. A similar

conclusion is reached by a mathematical treatment of the problem.(Ref. 22)

The results of the experiment are

shown in Figs. 7 and 8. Figure 7 gives

o

Fig. 7

_4 OJB

I Z 3 •

Phase m mul_pl_ of _r

The measured photon counting

rate R_ as a function of the displace _

ment of BSa. [Reproduced from Ref.

22.]

Displacement of BS z in Izm

0.2 0,4 0.6 O.S

c.

_e

T-_ Io I

I ± _- i

2 3 4;

Pha_ in mulUpl_ of _

Fig. 8 The measured two-photon

coincidence counting rate as a function of BS_ displacement. [Reproduced

from Ref. 22. ]

I0
the measured photon counting rate R,

as a function of the displacement _x

of BS 2. As expected, this exhibits

interference attributable to the two

alternative paths of s to D_. But this

is predicted by all theories, by

quantum mechanics, by classical wave

theory and by the guided wave theory.

Figure 8 gives the measured twophoton coincidence rate R,2 , after

subtraction of accidental counts, as a

function of BS 2 displacement• This

time there is no evidence of any

interference, in agreement with quantum mechanics, but in violation of the

guided wave theory. We have therefore

disproved one prediction of the guided

wave theory. Needless to say, this

conclusion applies only to the particular form of the theory described

above, in which probabilities are

calculated very much as in semiclassical radiation theory.

The fourth order interference

technique is capable not only of very

high accuracy, such as the measurement

of the time separation between two

photons to I fsec accuracy, but it

also lends itself to the exploration

of quite fundamental questions about

our quantum world•

REFERENCES

I •

.

•

•

.

Glauber, R.J., Phys. Rev. 130, p.

2529 and 131, p. 2766 (1963).

Mandel, L., Phys. Rev. A 28, p.

929 (1983).

Ghosh, R., C.K. Hong, Z.Y. Ou and

L. Mandel, Phys. Rev. A 34, p.

3962 (1986)•

Richter, Th. , Ann. Phys.

(Leipzig) 36, p. 266 (1979)•

Ghosh, R. and L. Mandel, Phys.

Rev. Lett. 59, p. 1903 (1987).

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•

•

10.

11.

12.

13.

14.

15.

16.

Ou, Z.Y. and L. Mandel, Phys.

Rev. Lett. 62, p. 2941 (1989).

Burnham, D.C. and D.L. Weinberg,

Phys. Rev. Lett. 25, p. 84

(19701) •

Ou, Z.Y. , C.K. Hong and L.

Mandel, Opt. Commun. 63, p. 118

(1987)•

Hong, C.K. , Z.Y. Ou and L.

Mandel, Phys. Rev. Lett. 59, p.

2044 (1987)•

Ou, Z.Y. and L. Mandel, Phys.

Rev. Lett. 61, p. 54 (1988).

Ou, Z.Y., X.Y. Zou, L.J. Wang and

L. Mandel, Phys. Rev. A 42, p.

2957 (1990).

Rarity, J.G., P.R. Tapster, E.

Jakeman, T. Larchuk, R.A. Camp•s,

M.C. Teich and B.E.A. Saleh,

Phys. Rev. Lett. 65, p. 1348

(1990)•

Ou, Z.Y., X.Y. Zou, L.J. Wang and

L. Mandel, Phys. Rev. Lett. 65,

p. 321 (1990).

Kwiat, P.G., W.A. Vareka, C.K.

Hong, H. Nathel and R.Y. Chiao,

Phys. Rev. A 41, p. 2910 (1990)•

Franson, J.D., Phys. Rev. Lett.

62, p. 2205 (1989)•

Ou, Z.Y. and L. Mandel, J. Opt.

Soc. Am. B 7, P. 2127 (1990).

17. C arm i chael, H. , private

communication (1991).

20.

Franson, J.D., to be published.

Ou, Z.Y. , L.J. Wang and L.

Mandel, to be published (1991)•

de Broglle, L. , The Current

Interpretation of Wave Mechanics,

Elsevier, Amsterdam, 1969.

21. Croca, J.R., A. Garuccio, V.L.

I!
22.

Lepore and R.N. Morelra, Found.

Phys. Lett. 3, P. 557 (1990).

Wang, L.J., X.Y. Zou and L.

Mandel, Phys. Rev. Lett. 66, p.

1111 (1991).

12
6 2 go j

1oh 

N92"22047

THE ULTIMATE QUANTUM LIMITS ON THE ACCURACY OF

MEASUREMENTS t

Horace P. Yuen

Department of Electrical Engineering and Computer Science

Department of Physics and Astronomy

Northwestern University

Evanston, IL 60208

ABSTRACT

A quantum generalization of rate-distortion theory from standard communication and information theory is developed for application to determining the ultimate performance limit of measurement systems in physics. For the estimation of a real or a phase parameter, it is shown that

the root-mean-square error obtained in a measurement with a single-mode photon level N cannot

do better than ,-_ N -1, while -_ exp{-N} may be obtained for multi-mode fields with the same

photon level N. Possible ways to achieve the remarkable exponential performance are indicated.

INTRODUCTION

Given whatever physical constraints one has to operate with, what is the best possible system

one can build for the measurement or estimation of a physical parameter of interest? It is evident

that a systematic approach to the answer of this class of questions is of great interest in physics,

which is so much concerned with the detection and accurate measurement of various quantities,

from routine temperature gauging to the detection of very weak gravitational radiation. In this

paper, I will describe a systematic theory for answering these questions. Conceptually, this theory

is directly transplanted from ordinary (classical) information and communication theory, although

technically the new quantum issues may greatly complicate the actual workout of a solution. As

illustrations, I will provide the ultimate quantum limits on the accuracy of estimating a phase

parameter, and also an arbitrary real parameter, when an optical field of a given power level is

employed. Let N be the available number of photons of a narrowband optical field. For both

the estimation of a phase parameter and a real amplitude parameter, the following results will be

proved. For a single-mode field, the best root-mean-square error one may obtain is

1 1

5¢_, 5r ,,_ _ (1)

whereas for a multimode field with sufficiently many modes one may achieve

5¢ ~ e-N, 5r ~ e-N. (2)

Moreover, the theory provides various indications on how one may actually approach the problem

of realizing a multimode system that would yield the remarkable exponential performance given by

(2). In the following, the underlying information theoretic results will first be explained before the

quantum situation is discussed. Due to limitations in space-time, everything can only be briefly

outlined. Nevertheless, I hope the discussion is self-contained and comprehensible.

t This work was supported by the office of Naval Research.

13
RATE-DISTORTION LIMIT

The theory of information transmission pioneered by Shannon (refs. 1-3) can be immediately

adapted to provide a systematic answer to the above class of questions. For a system described by

classical physics, the solution goes as follows. First, we assign an a priori probability distribution

p(u) on the parameter u we are interested in estimating. This parameter would modulate a physical variable in whatever physical system we pick for extracting information about this parameter.

For example, if u is the amplitude of a gravitational wave, the system may be a Michelson interferometer with the physical variable a certain optical phase of an electromagnetic field mode.

Some measurement is to be made on the system, such as a determination of the field strengths

of the mode, to extract information on the physical variable through which an estimate of u is to

be obtained. Let x be the physical variable, and y the measurement variable which is in general

random with conditional probability p(y]x),with z itself a function of u depending on the specific

scheme. Both the cases of discrete and continuous variables will be included throughout with

proper interpretation of the probabilities as a distribution or a density function of the random

quantities under discussion.

The condition probability p(ylx) defines a channel in information theory, with x the channel

input and y the channel output. For any input probability p(x), the joint probability p(y, x) =

p(ylx)p(x) is specified from which one can evaluate the average mutual information between x and

Y,

I(x;y) / p(x[y)

-- p(x,y)log p---(-_dxdy. (3)

The entropy of a single random variable can be defined as average self-information

H(v) - I(v;v) (4)

in which p(v]v) is to be interpreted as a Kronecker or Dirac delta. The units in (3) and (4) are

given as bits per channel use (per channel input) and bits per source symbol if the log is taken

to be of base 2, and as nats per use if the log is of base e. Shannon's channel coding theorem

and its converse (ref. 1) state that successive independent samples of a random variable v can

be transmitted over a channel p(ylx) with zero probability of error if and only if H(v) _< I(x;y).

However, for a noisy channel, i.e., when y does not specify x uniquely, channel encoding and

decoding are required to get zero error probability which is only obtained in the limit of arbitrarily

long codes. Note that we have deviated somewhat from the standard notations in information

theory to avoid conflict with later quantum notations. Also, the coding theorem is usually stated

in terms of the capacity of the channel, which is defined to be the I(x;y) obtained by maximizing

over p(z) under whatever further constraints one may impose on z. Typically, one assigns a cost

function 13(x) and constraint the average cost to be under a given level B. The capacity C(B)

will then be an increasing function of B. Roughly speaking, C(B) is the maximum number of

information bits one can transmit error-free over a channel with an average resource level B.

The rate-distortion function R(D) of a random variable u is defined to be the minimum I(u;v)

between u and any another random variable v such that the average distortion between u and v

E[d(u, v)] - f

d(u, v)p(u)p(vlu)dudv (5)

is at or below a given level D, where the distortion function d(u,v) is a given measure of the

difference between u and v. When u is a continuous real parameter, d(u, v) is often chosen to be

lu- v] 2 or ]u - v I. The minimization of I(u;v) is carried over p(vlu ) subject to the constraint

E[d(u, v)] < D. (6)

14
Onemay think of v as a data-compressed version of the source variable u -- v represents u with

an average distortion D but it requires less bits to represent v than u. Shannon's source coding

theorem with a fidelity criterion and its converse (ref. 3) state that a source can be asymptotically

represented with an average distortion D if and only if at least R(D) bits per source symbol is

provided. Again, source encoding and decoding are in general required to achieve such minimum

distortion in the limit of long codes. Nevertheless, this result shows that roughly speaking, R(D) is

the minimum number of information bits required to represent a source with an average distortion

level D.

What is the minimum average distortion D one can get for transmitting a source variable u

over a channel with resource B? The answer is provided by Shannon's joint source-channel coding

theorem (refs. 4-5) in what is often called the rate-distortion limit or rate-distortion bound. By

combining the source and the channel coding theorems, the bound is

D >_ R-'C(B) (7)

where R -1 is the inverse of the monotone function R(D). It is important to emphasize that the

Shannon theorem and its converse state that (7) is the ultimate limit and can be approached by

an actual system that employs source coding and channel coding separately. It does not say that

it can only be approached by separate source and channel coding. In fact, the following example

illustrating the power of the rate-distortion bound also shows that it is sometimes possible to

achieve it (to get the actual minimum) without any coding or nonlinear modulation at all.

Let u be a zero-mean Gaussian random variable with variance au,2 and distortion measure the

squared error d(u, v) = [u - v] 2. The rate distortion function in this case is well known [refs. 3-5],

R(D) = 1_log_ a2 0 <__D < a 2

= 0 D > (S)

Consider an additive Gaussian noise channel

y = x + n (9)

where n is a zero-mean Gaussian noise variable with variance Af0 statistically independent of x.

With a given power level, E[x 2] < S, it is wellknown [refs. 1-2, 4-5] that the capacity is

C(S) = 1 S_S_)

_log(1 -t- Afo " (10)

The rate-distortion bound solves the following problem which cannot be solved in any other way, to

my knowledge. Suppose each sample of u matches each use of x, i.e., the rate that u is generated is

equal to the rate that x can be transmitted over the Gaussian channel. We are interested in finding

the best signal .processing scheme before transmission over the channel and after transmission in

receiver processing that would yield an estimate of u with minimum mean-square error. From

(7),(8) and (10) one gets

D > o-,, 2 (1 ,-.{...-_0)-1 (11)

It turns out that the right side of (11) can be obtained by simply letting x = o'ulS1/2u and

estimating it(y) = a_S-1/2y, i.e., direct linear transmission and estimation without coding or

nonlinear modulation is already optimal as verified from the rate-distortion limit. On the other

hand, a direct optimization approach to this or most other comunication problems is very difficult

to just formulate, not to mention writing down the optimization conditions.

Despite the power of rate-distortion theory, we are faced with two complications in its application to measurement problems in physics. The first is derived from the fact that in a measurement

15
system,one may have very little or no room at all for source coding, basically because the parameter u in this case may be entirely out of one's control for further processing before modulating

onto the physical variable x. Thus, while the bound (7) still remains a limit, one is no longer sure

that the limit can be achieved arbitrarily closely. This problem can be overcome by replacing R(D)

by 7¢(D), which is defined to be the number of bits required to represnet u to a distortion level

D given a specific simple source coding scheme such as uniform quantization, or no coding at all.

The second problem is similarly derived from the fact that no channel coding may be employed. In

the same way, one can replace C(B) by an average mutual information Z(B) which incorporates

whatever constraint one must face, including perhaps some modulation but no coding. In contrast

to the source case T_(D), in the evaluation of Z(B) it may be difficult to actually take into account

precisely the constraints one operates with. Anyhow, in a way exactly parallel to the Shannon

joint source-channel coding theorem, the following generalized rate-distortion limit applies with

whatever additional constraints in the present measurement situation,

D >_ Ti-IZ(B). (12)

Depending on the specific case, the limit (12) may be much higher than (7).

In addition to providing an answer on how accurately one may actually perform a measurement

through (7) or (12), this theory also indicates a way to approach the best performance, namely,

through channel coding or modulation to achieve C(B) or 2"(B), assuming source coding cannot

be carried out. Illustrations will be given in the following quantum problems.

MEASUREMENT WITH A QUANTUM SYSTEM

The development of squeezed and nonclassical lights [ref. 6-7] has been strongly motivated by

their possible applications to precision measurements. It is logical, in fact imperative, to ask for

the ultimate limit of measurements in quantum physics; quantum fluctuations, being an intrinsic

feature of nature as we understand it, would have to be taken into account in assessing such

ultimate limits. Contrary to what one may first think, the uncertainty principle does not provide

the answer either by itself or in conjuction with other additional considerations as discussed in

the following two prime cases of continuous parameter estimation. Consider first the case of a

real parameter A defined over the whole real line N. For simplicity, let A be a Gaussian random

variable and N be the average available number of photons in a single-mode field that one can

use to capture A. That is, we wish to design the best measurement system, which is represented

by the way A modulates the quantum state p_ of the mode and the quantum measurement one

chooses to make on the mode, subject to the constraint

/tr[p_ata]p()_)d_ <_ N (13)

where p()_) is the probability of _, a the model photon annihilation operator, and p_ a density

operator on the Hilbert space of quantum states 7-/. To include all possible quantum measurements

such as heterodyning, a general quantum measurement on the system _ is represented, as far as

the measurement probability is concerned, by a positive operator-valued measure (POM) [ref. 8-10]

generalizing the usual selfadjoint operator description. In a notation including possible operatorvalued distributions, a POM X with measurement value x C Nn is a function x _ X(x) such

that each X(x) is a bounded positive semidefinite selfadjoint operator and all the X(x) sum to

the identity operator, i.e.,

X(x)>_O, (14)

f X(x)dx = I. (15)

16
When X(x) = Ix >< x I are orthogonal projectors, the POM X can be desribed by a unique

selfadjoint operator obeying the functional calculus

f f(x)X(x)dx. (16) f(X)

For a general POM, (16) does not hold. When X is measured on a system in state p, the probability

that x is obtained is given by tr[pX(x)]. Mathematically, the problem is to find a mapping

A _-_ pa, A C _, a quantum measurement X, an estimate A(x) of A, such that the resulting meansquared error between A and A is as small as possible subject to the constraint (13). It should be

clear that the uncertainty principle is of little help in solving this problem, although some weaker

conclusion may be obtained with its help [refs. 11-14]. Thus, if one assumes that X is to be a

single field-quadrature operator, and the criterion is changed to average signal-to-noise ratio, then

the uncertainty relation between conjugate quadratures

1

< Aa_ >< Aa_ >_> 1---6 (17)

can be used to show that the use of two-photon coherent states (TCS) or squeezed states in the

narrow sense [refs. 6,15] is optimum. In fact, it yields a mean-square error given by, from (11),

Do = (1 _a__ .]2 (18) + 2N" "

As will be shown in the next section, this turns out to be very close to the best one can do.

In the second case, consider the estimation a real parameter defined over a finite interval,

which for simplicity we take to be a phase parameter ¢ C (-Tr, 7r]. Mathematically, the problem

is exactly the same as above except that p(A) is changed. The number-phase uncertainty relation

in whatever form or interpretation,

1

ANA_ > _ (10)

is of no help at all here. In contrast to (17), (19) does not even place a limit on how small A¢

may get under an average photon number constraint because AN may still be arbitrarily large.

More significantly, in an actual measurement problem it is not the quantum fluctuation alone that

is important in determining the limit. The total quantum state (the full statistics) and the way

energy is distributed could be just as important. We now show how the rate-distortion theory can

be generalized to provide the answers.

ULTIMATE QUANTUM LIMITS

To obtain the ultimate possible performance for the abowe two problems, we note that with

the mean-square error criterion the rate distortion function R(D) for a Gaussian random variable

A with variance a 2 is given by (8) while that for a uniformly distributed ¢ E (-_-, _r] is difficult

to evaluate exactly. However, the wellknown Shannon upper and lower bounds [ref. 3] on R(D)

gives a very accurate estimate in this case: in nats per symbol

0.419 - logv_ <_ Re(D) _< 0.595 - logv/-D (20)

If the magnitude distortion function d(u,v) = lu - v] is employed instead, the R(D) for the

uniform phase parameter is known exactly while that of a Gaussian random variable is known

parametrically [ref. 16]. In both cases, they are quite close to that given by the Shannon lower

bound, and are approximately the same as the mean-square case with the natural replacement of

D by v/-D. Moreover, for the uniform phase variable the T_(D) function obtained from a uniform

quantizer (digitizer) can be easily evaluated. For the mean-square-criterion,

17
he(D) -_ 0.595 - logv'D (21)

which is exactly the upper bound part of (20)! Thus, uniform digitization without coding is

quite close to optimum in this case. For the Gaussian case, uniform quantization also leads to

a 7_(D) with a similar functional form to R(D), but with a further fixed constant difference. In

fact, it is well known that for a large class of memonylas sources and distortion measures simple

quantization already leads to a performance close to the rate-distortion limit. The upshot of our

discussion is that independently of the exact distortion criterion one chooses and without the need

of coding, the T_(D) functions for our two cases can be accurately estimated and they are close to

the rate-distortion limit R(D).

Given 7_(D) or R(D), the quantum limitation on communication or measurement is determined by substituting the ultimate quantum information transmission capacity C into C(B) in

the bound (7). For a given system, the ultimate quantum capacity C is the maximum average

mutual information I(x;j) one may obtain by picking an input alphabet J, discrete or continuous,

probability pj on J, a map j _ pj,j C J, pj density operators on the system state space 7-(, and

a POM X(x) subject to whatever constraints one may have. It is clear that the channel coding

theorem and its converse hold for this capacity C. The actual evaluation of C can be very complicated due to the added optimization over pj and X which are entirely of quantum mechanical

origin. However, for certain cases including the following ones, the evaluation can be carried out

with the help of an entropy bound [ref. 10]. Thus, for a single-mode optical field with average

photon number constraint N,

pjtr[psata] < N, (22)

J

the ultimate quantum capacity is achieved by photon number eigenstates with the result [ref. 10]

C(N) = (N + 1)log(N + 1)- NlogN. (23)

For a narrowband optical field with m modes of approximately the same frequency and a constraint

N on the total number of average photons in all m modes, the ultimate capacity is [ref. 10]

N N m

C( N) = mlog(m + 1)+ mlog( :_ + 1).

We may also be interested in the capacity of TCS with homodyne detection [ref.17]

(24)

Crcs HOM mlog(2 N 1) (N)= +

m

and the capacity of coherent states with heterodyne detection

(25)

cHET(N) = mlog( N + 1). (26)

Going back to our single-mode optimal measurement problem, it follows from (8) and (23) that

for a Gaussian parameter r with variance a 2, the best root-mean-square error 6r = _ one may

obtain is

a (1+ 1) -N,_ a N>>I (27)

6r- N+I N e--N '

The suboptimum TCS and CS performance are close to the optimum (27); from (25)-(26) with

re=l,

18
_rTC S _ (7 _rCS _ (7

2N + 1' N + 1 (28)

Note that 5r TCs can be achieved without coding or nonlinear modulation from (18) as discussed in

the previous section, while 5r cs = _ without coding or modulation. Thus, the use of TCS can

be viewed as an alternative to codin 9 or nonlinear modulation in at least the single-mode case. For

the phase parameter ¢ with uniform distribution, it follows from (21) and (23) that the ultimate

limit is

while with TCS and coherent states

1

5¢ ,,_ e--N' N >> 1 (29)

5¢TC S ,,_ 1 5¢cs ,,_ __1 N >> 1. (30)

2N' N'

Again, it is known that the use of TCS or other phased-squeezed states would lead directly to

5¢ ,-- _ [ref. 18], while the use of coherant states without coding or modulation yields only

5¢cs ,

,'_ :_.

Consider now the multimode limit under the constraint of the same number of photons N.

From (8) and (24), we have

5r=a 1+

which implies that 5r would go to zero at least as quickly as e -y for rn _> 0.1N.

coherent states,

(31)

For TCS and

5r TCS = cr 1 + , 5r cs = a 1 + (32)

which implies that they would go to zero as e -g for m >_ N. Similarly for the phase parameter ¢,

5¢ ,,_ (N)-m (1 + N) -N

5¢ TCS '_ (1 + -- , 5r cs ,'_ (1 +

(33)

raN---)-m (34)

This multimode behavior as indicated by equ (2) is not unexpected from communication theory, as

a larger number of modes is equivalent to signal space of higher dimension which means that the

different messages can be placed farther apart in signal space to combat the effect of noise [refs.

2,19]. This is familiar in what is called FM quieting in fi'equency modulation, and is commonly

referred to as the exchange of bandwidth with signal-to-noise ratio. The remarkable feature is that

a large number of modes moves the N -1 dependence of the ultimate limit to exp {-N} which is

so much more accelerated!

There are several approaches one may consider for obtaining such exponential performance,

although in a measurement rather than a communication situation one cannot be sure that the

above capacities can be actually obtained. Since the number-states channel is noise-free, its

capacity can be achieved without coding. Indeed it is achieved by a rather simple modulation

scheme and the effect of a small nonideal residue noise is not expected to affect the resulting

performance too much. The problem remains to find a scheme which, for a measurement system,

19
would naturally capture the parameter _ (either r or _b)of interest in such a modulation scheme

or another one which is nearly as good. On the other hand, one may consider the use of nonlinear

modulation on TCS or coherent states; different nonlinear modulation schemes are known to get

quite close to the rate-distortion limit in many classical communication situations [ref. 20]. In

particular, if nonlinear modulation or coding is to be employed, one may consider dispensing with

the use of TCS and staying with coherent states, with the resulting loss of a factor of 2 in the

exponent but a tremendous gain in practicality. Many different nonlinear modulation schemes may

be employed. For example, it is wellknown that a simple pulse frequency modulation in which the

modulated signal is given by

s(t,_) = sin(wo +/3)Qt, 0 < t < T (35)

where /3 is a known constant and E the energy of the signal, could lead to an increase in the

signal-to-noise ratio for the estimation of a phase parameter in the presence of additive Gaussian

noise by a fact m 2, where m = WT is the total number of modes in s(t, ,_) with W the frequency

bandwidth of the signal. While such a simple scheme may not lead to exactly an exponential

performance (2), it may still be a large improvement as the N -1 performance of (1) becomes

(raN)

In conclusion, the quantum generalized rate-distortion theory and the possible actual systems

it may suggest seem to hold much promise for greatly improved precision measurements in physics,

as our two important examples discussed in this paper amply demonstrate.

o

.

3.

Shannon, C.E., 1948,

pp. 379-423, 623-656.

Shannon, C.E., 1949,

REFERENCES

"A Mathematical Theory of Communication", Bell Sys. Tech. J. 27,

"Communication in the Presence of Noise", Proc. IRE 37, pp. 10-21.

Shannon, C.E., 1959, "Coding Theorems for a Discrete Source with a Fidelity Criterion",

IRE Nat. Cov. Rec., Part 4, pp. 142-163.

4. Gallager, R.G., 1968, Information Theory and Reliable Communication, Wiley, New York.

5. Blahut, R.E., 1987, Principles and Practice of Information Theory, Addison-Wesley, Reading, Mass.

6. Loudon, R. and Knight, P.L. 1987, "Squeezed Light", J. Modern Opt. 34, pp. 709-759.

7. Teich, M.C. and Saleh, B.E.A., 1988, "Photon Bunching and Anti-Bunching", Progress in

Optics XXVI, ed. by E. Wolf, Elsevier, New York, pp. 1-104.

8. Davies, E.B., 1976, Quantum Theory of Open Systems, Academic Press, New York.

9. Yuen, H.P. 1982, "Generalized Quantum Measurements and Approximate Simultaneous

Measurements of Noncommuting Observables", Phys. Lett. 91A, pp. 101-104.

10. Yuen, H.P., and Ozawa, M., 1991, "The Ultimate Information Carrying Limit of Quantum

Systems", submitted to Phys. Rev. Letters.

11. Yuen, H.P., 1976, "States that Give the Maximum Signal-to-Quantum Noise Ratio for a

Fixed Energy", Phys. Lett. 56A, pp. 105-106.

20
12. Yuen, H.P., and Shapiro, J.H., 1978,"Optical Communication with Two-Photon Coherent

StatesPart I: Quantum-State Propagation and Quantum-NoiseReduction", IEEE Trans.

Inform. Theory, vol. IT-24, pp. 657-668.

13. Shapiro, J.H., Yuen, H.P., and Machado Mata, J.A., 1979, "Optical Communication with

Two-Photon Coherent States - Part II: Photoemissive Detection and Structured Receiver

Performance", IEEE Trans. Inform. Theory, vol. IT-25, :pp. 179-192.

14. Yuen, H.P. and Shapiro, J.H., 1980, "Optical Communication with Two-Photon Coherent

States Part IIh Quantum Measurements Realizable with Photoemissive Detectors", IEEE

Trans. Inform. Theory, vol. IT-26, pp. 78-92.

15.

16.

17.

18.

19.

20.

Yuen, H.P., 1976, "Two-Photon Coherent States of the Radiation Field", Phys. Rev. A, vol.

13, pp. 2226-2232.

Viterbi, A.J., and Omura, J.K., 1979, Principles of Digital Communication and Coding,

McGraw Hill, New York.

Yuen, H.P., 1975, "Generalized Coherent States and Optical Communications", Proc. 1975

Conf. Information Sciences and Systems, John Hopkins Univ., Baltimore, MD, pp. 171-177.

Shapiro, J.H., and Shepard, S.R., 1991, "Quantum Phase Measurement: A System-Theory

Perspective", Phys. Rev. A, vol. 43, pp. 3795-3818.

Wozencraft, J.M., and Jacobs, I.M., 1965, Principles of Communication Engineering, Wiley,

New York.

Van Trees, H.L., 1971, Detection, Estimation and Modulation Theory, Part II, Nonlinear

Modulation Theory, Wiley, New York.

21
N92-22048

VIOLATIONS OF A NEW INEQUALITY FOR CLASSICAL FIELDS

J.D. Franson

The Johns Hopkins University

Applied Physics Laboratory

Laurel, MD 20723

ABSTRACT

Two entangled photons incident upon two distant interferometers can give

a coincidence counting rate that depends nonlocally on the sum of the phases

of the two interferometers. It has recently been shown that experiments of

this kind may violate a simple inequality that must be satisfied by any

classical or semi-classical field theory. The inequality provides a graphic

illustration of the lack of objective realism of the electric field. The

results of a recent experiment which violates this inequality and in which the

optical path length between the two interferometers was greater than i00 m are

briefly described.

INTRODUCTION

It has been shown I'2 that two-photon interferometer experiments can

violate Bell's inequality 3 and a number of experiments 4-7 have demonstrated

effects of that kind. Several experiments 4,6 based upon the two-photon

interferometer of Ref. I have not, however, violated Bell's inequality due to

the limited visibility (50%) of the interference fringes that results when the

resolving time of the photon detectors and electronics is not sufficiently

fast.

Those experiments may 8, however, violate a surprisingly simple

inequality that must be satisfied by any classical or semi-classical field

theory. The inequality follows directly from the assumption that the classical

field has some well-defined value and thus illustrates the lack of objective

realism exhibited by the quantum-mechanical field.

This paper will briefly review the nature of two-photon interferometry

and then derive the new inequality; the derivation closely follows that of

Ref. 8. Some additional details of the derivation that are not contained in

Ref. 8 but are required for applications to actual experiments are presented

in the Appendix. The results of a recent two-photon interferometer experiment

performed over a distance of I00 meters will be briefly described. Finally,

some comments will be made with regard to the connection between uncertainty

relations and inequalities of this type.

TWO-PHOTON INTERFEROMETRY

The experiments of interest 4,6,7 are outlined in Figure i. Two

coincident photons are emitted by parametric down-conversion and travel in

different directions toward two identical interferometers. Each interferometer

contains a shorter and a longer path, and the difference AT in transit times

over the two paths is taken to be much larger than the coherence time of the

photons. Nevertheless, interference between the quantum-mechanical amplitudes

for the photons to have both traveled the shorter paths or the longer paths

produces a modulation in the coincidence counting rate Rc given I by

PRECEDING PAGE BLANK NOT FILMED

23
Rc = _1 R coc°s2 I@i + @2 2+ _oAT 1 (I)

Here Re0 is the coincidence rate with the beam splitters removed, 81 and 82 are

phase-shifts introduced into the two longer paths, and _0 is the frequency of

the pump laser. Eq. (i) violates Bell's inequality but is only valid if the

resolution of the coincidence measurements is better than AT. The maximum

visibility is 50% for time resolutions much worse than AT.

There has been some question as to whether or not the experiments with

visibilities of 50% or less are nevertheless inconsistent with any semiclassical field theory. Ou and Mandel 9 have suggested that that is the case

but counter-examples to their argument have been given by Carmichael I° and by

Chiao and Kwiat1_ Although their semi-classical models are able to reproduce

the modulation in the coincidence rate, they are not able to represent the

fact that the photons are known from other experiments 12 to be coincident to

within a time interval much smaller than AT. That provides the physical basis

for the inequalities derived below.

BASIC INEQUALITY

The basic inequality that must be satisfied by any classical field is

based on Cauchy's inequality 13, which follows from the fact that

(a - b) 2 _ 0 (2)

where a and b are any two real numbers. Multiplying the two factors and

rearranging gives Cauchy's inequality:

2ab _ a 2 + b 2

(3)

When a and b are complex it is still the case that

labl = lal Ibl < lal= + Ibl=

2

(4)

The modulation of the coincidence rate will be found to be proportional

to the quantity Q defined by

Q m <lEa(t) E_(t) E2(t - AT) E1(t - AT> I> (5)

Here EI and E2 refer to the fields at the positions of detectors 1 and 2

(which will be assumed to be equidistant from the source) with the beam

splitters removed and O denotes an average over a long time interval.

It should be emphasized from the start that the angular brackets denote

an average over time and not an ensemble average. That is what the

experiments actually measure, since the results from a single system are

simply averaged over time. In addition, no assumption of ergodicity is

required in the proof that follows; the average over an ensemble is not

considered and it therefore makes no difference whether or not the time

average is equivalent to an ensemble average. It will also be found that the

proof does not assume stationarity, either.

24
The basic inequality can be obtained by choosing

a = E_(t) E2(t - AT) (6)

b = E$(t)E,(t - AT)

Inserting eqs. (6) and (7) into eq. (4) gives

<IE_(t) E$(t)E2(t - AT9 E1(t - AT)I>

< <E_(t) E_(t - AT) E2(t - AT) El(t)>2

+ <E_(t) E_(t - AT) EI(t - AT) E2(t)>/2

(7)

(8)

The physical significance of the above inequality can be seen in Figure

2, in which both fields E1(t ) and E2(t ) correspond to narrow pulses emitted at

the same time. If EI is evaluated at time t and E2 :is evaluated at time

t ± AT, as illustrated by the arrows in the figure, then one or the other of

the fields must be zero and their product vanishes. The right-hand-side of

eq. (8) is then zero, which requires that the left-hand-side also vanish.

Although this inequality may seem trivial in nature, it is a consequence of

the fact that the classical fields are well-defined (complex) numbers and the

inequality is violated by quantum fields, as will be discussed below.

INEQUALITY FOR THE VISIBILITY

The inequality of eq. (8) can be used to set a limit on the amount of

modulation that can occur in a classical treatment of the two-photon

interferometer experiments. Once again, let E1(t ) be the classical field that

would arrive at detector I in the absence of the two beam splitters and assume

for the moment that the half-width w of the coincidence window is negligibly

small. The corresponding coincidence rate as a function of the time offset

is then

Rco(z) = _<11(t) I2(t + z)> (9)

= _<E_(t) E$(t + r)E2(: + z)E1(t)>

where 11 and 12 are the intensities of the two beams and the constant _ is

related to the detection efficiencies and w. With the insertion of the two

beam splitters, the total electric field ETI(t ) at detector i becomes

1

ETI = _ [El(t) + e=eIE1(t - AT)] (i0)

A similar expression exists for the total field at detector 2 and the

classical coincidence rate R c with the beam splitters inserted and z = 0 is

given by

R c = _6_<l[E1(t) + eleIE_(t - AT)] [E2(t) + eie2E2(t - _r)] 12>111)

Multiplying out all the factors in eq.(ll) gives a total of sixteen terms:

25
R_ -- i--_i <E/(t) E;(t) E_(t) E2 (t)

+e_%E[(t) E;(t) E_(t) E2(t - AT)

+ eieiE[(t) E_(t) EI(t - AT) E2(t)

+ei[e1+ej E[ (: ) E] ( t) EI (: - AT) E2 (: - AT)

+ e-ie, E_(t) E2(t - AT) El(t) E2(t)

+E[ (t ) Ei (t - AT) El(t) E2 ( t - AT)

+ e i[e1-ej E[(t) EI(t - AT) EI(t - AT) E2(t)

+eie*Ei(t) Ei(t - AT) E_(t - AT) E2(t - AT) (12)

+ e-ie_E[(t -AT) E2(t) El(t) E2(t)

+e sc%-ej E_ (t - AT) El(t) El(t) E2 (t - AT)

+ E[(t - AT) El(t) E_(t - AT) E2(t)

+eie. Ei ( t - AT) E2 ( t) E_ ( t - AT) E2(t - AT)

+ e _E-e_ -e'_E[(t - AT) Ei(t - AT) E,(t) E2(t)

+ e-_%E[(t - AT) E2"(t - AT) E_(t) E2(t - AT)

+ e-_%E[(t - AT) E_(t - AT) E,(t - AT) E2(t)

+E[(t - AT) E2"(t - AT) E_(t - AT) E2(t - AT)>

As suggested by eq. (i), the experiments can be performed in such a way

as to measure the averaged coincidence rate as a function of OT = 01 + 82 :

-Rc(OT) - i_JO lJ0 d_2Rc(el'e2)8(el + 02 - eT) (13)

The averages over 81 and 02 were explicitly performed in one of the

experiments 6. In the remaining experiments the individual phases were not

directly measured and had essentially random values from one run to the next,

since variations in the temperature of the laboratory would have shifted the

phase of both interferometers by several fringes from one day to the next.

Thermal drifts during the course of an experimental run would have a similar

effect on the individual phases while leaving the modulation of the

coincidence rate unaltered.

In any event, the terms in eq. (12) with phase factors of exp(i81),

exp(i82) , exp[i(81 - 82)], etc., average to zero, leaving only those terms

with no phase dependence or a dependence on 81 + 82 . The remaining terms can

be written as

--Rc = _{_1 <E_ ( t) E_ (t) E2 (t) E_ (t) >

+ 1 <E_(t) E2(t ATgE2(t - AT) El(t)> (14)

+ 1

l--@n[ei%<E_(t) E2,(t)E2(t - AT)EI(t - AT)> + c.c.]

where the average over a long time interval ensures that

<E_ Ct - A T) E2 (t - A T) E2 (t - A T) EI (t - AT)> (is)

: <E[(t) Ei(t)E2(t)E1(t)>

and the symmetry of the two beams gives

26
<E[ ( t)E2 (t - A T) E2 (t - AT) El(t)> (16)

= <E](t) E[(t - AT) E_(t - AT) E2(t)>

The assumption inherent in eq. (16) is not essential and can be avoided by

simply replacing Rc0(AT) with [R=0(AT) + R=0(-AT)]/2 in what follows.

The maximum and minimum coincidence rates from eq. (14) satisfy

i <E_(t) E_(t)E2(t)E_(t)>

1 <E[(t) E_(t - AT) E2(t AT) Era(t)> (17)

+ -{'l

+ 1

_ 9<I E[ (t) E_ (t) E2 (t - A T) EI (t - AT)I>

1 <E_(t)E_(t)E2(t)E_(t)>

_in _ 8_

1 <E_(t) E_(t - AT) E2(t AT) El(t)> (18)

+ -{n

- i_<lE[(t) E_(t)E2(t - AT) EI(t - AT)l >

8

The visibility is defined as usual by

V = amax - amin (19)

%ax +bin

Using the inequality of eq. (8) and expressing the right-hand-side in terms of

Re0 (AT) gives

Rco (A T)

v < (20)

Rco(O) + Rco(AT)

Eq. (20) gives the maximum visibility that can occur in any classical field

theory and gives zero modulation for the case in which the fields correspond

to coincident pulses.

If the experiments are performed using detectors with limited time

responses and large coincidence windows, as is often the case, then the above

inequality can be generalized to

'r/2Rco(Z) dz + _A-I_'T/2 1 ;-3AT/2Rc°(Z) dz

V < (21)

2/oRco(r) dz

as is shown in the Appendix. Re0 is again the coincidence rate that would be

obtained using detectors with a negligible time response and a negligiblysmall window.

COMPARISON WITH EXPERIMENT

Earlier experiments 12 have shown that the down-converted photons are

coincident to within a time interval much less than the value of AT in at

least three 4'6'7 of the two-photon interferometer experiments, in which case

the inequalities of eqs. (20) or (21) show that there is no classical or semi27
classical field theory consistent with all of the available observations.

The author has recently completed an experiment 6 in which the optical

path length between the two interferometers was larger than I00 meters. The

main goal of the experiment was to investigate these effects in the limit of

large distances. Furry has suggested that the collapse of the wavefunction may

be degraded in some way when it occurs over sufficiently large distances,

leading to an eventual modification of the quantum-theory predictions. The

visibility of the interference pattern observed agreed with that expected from

the quantum theory to within the experimental uncertainty of 4% and violated

the inequality of eq. (21) by four standard deviations. This provides some

indication that the collapse of the wavefunction is unaffected even when it

occurs over relatively large optical path lengths.

CLASSICAL MODELS

In the classical models suggested by Carmichael 9 and by Chiao and

Kwiat I°, the fields E I and E 2 have well-defined frequencies that sum to the

pump laser frequency for a time interval larger than AT or the time resolution

of the coincidence circuits. In that case the coincidence rate of eq. (14)

simplifies to

Rc = 4 co 2 °A

This differs from the quantum-mechanical result by the additional factor of

1/2 and corresponds to a visibility of 50%.

Such models cannot simultaneously localize the fields into coincident

pulses whose widths are less than AT, however. Any classical model that does

would have the visibility reduced accordingly as required by the inequalities

of eqs. (20) or (21).

VIOLATION OF THE INEQUALITY IN QUANTUM OPTICS

The intensity operator is given by I(t) = E-(t) E÷(t) , where E + and Eare the positive and negative-frequency components of the electric field

operator 13. As a result, the quantum-mechanical equivalent of eq. (8) is

l<E_(t) E_(t)E_(t - AT) E_(t - AT)> I

< <E_(t) E_(t - AT) E_(t - AT) E_(t)>/2 (23)

+ <E_(t) E_(t - AT) E_(t - AT) E_(t)>/2

It has already been noted I that in experiments of this kind the coincidence of

the photons requires

E[(t) Ef(t ± AT) = 0 (24)

while conservation of energy in the parametric down-conversion process

requires that

E[(t - AT) Ef(t - AT) = e iC'_+-21AzE[(t) E_(t) (25)

where the sum of the two photon frequencies _I and _2 is equal to _0. (Eq.

(24) is only valid when AT is small compared to the pump laser coherence

time.) Inserting eq. (24) into the right-hand-side of eq. (23) gives zero,

whereas inserting eq. (25) into the left-hand-side gives

<E_(t)E](t)E](t)E_(t)>, which is the product of the individual beam

28
intensities and a nonzero quantity. Thus the inequality is violated in

quantum optics.

The quantum-mechanical situation is shown in Figure 3. The field

corresponds to an entangled state in which there is a superposition of times

at which the pair of photons may have been emitted, as indicated by the

existence of both the solid and dotted curves. Although the product of El and E$

at two different times is zero, that does not imply that the left-hand-side of

eq. (22) must vanish. Equations (24) and (25) would be logically

inconsistent if the fields were well-defined complex numbers and the violation

of this inequality provides a graphic demonstration of the lack of objective

realism of the electric field.

CONNECTION WITH UNCERTAINTY RELATIONS

The main topics of this conference are squeezing and uncertainty

relations. It may thus be useful to make some general comments about the

connection between the inequality derived above and the uncertainty relations

associated with the quantized field.

The inequality derived above is a result of the fact that the fields are

not just complex numbers and thus have no well-defined value. In particular,

the field operators are non-commuting and satisfy

[A_(X),A.(x/)] = -ich6_,D(x - x/) (26)

A variety of uncertainty relations can be derived from this commutation

relation, which illustrates the fact that the quantized field has no welldefined value. As a result, there is an unavoidable uncertainty in the lefthand-side of the classical inequality and this uncertainty is evidently large

enough that the left-hand-side can exceed the right-hand-side. Thus it seems

apparent that the violations of these classical inequalities in quantum optics

are related to the uncertainty relations for the quantized fields. More

detailed uncertainty relations for the actual quantities involved in the

classical inequality could be derived, if desired.

SUMMARY

Two-photon interferometer experiments with a sufficiently large

visibility will violate Bell's inequality and are thus inconsistent with any

local hidden-variable theory. Those experiments with smaller visibilities may

nevertheless violate an inequality for classical fields if the degree of

coincidence of the photon counts is taken into account. A recent two-photon

interferometer experiment with a large optical path length between the two

interferometers gave a visibility in good agreement with the quantum theory

and also violated the classical inequality, indicating that the effects

observed were quantum-mechanical in nature.

APPENDIX

When finite coincidence windows are used, Eq. (9) must be replaced by

Rico = 11jf_ww<I1(t) I2(t + z)>dr (27)

The modulation in the coincidence rate then involves

29
Q/ = -{_] 1 /_wwd_<IE_(t)E_(t+_)E2(t - AT + _)E_(t - AT)I> (28)

The range of the integral can be divided into two regions depending on the

value of IzI. For Izl _ AT/2, a and b can be chosen as

a = E_(t)E2(t - AT + _) (29)

b = E_(t + z)E1(t - AT)

and the analysis proceeds as in the text. For Izl a AT/2, a and b are chosen

instead as

a = E_(t) E_(t + _) (30)

b = E2(t - AT + z)E1(t - AT)

The inequality of eq. (21) then results from the use of eq. (8) in the limit

that w is much larger than AT.

REFERENCES

I.

2.

°

.

.

.

.

.

.

Franson, J. D., 1989, "Bell Inequality for Position and Time", Phys.

Rev. Lett. 62, pp. 2205-2208.

Horne, M. A., Shimony, A., and Zeilinger, A., 1989, "Two-Particle

Interferometry", Phys. Rev. Lett. 62, pp. 2209-2212.

Bell, J. S., 1964, "On the Einstein Podolsky Rosen Paradox", Physics i,

pp. 195-200.

Kwiat, P. G., Vareka, W. A., Hong, C. K., Nathel, H., and Chiao, R. Y.,

1990, "Correlated Two-Photon Interference in a Dual-Beam Michelson

Interferometer", Phys. Rev. A 41, pp. 2910-2913; Ou, Z. Y., Zou, X. Y.,

Wang, L. J., and Mandel, L, 1990, "Observation of Nonlocal Interference

in Separated Photon Channels", Phys. Rev. Lett. 65, pp. 321-324.

Rarity, J. G., and Tapster, P. R., 1990, "Experimental Violation of

Bell's Inequality Based on Phase and Momentum", Phys. Rev. Lett. 64, pp.

2495-2498; Rarity, J. G., Tapster, P. R., Jakeman, E., Larchuk, T.,

Campos, R. A., Teich, M. C., and Saleh, B. E. A., 1990, "Two-Photon

Interference in a Mach-Zehnder Interferometer", Phys. Rev. Lett. 65, pp.

1348-1351.

Franson, J. D., 1991, "Two-Photon Interferometry Over Large Distances",

submitted to Phys. Rev. A.

Brendel, J., Mohler, E., and Martienssen, W., 1991, "Time-Resolved DualBeam Two-Photon Interferences with High Visibility", Phys. Rev. Lett.

66, pp. 1142-1145.

Franson, J. D., 1991, "Violations of a Simple Inequality for Classical

Fields", submitted to Phys. Rev. Lett.

Ou, Z. Y., and Mandel, L., 1990, "Classical Treatment of the Franson

Two-Photon Correlation Experiment", J. Opt. Soc. Am. B 7, pp. 2127-2131.

Carmichael, H. J., 1990, private communication; and to be published.

Chiao, R. Y., and Kwiat, P. G., 1990, private communication.

30
12.

13.

Friberg, S., Hong, C. K., and Mandel, L., 1985, "Measurementof Time

Delays in the Parametric Production of Photon Pairs", Phys. Rev° Lett.

54, pp. 2011-2013; Ou, Z. Y., and Mandel, L., 1988, "Observation of

Spatial QuantumBeating with Separated Photodetectors", Phys. Rev. Lett.

61, pp. 54-57.

Loudon, R., 1983, The Quantum Theory of Light, Oxford University Press,

New York.

31
02

I_ ?2

\ /

]

Figure i Two-photon interferometer.

Figure 2

Field

amplitudes

E_ andE 2

t t + AT

Time

A pair of classical coincident pulses.

Field

amplitudes

E; and E_

t t+AT

Time

Figure 3 Quantum-mechanical field corresponding to an entangled

pair of coincident photons, with a superposition of

times at which the pair may have been emitted.

32
N92-22049

USE OF ENTANGLEMENT IN QUANTUM OPTICS*

Michael A. Home

Stonehill College, North Easton, Massachusetts 02357

Herbert J. Bemstein

Hampshire College, Amherst, Massachusetts 01002

Daniel M. Green0erger

City College of the City University of New York, New York, New York 10031

Anton Zeilinger

Atominstitta der Osterreichischen Universitaten

Schflttelstrasse 115, A-1020 Vienna, Austria

Abstract

Several recent demonstrations of two-particle interferometry are reviewed and shown to be

examples of either color entanglement or beam entanglement. A device, called a number filter, is

described and shown to be of value in preparing beam entanglements. Finally we note that all three

concepts (color and beam entanglement, and number filtering) may be extended to three or more

particles.

Introduction

In recent years there has been a variety of demonstrations of two-particle interferometry.

By a two-particle interferometer we mean an arrangement without polarizers whereby the

coincident count rate in a pair of detectors exhibits sinusoidal oscillations (two-particle fringes) as

some apparatus parameter is uniformly varied, but the singles rate in each detector is constant.

Whereas the earliest demonstrations 1, 2 of two-particle fringes employed pairs of photons from

atomic cascades and employed polarizer orientation as the parameter, the new experiments employ

photon pairs produced by down-conversion and usually employ mirror translation as the

parameter.

Two-pal_icle interference fringes occur only when the quantum mechanical state of the

particles is entangled. By entanglement we mean that the two-particle state does not factor into a

product of single particle states, but is a sum of at least two terms, each of which is a product.

Note that when two-particles are so entangled, neither particle separately has a state. Because

particles in an entanglement do not have states or even some properties, independently of each

other, we will often refer to them not as two particles, but simply as a two-particle, i. e., a single

entity.

The present paper reviews a selection of the recent demonstrations of two-particle

interferometry, in order to point out the central role of entanglement. The experiments are selected

so as to especially emphasize two important types of entanglement: color entanglement and beam

entanglement. Although each of these types of entanglement have previously been separately

discussed (but without these names) in earlier papers and conference proceedings, we thought this

an appropriate place for a review. In the course of the review of existing experiments, we also

describe a device, which we call a number filter, that may be of use in experimentally preparing

entanglements in the future. Finally, we note that all three ideas discussed here (color

33
entanglement,beamentanglement,andnumber-filtering)also applyto athree-particle,afourparticle,etc.

Beforestartingthe review, weemphasizetwo other aspects of our point of view, both

essential to the way we use entanglement. First, we distinguish beams or paths (labelled A, B,

C .... ) from particles or detectors (labelled 1,2,3,...). Second, we apply elementary quantummechanical concepts as follows. Amplitudes and kets will be assigned to the particles and not to

the beams. Total amplitude is the sum over all contributing amplitudes (i.e., the Feynman-Wheeler

rule). Note then that our approach is unorthodox in that quantum optics usually employs a

quantum field theory in which states (e.g., kets) are assigned not to the particles, (i.e., the

photons) but to the beams (i.e., the field modes).

A Color-Entangled Two-Photon

When a single particle decays into two, as for example in a down-conversion of an

ultraviolet photon into a pair of red photons, energy conservation, together with a suitable

apparatus, produce entanglement. Fig. (1) depicts an arrangement for producing entanglement in

this way during down conversion .3,4 Suppose, for simplicity that the incident photon is ideally

monochromatic with wavenumber 2ko, so that its state is

tb(k) = (5(k- 2k0) (1)

Suppose that the outgoing pair, 1 and 2, are selected as to direction by the symmetrically placed

slits and as to color by filters of wavenumber width c, centered at ko, and that _ is narrower than

any feature in the down-conversion spectrum. Then from eq. (1) and energy conservation, the

state of the down-conversion photons after the f'flters is

_(_,k,) . = (5(kI + k 2 - 2k 0) e-(k_-k°)212°_e-(k2-k°)2/2a2 (2)

Because of the 15-function, this state cannot be factored, i. e., the two red photons are actually a

color-entangled two-photon.

In general, a state entangled in k-space is also entangled in x-space. For example, eq. (2)

in x-space, with the time dependence included, becomes

I//(x r x:, tr tz) = e a 0_51-,_ )et_0(_2- -2)e- _' t(_1-_,,)-(_2-_ 2)15' (3)

along the outgoing beams. Here it is the real exponential that does not factor, i. e., the photons are

still entangled and with spectacular consequences. If detectors are placed in the beams equally far

from the source (Xl=X2),the joint probability density is, from eq. (3),

* o/= e -°_(_)_n, (4)

where "_ -_ t2 -tl is the time difference in the arrival of the photons. In short, the color

entanglement implies that the distribution in time separation of the photons is dictated by the filter

width .4

For even more spectacular consequences, consider the expanded arrangement first

proposed by Franson 5 and shown in Fig. (2). Here each beam of Fig. (1) has been fed into a

single particle interferometer. Kwiat, et al., 6 and Ou, et al.,7 have confirmed Franson's prediction

that two-particle fringes can be exhibited with this arrangement. These Franson fringes follow

easily from the color-entangled two-photon state (2) and (3), as has been shown in an earlier

conference proceedings. 8 The argument is as follows. For simplicity, suppose that the path

lengths are adjusted so that the two interferometers are identical, with the long path longer by A

than the short path. Place detectors in the corresponding output beams of the interferometer and

monitor for coincidences. From Fig. (2), the state failing on these detectors is

I/t(0,0,t_,t2) + tlt(A,A,t_,tz) + l/t(A,0,t_,t2) + IV(0, A,t_,t2), (5)

34
where_ is givenby eq.(3).

It foUowsfromthestate(5)thatthecoincidentcountrateis

A (1 + cos 2k0A) + Bcos k0A + C (6)

where A, B, and C are elementary (error) functions of c_, A, and the coincidence "window" T. In

the ideal limit that

era >> 1 (7a)

and

cT << A, (7b)

the third and fourth terms of the state (5) do not contribute and as a consequence, B=C=0 in

expression (6). That is, (6) reveals that under ideal conditions two-particle fringes of visibility

unity can be exhibited by varying A or ko. Although experiments cannot achieve such ideal

contrast, several groups at the present conference report continuing investigation of fringe contrast

in Franson interferometry. We propose that some of the relationships in 16) be compared with

experimental data.

A Beam-entangled Two-photon

The collapse of the state (5) to just two terms, when the conditions of the inequalities (7)

are satisfied, has already produced an example of beam entanglement. That is. the beams taken by

the detected pair of photons were either both short or both long, the other two cases being

impossible because of the choices of filter width cy, path difference A, and coincidence window T.

Beam entanglement may also be produced by directly exploiting momentum conservation 9 during

the decay process, instead of the energy conservation that was built into the color entanglement of

eq. (2). In general, this approach requires that the two beams of Fig. (1) be brought together.

Fringes were first produced in this way by Alley and Shih, 10, 1 1 but since their arrangement

involved polarization manipulation, we will review instead the simpler arrangement of Ghosh and

Mandel, 12 shown in Fig. (3_.

As indicated in that figure, Ghosh and Mandel uncovered some of the fun of two-particle

quantum mechanics in the small region of beam overlap. Consider, as shown in the Fig. (3)

insert, two small detectors placed in this region. When coincident counts occur in these detectors,

the count in detector 1 could have been caused by a photon 1 that took route A, in which case, by

momentum conservation at the source, photon 2 took route B. Equally likely, the routes taken

could have been reversed. Consequently, the state falling on the detectors is the beam-entangled

two-photon state

1 ,i_ I\ \

-_[ A/,iB/2 +,B/,A/:] (8)

where ket tA)t denotes particle 1 in beam A, etc.

Assume, for simplicity, that each beam is monochromatic and monodirectional. Then,

from state (8), the stationary two-particle amplitude at the detectors is proportional to

ea,Ar,ea,,.,_ + ea,,.,_ea, Ar_, _9)

where kA and kB are the wave-vectors of the beams and rl and r2 are the positions of the

detectors. It follows fi'om the amplitude (9) that the probability of joint detection is proportional to

1 + cos[(k a - k B) -(r: - r0], (10)

as confirmed by Ghosh and Mandel, who varied the separation of the detectors in Fig. (3).

35
Note that manipulation of the phasesof beamsA and B, either by changing their

geometricalengthswithmirror motionor theiropticallengthswith phaseplates,hasnoeffecton

thefringes(10). Thisis becauseeachtermof theentangledstate(8) hasbothanA-beamandaBbeamfactor,andhencethephasemanipulationsintroduceonly anunobservableoveraUpha_ in

the amplitude (9).

Fig. (4) depicts an extended arrangement of Rarity et a113 that is responsive to such phase

manipulations. Here, by simply relabeling the kets in (8), the state before the first beam splitters

is

which evolves into

+e ItJilo/2) (12)

after the phase plate and before the final beamsplitter. That is, now both particles either take route

A, or both take route B, and hence the phase 2cp enters. The derivation of (12) from (11) uses

only the elementary rules for transmission and reflection at beam splitters, i. e.,

\ ' " \ tB),), (13) Ic,, +

etc. Applying this rule again at the last beam splitter, the state (12) evolves into four terms, but

two of these describe both particles going into the same detector, and hence are of no interest in a

coincidence counting experiment. From either of the other amplitudes, the coincident count rate is

proportional to

1 + cos2tO (14)

i. e., two particle fringes appear when tp is varied, as was indeed observed by Rarity et al. 13

Fig. (5) proposes another arrangement for preparing the entangled state (12). Here a beam

of wave number k impinges on a beam splitter. Outgoing beam A contains two non-linear crystals

separated by a 2k filter. Clearly the only way k radiation can pass this three-element device is for

an incident two-photon to up-convert in the first crystal to a single 2k photon which, after passing

the filter, downconverts back to a two photon in the second crystal. Consequently, we call the

device a number filter, since only a two-photon can pass. Of course, the two-photon could avoid

the device entirely and take route B, which contains phase shifter qa. Thus the state (12) is

prepared and the two-particle fringes of (14) can be observed. Although this experiment has not

been performed, we note the similarity of Fig. (5) to an arrangement ofWu, et al. 14 It seems that

the only significant difference is that we assume the discrete counting of photons in coincidence,

whereas they continuously monitored the current difference of two photo-diodes. Clearly there

must exist interesting relationships between our point of view (i.e., entanglement of particles) and

theirs (i.e., field quantization with "squeezing").

A Two-Photon In Four-Beam Entanglement

Figs. (6) through (I0) depict various performed or proposed experiments for exhibiting

two-particle fringes by manipulating four beams of down-conversion radiation. Fig. (6) is a

proposal of Home, et al, 15 and an actual experiment of Rarity and Tapster 16 in which the four

beams are taken directly from the down-conversion crystal. Fig. (7) is a proposal of Reid and

Wails, 17 in which only two beams are taken directly from the crystal and each of these is split

before suitable recombining. Fig. (8) is an experiment of Ou et al. 18, in which the splitting is

done before the down-conversion. Fig. (9) is a proposal for a four-beam experiment employing

number-filtering. Fig. (10) is (the completion of) a figure in a recent proposal of Tan et al. 19

36
Themain point we wish to makehereis that, from an elementary entanglement point of

view, these five experiments are, essentially, identical. That is. in each case. the arrangement

prepares the four-beam two-photon entangled state

I r' 'i ' ' "1

_LIA'/'iB): + IA')'[B')2 J (15)

which, in an appropriate pair of the four detectors, produces fringes, as in (14). when the phases

are manipulated. For figs. (6) and (7), the state ¢15) is an immediate consequence of momentum

conservation at the decay. In Fig. (8), a single photon has its amplitude split at the beam splitter

and then each of these amplitudes down-converts (in different crystals) to produce the state (15).

Note that our description does not employ the "entanglement with the vacuum" description of Ou et

al. 18 In Fig. (9), the state (15) is an immediate consequence of the number filter, and Fig. (10), if

one ignores the extra detector marked 3, is identical to Fig. (9).

In Fig. (10), only the portion below and to the right of the dash-dot line appeared in the

figure of Tan, et al. 19 The remainder of Fig. (10) is drawn from their verbal description, i.e.,

their text. It isn't clear from their text whether they propose to monitor for two particles or for

three particles in coincidence. Consequently, we have included the (optional) detector 3 in order to

discuss both cases here. If detector 3 is ignored, and coincidence counts are monitored in various

pairs of detectors at stations 1 and 2, then the state is given by (15), since, just as in Fig. (9), the

particles must either both come through the number filter, or both avoid it. Thus the phases q_1 and

tp2 both enter in the second term of (15). and the joint probability of coincidence counts in

appropriate detectors at stations 1 and 2 is proportional to

1 + cos(tp: + tp_). (16)

On the other hand, if one does record a particle at 3, the other particle coming through the

number filter can either be 1 or 2, but not both. Consequently, the phases q_I and q_2 enter into

different terms and in fact the state approaching the final beam splitters is now

ei_2tA,/llB', 2 +e IA /1',z_/2 ' (17)

instead of(15). Then the fringe pattern

1 + cost_pz - tpl)... (18)

will occur in an appropriate pair of the detectors at 1 and 2. Since the fringes in (18) depend on the

phase difference, as in the fringe equations exhibited by Tan, et al.,it appears that they are

proposing a three-particle coincidence experiment. In either case, it is clear that our elementary

entanglement description is not compatible with their talk of a "single photon": their arrangement

studies at least a two-photon and, if detector 3 is used, a three-photon.

A Three-Photon

Clearly the concepts of color-entanglement, beam entanglement, and number filtering may

be applied to three or more particles. Fig. (11) depicts an arrangement producing a color-entangled

three-photon, i.e., the generalization of Fig. (1). Here, the analogs to eqs. (1) through (3) are

¢_(k) = tS(k - 3ko), (191)

_(ka,k2,k3) ¢5(k1 +k, + k 3 2k 0) e-_k'-k°_2°_e-_k_-k°_t2'r_e-(k_-k°_/2_ = . . - • (20)

and

Note that the real exponential in

among the three photons.

2

_ (x_,x2,x3,t_,t2,t3 ) = e_,Ot_,_,,) e_o__,_a,) e_ot _,_a, _e- ° [{2.1}+{3,2)+{1,3)],

where {2.1} = {(x: -ct 2) -(x_ - ctl)} 2, ac. (21)

this last equation implies remarkable space-time correlations

37
Fig.(12) depicts a three particle generalization of the Franson interferometer of Fig. (2).

From eq. (21) and the eight-term three-photon analog of the state (5), one easily finds that the three

photon equivalent of the counting rate (6), in the limit of ideal filters and detectors, is

1 + cos3k0A. (22)

In another paper in these proceedings 20, we also consider other higher-order entanglements: the

three photon generalization of the direct-beam interferometer of fig. (6) and of the number filter

intefferometer of Fig. (9).

Some aspects of three particle interferometry have also been explored by Choi. 21

Comments

We conclude with two comments. First, we have not attempted here a comprehensive

review of all of the recent demonstrations of two-particle interferometry but have selected enough

examples to exhibit the usefulness of entanglement. Consequently many beautiful experiments

(and theoretical papers) have not been discussed. However, we have found that our point of view

does provide simple, direct, and yet complete descriptions of all the experiments, either via color

entanglement, beam entanglement, or a combination of the two. For example, one may imagine

that the beams of Fig. (1) are brought together at a beam splitter, that the two filters are not inserted

until downstream of the beam splitter and, moreover, that the filters are now centered on different

colors, kl0 and k20. This is the arrangement of the "quantum beating" experiment of Ou and

Mandel. 22 Our description consists of two steps. First, generalize state (3) to include two

different filter colors. Second, superpose two of these generalized states to accommodate the beam

entanglement aspect of the arrangement. In this way, one reproduces the key result of Ou and

Mandel (their eq. (10)).

Second, note that a single quantum mechanical particle in an elementary plane-wave state

has only three adjustable properties: wavenumber, propagation direction, and polarization.

Consequently we claim that for two or more particles there are only three basic types of

entanglement: wavenumber, propagation direction, and polarization. Clearly color entanglement is

just the optical realization of wavenumber entanglement and our beam entanglement is intimately

related to propagation direction entanglement. We say "intimately related to" instead of "is"

because one must be on guard when idealizing a beam as monodirectional. A beam, unlike a

spatially unlimited plane wave, has a finite transverse width and hence can't ever be strictly

monodirectional, because of diffraction.

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This work was supported in part by grants from the National Science Foundation, and (for

DMG) the City University of New York CUNY-FRAP program.

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40
. I -

2k ° m x2

F/g. (1). An incident monochromatic UV photon of wavenumber 2ko downconverts in a crystal.

Two beams of the downconversion radiation are selected by slits and by filters of center ko

and width _, thereby preparing the color-entangled two-photon state ofeqs. (2) and (3).

Consequently, the distribution in time separation, %, of the two photons is given by eq. (4).

2

Fig. (2). Franson's two particle interferometer. The two beams of Fig. (1) are each fed into a

single-particle interfemmeter in which one path is A longer than the other and adjustable.

As zt is varied in both interfemmeters, the coincident count rate in two corresponding

outgoing beams (one at station 1 and one at station 2) exhibits the oscillations (two-particle

fringes) given by (6).

41
Fig. (3). The crossed-beam two-particle interferometer of Ghosh and Mandel, ref. (12). The

beams of Fig. (1) here intersect so that the two-photon falling on the pair of small

detectors, 1 and 2, is the beam-entangled state (8), i. e., a superposilion of particle (1) in

beam A and particle 2 in beam B, and vice-versa. Consequently, the coincident count rate

exhibits the two-particle fringes (10); r2 and rl are the positions of the detectors.

• f ose Shifter

Fig. (4). A two-beam two-particle interferometer of Rarity et al., ref. (13). The beams C and D

intersect on the central beam sputter and thereby prepare the beam-entangled state (12).

Consequently, the coincident count rate exhibits the two-particle fringes (14), where q) is

the phase shift imparted by the glass plate in beam B.

42
2k- Filter

Fig. (5). A two-beam two-particle interferometer employing a number filter. Incident radiation of

wavenumber k can transit beam A only ira two-photon upconverts in the first crystal,

passes the filter as a 2k single photon, and then downconverts in the second crystal.

Alternatively, the two-photon can take route B. Consequently the state 12 is prepared.

///11//////////////,

A _2

_ 2

- ._2

,'///////////////////

Fig. (6). Prepara_on of the two-particle beam-entangled state (15) by selecling four direct beams

of downconversion radiation.

43
2

Fig. (7). Preparation of state (15) by first downconverting and then beam splitting.

8

Fig. (8). Preparation of state (15) by first beam splittin 8 and then downconvertin8.

44
. 13t

k " 2

Fig. (9). Preparation of state (15) by using a number filter.

I A

E

B

R'

Fig. (10). An arrangement proposed by Tan, et al., ref. (19). If detector 3 is ignored, the state

(15) is prepared. If detector 3 is monitored for coincidences with detectors at 1 and 2, the

state (17) is prepared.

45
I

3k°_- I > 2

-3

Fig. (11). Three particle downconversion to produce the color-entangled three-photon state (20)

and (21). Each outsoin 8 beam contains a filter (not shown) of width _ and centered at ko.

3ko

_2

3

Fig. (12). A three-particle Franson type interferometer. Under ideal conditions the triple

coincident count rate at corresponding outgoing beams is given by (22), where A is the

path difference in each of the three branches.

46
N92-22 050

EPR Experiment and Two-Photon Interferometry

-- Report of A Two-Photon Interference Experiment --

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

Department of Physics

University of Maryland at Baltimore County

Baltimore, MD 21228

ABSTRACT

After a very brief review of the

historical EPR experiments, this paper

reports a new two-photon interference

type EPR experiment. A two-photon state

was generated by optical parametric down

conversion. Pairs of light quanta with

degenerate frequency but divergent

directions of propagation were sent to

two independent Michelson interferometers. First and second order interference effects were studied. Different

than other reports, we observed that the

second order interference visibility

vanished when the optical path difference of the interferometers were much

less then the coherence length of the

pumping laser beam. However, we also

observed that the second order interference behaved differently depending on

whether the interferometers were set at

equal or different optical path differences.

1. Historical EPR Experiments

In May 1935, Einstein, Podolsky and

Rosen published a paper in the form of a

paradox to show quantum mechanics fails

to provide a complete description of

physical reality. They put a question

as the title of the paper: "Can QuantumMechanical Description of Physical

Reality Be Considered Complete? "(_

It seemed to EPR that a necessary

requirement for a complete physical

theory was the following:

(1) Every element of physical

reality must have a counterpart in a

complete physical theory.

EPR also suggested the following

criterion for recognizing an element of

reality, which seemed to them a

sufficient criterion:

(2} If, without in any way disturbing the system, we can predict with

certainty (i.e., with probability equal

to unity} the value of a physical quantity, then there exists an element of

reality corresponding to this physical

quantity.

What EPR wished to do with their

criteria for reality was to show that

the quantum mechanics wavefunctlon

cannot provide a complete description of

all physically significant factors (or

"elements of reality"} existing within a

system.

A clear example of such system was

proposed by David Bohm in 1951. (2)

Bohm's gedankenexperiment concerned a

pair of spatially separated spin-l/2

particles produced somehow in a slnglet

state, for example, by disassociation of

the spin-0 system. The spin part of the

state may be written as:

1

-I >®1 > 1

1 2

(1)

A±

where I n > quantum mechanically

!

describes a state in which particle 1 or

2 has spin "up" or "down" respectively A

along the direction n. Since the

singlet state I @ > is spherically A

symmetric, n can be specified to be any

direction. Suppose one can set up his

47
experiment to measure the spin of the

particles in any direction and he wants

to measure the spin of particle 1 along

A

the x axis. What he can measure is not

predetermined by the quantum state

I @ >. However from [ @ > one can

predict with certainty that if particle

I is Found to have its spin parallel to

A

the x axis, then particle 2 will

immediately be found to have its spin

A A

antiparallel to the x axis if the x

component of its spin is also measured.

Thus one can arrange his experimental

apparatus in such a way, that he can

predict the value of the_ component of

spin of particle 2 presumably without

any way disturbing it. According to the

criterion, the x component of spin of

particle 2 is an element of reality.

Likewise, one can also arrange his

apparatus so that he can predict any

other component of the spin of particle

2 without interacting with it. The

A A A

conclusion would be all the x, y, z

components of the spin of each particle

are the elements of physical reality,

and of course all the v , v , v , must

x y z

exist without considering which component is being measured. But this is

not true in quantum mechanics, the

wavefunction can specify, at most, only

one of the components at a time with

complete precision. The conclusion is

that the wavefunction does not provide a

complete description of all elements of

physical reality.

The existence of an entangled

quantum state is the heart of the E.P.R.

argument. It must be a entangled

pure state There must be a definite

phase relation among the amplitudes of

the state. Does any such quantum state

exist ? Yes, experiments have demonstrated the existence of such quantum

states.

(1). Positronium Annihilation

The existence of the pure two photon

singlet state of the positronium annihilation was predicted by J. A. Wheeler in

late 40's and experimentally pEoyed by

C. S. Wu and I. Shaknov in 1950. TM

(2). Atomic Cascade Decay

Atomic cascade decay were introduced

to EPR experiments in 1970's. Several

groups of researchers have demonstrated

the existence of the pure two photon ErR

state from the atomic cascade decay.

Since 1965, when J. Bell provided a

theory to show that the local

deterministic hidden variable theory has

different predictions from those of

quantum mechanics in some special

experimental situations, experiments

have been performed to test his

inequalities using the light quanta pair

prepar_ from the atomic cascade

decay. Even though it is hard to

believe that the photon pair emitted

from the atomic cascade decay are phase

correlated when considering the rather

long life time intermediate state of the

atom, the experimental results seemed to

show that the phase correlation is

really there. Bell's inequalities are

violated in most of the experiments.

However, none of the above experiments has completely satisfied the

serious physics community. One of the

problems is the efficiency "loophole"

The emission of the photon pairs do not

have a defined K vector direction in

both the positronium annihilation and

atomic cascade decay experiments. The

emission is symmetric in 4_ solid angle

and the coliection angle can not be very

large. The low collection efficiency in

these experiments has been criticized by

dozens of physicists and philosophers.

It was concluded that none of these

experiments was a compelling test of

Bell's inequality, or in other words

that none of these experiments has

really demonstrated the phase correlation of the EPR state.

(3). Parametric Down Conversion

The first EPR experiment using light

quanta pair generated by (_ptical

parametric down conversion is

illustrated in figure 1. The two quanta

polarization pure quantum state is

prepared with the help of beam splitter.

48
Parametric down conversion generates

photon pairs with definite K vectors.

The collection efficiency could be I00%.

It is also different than all the other

EPR experiments in that the enta_ngled

pure quantum state is "made" by people

instead of God. The down conversion

state starts from a circular or linear

polarized eigenstate depending on

whether quarter wave plate or half wave

plate are used. It seems like "nothing

hidden" in this experiment. With the

help of a 50-50 beam splitter, the

following quantum states can be "made",

I ICe+B) >®l R >

I @> =5. e [ I R i z

-IL >oiL >]

1 2

1 i ((X1+_1) >®I L >

+_e { R1 I

--e1 i(_2+_2) I R >el L >

2 2 2

or,

I i(_+{_) >®{ y >

I @>=_e [ I X1 2

+ I YI >®I X2 > ]

1 i(_ ) >®I Y >

+ _. e i+a, I X i

I '(_z+_2 ) >®1 X >

+5. e { Y2 a

respectively. For the coincidence

measurement, only the first two terms

contribute. They are the singlet states

needed for the EPR experiments. For the

coincidence measurements, one would

have:

= X I @ >12 = 50% I< X i Y21 @ >12 I< Yi 2

2 @ >12 = O. I< X 1X21 @ >1 = I< Y1 Y21

and

I < x,(e t) X2(e 2) I • >21

= _1 sin2(81+02 ) = _1 sin 2

The experimental results agreed with

the Quantum mechanics prediction very

well.¢S), (s)

2. Two Photon Interference Experiment

A11 the above historical EPR

experiments are concerned polarization

correlation measurements. J. D. Franson

• _(7)

proposed a new type EPR experlment

for measurement of position and time

correlation in contrast to the

historical measurement of polarization

correlation. This proposed experiment

is also concerned to be a two-photon

interference experiment. This experiment may be simply i 11 ustrated in

Fig. 2: a pair of time and frequency

correlated photons is generated. One

travels to the left, another travels to

the right and both goes through a

independent interferometer. The optical

= L- S and path difference ALl i i

AL = L- S can be arranged to be

2 2 2

shorter or longer then the coherence

length of the down converted field.

Case I. AL < coherence length

!

Both interferometer I and II (or one

of them, if only one interferometer

satisfy the condition) will have

independent first order interference,

R i= RolCOS2(_l/2 ), (2)

where R is the counting rate of the ith

!

detector, 8 is the phase difference

l

between the L and S optical paths of i !

the independent interferometer. The

classical coincidence rate is expected

to be,

R = R cos2(al/2)cos2(a2/2). (3)

C OC

The same result comes from quantum

calculation.

49
Case 2. AL > coherence length i

The first order interference

disappears from both interferometers.

It was suggested by Franson that the

following coincidence detection probability amplitudes can be treated coherently,

(photon #I travel from path S )

l

® (photon #2 travel from path S )

2

and

(photon #1 travel from path L )

I

® {photon #2 travel from path L ), 2

if the travel time difference between

the long and short paths of the two

interferometers are equal.

The amplitudes:

{photon #I travel from path S )

I

® (photon #2 travel from path L )

2

and

{photon #I travel from path L )

1

® (photon #2 travel from path S )

2

will be cut off by the time window of

the coincidence circuit if the travel

time difference between the long and

short paths is larger then the time

window or will contribute to the noise

if the time window of the coincidence

circuit is not short enough.

The coincidence counting rate was

predicted to be

1 RoCOS2{[ ((_1 + _2).AT + ¢I + ¢2 ]/2}

= _ cos 2{ ,/2 + (_ /2 } (4)

where AT is the travel time difference

between the long and short paths of the

two independent interferometers and ¢i'

any other phase shift. Eq. (4) shows a

I00% interference modulation for an

arbitrary time difference of AT, in

other words, the interference pattern

will be the same even when the optical

path difference of the interferometer is

much longer (infinite) then the

coherence length of the field. It was

suggested that this prediction leads to

a violation of Bell's inequality and a

quantum non-local effect. Compared to

the historical E.P.R. experiments, which

used polarization as a measured quantity, this experiment is looking at the

direct phase correlation between the

long-long and short-short path amplitude. Unlike the other second order

interference experiments which superpose

the two photons at a beamsplitter, the

photon pair never "come" together in

this proposed experiment. The "interference" can not be explained by the

idea of definite field phase relation at

the beamsplitter as usually do. The

experiment simply counts the timing of

the detections and through the timing

analyzer to distinguish the coincidence

detection and the noncoincidence detection, i.e., the phase relation will be

explored through the timing of detection

and the width of the time window of the

timing analyzer.

Since then, two experiments have

reported the observation of the quantum

mechanical effect. (8)'(s) However, it

seems that these two experiments did not

provide enough data and information to

support the conclusion that the quantum

non-local effect was detected. Both

experiments reported only one visibility

measurement for one setting of the

optical path difference of the interferometers. More measurements are

required to test Franson's calculation.

We report a similar two-photon interference experiment with more measurements and different results.

The experimental arrangement is

shown in Fig. 3. A 351 nm CW Argon

laser line was used to pump a SO mm long

50
potassium dihydrogen phosphate (KDP)

nonlinear crystal for optical parametric

down conversion. Nonlinear optical

parametric down conversion produces

correlated pairs of photons which

satisfy the phase-matching condition:

= _I+ _2' k = kl + k2' (S)

where _ and k are the frequency and the

wave vector of the pumping beam, w1' _2

and kl, k2 are the frequencies and the

wave vectors of the generated light

quanta. The KDP crystal was cut at

TYPE I phase-matching angle for degenerate frequency but divergent propagation

direction of signal and idler light

quanta. The 702 nm photon pair was

selected by pinholes and traveled to two

independent Michelson interferometers (I

and II). Two detectors D and D with

I 2

I0 _ spectral filters (centered at

702 nm) were placed after the

interferometers. The detectors were

avalanche photodiodes operated in Geiger

mode with less then I nonasecond rise

time and less then SO picosecond time

jitter. The output pulses from D and

1

D were sent to a coincidence counting

2

circuit which had a lO0 picosecond time

window to record R , the countin E rate

c

of coincidence and R{, the counting rate

of single detector.

Before the experiment, we first

measured the coherence length of the

down converted field by using our

Michelson interferometer. It was

concluded by direct observation with out

any spectral filter that the first order

interference pattern disappeared at

about SO _ from the white light

condition. The coherence length of the

pump laser beam was measured to be much

much longer than SO mm (limited by the

interferometer).

The experiment was done by two

steps:

First, interferometer II was set

with AL = S mm from white light

1

condition and interferometer II was

scanned from the white light condition

to S mm. 96% second order and 82% first

order interference visibilities were

observed at the beginning of the

scanning (near white light condition),

see Fig. 4 and Fig. 5. The fist order

interference visibility dropped to 0 at

400 p (with 10 _ spectral filter). The

second order interference visibility is

reported in Fig. 6. It is important to

mention that the noise counting rate was

not subtracted from the visibility

calculation (the same as the other

reports)

R - R

y = maX min (6)

R + R

max mi n

Because the short time window of the

coincidence measurement, the noise

counting rate for the second order

interference measurement was almost

zero. On the other hand, the noise

counting rate from single detector

(first order interference measurement}

was significant. It is clear from

Eq. (6} that the contribution of the

noise counting rate will result a lower

visibility. It can not be concluded

that the "second order coherence length

is longer than the first order coherence

length", or "the visibility of second

order interference is better than that

of the first order interference" as in

some of the early reports.

The second order visibility was

measured to be zero at AL = AL = S mm,

2 1

this is different than Franson's

prediction.

Second step of the experiment,

interferometer II was moved 400 p at a

time from AL = 400 _ to AL = 6 mm and 2 2

interferometer I was scanned around the

position of equal path difference,

S1
AL2= AL I, for 50 _ and the visibility of

the second order interference was

measured. Fig. (7) reports this

measurement. It is clear that the

second order interference visibility

(for AL _ AL ) did drop to zero at about

2 I

4 nun from the white light condition

which is much shorter than the coherence

length of the pumping laser beam.

However, it is also true that the

visibility for equal optical path

difference measurement did not drop to

zero as quick as that for non-equal

optical path difference measurement

which was reported at step one. It

takes six to seven times longer distance

to approach fOX visibility when the

optical path difference are equal

(compare Fig. (6) and Fig. (7)).

The alignment of the optical system

is important. The alignment of the

interferometers were checked before

taking of date. We use He-Ne laser and

sodium discharge light to check the

alignment for AL from white light

condition to I0 mm.

A classical model predicts that the

visibility of second order interference

in the case of long coincidence time

compared to the coherence time of the

down converted beam approaches

1

l/ = _ exp -(AL / L) (7)

where AL = ALI = AL 2, and L is a

constant in length which expresses the

precision to which the phase matching

condition in Eq. (5) is satisfied. The

same result may be obtained from a

quantum mechanical model. The details

of these models will be presented later

elsewhere.

Reference

1. A. Einstein, B. Podolsky and N. Rosen,

Phys. Rev. 47, 777 (1935).

2. D. Bohm, "Quantum Theory", Prentice

Hall Englewood Cliffs, (19SI).

3. C.S. Wu and I.Shaknov, Phys. Rev. 77,

136 (1950).

4. For a review, see J.F. Clauser and

A. Shimony, Rep. Prog. Phys. 41, 1981,

(1976).

A. Aspect, P. Grangier and G. Roger,

Phys. Rev. Lett. 47, 460 (1981).

A. Aspect, P. Grangier and G. Roger,

Phys. Rev. Lett. 49, 91 (1982).

A. Aspect, J. Dalibe_rd and G. Roger,

Phys. Rev. Lett. 49, 1804 (1982).

5. Y.H. Shih and C.O. A11ey, Phys. Rev.

Lett. 61, 2921 (1998).

6. Z.Y. Ou and L. Mandel, Phys. Rev. Lett.

61, SO (1988).

7. J.D. Franson, Phys. Rev. Lett. 62,

2205, (1989).

. P.G. KwIat, W.A. Vareka, C.K. Hong,

H. Nathel, R.Y. Chlao, Phys. Rev. A,

41, 2910 (1990).

9. Z.Y. Ou, X.Y. Zou, L.J. Wang L. Handel,

Phys. Rev. Lett. 65, 321 (1990).

52
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Figure 1. First EPR experiment using parametric down conversion.

53
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59
N92-22051

The Energy-Time Uncertainty Principle and the EPR Paradox:

Experiments involving Correlated Two-Photon Emission

• $

in Parametric Down-Conversion

Raymond Y. Chiao, Paul G. Kwiat, and Aephraim M. Steinberg

Department of Physics, University of California, Berkeley, CA 94720

ABSTRACT

The energy-time uncertainty principle is on a different footing than the momentumposition uncertainty principle: In contrast to position, time is a c-number parameter, and

not an operator. As Aharonov and Bohm have pointed out, this leads to different

interpretations of the two uncertainty principles. In particular, one must distinguish

between an inner and an outer time in the definition of the spread in time At. It is the

inner time which enters the energy-time uncertainty principle. We have checked this by

means of a correlated two-photon light source in which the individual energies of the two

photons are broad in spectra, but in which their sum is sharp. In other words the pair of

photons is in an entangled state of energy. By passing one member of the photon pair

through a filter with width AE, it is observed that the other member's wave packet

collapses upon coincidence detection to a duration At, such that AEAt=_, where this

duration At is an inner time, in the sense of Aharonov and Bohm. We have measured At

by means of a Michelson interferometer by monitoring the visibility of the fringes seen in

coincidence detection. This is a nonlocal effect, in the sense that the two photons are far

away from each other when the collapse occurs. We have excluded classical-wave

explanations of this effect by means of triple coincidence measurements in conjunction with

a beam splitter which follows the Michelson intefferometer. Since Bell's inequalities are

known to be violated, we believe that it is also incorrect to interpret this experimental

outcome as if energy were a local hidden variable, i.e., as if each photon, viewed as a

particle, possessed some definite but unknown energy before its detection.

*This work was supported by ONR under grant N00014-90-J-1259.

61

PRECEDING PAGE BLANK NOT FILMED
INTRODUCTION

The momentum-position and the energy-time uncertainty principles have very similar forms:

ApAx > fi/2, (1)

AEAt >__fi/2. (2)

One expects this on the basis of relativistic considerations, since both momentum-energy and positiontime form four-vectors. However, in the usual formulation of quantum mechanics, there is an

important difference between the two uncertainty principles, since time is not an operator but a cnumber parameter, in contrast to position. Hence the standard method of derivation of the uncertainty

principle for momentum and position from the fundamental commutator of quantum mechanics,

[p, x] = R/i, (3)

does not work for energy and time.

quantity with a definite lower bound.

out by Aharonov and Bohm [Ref. 1].

2].

Furthermore, in contrast to momentum, energy is a physical

These difficulties are not merely mathematical ones, as pointed

There have also been many recent papers on this subject [Ref.

Aharonov and Bohm made a distinction between inner and outer times. Inner time refers to an

intrinsic time defined by the system itself, whereas outer time refers to a duration of measurement made

by some external apparatus. They showed by construction of an explicit counterexample that the

"usual" statement of the energy-time uncertainty principle in terms of an outer time, such as, "if the

duration of a measurement by an external apparatus on a system is restricted to At, then there exists an

uncontrollable amount of energy AE--_/At imparted to the system by the apparatus," is incorrect.

However, the standard example of the energy-time uncertainty principle in terms of energy broadening

AE of an atomic energy level due to its finite lifetime x, such that AE=fi/x, is correct, but here the

lifetime x refers to an inner time of the system. The latter example of the energy-time uncertainty

principle can be understood in terms of a classical Fourier analysis of a finite wave train of duration x,

i.e., Ac0=I/x.

Here we point out a nonclassical aspect of this uncertainty principle, which arises from the

nonlocal collapse of the wave packet upon coincidence detection of a correlated pair of photons. The

two correlated photons (conventionally called "signal" and "idler" photons) are prepared by

spontaneous parametric down-conversion of a uv photon in a Z (2) nonlinear crystal. When one member

of this pair (the "remote" one) is detected through a filter with width A_, the other member (the

"nearby" one) immediately collapses into a wave packet of duration x=l/Ac0. If the remote filter is

broad, the nearby photon wave packet becomes narrow upon collapse; if the remote filter is narrow, the

62
nearbywavepacketbecomesbroad,uponcoincidencedetection.In this sense,thereexistsanonlocal

actionat a distance.Henceit is closelyrelatedto theEinstein-Podolsky-Rosenparadox.The waywe

measuredx is to pass the nearby wave packet through a Michelson interferometer. If this wave packet

overlaps with itself after reflection from the mirrors of the Michelson onto the recombining beam

splitter, then there will be interference fringes detected in coincidence with the remote photon; otherwise

interference fringes will not be visible. The wave packet duration thus measured is clearly an inner

time of the system, since it is self-referential.

EXPERIMENT

In our experiment the incident light was prepared in an entangled state consisting of a pair of

photons whose energies, E s = ficos and E i = ficoi, although individually broad in spectrum, sum up to a

sharp quantity Ep = ficop because they were produced from a single pump photon whose frequency cop

was sharp. This entangled state is given by

I_t> =fdcosA(cos) [1>_ ]l>op-Os (4)

where A(cos)= A(cop- cos) is the complex probability amplitude for finding one photon with a

frequency cos, i.e., in the n=l Fock state ] l>o_s, and one photon with a frequency con- cos, i.e., in

the n= 1 Fock state I 1 > COp_os. According to the standard Copenhagen interpretation, dae meaning of

this entangled state is that when a measurement of the energy of one photon results in a sharp value E s,

there is a sudden collapse of the wavefunction such that instantly at a distance, the other photon, no

matter how remote, also possesses a sharp value of energy Ep - E s. Thus energy is conserved.

Entangled states, i.e., coherent sums of product states, such as the one given by Eq. (4), result in

Einstein-Podolsky-Rosen-like effects which are nonclassical and nonlocal [Refs. 3-4].

We prepared the entangled state of energy, Eq. (4), by means of parametric fluorescence in the Z(2)

nonlinear optical crystal potassium dihydrogen phosphate (KDP), excited by a single mode ultraviolet

(uv) argon ion laser operating at E=351.1 nm [Ref. 4]. The uv laser beam was normally incident on the

KDP input face. In this fluorescence process, a single uv photon with a sharp spectrum is

spontaneously converted inside the crystal into two photons with broad, conjugate spectra centered at

half the uv frequency, conserving energy and momentum. We employed type I phase matching, so that

both signal and idler beams were horizontally polarized. The KDP crystal was 10 cm long and cut such

that the c-axis was 50.3* to the normal of its input face. We selected for study idler and signal beams

both centered at _.=702.2 nm which emerged at +1.5" and -1.5 °, respectively, with respect to the uv

beam. Coincidences in the detection of conjugate photons were then observed.

In Fig. 1, we show a schematic of the experiment. The idler photon (upper beam) was transmitted

through the "remote" filter F1 to the detector D1, which was a cooled RCA C31034A-02

photomultiplier. The signal photon (lower beam) entered a Michelson interferometer, inside one arm of

63
whichweresequentiallyplacedtwo zero-orderquarterwaveplatesQ1andQ2. Thefastaxisof the first

waveplate Q1 was fixed at 45* to the horizontal, while the fast axis of the second waveplate Q2 was

slowly rotated by a computer-controlled stepping motor. After leaving the Michelson the signal beam

impinged on a second beamsplitter B2, where it was either transmitted to detector D2 through filter F2,

or reflected to detector D3 through filter F3. Filters F2 and F3 were identical: They both had a broad

bandwidth of 10 nm centered at 702 nm. Detectors D2 and D3 were essentially identical to D1.

Coincidences between D1 and D2 and between D1 and D3 were detected by feeding their outputs into

constant fraction discriminators and coincidence detectors after appropriate delay lines. We used EGG

C102B coincidence detectors with coincidence window resolutions of 1.0 ns and 2.5 ns, respectively.

Also, triple coincidences between D1, D2 and D3 were detected by feeding the outputs of the two

coincidence counters into a third coincidence detector (a Tektronix 11302 oscilloscope used in a counter

mode). The various count rates were stored on computer every second.

Our arrangement of quarter waveplates inside the Michelson interferometer generates a geometrical

(Berry-Pancharatnam) phase, proportional to the angle 0 between the fast axes of the quarter wave

plates. We shall not go into detail concerning this phase here, except to say that it affords a convenient

way to see interference fringes without changing the difference in ann lengths of the interferometer

[Ref. 5].

We took data both outside and inside the white 'light fringe region where the usual interference in

singles detection occurs. We report here only on data taken outside this region, where the optical path

length difference was at a fixed value much greater than the coherence length of the signal photons

determined by the filters F2 and F3. Hence the fringe visibility seen by detectors D2 and D3 in singles

detection was essentially zero.

THEORY

First we present a simplified quantum analysis of this experiment. In the Appendix, we will

present a more comprehensive analysis based on Glauber's correlation functions. The state of the light

after the Michelson interferometer is given by

IV>out=fdc0sA(o s)11> 11> _o + (5)

where O(0_p - cos) = 2_AL/_.c0 _ co + CBerry is the phase shift arising from the optical path length

difference AL of the Michelgon _or the photon with frequency co - cos, plus the Berry's phase P

contribution for this photon. The coincidence rate N12 (N13) between detectors D1 and D2 (D1 and D3)

is proportional to the probability of finding at the same time t one photon at detector D1 placed at r 1, and

one photon at detector D2 (D3) placed at r 2 (r3). When a narrowband filter F1 centered at frequency

cos is placed in front of the detector D1, N12 becomes proportional to

[V'out(rl, r=,0[2=l <rl,r2,tlV>'out[2o< l+cos , (6)

64
wheretheprimedenotestheoutputstateafteravon Neumannprojectionontotheeigenstateassociated

with the sharpfrequency cos upon measurement. Therefore, the phase _ is determined at the sharp

frequency cop - cos, or equivalently, the sharp energy Ep - E s. In practice, the energy width depends

on the bandwidth of the f'llter F1 in front of D 1, so that the visibility of the fringes seen in coincidences

should depend on the width of this remote filter. This fringe visibility will be high, provided that the

optical path length difference of the Michelson does not exceed the coherence length of the collapsed

signal photon wave packet, determined by F1. If a sufficiently broadband remote filter F1 is used

instead, such that the optical path length difference is much greater than the coherence length of the

collapsed wave packet, then the coincidence fringes should disappear.

RESULTS

In Fig. 2, we show data which confirm these predictions. In the lower trace (squares) we display

the coincidence count rate between detectors D1 and D3, as a function of the angle 0 between the fast

axes of waveplates Q1 and Q2, when the remote filter F1 was narrow, i.e., with a bandwidth of 0.86

nm. The calculated coherence length of the collapsed signal photon wave packet (570 l.tm) was greater

than the optical path length difference at which the Michelson was set (220 ktm). The visibility of the

coincidence fringes was quite high, viz., 60%+5%.* This is in contrast to the low visibility, viz., less

than 2%, of the singles fringes detected by D3 alone (not shown). For comparison, in the upper trace

(triangles) we display the coincidence count rate versus 0 when a broad remote filter F1, i.e., one with

a bandwidth of 10 nm, was substituted for the narrow one. The coherence length of the collapsed

signal photon wave packet was thus only 50 I.tm. The coincidence fringes in this case have indeed

disappeared, as predicted.

DISCUSSION

In light of the observed violations of Bell's inequalities [Ref. 6], it is incorrect to interpret these

results in terms of an ensemble of conjugate signal and idler photons which possess definite, but

unknown, conjugate energies before filtering and detection. Any observable, e.g., energy or

momentum, should not be viewed as a local, realistic property carded by the photon before it is

actually measured.

The function of the second beamsplitter B2 was to verify that the signal beam was composed of

photons in an n=l Fock state. In such a state, the photon, due to its indivisibility, will be either

transmitted or reflected at the beamsplitter, but not both. Thus coincidences between D2 and D3 should

never occur, except for rare accidental occurrences of two pairs of conjugate photons within the

coincidence window. However, if the signal beam were a classical wave, then one would expect an

equal division of the wave amplitude at the 50% beamsplitter, and hence frequent occurrences of

*The slightly nonsinusoidal component in Fig. 3 (lower trace) can be explained by a slight wedge in

Q2, in conjunction with the fact the signal beam was incident on Q2 off center.

65
coincidences.An inequality,whichwas stronglyviolatedin ourexperiment,placesalower boundon

thiscoincidenceratefor classicallight (seebelow). Thisverifies theessentiallyn=l Fock statenature

of thelight, and confirms the previous result of Hong and Mandel [Ref. 7].

The vertical arrows in Fig. 2 indicate the points at which triple coincidences were measured. Let

us def'me the anticorrelation parameter [Ref. 8]

a - N123 N 1 / N12 N13 ,

(7)

where N123 is the rate of triple coincidences between detectors D1, D2 and D3, N12 is the rate of double

coincidences between D1 and D2, N13 is the rate of double coincidences between D1 and D3, and N 1 is

the rate of singles detections by D1 alone. The inequality a>l has been shown to hold for any classical

wave theory [Ref. 8]. The equality a= 1 holds for coherent states I or), independent of their amplitude

tx. Since in our experiment the amplitude fluctuations in the double coincidence pulses led to a triple

coincidence detection efficiency rl less than unity, we should reduce the expected value of a

accordingly. The modified classical inequality is a>rl. We calibrated our triple coincidence counting

system by replacing the two-photon light source by an attenuated light bulb, and measured

rl=0.70-L--0.07. During the data run of Fig. 2 (lower trace), we measured values of a shown at the

vertical arrows. The average value of a is 0.08+0.04, which violates by more than thirteen standard

deviations the predictions based on any classical wave theory. It is therefore incorrect to interpret these

results in terms of a stochastic ensemble of classical waves, in a semiclassical theory of photoelectric

detection [Ref. 9]. Classical waves with conjugate, but random, frequencies could conceivably yield the

observed interference pattern, but they would also yield many more triple coincidences than were

observed.

We have therefore verified the energy-time uncertainty principle for pairs of photons in essentially

n=l Fock states, in a way which excludes with very high probability any possible classical explanation.

These results can be understood in terms of the nonlocal collapse of the wavefunction.

ACKNOWLEDGMENTS

We thank I. H. Deutsch for helpful discussions, J. F. Clauser, E. D. Commins and H. Nathel for the

loan of electronics, and R. Tyroler for the loan of the Michelson.

APPENDIX

A more rigorous theoretical description of the experiment can be carded out within the Glauber

correlation function formalism [Ref. 10]. We start with the entangled state of the down-converted light

IV) =fd°)sA(°)s) 11)% [1)0_-_s

(A.1)

66
where A(cos)=A(cop- COs)is the complex probability amplitude for finding one photon with a

frequency COsand one photon with a frequency COp- COs" For simplicity we have assumed that the

pump photon is monochromatic. The second order correlation function relating the field amplitudes for

the signal and idler modes at the times ts and ti, respectively, is defined as

G(2)(ts,ti; ti,ts) = (V[ E(s-)(ts) E_-)(ti) E_+)(ti) _(+)" I_s (ts)" ]_). (A.2)

.-- ^(+)

In this expression, E(s_!)(ts(i)) and Es(i)(tsci)) are the negative- and positive-frequency parts of the electric

field for the signal (idler) mode. Assuming, as in Fig. 1, that the idler photon is directed to detector D 1,

the field operators for this mode at the time t i may simply be Fourier expanded in terms of a frequency-

^t ^

dependent detection efficiency Illl (COi)12, and creation and destruction operators a_ (coi) and ai(COi):

t

^_E +)(ti) = | dCOi Tll(COi) * ^ai(COi) e -i°°iti

J

(A.2a)

E (ti)= (t (A.2b)

The effects of filter F1 are included in the factor 11. Similarly, the signal mode field operators may be

expanded, but these require an additional factor to account for the interferometer:

E_+'(ts) = I dCOsTl;(COs) as(C0s)e-i°°sts _ {1 -eiC°s x ei_B}

(A.3a)

E )(ts)= )(ts , (A.3b)

where x = AL/c is the optical delay time between the arms of the interferometer, and @B is the

geometrical/Berry phase.

The probability of joint detection of a signal-idler pair within the detector resolution window AT,

after a total time q;, is then given by

P = dts dti G(2)(ts,ti; ti,ts)

T/z .hs-aT

2

(A.4)

In practice, the duration time 'T of any data point is essentially infinite (with respect to all relevant

time-scales in the problem). In addition, for our experiment AT (=Ins) was much greater than x

(=730fs, for AL=2201.tm) and the reciprocal bandwidths, 1/ACOi and 1/ACOs, of the filters F1 and F2.

67
Hence,we are justified in setting the limits of integration to infinity:

v

X 111((0 _ 1"!1(0) i) TI2(0)s) 1];(0)'s) ei(toi- °Yi)ti ei(°s- O's)ts

x 21(1- e-i_xe-i¢B)(1 - ei°'sXe i¢B)

ai(0) i) as(o) s) [ 1, 1> o. _-o (A.5) X o..__,0.<1, 1

^ . ^ .

If we assume that the probability amplitude is essentially constant (A(0)) -- A o) over the filter

bandwidths A0) i and A0) s, and that rl2(0)s) = T120 over the bandwidth A0) s >> 1/x (i.e. a broad square

ban@ass filter F2 in front of detector D2), then (A.5) simplifies considerably:

P = IAol 2 20121d0)i _1(0)i_ 2 {1 - cos ((COp- 0)i)'_ + (_B)}. (A.6)

i1,

We now examine the behavior of this detection probability in two limiting cases of falter FI:

1. If lrh(0)i)] 2 = _ld 2 8(0)i- 0)i0) (i.e. a very narrow filter in front of detector D1), then

P = [A0[2 _2012 _'110_ {1 - cos ((COp- 0)i0)_ + 0B)}. (A.7)

It should be clear from (A.6) that in order to observe these fringes, it suffices to have A0)i << 1/x. This

is the situation in the lower trace of Fig. 2.

2. If _1(0)i)_ = _ld 2 e-(°_i-°_i0)2/A_i , where A0) i >> 1/_ (we have previously stipulated the

experimental condition A0)s >> 1/x ), then

p = I 12 _20[ 2 _xd 2 , (A.8)

a constant, with no fringes. The experimental results (top trace, Fig. 2) corresponding to a broad filter

F1 are in agreement with these predictions.

Note that since the filter F2 is relatively broadband (i.e. A0) s >> 1/_ ), there are no fringes visible

in the singles rate of detector D2, even though fringes in coincidence may be present.

68
REFERENCES

1. Y. Aharonov and D. Bohm, Phys. Rev. 122, 1649 (1961).

2. P.E. Hussar, Y. S. Kim, and M. E. Noz, Am. J. Phys. 53, 142 (1985), and references contained

therein; B. A. Orfanopoulos, Phys. Essays 3, 368 (1990).

3. M.A. Home, A. Shimony, and A. Zeilinger, Phys. Rev. Lett. 62, 2209 (1989).

4. P. G. Kwiat, W. A. Vareka, C. K. Hong, H. Nathel, and R. Y. Chiao, Phys. Rev. A41, 2910

(1990), and references therein.

5. P.G. Kwiat and R. Y. Chiao, Phys. Rev. Lett. 66, 588 (1991).

6. J.G. Rarity and P. R. Tapster, Phys. Rev. Lett. 64, 2495 (1990), and references therein.

7. C.K. Hong and L. Mandel, Phys. Rev. Lett. 56, 58 (1986).

8 P. G-rangier, G. Roger, and A. Aspect, Europhys. Lett. 1, 173 (1986).

9 J.F. Clauser, Phys. Rev. D9, 853 (1974).

10 R. J. Glauber, Phys. Rev. 130, 2529 (1963).

69
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N92-22052

HIGHER-ORDER QUANTUM ENTANGLEMENT

Anton Zeilinger

Institut ffir Experimentalphysik

Universit_t Innsbruck

Technikerstrasse 25

A-6020 Innsbruck, Austria

Michael A. Horne

Stonehill College

North Easton, MA 02357

Daniel M. Greenberger

Department of Physics

City College of the City University of New York

New York, NY 10031

ABSTRACT

In quantum mechanics the general state describing two or more particles is a linear superposition of

product states. Such a superposition is called entangled if it cannot be factored into just one product.

When only two particles are entangled the stage is set for Einstein-Podolsky-Rosen discussions and Bell's

proof that the EPR viewpoint contradicts quantum mechanics. If more than two particles are involved new

possibilities and phenomena arise. For example the GHZ disproof of EPR applies. Furthermore, as we point

out in this paper, with three or more particles even entanglement itself can be an entangled property.

INTRODUCTION

Of the many conceptual innovations of quantum mechanics the notion of entanglement is gaining

increasing attention over the last few years. This is because entanglement implies the quantum

nonlocality as discussed by Einstein, Podolsky, and Rosen (Ref. 1) and Bell (Ref. 2). The increased

attention has also led to a somewhat loose discussion of these topics, quite often lacking care with

respect to the fundamental issues involved. It is therefore one of the purposes of the present paper to

give a detailed discussion of the notion of entanglement.

The term "Entanglement" (in Schr6dinger's original German "Verschr/inkung") in quantum mechanics

goes back to Schr6dinger's famous 1935 paper (Ref. 3) where he gives a general confession, as he calls

it, of his understanding of the situation of quantum mechanics at that time. Most of the analysis of the

PRECEDING PAGE BLANK NOT FILMED

73
measurementproblempresentedin SchrSdinger'spaperrests on the properties of nonfactorizable states of

two-particle systems as first discussed in the same year by Einstein, Podolsky and Rosen (EPR). Here we

shall first briefly review entanglement for the case of two particles. Then, focussing on situations

where three particles are entangled, we will see that entanglement itself can be an entangled property.

TWO PARTICLES

For two-particle systems the best known entangled states are those which exhibit entanglement of

spin variables, e.g. in the case of two spin-l/2 particles with total spin zero this is the singlet state

(Ref. 4)

kb> = _/I---_ [ l+>xl-> 2 - l->xl+>2]. (1)

Here, I+> describes particle one with spin up etc. Because of the rotational symmetry of the singlet 1

state the direction along which the spins are defined in Eq. (1) need not be specified.

As can easily be seen, the state of Eq. (1) does not make any specific predictions for spin

measurement results on either particle, but it makes the definite prediction that, as soon as the spin of

one particle is found to be oriented along one direction, the spin of the other one will be found to be

oriented along the opposite direction should it be measured along that direction. SchrSdinger calls this

property "entanglement of predictions" or "entanglement of our knowledge of the two bodies". As in

classical physics, one might draw up before the measurement an expectation catalog which gives the

possible measurement results together with the probabilities of these various results. In quantum

mechanics, the expectation catalog has to be calculated from the quantum state of the system.

While in classical mechanics the combined expectation catalog for two objects (bodies, particles,

pieces...) can always be written as the logical sum of the expectation catalogs of the individual

systems, this is not possible anymore in quantum mechanics for the case of entangled states. Or, in other

words, while in classical physics (and certainly in the case of factorizable quantum states) disjoint

catalogs for two bodies that once did interact exist, in quantum mechanics this is generally not true

anymore. Or, in SchrSdinger's original words: Maximal knowledge of a total system does not necessarily

include total knowledge of all its parts, not even when these are fully separated from each other and at

the moment are not influencing each other at all. This results in the interesting nonlocality questions

in quantum mechanics.

Consider a measurement on particle 1 along some direction. The experimenter is certainly free to

choose this direction at will, call it direction n. The experimental result for particle 1 along that

74
direction can either be + ("up") or - ("down"). The entangled state (Eq. 1) implies that particle 2 is

then either in the state I-n> if particle 1 was found to be up along n, or particle 2 is in the state

I+n> if particle 1 was found to be down along n. This is simply a consequence of the von Neumann

reduction of the state vector which is equivalent to the property that, upon measurement of particle I,

the expectation catalog for particle 2 changes to be in agreement with the result for particle 1 and the

predictions obtained from the entangled state (Eq. 1).

In the case of two particle-entanglement just discussed it is the experimenter's choice of the

direction along which she measures the spin of particle 1 which determines that particle 2 will be in an

eigenstate along that direction. Which specific eigenstate of the two possible ones it will be is

completely random and outside the influence of the experimenter, it is "Nature's choice". We might also

express this as the property that after measurement of particle 1, the expectation catalog for particle 2

gives a definite prediction for measurement along the same direction, the specific result being

objectively undefined until the measurement on particle 1 is actually performed.

To summarize, for two particles, entanglement implies that no disjoint catalogs for all observable

properties of the two particles exist and that the specific result of the measurement of an entangled

quantity instantly permits prediction with certainty of the result of a measurement of the related

quantity on the other particle.

THREE PARTICLES, TWO TERMS

All the discussion on entanglement in quantum mechanics until recently exclusively focused on

two-particle states only. Yet it is evident that correlations between three and more particles provide a

richer abundance of new quantum phenomena. For example, while any entangled state of two particles can

always be written as a sum of just two terms (see e.g. Eq. 1) this in general is not true anymore for

three particles. Specifically, there may be experimental situations where the state of the three-particle

system consists of two terms or maybe even one term only, while in other situations three or more product

terms are necessary for a complete description. Here we shall first analyze a specific case of a

three-particle experiment where the state contains two terms.

The introduction of three-particle correlations into discussions of the EPR-paradox (Refs. 5 - 7)

and related questions not only did lead to more stringent contradictions between local realistic models

and quantum mechanics than in two-particle situations, it also provides qualitatively new entanglement

phenomena. Let us consider a three-particle interferometer experiment of the type recently proposed

(Ref. 7). A suitable source, say a nonlinear crystal exploiting a second-order nonlinearity of the

75
electric polarizability, emits three photons in the entangled state

l_b>= _/-i--_[IA>IIB>21C>s + [A'>I[B'>21C'>s]. (2)

Here, e.g.,[A>I describesphoton one in beam A etc.(seeFigure l).We now considertwo possiblechoices

the experimenter has:

(a)She might determine which path photon l takesby placingdetectorsintobeam path A and beam

path A'. As soon as one of these detectorsfires,the stateof the system collapsesdue to von Neumann

wave packet reduction.This impliesthat depending on whether the detectorin beam path A firesor the

one in beam path A' the stateof the remaining two particlesisdifferent.IfdetectorA registersphoton

I (we assume, as is customary in photon experiments, that the photon is absorbed by a detector

registering it) that state is

I_b>' = IB>21C>s.

But, if detector A' registers photon 1 the state of the remaining two photons is

(3)

I_b>' = IB'>_IC'> s. (4)

In either case, after registration of photon 1 the remaining two photons are left in a product state,

i.e. they are not entangled. In other words, registering photon 1 in either beam A or A' did untangle'_the

other two photons. From a complementarity point of view this might readily be understood on basis of the

fact that registering any of the photons in a beam path before it encounters the recombining beam

splitter instantaneously provides information not only in this photon's path but, because of the momentum

correlations implied in state (2), also on the paths taken by the other two photons.

(b) The experimenter might alternatively decide not to insert detectors into any of the beam paths

before the recombining beam splitter but measure the interference fringes instead. In order to simplify

the analysis we assume that the phase shifter phases (Fig. 1) are all chosen to obey the condition

_1 + ¢2 + ¢s = r/2. (5)

Let us now call D (E) the beam path of particle 1 leading to detector R 1 (L1) and likewise F (G) and H

(J) the beam paths of particles 2 and 3 leading to their detectors. Without loss of generality we assume

that photon 1 is registered in detector D: The state of the remaining 2 photons is then (see Ref. 7)

l_b> = v/1 /_ [IE>21F> s - IE'>2IF'>3]. (6)

Thus photon 2 and photon 3 are clearly in an entangled state now. This holds always for the remaining two

photons if a photon was registered after any of the other beam splitters. In other words, registering a

photon in a detector behind its recombining beam splitter does not untangle the other two photons.

76
Clearly, the experiment can be set up in such a way that the detection events for the three photons

occur at spacelike separation. Nevertheless, the experimenter's decision as to which measurement is to be

performed at one of the photons determines - upon registration of the measurement result - whether or not

the other two photons are left in an entangled state.

In SchrOdinger's terms this means that the experimenter, simply by deciding whether to measure

photon 1 before or after its recombining beam splitter, also decides whether or not the other two photons

each enjoy their own disjoint expectation catalog, no matter how far these other photons might be away at

the time that decision is made or at the instant of registration of photon 1. For completeness we simply

remark here that, as in two-particle entanglement, the detection events of the three photons might be

arranged in any time order.

A THREE-TERM STATE

For three or more particles with each particle enjoying its own two-dimensional Hilbert space there

are evidently a number of different three-term states. Clearly the details of the experimental situation

determine which state is present. To be specific, let us analyze the experiment represented in Figure 2.

There, the incident beam A bearing radiation with wave number k = 2_'/_ is split by a series of successive

partially reflecting mirrors into the beams B, C, and D and a through going beam. This latter one is

assumed to enter a nonlinear crystal where it is upconverted into radiation with wavenumber 3k and then

passes a filter set at the wavelength )d= 2_r/3k. Afterwards it is downconverted again. We shall call such

a device consisting of an upeonverter, a monochromatic filter and a downconverter a "number filter"

because it lets only pass states with a certain number of photons, in our case three. This beam

subsequently encounters a partially reflecting mirror again where some of it is deflected into a beam

which we call A' towards a detector set to count a particle, call it particle 1. Registration of particle

1 in that detector acts as a trigger signal to indicate that a three-particle state has passed the number

filter.

The transmitted beam again encounters some mirrors such that the beams B', C', and D' result as

indicated in Figure 2. These latter beams are then superposed at a set of three semireflecting mirrors

with the corresponding beams B, C, and D and we assume that detectors are placed into the outgoing beams

of these semireflecting mirrors. We arrange the experiment such that the amplitudes incident on either

port of any recombining mirror are equal and we look only for such events where in one and only one of

the outgoing beams of each semireflecting mirror a photon is found (assuming 100% efficient detectors).

Let us call these photons by their numbers 2, 3, and 4 respectively and let us agree to look only at the

detectors if particle 1 has been registered in its detector.

77
The state of these photon quadruples is

I_b>= _/1/-_ [A'> 1 IIB'>2IC'>aID>4 + [B'>2IC>aID'>4 + IB>2IC'>3ID'>41 (7)

because this describes exactly the situation where photon 1 triggers the detector in beam path A' thus

indicating that a three-photon has passed the number filter and one of these is photon 1. The other two

photons coming through the filter might either be photon 2 and 3, or photon 2 and 4, or photon 3 and>4 as

indicated by the primes in the terms above. For each term one other photon must have used the unprimed

beam completing the quadruple.

We immediately note that in the state (7) photons 2, 3, and 4 are entangled with each other, while

photon 1 is not entangled with any of the others, it enjoys its own disjoint catalog of predictions. More

importantly, let us now consider what happens if we place a detector into any of the primed beam paths of

photon 2, 3, or 4. Suppose explicitly and without loss of generality that a detector is placed into path

B' of photon 2. Still we assume that we only look at such cases where the detector in beam A' has

registered photon 1. Then two different possibilities arise: Either the detector in beam path B' fires or

it does not fire. If it fires, the state of the remaining two photons, the photons 3 and 4, is

I_b'> = _/1/-_ [IC'>zlD> 4 + IC>zlD'>4] (8)

which is clearly an entangled state for these two photons. On the other hand detector B' might not fire.

In that case the predictions for the quadruple of photons can be described by the last term of Eq. (8)

I_b'> --IA'>IlB>2IC'>sID'> 4

(9)

which now implies that none of the photons is entangled with any of the other photons. Most remarkably,

if the detector inserted into the beam path B' of photon 2 fires, photon 3 and 4 are entangled. If that

detector does not fire, photon 3 and 4 are not entangled anymore! The absence of a registration event of

the detector in beam B' untangles the other two photons. Whether or not such a registration event happens

is completely and objectively random, at least within the standard interpretation of quantum

probabilities. In this situation it is therefore a totally random event happening to photon 2 which

determines whether or not photons 3 and 4 are entangled.

We point out here too that the apparatus might easily be arranged in such a way that the detection

events on particles 2, 3 and 4 are spacelike separated from each other. Again we might call the specific

random event "Nature's choice" and we find it quite remarkable that a spacelike separated random event

happening to particle 2 decides whether or not particle 3 and 4 are entangled with each other.

78
CONCLUDINGCOMMENTS

Several comments might be in order. Firstly we remark that the analysis given above is not

restricted to multiparticle interferometry. In fact, it is rather straightforward to give an example in

terms of spin correlations. We might also point out that such experiments, though they have not been

performed yet, should be feasible given further development in our knowledge of the nonlinear conversion

processes in quantum optics. Furthermore, it was implicitly assumed that the incident radiation is rich

enough to contain the multiparticle states exploited in the various experiments. Finally, entangled

entanglement also is consistent with special relativity in the sense that it does not permit information

to be transmitted with a speed larger than that of light.

This work was supported by the Austrian Fonds zur F6rderung der wiss. Forschung projects No.

S 42-01 and P 6635 and by the US NSF grants No. DMR-87-13559 and INT-87-13341.

REFERENCES

1. Einstein, A., Podolsky, B., and N. Rosen, 1935, "Can Quantum Mechanical Description of Physical

Reality Be Considered Complete?" Phys. Rev., 47, p. 777.

2. Bell, J.S., 1964, "On the Einstein-Podolsky-Rosen Paradox" Physics, 1, p. 195. Reprinted, 1983, in

Quantum Theory and Measurement, J.A. Wheeler and W.H. Zurek, eds., Princeton, p. 403.

3. Schr6dinger E., 1935, "Die gegenw_irtige Situation in der Quantenmechanik," Naturwissenschaften, 23,

pp. 807-812, 823-828, 844-849. English Translation, 1980, "The Present Situation in Quantum

Mechanics," Proceedings of the American Philosophical Society, 124, p. 323; reprinted, 1983, in

Quantum Theory and Measurement, J.A. Wheeler and W.H. Zurek, eds., Princeton, 19. 153.

4. Bohm, D., 1951, Quantum Theory, Prentice-Hall, Englewood Cliffs, NJ, pp. 614-623.

5. Greenberger, D., Horne, M. A., and A. Zeilinger, 1989, "Going beyond Bell's Theorem," in Bell's

Theorem, Quantum Theory, and Conceptions of the Universe, M. Kafatos, ed., Kluwer Academic, Dordrecht,

The Netherlands, 19. 73.

6. Mermin, N. D., 1990, "Quantum Mysteries Revisited," Am. J. Phys., 58, p. 731; and "What's Wrong with

these Elements of Reality?," Physics Today, 43(6), p. 9.

7. Greenberger, D. M., Horne, M. A., Shimony, A., and A. Zeilinger, 1990, "Bell's Theorem without

Inequalities," Am. J. Phys., 58, p. 1131.

79
(I)1

Source

_3

B

Fig. 1: Principle of a 3-particle interferometer experiment.

8O
_ D

A , ._

s _

,- ","" '' :i_ I" 7'

3k 1 1A'

C'ID'

Fig. 2: Principle of an interferometer experiment exhibiting entanglement of more than two terms.

81
N92-22053

Algorithmic Information Theory and the Hidden Variable Question _ _Y

! J

Christopher Fuchs

Dept. of Physics & Astronomy, University of North Carolina

Chapel Hill, NC 27599

Abstract

This note explores, via information theory,

the admissibility of certain nonlocal hidden-variable

theories. Consider a pair of Stern-Gerlach devices

with fixed nonparallel orientations that periodically

perform spin measurements on identically prepared

pairs of electrons in the singlet spin state. Suppose

the outcomes are recorded as binary strings 1 and r

(with !n and r n denoting their n-length prefixes). The

hidden-variable theories considered here require that

there exists a recursive function which may be used to

transform 1n into r n for any n. This note demonstrates that such a theory cannot reproduce all the

statistical predictions of quantum mechanics. Specifically, consider an ensemble of outcome pairs (1,r).

From the associated probability measure, the Shannon

entropies H n and N n for strings In and pairs (in,rn)

may be formed. It is shown that such a theory requires that [ H n - Hn [ be bounded - contrasting the

quantum meehanical predietion that it grow with n.

I. Introduction

The class of inequalities initiated by Bell 1 do

not absolutely exclude the possibility of hidden variables underlying the phenomena statistically described

by quantum mechanics. Hidden-variable theories of

the so-called nonlocai variety are not constrained by

Bell's theorem. Although there is no pressing theoretical reason for taking the existence of such a theory

seriously, it is clear that one can only truly begin to

understand quantum mechanics when one first understands what it is not. This note will attempt to make

a contribution to this end. Here a seemingly not- a

priori unreasonable class of nonlocal hidden-variable

theories called the "computable hidden-variable theories" (CHV's) will first be defined for a particular

thought experiment and then shown to be inconsistent

with certain statistical requirements of quantum mechanics. The reason for this procedure is to make explicit, through the language of algorithmic information theory, 2'3 an aspect of quantum theory hitherto

seldom discussed and then demonstrate the practical

use of this aspect in answering foundational questions.

This aspect is that the data obtained from identical

measurements performed on identically prepared systems is generally "algorithmically incompressible."

H. The Thought Experiment and the Result

The thought experiment described is a modification of the standard one used for discussions of

Bell's theorem. Consider a pair of distantly separated Stern-Gerlach (SG) devices situated so to (flawlessly) measure the spins of a pair of correlated electrons. These are called the left and right devices, respectively. For definiteness, suppose that the correlated electrons are in the singlet spin state

Assume that the left and right SG devices, respectively, are oriented so that they invariably measure

spin along _ and an axis that differs from _ by a computable angle 0. "0 computable" simply means that

there is an algorithm for generating the decimal expansion of 0. E.g., 0--lr/6 is clearly computable.

Suppose that at periodic time intervals these devices

are supplied with identically prepared pairs of correlated electrons. (This would allow the measurement

outcomes to serve as a "window" into the "hidden"

dynamics of the devices, if such a dynamics did indeed

exist.) Finally, imagine that each SG device is endowed with the capability of recording its measurement outcomes as a string of binary digits - 0 and 1

denoting down and up outcomes, respectively. Denote

the left and right strings, respectively, by ! and r, and

their n-length prefixes by in and r n. E.g., a typical

run of the devices might give l-01101011.., and

r = 10110100... ; the length-4 prefixes for these

strings are i4 = 0110 and r4 = 1011. To cap off the

description of the thought experiment, assume that

there is in fact an ensemble of such devices: each

macroscopically identical to the next, each with its

own supply of electrons, and each performing the operation described. Associated with this ensemble will

be an ensemble of ordered pairs (l, r) and consequently

ensembles of pairs (In, rn).

With this as a scaffold, the CHV notion can

be formalized. Simply put, a CHV is said to be responsible for the measurement outcomes if for every

pair (1, r) in the ensemble, there is at least one of a finite set of computer programs (more formally recursive functions) that, for any n, produces the string r n

as output whenever given In as input. Note that each

string ! can have as its origin any process whatsoever:

83

PRECEDING PAGE BLANK NOT FILMED
deterministic or indeterministic. This definition only

requires that a rigid, mechanistic relation between l

and r be maintained. Furthermore, such a theory is

inherently nonlocal; the program provides the "medium" for the instantaneous action of the one string on

the other. The finiteness of the set of programs is

meant to allow for the possibility that the SG devices

in the ensemble might have differing microscopic

initial conditions.

The main result may now be stated. Because

a CHV provides a compression scheme for the measurement data, it must contradict the statistical predictions of quantum mechanics. Suppose the probability distributions for the ensembles of strings in and

pairs (ln, rn) are Pn(in) and qn(ln, rn), respectively.

The Shannon entropies for these distributions are:

Hn = - E tn Pn(ln)log [Pn(ln)]

and (2)

= - Ez Er q.(l.,rn)log[q.(l.,r.)],

where log denotes the base-2 logarithm. Consider the

quantity [ Ha- Hn I • Standard quantum theory requires that this be proportional to n. For CHV's,

however, this quantity is necessarily bound by a constant independent of n. The remainder of this section

will be devoted to justifying the quantum mechanical

result; the corresponding result for a CHV will be

derived in the next two sections.

Suppose that standard quantum theory does

indeed hold in the thought experiment. In that case,

the required Shannon entropies are straightforward to

derive. The essential ingredient in this derivation is

simply noted: quantum theory declares that the only

condition determining the measurement outcomes is

the probability distribution derivable from (1).

Hence, the probability of a 0 or a 1 occurring in the

k'th place of a string I n must be independent of k.

Furthermore, this probability is independent of which

left-hand SG device in the ensemble produced in.

Analogous results hold for any string r n and for the

correlation probabilities between the k'th places of !n

and r n. With these considerations, it is a simple

exercise in quantum mechanics to show that

Itln - Hal = -f(O) n, where

f(O)=(sin2_iog[sin2_] + cos2_log[cos2_]).

Therefore, for any 8 other than 0 = 0 or O = r,

IHn - Hal o¢ n. (3)

IV. Algorithmic Information Theory

This section introduces enough of the apparatus of algorithmic information theory that the main

result can be proven. It does not purport to be a general introduction to the subject. The notion of a "recursive function" is taken as primitive. For the most

part, this section follows the development of algorithmic information theory found in Ref. 3.

Notation and Definitions:

Let X = {A, 0, 1, 00, 01, ...} be the set of finite binary strings in lexicographic order, where A is

the empty string. Elements of X may be thought of

dually as strings and natural numbers. Let X n be the

set of n-length strings. O(1) denotes a bounded function. The variables a, t, v, z, and y denote elements

of X. The length, n-length prefix, and k'th digit of a

are denoted by Is[, sn, and s(k), respectively. A set

S C X is called an instantaneous code if for any z, y in

S, neither x nor y is a prefix of the other. Elements of

S (denoted generally by r) are called programs. A

computer C is.a recursive function C: S x X--_X. A

computer U is said to be universal iff for each computer C there is a constant k C such that: if C(r,v) is

defined, then there exists a program r'E S such that

U(r',v)=C(r,v)and Ir'l < Irl +kc" Let aparticular countably infinite instantaneous code S and universal computer U be chosen as standard. Finally, let

(, }: X x X --*X be a recursive bijection with the property that if[ s [ = [ t I, then (s,t) = s(1)t(1)s(2)t(2) ....

E.g., if s = 011 and t = 101, then (s,t) = 011011.

The algorithmic complexities are defined by:

Kc(s/t ) = min{ Ir I : c(r, t) = s}

Kc(s ) = Kc(s/A )

Kc(s ,t) = Kc((S ,t))

K(s/t) = Ku(s/t )

The canonical program s* for s is defined by s*=

min{r: U(r,A) = s}. Clearly Is*] = g(s).

Now let p:X_[0,1] and let Pn denote the

restriction of p to X n. p is said to be a probability

measure for a stochastic process if it satisfies:

E Pn(X) ----1 _ Pn- I(Y) -- Pn(Y 0)'_ Pn(Y 1)

Ixl =-

for any n and any y G Xn _ 1- If p is recursive, then p

is said to be a computable measure. In this section,

only computable measures are considered. The Shannon entropy Hn for p over n-length strings is:

H. = - p.( )iog

Ixl =-

Finally, with the measure p, the average complexities

(g/y)p and (g)p for n-length strings are defined by:

(KlY)$ = E Pn (x)K(z/y) & (K)_- (g/A)_.

I_:1 =,,

84
Theorems:

Theorems (a)-(f), from Ref. 3, are listed so

that the present treatment will be self-contained.

Theorem (i), from Ref. 4, provides the link relating

complexity to entropy. Theorems (g), (h), and (j) are

simple results due to the author. When crucial, rather

than relegating bounded terms to a term written as

O(1), a constant written in the form D..., where the

ellipsis symbolizes a set of subscripts, will be used so

that all dependencies are clear. (E.g., Dp denotes a

constant that depends only on the measure p.)

(a) For any computer C, K(s/t) < Kc(s/t ) + kC.

(b) K(s) < g(s, t) + O(I).

(c) K(s/t) <_ K(s) + 0(1).

(d) K(s,t) = K(t,s) + O(1).

(e) g(s,t)= K(s)+ g(t/s*)+O(1). (To make the

0(1) term's dependence on U explicit, this can be

written as I K(s, t) - K(s) - K(t/s*) ] <_D U.)

(0 If f:N--* N is a recursive function and _(l_/(n)

converges, then K(n) < f(n) + (5(1).

(g) K(t/s*) < K(t/s) + O(1).

Proof: Consider a computer C such that C(r, v)=

U(r, U(v,A)). Then C(r,s*) = U(r,s). Hence it must

be the case that Kc(t/s* ) =K(t/s) for any t.

By (a) then, K(t/s*) < K(t/s) + O(1). D

(h) -D v_< K(s)-K(s/lsl') <- 21oglsl +Du.

Proof: A similar result is derived in Ref. 4. The lefthand inequality is a consequence of (c). By (f), there

is a constant D U such that K(n)< 2 log(n)+ D U for

all n. The right-hand inequality then follows from

successive applications of (b), (d), and (e). E!

(i) There is a constant DU, p such that, for all n,

0 < (g/n*)_-Hn _< DU, p.

(j) There are constants D U and D U p such that, for

alln, - D U < (KI_ - U n <_ 21o'gn + Du, p.

Proo_....._f: This is a simple consequence of (h) and (i). []

V. Computable Hidden-Variable Theories

Armed with the last section's tools, a precise

definition of a CHV can now be formed. Let :g denote the set of all possible pairs (1, r) in the ensemble

of strings produced by the thought experiment.

Def: A CHV V is said to be responsible for

the measurement outcomes if there is a finite subset

V C S such that for each (1, r) E 3£ there exists a v E V

for which it is the case that U(v, in) = r n for every n.

Notice immediately that if V is responsible for the

outcome strings, then for each (l,r)E :g it follows

that K(rn/ln) <_max{ I v I:v E V} for all n. But then

by (g), K(rn/ln)< DU. v for all n. This, coupled

with (e), leads to th'e following conclusion.

_hm 1: If V is responsible for the measurement outcomes, then for each (1, r) E :g, it follows

that [K(In, rn) - K(In)I <- DU, V for all n.

Now consider the probability measures for the

outcome strings using the notation introduced in Section II. It is assumed that these measures are computable. (This will be the case if standard quantum

theory is valid since O is required to be computable.

If it were not the case here, by being noncomputable,

p and q would trivially differ from the values predicted by quantum mechanics and there would be no

need for further discussion.) For these measures:

_r qn(ln, rn) = Pn(ln) for all n. An important

factn to note is that qn(Sn, tn) vanishes iff (s, t)_ %.

Hence from Theorem 1 it follows that, if V generates

the measurement outcomes, for all n, the quantity

[ Eln,rnqn(ln, r,)K(ln, rn) - ElnPn(In)K(In) I

will be bounded by a constant D U v. Now because of

the form of the bijechon (ln, rn) , the double sum in

this expression may be construed as a single sum over

strings of length 2n. This fact leads to the following:

Thm 2: If V is responsible for the outcome

strings, I(g)2q n- (g)_l _<DU, v for all n.

Combining Theorems 2 and (j), the following emerges:

Thm _: If V is responsible for the outcome

strings, I B'n - Hn I < Dtr, v, p, q for all n.

This is the sought after identity; for, although

DU, V. p q depends on the the CHV explicitly (through

V and possibly p and q), It m independent of n.

I wish to thank D. Brown, G. Comer, M. Fuchs, J. KinseUa,

and J. W. York, Jr. for helpful discussions and encouragement.

This work was supported by NSF Grant PHY-8908741.

Refel_nc_!

1For a more recent effort see S. L. Braunstein and C.

M. Caves, Phys. Rev. Lett. 61, 662 (1988).

2G. J. Chaitin, Information, Randomness 8_ Incompleteness (World Scientific, Singapore, 1987); W. H.

Zurek, Phys. Rev. A 40, 4731 (1989).

3G. J. Chaitin, J. ACM 22, 329 (1975).

4S. K. Leung-Yan-Cheong and T. M. Cover, IEEE

Trans. Inf. Theory IT-24, 331 (1978).

85
N92-22054

Photon Polarization Version of the GHZ-Mermin Gedanken

Thomas E. Kless

Department of Physics

University of Maryland

College Park, MD 20?42

ABSTRACT

We have defined a photon

polarization analog of the GHZ

experiment that was initially

proposed for spln-1/2 quanta.

Analogs of the ket states and Pauli

spin matrix operators are presented.

DISCUSSION

We have developed an explicit

photon polarization version of the

three-quanta GHZ experiment (Ref. I)

discussed by Mermin (Ref. 2). We

define operators, eigenkets, and

measurements in the two-dimensional

space of photon polarization that

map directly onto the Pauli spin

matrix representation of a spin-I/2

system. This construction enables

us to represent the GH'Z experiment

in terms of photon polarization

measurements on a three-photon

quantum state.

This photon analog of the

spin-i/2 counterparts is developed

by using retardation plates, which

rotate the polarization of an

incident photon by producing an

associated change of phase. A A/4

retardation plate with optic axis at

45 ° to an imposed X-Y coordinate

system will act analogously to _ ,

x

up to a phase representing a

polarization-independent

translation. A A/4 plate with optic

axis parallel to one of the X-Y axes

produces the v operation, again up

Z

to an inessential phase factor. The

product operation defined by both

retardation plates accomplishes the

operation _ =I_ -_ .

y z X

The polarization states that

represent eigenkets of _ are [X>

z

and [Y>, measured by using a

birefringent polarization analyzer

(e. g. a Wollaston prism) oriented

to send each of these polarizations

into a distinct direction, en route

to one of two separate

photomultiplier tube detectors. The

polarization states that represent

eigenkets of v are defined

X

similarly, to be light linearly

polarized at 45 ° and 135 ° to the

X-axis. These polarizations are

measured by rotating the analyzer at

45°to its _ orientation, and

Z

recording which of the two

phototubes generated an output

pulse. Finally, the elgenkets of

y

are found to be left- and rightcircularly polarized light in this

representation. These are measured

(i. e. ±I is determined) by

inserting a quarter-wave plate in

front of the prevlously-defined

analyzer, to generate In one output

direction light that was originally

left circular polarized, and in the

other direction the originally right

circular polarization.

The experiment consists of

first verifying that the

three-photon state being studied is

a +I eigenstate of each of the

! 2 3 1 2 3

operators A= _ "_ "_ , B= _ "_ "_ ,

1 2 _ x x x y x

and C= _ ._ ._ . A measurement of

X X y

87

PRECEDING PAGE BLANK NOT FILMED
operator 0 = A.B.C represents the

decisive test between quantum

mechanics and local reality

theories. The quantum mechanical

prediction for the observed

eIgenvalue of 0 is -1, whereas the

local reality prediction can be

easily shown to be +1.

The most difficult element of

any non-gedanken experiment similar

to the one presented here is the

construction of the three-photon

quantum state. The requislte state

vector is a perfectly

anti-ferromagnetic "entangled" ket

proportional to

[IX_IX_IX>-IY>IY_IY_]3 I which is

difficult to manufacture (Ref. 3) by

any physical process, such as a

three-photon emission of an excited

species. We have proposed a method

of creating this state by hand, a

task not obviously possible because

most operators on optical photons do

not restrict themselves to the

Hilbert space of any one photon

exclusive of the others.

apparatus to cause the amplitudes of

some components of the total ket to

sum to zero. This cancellation of

undesired "cross terms" produces the

anti-ferromagnetlc state needed in

the CHZ test.

REFERENCES

i,

.

3.

Greenberger, D. M., M. Home,

and A. Zeilinger, 1989, "Going

Beyond Bell's Theorem," in

Bell's Theorem, Quantum Theory,

and Conceptions of the Universe,

M. Kafatos, ed., Dordrecht, The

Netherlands, p. 73.

Mermin, N. D., 1990, "What's

Wrong with These Elements of

Reality?," Physics Today

(June 1990), p. 9.

Greenberger, D. M., M. A.

Horne, and A. Zelllnger, 1990,

"Be11's Theorem Without

Inequalities," Am. J. Phys.,

58(12), p. 1131.

We assemble the needed

three-photon ket by sending a

productIX IX IX of three

one-photon number states (obtainable

by using an attenuated Glauber state

for each one) onto an array of beam

s;_!tters and interferometers

(_ither Mach-Zehnder or Michelson

will work here) to achieve a

summation of correlated amplitudes

at the end of the apparatus (see

Figure). The total apparatus is a

three-tiered structure, with one

tier for each photon. For clarity,

only one tier is shown in the

figure. The boxed interferometers

are specific to each tier; all other

optical components (beamsplitters

and mirrors) are common to all three

tiers, a requirement that can be

satisfied for sufficiently large

components. The entangled state is

obtained by using phases in the

88
0 - beamsplitter

I - half wave plate

I = mirror i_

D2 - phototube ......

pump

_'-KDP crystal

Apparatus Definin8 the Three-Photon State

89
°

N92-22055

PROPOSAL FOR A QND WHICH-PATH MEASUREMENT USING PHOTONS*

M. G. Raymer and S. Yang

Department of Physics and Chemical Physics Institute

University of Oregon, Eugene, OR 97403

A scheme is proposed for experimentally realizing the famous two-slit

ged&nken experiment using photons. As elegantly discussed for electrons by

Feynman, a particle's quantum pathways interfere to produce fringes in the

probability density for the particle to be found at a particular location. If the

path taken by the particle is experimentally determined, the complementarity

principle says that the fringes must disappear. To carry out this experiment

with photons is difficult because normally the act of determining a photon's

location destroys it.

We propose to overcome this difficulty by putting a type-II optical

parametric amplifier (OPA) in each arm of a Mach-Zehnder interferometer, and

observing fringes at the output, as shown in Fig.1. An OPA responds to an input

photon by increasing its probability to produce a pair of photons, one having

(vertical) polarization orthogonal to the (horizontal) polarization of the input

photon. A polarizing beam splitter is used to eject only those photons with

polarization orthogonal to the input, the detection of which allows partial

inference about the path taken by the input photon without destroying it. The

measurement is thus of the quantum nondemolition (QND) type.

1 Photon Input

Pumping ,2_'-T

.......... _ |, v I OPA

Vacuum _ OPA

Detector

Fig.1 Apparatus for which-path measurement.

PRECEDING PAGE BLANK NOT FILMED

91
The price paid for this inference is at least one noise photon in the

interferometer, which degrades the fringe visibility, in accordance with the

complementarity principle. Information theory is used to show that the

visibility is connected to the amount of information available to be collected,

regardless of whether or not anyone looks at it. In this sense information

should perhaps be regarded as a physical quantity, rather than a subjective

concept.

The calculation treats the signal (horizontal polarization) and idler

(vertical polarization) modes of the OPA quantum mechanically and the pump

mode as a given classical field. The one-photon input state is transformed on

the 50/50 beam splitter by a unitary transformation (Ref.1), and then is acted

upon by a factorized two-mode squeezing operator (Ref.2) for each OPA crystal.

From the resulting probabilities for mode occupation, Bayes theorem is used to

infer the probabilities P(upper I nv,mv) and P(Iower I nv,mv) for each path (upper

or lower) that the input photon may have taken. This inference is possible

because the probability distributions for numbers of generated idler photons,

nv,mv, depend on the number of photons (0 or 1) entering each OPA.

Information gain is defined in the following way knowing the prior

probabilities (1/2,1/2) and the final probabilities inferred from Bayes theorem:

AI=[prio r-I_,_,where lprio _=| bit is the initially missing information, and the

final information after the measurement is

I final = P(upperl n_, m_ ) log 2 P(upperl n_, m v ) +

P(lowerl n_, m_ ) log 2 P(lowerl n v , m_ ).

For a given sub-ensemble of trials in which nv (my) idler photons are generated

in the upper (lower) arm, the fringe visibility is found to be

V,,_.,,,_ = 2(m,, + 1)(n v + 1)/[(m_ + n_ + 1)(m_ + nv + 2)].

The sub-ensemble fringe visibility and information gain are plotted in

Fig.2 for different values of nv and mv. When nv=mv the information gain is zero,

and the visibility approaches unity for small nv and my. For larger values of nv

and mv the noise in the OPA degrades the visibility even though no information

is imparted. For nv=O and mv >>1, the information increases and the visibility

decreases, in accordance with the complementarity principle. See related

discussions (Refs.3-5).

92
Fig.2 Information gain and fringe visibility versus measured photon numbers.

Thus, partial measurements of the photon path can be made, but noise is

added, degrading the visibility, and thereby enforcing the complementarity

principle. If nv and my are not measured, and the total ensemble is used to

calculate visibility, it can be shown that there is still an inverse relation

between average information gain and visibility. Thus, it is not necessary to

collect the information, only that it be "out there" available to be collected.

This suggests that information has an objective, rather than a subjective,

physical reality. It should be considered whether information plays a l

unrecognized role in physical processes, and as such should be incorporated in a

more explicit, dynamical way into the theory of quantum mechanics.

* Research supported by the National Science Foundation.

1. Prasad, S., M. O. Scully, and W. Martienssen, 1987,0pt. Commun.62, p.139.

2. Schumaker, B. L., and C. M. Caves, 1985, Phys. Rev. A 31, p. 3093.

3. Glauber, R. J., 1986, "Amplifiers, Attenuators and the Quantum Theory of

Measurement," Frontiers In Quantum Optics, E R Pike and S. Sarkar, eds.,

Malvern, pp 534.

4. Sanders, B. C. ,and G. J. Milburn, 1989,Phys. Rev. A39, p. 694 .

5. Scully, M. O., and H. Walther, and B. G. Englert, and J. Schwinger,1990,

"Observatior_ And.Complementarity In Quantum Mechanics: New Tests And

Insights," Proceedings of the Matter Wave Interferometry Workshop, Santa Fe, New

Mexico, Jan. 15-16.

93
N92-22056

LOCALIZATION OF ONE-PHOTON STATE IN SPACE AND _ 2_//_3

EINSTEIN-PODOLSKY-ROSEN PARADOX IN SPONTANEOUS

PARAMETRIC DOWN CONVERSION (0 ff

A.N.Penin, T.A.Reutova, *)A.V.Sergienko

Department of Physics, Moscow State University,

Moscow 119899, USSR

*)Present address: Department of Physics and Astronomy,

University of Maryland, College Park, MD 20742

An experiment on one-photon state localization in space using a

correlation technique in Spontaneous Parametric Down Conversion

(SPDC) process is discussed. Results of measurements demonstrate

an idea of the Einstein-Podolsky-Rosen (EPR) paradox for coordinate

and momentum variables of photon states. Results of the experiment

can be explaned with the help of the advanced wave technique

developed by D.N.Klyshko /1,2/.

The experiment is based on the idea that two-photon states of

optical electromagnetic fields arising in the nonlinear process of the

spontaneous parametric down conversion (spontaneous parametric

light scattering) can be explained by quantum mechanical theory

with the help of a single wave function. The interaction of

monochromatic laser radiation with a nonlinear crystal without a

center of symmetry results in the spontaneous emergance of twophoton states with a broad set of different coordinate-momentum

and energy-time pairs of variables. The radiation after the nonlinear

crystal has a continuous distribution of wavevectors in space

depending on the nonlinear properties of crystal and phase matching

conditions t,01+f,02=O)L, kl+k2=kL. This forms the main reason why we

can easily measure coordinates or wavevectors of photons. The

typical experimental setup for the measurement of the distribution

of scattered radiation in space as a function frequency that have

being used in our earlier works /3-5/ is illustrated in Fig.1.

Ultraviolet radiation at Z,=325 nm from a He-Cd laser interacted with

a 2 cm long nonlinear LilO3 crystal and created broad band scattered

radiation centered at _,=650 nm. The radii of rings in the focal plane

of the collecting lens are defined by phase matching conditions. The

95

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j •

I I

I

I

I

I

1 1

Fig.l. Outline of the experimental setup for the investigation of

correlation properties of radiation in the spontaneous parametric

down conversion process. 1- nonlinear LiIO3 crystal, 2- collecting

lens, 3-spectral device, 4- photodetectors, 5- coincident circuit, 6-

counters.

Fig.2. Frequency-angular dependence of scattered radiation, the

symbols • and o denote the photons conjugated by phase matching

conditions.

96
thickness of rings of different frequency depends on the parameters

of spacial coherency and the focal length of collecting lens 2.

The frequency-angular dependence of scattered radiation for

the different orientations of the optical axis (z) of the crystal with

respect to the laser beam wavevector k L is shown in Fig.2. This

dependence was measured with the help of a spectral device placed

in the focal plane of the collecting lens.

The region of one-photon state localization was determined

from measurements of the fourth-order space correlation function

G(2)(E(-)(rl)E(-)(r2)E(+)(rl)E(+)(r2)). The three-dimentional shape of

that function was measured by scanning in space using micro holes

(see Fig.3). The micro holes had a diameter much smaller than the

space coherence area of radiation and were connected with

photodetectors by fibers. The point of maximum probability of onephoton state localization along the z-direction was calculated by

using a Gaussian approximation to the shape of the space correlation

function and projecting the half-width dependence onto the x-z

coordinate plane (see Fig.4).

It was found in our first experiment that the location of the

point giving the maximum probability of one-photon state

localization dependeds on the location of the reference photodetector

in space. This result demonstrates the EPR paradox conditions for

coordinate and momentum variables. We note here that the indirect

measurement technique used here gave only a qualitative result.

The accuracy of the first experiments was about 30-40%. We had to

use a method of interpolation of the space correlation function shape

because the time resolution of our electronics correlation circuit (x

Ins) could not allow us to make a direct measurement of the precise

space point of photon localisation.

The result of the experiment could be easily interpreted with

the help of the theory of hypothetical advanced Green functions

/1,2/ and classical lens equations if the nonlinear crystal is

considered as a mirror. However, it does not mean that real

advanced electromagnetic waves exist.

We look forward to improvements of time parameters of our

experimental apparatus to provide a quantitive result in the

measurement of coordinate and momentum variables of optical fields

generated in the SPDC process. Such work is in progress.

97
/

;/ • .-.loo ool k l

c....-.F ..=_._

Fig.3. Outline of measurement of fourth-order space correlation

function G(2)(rl,r2).

Fig.4. Result of the G(2)(rl,r2) space distribution measurement. The

point of maximum probability of photon localization was calculated

by interpolation of projection of half-width of correlation function

value in a Gaussian shape approach on the x-z- plane.

98
1.Klyshko, D.N., 1988, "Combined EPR and Two-Slit Experiments:

Interference of Advanced waves", Phys.Lett.A v.132, p.299 (1988).

2.Klyshko, D.N., 1988, "Effect of Focusing on Photon Correlation in

Parametric Light Scattering", Sov.Phys.JETP, v.94, p.82 (1988).

3.Malygin, A.A., A.N.Penin and A.V.Sergienko, "Efficient Generator of

Two-Photon Field of Visible Radiation," 1981, Sov.J.Quantum

Electron., v.11(7), p.939.

4.Malygin, A.A., A.N.Penin and A.V.Sergienko, "Absolute Calibration

of Sensitivity of Photodetectors Using a Biphotonic Field," 1981,

Sov.Phys.JETP Lett., v.33, p.477.

5.Malygin, A.A., A.N.Penin and A.V.Sergienko, "Spatiotemporal

Grouping of Photons in Spontaneous Parametric Scattering of Light,"

1985, Sov.Phys.Dokl., v.30(3), p.227.

99
N92-22057

GENERAIJZED PARAMETRIC DOWN CONVERSION, MANY PAR'IICI.E

INVI'ERFEROMF/IRY, AND BELI.'S "I'IIEOREM

Hyung Sup Choi

The City College of The City University of New York

New York, NY 10031

ABSTRACT

A new field of multi-particle interferometry is

introduced using a nonlinear optical spontaneous

parametric down conversion (SPDC) of a photon

into more than two photons. The study of SPDC

using a realistic Haliltonian in a multi-eK×le

shows that at least low conversion rate limit is

possible. The down converted field exhibits many

stronger nonclassical phenomena than the usual

two photon parametric down conversion.

Application of the multi-particle interferometry

to a recently proposed many particle Bell's

theorem on Einstein-Podolsky-Rosen problem is

given.

INTRODUCTION

A two photon spontaneous parametric down

conversion (SPDC) 1 has been known to be an

effective source of highly correlated photon

pairs that exhibit many interesting nonclassical

properties, such as squeezed states,

antibunching, violation of classical

inequalities, etc. Our study, which starts with a

realistic Hamiltonian not only shows that the

divergence problem 2'3, which occured in the usual

parametric approximation, does not occur when the

pump is quantized, but also shows that the phase

matching problem, in principle, doesn't prohibit

the phenomena to occur.

It is possible 4 to have the phenomenon at least

in the low conversion rate limit. Since we know

that quantum interferometers do not require a

high covereion rate (indeed we like to have only

one set of photons in the entire setup at any

time), we can introduce a multi-particle quantum

optical interferometry in which one measures the

quantum correlation properties among more than

two particles. One can construct three-photon

coherent state interferometers in the form of a

generalized momentum-position interferometry, a

generalized form of a Franson-type position-time

interferometry, and a generalized polarization

correlation experiments, and look for their

nonclassical behaviors.

I. CENHRALIZED PARAHETRIC DOWN CONVERSION

Starting with an interaction Hamiltonian for

three photon SPDC in the parametric approximation

which allows multiple mode down conversion from

the pump with wavevector k o and frequency rio:

HI = fdv XZZ K {&l&2&3 e-iA k'r+i_)°t

+ al+&2÷&3+e iA k'r-i_ot}, (1)

we obtain the expressions for the time

÷

development of the operators &, & :

aka = -l¢°k &ka -i Y-_K &l÷az+e It°o÷

• _(k0-k-kl-k 2 ) (2a)

'÷

a ka = i_k A ka +i ]_ _-'K &la2e i¢_ot

• _(k0-k-kl-k2) (2b)

A major difference between Eqs. ( 2 ) and the

equivalent two photon case is that in this case

the a function at the end of Itqs.(2) cannot

eliminate the summations (or integrals, for a

continuua) over k 1, k2 unless we have a special

selection mechanism such as ideal phase matching,

or photon resonances, for the spocif ic down

converted frequencies.

But in any case, the equations can be

solved and yield the same type of curves for the

photon number, although in non- idea 1 phase

matched cases, we have much miler values. For

example, for the 3 photon deaenerate case we

hRve

0e

N = 18K 2" (3N2+3N÷2) , etc. (3)

Except for the two photon case, which has a well

known analytic solution N = sinhZKt that diverges

101

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at infinity, it can be shown that in all higher

order cases the photon number diverges at a

finite time. On the other hand, if we quanttze

the pump field, the interaction Hsumiltonian HIQ

becomes

NIQ = fdv X X X X _q{glg2a3go+e -iA

k.r

+ ll+g2+g3+goeiAk'r}, (4)

where KQ is a quantum pump equivalent to K in

parametric approximation.

Frca this we have a time development of

the down converted photon number for the three

photon degenerate SPDC:

2 3

N = 18_Q{(3N2+3N+2)No(t)-N"- +5N+2)}, (5)

where N0(t) gives the expression for the depleted

pump beam and is related to the down converted

beam as <N0(t))=<N0(O)>-<N(t)>. The extra term

with a negative sign in gq.(5) will slow down the

change of the slope of the curve when the pump

depletion becoles significant. Notice that the

expression in the quantum pump reduces to the

parametric approximation for N0(t)>>N. The photon

number will eventually oscillate greatly for

large Kt. This is true even in the case when we

don't have an ideal phase matching.

II. HULTIPARTICLI INTERFERONETRY

Two fundamental relations for a multi-photon

spontaneous parametric down conversion, i.e.

k 0 = kl+k2+ "'" +kn, (6)

_0 = _I+_2 + "'' +Wn' (7)

along with the facts that the pump beam with k o

and _o is a coherent one and that the n-photon

dovm converted state is represented by the

product of the individual photon states, tell us

that each individual down converted photon

doesn't have a definite phase, while the total

system carries the phase information. This nphoton correlation property opens a new field of

multi-photon interferometry in which one measures

the joint detection probability of n photons.

Our scheme for multi-photon interferometry

starts with foming a quantum mechanically

entangled state.

[_> = 2 -1/2 { I+l+2'''+n> +

ei(_)l+_2+''+_n) I-l-2"''-n > }, (8)

where |+i > and _i ) refer to the two different

possible states of photon i and _i represents

the phase difference between those two states, It

is a matter of indifference whether J+l ) and J-i >

states are switched for any particle(s) i.

Now if our maasurement N on the system

involves off-diagonal matrix elelents, i.e., a

mixing of the two possible states, then the

quantum mechanical expectation value <_J N J_>

for the measurement will generally contain terms

that oscillate sinusoidally with (_1*(_2+'''+_)n .

These off diagonal elements or the mixing of the

states may be achieved by making use of beam

splitters or a polarization analyzer whose axis

lies in between the two orthogonal polarization

axes. We stay with three particle systems

because we would have an extremely small chance

of getting a right set of correlated photons in

higher order.

(1) Generalized Horne- Shimony- Zet 1 infer 5

interferometer: This two-photon momentumposition interferometer was implemented by Rarity

and Tsbster 6. Recently, a three-particle version

of the experiment was proposed by Greenberger

et.al 7 to test against a family of local realism.

Their gedanken three particle setup can be

realized through the three photon SPDC which we

described in the previous section. One would have

an expectation value for the three photon joint

detection that oscillates sinusoidal ly with

01.02.0 s.

(2) Generalized Franson lnterferoneter:

-- MS1

AX1

D1

: . ,

1 ML1

,_AX3

/ Ms2

D3 M /'/Ax2

Fig, 1 Three arm Franson tnterferoneter

102
Franaon 8 devised a two particle gedanken

interferonetor that uses the interference between

two possible states each of which belongs to a

different emission time.

The experiment was implemented by Ou et al. 9 and

by Kwiat et al. 10 A generalized three-photon

Franson interferometryma7 use the Ou et al.-type

of setup with three Nichelson interferoneters as

in Fig.1. The same analysis should go through as

in the two photon case and the expectation value

for the coincidence counting rate of three

photons will exhibit a sinusoidal oscillation

that depends upon the accumulated optical path

differences between two possible paths,

(3) Polarization Interfercleter: Finally, we

construct a third ty_e of entangled state formed

by two orthogonal polarizations of photons for a

three photon polarization interferometer. Suppose

all three down converted photons are x-polarized.

(one can in principle enforce this by placing xfilters after the apertures) In one set of paths

(primed ones) we place half wave plates and in

the other set of paths (unprimed ones) we place

compensators and the variable phase shifters _i'

Then we combine the beams on the beam splitters

so that the polarization states may be mixed

before they are registered by two channel linear

polarization analyzers. If we count a detection

of an x-polarized photon as +1 and a y-polarized

photon as -1, using two channel analyzers, then

we would have a three-photon joint detection

probability: 11

E(_1,_2,_3 ) = _3 cos(_1+_2÷_3). (9)

llI. BELL'S THEOREM AND MORE

In general, the many particle correlated system

we discussed here is not a mere generalization of

two particle correlated system. It exhibits much

stronger nonclassical effects than the usual two

particle correlated system through its additional

degree of freedum. Some found a stronger

squeezing 3 and a sore prominent _ntibunching 12 .

We found a stronger violation of classical

generalized Cauchy-Schwartz inequality by a

factor of (n-l)/n in a simple higher order system

which can be easily generalized to other systems.

We also found that in a Franson-typa time-energy

interfermmeter classical stochastic

electrodynamics fails rapidly to reproduce

quantum mechanical result in visibility by a

factor of 1/2 for each additional order.

Finally, we saw the dramatic breakdown

of local realism in many particle system due to

Creenberger et.al (GHZ) 7. It has shown that any

local theories that is based on EPR type realism

faces contradiction as it tries to immitate

quantum sochanical results in a many particle

correlated system. This theorem can be

implemented by multiphoton interferumetries which

we discrtbed in Section II. Nermtn 13 also has

shown that the violation of Bell type inequality

in a many particle system increases

exponentially as it goes to a higher order. This

is just an another example of a strong violation

of classical limits by a many particle system

through its additional quantum degree of freedom.

REFERENCES

I. C.Hong, L.Mandel, Phys.Rev.A 31, 2409 (1985)

2. H.Hillery, Phys.Rev.A 42, 498 (1990)

R.Ftsher et.al Phys.Rev.D 29, 1107 (1984);

3. S.Braunsteln, R.NcLachlan, Phys.Rev.A 35,

1659 (1987)

4. H.S.Chot, Submitted for publication

5. N.Horne et.al, Phys.Rev. Lett. 62, 2209 (1989)

6. J.Rarity, P.Tapster, Phys.Rev.Lett.64, 2495

(1990)

7. D.H.Creenberger et.al Am.J.Phys. 58,

1131 (1990)

8. J.D.Franson, Phys.Rev. Lett.62, 2205 (1989)

9. Z.Ou et.al, Phys.Rev. Lett.65, 321 (1990)

10. P.Kwtat et.al, Phys.Rev.A 41, 2910

(1990)

11. H.S.Choi, Phys. Lett.A 153, 285 (1991)

12. C.T.Lee, Phys.Rev.A 41, 1721 (1990)

13. M.P.Nermin, Phys.Rev. Lett. 65, 1838

(1990)

103
II. SQUEEZED STATES OF LIGHT

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105
 221o 

N92-22058

GOING THROUGH A QUANTUM PHASE

Jeffrey H. Shapiro

Department of Electrical Engineering and Computer Science

Massachusetts Institute of Technology

Cambridge, Massachusetts 02139-4307

Abstract

Phase measurements on a single-mode

radiation field are examined from a

system-theoretic viewpoint. Quantum

estimation theory is used to establish

the primacy of the Susskind-Glogower

keSG) phase operator; its phase eigents generate the probability operator

measure (POM) for maximum-likelihood

phase estimation. A commuting observables description for the SG-POM on a

signalxapparatus state space is derived.

It is analogous to the signal-bandximageband formulation for optical heterodyne

detection. Because heterodyning realizes

the annihilation operator POM, this analogy may help realize the SG-POM. The

wave function representation associated

with the SG-POM is then used to prove

the duality between the phase measurement and the number operator measurement, from which a mnnber-phase uncertainty principle is obtained, via Fourier

theory, without recourse to linearization.

Fourier theory is also employed to establish the principle of number-ket causality, leading to a Paley-Wiener condition that must be satisfied by the phasemeasurement probability density function

(PDF) for a single-mode field in an arb/trary quantum state. Finally, a twomode phase measurement is shown to afford phase-conjugate quantum communication at zero error probability with finite

average photon number. Application of

this construct to interferometric precision

measurements is briefly discussed.

1. INTRODUCTION

A classical single-mode radiation field

is characterized by a spatial-mode pattern

_'(_'), an oscillation frequency w in rad/s,

and a c-number phasor a. The latter specifies both the energy and the initial phase

shift of the field--we can take H = Nhw

to be the mode energy, and ¢ to be the

mode phase, where

a---- V_e _, (1)

is the polar decomposition of a. When

the single-mode classical field is quantized,

its mode pattern and frequency are unchanged, but a is replaced by the annihilation operator &. The phase problem,

for this single-mode quantum field, has

long been taken to mean finding a satisfactory quantum version of Eq. 1.1 However, owing to the noncommutative nature

of the quantum theory's operator algebra,

no such decomposition exists, i.e., there is

no observable ¢ such that:

a = y/ e (2)

One may quibble about the order of the

amplitude and phase terms on the rightside of Eq. 2, or prefer the use of N +

] = h&? in lieu of _/ = a?a, etc., but the

essential issue is the nonexistence of the

observable ¢.

Until recently the Susskind-Glogower

(SG) phase operator, 2

e%-- (aat)-l/ a, (3)

has seemed to provide the best quantum

description of phase. The SG operator is

non-Hermitian, and its quadratures,

c_"¢) = Re(ei'¢), and (4)

sin(C) -- Im(ei_b), (5)

are noncommuting observables which fail

certain reasonable conditions that the cosine and sine of a phase should meet. For

example, it turns out that

(¢lcos(¢) I¢) + (¢lsin(¢) l¢) < 1, (6)

107

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unless the state [01 is orthogonal to the

vacuum, [0). On the other hand, the SGbased commutator

,si = icos(_b), (7)

does lead to the oft-employed number

phase uncertainty principle, ANA¢ >

1/2, under a high-mean-field linearization.

Lately, there has been intense renewed

interest in the quantum phase problem. In

what follows we will review some of the

recent quantum phase work of Shapiro,

Shepard, and Wong, 3' 4 and present some

new results. Because the effort of Shapiro

et al. originates from a quantum estimation theory viewpoint, that tack will

taken here as well. Because the formalism of Shapiro et al. relies on the probability operator measure (POM) description of quantum measurement--a generalization of observables not well known in

the physics literature---we will begin with

a brief tutorial on POM's.

2. POM REVIEW

The textbook approach to quantum

measurement is through observables. 5 For

example, consider the quadrature components of the single-mode field's annihilation operator, i.e.,

al - Re(a), and (8)

a2 - Im(&). (9)

These are continuous-spectrum observables. In other words, they are Hermitian

operators

&J=aj, forj=l,2, (10)

with complete orthonormal (CON) eigenkets,

ajlaj)j = ajlas)j,

for -oo < a s < oo,(11)

S(a_laS)S = 6(a_ -aS), (12)

] = /2daslas)ss(as[,(13)

where ] is the identity operator, and 6(.)

is the Dirac delta function.

Measurement of a quadrature operator,

when the system is in state I¢), gives a

continuous-valued, classical random variable with PDF

p(as I1¢)) -Is(asl¢)l 2,

for -oo < a s < oo, j = 1, 2. (14)

For this classical probability density to be

correct, for all possible 1¢), it must satisfy

p(as I I¢)) >-0,

for -oo < a s < oo, j = 1,2, (15)

and

f?das P( aS [ 1¢) ) = 1,

O0

for j = 1, 2. (16)

These conditions are ensured by Eq. 13,

which leads to the familiar quadrature

representations--essentially the position

and momentum wave functions--given by

I¢) = i1¢) = /?da, la,)l,(axl¢)

O0

= £ dal ¢(al)lal)l, (17)

OO

and

1¢) = ]1¢) = /?da21a2)22(_21¢)

: /2da2 II/(a2)la2)2, (18)

OO

with the obvious identifications for ¢(al)

and _(a2). Of course, the quadratures are

noncomrnuting observables,

i- (19) [al, a2]= _I,

so they cannot be measured simultaneously. 5

The preceding review demonstrates that

the full specification of observables, i.e.,

Hermitian operators with CON eigenkets,

is not needed to produce a consistent

statistical characterization of a quantum

measurement. For an arbitrary quantum state, a resolution of the identity--an

outer-product sum like Eq. 13--generates

a proper classical-probability description

of a quantum measurement. This is the

108
essenceof the POM concept. Our principal purpose for introducing POM's is to

accommodate measurements that are not

observables on the state space, 7_, of the

_-mode. The best way to introduce such

nonobservable POM's is through an example. It is well known that the annihilation operator, a, is not an observable--it

is non-Hermitian,

at # a. (20)

Furthermore, its real and imaginary

parts, al and fi2, are noncommuting

observables--a cannot be measured in the

usual textbook sense. However, the annihilation operator does have eigenkets--the

coherent states, 6

ilia) -- ala),

for a E C, (21)

where C is the complex plane. These states

are not orthonormal, i.e.,

=

exp(-liaq2 - 2]a12 + a'*a) ,(22)

which is a consequence of the nonvanishing

commutator

implied by Eq. 19. Nevertheless, the

coherent states are complete, in fact

overcomplete--they form a resolution of

the identity

1

i = J .fcd2°'Io,)(,-,,I,

hold, for all 1¢).

The preceding POM has long been

known.7, s, 9 It represents a measurement, in the POM sense, of the annihilation operator a: the _-eigenkets generate the measurement statistics, and the _-

eigenvalues are the resulting observation

values. This parallels the usual observables description: the observable's eigenkets generate the measurement statistics,

and the associated eigenvalues are the observation values, cf. Eq. 14.

The POM formulation is not in conflict

with the conventional dictum that only observables can be measured. Any nonobservable POM on 7"i can be represented as

a collection of commuting observables on

some larger state space which describes

the original system interacting with an

appropriate apparatus. 7 The most familiar example of this genre is optical heterodyne detection of a single-mode signal

field, which provides both a commutingobservables description and a physical realization for the _-POM.

In optical heterodyne detection, s' 9, z0

a signal field of frequency v is mixed with

a strong local-0scillator (LO) field of frequency v - vlF on the surface of a pho-

(23) todetector. With a unity-quantum efficiency detector, and two-channel lockin amplification at the intermediate frequency (viE), this arrangement produces

a complex-valued, classical random variable, y, whose measurement statistics are

iatentical to those of the operator 9

which defines the _-POM. The outcome of

the _-POM is a complex-valued, continuous classical random variable with PDF

1

p(a I1¢>) =

for a E C,

when the field state is 1¢). Because of

Eq. 24, it follows that

and

p(a II¢)) ->o, for a EC, (26)

d2ap(_ II,P))= 1,

EC

Here aj and ]j, forj = S, I, are the annihilation and identity operators for the signal

mode (frequency v), and the image mode

(frequency v- 21liE), respectively. Both

of these modes beat with the LO to pro-

(25) duce IF waveforms, just as is the case in

classical superheterodyne radio reception.

It is easily verified that the real and imaginary parts of Z)are commuting observables

on the joint state space, 7_s @ 7-/i, and

so are simultaneously measurable in the

usual sense. Ordinarily, only the signal

mode carries information, i.e., the image

mode is unexcited. Under these circum-

(27) stances, the PDF for the observed y-value

109
reduces to 8, 9

p(yll¢)s) = ls(yl¢)sl =,

for y E C, (29)

where [Y)s is the signal-mode coherent

state with eigenvalue y, and [¢)s is an

arbitrary signal-mode state. Comparison

of Eqs. 29 and 25 completes the demonstration that heterodyning--a pair of commuting observables on an extended state

space with an unexcited image mode--

realizes the 5s-POM.

3. PHASE ESTIMATION

Rather than seeking a quantum formalism for phase by pursuing a quantum

version of Eq. 1, Shapiro, Shepard, and

Wong3, 4 approached the problem from

the estimation theory viewpoint. Consider the following abstract quantum estimation problem. A single-mode input field

of annihilation operator alN and quantum

state I¢)IN undergoes an unknown, nonrandom, c-number phase shift (I), yielding

a single-mode output field of annihilation

operator

= ei_alN , (30)

in state

10) = exp(i*atmam)I¢)m. (31)

By making an appropriate quantum measurement on the 5 mode, and knowing the

input state 10)l/v, we are to estimate the

phase shift (I). An interferometric phase

measurement can be embedded into this

scheme by placing appropriate constraints

on the allowable quantum measurement.

Optimizing a phase measurement within

this more restricted environment cannot

outperform the behavior obtained from

an unfettered measurement optimization.

Indeed, we should expect that joint optimization of the quantum measurement

and the input state will yield superior

phase estimation performance.

Without loss of generality, we can confine the phase shift to a 27r-rad interval,

i.e., we can assume that -lr < _ < lr. The

class of POM's we must optimize over, in

order to find the best phase measurement,

can be taken to be {dl](¢) : -_r < ¢ <

_r }, where

dII(¢) = dl](¢) t, (32)

and

i = Fdl](¢), (33)

on the state space of the output mode, h.

The conditional probability density, given

_, for obtaining a phase value ¢ from this

POM is

- (¢Idfi(¢)1¢)

de '

for -lr < ¢, (I) < It, (34)

where [0) is the state of the 5-mode.

In classical estimation theory, the

maximum-likelihood (ML) estimate _ML

of an unknown, nourandom, phase shift (I),

based on a noisy phase-shift observation ¢,

of known PDF p( ¢ [ • ), is the phase shift

which maximizes the likelihood of getting

the observed datum, i.e.,

CML(¢) = arg max p(¢10). (35)

-Tr<O_<_r

Often, the ML phase estimate equals the

observed phase shift, because p( ¢ I 0 ) has

its peak at 0 = ¢, for -lr < ¢ < 7r. Such

is the case for phase estimation in additive

white Gaussian noise.11 It then follows the

p( ¢ [ ¢), the peak likelihood, is a simple,

but meaningful, performance measure for

_ML. Indeed, its reciprocal,

1

6¢ = p(¢ I ¢)' (36)

is the PDF's width for the case of a uniform distribution; if the distribution is

Gaussian, then we have _f¢ = vf2_A¢,

where A¢ is the root-mean-square (RMS)

error.

Our problem is one of quantum estimation theory, namely, choosing the POM,

dl_(¢), and the input state, I¢)lN, to optimize our estimate of the phase shift (I). For

a given POM and input state, Eq. 34 supplies the PDF needed to perform classical

ML estimation. In this quantum setting,

however, the observed phase value ¢ is, by

presumption, our estimate of (I). Thus, in

110
orderfor this estimateto beoneof maximum likelihood, we can restrict our attention to POM's satisfying

for -_r < ¢ _< _', (37)

and optimize our estimate over dl_

and ]¢)IN by maximizing the peak

likelihood--minimizing 6C---averaged over

all possible & values. Here it is known

that, for the input state whose number

representation, Cn -- (nit), is

Cn -- ]Xbn]eix_, for n = 0, 1,2,..., (38)

6¢ is minimized by the following POM, z

=

for -lr < ¢ _< lr, (39)

where

le%¢) = (40)

n----0

Moreover, the reciprocal peak-likelihood

that results when we use this optimum

POM to estimate _ is easily shown to be

= 21r (£), {¢n , (41)

which is independent of the phases (X,_}.

In fact, p( ¢ { _ ) is independent of the

{x-}.

We can exploit the (Xn} independence

to good purposes by assuming, without

loss of generality, that the input state has

positive real Cn. Equation 40 then reduces

to

Oo

le%¢) = le --

n_0

for -lr < ¢ _< lr, (42)

which is the number-ket expansion of

the SG phase operator's (infinite-energy)

eigenkets, viz.

for -_r < _b_< ,r. (43)

This says that the SG-POM is the quantum measurement for ML phase estimation in the general measurement configuration when the input state has a positive real number representation. In other

words, the phase eigenkets of the SG operator generate the resolution of the identity,

l = _--_/;_rd¢ [ei¢)(ei¢[, (44)

needed for ML quantum phase estimation

in this case. For an arbitrary input state,

the optimum POM from Eq. 39 is equivalent to performing the unitary state transformation

oo

n----O

followed by the SG-POM.

To achieve the goal of jointly optimizing

phase-estimation performance over both

the measurement and the input state, it

only remains for us to minimize 6¢, from

Eq. 41, by appropriate choice of [¢)ZN.

This problem has been addressed, 3, 4 and

the state

A

= 1

for n -- 0,1,2,...,M < c_, (46)

where A is a normalization constant and

M is a truncation parameter, has been

shown to achieve

i_¢,,_ 1/N 2, (47)

in terms of its average photon number,

N = (ht&). Th_is performance is far supe- rior to the 6¢ 1/N reciprocal peak likelihood capability of optimized squeezedstate interferometry. However, the phase

measurement PDF for the Eq. 46 state

is a heavy-tailed distribution, viz., its

RMS phase error, A¢, is essentially independent of N. Thus, the degree to

which this reciprocal peak likelihood advantage can be usefully exploited has

yet to be established. 12 In what follows,

therefore, we will concentrate on the SGPOM, in that it constitutes the maximumlikelihood quantum phase measurement

for ogl quantum states.

111
4. SG POM resolvethe identity is alsoeasilyshown,

TheSusskind-Glogower operator,2, 13

ei_'__ (&&t)-l/25, (48)

affords a well-defined polar decomposition

of h,

h = _ ] e_, (49)

in terms of energy (number) and phase operators. Using the number-ket expansions

of N and h we have that

j, d¢ le_¢)(e':'¢'t

O0 O0 /,,,ff .

: Z Z J_

n--'-0m---0 -n

OO

= 2__ In)(nl : 2_Z. (57)

rL=0

That they are not orthonormal can be

demonstrated from some simple Fourier

transform manipulations, 14

OO

A

e'¢= _ {n><n+ II,

n_O

from which it follows that

and

OO

(50) (e'¢'le'¢) = _ e-'"(¢'-¢)

n----0

oo _ + sgn(n)e -in(¢'-¢) + 1

In + l><nl ¢ e_*, (51) .=-_

.=0 = _(¢, _ ¢)

_ _i cot (¢'- _ ¢) +_1 (58)

[e_%,e-'_¢]= IO>(Ol. (52) 2

In words, the SO phase operator is not

Hermitian, and does not commute with its

adjoint. Thus, as was seen earlier for h itself, the quadrature components of the SG

operator,

A A

cos(C) = Re(e_¢), and (53)

sin(C) = Im(e'¢), (54)

are noncommuting observables,

(55)

i

_-

The SG-POM derives from the fact that

ei¢ has an overcomplete set of eigenkets,

cf. the Sect. 2 discussion of the h-POM.

By direct substitution of Eq. 50, we can

verify that

for-Tr<¢<Tr, (56)

Here, 6(.) is the Dirac delta function, and

-1, n < 0, sgn(n) = 0, n = 0, (59)

1, n>0,

is the signum function.

5. COMMUTING OBSERVABLES

Recall that a POM on 7-I which is not

an observable on that space can be represented as a collection of commuting observables on a larger, signal×apparatus

state space, 7_ ® 7-/A, with the apparatus

placed in some appropriate state. We now

develop such a representation for the SGPOM. Aside from alleviating the qualms

of those who believe only in observables,

this representation may guide us to a realization of the SG-POM--the commutingobservables description of the h-POM is

intimately connected with its heterodynedetection realization.

where [e i¢) has the number-ket represen- Let aA be the annihilation operator of

tation given in Eq. 42. That these kets an apparatus mode, whose state space,

112
7"/A,isspanned by its number kets, { [n)A :

n = 0, 1, 2,... }. The non-Hermitian operator

+ ({o><o{,e_A), (60)

where

e=-C*A--a_(aAa_)-'/2, (61)

is easily shown to commute with its adjoint. Here, 1, is an operator on the joint

state space 7_ ® 7_A, and ei_ A is the apparatus mode's SG phase operator.

Because [1,, 1,t] _- 0, the quadrature

components of 1,--denoted 1,1 and 1"2-

are commuting observables, which can be

measured simultaneously, i.e., 1, = 1,1+1,2

can be measured in the usual sense. Solvz

ing for the eigenkets and eigenvalues of Y

we find that signalx apparatus number ket

IY)-In)Ira)A, fornrn > 0, (62)

is a 1,-eigenket with zero eigenvalue, and

1

}Y) = _ (}0>]0)A

) n----I

for -lr < ¢ < lr, (63)

is a 1,-eigenket whose associated eigenvalue is e i¢. Collectively, these comprise a CON set from which we have

that measurement of 1:I, when the

signalxapparatus state is I¢)SxA E T( ®

7_A, yields a mixed classical random variable, Y, which takes on either the discrete

value 0, or a value from the continuum

{ e_ : -lr < ¢ < lr }. The former occurs

with discrete probability

OO OO

Pr(OII¢)s×a) = _ _ 1¢..,12; (64)

n--I m----1

the probability density for the latter is

1

p( ¢ II¢)sxA) = _ I¢oo+

+

n-----1

for -lr < ¢ _< % (65)

where Cnrn -- A(ml(nlCsxA). These two

are properly normalized in that for all

I_b)SxA we have that

Pr(o II¢)s×A)

+ /2dCp(¢IIC)s×A) = 1, (66)

as required by classical probability theory.

Now, the commuting-observables representation of the SG-POM is at hand. Suppose we measure 1, when the apparatus

mode isunexcited,i.e.,l_)SxA= I¢)I0)A,

where [¢) E 7"/is an arbitrary signal-mode

state and I0)A is the apparatus mode's

vacuum state. Then the discrete value

zero is never obtained, and the PDF for

obtaining Y -- e i¢ reduces to

p(¢ I I¢)IO>A)

m_

1

2_1<e_¢1¢)12,

for -lr < ¢ <_% (67)

realizing the SG-POM statistics for an arbitrary state of the 5-mode.

The equivalence of the SG-POM to the

1, measurement with an unexcited apparatus mode allows us to clarify some basic points. First, because the SG operator does not commute with its adjoint,

it is really the operator analog of the cnumber e i¢ from the classical single-mode

field. In other words, there is no Hermitian phase operator, ¢, on 7_ such that

exp(i¢) = (aat)-l/2a. Restated in terms

of the quadratures of the SG operator.,_, this

means that cos(C) ¢ cos(C), and sin(C)

sin(C). As a result, the classical trigonometric identity,

cos(C) 2 + sin(C) 2 : 1,

for -Tr < ¢ < 7r, (68)

does not apply to the quadratures of the

SG phase operator, e.g., because

co_¢)2+si_¢)2= i IO)<Ol 2 ' (69)

113
any {0) with ¢0 ¢ 0 gives

(¢1c_-_C)21¢) + (¢[siL'_C)21¢)< 1. (70)

However, the outcome of the SG-POM /s

a phasor ei¢--the 1? measurement with

a vacuum-state apparatus mode yields a

complex-valued, continuous classical random variable Y = e i¢, where -lr < C < lr.

Thus, we have that {y]2 = 1, w/th probability one.

The second point to note regarding the

SG-POM is the relation of its mean value

to those of the SG quadratures. Using

_'1 _-" (co'C)_ [0)AA(0{)

+ (10)(0l®co_-¢)A), (71)

and

(72)

and assuming an unexcited apparatus

mode, we find that

A(0I('/'I?_I¢)I0)A = ('/'ICO-'_'C)I'P), (73)

and

A(01(¢IY21¢)I0)A --(¢Isin('"¢)l¢), (74)

for 021 h-mode states, 1¢). What this says

is that averages of the classical cos(C) and

sin(C) random variables obtained from the

SG-POM coincide with averages of the

SG phase operator's quadratures. To the

extent that the quadrature mean values

comprise the information of interest, we

can conclude that the SG-POM provides a

proper quantum measurement description

for simultaneous extraction of this information from both quadratures.

6. UNCERTAINTY PRINCIPLE

Number-ket expansions of Athe quadrature operators cos(C) and sin(C) lead to

the commutator

,si = icon(C), (75)

and the associated uncertainty principle

1 l(c_'i-C)> 2 (AN2)(Asin'('C)2) -->3, (76)

Equation 76 is valid for arbitrary states,

but its utility, in this general form, is

somewhat limited. First, the minimum

uncertainty product is state dependent--

a consequence of Eq. 75 not being a cnumber commutator. Second, the principle does not directly address the variance

of a phase measurement--it is the sin(C)

operator whose variance appears.

It is common practice to use the linearized form of Eq. 76,

1

ANAC -_ 2' (77)

which applies for states meeting the highmean-field condition,

(_> -_ I<a>l2 >> 1. (78)

The linearized result, while useful, can be

abused. Number kets have zero numbermeasurement uncertainty, and (_) = 0,

A

(cos(C)) = 0, hence the general result

leads to the correct number-ket limit,

(AN2)(Asin_-C)2> > 0, (79)

whereas the linearized form is inapplicable.

Although the SG-POM does not alleviate the state-dependent nature of numberphase uncertainty limits, it does lead to

an uncertainty principle which directly

addresses phase variance. Our route to

this principle--through Fourier theory--

has the following motivation. The

time-bandwidth uncertainty principle for

the continuous-time Fourier transform

(CTFT) 14 can be applied to the normalized position and momentum wave functions, ¢(al), and _(a2), because they satisfy the Fourier transform relations 5

_(a:) =

oo d_l i2a a

oo_ ¢(_)e- , _, (80)

and

¢(Ol) =

°Oda2

_ _I't(_2)e i2°la2 . (81)

114
The result of this procedure can be reduced to

1

(Aa_)(Ag) > i_' (82)

which is the Heisenberg uncertainty

principle for the annihilation operator's

quadratures.

Because of Eq. 44, any state ]O) has a

phase representation

¢(e_*) - (e_*10),

for -_" < _b< lr, (83)

such that

I¢) = i1¢)

1 t'_

= _ j-ldC_(e_4')le_4' ). (84)

The phase representation of [¢) is intimately related to its number-ket representation, Cn = (n[¢}--they are a Fourier

transform pair

oo

n=O

and

(85)

f

1 dCaY(e_4,)ein4,,(86)

as can be seen from Eqs. 83 and 42. In

other words, _(e i¢) and Cn constitute

phase and number wave f_netions, which

are capable of representing arbitrary

states. The complementarity of the number operator measurement--whose probability distribution is Pr(N = n) = ICnl 2-

and the SG POM--whose probability density function is p(¢) = [_(ei¢)[2/2_r--then

follows from the Fourier relations, Eq. 85

and 86. Thus, to obtain a number-phase

uncertainty principle for the product of

the number-operator variance and the SGPOM variance, we shall exploit this complementarity by paralleling the standard

Fourier derivation of Eq. 82.

With (A/V 2) denoting the numbermeasurement variance and {A¢ 2) the SGPOM variance, when the field is in an arbitrary state [g)), we have that

(A._,2)(A¢2)

= _ ("- _,)=1,/,.I 2

1,1-----0

× (¢- &l_(e'*)l 2 (87)

× ._ (¢ - _)21_,'(_*)1_ (88)

_> ._ (¢ - _)_,'(e_*)°

× d_'*) [2 (89)

_> {ReretJ-_ do2,_(¢ - ¢)_'(_)*

× (90)

= _lp(,_ I1¢)) - 1]2. (91)

In this development: fi and ¢ are the

mean values of the number and SG-POM

measurements, respectively, on the state

}_b); _'(e i¢) _. _(ei_)ei_¢; the Schwarz

inequality has been used in Eq. 89; and

the integration necessary to obtain _p .:_ follows from the SG-POM's PDF,

I¢)) = I_(e_'_)12/2"x.

Unlike the usual number-phase uncertainty principle, i.e., Eq. 76, our result

does not require any linearization before

it can be applied to phase variance. Equation 91 is still state dependent, but this is

unavoidable. When I¢) is a number state,

we have

1

p( ¢ II-)) = _, for --It < q__< r, (92)

a uniform distribution, which is maximally random, but still has finite variance.

There is no contradiction with Eq. 91 in

this case, even though AN = 0 for a number ket; the uniform PDF causes the right

member of Eq. 91 to vanish. On the other

hand, when ¢) is a high-mean-field state,

we will have p(Tr ¢)) << 1, so that

Eq. 91 reproduces the standard linearized

formula, Eq. 77. Indeed, for any state

satisfying p(lr [ I¢)) << 1, we have that

Eq. 77 holds. This makes the SG-POM

115
derivationof Eq. 77 more robust than linearization of Eq. 76.

7. Number-Ket Causality

The SG-POM underlies maximumlikelihood quantum phase measurement

for all quantum states. Given the problems associated with minimizing the SGPOM's reciprocal peak-likelihood _¢, by

choice of input state, 3, 4, 12 a different

state-selection criterion may be worth considering. In this vein, it is germane to ask

the following question. What SG-POM

phase PDF's can be realized by choice of

input state [¢)? It turns out that linear

system theory has the answer.

The Susskind-Glogower probability operator measurement on a state [¢) results

in a classical random variable ¢ with probability density function

- I (e' )12

for -Tr < ¢ < It. (93)

Here, _I,(e '_) is the phase representation

of the state |¢). According to Eq. 86, the

phase representation is the Fourier transform of the number representation. The

latter is a one-sided, discrete-parameter

sequence that is the inverse Fourier transform of ¢_(e_¢), i.e.,

1 '_

¢. = f d¢

= _ (n]¢), for integer n_> 0, (94)

t 0, for integer n < 0.

In system-theory parlance, the Fourier

pair Cn 4---* _(e i¢) is analogous to

that for a discrete-time waveform on

an unbounded interval and its periodic,

continuous-frequency Fourier transform,

i.e., the discrete-time Fourier transform

(DTFT). 14 More importantly, saying that

_bn is one-sided is equivalent to saying that

a discrete-time waveform is causal, viz. it

could be the impulse response of a causal,

linear time-invariant system. Determining what p(¢ [ I_b)) are possible from

Eq. 93 is then the same as determining

what [#(ei¢)[ are Fourier-transform magnitudes of one-sided {Ca}. To emphasize

the connection with causal waveforms, we

introduce the term number-ket causality

for the condition Eq. 94. This is a wellstudied problem in linear systems, so results are immediately available. 14, 15

From Eq. 93 and the Paley-Wiener

theorem t5 we have that p(¢ [ ]¢)) must

satisfy

/_ dClln[p(¢ [ [¢))][ < o% (95)

for all number-ket causal $(ei¢). From

this condition, it follows that no state can

confine the phase-measurement PDF to a

subinterval of (-r,_r], e.g., the uniform

density,

1

v(¢11¢) ) =

for I¢1-< < (96)

is impossible. The Paley-Wiener condition

is both necessary and sufficient, i.e., if a

PDF obeys Eq. 95, then there /s a state

which gives this density through Eq. 93.

Indeed, there are an infinite number of

such states, because Eq. 93 only constrains

the magnitude of the phase representation.

One such state can be obtained explicitly

via the discrete-parameter Hilbert transform. The procedure is as follows. For a

PDF obeying Eq. 95, set

=

for -Tr < ¢ _< _r. (97)

Next, find the discrete-parameter Hilbert

transform, 14

arg[*(e'*)] pf_r de' ln[l_(e,¢,)l ]

j__-_

× cot(¢'- ¢_

\ 2 /'

for -_r < ¢ < _r, (98)

where :P denotes Cauchy principal value.

Equations 97 and 98 then comprise the

magnitude and phase--the polar form--

of a properly-normalized phase representation _(e i¢) with the prescribed phasemeasurement statistics and a number-ket

causal inverse Fourier transform.

116
The preceding phase representation

construction for a state with prescribed

SG-POM statistics is by no means unique.

Equation 97 constrains [_(ei_)[, but no restriction is placed on arg[_(ei#)]. Consider a number-ket causal function, { hn :

n = 0, 1,2,... }, whose Fourier transform

has unity magnitude, viz.

H(e'*) -

oo

E hne-in4 _,

n=O

for-_" < _b < lr, (99)

obeys

IH(e'¢)l = 1, for -Ir < _b< lr. (100)

Such a function is known in digital-filter

theory as an all-pass filter; were { hn :

n = 0, 1, 2,... } the impulse response of a

discrete-time, linear, time-invariant filter,

the associated frequency response would

pass all frequencies with neither attenuation nor gain. The prototypical example of an all-pass filter is obtained--in the

poles z-transformwithin domain--by the unit circle balancing with f__//_/

zeros outside the unit circle to achieve 14

K e-i¢ _ p_

IIi :?;S--r*'

k=l

for -lr < ¢ <_ tr,

where IPkl < 1,

for k = 1,2,... K.(101)

Now, suppose we assemble the phase representation

for -lr < ¢ _< r, (102)

where _(e i_) is constructed according

to Eqs. 97 and 98 for a desired phasemeasurement PDF, and H(e i4') is an allpass phase representation from Eq. 101.

The convolution-multiplication theorem of

Fourier analysis, plus the fact that convolving two causal functions produces a

causal function, 14 guarantees that _(e _)

is a properly normalized, number-ket

causal phase representation; the all-pass

nature of H(e i¢) implies that _'(e i¢) has

the desired SG-POM statistics. Because

this process holds for all K >_ 1 and for

all pole locations within the unit circle,

there is an uncountable infinity of states

which have the same SG-POM statistics.

Nevertheless, the state constructed via the

discrete-parameter Hilbert transform has

a unique advantage--it is the minimum

average photon-number state with the prescribed phase-measurement PDF.

The proof of the minimum average

photon-number property follows almost

immediately from available linear-system

results. Let {_(e i_) : -lr < ¢ < 7r}

and {¢n : n = 0,1,2,...} be the phase

and number representations of the state

]¢), obtained via Eqs. 97, 98, and 94,

that realizes a particular phase PDF. Similarly, let { qF(e i_) : -It < ¢ < _r } and

{¢_n : n = 0, 1,2,...} be the phase and

number representations of any other state,

[¢_), with the same SG-POM statistics.

Then, we have that 14

M-1

E ->0,

n=0

for M = 1,2, 3,... (103)

Physically, this says that, of all states with

the desired phase behavior, the Hilberttransform generated state concentrates its

number-ket content closest to the vacuum.

Because both [¢) and [¢') are normalized,

i.e., unit-length, states, Eq. 103 is equivalent to

A M

oo

E (l¢.t I¢'1 -<0,

n=M

for M = 0,1,2,... (104)

Proving the minimum average photonnumber property is now straightforward:

-

oo

--

n=0

oo

= E AM<0.

M=I

(105)

Thus, Eqs. 97 and 98 provide the means

for choosing a state of minimum average

energy and prescribed phase-measurement

PDF.

117
8. PHASE COMMUNICATION

Sections 1-7 constitute an abridged version of Shapiro and Shepard. 4 That paper presents additional details regarding the state that achieves 6d_ ,_, 1/N 2,

as well as substantial material on new

classes of quantum states--coherent phase

states, squeezed phase states, rational

phase states--that are closely associated

with the SG-POM. Furthermore, it proves

that the Pe_-Barnett Hermitian phase

operatorl6,11 which existson a truncated state space and provides phasemeasurement statisticson the fullstate

space through a limitingprocedure--is

included within the SG-POM formalism,

i.e.,these two schema produce identical phase measurement statisticsfor all

quantum states. Neither of these topics will be considered herein. Instead,

we shall move away from the singlemode case and develop new resultsfor

two-mode quantum phase measurement.

Our objective will be to exploit the I_-

measurement--developed en route to the

commuting observables form of the SGPOM--when the signal and apparatus

modes are quantum-mechanically correlated and a phase-conjugate modulation is

applied to them.

Consider the phase-conjugate quantum

measurement setup shown in Fig. 1. This

is a phase-conjugate system because whatever c-number phase shift (I) is applied to

the signal mode, leading to the annihilation operator transformation

a_ N ---* e'*a_ N, (106)

the conjugate phase shift, -(I) is applied to

the apparatus mode, viz.

a_ N ---* e-'*a_ N, (107)

cf. Eq. 30. If we take the signal and

apparatus modes to be the appropriate

linear polarizations, a transverse electrooptic modulator can be used to induce

the necessary conjugate phase shifts, ts

Phase-conjugate shifts also appear, prototypically, in gravity-wave detecting interferometers. In fact, there are fundamental advantages to operating a phasesensing interferometer in phase-conjugate

fashion. 19 Our work does not depend explicitly on the means by which this modulation is accomplished. Its principal

motivation is to circumvent the PaleyWiener restriction that encumbers phas_

measurement PDF's for single-mode fields.

As we shall see, some startling new possibilities arise with two modes.

The Paley-Wiener condition applies to a

single-mode phase PDF because this density is proportional to the squared magnitude of the Fourier transform, { _(e i_) :

-lr < ¢ < lr }, of a one-sided sequence,

{0n : n = 0,1,2,...,}. We shall break

out of this limit, in the two-mode case,

through quantum correlation. On Ks ®

_/A, the joint state space of the signal and

apparatus input modes, we can construct

number-product vacuum states of the form

[¢)J'N

O0

= ¢0[0)SI0)A + _(¢_In)Sl0)A

n=l

+ ¢-_lO)sln)A), (108)

where

ICnl = 1. (109)

n_--oo

The term number-product vacuum is appropriate for such I_b)lN because, when

the signalxapparatns state is of this class,

a measurement of the number-operator

product--Ns ® NA--yields outcome zero

with probability one. Thus, for I¢)IN

a number-product vacuum state, Eqs. 62

and 63 imply that measurement of

yields a classical phasor e_, with -_r <

¢ _< _r. Moreover, ¢ in this case has PDF

I )

2

p(¢1¢) =

21r

for --lr < ¢, (I) < lr, (110)

in terms of

OO

-- ¢.e

n oo

for -lr < ¢ < lr. (111)

Note that {¢n : In[ = 0,1,2,...,}

and {_(e i_) : -lr < ¢ _< lr} are

not the number and phase representations, respectively, of any single-mode

field state. They are, however, the number

and phase representations, respectively,

for a two-mode, number-product vacuum

118
state. The {¢n, _(ei¢)} notation is convenientbecause,asshownby Eq. 111and

its inverse,

f_ de ..., i4,, in_

forln]=0,1,2,..., (112)

these functions are a Fourier transform

pair. More importantly, this notation

makes clear the fact that number-ket

causality does not restrict the possible

two-mode phase PDF's. In particular,

there are number-product vacuum states

that satisfy

I*(e'¢)[= 0, for [¢l-> ¢c,

with ¢c < lr. (113)

i.e., two-mode phase PDF's can be confined to subintervals of (-_r, it], a situation

that is forbidden to the single-mode case,

cf. Eq. 96. This possibility is of great significance for phase-based digital communication and phase-based precision measurement, as we shall see.

To cast the Fig. 1 structure into a digital communication mold, let us assume

that transverse electro-optic modulation is

used to transmit a randomly-selected digit

k, satisfying 1 < k < K, by using _ = Ck,

where

(2k- 1)7r (114) @_ -- -_r + K

Our objective is to make a minimum error probability decision as to which _k

was sent, based on the result of the Ymeasurement when the signal xapparatus

state is a number-product vacuum, characterized by {¢n, _(C_)}.

Hall and Fuss have considered the

single-mode version of this K-ary digital communication problem 2°. They

optimized a single-mode state--in conjunction with the SG-POM--to obtain

a phase-based quantum communication

setup whose error probability vs. average

photon number is significantly better than

that for optical heterodyne detection. Hall

and Fuss found a nonzero error probability

at finite average photon number, N, which

approached zero as N _ oo. Surprisingly, in our two-mode problem, zero error

probability can be achieved at finite rootmean-square (RMS) photon number. 21

In order to achieve zero error probability

in phase-conjugate quantum communication, we need only use a number-product

vacuum state which enforces Eq. 96, with

¢c <_ _r/K. Under this condition, we have

that the observed phase, ¢, satisfies

Pr ¢--_kl<_[_=0k =1.

(115)

W_ also have that [_k - Cj[ -> 2_r/K, for k. So, for any observed ¢ in the interval (-lr.lr], we know that

Pr(_=¢j[[¢-¢k[<K) = 0,

for all j ¢ k. (116)

This means we can unambiguously determine which digit was sent by choosing the

index associated with the unique C-value

that is within r/Krad of the observed

phase. Via this procedure we decode k

from ¢ with zero probability of being incorrect.

To make this technique for zero error

probability communication more explicit,

we shall introduce a specific input state

which has the desired property. The number representation we shall presume is

1

Cn = _/--_--_(1+2__[ )

sin[2 (1- _-_)1

×

lr 1

2

for Inl = 0, 1,2,...,

and K = 2, 3, 4,... (117)

The associated phase representation for

this state is easily computed to be

=

0,

In Fig. 2 we have plotted p( ¢ [ • = 0 ) vs.

¢ for this state when K = 4; we see that

for I¢I---

(118)

for < I¢I-<

119
thenonzerosupportof this PDF is the interval (-_r/4,1r/4). In Fig. 3 weindicate

howthisPDF leadsto zeroerrorprobability phase-conjugatecommunication;this

figure plots the four conditional PDF's,

{P(¢I ¢_) : 1 < k < 4}, which apply

when K -- 4. For any observed C-value

we must have that p( ¢ I _) > 0, otherwise that C-value could not have occurred.

Figure 3 shows that, for any given ¢,

there is only one possible ¢bk-value which

satisfies the nonzero PDF requirement--

zero error probability communication results from deciding that this value was the

transmitted phase.

The next question to address is the photon number statistics associated with our

phase-conjugate communication scheme.

For the general number-product vacuum

state we have that the total---signal plus

apparatus--photon number measurement

has the following probability distribution,

1¢012, I¢_.I 2 + I¢.I 2,

For the particular state given by Eft. 117,

it is then a simple matter to show that

K

- 2 (120)

In other words, we can achieve zero error

probability K-ary phase-conjugate quantum communication with an RMS total

photon number of K/2. Figure 4 is a plot

of Eq. 119 for the K = 4 case.

for n ----O,

(119)

for n -- 1,2,3,...

The preceding quantum communication result is, of course, idealized. We

have presumed a state generator--to produce a specific number-product vacuum

state--that as yet has no explicit realization. Likewise, our scheme uses the ]Ymeasurement; again, no explicit realization is yet available. At least we can say

that electro-optic modulation will impress

the phase information on the input state,

once that state can be produced. On the

other hand, we have implicitly assumed

lossless transmission; inclusion of loss will

inevitably lead to nonzero error probability.

Our main purpose in going to the twomode construct was to develop potential quantum-phase measurement schemes

that promise substantial benefits, i.e., benefits that warrant the effort to bring

them to fruition. This motivation is

very much in line with the starting point

for squeezed-state research. 22 In this regard, it is instructive to compare our

phase-based scheme for zero error probability quantum communication with a

more well-known approach based on number kets. For a single-mode field with

annihilation operator h, lossless transmission of one of the number kets { Ik - 1) :

1 _< k _< K } followed by ideal direct detection, viz. the N = h?5 measurement,

also yields K-ary digital communication

without error. For k equally likely to be

any digit between 1 and K, the avcrugeenergy efficiency of such a single-mode,

number-ket system is roughly the same

as that of our two-mode, phase-conjugate

system, i.e., both need slightly less than

K/2 photons on average. The numberket system has the advantage that its state

generator may be approximated via feedforward control using photon-twin beams,

and its measurement only requires a high

quantum-efficiency photon counter. Also,

the number-ket approach uses less bandwidth; only one mode is needed. Alternatively, number-ket direct detection on

a two-mode field can be used for errorfree K-ary communication at significantly

less than K/2 photons on average. However, if we shift our attention from phasebased communication, to phase-based precision measurements, the Fig. 1 arrangement has a capability that number kets

cannot match--phase sensing with controlled precision.

Suppose that we use the Fig. 1 arrangement for phase-conjugate precision

measurement. Specifically, let us use the

number-product vacuum state Eq. 117 in

conjunction with a phase-conjugate interferometer (see, e.g., Bondurant and

Shapiro xg) and the Y measurement. Now,

the phase shift ¢ takes on any value from

the continuum (-Tr, 7r]. Nevertheless, except for 2r-modularity effects which come

into play when ¢ is within 7r/K of iTr,

the observed phase will lie within 7r/K rad

of the true phase with probability one.

Thus, using less than K/2 photons on average, we can guarantee a phase measure120
meritwhich is within lr/K rad of the exact 14.

value. In other words, unlike more conventional schemes--which only ensure an

acceptable RMS phase-estimation error--

our phase-conjugate interferometer pro- 15.

.vides exact phase determination to a prescribed number of decimal places. 16.

9. REFERENCES 17.

1. P.A.M. Dirac, Proc. R. Soc. Lond. Ser.

h 114, 243 (1927).

2. L. Susskind and J. Glogower, Physics

1, 49 (1964).

3. J.H. Shapiro, S.R. Shepard, and

N.C. Wong, Phys. Rev. Lett. 62, 2377

(1989).

4. J.H. Shapiro and S.R. Shepard, Phys.

Rev. A 43, 3795 (1991).

5. W.H. Louise[l, Radiation and Noise in

Quantum Electronics (McGraw-Hill,

New York, 1964).

6. R.J. Glauber, Phys. Rev. 131, 2766

(1963).

Quantum Detection

Theory (Academic,

7. C.W. Helstrom,

and Estimation

New York, 1976).

8. H.P. Yuen and J.H. Shapiro, IEEE

Trans. Inform. Theory IT-26, 78

(1980).

9. J.H. Shapiro and S.S. Wagner, IEEE

J. Quantum Electron. QE-20, 803

(1984).

10. J.H. Shapiro, IEEE J. Quantum Electron. QE-21,237 (1985).

11. J.J. Stiffier, Theory of Synchronous

Communications (Prentice Hall, Englewood Cliffs, 1971).

12. S.L. Braunstein, C.M. Caves, and

A.S. Lane, "Maximum Likelihood

Statistics of Multiple Quantum Phase

Measurements," Technical Digest Opt.

So(:. Am. 1990 Annual Meeting, (Opt.

Soc. Am., Washington, D.C., 1990) p.

114.

13. P. Carruthers and M.M. Nieto, Rev.

Mod. Phys. 40, 411 (1968).

18.

19.

20.

21.

22.

A.V. Oppenheim and R.W. Schafer,

Digital Signal Processing (PrenticeHall, Englewood Cliffs, 1975).

A. Papoulis, Signal Analysis

(McGraw-Hill, New York, 1977).

D.T. Pegg and S.M. Barnett, Phys.

Rev. A 39, 1665 (1989).

S.M. Barnett and D.T. Pegg, J. Mod.

Opt. 86, 7 (1989).

A. Yariv, Optical Electronics, Third

Edition (Holt, Rinehart, Winston,

New York, 1985).

R.S. Bondurant and J.H. Shapiro,

Phys. Rev. D 30, 2548 (1984).

M.J.W. Hail and I.G. Fuss, "Quantum

Phase Detection and Digital Communication," Quant. Opt. to appear.

J.H. Shapiro, "Phase Conjugate Quantum Communication with

Zero Error Probability at Finite Average Photon Number," in preparation.

H.P. Yuen, Phys. Rev. A 13, 2226

(1976).

121
TWO-MODE

INPUT FIELD

_ASE- _9

CONJUGATE

SHIFT

t

MEASUREMENT ,._ PHASE

v ESTIMATE

Figure 1: Phase-conjugate quantum communication system.

p(¢ I_=0 )

Figure 2: Conditional phase-measurement PDF, given _ = O, for the state F-_I. 118

when K = 4.

122
P(_ I_1) P(_ I 2) I_3) P(_ I_4)

4 4 4 4

Figure 3: Conditional pha.se-mea.surement PDF's for the state F,q. 118 when K = 4.

Pr( N s* _i,= n)

K=4

012345678910

n

Figure 4: Signal-plus-apparatus number distribution for the state F,q. 117 when K = 4.

123
1 2 /o9

N92-22059/z

QUADRATIC SQUEEZING: AN OVERVIEW*

by M. Hillery, D. Yu, and J. Bergou

Dept. of Physics and Astronomy

Hunter College of CUNY

695 Park Ave

New York, NY10021

I. Introduction

The amplitude of the electric field of a mode of the electromagnetic field is not a

fixed quantity, there are always quantum mechanical fluctuations. The amplitude,

having both a magnitude and a phase, is a complex number and is described by the

mode annihilation operator a. It is also possible to characterize the amplitude by its

real and imaginary parts which correspond to the Hermitian and anti-Hermitian parts of

a,

Xl=_(a++a) X2=_(a+-a) , (1.1)

x

respectively. These operators do not commute and, as a result, obey the uncertainty

relation (h=l)

AX_AX2 _> 1 (1.2)

From this relation we see that the amplitude fluctuates within an "error box" in the

complex plane whose area is at least 1/4. Coherent states, among them the vacuum

state, are minimum uncertainty states with AXz = AX2 = 1/2. A squeezed state,

squeezed in the X1 direction, has the property that AX1 < 1/2 (Refs.l-3). A squeezed

state need not be a minimum uncertainty state, but those that are can be obtained by

applying the squeeze operator

S(O = e ;" az"; a+2 , (1.3)

to a coherent state(Ref.1). The phase of the complex parameter ; determines the

*This work is supported by National Science Foundation under Grant No. PHY-900173 and by a

grant from the City University of New York.

PRECEDING PAGE BLANK NOT FILMED

125
direction of squeezing and its magnitude determines the extent of the squeezing.

Squeezed states are examples of nonclassical states, that is they cannot be

described in terms of a nonnegative definite P representation(Ref.3). This means that a

field in a squeezed state cannot be modeled as a classical stochastic field. It should be

noted that even though a squeezed state is nonclassical it can have a large number of

photons. In fact, a highly squeezed state must have a large number of photons(Ref.4).

Thus we see that the usual association of large photon number with classical behavior

is not correct.

It is possible to generalize the idea of squeezing by looking at fluctuations in

variables more complicated than the mode amplitude. The simplest generalization

involves variables quadratic, rather than linear, in the amplitude. In the case of a single

mode the square of the amplitude, which corresponds to a2 , is one such observable. If

one considers two modes with annihilation operators a and b, then products such as

ab and a+b can be considered. At first glance this procedure appears more

mathematically than physically inspired. However, fluctuations in these quadratic

quantities can be converted into fluctuations of a single mode amplitude by certain

nonlinear optical processes after which they can be measured by standard

techniques. We shall now disscuss the kinds of higher-order squeezing to which

consideration of these quadratic variables leads and the properties they possess.

I1. Amplitude-Squared Squeezing

This is perhaps the simplest example of quadratic squeezing, i.e. squeezing in

a variable quadratic in the mode amplitudes. It describes the fluctuations in the square

of the amplitude of a single mode, a2 (Refs.5,6). Following the example of standard

squeezing we break this variable into its real and imaginary parts

Yl=l(a+2+a 2) Y2=_-(a+2-a2) (2.1)

The commutator of these operators is [Y1, Y2] = i(2N+]), where N=a+a, and this leads

to the uncertainty relation

AYIAY 2 > <N+l> (2.2)

A state is amplitude-squared squeezed in the Y1 direction if (AYI) 2 < <N+l/2>.

126
States with this property are nonclassical. This follows from the fact that (AY1) 2

can be written as

<" (Y1 - <YI>) 2 : > = (AY1) 2 - <N+_-> ,

(2.3)

where the double dots indicate normal ordering. For a classical state the normally

ordered term is always nonnegative so one can see that the onset of amplitudesquared squeezing corresponds to the onset of nonclassical behavior.

Amplitude-squared squeezing was first disscussed in a paper by Wodkiewicz

and Eberly under the name SU(1,1) squeezing (Ref.7). The reason for this name is

that commutation relations of the operators Y_, Y2, and N are closely related to

those of the Lie algebra SU(1,1 ). In particular this Lie algebra is described by three

operators El, K2, and K3, whose commutation relations are given by

[K1, K2] = -iK3 [K2, K3] = iK1 [K3, K1] = iK2 (2.4)

If one makes the identification KI= Y1/2, K2= -Y2/2 and K3= (N+I/2)/2, the above

commutation relations are satisfied. This means that the representations of SU(1,1)

can be used to study higher-order squeezing and this has been done by a number of

authors(Refs.8-10).

It is possible to find minimum uncertainty states for amplitude-squared

squeezing, i.e. states for which the inequality in Eq.(2.2) is replaced by an

equality(Ref.11). This is done by solving the eigenvalue equation

(Y1 + i3.Y2)IW> = 13IqJ> (2.5)

where 7Lis real and positive, and 13is complex. The states Iq_> which satisfy this

equation have the property that

(AY1)2 = _, <_1N+_- IW> (AY2)2 = _L <xlLIN+_ IW> . (2.6)

From these equations it is clear that ;L plays the role of a squeezing parameter. If

0 < 7L< 1, then Y1 is squeezed and if 7L> 1, then Y2 is squeezed. The real and

imaginary parts of 13are related to the mean values of Y] and Y2, respectively.

A particularly simple subset of these minimum uncertainty states occurs when 13

127
and X are related. If X> 1 and 13= (X2- 1) 1/2 (m+1/2), where m is a nonnegative integer,

then the minimum uncertainty states are of the form

I_> = Cm(_,) S(_) Hm(i'/(_.) a+) I0> (2.7)

Here Cm(_L) is a normalization constant, S(_) is a squeeze operator where the squeeze

parameter _ depends on X, Hm is the ruth Hermite polynomial and _;L) = [(;_2_1)1_/2_.]1/2.

The cases m=0 and m=l correspond to the squeezed vacuum and squeezed onephoton states, respectively. Note that this implies that the squeezed vacuum state is a

minimum uncertainty state for both normal squeezing and for amplitude-squared

squeezing.

A second kind of minimum uncertainty state is the amplitude-squared squeezed

vacuum 10,_,>. These states satisfy Eq.(2.5) with 13=0 which implies that they have the

property that <Y1 >=<Y2 > = 0. Such states are superpositions of photon number states

whose numbers are multiples of 4.

We now come to the conversion of fluctuations in a2 into fluctuations of the mode

amplitude of a second mode, b. This is accomplished by means of second harmonic

generation(Ref.5). If the mode described by a has frequency o_ and that described by b

has frequency 2o) then the Hamiltonian which corresponds to this process is

H = (oa+a + 2cob+b + k2(a+2b + a2b +) (2.8)

From this Hamiltonian, using perturbation theory, one can find how fluctuations are

transferred from mode a to mode b. First define the slowly varying operators

A(t) = ei_a(t) B(t) = e2i°_tb(t) , (2.9)

and

XlB(t) = 1[ B+(t) + B(t) ]

YIA(t) = 1 [ A+(t)2 + A(t)2 ]

X2B(t) = _-[ B+(t) - B(t) ]

Y2A(t) = _- [ A+(t) 2- A(t) 2 ]

(2.10)

We then find, if the b mode is initially in a coherent state, that after a time t

128
(AXIB(t))2=4I-+ (k2t)2[(AY2A)2- <NA+ _->]

(ttX2B(t))2=1 + (k2t)2[(AYIA)2_<NA + I>]

(2.11)

where quantities without a time argument, e.g. (AY1A) 2, are assumed to be evaluated

at t=0. What these equations tell us is that if the a mode is initially amplitude-squared

squeezed in the Y2 direction then the b mode will become squeezed in the normal

sense in the X1 direction. Similarly, if the a mode is amplitude-squared squeezed in

the YI direction the b mode will become squeezed in the X2 direction. Therefore, the

second harmonic generation process converts amplitude-squared squeezing into

normal squeezing.

Because normal squeezing can be measured via homodyne detection the

preceding results suggests how amplitude-squared squeezing can be detected. One

first sends the signal into a frequency doubler and then measures the squeezing of the

second harmonic. If it is squeezed, then the original signal was amplitude-squared

squeezed.

Finally, let us see how amplitude-squared squeezed states can be produced.

The fact that the squeezed vacuum state is also amplitude-squared squeezed shows

that a degenerate parametric amplifier can produce amplitude-squared squeezed

states. As one of us (D. Yu) has shown, a degenerate parametric oscillator can as

well(Ref.12). Well above threshold the field inside the cavity can reach a maximum

level of amplitude-squared squeezing given by (AY1) 2 / <N + 1/2> = 1/2, but just below

threshold the amount of amplitude-squared squeezing in the output field can, in

principle, be arbitrarily large. The fourth subharmonic generation process, which has

been studied in connection with generalized squeezed states(Ref.13), can also

produce amplitude-squared squeezing(Ref.6).

III. Sum Squeezing

Sum squeezing, as opposed to amplitude-squared squeezing, is a two mode

effect(Ref.14). In fact, amplitude-squared squeezing is the degenerate limit of sum

squeezing. Let us consider two modes with annihilation operators a and b and

frequencies tOa and cob. The variables involved in sum squeezing are the real and

imaginary parts of the product ab, i.e.

i + +

Vl =_-(a+b++ab) V2=_-(a b - ab) (3.1)

129
_i

The commutator of these operators is [ Vl, V2 ] - _- ( NA + NB + 1 ), where NA = a+a and

NB = b+b, which yields the uncertainty relation

AV1 AV2 > 1 < NA + NB + l > (3.2)

-4

A state is said to be sum squeezed in the Va direction if

(AV1) 2 < I<NA + NB +1> (3.3)

Such a state is nonclassical.

The commutation relations of the operators V], V2 amd NA + NB +1 are also

closely related to those of the SU(1,1) Lie algebra. In fact if one sets K] = V1, E2= - V2

and E3 = I(NA+ NB +1) one obtains the SU(1,1) commutation relations given in

Eq.(2.4).

The name, sum squeezing, comes from the fact that this kind of squeezing is

converted into normal squeezing by the process of sum frequency generation. Sum

frequency generation is a three-mode process which is described by the Hamiltonian

H = COaa+a + O)bb+b + COcC+C+ ks ( ca+b + + c+ab ) , (3.4)

where o3c=coa+o_o.As before we define the slowly varying operator A(t)=eiO_ta(t), and

similarly for B(t) and C(t). We also define

l + + i (C +_C) (3.5) Vl(t) = _-(A B + AB) Xc2 =

If the c mode is initially in a coherent state then to second order in ks we find

(AXc2(t))2=I+(kst)2[(AV1)2-41-<NA+NB+I>] , (3.6)

where, as before, quantities without a time argument are evaluated at t=0. Comparing

this equation to Eq.(3.3) we see that the c mode will be squeezed in the X¢2 direction if

the a and b modes are sum squeezed in the Vl direction.

If the a and b modes are uncorrelated, then there is a connection between

squeezing in the individual modes and sum squeezing. In particular, if neither mode is

130
squeezed, then the state is not sum squeezed. If one of the two modes is squeezed

and the other is in a coherent state, then the state is sum squeezed. Finally, if both

modes are squeezed, then the resulting state may or may not be sum squeezed.

This connection disappears if the modes are correlated. This can be seen by

considering the state produced from the vacuum by a parametric amplifier. This system

is described by the Hamiltonian

H = O)aa+a + O)bb+b + g (e-i_ta+b + + eiOotab) , (3.7)

where, again, coc= (o,+ _b. This Hamiltonian is an approximation to that in Eq.(3.4)

when the c mode is in a highly excited coherent state. Using this Hamiltonian we find

that if both the a and b modes are originally in the vacuum state, then

(AVI(t)) 2- I<NA(t) + NB(t) + 1> =- 2Lsinh 2 (gt) (3.8)

Therefore, the amount of sum squeezing increases with time and this device is a

possible source of sum squeezed light. A further calculation shows that neither of the

two modes is squeezed in the normal sense. Therefore, for correlated modes normal

squeezing is not a prerequisite for sum squeezing.

IV. Difference Squeezing

Difference squeezing is also a two-mode effect(Ref.14). Its name comes from its

close connection to difference-frequency generation. We again describe the modes by

annihilation operators a and b, and we assume that COb> tO,. The observables which

describe it are

Wl = _'-(ab + + a+b) W 2 = _- (ab + - a+b) (4.1)

Their commutator is given by

[Wl, W2] = _ (NA - NB) (4.2)

which yields the uncertainty relation

AWl AW2 > II<NA - NB>I •

(4.3)

131
A state is said to be difference squeezed in the W1 direction if

(AWl)2 < 1⁄4<NA - NB>

(4.4)

Note that for a state to be difference squeezed we must have <NA > > <NB>.

Difference squeezed states are nonclassical but there is a difference in this

regard between them and sum or amplitude-squared squeezed states. For both of the

latter, the condition for squeezing and the condition for being nonclassical are the

same. For difference squeezing this is not true. A state is nonclassical if

(AWl)2 < I<NA + NB>

(4.5)

which is not the same as the squeezing condition Eq.(4.4). Therefore, difference

squeezed states are well within the nonclassical regime.

Difference squeezing is also related to a Lie algebra but this time it is SU(2)

instead of SU(1,1 ). In fact, the operators which describe difference squeezing are

those used in the Schwinger representation of the angular momentum operators

(Ref.15). The SU(2) Lie algebra consists of three operators Jz, J2 and J3 whose

commutation relations are

[Jk, Jl] = iEklmJm , (4.6)

where all indices run from 1 to 3 and ekZ= is the completely antisymmetric tensor of

rank 3.

If the modes are uncorrelated, then at least one of them must be squeezed for

difference squeezing to be present. If the b mode is squeezed and the a mode is in a

coherent state le_>,the state wil be difference squeezed but only if[a[ 2 is large enough.

A necessary, but not sufficient condition is that <NB> < 91-[e_[2. If the modes are

correlated, then it is no longer true that squeezing in the individual modes is required

for difference squeezing.

Finally, as might be suspected from the name, difference squeezing is turned

into normal squeezing by difference frequency generation. The Hamiltonian

describing this process is

132
H = O)aa+a + obb+b + (0cc+c + kd(a+bc + + ab+c) , (4.7)

where ¢.oc= rob- ma. We define slowly varying A(t), B(t), and C(t) as in the previous

section and then set

W](t) = 1 (A(t)B+(t) + A+(t)B(t) ) Xc2 = _- (C+(t) - C(t) )

(4.8)

Using perturbation theory we find that if the c mode is originally in a coherent state,

then

(AXc2(t))2 = 1 + (kdt)2[(AWl)2 _ I <NA- NB >] (4.9)

This equation shows us that Xc2 becomes squeezed if W1 is difference squeezed.

Therefore, difference frequency generation can be used to detect difference squeezed

light.

V. Amplification of Higher-Order Squeezing

An amplifier consists of a collection of two-level atoms NI of which are in their

ground states and N2 of which are in their excited states where N2 is greater than NI.

We shall assume that we are in the linear regime of this system. An input signal is put

into the amplifier at t=0 and emerges at the output at time t. The signal amplitudes at

the input and the output are related by <a(t)> = G <a(0)> where G is the amplitude gain.

This system was analyzed rather thoroughly by Carusotto(Ref.16).

Hong, Friberg, and Mandel examined the effect of amplification on subPoissonian photon statistics and normal squeezing(Ref.17). They found that both of

these effects disappear at the output, no matter what the input state is, if the intensity

gain, IGI 2, is greater than two. The gain IGI2= 2 is known as the photon cloning limit

because one gets two photons out for every one that goes in. This gain has stood as

an upper limit for the amplification of nonclassical behavior.

Recently two of us looked at the situation for amplitude-squared squeezing

(Ref.18). We found that it can survive amplification for gains slightly greater than two. In

particular, amplitude-squared squeezing will be present at the output if

1 (5.1) IG[2 < 2 + <No> + 1/2 '

133
where <No> is the photon number of the input state. Because the right-hand side is

greater than two this suggests that there are states which will still be amplitudesquared squeezed at the output if IGI 2 is slightly greater than two. Further investigation

shows that the amplitude-squared squeezed vacuum, 10,X>, with _,<<1 is such a state.

Therefore, the photon cloning limit does not, at least in principle, represent a barrier to

nonclassical behavior. It would be of considerable interest to know if there are

nonclassical states which can remain nonclassical when they are amplified at gains

substantially larger than two.

VI. Conclusion

Quadratic squeezing represents a new class of nonclassical effects. States with

this property have fluctuations smaller than is possible for classical light in a variable

which is quadratic in mode creation and annihilation operators. As we have seen,

quadratic squeezing can be converted into normal squeezing by _(2) type nonlinear

interactions.

A direction for further investigations into quadratic squeezing is its connection to

interferometry. Interferometers, both with and without nonlinear elements, can be

described in a natural fashion in terms of the variables which describe quadratic

squeezing(Ref.19). This suggests that interferometers can be used to measure

quadratic squeeezing and that quadratic squeezed states may be of use in

interferometric measurements. We are currently studying these issues.

134
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M. Hillery, Opt. Commun. 62, 135 (1987).

M. Hillery, Phys. Rev. A 36, 3796 (1987).

K. Wodkiewicz and J. H. Eberly, J. Opt. Soc. Am. B 2, 458 (1985).

C. C. Gerry, Phys. Rev. A 31,2721 (1985).

P. K. Aravind, J. Opt. Soc. Am. B 5, 1545 (1988).

V. Buzek, J. Mod. Opt. 37, 303 (1990).

J. Bergou, M. Hillery, and D. Yu, Phys. Rev. A 43, 515 (1991).

D. Yu, submitted for publication.

S. Braunstein and R. McLachlan, Phys. Rev. A 35, 1659 (1987).

M. Hillery, Phys. Rev. A40, 3147 (1989).

See, for example, H. M. Nussenzveig, Introduction to Quantum O0tics

(Gordon and Breach, New York, 1973), P. 219.

S. Carusotto, Phys. Rev. A 11, 1629 (1975).

C. K. Hong, S. Friberg, and L. Mandel, J. Opt. Soc. Am. B 2, 494 (1985).

M. Hillery and D. Yu, submitted for publication.

B. Yurke, S. McCall, and J. Klauder, Phys. Rev. A 33, 4033 (1986).

135
N92-22060

QUANTUM SPATIAL PROPAGATION OF SQUEEZED LIGHT IN A

DEGENERATE PARAMETRIC AMPLIFIER*

Ivan H. Deutsch

Department of Physics, University of California Berkeley, California, 94720

John C. Garrison

Lawrence Livermore National Laboratory, University of California, Livermore,, California 94550

Abstract

Differential equations which describe the steady state spatial evolution of nonclassical light are

established using standard quantum field theoretic techniques. A Schrodinger equation for the state

vector of the optical field is derived using the quantum analog of the slowly varying envelope

approximation (SVEA). The steady state solutions are those that satisfy the time independent

Schrodinger equation. The resulting eigenvalue problem then leads to the spatial propagation

equations. For the degenerate parametric amplifier this method shows that the squeezed state is the

ground state of the squeezing Hamiltonian. The magnitude and phase of the squeezing parameter

obey nonlinear differential equations coupled by the amplifier gain constant and phase mismatch. The

solution to these differential equations is equivalent to one obtained from the classical three wave

mixing steady state solution to the parametric amplifier with a nondepleted pump.

1. Introduction

The standard approach in quantum optics for dealing with nonclassical light is to introduce a

normal mode expansion for the field. This naturally leads to time evolution equations for the mode

amplitudes (creation and annihilation operators) which satisfy the canonical equal time commutation

relations. Such an approach is well suited for dealing with systems in optical cavities, such as an

optical parametric oscillator, but is not appropriate for open systems such as a parametric amplifier.

For the amplifier we are usually interested in the spatial dependence of temporally steady state fields.

This problem has been treated for quantum fields either by identifying spatial evolution with temporal

evolution for an interaction time set by the effective interaction length [1-4], or through the use of new

quantization methods which require nonconventional commutation relations that are generally

inconsistent with the fundamental equal time commutators for interacting fields [5-9]. In addition, the

problem has recently been studied using the positive-P representation and stochastic differential

equations [10]. In this letter we propose an alternative scheme which employs only standard quantum

field theoretic techniques, and apply this analysis to the degenerate parametric amplifier (DPA) in

137

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order to study the spatial evolution of squeezed light.

In the classical three wave mixing analysis of the DPA, the positive frequency component of the

electric field is written as a slowly varying time independent envelope multiplying a carrier plane

wave. Such an ansatz physically represents steady state propagation and leads to spatial differential

equations for the envelope. Quantum mechanically, this ansatz cannot hold as an operator identity in

the Heisenberg picture. Upon closer inspection, we see that the requirement of steady state

propagation is enforced through the choice of state vector for the field. More precisely, steady state

fields are the stationary states of the appropriate Hamiltonian. The resulting eigenvalue problem is

then a time independent Schrodinger equation, and leads to spatial differential equations for the

functions which parameterize the global state vector. We will apply this technique to the DPA in one

dimension with a classical nondepleted pump. The stationary state solutions are squeezed states

specified by a set of coupled nonlinear differential equations for the magnitude and phase of the

squeezing parameter driven by the coupling constant of the three wave mixing.

2, Steady State Propagation Condition

Using the classical wave mixing analysis as our guide, we formulate the quantum theory in terms

of the SVEA. The positive frequency electric field operator is written as an envelope field modulating

a cartier plane wave of a given polarization I_. For systems in which the medium has no transverse

dependence over the beam diameter and is longitudinally short compared to the Raleigh range, the

theory becomes essentially one dimensional so that,

E_+}(x,t) = _/ 2retire e U?(z,t) e i(kz-°)t) ,

A n2(m)

(1)

where A is the beam area, and n(c0) is the linear index of refraction. The effective field theory in the

SVEA formally resembles a nonrelativistic many body theory for a complex scalar field qs [12],

[U?(z,t), Wt(z',t)] = _i(z-z'). (2)

The Hamiltonian (in the Heisenberg picture) can be written as,

H = He + I-Ienv +Hint (3a)

(3b)

138
-m [ /vtovovt.,3

Henv =-2--n((o)Jdz/ -_z _ ]

(3c)

(3d)

where we have decomposed the "free" field Hamiltonian into the carrier wave Hamiltonian H c, and

the Hamiltonian governing the free (plus linear) evolution of the envelope Hen v. The effective

interaction Hamiltonian for the DPA is given by Hin t [14] representing the mixing through the

nonlinear susceptibility, with pump frequency COp. The interaction coupling is determined by

l_(z) = lgo(z)eiO(z),

_ 4no_ ix(2>(z lsp(z go(z) n(o3)c

_(z) = - _ + Ak z + _(z)

(4a)

(4b)

(4c)

where g0(z) is the standard power gain coupling constant [13] (allowing for possible z-dependence),

Ak is the phase mismatch at the degenerate frequency, and a_(z) is the remaining phase stemming from

the pump and _(2).

We seek the quantum field state vector corresponding to steady state propagation. Maintaining

our close analogy with the classical analysis, we remove the carrier wave oscillation from the

dynamics of this state. Quantum mechanically we use the envelope picture (E.P.) originally

introduced by Caves [11], in which the dynamics of the operators are dictated by the carrier wave

Hamiltonian, and the states evolve by a Schrodinger equation with the Hamiltonian Hen v + Hin t. If

we denote objects in this picture by the superscript (E), then the dynamical equations in this picture

are

IlI_=[A (E),

1 I_I(I) > (E) _ (I-I_en) E) (E) • + n )lO> •

Ot

(5a)

(5b)

For steady-state applications there will be exact frequency matching between the carrier frequencies of

the various waves which mix through the nonlinear interaction, so that the Hamiltonian in Eq. (5b)

will be independent of time. The steady state (ss) solutions are thus identified with the stationary

solutions to Eq. (5b),

(E) I_I(E),_ith'5(E) kl(i)>_Es). (Hen v + .lintS,-_zss :(6)

139
Henceforth, the labels (E) and (ss) will be omitted. Eq. (6) is the key result of our analysis which we

now apply to the parametric amplifier.

3. Propagation Solution to the DPA

It is well known that the output produced during a parametric interaction is a squeezed vacuum

state [11]. In an oscillator configuration one can restrict attention solely to a single mode of the cavity

(for a doubly degenerate oscillator). In the amplifier configuration under consideration here, we must

generalize the squeezing operator to take into account the many modes associated with the envelope

field. We define a functional squeezing operator,

ex¢I (7)

with z-dependent squeezing parameter _(z) = r(z)ei°(z). One can easily show that the Bogoliubov

transformation generalizes to

qJ(z) - Sf[_lW(z)S[_] = cosh(r(z)) W(z)- e i°(z) sinh(r(z)) Wf(z). (8)

The squeezed vacuum,

=-sK] o), (9)

satisfies the well known eigenvalue equation [ 15],

A

Ud(z)l{¢} > = O, (lOa)

_(z) --- S[_]qJ(z)S't[_] = cosh(r(z)) _(z) + e i0(z) sinh(r(z)) _f(z). (10b)

Using the formalism developed in Section 2, we deduce the steady steady solutions by solving

the time independent Schrodinger equation (6). We take as our ansatz for the eigenvector a squeezed

vacuum state Eq. (9), i.e.

(Henv +Hint)l{¢}) =_LI{¢}). (11)

Our goal is to prove Eq. (11) and thereby obtain differential equations for the spatial squeezing

140
parameter_(z). Together,Eqs.(10a)and(11)imply

[Henv+ Hint, _(z)] I{;}) =0. (12)

SubstitutingEqs.(3) and(10b)into Eq.(12),usingEq.(8),andtherelation(4a)for thepolar

decompositionof _:gives,

[(_ - lg°c°s (0-_)) + i/d0 c°sh(2r)2,dz sinh(2r) + g° sin(0-_))} Ftl0_ =0. (13)

Thus, the eigenvalue condition requires the real and imaginary parts of the coefficient of W f to vanish,

yielding the desired propagation equations:

-= lg0cos (0-0)

dO = coth(2r) sin(O-C).

dz - go

(14a)

(14b)

The solution to (14) produces a relation for the squeezing parameters {r(z), 0(z)} in terms of the

experimental parameters {g0(z), qb(z)}. In the next section we give an interpretation of the meaning

of these equations and analyze their solutions.

4. Solutions and Interpretation of Propagation Equations

In solving Eqs. (14) we take as our boundary conditions,

r(O) = 0 (15a)

0(0) = _(0) (15b)

to ensure the continuity of the squeezing parameter. In order to get a better understanding of the

nature of the equations we examine them for various regimes. Consider first the situation in which

the phase of _, Eq. (4c), is a constant ¢(z) = 00 (i.e exact phase matching). With boundary

conditions (15), the solution to (14) is immediately seen to be

* 1 , (16a) r(z) = dz' _-g0(z ),

0(z) = ¢0. (16b)

141
For a uniformcrystalwith planewavepump,thisyieldstheexpectedresultrout= lg0L (for z>L),

whereL is thelengthof thecrystal.

In limit of shortcrystals,Eqs.(14) canbeexpandedfor smallr to yield

==_Dzz= _(z). (17)

Thus,in thislimit thecomplexsqueezingparameterisjust theintegralof theinteractioncoupling

constant.Henceforthweconsidercrystalswith uniformnonlinearityandaplanewavepump,sothat

gois aconstantand¢(z) = - _ + Akz. Thesolutionto Eq.(17)for z_>Lis of afamiliar form,

rout= lgoL sin( AkL/2 )

(18a)

0out = - _ + AkL/2. (18b)

If in addition Ak<<I/L the amplitude and phase essentially decouple. However, as the phase

mismatch increases, the magnitude of the squeezing decreases. Physically, large phase mismatch

would cause squeezing along wildly different quadratures as the beam propagates through the crystal,

thereby degrading the net squeezing.

For crystals containing many gain lengths, we expect strong sensitivity to the phase mismatch

since the nonlinear coupling between r and 0 causes rapid oscillations that destroy the magnitude of

the squeezing at some value of Ak. To see this we numerically integrate Eq. (14). In presenting the

data, we measure all distances in units of the natural length scale 1/g 0. Fig.(1) shows the magnitude

of the squeezing parameter as a function of z for various phase mismatches. Fig. (2) shows the

squeezing parameter of downconverted light as function of phase mismatch for various values of the

crystal length L, plotted simultaneously with the small r expression, Eq. (18a). For short lengths,

Eq. (18a) is an excellent approximation. However, for crystals many gain lengths long (as can be

obtained using intense pumps [ 16]) we see a sharp cutoff in the squeezing at the critical value Ak/g 0

=1. The sharp transition in output when Ak=g 0 is also characteristic of the classical three wave

mixing solution to the parametric amplifier. This leads us to hypothesize that the solution to Eqs.

(14) can be obtained from the classical propagation equations.

The classical solution for the envelope in a DPA is [13]

A(z) = (p(z)A(0) + iv(z)A*(0))e iakz/2 (19a)

142
=cos lgz) i sin z); v z =Vl = sin z)

g = _/g__ (Ak)2. (19c)

If we replace the c-number fields with creation/annihilation operators A*(0)---)af, A(z)---)b

representing "in" and "out" fields respectively, we recover the solution obtained by Caves and Crouch

[5], and Huttner et. al. [61,

b = (g(z)a + iv(z)a?_ iAkz/2. (20)

The unitary operator relating a to b, b = UfaU, is a generalized squeezing operator

U = Rf(_Arg(g)+ Akz))S(-isinhq(v)),

(21)

where S is the standard squeezing operator and R is the phase rotation operator [ 11 ]. This unitary

operator transforms the vacuum into a squeezed state with squeezing parameter _ = re i° given by,

r(z) = sinh-l(v(z))

0(z) = - _ + Arg(l.t(z)) +Akz.

(22a)

(22b)

Figs. (1) and (2) are now easily understood in terms of limiting cases of (22a). In fact, direct

substitution reveals that (22) is indeed the analytic solution to Eq. (14).

To understand the meaning of this result consider the nature of the interaction more carefully.

Because we have assumed a classical nondepleted pump, the propagation equations are linear in the

signal and idler amplitudes. In such a situation one expects the q-number and c-number solutions to

be equivalent, i.e., the quantum fluctuations should propagate like a classical signal. This provides a

substantial test on our formalism. The successful use of nonconventional commutation relations by

previous workers is explained by a combination of good physical intuition and the special linear

character of this problem. Indeed, Huttner et. al. showed that for this particular interaction the equal-z

commutation relations are consistent with the fundamental equal-t commutators. However, for a

general interaction in which the wave mixing equations are truly nonlinear we expect this method to

fail. For example, if the pump is treated as a quantum mechanical field, then we find that the equal-z

commutation relations are no longer consistent with the fundamental equal-t commutators. Thus, for

more complex systems there will be qualitative differences between the classical and quantum

propagation solutions. The formalism we have presented here is sufficiently general to handle these

situations.

143
Acknowledgements:

The authors would like to thank Aephraim Steinberg, Paul Kwiat, and Raymond Chiao

for many useful conversations. I. H. D. acknowledges support by a Department of Education

Fellowship.

References

* This research was performed under the auspices of the U. S. Department of

Energy at the Lawrence Livermore National Laboratory under contract no.

W-7405-Eng-48.

1. Shen,Y.R., 1967, Phys Rev. 155, p. 921.

2. Tucker, J., and Walls, D.F., 1969, Phys. Rev. 178, p. 2036.

3. Lane, A., Tombesi, P., Carmichael, H.J., and Walls, D.F., 1983, Opt. Com. 48, p.155.

4. Hong, C.K., and Mandel, L., 1985, Phys. Reav A 31, p. 2409.

5. Caves, C. M., and Crouch, D.D., 1987, J. Opt. Soc. Am. B 4, p. 1535.

6. Huttner, B., Serulnik, S., and Ben-Aryeh, Y., 1990, Phys. Rev. A 42, p. 5594.

7. Imoto, N. in Proceedings of the 3 ra International Symposium on the Foundations of

Quantum Mechanics, Tokyo, 1989, pp. 306-314

8. Yurke, B., Grangier, P., Slusher, R.E., and Potasek, M.J., 1987, Phys. Rev. A 35,

p. 3586.

9. Potesek, M. J., and Yurke, B., 1987, Phys. Rev. A 35, p. 3974.

10. Drummond, P.D., and Carter, S.J., 1987, J. Opt. Soc. Am. B4,1565, p. 6845.;

Raymer, M.G., Drummond, P.D., and Carter, S.J., 1991, (unpublished).

11. Caves, C. M. and Schumaker, B. L., 1985 Phys. Rev. A 31, p. 3068.

12. Deutsch, I.H. and Garrison, J.C., 1991 Phys. Rev. A 43, p. 2498.

13. Yariv, A., 1989, Quantum Electronics, 3rd Ed.John Wiley and Sons, Inc., New York.

14. Hillery, M., and Mlodinow, L., 1984 Phys. Rev. A 30, p. 1860.

15. Yuen, H., 1976, Phys. Rev. A, 13, p. 2226.

16. Slusher, R. E., Grangier, P., A LaPorta, Yurke, B., and Potasek, M. J., 1987,

Phys. Lett. 59, p. 2566.

144
(a)

r

5_

4

3

2

1

, , , , , gOz

2 4 6 8 I0

(b)

r

J

2

1.5

I

0.5

(c)

, , , , -gO Z

2 4 6 8 I 0

Oo:/ '

2 '_' 6 /// 8 I 0 gO z

',j

Squeezing magnitude as a function of propagation distance z measured in gain lengths go,

for different values of phase mismatch &k: (a) Ak=0.1g o , (b) Ak=g o , (c) Ak=2g o .

145
(a)

1

0.8

0.6

0.4

0.2

-0.2

N

",\

8 "_o _k/g°

(b)

-1

r

5

4 \

3- \_

2

1 \\ , / / _/_',, __ .

_k/gO

(c)

25

20

15

10

r

\ /' "-d.5\_ '-'iVv---_,,-,,- 1.5-.v- 2 Ak/gO

-5 \/

Squeezing at the output face as a function of Ak: (a) L=2g 0 ,(b) L=10g0, (c) L=50g 0. The

solid curve is the solution to Eq. (14), the dotted curve is given by the asymptotic result Eq. (18a).

146
N92-22061

SQUEEZED PULSED LIGHT FROM A FIBER RING INTERFEROMETER

K. Bergman and H. A. Haus

Department of Electrical Engineering and Computer Science

and Research Laboratory of Electronics,

Massachusetts Institute of Technology, Cambridge, MA 02139

ABSTRACT

Observation of squeezed noise, 5+0.3 dB below the shot noise level, generated

with pulses in a fiber ring interferometer is reported. The interferometric geometry is used

to separate the pump pulse from the squeezed vacuum radiation. A portion of the pump is

reused as the local oscillator in a homodyne detection. The pump fluctuations are

successfully subtracted and shot noise limited performance is achieved at low frequencies

(35-85 KHz). A possible utilization of the generated squeezed vacuum in improving a fiber

gyro's signal to noise ratio is discussed.

INTRODUCTION

Squeezing in a fiber was first demonstrated by Shelby et al. [1] who showed that

the fiber's Z(3) nonlinearity could be used to shape the field's fluctuations through self

phase modulation. The group's experimental results were, however, severely limited by

acoustic and thermal noise processes. First, the low Stimulated Brillouin Scattering (SBS)

threshold in fibers forced the IBM group to separate their CW pump into 25 frequency

components. The second noise source, more difficult to avoid, was the so called GAWBS

(Guided Acoustic Wave Brillouin Scattering) excitations [2]. GAWBS is caused by

thermal fluctuations that modulate the fiber's refractive index at high frequencies

(approximately between 20 MHz and 10 GHz) and thereby scatter acoustic waves which

are guided in the forward direction. Because the experiment was performed in a traveling

wave geometry, it was necessary to detect the squeezing at high frequencies (50-100 MHz)

where the laser noise is negligible. The measurement center frequency was within the

frequency range of GAWBS, and only about 12% of noise reduction was thus observed.

To improve upon these earlier results, we used short pulses in a fiber interferometer

configuration. Large nonlinear phase shifts of the order of several x are easily achieved

with short pulses of high peak power, while the SBS threshold is avoided. The fiber ring

geometry is used to separate the pump from the squeezed vacuum fluctuations. Subsequent

detection with a balanced detector permitted complete cancellation of the pump fluctuations

at frequencies as low as 35-85 KHz. Although the GAWBS noise can be frequency

down-converted into the detection "window" when pulses are used [3], it did not seem to

prevent the noise reduction in our experiment. The investigation of the down-conversion

of GAWBS requires further study.

147
The experimental results reported' in this paper have been recently published [4].

This paper begins with a brief review of the broadband squeezing process in a fiber.

Following the approach of Shirasaki and Haus [5], we then show how the pump is

separated from the squeezed vacuum when the squeezing is performed within a MachZehnder interferometer. The squeezed vacuum is observed with the aid of a local oscillator

derived from the pump. We describe some of the theoretical limitations on the observed

squeezing using homodyne detection and a gaussian shaped local oscillator. The

experiment and results are then described in detail.

SQUEEZING IN THE NONLINEAR MACH-ZEHNDER

We consider the simple propagation equation for the field operator, 3(z), in a

nonlinear Kerr medium. Assuming no dispersion this equation becomes:

d_(z) = itc[3+(z)&(z)l?#z)

dz

where _¢is the Kerr nonlinearity. The Kerr nonlinearity is calculated as follows,

(1)

2/i:

lc= T n2 (2) ae#

where A, is the wavelength, n2 the nonlinear index, P.p the peak power, and aeff the

effective coupling area. Equation (1) may be integrated dwectly, yielding a solution for the

field operator, _(z), after a propagation distance, L:

_( L) = e_+ (°)_(°)_(0) (3)

The effect of the Kerr medium is to add a nonlinear phase shift proportional to the photon

number, propagation distance, and Kerr coefficient. The mean square fluctuations of the

field are shaped by this nonlinear process as shown in figure 1. The X(3) process couples

the amplitude and the phase fluctuations, causing the phase insensitive mean square

fluctuations of the incoming field, represented by a circle, to stretch into an ellipse along the

amplitude tangents. The area of the resulting squeezed noise ellipse is equal to the area of

the initial circle.

Next, we place two equal lengths of Kerr media symmetrically in the two arms of a

Mach-Zehnder interferometer, as was analyzed in reference [5]. The input field and its

associated fluctuations enter one input port of the interferometer's fin'st beam splitter in

figure 2. Into the second, unexcited port enter the zero point fluctuations of the vacuum

field. The field's amplitude split coherently in two by the 50/50 beam splitter. The

fluctuations add and subtract incoherently.

As each field-half propagates through its Mach-Zehnder arm, it accumulates a

nonlinear phase shift (not shown on the phasors in figure 2) and its fluctuation circles are

stretched into ellipses. At the second beam splitter, the two half-field amplitudes interfere

coherently. Under a linearized f'wst order analysis, the squeezed fluctuations will again add

and subtract incoherently. As illustrated with the phasors in figure 2, the original mean

148
field amplitude emerges from the constructive interference port, and the squeezed vacuum

exits from the destructive interference port. The interferometer has made possible this

isolation of the squeezed vacuum.

The above explanation for a single frequency phasor (CW pump) also holds for the

case of a pulsed pump, as long as dispersion may be neglected. Without dispersion the

pulse may be divided into short time segments and each segment analyzed independently.

At the output the segments are superimposed to reconstruct the pulse.

DETECTION

The function of homodyne detection is to measure one quadrature of the incoming

signal, amplified by the local oscillator. This detection technique is used to observe the

phase sensitive fluctuations of the squeezed vacuum signal. If the local oscillator phase is

properly adjusted, the squeezed (reduced noise) quadrature is measured. The fluctuations

accompanying the pump are completely subtracted by the balanced detection. When the

signal arm is blocked, the homodyne detection measures the vacuum state noise. This is a

part of the experimental shot noise calibration.

In the Mach-Zehnder squeezing geometry, the exiting pump, shown in figure 2,

may be reused as the local oscillator. In this way, the local oscillator pulse shape matches

the squeezed vacuum pulse. However, to detect the full pulsed squeezing magnitude a

finer matching of the local oscillator phase is required since the squeezing direction as well

as magnitude will vary along the pulse duration. Thus, undei" ideal conditions the local

oscillator should be phase shifted in one direction at the pulse center but in a different

direction at the pulse wings. Experimentally the local oscillator is shifted by one constant

phase leading to a nonideal measured noise reduction magnitude. We have plotted the

expected noise reduction for the ideal and single phase adjusted gaussian pulse cases in

figure 3 (a). In figure 3(b) we show that for the gaussian pulse case, the amplified noise

amplitude is larger than the attenuated noise amplitude for the same peak nonlinear phase

level.

EXPERIMENT

The experiment was implemented by replacing the Mach-Zehnder with a fiber ring

interferometer, to take advantage of the ring's stability to vibrational and thermal

disturbances. Figure 4 is a schematic of the experimental arrangement. A mode-locked

1.3l.tm Nd:YAG producing 100 psec pulses at a repetition rate of 100 MHz was used as the

pump. This pulse train pump is coupled into the fiber ring reflector composed of 50 m of

PM (polarization maintaining) fiber which is spliced to the two pigtails of a 3 dB PM fiber

coupler. The coupler's splitting ratio was variable, and was carefully adjusted to 50/50

within 1 part in 500. The pump pulses are thus completely reflected from the ring. A

90/10 beam splitter picks off a portion of the reflected pump to be reused as the local

oscillator. The squeezed vacuum signal emerges from the unexcited transmission port of

the ring.

The local oscillator and the squeezed signal are then mixed in a 50/50 beam splitter

(BS2), and detected with a balanced dual detector receiver. The output difference current is

directly fed into a spectrum analyzer. The Power Spectral Density (PSD) reading

149
correspondsto a measureof the fluctuations magnitudealong one quadrature of the

squeezed vacuum. The amplified and reduced noise quadratures may now be measured by

adjusting the relative phase of the local oscillator and squeezed signal.

The reduced noise will increase with the pulse peak nonlinear phase shift, which

can be estimated from the following CW equation,

.ONE =_--_n2L_ (4)

where n 2 is the nonlinear index, L the fiber length, Pp the pulse peak power, and Aeff the

effective coupling area. In this experiment, a peak nonlinear phase shift of 1.4 radians

corresponds to approximately 100 mW of average power in each ring direction. The PSD

measurements were performed at a narrow low frequency window between 39 and 41

KI-Iz. Data was taken with an integration time of 400 msec and a frequency resolution of

2.5 Hz. Before discussing the results, we describe the methods used to calibrate the shot

noise level.

IV SHOT NOISE CALIBRATION

The shot noise level was calibrated in order to confirm that the reduced quadrature

fluctuations of the squeezed vacuum dropped below the zero point fluctuations level. Two

methods were used. First, direct excitation from the laser was sent through the homodyne

detection system and the detector current along with a corresponding PSD level were

recorded for a range of input power levels. To cross check this calibration, two white light

sources were used to generate detector current levels similar to those obtained with the laser

excitation. Again, the PSD levels were recorded along with the current readings for the

same range of power levels. The spectrum analyzer's noise floor was measured at -155

dBm/Hz and typical shot noise levels ranged from -120 to -125 dBm/Hz, so that electrical

noise was not a factor. The two curves plotted on a dB scale in figure 5 overlapped well

with a 45 degree slope. Thus, the laser noise has been successfully subtracted and the

detection's response has been shown linear within the measurement current range.

V E_ERIME_AL _S_TS

Having accurately established a shot noise level, we proceed with the squeezing

measurement. The local oscillator pulse and the squeezed vacuum pulse are aligned to

overlap spatially and temporally at the detection beam splitter (BS2). For alignment

purposes only, the signal magnitude may be temporarily increased by changing the

coupler's splitting ratio. The coupler is adjusted back to 50/50 and the relative phase

between the local oscillator and squeezed signal is allowed to drift. While the phase is

drifting, PSD measurements are taken continuously with an automated data acquisition

system.

150
The PSD level, a measurementof the squeezednoise will vary from some

maximumvalue to someminimumvaluecorrespondingto the amplified andattenuated

quadraturesof thesqueezedvacuum. Theresultinghistogramsfrom thesemeasurements

taken at three separatepump power levels: 60, 85, and 110 mW (in eachfiber ring

direction)areshownin figure 6 (a),(b), and(c) respectively.In thefigure,theblackbars

arePSDreadingstakenwith thesqueezedvacuumarmblocked,andarethustheshotnoise

calibrationfor the specificpowerlevel. Thewhite barsarea collectionof PSDreadings

takenafterthemixing of thelocaloscillatorandsqueezedvacuum.Thereducednoiselevel

or squeezingmagnitudeisdefinedasthedistancebetweentheblackbarsdistributioncenter

andtheleft edgeof thewhitebarsdistribution.We notethataspreviouslypredictedfor the

caseof agaussianlocaloscillator,theamplifiedquadraturenoiseis largerthanthereduced

quadraturefor thesamepowerlevel.

In figure 7 we plot the experimental maximum and minimum PSD readings (in dot

scatter format) obtained for a collection of power levels, on top of the theoretically

predicted limits for the gaussian local oscillator case. The experimental point plot was

adjusted horizontally, along the peak nonlinear phase axis for a best fit. This fitting

compensates for coupling losses, detector quantum efficiencies, and nonlinear phase

estimation.

VI SQUEEZED VACUUM IN A FIBER GYRO

It has been shown both theoretically [6,7,8], and experimentally [9,10], that the

sensitivity of a phase measurement device can be improved with the injection of squeezed

vacuum into the unexcited port. Normally, zero point fluctuations enter this unexcited port.

We briefly consider the circumstances of utilizing the squeezed vacuum generated by the

interferometric fiber ring to improve the performance of a second fiber ring functioning as a

fiber gyro. In principle, all of the pump power reflected from the ring may be reused in the

gyro.

If the same power levels and fiber lengths that are used in the squeezer ring are also

used in the gyro, nonlinear effects in the gyro must be considered. We have explored this

issue in detail in a separate paper [11]. Here we shall merely point out that the nonlinearity

in the fiber gyro will cause additional squeezing but in an opposite direction. The

squeezing that occurs in the gym will undo some of the squeezing initially injected.

The analysis is shown diagrammatically in figure 8, using a Mach-Zehnder

configuration. If the gyro is linear (figure 8(a)), it is clear that the squeezed vacuum should

be oriented along the horizontal direction. In this way, at the gyro's output signal port, the

reduced quadrature is along the signal direction.

For a nonlinear gyro, the effect of the Z(3) nonlinearity will be to pull the elliptical

locus of fluctuations toward a circular shape, thereby destroying the squeezing. As shown

in the inset of figure 8(b), one must prepare the squeezed vacuum to counterbalance some

of the effect of the gyro nonlinearity. The gyro's signal to noise ratio improvement will

then be diminished, but not destroyed completely. In fact, much of the noise reduction

advantages may be regained by properly designing the relative nonlinearity of the gyro and

squeezer rings. For example, if we set the squeezer's nonlinear phase at _ the ideal noise

reduction will be approximately -15 dB. If the gyro's nonlinearity is then half of the

squeezer's, the noise improvement will reduce to -6 dB.

151
VII CONCLUSIONS

We describedthe successfulobservationof squeezedpulsedvacuum,5+0.3 dB

below theshotnoiselevel, generatedfrom a fiber ring interferometer. Noise generated

from GAWBS excitationsdid not appearto damagethe squeezingmeasurementsat low

frequencies.FurtherstudyshoulddeterminetheexactGAWBSspectraandmagnitudefor

ourexperimentalconfiguration. If thesqueezedvacuum,properlyoriented,is injectedinto

alosslesslinearinterferometer,theinterferometer'signalto noiseratiowill improvebythe

noise reduction factor. We consideredthe utilization of squeezedvacuum in the

improvementof the sensitivityof a shotnoiselimited fiber gyro. In this casethe gyro's

nonlinearitymustbereducedin proportionto thesqueezer'snonlinearity.

VIII ACKNOWLEDGEMENTS

TheauthorsthankLincoln Laboratoriesfor providingthebalancedreceivercrucial

to thisexperiment.Oneof theauthors[KB] gratefullyacknowledgessupportfrom Kevin

ChampagneandRobertDahlgrenof DraperLaboratories,andthegeneroushelpof Steve

Alexanderof Lincoln Laboratoriesrenderedthroughthecourseof this experiment. The

supportof DrapergrantDL-H-404179,of NSFEET 8700474,andJSEPcontractnumber

DAAL03-89-C-001aregratefullyacknowledged.

152
.

.

.

.

5.

6.

7.

8.

9.

10.

11.

REFERENCES

R. M. Shelby, M. D. Levenson, S. H. Pedmutter, R. G. De Voe, and D. F. Walls,

Phys. Rev. Lett., 57, 691, (1986).

R. M. Shelby, M. D. Levenson, and P. W. Bayer, Phys. Rev. B, 31, 5244,

(1985).

R. M. Shelby, P. D. Drummond, and S. J. Carter, Phys. Rev. A, 42, 2966,

(1990).

K. Bergman and H. A. Haus, Opt. Lett., 16, 663, (1991).

M. Shirasaki and H. A. Haus, J. Opt. S,c. Am. B, 7, 30, (1990).

C. M. Caves, Phys. Rev. D, 23, 1693, (1981).

R. S. Boudurant and H. J. Shapiro, Phys. Rev. D, 30, 2548, (1984).

M. Shirasaki and H. A. Haus, J. Opt. S,c. Am. B, 8, 681, (1991).

M. Xiao, L. Wu, and H. J. Kimble, Phys. Rev. Lett., 55, 2520, (1986).

P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta, Phys. Rev. Lett., 59,

2153, (1987).

H. A. Haus, K. Bergman, and Y. Lai, "Fiber Gyro with Squeezed Radiation,"

accepted for publication in J. Opt. S,c. Am. B, (1991).

153
drawn with the nonlinear

phase shift subtracted

Figure 1 The field's phase insensitive quantum fluctuations are elongated into an ellipse of

squeezed noise by the self phase modulation process.

154
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Ideal Local Oscillator

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0.0 0.5

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SHOT NOISE LEVEL

attenuated quadrature noise

I I I I I I

o.o os 1.o 1.5 zo zs 30

Pulse Peak Nonlinear Phase Shift (rad)

(b)

Figure 3

(a) Expected noise reduction for the ideal and single phase adjusted gaussian pulse

cases. Co) Predicted maximum and minimum noise amplitudes for the gaussian

local oscillator case, plotted on a dB scale.

156
i.._ ;hi o.m

4dm

_u_n

:SG) C

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-112 -

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Figure 5 Shot noise calibration using both white light and direct laser excitation.

158
,o0 

eo40-

20-

0

-130 -12S -120 -11S

PSD CdBm/Hz)

I

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(c)

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-110 -105

Figure 6 Histograms of PSD readings taken at three separate power levels (a) 60, (b) 85, and

(c) 110 mW in each ring direction. The black bars are counts taken with the

squeezed signal arm blocked. The white bars are the squeezing PSD

measurements.

159
N

-IE

era

"0

e'-

cO

D

£0

m

m

2-

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nonlinear phase (rad)

Figure Maximum and minimum values of the PSD readings obtained experimentally (black

dots) overlaid on the theoretically predicted curves for the case of a gaussian local

oscillator.

160
gyro

r

\

(a)

gyro )

locus of motion of

ellipse due to

nonlinearity in gyro

Co)

Figure 8 (a) The linear gyro with squeezed vacuum injected. (b) The nonlinear gyro with

phase compensated squeezed vacuum.

161
N92-22062

EQUILIBRIUM TEMPERATURE OF LASER COOLED ATOMS

IN SQUEEZED VACUUM

Y Shevy

Applied Physics, California Institute of Technology, 128-95

Pasadena CA-91125

ABSTRACT:

It is shown that by squeezing the vacuum fluctuations of the electromagnetic field the

quantum fluctuations of the optical forces exerted on laser cooled two-level atoms, can be

dramatically modified. Under certain conditions, this modification in concert with the enhanced

average forces can lead to equilibrium temperatures below those attained under normal vacuum

fluctuations.

INTRODUCTION:

Laser cooling of atoms in a quasi-resonant standing laser wave has been attracting

considerable attention during the past few years 1. Another exciting subject has been the

modification of the statistical properties of the vacuum fluctuations of the electromagnetic field.

Reduction of these fluctuations in one quadrature phase of the field by almost an order of

magnitude has been already realized in the laboratory. It is well accepted that the minimum

equilibrium temperature of laser cooled two level atoms is determined by the interaction with the

vacuum fluctuations of the electromagnetic field.This raises the question whether the equilibrium

temperature of two level atoms in squeezed vacuum can be lowered below the normal vacuum level

and in particular below the so called "Doppler limit" of KbT= h F/2 for two level atoms.

In the following, the physical origin of the optical forces in a standing laser wave is

described and an intuitive model of the effects in a squeezed vacuum is offered, the modified force

in squeezed vacuum is presented and compared to the force in a normal vacuum. In order to find

the equilibrium temperature the modification of the fluctuations of these forces in squeezed vacuum

is calculated. This calculation show, under certain conditions, a dramatic modification of these

fluctuations relative to the normal vacuum state, it is found that, in an intense standing wave, the

reduced fluctuations in concert with the enhanced average cooling force can lead to equilibrium

temperatures below those obtained under normal vacuum fluctuations. Moreover, under certain

ideal conditions even sub-Doppler temperatures may be reached. In the running wave case,

however, the temperature can not be lowered below the normal vacuum level. In addition to being

of potential use for laser cooling, these results offer an interesting glimpse into the quantum nature

of the momentum exchanges between the atoms and the field.

163

PRECEDING PAGE BLANK NOT FILMED
A slowly moving atom (kv<F) in a low intensity standing laser light wave experiences a

velocity dependent force. This "radiation pressure" force is well understood in terms of absorption

and spontaneous emission. As first envisioned by Hgnsch and Schawlow 2, the atom experiences

an increased absorption of photons from the laser beam which is shifted closer to resonance due to

the Doppler effect. This velocity dependent differential absorption can provide a cooling force for

laser detuning to the red side of the atomic transition or a heating force for blue detuning. At high

intensity, however, stimulated emission can change the sign of the force to a heating force at red

detuning and to a cooling force at the blue side of resonance 3-4.

M

v

Z

o

E-,

O

gl

gl

"al

o

p_

0.005"

0.004 t ""

0.003- _ /_

iiii

--5 0 5

PROBE D_G (1/11)

Figure 1: Probe absorption as a function of its detuning from a pump tuned 20F to the red of an atomic

transition.A)at low pump intensity the probe sees higher absorption at positive detuning closer to the atomic

transition B)In normal vacuum at high pump intensity the TWM process is induced leading to less absorption for

frequency shifts closer to the atomic transition. C) At the same high intensity as in trace B but in squeezed vacuum

the TWM process can change its lineshape leading to an additional cooling force.(Ref. 8 eq. 12 with N =0.1,

M=0.33, gO= rc and _=81-3.

This stimulated force(or "dipole force") has been explained within the framework of the

dressed atom model 5 and equivalently as resulting from Two Wave Mixing 6 (TWM). The TWM

resonance appears in pump-probe spectra as a dispersive lineshape (as a function of the probe's

detuning from the pump). This feature has a width of the excited state decay rate, F, and shows

decreased absorption at probe detuning from the pump closer to the atomic transition (see figure

lb). In this process the atom absorbs one photon from one wave and emits a photon into the

164
counter-propagatingwave,thusacquiringamomentumkick of 2hk. Thisprocessusuallyrequires

high laserintensity ; however,it hasbeenshownto occur at lower intensity when the normal

relationbetweenthedipoledecayrateF2 andtheexcitedstatedecayF ( 1"2=0.5F ) is modifiedby

theinclusionof phaseinterruptingevents( 1"2>0.5F ). This effectis dueto theappearanceof a

TWM term atlower orderin laserintensityproportionalto [F2/1-'-0.5]PwherePis the saturation

parameter.This phenomenonis closelyrelatedto thedephasinginducedextraresonancesin Four

WaveMixing. Theseresonances,which originally havebeenstudiedby Bloembergenand coworkers, are induced wheneverthe normal decay ratesof the the atom aremodified. Their

relevanceto thestimulatedforceis discussedin moredetailin reference6.

Armedwith this insightinto theconnectionbetweenTWM andthe stimulatedforce,it is

instructive to find the effect of squeezingon the TWM process.Gardiner7 has shown that in

general squeezingthe vacuum fluctuations results in two different decay rates for the two

quadraturesof the atomicdipole,oneof whichis largerandtheothersmallerthanthenormal1-'/2

valuein ordinaryvacuum.Hence,afterthedecayof thefastdecayingquadraturecomponentof the

atomicdipoleone is left with theslowly decayingcomponentwhichmeansthat F2 canbemuch

smallerthan0.5 F. Onecanthereforeimmediatelyseethatthe"extra resonant"TWM processcan

alsobeinducedin squeezedvacuum.Calculationof thelineshape8-9of this processshowsthat

indeedthe TWM lineshapebecomesphasedependent and canevenchangeto a "dispersive"

lineshapewith oppositesign(largerabsorptioncloserto theatomictransition)asdemonstratedin

figure lc. This indicatesthatin squeezedvacuumthestimulatedforcecanbeinducedatlow laser

intensity. Moreover,it canchangesignto provideanadditionalcoolingforce insteadof heating

for redlaserdetuningfrom resonance.

THE AVERAGE OPTICAL FORCES IN SOUEEZED VACUUM

The physical system under investigation is a slowly moving two level atom (k v<< F) in

either a standing or a running wave (a motionless atom is considered in the fluctuations analysis).

The atom is embedded in a broad band squeezed light, so that all of the modes coupled to the

atoms are squeezed. The bandwidth of the squeezing is broad enough to appear to the atom as a _5

correlated squeezed white noise. The correlation functions for the multi-mode squeezed field can

then be written as 7.

<bt(t)b(t')> = FNS( t- t' ) , <b(t)bt(t')> = F(N+I)5(t - t' ) Eq.1

<b(t)b(t')>=<bt(t)b]'(t')>* = FMe(-2ico0t + 2ik.r) 6( t- t' )

165
Whereb,bt aretheoperatorsdefinedin termsof thepositiveandthenegativefrequencypartsof

thisfield, N andM arethesqueezingparameters,N is proportionalto thenumberof photonsin the

squeezedvacuum,while M< N(N+1)signifiesthe amountof correlationbetweenthesidebands

andtheequalitymaximumsqueezing.In thefollowing wewill chooseM to berealandpositive.

The Hamiltonian describingthe interactionof the atom with the quantizedmultimode

radiation field and a classicalcoherentfield is given in the electric-dipoleand rotating-wave

approximationby 10. Eq.2

H = _-h _0_22 + Had - (blF.oe-imttjt + tJ bt*E;e+it°t) + h ((Itb+ bt_)

where m0 is the atomic resonance frequency, t_22 , t_=t_12 and t_t=_ 21 are the atomic operators,

is the atomic dipole moment, H0 is the free Hamiltonian of the field, and E 0 is the amplitude of the

coherent field. One can then find the master equation for an atom in squeezed vacuum and derive

equations of motion for the atomic operators whose expectation value is given by:

< t_'12 > = - T< t_12 > - FM< o12 >* + _')< D > eq.3

< I) > = - F(2N +1)< D >+ F - 2[f_*< (_12 >+_<t_12 >*]

The average force can now be found by calculating, the expectation value of the atomic variables

and subsequently the first order corrections due to the atomic motion 3. This gives rise to the

following expressions 11 for the expectation value of the optical forces acting on the atom in the

running<F> and standing wave<FS> :

(F) = 2(2N+l+P) 1+ ( k.v )

x(2N+I+P) Eq.4

cthAP

__(2N+I+P)

4zMcos( 2_)P + 1-_IO_(2N+I-P) - 2xP2

1-

FZO_(2N+I+P) v )]

where P is the modified saturation parameter in squeezed vacuum given by P= 21f212O_/X ,

f_=ei*btE/h and the other quantities are defined by: FI=(2N+I)F, __ = 2N+l - 2M cos(2_),

= I_ 2 - F2M 2 , 3,=F1/2 - tA, A = A + FMsin(2_), where A is the laser detuning from the atomic

resonance.

166
It is instructive to examinethe new expressionof the force in the standingwave by

comparingit to theforce in ordinaryvacuum.In thislimit (N =M=0) theforceis reducedto the

well knownexpressionof theforce(ref.3eq.18)givenby:

_-_= -" P [ F2(a-P)- 2_2PZ_. v] "0_A-i-+--ff1- F_P)_-_ eq.5

Notethatin thislimit thefirst term in thenumeratorof thevelocity dependentpartof eq.4is zero

while theothertwo termsarereducedto thoseof eq.5.Thestriking appearanceof the additional

term in squeezedvacuum is analogousto the result of ref.6. In this case,the introduction of

classicalphasenoiseresultsin theappearanceof anextraterm -41_2[1-'2/1-'- 0.5]P , (F2=F/2+F_

whereF_ is therateof thephaseinterruptingevents).Thisterm cangive the stimulatedforce at

lowerintensitywhen1-"2/ F>0.5 as phase noise is added.

Notice that in the case of F 2 / 1"< 0.5, this term can also be induced but with opposite

sign. This is indeed the case with quadrature squeezing, which can result in either larger or smaller

phase noise than the vacuum level.This in turn introduces two different decay rates for the two

quadratures of the atomic dipole.One of these, F 2 x = F(N+M+0.5), is larger and the other, F 2

x=F(N - M+0.5 ), is smaller than the normal F2=F/2 value. Therefore, the sign of the extra term in

eq.2 4[Iy12-F2M2]M cos(2_) P can be controlled by the relative phase ¢ of the driving field with

respect to the squeezed vacuum. Hence, the stimulated force can not only occur at lower laser

intensity, it can change sign to provide an additional cooling force at red laser detuning from

resonance.This modification of the force can be further correlated with the TWM lineshape which

becomes strongly dependent on the laser phase q_and can even change sign as indicated by our

intuitive analysis.

Other important modifications of the force in squeezed vacuum are described by the term,

A+FMsin2_. This term gives rise to a force at zero detuning as well as strong variations of the

force at small detuning (A<FMsin2¢). These effects can be understood by noting that the

dephasing induced lineshape of TWM at resonance is absorptive in normal vacuum, but it can be

transformed to a dispersive lineshape in squeezed vacuum, giving rise to a force at resonance. In

addition it has been shown that the TWM can have sub-natural linewidth at small detuning 7-10

This indicates that one can obtain arbitrarily large cooling forces at small detuning as the number of

photons in the squeezed vacuum N, and therefore the amount of squeezing, is increased. This

can be understood by noting that F2y=F ( N - M+0.5 ) in the limit of N>>I becomes arbitrarily

167
small, F2y= F/8N.

In the analytic solution shown above the force is calculated only to first order in velocity

(i.e a linear velocity dependence is assumed), this is correct only for small velocities kv<<F.

Numerical solution of the OBE, however, can provide the full velocity dependence of the force.

This solution is shown in figure 2 for ordinary (trace A)and squeezed vacuum (trace B). This

figure demonstrates that the stimulated force which gives a heating force in normal vacuum for

velocities in the order of kv<F/2 (as expected from the TWM lineshape, fig.lb), can be

transformed to a cooling force in squeezed vacuum. The dashed lines in the figure are the results of

the spatially averaged analytic solution which show good agreement with the numerical solution at

small velocities.

O_

(C_IS_)

Figure 2: The velocity dependence of the spatially averaged force, in normal vacuum trace A and squeezed vacuum

trace B, obtained by numerical solution of the OBE. The dashed lines are the result of the analytic solution. The

parameters used for this figure are: A=-3F, _=1. 36F, F=10 7 Hz and k=5890 A for both traces and N=I., M=_/2

and ¢ =0 for trace B.

Figure 3 demonstrate the interesting dependence of the force on the driving laser phase

(using the analytic solution eq.2 with kv=F/2 ).This is shown for a constant number of photons in

the squeezed vacuum, N=I, but for various values of the squeezing parameter M. Trace A plots the

force for thermal light M=0 (i.e no correlation between the sidebands) with no variation on the

phase, as expected. Traces B-D, however, show large variations of the force for increasing

degree of squeezing up to the maximum value of M, (M2=N[N+I]). This dependence is due to the

different amount of noise that the induced dipole sees at different quadrature phase. Figure 3 also

shows that even a modest amount of squeezing induce large effects on the force.

168
o.t7

o

0

0

D

I

I 2

_ _ ( _d./,_ )

Figure 3: The spatially averaged force as a function of the laser phase _ for increasing amount of correlation between

the sidebands A)M=0 (thermal light no correlation), B) M-0.5, C)M=I and D)M='_2 (maximum squeezing). Other

common parameters used are: A=-3F, D.=I. 5F and N=I.

THE QUANTUM FLUCTUATIONS

It was recognized by Einstein 12 as early as 1917 that the fluctuations of the optical force,

originating from both spontaneous and induced absorption and emission processes, are important

in determining the Maxwellian distribution of the atomic velocity in thermal equilibrium. A simple

momentum diffusion process which comes readily to mind is due to the random direction of the

spontaneous emission recoils. In addition to this geometrical source of fluctuations one should

also consider the fluctuations in the number of photons emitted in a unit time. However, this

process can have sub Poisson statistics, as shown by Mandel 13 in resonance fluorescence, and

give rise to an anomalous contribution 3,14 which can decrease the momentum spread, as

discussed by Cook 14. An additional momentum diffusion mechanism becomes dominant at high

intensities in a standing wave due to fluctuations in the stimulated emission process between the

counter-propagating waves 3,14,5. This photon exchange between the waves, results in a random

transfer of 2h k units of momentum to the atom. Finally as shown by Gordon and Ashkin 3 an

atom even in its ground state can have random recoils due to the zero point vacuum fluctuations.

In the following it will be shown that the dynamics in squeezed vacuum modifies the fluctuations

J

of all of these processes.

We are now interested in finding the force fluctuations on a stationary atom which are

given by the diffusion constant, 2Dp •

169
2Dp =2 Re I dt [<F(0) F(t)> - <_>2 ]

Eq.6

Insertion of F(t) = -toVG + H.c. for the force (where G is the freely propagating field), using

the correlation functions for squeezed vacuum,<G>=f2 (since<b>=0) and the commutation relation

oij Okl=Oit _(j,k) for the atomic operators gives •

_(0)F(t))=h2[(°t(0)°(t)+°(0)Gt(t))l(Vf2)21-(°t(0)cyt(t)_Vf2) 2 (cY(0)(y(t))( _f_*)z] Eq.7

+(hk)2[ FN ((Yll + 022)+ F((Y22)] 8(0

Consider first the last terms which depend directly on the quantum fluctuations of the field. These

terms in the limit of normal vacuum(N=0) can be transparently modeled 3 as the random

instantaneous emissions of momenta hk at an average rate of F<cY22>. In the squeezed vacuum

case, this effect is enhanced, by the absorption of squeezed photons FN<_ 11> and spontaneous

emission FN<o22> due to the larger number of photons in the squeezed vacuum.

In order to evaluate the remaining terms, which describe the effects of the interaction of the

coherent field gradient with the atomic dipole fluctuations,we need to find the integral of the atomic

dipole autocorrelation functions which after

the diffusion in squeezed vacuum:

D0+h2_ zFx P [1 + I_] + h2_rl 2Dp=

2 (2N+I+P)

some algebra leads to the following expression for

2"- (2N+ I+P) 1)2 +

Eq.8

Where 0t=0, [3=k in a running wave; and 0t=-ktan(k x) ,

other terms are defined by" O+ = 2N+l + 2M cos(2q_)

D°=h2k2 (2N+I+P) 2N+ + 2 I)1=

p O+

(2N+ 1,q') _ (2N+ 1)_-

[3=0 in a standing wave while the

and the standing wave terms;

170
4P [(2N+1)%

D[ = (2N+1_-_)20_O+ _ 1..2 (1 ) D2= (2N+1--+P-)20__+ rrlo_

FF1

4% p3

FF1___+(2N+I+P)2

In thelimit of normalvacuum(N=M=0) Eq.8 correspondsto thetheresultsofref.3 eq.30

2Dp=h2_2 1+ P 1- + hZ0_2 1 - 3 _ +"

(I+P) 2 I_ (1-_ -7 (1 +P)2 F2 (1 +P)2-

P

+ h2k2F--

2(l+P)

Let us first examine the terms in the running wave case by associating them with the normal

vacuum limit. As we discussed previously spontaneous emission in squeezed vacuum, represented

by the diffusion term D 0, gives rise to increased fluctuations as a consequence of the increased

number of photons in this state. However, as can be clearly seen, the D 1 induced absorption

contribution, even in the normal vacuum limit, can reduce the spread. This term has been shown

to originate from sub Poisson statistics of the emitted photons. In fact in the normal vacuum limit

D 1 coincides exactly with Mandel's Q parameter 13,14.

Q=((Any)- (n)

<n>

Q--0 indicates a variance of <I1> 1/2 in the number of the emitted photons(i.e no correlation between

the photons) and negative Q sub-Poisson statistics. Figure 4 plots D 1 as a function of the laser

detuning. In squeezed vacuum, with an appropriate phase between the laser and the squeezed

vacuum, D 1 can reach a value of - 1, whereas in normal vacuum the maximum effect gives D 1= -

3/4. This indicates that in squeezed vacuum the photons can be emitted in an orderly manner, thus

eliminating this source of fluctuations. Unfortunately, in the traveling wave case, one can not take

advantage of this phenomenon due to the increased spontaneous emission term D 0. This is

demonstrated in figure 5a, which shows that the equilibrium temperature, given by

KbT= Dp/(-bv<F>), increases with the amount of squeezing.

We now turn our attention to the more complicated case of the standing wave, as in the

171
running wavecase,we still haveenhancedfluctuationsdue to thelarger spontaneousemission

term,Do. However, the muchlarger average cooling force, in conjunctionwith the smaller

fluctuations,of thehigherorderterms,makeit possibleto reachtemperatureslowerthanotherwise

obtainedin normalvacuum.Unfortunately,onecannottakefull advantageof bothof theseeffects

W

I--

W

<

_C

<

n

o

0.2

0.0

-0.2

-0.4

-0.6

-0.8

-I .0 I

-3 -2

J

t

0 1 2

DETUNING (F)

Figure 4: The deviation from Poisson statistics as a function of the laser detuning from resonance.The dashed line

shows the maximum effect in normal vacuum, while the solid line indicates that almost no spread in the number of

photons emitted in a unit time in squeezed vacuum can be achieved. The parameters used are _ =0.35F in the normal

vacuum while f_=0.26F, N=2, M2=N(N+I) and _ = 0.5 n in the squeezed vacuum.

a_

v

v

t.l.l

w

13..

Z_

ILl

b10

I / I I

1 2 3 4

SQUEEZING PARAMETER (N)

Figure 5: The equilibrium temperature as a function of the squeezing parameter N in: a) running coherent laser

cooling wave with A=0.5F, f_=F, 0 = 0.25 ft. b) a standing wave with A=0, f_=5F, 0 = 0.6 rc .The dashed line is

the Doppler limit temperature in normal vacuum.

at the same laser phase. Nevertheless, one can choose a particular phase which will minimize the

temperature as the squeezing increases. Figure 5b demonstrates the lowering of the equilibrium

temperature below the normal vacuum level (N=0) as the amount of squeezing increases.

172
Moreover, the temperature can be decreased to values slightly below the Doppler limit (which in

normal vacuum is achieved at low intensity). Lowering the equilibrium temperature is not the only

benefit of squeezing, in the above example, the average force becomes larger than the maximum

value in the normal vacuum giving rise to a shorter equilibrium time. Notice, however, that high

degree of squeezing is needed in order to reach sub-Doppler temperature, in addition, degradation

from ideal squeezing (M2<N(N+I)) results in a temperatures higher than the Doppler limit. This

unfortunately makes the experimental demonstration of this effect rather difficult.

A few words are now in order to get some insight into the modification of the fluctuations

in the standing wave case. We first discuss the various terms in the normal vacuum case. The most

notable difference from the running wave is the appearance of higher order terms in P. These terms

were interpreted as resulting from the fluctuations of the dipole force 3,5, 14 and become important

at high laser intensity. In particular the p3 term is the only term that does not saturate at high P.

Hence, although one can use the stimulated force in normal vacuum at the blue side of resonance to

give a very large average cooling force(with the advantage of very short equilibrium time) the large

fluctuations make the equilibrium temperature much higher than the Doppler limit 5

With regard to the modification of these processes in squeeze vacuum, we begin by

comparing the D1 s term to its counterpart in normal vacuum. As discussed in the running wave

case, this term can be interpreted as the deviation of the fluctuations from Poisson statistics. We

found that the modified D1 s in squeezed vacuum can reach values close to -1. However, the

behavior in a standing wave is quite different from the running wave, as is the case in the normal

vacuum.

The next term, D2 s , is not present in a running wave and we assume that it describes the

fluctuations of the lower order stimulated force. In a previous publication 6, it was shown that

while the velocity dependent stimulated force does not usually occur at low intensity, the inclusion

of phase interrupting events (which increase the dipole decay rate), gives rise to a stimulated force

term at lower order in laser intensity. In order to show that this identification is correct and to get

some insight into this term De s was calculated in normal vacuum but with increased dipole decay

F2=F/2+F0, where 1-'0 is the rate of phase interrupting events, which gives:

I_- (l+p')2 2P'2 I_2-_ _1/(_ + 2-_1

As can be seen an additional diffusion occurs as phase noise is added, F 2 >1-'/2. Note, that

when F2<F/2, this term becomes negative. Analogously, De s in squeezed vacuum may be

associated with the fluctuations of the extra stimulated force in squeezed vacuum (the first term in

173
the averageforce in squeezedvacuum, <FS>. Moreover, the fact that one of the dipole's

quadraturecomponentscandecayataratesmallerthanF/2 suggeststhatD2sin squeezedvacuum

canbecomenegative,asindeedis thecase.

Finally weturnourattentionto thehighestordertermin P, D3s,which is associatedwith

thefluctuationsof the normal stimulated force.The modification of this diffusion term is critical for

achieving lower temperature at high laser intensity where the stimulated force becomes dominant.

Figure 6 shows the spatially averaged total diffusion constant 2Dp, in a high intensity standing

wave, as a function of the laser phase, in normal and squeezed vacuum.This comparison

demonstrates the dramatically reduced fluctuations in a high intensity standing wave under

squeezed vacuum conditions.

200

Q.

a 15o

o,i

iz

Z 100

o

o

Z

Q

U) 50

LL

LL

a

0

0.0

I I I I

0.2 0.4 o.6 o.8

LASER PHASE ( lq tad)

Figure 6: The diffusion constant in a high intensity standing wave as a function of the laser phase in the case of

normal vacuum (dashed line) and squeezed vacuum(solid line) .The parameters used are: A=0, f_=5F, N=5.

As to the experimental verification of these interesting phenomena. Although 90% squeezing has

already been achieved in the laboratory, it is important to note, that the calculation presented here is

carded out with the assumption that the atom is embedded in squeezed vacuum. In practice, the

output of present sources of squeezed light (degenerate parametric oscillators) can couple only to

part of the 4n steradians enveloping the atom. A possible solution to this problem has been

proposed by Gardiner who suggested coupling the squeezed modes to the atom in a micro cavity.

The other important assumption here is that the spectrum of the squeezing is much broader than

that of the atomic transition. Theories which include finite bandwidth of squeezing 15 have shown

that the essential features due to squeezing are preserved, even for a bandwidth of squeezing only a

few times larger than the width of the atomic transition. It should also be noted that it will be

interesting to develop the theory with bandwidth of squeezing larger than F but smaller than the

Mollow sidebands separation. This can introduce the possibility of controlling all three of the

174
decayratesof theatomandthereforemightreducetheproblem thediffusion dueto of thehigher

rateof spontaneousemissionin abroadbandsqueezedlight 16.

In conclusion, this paper demonstrates a dramatic modification of the quantum fluctuations

of the mechanical effects of light on atoms which are also embedded in squeezed vacuum. In the

running wave case the temperature can not be lowered below the normal vacuum level. However,

in conjunction with the modified average cooling force even sub-Doppler temperatures may be

reached, under ideal conditions, for atoms cooled in a standing wave. These interesting results, in

addition for being of potential use, offer some insight into the quantum statistics of the photon

exchanges between the atom and the field under squeezed vacuum conditions. Further investigation

of other schemes of cooling with squeezed light might also be beneficial

REFERENCES:

1) For recent advances in this field see JOSA B November (1989).

2) T. W. Hansch and A. L. Schawlow, Opt. Commun. 13, 68 (1975).

3) J. P. Gordon and A. Ashkin, Phys. Rev. A21 ,1606 (1980).

4) A. Aspect et al, Phys. Rev. Lett., 57, 1688 (1986).

5) J. Dalibard and C. Cohen-Tannoudji, JOSA B, 2, 1707, (1985)

6) Y. Shevy, Phys. Rev. A 41, 5229, (1990).

7) C. W. Gardiner, Phys. Rev. Lett., 56, 1917, (1986).

8) H. Ritsch and P. Zoller, Opt. Commun. 64, 523 (1987).

9) S. An and M. Sargent III, Phys. Rev. A39,3998 (1989).

10) H. J. Carmichael, et al, Phys. Rev. Lett., 58, 2539, (1987).

11) Y. Shevy, Phys. Rev. Lett., 64, 2905 (1990).

12) A. Einstein, Phys. Z.18, 121, (1917).

13) L. Mandel, Optics Lett., 4, 205, (1979).

14) R. J. Cook, Phys. Rev. A 22,1078, (1980); Optics Commun., 35,347, (1980), S. Stenholm,

Phys. Rev. A 27, 2513, (1983).

15) A. S. Parkins and C. W. Gardiner, Phys Rev. A37,3867 (1988); H. Ritsche and P.Zoller

Phys. Rev. Lett. 61,1098, (1988).

16)A. S. Parkins, Phys. Rev. A 42, 4352, (1990).

17) The magnitude of the force is different from Ref. 11 due to an error by a factor of 5.89

175
N92-22063

On the Measurement of Time for the Quantum Harmonic Oscillator _5__

Scott R. Shepard

Massachusetts Institute of Technology _f

Cambridge,MA

Abstract

A generalization of previous treatments

of quantum phase[q is presented. Restrictions on the class of realizable phase statistics are thereby removed, thus permitting

"phase wavefunction collapse" (and other

advantages). This is accomplished by exciting the auxilary mode of the measurement apparatus in a time-reversed fashion.

The mathematical properties of this auxilary mode are studied in the hope that

they will lead to an identification of a

physical apparatus which can realize the

quantum phase measurement.

1. The SG phase statistics

A satisfactory description of the phase

of the quantmn harmonic oscillator has

recently been achieved by considering

the realizable measurement[q of the nonHermitian Susskind-Glogower (SG) phase

operator [2]

ei'_ - (fi + 1) -1/2& (1)

Although it is not Hermitian, the SG

operator does correspond to a realizable

quantum measurement. It's measurement statistics, however, can not be calculated from the familiar Hermitian operator rules (e.g. moments calculated via

(,/,l(_)kl,/,),J: = 1,2,... do not correspond to the SG operator's realizable measurement statistics). We have demonstrated a variety of ways in which the measurement statistics of the SG operator can

be accessed [1],[3]. Perhaps the simplest of

these is to form the phase waveftmction

¢'(¢)= (¢1¢), (2)

from which the phase probability distribution, p(¢) = [¢,(¢)[2/2% and it's associated moments follow directly. This proceedure is justified formally by the fact that

the infinite energy eigenkets of the SG operator

Oo

I¢) -=_ _'"*1") (3)

11.----0

resolve the identity, i.e. 1 = f__, _,l_b)(¢l.

This permits the extremely useful phase

representation of an arbitrary quantum

state:

analogous to the familiar number representation of a state:

OO

I¢) = _(nl¢)ln). (5)

n=O

The number-ket expansion coefficients,

¢, _= (nl¢/, may be viewed as a wavefunction in discrete n-space. The Fourier

transform relationship of the number and

phase wavefunctions

Oo

¢(¢) = _ ¢.e-'"_ (6)

n----O

_r

(7)

demonstrates the complementarity of photon nmnber and quantum phase.

Position and momentum are familiar examples of complementary quantities, whose wavefunctlon representations,

¢(_) - (_1,/-')and _(p) -- (pie), are

also related via Fourier transform. Indeed, several relations among ¢, and ¢(¢)

are renfiniscient of those encountered in

Schrodinger's wave mechanics. Analogous

to the position representation of the momentum operator, 15 _ -ih_, for exampie, we have a phase representation of the

.d

nmnber operator, h _ t_, viz:

(,/'l(,_)"l,/,)= (8)

. _¢'(¢)(i )_¢(¢) (9)

177

PRECEDING PAGE BLANK NOT FILMED
(where k -- 0,1,2._). These relations yield

the correct form of the number/phase uncertainty principle[S]:

> 1⁄4(I --

(10)

Since we are dealing with a single harmonic oscillator (of frequency w), phase

is related to time (_ = wt) in a mod-2x

sense, and number is directly related to energy (h = f//_ - I/2). In this sense, the

above constitutes a rigorous energy/time

uncertainty principle for the quantum harmonic oscillator.

The class of realizable SG phase statistics, however, is restricted (by a PaleyWiener theorem) due to the fact that _b(_)

is a one-sided Fourier series, i.e. this restriction stems from the absence of "negativenumber states" (_b, = 0 Vn < 0). One

aspect of this restriction is that _,(_,) is

prohibited from identicaly vanishing over

a non-zero interval -- thus, delta-functions

in phase are not allowed. In as much as we

may desire a "wavefunction collapse" view

of a phase measurement, the SG statistics appear to be incomplete. This dilema,

however, can be resolved by generalizing

an alternate route (the product space forrealism)to the SG statistics.

Fundamental to the reMizable measurement of any non-Hermitian operator is

the exJstance of an auxi]ary noise source.

Zero-point fluctuations from this au.xilary mode prevent a simultaneous, perfectly precise, measurement of the nonconunuting real and imaginary parts of the

non-Herndtian operator (so that the uncertainty principle is not violated). We

can study the interaction of our original

system of interest (Hilbert space 7_,) with

this auxilary system (Hilbert space 7_o)

by working in the product space 7_ =

7_, ® 7_°. The ertension of the SG operator onto 7_ is [s]

f = (e_)o @P. + P. @ (_)t, (11)

where f_ _= [0)(01. This extension

has eigenkets (of non-zero eigenvalue),

fl_)' = ei_[_b)', given by

OD

Id,)'--IO).1o).+ (12)

lrt# P1

+ e-=""blO).ln,)=. (13)

rt.-----I

These reside on a subset, 7(', of T_ which is

defined by the property n, na = 0. When

the auxilary mode is in the vacuum state

,= 0), the _> measurement yields the statistics and their attendant PaleyWiener restriction.

2. Beyond the SG statistics

We can go beyond the SG statistics by

exciting the auJ.ilary mode to create an

arbitrary state on 7_':

= ¢.o,olno).Io). (14)

n0 =0

oo

+ eo,,,.IO).ln.)..

'lrt.,= 11

(15)

For simplicity, let N --- n. - n., _bN -----

tbN,o (VN > 0), and _lv = _'o,-.,v (VN <

0). The generalized phase wavefimction,

is a two-sided Fourier series. The the

Paley-Wiener restriction is removed and

_,'(_b) can "collapse" to a delta-function.

The fact that the class of tb'(_b) is more

general than (and includes) the class of

_b(_) should prove useful for various optimizations. Indeed, Shapiro[ 4] has pointed

out that error-free communication could

in principle be achieved by exploiting the

newly aquired generality described herein.

Provided that niether of our two modes

is purely in the vacuum state, the excitation which creates a state on 7_° is

not arbitrary in that the n,n, = 0 property creates an entanglement. Thus, in

general, the original system and aux.ilary

modes are not statistically independent

on H', i.e. I_b)' _: I_/'°)°[_/'.},,. Denoting the probabilitythat a measurement

of fi°yieldsthe outcome n by I%_l2, we

see that l_.'l_ = I_.,oI2 (Vn >_1),whereas

I_I 2 = I_o,o[_+_=_ [_o,.[_ (similarlyfor
[¢_]2). In spite of the lack of statistical

independence, we can therefore assign any

individual probability distributions for n,

and na that we wish, provided that

I¢o,ol= = (1¢_,1_+ I¢?,1_- 1)> o (17)

is satisfied.

The auxilary mode can be interpreted

as a time-reversed mode in the following

sense. Consider the case of the auxilary

mode being in the vacuum state (ha = 0).

Denote the initial state by I¢o)'. The

state (in the Schrodinger picture) after

time evolution of an amount r is

I¢,)' = e-_"'_"l¢o)', (18)

so that the relation of the phase representations of the initial and delayed states is

simply

¢:(¢) = ¢'0(¢+ _) (-o = 0). (19)

Now consider the case of the original system being in the vacuum state (n, = 0).

The Schrodinger picture of the delayed

version of an initial state [¢0)' is

I¢,)' = e-_'_'"l¢o)'. (20)

The initial and delayed phase representations for this case are related by

¢'_(¢)=¢'0(¢-_) (n.=0). (21)

Thus the two modes are time-reversed in

that, under time evolution, the no _> 1

portion of the generalized phase wavefunction "moves backwards" with respect to

the n, _ 1 portion.

Consistent with the time-reversal property, the auxilary mode serves the topological role of a "negative energy" mode in

Hilbert space. The SG operator is a pure

lowering operator which stops at the vacU UIll:

ei¢In) = ]n-1) (n > l) (22)

e_10) = o. (23)

It cannot lower below the vacuum since we

have not allowed negative number (negative energy) states for the quantum harmonic oscillator. It's extension, Y, however, lowers the original system mode

number

f'ln,),lO)_ = In,-1),]O)_ (n. > 1), (24)

then continues through the vacumn

_1o),1o)_= I0).11)_, (25)

and raises the auxilary mode number

i>10).Ina)a = 10).In=+ 1)a. (26)

Topologicaly, it is as if Y continues to lower

below the vacuum into the auxilary ("negative energy") mode. The visualization of

this behavioral aspect can be facilitated

by simply relabeling the 7"/' number states

according to the value of N _ no - n_.

The auxilary mode has to be an irrevocable part of the physical apparatus which

realizes the quantum phase measurement

(so that the uncertainty principle is satisfied and so that the phase wavefunction

can collapse). All of the aforementioned

mathematical properties must be physicaly realized in the measurement apparatus. These restrictions should prove useful

in determining an apparatus which willrealizethe quantum phase measurement.

4

3. REFERENCES

S. R. Shepard and J. H. Shapiro,

OSA'88 Tech. Digest (Opt. Soc. Am.,

Washington D.C.,1988); see also: J.

H. Shapiro, S. R. Shepard, and N.

C. Wong, Phys. Rev. Lett. 62,2377

(1989).

L. Susskind and J. Glogower, Physics

1, 49 (1964).

J. H. Shapiro, S. R. Shepard, and N. C.

Wong, in Coherence and Quantum Optics VI, L. Mandel, E. Wolf, and J. H.

Eberly, ed.s (Plenum, NY, 1990}; see

also: J. H. Shapiro and S. R. Shepard,

Phys. Rev. A 43, (1991).

J. H. Shapiro, "Going Through a

Quantum Phase", Squeezed States

and Uncertainty Relations Workshop,

NASA CP- , 1991. (Paper of this compilation).

179
N92-22064

THE ORIGIN OF NON-CLASSICAL EFFECTS IN A 2gl(,2

ONE-DIMENSIONAL SUPERPOSITION OF COHERENT

STATES

V. Bu_ek, P. L. Knight and A. Vidiella Barranco

Optics section, The Blackett Laboratory

Imperial College, London SW7 2BZ, England

Abstract

We investigate the nature of the quantum fluctuations

in a light field created by the superposition of coherent

fields. We give a physical explanation (in terms of

Wigner functions and phase-space interference) why

the one-dimensional superposition of coherent states in

the direction of the x-quadrature leads to the squeezing

of fluctuations in the y-direction, and show that such

a superposition can generate the squeezed vacuum and

squeezed coherent states.

1 Introduction

The coherent states are always associated with the

"most" classical states one can imagine in the framework of quantum theory [1]. In the present Lecture

we will study the quantum interference between coherent states and how such interference leads to generation of states whose properties are as far as one

can imagine from "classical" states. In particular a

one-dimensional superposition of coherent states can

exhibit sub-Poissonian photon statistics or squeezing.

In our Lecture we will concentrate mainly on squeezing

which appears as a consequence of quantum interference between coherent states.

Light squeezing (for recent reviews see [2] as well as

topical issues of JOSA B [3] and J. Mod. Opt. [4])

remains a central topic in quantum optics. Generation

of squeezed light has been reported by various groups

[5-11] and offers new opportunities for the utilization of

light with reduced quadrature noise in interferometry,

fiber optics communications and high-precision experiments. Most studies of squeezed states have concentrated on those states generated by quadratic field interaction (e.g. parametric amplification). Recently it

has been shown by W6dkiewicz and coworkers [12] that

a superposition of two number states (for instance, the

vacuum state and the one- or two-photon states) of

a single mode electromagnetic field exhibits interesting non-classical properties. In particular, squeezing

of the variances of the quadrature operators can be

seen (although not necessarily of the quadratic, minimum uncertainty state quality). A superposition of a

181

PRECED;NG PAGE BLADIK NOT F_LMED
finite number of coherent states has also been studied [13-17]. In particular, Hillery [13] has studied

the superposition of two coherent states la) + [ - c_)

(the so-called "even coherent state" [14]) in connection

with amplitude-squared squeezing. Yurke and Stoler

[15] have shown that such a superposition of coherent

states can arise as a consequence of propagation of coherent light through an amplitude-dispersive medium.

It has been shown that the even coherent states exhibit

ordinary (second order) squeezing as well as fourth order squeezing [16]. In a recent paper, Janszky and

Vinogradov [17] extended the idea of superposition of

coherent states and investigated the quadrature variances of a continuous one-dimensional superposition

of coherent states. They have shown that such a superposition of coherent states can lead to significant

reduction of fluctuations in one of the quadratures.

2 Simple example

We start our Lecture with a simple example considering a superposition of two coherent states [al) and

1_2)

[_) = A 1/2 {[al) + [as)}, (1)

where A is a normalization constant

A -1 = 2(1 + Re(alia2)).

The coherent state [a) can be obtained by shifting the

vacuum state 10) by the displacement operator D(a) =

exp(aht - a'h):

I-) = b(.)10),

where fit (h) is the creation (annihilation) operator of

a photon.

At first sight, the result of Janszky and Vinogradov

seems quite remarkable, wimn reinterpreted in terms

of interference in phase space: a superposition of coherent states in the direction of the x-quadrature leads

to a suppression in the fluctuations in the y-direction,

whereas naively one would expect that the quantum

interference relevant to this superposition would modify the fluctuations in the original x-direction.

In the present Lecture we give a physical explanation

(in terms of the Wigner function and a phase-space forrealism [18-20]) of the origin of this noise suppression

and squeezing for a one-dimensional superposition of

coherent states. We further demonstrate that a suitable Gaussian superposition of coherent states not only

can be squeezed, but is actually a representation of the

minimum-uncertainty squeezed vacuum state.

The density matrix corresponding to the superposition of coherent states (1) is given by the following

expression

= A(l°q)(_ll + 1_2)(_21+ 1_1)(_21+ 1_2)(_,1), (2)

while the density matrix describing statistical mixture

of two coherent states lal) and la2) is

#M = PXI_l)(_a I + P21_)(a21 (3)

where pi is the probability to find the system in the

state lai). These probabilities are normalized to 1.

2.1 Wigner functions

Now we introduce the notion of the Wigner function

through the characteristic function C(w)(_), which is

182
associated with the symmetrical order of the bosonic

(photon) operators and is given by the relation [21]

c(w)(_) = Tr [/iexp(i_h t + i_*h)]. (4)

The Wigner function is defined as the Fourier transform of the characteristic function C(w)(_):

W(/3) 7r-2 / d2_ exp[-i(_fl* + _*fl)]c(W)(,_). (5)

d

The Wigner function corresponding to the superposition of two coherent states (1) can be written in the

form

where

and

W(fl) = A (Wl + W2 + W12) (6.a)

Wi = 2 exp(_21a i _ DI2); (6.b)

71"

× {exp [a2a_ - 2(fl - o_2)(_* - aT) ]

+ exp - - 1)O* -

(6.e)

The terms Wi are the Wigner functions corresponding to the coherent states lal), while the term W12

arises due to the quantum interference between coherent states under consideration.

The Wigner function for the statistical mixture (3)

is given by the relation

WM = pl Wl -t- p2W2 (7)

and it does not contain the term describing the quantum interfence between coherent states.

2.2 Even coherent states

To simplify our task we will suppose that al = -a2 =

a, where a is a real parameter. In this case we obtain

- _-_.,._ _ _ i¢1_ -

Figure 1: Wigner function corresponding to the even

coherent state (8) with a = 2. The r61e of the interference term is transparent.

from (1) the following state

I$) = A1/_ {I a) + I- a)}, (s)

with the normalization constant

A -1 = 211 + exp(-2a2)].

The state (8) is called [13,14] the even coherent state.

The Wigner function corresponding to this state can be

found using the general expression (6) and is presented

in Figure 1, where x = Re_ and y = Imfl. If we

compare this function with the Wigner function (see

Figure 2) corresponding to the statistical mixture of

states la) and I- a) described by the density matrix

1

PM = _ (l_)(al + I-- a)(--'_l) (9)

we can directly observe that the term W12 corresponding to the quantum interference between states [a) and

I-a) should play an important rOle in statistical properties of superpositions of coherent states.

183
¢i

0.40

_1.._0-35

(,D

_0.30

r'(

<: 0.25

>

W

0.20

I--

<_ 0.15

t-Y

<o.10

0

0.05

/

/

!

/

!

/

/

o.o_,_,, ....

O •o_ ................................... 0.50 1.00 1.50 '_:fb L'_'......_/_o

ALPHA

Figure 2: Wigner function corresponding to the statistical mixture (9) with a = 2 and pl = p2 = 1/2

2.3 Quadrature squeezing

The quadrature operators fl and h2 corresponding to

the creation and annihilation operators fit and h are

defined as:

f + fit f - at

fl- 2 ; f2=_. (10)

We can easily find that variances of these operators

((Ahi) 2) = (f_)- (hi) 2, (11)

in the statisticalmixture (9)are:

1

((Afl)2)M __4 + _2,

and

1

((Aa2)_)u = _.

From above it follows that the fluctuations in the dl

quadrature are larger in the case of the statistical mixture compared to the vacuum-state (or the coherentstate) value, which is equal to 1/4. Fluctuations in f2

remain the same for both the statistical mixture (9) as

well as for the coherent state Io_) (or l- a)).

Figure 3: Quadrature variances given by equations

(12) versus parameter a. The clashed line corresponds

to ((Aft1)2) and the solid line corresponds to ((Ah2)2).

On the other hand, for the even coherent state (8)

we find the reduction of fluctuations in f2 quadrature

(i.e. in the y-direction in the phase space - see Figure

1):

1 _2 exl)(_2a2)

((Af2)2) = 4 -- 1 + exp(--2(_2)" (12.a)

Simultaneously fluctuations in fl are enhanced:

1 a 2

((Aa')2) = _ + 1 + exp(-2_)" (12.b)

Variances ((Aa_) =) versus the parameter a (which is

related to the intensity of the even coherent state) are

plotted in Figure 3. We see that the maximum reduction in the fluctuations can be obtained for quite small

values of a. Reduction of fluctions below the vacuumstate (or coherent-state) level is called quadrature

squeezing. From the above it follows that quadrature

squeezing can emerge as a consequence of the quantum

interference between coherent states. We should note

that even coherent states (8) exhibit not only quadrature squeezing, but also higher-order squeezing as well

as amplitude squared squeezing [13,16].

184
0.40 q

-I

0.35

z

0

0.30

123

p_ 0.25

t--

U3

_,j 0.20

0.i5

_DO.IO

0.05

0.00

I

II \ \

k

0 2 ..5 4 5 6 7 8 9 10 11 12 15 14 15 16

Iq

Figure 4: Photon number distribution for a superposition of two coherent states (8) with amplitude (_ = 2.

The dashed line is the distribution of the corresponding statistical mixture (9).

2.4 Photon number distribution

distributions corresponding to two independent coherent states I(_1) and [a2). On the other hand, in the case

of a superposition of coherent states lal) and la2) the

term corresponding to the quantum interference plays

an important r61e. To see this clearly we wil_ assume

that _1 = -as = _. In this case the statistical. ,lixture

(9) has just the Poissonian photon number distribution

pM_ l'_123e-I,_t _ (16)

-- "-_I

The superposition of coherent states under consideration (i.e. the even coherent state) has the following

photon number distribution:

2exp(-,_}2)_ ifu=2m

p, = X+exp(- ,_ _) ,. (17)

0 if n=2m+l,

The oscillations in P, are very similar to those in the

case of the squeezed vacuum discussed by Schleich and

Wheeler [18]. Generally, these oscillations are due to

quantum interference in the phase space.

Here we discuss briefly properties of the photon number distribution of the statistical mixture (3) as well

as the superposition (2) of two coherent states. The

photon number distribution is defined as

Pn = (.lAin). (13)

and can be evaluated easily for both the statistical

mixture (3)

P,M = n--1_ {p, la,12.e_,_,, P +P21a21="e-'"='* } (14)

and for the superposition of coherent states (2):

A

P" = _.1 {1'_1='e-I''P + I_=1='e-t"=P

+ ,,, 1 = t]exp[-5(l' tl + (is)

2

In the case of the statistical mixture the photon number distribution (14) is just the sum of two Poissonian

3 Continuous superposition of

CS

Now we will discuss the properties of one-dimensional

continuous superposition of coherent states. Recently

Janszky and Vinogradov [17] defined the continuous

superposition I'_) of coherent states Ia) in the following

way

I_> - CF F(_,,f.)lc,)do_, (18)

--(DO

where the coherent state amplitude a is supposed to

be real. The normalization constant CF is defined as:

CF 2 = /F F(a, _)F(a', _) exp[-(a-a')2/2]dadc/.

o0

(19)

185
Withthesuperposition state 1_} given by equation (18)

one can find expressions for the mean values of products of the creation tit and the annihilation h operators

of the field mode in the following form:

= /F

× exp[-( - '. (2o)

In particular, if F(a,_) is taken to be the Gaussian

function

F(a,_) = exp [ (1_5) a2] (21)

with _ E (0, 1) and

C/1 = _r_1/2(1 _ _)1/4 ' (22)

then one can find for the variances of the quadrature

operators al, ti2 given by eq. (10) the following expressions

(_) (1+_). (23.a)

= (1)(1-0 (2a.b)

From this one can conclude that the states [{): i) belong to the class of the minimum uncertainty states;

ii) the fluctuations in the "second" quadrature are reduced below the shot noise limit.

on the vacuum state 10) of ttle field mode. To do so

we decompose the coherent state [c_) in equation (18)

into number states:

a n

I_} = exp(-a2/2) k ---_/_ In},

n=0

and exchange the order of integration and summation

procedures, i.e. we rewrite equation (18) in the form

oo

I_} = CF Z 1---_l,ln)

n=0 V(n_)

x d_a nexp [iq- a . (25)

After performing the integration on the r.h.s of equation (25) we find

[(2n)!lllZ,f,"12n), (26) [e} = ¢ X K'kl_e2]I/4Z. _

2nn[

n=O

from which it follows that the one-dimensional superposition of coherent states (18) with the distribution

function (21) is identicalto the squeezed vacuum state:

I_) = CF F F(a,5)dab(a)10) = S({)I0). (27)

We should stress here that the last equation describes

the relation between the states, but not between the

displacement and the squeeze operators themselves.

We see, therefore, that the one-dimensional superposition (with Gaussian distribution) of coherent states

leads to states exhibiting a large degree of squeezing

(in the limit _ _ 1). We now demonstrate by direct

calculations that if F(a,_) is the Gaussian function

(21), then the state [_} is equal to the squeezed vacuum

state generated by the action of the squeeze operator

s(at,a,O:

oS(ti, h,_) = exp [i(ht) u r^=] , --_a ,

= tanh r, (24)

3.1 Origin of squeezing

We next provide a physical explanation of the origin

of the squeezing generated by such a superposition.

We address the question of how the one-dimensional

superposition of coherent states in direction of the xquadrature (corresponding to tim operator til) leads to

squeezing of the fluctuations in y-direction (associated

with the quadrature operator /t2). To do so we use

the Wigner-function phase-space formalism: we define

the Wigner function W(fl) in the following way. First,

186
weintroducea

_(w)(4, 4', O:

_'(w)(4, 4', ¢) _=(4'1D(_)14)

= exp [1- I_12+ i_4' + i_*. - _(4 -- 4') 2],

and the "generalized" Wigner function I)V(4, 4',/_):

I_(4, 4', fl) : r -2 / d2(

× exp[-i((*fl + ¢/3")]c'(w)(4, 4', ¢) (29)

= -- exp _* - _ - 4' + .)_ ]

- 5(4 - .

The Wigner function W(B) can now be expressed in a

very simple form:

w(_) ; c_ /F a4e4'F(4)F(4')_(4, 4', _).

(30)

"generalized" characteristic function

(28)

To make our discussion more transparent we will

first analyze in detail the simple superposition of two

coherent states 14) and [- 4) and the vacuum state

10), i.e. we will study the state [17]

I_} = CF {14) +plO) + I -- a)}, (31)

which can be obtained from equation (18) with a

weight function F(x) = 6(_ - 4) + p6(x) + di(x + 4).

The normalization constant in this case is given by the

relation:

C_ _ = 2 + p_ + 4pexp(-,_/2) + 2exp(-242). (32)

One can easily find the variance of the quadrature operator a2 for the state (31):

1

((a_) _) = _ {1 - 4c_4 _

x[2 exp(-242) + vexp(--_)]}, (33)

from which it follows that a high degree of squeezing

(up to 74%) can be obtained for the optimum case,

a = 1.57 and p = 1.35.

The Wigner function of the state (31) can be expressed as the sum of two terms:

where

W(fl) =Wcl(fl) +Wquant(fl), (34)

We1(/3) = 2C_, {exp[-2(x - 4) _ - 2y 21

7f

+ exp[-2(_ + .)_ - 2y2]+ p2exp[-2_ 2 - 2y_]}

(35)

and

W, uam(13) = 2C_ {2 cos(4c_y) exp[-2x _ - 2y _1

+p cos(24y)[exp (-2(_ - 4/2) _ - 2y_)

+ exp (-2(x + ,/2) 2 - 2y_)] } . (36)

The normalization constant CF in this case is given

by equation (32) and x = Re_; y = Im/_. The function

W,l(]3) is equal (up to normalization factors) to the

sum of the independent Wigner functions of the vacuum state and two coherent states and can be identified with the Wigner function of the statistical mixture

of coherent states and the vacuum state described by

the density matrix

PM = p_lO)(Ot + p21o,)(41+ pal - 4)(-41 (37)

with properly chosen parameters pi.

This function is plotted in Figure 5a, from which it

is obvious that Wel(_) is positive for any value of z

and y. The phase-space contour lines of this function

are plotted in Figure 5b. In contrast to the function

187
"'%o "_

o

__4o o

_o. o° .

_.oO _

toooo

-aO0 -2.00 0.00 200

400 lilllllllJlll

b

200

0.00

--4.00 llll'lllllllllll

-4.00

-2.00

4.00

r I 4.00

2.00

0.00

-2.00

-4.00

-2.00 0.00 2.00 4.00

Figure 5: Function Wct(x,y) given by Equation (35)

representing the part of the Wigner function of the

superposition state (31) is plotted for a = 1.57 and

p - 1.35 (a). In Figure 5b the phase-space contours

corresponding to this function are plotted.

Wcz(/_), the function Wq,,,,,,t(fl) can be negative. This

function describes in phase-space the quantum interference effects between the states la), I- or) and ]0).

The quantum interference is responsible the appearance of the cosine terms in the y-direction, and these

oscillating terms are responsible for: 1) negative values of the function Weuant(fl ) (see Figure 6a) as well

as the total Wigner function W(/_) (Figure 7a); 2)

squeezing of the variance of the quadrature operator

in the y-direction, which is clearly seen in Figures 6b

and 7b.

This simple example helps us to understand the nature of squeezing in the one-dimensional superposition

of coherent states. The squeezing arises as a consequence of quantum interference between the macroscopically distinguishable states. Generally, if more

states are involved in the superposition, a higher degree of squeezing (depending on the appropriate shape

of the distribution F(a)) can be obtained for the same

mean value of photons in the mode.

Now we turn our attention to the displacement of

the superposition state such that there is a mean field

amplitude. We show that a one-dimensional superposition of coherent states with the distribution function

F((_, _,/_) centered at a non-zero value of a is equal to

the squeezed coherent state. We take for our distribution function F(a, ¢,/_) the displaced form

F(a,_,_)=exp[ (l_)(a-zo)_], (38)

with the normalization constant CF given by equation

(22) and with a displacement

x0

In this case from equation (18) we obtain for the state

188
".00 _ "_._0 0

.Y _.% _.000°

-4.00 -2.00 000 2.00 4.00

4.00 I I I I [ I I 1 1 I l I 1 l l 4.00

b

2.00 2.00

0.00

-2.00

-4.00

4.00

0.00

-2.00

--4.00 [ I _ I 1 I I I I t I I 1 I I 1

-4.00 -2.00 0.00 2.00

a

- Hill

.0

11%

_4'_ O0

_0. O0

,0 o $

-4.00 -200 OCS

40C f f i

b

20S

O.OC

-2 SS

200 4 00

, r ] , r 4.00

200

0-00

--200

I I , I 1 --4.00

2.00 400

-400 , J I I J i I I

-400 -2.00 0.00

Figure 6: Function Wq_,a,_(z, y) (36) representing the

interference between states in phase space is plotted

for a = 1.57 and p = 1.35 (a). In Figure 6b the phasespace contours corresponding to this function are plotted.

Figure 7: The total Wigner function W(x, y) is plotted

for a = 1.57 and p = 1.35 (a). In Figure 7b the phasespace contours corresponding to this function are plotted.

189
16) the following expression:

,,>, [

x E (X_'l)[(1-6)Zo]n

rt=O

IIn/zll 1 /''2(1 )'}In), (39)

x = (n-2k)!k! -'_)2x0_

where I1_11denotes the greatest integer less than or

equal to z. Using the new parametrization:

1 -6

/_- (1-62)1/2 ; v= (1-62)'/2, (40)

with/_2_v2 = 1 and > Ivl,we can rewrite equation

(39) in the form:

[6)=/_l/2exp [ (1--_v/.)fu ]

oo 1 2

_ \20] H"(#/ 2V/_)ln)' (41)

n.._O

where H.(z) is the Hermite polynomial. It is obvious

that the last expression obtained describes precisely

the squeezed coherent state as defined by Yuen [22],

i.e., we have explicitly proved that

•9(6)D(_)Io) = Cr daF(a,6,#)b(a)lO). (42)

o_

In other words, we can construct, through a onedimensional superposition of coherent states with a

properly chosen distribution function, the squeezed coherent state. Obviously, the physical reason for squeezing is the same as for the case of squeezed vacuum

state discussed earlier. It is amusing that a superposition of the most classical of field states, the coherent

states, can through the action of quantum interference,

generate the archetypal nonclassical field states - the

squeezed vacuum and the squeezed coherent state.

It can also be shown that the squeezed number state

[23] defined as a result of a action of the squeezing operator S(6) on the number state In) can be constructed

as a one-dimensional superposition of displaced number states [24], the states obtained through the action

of the displacement operator on the number state, that

is

S(6)ln) = CF F(a,QdaD(a)ln), (43)

--OO

where function F(a, 6) and the normalization constant

CF are given by the Equations (21) and (22), respectively.

4 Discussion

In our Lecture we discussed the r61e of the quantum interference in the origin of squeezing in the onedimensional superposition of coherent states. With

the aim to make the discussion as clear as possible

we started our Lecture with a simple example of superposition of just two coherent states Is) and l- a).

Finishing the Lecture we return to this simple example

but we take into account the relative phase between the

coherent states under consideration, i.e. we will study

the following superposition

1_) = A 1/2 {Ic_) -t- eiS[- a)}, (44)

with the normalization constant

A-1 = 2 (1 + cos ¢e-2_2).

We will show that the phase ¢ plays a crucial r61e

in the character of the quantum interference between

coherent states.

First of all we write down the corresponding Wigner

function for the state (44). This function can be expressed as a sum of two terms:

w(z)=wct(Z)+wq.o.,(Z), (45)

190
J

Figure 8: Wigner function corresponding to the superposition (44) of two coherent states with the relative

phase ¢ equal to 0 (a); a" (b) and 7r/2 (c); o_ = 2.

where

wc_(_) = -7- 4A {e_2(.__)2__y2

_be-2(x+a)_-2Y_ }

(46)

and

w,u..,(Z)= 4Acos(4 y+ ".

7r

As earlier we use the notation x = Rej3; y - lmiL

To investigate the dependence of the quantum interference on the value of the parameter ¢ we will employ two parameters describing nonclassical properties

of light fields. Namely, we will study the Mandel Q

parameter, defined as

Q = ((Z_h)_) - (h) (48)

(a)

which is related to the degree of sub-Poissonian photon statistics. In particular, if Q = 0 the state has

Poissonian photon statistics, while for Q < 0 (Q > 0)

the state has sub-Poissonian (super-Poissonian) photon statistics. The second parameter we will study is

the squeezing parameter

s, = 4((_a_) 2) - 1 (49)

describing the degree of quadrature squeezing. A state

is said to be squeezed if $1 or $2 is less than zero. In

what follows we will suppose three values of _b.

1) Let the phase ¢ be equal to zero. In this case the

state (44) is equal to the even coherent state (8) and

we find

4a _ exp(-2a 2)

Q-- 1- exp(-4a 2) > 0; (50)

4o_ 2

S_ = > 0; (51)

1 + exp(-2a 2)

4a 2 exp(-2a 2)

$2 = 1 +exp(-2a 2) < 0, (52)

191
from which it follows that the even coherent state has

super-Poissonian photon statistics and simultaneously

is squeezed in the h2 quadrature.

2) If ¢ = 7r then the state (44) is an odd coherent state [14]. This state has sub-Poissonian photon

statistics, i.e.

452 exp(-2a 2)

Q = 1 - exp(-4a 2) < 0; (53)

but is not squeezed

4_ 2

'S'I = 1 - exp(-2a u) > O; (54)

45 _exp(-2a _)

$2 = 1-exp(-2a _) > 0. (55)

3) Finally, if ¢ = 0, then the state (44) has Poissonian photon statistics

5 Conclusion

The main information carried in this Lecture is: A superposition of the most classical of field states

can through the action of quantum interference, generate the archetypal nonclassical field

states: the squeezed vacuum and the squeezed

coherent state.

Acknowledgements

This work was supported in part by the U.K. Science

and Engineering Research Council. One of us (VB)

would like to thank professor Y.S.Kim for hospitality

at the University of Maryland.

Q = o, (56)

and simultaneously we can observe squeezing in the h_

quadrature

S, = 4_; (57)

$2 = -452 exp(-4a _) < O. (58)

The dependence of the statistical properties of superpositions of coherent states on the value of the relative phase is caused by the character of the quantum interference, that is whether this interference is

constructive or destructive in various regions of phase

space. This can be clearly seen from Figure 8 in which

the Wigner function corresponding to the state (44) is

plotted for ¢ = 0; lr and r/2. We see significant differences in the shape of Wign_r functions for various

values of ¢, which is related to the completely different

statistical properties of the corresponding states.

6 REFERENCES

[1] R.LOUDON: "The Quantum Theory of Lighf'

(Clarendon, Oxford, 1983).

[2] R.LOUDON and P.L.KNIGHT: J. Mod. Opt.

34, (1987) 709; K.ZAHEER and M.S.ZuBAIRY:

in "Advances in Atomic, Molecular and Optical Physics", vol. 28. Edited by D.Bates and

B.Bederson (Academic Press, New York, 1990),

p.143.

[3] Special issue of JOSA B 4 No.10 (1987).

[4] Special issue of J. Mod. Opt. 34 No. 6/7 (1987).

[5] R.E.SLUSHER, L.W.HOLLBERG, B.YURKE, J.C.

MERTZ and J.F.VALLEY: Phys. Rev. Left. 55,

(1985) 2409.

192
[6] R.M.SHELBY, M.D.LEVENSON, S.H. PERLMUTTER, R.G. DEVOE, and D.F. WALLS: Phys.

Rev. Left. 57, (1986) 691.

[7] LING-AN WU, H.J.KIMBLE, J.L.HALL and

H.Wu: Phys. Rev. Lett. 57, (1986)2520.

[8] M.W.MAEDA, P.KUMAR and J.H.SHAFIRO:

Opt. Lett. 3, (1986) 161.

[9] S.MACHIDA, Y.YAMAMOTO and Y.ITAYA: Phys.

Rev. Lett. 58, (1987) 1000.

[10] A.HEIDMANN, R.HoRowlcz, S.REYNAUD, E.

GIACOBINO, C.FABRE and G.GAMY: Phys. Rev.

Left. 59, (1987) 2555.

[11] R.MOSCHOVICH, B.YURKE, P.G.KAMINSKY,

A.D.SMITH, A.H.SILVER, R.W.SIMON and

M.V. SCrINEIDER: Phys. Rev. Left. 65, (1990)

1419.

[12] K.WdDKIEWICZ, P.L.KNIGHT, S.J.BUCKLE

and S.M.BARNETT: Phys. Rev. A 35, (1987)

2567.

[13] M.HILLERY: Phys. Rev. A 36, (1987) 3796;

see also K.VOGEL and H.RISKEN: Phys. Rev.

A 40, (1990) 2847; W.SCHLEICH, M.PERNmO

and FAM LE KIEN: "A nonclassical state from

two pseudo-classical states." to appear in Phys.

Rev. A (1991); W.SCHLEICH, J.P.DOWLIN6,

R.J.HoRowIcz and S.VARRO: in "New Frontiers

in Quantum Electrodynamics and Quantum Optics, edited by A.Barut (Plenum Press, New York,

1990), p.31.

[14] J.PERINA: "Quantum Statistics of Linear and

Nonlinear Optical Phenomena" (Reidel, Dordrecht, 1984).

[15] B.YURKE and D.STOLER: Phys. Rev. Left. 57,

(1986) 13.

[16] V.BU2EK, I.JEX and TRAN QUANG: J. Mod.

Opt. 37, (1990) 159.

[17] J.JANSZKY and AN.V.VINOGRADOV: Phys.

Rev. Lett. 64, (1990) 2771.

[18] W.P.SCHLEICH and J.A.WHEELER: JOSA B

4, (1987) 1715; W.SCHLEICH, .F.WALLS and

J.A.WHEELER: Phys. Rev. A 38, (1988) 1177.

[19] W.P.SCHLEICH, R.J.HoRowlcz and S.VARRO:

in "Quantum Optics V", edited by J.D.Harvey

and D.F.WalIs (Berlin, Springer-Verlag, 1989),

p.133.

[20] J.P.DowLING, W.P.SCHLEICrl and J.A.

WHEELER: Ann. Phys., (1990) in press.

[21] W.H.LoUISELL: "Quantum Statistical Properties

of Radiation" (Wiley, New York, 1973).

[22] H.P.YUEN: Phys. Rev. A 13, (1976) 2226.

[23] M.S.KIM, F.A.M.DEOLIVEIRA, and P.L.

KNIGHT: Phys. Rev. A 40, (1989)2494.

[24] F.A.M.DEOLIVEIRA, M.S.KIM, P.L.KNIGHT,

and V.Bu2EK: Phys. Rev 41, (1990) 2645.

193
N92-22065 /1 f

LIMITATIONS ON SQUEEZING AND FORMATION OF THE SUPERPOSITION OF TWO 7 _5

PROCESS OF SECOND HARMONIC GENERATION.

Nikitin S.P., Masalov A.V.

Lebedev Physical Institute

Moscow, USSR

In this paper the results of numerical simulations of quantum state

evolution in the process of second harmonic generation (SHG) are discussed.

It is shown that at a particular moment of time in the fundamental mode

initially coherent state turns into a superposltlon of two macroscopically

distinguishable states. The question if this superposition exhibits quantum

interference is analyzed.

To describe the SHG we use the following Hamtltonian:

H = h_ a t a + 2h_btb + gh(atatb + aab t)

¢ b ¢ Here a, a , b, are annihilation and creation operators of the fundamental

mode and harmonic mode respectively, and g is a coupling constant

proportional to the nonlinearity of the medium. The nonlinear interaction is

described by the last term in the Hamiltonian. This Hamiltonian corresponds

to the case when there is no absorption loss in the medium. The initial

quantum state was taken to be a coherent state in the fundamental mode and

vacuum state in the harmonic mode.

In our calculations we have used a number-state basis in which a quantum

state is just a vector and operators are matrices of c-numbers. Details of

our calculations are described in Ref. 1. Earlier similar calculations have

been made by Walls and Barakat. It is known that squeezing in the SHG has a

minimum. It is shown in Ref. 1 that this minimum appears due to the formation

at the fundamental frequency of the superposition of macroscopically

distinguishable states. It is the formation of this superposition that is the

limiting factor of the largest squeezing achievable in the process.

Fig. 1 represents the dependence of amplitude squeezing in the

fundamental mode versus the dimensionless time _=gt¢_. N is the initial

average number of photons in the fundamental mode. Fig 2 represents the

quasiprobability distribution for the fundamental mode Q(a)=<alpl_>/u when

this superposition is formed. Here p is the density matrix of the quantum

state and la> is a coherent state described by a c-number a. Earlier, in

Ref. 2 it was shown that superposition of two coherent states can be obtained

using Kerr nonlinearity. The SHG process appears to be alternative nonlinear

process in which the superposition can be obtained.

The question of the origin of this superposition is discussed in Ref. 1

where this phenomenon is attributed to the instability of the SHG process

with respect to the initial harmonic phase which is completely uncertain for

the initial vacuum state in the harmonic mode. This instability was

illustrated by a classical equation solution where quantum uncertainty of the

harmonic state and the fundamental state was imitated by randomized initial

conditions distributed by the normal law with the same dispersion as quantum

states.

195

PRECEDING PAGE BLAP_K NOT FILMED
Here we would like to pay more attention to the question of whether the

superpositlon is coherent, that is, a pure quantum state, or whether it is a

statistical mixture of two coherent states. In order to answer this question

one usually uses simple numerical criteria such as T = Trp 2 For a pure state

T = I while for a statistical mixture T < I. The dependence of T versus T is

shown on the Fig. 3. If N=IO the superposition appears at T = 4. It is

clearly seen on Fig. 3 that T at this time is very far from parameter

specific to the pure state. So, one can expect that no quantum interference

effects could be seen in this state. However, we may check it directly using

the density matrix.

To see quantum interference we may consider the function P(x)=<x|p[x>.

Here ]x> is an eigenstate of a quadrature operator x = (a+a')/V_.

Experimentally this function P(x) can be obtained using homodyne

measurements. It is known that for a coherent state this function is a

gaussian. If we calculate this function for a statistical mixture of two

coherent states then we get the sum of two gaussians and no quantum

interference. For a quantum superposition of two macroscoplcally

distinguishable state this function exhibits an interference pattern. It is

therefore interesting to check if the superposition formed in the process of

the SHG exhibits quantum interference pattern in P(x).

FiE. 4 represents P(x) calculated from the density matrix of the

superposition at T=4 and N=iO. This function obviously exhibits quantum

interference, though visibility of the interference pattern is less than for

a pure superposition of two coherent states. This result could be explained

if we assume that the main portion of the statistical mixture, which in fact

the above-mentloned superposition is, is a quantum superposition of two

coherent states. Other states which the mixture contains reduce visibility of

the interference but can not destroy it completely. Thus the superposition

formed in the process of the SHG can exhibit quantum interference though,

generally speaking this superposition is a statistical mixture rather than a

pure state.

Conclusions

Squeezing in the process of the SHG is limited because of the formation

of the superposition of macroscopically distinguishable states at the

fundamental frequency. This superposition forms because of the quantum phase

uncertainty of the initial harmonic state. Though this superposition is not a

pure quantum state, it does exhibit quantum interference in P(x). This fact

illustrates that analysis of simple numerical criteria such as Tr p_ is not

enough to decide whether quantum interference appears or not.

REFERENCES:

1Nikitin S.P.,Masalov A.V.,1991 accepted for publication in Quantum Optics.

2.Yurke B. and Stoler D., 1986, "Generating Quantum Mechanical

Superpositions of Macroscopically Distinguishable States via Amplitude

Dispersion," Phys. Rev. Lett., H7(1), pp. 13-16

196
i !

!

.I /

.ill II

• fl t

. ii tl

0 l I I,

0 1 2 3 4

Fig. I Squuezlng in the fundamental mode vs T=gt 2vr2-N. N=IO (solid line)

N=20 (dotted line), N=40 (dashed line)

)1

Fig. 2

Quaslprobabillty Q(_) = <¢x[p[cx>/n

for funda_aental mode at T=4; N= I0.

Contours at O. I/n, 0.2/n, 0.3/n.

0.6

1,5 I I

1

0.5

0

T

0 2 4

Fig. 3

T = Trp 2 versus T for the fundamental

mode. Average photon number N = 10.

I I

6

0.4

0.2

Ii

-5 0 5

Fig. 4 Quantum interference in P(x) = <x[p[x> for the fundamental mode at

T = 4. Average photon number N = 10.

197
N92-22066

FIELD QUANTIZATION AND SQUEEZED STATES

GENERATION IN RESONATORS WITH TIME-DEPENDENT

PARAMETERS

V.V.Dodonou, A.B.El_moo, D.E.N_konoo

Moscom Physics Technical InstCtute,

16. Gagartn St., Zhukousky 140i60

Lebedev Physics Institute, •

53, Len£nsky Prospect, Moscom 1t7924 USSR

The problem of electromaqnet_c i£eld quant_zat£on £s usually

considered £n text-books under the assumption that the i£eld occufi_es so@e empty box. The case mhen the box _s t_ILed m_th a

uniform dielectric medium mas considered £n (Refs.l$2).2 The quan- t_zat_on of the f_eld _n the medium consisting tmo untiorm

d_electr_cs w_th d_fferent perm_tt_v_t_es was stud_ed _n

(Refs.3-5). The case of an arbitrary _nhomogen_ous d_electr_c

medium was _nve.st£gated 6n (Re/s.6,7) and especially _n (Refs.

8,g). However, _n all mentioned papers the properties o/ the meatum were beZieued t_me-_ndeRendent. Here we want to consider the

most qeneral case of non-un_Eorm and t_me-dependent media. Earlier t_s problem was _nvestigated _n (Re/.iO)_ but its authors

Eons_dered only approximate solutions of the Hetsenberg equations

tot Y_eld operators £n the case o/ small #olar_zat_on of the medium. Our approach d£/fers from that of (Re1.10) and enables to

study the case when non-uniform t_me-dependent d£eIectrlc med£um

£s con/lned Ln some space region mlth t_me-dependent boundar£es.

The bas£s of the subsequent conslderat_on _s the system of

Maxwell's equations in I_near passLve t_me-dependent dleIectr£c

and magnet£c med£tun m£thout sources:

rot H = llc DDIat,

d_v B = 0

B = _(r,t_H.

rot E = -ilc DBIDt,

D = e(r,:

Introducing the vector potential according to the relations:

E = -llc DA/Dt

(I)

B = rot A,

and _mpos_ng gauge conditions

d_u(eaAlat) = O,

we can replace the s_stem

the s_ngle second-order one:

rot(1/_ rotA) = -1/c _

(2)

9 = o (3)

o/ the /_rst-order equations (i) m_th

#IDt(eDA/at) = ilc #D/at. (4)

One can check that the vector equation (4) co_ncLdes w_th the set

of Euler's equations

199

PRECEDING PAGE BLANK NOT FILMED
_lSt _LI_(_rA_) + _l_z_ 8LIO(_A_) - c)Li_A_ = 0

for the Lagrang_an density

L = 1/2 [ E(r,t)(_A/_t)2/c 2 - (rotA)2/p(r,t) ]

(5)

(6)

in the case of qu_te arbitrary t_me and s_ace dependences of the

d_eLectric and maqnet_c p.ermi-tt_.v_t_es. 1nan introducing the canon_catlg conjuga_ted variable

P = _Li_)(_TA) = E(r,t)/c 2 _All)t = -lic O (7)

one can construct the ttoin_Lton_an density

E = P aAlat - L = 1/2 [ D21s(r,t) + B21_(r,t) ] (8)

which Leads again to eq.(_). But _n the general case the ezpress_on (8) _s by no means the energy of the system due to possi_bLe

t_me-dependences of the coeffgc_ents. This fact comp.L_cates the

quantLzat_on procedure. The usual procedure consists _n _ntroduci.ng the field ezpans_,on over mode functions

D(r,t) = T.qN(t)u,(r), B(r,t) = ZpN(t)uN(r). (9)

Subst_tut_nq these ezpansions _nto the Hom_Iton_an density (8)

end _ntegra_b_q it over the space variables one gets usually (due

to certacn orthogona_ty prop.ert_es o/ /unct_ons u N and _)M) a sum

o/ Lndependent oscLI_ator-[lke Ham_Itonlans

H = _ E(r,t)dar = I/2 Z(_INPN2 + r_NqN2). (10)

After thLs the coefficients PN and q_ are procl,a_med operators

satisfying canonica_ cominutat_on re,aryans, so that the Iie_ds

become quantLzed. But this sketch of standard quant_zat_on scheme

sho_s dgsti.nctly that _t can be used onI.u _n the case when the

sol.uti.ons of Maz_eZg's equations can be Factorised _nto the p.roducts of two funct_.ons: one dependent ongy of ti.me, and another

dependent on space coord_.nates onIg. In the generag case of nonun_.forrn and ti.me-dependent medium such so-guti.ons do not ez_st,

and the usua_ scheme of quanttzat_on ks gmpossibI.e. This means _n

partgcu_ar, that _e cannot ohtai.n any Eumtgtonian end, consequent_y, any uni.tary evolution operator. Therefore the Schrod_nger p_cture does not ez_st _n the general, case. But the Ee-

_senb_r_ _escript_on _s st_.l._ possi.b(.e. _It can be _.ntroduced as e

qen_rag_sotgon "of the a_proach used earI_er bq Moore (Ref.11) for

_he. .f_eld quant_zati, on. '_.n the erupt. .g space re_on confined _Lth

mov_ng boundaries.

First of aI[ we notice the importmit property, of equati.on

(4)" gt admLts a time-i.ndependent scal.ar product of miy t_o dLfferent sol.ut£ons _n the fogI.ow_ng form:

((A,B)) = -I/2 i _d_r _(r,t)[A#B"/at - B"aA/#t]. (11)

20O
It _s essential that the d_eLectr_c permitt_v_ty is a real

function, ._.e. the medium is assumed Lossless. Besides, the vector potential has to turn into zero at the surfaces confining

the \inteqrat_on dqmain. The case of mov_ne boundaries (considered _n ReT.11) _s _ncLuded to general situation automatically.

SupRose that before some _nstant of t_me (Let it be t=O) both

the mea_um and the boundaries mere time-_ndependent. Then soLutions of (4) could be factorized:

A(r,t) = g(r)ezp(-i_t),

rot[I/l_(r) rotg] -(o2/c 2 e(r)g = 0.

(12)

(13)

The scalar product (11) mas proportional to the usual scalar product:

((A,B)) = i12(_, + _)exp[_(_b-_.)t](g_,g,), (14)

(g_,g_) = _dar s(r)g_"g_. (15)

But it is knomn that solutions o/ eq.(13) form the complete

orthqgonaL m£th respect to scalar product (15) set of vector

tunc_cons. Therefore any real vector f£eLd can be decomposed

over this set of functions:

A(r,t) = r.[awgw(r)exp(-i(_Nt) + aw"gN"(r)exp(i(owt)]. (16)

ompar_ng (14) and (15) me can see that the basis functions can

e normac£sed _n such may that they w£LL satisfy the relations

((AN ,AN) ) = 6N,, ((AN,AN")) = 0. (17)

After the instant mhen the properties of the medium became t_medependent the bas_s funct£ons change the£r expL£cLt ezpress£ons.

but the scaZar products (t7) miLL not change, and _nstead of (16)

me can mr£te the decompos_t£on

A(r,t) = £[aNAN(r,t) + aw"AN"(r,t)]. (18)

Then.me proclaim the (t£me-_ndependent) coefficients of th£s expansion operators satisfying bosonic commutation reLatLons and

thus obtain the quant_zed-f_e_d from a cLassicaL one.

If in some time the medi.u@,miLL become again time-independent

then the #hgs£caL states _LL be describe_ m£th monochromatic

mode functions of the t_e (12), mh_ch m£LL not coincide _n general mith the bas£s functions of ezpansion (18). Therefore me miLL

have tmo d_fferent ezpans_ons .of the f_eLd o_erator: ezpans_on

(18) over the states corresponding to the physical photons in remote past and expansion L_ke (163 over the physical states ar£-

201
sf, ng _n future. Des_gnati, ng the "phusi, cal" states wi, th the superscrtpt "zero", we can ezpand eacT_ set of basi, s functions _nto

a ser_es m_th respect to another one:

(t9) A.. ( o) + 8..A. ( o)- ].

The correspond_na expansion odi,,"new" creation and operators ouer tl_e set of "el ones is as follows:

A #% #%

aN (°) = awaNN + aw+_wN ].

ann_h_latLon

(20)

The £n£t£at state of the auant£zed f£etd was determined w£th resnect to the set of "ogd" operators ( w_thout the superscript

"zero".). Then using expansion (20) we can calculate all quantum

s_ar£sr£cat characteristics of the f£etd £n the final state. Taking {nto account conditions (i7) and the evident properties of

the scalar product (li)

((A,B)) = ((B,A))" = -((B',A")) (21)

one can express the coefficients of expansions (lg) or (20) as

folloms"

a,, = ((Aw,A.(°_)), _N. = ((AN",A, (°_))"- (22)

The quant_zat_on scheme sketched aboue can be applied to the

most general s_tuat_on of an arb_trar_ space-time nonuniform medium and mou_ng boundaries. Eo_euer, the explicit calculation of

the mode functtons and coefficients of the canonical transformation (20) can be performed only for rather s_mpte special cases.

The f_rst of them corresponds to the med_a _th factorized electr£c and magnetic perm_tt_t_es:

e(r,t) = _(r)x(t), _(r,t) = _(r)v(t) (23)

( the boundaries do not moue). Then mode functions can be also

sought for £n a factorized form"

A(r,t) = g(r)_(t) D(r,t) = _(r)g(r)q(t). (24)

Let us demand the function g(r) to satisfy the following equation

rot(I/_ rotg) = k=_(r)g, k = const. (25)

Then eqs.(2) and (4) result £n the following ord£nar_ d£fferent£-

al equations for t£me-dependent factors oY the oec]or potential

and electric displacement:.

= k=c_/v(t) _ = -cq/x(t) (26)

Ea_s (26) resemble equations of motion of an oscillator _th t_med_pendent mass and frequencies. The role of qeneraI£zed coordinate _s.played by the electric, d£splacement. ]_me-dependent factor, mhtIe the uector potenttal ttme-dependent factor plags the

202
role of generalised momentum. Eqs. (26) can be replaced bg the

/oLZowLng second-order d_//erent_aZ equation:

em •

q + r(t)q + Q=(t)q = O, r = v/v, Q2 = k2c2/v(t)x(t). (27)

We shaZl consLder the /Leld _ns_de a resonator. Then solutions o/

eq.(25) can be chosen real vector /unct¢ons sat_sfg_ng the orthogonaL_g conditions"

_dar s(r)g_(r)gL(r) = k26[L. (28)

CompLex solutions of eq.(27) can be normalized as foLLows"

v(t)[Gn"- G'n]=

Th_s means that we choose the solution of

nary case _n the form of

qo(t) = (vo_o)'I/2ezp(-_Qot). (30)

Due to (28) coel/_c_ents (22) are not equal to zero only /or

co£nc_d_nq tnd_ces (£ntermode tnteract£ons are absent) so we can

omit the _nd£ces. Taking _nto account eqs.(ll),(26),(3_) one can

represent these coefficients as /oLLows:

a = 1/2(Vo/go)I/2(goq + _q)exp(_got ) (3i)

= 1/2(Vo/_o)_/2(_oq - _q)ezp(_Qot). (32)

Let us _ntroduce the quadrature components and their variances as

follows:

X_ = (2)-_]2(ao +ao +) X_ = £(2)-¢/_(ao* - ao) (33)

o_ = t/2<XxX _ +X_X_> - <X'_><X'_>. (34)

Suppose for s_mp_c£ty that _n_t_aLLy the f£eLd was £n the coherent quantum state. Taking _nto accoun_ eq.(20) one can eas_Lg

obtain the fo_ow_ng ezpress_ons"

o_ = 1/21a+_1 _ = i/2Vo_olql _, (55)

o_ = 1/21a-_l _ = t/2voOo-ilql _, (36)

o_ = Im(a_") = l/2voRe(qq*). (37)

We see that t_me-dependent medium transforms an [n_t[aL_g coherent state to a "correLated quantum state" characterized bg

a nonzero couar_ance (37) and unequa_ variances (55) and (36).

Th_s state m_n_m_zes the aeneraL_zed uncertainty reLatLon bg

Schrod_nger and Robertson (ReTs.12,t3)"

(29)

eq.(27) _n the star,o203
2

o_io22 - o_2 ) 1/4 (38)

29)the) equaIi'tqet_ tor takes place f,n the case under stud q due to eq. • detai, led reu_,ew of various forms o7 uncertainfg

relations see (Ref.14). Properti, es of correlated and squeeze_

quantum states were i,nuest_gated i,n (Refs. 15-19).

Let us consLder as an ezampLe the case of a paroJnetr_c ezc_-

tat_on when the propertLes of the medLumharmon_caLLg osciLLate

w_th twice frequency w_th respect to some (resonance) mode. Th_s

can be achLeued, for ezampte I by means of changing the dens_tg of

the medium. S_nce the magnetic effects are eztremetg weak, we can

wrLte

Q2(t) = _o 2 (t + =cos2got), T = 0. (39)

We took for the sotution of eq.(27) Ln the form

q(t) = (VoQo)-4/2 [u(t)ezp(i, Qot) + u(t)ezp(-iC_ot)] (40)

w_thstoWt_ntoUary27_Ln t_me deependent ampt_tudes. Subst_tut_nq (39)

ons we arrLue at the equations (neffLect_ng _he second order der_-

ua_ues ot s_ow_g uary_ng amp_i_uaes)

u = _o_U14, u = -Lgo_ul4 (4i)

whose sotut_ons are

u(t) = cosh(_o_t/4), u(t) = -_ sinh(Qoatl4). (42)

The uar£ances (35),(36) oscittate with the twice resonance frequency, but their ratio (the so catted squeezing coefficient) Ls

conf£ned at euerg instant between the uatues

ezp(-C_ozt) < _o_0_/o22 < ezp(gozt). (43)

Certain _nequatities for the squeezing coelfLcients can be found

for arb_trqrg t_me dependence of the frequencg in eq.(27). ConsiderLng again nonmagnetcc medLurn one can proue the unequatitLes

(Refs.i9,20)

[(I-R_/=)/(t+R_=)] _ < _o_O_/o_ < [(l+R_/=)/(l-R_/=)] _ (44)

where R is the energg reftect_on coefficient

potent_at barrLer corresponding to eq.(27).

from the effect_ue

We woutd t_ke to emphasize once again that we haue used the

Eeisenber. .picture for the descr£ptLon, of the .quant_zed etectro-.

maanet_c _etd. Eoweuer, s_nce for a factor_zabte medium (23)

Ma_wett's equatLons can be der£ued, from the Eamittqnian (8) or

(tO). the SchrodLnger descrLpt£on _s.atso poss£b_e _n th£s case.

We shatt _ttustrate _t for the most s_mpte case wnen

2O4
v(t) = 1, X = x(ht), %(0) = i. (45)

Here h i,s a characteri, sti, c trequency of the medium properties

changing. The method used below can be eas_Lq appl_edto a more

_qeneraI case. Let c = i and the di, mensionless "ti, me to = kt. So

_27) results i,n

q + co2(Zto)rl = 0, (46)

where

_2(zto) = e2(ht) = l/%(ht). z = h/tt. (47)

The quaqt¢zat¢on o/ a harmonic oscillator w¢th a variable frequency _s done bg introducing integrals of motion operators (Ref.

7):

A A _ A

a(to) = _(2) -_/2 (g(to)_ - 0(to)q), a'(to) = [a(to)]'. (48)

Here g(to) _s a "ruling solution" o/ eq.(46) satisfg_ng the timeindependent condition

g(to)g'(to) - g"(to)g(to) = 2i (49)

to ensure the foLLowing commutation relation

A A

[aCto),a+(to)] = 1. (50)

We want to stress the d¢//erence in s_gns o/ r¢ght-hand sLdes o/

(29) and (49) due to the d_f/erence between He_senberg and Schrod¢naer pictures. For the _nteoraLs o/ motion to coincide at t o =0

with creation and ann¢h¢Lat¢o5 operators, Lt ks necessarg (in accordance w_th (49)) to choose the _nitiaL conditions

g(o) = l, _(o) = i.

Alter the ¢nstant to = tr, when the medi, um properti, es

i,ng. "new" creati, on and anni, h_Lati, on operators are to

ced:

(51)

stop changbe £ntroduA A A

a(o _ = i(2(o(tr)) -If2 (E,- f,q(o(tr)).

Then the expansion ol "new" operators over "old" ones _s

(52)

A A _,

a(o, = aa(tr) + _a'(tr)

and we can get

J3(tr) = 1/2 (co(tr)) -_/2 (_(t_)y(t¢) + i_t(tr)).

(53)

(54)

Creation and annihilation operators mixing resutts in a change of

occupatLon numbers _n a g_oen mode• Generation o/ photons from

vacuum due to the medium properties change is worth considering.

I/ at to = 0 the number ofphotons _as zero then at an _nstant to

205
>0 the average number of photons _s

= 13"(to)IB(to). (55)

YabLonov£tch (Ref.21) stated that photons w[LL have a thermal

d£str_but_on at a temperature proportional to the rate of the medium properties change t _.e. proportional to h. Then for h_gh

photon energies (proportional to k_ there w_LL be

~ exp(-const k/h). (56)

The solution of (59) at a L_ttLe to can be found for an arbitrary

Law of change of %(t). At to = 0 we shaLL have

dldto (((o)112 13)= 112 d(o((aeto))/dto (57)

~ to/2 d(u(_to))Idto

~ (t/2 d(u(ht))/dt).

t -> 0

(58)

(59)

For ang _nstant t a decomposition of the solution can be found _n

the Lcm£t _ -> 0 ( for h_gh photon frequencies ) bu means of the

method of muLti-scale asymptotLc decompos_t£ons (Ref.22) for an

arb£trary Law o7 change:

y = ((_(Zto)) -t/2 { exp(_s) + z [E(=to)exp(£s) +

+ F(_to)exp(-f,s)] + ... } (60)

where

s = 2/z [ezp(zto/2) - t] (6t)

and functions E and F are determined by

F = const

cLEld(=to) =-i,l(4(o2) d2_old(zto)2 +

+ i,3/(8¢0_) [dco/d(zto)]2

(62)

w£th £n£t£aL cond£t_ons

E(O) + F(O) = O, i, (E(0) - F(0)) = t/2 d(o/d(zto). (6S)

For the case %.= ex_(Tht), which approz_mateLy describes d_eLectr_c _erm£_£v_y ta_ng ach£evabLe £n ezper£ments (Ref.21), one

can f_nd the exact soLut£on - a L£near comb£nat_on of two RankeL's functions Ro(_)[2/z exp(=to/2)] and Eo(2)[2/= ezp(=to/2)].

NevertheLess an asgmptot_c decomRos_t_on _s st_LL more useful and

can be expressed _n an explicit iorm:

y = exp(-=to/4)exp(_s) + • ezp(-=to/4) { (-L/16) x

x [1 + ezp(-_to/2)]exp(_s) + _/8 exp(-_s) } + ... (65)

206
Now we haue

= {=18 [exp(-i,s) - exp(-atol2)exp(_,s)] (66)

and the number of photons

= =2/64 [1 + ezp(-=to) - 2cos(2s)ezp(-=to/2)] (67)

Th£s number oscillates w£th a grow£nq frequency and a decreas£ng

empL£tude and /,n the LLm£t to -> ® -tends to a constant

-> h2/fkk 2 .

As we can see _t £s not £n agreement w£th the statement (38) from

(Ref.21). The energy of photons £n the mode £s

5_k -> h2164E exp(htl2), t -> ® (6g)

and £t _rows w£thout a l£m£t. ALso at any t£me the sum of enerq£es oY al_ modes d_uerges. Th£s can be ezpta£ned bg the fact

-that £t £s £mposs£ble to _ecrease d_eLectr£c perm£tt£u£-ty to zero

for nond£spers£ue med£a; thus the assumpt£on that £t does not depend on a frequency £s not ual£d for h£qh frequencies. In the

other L£m£t • = h/k -> ® we £ntroduce ano-ther d_mens£onless t£me

t t = _t o. The soLut£on ezpans£on ouer O = i/z ( not uaL_d for

Large ti ) _s

y = I + 0£ttl2 + 0214 [tt + 1 - expt_] + ... (70)

and the f_rst term for the number of photons does not depend on

the mode frequency:

= ezp(t_12) - t + 0£12 [ttezp(t_/2) + t - ezpt_] (71)

= 112 [cosh(ht/2) - 1]. (72)

Another ezqmple of t£me dependent resonator wh£ch can be sotued £s an emntu resonator w_th a mou_na _deaL waLL. Moore (Ref.

11) .proposec[t_e foLLowing complete ort_onormaI set of solutions

(£n the special case of a s£nqLe space d_mens£on, _.e. conf£n£ng

w£th the modes w£th L£near poZar£sat£on paraLteL to the wall surface):

These functions de#end on the so_ut£on of the funct£ona_ equation

(L(t) £s the pos£t_on of the mou_ng wa_ another wa_ £s assumed

to be at rest3

R(t+L(t)) = R(t-L(t)) +2. (74)

207
An ap#roz_mate solution of th£s equation _n the case of the smaLL

velocities of the wail was found b_ Moore (Ref.ll) and later used

Ln (Refs.25z24). Eowever, that solution _s not valid _n the case

of parametric resonance, when

L(t) = Lo[! + a s_n(2Qot)], lal<<t, _o = _/Lo (75)

(the resonance at the lowest resonator e_qenfrequencg). The correspond_nff solution for small values of the percentage modulatLon

wus touna_n (Re/.25):

+ I/4QoF_ s/,n(2_o_)] +

Eqs.(11),(22),(45) result _n the foLlowi.ng ezpress_ons

trunsformati.on coeffi.c_.ents (2t)"

(76)

for the

= 112 (/fIN) I/2 IL°+lezp{_=[-NR(Lor)+_Jlz]}dz.

_N ILo-i

(77)

The cuLcuLat_ons are ruther s_mpIe for not ver_ Lurge uuLues of

t£me, when the second-order correct£on £n (763 rema£ns smalL.

Then the foIlowLng sLmpLe formula /or uur6unces can be found

(Re/.25)"

°4_=_ ll2ezp(±t/2 =uN), N>>I,

02

laNl<<l (78)

where N _s the number of sem_per_ods of wall's u_brut_ons. One

can check fhat the mazLmum squeez_nq coe_f_c{en_s g_ven bg eqs.

(45) and (78) coincide for equa_ ou_ues of percentage modulu_-

on _n two d_ferent methods of ezc_t_ng the f_eId v_u the parametric resonance.

Now let us consi.der the _ong-ti.me us qmptoti.cs

tLon under the condi,ti,on alL. << 1, _ >> L,,"

perLod_c motion of the wa_l L(t) : Lo + st(t).

rhe soluti.on o/ eq.(7k) _n the fot_owLng form:

R(t) =,,_=o_NRN(t). (79)

Substi.tuti.ng ezpansi.on (Tg) to eq. (74) we attai.n

O0 O0

_=oSNR.[t+Lo+sf(t)] = _=oSNRN(t-Lo-_f(t)] + 2.

Deuelopi,ng both si,des of thi,s equation _,nto power seri,es we haue

_:o_:o'"_"`_'__+_°_'''<_>'_'=_:o_:o_"_."'_ _o__

x (-1)¢s[f¢(t)/K! + 2.

for the R-funcfor an urb_trarg

We shall choose

It _s conuen_ent to use another sungnat_on _ndez fl=-N+E:

2O8
_-I= (K)(t-L°)]f_(t)/g! = 2

® _M[RN-_ (K)(t+L) o - (-I)_RM -

0 0

From th_s equat£on we obtaLn the)foLLow£ng system el equat£ons for the funcf£ons R,(f) (M=0,1,... •

_= 0[R,__ (K_(t+Lo)-(-I)KR, __(_>(t-Lo)]f_(t)/K! = 26,0 • (80)

Further we cons£der the s£mpLest Law of mot£on (75) w£th _ = aLe

andthodtheto socuefrequenCYeq.(_)'= Q_o. We use the Four6er-trans/ormat_on me7(o) = _exp(_ot)f(t)dt, f(t) = _ezp(-£_t)7(o)d_/2x,

i/2x _exp(£wT)dt = 6(_), (81)

_exp(Lot)Y(t+Lo)dt = exp(-£G_Lo)7(o),

_exp(£ot)f(N_(t)dt = (-_O)w7(O).

Then we get from (80) the foLlow£ng £ntegral equation"

_=11/N ! _du(_£u)NRN_N(U) [exp(_£uLo)_(_1)Nezp(£uLo)] x

x @__U(s£nN(_Ot)) = 2_6(_)6.0 (82)

@o(s£nN_ot) = _exp(£_t)s_nN(_Qt)dt =

I/(2x) N-I J_=0CN_ (_i)_6(_+(N-2J)o0). (85)

Subst£tut£ng expKess_on (85) to eq.(82) we cm_ eas£1y make Lntegrat£on ouer u ana arr_ue at the equation

-R"(_)2_s£n(_L°) = 26"°_(_) - g= xlI=0 (-1)W'_/2N CW_

x [o+(N-2J)oQ] N _N_N[(_'+(N--2J)OQ] x

x [exp(-_Lo)-(-1)Nezp(£_Lo)] (-1) oN • (_4)

Taking £nto account the formula for der£uat_ues of 6-funct_on

6(N)(z) = -N6(N-_)(_)/Z,

one can easily J£nd the express_on for R0(o)

Ro(_) = 2x6"(o)/_Lo. (85)

Then making _nuerse Four£er-translormat£on _e haue

R0(t) = t/Lo. (86)

To l£nd the _ong-t£me asymptot_cs we _£_ seek for the so_ut£on

o/ eq.(84) £n the term of a sum ouer 6-/unct_ons

209
R.(_) =_=oa_"6c">(o>+(M-2X)_,) (87)

Thts choCce corresponds to the representation o/ the /unction

R(t) 6n the form of a power sertes wtth respect to the parameter

(Et/L 2) and negZecttng terms L_ke s3tw w_th J>N. Then only terms

correspondtng to N=i are stgnl/tcant 6n eq.(84)"

-RN(_)2_stn(_L o) = 26N06(_) + (-l)Ocos(_Lo> x

x C(_+C0Q)RN_I(<0+C0Q)- (e-_Q)RN__(_-C0Q)]. (88)

Tak_ng into account the ezpress_on for _o(_) (85) we get

Rt(_) = 2x(-1)°/2Lo cotan(_L o) [(¢_+_Q)6'(C0+_Q) -

- (G>-_Q)6'(e-_.)] = 2X(-l) ° [6"(CO+COQ)- $'(e-_o)]/2Lo 2 (89)

WLth respect to expans_on (87) we haue

ao° = 2_If.Lo, aol

Substttut_ng ezpanston (87)

relation tor N)2

{ 0 2

= -a_ = 2_(-i) /21,0 .

we obtain the YoLLow_ng recurrence

a_ N = (-l)°6_/2NLo [(N-2_.E-i)a[ N-( - (N-2.K'+i)a__i N-_] (90)

Let us 6ntroduce the notatton a_ 2" = 6EC2", E_ 2"*_ = a_ _"÷_ , then

N_[ N = (--_)N_[_KN-I(N--2](--I) -- -aK_ _ N-I(N_2](+I)]

= (-l)°_./2Lo • (gl)

Maktng 6nverse Four6er-transformatton of (87) we get (N)2)"

M

RN(t) = 1/2x__=0aK N (tt)Nezp(tt(N-2K)_)

Then 6t 6s easg to see that we have got the same etpressLon /or

R_(t) that was gtven 6n eq.(76). Now we constder the sum

= = a_Ne_p(Ct_a(N-2_E))p_/2_ (92)

N=2 N=2 E=0

for p=t. Taktng 6nto account the evtdent s_mmetrg condttton _ N =

N

= -aN-K we obtatn

@ = 2® =2ZN(-I) n_=0a --K N S 6n(_(N-2E)t)p_/2_, z = st, (93)

where n=[N/2] and [ ] 6s the entire part o/ a number. Taktng _nto

account that (-i)n = (__)N(N-I)I2 _e _ntroduce the notation @ =

=-21mF. Wtth the hq_p o/ the recurrence re_at_on (91) we get the

dtl/erent_aZ equation for the lunctton F

aF/_z = _[pezp(-6_t) - erp(_Qt)][z_F/_z - 2paF/3p +

210
+ (zal_Q)ezpC_ot) ] (94)

wi, th the in_,tiaL condition F(z=O,p) = O. Its solution is

F = -(z_l_Q)ezp(_ot) + _,

where _ satisfies the equation

0_10z = a[pezp(-_(aot) - ezp(_coot)][zff_/Oz -

- 2p_lOp] + (alxQ)ezp(6eot).

The particular solution of th_s equation _s as foLLows:

(95)

_Zo =-(l/2xQ) tn[pezp(-icoot)l(pezp(-£_Qt) -

-exp(i_Qt) ) ]. (g6)

Then the genera_ soLutLon of eq.(95) is the sum of _, and an arbitrarg solution of the uniform equation (95) with b=O. The uniform equation has the first integraL:

C = ezp(-_oot/2) {zp _/2 - 1/2_ In[(pt/2ezp(-_oot/2) -

-ezp(ioot/2))/(pII2ezp(-_oot/2)+ezp(_oQt/2))]}.

Then we qet the generat sotut_on of eq.(gs) in the form of • =

= _o + f(C), where f(C) is an.arbitrary function of the f£rst _ntegraI. From the condition @(z=O) = 0 _e can determine the form

of the function f"

f(z) = (I/2xQ) _n{[l+ezp(-2oexpC_at/2)x)]_/

Ikezp(-2_exp ( L_ tl2)z) }.

After some a_gebra_c transformations we _ind the function

= -(i/2xQ) _n{4pezp(-i,G)ot)ezp(-2cxz)/[cos(o _t/2) +

+ 2is_n(_oot/2)exp(-2o_z) ]2}.

Taking into account the first and the second terms of the ezpansion of R(t) in a set of st me obtain the final ezpression:

R(t) = t/Lo - (2/RQ) Im _:n[i + Ej + ezp(_(oot)(i-_)], (97)

_here a notation _ = exp[(-1)a÷i(ooet/Lo] is introduced. Now _e

can compute some characteristics^of the etectromagnetic fie_d _n

a cauity in the presumptLon et/Lo">>l, e<<Lo. Let us eua_uate the

number of photons which w_L be generated in the resonator w_th

mou_ng wa_Ls in the _,ong time L_mit. For this purpose _e must

eua_uate the integra_ _{()

13N. = 1/2 (M/N)_2 p[-i_r((M+_)z+Nf(z))],

_hero the funct_,on f(x) is giuen bg (97):

211
f(x) = -21xQ Im In[l+[+exp(L_Qz)(1-_)].

Let us consider the case when Q=-2p _s an even number, then [<<1.

Due to _x/Lo<<l we can consider /(z) as a periodical function

w_th the per_od of oscillations T=2/Q. Then

_N"?= (M/N)'" 1-1 _-l+2/O _N,) =0 erp[-i_(N±H)2K/Q] _-1 exp[-i_ x

x ((N+M)z+Nf(x))]dx.

Anatgzing the structure of

L6near functions as follows:

f(x) we can

(98)

approximate _t by three

f(z) =

i-SQ) (x-N+i), _IO-25) (z-N+I-IIQ)15

-(i-6O)(x-N+l-21O), '

N-1 < r < N-i+ilQ-_,

N-l+ilQ-8 < x < N-i+ilQ+5,

N-l+tlO+_ < r < N-i+21Q,

6 = 2_tl21_q.

Then one can easLLg calculate _ntegraLs in (98) and obtain the

general express_on for the coeffLc_en_s _N, and _NM

x ezp[i_(N-M)(Q-1)/Q] sin[x(N-M)]/sin[_(N-_)lQ],

(99)

_NN = 2(MIN)t/2(-I)<N-_>("+"_I[x(SNQ+_)] s_n[(SNQ+_)_/O] x

x ezp[i_(N+M)(Q-1)/Q] s_n[_(N+E)]/sin[_(N_E)/O].

Hereafter _e consider on_g the ma_n resonance of Q=-2. After

some a_gebra_c transformgtions _e get the foZ_o_ing expression

lOT the coefiLcLent l_NNI':

I_WNI 2 = LLI_/(Nx 2) [1-(-1)"cos(2N6_)] [1+(-1)"÷"]/

I(M+2N6)=. (t00)

To find the totaf number of photons _n the mode _th number M we

need to calculate the sum ooer N. First, _et us eua_uate the fo_-

Lo_ing auxi_iarg sum:

S(z,z) = _=[os(xN)/[N(z+N)_], (i01)

_here z=26_<<1 and z=8126>>i.Then _e huue

= llz _ -I12 I gezp(-zg)_n(2coshg-2cosz) dg. (102)

Sgnce z<<! and the main.rqgion of integratgon _l/z<<l, _e can expand coshy and cosx Bnto po_er serBes of U a z. Thus up to

the second order terms _e haue

212
S(z,z) = tlz 3 -1/2 I gexP(-Zg)In(g2+z2) dg. (105)

The Last _ntegra_ can be easily evaluated, _f one takes _nto account the _nequat£t£es y_l/z<<.r<<l. Therefore

S(z,z) = -lnz/z 2 + O(z-_). (104)

The similar sum (see (i00))

] (-1)"

can be obtained from (102) bu means of the replacement z -> z+_.

Then we haue _n (103) Ln(4+g=-z z) < LnlzZl for t<<l and y<<i_ so

that the corresponding terms can be omitted. The ma_n contrtbutLon to the sum due to the f_rst term ( m_th un¢tg in the numerator ) {n ezpress_on (100) _s proportional to

S(z,O) = -I gezp(-gz)Ln[1-ezp(-g)] = Lnzlz 2. (105)

Thus the number of photons generated _n the M-th mode {s

p. = _w.i 2 = 4[Ln(H/Zb)-(-1)"Ln(1/26_)]/(Mx2). (106)

Since Ln the consLdered case (/2=-2) 6(t)=ezp(--_Et_to2)/_, we get

the foLLom{ng rate o/ photon generation /or _t/Lo'>>l:

dP.Idt = 4ago[l-(-l)"]/(_2M). (I07)

Here Qo ,T-x/Lo. _s the ma_n e_genfrequenc_ of the resonator a=E/L_

_s the a_mens_onLess amplitude o/ osciLZat_ons of the maL_ (mh£ch

u_brates at the /requencg 2_,). Eq. (107) _s valid _n /act onLg

/or not uerg Large numbers o7 ezc_fed modes M (due to L_m£taf_ons

arising _n a_pro_mat_ons made before). Besides, _n real situation • e should L_£t the t£me t by the reLazat£on t_me o/ the.resonator • (due to the d£ssipat_on on the maLLs). Then the mat_mum

number of photons generatedtn the H-th mode equals approz£mate_g

p ,,x ~ 4/(_=H) [2aQ(M)/M + O(Ln(H+I))], (108)

mhere Q(M) _s the.auaL_tg factor o! the resonator's H-th mode.

Th£s formula is uaL aprovided aQIM>>_n(M+I).

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215
N92-22 067

RADIATION FORCE ON A SINGLE ATOM IN A CAVITY

M. S. Kim

Physics Department, Sogang University,

CPO Box 1142, Seoul, Korea

ABSTRACT

We consider the radiation pressure microscopically. Two perfectly

conducting plates are parallelly placed in vacuum. As the vacuum field hits the plates they

get pressure from the vacuum. The excessive outside modes of the vacuum field push the

plates together, which is known as the Casimir force. We investigate the quantization of

the stantfmg wave between the plates to study the interaction between this wave and the

atoms on the plates or between the plates. We show that even the vacuum field pushes the

atom to place it at nodes of the standing wave.

INTRODUCTION

Cazimir showed that when two perfectly conducting plates are parallelly

placed in the vacuum they attract each other [1]. Although there has been considerable

interest in the Casimir effect within the field of quantum electrodynamics [2a], it was

Milonni et al. who introduced the concept of the radiation pressure from the vacuum field

to interpret the Casimir effect [3]. When there is a field of a mode separated by a

conducting plate the radiation pressure exerted by the field is same both sides. When the

distance between the plates is d, the mode separation is proportional to l/d. Since less

number of modes are accommodated between the plates than outside of the plates the

excessive outside modes of the vacuum field push the plates together.

Milonni et al. calculate the vacuum radiation pressure intuitively. In this

paper we study the vacuum radiation pressure in a microscopical view point. We briefly

review previous works on the Casimir effect followed by a classical study of radiation force

exerted by standing waves.

Quantum theory has given various applicabilities of the radiation pressure.

Laser trapping and cooling and isotope separation have been realized based on radiation

pressure. The radiation pressure can increase, decrease or deflect atomic velocities.

Different isotopes of the same atom generally have slightly different electronic transition

frequencies. The radiation pressure can be tuned to deflect only one kind of isotope in a

mixture. Atoms can be slowed down by radiation pressure of counter propagating laser

and the velocity distribution of the atoms is narrowed. This process cools down the atomic

kinetic energy. Orthogonal pairs of counter--propagating laser beams are used to trap

atoms for long periods of time [2b].

We investigate quantization of standing waves between the plates and find

the quantum mechanical lorm of the Maxwell stress tensor. We then check the principle of

momentum conservation in quantum physics. When the field interacts with a two-level

atom in the cavity the atom is forced to be placed at nodes of the field. We calculated the

size of the vacuum force exerted on the atom.

217

PRECEDING PAGE BLANK NOT FILMED
THE CASIMIR FORCE

This section reviews the work of Milonni et al with appropriate extensions.

As shown in Fig. 1, the perfectly conducting plates are parallelly placed in the vacuum.

We take the z-axis normal to the plates. When the radiation field strikes the plates with

the angle of incidence 0, the pressure exerted by the radiation is the force, which-is

projected on the plate, divided by the area of incidence. Appl2'ing Gaussls law we find that

the normal component of force l_er area is the energy per volulne of the incident field. The

radiation field is totally refl.cte:l by the perfectly conducting plates. The field between the

plates propagates either to or from the plate so that the photon pressure on the plate is a

half the photon energy. Thus the radiation pressure from a mode of the vacuum field is

P = cos26 (1)

&

where _a is the frequency of the mode and V the quantization volume. When the plates are

large enough x- and y---components of the wavevector k take continuous values while

kz = nx/d, where d is the separation of the plates and n integer. The total outward

pressure is the sum of pressure exerted by each mode.

P,-r--_d- kc _ j_odk_ _0 dky (nr

where the magnitude of the wavevector

,a)2

k = [k,,2 + ky2 + (nx/d)2]L (3)

Considering the modes outside the plates, al[the components including the z-component of

the wavevector take continuous values so that the total inward pressure is

P2= hc dkx dky dkz --k--r. (4)

The wave vector is now k = [kx 2 + ky_2 + kz2] 1⁄2. The difference between the inward and

outward pressure is physically meaningful:

P2- Pl - rhc/480d4. (5)

The attractive force is inversely-proportional to the fourth order of the separation of the

plates. This is the Casimir force in agreement with Power [4].

d Y

/

/

/

Fig.1 Electromagnetic field incident on one of two perfectly conducting plates separated by

distance d.

218
RADIATION PRESSURE OF CLASSICAL FIELD

We consider the classical field for the quasimicroscopic study of the

field-conductor interaction [5]. As shown in Fig.2 the classical electromagnetic field strikes

the conducting plate with the angle of incidence 0. Taking the polarization of the electric

field on the x--,_ plane. We write the incident electric field

E = g0(x,z,t)cos0 x_- E0(x,z,t)sinO z_ (6)

and the incident magnetic field

H = Ho(x,z,t)X. (7)

The electric field on the x-y plane of the conducting plate is zero thus the x

and y components of the electric field is zero while the z-component of the total field on

the plate

E, = - 2 Eo(x,z=0,t)sing.

The total magnetic field is then

Hy = 2 H0(x,0,t).

(8)

(9)

When normal incidence is concerned, Ez = 0 and Hy = 2H0. This is obvious from the

property of the standing wave that the electric field meets the conducting plate at its node

while the magnetic field-sees the plate at its peak.

The z--component of the electric field attracts electric charges which causes

the surface charge density

e0"2g0sin0. (i0) .

In the interiorof the conductorthereiscertaincharge flowdue to the magnetic fielc_.The

currentdensityis

YxH. (11)

The charge and current cause the Lorentz force from the radiation field on the plate. The

force per area on charges

P, = -2_0E0%in20 (12)

and that on current

Pi = [®dz[(V x H_), B_]z. (13)

J0

plate

EMrefl.

Z

EMinci.

Fig.2 Electromagnetic field reflected by the conducting plate.

219
The intensity of the current decays exponentially so that

Pj = 2/_H0 2 = 2_0E0 2 (14)

where we have used the relation between the electric and the magnetic field which is

obtained from the Maxwell equations and e0_ = 1]cL The total radiation pressure is then

P = Pe + Pj -- 2e0E0 2 cos 2g. (15)

For normal incidence, ie 0-- 0, the total radiation pressure is 2e0E0 2 which is solely from

the magnetic field.

The radiation pressure on the conducting plate is the normal component of

the Maxwell stress tensor [6]. The normal component of the Maxwell stress tensor is

Tzz = 1⁄2e0(Ex2+EyZ-Ez 2) + 1⁄2/_Hx2+Hy2-Hz2). (16)

Substituting the total field we have in eqs.(8, 9) we find

Tzz = 2e0E0 2 cos20 (17)

which agrees with the result (15) obtained using the quasimicroscopical interaction theory.

QUANTUM MAXWELL STRESS TENSOR

To simplify the problem we consider normal incidence on the plate. The

electric field polarized along the x-axis propagates through the z---axis in fig.2. If the field

is a travelling wave the vector potential operator for a mode k of frequency w is known as

[7a]

where i is the unit vector along the x-axis and the caret is to denote the operator. The

operator a and.a "1"are respectively the field annihilation and creation operators. The cavity

composed of the two parallel conducting plates accommodate standing_ waves rather than

travelling waves. With a similar analogy to find the vector potential m eq.(18) we obtain

the vector potential for the standing wave [¢'¢%]

.1.

[,0-zv7] " "

A_."= h 2i coskz(ae -la + aJ'e iwt) (19)

where k = nr]d, n = 1, 3, 5,-.. and d is the length of the cavity. From the

one--dimensional potential vector we obtain the electric E_ and magnetic ]3 field operators

as

l

= aA i [ h_'] _

- -mi = L o-Vji cosk ( e-i t - te i t) (2O)

and

1

13= aA 1[ t_l _ ^ "

_ i= isinkz( ae-'wt + a'l'ei ) (21)

From eq.(16) we may write the Maxwell stress tensor for the quantized field

in the cavity.

T.= 1⁄2(_oEx2+ _ Bye). (_)

With the use of eqs.(20, 21)

220
= _Er -I- + aa e-2iWt)cos2kz}. (23)

If the number state In > resides in the cavity the radiation pressure on the conducting

plates is

= < Tzz > = _r(n + 1). (24) P

For the vacuum state, ie n = 0, the radiation pressure in eq.(24) becomes P = b.w/2V

which is in agreement with eq.(1) for normal incidence. When the cavity is prepared with

the squeezed vacuum the radiahon pressure on the cavity wall is

= _ (cosh2r + sinh2r coskz cos2wt) (25)

.I.

P

where r is the real squeeze parameter [8]. The second term in the curly bracket

fast--oscillates around zero. Taking the time average of the radiation pressure we get

P = _V" cosh2r

As squeezing gets severe, the radiation pressure increases exponentially.

radiation pressure for the coherent state of amplitude a is

P =,_(a_ + 11.

(26)

The time average

(27)

The momentum density of the electromagnetic field is proportional to the

Poynting vector _. While the Poynting vector is clearly defined in classical theory as

_E ,, _,H that is not the case in quantum theory because .the operator--orderin. . g problem.

arises. When the Poynting vector is concerned in calculahon of the intensity of a field we

have the normal ordering of the field operators, ie $ _ ata [7]. For the problem in hand we

define the Poynting vector with the usual quantum mechanical symmetrization:

RADIATION FORCE ON AN ATOM

If a two-level atom is isolated in the cavity composed of two parallel

conducting plates the atom is subject to the radiation pressure from the standing wave. To

simplify the problem we assume that the field propagation is along the z---axis in Fig.2. We

consider the two-level atom with ground state Ig _> and excited state.le > in interaction

with the radiation field of frequency w. The atom of total mass M has the atomic centre---ol

mass momentum

E = - i 2.

The total Hamiltonian of the coupled system is[9]

H = I_I,tom(t) + Hti,ld(t) + l:'(t)2/2M + Hi(t) (30)

where under the electric---d_pole approximation the interaction Hamiltonian

I_Ii(t) = ikg coskz {a(t)-_t(t)} {_t) + _r_'(t)} (31)

with the coupling constant

g-e (32)

The operators _', _]" are the atomic transition operators.

The radiation force is dPJdt. Using the Heisenberg equation of motion for

221
the momentum operator

To calculate the time-dependent interaction Harniltonian

time-.dependence of operators. With the use of the Heisenberg equation of motion

- = i gcoskz + ;t(t)}. (34)

The equation is formally integrated to give [7]

where a = a(0). Similarly for the transition operator

kt)=e "-iw°t [_r...gcoskz [:[2xt(t,)kt,).._l][_t,).._'l'(t,)]ei_'dt, ]

(36)

where r = ,'r(0) and _w0 is the energy difference between the excited and the ground states.

The .operators a(!) a.nd _.t) !n eqs.(3,5, 36) still have integration to carry out. It is not

simple to solve the lteratlve integral equations. For the weak coupling between the field

and the atom we calculate the first---order perturbative solution

^

a(t) = e--ia"t(a + ax--fl_ t) (37')

_(t)=e--i_e't [_r + (2_r "l"r -- 1)( _a -- (a_f)] (.38)

where

(33)

we need find the

gC oskz

a = i( w0 - _) {exp[--i(w0 -- w)t] -- 1}

gcosk,, {exp[i(_0+ _)t]- 1} fl = ii wo + w)

gcoskz w)t] 1}

= -- i( _'o - _') {exp[i( wo - -

( = i(gc°skzwo + _,') {exp[i(a,'e + t,.,)'t] - 1}

We are interested in the radiation force on the atom when the cavity is

initially prepared with the vacuum state which is a limiting case of number states. "We

substitute eqs.(37, 38) and their Hermitian conjugates into eq.(31) to find the radiation

force on the atom with the use of eq.(33)

F'(t) = -21'_kg2sin2kz {[(2x t r- 1)aa. t + _'_rtlA

_ (3o)

where the first term in the curly bracket varies slowly with the parameter

A = 1 - cos(wo - w)t (40)

_0-

while the second term varies fast with the parameter

B =1 - cos(_o + _.,)t C41)

_0+ _

The second term is so called the counter-rotating term which is often neglected when the

field frequency is near resonant with the atomic frequency.

If the field is initially prepared with the number state ]n > and the atom in

its ground state the radiation force is

222_
_'(t) -- _xkg2sin2kz {nA + (n + 1)B}. (42)

Neglecting the fast oscillating term (n + 1)B, which is so called the rotating wave

approximation, we write

< _'(t) > = 2nhkg2sin2kz 1 - cos(_o - _)t. (43)

wO- o3

For the vacuum field, n = 0. While the slowly varying contribution is zero, we have the

radiation force from the counter---rotating term

< > = g2sin2kz1 - co,(,,,o+ ,,,)t

_0+ w

The time--average force is thus

1 (45)

< F > "- 2hkg2sin2kz w0 + w.

Eq.(47) shows that the time--average force is zero when the electric field is not only at its

nodes but also at its peaks, This reflects the fact that the radiation field exerts the force

proportional to the gradient of the intensity.

CONCLUSIONS

We have studied the radiation pressure from the field between the

conducting plates. We quantized the standing wave and introduced the Maxwell stress

tensor in the quantum mechanical form. The Maxwell stress tensor is calculated for the

radiation pressure on the plates. For the special case of the vacuum field the radiation

pressure shows the Casimir force which pushes the plates together.

_hen the atom interacts with the travelling wave the time--_verage

radiation force on the atom is zero [10]. The standing wave on the other hand exerts force

on the atom to push the atom to a node of the field. Since the force depends on the

gradient of the intensity, that is the electric field squared, the force is zero at the nodes and

peaks of the field.

ACKNOWLEDGEMENTS

for many

discussions.

The author wishes to thank Prof. R. Loudon for suggesting the problem and

stimulating discussions. The author also thanks to Dr. J. Dowling {or

He suggested to calculate the stability of the atom at a position.

KEFERENCES

[1] H.B.G. Casimir, 1948, Proc.Kon.Ned.AKad.Wet. 51, p.793.

[2a] G. Plunien, B. Muller and W. Greiner, 1986, Phys.P_p. 13....A,p.87.

[2b] S. Stenholm, 1988, Contemp.Phys. 2, P.105.

[3] P.W. Milonnl, R.J. Cook and M.E. Goggin, 1988, Phys.Repv.A 38, p.1621.

lilt E.A. Power. 1964, Introductory Quantum Eiectrodynamics, Longrnans, London.

- M. Planck (translated by M. Masius), i959, The Theory of Heat Radiation, Dover, NY.

J.D. Jackson, 1975, Classical Electrodynamics, John V_iley, New York.

tL Loudon, 1983, The Quantum Theory of Ligh_ Clarendon, Oxford.

R. Loudon and M.S. Kim manuscipt in preparation.

It. Loudon and P.L. Knight, 1987, J.Mod.Opt. 34, p.709.

223
[9] A. A1-Hilfy and lq.. Loudon, 1985, Opt.Acta 32, p.995.

M. Babiker, 1984,J.Phys.B 17, p.4877.

[10] J.P. Gordon, 1973, Phys.Rev.A 8, p.14.

224
N92-22068

Squeezed light from conventionally pumped multi-level lasers

T.C.Ralph and C.M.Savage

Department of Physics and Theoretical Physics,

Australian National University, Canberra, ACT 2601, Australia

Abstract

We have calculated the amplitude squeezing in

the output of several conventionally pumped

multi-level lasers. We present results which

show that standard laser models can produce

significantly squeezed outputs in certain

parameter ranges.

lasing level contains at least two steps with

approximately equal rates. One of these steps may

be the pump itself. Any other rates must be faster.

In previous multi-level treatments solutions have

been obtained by assuming the pump rate is much

slower than all other rates (Ref.6,7). Squeezing

will not be seen under these conditions.

Introduction

Production of non-classical light by lasers is an

active field both theoretically and

experimentally. Sub-Poissonian output has

been predicted and observed from lasers in

which a regular pumping mechanism reduces

the population fluctuations in the lasing levels

(Ref. 1-4).

We present here results of squeezing spectra

calculations for incoherently pumped 3-level and

4-level lasers and a coherently pumped 4-level

laser (figl). The results highlight the basic effect

and how it varies between the models. We also

discuss a simple statistical model which

illustrates the physical mechanism behind the

squeezing.

Recently we have found that rigorous solutions

of conventionally pumped 3 and 4-level lasers

predict amplitude squeezing in their output

(Ref.5). This is contrary to standard laser

theory which predicts Poissonian output far

above threshold when the pumping is

conventional (Ref.6,7). Our results are in

agreement with those of Khazanov et al

(Ref.8).

The basic requirement for sub-Poissonian

output, without regular pumping, is that the

sequence of levels involved in moving an

electron from the lower lasing level to the upper

I _ 13>

I iT2_ I g

FI 12>

I I

I I1>

_ I 14>

I i_ y34 _t3>

I w I w12>

I

'r I 1_2 I1>

Fig.1 Laser atomic level schemes. Incoherently

pumped 3-level on the left and incoherently

(E=O) or coherently (F=O) 4-1evel on the right.

The Yij are spontaneous decay rates, g is the

dipole coupling strength.

225
Squeezing Results

Using standardtechniques(Ref.6,7) a master

equationfor the reduceddensity operatorp of

the atoms and cavity is derived. We solve for the

full quantum mechanics of the master equation

by transforming it into an equivalent partial

differential equation for the generalized Pfunction of Drummond and Gardiner (Ref.9).

We make the usual approximation that the

quantum fluctuations are small perturbations on

the semiclassical steady state (Ref.2,9,10). The

amplitude squeezing spectrum, V, of the laser

output field is calculated in the usual way

(Ref. 10).

In Figure 2 we plot the spectral variance at the

zero frequency local minimum of the spectrum

as a function of pump rate for the three cases.

The full spectra are approximately Lorentzians

(in the region shown) with linewidths

corresponding to that of the laser cavity. Laser

phase diffusion has been ignored. Parameters

have been chosen to show maximum squeezing.

0 is perfect squeezing and 1 is the coherent state

spectral variance.

Squeezing is improved both by increasing the

number of levels with similar rates and by using

a coherent pump. The 3-level laser has

maximum squeezing of 50% when the

spontaneous decay rate ?'12 is double the pump

rate (F). The incoherently pumped 4-level laser

has maximum squeezing of 66% when

F = ?'34 = 0.57'12. The improvement due to the

coherent pump is more significant. If

-_/-8E = ?'34 = 0.57'12, where E is proportional

to the coherent field strength, 80% squeezing is

predicted.

0.6'

0.5.

0.4

0.3

0.2

O.

'°o° °

.°°.°°°°

°°..°"

°°°°

Fig.2 The zero frequency minimum of the

amplitude squeezing spectral variance versus

pump rate for the incoherently pumped 3-level

(dotted line) and 4-1evel (dashed line) and

coherently pumped 4-1evel (solid line) lasers.

For the incoherently pumped case P= F and for

the coherently pumped case P=E .

Parameters in units of ?'12 are:

?'23 = 10-6, ?'34 = 0.5, _ = 1, t¢ = 0.01. t¢ is the

cavity decay rate and _, is the scaled dipole

coupling constant (Ref_5).

Discussion

The origin of the squeezing can be understood in

terms of the temporal behaviour of the electrons

in individual atoms 1. The variance in the time

the pump cycle takes to place an electron in the

upper lasing level of an individual atom, At 2,

and the spectral variance at zero frequency,

Vmi n , are related by 2

1H.Ritsch, P.Zoller, C.W.Gardiner and D.F.Walls,

to be published, (1991).

2T.C.Ralph and C.M.Savage, to be published,

(1991).

226
A_2 At2

= - (1)

Vmin _ (i)2

where i is the mean time it takes for the electron

to arrive in the upper lasing level, A_ is the

photon number variance of the output and _ is

the mean number of photons. To obtain this

result we have assumed the laser is well above

threshold and has a strong enough dipole

coupling such that the lasing transition time can

be considered to have zero variance. Also we

assume spontaneous emission out of the upper

lasing level is negligible.

The right-hand side of (1) can be evaluated

exactly 2. For an (r+3)-level incoherently

pumped laser the independence of the noise

introduced in each step leads to the following

expression

(2/)'L) 2 +(1/)'1)2+ ..... (1/)'r) 2 + (l/F) 2

Vmin = , 2

(2/rL + 1/r1+ ..... l/r)

where YL is the decay rate out of the lower

lasing level, F is the pump rate and 2'1 ...... Yr

are the rates of the intermediate steps. The rates

are matched for optimum noise reduction when

F=Yl =. .... =Yr =0.5 YL. The minimum value of

Vmi n is then 1/(r + 2). If the pump rate is much

slower than all the other rates then Vmi n->l, ie

Poissonian. This is the limit in which previous

calculations were carried out.

For a certain range of pump rates a coherent step

introduces less noise into the pump cycle than an

incoherent step 2. This leads to superior

squeezing in the coherently pumped laser.

Summary

We have presented a brief report of results we

have obtained from rigorous solutions of

conventionally pumped standard laser models.

Contrary to established theory we find amplitude

squeezing of the output beam is possible in certain

parameter ranges. Physically we find the noise

suppression is due to the independence of the

noise introduced in the various steps involved in

inverting the atoms.

We see no fundamental reason why lasers could

not be built which operate in regimes meeting the

requirement for squeezing.

References

1 y. M. Golubev and I. V. Sokolov, Soy. Phys.

JETP 60, 234 (1984).

2 M. A. Marte and D. F. Walls, Phys. Rev. A

37, 1235 (1988).

3 S. Machida and Y. Yamamoto, Phys. Rev. Lett.

60, 792 (1988).

4 W. H. Richardson and R. M. Shelby, Phys.

Rev. Lett. 64, 400 (1990).

5 T.C.Ralph and C.M.Savage, Opt. Lett, July,

(1991).

6 W. H. Louisell, Quantum Statistical Properties

of Radiation (Wiley-Interscience, New York,

1973).

7 H. Haken, Laser Theory, in Encyclopedia of

Physics, XXX/2c, ed. S. Flugge (SpringerVerlag, Heidelberg, 1970).

8

A. M. Khazanov, G. A. Koganov, and E. P.

Gordov, Phys. Rev. A 42, 3065 (1990).

9 p. D. Drummond and D. F. Walls, Phys. Rev.

A 23, 2563 (1981).

10 M. D. Reid, Phys. Rev. A 37, 4792 (1988).

227
N92-22069

STATISTICAL PROPERTIES OF SQUEEZED BEAMS OF LIGHT

GENERATED IN PARAMETRIC INTERACTIONS"

Reeta Vyas

Department of Physics, University of Arkansas,

Fayetteville, AR 72701

ABSTRACT

Fluctuation properties of squeezed photon beams generated in three wave mixing processes such as second harmonic generation, degenerate and nondegenerate parametric oscillations, and homodyne detection are studied

in terms of photon sequences recorded by a photodetector.

Photon number fluctuations and photon number

correlations are fundamental properties of a light beam.

These properties are different for different light sources

and can be used to characterize photon beams. In this

short communication we discuss our work on statistical properties of squeezed photon beams generated in

three wave interaction processes in terms of counting

and waiting time distributions. We summarize some of

the interesting results obtained for these systems. Processes that we consider here include second harmonic

generation, and degenerate and nondegenerate parametric down conversion (DPO and NDPO). Squeezed state

of light have been realized in these systems experimentally (Ref.1). Homodyne statistics when squeezed

light produced by the DPO is mixed with coherent light

from a local oscillator are also discussed. A dynamical

model for these beams is used and photon sequences

recorded by a photodetector are calculated.

We use positive-P representation (Ref.2) to map

quantum mechanical equations of motion for the annihilation and creation operators onto a set of C-number

stochastic equations for the complex field amplitudes.

Using simple transformation of field variables it can

be shown that the field produced in these processes

can be described in terms of independent real Gaussian

stochastic processes (Ref.3-4).

We use a generating function technique to obtain the statistics of the photons emitted by these light

sources. The generating function G(s, t, T) for the photon statistics measured by a detector with a parameter

s is given by (Ref.5)

-so t+T I(t')dt'

G(s,t,T) = (e .,t ). (1)

Here 7} is detector efficiency and I(t) is photon flux

emitted by the source. Generating function G(1, t, T)

is simply the probability of detecting no photon in the

time interval t to t + T. In order to obtain generating

function we express I(t) in terms of the c-number field

variables. Statistical averaging is performed by making Karhunen Lo6ve expansion of the field variables in

terms of a set of orthogonal functions. Following the

method developed in our earlier investigations (Ref.3)

we derive an analytical expression for the generating

function G(s,t,T) for the photon statistics. From this

generating function various statistical quantities such as

factorial moments, photon counting and waiting time

distributions can be obtained.

The photon counting distribution p(m,t, T) is

the probability of counting m photons in the time interval t to t,+T. It can be obtained from the generating

function by using the relationship

p(m,t,T)-(-1)m [d_G(s,t,T)] . (2) Tn! s=l

The waiting time distribution w(t, T) is the probability

density for two successive photoelectrons to be separated by the time interval T and it is given by

d 2

w(t, T = t' - t) = -011(0) -1 d--/_G(1, 1, t' - t). (3)

In the stationary regime these quantities are independent of initial time t. Here we only summarize photon

statistics only in the steady state regime. The field from

the DPO can be expressed in terms of two independent

Gaussian random variables with mean zero and different variances

l l,_ ¢1 6ij e -x'r (4)

< ui(t)uj(t + T) >= _ ,_---_-.

Here n is the mode coupling constant, and E is the

dimensionless amplitude of the pump beam incident on

the cavity. The decay constants A1 and )_2 are given

by

= (7 -- In '1), AS= (7 + el). (5)

Here (1/27) is the cavity lifetime. Below threshold

A1 and As are always positive. Using the properties

of field variables ul and u2 we obtain the generating

function for the DPO as (Ref.3)

G(s, T) = Ql(s, T)Q2(s, T), (6)

229

PRECEDING PAGE BLANK NOT FILMED
where

Qi(s,t,T)= [cosh(ziT)_-_t)sinh(ziT)] , (7)

with

fi(t) = _ + , (8)

and

z_ = ,Xx_+ 2srm,_c, (9)

Mean photon number inside the cavity is given by

=_ 2-1,_i 2)

(10)

For the DPO f is equal to half. Once the generating

function is known photon counting and waiting time

distributions are obtained from Eqs. (2) and (3). For

small mean photon number and short counting time

p(m, T) decreases monotonically.

q

g.

Ii R = 0.95

0 1 30 45 60

Figure la. Photon counting probability distribution for

the DPO for /t = 0.95, unit efficiency, and counting

time interval 27T = 20.

For long counting times p(m, T) shows sharp

even-odd oscillations implying that the probability of

detecting odd number of photons is much smaller than

the probability of detecting even number of photons.

As the mean photon number h is increased these evenodd oscillations become smaller. Figure (la) shows

p(m, T) for R = _ _/7 = 0.95 near threshold corresponding to h = 4.63. These curves are meaningful

only for integer values of m. We see that near threshold even-odd oscillations become less pronounced and

p(m,T) develops a long tail. We have also studied

photon statistics for the DPO in the transient regime,

that is, during its evolution from vacuum state to the

steady state (Ref.4). For small transient time even-odd

oscillations in photon counting distributions are even

sharper than the even-odd oscillations in the stationary

regime.

With the degenerate modes of a parametric oscillator, nondegenerate modes are also present. We

consider the nondegenerate modes of parametric oscillator for which two nondegenerate photons have the

same frequencies. These fields can be expressed in

terms of four real random Gaussian variables. Here

we discuss two cases, one in which amplitudes of the

two nondegenerate modes are homodyned and second

in which intensities of the two modes are added together. For a given pump strength the NDPO the mean

photon number is much smaller than the mean photon

number for the DPO. Mathematical expression for the

generating function for the first case is similar to the

generating function for the DPO. Thus p(m, T) for the

NDPO also shows even-odd oscillations. However, for

the same pump strength, even-odd oscillations in the

NDPO are sharper then the oscillations for the DPO.

They are centered towards smaller values of m. The

difference between the DPO and the NDPO lies mainly

in the value of the mean photon number.

¢q

77"1,

Figure lb. Photon counting probability distribution for

the NDPO for R = 0.95, efficiency r] = 1, and counting

time interval 27T = 20.

The second case that we consider is when intensities of the two modes are added together. In this case

the power £ appearing in Eq. (7) is one (Ref.6). In this

case the expression for Q1 (s, T) is similar to the generating function for thermal light (Ref.5). The counting

230
distributionforNDPO,however, isvery different from

that for the thermal light. It shows even-odd oscillations as a function of m whereas such oscillations are

not seen for thermal light. Figure (lb) shows evenodd oscillations in p(m, t) for R = t¢e/7 = .95 for the

second case.

Next we discuss photon statistics of the fundamental beam from an intracavity second harmonic generation (SHG). Field for this system can be expressed

in terms of two real Gaussian random variables and a

coherent component (Ref.7). These results are obtained

by linearizing the field amplitude equations around the

deterministic steady state values. The generating function for the SHG can be written as (Ref.6)

G(s, T) = Ql(s, T)Q2(s, T) e -'f(s'T) . (10)

Here

2

f(s, T) = -2sqfinoT [ z_ +

2srlhV( Zl ) 2 I

z_(l+ ,x_ ) (2+ART) _,_ ] J

(11)

with

v(zl)= i+__ J ' (12)

Here Ql(s, T) and Q2(s, T) are given by equation (7)

with l = 0.5 and

•_1 = (1 + 35), A2 = (1 + fi), (12)

For the SHG zl and z2 are given by

(13)

Here h and no are the average and threshold photon numbers, respectively. From this generating function various statistical quantities of interest can be calculated. Photon sequences in the SHG can be antibunched. Although the antibunching effect is very

small riding on an intense coherent background it is

clearly reflected in the behavior of the waiting time

distribution for the SHG.

Using similar techniques we also obtain the generating function when l!ght from the DPO is homodyned with coherent light from a local oscillator. Depending upon the relative phase between coherent light

from the local oscillator and squeezed light from the

DPO we can see sub-Poissonian or super-Poissonian

statistics for the homodyned photon beam. Figure (2)

shows waiting time distribution when the relative phase

is 0° and 90 °. We see bunched light when coherent

component is added to the unsqueezed component and

antibunched light when coherent component is added

to the squeezed component.

O

iJ _ = 02

_0 = 2.0

b-o

.,dQ-

,k

I I I I

1 2 3 4

T

Figure 2. Waiting time distribution for DPO mean photon number fi = 0.2 and local oscillator mean photon

number no = 2. Dashed curve is for coherent light

REFERENCES

* It is a pleasure to acknowledge collaboration with S.

Singh and A. L. DeBrito for part of this work. This

work was supported in part by grants from Arkansas

Science and Technology Authority (ASTA).

1. Special issue on squeezed states of light, J. Opt. Soc.

Am. B4(No. 10) (1988).

2. P. D. Drummond and C. W. Gardiner, J. Phys. 13, 2353

(1980).

3. Reeta Vyas and S. Singh, in Coherence and Quantum

Optics vI, edited by J.H. Eberly, L. Mandel, and E.

Wolf (Plenum, New York, 1990), p. 1189; Opt. Lett.

14, 1110 (1989); Phys. Rev. A 40, 5147 (1989).

Reeta Vyas and A. L. DeBrito, Phys. Rev. A 42,

592 (1990); Nonlinear Dynamics in Optical Systems,

edited by N. B. Abraham, E. M. Garmire and P. Mandel

(OSA, Washington, DC, 1991), p. 589.

B. E. Saleh, Photoelectron Statistics (Springer-Verlag,

Berlin, 1973); Reeta Vyas and Surendra Singh, Phys.

Rev. A 38, 2423 (1988); H. J. Carmichael, Surendra

Singh, Reeta Vyas, and P. R. Rice, Phys. Rev. A 39,

1200 (1989).

Reeta Vyas, [to be published].

G. S. Holliday and S. Singh, in Coherence and Quantum Optics VI, edited by J.H. Eberly, L. Mandel, and

E. Wolf (Plenum, New York, 1990), p. 509.

.

.

.

7.

231
N92-22 070

NONDESTRUCTIVE MEASUREMENT OF INTENSITY

OF OPTICAL FIELDS USING SPONTANEOUS PARAMETRIC

DOWN CONVERSION

A.N.Penin, G.Kh.Kitaeva, *)A.V.Sergienko

Department of Physics, Moscow State University,

Moscow 119899, USSR

*)Present address: Department of Physics and Astronomy,

University of Maryland, College Park, MD 20742

Results of nondestructive measurements of intensity (photons

per mode) of light from different sources are discussed. The

procedure of measurement does not destroy the state of the optical

field. The method is based on using the second order nonlinearity of

crystal media lacking a center of symmetry and the nonclassical

properties of the process of Spontaneous Parametric Down

Conversion (SPDC).

The interaction of laser radiation with nonlinear crystal leads

to the spontaneous emerging of correlated photons in two modes (col

and 002) of the optical field connected by phase matching conditions

0)1 + t.02 = O)L , kl + k2 = kL. The quantum theory of the parametric

amplification process /1-4/ shows that if all initial modes of

radiation are in the vacuum state (except the pump radiation), then

photon flux after the nonlinear interaction in mode with , for

example, frequency COl(Fig.l) is:

Nl'(t) = sinh2(gt) (1)

Where g=(2_/h)z(2)E0, Z(2) = an effective value of second order

susceptibility, and E0 = amplitude of pump radiation. If we have

initial radiation from an external source S producing an intensity n20

in the mode of frequency t.o2, then after the nonlinear crystal the

intensity of radiation in mode of frequency e01 is given by:

NI"= (1 + n2 0) sinh2(gt) (2)

Thus the value of the intensity n2 0 of the initial radiation can be

easily calculated without destruction of initial optical state after two

233

PRECEDING PAGE BLANK NOT FILMED
measurements of average intensity nl of radiation in the

parametrically conjugated mode of frequency 031:

N1 t'

n20 = NI' - 1 (3)

This result is a reflection of the intrinsically quantum character of

the SPDC process. The main point of this method is the use of

universal properties of electromagnetic vacuum fluctuations, i.e. that

the brightness is equal to one photon per mode. This explanation

reflects the phenomenological approach in an effective treatment

/5/. Detailed quantum description of nondestructive measurement

of parameters of optical fields using third-order Kerr nonlinearity

was made in /6/. That results could be easily transformed for the

case of second order nonlinear susceptibility ;_(2).

The outline of the experimental setup for the nondestructive

measurement of intensity of optical fields is shown in Fig.1. The

radiation of argon ion laser _ = 488.8 nm interacts with a LilO3

nonlinear crystal. The scattered (spontaneously generated) radiation

of frequency o31 is registered by a photomultiplier tube. The

radiation from the external source S falls on the crystal in the

direction defined by phase matching conditions. The position of the

chopper disc divides the process of measurement into two stages:

measurement of nl' and nl" values. After the propagation inside the

crystal the external radiation of frequency 032 can be used for other

purposes. In the visible region of the spectrum we can neglect

absorption of radiation inside the nonlinear crystal and consider this

kind of measurement as nondestructive.

The accuracy of this method improves the closer the brightness

of the measured radiation is to the brightness of electromagnetic

vacuum fluctuations of the same frequency. It has advantages in

measurements of laser radiation and radiation from other bright

(high effective temperature) sources of light such as plasmas, high

voltage discharges etc.

For our first measurements we used an infrared region of

radiation from a high temperature tungsten spiral incandescent lamp

as a source of external radiation. In the IR region this thermal

radiation has an intensity of about 10 -2 photons per mode ( in the

visible region in accordance with Planck formula Ntherm--10 -4

234
Fig.l. The outline of the experiment for nondestructive measurement

of the intensity n20 of light from an external source S.

0.02

0.01

0 [ ! I

4

3 (_crons)

Fig.2. Result of measurement of intensity of thermal radiation source

(tungsten spiral incandescent lamp) in the IR region of spectrum.

Solid line corresponds to the calculation by Planck formula intensity

distribution for the temperature measured by pyrometric technique.

All results include correction on crystal absorption coefficient.

235
photons/mode). The results of measurement are presented in Fig.2.

The solid line corresponds to the result calculated by the Planck

formula for the temperature measured by a pyrometric technique.

The accuracy of these measurements was about 5-20%. Fresnel

reflection of radiation on the crystal borders gives an additional

source of systematical error in this measurement. This effect could

be eliminated by using of a correction coefficient.

The laser (in our case a CW He-Ne laser at _=3.39 _t) has a much

higher spectral density of radiation (106 photons/mode) and can be

measured with better accuracy. However, in this case a special

procedure of matching external and internal radiation space modes

inside the crystal is needed.

The same problem arises in measurements of intensity of

second harmonic radiation generated in a KDP crystal by a Nd:YAG

pulsed laser. For the radiation at _. =532 nm the measured intensity

was 4-10 3 photons/mode.

1.Louisell, W.H., A.Yariv and A.S.Siegman, 1961, "Quantum

Fluctuations and Noise in Parametric Processes," Phys.Rev. v.124,

p.1646.

2.Mollow, B.R., 1967, "Quantum Statistics of Coupled Oscillator

Systems," Phys.Rev. v.162, p.1256.

3.Giallorenzi, T.G., and C.L.Tang, 1968, "Quantum Theory of

Spontaneous Parametric Scattering of Intence Light," Phys.Rev.

v.166, p.905.

4.Zel'dovich, B.Ya., and D.N.Klyshko,1969, "Statistics of Field in

Parametric Luminescence," Sov.Phys.JETP Lett. v.9,p.40.

5.Klyshko, D.N., 1988, Photons and Nonlinear Optics, Gordon and

Breach, New York.

6.Yamomoto, Y., and H.A.Haus, 1986, "Preparation, Measurement and

Information Capacity of Optical Quantum States," Rev.Mod.Phys.

v.58, p.1001.

236
SHORT-CAVITY

N92-22071

SQUEEZING IN BARIUM _/f)_

D.M. Hope, H-A. Bachor, P.J. Manson and D.E. McClelland

Department of Physics and Theoretical Physics

Faculty of Science

Australian National University

P. O. Box 4, Canberra A.C.T. 2601, Australia

Abstract

Broadband phase sensitive noise and squeezing

have been observed experimentally in a system of

barium atoms interacting with a single mode of a

short optical cavity. Squeezing of 13+3% was

observed. A maximum possible squeezing of

45+8% could be inferred for our experimental

conditions, after correction for measured loss

factors. Noise reductions below the quantum limit

were found over a range of detection frequencies

60-170 MHz and were best for high cavity

transmission and large optical depths. The amount

of squeezing observed is consistent with

theoretical predictions from a full quantumstatistical model of the system.

indicate that a particularly favorable configuration

for squeezing in this model exists where decay

rates of the atomic polarization and the cavity

mode are approximately matched. To achieve the

matching of rates without degrading the cavity

finesse the cavity length must generally be reduced

to a few millimetres; hence the term 'short-cavity

squeezing'. Orozco et al. observed 30%

squeezing in atomic sodium in this regime (Refs.

5-6). The J=0 _ 1 553 nm transition of 138Ba is a

suitable medium to test detailed theoretical

predictions of the dynamics of the cavity-atom

interaction. Its simple structure allows us to avoid

the complications of optical pre-pumping,

necessary to restrict alkali atoms to two-level

behaviour.

Introduction

The model of the interaction between a cavity

mode and an ensemble of two level atoms is of

fundamental importance in quantum optics (Ref.

1). Theory (Refs. 2-4) and experiment (Refs. 5-6)

Experiment

A schematic diagram of the experimental

arrangement is shown in Figure 1. A cw ring dye

laser supplies 300 mW of light at 553 nm. The

laser is frequency stabilized to 1 MHz. The first -

Cw ring dye laser

Galvoplate

Data

Acquisition

System _ _D_0_ . M2

Spectrum I _ PD2

Analyzer L_____2 v _ _

Intensity _------_L_ AOM

stabilizationS, Mode-

_ matching

Local oscillator _ lenses

___B 1 Signal

a oven

Figure 1. Experimental arrangement

237
order diffracted beam from an acousto-optic

modulator(AOM) passesthrougha pair of modematchinglensesandis divided into local osillator

and signal beamswith a 50% beamsplitter.The

AOM serves to isolate the laser from cavity

feedback, and is also operatedas an intensity

stabilizer.The phaseof the local oscillator(LO)

with respect to the signal beamphasemay be

variedwith a scanninggalvoplate,andtheLO and

cavity output beams are combined on a 50%

beamsplitter.The outputports of the beamplitter

arefocusedonto photodiodesPD1andPD2in the

balancedhomodynedetection system.Typically

the detectorgives 2 dB of quantumnoiseabove

amplifier noiseat 130 MHz for 1 mA of current.

The noise spectrumis displayed on a spectrum

analyzerandrecordedonacomputer.

The optical cavity is comprisedof two dielectric

mirrors mountedon piezoelectricstacksseparated

by 4.3 mm. It is held in a stainlesssteelchamber

evacuatedto 10-6 torr. The input couplerM1 has

transmissioncoefficientT1 < 0.001andaradiusof

curvatureRzl of 1 meter.The outputcoupler M2

hasT2 = 0.036andRz2= 2 m. Thecavity finesse

is measuredasF=150 +10 and the throughput on

resonance as 1.3%. The cavity beam waist is

calculated to be 92 IXm.

A high-density collimated beam of barium is

generated and injected perpendicularly to the cavity

mode. Optical depths of up to or01 =3.5 can be

achieved for a beam of diameter 2.3 mm.The

FWHM of the absorption peak is 44 MHz for

o_01=3.5, where these values are measured using a

monitor beam perpendicular to both cavity axis

and atomic beam. The natural linewidth of the

atomic line is T=20MHz. The additional width can

be attributed to residual Doppler broadening. We

have T_I_/_:=0.08 for this experimental

configuration, where T±=27 is the transverse

atomic decay rate for purely radiative decay, and _c

is the decay rate of the cavity mode.

Results

Figure 2 shows a spectrum analyzer trace with

13+3 % squeezing, after corrections are made for

the electronic noise contribution. It was observed

for a cavity input intensity of 14 mW, atomic beam

optical depth oc01 = 2.7 and an atomic detuning

of 600 MHz below the 553 nm transition. The

-60.0-

E

v-60.5_

0

Q.

o-61.0-

z

v

tO

(_

E

O3

t-

>

o

2 r_ change in 4)1o

0.8--

(a)

IO0

5O

0

0 1 2 3

Cavity detuning (cavity linewidths)

Figure 2. Noise power (a) and cavity transmission

(b) recorded for optical depth or01=2.7, atomic

detuning 600 MHz below the transition, cavity

input power 14 mW, detection frequency 147 MHz.

detection frequency was 147 MHz. A 4-second

quantum noise recording was taken while

scanning the phase of the local oscillator and

simultaneously sweeping through the cavity

resonance. The modulated signal in Fig. 2a is the

noise power. The quantum noise limit (equivalent

to 1 Vacuum Noise Unit, or 1 VNU) is given by

the center of the solid bar. It is obtained by

recording a noise trace with cavity output blocked,

under the same experimental conditions. The

width of the bar represents the rms fluctuations of

the noise trace. The quantum noise limit trace has

been corrected for the non-negligible amount of

power in the cavity output (0.18 mW compared to

1.1 mW in the LO beam). Figure 2b shows cavity

transmission plotted in units of cavity linewidths

from the resonance peak.

We see large amounts of phase sensitive noise

near the peak of cavity transmission, together with

clear reductions below the quantum noise limit,

with both noise and squeezing decreasing with the

sweep though the cavity. Other data reveal that

squeezing exists for a broad range of frequencies

60-170 MHz, and for a bistable cavity. In the

bistable regime phase sensitive noise and

squeezing are seen predominantly on the upper

branch.

238
Figure 3 simulatesthe experiment,usinga plane

wave ring-cavity quantumtheory of squeezing in

optical bistability (Refs. 3-4). The cooperativity,

cavity characteristics and atomic detuning are those

of the results in Figure 2, within experimental

uncertainties; cooperativity C=_01F/(2_) = 64,

atomic detuning A=(coa-coL)/T_L=50 and

T_L/_:=0.08, where coa and coL are the frequencies

of the atomic transition and the signal laser

respectively. Other parameters are optimized for

best squeezing. Cavity detuning is measured from

the peak transmission in units of cavity linewidth

and is incremented through the cavity resonance.

The parameter corresponding to LO phase is

varied at approximately the rate used in the

experiment. Trace (a) is the squeezing spectrum

plotted on a logarithmic scale against the left

vertical axis, where a variance V of unity

corresponds to the quantum noise limit and zero

corresponds to perfect squeezing. Trace (b) is the

intracavity photon number (proportional to cavity

transmission), plotted on the right vertical axis and

given in units of the saturation intensity on

resonance (Ref. 3).

Figure 3 shows good qualitative agreement with

experiment. Maximum squeezing is located near

the peak of cavity resonance, and the squeezing

decreases for decreasing transmission. The best

squeezing at these parameters is 55%. Loss factors

that reduce the amount of observed squeezing

have been measured individually, as follows;

>

O

¢..

m 1

"r"

>

10 13000

F 2000

l 000

(b)

0.1 _0

- 0 1 2 3

Cavity detuning

Figure 3: Theoretical modelling of experiment

cl

E

rco

¢.-

°_

ccavity escape efficiency p = 0.88, detector

quantum efficiency o_=0.65, mode matching

efficiency 112=0.56, propagation efficiency

T0=0.86, where these quantities are defined as in

Refs. 6-7. We calculate that for ideal propagation

and detection efficiencies the observed squeezing

would equal 45+8%. This is in reasonable

agreement with the theoretical prediction of 55%.

Conclusion

It was found that the interaction of a barium beam

with a single mode of a short optical cavity

generated reductions below the quantum noise

limit of 13+3%. We infer that in the absence of

loss factors 45+8% squeezing would have been

observable. This is consistent with the amount of

squeezing (about 55%) predicted from a quantum

theory of optical bistability.

References

1. Carmichael, H.J., 1986, "Theory of Quantum

Fluctuations in Optical Bistability," in Frontiers in

Quantum Optics, E.R. Pike and Sarben Sarkar,

eds., Bristol, p. 120; and references therein.

2. Castelli, F., Lugiato, L.A., and M. Vadacchino,

1988, "Squeezing in Optical Bistability without

Adiabatic Elimination," II Nuovo Cimento , 10

D(2), p. 183.

3. Reid, M.D., 1988, "Quantum Theory of

Optical Bistability without Adiabatic Elimination,"

Phys. Rev. A, 37(12), p. 4792.

4. Hope, D.M., McClelland, D.E., and

C.M.Savage, 1990, "Squeezed-State Generation

in a Spatially Varying Field Mode without

Adiabatic Elimination," Phys. Rev. A, 41(9),

p.5074.

5. Raizen, M.G., Orozco, L.A., Min Xiao, Boyd,

T.L., and H.J.Kimble, 1987, "Squeezed-State

Generation by the Normal Modes of a Coupled

System", Phys. Rev. Lett., 59(2), p. 198.

6. Orozco, L.A., Raizen, M.G., Min Xiao,

Brecha, R.J., and H.J. Kimble, 1987, "SqueezedState Generation in Optical Bistability," J.Opt.

Soc. Am. B, 4(10), p. 1490.

7. Ling-An Wu, Min Xiao, and H.J. Kimble,

1987, "Squeezed States of Light from an Optical

Parametric Oscillator," J.Opt. Soc. Am. B, 4(10),

p.1465.

239
N92-22072

AMPLITUDE SQUEEZED LIGHT FROM A LASER

D.L. Hart and T.A.B. Kennedy

School of Physics, Georgia Institute of Technology

Atlanta, GA 30332-0430

I. Introduction

Intensity squeezed light has been successfully generated using semiconductor lasers with sub-poissonian

pumping. 1 Control of the pumping statistics is crucial and is achieved by a large series resistor which regulates

the pump current; its sub-poissonian statistics are then transferred to the laser output. The sub-poissonian

pumping of other laser systems is not so simple however, and their potential as squeezed state sources is

apparently diminished. Here we consider a conventional laser incoherently pumped well above threshold, and

allow for pump depletion of the ground state. In this regime subpoissonian photon statistics and squeezed

amplitude fluctuations are produced.

II. Theoretical model

The atomic level scheme for the laser is indicated in fig. 1. We follow the notation of Lax and

A A

Louisell, 2 where N i (i = 0,1,2 ) are the atomic population operators and n the laser mode photon number

operator. The quantum Langevin rate equations are given by

d ^ ^ ^ ^ ^

_--_N0 = -w20N0 + F1NI+w02N2 + GO

d A A A A A A

d i N1 = "(F1 + FIn)N1 + (w12 + l-In)N2 + G1

d A A AA A A A

_-_ N2 = w20N0 + YInN1 - (F2 + I-In)N2 + G2

d ^ ^ ^ ^ ^ ^

d---tn = -'in + Fin (N 2 - N 1 )+ Gp.

Above threshold the mean inversion D = N2 - N1, is fixed at a constant value independent of pumping.

241

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The calculation of the quantum noise properties proceeds by linearizing the equations of motion about

the semiclassical steady states to calculate the variance c 2 in photon number, about the steady state mean value

n. The Mandel Q-parameter, and amplitude squeezing spectrum normalized to unit shot noise can then be

constucted from the mean and variance 3

(_2 _ n 1

Q =-- V(co) = 1 + 2Q 2

n ' 1 + (°/T)

where Q > 0, = 0, and < 0, correspond to superpoissonian, poissonian and subpoissonian photon statistics,

respectively, and co is the spectral offset from the laser frequency. Subpoissonian photon statistics, and

concomitant intensity squeezing ( V < 1 ) in the output are signatures of the quantum mechanical nature of the

electromagnetic field.

For the case where all spontaneous emission from the excited state goes to ground (w12 = 0 ) adiabatic

elimination of the atomic fluctuations leads to the equation for photon number fluctuations, for n >> n s (the

saturation photon number), and dropping carets for notational simplicity

d

d---tAn =-yAn + G(t),

where

G(t) =

w02 2w20

1 F1 F1

2w20G2- w02 2w20G1 + Gp .

w02 1 + +

1+ -fi-1+ rl rl

We then find

2w20

F1

(:r2= n - n+O(ns)

(I+ w0-_12 +2_120)2

so that the intracavity photon statistics are sub-poissonian, with intensity squeezing ( V < 1 ) in the output.

Fig. 2 shows the intensity squeezing at the laser frequency ( co = 0, in the rotating frame of reference ), as a

function of pump rate, for three different values of the stimulated emission coefficient 17. With increase in I7,

the degree of squeezing saturates at around 45% below shot noise level. Analysis of this result indicates that the

predicted subpoissonian photon statistics and squeezing are due to a reduction in the role of pump noise and

242
spontaneous emissionfromtheupperatomiclevel,whenw20isincreasedfromtheundepletedpumpregime(

wherethephotonstatisticsarepoissonianand the output is shot noise limited ), towards F1. For larger pump

rates these noise terms continue to decrease, and one might expect the degree of squeezing to increase. However

spontaneous emission from the lower lasing level, which has little effect on squeezing in the undepleted pump

regime provided only that F1 >> F2, becomes increasingly important as the pump rate is increased ( even for F1

>> F 2 ), and this random noise causes the degree of squeezing to be reduced with yet further increase in pump

power. The two opposing tendencies may be seen by inspection of the pump rate dependence of the coefficients

of the noise terms in the equation for G(t).

The results presented are consisitent with fully numerical solutions of a three level laser obtained by

Ralph and Savage. Note that the squeezing properties of the laser were also recently considered in ref. 5.

After our conference presentation we received a preprint by Ritsch et al, 6 which contains closely related results.

W12

Wo2

W2o _"

>

1.1 _i

1.0

0.9

0.8

0.7

0.6

0.5 J

0.5 1 I .5 2 2.5 3

Pump Rate (w2o/F)

Fig.l: Atomic level scheme Fig.2: Amplitude squeezing versus pumping

References:

1. S. Machida, Y. Yamamoto and Y. Itacha, 1987, Phys. Rev. Lett. 58, 1000.

2. M. Lax and W.H. Louisell, 1969, Phys. Rev. 185, 568.

3. T.A.B. Kennedy and D.F. Walls, 1989, Phys. Rev. A40, 6366.

4. T.C. Ralph and C.M. Savage, private communication.

5. A.M Khazanov, G. A. Koganov and E.P. Gordov, 1990, Phys. Rev. A 42, 3065.

6. H. Ritsch, P. Zoller, C.W. Gardiner and D.F. Walls, private communication.

243
III. THEORETICAL DEVELOPMENTS

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245
N92-22073

SQUEEZED STATES AND PATH INTEGRALS

Ingrid Daubechies

AT& T Bell Laboratories

Murray Hill

New Jersey, 07974

and

John R. Klauder

Departments of Physics and Mathematics

University of Florida

Gainesville, FL 32611

ABSTRACT

The continuous-time regularization scheme for defining phase-space path integrals is

briefly reviewed as a method to define a quantization procedure that is completely covariant

under all smooth canonical coordinate transformations. As an illustration of this method,

a limited set of transformations is discussed that have an image in the set of the usual

squeezed states. It is noteworthy that even this limited set of transformations offers new

possibilities for stationary phase approximations to quantum mechanical propagators.

1. INTRODUCTION

For many years now it has been customary to define path integrals with the aid of

coherent states [1]. Such formulations have been developed not only for the canonical

coherent states suitable for the Weyl group (i.e., the Heisenberg algebra), but for coherent

states based on other groups as well, notably the unitary and orthogonal groups with

(in)definite signature, the affine group, etc. However, for the sake of convenience and to

focus on the relation with standard squeezed states, attention in this paper will be confined

to path integrals constructed with the aid of canonical coherent states. The construction

of coherent state path integrals is generally carried out in one of two standard procedures

[2]. To illustrate these two procedures let us first introduce a few standard definitions

involving coherent states [2]:

f

1

- / IP, q> (P, ql dpdq/27r,

*J

H (p, q) = @, ql lP, q),

H (P2, q2; 191, ql) -- (P2, q217-/IP1, ql}/(P2, q2lPl, ql),

= [ h (p, q) [p, q }(p, q dpdq / 2 7r,

It

"hl

J

247

PRECEDING PAGE BLANK NOT FILMED
where IP, q/, (P, q) efft2, denotes one of a collection of coherent states defined by

[P,q) -- e-iqPeipQ[o), [Q,P] -- i, (Q + iP)10) = 0,

all of which are normalized, (p, q [P, q/= 1. In addition, we have introduced two "symbols"

associated with a fairly general operator 7-/, namely H and h as functions on phase space.

In terms of these quantities, the propagator form time t p to time t" -- t' + T, T > 0, is

given by either of the two expressions

(P"' q"le-iT-tTlp" q') -- N ---,

×

n=N

_I (Pn+ l, qn+ l ]Pn, qn) e-isH(Pn+a'qn+a;P'_'q'_)

n=O

n=N

1-I

r_----1

(P"' q"le-iT-tTlP" q') = N a m/l _ _ "'" ×

n=N n=N

H (Pn+l,qn+llPn,qn) 1-_

n=0 n=l

e- iSh(pn ,qn) dpn dqn / 2 7r,

where we have introduced the notation e = T (N + 1), P",q" = PN+(,qN+I,

and p_, qt = P0, q0. In a formal limit, in which the order of integration and the limit

are interchanged and the integrand is evaluated for continuous and differential paths, the

formal result emerges, respectively, that

(p., q,,]e-iT-tT [pp, qt) = jViJe i i[p;t-H(p,q)ldtDpDq '

(p", q'tJe -i_T [p', q') = ./_,fJe i f[P4-h(p'q)JdtDp79q,

where, as conventional, we use a single standard integral sign here to represent a (formal)

functional integration. Since, in the general case, H (p, q) :_ h (p. q), we are seemingly

led to a paradox, namely that two generally different expressions can be given for the same

quantity. That these two expressions are different is just a dramatic reflection of the very

formal nature of such "equations" in the first place; each is correct if interpreted in the

manner indicated in the lattice regularized form given above.

In recent years, a very different regularization and formulation of coherent state path

integrals has been developed that is both rigorous in construction and does not exhibit the

248
paradox outlined above [3]. In this formulation, a continuous-time regularization scheme

is found that takes the form

(p",q"l q')= lim 27revT/2 /

u ----_ O0

e i f_(t)dq(t)-h(p(t),q(t))dt]d/.t_V (p, q),

where /Z_v denotes a planar two-dimensional Wiener measure with diffusion constant

u that is pinned so that p(t'),q(t') = p',q' and p(ttt),q(t") = p't,q". Since p(t)

and q (t) are (independent) Brownian motion paths, the integral fp (t)dq (t) is properly

understood as a well-defined stochastic integral [4]. In the present form the Ito and

Stratonovich formulations yield the same result; however, under coordinate transformations

the Stratonovich form is chosen and so it is convenient to adopt the Stratonovich form

from the outset. It must be appreciated that the expression above involving the Wiener

measure is rigorous and unambiguous; the marvel is that this genuine, i.e., continuous time,

path integral formulation actually provides the correct propagator for the Hamiltonian

7-( provided one adopts the symbol h (p, q) to use as the classical Hamiltonian in the

action even though it may, in general, contain a nonzero h. In a formal, but nevertheless

suggestive language, one may also say that

(p", q"le -rot [p', q') - ¢i f[pit-h(p,q)]dt e- _ f [[92+(t_]dtT)pT) q,

which shows the continuous-time regulatory nature of the indicated expression inasmuch

as the u-dependent factor in the integrand formally goes to unity as u _ oe. Although

formal in nature, the last equation may be understood as a short hand expression for the

former one when it is accepted that the various terms do not have independent meaning

but only in combination with one another. Thus they may be recombined into the proper

mathematical form at any time. (This is similar to how the "quotient" dy/dx should be

understood for the derivative.)

The expression for the propagator given above is not only well defined mathematically,

but it also enjoys a covariance under generally time-dependent canonical coordinate

transformations. Let two canonical coordinate systems be related according to the equation

p dq - h (p, q) = _ d-q +dG (_, _, t) - k (_, _, t)

that not only holds in the classical case but for Brownian motion paths as well thanks to the

choice of the Stratonovich rule [4]. Under the same canonical coordinate transformation,

the metric on fiat space that supports the two-dimensional Brownian motion changes to

dp 2 + dq 2 = da 2 (_,-_, t) = A (if, _, t) d_ d_ +B (if, _, t) d_ d_ +C (_, _, t) d_ d_

249
and as a consequence the propagator takes on the form in the new coordinates, which for

convenience we relabel p. q again,

lira M e.i( G'- c.;')/ e i [_pti-k(p;q.t)]dte- _ f[da2(p.q;t)/dt2]dt.Dp.Dq v ----* oc

This formula expresses --for the first time and after over sixty years of the theory of

quantum mechanics--a fully canonically coordinate covariant formulation of the process

ofquantization [5]. This expression clarifies the role of the Schroedinger quantization rule

in which coordinates act as multiplication while momenta act as derivatives; this rule of

quantization is valid in and only in Cartesian coordinates as often noted, but Cartesian

coordinates in phase space rather than in p-space and q-space separately as commonly

stated [6]. Of course, the arena for classical mechanics resides in a symplectic manifold and

it does not employ a (Riemannian) metric in its formulation. On the other hand, quantum

mechanics has a different and richer basis in which a metric structure appears. Indeed,

it is not unreasonable from a classical viewpoint that a metric structure is appended to

the classical phase-space manifold, not for purposes of defining the Hamiltonian equations

of motion, but rather to keep track of just what physics a given system refers to. For

example, an harmonic oscillator (centered at the origin) appears as an harmonic oscillator,

e.g., with a Hamiltonian given by 1⁄2 (ap '2 + 2bpq + _:q'2) . _ > O, b > O. ax: > b2., only in

Cartesian coordinates in phase space. In non-Cartesian coordinates an harmonic oscillator

assumes a different form from that indicated. Just what system actually corresponds to an

harmonic oscillator (or free particle, or quartic anharmonic oscillator, etc.) is coded into

the classical scheme by the implicit use of an auxiliary flat metric on the two-dimensional

phase space, and its expression in Cartesian coordinates. This same fiat metric space

actually enters the formulation of the quantization procedure as described in the present

article through its use as a carrier for the Brownian motion. Once it is decided which sets

of canonical coordinates are the Cartesian ones, so that the expression for a system which

represents (say) an harmonic oscillator is unambiguous, then the quantization procedure

itself is unambiguous in the approach advocated here. After the well-defined path integral

is set up, then one is free to make a variety of coordinate changes within that integral,

among which possibly time dependent canonical transformations are to be distinguished.

Indeed, one can go so far as to introduce a Hamilton-Jacobi transformation so that the new

Hamiltonian vanishes. This puts all the dynamics into the curvilinear coordinate system

that is used to track the two-dimensional planar Brownian motion. As a consequence, the

overall level of difficulties is conserved, as one would expect to be the case.

It is hard to illustrate this program in its full potential, but it can be shown in a sort

of small scale fashion. Indeed, it is squeezed states that can be used to provide a limited

illustration of this overall program, and it is this "miniature" illustration to which we now

25O
turn our attention. A convenient place to start the investigation is with the kinematics

rather than the dynamics, and this in turn can be done simply by looking at the propagator

for vanishing Hamiltonian.

2. CHANGE OF VARIABLES IN THE PATH INTEGRAL:

CONSTANT _ TRANSFORMATIONS

KINEMATICS

ff the Hamiltonian vanishes, or in the limit that T _ 0, the "propagator" reduces to the

reproducing kernel, an integral kernel representing a projection operator onto the relevant

subspace of all square integrable functions on phase space as given by

(P" q'; llpr' qt ;1) = v _ cx)limM f ei S p4dte-_ $(P_+4_)dt79pT)q,

which may be explicitly evaluated as

(p., qr,; liP', q'; 1) = ei1⁄2(p"+P')(q"-q')-_[ (p''-p')2+(q'-q')2].

In these expressions we have added a "1" to the label to emphasize that the coherent states

are those based on an harmonic oscillator ground state with a unit angular frequency,

a: = 1. In particular, for a general value of w, the configuration space representation of

the coherent states reads

Ixlp, q; a:) (_)1⁄4 1⁄2w(z-q)2+ip(z-q)

and it follows that the overlap of two such states for the same value of w is given by

(p",q";_Jp',q';w)- f

= exp { _ (p" -t- p')(q" - q') - 1 [w-1 (p,, _ p,)2 + _ (q,, _ q,)2J }.

i

It is clear therefore that the coherent state overlap obeys the identity

{p",q";_lp',q';w) = (_-1⁄2p",w1⁄2q"; llw-1⁄2p',_1⁄2q'; 1)

which relates dilation of the angular frequency to a corresponding dilation of the coherent

state labels, i.e., an expansion of one phase space coordinate and a contraction of the other.

This relation may be codified another way as well. Let

D=1⁄2(PQ+QP)

251
denote the self-adjoint dilation operator with commutation properties [Q,D] =

iQ, [P, D] = -iP, then it follows that coherent states for different angular frequencies are connected by

[p, q; ca) = e'_ln(_)Dlw-_p,_q; -1 1 1 1).

Thus the unitary transformation generated by D is nothing other than the squeeze operator

relating coherent states and squeezed states, or relating two sets of squeezed states with

different squeezing values. In forming the overlap illustrated above, the squeezing operator

drops out leading to the indicated relation.

A path integral expression for the coherent state overlap at angular frequency _; can

be readily obtained just by a coordinate change of the path integral appropriate for a unit

angular frequency. In particular, if one makes the change of integration variables given by

1 1

then it immediately follows that

(P", q"; _IP', q'; w) --

• t x

eif pqd c-_ f(w-_f +wq'_)dt_p_q

showing quite clearly the connection of the relative scale factor in the two-dimensional

Brownian motion and the parametric dependence in the coherent state representation. All

this has assumed that a; has been constant throughout; next we take up the case of a time

variable _.

3. CHANGE OF VARIABLES IN THE PATH INTEGRAL:

NONCONSTANT _ TRANSFORMATIONS

The overlap of two coherent states for two different values of _ is given by

q"; J'IP', q' ; w') = f (p", q" ;J'lx) (xlp', q' ;J)dx

252
This expression also exhibits an alternative form given by

• 1 I 0311 _,

{p",q",J'lp',q';J)" = {w"-1⁄2p",J'1⁄2q";1]e-'_ln{7)D]w'-1⁄2p ', _ _q'1);,'_

= {wn-1⁄2pn, w"}q"; lie -il _D]wt-1⁄2V' , _zq, '_ " 1)

! t = ll - f eet'l '- p, J1⁄2q';1)

where we have introduced a smooth but otherwise arbitrary function w (t), t _ < t <

tn,which interpolates between w n = w (t n) and w_ = w (t_). Of course, this expression

holds as well even in the special case that w n = J in which case w (t) goes smoothly

between equal initial and final values, but is otherwise arbitrary.

In the latter form the overlap of two coherent states for differing angular frequencies

has been expressed in terms of the matrix element of a kind of propagator between coherent

states of the same angular frequency. But the latter form admits a path integral expression.

In particular, it follows that

(p,,, qn; 1]e-if _ Ddtlp,, q, ; 1)

• d_ 1

lira M e i f[pq-_pq]dte-_ f[P_+i12]dt2")p_')q.

ll ----_ 00

Now, much as was the case earlier when w was constant, we next make a time-dependent

change of variables of the form

p(t)--, (t)-kp (t),

q(t)---,w(t)}q(t),

where wn = w (tn), w_ = w (tt). In making a time-dependent substitution of variables,

additional terms will arise in the path integral integrand on the right hand side. In particular,

the term

f _9(t - _pq] dt --* f [p_] dt,

the formal flat measure remains unchanged,

79p _Dq --, 79p _Dq ,

while the all important formal weighting factor

e- 2"_ 1 f[_2+O2]d t _ e- 2"_, f[ ,_(t) -, D2+w (t )_/21 dt.

253
Other terms might be contemplated in the exponent of the final expression such as those

involving time derivatives of _ (/); however all of these will be negligible in the limit that

v _ oc since they are not as singular as the indicated terms. While we prefer this heuristic

characterization of the transformed Wiener process one should bear in mind that only the

coordinate description of the planar two-dimensional Brownian motion is being changed

and the process itself is in no way effected. We are encoding this change of coordinates

by means of the change of coordinates of the metric on the plane. (A rigorous analysis of

such transformed Wiener processes is in progress by the present authors.) Thus it follows

after such a substitution of variables that

(P", q";J'lP', q'; w') = (w"-1⁄2p", w"1⁄2q"; 1 le-i1⁄2 f _ Ddtlw'-1⁄2p' , w _q ; 1)

fl,4 f e i f p_dt e-_ f['(t)-lP2+w(t)(t2]dtT)pT)q.

Consequently, the introduction of a smooth, time-dependent angular frequency that

interpolates between the initial and final values in the Wiener measure provides just the

fight ingredient to yield the overlap between two different coherent states based on two

different angular frequencies. This expression yields a simple but nontheless bona fide

example of how the classical Hamiltonian- here just _ (t)p(t)q (t)/2w (t) -- may be

eliminated in favor of a change of coordinates with which to describe the two-dimensional

Brownian motion on the phase space plane. Such an elimination additionally involves a

change of coordinates at the endpoints, as illustrated in the central equation, but in the case

of squeezed states, there is an alternative interpretation involving coherent states based on

differing angular frequencies as embodied in the first part of the equation. We now tum

our attention to the inclusion of dynamics in this example through the presence of a rather

general nonvanishing Hamiltonian.

4. CHANGE OF VARIABLES IN THE PATH INTEGRAL: NONCONSTANT

w TRANSFORMATIONS AND A GENERAL HAMILTONIAN

INTRODUCTION OF GENERAL DYNAMICS

Based on the earlier discussion it is quite straightforward to include a rather general

Hamiltonian h (p, q) into the problem. In particular, based on the initial discussion, let

us consider

(p,,, q,,; l lTe-i f[_+_ D]dt[p,, q,; 1)

m

m

lira M/e i f[P4-h(p'q)--_pq]dt e-_ f[lJ_+4_]dt'Dp'Dq

254
which after a change of integration variables becomes

where

lim j_4 / e i f[p_-k(p,q,t)]dt e -1 f[w(t)-l_2+o_(t)_]dtT)pT)q v.---_ O0

= @"-1⁄2p",w't1⁄2q"; llTe -i! [U+_D]dtlwt-1⁄2pt, wt1⁄2q'; 1)

(w"-_ _p " "_ " le-_lnJ'DTe-ifU'(t)dte_lnJDlw'-1⁄2p' , '_- '1) = ,w _q ;1 w2q;

-- (p", q" ;J'lTe -i f W(t)dtlp' , q'; o.,'),

(p (t) , q (t) , t) = h (w (t)Q p (t) ,w (t)1⁄2 q (t)) ,

k

which contains an explicit time dependence from the angular frequency as well as an

implicit dependence just from the time dependence of p and q themselves, and in addition

where

• 1 _(t)D ", w.__.D

7-t'(t) _-- e";_n--_ 7"re-';In tM •

The basic significance of the preceding equations can be summarized as follows:

(p",q" ; w"lTe -i f U'(t)dtlp' , q' ; w')

= (w"-1⁄2p",wn1⁄2q";l[Te-iI[U+_D]dt[j-ip ', w'_q; '_ ' I)

=[ • o) x 1

lira /vt e i I[pq-h(p,q)-_pq]dte-_ I[f +q_]dt79pT)q p(t)--,.,(0-1⁄2_,(t)

1]---.-_ (X)

1

q(t)-+_(t) 2 q(t)

fl"[ f ei I[p4-k(p,q,t)ldte-_ f[_(t)-llP+w(t)q_]dtT)p79q"

This is the most general form we are able to offer using squeezed states, and it shows, in the

first of the equalities, how part of the Hamiltonian can be absorbed quantum mechanically

by a change of the fiducial vectors -- indeed just like going to the interaction picture in

ordinary quantum mechanics, which is then responsible for the introduction of the timedependent "interaction" picture Hamiltonian 7-/_(t). The second pair of equalities just uses

the original form of the path integral as modified by a change of variables that effects

the end point conditions as well. The final equality just accounts for that very change of

variables as requested in the line above.

255
5. EQUAL END POINT ANGULAR FREQUENCIES

Let us return to the path integrals discussed at the beginning of this article, namely

to those for which the initial and final angular frequencies are the same. For the sake

of convenience, let us choose that value to be unity, i.e., w" = J = 1, and return to

the original notation for the coherent states with unit w, namely that IP, q) = ]P, q; 1).

However, this time we will retain the option of using a time-dependent angular frequency

w (t) to interpolate smoothly between the original and final values of unity. In this case

the formulas developed above simplify to become

{p",q"lTe-i f W(t)dt IP', q')

-- <p", q"lTe -i I[_t+_D]dtlp' ' q')

["m f

= M

b' ----_ (X2 e i f[pq-h(p'q)--_-_pq]dte-_ f[P2+q_]dt l)p'Dq] _I

p(t)---*w(t} 2p(t)

q(t)--._(t)1⁄2 _(t)

m e i f[Pdl-k(p'q't)]dte -2Aft f[w(t)-llp+w(t)q'_] dt "Dp79q.

With a slight generalization, we can now turn this equation around to read

(p,,,q,, le- iT-lTip,, q')

m

m

lira Jr4 / e i f[P(l-h(p,q)]dt e-_ f_2-I-q_]dt DpDq. V "'+ O0

f

= M

/J "--_ OO e i f[pq-h(p'q)]dte-_ f_+4_]dt T_pDq] p(t)--,_(t)-1⁄2p(O

1

q(t)-_(t)_ q(t) i

ei f[pO-k(p,q,t)+'_pq]dt e-_-_ f[w(t)-ilj_+w(t)q2] dt _Dp_q.

In this expression the third line holds because it simply corresponds to a change of

integration variables that does not have any effect on the values of the boundary labels

since w" = J = 1. The last line represents, just as before, the consequences of that very

change of variables.

256
Now observe that on the left side of this equation there is no reference to the function

(t), t' < t < t", _ (t") = w (t') = 1, while on the right side of the equation, in

the last part of the equation in particular, the function w (t) enters in a prominent way.

This becomes especially significant when an approximate evaluation of the path integral

is admitted, such as that which arises from a stationary phase approximation. Stationary

phase approximations for coherent state path integrals with Wiener measure regularization

of the kind considered here have been worked out previously [7] and we do not repeat

that discussion here. The point we wish to emphasize, however, is that the choice of the

angular frequency _ (t) will enter most probably in the form of the approximate solution,

and naturally some expressions will be better approximations to the real answer than other

expressions will. Just which will be the best approximation is, of course, not too easy to

establish. Perhaps one scheme is to ask that the result be stationary with respect to small

changes of the functional form of a; (t). In practice one might want to let w (t) depend on

just a few discrete parameters and to seek stationary variations with respect to just these

few parameters. This certainly seems easier to do than to ask for an extremal variation

with respect to the entire function _ (t).

One can actually see a miniature working of this general kind of procedure in comparing

the usual and the Maslov stationary phase approximations to the sharp position propagator;

see, e.g., [7]. As given earlier, the configuration-space form of the coherent states given by

1

makes clear that lim (ff)1⁄4(xlp, = which converts the coher- t.d----_ OO

ent state representation to the sharp position or configuration representation, while

lim 1 1 1

,.,_o (4-¢d) _ (x]p,q;w) -- eip(x-q) which converts it to the dual or momentum rep- x/2_r

resentation (up to an unimportant phase factor). These features can also be seen in the

relation

(P", q"; w"lP', q' ; w') = liP", q" ;J'Jx)(xlP', q'; J)dx

d

= ] __ exp[P"W'i( + p'w")(q" -q') (p,, _ p,)2 _ w"w' (q" _ ¢)2]

that gives the overlap of two coherent states based on differing angular frequencies.

Consider the limiting situation in which both w" ---, oc, a/---, 0. In that case it follows that

1

lim [ w" _ _ (p,,, l__.__eiP,(q,,_q, )

_"--*_ \16-_'] q";J'JP"q';J)- _ "

Wl---,0

257
Thus it should be no surprise that the two standard stationary-phase type approximations are

actually contained in the coherent state approach in the form of suitable limits. In a manner

of speaking, the usual configuration space approach just involves choosing a constant and

very large value of the angular frequency parameter _ (taken to infinity at the end of the

calculation) and making a stationary phase approximation to the resulting path integral.

On the other hand, the Maslov approach takes the propagator from a sharp configuration

initially to a sharp momentum finally, approximates that path integral by a stationary phase

approximation, and then returns the end point to configuration space by a Fourier transform.

This approach can be approximated in our method by taking an angular frequency history

(t) that is initially huge (tending toward infinity) and finally very small (tending toward

zero), approximating that path integral by a stationary phase approximation, and finally

making a change from coherent states based on a very small angular frequency to one based

on a huge angular frequency just by the kinematical factor given above. The coherent

state approximation developed in particular in reference [7] proceeds in yet another way,

namely, starting with a sharp configuration initially, propagating to a coherent state with

a finite nonzero value of the angular frequency, i.e., oJ = O (1), approximating that path

integral by a stationary phase approximation, and then passing from the final coherent state

representation to a sharp configuration one. This approach can also be approximated in our

scheme by having an _ (t) that initially is huge, and finally is finite and nonzero [ O (1) ],

approximating, in turn, that path integral by a stationary phase approximation, and then

passing back to a coherent state based on a huge angular frequency at the final point.

6. CONCLUSIONS

In this article we have attempted to show the reader what the authors believe is

the "latest" in path integral construction N the state of the art -- and illustrate how

variable changes can be rigorously carded out within the path integral formulation itself.

Squeezed coherent states have been used as convenient bases throughout in the illustration

of the general program by a "miniature" subprogram involving a fairly limited change of

integration variables. The resultant formalism is able to express a path integral in terms

of an essentially arbitrary function, the time varying angular frequency, _ (t), which lends

itself to various selections in case an approximation scheme is invoked. By illustrating

that the usual and the Maslov approaches are but two small examples of how such an

optimization can be used, it becomes clear that there are hidden in these formulas a whole

host of differing approximation schemes some of which, in certain applications at least,

may well be better than the schemes currently in use. It is left to the future to see just

how to exploit the vast number of possibilities that have been opened up here.

258
REFERENCES

1. Klauder, J.R., 1960, "The Action Option and The Feynman Quantization of Spinor

Fields without Anticommuting C-Numbers," Annals of Physics 11, p. 123.

2. Klauder, J.R. and B.-S. Skagerstam, 1985, Coherent States, World Scientific, Singapore.

3. Daubechies, I. and J.R. Klauder, 1985, "Quantum Mechanical Path Integrals with

Wiener Measures for All Polynomial Hamiltonians, II," J. Math. Phys. 26, p. 2239.

4. See, e.g., McShane, E.J., 1974, Stochastic Calculus and Stochastic Models, Academic

Press, New York.

5. Klauder, J.R., 1988, "Quantization Is Geometry, After All," Annals of Physics 188,

p. 120.

6. See, e.g., Dirac, P.A.M., 1956, The Principles of Quantum Mechanics, Third Edition,

Oxford University Press, Oxford, p. 114.

7. Klauder, J.R., 1987, "Global, Uniform, Asymptotic Wave-Equation Solutions for Large

Wave Numbers," Annals of Physics 180, p. 108.

259
N92-22074

SUPERCOHERENT STATES AND PHYSICAL SYSTEMS

B.W. Fatyga and V. Alan Kosteleck2_*

Physics Department, Indiana University

Bloomington, IN 47_05, U.S.A.

Michael Martin Nieto

Theoretical Division, Los Alamos National Laboratory

University of California, Los Alamo3, NM 87545, U.S.A.

D. Rodney Truax

Chemistry Department, University of Calgary

Calgary, Alberta, Canada T2N 1N4

A method is developed for obtaining coherent states of a system admitting a

supersymmetry. These states are called supercoherent states. The approach

presented in this talk is based on an extension to supergroups of the usual

group-theoretic approach. The example of the supersymmetric harmonic

oscillator is discussed, thereby illustrating some of the attractive features

of the method. Supercoherent states of an electron moving in a constant

magnetic field are also described.

1. Introduction

Over the past three decades, the notion of coherent state [1-6] has enjoyed a

significant role in diverse areas of physics. Several basic definitions are in use [7].

For example, among the possibilities for the simple harmonic oscillator are the

definition as eigenstates of the annihilation operator, the one as states having and

preserving minimum uncertainty, and the one via the displacement operator. All

these yield the same harmonic-oscillator coherent states, representing a gaussian

wavepacket preserving its shape while executing the classical motion.

This talk describes a generalization of the concept of coherent states to that

of supercoherent states, relevant for systems admitting one or more supersymmetries. A supersymmetry involves both bosonic and fermionic states, and the

* Speaker

PRECEDING PAGE BLANK NOT FILMED

261
corresponding symmetry generators close under a combination of commutation

and anticommutation relations into a superalgebra. The additional structure this

entails means that the physically appropriate generalization of coherent states to

supercoherent states is not immediately apparent.

Our solution to this problem involves a extension to supergroups of a generalized method [6] for ordinary coherent states that is based on Lie groups and

involves use of the Baker-Campbell-Hausdorff (BCH) relations [8-13] connecting

different group parametrizations. Supergroups can be viewed as extensions of

Lie groups with Grassmann-valued parameters. The theory of supergroups considered both as abstract groups and as superanalytie supermanifolds has been

developed [14-16], and methods for obtaining BCH relations for supergroups are

known [17-19]. A summary of our methods is provided in section 2.

As an example of the method, the supercoherent states for the supersymmetric harmonic oscillator are considered in section 3. The supersymmetry for this

case is generated by the super Heisenberg-Weyl algebra, containing the identity

and bosonic and fermionic creation and annihilation operators. It is closely related to supersymmetric quantum mechanics [20-29], which is applicable in several

physical situations. An example with relevance to the quantum Hall effect is the

ease of an electron moving in a constant magnetic field [28,29]. This situation is

considered in section 4.

The reader is referred to [30], on which this talk is based, for more information

about our general construction of supercoherent states, about its relation to other

approaches [31-33], and about applications in various physical situations.

2. Method

There is a close connection between group theory and coherent states. To

see this for the simple harmonic oscillator, consider the usual approach via the

displacement operator D, given by D(a) = exp(aa t --_a). This displaces the

annihilation operator a by a complex constant a: D-l(a)aD(a) = a + a. The

operator D is a unitary element of the harmonic-oscillator symmetry group, called

the Heisenberg-Weyl group, for which the associated algebra is [a, a t] = 1. By

definition, the coherent state parametrized by a is given by the action of D(a)

on the ground state ]0>. The correct normalization of la) is fixed by the unitarity

of D. The form of la) can then be explicitly exhibited using the BCH relation

eAe B = e(A+B+}[ A'B]), valid for any two operators A and B both commuting with

[A,B].

For a general system with an arbitrary Lie group G as symmetry group,

coherent states can be defined as follows [3,6]. Given a unitary irreducible representation T(g) of G acting in a Hilbert space H, set I_0) as some given element

262
in H. The coherent states are then the set {1_9}} = {T(g)l_0)}. This definition

is parallel to the displacement-operator approach for the harmonic oscillator.

For systems admitting supersymmetry, we extend this method to supergroups

using the construction of refs. [14-16]. In this approach, supergroups are defined in

analogy with the definition of Lie groups via analytic manifolds, using Grassmannvalued parameters instead of real or complex ones. The resulting supergroup

coordinates include both commuting and anticommuting variables. We refer the

reader to refs. [14-16] for details of the construction. A summary of the essential

points is contained in the paper [30] on which this talk is based.

To find supercoherent states via the group-theoretic method requires the

use of unitary supergroup representations. Introduce the supergroup generators

B j, F_, where the corresponding superalgebra* involves commutators among the

Bj and anticommutators among the F_. Choose a superhermitian basis [31], i.e.,

set B_ = Bj and F_ = -Fa. Then, a general unitary supergroup element is

T(g) = exp(AjBj + OaF_), where Aj is real Grassmann commuting and O_ is real

Grassmann anticommuting.

Supercoherent states are found by applying T(g) to an extremal state in

the (super) Hilbert space. To find explicit expressions requires the use of BCH

relations for the supergroup. A general method for determining these and specific

formulae for some frequently used supergroups may be found in refs. [17-19].

3. The Supersymmetric Harmonic Oscillator

By definition, the hamiltonian H of a supersymmetric quantum-mechanical

system [20-23] commutes with N supersymmetry operators Qj of which it is a

quadratic function: _SjkH = {Qj,Qk}. The superalgebra generated by H and

Qj is called sqrn(N). Choosing N = 2 gives sqm(2_, which appears in several

physical contexts [24-29]. Defining Q = (Q1 + iQ2)/x/2 and Qt = (Q1 -iQ2)/v/_,

the superalgebra sqm(2) is g = {Q, Qt}, [H, Q] = [H, Qt] = o.

The supersymmetric quantum harmonic oscillator can be defined in terms

of annihilation and creation operators a, at; b, bt generating a supersymmetric

extension of the usual Heisenberg-Weyl algebra: [a, a t] = {b, bt } = I. The corresponding super Hilbert space is spanned by states In, u}, where n = 0, 1,2... and

u = 0, 1. States with v = 0 are called bosonic and those with u = 1 are called

fermionic.

The sqm(2) superalgebra is generated by the oscillator hamiltonian H =

ala+ bib and by the supersymmetry operators Q = ab l, Qt = alb. It follows from

* For an overview of superalgebras, see ref. [34]

263
H[n,u) = (n -F u)[n,u) that [n,0) and In- 1,1) are degenerate states for all n

except n = 0. The ground state [0,0} is thus unique. Unbroken supersymmetry,

Q[0, 0) = Qt[0, 0) = 0, implies that the ground state has energy eigenvalue zero.

The generator Qt takes bosonic states into fermionic ones, while Qt takes fermionic

states into bosonic ones.

Following the method described in section 2, supercoherent states for the

supersymmetric oscillator are given in terms of a unitary representation T(g) of

the super Heisenberg-Weyl group. The supergroup element of relevance may be

taken as T(g) = exp(--Aa + Aa t + Obt + -Ob) where A is complex Grassmarm

commuting and 0 is complex Grassmann anticommuting. The necessary BCH

relation for the super Heisenberg-Weyl group, needed for explicit calculation of

the supercoherent states, is found using Lemma 1 of ref. [17]. The result is

T(g) = exp( l o-o - _]Al2)exp(Aat)exp(Obt)exp(-Aa)exp(-Ob) (3.1)

The supercoherent states ]Z) are obtained by applying T(g) to the ground

state [0, 0). They are given by

[Z) =(I+200)IA, 0)+0IA,1 } , (3.2)

where for convenience we have defined ]A, u} = exp(-lAI 2/2)exp(Aa*)lO, u).

The supercoherent states IZ) have the following attractive properties, all

of which are natural generalizations of the correesponding features of ordinary

harmonic-oscillator coherent states.

• They are defined via a natural extension of the usual displacement operator

approach.

• They are eigenstates of the annihilation operators a and b: a]Z) = Alz),

blz) : -olz).

1

• They maintain the minimum-uncertainty value AqAp = _ in time.

• They are unity normalized, (Z]Z) = 1.

• They are not orthogonal and form an (over)complete set. The identity is

resolved by f IZ)(Zld-OdOdA = 7rI.

• They yield the usual harmonic-oscillator coherent states IA) when 0 = 0.

• They contain as the subset A = 0 the usual fermionic coherent states [35] for

a single anticommuting fermionic degree of freedom.

4. A Physical Example

The quantum system consisting of a nonrelativistic electron of mass M and

charge e moving in a constant uniform magnetic field B = B;? provides a physical

254
realization of supersymmetric quantum mechanics [28,29]. The wavefunctions

e-iEt_b(r -*) for this system obey the two-component Pauli equation, which reduces

to He = E¢ with H - 2M1 [_. 07 - e._)] 2. The use of cylindrical coordinates

is natural, as is the choice of cylindrical gauge A, = -TBy,1 Av = 1⁄2Bx. For

simplicity, we restrict the analysis to the two-dimensional problem, so that p, = 0.

The explicit realization of the super Heisenber_-Weyl algebra is as follows.

Define the dimensionless quantities H = MH/eB, E = ME/eB, and introduce

the annihilation operators

1

1 ei_,(O r + !0_ + _eBr) (4.1)

a- vfff_

and

b= [00

Then, the Pauli equation takes the manifestly supersymmetric form

.f-I¢ - (ata + btb)¢ = E¢ (4.3)

All the features of the supersymmetric harmonic oscillator discussed in section 3

are reproduced. Note that the fermion annihilation operator b acts to reverse the

electron spin, and therefore the sqm(2) generator Q does also.

Equation (4.3) is equivalent to a confluent hypergeometric equation with twocomponent solutions labeled by two quantum numbers, one related to the energy

eigenvalue E and one labeling degenerate eigenstates. The explicit solution is

given in our paper [30]. We write ¢ = [n,m;u), where the upper and lower

components of ¢ are labeled by u = 0 and v = 1, respectively. The operators a

and a t act as canonical lowering and raising operators on the quantum number n,

while b and bt act on u. To form a complete set, introduce

ct - 1 i i 1

V/- e -_eBr) (4.4) (Or + -

acting as a canonical lowering operator on m and satisfying [c, c*] = 1. The full supergroup for this physical system is therefore the product of the super HeisenbergWeyl group (generated by a, b, and conjugates) with another Heisenberg-Weyl

group (generated by c and conjugate).

The supercoherent states can now be constructed via the method of section

2. Their explicit form is quickly found from eq. (3.2) by noting that coherent

states with respect to c and c t are the usual harmonic-oscillator coherent states

and that c and c t commute with all other generators. The result is

1 [A__ ex , [C[2 , AnCm ( ) [Z) =exp( O-O)exp(- ) pt,--_--) E v/_.n_.x/_mt. In, m;0)+0[n,m;X)

n_m

(4.5)

265
These supercoherent states depend on three Grassmann-valued variables, A, C,

and 0. It can be shown that all the attractive features of the oscillator supercoherent states discussed in section 3 are reproduced.

The expectation values of the hamiltonian H, (ZIHIZ } = _(A-A- 0-0),

and of the magnetic-moment interaction energy U = -eB(rz/2M, (ZIUIZ) =

_U (1 + 200) provide insight into the role of the Grassmann-valued variables 2M

in Eq. (4.5). The difference (ZIH - UIZ) = _(A-A + 1⁄2) represents the energy

expectation in the absence of the magnetic moment. It is independent of 00 and

the value of AA is shifted by one half. Since the magnetic moment U distinguishes

between eigenstates with v = 0 and v = 1, it follows that the term with 00 contains

the information about the energy splitting between the two sets of eigenstates.

As we have seen, the supersymmetry present in this physical system ensures

a group-theoretical and natural incorporation of the electron spin. This feature

of supersymmetry is manifest in other physical systems. For instance, one key

aspect of atomic and ionic supersymmetry [25] is the natural appearance of the

Pauli principle.

Acknowledgments

This research was supported in part by the United States Department of

Energy under contracts DE-AC02-84ER40125 and W-7405-ENG-36 and by the

Natural Sciences and Engineering Research Council of Canada.

References

1. For an overview see, for example, J. R. Klauder and B.-S. Skagerstam, Coherent States (World Scientific, Singapore, 1985); W.-M. Zhang, D.H. Feng

and R. Gilmore, Rev. Mod. Phys. 62,867 (1990).

2. E. SchrSdinger, Naturwiss. 14,664 (1926).

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267
N92-22o75

SQUEEZED STATES,

TIME-ENERGY UNCERTAINTY RELATION, AND

FEYNMAN'S REST OF THE UNIVERSE

D. Hall

National Aeronautics and Space Administration, Goddard Space Flight Center,

Code 936, 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

A b st ract

Two illustrative examples are given for Feynman's rest of the universe. The first example

is the two-mode squeezed state of light where no measurement is taken for one of the modes.

The second example is the relativistic quark model where no measurement is possible for the

time-like separation of quarks confined in a hadron. It is possible to illustrate these examples

using the covariant oscillator formalism. It is shown that the lack of symmetry between the

position-momentum and time-energy uncertainty relations leads to an increase in entropy

when the system is measured in different Lorentz frames.

1. Introduction

In his book on statistical mechanics [1], Feynman makes the following statement on the density

matrix. When we solve a quantum-mechanical problem, what we really do is divide the universe

into two parts - the system in which we are interested and the rest of the universe. We then usually

act as if the system in which we are interested comprised the entire universe. To motivate the use

of density matrices, let us see what happens when we include the part of the universe outside the

system.

The purpose of this paper is to discuss two physical examples of Feynman's rest of the universe.

We shall consider first the case of the two-mode squeezed state. In 1987, Yurke and Potasek

observed that the failure to make measurements on one of the two modes will lead to non-coherent

excitation of the first mode, as in the case of Einstein's calculation of specific heat in the harmonic

oscillator model [2]. They observed further that this excitation is just like the thermal excitation

of the ground-state harmonic oscillator. From the measurement theoretic point of view, this

non-coherent excitation corresponds to an increase in entropy [3].

Let us next consider the quark model in which two quarks are bound together inside a hadron

[4]. This system has a time-like separation between quarks as well as a spatial separation between

them[5]. While there is no place for the time-separation variable in nonrelativistic quantum

269

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mechanics, it plays an essential role when observations are made in different Lorentz frames. For

this time-like separation, there is a time-energy uncertainty relation. It is of interest to see how

this uncertainty relation is combined with the position-momentum to an observer in a different

Lorentz frame.

We show in this paper that the longitudinal and time-like excitations in the relativistic quark

model are exactly like two photon modes in a two-mode squeezed state [6]. We shall study how the

non-measurement of the time-separation variable affects measurements along other coordinates.

In Sec. 2, we study the statistical effect on measurement and density matrices. In Sec. 3, we

derive the result of See. 3 using the shadow coordinate system commonly used in thermo-fielddynamics [7]. In See. 4, the concept of entropy is introduced as a measure of our ignorance [3]

[8]. In Sec. 5, the formalism of Sec. 4 is applied to the two-mode squeezed state of light.

The rest of thi's paper consists of the application of the concept of entropy to the relativistic quantum system in which the time-energy uncertainty relation is coupled covariantly to the

position-momentum uncertainty, using the same mathematical formalism developed in Secs. 3,

4, and 5. We start this discussion in Sec. 6 by studying the time-energy uncertainty relation

applicable to the time separation variable in the relativistic quark model. In this connection,

the covariant harmonic oscillator formalism is presented. In Sec. 7, Lorentz-squeezed hadrons

are discussed in terms of the covariant oscillator formalism. Finally, in Sec. 8, we note that the

present form of quantum measurement theory does not measure the time separation variable. This

incompleteness in measurement leads to an increase in entropy.

2. Statistical Decoherence

In measuring physical quantities, the accuracy of the measuring device is very important.

Often, we have to face the situation where the measurement is taken on many different objects.

For instance, in the case of the one-dimensional harmonic oscillator, the most general form of

normalized solution is

¢(x,t) = E

n

(2.1)

where _n(x) is the solution of the time-independent oscillator equation with the energy level

w(n + 1/2). The wave function _b(x,t)is normalized:

= IO,,12= 1. (2.2)

n

The expectation value <A>= (¢(x,t),A¢(x,t)) of an operator A(x) can be written as

<A>= + E C_nCneiW(m-n)t(¢m(x),A(x)¢n(x)) "

n n¢rn

(2.3)

If we take the ensemble average for many oscillators prepared independently with different

initial times, the net effect is same as that of taking the time average, and the second term in the

above expression vanishes. As a consequence, the ensemble average is

</i>= lC,12(¢n(x),A(z)¢n(x)).

n

(2.4)

270
We usethe word "mixed" or "non-pure" in order to describethis ensembleaverage.

It is very convenientto treat this problemif weintroduce the density matrix definedas [1] [6]

p(x,x') = E Ic7212_?2(x)¢n(x . ,), (2.5)

n

and

with

f dx'fA(x',x>(x,x'ldx, (2.6)

A(x',x)=$(x'-x)A(x).

The above expression is then the trace of the matrix A(x', x)p(x, x') often written as

</i,> = Tr(pA). (2.7)

If C,_ = 672m for a given value of m, we say that the system is in a pure state. Otherwise, the

system is in a mixed state. The information from the interference terms contained in Eq.(2.3) is

lost during the process of taking the ensemble average. This information lies in Feynman's rest of

the universe.

The best-known example is the thermally excited harmonic oscillator which was used by Einstein in his calculation of the specific heat of a solid. The density matrix takes the form [1]

[6]

pT(x,,') = _(1 - _-°'/kr)_-?2_'/_r¢?2(x)¢:(x'). (2.8)

n

In the zero-temperature limit, the system is purely in the ground state. As the temperature

increases, [Cn[ 2 becomes (1 -e-_°/kr)e -'*_/_T, but the above expression does not tell us anything

about the phase of Cn. The density matrix does not give any information about the coherence of

the system. In Sec. 4, we shall study how this ignorance is translated into entropy,.

3. Shadow Coordinates

We discuss in this section a method of deriving the density matrix, without taking the ensemble

average, by introducing an auxiliary Hilbert space consisting of ¢_(_) and attach it to Cn [1] [7].

Let us consider the wave function of the form

_)(X,X) = y_(Cn_)n(ffg))¢n(X). (3.1)

?2

The auxiliary coordinate k is called the "shadow" coordinate in the literature [7]. It is possible

to derive the result of Eq.(2.4) by treating ¢(x, k) as a pure-state wave function defined in the

total Hilbert space consisting both of ¢,_(x) and ¢,_(Y:). Because of the orthogonality relation for

¢,_(5:), the expectation value of A(x):

<A>= _ C_C_(¢m(Yc),¢_(k))(Vm(x),A(x)¢_(x)),

n,m

(3.2)

271
is the sameas the ensembleaverage<4> given in Eq.(2.4). It is possibleto obtain the density

matrix by integrating _b(x,k)_b*(x',5:) over the 5: variable:

p(x,x') = f g,(x,2)_/,_'(x',:?)dx. (3.3)

The evaluation of this integral leads to the expression for p(x, z') given in Eq.(2.5). The shadow

coordinate plays the role of taking the ensemble average discussed in Sec. 2.

Let us illustrate this again using the ground-state harmonic oscillator wave function

Oo(x,k) = _bo(X)¢o(k)= exp - (x 2 + k2) , (3.4)

where x is measured in units of 1/x/'rnw, and let us now make the coordinate transformation:

(;',) = (coshT/ \ sinh q coshr/) sinhr/ (;) ' (3.5)

where

cosh r/= 1/(1 - e-_'/kr) 1/2, sinh q = e-_'/2kT / (1 -- e-w/kr ) 1/2 (3.6)

which shares the same mathematics as a Lorentz boost as we shall see in Secs. 7 and 8. Then this

coordinate transformation leads to the wave function of the form

@T(X,:g): [___j 1/2 exp {-_[(tanh 4-_.T)(x 02 + _)2 + (coth 4--_)(x _2 - k)2]}.

The wave function of the two variables can be expanded as [6]

(3.7)

¢T(X,X) = [1 -- exp(-w/kT)] '/2 _ exp(-nw/2kT)_b,_(x)_n(k).

?2

(3.8)

The evaluation of the density matrix given in Eq.(3.3) with this form of the wave function

leads to the density matrix of the form of Eq.(2.8). The same evaluation with the wave function

of the form of Eq.(3.7) gives [6]

pr(z,x') = [1 tanh __],n exp { - _[ (x + x,)_tanh 2-__ + (x - x,)_coth _,..o]} . (3.9)

Then the probability distribution pr(x) = pr(x, x) becomes

: exp{ ( 10)

This expression is normalized. In the T = 0 limit, the probability distribution becomes

po(x) = (lift) '/2 exp(-x2). (3.11)

272
The increaseof temperature broadensthe probability distribution. It is possible to carry out

the same analysis for the momentum variable. The momentum distribution will also become

widespread. The net result is the increasein uncertainty. This increaseis due to our ignorance

about the shadow coordinate system. Feynman'srest of the universeconsists of the shadow

coordinate.

4. Entropy and Ignorance

The interpretation in termsof thermal excitation was possible because the expansion coincides

with the thermally excited oscillator state. There are, however, cases where the density matrix

does not correspond to any state in thermal equilibrium. For instance, if we start from one of

the excited harmonic oscillator states [9], the density matrix does not correspond to a thermally

excited state. What then will be the variable which measures our ignorance about the second

coordinate variable?

The answer to this question is the entropy defined as [8]

S = - y_" p,_ ln(p,_). (4.1)

n

In general, the density matrix is Hermitian and can be diagonalized, p,_ in the above expression is

the diagonal element. If the system is in a pure state, the entropy is zero. If the system is not in

a pure state, the entropy is positive. This definition of entropy does not depend on the question

of whether the system is in thermal equilibrium. The definition given in Eq.(4.1) does not depend

on temperature.

On the other hand, the above definition does not exclude a system in thermal equilibrium. In

the case of a thermally excited harmonic oscillator, the density matrix of Eq.(2.8) is diagonal and

its elements are

p,_ = (1 - e-_'lkT)e -_lkr.

Thus, according to the definition given in Eq.(4.1),

(4.2)

S = w/kT(e _/kT - 1) - ln(1 - e-_°lkr). (4.3)

This expression is the same as the one available from textbooks on statistical mechanics.

In Secs. 5 and 8, we shall study the examples which are not thermal excitations, but share the

same mathematical formalism. The concept of temperature is convenient but not essential in the

examples to be discussed in the following sections.

5. Entropy and Two-Mode Squeezed States of Light

As is well known, the mathematics of harmonic oscillators is the standard language for the

photon-number space. The energy level in a given oscillator system corresponds to the number of

photons, and the ground state corresponds to the vacuum or zero-photon state. The step-up and

step-down operators in the oscillator formalism are given by

at=_ 1(0) X-_xx , a-_ 1(0) x-F_x , (5.1)

273
respectively. Theseare now the creation and annihilation operators. Let us considertwo setsof

theseoperators: al, a and fit a for the first and second modes of photons respectively. We are

interested in the state of these photons where those created and annihilated by a_ and 5 are not

observed.

We construct the two-mode state by applying to the vacuum state the operator exp(-ir/G),

where [6] [10]

i (at at aa) - ,

where the subscripts 1 and 2 are for photons of the first and second kinds respectively.

two-mode squeezed state constructed from

(5.2)

The

17/>= e-i'alO, 0>, (5.3)

where In, fi> is the state with n photons of the first kind and fi for the second kind. According to

this definition, 10, 0> is tile vacuum state. The power-series expansion of the exponential factor

leads to

Iq>= (1/cosh q) _--_(tanh,Pln, n>.

n

(5.4)

In order to distinguish the photons of the first and second kinds, we write the above expression as

I,>: (1/cosh,)_-_(tanh r/)nS,_fitn, a> . (5.5)

The mathematics which led to the above expression is exactly the same as that for the harmonic

oscillator with a shadow coordinate given in Sec. 3. From the mathematical point of view, this

form is the same as the expansion given in Eq.(2.8), and they become identical if we use the

correspondence between T and r/ given in Eq.(3.6). In terms of the 7/ parameter, an element of

the diagonal density matrix is

pn : (tanh r/)2'_/(cosh 7/)2, (5.6)

which leads to the entropy:

S = ln(cosh r/) 2 -(sinh q)2 ln(tanh r/) 2. (5.7)

This form of entropy is determined directly from the squeeze parameter 71, and it is not

necessary to introduce the concept of temperature. The fact is that the measurement or nonmeasurement of photons of one kind affects the measurement of photons of the other kind. In

the present case, the non-measurement of the photon of the second kind increases the degree of

ignorance of photons of the first kind, and this degree of ignorance is measured in terms of the

entropy. The system of photons of the second kind is Feynman's rest of the world.

There are however special cases where the entropy can be associated with temperature. This

is one of those cases. As Yurke and Potasek observed [2], it is possible to define the temperature
of this system by using the connectionbetweenthe squeezeparameter and temperature. The

temperature T is related to the squeeze parameter by

tanh 7/= e -_/2kT. (5.s)

If T approaches zero, the squeeze parameter also becomes zero. As the temperature becomes very

high, the squeeze parameter becomes very large.

6. Time-energy Uncertainty Relation and Relativistic Quark Model

In order to study the role of the time-energy uncertainty relation in relativistic quantum

mechanics and relativistic measurement theory, we consider here a concrete physical example

which gives observable effects in high-energy laboratories. Let us consider a hadron consisting of

two quarks. If the space-time position of the two quarks is specified by xa and xb respectively, the

system can be described by tile variables:

x = (xa+ xb)/2, x = (xo- xb)/2v , (6.1)

The four-vector X specifies where the hadron is located in space and time, while the variable x

measures the space-time separation between the quarks. As for the four-momenta of the quarks p_

and Pb, we can combine them into the total hadronic four-momentum and the momentum-energy

separation between the quarks [4]:

P = Pa + Pb, q = V/2(P_ -- Pb), (6.2)

where P is the hadronic four-momentum conjugate to X. The internal momentum-energy separation is conjugate to x.

In the convention of Feymnan et al. [4], the internal motion of the quarks can be described by

the Lorentz-invariant oscillator equation:

1 x2" ¢(x) = A_,(x), (6.3)

where we use the space-favored metric: x" = (x, 9, z, t). The four-dimensional covariant oscillator

wave functions are Hermite polynomials multiplied by a Gaussian factor, which dictates the spacetime localization property of the wave function. The Gaussian factor takes the form

exp {-_(x1 2 + y2 + z 2 + t2)} . (6.4)

We are accustomed to the polynomial (x 2 + y2 + z 2 _ t2), but not with +t 2. What is the

physics of the Gaussian factor of Eq.(6.4)? If the hadron is at rest, it is possible to construct

three-dimensional harmonic oscillator wave functions with excited energy levels. This would be a

multiplication of the appropriate Laguerre polynomial with the Gaussian factor exp{-(x 2 + y2 +

z2)/2}. As for the time-like separation, Eq.(6.4) contains the factor exp(-t2/2). However, unlike

the position coordinates, there is no excitation along this axis, since the time variable is a c-number

275
[5]. The fact that the time-energy uncertainty relation is a c-number relation is well-known and

well-established. Figure 1 illustrates these features of the uncertainty relations.

t

" : Uncertainty %_ \

t Excitations

• =

Heisenberg: I Uncertainty z with Excitations

It u

s

=2 (t2-z a)

FIG. 1. Quantum mechanics and relativity. The left part of this figure illustrates

that the position-momentum uncertainty relation with excitations and the time-energy

uncertainty relation without excitations, as the time is a c-number variable. The right

part is special relativity. In the light-cone system, it is transparent that the Lorentz

boost is a squeeze transformation. One way to combine quantum mechanics with

special relativity is to superimpose these two figures, as is done in Fig. 2.

Since the three-dimensional oscillator differential equation is separable in both spherical and

Cartesian coordinate systems, ¢(x,y,z) consists of Hermite polynomials of x,y, and z. If the

Lorentz boost is made along the z direction, the x and y coordinates are not affected, and can be

dropped from the wave function. The wave function of interest can be written as

¢'_(z,t) = exp(-t2/2)¢_(z), (6.5)

with

1 _ 1/2

¢"(z) = \-V,__72,_n!) H_(z)exp(-z2/2),

where ¢'_(z) is for the n th excited oscillator state. The full wave function ¢_(z, t) is

¢:(z,t) = (_) 1/2 H_(z)exp{-_(z2+t2)}. (6.6)

The subscript o means that the wave function is for the hadron at rest. The above expression is

nokLorentz-invariant, and its localization undergoes a Lorentz squeeze as the hadron moves along

the z direction [5].

7. Lorentz-squeezed Oscillator Wave Functions

Let us next consider special relativity and Lorentz transformations. It is important to note

that the Lorentz boost is a squeeze transformation in the zt coordinate system if the boost is

made along the z axis. In order to see this point, let us use the light-cone variables, which are

276
definedas

u = (z + t)/v_, v = (z - t)/v_. (7.1)

The u and v axes are perpendicular to each other. In terms of these variables, the Lorentz boost

along the z direction,

takes the simple form

(z') (coshr/ sinhr/)(_) (7.2) t' = k,sinhT# coshr/

?J,# _ e'r/ti_

This transformation is illustrated in Fig. 2.

v' = e-'v. (7.3)

This is an area preserving transformation where

one side becomes contracted while the other side is expanded in a manner that their product is

constant. This is a squeeze transformation.

QUARKS , PARTONS

I,-

z_

Z

..ZSPACE-TIM__ (*'::':"gino) ,_

I DEFORMATION Quarks become II

I I (elmoit) free

z

uJ qo

=_

).-

E<)

I

J I

qo

_.o .o--_5-E_-- _ •o.e

\

• qZ

I

<

' I

\,, _::

t

i j__

t

I Porton momentum I

-'I( d,,,r,bu,,o,)i'--

II becomes wider I

FIG. 2. Lorentz-squeezed space-time and momentum-energy wave functions. This

figure is a result of combining quantum mechanics and special relativity described in

Fig. 1. The physical significance is that this figure gives a unified picture of the quark

model for slow hadrons and the parton model for rapid hadrons. This figure is from

Refs. [5] and [6].

277
In the light-cone coordinate system, the oscillator wave function in the rest frame takes the

form

1 ] 1/2 I7.4)

If the system is boosted, the wave function becomes

¢_(z,t) = [1_r(n_)2,_. 1/2 H_((e-nu + e'v)/v/2)exp ¢_-_(e-1 2_2u + e2'Tv2) } .

(7.5)

This wave function can be expanded as [5]

¢'_(z,t) = (1/coshr/) TM y_(C_,k)l/2(tanhrl)k¢:+k(z)_b_(t),

k

(7.6)

where

Cn,k = (n ÷ k)!/n!k!.

Since the space-time localization property is dictated by the Gaussian factor,

detail the ground state with n = 0. In this case, the boosted wave function is

let us study in

(7.7)

The quantum space-time distribution of Fig. 1 is squeezed to an ellipse described in the upper

half of Fig. 2.

Let us next consider the momentum-energy wave function, which is the Fourier transform of

¢,(z,t):

¢,(qz,qo) = 1 f ¢,(z,t)exp{i(p_z-pot)}dzdt, (7.8)

where qz and qo are defined in Eq.(6.2). Since the integration measure is invariant under the boost,

the evaluation of the integral is straight-forward, and the momentum-energy wave function takes

the form

"'rl(qz,qo) = (1) 1/2 exp{- 17(e2_'2_,+ e- 2' q_) }

(7.9)

with

q_, = (% - qo)/V_, qv = (% + qo)/x/'2.

The Lorentz-squeeze property of the momentum-energy wave function is the same as that of

the space-time wave function, as is illustrated in the lower half of Fig. 2. The significance of

the Lorentz-squeeze property is that it gives observable consequences in high-energy laboratories.

278
By now the quark model for hadrons is firmly established. The proton consistsof three quarks

bound together by a oscillator-like force, accordingto an observerin the Lorentz frame in which

the hadron is at rest. On the other hand, to an observerin a moving frame, the wavefunction

appearssqueezed.If the frame moveswith a speedcloseto that of light, the hadron appearsasa

collection of an infinite number of partons [5][11]. This is called geynman'sparton picture. This

phenomenonis now universally observedin high-energylaboratories,and the squeezedpicture of

Fig. 2 givesan explanation of Feynamn'sparton picture.

Oneof the most uncomfortableaspectsof the presentdiscussionis the time-separationvariable.

Without this variable, it is not possibleto perform Lorentz boosts. On the other hand, there is

no time-separation variable in any of the existing measurementtheoriesof quantum mechanics.

In order to reconcile this difference,we haveto concludethat the time-separationexists, but is

not a measurablevariable. This variableis in Feynman'srest of the universe.

8. Entropy and Lorentz Transformations

Entropy is a measure of our ignorance and is computed from the density matrix, as was noted

in Sec. 4. The density matrix is needed when the experimental procedure does not analyze all

relevant variables to the maximum extent consistent with quantum mechanics. For the bound

state of two particles, the present form of quantum mechanics does not tell us how to measure the

time-separation variable, as is illustrated in Fig. 3.

,.Q

k_

Ol

Q

2;

t

V

-_=0

It

2.c_ffi0"8L_

z

Measurable

FIG. 3. Localization property in the zt plane. When the hadron is at rest, the

Gaussian form is concentrated within a circular region specified by (z+t)2+(z-t) 2 = 1.

As the hadron gains speed, the region becomes deformed to e-2n(z+t)2+e_n(z-t) 2 = 1.

Since it is not possible to rnake measurements along the t direction, we have to deal with

information that is less than complete. The time-separation variable lies in Feynman's

rest of the universe.

If the time-separation were a measurable variable, the pure-state density matrix would be

279
p_(z,t;z',t') = _p_(z,t)[_p_(z',t')]*, (8.1)

which satisfies the condition p2 = p:

= t/)prl(Zrl. 11 Z t ' ,t"; (8.2)

This pure-state density matrix is possible only if both the z and t coordinates are measurable

space-time variables. On the other hand, there are at present no measurement theories which accommodate the time-separation variable t. Indeed, this time separation variable is the coordinate

in the part of the universe outside the system. We do not observe the distribution outside the

system. What we do then is to take the trace of the p matrix with respect to the t variable. The

resulting density matrix is

= z',t)dt =

= (1 / cosh r/) 2(n+1) _ Cn,k(tanh q)2k _,o_+k(z) [_2+k (z')]'. (8.3)

k

The trace of this density matrix is one, but the trace of p2 is less than one, as

Tr(p 2) = p_(z,z l n I)p,(z ,z)dz'dz

= (1/cosh ?_)4(n+l) E(Cn,k)2(tan h q)4k,

k

(8.4)

which is less than one. This is due to th," fact that we do not know how to deal with the time-like

separation which lies in Feynman's rest of the universe. Our knowledge is less than complete.

We can now go back to See. 4 on entropy, and write Eq.(4.1) as

S = -Tr[pln(p)]. (8.5)

If we pretend to know the distribution along the time-like direction and use the pure-state density

matrix given in Eq.(8.1), the entropy is zerc. However, if we do not know how to deal with the

distribution along t, then we should use the density matrix of Eq.(8.3) to calculate the entropy,

and the result is

S = 2(n + 1){(cosh r/)2 ln(cosh r/)- (sinh r/); ln(sinh r/)}

-(1/cosh q)2t_+l)}--_[C_,k ln(C_,k)](tanh q)2k.

k

(8.6)

In terms of the velocity v of the ha&on, where v/c = tanh r/,

S= -(n + 1)/ln[1 -(v/c) 2] + (v/c)21n

[

280
-[1 - (v/c)2] ('_+') Y_[Cn,k ln(C,,,k)](v/c) 2k.

k

(8.7)

Here again, entropy is derived as a measure of ignorance. It does not depend on the question

of whether or not the system is in thermal equilibriu in. The expression for S in Eq.(8.7) does

not depend on temperature.

It was noted in Sec. 7 that the ground-state wave function occupies an important place in the

oscillator formalism, and it will undoubtedly give a simpler and more transparent expression for

the entropy. In terms of the z and t variables, the Lorentz-boosted wave function of Eq.(7.7) takes

the form

which can be expanded as

_b,7(z, t) = (1 / cosh q) _--_(tanh q)_bn(z)¢_(t).

n

(8.9)

The density matrix is

p_(z,z')= rrcosh271 exp -[(z+z')2/cosh2q+(z-z')2cosh2q] ,

(8.10)

and the entropy becomes

S = ln(cosh q)2 _ (sinh 7/)2 ln(tanh 7/)2. (8.11)

As a consequence of Eq.(8.10), the quark distribution p(z, z) becomes

_Z 2

The width of the distribution becomes cosh 2q) 1/2, and becomes wide-spread as the hadronic speed

increases. Likewise, the momentum distribution becomes wide-spread [5] [11]. This sinmltaneous

increase in the momentum and position distribution widths is called the parton phenomenon in

high-energy physics [11]. The position-momentum uncertainty becomes coshr/. This increase in

uncertainty is due to our ignorance about the physical but unmeasurable time-separation variable.

For the special case of the groun.d state, it is possible to convert the entropy into the temperature scale, exactly as we did for the case of two-mode squeezed states in Sec. 5. The squeeze

parameter 7/ used in See. 5 is now the boost parameter. We can use Eq.(5.8) to establish the

correspondence between the temperature and squeeze parameter [12].

281
References

[1] R. P. Feynman, Statistical Mechanics (Benjamin/Cummings, Reading, MA, 1972).

[2] B. Yurke and M. Potasek, Phys. Rev. A 36, 3464 (1987).

[3] J. Von Neumann, Mathematical Foundation of Quantum Mechanics (Princeton Univ. Press,

Princeton, 1955).

[4] R. P. Feynman, M. Kislinger, and F. Ravndal, Phys. Rev. D 3, 2706 (1971).

[5] Y. S. Kim and M. E. Noz, Theory and Applications of the Poincard Group (Reidel, Dordrecht,

1986).

[6] Y. S. Kim and M. E. Noz, Phase Space Picture of Quantum Mechanics (World Scientific,

Singapore, 1991).

[7] H. Umezawa, H. Matsumoto, and M. Tachiki, Thermo Field Dynamics and Condensed States

(North-Holland, Amsterdam, 1982).

[8] E. P. Wigner and M. M. Yanase, Proc. National Academy of Sciences (U.S.A.) 49,910 (1963).

[9] Y. S. Kim and E. P. Wigner, Phys. Lett. A 147, 343 (1990).

[10] P. A. M. Dirac, J. Math. Phys. 4, 901 (1963).

[11] R. P. Feynman, in High Energy Collisions, Proceedings of the Third International Conference,

Stony Brook, New York, edited by C. N. Yang et al. (Gordon and Breach, New York, 1969).

[12] D. Han, Y. S. Kim, and M. E. Noz, Phys. Ray. A 41, 6233 (1990). Phys. 193, 255 (1989)

282
o .

N92-22976

WAVELETS AND THE SQUEEZED STATES

OF QUANTUM OPTICS*

B. DeFacio

Department of Physics 8z Astronomy

Missouri University

Columbia, MO 65211

ABSTRACT

Wavelets are new mathematical objects which act as "designer trig functions." To

obtain a wavelet, the original function space of finite energy signals is generalized to a

phase-space, and the translation operator in the original space has a scale change in the

new variable adjoined to the translation. Localization properties in the phase-space can

be improved and unconditional bases are obtained for a broad class of function and distribution spaces. Operators in phase space are "almost diagonal" instead of the traditional

condition of being diagonal in the original function space. These wavelets are applied to

the squeezed states of quantum optics. The scale change required for a quantum wavelet

is shown, with Prof. G.M. D'Ariano, to be a Yuen squeeze operator acting on an arbitrary

density operator.

1. INTRODUCTION

Wavelets were created in France less than a decade ago 1-_ when J. Morlet 1'4 generalized the phase-space of Gabor 6 by adding a scale change to the frequency (wavenumber)

axis for applications to geophysical exploration. Grossmann 2,5 and Meyer a'5,7 immediately

saw the importance of wavelets for mathematical physics and to deep questions in harmonic analysis, respectively. There are a number of review articles 4,5'7-12 available today,

each specializing in different aspects of wavelets.

In terms of this paper, which applies wavelets to the squeezed states of quantum

optics, two long mathematics papers are the most important. The author is convinced

that they will also be the most important for physics, applied mathematics, engineering

and industrial problems. The two key papers are those of Daubechies is, and Frazier and

Jawerth.14 Daubechiesla first constructed a large family of orthonormal bases of compactly

supported wavelets in L2(Rn). Frazier and Jawerth 14 gave a thorough, complete treatment

of sampled wavelets which is valid both in the classical function spaces and in the modern

distributional spaces.

The approach to squeezed states and quantum optics 15-22 will be through the coherent

states. 23-25 The three main approaches to coherent states are those due to Klauder, 23'26-29

to Perelomov, 2s and to Onofrio. 29 The Klauder construction starts with an arbitrary representation of a Lie group G on a complex separable Hilbert space 7-/and induces a representation of G on itself with "H as a closed subspace of L2(R a, d#) This yields a subrepresentation of the regular representation in the sense of Mackey. 3° It works equally well with

states or fl'ames. The approach of Perelomov starts with a "Little vector" and requires a

multiplier to add enough structure to force projective representation to be unitary. There

is additional subtlety in obtaining an invariant measure on the coset spaces used to reduce

G in that G/H1 can have all invariant measure d#l whereas G/H2 may not. Thus, the

choice of a "Little group" or "stability subgroup" is a sensitive issue in the Perelomov approach. The Onofrio construction yields a holomorphic representation of the Lie group. At

* Supported in part by AFOSR grant 90-307

283
least in simple cases it gives a complexification of the real homogeneous space 2_¢ = G/H,

of Perelomov. For additional structure and the proofs see the nice new monograph of

Kaiser. aa

2. SQUEEZED STATES OF QUANTUM OPTICS

The coherent states for each complex number a are generated from the unique, translationally invariant Foek vacuum [ 0 > using a unitary displacement operator D(a) which

is defined below. Let a,a + be the Bose destruction and creation operator which satisfy

the canonical commutation relations

[a, a +] = 1 ,

[a,a] = [a+,a+] = 0

and define D as a Weyl-Heisenberg operator,

(i)

D(a): = exp(aa + - a'a) (2)

Then

[a >= D(a) [ 0 > (3)

is the ordinary coherent state. In terms of an additional complex parameter _, the twophoton squeezed states [ _, a > of Stoler 15 and Yuen 1° can be generated using the squeezing

operator S(C )

S(C): = exp((a +2 - C'a2) (4)

through the action

I¢,- >= s(¢) I- >

= S(()D(a)[O > (5)

The states generated in Eq. (5) will be called amplitude squeezed states. These coherent

states satisfy the uncertainty principle but squeeze one side, say time or frequency,

exponentially;. Naively, it would seem that higher order squeezed operators S (k), k > 2,

can be defined through the definition

S(k)(_): = exp((a tk - _*a k) , (6)

but a neat paper by Fisher, Nieto and Sundberg 32 has shown the matrix element divergence

< o Is(k)(¢) IO>_ _ , (7)

for all k > 2! This can be interpreted as either non-analyticity of the vacuum or as operator

domain problems. The task of defining k-photon squeezed wavelets will be relegated to

future works. Here a new quantum or operator-valued wavelet of D'Ariano and the

author a3 will be presented. Let A be an observable

Ala>=ala>

where the states [ a > give a resolution of the identity

(8)

1 = fd#(a) Ia >< a I , (9)

284
where d#(a) is the invariant measure. The generating function of moments of the observable A in a state whose density operator is _ is given by

< _;_a >:= Tr[d_A_] (10)

The probability distribution function P(p, a) is defined as

P(/_,a) : = Tr[I a >< a I P]

and is the Fourier transform of the generating function of moments with respect to the

measure d#(a )

< e i_A >= f d#(a)ei_aP(fi, a) (11)

d

A filtered Fourier transform with window function -),(a) for Eq. (11) can be defined naturally as

(e i_A ) : = / d#(O)ei_a7(a)P(_,a ) (12)

7

A c-number wavelet transform analogous to Eq. (12) is given by

w(_,,7,_): - I_ Ii/2 dv(O)x P(_,a) (13)

In the next section, additional discussions of wavelets will be given.

3. WAVELETS

For simplicity of exposition, let feL2(R 1) be a real or complex-valued finite energy

signal and denote its Fourier transform by f(k). In L z, f is guaranteed to exist and by

Parseval's theorem Ifl 2 = Ill 2 with proper normalization. Choose conventions s.t.

and

](k)=f_ :¢

CO

f(x) =//

(14)

(15)

If supp(.f) C (-1/2,1/2) and/eL2(/_ ')

f(x) = _{ f(k) _in(,_(_ _(x - -_)k) }

Inverting Eq. (16) yields

f(k) = {_r f(r)e2_rirk }X(_l/2,1/2)(r)

(16)

(17)

285
where XA is the characteristic function of the interval A C R 1 . With discretization the

mn th coefficient of f (m, n integers) with "window function" g(x-nxo), which is one when

(x - nxo) is positive, is given by

Cmn(f) = £ e27rimk°X°g(x -- nxo)f(s)dx

OD

(18)

Observe that the n-index is a spatial translation of units of x0 and the m-index is a wavenumber translation in units of k0. The joint appearance of (re, n) indicates that Cmn

lives in 2a ® Z d, a phase-space. The scale change x _ x/2 _" plays an important role in

the Calder6n complex interpolation approach. A dilation is a translation in x (space)

with a scale change in Fourier transform variable k. Calderdn 35 published his famous

reproducing formula in 1964. The conditions required are the following: (i) Let ¢ and

be radial, smooth L2(R a) functions whose Fourier transforms ¢ and _b in L2(/_ z) with

support in a set A,

A := supp((b,_) = {k [ O < Ca <[ k [< Cz < cx_} (19)

(ii) For each [k IS 0

(:X)

Y_ _(2_lk I),_(2" I k I)= 1

Let feL2(R d) with ](0) = 0 and then

(20)

Let

](k) = _ ](k)_(2-i k 1)_(2-i k l)

V

a.(k) = ](k)_(2" Ik I)

which implies that rearrangement of Eq. (21) into

(21)

(22)

E c,_,x)e /.\ 2_ri2Vk

8

(23)

has an obvious parallel to Eqs. (14-18). Define the quantities

_v,r(x):=2-"/2_(2-_'x-r)

¢,,,r(x) : = 2-v/2¢(2-"x - r)

_,.(x) : = 2__/2_,(x)

(24)

(25)

(26)

and use these with Eq. (23) to obtain

Now

cs(k) = qok */(s. 2k): =< f, qok,s > (27)

286
k s

(28)

is a continuous wavelet expansion with wavelet coefficients given by the cs(k)'s of Eq. (27).

The mathematical importance of the wavelet expansion over the Fourier method, is that it

generalizes to many function and distribution spaces where Fourier analysis is inapplicable.

The potential physical importance of wavelet methods is to make possible new formulations

and calculations of physical models. For computation or for experimental signal processing

the discrete wavelet transform of Frazier-Jawerth called the _0-transformation is required.

The _0-transform of f is

(f,%¢) ----+ _ _ < f,_0k,, > Ck,,

k s

(29)

and holds in general function and distribution spaces. The requirements on the L 2 function

which _ must satisfy are:

(i) £ v(x)dx=0 , (30)

oo

x-_ (31) (ii) qo(x)=qor,,(x)= qo(----_) ,

// (iii) c,.,_,(_) = 27r ]_(k) ]2 dk k---5 i < _ (32)

The inverse problem of reconstructing f(x) from coefficients 1'2'a'5'12 can then be reduced

to the matrix problem

(33)

Observe the correspondence of 2 v with a scale change and r(r/s) with a spatial translation. A major improvement of wavelets over Fourier methods is apparent from Eq. (16).

Whereas, ](k) has great localization with compact support, f(z) has terrible localization

properties in x since as x --+

sin(Trx) 1

-- (34)

The Prazier-Jawerth improvement in localization via the v-transform is easily seen for

¢(k)eC_(R), where the integration by parts

foo (Of_(k)e_21rik,xdk

(27fix). f(x)= -J__\Ok)

(35)

can be repeated r-times to obtain

(2rix)"f(x) = (-1)" oo -ff'£7 (k)e dk

(36)

287
Hence, asx _ oo

0/" _1

If(x) ,= _,_--_) (37)

The size of cr,s(f, _p) depends largely on f in a neighborhood of the point (0, #) with a

spread (2-") and far from this point the coefficient decays as I x [-s. The simplest way to

obtain wavelets is to decompose the space of interest V1 into a closed subspace V0 and its

orthogonal complement W0 according to the direct sum

v1 = v0 • w0 , (3s)

or schematically

yl

v0 w0

In order to maintain simplicity let V1 denote either L 2 or g2. Let _ be a given function in

V1 which satisfies the relation

eft(x) = Z h(n)21/2¢fl(2x - n) , (39)

n

where the set of coefficients {h(n)}N=l are a collection of constants, called "masking coefficients" and the 21/2 factor in front of W is for L 2 normalization. If V1 is the closed, linear

span of all functions {21/2. V(2x - n)} ,

V1 -- Yn {21/2_p(2 • -n}

and V1 -_ g2 and is a (useful) special case of Frazier-Jawerth.

Proposition: If the masking coefficients satisfy the condition

sup{_lh(k-2n) l2}. <_A

(40)

(41)

for 0 < A < cx) and AeR 1 then the space V0 of Eq. (39) is given by

v0 = v._(.- n)

with V0 -_ g2 and V0 C V1.

Proof: Any f(x)eV1 can be expanded in _,(-- n)'s

(42)

Let

f(x) = E cncp(x - n)

n

= 21/2 c.h(m) (2(x- n) -

n m

= 2'/2 E E c.h(rn)_p(2x - rn - 2n)

n m

(43)

288
where

and take

k.=m+2n

f(x) = 21/2 E bk_(2x - k)

k

bk= cnh(k- 2-) ,

n

 lbk I

k k

< _--]. y]lc. 121h(k - 2n)12

k n

The requirement in Eq. (42) suffices for

k

(44)

(45)

(46)

_. 12) , (47)

for every _eV1. In the event that {h(n)}n=le _2, eq. (42) is automatically satisfied and V0

is a closed subspace of V1.

Question: Does a function CeV_ exist s.t.

(i) W0 = V0_ = V¢(.- n) and

(ii) {¢(.- n)}_= 1 is an orthonormal basis.

Answer: Yes; Daubechies 13 in L 2, by Frazier and Jawerth la in _2, Besov spaces, Sobolev

spaces, bounded mean oscillation (BMO) spaces and Triebel-Lizorkin spaces. Such a function ¢ is a wavelet and

{2v/2¢(2V.-n):v, neZ} , (48)

is an orthonormal basis for L2(R). This reduces the problem to that of finding a finite set

of masking coefficients {h(n)}_ N_ 1. The easiest method for finding these coefficients is due

• #t _ , o • o

to Daubechms in Ref. (13). Assume that the set of non-zero masking coefficmnts is a fimte

set and let _eL 2 s.t.

_(0) = 1 (49)

Given the wavelet expansion

_(x) = E h(n)21/2_(2x - n) ,

take its Fourier transform to obtain

_(k) = _ h(n)e2"'"kl2_(kl2)12112

=: m(k/2)_(k/2) (50)

It is now necessary to show that

N

_(k)= lim _]-[m(k/2J).¢(k/2")}

N---,c¢ [,___

= H m(k/2J) (51)

j=l

289
t

1.0

,_x 0.6

D_

0.2

-0.2

Phi2(x) vs. x

-" I ........_L I .... I ....

| ,=,,I .... I .... 1 ....

0 1 2 3 4

1.0

._ o.s

rO0.0

Phi4(x) vs. x

I .... | A , . = I , m . . I .... I ....

0 1 2 3 4

X

v

cQ_

1.0

0.5

0.0

Phi6(x) vs. x

t , , a .I .... I .... I .... I .... I

0 1 2 3 4 5

X

1.0

X

_" 0.5

c--

D.

0.0

Phi7(x) vs. x

I .... I .... I .... I .... I

0 1 2 3 4

X

Fig. 1

The second, fourth, sixth and seventh iteration of the Box function for T(x).

makes sense .in L 2 (or V1). This suggests a method of finite approximation providing the

masking coefficients are known. Let 7lo(x)eL2(V1) s.t.

and iterate

 0(0) = S o(=)dx = 1 , (52)

OO

= - , (53)

n

to generate _ which is a wavelet, but is not the wavelet ¢ of Eq. (25), but rather is that of

in Eq. (24) instead. In Daubechies nomenclature _ is called a father wavelet and if x is

identified as a "time" variable the dilations (= scale changes and translations) of _ span V0

which acts as the high frequency, k = w, content of the full space V1. The function _ can

be thought of as a "pixel shape" in V0 as pointed out by Kaiser. 31 Similarly, ¢ is called the

mother wavelet and the dilations of ¢ span W0, which contains the low frequency content

of V1. In Figs. 1, and 2 a mother and father wavelet generated by the choice T/0(x) = b(x),

the Box function

b(x)= 0,1, otherwise 1/2 < x < 1/2 (54)

290
Parent Wavelets

1.5

1.0

x

v

O9

" 0.5

"O

cx

0.0

.(::

Q.

-0.5

-1.0

I I I I I I

Father Wavelet

---t.--- Mother Wavelet

I , , , 1 , , , • I . • I , , , , I , I

0 1 2 3 4 5

X

. o

Fig. 2

The father wavelet _o(x) and the mother wavelet ¢(x) for r/0 equal to the Box function.

Since CeWo C 1/'1 ,

¢(x) = Z h(_)21/2v(2x - _)

_(k) = _(k/2). 4(k) (55)

where {h(n)}N=l is a set of masking coefficients for W0 and _(.) is a function analogous

to rn(.) in Eq. (47). Let v -- _r be a translation parameter and observe that finding _ and

¢ is equivalent to finding two trigonometric polynomials s.t. the 2 x 2 matrix

u(k)= (re(k) m(k+r) _(k+r) rh( k ) ) (56)

is unitary. It is useful to consider re(k) as a phase function which partitions by translations,

Im(k)l 2 + Im(k + 7)12 = 1 (57)

To solve for Ira(k)]2 = p([ sin x[ 2, [cos x[ 2) treat P as the probability of a binomial process

with possible outcomes

pl = ]sinx[ 2

p_= Icosxl= (5s)

291
Then re(k) is the square root of P. There are many solutions since P is an even, positive

polynomial but only one is needed. Then using m, it is straightforward to find rS. This

completes the discussion of simple wavelets.

For mathematical physics, operators and their expectations are the objects of interest.

The spectra of operators give infinite dimensional "diagonalizations" in terms of generalized

eigenfunctions. In a wavelet basis an operator is "almost diagonal" in a sense discussed

next.

Let T be an operator, f a function in a normed space which is in the dense domain

of T,

T:f_Tf

(59) (Tf)(x) = [ K(x,v)f(v)dv

d

s.t.

and

C

(i) [K(x,y)l <_ ix _ y------_, (60a)

(ii) ]og(x'V)lOx+ IOK(x'Y) <or I - Ix-cvl2 (6oh)

Let

f(x) = E cjkCjk(x) (61)

j,k

where the Cjk(x)'s are a wavelet basis. Then the kernel of T can be written as

j,k

(62)

where the _)jk'S satisfy the estimate

1 (1 + 12J.x - k]) -1-e I¢;,(x)l < 2-777 (63)

In order to prove that K satisfies the conditions (i) and (ii) split the sum according to

E=E÷E

j,k j<jo ,k j>_jo,k

(64)

with j0 chosen so that

Using the decomposition of Eq.

follows that

The same techniques yield

2_°< Ix- yl < 2J°+1

(64), the estimate of Eq.

C

IK(x,v)l __) _ vl

(65)

(63) and geometrical sums, it

(66)

292
and

OK(x,y)

OX

OK(x,y)

Oy

C

(67) Ix_yl 2 '

C

_< (68) Ix- yl2

There are several consequences of conditions (i), (ii) and Eq. (63). One is that such

operators map L p ---+ L p for all 1 < p < oo, solving deep, old problems. Another is

that {¢jk} are an unconditional basis for L p, 1 < p < co. The proofs work because of

phase-space localization; if two frequencies are well separated their wavelet coefficients

are small and if two times are well separated their wavelet coefficients are small. The

localization structure in Eq. (64) is the reason that the disadvantages of "almost diagonal"

are outweighed by the advantages.

4. WAVELETS FOR SQUEEZED STATES

It is clear that in order to define a wavelet for the squeezed states of quantum optics,

it is necessary to define an operator which changes the scale. This has been accomplished

in Ref. (33) in a project with G.M. D'Ariano which was initiated at this workshop.

Let X(') be an analytic function of the observable .4 defined s.t.

and

x(A)la >= ala >

x(A): = [dp(a)x[a >< a[

J

(69)

(70)

This function satisfies the relation

< X(f) > = Tr[x(h)fi]

= / d#(a)x(a)P(fi, a) (71)

A dilation operator in the Heisenberg picture is defined on an observable A as

Z_n,(A).-- A- '7 (72)

The time picture is suppressed since no other picture will be used here although SchrSdinger

picture operators are given in Ref. (17). For some observables, .,t, the dilation operator

is unitary but D'Ariano 21'a4 has shown that there are important operators of quantum

optics that are completely positive maps, abbreviated CP, and are non-unitary. In the

unitary case, which is the only case discussed here, all products of operators are preserved

and

v,,(A) = x{v,,(A)}

=-X e

(73)

293
Using Eq. (73), the operator-valued wavelet transform can now be written as

7, : - 1 Tr{D,,(A)_}

1 11/ 

1 (D,7_ (X(-'_))) Jell/2

(74)

Thus, the dilation operator squeezes any state described by a density operator/3. In Ref.

(34) two examples are presented:

(i) The unitary dilation of one quadrature of the electric field. This case is applicable to

a phase sensitive amplification.

(ii) The CP dilation map of the particle number which is applicable to improving noise

sensitivity in squeezed light signals.

and analogies of these have been obtained for a quantum wavelet in Ref. (33) with Prof.

G.M. D'Ariano.

5. CONCLUSIONS AND OUTLOOK

The scale change part of the wavelet dilation is accomplished by the Yuen 16 squeeze

operator. The application of wavelets to quantum optics is an idea with some potential.

For example, nonlinear modes and mode-coupling using wavelets should prove useful. The

quantum squeezed wavelet with D'Ariano should be a good candidate for highly dispersive

biological media. Future work will focus on these ideas.

6. ACKNOWLEDGEMENTS

Useful conversations with Professors Grant Welland, BjSrn Jawerth, and G.M.

D'Ariano and with Mr. Randy Thompson are appreciated. Thor is thanked for his company.

294
REFERENCES

1. J. Morlet, G. Ahrens, I. Fourgeau and D. Giard, "Wave Propagation and Sampling

Theory," Geophysics 47, 203-236 (1982).

2. A. Grossmann and J. Morlet, "Decomposition of Hardy Functions into Squareintegrable Wavelets of Constant Shape," SIAM J. Math. Anal. 15, 723-736 (1984).

3. Y. Meyer, "Principle D'incertitude, Bases Hilbertiennes et Alg6bras D'op_rateurs,"

S_minaire Bourbaki, 1985-1986, nr. 662.

4. J. Morlet, "Sampling Theory and Wave Propagation," NATO ASI Series Volume F1;

C.H. Chen, Editor Springer-Verlag, Berlin Heidelberg, 1983.

5. J.M. Combes, A. Grossmann and Ph. Tehamitchian, Editors, Wavelets TimeFrequency Methods and Phase-Space, (Springer-Verlag, Berlin and New York,

1989).

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614-627 (1989).

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Rev. 31,628-666 (1989).

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AL, summer 1989, to be published by SIAM.

11. A. Daubechies, NSF-CBMS Conference at Lowell Unviersity MA, summer 1990, to be

published by SIAM.

12. B. DeFacio, C.R. Thompson and G.V. Welland, "Wavelet Version of Fourier Optics,"

SPIE 1351, Digital Image Synthesis and Inverse Optics, (Soc. Photo-Opt Instr.

Engr Bellingham WA, 1990).

13. 1. Daubechies, "Orthonormal Basis of Compactly Supported Wavelets," Comm. Pure

Appl. Math. 41,909-996 (1988).

14. M. Frazier and B. Jawerth, "A Discrete Transform and the Decomposition of Distribution Spaces," J. Funct. Anal. 93, 134-170 (1990).

15. D. Stoler, "Equivalence Classes of Minimum-Uncertainty Packets II," Phys. Rev. D

4, 1925-1926 (1971).

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2226-2242 (1976).

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Coherent States," Phys. Rev. D 32, 1034-1037(1985).

D. Han, Y.S. Kim and M.E. Noz, "Linear CanonicalTransformations of Coherentand

SqueezedStates in the Wigner Phase-spaceIII Two-mode States," Phys. Rev. A 41,

6233-6244 (1990).

R.J. Glauber, "The Quantum Theory of Optical Coherence," Phys. Rev. 130, 2529-

2539 (1963).

G.M. D'Ariano, "Number-phase Squeezing Through Non-unitary Scaling," Phys. Rev.

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Amer. J. Phys. 58,439-448 (1990).

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296
--

N92-22077

COHERENT STATES AND

PARASUPERSYMMETRIC QUANTUM MECHANICS

Nathalie DEBERGH

Physique th6orique et math6matique,

Institut de Physique au Sart Tilman (B.5), Universit6 de LiSge,

B-4000 LIEGE 1 (Belgique)

It is well known (Refs. 1,2) that Parafermi and Parabose statistics are natural extensions of the

usual Fermi and Bose ones, enhancing trilinear (anti)commutation relations instead of bilinear ones.

Due to this generalization, positive parameters appear : the so-called orders of paraquantization

p (= 1,2,3,...) and h 0 (= 1/2,1,3/2,...), respectively, the first value leading in each case to the usual

statistics. The superposition of the parabosonic and parafermionic operators gives rise to

parasupermultiplets (Refs. 3-5) for which mixed trilinear relations have to be envisaged. In the

particular case of quantas of the same order (p = 2h 0), these relations have already been studied

(Ref. 6) leading to two (non equivalent) sets : the relative Parabose and the relative Parafermi ones.

For the specific values p = l = 2h 0, these sets reduce to the well known supersymmetry (Refs. 7,8).

Coherent states associated with this last model have been recently put in evidence through the

annihilation operator point of view (Ref. 9) and the group theoretical approach or displacement

operator context (Refs. 10-12). We propose here to realize the corresponding studies within the new

context p = 2 = 2h 0, being then directly extended to any order of paraquantization. Even if we have

to take account of the two relative sets separately, the arguments are so similar in both cases that we

just concentrate on the Parabose set in the following. Within the relations characterizing such a

1 atbt2 model, it is easy to prove that the operator A = a + 2 [ a,at ( b,b t) are the usual bosonic

(fermionic) annihilation and creation operators] exactly plays the role of a generalized annihilation

operator i.e. satisfying the expected commutation relation with the hamiltonian and displaying the

right action on the state basis-of the Hilbert space (Refs. 13,14). Parasupercoherent states (Refs.

13,14) ] z > are then defined as eigenstates of this operator A with eigenvalues being arbitrary

complex numbers. The corresponding uncertainty relation is found to be nearly 1 ( h = 1).

The group theoretical approach asks for the consideration of a specific representation of the

para-operators : the Green-Cusson Ans_itze (Refs. 1,15) in which each operator is decomposed into a

sum of two other ones related to the usual bosonic scaling operators and the Pauli matrices. By

introducing parameters realized through two by two matrices with Grassmannian elements, we are led

to the corresponding color supergroups (Ref. 16) and we are thus able to define the associated

coherent states by a unitary representation of these groups. Convenient Baker-Campbell-Hausdorff

relations (Ref. 12) are of particular interest in this study. Moreover the states obtained in this way are

297
effective eigenstates of the operator A introduced before. The three usual definitions of ordinary

coherent states are thus satisfied in this parasupersymmetric context.

REFERENCES

1. Green, H.S., 1953, "A Generalized Method of Field Quantization", Phys.Rev., 90, p.270.

2. Ohnuki, Y., and Kamefuchi, S., 1982, Quantum Field Theory and Parastatistics, University

of Tokyo Press.

3. Beckers, J., and Debergh, N., 1990, "Parastatistics, Supersymmetry and Parasupercoherent

States", J.Math.Phys., 31, p. 1513.

4. Beckers, J., and Debergh, N., November 1990, "On Supersymmetric Harmonic Oscillators

and Green-Cusson's Ansatze", Preprint Univ.Li_ge PTM 90/14.

5. Beckers, J., and Debergh, N., 1990, "Parastatistics and Supersymmetry in Quantum

Mechanics", Nucl.Phys. B340, p.767.

6. Greenberg, O.W., and Messiah, A.M.L., 1965, "Selection Rules for Parafields and the

Absence of Para Particles in Nature", Phys.Rev B 138, p. 1155.

7. Fayet, P., and Ferrara, S., 1985, Supersymmetrv in Physics, North Holland, Amsterdam.

8. Witten, E., 1982, "Constraints on Supersymmetry Breaking", Nucl.Phys. B202, p.253.

9. Aragone, C., and Zypman, F., 1986, "Supercoherent States", J.Phys. A (Math. and Gen.)

19, p.2267.

10. Balantekin, A.B., Schmitt, H.A., and Barret, B.R., 1988, "Coherent States for the Harmonic

Oscillator Representations of the Orthosymplectic Supergroup OSP (1/2 N, _ )", J.Math.

Phys. 29, p.1634.

11. Fatyga, B.W., Kostelecky, V.A., Nieto, M.M., and Truax, D.R., 1991, "Supercoherent

States", Phys.Rev. D43, p. 1403.

12. Kostelecky, V.A., Nieto, M.M., and Truax, D.R., 1986, "Baker-Campbell-Hausdorff

Relations for Supergroups", J.Math.Phys. 27, p. 1419.

13. Beckers, J., and Debergh, N., 1989, "On Parasupersymmetric Coherent States", Mod.Phys.

Lett. A4, p.1209.

14. Beckers, J., and Debergh, N., 1989, "Coherent States in Parasupersymmetric Quantum

Mechanics", Mod.Phys.Lett A4, p.2289.

15. Cusson, R.Y., 1969, "Examples of Parastatistics", Ann.Phys. 55, p.22.

16 Rittenberg, V., and Wyler, D., 1978, "Generalized Superalgebras", Nucl.Phys.B 139, p. 189.

298
N92-22078

A GAUSSIAN MEASURE OF QUANTUM PHASE NOISE

Wolfgang P. Schleich*

Abteilung ffir Theoretische Physik III, Universit/it Ulm

Oberer Eselsberg W-7900 Ulm, Germany

and

Jonathan P. Dowlingt

Weapons Sciences Directorate, AMSMI-RD-WS

U.S. Army Missile Command

Redstone Arsenal, Alabama 35898-5248

ABSTRACT

We study the width of the semiclassical

phase distribution of a quantum state in its

dependence on the average number of photons

(m) in this state. As a measure of phase

noise, we choose the width A_ of the best

Gaussian approximation to the dominant

peak of this probability curve. For a coherent

state this width decreases with the square root

of (m) , whereas for a truncated phase state it

decreases linearly with increasing (m). For

an optimal phase state, A_ decreases

exponentially -- but so does the area "caught"

underneath the peak: All the probability is

stored in the broad wings of the distribution.

I. INTRODUCTION

The ultimate quantum limit in the goal of

optically detecting gravitational waves is to

operate a Michelson interferometer with light

in a quantum state that minimizes the phase

noise at a given mean number of photons

(Ref. 1). But what is a good measure for phase

noise? Should we consider the inverse peak

height of the probability distribution (Ref. 2)

-- the so-called reciprocal likelihood -- or

perhaps the second moment of the phase

distribution (Ref. 3) or even the periodic

measure advocated in Refs. 4 and 5? These

are all based on the idea of a phase

distribution -- but we recall that this in itself

is not a trivial construction since the concept

of a Hermitian phase operator is not without

complications (Ref. 6).

In the present paper we therefore start

from the serniclassical phase distribution

W_[l_/)] of a quantum state I_/) (Refs. 7 and

8). We consider states that have a single

pronounced maximum of W_0 at a phase value

that, without loss of generality, we take at _0=

0. We approximate (Ref. 9) this peak by a

Gaussian distribution with an identical

height of Wq_=0. The distribution's width h_0

is determined by the curvature of W_0 at _o=0,

together with W_o=o. The examples of a

coherent state l ycoh) and a truncated phase

state Ivp) will illustrate this scheme: Their

widths A_ [I _coh)] and ACp[I _/p)] decrease as

the square root and linearly with increasing

average photon number (m), respectively. In

the case of the optimal phase state -- the state

(Ref. 2) that minimizes the reciprocal

likelihood -- we find (Ref. 9) even an

exponential decay for 5(p. However, in

contrast to the coherent or the truncated phase

state, the area underneath this maximum is

not normalized but decays as well (Ref. 10).

* Also at: Max-Planck-Institut ftir

Quantenoptik, D-8046 Garching bei Miinchen,

Germany.

? National Research Council Associate.

299
II. FROM PHASE FUNCTIONAL TO

GAUS SIAN-APPROXIMATED

PHASE DISTRIBUTION

In this section we start from the

semiclassical phase amplitude functional

w_ [Iv>] of a quantum state iV> and derive a

Gaussian approximation to the dominant

maximum of this probability curve.

In the semiclassical limit the phase

distribution W_o [ IV>] of a quantum state

l_v>= N 2_ _m tin)

m=0

(2.1)

represented as a superposition of photon

number states Im), with expansion

coefficients _/m - <mlv) and the

normalization N such that <VIv) = 1, follows

from the phase functional

im_o

w_0[iV>] ---_'2_ m_=oWme

(2.2a)

via

. [,.>1-lw[,.>ll

o_ i(m- n_

=_'_ Z _mVn e

nm,n=O

(2.2b)

For the sake of simplicity, we consider in this

article only quantum states such that Vm =

Vm -> 0. Hence, for the phase value _o= 0 the

terms in the sum of Eq. (2.2b) add

constructively, whereas for _ ¢ 0

cancellations occur. This results in a

maximum at q_= 0.

An approximate analytical expression for

W m [IV)] in the neighborhood of the origin

follows from an expansion of W_o into a

Taylor series around ¢p= 0, that is

1 " 2

W =W + _W_0=0 ¢P ¢p=0

-4- .,,

(2.3)

Here primes denote differentiation with

respect to _0 and we have used the property

• _9K2 L i (m-n)V/mV/n = 0,

WcP 0= 2_ m,n=O

following from Eq. (2.2b). With the help of

Eq. (2.3), we arrive at

W(:ak) = exp ( In %) -- Wq)=0 exp [- ( _0-_-) A 2]

(2.4a)

where the width A_ of this Gaussian is given

by

A_o2 --

2W

¢p=0

W" I

_=01

m=O

(2.4b)

We emphasize that this procedure isvalidfor

any statewhose phase probability W_o [Iv>]

enjoys a maximum at _ = 0.

The area underneath this Gaussianapproximated peak reads as

3OO
oo

SS

--oo

r=

= 4_ A_0 w (2.5)

_-o

This clearly demonstrates that the Gaussian

fit of the peak, Eq. (2.4), is different from the

properly normalized Gaussian G(q)) = _-1/2

W (peak) is

(Acp)-1 exp [-(q_/A_o)2].Whereas

tailored to have a height identical to

W_0=0 [IV)] , the height of G, that is,_-1/2

(Ag) -1, adjusts itself to the width of the

Gaussian, so as to keep the area normalized.

In

o

Z 1 I m) I_¢s) - N l+---m"

m=O

(3.3)

recently proposed in Ref. 2.

These states are normalized to unity.

For a coherent state, Eq. (3.1), we arrive at

_/m-a2_ 2

)

In=O

=_ g

-1/2 dme -m_2

III. PHASE NOISE AND AVERAGE

NUMBER OF PHOTONS

In this section we apply the Gaussian

approximation, Eq. (2.4a), in order to discuss

the width A_ , Eq. (2.4b) of the phase

distribution of:

(i) a coherent state of large displacement

a>>l,

I (2_)-1/4a-1/2 _ _ 2u J %oh )= X..e ira>

m--0

(3.1)

(ii) a truncated phase state

In

I_= (mo+l) -1/2 _ Im),

m=O

(3.2)

and

(iii) the optimal phase state

= 1 (3.4)

where we have replaced the summation by an

integration. For the truncated phase state

IVp ) we find directly

In

-1 o

(¥plVp) = (ino +I) Z 1 ffiI,

mffi0

whereas for the optimal state IVs), Eq. (3.3),

the normalization constant N is given

implicitly by

m o

@/sl_s) = Pt2 Z (1+m)-2

m=O

= _2 {k_=l k-2 -k___(m° '_ + 2 +k)-2 )

that is, (Ref. 11)

301
= mo+l1

Here,

_(s,v) = _(v+k) -s

k--O

denotes the generalized Riemann

function (Ref. 12).

(3.5)

zeta

and the width A_0, given by Eq. (2.4b) reduces

to

A_02[IWcoh)] --

2

(r m) dm

(fdm m2 Ym ) _dm Wm ) - (_dm m _/m) 2

(3.6)

A. Coherent State

For the coherent state, Eq. (3.1), the

expansion coefficients Vm read

_m= L_ojj

Here, we have once again replaced

summations over m by integrations. We

evaluate the Gaussian integrals most

economically by appl_(ing the symmetry of

_m with respect to a z before performing the

integrals. This yields

j'dmm2Wm = dm(m-a2)2Wm+2a 2 j'dm(m-a2)_/m

•--oo _

+at

4

fdraw m = dm(m-a

2

)

2

Win+ a4 Tdm_/m

---oo --oo

and

°?dm m Wm= fdm (m - ct2) _/m

---¢o

+ a amy m ffi a amy m .

This result reduces Eq. (3.6) to

302
T dyexp

_ dyy2 exp

--oo

We now evaluate the area underneath

the Gaussian approximation, Eq. (2.5), for

the coherent state, Eq. (3.1). The maximum

value of the phase distribution reads

=N/_ (m) (3.8)

1

m

2a 2

(3.7a)

The average number of photons (m)

follows from the normalization condition,

Eq. (3.4), as

(m) -= ;dmmW2m

.-=oo

2

-- Ct

and hence

(3.7b)

for large (m). Asymptotically, the width

A_o [IVcoh)] of the phase distribution of a

coherent state decreases inversely as the

square root of the average photon number.

and hence the area of the Gaussian is

•w 0[' od --

Thus the Gaussian approximation, Eq. (2.4)

for a coherent state is properly normalized.

Its width A_ [IVcoh)] decreases linearly with

a but its height We=0 [IVcoh)] increases

linearly with a- _](m) to leave the area

AGauss constant.

B. Truncated Phase State

We now turn to the discussion of the

truncated phase state I Vp) , Eq. (3.2).

According to this equation, the expansion

coefficients _gm read

/

=Jl , for O<-m<m o

\ 0 , form> m °

which reduces A_ [I _p)] to

303
2

(mo+l)

12

=--+0 ,2(0m 3).

2

m

o

(3.10)

Here, we have made use of the summation

formulae (Ref. 11)

m

o

Z l=mo '

m=l

m

ITl = _'rrlO

m=l

and

In

o

Z_

m=l

=_o(_o+1)(_o+,) (3.11)

When we express m o in terms of the

average number of photons

m o

<_>(,÷_o) -1Z

m=l

1

m=--m ,

2 o

we find

(3.12)

again for large (m). The width Acp [l_'p)] of

the phase distribution of the truncated phase

state decreases linearly with the average

photon number (m).

The value of W_o at _=0 reads

_o.<_=>1Cr"o+') .-1l_o /Y--,' )5

I,m=O

= (2_)-1(mo + 1)

~i

=- (rn), (3.13)

and yields for the area underneath the

Gaussian approximation, Eq. (2.4), of the

truncated phase state, Eq. (3.2),

i.e., almost perfect normalization. Again, we

note from Eqs. (3.10) and (3.13) that the width

Aq_ decreases with m o in the same manner

as the maximum height We=0 increases --

keeping the area AGauss normalized and,

more importantly, keeping it independent of

m O .

3O4
C. Optimal PhaseState

We conclude our illustration of the

Gaussian-approximated phase distribution by

applying it to the optimal phase state Ivs>,

Eq. (3.3), which enjoys the expansion

coefficients

t(_+m) -1 for 0 < m < In

' O

_in=

, for In>m o .

The width A_0 [IVs>], Eq. (2.4), then reads

Acp2[ IYs)] =

In=O

1....L_ -

l+m] _m=O l+m /

When we use the summation formulae, (Ref.

11)

In

o

f(mo) - E 1

In---o_

=C+ln(ino+l)+O[(mo+l)-l], (3.15)

where

= 0.577215...

is Euler's constant, and from Eq. (3.11)

In

o

E (l+m)

In=0

=(mo+l)+lmo(mo +1)

1 (mo+l)(mo+2) ='_"

we find

[ )] Ino m

A_02[iWs)] f2 (mo) f(mo) (m+l)2_ 2(m+1)+ 1 o

= _ m+l-I

m=O In+ I In+l

--f2 _mo) m+1,-2(mo÷ 1)

=

-I

+f(m o) f(rno)- mo+l)-f(in o)

= f2 (mo) [2(rno + I) (mo + 2)

-I

_mo,-(mo+X) 2]

In the large mo limit we arrive at

305
A¢02[IWs)]__.2f (mo) 14 1 2 t -1

(:o_1) 2 m°'+l f (m o

o mo+ )

and with Eq. (3.15) we then have

The average number of photons in this state

reads

m

o

(m+l)=N 2 (m o) _ 1

m--O m+l

= _2(m o) moJJ

(3.17a)

that is,

(3.17b)

where we have defined ? = e C in a standard

notation. In determining the remainder, we

have applied in the last step of Eq. (3.17a) the

asymptotic expression for N, Eq. (3.5). With

the help of Eqs. (3.17a) and (3.17b), Eq. (3.16)

reads

Aep[ [Ws)] = _ (m+l) I]2

2

- _ (m+l) 6

e

+ O[(m+l) -1/2

(3.18)

which shows that the Gaussian-approximated

width h_o of the phase state I _s) decreases

exponentially with the average number of

photons. The height

__

= _ (m+l) 2 (3.19)

12

of the distribution increases only

quadratically. Here, we have also made use

of Eq. (3.17a). Hence, the phase probability or

area caught underneath the peak,

1-_ (re+l) 5/2 e + O (re+l> 3/2 e-- _- (m+l) ,

3O6
decreases rapidly to zero with increasing

average photon number. All the probability is

stored in the broad wings of the distribution.

IV. SUMMARY AND OUTLOOK: A NEW

VARIATIONAL PROBLEM

In this article we have presented a

Gaussian approximation to the maximum of

the semiclassical phase distribution of an

arbitrary quantum state. We have illustrated

this scheme using the example of a coherent

state, a truncated phase state, and the

intriguing optimal phase state. Our main

results are summarized in Table 1. The

Gaussian-approximated width A9 [l_coh)], Eq.

(3.7), of the coherent state IVcoh) decreases

as the square root of the average number of

photons, whereas for a truncated phase state

_p) the width A9 [Ntp)], Eq. (3.12), decreases

linearly with (m). In both cases the Gaussian

is properly normalized, that is, the probability

caught underneath the peak is almost

identicalto unity, Eqs. (3.9)and (3.14),and

independent of (m).

The situationis quite differentfor the

optimalphase state l_s).Here, A9 [l_/s)],Eq.

(3.18),decreases exponentiallywith (m), but

the maximum W_0=o [l_s)], Eq. (3.19),

increases only quadratically with (m),

leading to vanishing probabilityin the peak.

All probabilityin this case is in the broad

wings of the distribution,as is discussed in

detail in Ref. 9.

We conclude by noting that the

Gaussian approximation might lead to

insight into questions such as the existence of

a lower bound of A9 for a given fixed

number of photons (m). For that purpose we

would like all the probability to reside in the

peak, that is, AGauss = 1. From Eq. (2.5), we

find

-1/2 -1

W =n acp

q,=0

which, when substitutedintoEq. (2.4b),yields

Aq) 3 = = 2'1"_'_N-2 [ (m_=0m2 Wrn) (m___0Wrn)-- (m___0mWm )_1

-1

= 44x (m-n) 2 _mVn •

0

(4.1)

Two strategies offer themselves: (i) Use

appropriate inequalities to rewrite the

expression in square brackets in Eq. (4.1) in

terms of the average number of photons and

its variance. This might lead to a lower

bound of Ag. (ii)Minimize A93, that is,

maximize the expression in square brackets,

Eq. (4.1),under the constraintofconstant (m)

and phase statenormalized to unity.

ACKNOWLEDGEMENTS

We would like to thank R. E. Slusher and

B. Yurke for drawing this problem to our

attention. In particular, we thank V. Akulin,

R. Bruch, C. M. Caves, R. Hellwarth, and Y.

Yamamoto for many fruitful discussions.

One of us (JPD) acknowledges H. Walther

and the Max-Planck-lnstitutfiirQuantenoptik

for hospitalityand support. We also would

liketo express our deepest appreciationto D.

Han, Y. S. Kim, and W. W. Zachary for

organizing a most splendid conference.

REFERENCES

° Caves, C. M., 1981, "Quantummechanical noise in an interferometer,"

Phys.Rev. D, 23 (8),p.1693.

. Shapiro,J. H., Shepard, S. R., and Wong,

N., 1989, "Ultimate Quantum Limits on

Phase Measurement," Phys. Rev. Lett.,62

(20),p.2377; Shapiro,J. H. and Shepard,

307
.

.

.

.

.

.

S. R., 1991, "Quantum phase

measurement: A system-theory

perspective," Phys. Rev. D, 43 (7), p. 3795;

for a detailed discussion of related

problems, see the articles:

Shapiro, J. H., Shepard, S. R., and Wong,

N., 1989, "A New Number-Phase

Uncertainty Principle" and "Coherent

Phase States and Squeezed Phase States,"

in Quantum Optics VI, J. H. Eberly, L.

Mandel, and E. Wolf, eds., Plenum, New

York, p. 1071 and 1077; Shapiro, J. H.,

"Going Through a Quantum Phase,"

Workshop on Squeezed States and

Uncertainty Relations, NASA CP- _,

1991. (Paper __ of this compilation).

Summy, G. S. and Pegg, D. T., 1990,

"Phase Optimized Quantum States of

Light," Opt. Commun. 77 (1), p. 75.

Bandilla, A. and Paul, H., 1969, "LaserVerst_irker und Phasenunschfirfe," Ann.

Phys. (Leipzig), 23 (7), p. 323; Paul, H.,

1966, "Ein Beitrag Zur Quantentheorie

der Optischen Koh_irenz," Fortschr.

Phys., 14, p. 141.

Bandilla, A., Paul, H., and Ritze, W. R.,

"Realistic Quantum States of Light with

Minimum Phase Uncertainty," to be

published.

For a review, see: Pegg, D. and Barnett,

S., 1986, "Phase in quantum optics," J.

Phys. A, 19, p. 3849.

Schleich, W., Horowicz, R. J., and Varro,

S., 1989, "Bifurcation in the phase

probability distribution of a highly

squeezed state," Phys. Rev. A, 40 (12), p.

7405; also, 1989, "A Bifurcation in

Squeezed State Physics: But How?" in

Quantum Optics V, D. F. Walls and J.

Harvey, eds., Springer, Heidelberg;

Schleich, W. P., Dowling, J. P.,

Horowicz, R. J., and Varro, S., 1990,

"Asymptotology in Quantum Optics," in

New Frontiers in QED and Quantum

Optics, A. 0. Barut, ed., Plenum Press,

New York.

The phase distribution resulting from the

newly proposed phase operator [Pegg,

D.T. and Barnett, S. M., 1988, "Unitary

Phase Operator in Quantum Mechanics,"

Europhys. Lett., 6 (6), p. 483; 1989, "Phase

properties of the quantized single-mode

electromagnetic field," Phys. Rev. A, 39

(4), p. 1665.] yields results identical to

those obtained from semiclassical

quantum mechanics, as in Ref. 7.

. Schleich, W. P., Dowling, J. P. and

Horowicz, R. J., 1991, "Exponential

decrease in phase uncertainty," Phys.

Rev. A, to be published.

10. Similar results have been reported by

Caves, C. M., Lane, A. S., and

Braunstein, S. L., "Maximum-Likelihood

Statistics of Multiple Quantum Phase

Measurements," Workshop on Squeezed

States and Uncertainty Relations, NASA

CP- __, 1991. (Paper _ of this

compilation). Moreover, simulations of a

sequence of phase measurements based

on the scheme of Ref. 2 have been

reported by, Caves, C. M., Lane, A. S.,

and Braunstein, A. S., 1991, "Maximumlikelihood Statistics of Multiple Quantum

Phase Measurements," Proceedings of

NATO Advanced Research Workshop on

Quantum Measurements in Optics, Cortina

d'Ampezzo, 1991, P. Tombesi and D. F.

Walls, eds., Plenum Press, New York.

11. Hansen, E. R., 1975, A Table of Series

and Products, Prentice-Hall, Englewood

Cliffs.

2. Magnus, W., Oberhettinger, F., and Soni,

R. P., 1966, Formulas and Theorems for

the Special Functions, Springer Verlag,

Heidelberg.

308
Table 1. Gaussian approximation WV = W_o=0 exp [- (_/Aq))2]for dominant maximum ofthe phase

distributionW¢ = (2_)-1 N 2 [Zm__0 _m exp (imp) [2 fora coherentstate [_coh),a

truncated phase state I_p) and an optimalphase state IWs) •

Width = A_o

Maximum = Wv=0

Area = _44 Aq_ W_o=0

Coherent

State

2-1/2 (m)-l/2

<m>I]2

Truncated

Phase State

-_ (m)-I

=-:<m>

= 0.98

Optimal

Phase State

(_/_/-3) (m+l) 1/2 exp [__2 (m+l)/6]

7C

_" (re+l)2

rc 5/2 _Z_ (m+l) 5/2

_ exp [__2 <m+l>/6]

309
N92-22 079

IDEAL PHOTON NUMBER AMPLIFIER AND DUPLICATOR

G. M. D'Ariano

Dipartimento di Fisica 'Alessandro Volta', via Bassi 6, 1-27100 Pavia, Italy

ABSTRACT: The photon number-amplification and number-duplication mechanisms are

analyzed in the ideal case. The search for unitary evolutions leads to consider also a numberdeamplification mechanism, the symmetry between amplifcation and deamplification being

broken by the integer-valued nature of the number operator. Both transformations--

amplification and duplication--need an auxiliary field which, in the case of amplification,

turns out to be amplified in the inverse way. Input-output energy conservation is accounted

for using a classical pump or through frequency-conversion of the fields. Ignoring one of the

fields is equivalent to consider the amplifier as an open system involving entropy production.

The Hamiltonians of the ideal devices are given and compared with those of realistic systems.

1. INTRODUCTION

Squeezing and amplification are two intimately related concepts. A scaling of the quantum

fluctuations (A0 2) --_ G2(A02), independently on the state of the field, corresponds to the

amplification of the fluctuating observable (9 ---* GO. Such kind of ideal quantum amplification

rescales all the moments of 0 simultaneously, leaving the signal-to-noise ratio (SNR) unchanged

when detecting 0.

Ideal quantum amplifiers are key-devices in quantum optical applications, where, depending on

the particular circumstances, one would possibly change the levels of both signal and fluctuations

without degrading the SNR. For example, in local-area network (LAN) communications, strongly

subpoissonian fields with limited average number of photons are needed to exploit the ultimate

channel capacity of the field (which is constrained in the total power and the bandwidth). On

the other hand, a large signal is preferred just before the detection stage, in order to minimize all

the subsequent sources of disturbance. In both cases an ideal photon number-amplifier (0 - _)

would allow to change signal and fluctuations as desired, leading to significant improvements of

the network performance.

Another point which should be considered in any quantum amplification process is the role

played by the Heisenberg principle in defining the ideal behaviour of the amplifier. In fact, the

^

amplification of the observable O affects the statistics of the observables which do not commute

with 0. For a couple of conjugated variables (01,(92), analogous to the momentum and the

311

PRECEDING PAGE BLANK NOT FILMED
position of a particle, the quantum fluctuations are constrained by the uncertainty relation

1 [01,0=1} = (1) >__i{

according to which, when the 01 fluctuations are rescaled as {A0_) --+ G=(A0_}," the

corresponding 0= fluctuations become G-={A0_} or larger. An ideal 01 amplifier, namely

an amplifier performing at best, should preserve the minimum uncertainty product and, as a

consequence, it should simultaneously attain the two opposite amplifications

O1 + O01 , 02 --+ O-102 • (2)

Depending on the conjugated pair (61,0=) one has different kind of amplifiers and related different

kind of squeezing. For example, when the conjugated variables are two quadrature of a field

mode (al,a=)--< + iaa = a being the annihilation operator--the rescaling (2) defines the phase

sensitive linear amplifier (PSA). The ideal PSA (essentially a parametric amplifier) preserves the

homodyne SNR= {al}/_ and produces the squeezing in a quadrature of the field. In this

case the transformation (2) is realized by the Yuen's 1 unitary evolution 0*a_ r = #a + t,a*, with

# = (G + G-_)/2 and v = (G - G-_)/2, _] representing the usual squeezing operator.

The photon number-amplifier (PNA) is another example of ideal amplifier, which would

transform ideally r_ into Gh, preserving the direct detection SNR and the number-phase

uncertainty product. The corresponding kind of squeezing is the number-phase squeezing 2 (or

amplitude squeezing), in which the quantum noise is shared between the number h and the

phase 4. This kind of amplifier is a relatively new concept and is probably not simple to realize

concretely: it has been proposed by Yuen a-s, who also suggested physically viable approximate

schemes based on resonance fluorescence. PNA's would be particularly useful in direct-detection

receiver and transceiver in a LAN environment, where, as already mentioned, number states are

preferred to coherent or squeezed states, in order to achieve the ultimate channel capacity of the

field. Furthermore, a PNA (but also a PSA) can be used to realize a lossless optical tap, which,

in a LAN would enable a very large number of users to obtain the same performance as the first

user. 6

In this paper the number-phase amplification mechanism is analyzed in detail, in order to

find physical schemes for an ideal PNA. It is shown that, due to the peculiar role of the two

conjugated variables (h, 4) in the Fock representation, the requirement of a unitary transformation

leads to consider a second field in addition to the amplified one, the two fields being inversely

amplified by the transformation. Input-output energy conservation can be accounted for either

by adding a suited classical pump or by locking the frequencies of the two fields, attaining

simultaneously number-amphfication and frequency-conversion. The obvious constraint of integer

gain G (preserving the integer-valued nature of fi) must be relaxed, to consider the deamplification

case: as a consequence, the abstract amplifying transformation [n} _ [Gn} is replaced by

In} ---+ I[Gn] + n0}, Ix] denoting the integer part of m, and n0 being a constant as a function

of n and depending on the input state of the other field. The general SchrSdinger evolution of

the two fields is

gr(a-,)ln,,',z> = I[On] + G (a<m), [G-lm] + G -1 (Gn} > , (3)

312
(x) = x - [x] denoting the fractional part of x and the gain G being restricted to be either

integer or the inverse of an integer. Eq.(3) can be attained by means of a unitary transformation

involving a classical pump operating at the frequency fl = G-lw,, -wb, wa and wb being the

frequencies of the G-amplified and G-deamplified fields respectively. In the case of simultaneous

amplification/frequency-conversion one has the resonance condition wb = G-Xwa, and the two

fields are intertwined in (3) in order to preserve the total input energy E = (n + G-am)w,,.

In Sect.2 1 derive the transformation (3) and the related Hamiltonian. Apart from the eventual

classical pump, the ideal PNA in the present framework is a four-port nonlinear device (see Fig.l).

However, it can be regarded as a two port device by fixing one input port state (for example,

PNA

I[Gn] +G(G-'m) )

I[G- m] + G (Gn)>

Figure 1: Scheme of the ideal PNA.

using the vacuum) or by totally ignoring one field. In the last case the PNA should be regarded as

an open quantum system which changes the entropy of the input state of the field: the particular

case of coherent inputs will be examined in this respect.

In Sect.3 another device analogous to the PNA is analyzed, namely the ideal photon numberduplicator (PND). Instead of amplifying the number of photons, the ideal PND produces two

copies of the same input state for eigenstates of the number operator. Such a device would be

extremely useful in LAN applications, because it provides a convenient realization of the quantum

nondemolition measurement of the photon number, beside itself realizing lossless optical taps

superior to the amplifier tap s (both applications make possible sharing of information in a LAN).

Arguments related to unitarity--similar to those used for the ideal PNA--lead to the need of a

third auxiliary field, whereas input-output energy conservation can be taken into account either

by means of a classical pump or through frequency-conversion, in a way completely analogous to

the case of the ideal PNA.

In the last section I make some preliminary comparisons between the Hamiltonians of the

ideal devices and those of realistic systems, focusing attention on the gain-2 amplifier, in some

respects very similar to the duplicating device.

2. IDEAL PHOTON NUMBER-AMPLIFIER

2.1 The unitary transformation

In the Heisenberg picture the ideal PNA corresponds to a multiplication of the number operator

h by the amplification factor G

----, Gh. (4)

313
The requirement of an ideal--i.e, minimum-uncertainty preserving--behaviour reflects on the

Heisenberg transformations for the phase operator _, which should be the inverse of (4), namely

For highly excited states (i.e. states approximately orthogonal to the vacuum [0)) and for small

phase uncertainties (Aq, 2) << 1 the following simple definition for the phase can be adopted 7

E_ = e_'* , (6)

/_+ denoting the shift operators E_ = (ata + 1)-1/2a, E+ = (/__)t (/_+ln) = In + 1)). Eq.(6)

shows how the integer-valued nature of h reflects on the phase operator ,_: the amplification (5)

can simply be attained for G -1 = r integer, raising the shift operators to the power r, whereas,

for the number operator, preservation of its integer-valued nature requires G itself to be integer.

For noninteger G, the transformation (4) can be substituted with the following

which coincides with (4) for integer G. For the moment, I focus attention on the deamplification

case (G -1 = r integer), the integer-G case being naturally contained in the following framework.

Denoting by S_ ) the Heisenberg transformation corresponding to Eqs.(7) and (5), for integer

G -l=ronehas

(/_+)r now being represented on the Fock space as follows:

(8)

(E±)rln> = Inir> • (9)

From Eqs.(8) and (9) it turns out that the S_ ) acting on a generic operator 0 has the general

form

,X=O n=O

and the phase factors, being totally ineffective in the action (10), will be dropped in the following.

One can check that the Heisenberg transformation (8) attains the number-amplification (7)

s_)(_) = [,V,'I, (11)

and, formally, S_ ) achieves the phase amplification (5) according to Eq.(6). The transformation

(8) is not unitary and, as a consequence, there is no Hamiltonian producing it. The operators

S(_) in the definition of the map (10) satisfy the following relations

r-1

_(_(_'))t_(_) = 1, (12)

A---O

_[')(_(,))t =_., (13)

..6(_)_(,) A "tt = _(r,) "_s+t_ " (14)

314
Despite the map S_ ) is not unitary, the completeness relation (12) and the orthogonMity

conditions (13) allow one to recover a unitary evolution on an enlarged quantum system. I

postpone for the moment the construction of the corresponding unitary evolution and the related

Hamiltonians, to continue the discussion on the properties of the map.

Eq.(14) leads to semigroup composition of the maps S_ )

s(r)¢(.) v H = (15)

corresponding to the amplification of PNA's in series. On the other hand, as a consequence of the

completeness and orthogonality relations (12) and (13), S_ ) preserves the operator products and

the adjoint operation, thus transforming consistently the whole operator algebra. When applied

on the particle operator a the transformation S_ ) gives the result

S_)(a) _ I-,- 1)[n_/_ (n I _ (1 + [fi_/r] )hi] 1/2a r

n=l (_ "3t- r)[ J _ act) '

(16)

and for the creation operator one has S_)(at) = a_r). Eq.(16) shows that the transformed particle

operator S_)(a) coincides with the r-boson operator a(_) introduced by Brandt and Greenberg: l°

a(r) and a_,) annihilate and create r photons simultaneously and satisfy the commutation relations

[a(r),a_r)] = 1, [h,a(_)] = -ra(_). The preservation of the operator product implies that the

transformation S_ ) applied to a generic operator _) = 0(a, at) (Hermitian analytic function of a

and at) can simply be obtained substituting a and at with a(_) and a_), i.e. S_)(0) = 0(a(r), a_,)).

Therefore, S_ ) corresponds to the construction of the r-photon observables of D'ArianoJ 1'12

The completeness and orthogonality relations (12) and (13) are preserved by similarity

transformations

S(')' = VS(_')I_¢", (17)

I7 and lid being unitary operators. A general transformation (17), however, would destroy, the

ideal behaviour of the PNA: the only similarity transformations which preserve the Heisenberg

evolutions (7) and (5) are the permutations of the ),'s

_(,), /5_(,) _,(') (18)

where P denotes the operator representing a permutation of the A's, namely Pint + A) =

Inr + P(A)).

I return now to the construction of the unitary evolution corresponding to S_ ). From the

definition (10) and the completeness relation (12) it follows that S_) is a completely positive map

(shortly CP map): s'9 this is physically relevant, because the subdynamics of the open systems

are CP maps, the set of CP maps being closed under partial trace. Here I recall that a unitpreserving CP map has the general form T(0) = _o VJOI)'_, where _]o IYJIT_ = 1. The space

of the unit-preserving CP maps is closed under: i) convex combination _]iPiT/; ii) composition

TtT2; iii) tensor product _ ® _; iv) partial trace: namely, if T is CP on _'1 ® _'_ and/_ is a

315
density operator on .T'2,then T_(6) = Tr2[/_T(O ® i)] is CP on .T'I. The last point means that

if we have a unitary evolution in a closed system and if subdynamics on a (open) subsystem can

be defined--i.e, partial trace on the subsystem degrees of freedom--then these subdynamics are

CP maps. However, the converse is not true in general (namely not every CP map corresponds

to a unitary evolution on an enlarged system) and the additional orthogonality constraints (13)

are essential in guaranteeing also the converse assertion.

The unitary evolution corresponding to S_ ) can be constructed by using two different photon

fields in the amplification process and considering the following operator _)(r) acting on the Fock

space of the composite system .7" @ _"

r-1

0(,) = _ _/® (R_'))t, (19)

_=0

where/_(_) are similar to S(_) in the sense of Eq.(17). U(r)-unitarity follows from Eqs.(12), (13)

and (17). The subdynamics of the first photon field correspond to S_):

(0_)010(r)) _--- TY [(Pl ®P2)0_)(01 ® i)0(r)] : Tr 1 [piSS)(01)] , (20)

where the uncorrelated pair of states (_x ® _2) has been considered as the input of the amplifier.

The semigroup property (14) reflects on the composition law for the operators D'(_)

0¢,)0¢oi_-,, 0¢.) , (21)

the symbol __p denoting similarity under permutations (18).

Among all operators/)(_) the case of/t(_) = S(_') is particularly interesting, because the second

field undergoes the transposed transformation of S_ )

r-1

u(r/= _ s_)® (_))t. (22)

One should notice, however, that <ut)O2U(_)) depends on the first input state _x in general. In

fact, the action of the operator/)(r) in Eq.(22) on a number eigenstate is

_¢,)ln,m>= Iln/d,mr + (n/r)), (23)

and the second field undergoes an exact number-amplification only if the first field is in a rphoton state, (namely it contains only number of photons multiple of r), in particular if it is in

the vacuum state. Eq.(23) can be rewritten in the following more symmetrical form

0_-,)ln, m>: Itanl + a (a-_m), [a-'ml + a -_ (a_) >, (24)

which coincides with Eq.(23) for G -_ = r integer, whereas, for G integer, corresponds to (23) but

with the roles of the two fields interchanged, as a consequence of the identity _)(tG) = U(-J) = _)(G-,)

(notice that Eq.(24) leads to integer valued number of photons only if either G or G -_ is integer).

2.2 The Hamiltonian

316
I consider the operator _'(G-1)in Eq.(23) only for the caseG -1 = r integer, the integer-G case

f)(c) corresponding to the inverse operator 0:c_, ). I denote by at and bt the particle operators

of the two fields, namely

In, m>- (a+)" (bt)ml0,0 > . (25)

Comparing the transformation (24) with the action of the multiboson operator of Eq. (16)

a_,)]n) = v/In/r] + 1In + r} , (26)

one can see that the ideal amplification (23) can be attained by interchanging ai with b_,) and

then permuting at with b_ modes. The operator permuting at and b_ (apart from a sign) has the

form

and, as a consequence, the operator U(,) is given by

U(_) = exp [2 (a'b-b'a)] exp [-2 (a_,)b-b'a(,))] . (28)

The representation (23) of the operator _)(,)in Eq. (28) can directly be checked using Eqs.(25)

and (26). From Eq.(28) one can see that, apart from a permutation of the form (27) (which could

be attained by means of beam splitters), the ideal PNA is described by an interaction Hamiltonian

in the Dirac picture

_IiS=-ik(a_,)b-b*a(,.)) , (29)

and an interaction length L given by

71"

kL=-

2'

k being the interaction coupling per unfl length.

rewritten in terms of the a and b particle operators

(30)

Using Eq.(16) the Hamiltonian (29) can be

[-I_ = -ik ((a_)_ f(,.)(a_a)b - b _f(,.)(ata)a ")

S(")(ata) = (ata +1 +1) [ata/r :_.(-J-a I + r) ]

1/2

(31)

Regarding the energy conservation during the amplification process, one can now thinks to

the four-port device as a globally inelastic process in the time domain (Dirac picture), with the

free Hamiltonian

(32) [-Io = waa_a + wbb_b •

317
The interaction Hamiltonian/_D has the form (29) when in the SchrSdingerpicture reads

(33)

with f_ = rwa- Wb. As for the usual parametric frequency-converter the phase factor oscillating at

frequency f/can be attained by considering an additional classical (i.e. highly excited) pumping

field. On the other hand, also the first permuting operator in (28) requires a classical pump (at

frequency f_' = ¢0,, - wb) and it can be attained by means of beam splitters if Wa = Wb, otherwise

it corresponds to a parametric frequency-converter.

The case Wb = rw_ requires no pump at the second step in (28). In this case, the second

operator in (28) can be reviewed as a PNA/frequency-converter (PNAFC) described by the

equations

In/ I + a (an) I,

Ira/- Ila,q + a(a-lm)/,

CO _ COa ) ,

(CO _ COb _ 7'(.0 a) ,

(34)

The PNAFC conserves the total input-output energy E = (n + rm)w_, as it follows from Eq.(34).

On the other hand, for COb= rw, the first permuting operator in Eq.(28) needs a pump at frequency

f_' = (r - 1)co_ and represents now a parametric frequency-converter (FC) intertwining the two

amplified modes. In this fashion the ideal PNA can be viewed as the cascade of an ideal PNAFC

(an energy preserving four-port device) followed by an ideal FC (a four port device with pump).

2.3 The PNA as an open system: the amplification entropy

In practical applications it is useful to consider the ideal PNA as a two-port device, actually

ignoring not only the pumping field, but also one of the two amplified fields. This description

is equivalent to consider the PNA as an open quantum system, which no longer preserves both

the energy and the entropy of the input field. However, the amplification and the deamplification

cases now become quite different. This follows from the unitary transformation (24) where, despite

the apparent symmetrical roles of the two fields, the state of the amplified one depends on the

state of the other, whereas the deamplified field is always independent on the amplified one. This

strange unilateral dependence is due to the integer-valued nature of h, that breaks the symmetry

between amplification and deamplification. In the following I examine the two cases separately.

The number-amplification--ignoring the deamplified field--corresponds to the partial trace

(02) : Tr (tSa ®_2)0-_1(1 ®02)0"(,) = Tr2 ") ,

(35)

where

(36)

Therefore, the amplification corresponds to the CP map

r-1

S(_/')(O) = _--_(l)'_('))J'011_ (_) , (37)

)_=0

318
which, due to the form of operators 1)'_()in Eq.(36), dependson the state _1 of the other field,

(namely on the 'temperature' of the PNA). The case of _1 equal to the vacuum state ('zero

temperature' ) is particularly simple

S_/')(6)= (?0_'))t090 _') ,

oo

90(_)= (_o(_))t = _ I,-n>(,q, (38)

n----O

and corresponds to the exact number-amplification

s(_/')(f(_)) = f(,,_). (39)

In the SchrSdinger picture one has

(40)

Despite the evolution (40) is not unitary (it is only an isometry), it preserves the NewmannShannon entropy

S(_6) = -Tr_log_. (41)

The entropy conservation follows from the orthogonality conditions (13) which imply that

(I70(_))t$70(') = I (but I:0(')(V0(_))t ¢ 1). Thus, the physical picture of the abstract numberamplification In) _ ]rn) corresponds to an ideal PNA operating with the auxiliary field in the

vacuum (namely a PNA at zero temperature). As long as the number-amplification is attained

exactly, no entropy change of the field occurs.

_o

¢,1

I

I

' ' _ I ' ' ' ' I ' ' ' ' I ' '

, , s , I , , , , I , , , , I , , J , I

6 -8 0 8 16 0

I ' I ' I ' I

a.,

Lo

c5

lo

c_

.,,,Jlllll,,,,.,. o

40 80 f20 160 200

he(z) n

Figure 2: Q-function and number distribution of a coherent state having (fi} = 10 photons after

number-amplification with G = 10. The final moments are (5) = 100 and (Aft2) _/2 = 31.52.

319
In Fig.2 the effect of exact number-amphfication on an input coherent state is illustrated in

both the Q-function and number representations. The amplified coherent state

[a)(r) = e -1⁄21_ff _. a" ._-0 'trn> (42)

has a Poisson distribution of multiples of r photons. In the phase space the exact numberamplification corresponds to a symmetrical split of the quantum distribution into r identical

replicas having fluctuations enhanced in the number and reduced in the phase (the quantum

distribution becomes longer in the radial direction and narrower in the tangential one).

I now examine the case of number-deamplification. Ignoring the amplified field corresponds

to the partial trace

(43)

The deamplification thus corresponds to the CP map (10)

r-1

= , (44)

A:O

and is totally independent on the state of the amplified field. In the Schr6dinger picture one has

r-1

= Z (45)

namely, the number-deamplification corresponds to an isometric evolution which does not preserve

the entropy (41) in general. The entropy change depends only on the gain G -1 = r and on

the input state of the deamplified field (and not on the other field). It is worth noticing that

the entropy during deamplification can either increase or decrease as a function of r. As the

photon number-deamplification leads to the vacuum state for r = G -1 --, oo, the entropy is

asymptotically a decreasing function of r for large r. On the other hand, the evolution (45) would

in general transform a pure state into a mixed one (the only state which are left pure being the

number eigenstates and the v-photon states), and thus leads to an increase of entropy in this case.

Therefore, when a pure state is number-deamplified, the entropy exhibits at least one maximum

as a function of the inverse gain r. In Fig.3 the Newmann-Shannon entropy (41) is plotted as a

function of r, for two different input coherent states. One can see that for small average number

of input photons the entropy has only one maximum, whereas for intense input fields several

maxima appear and local very low minima can occur (corresponding to almost pure states). As

a rule, for coherent inputs the maxima are located approximately at r __ being the

average number of input photons and 1 = 1,2,...--the maxima weakening for increasing l and

the entropy S being always smaller than log 2, which is the entropy of two pure states mixing.

In Fig.4 the Q-function and the number distribution of a strongly deamplified coherent state

are reported, in order to illustrate the number-phase squeezing related to number-deamplification.

In a fashion which is exactly the opposite of that depicted in Fig.2 the number-deamplification

leads to spreading in the phase and narrowing in the number, thus converting highly excited

320
to

t,. t,.

¢5 c5

eq

cS

I ' I I ' I

to

4 8 12 16 20

L

0 0 40 160 200

,/

80

r=G -1 r=G -¢

Figure 3: Deamplification Newmann-Shannon entropy versus the inverse gain r = G -1 for a

coherent state with <fi) = 10 photons (figure on the left) and (fi) = 100 photons.

states into nearly Fock states. Asymptotic evaluations 12 for G --+ 0, keeping constant the average

number of output photons, leads to (fi) -,- Glal _ and (Aft2) ,,_ G21al 2. Therefore, the gain G

corresponds to the Fano factor F = (fi//(A_ / of the output distribution, as long as the input

state is excited before amplification in order to keep constant the intensity at the output: in this

way the PNA works as a device converting coherent states in nearly-number eigenstates.

3. IDEAL PHOTON NUMBER-DUPLICATOR

A photon number-duplicator (PND) is a device which, upon acting on a input field in a certain

number eigenstate, produces two output fields both of which are in the same number eigenstate

as the input. Such a device can be realized in principle, whereas a 'cloning' device producing

multiple copies of a (generally not orthogonal) input set of states would violate unitarity. 4'13 For

the ideal PNA the unitary transformation has been obtained starting from the amplifying CP

map defined by the relation

. (46)

The case of the ideal PND can be obtained in strict analogy with Eq.(46) by means of the

duplicating CP map

= E±® E+. (47)

The general transformation attaining the duplication of the shift operators (47) has the form

(48)

321
to

o

to

I

L10

I'''l['ll 'it'''

iiililiililllllill

-5 0 5

I ' I ' I ' I

to

b,.

_5

tO

to

ci

q)

0 20 2,2 24 26 2,8 30

Re(z) ,_

Figure 4: Q-function and number distribution of a coherent state having (_) = 5000 photons, after

strong number-deamplification G -- .005. The final moments are (_) -- 25 and (A/t2/1/2 ----1.16.

where the nonunitary operators S_ (S_ : .T" ® .T" _ .T') are given by

S_= __, 6,,.,,+_[nfin{n,m})(m,n[, (49)

n_m=0

and satisfy the orthogonality and completeness relations

_;,_J = _.i, (50)

oo

E SJS_ = i ®i. (51)

By adding a third photon field we can write a unitary operator U (U : _ ® .T @ .T" ---, .T"® ._ @ _)

as follows

oo

_=--00 _:--00 nl ,n2 ,Tr#,l ,m2 =0

Infin{nl,mi})(mfn{n_,m2}[ ® I_,.)<,-,11® Im=)<m_l. (52)

The operator _r is involutive (i.e. _-2 = 1) and produces the intertwining

0(#+ ® i ® i)0 = i ® #± ® #+,

0(i ® #± ® .E±)O= #+ ® i ® i,

which corresponds to the Fock representation

Ull, m,n) = {

Ira, l, l + n - m) n > m ,

Im, l-n+m,l) n<_m.

(53)

(54)

322
In particular, one has Ull, n,n) = [n,l,l), and for the practically interesting case of the second

and third fields in the vacuum state one obtains

 ln,0,0> = 10,n,n>, (55)

which As the is the required duplication. The scheme of the ideal PND is depicted in Fig.(5).

ideal PNA corresponds to intertwining the one-particle operator at with the r-particle operator

b_,), the PND performs a change between the one-mode operator at and the two-mode operator

b_l.x)

b_l,1) = btct (max{btb, ctc} + l) -1/2 , (56)

which It satisfies the commutation relations [b(1,1),b_l,1)] = 1 and [btb + ctc, b(l:)] = -2b0,1 ).

follows that the Hamiltonian in the Dirac picture is

ttD =-ik (a_b(1,1)-b_l,1)a) , (57)

with the same interaction length as in (30). Conservation of energy now requires a classical pump

at frequency _ = w,, - wb - we, apart from the case of frequency matching w,, = w_ + we, which

preserves the input energy E = wal + ¢,,bm + wen. The PND described in the present context is

more precisely a PNDFC (frequency-converter): in order to keep the frequency constant during

the duplication one can choose wb = we and put a parametric frequency-converter acting on the

input field a.

In conclusion of this section I remark that the ideal PND produces the same effect of a gain-2

PNA when the total number of photons of the two replica outputs is detected. In fact, one has

where

(_,l)(alf(btb + ctc)l_)(x,_)= (2)(_lf(ata)l_)(2) = (alf(2ata)la) , (5s)

oo

1_)(2) = _ anl2n) (59)

n-----O

(60)

denotes the gain-2 PNA output state and

n=0

= 1. (61)

the PND output state corresponding to input

[a) = Ean[n), Elan[ 2

n=0 n=0

(1,1)(o_[f(b_b)lo_)(1,1) : (_,x)<_lf(cte)l<_)(_,x): <_[f(a?a)J_), (62)

Moreover, one has

323
In) *

Io/

Io/

(pump) /

PND

10)

In>

In)

Figure 5: Scheme of the ideal PND.

and the SNR in detecting b'fb, ctc (or their sum) remains the same as at the input of the PND:

in this sense the PND can be regarded an ideal amplifier.

4. COMPARISON WITH REAL SYSTEMS

The Hamiltonians (31) and (57) are quite complicated, due to the presence of an interaction

strength which depends on the input number of photons of one field. For high average number of

photons (ata) >> r the interaction strength in (31) behaves asymptotically as follows

/ _ _-1⁄2(r-1)

f(o(ata) _ _a.a) . (63)

Alternatively, one can look directly at the asymptotic behaviour of the multiboson operator a_):

: ___ (at)r =

[ata/d 1/2(,_- _ + 1)-l/2r_/£v(_)a t

where tcr(_) denotes the function of the phase

_.(_) = _-_e-_(.-_)_.

_,(_,)at, (64)

(65)

Taking into account also the pumping field, the phase-number amplifier would require a medium

with a X (2) susceptibility and an interaction Hamiltonian of the form

H; _ ,_,(_2a)atbc + h.c., (66)

c denoting the annihilator of the pumping field. From Eq.(66) it follows that in order to attain

phase-number amplification one should use a X (2) medium having polarization which depends on

the phase of the field according to (65) only in a limited frequency range. The amplifier gain r is

involved only in the phase factor (65), and the interaction length has to be tuned at the complete

conversion value i = _r/[2,XIX_/2], I_ being the average power flux of the (classical undepleted)

324
pump, and Xc<X (2). For r = 2 one has _2(_) = 2 -1⁄2e-i_ = (2h)-_at. This case is approximated

by the usual degenerate four-wave mixing medium having Hamiltonian

[-I, _ _(3)(at)%c + h.c.. (67)

For )_ o¢ X(3)Ia 1/2 this medium attains gain-2 number amplification approximately in the average

values.

The ideal PND is quite similar to the gain-2 ideal PNA, the main difference being that the field

in the phase-dependent frequency range now splits into the two nondegenerate modes bearing the

replica-states (actually one can analogously define G-Cmultiplicator' devices, which then compares

to gain-G PNA's). When operating on two vacuua as in Fig.5, one can substitute the function

max{btb, ctc} in the Hamiltonian with either btb or ctc, without changing the output.

In conclusion, some remarks are in order, regarding the possibility of attaining the amplifying

CP maps (37) and (44) through interaction with atomic--instead of electromagnetic--fields. In

this case the nonunitary operators in Eqs.(37) and (44) should be regarded as partial trace

of the interaction over the atomic degrees of freedom. The relations (12-14) have no faithful

representation on a finite-dimensional Hilbert space, and one cannot realize them using atoms

with a finite number of levels. However, some similarities can be recognized between this case

and the PNA mechanism. For example, in the high-Q micromaser Fock state generation, the

role of the auxiliary field is played by an inverted two-level atom entering the cavity with a well

defined velocity: 14 the nonunitary reduction of the signal field is obtained by means of nonselective

measurements of the atomic variables. The CP map experimented by the electromagnetic field is

r--1

= y] (68)

or----0

where r = 2, /_D denotes the usual Jaynes-Cummings Hamiltonian, and I T) represents the

inverted state of the atom. As a matter of fact, the high-Q micromaser works as a 'number-phase

squeezer' (in this fashion the micromaser is a sort of PNA and the successive atomic passages

correspond to gain-2 open PNA's in series). One should notice, however, that the number-phase

squeezing in the micromaser strongly depends on the initial state of the field (which should have

less photons then the asymptotic 'trapping' state), and this feature does not depend on the

particular form of the interaction Hamiltonian /:/D, as long as a finite number of the atomic levels

is concerned.

5. ACKNOWLEDGMENTS

I am grateful to V. Annovazzi-Lodi and S. Merlo for discussions and suggestions. This work has

been supported by the Ministero dell'Universith e della Ricerca Scientifica e Tecnologica.

6. REFERENCES

1 H. P. Yuen, Phys. Rev. A13 2226 (1976)

325
2Y.

3H.

4H.

5H.

6H.

Yamamoto, S. Machida, N. Imoto, M. Kitagawa and G. BjSrk, J. Opt. Soc. Am. B 4 1645

(1987)

P. Yuen, Phys. Rev. Lett. 56 2176 (1986)

P. Yuen, Phys. Lett. All3 405 (1986)

P. Yuen, Opt. Lett. 12 789 (1987)

P. Yuen, in Quantum Aspects of Optical Communications, Ed. by C. BendjabMlah, O.

Hirota and S. Reynaud, Lecture Notes in Physics 3'/8, Springer, Berlin-New York (1991)

(in press)

7 Strictly speaking, according to Eq.(6) 4 is not Hermitian (/_+ are not unitary); one should use

the Hermitian operators sin 4 and cos 4. However, for states approximately orthogonal to

the vacuum and for small phase uncertainties (such that sin 4 ,-_ 4), 4 is approximately

Hermitian: the asymptotic commutation relation [h, 4] --_ i holds, assuring that (h, 4) can

be treated as a conjugated pair (see R. Loudon The Quantum Theory of Light, Clarendon

Press, Oxford (1983)

8 K. Kraus States, effects and Operations, Lecture Notes in Physics 190, Springer (1983)

9 G. Lindblad in Quantum Aspects of Optical Communications, Ed. by C. Bendjaballah, O.

Hirota and S. Reynaud, Lecture Notes in Physics 378, Springer, Berlin-New York (1991)

(in press)

10 R. A. Brandt and O. W. Greenberg J. Math. Phys. 10 1168 (1969)

11 G. M. D'Ariano and N. Sterpi Phys. Rev. A 39 1860 (1989)

12 G. M. D'Ariano Phys. Rev. A 41 2636 (1990)

13 W. K. Wootters, W. H. Zurek, Nature 299 802 (1982)

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

326
N92-22080

DECOHERENCE IN QUANTUM MECHANICS

AND QUANTUM COSMOLOGY

James B. Hartle

Department of Physics, University of California

Santa Barbara, CA 93106

ABSTRACT

In the search for a theory of the initial

condition of the universe, quantum mechanics must be applied to the universe as a whole.

For this the "Copenhagen interpretations"

of quantum mechanics are insufficiently general. Characteristically these interpretations

assumed that there was a division of the universe into "observers" and "observed", that

the outcomes of "measurements" were the

primary focus of scientific statement, and, in

effect, posited the existence outside of quantum mechanics of the quasiclassical domain

of familiar experience. However, in a theory

of the whole thing there can be no fundamental division into observers and observed.

Measurements and observers cannot be fundamental notions in a theory that seeks to

describe the early universe where neither existed. In a unified theory of cosmology there

is no fundamental basis for a separate classical physics. Copenhagen quantum mechanics

must, therefore, be generalized to apply to

cosmology.

This talk sketched a quantum mechanics for closed systems adequate for cosmology developed in joint work with Murray

Gell-Mann. 1'2'3 This framework is an extension and clarification of that of Everett 4 and

builds on several aspects of the post-Everett

development. It builds especially the work of

Zeh 5, Zurek 6, Joos and Zeh 7, and others on

the interactions of quantum system with the

larger universe and on the ideas of Griffiths 9,

Omn6s 1°, and others on the requirements for

consistent probabihties for histories.

Three forms of information are necessary

for prediction in the quantum mechanics of a

closed system. In an approximation in which

quantum spacetime is ignored, these are the

Hamiltonian, the initial density matrix of the

universe, and the information specifying particular histories. The most general objective of quantum theory is the prediction of

the probabihties of the individual histories

in a set of alternative histories for the universe. However, the characteristic feature of

a quantum theory is that not every set of

histories that may be described can be assigned probabilities bzcause of quantum interference. Probabilities cau be assigned only

to sets of histories for which there is negligible interference between the individual histories in the set as a consequence of the initial

density matrix of the universe and the Hamiltonian governing its dynamics. Such sets of

histories are said to decohere.

Histories described at an arbitrarily finegrained level do not decohere; some coarse

graining is necessary for decoherence. Coarsegraining was described in the talk and the decoherence functional that measures the level

of decoherence for sets of alternative coarsegrained histories was introduced. Mechanisms for decoherence were reviewed in simple models. Habitual decoherence was argued

to be widespread in the universe for coarsegrained histories defined by certain quasiclassical variables.

A quasiclassical domain is roughly a set

of alternative coarse-grained histories that is

as refined as possible consistent with decoherence and has individual branches that are defined by quantities that are similar from one

time to the next correlated in time mostly according to classical deterministic laws. The

problem of precisely defining quasiclassical

domains was discussed. The question of

whether or not the universe exhibits a quasiclassical domain like the one of familiar experience is a calculable one in quantum cosmology given a suitably precise definition, the

Hamiltonian of the elementary particles and

the initial density matrix of the universe. In

particular, the variables that describe classical physics and the form of its phenomenological equations of motion should be deriv327
able from that Hamiltonian and the initial

condition. 10

A measurement situation is one in which

a variable becomes correlated with a quasiclassical operator of the "quasiclassical domain". The theory of measurements in quantum mechanics was discussed from this point

of view. The recovery of the Copenhagen

formulation of quantum mechanics as an approximation to the more general framework

appropriate in measurement situations was

described. An "observer" (or information

gathering and utilizing system, IGUS) was

treated as a complex adaptive system that

evolves to utilize the relative predictability of

a "quasiclassical domain".

The talk concluded that resolution of

many of the problems of interpretation presented by quantum mechanics is not to be

found within the theory in general but rather

through an examination of the universe's initial condition and the emergent features that

it, together with the Hamiltonian of the elementary particles, implies. Quantum mechanics may be best and most fundamentally

understood in the context of quantum cosmology.

Acknowledgement

Preparation of this abstract was supported in part by the National Science Foundation under NSF grant PHYg0-08502.

References

1. Cell-Mann, M. and J.B. Hartle (1990)

"Quantum Mechanics in the Light of

Quantum Cosmology" in Complexity, Entropy and the Physics of Information,

Santa Fe Institute Studies in the Sciences

of Complexity, Volume VIII, edited by

W. Zurek, Addison Wesley, Reading or

in (1991) Proceedings of the 3rd International Symposium on the Fundations of

Quantum Mechanics in the Light of New

Technology, edited by S. Kobayashi, H.

Ezawa, Y. Murayama, S. Nomura, Physical Society of Japan, Tokyo.

2. See, e.g. Hartle, J.B. (1991) "The Quantum Mechanics of Cosmology" in Quanturn Cosmology and Baby Universes: Proceedings of the 1989 Jerusalem Winter

School, eds. S. Coleman, J.B. Hartle, T.

Piran, and S. Weinberg World Scientific,

Singapore.

3. Cell-Mann, M. and J.B. Hartle (1990)

"Alternative Decohering Histories in Quantum Mechanics", to appear in the Proceedings of the International Conference

on High Energy Physics, Singapore, August (1990).

4. Everett, H. (1957) Rev. Mod. Phys.,

21), 454.

5. Zeh, H.D. (1971) Found. Phys. , 1, 69.

6. Zurek, W. (1981) Phys. Rev. D , 24,

1516; (1982) ibid. , 26, 1862.

7. Joos, E. and H.D. Zeh (1985) Zeit.

Phys. , B59, 223.

8. Griffiths, R. (1984) J. Star. Phys. , 36,

219.

9. Omn_s, R. (1988) J. Stat. Phys., 53,

893; (1988) ibid., 53,933; (1988) ibid.,

53, 957; (1989) ibid., tiT, 357.

10. Cell-Mann, M. and J.B. Hartle (1991)

"Classical Equations for Quantum Systems". Preprint UCSBTH91-15.

328
Q

N92-22081

SQUEEZED STATES IN THE THEORY OF PRIMORDIAL GRAVITATIONAL WAVES

Leonid P. Grishchuk*

JILA, University of Colorado, Boulder, CO 80309-0440

and

Sternberg Astronomical Institute, Moscow University, 119899 Moscow, V-234, USSR

ABSTRACT

It is shown that squeezed states of primordial gravitational waves are inevitably produced in the

course of cosmological evolution. The theory of squeezed gravitons is very similar to the theory of

squeezed light. Squeeze parameters and statistical properties of the expected relic gravity-wave radiation are described.

INTRODUCTION

Squeezed quantum states of light have been successfully generated and detected under laboratory

conditions. It is known how much skill and effort by our experimentalist colleagues it requires to

achieve even a modest amount of squeezing, that is, to obtain the squeeze parameter r of order of 1.

The main purpose of my talk is to show that, in the cosmos, squeezed quantum states of gravitational

waves are produced inevitably and with a much much larger amount of squeezing, simply as a result

of expansion of the Universe.

In the context of gravity-wave research, the notion of squeezed quantum states has been often

referred to. However, what was always meant was the squeezing of a quantized vibrational mode of a

detecting device that could be implemented for a better detection of a classical gravitational wave. For

instance, it was shown (Ref. 1) that the performance of a laser interferometer gravity-wave detector

can be improved by using squeezed light. In another paper (Ref. 2) it was argued that any detectoroscillator can be specially "prepared" in a squeezed state and used for gravity-wave detection during

some interval of time before the thermal noise destroys squeezing and degrades the detector's sensitivity.

However, it is the squeezing of the gravitational waves themselves that will be discussed in my

talk today. I will show that the production of squeezed relic gravitational waves is an inescapable

consequence of the variability of cosmological gravitational field and the existence of zero-point quantum

fluctuations.

The mathematical theories of relic graviton production and squeezing of light are very similar. To

make this similarity especially transparent, I will begin by presenting Einstein's general relativity in the

form of a traditional field theory, such as the theory of classical electromagnetic fields. Those of you who

may feel uncomfortable, or even intimidated, with the notion of curved space-time, will, perhaps, find

it easier to deal with the concept of a gravitational field given in the usual flat Minkowski space-time.

(More details about this 'Held-theoretical" formulation of general relativity are presented in Ref. 3; it

is important to emphasize that we are dealing with a different mathematical formulation of general-

"1990-91 Visiting Fellow, Joint Institute for Laboratory Astrophysics of the National Institute of Standards and

Technology and University of Colorado.

329
relativity, not with some alternate physical theory, see Ref. 4.) This approach leads to manifestly

nonlinear field equations. In contrast to quantum optics based on the laws of linear electrodynamics,

in the case of a gravitational field one does not need any material medium in order to couple a "pump"

field with the "signal" waves; for gravity this is achieved automatically due to the nonlinearity of the

gravitational field itself. As is often done, we will present the total gravitational field in the form

of an approximate sum of a large "classical" contribution and a small quantized perturbation. This

approach will be applied to the cosmological gravitational field of the expanding Universe acting upon

zero-point quantum fluctuations of the gravitational waves. I hate to call the variable gravitational field

of the expanding Universe -the most grandiose and magnificent phenomenon we are aware of- just a

"pump" field, but, technically speaking, it plays precisely this role. As a result, the initial vacuum state

of gravitational waves will evolve into a strongly squeezed vacuum state with very specific statistical

properties.

We will discuss the expected characteristics of the relic gravity-wave background radiation and the

problem of its detection.

FIELD-THEORETICAL APPROACH TO GENERAL RELATIVITY

A gravitational field is fully described by a symmetric second-rank tensor hu_ (note that the gravitational field variables have just one extra index as compared with the electromagnetic 4-vector potential

Au, not a big difference!). For writing down the Lagrangian of the gravitational field it is convenient

to use also an additional set of variables: the tensor field P_,, symmetric with respect to the last two

indices. However, Pau, is not a new physical field but rather a combination of the first derivatives of

hu, , as follows from the field equations.

Gravitational field potentials hu_(x,y,z,t) are mathematically treated as components of one of

many physical fields immersed in the ordinary Minkowski space-time:

da 2 = c2dt 2 _ dz _ _ dy 2 - dz 2 . (1)

The metric tensor of Minkowski space-time will be denoted by 7_. With respect to this metric tensor

all covariant differentiations (denoted by the symbol ";') and lowerings or ralsings of indices are to be

performed. In the Lorentzian coordinates, like the ones implied in eq. (1) and which we will be using

in practical calculations below, %,_ acquires especially simple values: 700 = 1,711 = 722 = "/33 ---- --l,

with the rest of 7uv being equal to zero.

The gravitational part of the total action S = S 9 + S m is

f 8rG Sa = 1 d 4z L a t¢

2ct_ ' ca

where the gravitational Lagrangian La has the form

L9 = (-7) 1/2 [hU";,_P m, - (hu_' + 7m') (P_P_,_ - _

The nongravitational sources and fields and their interaction with the gravitational field are described

by

S, n = --1 [ d4z L m .

C J

330
Theenergy-momentum tensor tgv of the gravitational field itself and the energy-momentum tensor

vgv of the nongravitational matter and fields interacting with gravity are defined in the usual manner:

1 6/2 2 _fL'n

, r v- f7.

The precise expression for tct_, is as follows:

tct_,_ = P_BP_a - _2%,vT"PP_,aP a^lc, - "31( P.P_ - 17uvP_z P_/ + Q"" (2)

where P_, = Paau , Q.. = Qauu;a and

By varying the action 5' with respect to h uv and Pauv one can derive the following gravitational

field equations:

16rG

hgv'" h" -h _ - (t_. + ruv) (3) ,a + 7_v h_B ,a,B - u,_#, _,,a,u c4

where the comma means an ordinary derivative (,,_ = O/Ox _') and %,,, is assumed to be in the simplest

form corresponding to eq. (1). For reference purposes we will also write down the relationship between

the first derivatives of h_,_ and P_uv:

-h"U.,o-lh"_7 a# 7a,,,+(7"_+ h"_)P_,+(Tva+ hva)P_ o-_Pal [(Tua+ h_,_),: + (7._ + h._) _fX] = 0.

The theory possesses a gauge freedom quite similar to the gauge freedom of classical electrodynamics. One can apply the gauge transformations to the gravitational variables h_ and matter variables

without changing the field equations. At the expense of the gauge freedom one can impose some gauge

conditions which are normally used for diminishing the number of variables and simplifying the field

equations.

The transition to the usual "geometrical" formulation of general relativity is established by introducing the new functions g_,_ according to the rule

a.V = (-r + h.") (4)

and by identifying the g_v with the metric tensor of the curved space-time: ds 2 = ga_dxadx _.

GRAVITATIONAL FIELD OF THE EXPANDING UNIVERSE

Let us apply the developed formalism for description of the gravitational field of the homogeneous

isotropic Universe. From our new point of view this is just a specific gravitational field h.v(t, x, y, z)

given in Minkowski space-time (1). Let us take the nonvanishing gravitational potentials in the form

/too = a3(t) - 1 , hn = h22 = h33 -- 1 -- a(t) (5)

33i
wherea(t) is, as yet unspecified, function of time. One can calculate the gravitational energymomentum tensor tuv, eq. (2), and find that the nonvanishing components of at,_ are

3a. 2

at00 = -_a(a -1)-3h 2

.,

Ktll = -_a(3a a - a2 + a - 3) - a2(3a - 1) (6) t_t22 gt33

where the dot means the time derivative (for simplicity we choose the units where the velocity of light

c=1).

The nongravitational sources are assumed to be in the form of hydrodynamical matter with the

Lagrangian

1

L m = -_v/'_ [e + 3I) - (_ + P)gt,_ ut' ut']

where g_., is defined by eq. (4). One can find the nonvanishing components of the energy-momentum

3 2 1)(e p)

roo = ¢+_(a- -

1 2 1)(e p)

7"11 = 7"22 = T33 = p -- -_(a -- -- •

tensor%_:

(7)

By substituting expressions (5), (6), and (7) into the field equations (3) one can derive equations

governing the function a(t) and, hence, the gravitational field (5):

a

- 4_G,e( + 3p), = --e.

a 3 3

(In "geometrical" language, these are, of course, the Einstein equations for a spatially fiat cosmological

model: ds 2 = dt 2 - a2(t) (dx 2 + dy 2 + dz2).) By specifying the relationship between e and p ("the

equation of state") one can solve these equations and find a concrete function a(t).

AMPLIFICATION OF GRAVITATIONAL WAVES

Gravitational field (5) is just a main term of a more complicated and realistic cosmological gravitational field which includes the gravity-wave perturbations. Let us present the total field h._ in the

form

h,,,.= h(.°)+ ,x,y,z) , (S)

where h(°) is given by eq. (5). At the expense of gauge freedom one can always satisfy the conditions

h(1),"_ = 0. Moreover, in the case of gravitational waves, the perturbations of e, p, and u a are equal to

zero and one can, in addition, satisfy the requirements h(_ ) = 0, ,o,vI"(1)~"_/ = 0, so that one is left with

only two independent polarization components (designated by a = 1, 2 ) of h02. For a wave with the

wave vector n one can write down the nonzero components of the field:

hl?(,, (9)

where the constant polarization tensors Pi_ fulfill the conditions p_kn k = O, p'_i = O.

332
Nowoneshouldsubstitute(8) into the field equations(3) andlinearizethemwith respectto h !1)

tj •

It is clear that the left-hand side of eq. (3) is simply the usual D_Alembert differential operator applied

to h!_ ). At the same time, the right-hand side of eq. (3) will contain the products of h (°) and .o_,_h(1) since

all the nonlinearities are collected there.

For a given perturbation with the wave vector n and for each of the two polarization components,

the field equations reduce to a single equation for the time-dependent function #(t) (indices n and a

are omitted):

+ n2" = [ _i + -. + _,- _, (10)

where n 2 = n 12 + n 2_ + n 3_. If there is no "pump" field (5), that is a(t) = 1, the right-haxtd side of eq.

(10) vanishes. It is worth noting that in the gravitational case, in contrast to electrodynaznics, there

is not any dimensional coupling constant between the "pump" and "wave" fields, the strength of the

coupling is regulated by the rate of variability of the "pump" field.

It is convenient to introduce a new time coordinate _/related to t by d_l = a(t)-ldt, and to denote

the _?-time derivative by prime. Equation (10) gets an especially simple form (Ref. 5):

#"+(n2-£--_a) l_=O (11)

which makes it possible to treat the problem as a problem for a parametrically excited oscillator. The

notion of squeezing appears quite naturally.

SQUEEZED VACUUM STATES OF RELIC GRAVITONS

In some cosmologically interesting and realistic situations, the function a"/a goes asymptotically

to zero for T/--, -c¢ and _7_ +o¢. In the asymptotic regions _/--, -c_ and _/-* +_, solutions to eq.

(11) axe very simple: #(_/) ,,_ e+inÈ The general complex solution to eq. (11) can be presented in the

form

p(rl) = a_e(r/) + b+_*(rl) (12)

where _¢(rl) and _*(r]) are complex-conjugated normalized base functions. The same general solution

can be decomposed over other base functions x(r/) and X*(r]):

_u(r/) = cx(rl) -}-d+ X*(rl). (13)

One can choose the base functions in such a way that

and

1 --in-

_(,)-. _e ', for ,-.-_

1 -inX(r/) _ _nn e " for r/--++_.

Since (12) and (13) describe the same solution, their coefficients are related:

a = uc + vd +, b+ = v* c + u*d +, (14)

wherelul2-Ivl 2= 1.

333
For a quantized field, the coefficients a, b+ , c, d + have the meaning of the creation and annihilation

operators and the relations (14) are called the Bogoliubov transformations.

Complex numbers u, v can be parameterized by the three real numbers r, 6, q0, r _> 0:

u = e -i° chr, v = -ei(°+2_*)shr. (15)

The transformations (14) can also be presented in the form:

a = R+S+cSR, b+ = R+S+d+SR, (16)

where S(r, qa) and R(0) are unitary operators:

S( r, tp) = exp[r( e-2i_ cd - e2i_c+d+)]

R(O) = exp[-iOc+ c - iOd+ a_ .

In the theory of squeezed quantum states, the operator S(r, qa) is called the two-mode squeeze operator,

the operator R(O) is the two-mode rotation operator, r is the squeeze parameter and q0 is the squeeze

angle (see, for instance, Ref. 6).

As a result of evolution, the two-mode vacuum state 10,0) transforms into a two-mode squeezed

vacuum state:

ISS) =

The two modes under discussion are two waves with the same frequency but propagating in opposite

directions. In the field of quantum optics, one is usually interested in the temporal fluctuations of the

light field, so that the spatial distribution of the field is not always important. However, in cosmology,

we need to know the complete space-time distribution of the gravity-wave field. For this aim we augment

the time-dependent functions p(r/) with the spatial functions U(x). For every n-mode contribution and

a given polarization component, one has:

hn = Pn,1 Un,1 + Pn,2 Un,2 •

One can work, for example, with complex functions Un,1 = Ke inx, Un,2 = Ke -inx (and complex

conjugated p: pn,2 = P_,l) or with real functions UnA = K cos nx, Un,2 = Ksin nx (and real #),

where K is a normalization constant. Classically, this corresponds to the decomposition of the field

over traveling or standing waves. For the field operators, one writes

hn = (a_ + b+_ *) K e inx + (a+_" + b_) K e -inx ,

in the first case, and

hn -- (bi_ "Jr bl+_ *) v_K COS nx + (b2_ -_-b2+_ *) v_K sin nx, (17)

in the second case. Transition between the two descriptions is fulfilled by the transformation

a + b i(a - b)

V_ = bl, v/_ = b2. (18)

In terms of the theory of squeezed states, it means that, in the second case, one will be dealing with a

pair of one-mode squeezed states instead of a single two-mode state. Indeed, under the transformation

334
(18)the two-modesqueezeoperator factorizes into a product of the two one-mode squeeze operators

Sl(r,_o) , so that instead of (16) one will have

bl = R tStclSiR1

where S1 (r, _o) is the one-mode squeeze operator

Sl(r,_o) = exp r[2:,_,- 1_e,. ]

and R1(8) is the one-mode rotation operator

RI (0) ---- exp(--/0Cl Tel).

The operator b2 transforms in precisely the same manner and is associated with the other squeezed

mode.

As a result of evolution, the one-mode vacuum state transforms into a one-mode squeezed state.

In what follows, we will base our analysis on the representation (17) and on the one-mode squeezed

states. The annihilation and creation operators will be denoted by b and b+.

The classical equations of motion (11) can be derived from the Hamiltonian

]

1 2

H = _ + -/-(_,p+p.) + n_. _

where p is the momentum, canonically conjugated to the coordinate p, p = p' - _-p. "' In a standard

manner, by introducing the annihilation and creation operators b, b+:

one can present the Hamiltonian in the form

H = nb+b + a(,)b +2 + a*(,)b _ (19)

where the coupling function a(T}) is a(r}) = ia'/2a, and the Planck constant h = 1. Note that the

Hamiltonian (19) belongs to the class of Hamiltonians that characterize a number of physical processes

(Ref. 7). However, in most of them the function a(t) has a specific form a(t) = ae -2i_t where a is a

constant, albeit in our case a(_}) is a more general function of time.

The Heisenberg equations of motion, following from this Hamiltonian, have the form

i db a' . db+ a'

-_ = nb + i--b+a , -s.-_ = nb +-i-ha "

Their solution is

b(r]) -" u(rl)bo + v(r/)bo+ , b+(r]) = u*(rl)b+o -}-v'(r/)bo,

335
whereb0, b+ are the initial values of b(r/), b+(r/) (Schrhdinger operators) and the complex functions

u(r/) and v(r/) satisfy the equations

a I , a t

iu t=nu+i-v*, -iv*'=nv*-,--u, u(O)= 1, v(O)=O. (20)

a £1

It follows from these equations that u + v* satisfies the equation identical to eq. (11):

a II

(u + v')" + (n_- --)(. + v') = o .

As for the function u - v*, it can be found from the relation

-in(u- v*)= (u + v*)'- a' 7(,,+ v*).

By substituting eq. (15) into eq. (20) one can find equations for the time-dependent parameters r(r/),

0(_), _(_):

a t

r t = ---- COS2qO

a

a t

01 = n- -- sin2qo thr

{1

a t

qo_ = -n---sin2_octh2r. (21)

a

Solutions to these equations determine the precise form of evolution of the initial vacuum state into

a one-mode squeezed vacuum state. Statistical properties of the final state depend on the numerical

values of r and _ in a well-known way (see, for instance, Refs. 6,7).

A possible way of calculating r(r/) and qo(r/) (Ref. 8) for a given gravitational field a(r/) relies on

the observation that the complex function B(r/), where

u - v* _ * =_chr + e2i_ shr = 2-B(o) (22)

\ u + v* ] chr - e2i_shr n '

satisfies the equation

with the solution

B t .- z-- • n 2 - 2_ B- 2iB 2

2 a

i(p/a)'

B(_)= 2 _/_

where p(r/) obeys eq. (11). The properly chosen solutions to eq. (11) define B(r/) and allow one to find

r(r/) and _(r/) from eq. (22). The meaning of the function B(r/) is that it determines the Gaussian wave

function _(#,r/) ,,, exp(-B(r/)# 2) which is a solution to the Schrhdinger equation in the coordinate

representation.

The parameters r(r/), _(r/) can be calculated (Ref. S) for a cosmological model which indudes

three sequential stages of expansion: inflationary (a(t) ,,_ ell°t), radiation-dominated (a(t) ,_ t 1/2) and

matter-dominated (a(t) ,,, t2/3). It can be shown that the present-day values of the squeeze parameter

r range from r _ 1, for frequencies v _ l0 s Hz, up to r _ 120, for frequencies v _ 10 -is Hz. In the

336
frequencyinterval ix_ 10-1 - 10-3 Hz, accessible for the planned Laser Interferometer GravitationalWave Observatory in Space (Ref. 9), the squeeze parameter r reaches large values of order 40-50.

As for the parameter _(il), it can be shown to have the form _ _ -nil + _0, where _a0is a constant.

This behaviour can already be envisaged from eq. (21) for _d, since, in the asymptotic region il -+ +oo,

one has I_] << n and cth2r _ 1.

RELIC GRAVITONS: A STOCHASTIC COLLECTION OF STANDING WAVES

As we see, the cos nx and sin nx modes in the representation (17), evolve into a strongly squeezed

vacuum state. The mean number of quanta (N) and its variance ((AN) 2) are determined by the

squeeze parameter r:

(N) = sh2r, ((AN) 2) = 2sh22r .

"l

The mean values of #2 and 15are equal to zero, but their variances do not vanish:

=l(ch2r- h2rcos2 ), =2(ch2r+ h2 cos2 ).

In order to relate the rigorous quantum-mechanical treatment, described above, with the notions

of random classical waves, one can use the Wigner function formalism. It allows one to derive the

distributions of the random variables A and ¢ entering the classical expression for #:

# = A sin(-nil + ¢). (23)

It can be shown (see Ref. 8) that, in the limit of large r, the Gaussian distribution for ¢ is very

narrow, like a S-function. It is concentrated near the values

¢ = _ao+ _rt (24)

where _ao is a constant, the same for all unit vectors n/n, and t = 0,-4-1, ....

A similar conclusion can be reached in a simpler, though perhaps less rigorous way. Let us consider

the ratio ((A/i)2)/((A/i)2). For large r, this number is approximated by

1

2) tg2 •

For a classical expression (23), this ratio corresponds to the number n -2 tgl(-nil + ¢). From their

comparison, and taking into account the fact that _ _ -nil + q00, one can derive eq. (24). The very

small variance of the phase, A¢, is, of course, consistent with the large variance of the number of

quanta, ((aN)l).

The negligibly small variance of the phase distribution leads to an important result: every pair

of cos nx, sinnx modes forms together a standing wave. Indeed, let us consider a given n. The

corresponding terms, contributing to the total wave-field h(il, x,y,z), can be written in the general

form:

hn = A1 sin(-nil + ¢1) COS nx + A2 sin(-nil + ¢2) sin nx. (25)

337
TheamplitudesAx and A2 axe taken from a broad Gaussian distribution and axe, in general, different.

However, the phases 4h and 4_z are taken from a very narrow Gaussian distribution and are essentially

fixed and equal up to +Tr. Because of that, expression (25) can be written as a product of a function

of time and a function of spatial coordinates:

hn = + sin(-nr/+ qoo)(A1 cos nx + A2 sin nx). (26)

In other words, expression (26) describes a standing wave. A characteristic feature of a standing wave

pattern is that the field vanishes all over the space at every half of the period. The randomness of the

wave-field is displayed in its spatial functions At cos nx + A2 sin nx. This is why we say that the relic

gravitational waves are present now in the cosmos in the form of a stochastic collection of standing

waves.

The total field h(r/,x) is obtained by summing over all n-mode contributions (26). Of course, the

total field loses the property of vanishing at some moments of time, because the various sin(-nr/+

_o0) factors have different arguments. However, the difference in the arguments is not random, but

deterministic. For instance, if at some moment of time r/= r/0 the component hn(r/,x) vanishes, the

same will be true for all other components hra(r/,x), where m = n(1 + k/g) and k/g is an arbitrary

rational number. Hopefully, this property can somehow be used in a specific strategy of observational

discrimination of relic gravitational waves from stochastic gravitational waves of a different origin. I

think that the inevitable "squeezing" of relic gravitational waves (and other primordial fluctuations

of quantum-mechanical origin) can manifest itself in a variety of circumstances, not all of which are

foreseeable at the moment.

References

1. Caves, C.M., 1981, Phys. Rev. D 23, p. 1693.

2. Grishchuk, L.P., and M.V. Sazhin, 1983, Zh. Eksp. Teor. Fiz. 84, p. 1937 (Soy. Phys. JETP,

1983, 57, p. 1128).

3. Grishchuk, L.P., A.N. Petrov, and A.D. Popova, 1984, Commun. Math. Phys. 94, p. 379;

Grishchuk, L.P., and A.N. Petrov, 1987, Zh. Eksp. Teor. Fiz. 92, p. 9 (Soy. Phys. JETP, 1987,

65, p. 5); Grishchuk, L.P., and A.N. Petrov, 1986, Pis'ma Astron. Zh. 12, p. 429 (Soy. Astron.

Lett., 1986, 12, p. 179).

4. Grishchuk, L.P., 1990, Usp. Fiz. Nauk 160, p. 147 (Soy. Phys. Usp., 1990, 33, p. 669).

5. Grishchuk, L.P., 1974, Zh. Eksp. Teor. Fiz. 67, p. 825 (Soy. Phys. JETP, 1974, 40, p. 409);

Grishchuk, L.P., 1988, Usp. Fiz. Nauk 156, p. 297 (Soy. Phys. Usp., 1988, 31, p. 940).

6. Schumacher, B.L., 1986, Phys. Rep. 135, p. 317; Yuen, H.P., 1976, Phys. Rev. A 13, p. 2226.

7. See special issues: 1987, J. Opt. Soc. Am. 84(10); 1987, J. Mod. Opt. 34(6,7).

8. Grishchuk, L.P., and Yu.V. Sidorov, 1990, Phys. Rev. D 42, p. 3413; Grishchuk, L.P., and Yu.V.

Sidorov, 1989, Class. Quantum Gray. 6, p. L161.

9. Failer, J.E., P.L. Bender, J.L. Hall, D. Hils, R.T. Stebbins, and M.A. Vincent, 1989, Adv. Space

Res. 9, p. 107.

338
SQUEEZED STATES:

S. T. All a, J. A. Brooke b. J.-P. Gazeau c

a Department of Mathematics and Statistics

Concordia University

Montreal, Canada H4B 1R6

A GEOMETRIC FRAMEWORK

N92"22082

b Department of Mathematics

Florida Atlantic University

Boca Raton, FL 33431, USA

and

University of Saskatchewan

Saskatoon, Canada S7N 0W0

c Laboratoire de Physique Thdorique et Mathdmatique

Universitd Paris 7, F-75251 Paris Cedex 05, France

ABSTRACT

A general definition of squeezed states is

proposed and its main features are illustrated

through a discussion of the standard optical

coherent states represented by "Gaussian pure

states" (Ref. 1).

The set-up involves representations of

groups on Hilbert spaces over homogeneous

spaces of the group, and relies upon the

construction of a square integrable (coherent

state) group representation modulo a subgroup

(Ref. 2). This construction depends upon a

choice of a Borel section which has a certain

permissible arbitrariness in its selection; this

freedom is attributable to a squeezing of the

defining coherent states of the representation,

and corresponds in this way to a sort of gauging.

AN EXAMPLE: GAUSSIAN PURE STATES

Gaussian pure states (GPS's), as defined

by Schumaker (Ref. 3) and elaborated by Simon,

Sudarshan and Mukunda (Ref. 1), are functions:

¢(5) = (const) exp[P(_)]

where _ E [Rn and P(_) is a quadratic

(i)

*S.T.A. and J.A.B. gratefully acknowledge partial

support from NSEI_C.

polynomial in _ with complex coefficients:

p(;¢)= -i/2 xZx+lw. "" .- _¢+w o (2)

with Z a symmetric (Z t =Z) complex nx n

matrix, _v e _:n, w o e _:, and where, if

Z = V- iU is the decomposition into real and

imaginary parts, then U is positive definite

(v > 0).

Let G denote the semi--direct product of

the Weyl-Heisenberggroup H(2n + 1) with the

symplectic group Sp(2n;[R) of symplectic linear

maps of [R2n.

G = H(2n -t- I) o Sp(2n;IR). (3)

Multiplication in G is as follows:

glg2 = (clc2exp[if_(Ql,S 1Q2)/2],QI+s 1Q2,sls2)

(4)

where g = (c,Q,s) denotes a general element of

G; (c,.Q) eH(2n+ 1) where [c[ = 1, Q e[R2n;

s e Sp(2n;_); f_ is the symplectic structure on

IR2n defined for Qr = (qr'Pr)' r = 1,2, by

f_(Q1,Q2) = pl.q2-P2.ql.

Let U(g), g e G, denote the irreducible

unitary representation of G on the Hilbert

339
SQUEEZED STATES: A GEOMETRIC FRAMEWORK

space rid= L2([Rn) . whose restrictions to

H(2n + 1) and Sp(2n;IR) are:

[U(c,Q)¢](_) = c expp[p.x-p.q/2

and

[U(s)¢](_) = (2r)-n/2(detA) -1/2

x ] RneXp[iS(_,_)] _p(_)d]_

(6)

where

S(Tc,I_) =-1/2_¢CA-1_ + ]_A-I_c + 1/2]_A-1B]_,

¢ is the Fourier transform of ¢, and we assume

det A _ 0 if

U defines

Sp(2n;iR),

metaplectic

ignored here.

Some facts (see Refs. 1 and 4):

a double-valued representation of

i.e., a representation of the

group, but this subtlety will be

(i) Let ¢o = r-n/4exp[---x" E/2] be the special

-_ ...4

GPS with U=I n , V=O n , w=o, w o=o,

and suppose ¢ is any GPS. Then

¢ = (const)U(g)¢ ° for some g e G; moreover if

g" e G also satisfies this condition then

g" =gk ° for some koeKo:

K ° = Sp(2n;lR) N O(2n;lR) _ U(n) (7)

which is a maximal compact subgroup of

Sp(2n;IR), and conversely.

(ii) Sp(2n;iR) has a "block Iwasawa"

decomposition:

Sp(2n;IR) = N'AK o

(s)

where N" = °I n ]:vt-v} ,

A On]:At = A >0t,

J

Ko={[-_ b] :ata+btb=I n ,

Moreover, defining U = A -2 and

to 1/2

we have for ¢ c ,_":

[U(s(U,V).)¢](_)=

(detU)1/ 4exp[i_V_/21¢(U1/2_)

and if ¢= ¢o in(i), Z=V-iU

atb _ bta}.

(9)

then:

[U(s(U,V))¢o](_)

--n/4(detU)l/4exp[_i_Z_/2]

(lO)

(iii) A maximal compact subgroup

g = V(1)x K ° =_V(1)x U(n)

K of G is:

(11)

where U(1) = {(c,O) e H(2n + 1)}, and with

the same meaning for U(1) define also:

H = U(1)x Sp(2n;IR). (12)

By Y we mean the homogeneous space:

Y = H/K ._ Sp(2n;IR)/K o

(13)

which may be regarded as the set of positive

symplectic matrices; these are uniquely

t

expressible in the form s(U,V) -1 s(U,V) -1

thereby establishing an identification of Y with

the collection of GPS's centered at

= _ -_ [R2n. Q (o,o) e By X we mean the

homogeneous space:

X = G/H z [R2n (14)

the identification being through the map sending

(c,Q,s) eG to Qe[R 2n. In this way X inherits

340
SQUEEZED STATES" A GEOMETRIC FRAMEWORK

the symplectic structure _ which happens

therefore to be G-invariant.

SQUEEZED COHERENT STATES

The theory of square integrable

representations modulo a subgroup (Ref. 2) is

immediately applicable in the present situation,

and together with the notion of squeezing, to be

outlined below, provides a convenient picture of

GPS's. In what follows, a general discussion

(example of which is the case of GPS's, same

notation) will be given.

Let H be a closed subgroup of a Lie group

G. Let X = G/H and suppose du(x) is a left

G-invariant measure on X (in general, du

need only be quasi-invariant); let _:X-_ G

denote a Borel section. Now suppose U(g),

g e G, is an irreducible unitary representation

on Jd, and suppose there exists an admissible

vector 77e d_d such that, as a weak integral:

]X}_?/3,x> <r//3,x[dU(x ) = a/3 (15)

defines a bounded, positive operator A/3 with

(possibly unbounded) inverse; 77/3,x denotes the

vector U(/3(x))_? (example: notation as before

with 77= ¢o' x = Q, /3(x) = (1,Q,I), dr(x) =

_Q ^ f_Q ^ ... ^ f_ Q (n factors), A/3 is a

multiple of the identity on ¢g). In this case U

is square integrable rood (H,/3), and {r//3,x} is

a family of coherent states for U.

Let K be a closed subgroup of H of k's:

U(k)r/= p(k)'r/ (16)

where p is a 1---dimensional unitary

representation of K ([p(k)[ = 1). Let

7:Y = H/K-_ H denote a Borel section, and

define:

o:X ,, Y --, G, a(x,y)= /3(x)7(y) (17)

For fixed y c Y, a(.,y):X _ G is a Borel section

and letting Ya(.,y),x = U(cr(x,y))r/ it is easy to

verify that U is square integrable

rood (H,a(.,y)) for each y e Y; in fact:

/'x[r/a(.,y),x > <r/a(.,y)x[dU(x)= A_8iy )

defines a bounded, positive invertible operator

(which coincides with A/3 here). The collection

{_a(., ,y),x} ,, for y fixed, is a family of squeezed

coherent states associated to {r//3,x}; one

interprets U(7(y)) as the squeezing operator

defining a change of section /3(.) _/3(.)7(y)

(example: k = (c,O,ko), p(k) = c,

y = Z - V-iU, _y) = (1,O,s(U,V))).

In this manner, squeezing in its general

setting is describable in terms of changes of

Borel section of the associated coherent state

representation. Details of this general

construction with examples will appear

elsewhere.

REFERENCES

1. Simon, R.A., E. C. G. Sudarshan, and N.

Mukunda, 1988, "Gaussian pure states in

quantum mechanics and the symplectic group,"

Phys. Rev. A 37, pp 3028-3038.

2. Ali, S.T., J.-P. Antoine, and J.-P. Gazeau,

1991, to appear, "Square Integrability of Group

Representations on Homogeneous Spaces I & II,"

Ann. Inst. H. Poincard A.

3. Schumaker, B.L., 1986, Phys. Reports 13___.55,

317.

4. Folland, G. B., 1989, Harmonic Analysis in

Phase Space, Annals of Math. Studies #122,

Princeton U. Press, Princeton.

341
N92-22083

Wormholes and Negative Energy from the Gravitationally

Squeezed Vacuum

David Hochberg

Department of Physics and Astronomy, Vanderbilt University

Nashville, TN 37235

ABSTRACT

Minkowski-signature wormhole solutions of the Einstein field equations require the existence of

negative energy density in the vicinity of their throats. In this note, we point out that the

gravitational interaction automatically generates squeezed vacuum states of matter, which by

thier nature, entail negative energy and thus provide a natural source for maintaining this class

of wormholes.

1. Introduction

Wormholes are handles in the spacetime topology linking widely seperated regions of the Universe. Major insights have been made in the past few years in understanding general properties

and physical consequences of Minkowski-signature wormholes [1-3]. A key aspect of wormholes

discovered in [1] has to do with the type of matter and energy needed to thread the wormhole

throat: it must violate the weak energy hypothesis. Although no known form of classical matter

violates this energy condition, the squeezed vacuum does, and moreover, the coupling of matter to

gravity leads automatically to the production of squeezed vacuum states [4]. The negative energy

of the squeezed vacuum can be understood in simple terms. Consider a single mode oscillator. Its

vacuum state is represented in phase space (from the Wigner distribution) by a circle centered at

the origin. The squeezed vacuum state, by contrast, leads to an elliptical region. As this ellipse

rotates (with the angular frequency of the mode), its periodic profile exhibits quantum fluctuations both larger and smaller than the uniform profile characteristic of the unsqueezed vacuum

state. In field theory, the energy of the unsqueezed vacuum gets renormalized to zero. Thus, any

state having lower fluctuations than the ordinary vacuum must have a negative (renormalized)

energy.

2. Quantized scalar in a uniform gravitational field

We make these concepts explicit by showing that the interaction between matter and gravity

leads to a squeeze operator acting on the Fock space of particle states, which includes the vacuum.

We consider a scalar under the influence of a uniform background gravitational field. From the

equivalence principle, this can be handled by transforming to a uniformly accelerating frame (i.e.,

Rindler space). Take the background field pointing in the x-direction. The transformation from

Minkowski (t,x) to Rindler (T,X) is x = X cosh(T) and t = X sinh(T), and the scalar equation

to be solved is

[](I) + ra:_ = 0. (1)

The normalized solution is given by

4,k.5(T, X; xi ) = 7r-'[sinh(Trj)]l/2K_j((m: + k_)l/:X)e -ijTe ik±'x± , (2)

343

PRECEDING PAGE BLANK NOT FILMED
where K is a modified Bessel function of imaginary order, m is the scalar mass, k± = (kv, kz)

refers to the transverse momentum and j _> 0. Contrast these Rindler modes with the familiar

plane wave solutions of (1) for Minkowski space:

ei(k.x-0_kt)

Vk-- 2v i, (3)

where w k = (rrt 2 + k2) 1/2. The complete solution of (1) in Rindler or Minkowski space can be

expanded in terms of the sets (9.) and (3), respectively. The expansion coemcients become, after

canonical quantization, operators satisfying the algebras [5(j, k±), 5t(j', p±)] = 6(j, j')6(k± - p±)

and [a(k),at(k)] = 6(k- p) The Rindler and Minkowski vacua are defined by 5]0 >= 0 and

a[0 >= 0. Completeness of the two sets of modes (2) and (3) leads to nontrivial relations among

the Minkowski and Rindler creation/annihilation operators:

a(k) = f dj d2px a(j, px]k)5(j,p±) + t3*(j,p±]k)_tt(j,p±), (4)

together with the Hermit±an conjugate. The Bogolyubov coefficients in (4) are computed from the

inner product and measure the overlap between the Rindler and Minkowski modes: a = ((I)kj.jIUp)

and/3 = -((I)kzj[Ut_ ). A most important consequence of (4) is the inequivalence of the vacuum

states [0 > and ]0 >, and the Fock spaces built up from them. This inequivalence shows up

physically as squeezing.

3. The Squeezed Vacuum

The free Hamilton±an for a massive scalar in Minkowski space is

H = f dZkwkat(k)a(k), (5)

where Wk = (m 2 + k2) a/2. From the point of view of the Rindler modes, (5) is a quasiparticle

hamilton±an, and so the canonical transformation in (4) allows one to derive the exact interaction

hamilton±an acting on Rindler states. One can also start from the exact Rindler hamilton±an,

which has the same form. as (5) when expressed in terms of the Rindler operators, and applying

(4) leads to the exact Minkowski interaction hamilton±an. Since the intermediate momentum

integrations are easier to carry out in the Minkowski picture, we derive the squeeze operator for

Rindler states, but the equivalence principle guarantees the existence of an identical operator

(expressed in Minkowski momenta) acting on the Minkowski modes. Using (4) and performing

the intermediate integrations over the Minkowski momenta yields [4] H = Ho + H' where

and

Ho = f dj d_p± wpihl(j)?zt(j,p±)a(j,p±) , (6)

H' = J dj dj' d2p± Wp± (h2(j,j')[?L(j',p±)?z(j,-p±) + at(j,, px)at(j, _p±)]

344
+hl(j,j')O(j - j') + O(j'- j)_t(j, pi)_(j',p±)) , (7)

where

cosh[_(j + j')/2](1 + (j - j,)2)-1 _e-p_(j-j')/2(1 + (j + j,)2)-a

h1 (j, j') = and h2(j, j') = (8)

2_r[sinh(nj)sinh(Trj,)]x/2 ' 4_[sinh(_rj)sinh(_j,)]_/2 '

are functions computed from the Bogolyubov coefficients [4]. We begin to see the operator structure characteristic of squeezing. To make this precise, consider the SchrSdinger equation for a

state of the scalar formulated in terms of the interaction picture. The above splitting of the Hamiltonian suggests writing the full time evolution operator as U = U°U ' where U°(T) = e -iH°T. The

interaction Hamiltonian in the interaction picture is computed from H'I(T) = U°t(T)H'U°(T)

[4]. Then, the state of the scalar at any time T is simply given by

I¢(T) >1= Texp (/:) -i/h H_(T')dT' I¢(To) >, (9)

where the time evolution operator is a (multi-mode) squeeze operator, by virtue of (6) and (7).

This is the main result. If we now identify the initial state with the Rindler vacuum, ]O(To) >=

10 >, the final state is precisely the gravitationally squeezed vacuum. Since every quantum field is

equivalent to an infinite collection of (coupled) harmonic oscillators, it should come as no surprise

that the evolution operator for • is just a multi-mode generalization of the single mode squeeze

operator. If we specialize to two scalar modes having equal but opposite values of the transverse

momentum and with j = j', then the evolution operator in (9) reduces to

-i

S(z) = ezp (-_-[za(3, p±)a(j,-px)- z*at(j, pi)_t(j,-px)]) , (10)

ih2(J)[e-2iwP± hl(j)T 1] is the squeeze parameter for these modes. Apart from the where z = h1(3)

bounded T-dependent factor, we see that appreciable squeezing obtains for j _ 0. Expressing j

back in terms of physical quantities, we have j = _ where/_ is the mode wavlength and rs is the ), ,

Schwarzschild radius of the equivalent gravitational mass giving rise to the constant acceleration

at the point rs [4]. Thus, the interaction between matter and gravity leads to squeezed states

of matter, and these provide a natural source of negative energy for supporting wormholes in

Lorentzian spacetime.

1. M.S. Morris and K.S. Thorne, Am. J. Phys. 56 (1988) 395.

2. M. Visser, Phys. Rev. D39 (1989) 3182; Nucl. Phys. B328 (1989) 203; Phys.

Lett B242 (1990) 24; Phys. Rev. D41 (1990) 1116; ibid D43 (1991) 402.

3. D. Hochberg, Phys. Left. B251 (1990) 349.

4. D. Hochberg and T.W. Kephart, Lorentzian Wormholes from the Gravitationally Squeezed Vacuum, VAND-TH-91-1 (April 1991).

345
N92-'22 084

Exact Solution of a Quantum Forced Time-Dependent Harmonic Oscillator

Kyu Hwang Yeon

Department of Physics, Chungbuk National University

Cheong Ju, Chungbuk 360-763, Korea

Thomas F. George

Departments of Chemistry and Physics & Astronomy, 239 Fronczak Hall

State University of New York at Buffalo, Buffalo, New York 14260, USA

Chung In Um

Department of Physics, College of Science

Korea University, Seoul 136-701, Korea

The Schr6dinger equation is used to exactly evaluate the propagator, wave

function, energy expectation values, uncertainty values and coherent state for a

harmonic oscillator with a tlme-dependent frequency and an external driving timedependent force. These quantities represent the solution of the classical equation

of motion for the time-dependent harmonic oscillator.

I Introduction

It is well known that an exact solution of the Schr6dinger equation is

possible only for special cases. For this reason, approximate methods are needed.

Exact solutions provide important tests for these approximate methods and for

various models of physical phenomena. In general, the solution of the Schr6dlnger

equation for explicit time-dependent systems has met with limited success because

of analytical difficulties, although progress has been made during the past three

decades. I'5 Camiz et al 6 have obtained the wave functions of a time-dependent

harmonic oscillator perturbed by an inverse quadratic potential, using the

Schr6dinger formalism and a generating function. Further, Khandekar and Lawande 7

have evaluated the exact propagator and wave function for a time-dependent harmonic

oscillator, both with and without an inverse quadratic potential, using Feynman

path integrals. In addition, Jannussis et al 8 have calculated the propagator for

several quantum mechanical systems with friction.

In a previous paper, 9 we have evaluated the propagator, wave function, energy

expectation values, uncertainty values and transition amplitudes for a quantum

damped driven harmonic oscillator by using path integral methods. Also, we have

I0

obtained the coherent state for the damped harmonic oscillator and calculated the

ii

propagator for coupled driven harmonic oscillators.

347

PRECEDING PAGE BLANK NOT FILMED
In this paper we discuss the exact quantum theory of a forced harmonic

oscillator with a tlme-dependent frequency. In Sec. II we evaluate the propagator

using the Schr6dlnger equation and path integral methods, and in Sec. III we

calculate the wave functions using the propagator. In Sec. IV we define the energy

operator and calculate energy expectation values. In Sec. V we obtain the

uncertainty values. In Sec. IV we determine the coherent state and its properties.

Finally, in Sec. VII we present results and a discussion.

II. Propagator

We consider a system whose classical Hamiltonlan is of the form

H - 2LM p2 + _2 2(t ) x 2 . f(t)x , (2.1)

where x is a canonical coordinate, p is its conjugate momentum, w(t) is a frequency

as a function of time, M is a positive real mass, and f(t) is an external driving

force. The Lagrangian corresponding to the Hamiltonian (2.1) is

L - 2 M_2 - 2 M_2(t)x2 + f(t)x (2.2)

Here, the Hamiltonian H and Lagrangian L depend on time. The classical equation of

motion for our system is

d2

dt 2

-- x + w2(t)x - _ f(t) (2.3)

For the case where _(t) - w (constant), the solution of Eq. (2.3) represents

o

harmonic motion; otherwise, it is difficult to evaluate the exact solution.

The path integral formulation of Feynman provides an alternate approach to

solving dynamical problems in quantum mechanlcs. 12 In this approach, the usual

Schr6dinger equation is replaced by the integral equation

@(x,t) - I dx' K(x,t;x't') @(x',t') (t > t') (2.4)

with the initial condition _(x,t) - _(x',t). Here, _(x,t) is a wave function and

K(x,t; x',t') is a propagator. The propagator K(x,t; x',t') is defined by the path

integral 12

K(x,t; x',t') - lim (x, t) N-III dx_ exp[ S(x,t; x',t')]

_(x' ,t') j-i J

(2.5)

348
where the integration is over all possible paths from the point (x' ,t') to the

point (x,t), and S(x,t; x',t') is the action defined as

t

S(x,t; x',t') - df L(x,x,f) (2.6)

t'

For a short time interval ¢, substitution of Eqs. (2.2) and (2.6) into Eq. (2.5)

gives the normalizing factor Aj and the usual Schr6dinger equation:

A. - (2i_c)4/M) h (2.7)

3

(2.8)

2M ax 2

Since K(x,t; x't') can be thought of as a function of the variables (x,t) or of

(x',t'), it is a special wave function, and it satisfies Eq. (2.8):

_ _ _x_:x,_,>___ _-_ _._,_,_+__ x__x.t,x,._,_ ' ' 2M ax 2 '

f(t) x K(x,t;x',t'), (t > t') (2.9)

i)4 a__ K(x,t;x',t') - M2 _[2 i 2

at' " 2M 2 K(x,t; x't') + _ M_(t') x'

ax'

x K(x,t;x',t') - f(t') x' K(x,t;x',t') ,

(t' > t)

12,13

Because the Lagrangian is quadratic, the propagator has the form

K(x,t; x't') - exp[a(t,t')x 2 + b(t,t')xx' + c(t,t')x '2 + g(t,t')x

(2.10)

+ h(t,t°)x ' + d(t,t')] ,

(2.11)

where from Eqs. (2.9) and (2.10) we can easily deduce that the coefficient of the

third and higher powers in x is zero.

Substituting Eq. (2.11) into Eqs. (2.9) and (2.10), we obtain the differential

equations

349
d__ 2i_ a2 + M 2

dt a- M _ (t)

(2.12)

d__dt(bx'+g) - 21_M a(bx'+g) + _ f(t)

(2.13)

d_dt (cx,2+hx0+d) - i__ M a + _M (bx'+g) 2

(2.14)

d c - 2__ Mi c2 + _ w(t') 2 (2 15)

__ddt,(bx+h) - c (bx+h) + i_

(2.16)

dt'd (ax2+gx+d) " _2iM (bx+h)2 + M_i c (2.17)

_u ..... w(t) -

Equations (2.12) and (2.15) are nonlinear equaticnz. _o_ .... cazz _,,e_e o'

a solution is easily found, but in other cases it is difficult to find an exact

solution. If q(t) obeys the differential equation

d 2

-- q(t) + _2(t) q(t) - 0

dt 2

(2.18)

then the solutions of Eqs. (2.12)-(2.14) are

i_MMg_(__ (2.19)

a(t) - 2_ q(t)

b(t)x' + g(t) - _illt q [ ds f(s)q(s) + bo]

(2.20)

b 2 t ds

c(t)x'2+ h(t)x' + d(t) - lnq "_ + 2-_ f q(s)2

b ° ft. d s

-- + _-_ q(s)2

f, , f' f x dp f(p)q(p) + _ q(s) 2 dp f(p)q(p) dr f(r)q(r) + do

(2.21)

where b and d are constants of integration and do not depend on t, and the

o o

solutions of Eqs. (2.15)-(2.17) are

M

c(t') - q(t')

t'

b(t')x + h(t') - i_q if ds f(s)q(s) + bo]

(2.22)

(2.23)

350
ib '2 t' ib' t ds

a(t')x 2 + g(t')x + d(t') - Inq(t') "h + _ q(s) 2 q(s) 2

t #

x dp f(p)q(p) + 2--_ --- dp f(p)q(p) dr f(r)q(r) + d' (2.24)

q(s)2 o '

r

where b ° and d'o are constants of integration and independent of t' Since only t

is a variable in Eqs. (2.19)-(2.21), we have suppressed t' in a(t,t'), b(t,t'),

etc., and we have similarly suppressed t in Eqs. (2.22)-(2.24).

In polar form we may write

q(t) - _(t)e iT(t) , (2.25)

where _(t) and 7(t) are real quantities.

ea

_(t) _(t) _2(t) + _2(t) _(t) -0

From Eqs. (2.18) and (2.25) we note that

(2.26)

2_(t)7(t) + _(t) _(t) - 0 (2.27)

• 2

(t);(t) - _ (2.27')

where the constant _ is a time-variant quantity, From Eqs. (2.26) and (2.27), we

find another form for the solution of Eq. (2.18) as

q(t) - .(t) sin(_-_') (2.28)

q(t') - n(t') sin(7-7') , (2.29)

where 7 - _(t) and _' - 7(t').

Substitution of Eq. (2.28) into Eqs. (2.19) and (2.21) gives

a(t) - _ [_ + -_ cot(7-7')] (2.30)

ibo i ftds n(s)f(s) sin[v(s)-7'] (2.31)

b(t)x' + g(t) - X_sin(v-V) + ]_sin(7-V')

ib 2

c(t)x '2 + h(t)x' + d(t) - ln[. °_ sin'S(7-7')] + 2-_ cot(7-7')

351
+

b

Q

i_flM sin(7-7')

t

ds f(s)_(s) sin[7(s)-7')]

+ 2i_flM sin(v-7') ds dp f(s)f(p)_(s)_(p)

× sin[7(s)-7']sln[7(p)-7') (2.32)

Furthermore, substitution of Eq. (2.29) into Eqs. (2.22) and (2.24) gives

c(t') - 2_ [" _' + _' cot(7-7')] (2.33)

b #

o

b(t')x + h(t') - iHfi'sln(7-7') + i_sln(7-7') 1 ds f(s)_(_)si_n[_-_.(s)]

(2.34)

ib ,2

a(t')x 2 + g(t')x + d(t') - ln[_ '-h sln'h(7-7')] + _ cot(7-7 ° )

ib' r t'

0

I ds f(s)_(s) sin[7-7(s)]

MflMsin(7-7') J

t e t e

+ 2ingl'fsin(7-7')

' I ds I dp f(s)f(p)_(s)_(p)

X sin[7(s)-7]sin[v(p)-7] (2.35)

From Eqs. (2.31), (2.32), (2.34) and (2.35), we deduce that the constants b

o

and b' are given as

o

b - -M_'_'x' (2.36)

o

b' - M_Wx (2.37)

o

Also, from the normalization condition,

d o - in(2_)h (2.38)

From Eqs. (2.30)-(2.37) and (2.27'), we find that

352
(2.39)

(2.40)

IM J_' (2.41)

b(t,t') - -)4 sln(v-V')

- itds _ sin[7(s)-7]

g(t,t') )4 sln(7-7') t' _(s)

h(t,t') - )4 sln(7-V') , J_(s)

- ds _ sin[7-7(s)] dp f-_

d(t,t') - 2M]_ sln(7-7') t' J_(s) t' JT(p)

(2.42)

(2.43)

x sin[7(p)-7'] (2.44)

Inserting Eqs. (2.39)-(2.44) in Eq. (2.11), we obtain the propagator for the forced

time-dependent harmonic oscillator as

K(x,t;x't') - L2_i)4sin(?-7')J

X exp[_ (_ x2 - _,' x'2)]

x exp 2_ sin(7-7') i M [(7x• 2 + _'x' 2 ) cos(7 -7' ) - 2J7_' xx'

+ x ds _ sln[7(s)-7']

t' J_(s)

+ X' ds _ sln[v-7(s)]

t' ./_(s)

-- ds _ sin[7-7(s)] i dp _ sin[7(p)-7'] ]} M2 t' /_(s) t' _(p)

p

(2.45)

where the unprimed and the primed variables denote the quantities which are

functions of time t and t', respectively. It may be easily verified that for the

case where w(t) is a real positive constant Wo, we have _(t) - I and 7(t) - Wot,

353
and the propagator of Eq.

12

harmonic oscillator.

(2.45) reduces to the usual expression for a forced

III. Wave function

We now rewrite the propagator in another form in order to derive the wave

function:

r M (_,_)_

sin(?-?')J

] × x ds cos[v-V(s)]j

J:¢(_)

- II]

L'7' J_,(s)

X exp cot(7-7') x - _ ds f(s_ sin[7_7(s) ] 2

J_(s)

sin(_-_') x - ds fCs) sinIv_7(s) ]

J_(s)

]}]

X exP[2_ {cot(7-7') [J'tds _ sin[7-7(s)]]2

+ cot(7-7')_ dp _ sin[_'-_(p)] ]2

J_(p)

t t'

2 ; ds sin(7.Ts) I dp f___f_ sin[7,_7(p) ]

+ sin(_-_' ) j_(p)

ds _ sin[_-_(s)]

sln(7-7') t' ]_(s) /_(p)

x sin[7(p)-'7' ]}] (3. I)

354
-i(_-_')

M h e

. (_) <_, )k(_)k -2i(7-_')

1 - e

[_[_x2 2_ _ ds

x exPt2}_Lt,7 + M J%(s)

cos[_-_(s)]]

f

ni

,7'

x,2 + _-J_'x' I"ds

[,(s) __ cos[_'-_(s)} I}]

X exp -f x - 1M _ sin[_-.f(s)] 2

Jr(s)

t ]}_

[ -M {I_x - i I_ds _--_,'n_-_'l --

x exp -2i(_-_') Jr(s) _[1-e ]

+ I,_., _I_ds _ - sin[(7"7(s)]l 2

J-_(s)

- 2 [_ x-!M I t ds _ si_[v-_(s)]][J_x'

- ds _ sin[v'-?(s)]

M J_(s)

(3.2)

where

8(t') - 8(t) - _ { eot(_-_') _ tds f---fi_ -- s in[7-V(s) ]I 2

J:r(s)

+ cot(7-7' ) t .ds _ %---- sin[_/' -_f(s)] ]2 + 2 sin(7-7')I

J_(s)

X I_ds _-_ -- s_ni_-_s_1 _ dp s_i_,-_l

t ds _ ;----- sin[7-V(s)] I_,dp _ sin[7(p)-7']

t' J_(s) J_(P)

14

Let us introduce Mehler's formula,

(3.3)

355
exp[-(X2+ y2-2XY)/(I-Z2)]

Jl - Z2

cO

(X2+y 2 ) Zn

- e" > 2nn!

n-O

-- Hn(X)Hn(Y )

(3.4)

where

J (s)

]

J_(s)

Z - e "i(7-7')

(3.5)

(3.6)

(3.7)

Substituting Eqs. (3.4)-(3.7) in Eq. (3.2), we obtain

K(x,t; X' ,t) --

n-0

#_(x,t) _n(X',t') (3.8)

where

#n(X,t)- [___i_l (M__)h]1⁄2 exp{_[_ x2 - M_ _'x ftds _ cos[7-7(s)]]}

2nn! =_ ' /_(s)

X exp _M 2 x - _ ds _ si_[v-7(s)] 2

J_(s)

X x - - ds _ sin[7-7(s ]

i[0(t)-(n+1⁄2)7(t)]

xe (3.9)

Moreover, we may write

_n(X,t) - exp{i[8(t)

I

(n+_)v(t)]} _6n(X,t) (3.10)

where

_n(X't) - [2nn !--!-1.(_)M__ t_] exP{2_[_ x- M_ _Tq _tds _7(s)_ cos[7-7(s)]]}

356
x exp J7 x - _ ds _ sln[7-7(s)] 2

J;(s)

× H n x - _ Ids _ sin[v-7(s)] }] (3.11)

J_(s)

In Eq. (3.10), the wave function @n(X,t) is merely a unitary transformation of

4n(X,t), and thus 4n(X,t) satisfies all the properties associated with _n(X,t):

_dx _n " <sin> - _ dx _n " Sm,n

The expectation value of a given operator O is

IV.

(3.12)

<mlOln>- _ dx #* O# n - _ dx 4*nO4n (3.13)

Energy expectation values

For the forced time-dependent harmonic oscillator system, both the Hamiltonian

and Lagranglan have the units of energy but depend on time. We must therefore find

a time-invarlant energy operator. If _(t) is a particular solution of Eq. (2.3),

we have

d 2 2 (x-_) + (t) (x-_) - 0 (4 1)

dr2 ' •

and from Eqs. (2.26) and (2.27') we note that'

+ _2(t) 7 - 02/73 (4.2)

From Eqs. (4.1) and (4.2), we get the following expression for the energy:

Because Eq. (4.3) is time invariant, we can use it for the quantum mechanical

energy operator,

i_2 2 __ M.2 .2 2

" _('; +77 )x _2 (2XaLxx + I)

Eop - 2M ax 2 +

+ (#_-7_)(i_8_'-xx + M_x) + MT;2#x + _ 7;2# 2 + 2(#_-7_) 2 (4.4)

357
Equation (3.11) now simplifies to the expression

4n(X't) - [2n n! ___ e i_x2+iAx e-h62(x-_) 2

H [6(x-_)]

n

2nn?,/_ e Hn[ 6 (x- B) ]

(4.5)

where

6- (Mv/_)_ (4.6)

.(t)- _ _

2M

I It

_(t) - I_

A- tV - 62/2

B - iA + _62

(4.7)

ds _ cos[_-_(s)] (4.8)

J_(s)

ds _ sin[7-7(s)] (4.9)

J_(s)

(4.10)

(4.11)

Here, _(t) is a particular solution of Eq. (2.3).

In order to evaluate the energy expectation Em,n - <mIEopln>'

following calculations:

xln> - _6 [4_In+l> + 4_In'l>] (4.12)

x21n> - j_l [j(n+2)(n+l)in+2 > + (2n+l)in > + 4_-_In.2>] (4.13)

262

pln> - i_6 2(n+l)In> + _ Bin> + _(6+_) 4C_In-l> (4.14)

A 2

p2;n > - __2 [2A6T j(n+2)(n+l)in+2> + 2_ _ 4_In> + 2(A+_)(2n+l);n>

2

+ 24_ _ + 6B)_In-l> + 2(_-$o+2A+62)4_-(_?_In>]'- (4.15)

6_

we perform the

358
xpl_- _ {A_J(n+2)(n+l)In+2> ÷ _ _In+l> ÷ [A--(mn÷l)+nlI_

6 2 6 2

B /6In-l> + (A+l)_In-2>)

+_-6 52

pxl_>- xpl_> + In>

(4.16)

(4.17)

Substituting Eqs. (4.6)-(4.17) into Eq. (4.4), we directly obtain the energy

expectation values as

X 2.

E - En - _(N 7)(2n+I) n,n

- n_(n+_) (4.18)

This energy expectation value is a time-invarlant quantity.

V. Uncertainty values

The uncertainty product defined as

(AxaP)m, n - {[(<=Ix21n>- <mlxln>2)*(<mlx21n>. <_Ixl_>2)]h

*.

x [(<mlp2ln> - <=lpln>2)*(<=lp21n> - <mlpln>2)l_) _ (5.1)

Inserting Eqs. (4.12)-(4.15) into Eq. (5.1), we obtain

(A_P)n, n - (1 + ._2 )_ (n+_)_ (5.2)

7N

.2

(AxAP)n+2, n -- (i + .--_2 )_ J(n+2)(n+l)

7

(5.3)

(AxAP)n,n+ 2 - (i + .--_2) _ _

7

(5.4)

Vl. Coherent states of the time-dependent harmonic oscillator

First, we construct the creation operator a f and destruction operator a. For

a forced tlme-dependent harmonic oscillator, it is not possible to construct a and

a f , but we can construct a and a f for the time dependent harmonic oscillator. From

Eqs. (4.12) and (4.14), we obtain

359
at_

_-y M7

a- ( )_ [(i - i_?.)x + i_-].

_V M_

From Eqs. (6.1) and (6.2), we can represent (x,p) in terms of (at,a) as

x - (1)h (at+a)

2M_

p l (2_) _ [(_ + i)at+ (_

(6.1)

(6.2)

(6.3)

i)a] (6.4)

Also from Eqs. (6.1) and (6.2), if Ix,p] - iN we see that

[at,a] - i (6.5)

Conversely, from Eqs. (6.3) and (6.4), if [at,a] - i we note that [x,p] - i_.

The coherent state can be defined by the eigenstate of the nonhermitian

15

operator a,

al=> l _I_) (6.6)

Let us find the coordinate representation of the coherent state. From Eqs. (6.2)

and (6.3), we have

(2_)_[(I- i_)_+ _ a-_]<x'l=>- =<x'l=> (6 7)

177 M7

We solve this equation and change the variable x' into x for convenience,

(6.8)

We choose the constant of integration N such that

I dx l<xl=>l2 - i (6.9)

Then, we find the eigenvector of the operator a in the coordinate representation

x> as

360
(6.10)

Next, we show that a coherent state is a minimum uncertainty state. From Eqs.

(6.3), (6.4) and (6.6) and their adjoints, we evaluate the expectation values of x,

p, x 2 and p2 in the state I_>:

<:Ixl:>- (-_.)_ (:*+=) (6.11)

2H7

- [<%÷ +<%-

tT"}' 77

(6.12)

<=Ix21=> ---M-<ela+2+a2+aa++a+al=>

2M;

-_ (_'2+_2+2_*+I) (6.13)

2M_

<=lp21=>- _ {(_+ i) 2 a*2+ (L.. i)2e 2

77 _

+ [(L.)2 + l](2ee*+l)} (6.14)

The uncertainty value is

AxAp - [(<=Ix21=> - <=lxl=>)(<=lp21=> - <=ipl=>>]_

- M/2 [1+(_-_)2] & _ ,

which is the minimum value allowed by Eq. (5.2).

(6.15)

VII. Results and discussion

In the previous sections, we have obtained the propagator, wave function,

energy expectation values, uncertainty values and coherent state for a quantum

forced time-dependent harmonic oscillator. These quantities represent the solution

of the classical equation of motion for the time-dependent harmonic oscillator. If

we set f(t) equal to zero, then our solution is correct for the time-dependent

harmonic oscillator. Setting _(t) - _ gives results for the forced harmonic

O

361
oscillator. For the case where f(t) - 0 and w(t) - w our results are those of

o'

the simple harmonic oscillator.

For the explicit tlme-dependent system, we need to consider the quantum

mechanical operator. In our work, the Hamiltonian, Lagranglan and mechanical

energy have the units of energy, but these are not time Invariant. Yet, in order

to solve macroscopic physical problems, we use time-lnvariant operators. For this

reason, we have derived the energy operator from the classical equation of motion

and used it to calculate energy expectation values. Our energy operator is similar

to the Ermakov-Lewis invarlant operator. 1'2 Our quantum energy expectation vlaues

are time-independent quantities, and our uncertainty values are consistent with

Heisenberg's uncertainty principle. Yet, our uncertainty values are time

dependent, in contrast to those of tlme-independent systems.

Since it is not possible to construct a coherent state for the forced timedependent harmonic oscillator, we have constructed it for the tlme-dependent

harmonic oscillator. In general, the coherent state is a minimum uncertainty

state, which is also true for our system.

Time-dependent systems are observed in various physical experiments. Two

general types of such systems are: that which is formed through its own

environmental conditions, and that which is formed when external forces are added.

In regard to the second type, various experiments are being carried out to see how

an applied, time-dependent electric, magnetic o@'other field can alter the physical

properties of materials such as semiconductors and superconductors. Experiments

show that a system becomes time dependent when a time-dependent electric or

magnetic field (such as a.c.) is applied. However, obtaining the quantum

mechanical solution by a direct method is not easy mathematically. One way of

obtaining a solution is to use the propagator method as indicated in this paper,

where the relevant equations are those of a time-dependent harmonic oscillator.

Our results, which are exact for one dimension, can be extended to two or more

dimensions, and they can also be applied to time-dependent macroscopic systems.

One example of an extension to two dimensions would be to solve the motion of a

quantum electron in a time-dependent magnetic field.

Acknowledgments

KWY and CIU acknowledge financial support from the Korea Science and

Engineering Foundation and the Basic Science Research Institute (BSRI), Ministry of

Education 1989, Republic of Korea. This research was also supported by the Office

of Naval Research and the National Science Foundation under Grant CHE-9016789.

362
References

1. H. R. Lewis, Jr., Phys. Rev. Lett. 18, 510, 636 (1967).

2. H. R. Lewis, Jr., J. Math. Phys. _, 1976 (1968).

3. P. G. L. Leach, J. Math. Phys. 10, 1902 (1977).

4. J. R. Burgan, M. R. Felx, E. FiJalkow and A. Munier, Phys. Lett. A __4, 11

(1979).

5. J. G. Hartly and J. R. Ray, Phys. Rev. A 24, 2873 (1981).

6. P. Camiz, A. Oerardi, C. Marchioro, E. Presuttl and E. Scacclatelli, J. Math.

Phys. 12, 2040 (1971).

7. D. C. Khandekar and S. V. Lawande, J. Math. Phys. 16, 384 (1975).

8. A. D. Jannussis, C. N. Brodimas and A. Streelas, Phys. Lett. A 74, 6 (1979).

9. C. I. Um, K. H. Yeon and W. H. Kahng, J. Phys. A 20, 611 (1987).

I0. K. H. Yeon, C. I. Um and T. F. George, Phys. Rev. A 36, 5287 (1987).

Ii. K. H. Yeon, C. I. Um, W. H. Kahng and T. F. George, Phys. Rev. A 38, 6224

(1988).

12. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGrawHill, New York, 1965).

13. G. J. Papadopoulous, Phys. Rev. D ii, 2870 (1975).

14. A. Erdelyi, Higher Transcendental Functlons, Vol. 2 (McGraw-Hill, New York,

1953), p. 194.

J

15. W. H. Louisell, Quantum Statlstlcal Prope@tles of Radlation (Wiley, New York,

1973).

363
N92-220 5

Nonclassical Depth of a Quantum State*

Ching Tsung Lee

Department of Physics, Alabama A & M University, Normal, Alabama 35762

A measure is defined for how nonclassical a quantum state is, with values ranging

from 0 to 1. When it is applied to the photon-number states, the calculated value

is 1, the maximum possible. For squeezed states, it is a monotonically increasing

function of the squeeze parameter with values varying from 0 to 1/2. The physical

meaning of the nonclassical depth is found to be just the number of thermal photons

necessary to ruin the nonclassical nature of the quantum state.

In the coherent-state description of radiation fields, initiated by Glauber (Ref. 1) and

Sudarshan (Ref. 2) in 1963, there are P and Q representations corresponding to the normal and

antinormal ordering, respectively, of the creation and annihilation operators. Their distribution, or

quasi-distribution, functions in the complex plane are related to each other through the following

convolution transform (Ref. 3):

2 2

Q(z) = _We-'_-_' P(w), (1)

_r

where z and w are complex variables. We can introduce a continuous parameter v and define

a general distribution function as

R(z,T)= .... f d2Wexp( l[z wl2)P(w). (2)

We shall call R(z, T) the R function from now on. The original P and Q functions are two

limiting cases of the R function with _- = 0 and 1, respectively.

Our motivation for introducing the r parameter is to define a measure of how nonclassical

quantum states are. It is well known that the origin of the nonclassical effects is that the P

functions of all pure quantum states are singular and not positive definite, as shown by Hillery

(Ref. 4); hence it is called quasi-distribution function. On the other hand, the Q function is

always a positive definite regular function. The smoothing effect of the convolution transform

of Eq. (2) is enhanced as T increases. If T is large enough so that the R function becomes

acceptable as a classical distribution function, i.e., it is a positive definite regular function, then

we say that the smoothing operation is complete. Let C denote the set of all the T that will

complete the smoothing of the P function of a quantum state and let the greatest lower bound,

or infimum, of all the T in C be denoted by

T_ ----inf (T). (3)

rEC

We propose to adopt T,_ as the nonclassical depth of the quantum state.

According to this definition, we have Tr,, = 0 for an arbitrary coherent state Is) since its P

function is of the form of a delta function, 7r52(z - a). On the other hand, for T = 1 we have

R(z, 1) = Q(z), which is always acceptable as a classical distribution function for any quantum

state; hence, 1 is an upper bound for T,,. Therefore, we can specify the range of T,,, to be

o _< _<1. (4)

365

PRECEDING PAGE BLANK NOT FILMED
We shall try this definition on two of the most familiar types of nonclassical radiation states:

the photon-number (Fock) states and the squeezed states.

For a photon-number state In), we obtain

R_(z,r)= 1/ 1 )rT" exp (#)(z) -- L, T(1--T) '

(5)

where L, is the Laguerre polynomial. From Eq. (5), we see that, for 0 < T < 1, R,(z, T) is not

positive definite since the Laguerre polynomial has n real positive roots. However, for T > 1,

the argument of the Laguerre polynomial is negative and R,(z, r) becomes positive definite.

So we have rm = 1, which reconfirms our belief that the photon-number states are the most

nonclassical quantum states.

For the squeezed state generated from the vacuum state by the well-known squeeze operator

S(() with the complex parameter _ -- re _e, we obtain the Gaussian function

R¢(z,r)_ (sechr){ exp --_[ax1 = + 2bxy + bx _]} (6)

with

a = r + (1 - r)tanh =r- cos0tanhr,

c = r + (1 - r)tanh 2r + cos0tanhr,

b = sin 0 tanh r,

D = r 2 - (1 - T)_ tanh 2 r. (7)

For R¢(z,r) to be normalizable we must have

ac- b2 > 0 and D>0.

(8)

Both conditions lead to the same conclusion that

r,,, = tanh r/(1 + tanh r). (9)

This nonclassical depth can be expressed as a function of the squeeze parameter, s = e r, as

follows:

r=(s) = (s 2 - 1)/2s 2. (10)

From Eq. (10) we see that rr_ is a monotonically increasing function of s; it varies from 0 to 1/2

as s varies from 1 to c_.

In the first example, rm is determined by the requirement that R,(z, r) be positive definite;

while in the second example, it is determined by the condition that Re(z , T) be normalizable.

Is there a more systematic way to determine r,,? To answer this question, we need to study

more examples.

As a by-product of such calculations, we will also obtain new expressions for the P functions

as follows:

P(z) = limR(z, 7"). (11)

•r---* 0

Since the P function of a quantum state is typically highly singular, it is usually very difficulty

to visualize in its original form. Now we can visualize it as the limit of the regular R function

as 7- _ 0.

366
On theother hand, we consider the superposition of two quantum states with P1 (z) and P2 (z.)

as their P functions. According to Glauber (Ref. 5) the P function for the superposed state is

the convolution product of P1(z) and P2(z); namely,

P,u(z) = f ---_-P_(z- d2z . w)P_(w) (12)

It is well known that the P function for a single-mode thermal radiation is (Ref. 5)

1 (,3) P,h(z)- (nth) exp

where (nth) -= (e h_/kT - 1) -1 is the average photon number in the thermal radiation.

We now consider the superposition of the thermal radiation with an arbitrary state of singlemode radiation with P(z) as its P function. Then the P function of this quantum state with

thermal noise can be expressed as

P,u(z)- (nth) 1 j 7exp d2w (-]z -- wl2/(nth))P(w) (14)

Comparing Eqs. (1) and (14) we see that the superposed P function, P,u(z), is identical to

the Q function when (nth) = 1.The implication of this coincidence can be stated as follows:

One thermal photon is always sufficient to destroy whatever nonclassical effects any single-mode

radiation might have.

The R function for the superposed state of Eq. (14) can be obtained as

R,.(z,r)-r+in,h 1 ) i d2w-_ exp[-lz-wl_/(r+(n,h))]P(w) •

Therefore, we have

Ttmh m Tm -- (nth);

(15)

(16)

which means that the reduction in the nonclassical depth of a quantum state in the presence

of thermal noise is exactly equal to the average number of thermal photons present. This also

gives the following physical meaning to the nonclassical depth we have defined previously: The

nonclassical depth of a quantum state is the minimum number of thermal photons necessary to

destroy any of its nonclassical characteristics.

We have previously calculated the nonclassical depth of a Fock state to be exactly 1, so it

takes one thermal photon to ruin the nonclassical nature of a Fock state. We have also calculated

the nonclassical depth of a squeezed state to varies from 0 to 1/2 as s varies from 1 to _; so it

never takes more than 1/2 of a thermal photon to ruin a squeezed state.

*This work was supported by the U.S. Navy, Office of Naval Research, under Grant #N00014-

89-J-1050.

1. Glauber, R. J., 1963, "Photon Correlations," Phys. Rev. Lett., 10(3), p. 84.

2. Sudarshan, E. C. G., 1963, "Equivalence of Semiclassical and Quantum mechanical Descriptions of Statistical Light Beams," Phys. Rev. Lett., 10(7), p. 277.

3. Cahill, K. E., and Glauber, R. J., 1969, "Density Operators and Quasiprobability Distributions," Phys. Rev., 177(5), p. 1882.

4. Hillery, M., 1985, "Classical Pure States Are Coherent States," Phys. Lett., 111A(8,9), p.

409.

5. Glauber, R. J., 1963, "Coherent and Incoherent States of the Radiation Field," Phys. Rev.,

131(6), p. 2766.

367
N92-22 08,a

TOWARDS A WAVE THEORY OF CHARGED BEAM TRANSPORT:

A COLLECTION OF THOUGHTS

G. Dattoli, C. Mari, and A. Torre

ENEA - Area INN, Dip. Sviluppo Tecnologie di Punta,

P.O. Box 65 - 00044 Frascati, Rome, Italy

ABSTRACT

In this paper we formulate in a rigorous way a wave theory of charged

beam linear transport. The Wigner distribution function is introduced

and provides the link with classical mechanics. Finally the Von-Neumann

equation is shown to coincide with the Liouville equation for the linear

" transport.

INTRODUCTION

A formal "quantum" theory of charged beam transport has been recently

proposed. (Ref. i) Within such a context the possibility of viewing the

beam emittance as a kind of quantization constant has been considered

on the basis of some strong conceptual similarities existing with the

so-called "quantum" theory of light rays. (Ref. 2)

The proposed quantization procedure (described in some detail below)

cannot be thought as fully satisfactory and in particular the role played

by the "beam wave function" (b.w.f.) and its link with classical dynamics

have not ben thoroughly investigated and clarified.

In this paper we develop, in a more rigorous way, the fundamental

steps towards a "quantum" theory of beam transport through linear elements,

showing the relation with classical mechanics and discussing the connections with the Liouville equation which describes the time evolution

of non-interacting classical ensembles. We will show that classical

dynamics and the Courant-Snyder theory (Ref. 3) can be recovered by using

the formalism of the Wigner phase-space function, (Ref. 4) which therefore

can be seen as the natural framework to study the evolution of charged

beam transport through linear magnetic systems.

The paper is organized as follows. In Sect. 1 we review some aspects

of symplectic mechanics (Ref. 5) and introduce some definitions which

will be useful in Sect. 2 where the "quantum" theory of charged linear

transport will be completely developed in terms of the Wigner phase-space

function. Some comments and final remarks on the "quantum" theory of

nonlinear charged transport conclude this paper.

369

PRECEDING PAGE BLANK NOT FILMED
i. SYMPLECTIC MECHANICS OF QUADRATIC HAMILTONIANS

The most general time dependent quadratic Hamiltonian in one degree

of freedom can be written as

] l

H = _a(t)q z+ _b(t)p z+ c(t)qp

or in a matrix form as

1 T ^

where

(].])

(1.2)

2:= (q) Ro = (a(t) c(t)) (1.3)

- p ' c(t) b(t)

and the superscript " T" denotes transpose.

The equations of motion for the vector z are obtained from the Poisson

Brackets (P.B.) with the Hamiltonian (i.i) _The relevant rules are easily

derived. It is quite straightforward to realize that

and that

(l.4a)

{z,zT#z}=2S#z * (l.4b)

where S is the unit symplectic matrix in two dimensions. According to

the above rules the equations of motion of z are easily written, namely,

5".---"{ z, zT H z } = S H.z. (].5)

and therefore immediately integrated, thus yielding

z(t)= P(t)z(o) (1.6)

-- m

where

Note that {_T,_}=Oand therefore the above product cannot be, strictly

speaking, considered a conventional P.B. In the case that the second

term in the {,}-product is a scalar function [as in Eq. (l.4b)] then

the {,} is a conventional P.B.

370
U(t)= <[;0 e×p ,S fl(t')dt" ]>4"

(1.7)

and (, }+ denotes time ordering for the classical evolution operator which

is necessary when the commutator [S]](t),Sl_(t')] is different from zero.

The above formalism is particularly useful to introduce the

fluctuation tensor and its dynamical features, namely, let us pose

_(t) =<z_(t)z_T(t)>-<E><£T> (1.8)

where <,> denotes ensemble average.

In a matrix form we recover the usual expression

0 2 Opq

_(t) = 2 (1.9)

The equation of motion of _ follows from Eq. (1.5) and reads

d

_-_Z(t) = 9Y.+ Z9 T (1.10)

where V = SH, and can be immediately integrated in terms of the above

evolution operator U thus getting

£(t) = &(t)£(o)&T(t) (i.i])

Within this formalism it is easy to recover the well-known quadratic

invariant of Courant-Snyder, namely, let us write the most general

quadratic (time dependent) expression in z

l=zTT(t)z (1.12)

where

3_(t) = °)13 (1.13)

Then we impose that I is a "time-dependent" invariant, i.e.,

ctl 31

- +(I,H}=O (1.14) dt 3t

thus obtaining the following equations specifying _,_,y

371
li = -2ha + 2c13 13(0)= 13o

2aa-2cy y(O)=Yo

a13-13y a(O)=a 0

from which it is easy to verify that

BY = 1 + ct2

The Courant-Snyder invariant reads explicitly as

l =yq2 + 2apq+_p 2

and it is widely used in the theory of linear transport.

(l.lS)

(1.16)

(1.17)

2. TOWARDS A QUANTIZED THEORY

In the previous section we introduced the necessary background to

derive a kind of uncertainty principle in the theory of linear transport.

From Eq. (i. Ii) we get the relevant result

det Z(t) = det _:(0) (2.1)

i.e., det_ is an invariant quantity and the emittance A of the system

A(t) = [< q2 >< p2 > _ < qp >2] 1/2 (2.2)

is preserved in time.

Furthermore this means that

>A OqOp --

which represents a kind of uncertainty principle in the canonical variables

q,p and can be used as the starting point of our quantization procedure.

The rules are simple, the beam momentum is replaced by an operator

specified by

.,4---3

P = i ,)q (2.3)

where_d_is the beam reduced emittance and the following rule of commutation

is also assumed:

[p,q]:-i4" (2.4)

372
A "Hamiltonian" operator is finally associated to the longitudinal

coordinate of propagation s

# = i.4- a (2.s)

3s

so that the "Schr6dinger" equation for a beam passing through a quadrupole

of strength k(s) reads

a [ 4_f2 a2 1 1

*-k(s)q 2 _(q,s)

_4"_--s_(q's)= 2 3q 2 2

(2.6)

where _(q,s) denotes the beam wave function (b.w.f.). Clearly the b.w.f.

_(q,s) must be related in some way to the "classical" beam distribution

o(q,p,s) satisfying the Liouville equation for the time evolution of an

ensemble of single-particle systems. The link is not obvious since

depends only on q and eventually on s while p is a function of q,p and

s. The answer is given by the Wigner distribution function which is defined

as follows:

÷_

h/(q,p,s)=-_ i;dykU* (I)(l q+_y,s _ q-_y,s )0y -o0

(2.7)

and satisfies the Von-Neumann equation for a generic potential V(q) (Ref.

6)

i _s+p_ q W(q,p,s) V q+5. _pj-V q-_ _pj h/(q,p,s)

(2.8)

In the above-considered case of propagation through a quadrupole of

l 2

strength k(s) i.e., for an elastic potential V(q)={k(s)q , the Von-Nuemann

equation reduces to

--h/(q,p;s)=- 3s p q-k(s)q - p W(q,p;s)

(2.9)

which is equivalent to the Liouville classical equation. In this simple

case we can identify the classical distribution density 0 with the Wigner

distribution function h/ thus providing a completely consistent quantization scheme. Within this framework the physically meaningfulquantity

is not [_(q,s)[2dq but W(q,p;s)dqdp which ensures the consistency of

our procedure.

As a simple exercise it is straightforward to see that

373
1 exp[ 1 y(s)qZ+2(z(s)qp+F_(s)p z] h/(q'p;s)=(2x)(_/2) -2 (A/2) (2.10)

is a solution of (2.9) if and only if the "time-dependent" parameters

(u,_,y) satisfy the following system of differential equations:

!! = -2u

k(s)l -y

2k(s)u

(2.1])

which are the well-known equations defining the evolution of the Twiss

parameters in quadrupole lenses [see Eq. (1.15)].

3. CONCLUDING REMARKS

The extension of the developed theory to the case of nonlinear

transport of charged beams is not straightforward and the problems involved

can be illustrated in a simple example.

A sextupole term introduces, in the single-particle Hamiltonian, a

contribution of the type

X(s) 3

V(q,s)- 3 q (3.])

Inserting the above potential in the Von-Neumann equation (2.8) we easily

get the following evolution equation for the Wigner distribution function:

as_---W(q,p;s)=- pTq-k(s)q l_(q,p;s)

L

I

c)p 314/( q , P , s) (3.2)

where [ denotes the Liouville operator associated to the potential (3.1)

and given by [=-p_+ kq 2 . The extra term in Eq. (3.2) is a purely

"quantum" contribution and it is not present in the classical Liouville

equation for p(q,p;s).From this point of view_and p cannot be identified

and since they coincide in the limit_ 0,p may be viewed as the "classical"

counterpart of the Wigner distribution function W.

374
lo

.

3.

4.

.

.

REFERENCE8

Dattoli, G., Gallerano, G.P., Mari, C., Torre, A., and Richetta, M.,

"Twiss Parameters and Evolution of Quantum Harmonic Oscillator

States," accepted for publication in II Nuovo Cimento.

Fedele, R., Miele, A., and Ricciardi, G., 1989, Thermal Wave Model

behind the Envelope Equa_iQn fQr the Final _O_US in a Linear Collider,

INFN Report IFUP-TH 11/89.

Gloge, D., and Marcuse, D., 1969, JOSA, 59, p. 1629.

Courant, E., and Snyder, H., 1958, Ann. Phys., 3, p. i.

Wigner, E.P., 1932, Phys. Rev. A, 40, p. 749.

Han, D., Kim, Y.S., and Noz, M.E., 1988, Phys. Rev. A, 37, p. 807;

1989, P_ys. Rev. A, 40, p. 902 and 1990, Phys, Rev. A, 41, p. 6233.

Dattoli, G., Mari, C., and Torre, A., 1991, _ _uovo Cimento B, 2,

p. 201.

Carruthers, P., and Zachariasen, F., 1983, _ev, Mod, Phys., 55, p.

245.

375
N92-22087

Entropic uncertainty relation at finite temperature

Sumiyoshi Abe (a) and Norikazu Suzuki (b)

(a) Institute for Theoretical Physics, University of Erlangen-Niirnberg, W-8520 Erlangen, Germany

(b) Max-Planck-Institute for Nuclear Physics, W-6900 Heidelberg, Germany

and Department of Physics, College of Science and Technology, Nihon University, Tokyo 101, Japan

The uncertainty relation associated with the measurements of a generic noncommutstive pair of observables (A, B) in a normalized state I_b) is usually expressed as

1

Z_A. Z_B _> _[ ('/'l [A, B] [¢')[. (1)

For a canonically conjugate pair, the position and momentum of a particle (X, P), this equation gives the

original Heisenberg uncertainty relation AX. AP _> _, (h = 1). On the other hand, if the commutator [A, B]

remains as a q-number, the r.h.s, depends on the state [_b) and can be made arbitrarily small. For example,

if [_b) is chosen as an eigenstate of A, then Eq.(1) becomes trivial and no information can be extracted on

AB. Thus this formulation of the uncertainty principle has no practical meanings in general.

To improve the situation, the information-theoretic formulation of the uncertainty principle has been

repeatedly studied in the recent literature. Deutsch [al and Partovi[al discussed that the sum of entropies,

U[A, B: ¢] = SA[_] + S810], (2)

has an irreducible lower bound independent of the choice of I_b). Here, the information entropy associated

with the measurement of A is defined by

SA[_] = -Sol(al'_)laln [(al'_)[', (a Ia) -- _ Is)), (3)

where S_ stands for the summation (integration) over the discrete (continuous) spectra. This is a quantity

dependent on the choice of the representation Is) in general and is not expressed as s quantum mechanical

expectation value of a certain operator.

Prior to the authors of Refs.[1, 2], Bialynicki-Birula and Mycielski t'_ discussed the sum (2) for the pair

(X, P) and proved the optimal relation

U[X,P: ¢] _> 1 + lnTr. (4)

Here we discuss that how much information loses when a particle is in equilibrium with the thermal

reservoir of temperature T(= 1//3) 1'J. The universal temperature correction to the r.h.s, of Eq.(4) is determined.

For this purpose, it is convenient to employ the framework of thermo field dynamics (TFD) formulated

by Takahashi and Umezawa [sl. This formulation of finite-temperature (T _ 0) quantum theory utilizes the

doubled Hilbert space 7/@7_ I,I the normal operator (A) acting on the objective space 7"[and its corresponding

tildian operator (4) on the fictitious space 7_.

A thermal state [¢, 6;/3) in _ @'_ is not a physical state. The physical probability density associated

with the measurement of the normal operator A is given by the reduced one

P.('_) = S,_ l(_, &[,_, 6;/3) I', (5)

where [c_, &) is the complete eigenbasis of A and +i. With this quantity, we define the information entropy

at T +6 0 as follows:

SA[¢', 6;/3] = - Sc, pR(ol)lnpR(oO. (6)

Now we wish to find the stationary value of the functional

V[X,V: _, 6;/3] = Sx[_, 6;_] + Sv[_, 6;/3], (7)

at given T. In what follows, we propose a variational approach.

We are not concerned with the whole system including the tildian but only with the reduced one.

Therefore, the minimum value of the functional U at given T can be determined completely within the

reduced subsystem. This philosophy should be also respected by the variational operation itself. The

operation proposed here is as follows:

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I_,,_;_/-_ I_,_; _) + _1_,_;/3), (s)

where e and _ denote an infinitesimal variation parameter and an arbitrary deformation of the ?_ component,

respectively. Under this operation, the functional U of the normalized thermal state [¢, _;/3) varies as

v[x,v: _, _;/3] -_ v[x,v: ¢, _;/3] + _r + o(d), (9)

f /,

F

=-[I dzpR(z)lnpR(z) + I dppa(p)lnpR(p)](¢, _;/3)

d d

(10)

f f 6; f f d/31n [PR(P)](¢, ¢;/3[p,l_)(p,_[_, _;/3).

We do not know how to solve generally the equation r = 0 with respect to the unknown state [¢, _;/3).

Here, instead, we examine the thermal coherent state (TCS) t'_, which is the oscillator coherent state at

T _ 0. This is based on the following viewpoints; (i) the information entropy is the measure of uncertainty,

and (ii) at T = 0, the coherent state saturates the Heisenberg uncertainty (AX. AP = 1⁄2).

Let us consider a harmonic oscillator with a frequency w in TFD. The thermal vacuum state is generated

from the T = 0 Fock vacuum state 10, 8/ by the Bogoliubov transformation

10(/3))= exp (-iC)I0, ()), -it(3) = 0(/3)(atfi ! - fa), (11)

cosh 0(/3) = [1 - exp (-/3w)] -1/', (12)

provided that the creation and annihilation operators satisfy [a, a t] = [¢i,a t] = 1, [a, a] = 0, and so on. With

this state, the TCS is defined as foUows:

I=, ,_;/3) = exp [zat (/3) - z'a(/3) + i'at (/3) - ,/a(/3)] 10(/3)), (13)

a(/3)1_,./;/3) = zlz, ./;/3), a(/3)1_,_;/3) = i* l=,i;/3),

where the operators at T ¢ 0 are given by

a(/3) = exp (-iG)a exp (iG) = a cosh 0(/3) - _t sinh 0(/3),

(14)

(15a)

a(/3) -- exp (-iG)fi exp (iG) = a cosh 0(/3) - a t sinh 0(/3), (15b)

and so on. The self-tildian condition 1,1states z = i.

One can find that the TCS actually gives the desired result r Tcs = 0, and, therefore, Eq.(9) becomes

U[X, P: z, i;/3] _ 1 + In 7r + ln[cosh 20(/3)] + o(e2). (16)

Thus we have the thermal information-entropie uncertainty relationl'_

U[X, P: ¢, _;/3] _> 1 + In 7r + ln[cosh 20(/3)]. (17)

The third term in the r.h.s, determines the minimum loss of measurement information due to the thermal

disturbance effects.

The Heisenberg uncertainty relation at T _ 0 can be derived from gq.(17). To see this, let us tlnd the

maximum value of the concave entropy functional Sx with fixing the variance ((X - IX))') = (AX)L ({')

denotes the expectation value with respect to the normalized probability density pa(=)/(¢, _;/3[¢, _;/3).)

This is just the constrained variational problem characterized by the functional

¢[_, ,_;/3] = Sx[¢, _;/3]- _[<(X - <X>)') - (ZXX)'], (18)

where )_is Lagrange's multiplier.Applying again the variationaloperation (8),we can find the maximum

Value

1

SxmaX[¢, 1_;/3] = _ In [2a-e(AX)=].

Therefore we have an inequality

Sx [¢, _;/3] < _ In [2a'e(AX)=].

Repeating a similar discussion for the momentum P, we also get

(19)

(20)

378
Sp [_b,¢3;/31< in [2t.¢(Ap)']. (21)

The combination of Eqs.(20) and (21) leads to

2(AP) 2 > exp (--1 -- Int" + 2sp[_, _;/3])

> exp (1 + In t" + 21n {cosh[28(/3)]} - 2Sx[_, _;/3]) (22)

_>

Thus we obtain the thermal Heisenberg uncertainty relation

AX. AP > _Icosh [20(8)I. (23)

-2

We have used Eq.(17) in the second inequality of Eq.(22). This shows that the information-entropic uncertainty relation is stronger than Heisenberg uncertainty relationt'l.

Finally, we comment on squeezing of the thermal uncertainty relation. The thermal squeezed state is

defined by

Iz, i: ,, @;_) = exp [zat (/3) - ra(/3) + i'at(/3) - ia(/3)]

X exp [l{nat2(/3)- I/*a2(/3)+ @*at2(/3)- @_i2(/3)}]10(/3)). (24)

z

Straightforward calculation gives

Sx 7h /3] 1(1 + In t" + In {cosh [28(/3)]} + In [cosh (2r) + sinh (2r) cos (_a)]),

Z

(25a)

Sp[z,g':7/,@;/31= _(I + In t"+ In{cosh [28(/3)I} + In [cosh(2r)- sinh(2r)cos(_)]), (25b)

U[X, P: z, i: 7/, @;/3] = 1 + In t" + In {cosh [28(/3)]} + In [1 + sinh2(2r)sin'(_a)] ] , (26)

AX = {_1 cosh [28(/3)](cosh (2r) + sinh (2r) cos (_a))}|

1

AP = {_ cosh [28(/3)](cosh (2r) - sinh (2r) cos (_a))} i ,

AX. AP = I cosh [28(/3)](1 + sinh'(2r)sin_(_)) i ,

(27a)

(27b)

(28)

2

where we have employed the self-tildian condition for a squeeze factor (i.e., 7/ = f/), and r/ ==_rexp (i_a).

These results describe how the thermal disturbance effects in Sx or Sp (AX or AP) can be suppressed by

squeezing with keeping U = Sx + Sp (AX • AP) its minimum value.

One of us (S.A.) acknowledges the Alexander yon Humboldt Foundation for support. Another author

(N.S.) acknowledges Profs. J. Hiifner and H. A. Weidenmliller, and Max-Planck-Institute for Nuclear Physics

for the hospitality extended to him.

REFERENCES

1. D. Deutsch, Phys. Rev. Lett.SO (1983),631.

2. M. H. Partovi, Phys. Ray. LetS. 50 (1983), 1883.

3. I. Bialynicki-Birula & J. Mycielski, Commun. Math. Phys. 44 (1975), 129; see also H. Everett III,

in The Many-worlds Interpretation of Quantum Mechanics, edited by B. S. De Wits & N. Graham

(Princeton University Press, Princeton, N J, 1973).

4. S. Abe & N. Suzuki, Phys. Rev. A41 (1990), 4608.

5. Y. Takahashi & H. Umezawa, Collect. Phenom. 2 (1975), 55.

6. I. Ojima, Ann. Phys. (N.Y.) 137 (1981), 1.

7. A. Mann, M. Revzen, K. Nakamura, H. Ume,awa & Y. Yamanaka, 2. Math. Phys. 30 (1989), 2883.

379
N92-22088

THE HAMILTONIAN STRUCTURE OF DIRAC'S EQUATION IN TENSOR FORM

AND ITS FERMI QUANTIZATION

Frank Reifler and Randall Morris

Government Electronic Systems Division

General Electric Company

Moorestown, NJ 08057

ABSTRACT

Currently there is some interest in studying the tensor

forms of the Dirac equation to elucidate the possibility

of the constrained tensor fields admitting Fermi quantization. In this paper, we demonstrate that the bispinor and tensor Hamiltonian systems have equivalent

Fermi quantizations. Although the tensor Hamiltonian

system is noncanonical, representing the tensor Poisson brackets as commutators for the Heisenberg operators directly leads to Fermi quantization without

the use of bispinors.

I. CLASSICAL DERIVATION

We apply the double covering map from bispinors to

their tensor equivalents. This map 1, an extension of

Cartan's spinor map [Ref. 4], maps the bispinor _ to

a constrained set of SL(2,C) x U(1) gauge potentials A K

and a complex scalar field p, where a -- 0, 1, 2, 3 is a

Lorentz index and K -- 0, 1, 2, 3. 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(1) gauge potential A o is

real. A_ and p satisfy the following constraint:

A_ Axg = --[9[2gag (1)

where K is contracted using the SU(2) × U(1) Killing

metric and g_ is the space-time metric.

With this constraint, the Dirac bispinor Lagrangian

comes from the following Yang-Mills tensor Lagrangian L, in the limit of a large Yang-Mills coupling

constant g:

1The Cartanmap [Refs.I,2]maps thebispinory to a tripletof

complexantisymmetrictensors_I_ (wherej = 1,2,3)of Carmeli

classG [Ref.3].Such F_ can be expressedasjl_a--p(AoaA_-A_Ao _

+i_jkraA_A_m)where p is a complex scalar and A_tfor K = 0,1,2,3 are

SL(2,C) × U(1) gauge potentials satisfying (1). (_kra for j, k, m = 1,

2, 3 is the permutation symbol.)

L = - _ Re [A_ A_0 ] + (D"_) (Dap) - ]p + 2mJ 4

(2)

where m denotes mass, D a denotes the Yang-Mills

covariant derivative with connection coefficients A K,

and AKg is the Yang-Mills curvature tensor associated

with the gauge potentials A K. The indices arc contracted using the Killing metric as well as the spacetime metric. All bispinor observables can be derived

from L using Yang-Mills formulas. Although previous

authors [Refs. 5, 2] derived the tensor form of the

Dirac Lagrangian, they did not put it in the gauge

symmetric Yang-Mills form (2).

The Dirac equation can be derived from the Lagrangian L by ascribing to the Yang-Mills field A K a large

self-coupling constant g. To be consistent with observation, the fields A K and p must couple more weakly

(by the factor l/g) with other fields. In particular, Einstein's equation becomes Gag -- kTa_ where Gag is the

Einstein tensor, k is the gravitation constant, and Tag

is the energy-momentum tensor derived from the Lagrangian L. In the limit of large self-coupling g (neglecting terms in Tag not containing g) we have TQ_ =

g T'a_ where T'_g is exactly the usual Dirac energymomentum tensor [Ref. 5]. Hence k' = kg is the observed gravitational constant, not k. Note also that in

the Lagrangian L, the observed mass is m' = rag, not

m. Then, as g tends to infinity, the Lagrangian kL is

independent of g. In this limit, which we henceforth

assume, we have:

Lira k L = k' L' (3)

g---,_c

where L' is exactly equal to Dirac's bispinor Lagrangian [Ref. 5]. Thus, as previously stated, Dirac's bispinor Lagrangian is a limiting case of the Yang-Mills

Lagrangian (2), in which the self-coupling constant g

tends to infinity.

II. FERMI QUANTIZATION

We quantize A K and p by defining the classical Hamiltonian to be: (Let SCR 3 be a large cube.)

381

PRECEDING PAGE BLANK NOT FILMED
H= _s T°°dx (4)

where T al_ is the energy-momentum tensor derived

from the fermion tensor Lagrangian (3). We make a

classical change of variables that simplifies H. The

resulting Hamiltonian equations are then formulated

as Heisenberg operator equations.

Because the SL(2, C) × U(I) gauge group is not compact, H is not bounded from below. This has the consequence that any quantization of the fields A_and p

must obey the exclusion principle; otherwise fermions

descend forever to lower energy states.

By the Cartan map [Refs. 1, 5] the energy-momentum

tensor has an expansion of the form:

Tal3(x,t) = E E Tp_q(x)apq(t) (5)

P q

where the sum is over all pairs of fermion modes p and

q, and where Tp_ (x) are fixed functions of x e S, and

apq(t) are complex functions of time t satisfying apq =

_qp. The bimodal expansion (5) is irreducible because

it cannot be expressed in tensor terms as a sum over

products of single modes, as is the case with bosons.

The Hamiltonian (4) can be written in terms of the

amplitudes apq(t) as follows:

H = E O)p app (6)

P

where O)pis the frequency of the mode p. Note that for

simplicity, the amplitudes apq(t) are defined to be consistent with the hole theory.

The classical Hamiltonian equations (which are equivalent to the constrained Euler-Lagrange equations for

A_and p) are given by:

dapq= (apq, H) (7)

dt

where the Poisson brackets ( , ) are defined for the

classical amplitudes apq(t) as follows:

(apq, ap,q,) = -i (apq, _p,q -- ap,q _pq,) (8)

where _pq equals one if p = q and zero otherwise.

Formulas (6), (7), and (8) are noncanonical tensor

Hamiltonian equations which cannot be formulated as

canonical equations in tensor terms. Nevertheless,

they are easily quantized by replacing the classical

amplitudes am(t) with Heisenberg operators, denoted

as _w(t), and the Poisson brackets (8) with (equal time)

commutators [, ] as follows:

[apq, ap,q,] = apq, 5p,q - ap,q 5pq, (9)

The Heisenberg equations become:

da_a = --i [apq, if'I] (10)

dt

where I=Iis the operator version of the Hamiltonian (6).

To further simplify these equations, we attempt to

factor _,pq(t) into a product of operators:

apq(t) = _tp(t) Cq(t) (11)

where the dagger (t) signifies adjoint. Since they do not

occur explicitly in the Hamiltonian I=I, the new operators _p(t) a priori could satisfy any relations consistent with the commutation relations (9). We exploit

this arbitrariness in order to satisfy the exclusion principle, previously discussed. At time t we define:

_tp _q + _q_tp : _pq (12)

All other equal time anti-commutators of 6p(t) are defined to be zero. Formulas ( 11) and (12) are consistent

with the commutation relations (9) as required.

It is clear that equations (9), (10), (1 l), and (12), while

derived from the tensor Hamiltonian equations, are

equivalent to Fermi quantization via bispinors. Thus,

the tensor Lagrangian (3) leads to Fermi quantization

without the use of bispinors.

Again, without the use ofbispinors, we may extend the

tensor Lagrangian (3) to include the electromagnetic

field. Quantization is straight forward due to the fact

that the interaction term is a function of the fermion

amplitudes apq(t), as well as boson amplitudes b.(t).

III. QUANTUM GRAVITY

Spinor structure can be defined on a noncompact

space-time manifold M by specifying, at each point

xeM, a set of Pauli spin-half matrices O_B,(X) satisfying

[Ref. 3]:

(_B,O-I]AB' __. gad (13)

Formula (13) has a topological as well as a metric

consequence. The topological consequence of (13) is

that M must be parallelizable [Ref. 6]. The metric

consequence is that g_a is constrained as in formula ( 1).

Since, for noncompact parallelizable space-times for382
mulas(1)and(13)areequivalent,spinorstructureis

nothingbutanindirectwayof constrainingthemetric

gQ_onsuchspace-times. However,thetensorfieldsAK

andP satisfying the constraint (1) are more general

than (13), since they can be defined on general spacetimes.

Formula (13) presents a dilemma [Ref. 7] for quantizing both gravity and the Dirac field, since the definition of the Pauli matrices (_,B' depends on the gravitational field g_. The problem is resolved by

identifying the degrees of freedom in the constraint (1)

as follows.

Consider a fixed metric g_ on M and define Pauli

matrices 8_,B,with respect to _,al_.The metric g_ on M

is expressed by:

gal3 = gttl3+ h_l_

We also express the gauge potentials A_ by:

(14)

A_ = ¢ A_ (15)

where AK satisfies the constraint (l) with respect to the

fixed metric _. The dynamical fields are then A_, p,

and hal_provided that the matrix f= _ can be uniquely

solved as a function of h_a. Since AK and p have bispinor coordinates with respect to the fixed spinor

structure on M, the fields AaK p, and haa can be quantized as in Section II.

It remains to solve for the matrix f in formula (15)

using the constraint (1). Since

formulas (14) and (15) give:

(16)

gv¢, _ _ = g=l_+ h_,a (17)

The solution of (17) is given by:

0¢

f = Z _,1/2 h n

v n (18)

n=0

where C_ denote the binomial coefficients, and the

matrix h is defined by:

h = h_ = $1_vhva (19)

where _ is the inverse matrix of _,_1_"

For the power series (18) to converge, the eigenvalues

of h must lie within the unit circle. This restricts the

validity of quantum gravity to small fluctuations ofg_l 3

about the fixed metric _a.

IV. CONCLUSIONS

In this paper we have adhered to the program of first

defining all fields, Bose and Fermi, as classical tensor

fields, and then quantizing them using Hamilton equations and Poisson brackets. From this vantage point,

the Dirac equation becomes a classical tensor equation

on the same level as the electromagnetic and gravitation tensor equations. Fermions, photons, and gravitons are obtained by quantizing the degrees of freedom

allowed by the tensor constraint (1). We have shown in

Section III that the constraint (1) implies that we cannot, in general, separate fermion and graviton degrees

of freedom, except when the power series (18) converges.

We also found that the fermion degrees of freedom

require the use of noncanonical Hamilton equations

(6), (7), and (8). Since the free Dirac tensor equation is

completely integrable, we have shown that current usage of only canonical Hamilton equations is too restrictive for quantizing integrable tensor fields.

REFERENCES

.

,

Reifler, F. and R. Morris, 1986, "A Gauge Symmetric Approach to Fierz Identities," J. Math. Phys.,

27(11), pp. 2803-2806.

Zhelnorovich, V. A., 1990, "Complex Vector Triplets in the Spinor Theory in Minkowski Space,"

Proc. Acad. of Sciences of USSR, 311 (3), pp.

590-593.

. Carmeli, M., Kh. Huleihil, and E. Leibowitz, 1989,

"Gauge Fields," World Scientific, Singapore, pp. 21

and 40.

4. Caftan, E., 1981, The Theory of Spinors, Dover,

New York, p. 41.

. Takahashi, Y., 1983, "The Fierz Identities--A Passage Between Spinors and Tensors," J. Math. Phys.,

24(7), pp. 1783-1790.

. Geroch, R., 1968, "Spinor Structure of Space-times

in General Relativity I," J. Math. Phys., 9 {11), pp.

1739-1744.

. Ashtekar, A. and R. Geroch, 1974, "Quantum Theory of Gravitation," Rep. Prog. Phys., 37, pp.

1211-1256.

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4. TITLE AND SUBTITLE

Workshop on Squeezed States and Uncertainty Relations

6. AUTHOR(S)

D. Han, Y. S. Kim, and W. W. Zachary, Editors

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

NASA Goddard Space Flight Center

Greenbelt, Maryland 20771

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National Aeronautics and Space Administration

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11. SUPPLEMENTARY NOTES

D. Han, NASA-GSFC, Greenbelt, Maryland, 20771

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

W. W. Zachary, COMSERC, Howard University, Washington, D.C. 20059

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13. ABSTRACT (Maximum 200 words)

This conference publication contains the proceedings of a workshop on squeezed states and uncertainty relations held at the University of Maryland, College Park, Maryland, on March 28-30, 1991.

Squeezed states were predicted theoretically in the 1970s. They became a physical reality during the period

1985-1988. Efforts are being made to produce more efficient squeezed-state lasers. More refined theoretical

tools are being developed for this new physical phenomenon. One of the pressing questions during the 1990s will

be: What should we do with squeezed states? This is the main question we wanted to address in this workshop.

There are many who say that the potential for industrial applications is enormous, as the history of the

conventional laser suggests. All those who worked so hard to produce squeezed states of light are continuing

their efforts to construct more efficient squeezed-state lasers. Quite naturally, they are looking for new experiments using these lasers. The physical basis of squeezed states is the uncertainty relation in Fock space, which

is also the basis for the creation and annihilation of particles in quantum field theory. Indeed, squeezed states

provide a unique opportunity for field theoreticians to develop a measurement theory for quantum field theory.

14. SUBJECT TERMS

Squeezed States, Quantum Optics, Uncertainty Relations, Poincare Group

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