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NIST Technical Note 1353

NIST

jpUBllCATlONS*;:;

U.S. DEPARTMENT OF COMMERCE / National Institute of Standards and Technology

Selected publications of the

Storage Group

e and Frequency Division

J, Boulder, Colorado

Edited by: James C. Bergquist

John J. Bollinger

Wayne M. Itano

David J. Wineland

/S & 3

'^1

Trapped Ions and Laser Cooling

Selected publications of the Ion Storage Group of the Time and Frequency Division

Edited by

James C. Bergquist

John J. Bollinger /

Wayne M. Itano ^ , David J. Wineland"^

Time and Frequency Division

Physics Laboratory

National Institute of Standards and Technology

Boulder, Colorado 80303-3328

Supported in part by

U.S. Office of Naval Research

800 North Quincy

Arlington, Virginia 22217

April 1992

*^TE3 o*

*"

U.S. DEPARTMENT OF COMMERCE, Barbara Hackman Franklin, Secretary

TECHNOLOGY ADMINISTRATION, Robert M. White, Under Secretary for Technology

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY, John W. Lyons, Director
National Institute of Standards and Technology Technical Note

Natl. Inst. Stand. Technol., Tech. Note 1353, 204 pages (April 1992)

CODEN:NTNOEF

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1992

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402-9325
PREFACE

This collection of papers represents the work of the Ion Storage Group, Time and

Frequency Division, National Institute of Standards and Technology from October 1988 to February 1992. It follows the collections of papers contained in NBS Technical Note 1086,

Trapped Ions and Laser Cooling (June 1985) and NIST Technical Note 1324 Trapped Ions

and Laser Cooling II (September 1988). Although the primary goal of this work has been

the development of techniques necessary for achieving high resolution spectroscopy, we have

also been able to investigate related areas of research.

Papers listed on page vii were published or prepared during the period October 1988

to February 1992 but are not included here. They can be obtained on request. We intend to update this publication periodically to include new work. We hope this collection of papers

will be useful to our colleagues in this and related fields.

We acknowledge the important contributions of several colleagues to this collection.

In particular, we thank Larry Brewer, Claude Cohen-Tannoudji, Jean Dalibard, Frank

Diedrich, Frank Eisner, Sarah Gilbert, Jon Gilligan, Dan Heinzen, Dan Larson, Chuck

Manney, Fred Moore, Mark Raizen, and Carl Weimer. We gratefully acknowledge the

continued support of the U. S. Office of Naval Research and the support of the Air Force

Office of Scientific Research through September 1991.

James C. Bergquist

John J. Bollinger

Wayne M. Itano

David J. Wineland

Boulder, Colorado

March 1992

111

CONTENTS Page

ADDITIONAL PUBLICATIONS viii

LASER COOLING

1. "Perpendicular Laser Cooling of a Rotating Ion Plasma in a Penning TN-1

Trap," W. M. Itano, L. R. Brewer, D. J. Larson, D. J. Wineland, Phys.

Rev. A 38, 5698 (1988).

2. "Laser Cooling to the Zero-Point Energy of Motion," P. Diedrich, J. C. TN-10

Bergquist, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 62, 403

(1989).

3. "Quantum-Limited Cooling and Detection of Radio-Frequency Oscillations TN-1

4

by Laser-Cooled Ions," D. J. Heinzen and D. J. Wineland, Phys. Rev. A

42, 2977 (1990).

4. "Sisyphus Cooling of a Bound Atom," D. J. Wineland, J. Dalibard, and C. TN-32

Cohen -Tannoudji, J. Opt. Soc. Am. B 9, 32 (1992).

SPECTROSCOPY AND FREQUENCY STANDARDS

1. "Hg"^ Single Ion Spectroscopy," J. C. Bergquist, F. Diedrich, W. M. TN-43

Itano, and D. J. Wineland, in Laser Spectroscopy IX. ed. by M. S. Feld,

J, E. Thomas, and A. Mooradian (Academic, San Diego, 1989), p. 274.

2. "Frequency Standards in the Optical Spectrum," D. J. Wineland, J. C. TN-47

Bergquist, W. M. Itano, F. Diedrich, and C. S. Weimer, in The Hydrogen

Atom. Ed. by G. F. Bassani, M. Inguscio, and T.W. Hansch (Springer

Verlag, Berlin, Heidelberg, 1989), p. 123.

3. "Progress at NIST toward Absolute Frequency Standards Using Stored TN-58

Ions," D. J. Wineland, J. C. Bergquist, J. J. Bollinger, W, M. Itano,

D, J. Heinzen, S. L. Gilbert, C. H. Manney, and M. G. Raizen, IEEE

Trans. Ultrason. Ferroelec. Freq. Contr. 22, 515 (1990).

4. "A 303-MHz Frequency Standard Based on Trapped Be+ Ions," J. J. TN-67

Bollinger, D. J. Heinzen, W. M. Itano, S. L. Gilbert, and D. J. Wineland,

IEEE Trans. Instrum. Meas. 40, 126 (1991).
5. "Single Ion Optical Spectroscopy," J. C. Bergquist, W. M. Itano, F. TN-70

Eisner, M. G. Raizen, and D. J. Wineland, in Light Induced Kinetic

Effects on Atoms. Ions and Molecules, ed. by L. Moi, S. Gozzini, C.

Gabbanini, E. Arimondo, and F. Strumia (ETS Editrice, Pisa, 1991), p. 291.

6. "Linear Trap for High-Accuracy Spectroscopy of Stored Ions," M. G. TN-79

Raizen, J. M. Gilligan, J. C. Bergquist, W. M. Itano, and D. J. Wineland,

J. Mod. Opt. 29, 233 (1992).

FUNDAMENTAL TESTS

1. "Test of the Linearity of Quantum Mechanics by rf Spectroscopy of the TN-89

^Be^ Ground State," J. J. Bollinger, D. J. Heinzen, W. M. Itano, S. L.

Gilbert, and D. J. Wineland, Phys. Rev. Lett. 63, 1031 (1989).

2. "Quantum Optics of Single, Trapped Ions," W. M. Itano, J. C. Bergquist, TN-93

F. Diedrich, and D. J. Wineland, in Coherence and Ouantum Optics VI. ed. by J. H. Eberly, L. Mandel, and E. Wolf (Plenum, New York, 1990)

p. 539.

3. "Quantum Zeno Effect," W. M. Itano, D. J. Heinzen, J. J. Bollinger, and TN-98

D. J. Wineland, Phys. Rev. A 41, 2295 (1990).

4. "Reply to 'Comment on the Quantum Zeno effect'," W. M. Itano, D. J. TN-I04

Heinzen, J. J. Bollinger, and D. J. Wineland, Phys. Rev. A 4^, 5168

(1991).

5. "Search for Anomalous Spin-Dependent Forces Using Stored-Ion TN-106

Spectroscopy," D. J. Wineland, J. J. Bollinger, D. J. Heinzen, W. M.

Itano, and M. G. Raizen, Phys. Rev. Lett. 67, 1735 (1991).

6. "Atomic Physics Tests of Nonlinear Quantum Mechanics," J. J. Bollinger, TN-UO

D. J. Heinzen, W. M. Itano, S. L. Gilbert, and D. J. Wineland, Atomic

Physics 12 (proc. 12th International Conference on Atomic Physics), ed.

by J. C. Zom and R. R. Lewis (Amer. Inst, of Physics Press, New York,

1991), p. 461.

VI
NONNEUTRAL PLASMA STUDIES

1. "Coulomb Clusters of Ions in a Paul Trap," W. M. Itano, J. C. Bergquist, TN-130

and D. J. Wineland, Proc. Workshop on Crystalline Ion Beams, Wertheim,

W. Germany, ed. by R. W. Hasse, I. Hofmann, D. Liese (GSI report GSI89-10, ISSN 0170-4546, 1989), p. 241.

2. "Observation of Correlations in Finite, Strongly Coupled Ion Plasmas," in TN-144

Strongly Coupled Plasma Physics , ed. by S. Ichimaru (Elsevier Science

Publishers B.V. / Yamada Science Foundation, 1990) p. 177.

3. "Microplasmas," J. J. Bollinger and D. J. Wineland, Sci. Am. 262, 124 TN-155

(1990).

4. "Rotational Equilibria and Low-Order Modes of a Non-Neutral Ion TN-162

Plasma," D. J. Heinzen, J. J. Bollinger, F. L. Moore, W. M. Itano, and

D. J. Wineland, Phys. Rev. Lett. 66, 2080 (1991).

GENERAL ARTICLES

1. "Cooling Methods in Ion Traps," W. M. Itano, J. C. Bergquist, J. J. TN-166

Bollinger, and D. J. Wineland, chapter in Physics with Charged Particles

in a Trap , ed. by G. Gabrielse and G. Werth, submitted January 1990.

2. "Recent Experiments on Trapped Ions at the National Institute of Standards TN-182

and Technology," D. J. Wineland, J. C. Bergquist, J. J. Bollinger, W. M.

Itano, F. L. Moore, J. M. Gilligan, M. G. Raizen, D. J. Heinzen, C. S.

Weimer, and C. H. Manney, Proc. Enrico Fermi Summer School on

"Laser Manipulation of Atoms and Ions," July 1991, Varenna, Italy,

submitted September, 1991.

Vll
ADDITIONAL PUBLICATIONS

Reports published or prepared during the period October 1988 to February 1992,

but not included in this technical note.

1. "Liquid and Solid Phases of Laser Cooled Ions," S.L. Gilbert, J.C.

Bergquist, J. J. Bollinger, W. M. Itano, and DJ. Wineland, in Atomic

Physics 11 . ed. by S. Haroche, J. C. Gay, and G. Grynberg (World

Scientific Press, Singapore, 1989), p. 261.

2. "The Digitized Atom and Optical Pumping," D.J. Wineland, W.M. Itano,

J.C. Bergquist and R.G. Hulet, in Atomic Physics 11 . ed. by S. Haroche,

J.C. Gay, G. Grynberg (World Scientific Press, Singapore, 1989), p. 741.

3. "Progress at NIST Towards Absolute Frequency Standards Using Stored

Ions," D. J. Wineland, J. C. Bergquist, J. J. Bollinger, W. M. Itano, D.

J. Heinzen, S. L. Gilbert, C. H. Manney, and C. S. Weimer, Proc. 43rd

Annual Symposium on Frequency Control, Denver, June 1989, IEEE

Catalog no. 89CH2690-6.

4. "Observation of Shell Structures with Ions Stored in Traps," J. J. Bollinger, S. L. Gilbert, and D. J. Wineland, Proc. Workshop on

Crystalline Ion Beams, Wertheim, W. Germany, Ed. by R. W. Hasse, I. Hofmann, D. Liese (GSI report GSI-89-10, ISSN 0170-4546), p. 231.

5. "Trapped Atoms and Laser Cooling," D. J. Wineland, "essay" for

undergraduate text Physics , by Paul Tipler (Worth, New York, 1991).

6. "Progress at NIST on Absolute Frequency Standards Using Stored Ions,"

D. J. Wineland, J. C. Bergquist, J. J. Bollinger, W. M. Itano, D. J. Heinzen, S. L. Gilbert, C. H. Manney, M. G. Raizen, and C. S. Weimer,

Proc. 4th European Frequency and Time Forum, Neuchatel, March 1990,

p. 267.

7. "Liquid and Solid Plasmas," J. J. Bollinger, S. L. Gilbert, D. J. Heinzen,

W. M. Itano, and D. J. Wineland, in Atomic Processes in Plasmas , ed. by

Y. K. Kim and R. C. Elton, AIP Conference Proceedings 206 (American

Institute of Physics, New York, 1990), p. 152.

8. "Test of the Linearity of Quantum Mechanics by RF Spectroscopy of the

^Be^ Ground State," D. J. Heinzen, J. J. Bollinger, W. M. Itano, S. L.

Gilbert, and D. J. Wineland, in Coherence and Ouantum Optics, ed. by J. H. Eberiy, L. Mandel, and E. Wolf (Plenum, New York, 1990), p. 479.

Vlll
9. "Trapped Ion Frequency Standards," D. J. Wineland, J. C. Bergquist, J. J. Bollinger, W. M. Itano, D. J. Heinzen, C. H. Manney, F. L. Moore, M.

G. Raizen, and C. S. Weimer, Proc. 22nd Ann. Precise Time and Time

Interval (PTTI) Applications and Planning Meeting, ed. by R. L. Sydnor

(NASA Conf. Publ. 3116, 1991), p. 53.

10. "Atomic Ion Frequency Standards," W. M. Itano, Proc. IEEE 79, 936

(1991).

11. "High Resolution Atomic Spectroscopy of Laser-Cooled Ions," D. J. Wineland, J. C. Bergquist, J. J. Bollinger, W. M. Itano, F. L. Moore, J. M. Gilligan, M. G. Raizen, C. S. Weimer, and C. H. Manney, Proc.

Enrico Fermi Summer School on "Laser Manipulation of Atoms and Ions,"

July 1991, Varenna, Italy, submitted.

12. "Laser Cooling of Trapped Ions," W. M. Itano, J. C. Bergquist, J. J. Bollinger, and D. J. Wineland, ibid.

13. "Ionic Crystals in a Linear Paul Trap," M. G. Raizen, J. M. Gilligan, J. C. Bergquist, W. M. Itano, and D. J. Wineland, Phys. Rev. A, to be

published.

14. "Atomic Physics Tests of Nonlinear Quantum Mechanics," J. J. Bollinger,

D. J. Heinzen, W. M. Itano, S. L. Gilbert, and D. J. Wineland, Santa Fe

Conference on Quantum Measurements, July 1991, to be published.

15. "Single Ion Spectroscopy," J. C. Bergquist, D. J. Wineland, W. M. Itano,

F. Diedrich, M. G. Raizen, and F. Eisner, Proc. Soc. Photo-optical

Instrumentation Engineers, to be published.

16. "Low Order Modes of an Ion Cloud in a Penning Trap," J. J. Bollinger,

D. J. Heinzen, F. L. Moore, W. M. Itano, and D. J. Wineland,

Proceedings of the Workshop on Physics with Penning Traps. Lertorpet,

Sweden, Physica Scripta, to be published.

IX

PHYSICAL REVIEW A VOLUME 38, NUMBER 11 DECEMBER 1, 1988

Perpendicular laser cooling of a rotating ion plasma in a Penning trap

Wayne M. Itano, L. R. Brewer,* D. J. Larson,' and D. J. Wineland

Time and Frequency Division, National Bureau ofStandards, Boulder, Colorado 80303

(Received 13 June 1988)

The steady-state temperature of an ion plasma in a Penning trap, cooled by a laser beam perpendicular to the trap axis, has been calculated and measured. The rotation of the plasma, due to crossed E and B fields, strongly affects the minimum attainable temperature. This is because the velocity distribution of the ions, as seen by a laser beam intersecting the plasma at some distance from

the axis of rotation, is skewed, and this leads to a change in the velocity distribution (and hence temperature) at which a steady state is attained. The calculated temperature is a function of the intensity, frequency, and position of the laser beam, and of the rotation frequency of the plasma. Temperatures of 'Be^ plasmas were measured for a wide range of experimental parameters. The lowest and

highest temperatures were approximately 40 mK and 2 K. The measured and calculated temperatures are in agreement.

I. INTRODUCTION

Laser cooling is a method by which radiation pressure

is used to reduce the temperature of atoms.' It has been

demonstrated with free and bound atoms and also with

ions confined in Paul and Penning traps. Laser cooling of

atoms confined by a harmonic potential has been studied

theoretically by many methods.' The Paul (radiofrequency) trap uses an oscillating, inhomogeneous electric field to create a harmonic potential for the average

motion of an ion.^'^ However, motion in a Paul trap

differs from that in a harmonic trap, since there is a forced oscillation (micromotion) at the frequency of the

appHed field. The micromotion has not been included in calculations of laser cooling, except by numerical simulation.4 In a Penning trap, static electric and magnetic fields confine the ions. In some ways, the Penning trap is more

difficult than the Paul trap to treat theoretically. Some of

the difficulties arise because the electrostatic potential en- ergy decreases as an ion moves radially outward from the

trap axis.^'^ A magnetic field along the trap axis is re- quired to confine the ions radially. The combined effects of the magnetic field and of the radial electric field lead to

a circular EXB drift of the ions about the trap axis. This

rotation leads to a basic difference between laser cooling

in a Penning trap and in a harmonic trap. It can be

shown that conservation of energy and of the axial component of the canonical angular momentum L^ guarantees confinement of the ions.^ However, external torques,

due to colhsions with neutral molecules or to asymmetries of the trap fields, change L^ and may cause the

ions to move outward radially. ^'^ Thus long-term

confinement is not guaranteed even if laser cooling

reduces the random motion of the ions. A laser beam of

appropriate tuning and spatial profile can be used to apply a torque to the ions in order to prevent this radial

drift, while also cooling their random motion.^ The axial motion in a Penning trap does not suffer from these complications, since the axial well is harmonic, and since

there are no axial magnetic forces. A laser beam which is

not perpendicular to the trap axis can be used to cool this motion.

The previous theoretical treatment of laser cooling in a Penning trap considered only a single ion.^ When many

ions are confined, the additional electric fields lead to an

increased rotation frequency of the ions around the axis,

which affects the cooling. Also, detailed calculations were carried out only for the case where the Doppler

broadening was much smaller than the natural linewidth

of the optical transition. This is often not the case in practice.

Several reports of laser cooling of ions in Penning traps

have appeared previously.^" '^ While temperature measurements were reported in all of these references, and

Ref. 9 reported a measurement of the cooling rate, in rough agreement with theory, very little has been done in the way of quantitative comparisons of theory and experi- ment. Simple theoretical calculations yield a predicted

minimum temperature T^^j^=-fiYQ/2kg, where Yq is the

radiative linewidth of the atomic transition in angular

frequency units and kg is Boltzmann's constant.^ This

minimum temperature has been approached within about

a factor of 10 when the laser beam was perpendicular to the trap axis, and within about a factor of 2 when the

cooHng laser was not perpendicular to the axis.'^ More

typically, the lowest observed temperatures were higher

than r^^in by two orders of magnitude, and no explanation of this discrepancy has been given. In this paper we present a calculation of the perpendicular (cyclotron) temperature in a Penning trap for the case in which the cooling laser beam is perpendicular to the trap axis. This geometry is often used experimentally 9,10,12-14 ji^g calculations are compared to experiments in which temperatures of ''Be'*' plasmas were measured for a wide range of experimental conditions. For

typical experimental conditions, the minimum perpendicular temperature is found to be much greater than

T^^^=^^YQ/2kg. The reason stems from the condition

that the work done by the laser on the ions be zero in the

steady state. The rate at which work is done on an ion of

velocity v by a force F, which in this case is parallel to

38 5698 TN-1
38 PERPENDICULAR LASER COOLING OF A ROTATING ION . 5699

the direction of propagation of the laser beam, is F-v. If the laser beam intersects the ion plasma on the side which is receding from the laser due to the rotation of the

plasma, F-v would tend to be positive. (If it intersects on

the other side, the plasma radius expands and there is no

steady state. '°) In order to make the average value

(F-v) wO, the frequency of the laser is tuned below reso- nance, which, because of the Doppler effect, makes F

stronger for ions with F-v <0 than for ions with F-v>0.

In order to get (F-v)wO, due to cancellation between

negative and positive contributions to the average, the

width of the velocity distribution must have a certain

value, which corresponds to a temperature higher than

II. THEORY

A. Simple theory

In this section a simplified theory of perpendicular

laser cooling of an ion plasma in a Penning trap is presented. Some of the simplifying assumptions will be

modified in the following sections, in order to obtain

quantitative predictions that can be compared with ex- periment.

The trap is assumed to approximate the ideal Penning

trap, which consists of a uniform magnetic field B = Bz. superimposed on a static electric potential

(f)j-(r,z) =

mo).

Aq

[2z^-r' (1)

where m and q are the mass and charge of an ion, and o)^

is the frequency of axial motion of a single ion in the trap.

The actual frequency of motion of an individual ion is shifted by the presence of the other ions. Here, r and z are the cylindrical coordinates of a point in the trap.

It is assumed that the plasma rotates about the z axis with a uniform angular frequency o, that is, without

shear. The velocity distribution of the ions is assumed to be a Maxwell-Boltzmann distribution superimposed on

this uniform rotation. A Maxwell-Boltzmann velocity

distribution superimposed on a uniform rotation is a

characteristic of a non-neutral plasma in thermal equilibrium in cylindrically symmetric fields. '^''^ If there is shear within the plasma, a resulting frictional force tends

to reduce the shear, leading to uniform rotation in ithe steady state. Approximate uniform rotation of the plas- mas has been verified experimentally. '^'^'^•^^ The temperatures T^ and Ty describe the velocity distributions in the x-y plane in the z direction, and differ because cooling

of the z motion occurs only by ion-ion collisions. If there were no relaxation between T^ and Tj , then T^^ would in- crease without limit, due to heating by photon recoil. We

assume that the relaxation is sufficient that T^^ reaches a steady value somewhat higher than Ty. Under the as- sumptions stated in this section, the calculation of T

does not require any knowledge of T, . If the beam is not perpendicular to the axis, the temperatures depend on the

relaxation rate between the perpendicular and parallel ki- netic energies, which could be measured by a separate ex- periment.

A key assumption in the calculation of T^ is that, ex- cept for the change in energy due to the scattering of

photons from the laser, the total (kinetic plus potential)

energy of the ion plasma is conserved. This is true to the

extent that the plasma is isolated from the rest of the universe and can be described by a time-independent

Hamiltonian. These assumptions would be violated by

collisions of the ions with neutral atoms or by interac- tions with time-varying external electric and magnetic

fields. In practice, the pressure in the trap and the timevarying fields (due, for example, to thermal radiation or

to charges moving on the trap electrodes) can be made

low enough that the interaction with the laser is the main

source of energy change of the ion system. The rapid

heating of the ion plasma that was observed in one exper- iment after the coohng laser was shut off^' can be ac- counted for by a conversion of the electrostatic potential

energy of the ions to kinetic energy as the plasma expanded radially.

On the other hand, it is not assumed that L^ is conserved in the absence of the torque due to the interaction

with the laser. If the trap fields were perfectly axially

symmetric, and if the ions were totally isolated from collisions with neutral atoms and the radiation field, then L^

would be conserved. However, static electric or magnetic

fields which violate axial symmetry can apply a net torque to the ion plasma and change L^, even though, be- ing time independent, they cannot transfer energy. In

fact, such torques are required in order for the plasma to attain a steady state (a state in which all of the observable

properties are constant in time) while the laser beam is applying a finite torque to the plasma. It has been experimentally observed that a steady state can be obtained even when the radial displacement of the laser beam from

the trap center is greater than the beam radius. In this case, the torque due to the laser must be balanced by the

other external torques. The oscillations that have been

observed in the fluorescence of a plasma of laser-cooled Mg^ ions''^ may be due to the fact that the nature of the

external torque is such that the energy and the angular

momentum cannot be balanced at the same time, for those experimental conditions.

The calculation of the torque applied by the laser is straightforward.^ The torque due to static field asymmetries is not well understood in general and is a topic of

current experimental^' ^'^^"^^ and theoretical'^^'^^ research. In studies of confined electron plasmas, it has

been observed that, at sufficiently low pressures, the radial expansion rate, which is a measure of the external

torque, is independent of the pressure.^ The expansion

rate decreased when care was taken to reduce asymmetries in the apparatus. ^^ Some plasma instabilities which can be driven by static field asymmetries have been

studied experimentally. ^^'^"^ No attempt will be made in this article to calculate the external torque. Rather, it is assumed that values of co and T^ that ensure a steady

state can be found. The point of the calculation is to determine the value of T^ which is consistent with given

values of co and the other parameters. In the experiments, (i) and T^ were varied by varying the position, in- tensity, and frequency of the cooling laser beam.

TN-2
5700 ITANO, BREWER, LARSON, AND WINELAND 38

The ion plasma is assumed to interact with a monochromatic laser beam, whose frequency w^ is close to res- onance with a strong optical transition of frequency coq. We assume that the upper state of the transition decays

only to the ground state. The center of the beam lies in the plane defined by z=0, and passes through the point

(x,y,z) = (0,d,0), parallel to the x axis (see Fig. 1). The

coordinate system is such that, for d >0, the rotation of

the plasma causes the ions to recede from the laser. The

laser intensity is assumed to be so low that saturation of

the transition can be neglected. The interaction of the

ions with the laser radiation is assumed to consist of a

series of photon-scattering events. The events are as- sumed to occur at random times; hence small effects such

as photon antibunching^^"^^ are neglected here. The delay between the photon absorption from the laser beam

and the subsequent photon emission is neglected. This is justified as long as the natural linewidth of the transition

is much greater than the frequencies of motion of the ion. (This case has been called the "weak binding" or "heavy

particle" Hmit.') The change in kinetic energy due to scattering a photon depends on the direction of the emitted photon. The angular distribution of the emitted photons, in a frame moving with the atom, is such that the

probability of emission in the direction k is equal to the

probability of emission in the direction — k. The energy

change per scattering event, averaged over all possible

emission directions, is then

{^EK)=-nk-v + 2R (2)

where k is the wave vector of the absorbed photon

(|k|=a)£^/c), V is the velocity of the ion before the

scattering event, and the recoil energy R={-hk)^ /[2m)

[see Eq. (3) of Ref. 34]. This quantity can be thought of

as the time integral for one scattering event, of F-v, in- cluding the average effect of the photon recoil. The rate of energy change of an ion while it is within

the laser beam is equal to the product of the average en- ergy change per photon scattering [Eq. (2)] and the rate of photon scatterings. The rate of photon scatterings (in the low-intensity limit) is

(a)

^ Laser Beam

Laser Beam

FIG. 1. Diagram showing the laser beam intersecting the ion plasma in (a) the x-y plane and (b) the y-z plane. The laser beam

is parallel to the x axis and lies in the z =0 plane. Its distance of

closest approach to the z axis is d. The direction of the plasma

rotation (at angular frequency co) is indicated.

lOc

Yl

iron)'

^^L [(ro/2)' + A^] ' (3)

where / is the light intensity, cjq is the scattering cross section at resonance, Yo i^ the natural linewidth of the

transition, and i^ = coi^—o}'Q — kVj^ is the detuning of the

laser frequency from resonance, taking into account the

Doppler shift. Here, o)'q = (i)q + R /'I'l is the resonance ab- sorption frequency of the ion, which is slightly shifted by

the recoil term R /fi. The velocity-averaged rate of ener- gy change of an ion in the beam is

ldE\^l^ r +

\ dt Ho), I.

(fikv^+2R)exp[-(v^-o)d)^/u^]

;

T^^d^x

\ + [{2/ro)(cOi^-o}'o-kv^)Y]VTTu

(4)

where u ={2kgT^/m )'^^. Equation (4) is the same as Eq. (7) of Ref. 34, except that the center of the velocity distribution is shifted from v^=Q to cod due to the uniform rotation of the plasma. The finite width of the beam has been

neglected here, but will be included later. Below, we solve Eq. (4) for the steady state given by {dE /dt ) =0.

Equation (4) can be put into a form which is more suitable for computation by expressing the integral in terms of the

real and imaginary parts of the complex error function w(X-\-iY), which can be defined, for 7 >0, by

_,2

w{X+iY)-- -I -dt

The real and imaginary parts o{w{X-\-iY) are

(5)

+ 00

TT " -co {X-tf^Y^ Reu;(X+/T)= —

/

Imu;(A -1-/7)=— : 77 •'-00 {X-tf+Y^

dt ,

-dt .

(6)

TN-3
38 PERPENDICULAR LASER COOLING OF A ROTATING ION 5701

Tables^" and methods for the numerical approximation^^ of wiX +iY) have been published. Let the integration variable in Eq. (4) be changed to t = (v^—cod)/u, and let Y = YQ/{2ku) and X

-co'n — kcod)/(ku). Then

ldE\_ I^^qY' r +

\ dt J ijcoi^ V^ ^ -^

00 {Fik{ut+a)d) + lR]e~''

dt Y^ + (X-t)-

Replacing t in the numerator of the integrand by X — ( X — f ), we have

(7)

dE\_ ItrpkYdt I coi^Vtt _ lao^^kY

uX + cod IR

m /

+ 00 -dt-uj + 00 {X-t)e

uX + cod +

Hk

m

Y^ + (X-t)'^"' "-^-00 Y^ + {X-t?

Ktw{X+iY)-uYlmw[X+iY

dt

If the values of the rotation frequency co, the spatial offset d, and the laser frequency detuning (oi^—co'q) are fixed,

then ( dE /dt ) is a function only of u, which is a function

of T^. The steady-state value of T^ (if it exists) is found

by solving the equation (dE /dt)(T^) — for Tj_. This

equation has been investigated numerically, and solutions

have been found to exist for (od^O and (w^— a)o)<0.

The steady-state value of T^ is a function of co, d, and

Acoi =coi —(o'q. Since a and d occur only in the combination (lid, T^ is a function of only two variables, cod and

Afu^, for a given transition in a given ion. There is no

dependence of Tj^ on the intensity /, since it enters the ex- pression for ( dE /dt ) as an overall factor and thus does

not change the solution of the equation {dE /dt){T^) =0.Figure 2 shows T^ as a function of Aw^, for (y/27r = 50

kHz and for several values of d. The values of m, yg, and

(On are for the 2s 'S 1/2 to 2p^P3/2 [A. = 313 nm.

yo = (27r)19.4 MHz] transition of ^Be"^. For a given

value of Aw^, T^^ increases as d increases. Hence the

lowest temperatures are obtained by keeping d as small as

possible. The curve of Tj^ for d =0 is the same as what

would be obtained for a nonrotating plasma {(o = 0), be- cause of the way T^ depends on (od. It attains a minimum value T^»-fiYQ/2kg for Ao)^ w — 7o/2. just as predicted by theories developed for free atoms or for

0.0001

-100 -80 -60 -40

Awl /27T (MHz) FIG. 2. Graph of the steady-state value of T^ for 'Be^ [Yo=i2ir)l9A MHz, X=313 nm] as a function of the laser fre- quency detuning for several values of the spatial offset d. The

rotation frequency o)/2n is 50 kHz. This calculation neglects the finite beam radius and saturation of the optical transition.

(8)

atoms trapped in harmonic wells, where rotation does not have to be considered. ''•^'* We can give a simple explanation for the increase in T^^ as d increases. In the steady state, ^k-v, averaged over

all photon-scattering events, must be negative, in order to balance the 2R term in Eq. (2), which is positive. This is done by making A(y^<0, so that, due to the Doppler

shift, there is a preference for ions to scatter photons

when k-v<0. However, for d >0, y^^imax), the most

probable value of y^, is shifted to i;^ (max) = W > 0, so

that kv^ (max) > 0. If the width of the v^ distribution, due

to the finite value of Tj_, were narrow compared to u^(max), this would tend to make the average value of

^k-v positive. The steady state can come about only

when the velocity distribution is wide enough that there

is a significant probability for an ion to have ^k-v<0.

This corresponds to a higher value of T^^. Here we make a few remarks about the motion in the z

direction. This motion is affected by the recoil of the

photons emitted from the ions and by ion-ion collisions. We have assumed that it eventually reaches a steady

state, at some Ty, and that there is no additional source

of heating or cooling for the z motion. Then Eq. (4) contains all of the contributions to dE /dt, since all of the

recoil energy is contained in the term iik-\ + 2R, and

since the total energy of the ion plasma is conserved in collisions between the ions. Thus Eq. (4) can be used to derive the steady-state value of T^, and this value does

not depend on the final value of Ty.

B. Saturation effects

Equation (4) is valid only in the limit that the laser in- tensity / is low. To generalize it to arbitrary intensity,

the expression for the average scattering rage [Eq. (3)] is replaced by^^

Yl (9)

where y is the power-broadened linewidth, given by

r = y,x'\+2S . (10)

The saturation parameter S is related to the intensity by

TN-4
5702 ITANO, BREWER, LARSON, AND WINELAND 38

lar

HmoYo

:ii)

As stated previously, we ignore the correlations be- tween photon emissions of an ion, which result in photon

antibunching. In more sophisticated treatments of laser cooling, which have not yet been applied to the Penning

trap, these correlations lead to a modification of the

diffusion constant which enters into the equation that describes the evolution of the distribution function of the

ions.^^~^^ Javanainen^^ has shown that this correction to the diffusion constant can affect the final temperature of

laser-cooled ions in a harmonic trap by as much as 20%

at high intensities. It has no effect in the limit of low in- tensities. We do not include it here, since it would complicate the calculations without greatly increasing the ac- curacy. Using the power-broadened line shape [Eq. (9)] in place

of the low-intensity line shape [Eq. (3)], we find that the

average rate of energy change of an ion in the beam is

dE

dt

I(TqYq p+oo (itkv^+2R)exp[ — {v^—cod)^/u^]

r I :; 7=

dv^ ;i2)

Aside from an overall multiplicative factor, which does not affect the solution of the equation {dE /dt )(T^) = Q, the

only difference between Eq. (12) and Eq. (4) is the replacement of the natural radiative linewidth Yq by the powerbroadened linewidth y. Figure 3 shows T^ as a function of Aw^ for several values of 5. The other parameters are a)/27r=50 kHz and d = 50

jim. The lowest temperature for a given frequency detuning is obtained by minimizing S.

C. Finite beam geometry

Up to now, the finite radius of the laser beam has been neglected. In this section we investigate its effect on the

steady-state temperature.

The average scattering rate at each point in the plasma depends on the intensity at that point through Eq. (9). Let

n {x,y,z) be the time-averaged density (ions per unit volume) at the point {x,y,z). Then the rate of change of the total energy of the ion plasma is

dE,

dt

-) = — dx \ dy \ dz \ dv^I{x,y,z)n{x,y,z)-

I nCOr •'-00 •'-00 •'-00 •'-00

(nkv^+2R)exp[-{v^-coy)^/u^]

\ + [{2/Y)icoL~a)'o-kv,)f]VnuY^

;i3)

The linewidth 7 is a function of position through its dependence on I, and the fixed spatial offset d of Eq. (4) is replaced by the position variable;^. In the following, we assume that the laser beam is Gaussian, propagates in the x direction, and has its focus

I

in the radial iy) and axial (z) directions to be different,

since this was sometimes the case experimentally. The

intensity is then

I{y,z) = lQ exp[-2{y-d)'/w^-2z^/w^] , (14)

at {x,y,z) — {0,d,0). We allow the beam radii Wy and w^ where/ =2P /{ttw w

-60 -40 -20

Awl /2tt (MHz) FIG. 3. Graph of the steady-state value of T^ for ''Be^ as a function of the laser frequency detuning for several values of the saturation parameter S. The rotation frequency co/2n is 50 kHz

and the spatial offset is 50 ;um. This calculation neglects the

finite beam radius.

Q — .i.i / KiiLUyVu^ I and where P is the total power in the beam. We neglect the variation of w^, and w. with x, since, in our experiments, it was small over the size of the plasma. The confocal parameter b=Trw^/X, which is the distance from the focus at which the beam radius w

grows by a factor of ^2, was typically much greater than

the radius of the plasma.

In a continuum model, at low temperatures, the density of a non-neutral plasma is uniform, except for a region on the outer surface whose thickness is on the order of

the Debye length."^ For a non-neutral plasma, the Debye

length is defined by'^

1/2 A,

k,T

4iT/;o9"

(15)

where //q is the ion density and Tis the temperature. For

the plasmas studied here, we had TS2 K and

«o-2X10^ ions/cm\ so that Ao£20/im. Since the plasmas studied typically had diameters greater than 500

TN-5
38 PERPENDICULAR LASER COOLING OF A ROTATING ION . 5703

fim, the density can be considered to be uniform. The

plasma is a spheroid, whose aspect ratio can be calculated

from o).^'^ It has been shown theoretically^* and recently

confirmed experimentally^' that, at very low temperatures, ions in a Penning trap form concentric shells. However, this spatial ordering should have little net

effect on the cooling and is ignored here. We assume that

the plasma has a uniform density Hq inside a spheroid of

diameter Ir^i in the z=0 plane and axial extent 2z^^ and

is zero otherwise.

We also assume that the beamwidth is small compared

to the plasma dimensions. With these approximations,

Eq. (13) is reduced to

dt

2/ocroro«o — CO — 00 — CO

( Mu^ + 2/? ) exp[ - ( y^ - &)>; ) Vw ^

X

[\ + ['^2/r){(Oi^-oy'Q-kv^)f]VTrur^

(16)

Here, 2(r^, —d^)^^^ is the length over which the laser beam intersects the plasma. The v^ integration can be expressed

in terms of the complex error function, leaving a two-dimensional integration to be done numerically:

dt

2IoaonQViTkuYl

CO, (rli-d^)^^^ f^

'^ dy f^

^

dz exp[-2{y -d)^ /w^-2z^ /wj]

(uX + coy+Hk/m) X

uY{y,z) Rew{X+iY)-lmw(X +iY) (17)

Here, YQ = yQ/{2ku), X(y) = (coi^—co'Q — k(oy)/{ku), and

Y(y,z) = Y{y,z)/{ 2ku ). For a given set of parameters, the

equation {dE^o^/dt )(rj^) = is solved for T^. For some

ranges of parameters, no solution can be found. This

occurs when d is too small relative to Wy, so that the

beam has too much intensity on the_v <0 side of the plas- ma.

Figure 4 shows T^ as a function of Aw^ for co/2Tr = 50

kHz and for several values of d in the limit of low laser

intensity. The beam radii are given by Wy = W2 = 50 [xm.

The values of T^ calculated from the simple theory of

Sec. II A are shown as dashed lines. The effect of the

finite beamwidth is to reduce the effective value of d, re- sulting in a lower value of T^. The effect decreases as d

increases. For this example, there is a range of values of

Acoi for which no solution of the equation

{dE^^^/dt )(T^) = can be found, when d is less than or

equal to about 30 fivn. Figure 5 shows the same for a laser power of 50 juW,

which is a typical experimental value. This corresponds

to a saturation parameter at the center of the beam of

5 = 3. 85. The effect of saturation is to increase T^. This can be seen by comparing the solid curves of Figs. 4 and

5.

1 u

d = 300 /^m^,-^ /I:

1 d = 1 00 iin\^^^ 1 \

0.1 4// ——^-_^_d = 40 /xm y

-100 -80 -60 -40 -20

AojL /27T (MHz) FIG. 4. Graph of the steady-state value of T^ for '^Be''^ as a

function of the laser frequency detuning for several values of c/, in the limit of low laser intensity (solid lines). The laser beam is assumed to be Gaussian, with u;^ = id^ = 50 /xm. The rotation frequency w/Itt is 50 kHz. The results of the simple theory,

which neglects saturation of the optical transition and the finite beam radius, are shown as dashed lines.

III. EXPERIMENT

For the most part, the experimental techniques have

been described previously.'^' '^ One exception is the absolOr

^^

0.1

0.01

d = 300 /im

- d

d

=

=

40

1 00

^m

yLtm ^

—^ZIIIZIl.

^"

;

•100 -80 -60 -40 -20

Acj|_ /27T (MHz) FIG. 5. The same as Fig. 4, except that the laser power is 50

/iW. This corresponds to a saturation parameter S at the center of the beam of 3.85. TN-6
5704 ITANO, BREWER, LARSON, AND WINELAND 38

lute calibration of the laser frequency detuning from reso- nance ^o)i , which will therefore be described in some detail.The Penning trap has been described previously.'^' '^ It was referred to as trap II in Ref. 15 and has characteristic dimensions /-q = 0.417 cm and Zq =0.254 cm. The

operating parameters were 5 ~ 1.42 T [cyclotron fre- quency o)^ w(277-)2.42 MHz] and a)2«(27r)213 kHz

(ring-to-endcap voltage Fo«1.5 V). The ^Be"'" ions were

loaded into the trap by ionizing neutral Be atoms, eva- porated from a filament, with an electron beam. The

electrons and the neutral Be atoms entered the trap

through holes in the endcaps which were centered on the

trap symmetry axis. Two cw single-mode dye lasers were used in the experi- ment. Both were frequency doubled to 313 nm in a temperature phase-matched rubidium-dihydrogen-phosphate crystal. The two laser beams were directed though

the ion plasma, in opposite directions, perpendicular to the magnetic field. The beams passed through 0.25-cmdiam holes on opposite sides of the ring electrode. One of

the laser beams, with a power of approximately 50 /iW at 313 nm, was used to cool the ions. The other laser beam,

which had much lower power at 313 nm (typically less than 1 ^W), was used as a probe, inducing changes in the

313-nm resonance fluorescence light from the cooling

beam, from which the temperature and rotation frequen- cy could be derived.'^' '^ The backscattered resonance

fluorescence light in an //6 cone centered on the cooling

beam was collected by a mirror (which had a hole to al- low passage of the cooling beam) and focused by a system

of lenses onto a photomultiplier tube. The beams were displaced laterally by lenses attached

to micrometer-driven stages. The position of the cooling

beam relative to the center of the trap was determined by

monitoring the intensity transmitted through the trap as the beam was cut off by the edges of the holes in the elec- trode. The uncertainty in this determination was estimated to be about 13 fim.

The average frequency of the laser beam used for cool- ing was kept fixed to about 1 MHz by a servolock to a saturated-absorption resonance in molecular iodine. The

laser frequency was modulated over a range of about 16 MHz (at 313 nm), in order to obtain the feedback signal necessary for the lock. To simulate this frequency modulation, an additional integration over Aw^ could be per- formed. This would improve the accuracy of the calculations only for small values of Aw^.

An optical double-resonance method described in detail elsewhere'^' '^ was used to measure Tj^. The cooling

laser was tuned to the 2s^S^/2^mj = +j, mj = + j) to 2p^P^/2(nij = +j, mj = +j) transition, hereafter called the cycling transition. This yielded a steady fluorescence

signal, since this excited-state sublevel always decayed

back to the same ground-state sublevel. The weak probe

laser was tuned to drive the depopulation transition from

m , = + 4- ) sublevel to the

of

2

2'— -) sublevel, which decayed j

the 25^Si/2(m/= +

2p^Pi/2{mj = ^, mj J the time to the 25 ^5

] /2 ( m^ = + y, my = — y ) ground-state

sublevel. This sublevel did not fluoresce^ since it was far from resonance with either laser. Population in the

25 ^5'[/2(m; = +y, mj = —j) sublevel was transferred

back to the 2s^Sy/2imj = +j, mj = +j) sublevel in about 1 s by off"-resonance optical pumping from the cool- ing laser. '^''^ The probe laser resonance was observed in

TABLE I. Comparison of calculated and measured values of T^ for perpendicular laser cooling of 'Be^ in a Penning trap, for experimentally determined parameters co, d, Acoi, P, Wy, and w^.

co/lir (kHz)d

(/im) (MHz)

P

(^l'W) (jum)

T* icalc (K)T^ Imea; (K)

67.5 127 -29 30 24 24 3.0+H 1.9+0.2

99.7 127 -44 49 82 31 2.oii:? 1.6±a.3

65.7 127 -44 38 82 31 o.9ilg:3^? 1.27±0.10

35.9 165 -78 30 24 24 0.82lg:lt 0.96±0.11

36.7 165 -46 30 24 24 i.05+g:^i 0.96+0.11

36.3 165 -61 30 24 24 0.89l§:?^ 0.91 ±0.08

38.6 51 -13 30 24 24 0.62+i°47 0.69±0.07

40.0 127 -46 30 24 24 0.77lj§ 0.61+0.05

32.6 127 -80 30 24 24 OA9tro^ 0.55+0.05

35.9 127 -62 30 24 24 O.SlirA 0.47+0.05

35.0 89 -29 30 24 24 0.48l8i? 0.27+0.03

28.5 89 -63 30 24 24 0.241°°^ 0.25+0.03

28.1 89 -79 30 24 24 0.25l8:gt 0.21+0.03

31.0 89 -45 30 24 24 0.29tr^ 0.20+0.02

29.2 76 -55 49 82 31 0.087+g;gJ! 0.13+0.04

23.1 76 -55 49 82 31 O.OlStroll 0.11+0.05

21.6 51 -79 30 24 24 OAOOtmi 0.097+0.020

38.1 64 -44 38 82 31 0.044ti°^l 0.085+0.009

29.4 76 -44 49 82 31 0.084^°:°^^ 0.081+0.011

23.8 51 -46 30 24 24 0.092+g^^^ 0.080+0.020

17.3 64 -44 38 82 31 0.037l°«|? 0.043+0.040

TN-7
38 PERPENDICULAR LASER COOLING OF A ROTATING ION . 5705

the form of a decrease in the fluorescence from the cy- cHng transition as the probe laser was tuned in frequency.

The contribution to the width of this resonance from the

Doppler broadening was used to derive T^. The optical

double-resonance method has the advantage of not greatly perturbing the temperature of the plasma. The rota- tion frequency co was measured from the shift in the fre- quency of the center of the depopulation resonance, due

to the Doppler shift, for a given translation of the probe

beam in the^' direction.'''^

The absolute calibration of the cooling laser detuning

t^coi was made by the following method. The magnetic

field was fixed at a particular NMR resonance frequency,

and the cooling laser was locked to the iodine signal. The

edges of the plasma were located with the probe beam by

finding the positions at which the double-resonance signal

disappeared.'^'^ The probe beam was then positioned at the center of the plasma and tuned to the cycling transition. At the center of the plasma, d=0, so the average

Doppler shift due to rotation is zero. The increase in fluorescence as the probe laser was tuned across the cycling transition was observed while the cooling laser kept

the plasma at a constant temperature. For this measure- ment the cooling beam was chopped, and the fluorescence was detected while it was off", in order to reduce the background. The probe laser was then tuned to the frequency

at which the peak of the cycling resonance occurred.

Some of the radiation from each of the two lasers was combined on the surface of a siHcon photodiode by means

of a beamsplitter. The beat frequency was measured with

a radio-frequency spectrum analyzer. This measurement

yielded the value of Aco^ /In with an uncertainty of

about 10 MHz (at 313 nm). The value of Ato^ at other

magnetic fields could be calculated from the known Zee- man shift of the resonance line.

IV. RESULTS

Table I summarizes the results of the measurements.

The values of T^ calculated from the various experimental parameters are shown in the column labeled r^caic- They are in good agreement with the values of T^ measured from the Doppler broadening of the optical double

resonance, which are shown in the column labeled Tj^measThis agreement is maintained over a wide range of values

of d, CO, and ^co^. The range of values of T^ is from

about 0.04 K to about 2 K. All of the parameters that go

into the calculation are measured, rather than fitted. The

uncertainties in Tj^caic ^^^ due to the uncertainties in the

parameters co, d, ^coi, P, cOy, and co^. In most cases, the

uncertainty in d of about 13 fim makes the largest contribution to the uncertainty in Tj^caic- Po*" small values of

^0), , the uncertainty in Aty^ of about (27r)10 MHz

makes the largest contribution. The uncertainty in co of

about 5% and the uncertainties in P, w^, and w^ of about

20% make relatively minor contributions. Figure 6 is a

scatter plot of the measured values T^^^^^^ versus the calD

0) H

0.01

'_L calc

FIG. 6. Scatter plot of measured values of T^ (vertical axis) vs calculated values of T^ (horizontal axis). The dashed line corresponds to rx„,eas = T'lcak •

culated values T^^^.^^^.. The data points lie close to the

dashed line, which denotes the condition T'lmeas ~ ^icaic The observed values of T^ are much higher (up to three orders of magnitude) than those predicted by a theory which does not take rotation into account (the

d=0 curve in Fig. 2). In order to obtain very low temperatures, the cooling laser beam should not be perpendicular to B. Low temperatures, approaching the

T^^j^—fiyQ/lkg hmit, have been attained for ^Be"*" plas- mas in Penning traps by supplementing perpendicular

cooling with cooling by a beam which was not perpendicular to B.'^'^'^ A possible configuration for obtaining the lowest possi- ble temperatures is a combination of two laser beams, one

to cool the ions and the other to maintain the radial size of the plasma. The strong cooling beam, with a frequency detuning Acu^ = —Yq/2, would be directed along the z

axis. If the power in this beam were not too high, then

Coulomb coupling between ions would ensure ^i ~ ^11 ~^7o/2/c5. The second beam, in the x-y plane,

would be positioned to apply a torque to counteract the

radial expansion due to external torques and to photon

recoil from the cooling beam. The photon-scattering rate from this beam could be made low enough so as not to

significantly aff'ect the overall temperature.

ACKNOWLEDGMENTS

We gratefully acknowledge the support of the U.S.

Office of Naval Research and the U.S. Air Force Ofl[ice of

Scientific Research. One of us (D.J.L.) acknowledges

partial support of the National Science Foundation. We

thank J. C. Bergquist, J. J. Bollinger, and S. L. Gilbert

for carefully reading the manuscript.

TN-8
5706 ITANO, BREWER, LARSON, AND WINELAND 38

'Present address: L250, Lawrence Livermore National Laboratory, Livermore, CA 94550.

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1961); B. D. Fried and S. D. Conte, The Plasma Dispersion

Function (Academic, New York, 1961). 3^See, for example, A. K. Hui, B. H. Armstrong, and A. A. Wray, J. Quant. Spectrosc. Radiat. Transfer 19, 509 (1978); J. nvLxnMcek, ibid. 21,309(1979).

^''See, for example, W. Demtroder, in Laser Spectroscopy, Vol. 5 of Springer Series in Chemical Physics, edited by V. I. Goldanskii, R. Gomer, F. P. Schafer, and J. P. Toennies

(Springer, Bedin, 1982), p. 105. 3«D. H. E. Dubin and T. M. O'Neil, Phys. Rev. Lett. 60, 511

(1988).

-'''^S. L. Gilbert, J. J. Bollinger, and D. J. Wineland, Phys. Rev.

Lett. 60,2022(1988).

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Volume 62, Number 4 PHYSICAL REVIEW LETTERS 23 January 1989

Laser Cooling to the Zero-Point Energy of Motion

F. Diedrich/"^ J. C. Bergquist, Wayne M. Itano, and D. J. Wineland

Time and Frequency Division, National Institute ofStandards and Technology. Boulder. Colorado 80303

(Received 28 July 1988)

A single trapped "*Hg "* ion was cooled by scattering laser radiation that was tuned to the resolved lower motional sideband of the narrow ^5i/2-^£>5/2 transition. The different absorption strengths on the upper and lower sidebands after cooling indicated that the ion was in the ground state of its confining

well approximately 95% of the time.

PACS numbers: 32.80.Pj, 32.30.Jc, 35.10.-d

The subject of laser cooling of ions and neutral atoms

is currently of great experimental and theoretical interest. ' It has been applied to high-resolution spectros- copy, low-energy collisions, quantum jumps, and photon

antibunching.^ In all cooling experiments done so far, the oscillation frequency (Oo of the particle in its confining well was less than the linewidth T of the cool- ing transition. This condition also applies to free-atom

experiments, where 6)„—0. The lowest temperatures T

have been obtained in recent free-atom experiments, '•* where kinetic energies near or below that corresponding

to the recoil of a single photon from an atom at rest have

been achieved (ra* 1 //K). In this Letter we report, for the first time, laser cooling of a single bound atom in the

resolved sideband regime r^(Uo. An ion has been

cooled so that it occupies the ground state of its confining potential most of the time.

The idea of laser cooling in the resolved sideband re- gime is as follows^: Let the rest frequency of the atom's

cooling transition be o>o. If the atom oscillates at fre- quency Q)v in its confining well, the atom's absorption

and emission spectrum (as viewed in the laboratory) has

resolved components at cdq and cdq ± mo)u int an integer).

If we irradiate the atom with narrow-band radiation

tuned to the first lower sideband at (Oo — q)v, the atom absorbs photons of frequency h (too ~ <»» ) and reemits photons of average energy hmo. Hence, on the average, each scattered photon reduces the atom's vibrational en- ergy by hcoc, or reduces the atom's vibrational quantum

number n,, by 1. In this way, we can obtain (/ii,)<K 1 and

have the atom most of its time in the ground-state level of its confining potential.*"* When <«„>«!, T is no

longer proportional to </»„) but depends logarithmically'* on (rtp). The technique of sideband cooling has previously been applied to cool the magnetron motion of trapped

electrons with an rf electronic excitation of the axial motion that was coupled to a cooled resistor.' The final temperature in this experiment was limited by thermal

excitation and the energy of the magnetron motion cor- responded to (n„)»l. In the experiments described

here, we achieve </i„>«l by optical sideband cooling.

For our value of a>„, T was about 50 /iK. However, to the extent that the particle is in the ground state of its

confining potential (about 95% of the time here) the fundamental limit of laser cooling for a confined particle has

been reached.

Our experiments were performed with a single "'Hg"*"

ion stored in a Paul (rf) trap'®" which had a)„/2;r"2.96

MHz (see Fig. 1). In order to optimize the cooling, a

two-stage process* was used. First, the ion was cooled to near the Doppler cooling limit' {T^hT/lkB, where kg

is the Boltzmann constant) by scattering light of wavelength 194 nm on the strong ^Sm-^Pin transition 1(A)

in Fig. l(a)l.'^ At the Doppler cooling limit, <«„)« 12

(r=1.7 mK) for each degree of freedom.* In the next

stage of cooling, the 194-nm radiation was turned off and the narrow ^Sm-^D5/2 electric quadrupole transition [(B) in Fig. 1 (a)] was driven on the first lower sideband

frequency oao — cov. Since the natural lifetime of the

^Z>s/2 state limits the maximum scatter rate to approximately yr(^Z)5/2)as6 photons/s,'°'^ a cooling time of

at least 6 s is required to reach </i(,)— for all degrees of

freedom. This time becomes even longer, or cooling is prevented, if external heating is present. Therefore, in order to enhance the sideband cooling rate, the lifetime of the ^Z>5/2 state was shortened by coupling it to the fast decaying ^P^n state by 398-nm radiation [(C) in Fig.

1(a)]. From the ^P^n state, the ion has high probability

398 nm

281.5 nm-IDPaul Trap

194 nm

FIG. 1. (a) A simplified energy level scheme of Hg 11 showing the optical transitions involved in our sideband cooling ex- periment, (b) The geometrical arrangement of the laser beams. The 194-nm fluorescence is detected normal to the plane of the figure. Mirror M reflects 194-nm radiation while

transmitting 398-nm radiation.

403

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Volume 62, Number 4 PHYSICAL REVIEW LETTERS 23 January 1989

of rapidly decaying to the ground state. When <rti,><K 1, a quantitative measurement of </ii,) from the absorption

spectrum becomes very simple. The strength of absorption Sl on the lower sideband is proportional to (/ii,), while the strength Su of the upper sideband is proportional to (/!,,> -1-1.^ When (/it,) approaches zero, the lower

sideband disappears because no more vibrational quanta

can be extracted from the ion. If the sideband absorption spectrum is probed with saturating power, the ratio of the strengths of lower to upper sidebands becomes in- dependent of power'"* and directly gives («j,). To simplify our discussion, we have assumed the trap

states and associated wave functions are those of a harmonic potential which is equal to the classical pseudopotential of the rf trap. In the quantum treatment of the rf trap,'^'^ the relevant states are not energy eigenstates

because of the time dependence of the potential. However, when the trap drive frequency ft »(Ui,, the atom's optical spectrum and transition matrix elements relevant

for cooling closely approximate those for a harmonic potential equal to the classical pseudopotential. The states which represent the cooled ion look like harmonicoscillator states whose dimensions oscillate with small

amplitude at frequency ft. These states are of the form

exp[ — /g)i,(«+ 2" )/]/n(x,/) where the /„ are periodic in time with period 2;r/ft.

'^ Our trap'"" (ro=466 /im, zq— 330 /im) was operated at a trapping field frequency ft/2;r="23.189 MHz.

With an rf peak voltage amplitude Ko« 1.2 kV and a

static potential (/o"" + 71.4 V applied to the ring elec- trode, the trap potential was approximately spherical. In order to cool all motional degrees of freedom to near the

Doppler cooling limit for the (.A) transition, two orthogonal beams of 194-nm radiation, both at an angle of 55°

with respect to the trap symmetry (z) axis, were used

(Fig. 1). The radiation to drive the iB) transition was

derived from a frequency stabilized dye laser (A. "563

nm) with a linewidth less than 20 kHz. The output radi- ation from this narrow-band laser was frequency doubled

and focused to as much as 25 W/cm^ at the position of

the ion; this allows strong saturation on the cooling transition. The radiation to drive the (C) transition was derived from a frequency stabilized LD 700 dye laser whose output radiation was frequency doubled and fo- cused to give approximately 1 mW/cm^ at the position of

the ion. Before the sideband cooling experiment was started, an absorption spectrum of the iB) transition was taken '^

to determine the carrier frequency as well as the side- band frequencies (inset Fig. 2). We made sure that the 282-nm source had equal power at both the upper and

lower sideband frequencies. For the sideband cooling

and the probing of the absorption spectrum, the follow- ing computer-controlled sequence was run repeatedly.

First, the 398- and 194-nm radiation were turned on

simultaneously for a 20-ms interrogation period. If the 194-nm fluorescence exceeded a preset value during this 404

100 F—'—

r

t 50 Sl Su —»A \

J 1 i__j. -3.0 -2.9 2.9 3.0 Frequency Detuning (MHz)

FIG. 2. Absorption spectrum of the ^S\n-^Dsn electric quadruple transition of "^Hg"*". The inset spectrum was taken

before sideband cooling was applied. It shows the carrier at zero detuning (frequency (uo) and the first sidebands (at fre- quencies (uo~<Up and tuo+Wo) generated by the ion's motion in the approximately spherical well. For this spectrum, the bandwidth of the 282-nm radiation was broadened to 120 kHz to reduce the number of required data points and the laser power was reduced in order to avoid saturation. The enlarged part of

the figure shows the absorption strength Sl (Su) on the lower

(upper) motional sideband 1 5 ms after the end of the sideband

cooling. Values for Sl and Su were obtained from Gaussian

fits to the data points which are averaged over 4 1 sweeps.

period, it could be assumed that the ion was laser cooled

and cycling between the ^5 1/2 and the ^P\/2 states. This

20-ms interrogation period was repeated until this condition was satisfied. Then the 194-nm radiation was

switched off and the 282-nm radiation, tuned to the first lower sideband at coq — oji,, was switched on for a cooling

time Tc (typically 200-500 ms). After the 282-nm radi- ation was switched off, the 398-nm radiation was kept on

for a relaxation time t, (typically 5 ms) in order to empty the ^D5/2 state. After this, the cooled ion was in the

electronic ground state and the probing of the absorption

spectrum was done as follows: The 282-nm source was

switched on at saturating intensity for 10 ms at a fre- quency corresponding to one point near the upper or lower sideband frequency. After this, the 282-nm beam

was switched off and the 194-nm radiation was switched on to see if the ion had made the transition to the ^Ds/2

state.

'

' The result was averaged with the results of pre- vious measurements at the same probe frequency. The

frequency of the 282-nm source was stepped to the next value and the cooling and probing cycle was repeated until about 40 cycles for each value of the probe frequency were completed. The results of a typical run are shown

in Fig. 2. In order to deduce (/ly) for the different motional de- grees of freedom, the geometry of our experiment (Fig.

1) has to be considered. The 282-nm beam enters the

trap at an angle of 55° with respect to the z axis. The x

and y directions were previously determined by the fixed

spatial alignment of two simultaneously stored ions,

'^ which we take to be along the x axis. From these data, the squares of the projections pi of unit vectors along the

TN-11
Volume 62, Number 4 PHYSICAL REVIEW LETTERS 23 January 1989

trap axes onto the 282-nm beam axis are calculated to be

pi""0.03, py'~0.64, and pi '^0.33. Since we make the

differences between the x, y, and z frequencies bigger

than I/tc, all directions are cooled simultaneously.^

However, in the analysis, we assume that the probing absorption strength is due only to the ion's motion in the y

and z directions. Since the x axis is nearly perpendicular

to the 282-nm beam, no meaningful statement abut the

energy in this degree of freedom can be made. By

neglecting the contribution of the x motion to the side- band strength we overestimate (^i,) for the y and z directions. In order to deduce inj for the y and z oscillations from our data (Fig. 2), an assumption about the energy

distribution between the two directions has to be made.

If we assume temperature equilibrium between the y and

z degrees of freedom, both contain an energy corresponding to (/i„)-(l-5L/St/)"'/2- 1-0.051 ±0.01 2 quanta. Therefore, for the y and z degrees of freedom, the

ion is in the «i, "0 state 95% of the time. The corre- sponding temperature given by* kaT^hMv/lnil + l/

iriv)) is r—47±3 /iK. For any other energy partition,

(/ii,) and T for one degree of freedom would be less than

these values. Independent of the energy distribution, for both degrees of freedom the temperature is much lower

than the 194-nm Doppler cooling limit and the ion

spends most of its time in the harmonic-oscillator

ground-state level. The theoretical sideband cooling limit* gives a value of

(/!„)« 10 "^ However, since the probing in the experiment is done at saturating power, the measured (n„) cor- responds to the energy of the ion at the end of the probing interval, which is typically 1 5 ms after the end of the

sideband cooling, and external heating might have oc- curred. In order to check for external heating processes, we extended t, up to 100 ms and measured SjSv as a

function of Xr. We determined a heating rate (due apparently to pickup of stray noise fields at radio frequencies) of (/ii,)«6/s. If we assume the heating is due to thermalization of the ion to room temperature by noise

at frequency (t)^,, the heating time constant is 95 h.

'* This rate varied slightly with w^ around Wi, "3 MHz,

but was substantially higher for (0^^2.5 MHz. From

these heating data, the measured <«„) is consistent with

the theoretical limit at the end of the sideband cooling

period. We also measured </!„> for a single degree of

freedom directly by changing Uq to —25 V in order to

split the radial and axial sideband frequencies. The

282-nm radiation was tuned so that only the first axial sideband at (Ui,/2;r"4.66 MHz was cooled and probed.

From the results, shown in Fig. 3, we calculate iriv) "0.049 ±0.045 at the end of the cooling period, consistent with the theoretical cooling limit. The

confinement of the axial motion is given by the spread of

the zero-point wave function z(rmi>) «= 2.4 nm.

For our data, the uncertainty in the second-order

Doppler shift is dominated by the uncertainty in («„) and

cV

i..J

_

1

1

1

1

1

1

1—1—1—1—

~

1

[ _

1.0 - HH

y

/-

: /

Y

~

0.5 - y\ -

~ yX. "

-

1 1 1

1 1 1 1 > 1 I 1 50 100

Delay Time (ms)

150

FIG. 3. Vibrational quantum number </»,,) for the axial motion (ft)i,/2;r""4.66 MHz) as a function of time delay be- tween the end of the sideband cooling and probing. A linear extrapolation of the data points (circles) to zero delay time

yields <«»> (triangle) consistent with the theoretical expectation.

amounts to Av/v< 10"^** (Ref. 8). It can be made substantially lower by adiabatically lowering the potential well depth after the ion is cooled into the ground state. With our experiment, the absorption of a single quantum

of energy at a (tunable) frequency in the MHz range can be detected with an efficiency of nearly 100%. With

appropriate coupling to the ion's motion (for example,

via one of the endcaps), a similar apparatus could serve

as a very sensitive spectrum analyzer. In another possi- ble application, the motion of a trapped charged particle

could be damped by coupling it electronically'' to a second laser-cooled ion in a separate trap, thereby reducing the first charged particle's kinetic energy to near the

zero-point energy. Resonant excitation of the first particle's motion could then be detected very sensitively

by its influence on the laser-cooled ion. Such a device

might be useful in mass spectroscopy.

In summary, we have realized laser cooling in the

resolved sideband regime for the first time. The kinetic energy of a trapped atomic ion was reduced to a value

where it spent most of its time in the ground-state level of its confining well. To the extent that the ion is in the

zero-point energy state of motion, this realizes for the

first time the fundamental limit of laser cooling for a bound particle and the ideal of an isolated atomic particle at rest to within the quantum-mechanical limits imposed by the surrounding apparatus.

We acknowledge the support of the U.S. Air Force

Office of Scientific Research and the U.S. Oflfice of Naval Research. F. D. thanks the Deutsche Forschungsgemeinschaft for financial support. We thank C. Wieman,

J. Bollinger, and J. Helffrich for comments and suggestions on the manuscript. We thank R. Cook and R.

TN-12

405
Volume 62, Number 4 PHYSICAL REVIEW LETTERS 23 January 1989

Bliimel for helpful discussions on the quantum-mechanical Paul trap.

^^^Present address: Max-Planck-Institut fiir Quantenoptik,

Garching, West Germany.

'See, for example, S. Stenholm, Rev. Mod. Phys. 58, 699

(1986); D. J. Wineland and W. M. Itano, Phys. Today 40, No.

6, 34 (1987), and references therein. ^See, for example. Laser Spectroscopy VII, edited by T. W.

Haench and Y. R. Shen, Springer Series in Optical Sciences

Vol. 49 (Springer-Verlag, Berlin, 1987); "Atomic Physics 11,"

edited by S. Haroche, J. C. Gay, and G. Gynberg (World

Scientific, Singapore, to be published).

3p. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P.

L. Gould, and H. J. Metcalf, Phys. Rev. Lett. 61, 169 (1988). ^A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and

C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988).

^D. J. Wineland and H. Dehmelt, Bull. Am. Phys. Soc. 20, 637 (1975).

^W. Neuhauser, M. Hohenstatt, P. Toschek, and H.

Dehmelt, Phys. Rev. Lett. 41, 233 (1978). 'D. J. Wineland and W. M. Itano, Phys. Rev. A 20, 1521

(1979). *D. J. Wineland, W. M. Itano, J. C. Bergquist, and R. G.

Hulet, Phys. Rev. A 36, 2220 (1987). 'R. S. Van Dyck, Jr., P. B. Schwinberg, and H. G. Dehmelt,

in New Frontiers in High-Energy Physics, edited by B. M.

Kursunoglu, A. Perlmutter, and L. F. Scott (Plenum, New

York, 1978), p. 159.

"'J. C. Bergquist, D. J. Wineland, W. M. Itano, H. Hemmati,

H. U. Daniel, and G. Leuchs, Phys. Rev. Lett. 55, 1567

(1985). "J. C. Bergquist, W. M. Itano, and D. J. Wineland, Phys. Rev. A 36, 428 (1987).

'^H. Hemmati, J. C. Bergquist, and W. M. Itano, Opt. Lett.

8, 73 (1983).

'^W. M. Itano, J. C. Bergquist, R. G. Hulet, and D. J. Wineland, Phys. Rev. Lett. 59, 2732 (1987).

'''Ref. 8 is in error on this effect of saturation. '5R. J. Cook, D. G. Shankland, and A. L. Wells, Phys. Rev. A 31, 564(1985).

'^M. Combescure, Ann. Inst. Henri Poincare 44, 293 (1986).

''D. J. Wineland, J. C. Bergquist, W. M. Itano, J. J. Bollinger, and C. H. Manney, Phys. Rev. Lett. 59, 2935 (1987). '^The quantity actually measured is the probability, after a time Tr, of finding the ion in a state with quantum number n^, where 1 <«„<«* = 10*. Strong, discontinuous heating, such

as collisions with neutral atoms, would result in n„ » 1 after each heating event. Hence, a measurement of Sl/Su vs r, would yield a heating rate lower than the actual one. In contrast, a continuous heating process, such as rf noise, would

cause inv) to increase smoothly with time, and the analysis should be valid. "D. J. Wineland and H. G. Dehmelt, J. Appl. Phys. 46, 919

(1975).

406

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PHYSICAL REVIEW A VOLUME 42, NUMBER 5 1 SEPTEMBER 1990

Quantum-limited cooling and detection of radio-frequency oscillations by laser-cooled ions

D. J. Heinzen and D. J. Wineland

Time and Frequency Division, National Institute ofStandards and Technology, Boulder, Colorado 80303

(Received 13 March 1990)

A single trapped ion, laser cooled into its quantum ground state of motion, may be used as a very-low-temperature detector of radio-frequency signals applied to the trap end caps. If the signal source is a resonant oscillator of sufficiently high Q, the source may also be placed in its quantum

ground state by coupling to the ion. Parametric couplings may be used to cool and detect source modes other than the mode directly coupled to the ion. A theoretical analysis of these cooling and

detection processes is presented, and as an example, their application to single trapped electron and

proton spectroscopy is examined. Squeezing and low noise detection of one quadrature component

of the source oscillation are also discussed. The techniques discussed here may lead to radio- frequency measurements of improved accuracy and sensitivity. Cooling and detection of vibrations of macroscopic oscillators also appear possible.

I. INTRODUCTION

Frequently, measurements depend on the detection of

weak signals at radio frequencies. An example to be dis- cussed in this paper is the detection of rf currents in pickup electrodes induced by charged-particle motion. Weak

signals must first be amplified to detectable levels. This

amplification may be linear, in which case the fundamental limit to signal detection is set by zero-point fluctuations.'"^ Alternatively, signals may be amplified by

"quantum multiplication," in which an absorbed quantum from the signal-carrying field generates a detectable

number of secondary quanta, such as occurs in a photomultiplier tube. In measurements using quantum multiplication,^"^ there is no fundamental lower limit on

noise power. However, the signal acquires shot noise

since the absorbed energy is quantized; moreover, phase

information is lost. At radio frequencies, sensitivity at the quantum level cannot be achieved with present techniques. That is, linear rf amplifiers add noise far in excess of the

quantum-mechanical minimum;'"^ also, "quantum multipliers" capable of detecting single rf quanta cannot be

constructed with conventional techniques. Essentially,

this is due to the smallness of Hco/kg (48 fiK at a>/2-rr= 1 MHz), and to coupling of dissipative elements in the

amplifier to thermal reservoirs of temperature

T»iia>/kg, which introduces excess thermal noise. Here, l-rrir is Planck's constant, co is the signal-carrier fre- quency, and kg is Bokzmann's constant. The best results have been obtained with a superconducting quantum in- terference device^' (SQUID) and field-effect- transistor'"'" (FET) amphfiers, both of which exhibit

effective noise temperatures'"^ T^ff>0.l K»iico/kg,

that preclude quantum-limited sensitivity. In order to improve the performance of rf amplifiers,

circuit elements with much lower dissipation and noise

temperature are required. Such low temperature and dis- sipation may be achieved in the motion of laser-cooled

42

trapped ions. Recently, a single Hg"*" ion harmonically

bound in an rf Paul trap'^'^ has been laser cooled to an

extent where it spends most of its time in its zero-point

(ground) state of motion.'"* (Although this cooling was

demonstrated only for two degrees of freedom of the ion's motion, it can straightforwardly be extended to all three

degrees of freedom.) As a by-product of this experiment,

single rf quanta absorbed by the ion could be detected

with high efficiency. In this paper, we further examine

how such a laser-cooled ion can function as a sensitive detector of rf signals applied to the trap electrodes. We

examine detection based on both Hnear amplification and

quantum multiphcation. For ions which are cooled to temperatures less than fico/kg, sensitivity at the quantum

level may be achieved.

For such a detector to be useful, the signal source must

also have a low noise temperature. Most rf sources do

not satisfy this condition, but a source consisting of a res- onant oscillator mode of very high Q can be cooled to a very low temperature by coupling to the ion. Examples

of such resonant source modes might include modes of a

bulk wave resonator such as a piezoelectric crystal, elec- tromagnetic cavity or tuned circuit modes, or an ion

confined in another trap. We also show that parametric

couplings may be used to cool and detect source modes

other than the mode directly coupled to the ion. This

may be useful if the desired source mode is not easily coupled out electrically or has a much higher or lower fre- quency than is convenient to match to the ion's frequency.Since most rf signals are contaminated by noise far in excess of the quantum-mechanical minimum, we envision

the primary applications of these ideas to be the cooling

and detection of small excitations of such high-g oscillators. There are at least two situations in which this is useful. First, the oscillator may be driven by a classical

field that is strong (containing many quanta per mode)

but couples very weakly to the oscillator. Such couplings

might include gravity waves,'^"'^ or other weak

forces. '^'^ Second, it may be that a precise measurement

rpvr_ 1 / Work of the U. S. Government

2977 Not subject to U. S. copyright
2978 D. J. HEINZEN AND D. J. WINELAND 42

of the source oscillator's frequency is desired, and that

large amplitudes are undesirable because of anharmonic

frequency shifts. When the source mode is the harmonic oscillation of an ion or ions in a separate trap, this technique of "coupled traps" can provide another means to extend laser cooling to ions which cannot be directly laser cooled. Although sympathetic laser cooling'^ has been successfully

applied to ions of the same charge in a Penning trap'^'^^

and to small numbers of ions in a Paul trap,^''^^ the coupled trap configuration has the potential advantage that the coupling between different ion species can be easily

turned on and off (by changing the resonant mode fre- quencies) and therefore perturbations between ion species can be avoided.

An interesting application of these ideas may be the

coohng and detection of excitations of a single electron or proton (or their antiparticles) confined in an ion trap. We

show that these particles can be placed in their quantum-mechanical ground state, and single-quantum

excitations to any of the particle's degrees of freedom

detected. In addition to the aesthetic appeal of finding

such a particle in its ground state, this dramatically

reduces perturbations to the particle's frequencies of

motion. ^^"^^ This has important implications for the

measurement of the electron g factor,^^'^^ electron- positron g-factor ratio^^ and mass ratio, ^* protonantiproton mass ratio, ^^ and mass ratios of other ions.^^'^^ In addition, a high-precision determination of the proton

g factor (and proton-antiproton g-factor ratio) should be

possible with use of this technique; this measurement is very difficult with already proven techniques.

It should also be possible to parametrically drive the

ion's motion, resulting in noise-free linear amplification"*

of one of its quadrature amplitudes. Thus, if a signal is present in this quadrature, it can be detected with no

added noise. In addition, it should be possible to prepare

the laser-cooled ion in a squeezed state'*'^'^"^^ by a nonadiabatic change in the ion trap potential or by a parametric

drive. A source consisting of a high-^ oscillator mode

may then also be prepared in a squeezed state by coupling

to the ion. Thus one of the source mode's quadrature

ampUtudes may be detected with extremely low noise,

limited only by the initial degree of squeezing of the ion. Because this system is conceptually simple, it could

serve as a useful paradigm for quantum measurements.

Certain concepts, such as squeezing, can be readily visualized. As a practical matter, we think that these ideas

will enable rf measurements to be made with improved

accuracy and sensitivity.

This paper is organized as follows. In Sec. II, we begin

by discussing signal detection by a single ion, in the

simplified case in which the ion trap is driven by a volt- age source with added thermal noise. We discuss the

detection of a single rf quantum by laser sideband cooling

and detection of the ion's motion, and derive expressions

for the number of signal-induced and thermally induced

quanta. In Sec. Ill, we extend these results to the case in which the signal source is a high-Q oscillator, showing

that cooling and detection at nearly the quantum limit may be achieved by coupling to the ion. Cooling and

detection of source modes other than the one directly

coupled to the ion via parametric couplings is discussed

in Sec. IV. The technique of "coupled-trap spectroscopy," in which the laser-cooled ion is used to cool and

detect excitations of a charged particle in a second trap,

is introduced in Sec. V. Linear amplification and squeezing of the ion or source oscillator's motion is discussed in Sec. VI. We conclude in Sec. VII.

In Appendices A-C, we further examine the experimental possibilities. A technique for single-quantum

detection using Raman transitions is discussed in Appendix A. This technique allows the laser sideband cooling

and detection method to be extended to low-mass ions

such as ^Be"*". This increases the coupling between the source oscillator and the ion, since low-mass ions couple more strongly than high-mass ions. In Appendix B, we

analyze an experiment to cool and detect excitations of an electron in a Penning trap with the coupled-trap

method. Finally, in Appendix C we examine continuous

cooling and the possibility of cooling a mode of a quartzcrystal oscillator.

II. SENSITIVE SIGNAL DETECTION

BY A SINGLE ION

We first consider the simplified case in which the ion is driven by a source with purely resistive impedance, as il- lustrated in Fig. 1. A single ion of charge 9, and mass m,

is confined in a rf trap or Penning trap'^'^'^^'^^ of endcap separation c?,. For brevity, we consider the detection

of a signal by its influence on the ion's motion along the

trap symmetry (z) axis. Similar considerations would apply for detection with motion in the x-y plane using a

split-ring electrode. The trapping potential along the z axis is assumed to be harmonic and results in an ion axial

oscillation frequency cj,. We model the source as a volt- age source u^, with dissipation in the source represented

by a series resistance R^ at temperature T^. Associated

with the source resistance is a noise voltage source u„^.

We also assume that the amplitude and phase of the source u^ is classically well defined, but that the source

excites the ion only weakly. This will be a valid assumption if «j is applied for a very short time or is derived

(3) endcap

ion

Ci:> endcap

Ct

FIG. 1. (a) Single trapped ion driven by a signal source at- tached to the trap end caps. For simplicity the ring electrode of

the ion trap is not shown, (b) Electrical equivalent circuit of the arrangement in (a), where the trapped ion is represented as a

series /,c, circuit in parallel with the end cap-to-end cap ca- pacitance C, (Ref. 34). TN-15
42 QUANTUM-LIMITED COOLING AND DETECTION OF RADIO- 2979

from a classically strong source which couples very weakly to the ion (due to shielding, for example).

The ion's charge is coupled to the electrical circuit of

Fig. 1(a) via the image charges it generates in the end

caps. This induced charge is given by^^'^'*

. ind- (1)

where z, is the axial displacement of the ion from its equilibrium position, a, is a geometrical factor of order unity,

and we have assumed the rf wavelength is much larger

than the trap dimension. (A voltage V applied across the

trap end caps generates an electric field ajV/dj at the

ion. For traps with hyperbohc-shaped electrodes

a, =^0.8.^^) Because the ion is harmonically bound, it can

be shown that the ion and its image current are electrically equivalent to a series /,c, circuit which shunts the end

caps. 3* with

/,=m,(£f,/a,9, (2a)

(2b)

Additional forces on the ion from the induced charge qj"'^ are assumed to be neghgible.^'* The circuit of Fig. 1(a) is therefore electrically equivalent to that shown in Fig.

Kb), where Cj- is the capacitance between the trap end

caps. The effect of the trap capacitance can be neglected

if R^ «(cy,C7-)~'; we assume that this is the case in the

remainder of this section. Neglecting the trap capacitance Cj, the circuit of Fig.

Kb) is a driven simple harmonic oscillator, with the resis- tance R^ and noise source «„j representing the coupling

of the circuit to some thermal reservoir. The behavior of

such a damped oscillator is described quantum mechanically by^^

~dt'

dt

iO): + il S,+/„(r)+/,(r),

ICO: 2 sj+/l(r)+/;(r

(3a)

(3b)

where S ] and S, are the creation and annihilation operators for a quantum of oscillation; they are related to the

charge ^

-""^ on the capacitor c, by

n)

ind loi.l,

1/2 (S +s,- (4)

Here y = R^/lj is the damping rate for the oscillator, and

/„ ( ? ) and / „ ( f ) are noise operators, which satisfy^*

(fjt)) = {fl(t))=0,

^r,^t,)f„(t,])=Ynb(t,-t^),

(5)

(6a)

(6b)

where the brackets denote an average over the thermal

reservoir states, and

« =

^f>0>,/kgT,_^

(7)

is the number of quanta in the circuit in thermal equilibrium. The effect of the noise voltage source «„j is included via the terms f„{t) and f„(t). The drive is included

via the term fs(t), which is assumed to be classical. For the case of a resonant drive of the form u^ — ieo

= Re(Mjoe ),

/,=/u,o^~'"''(8^a,,/,)-•/^

From Eq. (3a), we find that

+/;[/„(M+/.(^,)>"'"'''''''""'''^r., (8)

and the corresponding conjugate equation for 2 . For simplicity, we assume that u^ is a resonant pulse of

duration t^ . We find that

%(t^) = n,{Q)e

-Uoi,+y/2U„

+ />"< t,)e

-(ia.,.+r/2)(r„-f,)

dt,

+

iu,Qe ""(1-e '")

2/ \\/2 ilfioijHi)

(9)

From Eq. (9) we see that the oscillator's amplitude contains a term proportional to the drive amplitude u^q. This amplitude contribution may be detected directly by

a linear parametric amplification technique discussed in Sec. VI.

Alternatively, we may detect the absorbed vibrational

quanta. We find that the mean number of quanta

( f?, ) = { 3 |3, ) in the circuit at time t^ is given by

(/j,(r^)) = <^,.(0)>e

-r'm _i_ + IT«(l-e -r'„

+

'jOI

UrcOiY^li-d-e -r'„/2 )^ (lOa)

where we have neglected possible coherence of the initial state; that is, we assume that < S(0) ) = < S ^(0) ) =0. The

three terms on the right-hand side of Eq. (10a) describe

the decay of initial quanta stored in the circuit, the buildup of "blackbody" quanta in the circuit, and the buildup

of "signal" quanta. For t^ «7~',

(/r,.(r^)>^<^,(0))+r«r^ + - 8^a>,/, (10b)

Single-quantum excitations in the circuit representing

the ion's motion may be detected via the method outlined

in Refs. 14 and 37 and as illustrated in Fig. 2. To summarize the method, we first assume that the ion's internal structure has a ground state \g ) which is weakly coupled

to an excited state \w), which slowly decays to \g) at a

rate yw- Also, \g ) is strongly coupled to an excited state

\s ), which rapidly decays to \g) at a rate Ys »7u)- ^^

assume that y w «^/> so that the combined internal- translational quantum states of the ion for levels \g ) and

\w ) consist of well-resolved ladders of states |g,n, ) and

liD.n,), with «,= 0,1,2,... . We also assume that R «fi(Oj, where R=(-hk )^/2m is the recoil energy, with

TN-16
2980 D. J. HEINZEN AND D. J. WINELAND 42

FIG. 2. Sideband cooling and detection of a single trapped

ion. The sideband cooling laser (1) is tuned to the first lower

sideband of the \g)—^\w) transition and primarily induces

transitions of the type \g,n,)^\w,n,~l). The detection laser

(3) primarily induces transitions of the form |g,n, )—>-|s,n, ) Observation of the absence of fluorescence from this transition in- dicates that the ion is in the \w) state. (Dehmelt's electron shelving scheme, Ref. 7.)

k the wave number of the weak transition. The ion is cooled by tuning a laser to the first lower sideband at (0,„„—c0j, where a)„,„ is the \g)^>-\w) transition freqiien- wg cy of an atom held at rest. This induces A«, = — 1 transi- tions in the ion, with spontaneous decays back to level Ig) occurring predominantly with A«,=0. The ion is therefore optically pumped down the ladder of states until it reaches «,=0 with high probability, at which point

it interacts very weakly with the laser. ^^~^^ A more detailed analysis of the sideband cooling process leads to a

limit on {nj{0))=fyl,/(oj, where / is a constant of or- der unity. ""^^ Single-quantum excitations from |g,«,=0) to

Ig, «, = 1 ) may be detected by first applying a v pulse"^" of

radiation to the |g,«, = l ) to |u;,tt,=0) transition. If the

ion was initially in the |g,n, = l) state, it is therefore

transferred to the excited state \w), where it remains for a time of order y~K If it was initially in the |g,«,=0)

state, it remains in the state \g ).

''''^^ Then a strong pulse

of radiation is applied on the \g )-<r^\s ) transition of duration t^ such that Y^^ «tj «y~^. During this time,

many photons are scattered and detected if the ion was

initially in |g,«,=0). If it was initially in the state

|g,«, = l), it remains in the state |u;) during the time tj and scatters no photons. High sensitivity of detection of

the \w ) state is achieved because of the absence of many

scattered photons during the time t^. This is an example

of the quantum multiplication provided by "electron

shelving" of the ion.

^"^ '"'*'* This sideband cooling and

detection scheme has recently been demonstrated experimentally.''* (The work of Ref. 14 differs slightly in that the 77 pulses on the |g, «, = l) to |u;,/i,=0) transition were not used.)

For this single-quantum detection method to be useful,

it is necessary to avoid saturating the detector with

thermally induced quanta. Since typically « » 1, this implies that we must have yt^ < \ /n « 1. In this case, the signal-to-background ratio is

</t,(0))+«jh

(11)

12,2 where «j = IujqI 'm /8^w,/, and nf^^ = ynt„. The noise

will be due to fluctuations (shot noise) in the number of

either the signal or background quanta. For example, as- sume that (fi, iO)), n^^^«n^ «l. Then the noise is dominated by the shot noise of the signal. After A'^ measure- ment cycles, on the average we detect Nn^ excitations to

«, = 1, where the rms fluctuations in this number are given by [Nn,(\-n,)y^^^{Nn^)^^^. Therefore the

signal-to-noise ratio after A'^ measurements is (S/N)!^^(Nn^)^''^. We note that this shot-noise limita- tion may be overcome by using the parametric

amplification techniques discussed in Sec. VI.

If n^^^{t^)>\ or n^{t^)>\, it may be desirable to reduce the interaction time to a time t^, such that

rt,h(^'„ ) « 1 and rtj(r^ ) < 1, although this reduces the ratio «i/«th ^y the factor t'^ /t^. Alternatively, it may be

desirable to consider methods to detect final states with

{«,(r^))»l. One method would be to monitor a de- crease in fluorescence from the ion due to increased

Doppler broadening or to reduced spatial overlap with

the exciting laser beam.^"* Although for simplicity we have considered the ion to be directly driven by a signal source, similar considerations hold when a separate high-^ oscillator is driven by

an oscillatory signal and the resulting excitation

transferred to the ion for detection, as will be discussed in Sec. III. In this case, the buildup of excitation in the

separate oscillator is also described by Eqs. (3) -(10),

where y and n now refer to the damping rate and equilibrium number of thermal quanta of the oscillator, and we

replace /, by the oscillator's eff"ective inductance /j. The

oscillating drive need not be electromagnetic; the example of a piezoelectric crystal is discussed in Appendix C. The use of such high-^ oscillators as sensitive detectors

of oscillating forces has been discussed in Refs. 15-18.

III. REFRIGERATION AND SENSITIVE DETECTION

OF EXCITATION IN A SECOND RESONANT SYSTEM

In this section, we consider a source consisting of a sin- gle oscillator mode of sufficiently high Q, such that damping into external reservoirs may be neglected over

times of interest. In this case, the source and laser-cooled ion behave simply as a pair of coupled oscillators, and, in the weak damping limit considered here, oscillatory ex- change of energy between the two oscillators may be ex- pected. Under these conditions, the ion itself may be

used to both cool and detect excitations in the source. This process may be modeled by the equivalent circuit shown in Fig. 3. As before, /, and c, are the effective in- ductance and capacitance of the trapped ion. The source

oscillator may be modeled by a series /^c, circuit, with

resonant frequency w^ =(/^cJ~'^". Any parallel source

reactance may be included in the capacitance C, . In gen- TN-17
42 QUANTUM-LIMITED COOLING AND DETECTION OF RADIO- . . 2981

=FCt

FIG. 3. Equivalent circuit for a single trapped ion coupled to a particular mode of a source oscillator represented by a series

era], the source oscillator will in turn be driven by a sig- nal represented here by the voltage source u^; for the moment, we assume that u^ =0.

The Hamiltonian for the circuit of Fig. 3 may be writ- ten as

where

(12)

(13)

Cj = Cj Cj /( Cj -\-Cj), (x)'j = ( IJ cj ) ~

' ''^, with j =s or i, and it

is assumed that the total charge in the circuit is zero. As

usual, we define operators Pj and Xj in terms of the rais- ing and lowering operators for oscillator j: a

Inijco'j

firrijCo'j

1/2 {a]+dj:

1/2

-a^)

(14a)

(14b)

ind The "mass" m is related to the charge § j on the capacitor Cj by ^ j={ rrij /Ij

' ''^Xj . (For an ion's axial oscilla- tion, 171:= nil 's the ion mass.) The Heisenberg equations

of motion for Xj with the Hamiltonian of Eq. (12) are

d^x

dt'

d^Xi

dt

+ ((0'l)^Xi^-g^

mi

1/2

1/2

(15a)

[5b)

The general solution to Eqs. (15) is oscillatory, with fre- quencies C0+ given by

col = {[((o[)' + {co])^]±\\[{(o:)'-i(o])^]^ + 4g'

(16) We focus here on the case ( IwJ — Wjlw)'^^ «g «w-,(ys,

where oj = ioj++(o^)/2^co'i^(o',. The solutions of Eqs.

(15) become

xAt)-- xAO) cos( cot )-\ sin((yr

m,(o

cosico^j)

+

A(0)

-x,(0)sin((yf )H — cosicot)

nijCO

X

1/2sin((y„r) , (17a)

x,{t)--

A(0)

x,(0)cos((i)r )H — sinicot

m:0)

cos(co^J)

+ -xJO)sin((yr)H cosicot)

m,(o

X

ru:

1/2

where

(^ex-V(^+-^-

sinico^J) ,

7-J^'^'\W2 CoiCjC^) 1 2Ct 2a)(/,/,)'/2Cr

and where we define

t,, = TT/2co,, = rrmiil,)'''Cr .

(17b)

(18a)

(18b)

Equations (17) describe an oscillatory exchange of exci- tation between the two oscillators. In particular, for time

t «r,, we find that

x^{t)^x^(0) cos{cot)-\ —sm{o)t) , m,(o

x,(r)^x,(0)cos(6;r)H — sinicot) . niiO)

Whereas, for times \t —t^^\ «t^^, we find that

1/2

(19a)

(19b)

X,(t): x,(0)cos (ot + —

2

H sm

nijCo

0}t +

17 (20a)

xAt):

m,

1/2 r Xj(0)cos a)t + —

H sm

m.co

cot + —

2

(20b)

At time t=t^^, the Xj operators have exactly inter- changed values, apart from an amplitude factor and

phase shift of Tr/2. Further, the pj operators have also interchanged values, apart from an amplitude factor and

identical phase shift. It then follows that at time t^t^^,

the source and ion oscillators have exactly interchanged wave functions, apart from a phase shift of v/2. (This

may be proved explicitly in the Schrodinger picture.)

This oscillatory exchange between the two oscillators suggests the following scheme for cooling the signal

TN-18
2982 D. J. HEINZEN AND D. J. WINELAND 42

source, as illustrated in Fig. 4(a). The coupling between

the two oscillators may be switched on and off by rapidly

(compared with t^^ ) switching the resonant frequencies of

the ion oscillator on or off resonance with the source oscillator. Initially both ion and source are assumed to be

hot (< «^ ),<«,)» 1 ) [Fig. 4(a) (i)]. First, with the oscil- lators decoupled, the ion is cooled to the zero-point state using sideband cooling [Fig. 4(a) (ii)]. The cooling laser is then turned off, and the ion and source coupling is switched on for a time t^^, at which time the ion's zeropoint state has been transferred to the source [Fig. 4(a)

(iii)]. Finally, at t =t^^ the coupling is switched off, and

the ion is again cooled to the zero-point state, leaving

both ion and source in the ground state [Fig. 4(a) (iv)]. Because the oscillators may be anharmonic, this may take

several such cooling cycles depending on the initial values

of («5 ) and ( «, ). The same idea may be used to detect single-quantum

excitations to the source oscillator, as illustrated, in Fig.

4(b). The source and ion are initially cooled into the

(a) (b)

(i) T

source ron ion

T f

system

initialized

(ii) ?

sideband

cooling

laser

(ii) T

i

T ?

single

quantum

excitation

of source

(iii) T ?

^ energy

cr:- exchange

(iii) ^

cb

T ?

~ energy

;r=. exchange

(iv)

sideband

cooling aser

sideband

cooling and

detection

detector

FIG. 4. (a) Cooling of a source to the zero-point state by exchange of energy with an ion which has been laser cooled to the zero- point state, (b) Detecting single-quantum excitations of the source. TN— 1

9
42 QUANTUM-LIMITED COOLING AND DETECTION OF RADIO- . 2983

zero-point state with the coupHng turned off, as in Fig.

4(b) (i). We suppose that the external drive u^ is appHed

to the source oscillator, which puts at least one quantum

of energy into its motion [Fig. 4(b) (ii)]. Then the source and ion are recoupled for a time t^^, resulting in the

transfer of the source excitation to the ion [Fig. 4(b) (iii)]. Finally, the excitation of the ion is probed using the side- band method [Fig. 4(b) (iv)].

It should be emphasized that if the signal u^ is continu- ous and monochromatic, the signal-to-noise ratio is maximized when the source oscillator is driven for times of

the order of its relaxation time [see Eqs. (10)]. Then, the source oscillator is contaminated by thermal noise at its equilibrium temperature T^, and the single-quantum

detection technique is not useful. However, this tech- nique should be of advantage whenever the bandwidth of

the drive signal is large compared with the source oscilla- tor damping rate. For example, the signal incident on a resonant gravity wave detector may consist of finite duration pulses, rather than a steady drive. "''^ This tech- nique should also be useful when the signal energy comes

from a spin system, which can store only a finite number

of quanta (see Sec. V). In each of these cases, the signalto-noise ratio can be increased by detecting the driven source oscillator's energy against a much lower thermal

background.

An additional consideration is that a large source ener- gy may introduce undesirable perturbations to the

source. An example, to be discussed further in Sec. V, is the measurement of the cyclotron frequency of a second

trapped particle. Most of the important perturbations to the cyclotron frequency, such as anharmonic and relativistic shifts, scale in proportion to the particle energy. Thus, even though a large signal-to-noise ratio is in principle available by applying a large steady drive, the accu- racy of the measurement may be degraded by the large energy. The technique described here allows for efficient signal detection at the lowest possible energies, minimizing such perturbations.

The buildup of thermal quanta in an oscillator is de- scribed by Eqs. (10). If the present cooling and detection

technique is to prove useful, the number of thermally in- duced quanta ynt^ must be less than one, where t^ is the

duration of a cooling or measurement cycle. This means

that the quality factor Q =(x)/y of the oscillator must

satisfy

Q>ff(ot^={kT,mt„ (21)

where the equality holds for kT»fio). For example, at 7, =4.2 K and /„ =0.02 s, we require Q > 10'°. Such

high Q's can be achieved in the motion of trapped

ions Given the progress that has been made with 12-14 superconducting cavities and crystal oscillators,'^'''' it may be possible to satisfy inequality (21) for these systems

as well. Note that if Q is sufficiently large, the large heat

capacity of a macroscopic object such as a crystal oscilla- tor is irrelevant, since only vibrations of a single mode

are cooled, not the crystal as a whole.

IV. PARAMETRIC COUPLING OF INTERNAL

SOURCE MODES

Parametric coupling of two modes can exchange ener- gy between them with no added noise.' Thus the cooHng

and detection techniques described in the previous sec- tion may be extended to other source oscillator modes

through parametric couphngs. To this end we assume

that during a time over which the parametric drive is applied, the Hamiltonian of the source may be written as'

// = (S|a,+i-)^W, + (a ^22 + 1)^^2

+ (mim2)'^^g^cos(&)jf )xiX2 , (22) where Xj and Pj are given by Eqs. (14) with co'j=cOj, and

the third term represents the parametric coupling be- tween two of the source modes, modes 1 and 2, with a frequency co^ and strength g^. The parametric drive may

be assumed to be classical.' It is assumed that the couplings between modes 1 or 2 and all other source modes

may be neglected. The Heisenberg equations of motion

of x^ for the Hamiltonian (22) may be written as

dt'

+ a)]xi = -g]

mi

1 2^ +(^2X2 = -gl

m,

1/2 COS( COjt)x 2 '

1/2 COsCw^f )X]

(23a)

(23b)

where, in the present case, gi=g2=g^ (a case where

gi^gl is treated below). For maximum parametric coupling of the two modes, we require coj=o}i — co2- Defining new slowly varying operators x (/) by

Xj(t) = Re[xj(t)e ' ], substituting for Xj and co^ in Eqs.

(23), assuming g «<y,,6;2, and keeping only secular

terms, we find that

~dt

dxj

dt ~ Aoi^

ig\

mj

/"•

m,

^1

1/2

1/2

2 ' (24a)

(24b)

Thus noting that Xj{Q) =Xj{0) — ipj(0)/mjCOj, we find that the solutions for 3c, and 3c 2 are

xAt)'. x,(0)cos<y,f +

Pi(0 ) m,a;,

-%\no}^t cosco^J

+ -X2(0)sinw,rH cosft>,f niTCo 2UJ2

X

^2^2^!

m,a;,g2

1/2 smco^j (25a)

[t)-- X 2(0) C0S(02t +

m2C02

sm(02t cosco,J

+

p,(0)

(0) sina)2^ ^ cosco2t m \C0\

X

/?I|W,g2

m2(U2gi

1/2 smco^J (25b)

TN-20
2984 D. J. HEINZEN AND D. J. WINELAND 42

where co^^=gig2/Mo)^o)2)^^^.

These equations are essentially the same as Eqs. (17), apart from the factor imjOj /cojCOj)^^'^ multiplying the

sinico^^t) terms. Thus the same conclusion may be

reached regarding the exchange oscillation: Apart from a phase shift of n/l, oscillators 1 and 2 exactly exchange wave functions when the drive has been applied for a time t^J^=^/2co^^. The factor (m,a), /myWy )'^^ normalizes the exchanged wave functions so that the number of

quanta are conserved; that is, if \xp\(t =0)) has an expansion in terms of the number states of oscillator 1, then

1^2^^ =?e^)) has the same expansion in terms of the number states of oscillator 2, apart from a phase shift. Equations (25) have important implications for a source coupled to a trapped ion. Suppose that the source

is such that only source mode 1 can be efficiently coupled

to the ion, but it is desired to measure excitations in source mode 2. Then, according to Eqs. (25), source mode 1 can first be cooled to the ( « i ) =0 state, and this state parametrically transferred to source mode 2, so that (n2)=0. We assume that source mode 1 is then

recooled to {«,)=0. With the parametric drive off, mode 2 may be excited by a signal drive. The excitation

in mode 2 may be parametrically transferred to mode 1 and then to the laser-cooled ion for detection. Thus para- metric drives may be used to extend the single-quantum

detection to source modes other than the one directly

coupled to the ion.

V. APPLICATION TO COOLING AND DETECTION

OF AN ELECTRON, POSITRON, PROTON,

OR ANTIPROTON IN A PENNING TRAP

As an application of the preceding ideas, we consider

the measurement of the cyclotron frequency of an elec- tron, a proton or their antiparticles in a Penning trap.

The Penning trap consists of a ring electrode and two

end-cap electrodes, which to a good approximation pro- duces a potential (in cylindrical coordinates) of the form

mcol . <l>{T)=^r^(z^-\r^) , (26) 2q

where m, co^, and q are the mass, axial frequency, and

charge of the trapped particle.'^-'^'^^'^^ Superimposed on

the electrostatic potential is a uniform magnetic field B =5z, which provides radial confinement of the particle.

The Hamiltonian of a single electron in a Penning trap

may be written as^^ Hr fi^^\)-h(o,+(n^+\)-Fico[-{n^^-moi^+\fi(o,d,

where fij =3 Js^ (j =z, c, or m ), and

co,=

CO.,

+

4

2

2

2

1/2

1/2

(27)

(28a)

(28b)

Here, «, — (g/2)a)^,, with g the electron g factor, a, is the

electron's z component of spin, with eigenvalues ±1, and

co^=eB /mc is the unperturbed cyclotron frequency of

the electron. The electron has four independent degrees of freedom: an axial oscillation at frequency (o,, a cyclotron motion

(perturbed by the trap electric field) at frequency w^, a circular magnetron drift about the z axis at frequency

£t),„, and the spin. The magnetron oscillator is inverted,

corresponding to the instability of the magnetron motion

in the trap. In terms of the Oj and 9

J, we may also define the operators''

45

fl 2mo)^

amco.

1/2 (s!+9.

Pz=^

1/2

a

1/2 Pc- mil

^,=iCD[

Urn

CI

(s -sj.

3^+9^

1/2 {al~a.

mil

ICO,

1/2

1/2 Hm

~ir ^^ m-^m^ '

X=Mpc+Pm) .

y

1 2m

+ •

(O, CO,

(29a)

(29b)

(29c)

(29d)

(29e)

(290

(29g)

(29h)

where CL = (co[ —co^ )/2.

For reasonable experimental parameters, the electron's cyclotron frequency will lie in the GHz range, while the

laser-cooled ion's axial frequency will lie in the MHz

range. Therefore the desired cyclotron mode cannot be

coupled directly to the ion. In order to cool and detect the electron, we connect the end caps of the electron trap

to the end caps of the ion trap and adjust the trap volt- ages, so that the axial frequency of the electron co, equals

the axial frequency of the ion co,. (We might also consider coupling the electronic axial resonance to the ion's cy- clotron resonance by splitting the ion trap's ring elec- trode.) In this case, the two axial oscillators are equivalent to the circuit of Fig. 3, where u^ =0, Cj^ is the combined capacitance of the two traps in parallel, /, and

c, are given in Eqs. (2), l^=m{d^/a^,e)^, Cj=(a)?/J~', d^,

is the electron end cap separation, and a^, a constant of

order unity. "^^ Therefore the discussion of Sec. Ill applies; and it follows that if the ion is prepared in an initial (^,)=0 state, this state is transferred to the electron's

axial oscillation by allowing them to couple freely for a time t^^=7rco^{lJi)^^^Cf. The axial oscillators can be

decoupled by rapidly (but adiabatically) changing the ion trap potential, so that iw, —w. I »[(L)_.(/,/, )' 'C,] '. The electron's other two degrees of freedom, the cycloTN-21
42 QUANTUM-LIMITED COOLING AND DETECTION OF RADIO- 2985

tron and magnetron motion, may each be coupled

parametrically to the axial motion in the manner dis- cussed in Sec. IV. This problem has also been considered

and demonstrated in the classical regime by Cornell

et al.'^^ We consider an applied field of the form

^0

(6( r ) = ——xz coscoj t a

(30)

with E=— V(/). Again, the drive may be assumed to be

classical.' The applied potential gives rise to an additional contribution to the Hamiltonian,

HmleEr

-xz COSCOjt (31)

The Heisenberg equations of motion of p^,, p^, and z for the total Hamiltonian H =HQ + H^J^^ are

dt'

d'-z

+(«;)^p, = -

+a)i

o/^eEc

Clmd

Clmd

z cosco^t

-Z COSCOjt

2 ^^0

(32a)

(32b)

(32c)

As before, we introduce slowly varying operators

by definmg z = Re(ze ' ), p^ = Rt(p^e ^ ), and

__ —i(Dt

p,„=Re(p,„e '"). Parametric coupling of the axial and cyclotron motion occurs for a)^ =&>^ — w^. Substituting for oj, z, p^, and p^ in Eqs. (32), and neglecting nonsecular terms, we find that

ieE, _

zdt AmCld'

dz _ JeEp __ dt Smco^d^'

dpn

Aoi,

dt =

(33a)

(33b)

(33c)

Equations (33a) and (33b) are of the form of Eqs. (24), with X|=p^, ^2=2, 6)]=&>c, (02 = 0^2, g]=a)[eEQ/ilmd,

and g\=eEQ/2md. Therefore the general solution is of

the form of Eqs. (25), where p^—rr^, P2~Pz>

m,=/rj2 = m, (co2g]/o)^gj)^''^ = {2(o^/CL)^^^, and

a)^^ = (eEc)/Amd)(2noj^)~^^'^. Again, the solution is ex- actly analogous to Eqs. (17), apart from the factor ilco^/il)^^^^ multiplying the sin^y^^r terms. Thus we find that for t^^ =Tr/2oj^,^, the axial and cyclotron wave functions have exactly interchanged, apart from a phase shift of 77/2. The factors (2w^/ft)^'^^ normalize the wave

functions, so that the number of quanta are conserved,

and so that the exchanged wave functions have the same

expansion in a basis of mimhcr states, apart from a phase

shift. A drive at o}j = a>^+(i)„, will couple the axial and magnetron motions. The sum, rather than the difference, of

the frequencies appears because of the inversion of the magnetron energy. By again substituting for mj, p^, p,„,

and i in Eqs. (32) and neglecting nonsecular terms, we get

dp = . dt dp„

dt 4m D.d " 4co

ieE ^ig\ z = z ,

dz ieE _

dt %mco,d Aco,

(34a)

(34b)

(34c)

In this case also, Eqs. (34b) and (34c) are of the form of

Eqs. (24), with x-^=p^, X2=z, co^ = oi^, cl)2 = (o^, g]=CL)^eEQ/ ilmd, and g\=eEQ/2md. The general

solution is of the form of Eqs. (25), where p-^=fr^,

P2="Pz, m,=m2=m, {(O2g]/o}\g\)^^^ = {2a)^/il)^^^,

o)g^ = {eEQ/Amd)(2flo)^)~^^^, and coi = —o}^. Again,

apart from a phase shift, the axial and magnetron wave

functions are exchanged after a time rg^=7r/2&)g^, with

the wave functions normalized so that the number of

quanta are conserved.

The electron can be cooled into its \n^,n^,nf„ ) = 1 0,0,0) ground state, by cooling to n^=0 via couphng

to the ion, as illustrated in Fig. 4(a) and by then parametrically driving n^ to zero, recooling to n^=0 again,

parametrically driving n^ to zero, and, finally, recooling

n^ to zero. Measurements of the electron's cyclotron fre- quency may be carried out by first applying radiation

with a frequency near co'^ to drive the |0,0,0) to |0, 1,0)

transition, and then by parametrically exchanging the cy- clotron and axial wave functions, as illustrated in Fig. 5. If no transition to the |0, 1,0) state occurs after the radiation is applied, the final state of the electron after para- metric exchange is 1 0,0,0); whereas, if a transition to

1 0,1,0) does occur, the final state of the electron is

1 1,0,0). The transition may be detected by detecting the presence of a quantum in n^, by couphng to the ion, as il- lustrated in Fig. 4(b). As discussed in Sec. Ill, the abihty to cool and detect single-quantum excitations in this manner does not in- crease the inherent signal-to-noise ratio of the driven

electron, given that a certain amount of damping is present and that a spectrally pure drive is available. Thus

ICO7

nn=1

Hz =2

nz=1

n,=:0

flCOf

a b

:0-

n, = 2 n,=1

riz^O

FIG. 5. Lowest electron cyclotron and axial energy levels, il- lustrating the transfer of a single cyclotron quantum into the axial motion. Only the few lowest ,cyclotron and axial levels are shown for clarity, (a) \n,,n^ ) = |0,0) to |0, 1 ) transition at co'^. (b) |0, 1 ) to 1 1,0) transition at a)', —co^. TN-22
2986 D. J. HEINZEN AND D. J. WINELAND 42

one can also achieve an equivalent signal-to-noise ratio by

applying a strong continuous drive to the electron's motion, resulting in a large signal power, and by detect- ing the signal using conventional rf detection. ^^ However, this introduces perturbations to the particle's frequencies b)\, co^, and a)„, making this large signal-to-noise ratio less useful to precision experiments.

For a purely quadratic electric potential and uniform

magnetic field, the "true" cyclotron frequency of the particle, in terms of the measured frequencies (o[, co^, and

co^ , is given by^^

(O l = {colf + col + coi . (35)

This expression remains valid for arbitrary tilt angle of

the magnetic field with respect to the trap axis.^^ The

most important corrections to Eq. (35) arise from

second-order inhomogeneities in the magnetic field, fourth-order corrections to the electric field, and relativistic corrections. (The magnetic- and electric-field inhomogeneities may be parametrized by coefficients Bj and

C4, respectively.^^) These corrections are proportional to tne particle energies; so accuracy increases as the energies

are lowered.

For trapped electron spectroscopy, the Bj correction

has typically been the largest, since a "magnetic bottle"

(large B2) has been dehberately applied for detection purposes.

^^'^^ However, it should be possible to reduce this, either by use of a "switchable bottle""*^ or by using alter- native detection schemes without a magnetic bottle."^* If such schemes are successful, the most important correction will be the relativistic shift {i^co^/co^)^^^ = —E^/2mc^, where E^ is the axial energy. (We assume

that the cyclotron motion has cooled to ( «^ ) < 1 by cy- clotron radiation, so that relativistic shift due to cyclo- tron motion may be precisely corrected for, as discussed

below.) For E^ = ksT, with T=4.2 K, we find that

Aft>/«^ = — 3.5X 10~'°. Thus electron-positron mass

comparisons or g-factor anomaly measurements at or beyond a few parts in 10'° will require significantly lower

axial temperatures. We note that presently the electron- positron mass ratio has been measured to one part in 10^,^* and their anomalies measured to two parts in 10^.^^ The character of the systematic corrections changes

dramatically when the electron or proton can be cooled

to its lowest quantum state. In this case, the resolved

quantum level structure of the particle becomes apparent,

as was illustrated in Fig. 5, and the perturbations take the

form of energy-level shifts."*^ Also, since only singlequantum excitations are observed, the observed reso- nances will be sharp lines, not broadened by the perturbations. Since the relativistic level structure can be calculated with high precision, the "true" cyclotron frequency can be calculated from the observed level spacings, with

well-known corrections taken into account. In fact, this quantum relativistic level structure is of interest in its own right. The relativistic corrections will therefore

have essentially no effect on the measurement accuracy in the single-quantum regime.

One of the difficulties associated with mass comparisons""'^^ in the Penning trap is that the magnetic field

typically drifts between the loading of one particle and

the next. This has limited the accuracy of mass comparisons of atomic ions to a few parts in 10'°.^^'^^ We envision performing such experiments with a single ^Be^

ion as the detector ion (Appendices A and B). This near- by ^Be^ ion can also serve as a highly accurate magnetometer by measurement of its ground-state electron

spin-flip transition frequency, so that such drifts may be

corrected for to a high degree. The Bj, C4, and relativistic corrections are of comparable magnitudes for atomic

ions and will limit accuracies to approximately one part

in 10" without improved cooling techniques. The

present technique provides a means of dramatically

lowering these corrections. Mass-ratio comparisons at or beyond a few parts in lO''^ should be possible.

These techniques may be applied to a measurement of

the electron g factor. In this case, the largest systematic

effect has been a cavity shift of the cyclotron frequen- cy

23,49 jYiQ uncertainty in this effect could be substantially reduced, however, by choosing electrode shapes

which approximate a cylindrical cavity. ''^'^^ Parameters

could be chosen so that this cavity has a lowest-order

mode frequency higher than the cyclotron frequency, so

that the shift is much smaller and more tractable. The

cavity shift effect is less important for the electron- positron g-factor ratio and mass ratio, since it should be

the same for both particles. For the mass ratio, the limit- ing systematic error will most likely be due to an offset of

the trap center for the electron compared with that for the positron, due to stray contact potentials. This causes

the two particles to see a slightly different magnetic

field. ^' We think that the techniques discussed here

should allow improved electron-positron mass-ratio and

g-factor comparisons. Similar improvements in accuracy

should be possible for a direct measurement of the elec- tron g factor, assuming that the cavity shift problem can

be adequately addressed.

It should also be possible to make a high-precision

measurement of the g factor of the isolated proton or other atomic ions. This measurement has not been carried out using existing techniques of single trapped-particle

spectroscopy, because the proton spin and cyclotron

magnetic moment are too small to detect the proton-spin

flips with magnetic bottle techniques."'^ However, in the technique discussed here, the measurement sensitivity is great enough to detect the spin flips directly, as illustrat- ed in Fig. 6. The proton can be prepared in the |mj = +|,rt^=0) state by first preparing it in the rrl.O)

state as described earlier, and parametrically driving at co^—o)'^, where (o^=gpfiyB /fi is the spin-flip frequency,

and ^Y is the nuclear magneton. The proton will either remain in the |+y,0) state or be driven to the | + |, l)

state. The cyclotron quantum state is exchanged with the

axial in an !«_. =0) state, as previously described, leaving

the proton in the i + I.O) state. Spectroscopy of the

spin-flip transition at co, is carried out by driving the

i +1,0) —>

I — -S-,0) transition, followed by a drive at co^—(o'^, transferring the proton to | + |,l) if a spui-flip transition occurs, or leaving it in l + I.O) if a transition does not occur. The cyclotron quantum is then detected

TN-23
42 QUANTUM-LIMITED COOLING AND DETECTION OF RADIO- . . . 2987

— nn=1

r\r=3

nr=2

nr=1

nr.=0

nr=0

S = -

FIG. 6. Lowest proton cyclotron and spin energy levels, illus- trating the transfer of proton spin-flip energy into the cyclotron

motion. Only the few lowest cyclotron levels are shown for

clarity, (a) i + i,0)—>| — y,0) spin-flip transition at w^. (b) ^,0)—>-\ + j,l ) transition at co,—c

as described in Fig. 5. In this way, single-quantum spinflip transitions may be detected and the proton g factor determined by g^ = 2co^ /o^

.

The proton g factor enters into the determination of

the fundamental constants via the three quantities

gp/l=IXp/^i^, fip/fXsy and rrip/m^. The two most precisely measured of these quantities determine the third

and also appear as auxiliary constants in the least-squares

adjustment of the fundamental constants. ^^ At present,

Up //ig and rUp /m^ are the most precisely determined,

with fractional uncertainties 1X10~^ and 2X10~*, re- spectively.^^ All three of these quantities might be measured with improved accuracy by the techniques dis- cussed here. Also, the proton-antiproton g-factor ratio might be measured with these techniques.

VI. SQUEEZING AND SIGNAL DETECTION

BY PARAMETRIC AMPLIFICATION

We have so far described only measurement by quantum multiplication. Linear amplification using a single

stored ion is also possible: Degenerate parametric

amplification occurs if the ion trap's ring-end-cap volt- age is driven at twice the ion's axial resonance frequency.^'' This process can amplify one of the oscillator's two

quadrature components with no added noise."* The quadrature components X, and X-i are defined by'*'^"

(36a)

(36b)

In terms of A", and X^, the displacement operator z may

be written as

ma

1/2 (X. coso)? +A2sin«?) , (37)

so that A'l and X^ may be identified with the coscl)? and

sino)? components of the oscillation. X^ and X-^ satisfy the uncertainty relation AX,A^2>i-, (38) where A^,^ <(A^, )^)'^^ is the rms fluctuation of .^, about its mean. For the vacuum state or for a coherent

state, AX| and A^2 ^^e equal: A^j =A^2~7-

Consider the state in which the operators have initial value XM at time t—t\. Degenerate para- p'l and i'j^'i

metric ampHfication of this state occurs when an oscillat- ing potential Re( Fe^"^') is apphed between the ring and

end-cap electrodes of the ion trap, with V a complex constant, V = \V\e'^. This gives rise to an added potential

^ =

-Iz' dl

Re(Fe^''"') (39)

where dj is a characteristic trap dimension. The Hamiltonian describing the axial motion of the ion then be- comesH = ii(o{a^Q + \

lez

d\

-Re(Fe''"') . (40)

Neglecting nonsecular terms the Heisenberg equations of on f

d5t,

motion for Zj and 51^ become

dt dk,

~dt

= ^(l2COs6»+Xisin0) ,

= ^{X^co%e-XjSmd) ,

(41a)

(41b)

where ^ = e\V\/md^o). For B = tt/1, Eqs. (41) have the

solution

X,(r2) = G

X,{t,)^G

1/2X,(r,) , (42a)

(42b)

with G= exp[2|;(f2~'i )]• Thus the component a, is amplified by the factor G '^^; X^ is attenuated by the same

factor. Similarly, for B=—tt/1, X^ is amplified and a,

attenuated.

Suppose that a signal X^^, which results from a drive

applied from time / =0 to ?[, is present in a ,. We have

X,(r,)=X,(0)+X, (43) This equation follows from Eq. (8) and the corresponding

conjugate equation for a\ty). where X, ifso +f*o )?] /2, with / =fso^

-icot^ ^^^ where we assume that Yt\«\ and that the noise term may be neglected.

X\(t\) may be amplified according to Eqs. (42), giving a

signal 5 and noise A^ of

S = {)C^{tj)y = GX], , (44)

yV = <AJ?,(r2)^) = G(A^,(0)2) , (45)

where we assume that (.?|(0))=0. For G sufficiently

TN-24
2988 D. J. HEINZEN AND D. J. WINELAND 42

large, 5 can be detected with negligible additional noise

by monitoring changes in the laser-induced fluorescence

of the ion^"* or by detecting the ion's image current with

conventional electronics. (For example, in the work of

Refs. 26 and 46, the axial amplitudes of single ions are determined with a signal-to-noise ratio greater than unity

in a single measurement.) The resulting signal-to-noise

ratio S /N is just that of the state before amplification.

For an initial vacuum state at / =0, {A.X^(0)^)=J,

which gives 5 /A^ = 4Zf,.

The fluctuations in the signal quadrature X^ may be re- duced at the expense of increased fluctuations in quadrature Xj, by preparing the oscillator in a squeezed

state."*' ^° Such a state may be characterized by a wave

function (at t =0) given by^'

(a)

xb (z) = N

g(~'/2)[;3(mw/«)z^-2/36(2mw/«) sq (46)

Here 13 is the "squeeze parameter," 6 is a coherent amplitude in the X^ quadrature and A^^^^ is a normalization factor.In this state, the variance of ^, is decreased by the fac- tor p. Thus the noise is N^ = \ /4j8, and the signal-tonoise ratio S/N^=APX]^, an improvement by the factor

p. For /S»l, this implies that a signal energy fico/fi, which contains much less than one quantum, may be

detected with a signal-to-noise ratio of order unity in a

single measurement. Therefore signal detection by de- generate parametric amplification is in principle more

sensitive than signal detection by direct measurement of

absorbed quanta.

A very direct method of producing the state given by

Eq. (46), illustrated in Fig. 7, is to first prepare the ion in the ground state rpQ of a well with a different restoring potential (f)'(z') = m(co')^(z')^/2q (co'>(o) and different ori- gin A=z —z', and then nonadiabatically dropping the potential back to (f>iz) squeezed state — ^„2.,2 32 mo) z /2g.^ This produces the

^o(^') — ^ _(-l/2)(m<.j'/^)(z-A)^-

IT 1/4 A,(^) (47)

This implies that P=co' /co, and that 6 = A(m&j/2^)'^l

A second method of preparing an initially squeezed

state is to prepare an initial vacuum state, followed by a parametric drive as in Eqs. (41), with 0=—tt/2. The

variance in the X, quadrature is reduced by a factor G,

while that in the Xj quadrature is increased by the factor G. The wave function is of the form of Eq. (46) with

b=0 and 13= G. Squeezing and detection by degenerate parametric

amplification may be extended to source oscillators other

than the ion by the methods described in Sec. III. Thus,

if the ion is prepared in an initially squeezed state and

coupled to a source oscillator as in Fig. 3, after a time t^^, the oscillators have exchanged wave functions; so the source oscillator is prepared in a squeezed state. A signal

drive applied to the source generates a source excitation

analogous to that described by Eq. (43). The squeezed

and driven source oscillator wave function may then be

transferred back to the ion, and its X, quadrature amplitude measured by degenerate parametric amplification of

TN-25

FIG. 7. Preparation of a squeezed state by a nonadiabatic

change in the trapping potential, (a) Potential and wave function immediately preceding the potential change where the ion

is assumed to be cooled into its ground state of the potential

well represented by <f>'{z). (b) Restored potential and squeezed wave function immediately after changing the potential from

<t>'(z) to (t>{z).

the ion's axial motion. In an experiment to measure a very weak force by its excitation of a high-Q oscillator,

both quadrature amplitudes may be measured with high

sensitivity by using two oscillators, one for each quadrature

54 The buildup of thermal excitation in an oscillator is de- scribed by Eq. (9). For yf, «1, it follows from Eq. (9) and its conjugate that there is a contribution to the vari- ance ( AJf] ( r

, )^ ) of 7( « + y )r , /2. Thus, if thermal noise

introduced by the coupling of the oscillator to a thermal

reservoir is to be negligible, we require yin +-j)ty/2

«<AX,(0)^) = 1/4/3. For yS= 1 (no squeezing), this is essentially the same requirement on oscillator Q as the in- equality (21); for ;8> 1, the oscillator's Q must be still higher by a factor of )S.

VII. CONCLUSION

Sensitive detection of signals requires that both signal source and detector exhibit low damping rates, that the source and detector couple efficiently, and that some

means is available to cool both source and detector to a temperature of order fico/kg. In addition, the detector

should provide nearly quantum-limited gain. We have

shown that a single laser-cooled trapped ion, coupled to a high-^ source through image currents in the trap elec- trodes, can provide a realization of such high-sensitivity

detection in the rf regime.

Such a detector may have important applications to

^H^^^^^^
42 QUANTUM-LIMITED COOLING AND DETECTION OF RADIO- 2989

precision measurement. We have analyzed one application, the coohng and detection of motions of a second

trapped charged particle, and seen that improved accura- cy and sensitivity may be expected. These techniques

may improve the accuracy of measurements of the elec- tron g factor,^'' electron-positron g-factor ratio, ^^ and mass ratio, "^ and can in principle yield the proton g fac- tor and the proton-antiproton mass ratio^'^ and g-factor

ratio. By extension, these techniques may also find application to measurement of mass or internal degrees of

freedom of other atomic or molecular ions. These

methods may also prove useful in any experiment requiring very high sensitivity in the rf range or at lower fre- quencies, such as in gravity wave detection, or detection

of very weak forces.

'^"'^ Because this kind of experiment is conceptually simple,

it may prove useful as a testing ground for topics in quantum measurement theory. For example, studies of

squeezed states or quantum limits to signal detection may

be carried out this way. The quantum behavior of single

trapped particles may also be studied.

An important question is how practical the proposed

methods are. This has already been demonstrated to some degree in Ref. 14, and in Appendices A-C we fur- ther examine the experimental possibilities. We are currently developing a coupled-trap experiment similar

to that described in Appendix B. ACKNOWLEDGMENTS

We gratefully acknowledge the support of the U.S. Air

Force Office of Scientific Research and the Office of Naval Research. We also thank John Bollinger, Wayne

Itano, Jim Bergquist, Mark Raizen, Fred Moore, Don

McDonald, and John Thomas for comments and suggestions on this manuscript.

APPENDIX A: RAMAN SIDEBAND COOLING

AND DETECTION

Most of the experimental techniques discussed in this paper depend on resolved sideband cooling and detection.

As described in Sec. II, this requires an ion with the level structure shown in Fig. 2, with easily accessible "weak"

(7^,«a)^) and "strong" transitions. Since typically

(o^ ^ 1 MHz, this condition is so far experimentally

satisfied only in a few relatively heavy ions, such as Hg^

(Refs. 14, 32, and 43) or Ba+ (Refs. 41 and 44) where optical transitions to metastable levels occur. A technique

to cool lighter ions is desirable, since this maximizes the coupling between the ion and source.

In this section, we describe how a stimulated Raman

transition between two ground-state sublevels (or a ground state and metastable excited state) may serve as the "weak" transition. In this way, sideband cooling may

be extended to lighter ions, such as ^Be^, which is the

lightest ion which can easily be laser cooled by Doppler

cooling."'^'' Cooling by stimulated Raman transitions has been previously discussed for the case of Ba^."'^** The general principle of Raman sideband cooling for the axial energy of the ion is illustrated in Fig. 8. We

|0,n>

12,2 >

12,1 >

12,0 >

FIG. 8. Energy-level structure for Raman sideband cooling

and detection. The Raman cooling transitions are induced by

beams 1 and 2; beam 3 is used only for detection of Raman cooling and for Doppler precooling.

consider a four-level system with lower levels 1 1 ) and |2

)

coupled to a common upper level |0). We also assume

that level 1 1 ) is coupled to an upper level |3 ), which de- cays only back to level 1 1 ). The 1 1 ) ^- 13 ) transition will serve as the detector of Raman cooling and will also provide Doppler precooling. We denote the combined

internal-translational states by \J,n), where J denotes the

internal state {J =0, . . . ,3) and n the axial harmonic oscillator quantum number. Two classical fields,

ReiEQje

iik, -10,1)

), / = 1,2 (Al)

couple level |0) to levels |l ) and |2), with Rabi frequencies gQj = \fiQj-EQj\/2ii, where fiQj = {0\iJ,\J) is the di- pole matrix element between states |0) and \J) {J = 1,2). The frequencies of the two beams are assumed to be equal

to

CO, —CO,01

CO-, ^02~^ '

(A2a)

(A2b)

where (Oqj is the transition frequency between states

|0,« ) and \J,n ), and co^ is the axial oscillation frequency

of the ion. The difference frequency co^—coj is tuned to the first lower sideband of the 1 1 )*-*|2 ) stimulated Raman transition, inducing primarily i 1,« )-(-^|2,« — 1 ) transitions. For effective cooling, we require that

Pj

Hco, «1 {A3)

where Rj = (fikj)^ /2m is the recoil energy for the

|0)^-|j) transition (/ = 1,2). In addition, spontaneous

Raman scattering occurs. When 13 «\, spontaneous

scattering occurs predominantly with A« =0. Thus cooling can occur by means of stimulated Raman transitions

|l,«)—>-|2,«— l), followed by spontaneous Raman transitions |2,« — 1 )-^ 1 1,/? — 1 )." In this case, cooling by

stimulated Raman transitions is essentially the same as

laser cooling using a narrow optical transition, as illus- trated in Fig. 2. In the case of Fig. 2, there is no inter- mediate level, and the \J = 2) level of Fig. 8 is replaced

TN-26
2990 D. J. HEINZEN AND D. J. WINELAND 42

by the \w} level of Fig. 2. In the case of Fig. 2, the atom

is recycled back to the ground state by spontaneous emission \w)—>\g), rather than by spontaneous Raman transitions |2,« )^-|0,« )^-|l,« ). A quantitative description of this cooling process is fairly complex and requires a solution of the density matrix equations for the infinite ladders of states \J,n ), y =0, . . . , 3; n =0, . . . , 00 . In order for efficient cooling

to occur, the atom must make the |l)—^-12) transition

primarily by stimulated Raman scattering, and the |2)—>-|l) transition primarily by spontaneous Raman

scattering. One means of accomplishing this with cw ex- citation is to make gQj »go\ ^^ This case has been analyzed by Lindberg and Javanainen,^^ who obtain hmiting

values of the temperature. In this section, we discuss an

alternative method, based on Raman excitation in the

transient regime. The analysis of this method is some- what simpler than the cw case. In addition, this method

provides an efficient means of detecting the final temperature. We first suppose the ion is in the level 1 1,« ), and that both beams are turned on for a time t, where t is sufficiently small that the probability of a spontaneous

transition is small. The behavior of the system is adequately described by amplitude equations.

daJn dt n 2 ^rn'^

J\n'

^JnJ'n' Yj

'Jn (A4)

In Eq. (A4), aj„ is the amplitude for the state \J,n),

^^JnJ'n' is the energy of state \Jn ) minus the energy of

\j'n'), Vj^j.„' is the matrix element of the perturbation —/i-lEi + Ej) between states \J,n) and \J'n'), and

7y = 75yo. where y is the total spontaneous decay rate of

level 0. If the condition (A3) is satisfied, i^kjZQn '^^« 1, and if

1^01 I -1^021=^ ' A» Try 2|6feJzo«'/^ '

8'

«0),

(A5a)

(A5b)

(A5c)

then we can show from Eqs. (A4) that for times f ^ 1 /ft

only the |l,«)-<-^-|2,n — l) transition is appreciably excit- ed, and that the populations of these two levels are given

approximately by

l,n

2,n

-COS^ft? ,

, ^sin^ftr , where

ft = |6A:^|zo«

1/2

(A6a)

(A6b)

(A7)

Here bk, = k Iz A:,^ is the diff'erence between the z components of the wave vectors of beams 2 and 1, and

ZQ — {h/lmco^)^^^ is the spread of the zero-point motion

of the ion in the axial well. We see that a kind of

coherent "Rabi oscillation" between the two lower levels

is observed, and that, if the two beams are left on for a time r^ = 7r/2ft, a "Raman v pulse" is obtained, in which

the population of level 1 1,« ) is completely transferred to |2,«-1).

The first inequaUty (A5b) guarantees that the total spontaneous decay probability out of level |0) is small,

and the second inequality (A5c) guarantees that the population transferred by stimulated Raman transitions to

states other than |2,a7 — 1 ) state is small. This "Rabi os- cillation" is similar to that which has been previously an- alyzed for a three-level system;^" our analysis differs in that a minimum of four levels is involved, since the Ra- man transition proceeds equally through the two excited

levels |0,« ) and |0,«— l). l!" I»^i l/^lg^il/^ coherent

pyondla^onr exchange/still /)ccors Wut/witl( le/s tl^n ^0%

eflpxrie/cyisijnilar toM;he /hr^-le/el^as^'^).

The behavior given by Eqs. (A6) indicates that the Ra- man transition may be used to verify that the ion has

been cooled into the state 1 1, « ) , with ( « ) « 1 . The two

laser beams are pulsed on for a time t^ corresponding to « = 1. The ion is transferred to the |2,0) state with high

probability if it was initially in |l,l ); but if it was initial- ly in the state 1 1,0), it remains there. This behavior is il- lustrated in Fig. 9, which shows the result of a numerical

(a)i.

0.0

0.0 9<< O'er * time (fisec) ^

0.90

0.0 4 time ifj-sec) 9 IZ

FIG. 9. Level populations for stimulated Raman transitions as functions of time of the states | J, n )..». Parameters are |go,|/2n-=|go2l/27r = 40 MHz, A/27r=>^t}Hz. w,/2~ = 5 MHz, and A:|.Zo= — A2..Zo = 0. 115. (a) Populations of the il.l ) and !2,0) states, with all population initially in the Il.l) state.

(b) Population of the 1 1,0) state, with all population initially in the 1 1,0) state. TN-27
42 QUANTUM-LIMITED COOLING AND DETECTION OF RADIO- 2991

integration of Eq. (A4). In this simulation, couplings are considered only up to the first order in kjZQti '''^, and the

basis includes only the « =0, 1, and 2 levels of each elec- tronic state. For the parameters indicated in Fig. 9, the

|l,l)^|2,0) transition is driven with over 97% probability, whereas transitions out of the |l,0) state occur

with less than 1% probability. The amplitude equations

do not include the repopulation of the ground states due

to spontaneous emission from the excited states. For the

conditions of Fig. 9, (a) and (b), the integrated decay

probability during the time t„ is (g/A)^yr^«0.9%.

Thus, for the conditions of Fig. 9(b), most of the population loss from the |l,0) state occurs via spontaneous

emission, and most of this population returns to the 1 1,0)

state by the end of a cooling cycle, since the probabihty

for the ion to recoil into the n = 1 state for a single spon- taneous scattering event is »fi«\ (see discussion

below). The presence or absence of the initial n = 1 quanta may be determined by following the Raman pulse with

a pulse of radiation on the |l)-<-^|3) transition, which

here plays the role of the strong transition in a manner

analogous to that discussed in Sec. IL Equations (A6) also suggest one means of cooling the

ion. The ion is first precooled by ordinary Doppler cooling^^'^^ on the !l)—>|3) transition to a condition where

{n)=nj) — {Y-^/o)^ — \)/2.^'' Beams 1 and 2 illuminate

the ion, with IgoJ^ISozl' ^"d beam 1 is chopped on and

off, with the initial "on" period adjusted to give a "tt pulse" with « =1, and the "ofF' period sufficiently long

that the ion returns from state |2) to state 1 1 ) by spon- taneous Raman scattering with high probability. The ion

then is pumped down the ladder of states, losing, on average, approximately one quantum per on-off" cycle.

(Ramping the pulse length from t^n^^^^ to t^ would

shorten the cooling time.) This may or may not be more

efficient than a scheme in which both lasers are left on

continuously with |go2l » Is'oi 1 5 i^ has the advantage that the experiment will already be set up in this way for detection purposes, and that the cooling time may readily

be estimated. The limiting population of the n = 1 state

will be determined by the probability that an ion in the

1 1,0) level is promoted into the |l,l) level during one

cooling cycle. This can occur by an off-resonant stimulated Raman transition with probability of order

P^^p^=(g^/Ao)^f. Also, the ion can reach the |l,l)

state by spontaneous Raman transitions, either by directly undergoing the 1 1

, ) -^- 1 1

, 1 ) transition, or by first un- dergoing the 11,0)^-12,0 or l) transition during the beam 1 on period, and then the 1 2,0 or 1 ) —> 1 1, 1 ) transi- tion during the beam 1 off period. Both of these processes occur with a probability of order P^p ^ = (g/A)^7^/^.

An ion reaching the |l,l ) level returns to the |l,0) level during the following cooling cycle with high probability.

Therefore the average population of the 1 1, 1 ) level is ap- proximately given by the greater of P^lr or ^sp.R- ^^'^ the example of Fig. 9, P

4-4.

st.R ^4.6X10" and

P R = 3.9X 10 '*; so the limiting population of the n =1

state is ~10 \

In addition to cooling the axial motion, the particle's

two transverse degrees of freedom will heat because of

recoil. This will not affect the present discussion as long

as the transverse motion is not inordinately hot, since the

Raman transitions are sensitive only to axial temperature

if ki — k2 is oriented along z. The transverse temperature

may be kept reasonably low by periodic application of

Doppler cooling by beam 3, presumably during detection

periods. Also, the sideband coohng methods discussed

here may easily be extended to the particle's transverse degrees of freedom, so that a very low temperature in three dimensions may be obtained.

The Raman sideband cooling method discussed here

may be implemented in the ^Be"*" ion.^° We assume that the ^Be"*" ion Zeeman sublevels are split by a strong magnetic field, and that the hyperfine structure may be

neglected due to optical pumping^' into the nuclear spin

state mj = +j. The levels corresponding to Fig. 8 are

i) = |-s \/2'mj =+;)

|2) = PS,/2,m, = -i) ,

|0) = |2p3/2,m, = +l) ,

|3) = PP3/2,m, = +|) . The parameters given in Fig. 9 are representative of those

which may be obtained with ^Be"*" in a Penning trap with

a large magnetic field ~6 T. For these conditions,

nj)=2A; so only a few Raman cooling cycles are necessary to obtain ( n ) « 1

.

APPENDIX B: EXAMPLE DESIGN PARAMETERS

FOR A COUPLED-TRAP EXPERIMENT

In this section we discuss the design of an experiment

to cool and detect excitations of an electron in a Penning

trap. We assume that a ^Be"*" ion is cooled and detected

by the Raman sideband method discussed in Appendix

A. The electron is assumed to be confined in a Penning

trap of end-cap separation d^, and the ion in a second

Penning trap of end-cap separation c?,. We assume that

the two traps are arranged in a coaxial fashion, as illus- trated in Fig. 10, with a common central end-cap elecion —

1

T

EC

- R

Vol

1 1 EC

1 "^ Rb

C

R

> de • electron

1

|ec

-|lVoe

FIG. 10. Coupled-trap experiment: End-cap electrode (EC)

and ring electrode (R). The central end cap is common to the two traps and is connected to the shield by resistor Rg. The

trap electrodes and shield (outer boundary) are drawn to scale, with c/^, =0.4 cm, and are assumed to have cylindrical symmetry. TN-28
2992 D. J. HEINZEN AND D. J. WINELAND 42

trode, and with the two remaining end-cap electrodes

connected to an outer conducting shield. The central electrode may be connected to the shield potential via a

large resistor and small shunt capacitance, without intro- ducing significant thermal noise or excess capacitance.

Potential differences of Vq^ and Kg, are applied between

the shield and the electron trap and ion trap ring elec- trodes, respectively.

We have evaluated the trap design in Fig. 10 numerically, using a modified version of the computer code

POISSON.^^ Figure 10 is drawn to scale, and we assume

that dg=OA cm (c?,=0. 1 cm). We assume that the trap

capacitance Cj- is given by its upper limit, which is simply the sum of the capacitances of the central end-cap

electrode to all other electrodes and to the shield. We

find that

Ct-^O.SO pF , /,-m,(J,/a,e)2 = 905 H ,

/,=m,(c?,/a,e)^ = 9.86X105 H ,

(Bl)

(B2)

(B3)

where we calculate that 0^=0.793, and a, =0.772. If the

traps are placed in a magnetic field of 6 T, then the ^Be"*" cyclotron frequency v^ is 10.2 MHz, and the maximum

axial frequency is v^ ( max ) = v^ /2 ' ''^ = 7. 23 MHz. We as- sume that v^ = 5 MHz, which then gives

/ex=27rV,(/,/,.)'^^C7- = 0.88 s , ^B4)

Finally, we calculate that the axial potentials of the two

traps are given by

VAzJ=0.563{z,/dJ^V,Oe ' v,(zi)=\.mzi/di)%Oi »

(B5)

(B6)

where z^ and z, are the electron's and ion's axial displacements from the center of their traps. This implies that an

axial oscillation frequency of v^ = 5 MHz is obtained in each trap for Tq^ =79.8 mV, and Fq, =25.5 V. The time t^^ must be shorter than the time for absorption of one thermal quantum of energy on the axial motion of either the ion or the electron. This seems feasi- ble, since a heating rate of approximately six quanta/s was obtained in the Hg"*" experiments,'"* and this may

presumably be improved upon. For single-quantum exci- tation and detection of the electron's other degrees of

freedom, the excitation of that degree of freedom and

transfer into the axial motion must be shorter than the

time for single-quantum excitation on that degree of free- dom due to thermal radiation. The traps must be harmonic and tuned into resonance with each other to an ac- curacy better than Av^ = l/2?gx=0. 57 Hz, or a relative accuracy better than 1.1 X 10~^. This may be difficult but

also seems feasible.

APPENDIX C: CONTINUOUS REFRIGERATION

The previous discussions have concentrated on cooling

the source modes to the zero-point energy. For some applications, this degree of cooling may not be required and

a more modest form of the basic scheme can be employed.

As an example, consider that a cloud of A'^ trapped ions

which is laser cooled is used as a refrigerator. For simpHcity, we assume that the cooling transition's radiative de- cay rate y is larger than the ion's motional frequencies;

that is, we are in the Doppler-cooling regime. ^^'^^ We

will consider, as before, the axial motion of the ions, but

similar arguments apply to the other degrees of freedom.

For this case. Fig. 1 1 applies where we can determine Ry

from the following argument. Near the Doppler-cooling

limit, the cooling rate is approximately [see, for example,

Eq. (lO)ofRef. 39]

dE,

dt R

2E^

•Fty + 1 (CI)

where / is the laser beam intensity on the ions, Gq is the resonant scattering cross section, coi is the laser frequency, E^ is the total axial energy, and / is assumed to be

below saturation. We assume that the recoil energy R is such that R «-fiy, and that the laser is tuned to

(Oq — cO[^ —7/2 for minimum temperature, where coq is the

rest frequency of the cooling transition. Equation (CD

apphes as long as the Doppler width of the cooling transition is less than the natural width y. Equation (CD can

be rewritten

dt iE^-Hy /2)= -N,^(E,—ny /2 (C2)

where N^=IaQiia)i^ is the resonant scatter rate. {N^<y

by the assumption that the laser intensity is below saturation.) The energy decay rate for the equivalent circuit in the left-hand side of Fig. 11 is equal to R^^/lj^; so we make the identification that laser cooling can be

represented by a resistor R^ at temperature T^y where

R!^ = l^N,R/{fiy/2) , (C3)

and where^'* lj^ = lj/N, /, is given in Eq. (2a), and

Rn, Tn

Ct

V

ion in trap

FIG. 1 1 . Use of a cloud of A^ ions ( /„ = /, /N) as a continuous

refrigerator to cool a source mode represented by a series L,C,R, circuit The resulting source "temperature" T* is as- sumed to be given by (;';>/, /Arg, where /\ is the total rms

current in the inductor /,

.

TN-2Q
42 QUANTUM-LIMITED COOLING AND DETECTION OF RADIO- . 2993

Tf^- = 'fiy /2kg is the Doppler-cooling limit. ^^'^^ For

laser-cooled Be"*" ions as the refrigerator, y /2Tr=\9A

MHz assuming laser cooling on the 313-nm first reso- nance line. Then /? y -2.2X 10'*[iV,(s "')/iV] CL. Since

N^ can be as high as approximately 10^ /s, extremely

large values of resistance at the Doppler-cooling-limit

temperature [rylBe''" )==0. 5 mK] can be obtained.

The main problem in achieving strong damping of source modes is the shunting effect of the trap capaci- tance Cj. This is not a problem as long as the coupling

of the source oscillator mode to external heat sources is weak (R^—^0), in which case the source mode comes to the temperature Ty of the cooled ion. This would be a good approximation in the example where the axial motion of an electron is cooled by the axial motion of an

ion (Appendix B). In principle, the shunting effects of the

trap capacitance can be eliminated by using an inductor

in parallel with the trap capacitance. This might be used

to advantage in some cases, but the heating effects due to

losses in this inductor may outweigh the benefits of the

capacitance cancellation.

The refrigerator ions can also be used to cool a mode of

a macroscopic oscillator such as a quartz crystal. The

equivalent circuit of a quartz-crystal resonator is essentially the same as that of ions in the trap; that is, a series

LqRqCq circuit in parallel with electrode capacitance

Cq. To get an idea of the degree of cooling that could be

obtained, assume" that £^ = 19.5 H, Rq = 65 Q., (LgCg)"'^- = 27rX2.5 MHz, Q^5X10^ and Co^4pF.

Assuming the same Be"*" trap conditions as in Appendix

B, we require A'^^/,/Lg =^50 000 to make If^^lg. Cj is in parallel with the much larger quartz-resonator elec- trode capacitance Cq. Therefore Cj^-Cq in Fig. 11. Adjusting the laser cooling to make Ri^ = l/coC-p, we numerically evaluate the energy in the source mode {i^)L^

to be equivalent to a temperature {il)L^/kg^3>5 mK,

where we have assumed that T^=4.2 K. Since crystals

with much higher ^'s (lower values of Rq) might be obtained, perhaps much lower values of temperature might

eventually be reached. For example, Ref. 64 reports a 1- MHz crystal with a Q of 4 X 10^ at a temperature of 2 K.

If this same Q could be obtained on our example crystal,

then R^ ->0.08 il, and ( i^ )L^ /k^ =0. 54 mK ^ T^.

By use of suitable mechanical transducers and/or of in- termediate transfer oscillators, such as quartz crystals,

macroscopic oscillators of much larger mass, such as gravitational antennas, may possibly be cooled. Detection of energy changes in these macroscopic resonators can then be detected by the reaction back on the ions as described previously.

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2994 D. J. HEINZEN AND D. J. WINELAND 42

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TN-31
32 a reprint from Journal of the Optical Society of America B

Sisyphus cooling of a bound atom

D. J. Wineland

Time and Frequency Division, National Institute of Standards and Technology, Boulder, Colorado 80303

J. Dalibard and C. Cohen-Tannoudji

Laboratoire de Spectroscopic Hertzienne de TEcole Normals Superieure, and College de France, 24, Rue Lhomond,

F-75231 Paris Cedex 05, France

Received May 28, 1991; revised manuscript received September 6, 1991 Cooling that results from optical dipole forces is considered for a bound atom. Through optical pumping, the atom can be made to feel decelerating optical dipole forces more strongly than accelerating optical dipole forces. This effect, which has previously been realized for free atoms, is called Sisyphus cooling. A simple model for a bound atom is examined in order to reveal the basic aspects of cooling and heating when the atom is confined

in the Lamb-Dicke regime. Results of semiclassical and qviantum treatments show that the minimum energy

achieved is near the zero-point energy and can be much lower than the Doppler cooling limit. Two practical

examples that approximate the model are briefly examined.

1. INTRODUCTION

Interest in laser-cooled atoms and atomic ions has continued to increase over the past few years. The simplest

cooling scheme that has been proposed is called Doppler

cooling, in which the atom experiences a damping force owing to spontaneous scattering of light.' This cooling

applies to a two-level system. That is, it can be explained

by a consideration of scattering between a single ground

state and a single optically excited state. It leads to minimum temperatures (the Doppler cooling limit)

Td — fiT/2kB when hT » R. Here 2Trh is Planck's constant, r is the decay rate from the excited state, ks is Boltzmann's constant, and R = (fikf/2m is the recoil energy, where A = 2TT/k is the wavelength of the exciting ra- diation and m is the mass of the atom. For strongly

allowed (large-F) electric dipole transitions in atoms and

ions, this leads to temperatures of approximately 1 mK.

In 1988 an important experiment by the group of

Phillips at the National Institute of Standards and

Technology, Gaithersburg, Maryland,^ showed that temperatures of laser-cooled sodium atoms were substantially

below the minimum temperature predicted theoretically

for two-level systems. A theoretical analysis and subse- quent experiments^"^ showed that the new cooling mechanisms responsible for these low temperatures were based on a combination of several effects such as optical pumping between more than one ground-state sublevel, light

shifts, and polarization or intensity gradients.

The idea of the new cooling mechanisms is that laser beam electric fields E create a polarization d in an atom

that depends on the ground-state sublevel and on the laser polarization. This dipole interacts with the laser field, and the reactive part of this interaction induces an energy

shift of the ground-state sublevels, which varies from one

ground-state sublevel to the other. This shift is called the light shift or the ac Stark shift. If the laser intensity

or the laser polarization varies in space, these light shifts,

which are now position dependent, give rise to a dipole or

ponderomotive force equal in magnitude to the gradient of the atomic energy. Optical pumping takes place between

the sublevels, so if the laser polarization/intensity varies

in space, the relative populations of the ground-state sublevels are also position dependent. For an atom moving

arbitrarily slowly in a spatially periodic light field, the

averaged force is equal to zero, because the deceleration

force experienced by the atom in the ascending parts

of the potential curves (associated with the positiondependent light shifts) is offset by the acceleration felt in the descending parts. However, optical pumping takes a

finite time to occur (the pumping time Tp is inversely pro- portional to the laser intensity /). Therefore, in general,

for a moving atom, it is possible to arrange the laser tun- ing and the polarization/intensity gradients so that the

atom has time to run up one of the dipole potential hills (thereby losing kinetic energy) before being optically

pumped into another ground-state sublevel where the

dipole potential has less effect on the kinetic energy. After a while, the atom is returned by optical pumping (or by some other relaxation mechanism) to the original

ground state, and the process is repeated. Since the atom

appears to be running up the potential hills more than

down, this cooling has been called Sisyphus cooling after the Greek myth® (see also Refs. 10 and 11). Cooling that

is due to static electric and magnetic fields in combination

with optical pumping has also been considered. '^'^ The Sisyphus cooling mechanism was first introduced

for a two-level atom moving in a high-intensity laser standing wave.® The required multilevel structure of the atom was then provided by the dressed states of the atom

in the strong laser field. However, the time lag appearing

between the dressed-state populations of a moving atom

and the corresponding populations of an atom at rest is, for a two-level atom, of the order of the radiative lifetime

Tfl - r~' of the atomic excited state. At low intensity the

optical pumping time tp for a multilevel atom can be much

longer than tr. Because of this, the new cooling mechanisms based on optical pumping and light shifts can lead

0740-3224/92/010032-ll$05.00 TN-32
Wineland et al. Vol. 9, No. 1/January 1992/J. Opt. Soc. Am. B 33

to friction coefficients as high as Sisyphus cooling for twolevel atoms but at lower intensity, since, with longer time

lags [tp » tr), smaller velocities can give rise to comparable decelerations.

Doppler cooling exhibits the damping of the sponta- neous scattering force, whereas Sisyphus cooling uses the

decelerating effect of the dipole force combined with a dis- sipation of potential energy by spontaneous Raman processes. For completeness, it is useful to identify a third

kind of cooling, which may be regarded as due to optical

pumping of the atom into a low-energy state that no longer

interacts with the laser. Velocit5'-selective coherent

population trapping®''' and resolved sideband cooling of

trapped ions'^'*^ are of this category. In both cases the

atoms (ions) are pumped into low-energy states that inter- act only weakly with the laser. This interaction becomes

weaker the smaller the velocity becomes or, in the case of sideband cooling, the narrower the cooling transition is. Related schemes, which put atoms into a low-energy state that no longer interacts (or interacts weakly) with the

laser light, have been proposed."

Sisyphus or polarization-gradient cooling® has been an

important development for cooling and manipulation of atoms. For free atoms Sisyphus cooling gives temperatures down to values where the kinetic energy of the

atoms is approximately equal to the depth of the potential

wells associated with the position-dependent light shifts. This limit applies down to the point where the dipole well energies are of the order of the recoil energy R, in which

case the limiting energy is approximately equal to R.

In this paper we investigate the possibility of extending

the Sisyphus cooling mechanism to trapped atoms or ions. We propose a specific scheme and show that it can lead to temperatures much lower than the Doppler limit. Instead

of permitting the atom to move freely in space, we assume

that the atom is bound to a localized region of space by some external potential. We assume that the intensity

gradient is due to a standing-wave laser field. By a local- ized region of space we mean a region whose spatial extent

is much smaller than the wavelength of the cooling laser divided by 27r (e.g., k'^). That is, the atom is confined in the Lamb-Dicke regime. For simplicity of discussion, the

atom is assumed to be bound in one dimension by a har- monic well characterized by the oscillation frequency a>„. The spread of the minimum-energy state is given by Xq —

{h/2mwS'^, where m is the atomic mass. The spatial ex- tension of the wave functions then scales as Xo, and the condition Xo < (spatial extent) « 1/k is equivalent to

ftoj,. » R. When R » hw^., the atom, even when it is near its cooling limit, will always sample many peaks and

valleys of the standing-wave laser field, and the cooling

limit should be adequately described by the theory for free atoms. For a two-level bound atom the Doppler cooling limit is achieved when the natural width F of the optical transition is much larger than the oscillation frequency oi, ' and

R/fi. The opposite condition, f « w,., corresponds to what is called resolved sideband cooling. In this regime it has been shown''''^ that it is possible to achieve conditions

in which the difference between the minimum energy and

the zero-point energy is less than the recoil energy as long

as w,. » R/ti. (For this case, however, /cbT > fiV.) In order to compare Doppler cooling and Sisyphus cooling in

the same conditions, we will therefore assume here that r » w^.. The main result of this paper is that, when

r » w,, and hoj^, » R, Sisyphus cooling leads to minimum temperatures T^ such that knT^ — ftw,, whereas

Doppler cooling leads to temperatures Tu such that

^bTd — hV, that is, higher than T, by a factor of r/co^. To illustrate the basic ideas, we consider a simple

model system, which is described in Section 2. Morecomplicated cases can be generalized from this example.

From a semiclassical treatment (atomic motion treated

classically) given in Section 3, we will be able to derive

cooling rates and limits. From this treatment we will see

that the minimum energies are approximately equal to

fiWu- At these energies we might expect quantum effects of the motion to be important, so we will examine the cool- ing limits, treating the motion quantum mechanically in Section 4. In Section 5 we briefly compare the cooling

rates and limits achieved here with those obtained from

Doppler cooling. Although the model was chosen to be

simple enough to show the basic effects, in Section 6 we

examine a couple of possible experimental examples that

closely approximate the simple model.

2. MODEL

A. Atomic Level Structure and Laser Configuration

We consider Sisyphus cooling of a single bound atom. We

assume that the atom's internal structure is represented

by three levels as shown in Fig. 1. We label these levels g

for ground state, e for excited state, and r for reservoir

state. We assume that the transition ^ -^ e is an electric dipole transition excited by a standing-wave laser beam

whose frequency w is tuned above the resonance frequency

Wo for this transition by an amount 8 (where all frequencies are expressed in radians per second). Level e spontaneously decays at rate T. It decays to level r with

branching fraction (3 and to level g with branching frac- tion 1-/3. We assume that the laser's intensity and de- tuning are such that the fraction of time the atom spends

in the excited state is negligible. We assume that level r

is transferred back to level g at a rate Rr^g—l/Tr, where

the duration of the transfer process is so short {«Tr)

that one can consider the position of the ion as remaining

unchanged during the transfer. We assume that the atom

is confined to one dimension (the x direction) by a har- monic well f/o [Fig. 1(b)] characterized by the atom oscilla- tion (or vibration) frequency w, . Here, Uq is assumed to be an external potential that is independent of the laser beam. When the atom spontaneously decays from level e, the direction of photon emission is assumed parallel (or antiparallel) to the atom's motion. We assume that

E,, - Eg » \Er - Eg\, so that the effect of recoil heating

is mathematically simple. These simplifying assumptions will not qualitatively affect the results for other, more complicated cases. A key element in Sisyphus cooling is that the energies of the dressed states or of the light-shifted ground-state sublevels vary along the direction of the atomic motion.

Such a situation will be achieved here if we take a laser intensity that varies linearly with .v over the range of atomic motion. We are concerned mainly with the case in which this intensity gradient is due to a laser standing wave along the .r direction (see also Appendix A). We then

TN-33
34 J. Opt. Soc. Am. B/Vol. 9, No. 1/January 1992 Wineland et al.

(a)

(b) 1.3,-^

Fig. 1. Model system, (a) The atom has three internal energy

levels: g,e, and r. A standing-wave laser beam drives g ^ e above its resonance frequency. Level e decays with branching

fraction /3 to level r and with branching fraction 1 - yS to level g. Transfer from level r to ^ occurs at a rate Rr-g- (b) The atom is assumed to be confined by a harmonic well in the x direction. The maximum extent of the atom's motion is assumed to be less than A/2— (Lamb-Dicke regime).

write the laser electric field (classically) as E = (Eniax/2)[C0S(/jX - Cot) + COS(/jX + (Ot)] = Emax COS wt COS kx , (1) where x is the atom's position and the wave vector

k = 2tt/\, where A is the laser wavelength. According to Eq. (1), the laser intensity I{x) varies as cos'^(^x), as shown

in Fig. 1(b). Sisyphus cooling will be most efficient when

the intensity gradient is a maximum; this occurs at the

half-intensity point of the standing wave. Therefore we

assume that the atom's average position is near x - n/Ak

[Fig. Kb)]. The linear variation of the light shift of level g with X over the range of the atom's motion follows from

the localization assumption: The extent of the atom's motion is less than A/27r (Lamb-Dicke regime).

In this paper we focus on the regime

Rg^ryRr-H « W„ « r « 5, (2) where R^^r and Rr-i^ are, respectively, the rates of transfer from g to r when the atom is in level g and from r to ^

when the atom is in level r. We also assume that the laser intensity is weak enough to avoid any saturation of the g —' e transition. That is,

s =

2a>,^ 4^^ + V

« 1. (3)

In Eq. (3), s is the saturation parameter and wi =

{e\d\g) Emax//' is the Rabi frequency. In this lowsaturation regime the population of the excited state is small, and we can derive rate equations for the evolution

of the lower states r and g. The condition F « 5 implies that the light shift of g

[Eqs. (13) below] is larger than the contribution to the

width of this level that is due to the photon scattering

rate. The condition w,. « T has already been discussed

in Section 1 as defining the regime in which the Doppler

limit is achieved for a two-level bound atom. In a point of view in which the atomic motion is treated classically

(semiclassical approach), the condition Rg^r,Rr-^g « dv means that the bound atom makes several oscillations in the harmonic binding potential before being transferred

from g to r or from r to g. It is then possible to neglect the variation with space of the populations of levels g and

r and to consider only their average over an oscillation period. This greatly simplifies the calculation of the rate of variation of the atomic external energy in the semiclassical approach. From a fully quantum point of view, in which the atomic motion is quantized, the condition Rg^r,

Rr^g « CO,, means that the width of the vibrational levels in the binding potential, owing to the excitation rates from g or r, is smaller than the spacing hcoi, between these

levels. This introduces a similar simplification into the

calculations, since it permits a secular approximation to be used. The secular approximation consists in neglecting, in the master equation describing the evolution of the atomic density matrix a, any coupling between the populations of the vibrational levels and the off-diagonal ele- ments of a. In this way we can obtain a set of equations

involving only the populations of the vibrational levels, which has a simple interpretation in terms of transition

rates. This is an example of a situation in which the rate of variation of atomic internal variables (Rg^r, Rr^g) is slower than the rate of variation of external variables {mu).

A similar example was recently studied in connection with

the quantization of atomic motion in optical molasses.^*

B. Qualitative Explanation of the Cooling For 6 > 0, as shown in Fig. 1(a), the electric field of the

laser shifts the level g to higher energy. Because s « 1, the atom is primarily in g or r. When it is in the state g

(more precisely, in the dressed state, which contains the

larger admixture oi g), it is subject to an additional potential energy Ui{x), which is a function of x proportional to the laser intensity. For the conditions of Fig. 1(b) the ef- fect of this potential for the limited range of the atom's motion is to give an additional force in the -\-x direction. A crude explanation of the Sisyphus cooling is the fol- lowing: As the atom oscillates back and forth (with fre- quency w,,) in the well U{){x), it experiences an additional

potential hill Udx) sloping down toward the +x direction. The atom is more likely to be transferred to level r by

spontaneous Raman scattering when it is in regions of higher intensity. Therefore the predominant effect is that, after the atom runs up the laser hill f//,(x) in the -x

direction (therefore losing kinetic energy), it is transferred to level r, where UiXx) = 0. After a time of order

Tr it is transferred back to level g, where on the average it must run up the hill U/ix) again before being transferred

to level r. This leads to a net cooling effect.

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Winelande^a/. Vol. 9, No. l/January 1992/J. Opt. Soc. Am. B 35

Uq(x) (atom in level g)

Ur(x) (atom in level r)

Fig. 2. Qualitative explanation of the cooling. When the atom

is in state r, it experiences the confining potential t/,(x), which is unperturbed by the laser fields. When the atom is in level g, the

potential well is shifted in the +x direction by the dipole force of the laser. This shifted potential is represented by Ug(x). The

laser excitation is assumed to be weak enough that the atom

spends a negligible amount of time in level e. Therefore the spontaneous Raman transitions g ^ e ^>- r can be represented by

transitions between g and r such as those indicated by (I) and (II) in the figure. Process (I) is favored more than process (II) be- cause, when the atom is on the left-hand side of the well Ug, the chance of g^ ^ e excitation is higher since the laser intensity is higher [see Fig. 1(b)]. Process (I) leads to cooling because the atom drops to a lower part of the well Ur. Process (II) causes

heating and, along with the heating resulting from recoil, eventually balances the cooling, resulting in a minimum energy.

A more accurate explanation is the follow^ing: When

the atom is in the state g, it experiences the combined

potential Uq + Ui - Ug, which is a well with frequency co^ shifted by an amount Axq in the +x direction (upper part

of Fig. 2). When the atom oscillates in the -x direction

in the well Ug, it is more likely to be transferred to level r by spontaneous Raman scattering because the laser intensity is higher. It can therefore drop into a lower part of

the unshifted well Ur{x) = Uo(x), keeping the same posi- tion X since the duration T"' of the transfer process is much smaller than the oscillation period Wt,"^ in the binding potential. It therefore loses potential energy [process

(I) of Fig. 2]. After a time of order Tr (which is independent of the position x), the atom is transferred back to

level g and the process is repeated. Heating, as illus- trated by process (II) in Fig. 2, in which the atom gains

potential energy, can also occur but is less favored because

the laser intensity is less in the +x direction. Spontaneous Raman scattering (g ^ e —* r) occurs at a

rate proportional to the intensity, which according to

Eq. (1) can be expressed as

Rg^r « Emax I,, , ,1 — " cos {kx) = /I X = —

'-''^^-u

(4) The term -2kxl{x = -n/Ak) is responsible for the cooling.

The term that is independent of x causes heating because

of two effects. First, for each scattering event, photon

recoil causes an average increase in energy 2R}^ Second,

when the atom changes from level g to level r, the well switches from U^ to f/,.. As we will see in Section 3, the corresponding potential energy change, when it is averaged over one oscillation period, leads to heating. A similar effect occurs in the transitions from r to g. The

balance of the cooling with these sources of heating yields the minimum kinetic energy for the atom. Since 5 > 0, there is some additional Doppler or spontaneous force antidamping that tends to increase the atom's energy; this

is shown in Subsection 5.B to be negligible for the conditions assumed here. The cooling is the same when the atom's well is in a region of positive slope [e.g., x = ^-rr/Ak in Fig. 1(b)]. The

cooling effect goes to zero when the well is centered at points of zero slope [e.g., x = 0, Tr/2k, and v/k in Fig. 1(b)].

3. SEMICLASSICAL TREATMENT

In this section we treat the external motion of the atom

classically. Such treatment, often called semiclassical, is valid if the atomic de Broglie wavelength is small compared with the length scale (the light wavelength A) over which the field varies.^" In principle, it should also re- quire that the extension of the atomic wave packet be

much larger than the size of the ground state of the con- fining potential. Actually, we will see that the latter condition can be dropped, and the semiclassical and quantum

results are identical even if the atom is mostly in the

ground state of the well.

A. Internal Atomic Dynamics

The internal states of the atom are treated quantum mechanically. In the low-intensity domain of interest here

[relation (3)], the optical Bloch equations, giving the evolution of the atomic density operator, can be simplified by an adiabatic elimination of the optical coherences (off- diagonal matrix elements of the density operator between

ground and excited states). We are then left with a set of

rate equations for the populations tt, (7 = g, r, e) of the

atom's three internal levels^"'^^:

irg = -Rg^,{x)(7Tg - TTe) + Rr^gTTr + (1 " PWlTe , (5a)

TTe = Rg^e(x){'rrg - TTe) - TVe , (5b)

TTr = -Rr-g-rVr + PTVe , (5c)

where

Rg^Ax) = iTs/2)cos\kx). (6)

Since we have assumed that s « 1, the scatter rate

Rg-,e(x) is much smaller than T and the population of the

excited state, Ve, is small compared with 1. For times

long compared with F"', we have therefore approximately

TT, = Vg{s/2)cos\kx)

,

TTg = — Rg-.r{x)'TTg + Rr^gTTr,

TTr -71,«'

where

Rg .Ax) = f3Rg-.Ax) = l3(\s/'^)cosAkx).

(7)

(8a)

(8b)

(9)

In the limit Rr-g, Rg~.r « Wt. and when we use

Jig + TTr — 1, tne steady-state solution of Eqs. (8) is simply

TN-35
36 J. Opt. Soc. Am. B/Vol. 9, No. l/January 1992 Wineland et al.

Rr-

' {R,^r) + Rr

{R,-r)

TTr (i?,_.) + Rr-

(10a)

(10b)

where (Rg-r) stands for the average of Rg^r(x) over the

atom oscillation period for the atom in level g (recall that we have assumed that Rr^g does not depend on x). The

time constant Tint for the relaxation of these internal vari- ables is given by

1/T,„t = (Rg^r) + Rr- (11)

B. External Atomic Dynamics

When the atom is in level r, or when the atom is in level g

in the absence of the laser, the center of the atom's well is denoted Xro- Therefore the atom's harmonic well can be

described by

Ur{x) = Uo{x) = V2m(l)^!^{X - Xro) (12) When the atom is in level g in the presence of the laser

field, the center of the atom's well is shifted by the laser

light. Using the dressed-state formalism,® we find that the energy of the state g is shifted by an amount

Ul{x) = -^-[wi^ cos'(^x) + 5']''2 _ l^^u^^ cosHkx),

(13a)

where

Vu, = fta;,74S (13b)

is the maximum value of [/£,. Equations (13) are valid for

5 » Y,wi. The internal state 1^^) changes to the dressed

state 1^)', but |^)' — \^ in the limit a>i « 5. The center

of the atom's well, jr^o, when it is shifted by the dipole

potential, corresponds to the spatial minimum of the total potential for the atom in level g:

Ugix) = Urix) + Udx). (14)

Since the cooling will be a maximum where the intensity

gradient is maximum, we choose kxro = tt/4. Assuming

that the shift Axo between the two wells is small compared with A/27r, we linearize Ui{x) around x^o to obtain

Udx) = Ulo/2 - kUioix - Xro). The shift Axq is then

where

Axo = Xgo - Xro = 2^(kxo)xo

f = U,.o/hw,

(15)

(16a)

(16b)

and Xo = (h/2mo)^Y'^ is the spread of the zero-point wave

function in the harmonic well. We show in Subsection 3.D that the minimum kinetic energy is achieved for Uu) — ho),, that is, for ^ = 1. Therefore, since we have

assumed the Lamb-Dicke criterion kx^ « 1, the shift Axo

of the well U,,(x} that is due to the dipole potential is, for minimum kinetic energy, much smaller than xo and thus

much smaller than A/2tt, as we assumed in deriving

Eq. (16a).

C. Cooling Rate The variation of the external (kinetic -I- potential) en- ergy E:, can be expressed by the equation

i, = TTg{Rg^r{x)[Ur{x) - Ug{x)]) + 7rrRr^g{Ug(x) " t/,(x)) + TTg{Rg^Ax))2R

.

(17) The first two terms stand for the average change in potential energy as the atom goes from g to r and from r to g, respectively. They therefore contain both Sisyphus cool- ing and the heating that is due to the sudden switching of the wells. Recoil heating for the g ^> r transfer is con- tained in the third term of Eq. (17), and spontaneous force Doppler antidamping (heating) can be neglected (Subsection 5.B). The averages are taken over one oscillation period in the initial potential [Ug(x) for the first term of Eq. (17) and t/, (x) for the second].

As in relation (4), we linearize the rate Rg^r(x) [Eq. (9)] around x^o:

Rg^^r{x) - i3(rs/4)[l - 2^(x - x.o)], (18)

and we use the equipartition of energy for the atom oscil- lating in Ug{x):

E^ 2( —mco,\x - Xgof (19)

For times long compared with T,nt, we then obtain, using

Eqs. (10) for -n-g and tt-^, E,= -(1/t,)(£, - E^o),

where the steady-state energy E^^o is

(20a)

mw„^(Axo)

Exo = ~~ 1- R

2k kpiAxo) 2

1 + e^

2 ''h (20b)

and the Sisyphus cooling time constant Tj is given by

1 A{Rg^r)Rr^

{Rg-^r) + Rr~ R

(20c)

For the conditions of interest (^ = 1, R/hw^ « 1), one can

check that t^ » TjntD. Cooling Limit

In steady state the atom's external energy is reduced to E^o [Eq. (20b)]. E:,o is minimized when the laser intensity

is adjusted to make ^ = Uio/hwu - li'^'^, in which case

^xo"'" = hcojVp = Ul. (21) From the second expression in Eq. (21) the minimum ex- ternal energy is equal to the depth of the wells created by

the standing-wave laser beam. This agrees with the re- sult of Sisyphus cooling of free atoms.* When jS = 1, this predicts an energy near the zero-point energy for the

ion in its well. In such a situation we might question the

validity of this approach, which treats the atomic motion

classically. Therefore it will be useful to compare

the cooling limit derived here with the results from a quantum-mechanical treatment of the atom's motion.

TN-36
Wineland et al. Vol. 9, No. 1/January 1992/J. Opt. Soc. Am. B 37

Fig. 3. Diagram for the cooling when the motion is quantized.

Each electronic state has a sublevel structure of harmonicoscillator levels labeled by the quantum numbers rij, where^' = g,

e, or r. Cooling is represented by scattering processes of the form ng ^ rie ^- rir, where rir < rig. This may be viewed as due

to spontaneous Raman scattering off states |^, Ug)' and |e, ne)", which are dressed by the laser.

4. QUANTUM TREATMENT

A. Transition Rates

In this section we treat the atom's motion in its well quantum mechanically. We describe the cooling process as shown in Fig. 3, where laser photons undergo spontaneous

scattering from the dressed states, which now include the

perturbed harmonic-oscillator states (denoted by primes

and double primes).

Cooling results from spontaneous Raman scattering processes of the form \g, rig)' -* \e, n^)" -^ \r, rir), where

rir < ng. States \g,ng)' and |e, ^e)" are assumed to be

dressed by the standing-wave laser field. We can calculate the cooling and the cooling limit as was done in

Ref. 19. For practical purposes the cooling rate is ade- quately represented by the semiclassical treatment of

Subsection 3.C. Here we are interested primarily in the

cooling limit, since the semiclassical treatment predicts a minimum energy approximately equal to ^w„. We can

calculate the minimum energy, using Eq. (27) of Ref. 19.

It will be more instructive, however, to calculate the rates

for each process rig -^ rir separately. According to Ref. 19, these rates can be written as

r = r

(nr|exp(-jks • 'S.\ne)"{{ne\')f{X)\ng)' 8 - (Er,, - E,)/h + ir/2

(22)

where C is a factor that includes the laser intensity (at the

position X = v/Ak) and matrix elements for the g -^ e transition, X is the atomic position operator, and k, is the wave vector for the scattered photon. For simplicity we

will assume the scattered photon to be in either the +x

or the -X direction, so that k^ \ - ±kX. f(X) is a function representing the x dependence of the laser's electric field amplitude near x = w/Ak; from Eq. (1), f(X) —

1 - k(X - Xro)- £„,, and E,,^ are the energies of the indi- vidual harmonic-oscillator levels for the excited and

ground states, respectively. The sum is performed over

all possible excited harmonic-oscillator states. Since we have assumed that 8 » cju, the denominator

in the sum in Eq. (22) can be approximated by 8. This

amounts to neglecting Doppler antidamping; the validity

of this approximation is discussed in Subsection 5.B. With this approximation and the closure relation over the

states \ne)", Eq. (22) simplifies to

'g.le—r p{C/8')\{nr\exp{-i]f., X)fiX]\ng)'\ (23)

When )3 5!i 1, we must also consider scattering directly

back to the ground state from the excited state. These

direct rates give rise to additional recoil heating for the

g -^ e ^ g scattering process:

' g. ns^g. ng' (1 - p){C/8')\{{ng*\'}expi-iK " X)f{X)\ng)'\'

(24)

where rig* is the final harmonic-oscillator state in the di- rect scattering process. Finally, we must consider the

change in harmonic-oscillator energy levels in the transi- tions r —> g. As in the semiclassical treatment, this leads

to a heating resulting from the sudden switching of the

harmonic well (even without any spatial variation of the

transfer rate Rr^g)- This process is described by the rate

^r,nr^g.ng = C'|{(n,|'}l|n.) (25)

where C is a constant characterizing the rate of the

r ^ g process. In Subsection 4.D these rates will be used

in a master equation for the populations of the harmonicoscillator levels. Before that is discussed, two preliminary

steps are required. First we must find an expression for the perturbed harmonic-oscillator wave functions.

Then we will obtain simple expressions for the rates in Eqs. (23)-(25).

B. Dressed Harmonic-Oscillator Wave Functions

As we discussed in Section 1, the dipole force acts uni- formly over the extent of the wave functions of interest in the Lamb-Dicke limit. Therefore the effect of this dipole

force is simply to shift the center of the harmonicoscillator well for the^ state to x^o + Aatq, the value calcu- lated semiclassically. (This can be explicitly verified by

the use of second-order perturbation theory to dress the wave functions.) The wave functions \ng)' are simply

obtained by using the spatial translation operator

\ng)' = exp(-iAxoPt//!)|ng), (26)

where P^ is the x component of the atomic momentum

operator. In the following, we will require both the position and

momentum operators X and P, in terms of raising and low- ering operators a* and a: X = Xo{a' -h a),

Pj. = ixomw,.(a* - a)

.

C. Evaluation of the Rates Using Eq. (26) in Eqs. (23)-(25), we find that

Tg,ng-.r.nr = PiC/8')\{nr\A,\ng)\\

r«..,^,.V = (1 - li){C/8')\{ng*\A,\ng)\\

^r.nr^g.n, = C'|(ng| Asln,)] ^

(27)

(28)

(29a)

(29b)

(29c)

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38 J. Opt. Soc. Am. B/Vol. 9, No. l/January 1992

where

Ai = exp(-ik,X)f{X)exp(-i\xoPJh), (30a)

A2 = exp{iAxoPx/fi)exp{-iksX)f(X)exp{-iAxoPx/h),

(30b)

A3 = expi-ilxoPJfj) (30c)

and where k, — ±k, depending on the direction of the

spontaneously emitted photon (assumed to be along the + j: or the -x direction).

We now replace the position and momentum operators X

and Pt in Ai, A2, and A3 by their expressions in terms of a and a* [Eqs. (27) and (28)]. It is convenient to change the

origin of the x axis to Xro, in which case f{X) — 1 - kX.

Since kxo{n + if'^ « 1 (Lamb-Dicke criterion), we keep

only terms in first order in kxoa or kxoa\ This gives

Ai = 1 - kxo[a\l - ^ + ikjk) + c(l + ^ + ikjk)],

(31a) A2 - 1 - kxoia' + a)(l + iks/k), (31b)

A3 = 1 + ^kxo{a - a^)

.

(31c)

Therefore, in the Lamb-Dicke limit, the important rates are those for which the vibrational quantum number

changes by 1, and we have

C

8' ^g.n-T.n P o2 ~ ^g-Tt (32a)

r^,„^,,„,i = Rg^rikxofin + 1)[1 + (1 - 0'], (32b)

r^.„^,,„_i = Rg^rikxoMl + (1 + ^f], (32c)

^r.n^g.n = C' = Rr—g, (32d)

r,,„^g,„.i = Rr^,ikxof(n + 1)^' n + I

. n-»g, «-l >

(32e)

^g.n^g.n + i = ;:; Rg-.r2{kXQf{n + 1) = ( ]rg,n^g.n g,n-l •

(32f)

D. Rate Equations and Cooling Limit

A master equation for the populations takes the form

IT,g.n

n + 1 2 ^ i-^g.n-.t.n-rrg,n + ^i. n-^g.nTT^.n) , (33a)

n'=n-l i = g,r

n + l

'^r,n ^ \ ^ r,n-*g,n''^r,n ' * g,n'-T,n'^g,n) y ri=n-\ (33b)

where tt^^ denotes the population with internal state g

and external state n, etc. As expected, in steady state these equations show, after a little algebra, that the total population going from n to ^ + 1 is equal to the total population going from n + 1 to n. That is,

\^ g,n-*r,n^l ' ^ g,n~*g,n + l)'^g,n ' * r,n~*g,n + l'^r,n ~ (l#,n + l—r.n t" ^ g.n + l^g.nl^Tg.n + l ' 1 r,n + l—^, n'Tl'r, n+ 1 • (34)

In order to exploit this result, we now look for an approximate expre.ssion for tt^ „ and tt^,, as a function of the total population of level n (vn = tt^ „ -I- Vr.n)- Equations (33),

Wineland et al.

written for steady state, give a relation between iTg „, TTr,n, TTg.n+i, and TTr.nti. If we assume that ^ is not large com- pared with 1 and recall that kxo{n + ly^ « 1, then

Eqs. (32) and (33) give Vg,n^r.nVg,n /^r.n-^g.nTTr.n =

1 + 0[(n -I- 1) {kxofl so

r.,„- r*£, n—T, n +'^ r' r,nr + r

(35a)

(35b)

Substituting relations (35) into Eq. (34) and using

Eqs. (32), we obtain

7r„.i _ 1 + m^ - 1)

77, 1 + mi +

1)

(36)

where we have neglected terms of order {n + 1) (kxof and

higher. For ^ « 1, we have -n-n+i/vn -^ 1. Vn+i/vn is minimized for ^ = ULo/fi(tiv = P~^'^, in which case

TTVfl

"^n / min v^

2 + Vp

(37)

Since, according to Eq. (36), iTn+i/vn is independent of n, we can write

exp - knT (38)

which shows that the system reaches a thermodynamic

equilibrium. For such a thermodynamic equilibrium,

(n„) = [exp(^a>;,/^BT) - 1]"\ in which case

(n„) = [1 + /3^(^ - l)]/2/3^ (39)

This is minimized for £, = /3"^'^, in which case

(n„) = /3-^'^- 1/2. (40)

This corresponds to a minimum energy

£;,o'"'" = hwA{n:) + 1/2) = hio,^-"\ (41)

in agreement with the semiclassical answer [Eq. (21)]. This remarkable agreement between the semiclassical

and quantum theories can be interpreted in the following way. The motion of a quantum particle in a harmonic

well is governed by the same basic equations as those for the classical motion. (This is particularly clear in the Wigner function approach and is a particular feature of the harmonic potential.) Consequently, the only place

where Planck's constant plays a role in the problem here is in the recoil heating, where R - h'^k'^/2m, and this recoil

is handled in the same way in both treatments. Therefore it should not be surprising that both approaches lead

to the same minimum energy, even though this energy is close to the zero-point energy in the quantum well.

5. DISCUSSION

A. Cooling Rate

It is interesting to compare the rate for the Sisyphus cool- ing discussed here with the rate for Doppler cooling.

Doppler cooling can be represented in Fig. 1 by the conTN-38
Wineland et al. Vol. 9, No. 1/January 1992/J. Opt. Soc. Am. B 39

ditions p = and 8 < 0. The rate for Doppler cooling has

been calculated by many authors; for example, from

Eq. (19a) of Ref. 22 the cooling rate is

td'' = -2RsDfik^SD{m[(r/2f + 8o']}- (42)

where Rsd is the total scattering rate, the subscripts D

stand for Doppler cooling, and the intensity is assumed to be below saturation (s « 1). For minimum temperature

using Doppler cooling, we want 8d - -r/2. For this case,

td'' = ARsoR/hV. (43)

According to Eq. (20c), assuming, for example, that

Rr^g » (Rg^r) (that is, -n-g—l), the ratio of rates for Sisyphus cooling and Doppler cooling is

td RSD ^ (44)

Therefore, for the minimum-temperature case (^ = 1), the

ratio of Sisyphus cooling to Doppler cooling is approximately given by the ratio (F/wi. )((Rg^r)/i?sD)- For

(/?g^r)/jRsD=l, Sisyphus cooling is faster by F/w,,. Comparing the Sisyphus cooling rate with the cooling

rate in the resolved sideband limit ''^'^''•^^ is not so straightforward because the conditions in which each applies are

different. For efficient sideband cooling (in a two-level atom), we want 8 - -w,. and F « w,., and these conditions are incompatible with relation (2). Therefore a comparison of cooling rates depends on the specific atomic

system considered. The practical advantages of one cool- ing scheme over the other will also depend on the specific

system investigated; this will depend, for example, on

the availability and the ease of operation of the re- quired lasers.

B. Contribution of Doppler Antidamping in Sisyphus

Cooling For the model considered here {8 > 0), Doppler cooling is turned into an antidamping term. We can use Eq. (42) to check that this heating effect can be neglected compared

with Sisyphus cooling. For the situation of interest here

(8 » F), we have

Td 4ng{Rg^.)\ R

/3 hS

{Rg^r)Rr^g

pl\{Rg^r) + RrBy comparing with Eq. (20c), we find that

Tp"^ / 1 \ W,

, — « 1

.

(45)

(46)

Therefore we may neglect the effects of Doppler anti- damping compared with Sisyphus cooling except when

P « 1.

C. Cooling Limit

The minimum energy for Doppler cooling is tir/2. Therefore, according to Eq. (21) or (41), the minimum energy for Sisyphus cooling will be smaller than that for Doppler

cooling by the approximate ratio io,./r « 1. Consequently, the Sisyphus cooling rate is more efficient (at low

temperatures, where the Lamb-Dicke criterion is satisfied) and the Sisyphus cooling limit is smaller than that

for Doppler cooling. The minimum energy for sideband

cooling and Sisyphus cooling is approximately the same,

since the ion can be cooled to near the zero-point energy

(h(i)J2) in either case. However, for some applications it may be desirable to cool to (n^) « 1, in which case side- band cooling appears to be the appropriate choice.

6. POSSIBLE EXPERIMENTAL

CONFIGURATIONS

To give an idea of how Sisyphus cooling might be employed, we consider the example of a single, trapped ^''Mg" ion in a magnetic field B. To make the problem tractable, we

will consider only the cooling of one degree of freedom,

namely, the axial oscillation in a Paul rf trap or a Penning

trap. To be consistent with the notation above, we call this the X degree of freedom. We will also make the simplifying assumption that the scattering rate for cooling

the other degrees of freedom is adjusted so that, on the

average, the recoil heating of the axial degree of freedom

appears to be due only to reemission along the -i-x or the -X direction from the laser that cools the axial degree of freedom. This is consistent with the assumption made in the calculations of Sections 3 and 4. The relevant internal energy levels are shown in

Fig. 4(a). We assume that a linearly polarized standing-

'Mg*

B—

- (+3/2) - -

-(+1/2)- -

- (-1/2)- -'

- (-3/2) - -

-(+1/2)- -

^ 1/3

^1/2^ - -- -(-1/2)

1/3

(b)

Fig. 4. The cooling described in the text could be approximately

realized by a trapped "^Mg^ ion in a magnetic field. A specific case is illustrated for the levels labeled g.e. and r in (a). Transfer from level r to levels could be realized with microwave radiation tuned to the r -^ g transition. This transfer could also be ac- complished by spontaneous Raman transitions ;• — e' —» ^ by

using a traveling-wave laser beam (laser 2 in the figure) tuned to the r —> e' transition. In this case additional recoil heating must

be accounted for (see the text). Another case that would apply to an ion or an atom with an outer unpaired electron and a spin 1/2 nucleus (and no intermediate electronic states) is shown in (b). Here the magnetic field is assumed to be small enough that the Zeeman structure is unresolved. Transfer from r io g

through spontaneous Raman transitions (laser 2) is indicated in the figure.

TN-39
40 J. Opt. Soc. Am. B Vol. 9, No. 1 January 1992 ^/^6 b/z

Wineland et al.

wave laser beam (k vectors parallel and antiparallel to x)

is tuned above the -S^oimj = +1/2) -^ ^Ps-zimj = -1/2)

transition frequency by an amount 5 [laser 1 in Fig. 4(a)]. We assume that B (parallel to x) is large enough that

fi8 « 2/XB-B/3, where mb is the Bohr magneton. This en- sures that there will be negligible excitation on transi- tions other than those of interest. One situation for approximate realization of the cooling limit described in Subsections 3.D and 4.D is the following: First we leave

laser 1 on long enough that the ion is pumped to level r with high probability [from Fig. 4(a), p = 2/3)]. We then

turn laser 1 off and transfer ions from level r to g with a microwave - pulse. (To accomplish the transfer, we need

not as-sume that the duration of the tt- pulse is less than

Wi-"'.) Laser 1 is then turned on again, and the process is repeated. When laser 1 is turned back on, we assume

that it is accomplished in a time long compared with co,''' but short compared with Rg^r'\ so n^ is unchanged as \g) becomes dressed. As far as the cooling limit goes, this situation is actually somewhat better than the model that we have assumed in Section 2 because we avoid the heat- ing resulting from well switching in the r ^- g transition. Mathematically, this amounts to setting A3 = 1 in

Eq. (29c). In this case the steady-state energy would be

given by (n,) == [2 -I- /3^(^ - 2)]/4/3£. This would be minimized for e = (2//3)"-, in which case («.)„,„ - (2/3)"'' 2 - 1/2. For /3 = 2/3,(n,,)n,in = 0.37. Experimentally, it might be easier to accomplish the

r -* g transfer with a second, traveling-wave laser beam

tuned near the ^Sniimj = -1/2) -^ ^P3 2{mj = -Hl/2)

transition [laser 2 in Fig. 4(a)]. (Additional cooling could be obtained with a second, standing-wave laser beam, but

this would be more difficult to arrange experimentally.)

The r -* g transfer is then accomplished by spontaneous

Raman scattering with probability /3' = 2/3 for each scat- tering event from laser 2. This basic scheme could also be realized on "Si 2 -^ ^P\ 2 transitions in ^^Mg^ or other

ions (atoms) with similar energy-level structure. Using laser 2 will lead to a slightly higher minimum en- ergy than our model predicts because of the additional recoil heating in the r ^- g transfer. To account for this heating properly, we must modify Eq. (17). Therefore we write

£/ = E, + TTrRr^A2R), (47)

where E^ is given by Eq. (17), R^^r is the excitation rate from r to e' [e' is the ^P32{'n.j = +1/2) level] that is due to the second laser, and Rr-g - 2i?r-.e/3. In steady state this amounts to doubling the recoil heating term in Eq. (17). For this case, E,o = ft(oA( + 3/^)/2, which is minimized for ^ = V3, yielding E^o = \/3fi(i), or

<n,) = Va - 1/2 = 1.23. For ^'Mg^ we take r/27r = 43.0 MHz and A -

279.64 nm. If we assume that ojJ2it = 2 MHz, the

conditions of relation (2) are reasonably satisfied for

rt/277 = 1 GHz and 6j,/27r = 0.118 GHz (obtained from

^ = VS), in which case fl^ .. = 4.68 x 10"' s"'. For these

conditions and laser 2 adjusted to make Rr^r - Rg^^, we

have T, = 17.4 ^t8. For the transitions of interest the reso- nant cross section is \^/4v: If laser 2 is tuned to reso- nance the required intensity is 5.34 mW/cm^. For one of

the beams in the standing-wave laser beam, we require an int ensity of 5.78 W/cm^ To satisfy the condition

h8 « (Im flT^ we require that B » 0.107T. The effective temperature of the ion is given by

T = ^aj,/[^Bln(l + (n,)-i)]. (48)

For (n,) == 1.23, T = 0.16 mK. For the same transition the Doppler cooling limit is (^i) = 10, corresponding to T = 1 mK. For (n,) < 1, Eq. (48) shows that the temperature is only logarithmically dependent on (n„). To

achieve lower temperatures, we require Wi: to be lower, but

if we make w,, too low, we violate the condition kxo « 1 (in the example above, kxo — 0.23). Another scheme that might be used to cool an ion in a

rf trap in a low magnetic field is illustrated in Fig. 4(b). This would work for ^^^Hg"^ ions, for example. Laser 1 provides the Sisyphus cooling, and laser 2 [or microwave

radiation between the ^Si 2 {F - 0) and (F = 1) levels] provides the repumping into level g. Similar schemes

exist in other ions or atoms.

Experimentally, it may be difficult to make the center

of the atom's well coincide with the point of maximum

intensity gradient. One way around this problem for the standing-wave configuration would be to permit the

standing wave to run over the center of the atom's position. This could be accomplished by the construction of

a moving standing wave composed of two counterpropagating laser beams separated by frequency co^. When

Rg~.r « iOm « w„, the above expressions hold, with the magnitude of the intensity gradient replaced by its value averaged over the standing wave.

7. SUMMARY

We have examined Sisyphus cooling for a bound atom for the condition in which the radiative linewidth T is much

larger than the oscillation frequency cj^. of the atom in its well. We have studied the strong-binding limit, which is given by the condition that the recoil energy be much less than the zero-point energy of the atom in the well. We

have examined the cooling, using a simple three-level model for the internal states of the atom. We have as- sumed that the atom is confined in one dimension and

cooled by a standing-wave laser beam. This model can be

generalized to more-complicated cases. Since the minimum energies are near the zero-point energy of the atom

in its well, we have examined the cooling limit, treating

the atom's motion both classically and quantum mechanically. Within the approximations of the model the cooling

limits are found to be the same. We found that the minimum energy of the atom occurs when the depth of the dipole potential well created by the standing wave is approximately equal to the zero-point en- ergy of the atom. In this case the minimum energy of the

atom is given by the condition that the mean occupation

number (n„) of the atom in its well be approximately equal

to 1. If we compare Sisyphus cooling with Doppler cooling, which operates on the same type of broad transition

(F » Wi), we find that Sisyphus cooling is much more ef- ficient for both the cooling rate (as long as we are in the Lamb-Dicke limit) and the cooling limit. Sisyphus cool

TN-40
Wineland et al. Vol. 9, No. 1/January 1992/J. Opt. Soc. Am. B 41

ing leads to a temperature nearly as low as that result- ing from sideband cooling without the requirement of a narrow atomic transition. We have briefly examined

experimental examples that assume atomic ions stored in

traps. The same ideas should apply, however, for neutral atoms confined by magnetic traps, optical dipole traps, etc.

W M. Itano and W D. Phillips for helpful .suggestions on

the manuscript.

The Laboratoire de Spectroscopie Hertz ienne de I'Ecole Normale Superieure is associated with the Centre

National de la Recherche Scientifique and the Universite

Pierre et Marie Curie.

APPENDIX A: COOLING IN OTHER

INTENSITY GRADIENTS

In the main text we have treated Sisyphus cooling in a

standing-wave laser beam. The main ingredient in the

cooling is that the atom experiences an intensity gradient

along its direction of motion. Therefore it is useful to consider cooling in a more general intensity gradient.

The cooling treatment of Section 4 applies, except that we

must replace k in relations (4) and (15), Eq. (16a), and

relation (18) by [d//dx,]/27 evaluated at the equilibrium

position of the ion, x,(0). If we define

1 [d//d;c,],, (0) 2/[x,(0)]

(Al)

then we find that the cooling rate is equal to the cooling

rate of Eqs. (20) multiplied by ^^, and the cooling limit (from the condition E^ - 0) is given by

E. [/i 1 + 1/e^'p

i-o" (A2)

This limiting energy is minimized for ^ = (2//3f^)^'^; which case

in

E.r" = hwjC^. (A3)

As an example, consider cooling in the intensity gradi- ent of a focused traveling-wave Gaussian laser beam

whose intensity in the direction transverse to the axis of the beam is given by 7(x,) = h exp(-2x,^/u;o^), where Wo is the beam waist. If we assume that the atom is nominally

localized to the position where /[jc,(0)] = /o/2, then

x,(0) = ±[(ln 2)/2Y'^Wo. For this condition, C =

[(In 2)/2t7^]^'^(A/«;o)- For a given w„, the cooling rate is reduced by the factor f

^, and the minimum energy is in- creased by the factor C~^- However, in certain experiments this cooling might still be useful, because the region

over which this intensity gradient is linear is larger and a standing-wave laser beam is not required. Therefore the

condition hw^, » R need not be satisfied. On the other

hand, the minimum achievable energy is larger than the

recoil energy. This can be shown with the fact that the

relative variation of /(x) over the spatial distribution cor- responding to Eq. (A3) should remain much less than 1. ACKNOWLEDGMENTS

D. J. Wineland thanks the U.S. Office of Naval Research

and the U.S. Air Force Office of Scientific Research for support. J. Daiibard warmly thanks the National Insti- tute of Standards, Gaithersburg, for hospitality and partial support for the period during which this research was

completed. He also acknowledges financial help from Di- rection des Recherches, Etudes et Techniques. We thank

REFERENCES AND NOTES

1. For a review of early experiments and theory see

W. D, Phillips, ed., Laser Cooled and Trapped Atoms, Vol. 8 of Progress in Quantum Electronics (Pergamon, Oxford, 1984);

P. Meystre and S. Stenholm, eds. feature on the mechanical

effects of light, J. Opt. Soc. Am. 2, 1705 (1985); S. Stenholm,

Rev. Mod. Phys., 58, 699 (1986); D. J. Wineland and

W. M. Itano, Phys. Today 40(6), 34 (1987).

2. P. Lett, R. Watts, C. Westbrook, W D. Phillips, P. Gould, and

H. Metcalf, Phys. Rev, Lett. 61, 169 (1988).

3. J. Daiibard, C. Salomon, A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, in Atomic

Physics 11, S. Haroche, J. C. Gay, and G. Grynberg, eds. (World Scientific, Singapore, 1989), p. 199; J. Daiibard and

C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6, 2023 (1989).

4. S. Chu, D. S. Weiss, Y. Shevy, and P. J. Ungar, in Atomic

Physics 11, S. Haroche, J. C. Gay, and G. Grynberg, eds. (World Scientific, Singapore, 1989), p. 636; P. J. Ungar,

D. S. Weiss, E. Riis, and S. Chu, J. Opt. Soc. Am. B 6, 2058

(1989); D. S. Weiss, E. Riis, Y. Shevy, P J. Ungar, and S. Chu,

J. Opt. Soc. Am. B 6, 2072 (1989).

5. P. D. Lett, W D. Phillips, S. L. Rolston, C. E. Tanner,

R. N. Watts, and C. I. Westbrook, J. Opt. Soc. Am. B 6, 2084

(1989).

6. B. Sheehy, S-Q. Shang, P. van der Straten, S. Hatamian, and

H. Metcalf, Phys. Rev. Lett. 64, 858 (1990); S-Q. Shang,

B. Sheehy, P. van der Straten, and H. Metcalf, Phys. Rev.

Lett. 65, 317 (1990).

7. See S. Chu and C. Wieman, eds., feature on laser cooling and

trapping of atoms, J. Opt. Soc. Am. B 6, 2109 (1990); L. Moi,

S. Gozzini, C. Gabbanini, E. Arimondo, and F. Strumia, eds.. Light Induced Kinetic Effects on Atoms, Ions, and Molecules (ETS Editrice, Pisa, 1991).

8. C. Cohen-Tannoudji and W D. Phillips, Phys. Today 43(10), 33 (1990).

9. J. Daiibard and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 2, 1707 (1985).

10. A. P. Kazantsev, Zh. Eksp. Teor. Fiz. 66, 1599 (1974) [Sov. Phys. JETP 39, 784 (1974)]; A. R Kazantsev, V S. Smirnov,

G. I. Surdutovitch, D. O. Chudesnikov, and V. P. Yakovlev,

J. Opt. Soc. Am. B 2, 1731 (1985).

11. V. G. Minogin and O. T. Serimaa, Opt. Commun. 30, 123

(1977).

12. T. Breeden and H. Metcalf, Phys. Rev Lett. 47, 1726 (1981).

13. D. Pritchard, Phys. Rev Lett. 51, 1336 (1983).

14. A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and

C. Cohen-Tannoudji, Phys. Rev Lett. 61, 826 (1988); J. Opt.

Soc. Am. B 6, 2112 (1989).

15. F Diedrich, J. C. Bergquist, W. M. Itano, and D. J. Wineland,

Phys. Rev Lett. 62, 403 (1989); W Neuhauser. M. Hohenstatt,

P. Toschek, and H. Dehmelt, Phys. Rev Lett. 41. 233 (1978);

D. J. Wineland, W M. Itano, J. C. Bergquist. and R. G. Hulet. Phys. Rev A 36, 2220 (1987).

16. Sideband cooling in the resolved sideband limit could also be obtained by using stimulated Raman scattering. See. for example, M. Lindberg and J. Javanainen, J. Opt. Soc. Am. B

3, 1008 (1986); H. Dehmelt, G. Janik, and W. Nagournev,

Bull. Am. Phys. Soc. 30, HI (1988); P. E. Toschek and

W. Neuhauser, J. Opt. Soc. Am B 6, 2220 (1989);

D. J. Heinzen and D. J. Wineland, Phys. Rev A 42. 2977

(1990).

17. T. V Zueva and V. G. Minogin, Sov Tech. Phys. Lett. 7, 411

(1981); L. Moi, Opt. Commun. 50, 349 (1984); J. Liang and

C. Fabre, Opt. Commun. 59. 131 (1986): D. E. Pritchard. K. Helmerson, V. S. Bagnato, G. P. Lafgjatis. and

TN-4I
42 J. Opt. Soc. Am. B Vol. 9, No. 1 January 1992 Wineland et al.

A. G. Martin, in Laser Spectroscopy VIII, W Persson and 21. These results may be derived from more-general density maS. Svanberg, eds. (Springer-Verlag, Berlin, 1987), p. 68; trix treatments. See, for example, Ref. 20 and V S. Letokhov

J. Hoffnagle, Opt. Lett. 13, 102 (1988); H. Wallis and and V. P. Chebotayev, Nonlinear Laser Spectroscopy

\V. Ertmer, J. Opt. Soc. Am. B 6, 2211 (1989). (Springer-Verlag, Berlin, 1977).

IS. Y. Castin and J. Dalibard. Europhys. Lett. 16, 761 (1991). 22. W. M. Itano and D. J. Wineland, Phys. Rev. A 25, 35 (1982).

19. D. J. Wineland and W. M. Itano, Phys. Rev. A 20, 1521 (1979). 20. S. Stenholm, Foundations of Laser Spectroscopy (Wiley, New

York, 1984).

TN-42
Hg* Single Ion Spectroscopy*

J.C. Bergquist, F. Diedrich,' Wayne M. Itano, and D.J. WineLand

Time and Frequency Division

National Institute of Standards and Technology

Boulder, CO 80303, USA

Introduction

A single ion in an electromagnetic trap can be localized in a small

volume and held for a long period of time. A trapped ion also can be

isolated from collisions and other perturbations and laser cooled to low

temperatures. By monitoring the presence or absence of fluorescence from

a strongly allowed transition, one can detect each transition to a metastable state. These characteristics make it possible to study the

spectrum of a single ion with great precision. The vibrational energy

states of the bound ion are quantized. We have used laser-sidebandcooling to drive the ion to the lowest energy level of the harmonic well

of the trap.

Experiment

A mercury atom that is ionized by a weak electron beam is captured

in the harmonic well created by an rf potential applied between the

electrodes of a miniature Paul trap (1,2). The ring electrode has an inner diameter of about 900 ^m and is placed symmetrically between two

endcap electrodes that are separated by about 650 /xm. The frequency of

the rf potential is about 21 MHz. Its amplitude can be varied up to

1.2 kV. The classical motion of the ion consists of a smal 1 -amplitude

oscillation, at the frequency of the applied rf potential, superimposed

on a larger-amplitude harmonic motion, called the secular motion. The

frequencies of the secular motion depend on the strength of the rf field

(and any static field applied to the electrodes) and were typically 1-4

MHz. The ion is laser cooled to a few millikelvins by a few microwatts

of 194 nm cw laser radiation (3) that is frequency tuned below the

5d^°6s ^S^ - 5d^°5p ^ Pj^ first resonance line (1,4). In order to cool all

motional degrees of freedom to near the Doppler cooling limit (T = h^/lk^

==1.7 mK) the 194 nm radiation irradiates the ion from 2 orthogonal

directions, both of which are at an angle of 55° with respect to the

symmetry (z) axis of the trap. The 282 nm radiation that drives the narrow 5d^°6s ^ Sj^ - 5d^6s^ ^^5^2 transition is obtained by frequencydoubling the radiation from a narrowband cw ring dye laser. In the long

term, the laser is either stabilized by FM optical heterodyne

spectroscopy to a saturated absorption hyperfine component in ^^^l2 (5) or directly to a stable high finesse reference cavity (6) . The frequency

of the laser is scanned by an acousto-optic modulator that is driven by a computer controlled synthesizer. Up to a few microwatts of 282 nm radiation could be focussed onto the ion in a direction

counterpropagating with one of the 194 nm light beams.

Optical-optical double resonance (electron shelving) (1,2,4,7-11)

with a net quantum amplification in excess of 100 at 10 ms is used to

detect transitions driven by the 282 nm laser to the metastable ^05^2

state. The fluorescence from the laser-cooled ion is constant when it is

cycling between the ^S^^ and ^ P^ states and zero when it is in the

metastable ^D5/2 state (1,4,7,8). Thus the ^Sj^ - ^05^2 resonance

*Work of U.S. Government; not subject to U.S. copyright.

^Present address Gsanger Optoelektronik GmbH, Planegg,

Federal Republic of Germany

Laser Spectroscopy IX 274 TN-43
spectrum is obtained by first probing the S-D transition at a particular

frequency near 282 nm, then turning off the 282 nm radiation, turning on

the 194 nm radiation, and looking for 194 nm fluorescence. The two

radiation fields are alternately applied to avoid light shifts and

broadening of the narrow S-D transition. The absence of 194 nm fluorescence indicates that a transition into the metastable D state has

occurred; its presence indicates that the ion has remained in the ground

state. The 282 nm frequency is then stepped and the measurement cycle

repeated. Each new result at a particular 282 nm frequency is averaged

with the previous measurements at that frequency. The normalization of

the signal is 1 for each measurement of high fluorescence and for each

measurement of no fluorescence. The high fluorescence level makes it

possible to determine the state of the atom with almost no ambiguity in a few milliseconds. Thus, it is easy to reach the quantum noise limit of a single atomic absorber (1). Figure 1 shows the signal from an 8 MHz scan

of the 282 nm laser through a Zeeman component of the S-D transition in

i98j^g+ (1) i\^Q Doppler-free central feature (carrier) and the motional

sidebands due to the secular motion of the cold ion are fully resolved

(12,13). The number and strength of the sidebands are a direct measure

of the amplitude of the ion's motion and of its temperature.

In Fig. 2, we show a high resolution scan through the Doppler-free

resonance of the ^S. (F 0, fflc = 0) D5/2 (F = 2, mp =0) transition in

issHg"^ which is first-order field independent at a magnetic field

B = 0. The full width at half maximum (FWHM) is approximately 86 Hz at

563 nm (172 Hz at 282 nm) . This corresponds to a fractional resolution

of better than 2x10'^^ (Q = 5x10^^). For this trace the laser was spectrally narrowed by locking to a mechanically, acoustically and

thermally quiet reference cavity that had a finesse of about 60,000. The

5d^°6p

3/2 100 -

0.95

2^ 0.90 O

194 nm

10 5cl 6s Si/2

281.5 nm

o.as

g 0.80 o

T—I—I—I—I—I—I—I—I—I—I—I—I—

r

pfVvvM fAfM AWv^K^A^^

I I I I I

I I I L. -2 2 Frequency Detuning (in MHz)

Fig. 1. On the left is a simplified energy-level diagram for ^^^Hg'''.

The 282 nm transition can be observed by monitoring the 194 nm

fluorescence. If the ion makes a transition from the ^S^ to the ^Dj^j

level the 194 nm fluorescence disappears. For the figure on the right,

the relative detuning from line center is plotted in frequency units at

282 nm. On the vertical axis is plotted the probability that the 194 nm fluorescence is on immediately after the 282 nm pulse. The Doppler-free

recoilless-absorption-resonance or carrier (central feature) can provide

a reference for an optical frequency standard. (From Ref. 1)

275 TN-44
250 -125 125 250

Laser detuning (Hz)

375

Fig. 2. High resolution scan

through the Doppler-free reso- nance of the ^Sjj(F = 0, nip = 0)

D 5 / 2 (F = 2 = 0) transition

in a single laser cooled ^^^Hg"*"

ion. A frequency doubled and

stabilized 563 run laser (u ~

5x10^'' Hz) is stepped through the resonance in 25 Hz increments.

The full width at half maxiraum is

about 86 Hz. The integration

time per point is about 0.5 s.

frequency of the laser fluctuates less than 50 mHz relative to this

reference cavity. This is determined from the measurement of the noise

in the error signal. The actual frequency fluctuations of the laser are governed by the stability of the mechanically and thermally isolated

reference cavity (6). The natural linewidth of the S - D transition is

approximately 1 . 7 Hz (2). The measured linewidth in Fig. 2 is likely

limited by low frequency fluctuations in the length of the stable

reference cavity (14). Better mechanical isolation of the reference

cavity might permit us to reach the 1 . 7 Hz resolution limit.

The possibility of cooling the Hg"*" ion further, to the zero point

energy of motion, is intriguing for several reasons. Cooling so that the

average occupational number <n^> for the motional energy of the bound

atom is zero is a fundamental limit to laser-cooling for any bound

particle (1,15) and forces a quantum mechanical treatment of the atomtrap system. Driving a single atom in a macroscopic trap to the zero point energy of motion exploits the benign environment near the center of an rf trap. Finally, if an ion is prepared in the lowest vibrational

state, experiments such as squeezing the atom's position and momentum can

be demonstrated (16).

In a rigorous treatment of the ion-trap system, stationary states

do not exist since the trapping potential is time dependent. However,

quasi-stationary states, obtained by solving for the eigenvalues of the

Floquet operator, do exist (17,18). That these states correspond closely

to the stationary states of the pseudo-potential of the rf trap is

verified by the spectrum shown in Fig. 1. The carrier (at frequency Wg

)

results from transitions in which the vibrational quasi-energy is

unchanged. The upper and lower sidebands, spaced by multiples of the

secular frequencies (wg ± no)^ ) , correspond to transitions which increase

or decrease the quasi-energy. Recently, we have cooled a single ^^^Hg"*"

ion to the lowest vibrational state (n^ = 0) by a method called optical

sideband cooling (15). First, the ion was laser-cooled to a few

millikelvins by radiation scattered from the strong transition at 194 nm. This reduced the vibrational quantum number to a mean value of <n^> = 12.

The secular frequency was about 3 MHz. Laser radiation tuned to the

frequency of the first lower vibrational sideband (at frequency Wg " '^v

)

of the narrow S-D transition was then applied to the ion. For each

photon re-emitted at the unshifted carrier frequency, the vibrational

energy was reduced by fiu>^ . After the sideband cooling, laser radiation

of saturating intensity was applied at the lower sideband frequency.

276 TN-45
Absence of absorption indicated that the ion was in the n^ = state.

The ion was found to be in the n^ = state about 95% of the time.

Starting from the ground vibrational state (n^ = 0) , the absorption

of a single quantum of energy at a frequency corresponding to the secular

vibrational frequency (cj^ ) would raise n^ by one unit. This could be

detected with an efficiency of nearly 100%. Also, it should be possible

to produce squeezed states (16) of the aton.'s motion from the zero point

energy state by a sudden, non-adiabatic weakening (and, in general,

shifting) of the trap potential or by driving the atomic motion

parametrically at 2 w^ . If after some time the atom could then be

returned to the n^ = state by reversing the above procedures, the zero point energy state could be detected by the absence of the lower

sideband.

We acknowledge the support of the U.S. Air Force Office of

Scientific Research and the U.S. Office of Naval Research. F. Diedrich

thanks the Deutsche Forschungsgemeinschaft for financial support.

References

1. J.C. Bergquist, W.M. Itano, and D.J. Wineland, Phys . Rev. A 16. ^28

(1987).

2. J.C. Bergquist, D.J. Wineland, W.M. Itano, H. Hemmati, H.-U. Daniel,

and G. Leuchs , Phys. Rev. Lett. 55, 1567 (1985).

3. H. Hemmati, J.C. Bergquist, and W.M. Itano, Opt. Lett. 8, 73 (1983).

4. J.C. Bergquist, R.G. Hulet, W.M. Itano, and D.J. Wineland, Phys.

Rev. Lett. 57, 1699 (1986)

.

5. J.L. Hall, L. Hollberg, T. Baer, andH.G. Robinson, Appl. Phys.

Lett. 39, 680 (1981).

6. J. Hough, D. Hils, M.D. Rayman, Ma L.-S., L. Hollberg, and J.L.

Hall, Appl. Phys. B 33, 179 (1984).

7. W. Nagourney, J. Sandberg, and H. Dehmelt, Phys. Rev. Lett. 56, 2797

(1986).

8. T. Sauter, W. Neuhauser, R. Blatt, and P.E. Toschek, Phys. Rev.

Lett. 57, 1696 (1986) . 9. D.J. Wineland, J.C. Bergquist, W.M. Itano, and R.E. Drullinger, Opt.

Lett. 5, 245 (1980); D.J. Wineland and W.M. Itano, Phys. Lett. 82A . 75 (1981).

10. H. Dehmelt, J. Phys. (Paris) Colloq. 42, C8-299 (1981).

11. H. Dehmelt, Bull. Am. Phys. Soc . 20, 60 (1975).

12. R.H. Dicke, Phys. Rev. 89, 472 (1953).

13. D.J. Wineland and W.M. Itano, Phys. Rev. A 20, 1521 (1979).

14. D. Hils and J.L. Hall, in Frequency Standards and Metrology , edited

by A. DeMarchi (Springer-Verlag, Berlin, Heidelberg, 1989) p. 162.

15. F. Diedrich, J.C. Bergquist, W.M. Itano, and D.J. Wineland, Phys.

Rev. Lett. 62, 403 (1989).

16. J.C. Bergquist, F. Diedrich, W.M. Itano and D.J. Wineland, in

Frequency Standards and Metrology , edited by A. DeMarchi (SpringerVerlag, Berlin, Heidelberg, 1989) p. 287.

17. R.J. Cook, D.G. Shankland, and A.L. Wells, Phys. Rev. A 31, 564

(1985).

18. M. Corabescure, Ann. Inst. Henri Poincare 44, 293 (1986).

277 TN-46
Frequency Standards in the Optical Spectrum*

DJ. Wineland, J.C. Bergquist, WM. Itano, F. Diedrich**, and C.S. Weimer

National Bureau of Standards, Boulder, CO 80303, USA

1. Introduction

A long standing problem of spectroscopic measurements in the optical region

of the spectrum has been the difficulty of obtaining accurate frequency

reference standards to which spectral measurements can be compared. As will

be discussed in this symposium on the hydrogen atom, this affects the value

of the Rydberg constant as derived from optical measurements in hydrogen. As

the experiments on hydrogen improve, the need for better reference points

becomes more acute.

In this paper, we address two aspects of this general problem. First, we

discuss the problem of frequency standards in the optical spectrxim. (An

analogue in the microwave region of the spectrum is the cesium beam frequency

standard.) If one or a few of these reference frequencies can be accurately

calibrated (perhaps by a frequency synthesis chain-'-) then it may be possible

to compare optical spectra to these standards. As an example of the

precision that might be achieved, we discuss only optical standards based on stored ions. Second, we discuss the problem of frequency comparison of

unknown frequencies to the standards. Here we primarily restrict discussion

to generation of wideband frequency "combs".

2. Optical Frequency Standards

One way an optical standard could be provided is by harmonic multiplication

of a microwave frequency standard in a synthesis chain. By use of this

technique, a laser at 88 THz (3.39 ^m) has been made phase coherent with a microwave oscillator.^ The best optical frequency standards may be made by

locking a local oscillator (laser) to an atomic or molecular resonance line.

State-of-the-art accuracies are characterized by measurements on methane

stabilized He-Ne lasers in which reproducibilities in the 10"-*--^ range have

* Supported by the U.S. Air Force Office of Scientific Research and the U.S. Office of Naval Research. Contribution of the U.S. National Bureau of

Standards; not subject to U.S. Copyright. ** Supported by the DFG.

The Hydrogen Atom Editors: G.F. Bassani • M. Inguscio -T.W. Hknsch 123 © Springer- Vcrlag Berlin. Heidelberg 1989

TN-47
been reported. There are many other schemes for optical frequency

standards. " Here, we will only discuss possibilities with single stored

ions in order to give an idea of what stabilities and accuracies might be

achieved.

3a. Single Ions The use of single trapped ions as optical frequency standards was suggested

by Dehmelf^ in 1973. Since that time, demonstrations of laser cooling,^ 100%

detection efficiency,"'" and lasers with sub-Hz resolution, '^ »^ have increased

confidence that a single-ion frequency standard is viable. As an example, we summarize the current status^*^ and discuss future prospects using single Hg

ions. Similar experiments using different kinds of ions have been reported

at Seattle'^'" and Hamburg ,'' ^ ' and other labs are initiating experiments.

In the NBS work, the transition of interest for a frequency standard is the Hg 5d^"6s ^Si -> 5d^6s'^ ^5/2 q^^drupole transition at 281.5 nm shown in

Fig. 1. The D5/2 level has a lifetime of 86 ms, corresponding to a natural

width of 1.8 Hz. Use of the single photon quadrupole transition has an advantage over two-photon Doppler free transitions-'-'^ because ac Stark shifts

are negligible.

A mercury atom that was ionized by a weak electron beam was captured in a miniature Paul (radio frequency) trap that has internal dimensions of

r^ s A66 pm and Zq = 330 pm. The rf trapping frequency was 21.07 MHz with a peak voltage amplitude of about 730 V. The ion was laser cooled by a few

microwatts of cw laser radiation that was frequency tuned below the 6s •^Si -

6p ^Pi electric dipole transition near 19A nm. Vfhen the Hg ion was cold and

the 19A nm radiation had sufficient intensity to saturate the strongly

allowed S-P transition, 2 x 10° photons/s were scattered. With our collection efficiency, this corresponded to an observed peak count rate of

about lO-' s~^ against a background of less than 50 s"-*-. Optical-optical double- resonance utilizing quantum amplification^'""" was used to detect transitions of the ion driven by the 281.5 nm laser to the

metastable D5/2 state. This method makes use of the fact that the 19A nm

fluorescence intensity level is bistable; high when the ion is cycling

between the S and P states (the "on" state) and nearly zero when it is in the

metastable D state (the "off" state). The fluorescence intensity in the on state is high enough that the state of the atom can be determined in a few

milliseconds with nearly 100% efficiency. The full measurement cycle was as follows: A series of measurements of the 19A nm fluorescence was made, using

124

TN-48
a counter with a 10-ms integration period. As soon as the counter reading

per measurement period was high enough to indicate that the ion was in the on

5d 6s s

100

2f 0.90 o

•D 0.85

TM 080 O

0.75-4

1—I—n—I—

r

I

I I I I I I jy^ M|>V Nr*\^j'\!\i^

J I I I I I I I I I I I I I L

-2 2 Frequency Detuning (in MHz)

Fig. 1. On the left is a simplified energy-level diagram for ^'"Hg"*". The

281.5 nm quadrupole "clock" transition can be observed by monitoring the 19A

nm fluorescence. If the ion has made a transition from the '^Si to the D5/2

level the 19A nm flourescence disappears. For the figure on the right, on

the horizontal axis is plotted the relative detuning from line center in frequency units at 281.5 nm. On the vertical axis is plotted the probability

that the fluorescence from the 6s •^Si - 6p ^Pi first resonance transition,

excited by laser radiation at 19A nm, is on immediately after the 281.5 nm

pulse. The electric-quadrupole-allowed S-D transition and the first- resonance S-P transition are probed sequentially in order to avoid light

shifts and broadening of the narrow S-D transition. The recoilless

absorption resonance or carrier (central feature) can provide a reference for

an optical frequency standard. (From ref . 11)

state, the 19A-nra radiation was shut off and the 281.5 nm radiation was pulsed on for 20 ms. Then, the 19A nm radiation was turned on again, and the counter was read. If the reading was low enough to indicate that the ion had

made a transition to the D5/2 state (the off state), the signal was defined

to be 0. Otherwise, it was defined to be 1. The 281.5 nm laser frequency

was then stepped, and the cycle was repeated. As the 281.5 nm laser

frequency was swept back and forth through the resonance, the quantized

measurement of the fluorescence signal at each frequency step was averaged

with the previous measurements made at that same frequency. Since we could

125

TN-49
detect the state of the ion with nearly 100% efficiency, there was essentially no instrumental noise in the measurement process. Occasionally,

while the 19A nm radiation was on, the ion decayed from the ^Pi state to the

metastable D3/2 state rather than directly to the ground state. ' This

process led to a background rate of false transitions which was minimized by

the quantized data-collecting method described above and by decreasing the

19A nm fluorescence level thereby decreasing the Pi" D3/2 decay rate. Neglecting decay to the D3/2 level, the quantized measurement scheme removes any contribution to the signal base line due to intensity variations in the

19A-nm source. The 281.5 and 19A nm radiation were chopped so that they were

never on at the same time. This eliminated shifts and broadening of the narrow 281.5-nm resonance due to the 19A nm radiation.

Figure 1 shows the signal from an 8 MHz scan of the 281.5 nm laser through

a Zeeman component of the ion. The central feature or "carrier" can be the

reference for an optical frequency standard. The laser could be servoed to this resonance by probing on both sides of the resonance to develop an error

signal which forces the average frequency of the laser to the center of the resonance. Two primary requirements for good performance are: (1) immunity

from and ability to characterize systematic frequency shifts (accuracy) and

(2) sufficient signal-to-noise ratio and line Q to reach the accuracy level

in a reasonable length of time (stability).

3b. Stability

If we make the simplifying assumption that the time-domain Ramsey method is

used to interrogate the clock transition, then the frequency stability (twosample Allan variance^^) is given as-'--'

Oy(x) = [<(<u)k>x - <'^+l>x)^>k/2wS]^ = (2uj§Trt)"2 . (1)

In this expression, <(i>1q>x is the kth measurement of the frequency of the

locked oscillator averaged over time x, < >j^ denotes an average over many

measurements, wq is the nominal frequency of the clock transition, and Tj^ is the interrogation time between pulses in the Ramsey method. Thus o„(x) is a measurement of the rms fluctuations of the average frequency between adjacent

measurements of duration x. Eq. 1 assumes that optical state preparation

(cooling and optical pumping) and fluorescence detection takes a time much

less than Tj^. We have also assumed Tj^ « ^(05/2) ^'^^ that the frequency of

the 281.5 nm laser changes by much less than (2Tjj)~^ in time Tjj. For our Hg"*" example, assuming Tj^ = 25 ms, Oy(x) s 7 x 10~^^x~2, Therefore, at averaging

126

TN-50
times of 1 s, the measurement imprecision on a single ion should be on the

order of 1 part in 10^^ or 1 Hz. Even better performance is expected on ions

with narrower linewidths. To realize such stabilities, very narrow, tunable

lasers are required but this technology is now available.^

3c. Accuracy

The advantage of single ions over other experimental configurations is the

relative immunity from systematic frequency shifts. Laser cooling gives the

lowest second order Doppler shifts for single (as opposed to many) trapped

ions.-*- In the Doppler cooling limit,-' the magnitude of the second order

Doppler shift for Hg ions in an rf trap is fractionally 2.U x lO"-*-". Laser

cooling to the zero point energy, which has recently been achieved,^' gives

even smaller shifts. One must also account for all perturbing influences on

the ion's internal structure.^' ^'' For example, one must consider the

perturbations due to static and time varying multipole interactions for

electric, magnetic, and gravitational fields. These include atom trapping

field interactions, collisions with neutral atoms, ac Stark shifts due to

laser beams, stray electric and magnetic fields, gravitational red shifts,

etc.

9 9 +

For the •^Si -> D5/2 quadrupole transition in Hg , the limiting accuracy

may be due to the uncertainty in the interaction of the D5/2 atomic

quadrupole moment with static electric fields of quadrupole symmetry. '^"

The interaction with the quadrupolar electric fields of the trap can be

calibrated since they affect the ion oscillation frequencies of the trap in a known way. More difficult to control are the effects due to patch fields or stray charge build up. Shifts may be on the order of 0.1 Hz. We can significantly reduce the uncertainty due to these shifts, however, by taking

the mean quadrupole transition frequency for three mutually orthogonal

orientations of a quantizing magnetic field. In this case, the mean

quadrupole shift is zero. For the ^Sq ^ -^Pq transitions of the group IIIA

ions, such shifts are absent.

Although experimental verification is still lacking, it would appear that

single ion optical frequency standards will eventually yield extremely high

performance. Accuracies and stabilities better than 1 part in 10-'^-' seem

quite feasible; eventually they could exceed 1 part in 10^°.

A. Frequency Comparison

To appreciate the problem of frequency comparison in the optical region of

the spectrum one needs only to plot the electromagnetic spectrum on a linear

127

TN-51
frCTiuencT scaLe. l^.e rrjaber and spectral density) of accarate DpticaL

rreaisesacT -Standards is. 'err '^mal' to begin wlt±i; this is ro be -oaipared nth.

-h*^ sataiarxon zn the raicrowave spectrum (and below) where ^annerclalLy

available syinthesxzers provide state-of-the-art precision at any fxeqrtgncy.

A laser whose frequency is jnknawn can be aoiiq>ared to a reference laser by

"leterottyne aethoos to rj.gh precision. Beat frequency measur^nents op to Z-5c

TEx, -XL tte tLsible s-sectron Iiave already been niade.^' An alternative to sinple -ieterodjne ichesnes ^s riaxzionlc aixing by use: of synthesis: chains:. -

^fere, the jnknovr. " r.s.Kr rregtTpnrr is "onipareA '^via heterodyne tEthods:) tuc sharraoni-c :>f some 'ell .tEowni refi^remce line stich as the aethane stabilized 3ie- Se laser a.t 3.39 vsm.

Comparisons can also be siade by wavelength methods. These have thet advantage that two .as^rs 3cf qaite differeat frequency can be compared beet sre currently limited In accuracy to about: the L0~"^ to 10~^^ range.'-''

For tigh. accuracy coaiparxEoni, heterodyne detection and synthesis 'HaTTt^; are proven ciethods tut nave .lot been demonstrated in the visible \'et.

ioreover, since they .ire somewhat, cumbersome and their coverage in the

optical o^ectnan somewhat sparse, it. is useful to pursue alternative sEthods.

ja. -HtiebaEid Tomb Jensratiois

Precise :req'jencies relative to a frequency standard (at frequency 'jj»j) can be

provioed. by comb ^eEeratton. If we amplitude ar frequpncy aodulate at

frequ^Escy 1) a source at the standard frequency, then spectral components at

freau^ECi.es ukj t nil ' n ss integer; are g^ierated. If a and. i arsE rhtiKsufr soc that j«j -^ Tf2 s aiuuicEovRj then uCunknoMc) can be precisely detemined 'oy leterodyne riethods. 7h& cfcaxlesige _s to oake ', n2) rery _arge.

"r . inpj-i tLicB loarj. atiorr

.•4S a . pecific axample, :.et;

M'^tj = sintuQC f- jj,;) rCttj,, (Z)

»A»snEE: fttr) is periodic; with period Ip'/JL, Wc will assume f(t) is an earen function and "taar : ~ 3. has: the forrrr

r<t; = axpf-'t/-;^i (-|<t<iy,. (1)

MtJ in a roarirar

2&

TTI-jZ
^^^^ ^ ^V -i°<'^'^+*>

(4.)

-^ )ea3>(-(nQT/2)^) (sin[(oiQ+tifl)t-hpQj -^ sin{(u^-nfl)t * »qj)

<i;k^

t-=a x-2.Trm

Fig. 1. -k sxne-vave ::arrije£r (frequency ^} is assomea :o De perlixiicaj-ly

ampilttuie :ioclulared at frequency 2 with i laussian anveiope. 'zsjsjd: freaueicy

division uq/Q = integer) is obtained if the phase jf the -arr^eT is fixes, ro the phase of the inodxtiatlon envelope.

7o nalLe "rhe spectrai compcoaent: at jrj t nSL as laa-gp as. possible -'e -'ant

nfitr/Z dose co Jinty. Yor >.amonxc generato-on, -jf) = 1. Ttxr f^ithpr- 'lanaomix

gKieration or ciomb g«ierat±on around a Aigh rrequ^iiiy jq, the -nmh janoviottr

IS agproTi mately equal ta 1/ r. High order aoltip 1 1 ratT on to rte isxtaet or

largp; bandwxdth combs in the visible are difficult lo -x-hieana ''x>z- ~to reasons: (l.) it is ififficnlt to i^iake " small snon^ and (Z) t±Kt raodnlaroinnnstT- bEE snffici^ntly broadband to pass the rnnaanw^xtal amd its? "nanaoirics or

sixiebanos.

vie remark that 75.g. 1 is ::haracreristi-C or ihe output of a niode -orsea

iasrar^-^ or i free electron 'a«;t^T- with puxsed inptut electron oesm. '— "ioae- locked lasers, .xave a "i re-ady been su^esred as: denrixes fbrr frequsncy

~ansiii.ti.an m ±ie jptical- spectmm. ~^

TSr

Tl"
5c. Frequency Modulation

Consider sine wave modulation of a carrier at frequency ojq:

A(t) = AQsin[wQt + jT- sinQt + <p] (5) oo

= Aq y Jn'^f^) sin[(wo + "")t + cp],

n=-oo

where Aw is the frequency swing. Assume we are interested in the spectral

component at a particular (large) value of n. We make the substitution

Aoj/fi = nP, and will limit ourselves to the case where 3 < 1. From Meissel's

formula^-^ for Jj^(n3), for large n and 3 < 1, we can write

Jn(nP)

pj^-^')^

l+(l-p2)^

n [2Tin(l-02)5]-2 . (6)

We can make the spectral component at wq + nfl reasonably large by making 3 close to unity or the modulation index Aoj/Si close to n. This is the primary

practical problem for n large.

6a. Frequency Division

Using amplitude or frequency modulation of a carrier at frequency coq, we can achieve exact frequency division if we make sidebands of the carrier to such

low frequency that we can force the condition ojQ-nQ = (n+2)J2-(jjQ = Q so that

ojq/Q = n+1. For example, if we examine Fig. 2, we can achieve exact

frequency division by any means which locks the phase of the carrier to the

phase of the amplitude modulation; that is, the undulations of the carrier do not "slip" under the envelope of the amplitude modulation. A divider based

on these principles would be quite useful if Q is in the microwave region (or

below) where precise frequency synthesis is possible. Since Q and n could be

freely chosen, any value of wq could be measured in a single device.

6b. Single Electron Frequency Divider

The required extreme amplitude or frequency modulation might be accomplished

with the interaction of an electromagnetic wave (at frequency (dq) with a single electron oscillating at frequency Q. For example, Q might be the

cyclotron frequency of the electron in a magnetic field. As originally proposed, ^^ extreme amplitude modulation of the

electromagnetic wave (at the site of the electron) could be accomplished by

130

TN-54
focussing the wave with a fast lens or slow wave structure (miniature

waveguide) onto a portion of the electron's cyclotron orbit. The energy

coupled into the electron at frequency WQ-nJi = (n+2)n-WQ = D is balanced by

synchrotron radiation and phase locking can occur. Lens focussing requires

the electron to be relativistic, y B (l-(v/c)^)"2 -^ 2 to achieve a significant coupling*^^ (v = electron speed, c = speed of light). Use of a slow wave structure would allow use of lower energy electrons.

Driving the electron could also be accomplished using an inverse

synchrotron radiation geometry.^-* Here, a collimated laser beam is made to coincide with a tangent to the electron's orbit. We require the electron to

be relativistic to achieve a significant interaction. If the laser beam

waist is made much larger than the electron orbit dimensions, then we have

the case of extreme frequency modulation of a plane wave as first analyzed by

Kaplan.'^" Here, the electric field at the site of the electron is given by

Eqs. 5 and 6 with 3 = v/c. Again, we require the electron to be somewhat

relativistic to make J^ and J^+j sufficiently large (wQ-nQ = Q)

.

In all cases above, the center of the electron's orbit (in the plane of the orbit) must be held within ^(= c/wq). A specific scheme for

accomplishing this is discussed in Ref. 24. At NBS we have initiated a project along these lines. At first, low order division (n < 10) where ojq and Q are both microwave frequencies will be tried. Aside from the practical

application discussed here, such an experiment is interesting in terms of the

fundamental nonlinear interactions of simple elementary systems.

We thank L. Hollberg and D. Jennings for helpful comments on the

manuscript.

1. D.A. Jennings, K.M. Evenson, and D.J. Knight, Proc. IEEE 7±, 168 (1986);

S.N. Bagaev and V.P. Chebotayev, Usp. Fiz. Nauk 1A8, 1A3 (1986) (Sov.

Phys. Usp. 29, 82 (1986)), and references therein.

2. Proc. Fourth Symp. Freq. Standards and Metrology, Ed. A. DeMarchi,

Ancona, Italy, Sept. 1988 (Springer-Verlag, Heidelberg) to be published.

3. Proc. Third Symp. Freq. Standards and Metrology, J. de Physique, Vol.

A2, Colloque C-8, 1981.

4. H.G. Dehmelt, Bull. Am. Phys. Soc. ^S, 1521 (1973) and 20, 60 (1975); H.G. Dehmelt, IEEE Trans. Instrum. Meas . IM-31 , 83 (1982); H.G. Dehmelt,

in Advances in Laser Spectroscopy , Ed. by F.T. Arecchi, F.S. Strumia,

and H. Walther, (Plenum, New York, 1983), p. 153.

131

TN-55
5. See ioT example, E.J. Wine. land and M.K. Itano, Phys. Toda>-, jiL, no. 6,

p . 3- ( 193^ ) , and xeferences 1:heTein

.

6 M. laagoarne-y , Z. Sandbexs, E. Deimel-, Phys, Sev. istx, 56, 2797 (1936).

T. 1. Ssatar, Vi. NeiihairseT , E. Blatt, ?.E. Toschek, Phys. Ssv. Letrt. 57 , 1696 '1936). «. J.C. aexgquisc, S.G. HuLet, W.M. Itano, D.J. Wineland , Phys. Bev. Lett.,

52., 1699 (1986).

9. j.L Hall, E. Hxis, C. SaloBion, and J.M. Chart.ier, ir. -.aggr Specrroscopy

Vm. ed. by V. Perssor. and S. Svanberg^ (Sprii^er, AsrJLm Heioelbers,

1987, p. -37c.

If.. J.C ^aTgqtirsr, ^'.K. I-^^nc, E.J. Mmaiaud Pnys. 2sv Aj6, 4Z?. li'J^"*

.

JJ.. P.E. Toschek, in Tendances Actiielles en Physique Atomique, Les Houches,

Sessxon 3&, G. GrynDerg and E. Stora, eds., (North-Holland, 1984) p. 331.

12. J.C. Bergquist, S.J. Hineland, W.H. Xt. , fi. Hemati, &.1]. Baniel, G.

iisaccjas, Phrs. ^ss-j. Lsti. 5^, 156~ (1935;. n. ^.K Itano, II.C. Bergquist, E.G. Bulet and E.J. Hinelacid> Piiys. 5ev.

Lat-. 52, 2732 (1987).

14. l.k Barnes, A.?». Chi, L.S. Curler, E.J flea ley , L.£, Leesor.. *;.£. HcGunigal, J.A. Mullen, Jr., K.L. Smith, E.L. Sydnor, H^T.C. Vessot , asad G.K.A. \<ini:ier^ I22E Trans. Instrma Meas. IM-20 .. 105 (1971>.

15. E.J. ^iinelsna, Vt.h. Itano, J.C. Bergquis:: and 'F.L. Vtalls, in Proceedings

ot tfl£ Tairy-ii±th Annual Symposiun on Prequenr- Control (1931>, p. 601

I cop Les available fram Electronics Industry Assn., 2001 ^jb" St., K.*«. Washington, E.C. 20006).

16. E.J. Mineland, ir. Precision Measarenent and Tundanental Constants XI,

Natl. Bur. Stand. (L.S.) Special Pubi. No. 617, edited by B.K. Taylor

and Vi.E. Phillips, (U.S. GPO, Washington. D.C., 1984). p. 82.

J7. ?S. Biedrich, J.C. Bergquist, W.M. Itano, B.J. Wineland, tc be published.

18. L.J. Wineland, Science 226 , 395 (1984), and references therein.

19. S.X. IhrullingeT, K.M. Evenson, D.A. Jennings, P.R. Petersen, J.C. Bergquist, L. Barkins, and H.li. Daniel, Appl. Phys. Lett. 42, 137

(1983).

2C. E.G. DeVoe, C. Pabre, X.. Jungmann, J. Hoffnagle, and E.G. Brewer, Phys. Eev. A37, 1802 (1988).

21. S-E. Bxamell. E.M. Xane, and A.I. Perguson, Opt. Conun. 56, 12 (1985); J.A. Valdmanis, E.L. Pork, and J. P. Gordon, Opt. Lett. 10, 131 (1985); see also E.H. Xane, S.E. Braniell, and A.I. Perguson, Appl Phy^. B39,

171 (1986).

132

TO-56
22. S. ?ennsT ct al., in Proc. 9th Int. T.E-L. Coni .

, Williamsoorg , VA,

1987, X.C be published.

23 G.N. Watson, Theory of Bessel functions (Cambridge "Press, 1952 j, p. 227.

2i.. L.J. Wineland, J. Appl. Phys. 50. 252£ il979), J.C. Bergqais., Li.J. \iinel£na, IsatJ^. fior. Standi. (L.S.;, "Eedri liioije 1086, (i;.3. G.r.G. , 1935,

p. IN-SC.

li. li.ii Paarell amr J. A. Edighoffer, J Apol. Phys 5^ 1^5 119S0>.

.2i. A.i. iapxar., Ojjtir.s Lett. 1- , 4S9 (1937;.

133

TTv-S'
IEEE TRANSACTIONS ON ULTRASONICS. FERROELECTRICS. AND FREQUENCY CONTROL. VOL. 37. NO. 6. NCVEMBER 1990 515 Progress at NIST Toward Absolute Frequency

Standards Using Stored Ions

D. J. WINELAND, J. C. BERGQUIST, J. J. BOLLINGER, W. M. ITANO, D. J. HEINZEN,

S. L. GILBERT, C. H. MANNEY, and M. G. RAIZEN

Abstract—Experiments at NIST, whose goal is to realize frequency

standards of high accuracy using stored ions, are briefly summarized.

In one experiment, an RF oscillator is locked to a nuclear spin-flip hyperfine transition (frequency - 3.03 x 10* Hz) in 'Be* ions that are stored in a Penning trap and sympathetically laser-cooled. Stability

is better than 3 x 10"'^ .^-d/i) gpj uncertainty in Doppler shifts is estimated to be less than 5 x 10"'^. In a second experiment, a stable

laser is used to probe an electric quadrupole transition (frequency =

1.07 X 10" Hz) in a single laser-cooled '"Hg"^ ion stored in a Paul

trap. The measured Q value of this transition is approximately 10'^. Future possible experiments are also discussed.

Introduction

ONE GOAL of the ion storage group at NIST (formerly the National Bureau of Standards) has been to realize a stored ion frequency standard whose absolute in- accuracy is less than 1 part in 10*'*. Primary design considerations in these experiments have been influenced by

the assumptions that 1) for many ions in the trap, the uncertainty in the measurement of the second-order Doppler

shift will be the largest contribution to inaccuracy and 2)

the magnitude of the second-order Doppler shift decreases

as the number of ions in the trap decreases.

These assumptions are supported both by experiments

and theoretical analysis [l]-[6]. For a given number of

ions N in the trap the second-order Doppler shift is minimized when the secular motion in the Paul trap [2]-[5] or the cyclotron and axial motion in the Penning trap [1]-

[3] are reduced by some means to negligible values (for

example, by the use of buff"er gas collisions [4] or laser cooling [6], [7]). For the Paul trap, the second-order Doppler shift is then dominated by the velocity in the RF micromotion [l]-[5]. The basic idea is as follows: in the quadrupole Paul trap the RF micromotion velocity in- creases with the distance of the ion from the center of the

trap. As the number of trapped ions increases, space

charge repulsion holds ions farther from the center of the

trap thereby increasing the ion's micromotion speed and

second-order Doppler shift. For a spherical cloud of ions, the number of ions A^ is proportional to the fractional secManuscript received February 1, 1990; revised May 28, 1990; accepted

May 31, 1990. Thi.s paper is based on a paper by the authors in Proc. 43rd

Anna. Symp. Freq. Conlr., Denver, June 1989, pp. 143-150, (IEEE Cat.

no. 89CH2690-6, NO Nov. 1989). The authors are with the National Institute of Standards and Technology, Time and Frequency Division, Mail Code 576, Boulder, CO 80303. IEEE Log Number 9038497.

ond-order Doppler shift, averaged over the cloud,

{^vo2/vo) [2]-[4]:

A^ = -2.16 X \Q^^ r,,M{^viyi/vQ)/Z^ (RF trap)

(1)

where r^/ is the cloud radius in centimeters, M is the ion mass in atomic mass units and Z is ion charge in units of

the proton charge. The number of ions for a given secondorder Doppler shift can be increased by using a nonspherical ion cloud geometry in an elongated trap [5]. However, the basic idea still holds that as the number of ions

increases so does |< AfD2/''o)|- The second order Doppler shift is minimized for a single ion [6] or for more

than one ion in a trap with linear geometry (discussed below).

In the Penning trap, when the cyclotron and axial ki- netic energies are reduced to small values, the second or- der Doppler shift is dominated by the velocity in the ro- tation of the ion cloud. In this case, A^ and < AfD2/j'o)

are related by [l]-[3]

1/2 A^ = 3.10 X 10'^fi< -^vm/vQ)

• Zciirci - r;)/Z (Penning trap) (2)

where B is the trap magnetic field strength in teslas, Izd

and r^ are the cloud height and radius in centimeters, r^ = (5 < -Appj/^o > )^^^c/Qc i^c is the ion's cyclotron fre- quency and c is the speed of light ), and Z is the ion charge

in units of proton charge. Equations (1) and (2) are valid

for one species of ion in the trap.

Therefore, for both traps, | < Aj/o2/i'o > I increases as A^ increases and we are faced with a compromise in design.

For good signal-to-noise ratio (SNR) and therefore good

frequency stability, we desire large A^. However this in- creases |< Ai'o2/i'o>l and therefore decreases accuracy

because of our inability to measure precisely the velocity

distributions needed to determine < Ai>d2/i'q > . This tradeoff" between stability and accuracy has resulted in diff^erent approaches. In the work of one group [4], [8], a stored

'99j^g+ JQp standard with excellent stability has been re- alized. In these experiments A^ = 2 x 10^ and < Av^il vq > = -2 X 10"'^ so an accuracy of 10"''* would require a knowledge of < i?- > to 0.5% precision. This problem can

be reduced in the elongated trap geometry of [5] but, inU.S. Government work not protected by U.S. copyright TN-58
516 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS. AND FREQUENCY CONTROL. VOL. 37. NO. 6. NOVEMBER 1990

dependent of the trap geometry , the accuracy can generally be improved by using smaller numbers.

At NIST the primary goal of stored-ion frequency stan- dard work has been high accuracy. The preceding argu- ments have forced us to use small numbers of ions (approximately 10"* or less) that are laser-cooled. The loss in short-term stability due to reduced numbers can be partially regained by going to very long interrogation times.

These trade-offs are apparent from the expression for stability if we assume the Ramsey method of interrogation

and assume 100% detection efficiency. For these conditions [3],

(mi, mj)

(1/2) o,(t) = irNT,<.ir"'' (r > T,) (3)

where t is the averaging time, Tn is the Ramsey interro- gation time and coq is the clock transition frequency (in

radians per second). From this expression, we also see

the importance of making o^o large.

Of course, to achieve high accuracy, we must also ac- count for the perturbations due to static and time-varying

multipole interactions for electric, magnetic, and gravitational fields. These include atom-trap field interactions,

collisions, shifts due to uncontrolled electric and magnetic fields, and gravitational red shifts. However for more

than one ion in the trap, the dominant uncertainty appears

to be caused by the uncertainty in the second-order Doppler shift. Below, we briefly discuss experiments at NIST on

stored-ion frequency standards. More detailed accounts

are forthcoming.

^Be"^ Hyperfine Clock

In this experiment, an oscillator has been locked to the (w/ = -1/2, my = 1/2) ^ (-3/2, 1/2) nuclear spinflip hyperfine "clock" transition (a)o/27r = 303 MHz)

in the ground state of ^Be"^ (Fig. 1). The basic idea of this experiment has been described previously [9]-[12]; the current experiment works as follows [12]: Between 5000

and 10 000 ^Be^ ions and 50 000 to 150 000 ^^Mg+ ions were simultaneously stored in a cylindrical Penning trap

[12] with B = 0.8194 T under conditions of high vacuum

( < 10~* Pa). At a magnetic field B of 0.8194 T the clock

transition depends only quadratically on magnetic field fluctuations, and therefore the accuracy is not limited by

field fluctuations. To minimize second order Doppler

shifts of the clock transition, the ^Be"^ ions were cooled

to less than 250 mK in the following manner: The ^^Mg^

ions were directly laser-cooled and compressed by a nar- row-band ( ~ 1 MHz) laser radiation source at 280 nm. The ^Be^ ions were then sympathetically cooled [13] by

their Coulomb interaction with the cold Mg^ ions (see

Appendix I). A narrow-band 313-nm radiation source was

used to optically pump and detect the "^Be^ ions [9]-[12].

With the 313-nm source tuned to the 2s ^S\/2{m, = 3/2,

mj = 1/2) to 2p 'P3/2(3/2, 3/2) transition, 94% of the

'Be"^ ions were optically pumped into the 2s 'S,/2(3/2,

1/2) ground state. The 313-nm source was then turned

F=1

F=2

Fig. 1. Hyperfine energy levels (not drawn to scale) of the 'Be* 2s ^S^/z ground state as a function of magnetic field. At B = 0.8194 7 the 303 MHz clock transition is independent of magnetic field to first order.

off to avoid optical pumping and ac Stark shifts while the

clock transition was driven.

The clock transition was detected by the following

method: After the 313-nm source was turned off, the ions

in the (3/2, 1/2) state were transferred to the ( 1/2, 1/2)

state and then to the ( — 1/2, 1/2 ) state by two successive

RF 7r pulses. Each pulse was 0.2 s long and was resonant with the appropriate transition frequency (around 321

MHz and 311 MHz respectively). The clock transition was then driven by Ramsey's method of separated oscil- latory fields in the time domain with RF pulses of about

1-s duration and a free precession time on the order of

100 s. This transferred some of the ions from the ( — 1/2,

1/2) state to the ( —3/2, 1/2) state. Those ions remaining in the ( — 1/2, 1/2) state were then transferred back

to the (3/2, 1/2) state by reversing the order of the two

RF 7r pulses. The 313-nm source was then turned back

on, and the population of ions in the ( —3/2, 1/2) state was registered as a decrease in the ^Be"^ fluorescence, rel- ative to the steady-state fluorescence, during the first sec- ond that the 313-nm source was on. (The optical repumping time of the ions from the ( — 3/2, 1/2) state to the

(3/2, 1/2) state was about 10 s. ) The sympathetic cooling of the ^Be^ ions by the continuously cooled Mg^ ions

is necessary if long interrogation times are to be used,

since otherwise the ^Be^ ions would slowly heat up while

the 313-nm laser is off [11].

The Ramsey signal was used to steer the frequency of

a synthesized RF source [9]-[12]. Ramsey signal measurements were taken near the frequencies corresponding

to the half minimum points on both sides of the center frequency. The difference in the measured signal strengths on either side of the line center was used to electronically

steer the average frequency of the synthesizer to coq. Most

runs were taken with a commercial cesium beam clock

(fractional frequency stability ay{r) — 6 x 10"'" t~*'/~'

for measurement time t in seconds) as the reference oscillator, but a few runs were taken with a passive hydrogen maser {Oyij) — 2-3 x 10"'" t~"/"') as the refer- ence oscillator. Stabilities of the '^Be"^ clock are measured

to be better than 3 x 10"'" t~* '"* for the number of ions

used, which is within a factor of 4 of the theoretical max- imum stability given by (3). The systematic offset of our measurement due to the second-ovder Doppler frequency

shift was dominated by the velociiy in the rotation motion

TN-59
WINELAND « aL: PROGRESS TOWARD ABSOLUTE FREQUENO STANDARDS USING STORED IONS 517

of the ions about the trap axis. This shift was measured

tobe(-lJ2 ±0.5) X 10~'^.

An apparent pressure siiift witii an unexpectedly large

value was discovered when the background gas pressure was increased. The background gas pressure could be in- creased by moving the magnet of the sputter ion pump

that evacuated the trap region so that it oveiiapped fewer

pumping cells and reduced the pumping speed. (We

checked to make sure the magnetic field at the site of the

ions was not disturbed.) The composition of the gas was

not known since the pressure was measured with a Bayard-Aipert gauge. However, when the vacuum vessel

containing the trap was initially evacuated the dominant

bacl^TOimd gases were H; and He as detennined by a quadmpole mass analyzer. Therefore we expect that the

background gas during the frequency standard measure- ments was either Ht or He, or both. If the background gas was dominated by He. the fractional pressure shift was

measured to be about —3 x 10~^ Pa~'; if the background was dominated by H2, the pressure shift was measured to be about —9 x 10~^ Pa~'. Atomic ion hypeifine pressure

siiifts for He have previousiv been measured in ^^'Ba'

[14] and m '^g~ [8] to be 5 x 10"" Pa~' and 4 x

10~*' Pa~^ respectively. The authors of [14] show that the charge induced multipole interaction between the Ba"

and the noble gas atoms used in that study should give an

important contribution to the pressure shift. Since this in- teraction depends primarily on the polarizabiUty of the

neutral we would expect that the pressure shift for He at- oms on ^e"" ions would not be significantly different than

that for Ba~ or Hg^ . Similarly, since the polarizability of

H: is midway between Ar and Ne, we might expect the

pressure shift for H2 on Be" to be near those for Ar and

Ne on Ba*. which were measured to be —6 x 10~'" Pa~^

and —6 x 10~' Pa~' respectively.

The apparent large discrepancy between our data and

other measured pressure shifts is not understood at this time. One possible explanation is suggested by smdies

[15] of radiative association of C^ with H2 to form

CHJ. In the models of this process, it is assumed that the

H2 can stick to the C^ for a long enough time to allow

the C~-H2 complex to radiatively stabilize. This sticking

is possible because the collision eneig>' can be taken up

by the internal degrees of freedom in the H- molecule.

The sticking time can be orders of magnitude longer than

the interaction time during a simple elastic collision. If a

similar sticking mechanism is active in Hi-Be"^ colli- sions, it may account for the apparent large pressure shift. One way to check this hyjxjthesis is to measure the pres- sure shifts of Be* on Ne and Ar, where the sticking would

not be expected to occur, and on H2 or other p>olyatomic

gases, where sticking might occur. The uncertainty in this apparent pressure shift will probably limit the accuracy of the current ^e^ clock

measurements to around 1 part in lO'^. However we hope

t») more fully understand the shift before stating a final uncertainty We arc alsd continuing to search for other causes of systematic errors such as electronic offsets

However, we feel the accurac} couid be improved beyond

5 pans in lO' m the nmire. If the apparent coliisional

shift is real, it may be necessar*' to use liquid He ciyopuinping to reduce the background pressure.

Hg"^ Optical Clock

The velocity' in the micromotion ft>r an ion ir a quadrupole RF trap is proportiona] to the distance of the ion

from the center of the trap. For two or more laser-cooled

ions in the trap, the Coulomb repulsion between ions holds

them away ftom the trap center, and the second-order

Doppler shift is dominated b) the veiocm of micromotion. However, a smgie ion can be held near the trap center if sufficiently cooled. In this case the kinetic energy in the micromotion can be equal to that of the secular motion

[2], [3], [16], [17]. If the ion is iaser-cooied. resultuig

Doppler shifts can be extremely small; uncertainties can be less than 1 part in 10^° in some cases [17]. [18] However with N = \, stability' is marginal unless we make wr, high enough (see 3). One way to accomplish this is to let

0)0 correspond to an optical transition. Dehmelt suggested

this idea in 1973 [6], The reasons thai a clock basec on

an optical transition in an ion has not been realized yet

are: 1) it took several years tc isolate and manipulate smgie ions in the traps; 2) local oscillators (lasers) with the

desired spectral purit\' are still not available: and 3) accurate comparison of laser and microwave frequencies is extremely difficult and remains an important problem.

Nevertheless the potential accurac}' of single-ion optical

frequency standards is extremely high [6]. [19] ( — 10~'^

)

so it is important to pursue this research

At NIST we have investigated the use of the 5d^%s

%P_ ^ 5d%sr ^L

(a)o/2x == 1.07 X 10'^Hz)m ''*Hg

electric quadrupole transition

(see Fig. 2 1 as an

optical frequency standard [20]. The single mercur> ion

is confined in a miniature RF trap that has internal dimensions of r„ s 466 fixn and z„ = 330 fim [20], [21]. The

amplitude of the trapping field (ftequency I2/27r s 21-

23 MHz) can be varied to a peak of 1.2 kV. The ion is laser-cooled to a few millikelvins by a few irucrowatts of

CW laser radiation that is tuned below the ^5] /2-^fi /: first resonance line near 194 nm. In order to cool all motional

degrees of freedom to near the Doppler cooling limit [7]

(T = hy /Ikg = 1.7 mK) the 194 nm radiation irradiates the ion from 2 orthogonal directions, both of which are at an angle of 55° with respect to the symmetry (z ) axis of

the trap. The 282-nm radiation that drives the narrow

^S\/2-^D^/2 transition is obtained by frequency-doubling

the radiation from a narrowband CW ring-dye laser. The

frequency of the laser is stabilized by locking it to a stable Fabry-Perot cavity. The frequency of the laser is scaimed

by an acoustooptic modulator that is driven by a computer

controlled synthesizer. Up to a few microwatts of 282 nm

radiation could be focussed onto the ion in a direction

counterpropagaung with one of tint 194-iur iighi oeam^

Opticai-optical double resonance was used to detect

transitions driven by the 282-nm laser from the ion's ^Sj /2 ground state to the metastablc ^Ds/; state [20]. [21]. The

TN-6C
5U lEE^ TRANSACTI<»JS OJv ULTRAStMWlCS. FERROELECTJUCS A^aJ FREQUEKC"- CONTROL \'OL 37 NO t NOVEMBER 199C

2813 mil

Fig. 2. Simplified optical energy-level diagram for Hg"^. The lifetime of

the ^Dj,2 level is about 0. 1 &. which would give a linewidth of approximately 2 Hz on the electric quadnipole ^S,/2 ~' ^D^/i tniiisition. By

observing the presence of or lack of fluorescence from the ^S, /, ~* ^Pi n

oansiDoi:. the qQadrupoie "ciocl' ' transition can be detected with 1009^

efficiency

.

194-iim fluorescence rate from the laser-cooled ion is high

when the ion is cycling between the ^Si ji and ^Pi /2 states (Fig. 2) and neari} zero when the ion is in the metastable

^2)5/2 state. The ^5i/2-"X>5/2 resonance spectnim was obtained by probing the S-D transition at a particular fre- quency for the 282-nm radiation for 20 ms, then turning

off the 282-nm radiation and turning on the 194-nm ra- diation to look for the presence or absence of scattered

photons at 194 nm. (The two radiation fields are alter- nately applied to avoid light shifts and broadening of the narrow S^D transition by the 194-nm radiation.) If there was no fluorescence at 194 nm, a transition into the metastable D state had occurred; the presence of 194-nm flu- orescence indicated that the ion was in the ground state and no transition was recorded for this frequency of the

282-nm laser. The frequency of the 282-ruD radiation was

then stepped and the measurement cycle repeated. As the

frequency was swept back and forth each new result at a

particular frequency of the 282-nm radiation was aver- aged with the previous measurements at that frequency.

Normalization (or digitization) of the signal was obtained

by assigning a 1 to each measurement of high fluores- cence and a to each measurement of no fluorescence.

The high fluorescence level made it possible to determine

the state of the atom with almost no ambiguity in a few

milliseconds. Thus, it is possible to reach the shot noise

limit imposed by the single atomic absorber [21].

The quantized fluorescence signal obtained from an 8

MHz scan of the 282-nm laser through the ^Si/2(/nj =

— 1/2) — ^D^/iimj = 1/2) Zeeman component of the

electnc quadnipole transition in '"'Hg^ is shown in Fig.

3. The recoilless-absorption resonance (carrier) and the motional sidebands due to the secular motion in the harmonic well of the RF trap are completely resolved [21].

To avoid broadening of the quadrupole transitior due

to magnetic field fluctuations, we have recently [22] per- formed the same expenmeni on the ^^j/iCF = 0, mf =

0) -* ^Ds/2iF = 2, my = ) transmon m '^'Hg". which

becomes field indeoendeni a.s £> — . The earner is now. observed with a linewidth Ap ^ 100 Hz (limited b} laser spectral purity), which gives a line Q of about 10'\ the

highest reported ui atomic or molecular spectroscopy

o 090 -

Frequency Detuning (in MH;

'

Fi£. 3. Quantized signal showing the electric-quadiupole-allowed Sd'%s

Si/jimj = — 1 /2) — 5d%r ^Ds/2("h = 1/2) transition in a single, laser-cooled '^Hg^ ion. On the hoiizontal axis is plotted the reiaiive detuning from Ime center in frequency units at 282 mc. On the vcmcal

axis is plotted tne iJiobabiiiiy that the fluorescence from tne or '.

i /--6 .". ^P,/2 fiist resonance transition, excited by laser radiauon at 194 nm. is on. The electric-quadiupole-allowed S-D transition and the first-reso- nance S-P transition are probed sequentially to avoid light shifts and

broadening of the narrow S-D transition. Cleari> resolved are the re- coilless absoiption resonance (earner) and the Doppier siceoaaa> aoe k

the residual secular monon of the laser-coolea ion. Eacn poici i^ tne average of 230 measurement cycles (from [21]).

Current efforts are devoted to improving the 282-nm laser spectral purity by locking it to a more stable reference

cavity. If the laser's spexrtral purity can be made high

enough, then when the laser is locked to the ion transi- tion, stabilities are anticipated to be better than 10~''

T~^'^^' and accuracies could be 1 part in lO'^ or better

[1], [3], [6], [17]-[21].

Future Penning Trap Experiments

The 'Be"^ ion experiments have the primary disadvantage that 0)0 is relatively low and the resulting frequency

stabilities are modest. We might hope to substitute ^°'Hg~

ions in place of the ^e"*^ ions because ^"'Hg"^ has a higher

frequency clock transition (&)o/2x = 26 GHz), which is field independent ai B = 0.534 T [1]. However two dis- advantages compared to the ^e" case arise: (1) If the Hg^ is sympathetically cooled by lighter ions such as Mg~

or Cd^ it will reside in an annulus surrounding the lighter

ions [13]; this makes the second-order Doppier shift larger

for a given density and number of ions. (2) 'Be* and

^Mg^ have simple optical pumping schemes whereby a

single laser frequency can be used to optically pump into a single ground state sublevel [10]. For -°'Hg" or ' Hg~

in a strong field (required for a Penning trap) optica!

pumping schemes would require auxiliars laser Imes at 194 nm and microwave oscillators to manipulate the ground state subievels [1]; the simple optical pumping

schemes as in the case of ^e" and "^Mg* do not appear

possible. "'Hg' in a Penning trap would provide a veiy

interesting system wher magnet- of high enough field strength become available. For example, the («/ = iP..

n< = 1 /2 )

-^-^

( — 1 /2, I '2 I hyperfinc transiuor ir the ground state of '**Hg" (o>o/2t = 20. « GHz i> neic w

dependent at B = 43.9 T Ai present, wt: mus! awai ti»e requireti magnet
WINELAND et al.: PROGRESS TOWARD ABSOLUTE FREQUENCY STANDARDS USING STORED IONS 519

TABLE I Expected Frequency Stabilities for Various Ions of Interest for Frequency Standards

a,.(l s) M,{u) M.(u) COo 5(7-) i I'^>'D2/''oI ^, Z,,(cm) A', T* (T^ = 100 s)

9(Be") 26(Mg") 303 MHz'" 0.8194 0.1 10-" 125 ^m 0.5 6630 65 mK 6.45 X 10-"

9(Be") 26(Mg") 303 MHz'° 0.8194 0.01 10-" 1.25 mm 0.5 67400 65 mK 2.02 X 10""

67 (Zn-^) 113(Cd*) 1 GHz" 8 0.1 10-" 56 /xm 0.5 28500 484 mK 9.44 X

10-'" 67 (Zn*) inccd-") 1 GHz" 8 0.01 10-" 560 ^m 0.5 292000 484 mK 2.94 X 10-'" 199(Hg") 203 (Tr) 20.9 GHz 43.9 0.01 10-16 58 ^m 0.5 52400 144 mK 3.51 X 10-"

201 (Hg") 203(11*) 25.9 GHz' 0.534 0.1 10-" 1.48 mm 0.5 50100 1.45 K 2.75 X 10-"

201(HgM 203 (ir) 25.9 GHz' 0.534 0.01 10-" 1.48 cm 0.5 524000 1.45 K 8.49 X 10"'* 201 (HgM 203 (T^) 7.73 GHz' 3.91 0.1 10-" 204 Aim 0.5 50100 1.45 K 9.21 X 10-"

201 (Hg") 203(Tr) 7.73 GHz' 3.91 0.01 10-" 2.04 mm 0.5 524000 1.45 K 2.85 X 10-"

"The ions are assumed to be confined in a Penning trap and sympathetically cooled. M, is the mass of the "clock" ion in atomic mass units (u), M2 is the mass of the ion that is directly laser cooled and sympathetically cools M^. For all cases M, < A/j so that the cloud geometry of Fig. 5 applies. The clock frequency ojq is assumed to be "field independent" as for the 'Be* case described in the text. This choice determines the magnetic

field. From the magnetic field B and an assumed value of ^ (6), the density of species 1 can be determined from (5). If we desire a certain value of

the second order Doppler shift ( Acot/i'o). we then determine a value of the cloud radius t, and the ion number (7) for an assumed value of z^,,

.

From (3), we then determine 0^(7) assuming 7"^ = 100 s for the Ramsey interrogation time. We refer to a,,( 1 s) for convenience. 7"* is the temperature

at which the second order Doppler shift from the axial and cyclotron motion would be equal to the value listed in the table. Therefore we have

implicitly assumed the temperature of the cyclotron and axial motion is much less than 7"*. From this, we see the importance of laser cooling.

Within the limits imposed by today's technology, an

experiment similar to the ^Be"*^ experiment but with better expected performance might be provided by ^^Zn"^ ions

[23]. The clock transition could be the {m, = /I, rrij =

1/2) ^ (3/2, 1/2) transition (a;o/27r - 1 GHz), which

is field independent at fi = 8 7. Some other examples are summarized in Table I. Similar schemes are possible with

ions such as Ba^ but optical pumping is complicated.

Future Paul Trap Experiments

The advantage of the Paul trap is that a magnetic field

is not required for trapping. This allows us to be rid of a cumbersome magnet and allows use of transitions that are

field independent at fl = 0. The primary disadvantage is that up to the present time it has been impossible to laser- cool very many ions (A' > 100). As discussed above, the use of small numbers may not be a limitation if coq can be

made big enough. This is the basic philosophy behind the single-ion optical frequency standards. Even for ooo/27r = 40.5 GHz ( '^^Hg^ ) and N = 1 , from (3) we can expect

[3] Oyir) = 3.9 X 10"'^ 7--"/2) ^^en T^ = 100 s. Because the second-order Doppler shift is expected to be so

small for single ions, it is perhaps useful to look at this case a little more closely.

The main advantage of using a single ion in an RF trap

is that the kinetic energy of micromotion can be on the order of the secular motion energy. For a single '^^Hg"^

ion cooled to the Doppler-cooling limit [21], the secondorder Doppler shift would be [3] < Aj'o2/fo> = —2.3 x

10~'^. In a quadrupole ion trap, two or more ions in the

trap are pushed from the center of the trap by the mutual

Coulomb repulsion and the second-order Doppler shift is higher [2]. Consider the trap shown in Fig. 4. This design

is essentially the same as that described by Prestage et al.

[5] and Dehmelt [24] . In the trap of Fig. 4, the RF electric

fields are transverse to the trap axis for the entire z extent of the trap. If a single string of ions is trapped along the

z axis, then the kinetic energy of micromotion is about

IC^

'0-1 ^, c^^ s^' [- 2zo -I

Fig. 4. Linear trap configuration. The alternating RF voltage K„ cos fir is assumed to be applied to diagonally opposing electrodes as shown. We

assume the end portions of the electrodes are long enough that the re- sulting RF potential at the position of the ions is independent of z, so

that the RF electric fields are parallel to the x-y plane. To trap ions along

z, we assume the center four electrodes are at static ground potential and

the two sets of four electrodes on either end are at a static potential Uo (Uo > to trap positive ions). The average position of the ions could

be made to coincide with the RF electric field null by applying slightly different static potentials to the four central rods to correct for contact

potential offsets etc. This geometry would allow laser beams to be di- rected along the z axis. Such a trap might also be useful for studing fundamental atom-radiation interactions such as the statistics of fluores- cence from or the interaction of a cavity with two or more well localized atoms.

equal to the kinetic energy in the secular motion. Therefore, the fractional second-order Doppler shift could be

as low as 5kT/2mc^. This is 5/6 of the value for a quadrupole RF trap [2] because of the absence of RF micromotion along the z direction. At the Doppler-cooling limit,

this gives Apdj/^q — -2 x 10~'^ for all Hg^ ions in the

string.

Use of the trap of Fig. 4 would allow A^ » 1 giving

good stability and still yield a small second-order Doppler

shift. For the experimentally achievable conditions of Appendix II, A^ = 50 ions could be stored along the z axis

like "beads on a string" with ion spacings of approximately 5 ixm. With this spacing each ion could be independently detected by using an image detector [25], [26],

[39]. Therefore each ion could be treated as an independent atomic clock where the clock transition could be detected with 100% eflficiency [21]. From (3), for Tr = 100

s and coo/27r = 40.5 GHz ( '""Hg^) the frequency stability of this clock "ensemble" would be 0^(7) = 5.5 x

10

-14 -(1/2) For these long interrogation times, sympathetic cooling might be required to avoid heating while

TN-62

-rrr
520 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL. VOL. 37. NO. 6. NOVEMBER 1990

the Hg"^ optical pumping laser was turned off to avoid

light shifts during the Ramsey period. The ions used to cool the Hg"^ ions would also find a position in the string

of ions along the z axis. Arrays of traps have also been

proposed previously [27]. These trap arrays would have

the same advantages as above for increasing A^ and minimizing the second-order Doppler shift. For optical spectroscopy, lasers with high spectral purity are required (see the following). This is the current

limitation to reaching the natural linewidth resolution in the NIST Hg^ experiments. In future experiments, the

laser can be locked to the ion resonance line in a manner

similar to that described for the ^Be^ hyperfine transition. That is, the clock transition is alternately probed on either

side of the resonance (at the frequencies near maximum

slope). After several measurements an error signal is built up which is used to servo the average frequency of the

laser to the center of the clock resonance. A partial test of this lock can be made by briefly interrupting this servo, probing a portion of the clock resonance line (while as- suming that the center of the clock resonance is given by

the most recent servo measurements), and then returning

to servo operation. This process is repeated until a reso- nance line similar to that of Fig. 3 is obtained. This, of

course, does not test for a slow drift in the ion resonance

frequency, but should be a fairly good test of the ability

to servo the laser to the clock transition. The main systematic error for the Hg"^ optical experiment may ultimately be the uncertainty in the shift of clock

transition from static electric fields of quadrupole symmetry such as those caused by static potentials applied to the trap electrodes [6]. The basic idea is that the D state of the clock transition has a quadrupole moment that can

interact with static fields of quadrupole symmetry to give

a shift that must then be measured. Although trouble- some, it should be possible to remove this offset from a clock calibration to about the same precision as the measurement precision [3]. Dehmelt has pointed out the advantage of the singly ionized group IIIA atoms in this re- spect [6]; the interesting clock transitions in these ions are

the '5o -^ ^Pq intercombination lines that are not shifted

by static quadrupole electric fields. However, at low magnetic field these transitions have magnetic field dependence on the order of the nuclear Zeeman eff'ect. Therefore careful control of the magnetic field would be

required [6]. At higher fields, field-independent transi- tions could be used to advantage [28]. A linear trap or

trap arrays could be used to increase SNR as described

previously. For clock transitions involving a state with a quadrupole moment, such as for Hg^, the mutual Coulomb interaction between ions would cause an additional

quadrupole frequency shift that must be taken into ac- count [29]. For the group IIIA ions, this shift would be

absent.

Lasers and Laser Metrology

For optical frequency standards and spectroscopy ex- periments, the most outstanding current problem is that of

local oscillator (laser) spectral purity. Lasers that are locked to reference cavities have been shown to track the

cavity frequency to precisions of much less than 1 Hz [30].

The problem then remains that the length of the cavity and

therefore the frequency of the laser are modulated by

acoustic noise with deviations typically much greater than

1 Hz. It is desired to make this reference cavity stable enough (frequency deviations less than the linewidth of

the optical transition) over the attack time of the servo used to lock the laser to the ion line. A more general problem is that of cost, power requirements, and complexity of the lasers required for optical

pumping and laser cooling. At present, most experiments

use near-ultraviolet laser lines provided by gas laser pumped dye lasers that are frequency doubled or summed

to give a few microwatts at wavelengths corresponding to certain transitions in various ions. However, this tech- nology may simplify in the coming years. Partly because

of efforts by the optoelectronics industry, it is not unrea- sonable to think that such lasers will be replaced by cheap,

high-efficiency solid-state lasers. This is already happening at near-infrared wavelengths where diode lasers are used to optically pump and laser-cool neutral cesium and

rubidium atoms.

Finally, we call attention to the problem of frequency

metrology that is important in the generation of time from

optical frequency standards. The technology to connect

the microwave region of the spectrum to the optical spec- trum through the use of synthesis chains exists [31]. Other

schemes have been proposed and are being worked on

[32], [33], [40]. However, this metrology problem is sig- nificant and simpler schemes would help realize the full benefits of optical frequency standards.

Appendix I Sympathetically Cooled Ions in a Penning Trap

An approximate model of a two species ion plasma in a Penning trap is shown in Fig. 5. To make the problem

tractable, we will assume that the species of spectroscopic

interest is of lighter mass (species 1) and therefore resides

inside species 2, which is directly laser-cooled. The case where species 2 is sympathetically cooled by species 1 [13] could be treated in a similar manner. If we assume

A', « A^2' we can approximate the species 1 ions by a cylinder of radius b\, and height Izd If we can assume

that the cyclotron and axial motion of the ions have been

cooled to negligible values, the second-order Doppler shift averaged over ions of species 1 is due to the velocity of

cloud rotation and is given by

where co is the rotation frequency and c is the speed oi

light. The density of species 1 is

«i = /?;|a)(ni — (ji)/2-Kq\ (5)

where m\, fi,, and q\, are the mass, cyclotron angular fre- quency and charge of species 1 [341, \yf^\. .\n cxperiinonOlly useful parameter to characterize the density of spc- TN-63
WTNELaKD ri a/.: PROGRESS TOWARD ABSOLLTE FREQUENCY STANDARDS USMG STORED IONS 521

Biz

Hg. f . ApproMinaie model for a sympathetically cooled ion sample in a ^mmis Trap. We assmne /7, />r. > q->i m-. so Uiat species 1 which is

•.ae "cioct" ioectes) is appioxiniated by a mniorm density column of

.leight 2c^ and radios b^ . Species 2 is assimied to be contimioiisly laser cooled and by the Coulomb interaction continnonsly cools species 1 . All

ions aie assmned to be in thermal eqoiiibrnmi and theiefoie co-rotate aboat the c axis at frequency j. The second order Doppler shift of species 1 .s assumed to be commatec by the velocity .a tiiis rotation.

cies 2 is the parameter

^ = 2o}/ a. (6)

where Qi is liie cyciotron frequency of species 2. is the

ratio of the rotation frequency aj to the maximum allowable rotation frequency %/'2 given bv the Brillouin density [34], [35].

From (4)-<6), and the fonnula Vi = 2x^,"Cci/ii, we

have

.V, = %nh<h^{\ — fm,ft/2/nT^,)< -£^m/vo) /^Mitli

(7)

where /Hi and qj are the mass and charge of ions of species

2. From these expressions and (3) applied to species 1. we can generate the parameters of Table I. Accuracy of

10~'^ with reasonable stability seems f>ossibie. Of course

the local oscillator spectral punty must be good enough

to reach these conditions. The '^Hg" example is not now

realistic due to the lack of a suitable magnet.

Appendix H

Linear Paul Trap

To describe the trapping from pwnderoraotive fields we

assume "hat near the trap axis, ihe time varying potential

has the form

o = cos 0/ (8)

where R is approximately equal to the dimension tq indi- cated in Fig. 4. I /? = />) if the electrode surfaces lie along

equiixnentiais of (8). ) If Q is high enough (defined in the followmgj this results n the harmomc pseudopotential

well in the radial direction:

where a>^ = qV^/iyfl niSlR^) is the single-ion oscillation

ftequency in the radial direction. Expression (9) is ex- pected to be fairly accurate when Q » us^, which is the

condition for Q being large enough.

For the static potential due to voltage Ur. applied to the

end sections, we will make the approximation thai, near

the center of the trap, this potential is gi>'en by ,

<i), = fcDoir - X- ^ r)/2)

m

C«J. .-a 5fz^ - (x^ + /)/2) (10)

whoe ^ is a geometric factor and ci>- is the z oscillation

ftequency for a single ion in the trap. Equation 1 10) represents the lowest order term in the expansion of the static potential for the electrode configuration of Fig. 4 with the

bias assumed. If the length of the central rods is long

compared to their spacmg, then higher-order terms m the

expansion become more important so Jbai the :tatic potential changes significantly only near the point of separation between the end and center sections. For simplicity, we will assume that the length of the central rods is- approximately equal to the rod spacing and that the extent of the ion sample is small compared to R so that ( 10)

should be a reasonable approximation to the actual 30tential.The addition of this static potential weakens the strength

of the pseudopotential well m the radial direction so thai

the total potential in the radial direction is given by

m m

2^ 2^

(11)

where o)^ is the single ion radial oscillation frequency for

the combined potentials.

A possible advantage of such a linear trap may be for

laser-cooling large numbers of ions in an RF trap. As has

been noted in recent papers [36], [37], the RF micromotion m a quadrupole ?am rap perturbs the ion's spectrum

and can inhibit laser-cooling. This may be one of the mam

reasons large numbers of ions m an RF trap nave not been

laser cooled. If a cooling laser is directed along the z axis

of the trap in Fig. 4. the ion s spectrum is not perturbed

by micromotion ana laser cooling snould occur according

to the simple theory [38], [41] if RF heatmg is not too

severe.

'Beads on a Strfng

The most interesting case for accurate spectroscopy is when all ions are confined ;dong the z ixis. To inalyze

this case, first assume hat oj, s large enough to make the

ions lie along ihe :. a>:is. Hien. or a given V and j,, we

minimize the total ion potential lo obtam Zi , for / = ( \.

•

, <V). From the :; we can hen lind a condition on

00^ to insure all ions stay along the - axis. For ;ons pinned m he :. ^;is. he 'otai ion potential

TN-64
522 IEEE TRANSACTIONS ON ULTRASONICS. FERROELECTRICS. AND FREQUENCY CCPTTROU VXIL 37. NO. 6. NOVEMBER 390

energy is

^7- = — cd? 2 zf + q^ 2 I z, — z, 2 ~ .' '</ ' -I (12)

Using a computer, we have minimized <p7- as a fimction

of ion fwsitions for various experimental parameters. Near

c = (trap center), the ions are closest together and fairly

regularly spaced. The closer the spacing, the higher co^ must be to maintain all ions on the z axis.

To find the condition on m'^, consider an infinite string

erf ions ^.vith equal spacing d along ihe z axis. If one Ion

IS displaced a distance dx{ «<i t from ihe z axis, it experiences a force away from the z axis of strength

= (2 S /I n =l

-3 i

d

^ ax = (2.404) ^j&c. I (13)

d'

If we have only 5 ions spaced by d the analogous force on the center ion is given by (13) with the numerical factor equal to 2.250 rather than 2.404. Therefore even if the

ions are only approximately evenly spaced, <^i3) win be a

good estimate of F^ for a particular ion if we take <i to be

the mean of the spacing to the nearest neighbor ions. For

the ions to be stably trapped on the z axis, we want the

pseudopotential force inward to be larger than the force

outward as expressed by (13). From (11) and (13) we

therefore require that

(oi;)^ > 2.404 (^/md^ (14)

or c</2ir > 91.9 (Md^) ~- '^^' where M is the ion mass

n atoimc mass units, d is the spacing m micrometers, and

J, / Isr IS m megahertz.

As an example, for N = 50, uij 2ir = 50 kHz. and M

= 199 < "^g~ ions ), die computer calcuiauon gave an

overall stnng length oi 266 /xm. The spacings of adjacent

qairs of ions ;n _an, starting from the outermost pair of

ans on one end and ending with the center pair are: 10.48,

S.46, 7.43, 6.77, 6.30, 5.95, 5.68, 5.46, 5.28, 5.12,

4v95t, 4.88. 4.79, 4.71, 4.64, 4.58, 4.53, 4.48, 4.45,

.42, 4.39, 4.37, 4.36, 4.35, 4.35. Using 4 =4.35 ^m,

ve lind from 14), o^/2ir > 718 kHz, a}V2x > 719

kHz. If i? = 0.75 mm and Q/2x = 5 MHz, then we reluire V^ > 233 V. Acknowledgment

The authors gratefully acknowiedge the continued support of the Office 3f Naval Research and the Air Force

Office of Scientific Research. The authors thank R. Druilinger. H. Robinson. andM. Young for helpful comments

on the noanuscript.

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WINELAND et al.: PROGRESS TOWARD ABSOLUTE FREQUENCY STANDARDS USING STORED IONS 523

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TN-66
126 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT. VOL, 40. NO. 2. APRIL 1991 A 303-MHz Frequency Standard Based on Trapped

Be^ Ions

J. J. Bollinger, D. J. Heinzen, W. M. Itano, S. L. Gilbert, and D. J. Wineland

Abstract—\ 303-MHz hyperfine transition in the ground state of 'Be'^ was used as a basis of a frequency standard. The ions were stored in a Penning ion trap. Linewidths as narrow as 900 /xHz were obtained. The

frequency stability was measured to be better than 3 x 10 ~'^t~'^^. The inaccuracy in the second-order Doppler shift was reduced to 5 parts

in 10'^ by laser cooling. An apparent pressure shift with an unexpectedly large value was discovered which limits the accuracy of the current experiment to approximately 1 part in 10'^.

I. Introducton

ION TRAPS provide the confinement with low perturbations necessary for improved frequency standards. For a micro- wave (or RF) frequency standard, typically many ions are stored

in a trap in order to increase the signal-to-noise ratio and obtain good frequency stability. With many ions in a trap, the secondorder Doppler shift is one of the largest systematic contributions

to the inaccuracy of the standard [l]-[6]. In frequency stan- dards which aim at high accuracy, the technique of laser cooling can be used to reduce the second-order Doppler shift to an ac- ceptable level [l]-[3]. In this paper we investigated the use of

Be'*' ions stored in a Penning trap to make a microwave (or RF)

frequency standard with high accuracy. Be"*" is technically easy

to trap and laser cool and, therefore, is a good candidate for investigating the use of stored ions as a microwave frequency

standard. Other ions with even higher potential line Q's are dis- cussed in [3]. Fig. 1 shows the energy level structure of the ground state of ^Be^ as a function of the trap magnetic field B. At fl = 0.8194

T, the transition between levels 1 and 2, called the clock transition, depends only quadratically on magnetic field fluctuations and is, therefore, a suitable transition for a frequency standard.

In this experiment, an oscillator is locked to this clock transition. The basic idea of this experiment has been described pre- viously [7]-[10]. A description of the current experiment [10]

is given below.

II. Experimental Technique

Between 5000 and 10 000 "Be^ ions and 50 000 to 150 000

^^Mg"*" ions were stored simultaneously in a cylindrical Penning

trap [11] with B — 0.8194 T under conditions of high vacuum

( = 10^^ Pa). In order to minimize second-order Doppler shifts of the clock transition, the ''Be^ ions were cooled to less than 250 mK by the following method. The ^^Mg''^ ions were directly

laser cooled and compressed by a narrow-band ( ~ 1 MHz) raManuscript received June 13, 1990; revised October4, 1990. This work

was supported by the Office of Naval Research and by the Air Force Office of Scientific Research. The authors are with the National Institute of Standards and Technology.

Boulder, CO 80303. IEEE Log Number 9042018 .

Fig. 1. Hyperfine energy levels (not drawn to scale) of the 'Be"^ 2s ^S|/2

ground state as a function of magnetic field. At B = 0.8194 T the 303- MHz clock transition is independent of magnetic field to first order.

diation source at 280 nm. The 'Be^ ions were then sympathetically cooled [12] by their Coulomb interaction with the cold

Mg"*" ions. A narrow-band 313-nm radiation source was used to optically pump and detect the ^Be"" ions [13], [14]. Lenses were

used to image the ^Be^ fluorescence onto the photocathode of

a photon-counting imaging detector. The overall detection ef- ficiency was ~2 X lO""*. With the 313-nm source tuned to the 2r5,/2(w/ = 3/2, nij = 1/2) to Ip-P^i.O /l, 1>/1) transition, 94% of the ^Be"^ ions were optically pumped into the 25'^S,/2(3/2, 1/2) ground state. The 313-nm source was then

turned off' to avoid further optical pumping and ac Stark shifts. The sympathetic cooling of the 'Be^ ions by the Mg^ ions pro- vided a steady cooling source independent of the 313-nm radiation and therefore permitted the use of long transition times. The clock transition was detected by the following method.

After the 313-nm source was turned off, the ions in the (3/2,

1 /2 ) state were transferred to the (1/2, 1 /2 ) state and then to the ( —1/2, 1/2) state by two successive RF tt pulses. Each

pulse was 0.2-s long and resonant with the appropriate transi- tion frequency (around 321 and 311 MHz, respectively). The

clock transition was then driven by Ramsey's method of separated oscillatory fields with RF pulses of about 1-s duration and

a free-precession time on the order of 100 s. Free-precession

periods as long as 550 s were also used. This procedure trans- ferred some of the ions from the ( -1/2, 1/2) state to the

( -3/2, 1/2) state. Those ions remaining in the ( - 1/2. 1/2)

state were then transferred back to the (3/2. 1/2) state by re- versing the order of the two RF tt pulses. The 313-nm source was then turned back on, and the population of ions in the (-3/2, 1/2) state was registered as a decrease in the ''Be* fluorescence, relative to the steady-state fluorescence, during

the first second that the 313-nm source was on. The time for optical pumping from the ( -3/2, 1 /2) state to the (3/2. 1/2)

state was approximately 10 s. After each measurement the sys- tem was reinitialized by leaving the 313-nm laser on for about 25 s.

III. Results

Fig. 2 shows the Ramsey signal obtained with a 550-s free precession period. The 900-/iHz linewidth gives a line Q of 3.4

00 1 8-9456/9 1/0400-0 1 26501. 00 vc: 1991 IEEE TN-67
BOLLINGER ei «/.: FREQUENCY STANDARD BASED ON TRAPPED Be* IONS 127

FREQUENCY - 303016377.265

(Hz)

Fig. 2. Ramsey signal of the clock transition for a 550-s free precession

period. The data are the result of one sweep (that is, one measurement per

point). The sweep width is 2.4 mHz and the frequency interval between

points is 0.2 mHz. The dots are experimental and the curve is a least- squares fit.

X 10". Ramsey signals obtained with a 100-s free precession

period were used to servo the frequency of a synthesized RF

source [7]-[10]. Measurements were taken near the frequencies

corresponding to the half minimum points on both sides of the center frequency. The diflFerence in the measured signal

strengths on either side of the line center was used by a computer to servo the average frequency of the synthesizer to the clock transition frequency. Most runs were taken with a commercial cesium beam clock (fractional frequency stability a,, (x) = 6 X 10" '"t''^" for measurement time t in seconds) as the reference oscillator for the synthesizer, but a few runs were

taken with a passive hydrogen maser (a,, (r) = 2 — 3 x

10~'^

r"'''') as the reference oscillator. The stability of the ^Be"^ clock was measured to be better than 3 x 10"'' t"'''" for 10^ s < x < 10'* s, which is within a factor of 4 of the theoretical limiting

stability for the number of ions used [3]. For x > 10^ s the frequency stability was apparently limited by the pressure shift discussed below to - 3 x 10~ '". The largest contribution to the second order Doppler shift was due to the E x B rotation of

the ions about the axis of the trap. The rotation frequency and

ion cloud radius were measured by using a weak laser beam to probe the ion cloud [13]. The fractional second-order Doppler

shift was calculated to be ( - 1 .2 ± 0.5 ) x 10" '•*. An apparent pressure shift more than three orders of magnitude larger than expected was discovered when the background

gas pressure was increased. The background gas pressure could

be increased by moving the magnet of the sputter ion pump

which evacuated the trap region so that it overlapped fewer

pumping cells and reduced the pumping speed. (We checked to make sure the magnetic field at the site of the ions was not disturbed. In addition, the shape and apparent temperature of

the ion cloud could be monitored immediately before and after the Ramsey interrogation period from the image of the ion flu- orescence. The increased background gas had a negligible effect on the cloud shape and temperature.) The composition of the gas was not known since the pressure was measured with a Bayard-Alpert gauge. However, when the vacuum vessel containing the trap was initially evacuated, the dominant background

gases were H^ and He. If the background gas was dominated

by He. the fractional pressure shift was about -3 x 10 '' Pa ';

if the background was dominated by Hi, the pressure shift was

about -9 X lO"*" Pa '. Atomic ion hyperfine pressure shifts due to He have previously been measured in "^Ba^ [15] and in

''^''Hg* [16] to be 5 X 10"" Pa"' and 4 X 10"" Pa"', re- spectively. Vcttcr et al. [15| show that the long-range chargeinduced interaction between the Ba^ and the noble gas atoms

gives one of the largest contributions to the pressure shift. Since

this interaction depends primarily on the polarizability of the

neutral atom we would expect that the pressure shift for He

atoms on ''Be^ ions would not be significantly different than that

forBa"*" orHg"^. Similarly, since the polarizability of H, is midway between Ar and Ne, we might expect the pressure shift for Hj on Be"^ to be near those for Ar and Ne on Ba"^, which were measured to be -6 x 10"** Pa"' and -6 x 10"'° Pa"', re- spectively [15]. The apparent large discrepancy between our data and other measured pressure shifts is not understood at this time. One

possible explanation is suggested by studies [17] of radiative association of C"'" with Hj to form CH2^. In the models of this process, it is assumed that the Hj can stick to the C"^ for a long

enough time to allow the C^— H, complex to radiatively sta- bilize. This sticking is possible because the collision energy can

be taken up by the internal degrees of freedom in the H2 molecule. The sticking time can be orders of magnitude longer than

the interaction time during a simple elastic collision. If a similar

sticking mechanism is active in H, - Be"'" collisions, it may

account for the apparent large pressure shift. We plan to check

this hypothesis by measuring the pressure shifts of Be^ on Ne

and Ar, where the sticking would not be expected to occur, and on H2, where sticking might occur.

If the apparent collisional shift is real, the background pres- sure of 10"^ Pa in this trap will limit the accuracy of the Be"'" clock to about 1 part in 10'^. We are continuing to search for other causes of systematic error such as electronic offsets. We

feel the accuracy could be improved beyond 5 parts in 10'^ in the future. It may be necessary to use liquid He cryopumping

to reduce the background pressure.

Acknowledgment

The authors thank F. Moore and J. Wells for carefully reading the manuscript.

References

(1) D. J. Wineland, W. M. Itano, J. C. Bergquist, and F. L. Walls,

"Proposed stored '"'Hg* ion frequency standards," in Proc. 35th Ann. Symp. Freq. Control. Philadelphia, PA, pp. 602-611, 1981.

[2J D. J. Wineland, W. M. Itano, J. C. Bergquist, J. J. Bollinger,

F. Diedrich, and S. L. Gilbert, "High accuracy spectroscopy of

stored ions," in Proc. 4th Symp. Frequency Standards and Metrology, (Ed. A. DeMarchi) Ancona, Italy, pp. 71-77, 1989.

[3] D. J. Wineland et al., "Progress at NIST towards absolute fre- quency standards using stored ions," IEEE Trans. Ultrason. Ferroelec. Freq. Contr., vol. 37, pp. 515-523, 1990.

[4] M. Jardino, F. Plumelle, M. Desaintfuscien, and J. L. Duchene,

TN-68
128 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT. VOL. 40. NO. 2. APRIL 1991

"Experimental determination of the energy of stored ions from [11] the side-bands in their microwave spectrum," in Proc. 38th Ann. Symp. Freq. Control, pp. 431-434, 1984,

[5] L. S. Cutler, C. A. Flory, R. P. Giffard, and M. D. McGuire, [12] "Doppler effects due to thermal macromotion of ions in an RF

quadrupole trap," Appl. Phys. B., vol. 39, pp. 251-259, 1986.

[6] J. D. Prestage, G. J. Dick, and L. Maleki, "New ion trap for frequency standard applications," J. Appl. Phys., vol. 66, pp. [13] 1013-1017, 1989.

[7] J. J. Bollinger, S. L. Gilbert, W. M. Itano, and D. J. Wineland,

"Frequency standards utilizing Penning traps," in Proc. 4th [14] Symp. Frequency Standards and Metrology, (Ed. A. DeMarchi)

Ancona, Italy, pp. 319-325, 1989.

[8] J. J. Bolliri^er, W. M. Itano, and D. J. Wineland, "Laser cooled [15] 'Be* Acbiirate Clock," in Proc. 37th Ann. Symp. Freq. Contr.,

pp. 37-41, 1983.

[9] J. J. Bollinger, J. D. Prestage, W. M. Itano, and D. J. Wineland, [16] "Laser-cooled-atomic frequency standard," Phys. Rev. Lett., vol. 54, pp. 1000-1003, 1985.

[10] J. J. Bollinger, D. J. Heinzen, W. M. Itano, S. L. Gilbert, and [17] D. J. Wineland, "Test of the linearity of quantum mechanics by

RF spectroscopy of the 'Be"^ ground state," Phys. Rev. Lett.,

vol. 63, pp. 1031-1034,

'Be"^ ground state,' 1989.

S. L. Gilbert, J. J. Bollinger, and D. J. Wineland, "Shell-structure phase of magnetically confined strongly coupled plasmas,"

Phys. Rev. Lett., vol. 60, pp. 2022-2025, 1988. D. J. Larson, J. C. Bergquist, J. J. Bollinger, W, M. Itano, and

D. J. Wineland, "Sympathetic cooling of trapped ions: A laser- cooled two species nonneutral ion plasma." Phvs. Rev. Lett., vol. 57, pp. 70-73, 1986.

L. R. Brewer, J. D. Prestage, J. J. Bollinger. W. M. Itano, D.

J. Larson, and D. J. Wineland, "Static properties of a non-neu- trarSe* -ion plasma," Phys. Rev. vol. A38, pp. 859-873, 1988. W. M. Itano and D. J. Wineland, "Precision measurement of the ground-state hyperfine constant of ""Mg*," Ph\s. Rev., vol. A24,

pp. 1364-1373, 1981.

J. Vetter, M. Stuke, and E. W. Weber, "Hyperfine density shifts of '"Ba* ions in noble gas buffers," Z. Ph\s.. vol. A273, pp. 129-135, 1975,

L. S. Cutler, R. P. Giffard, and M. D. McGuire, "Mercury-199

trapped ion frequency standard," in Proc. 37ih Ann. S\mp. Freq.

Contr., pp. 32-36, 1983.

E. Herbst, "A reinvestigation of the rate of the C* + Hi radiative association reaction," Astrophys. J., vol. 252. pp. 810-813,

1982, and references therein.

TN-69
UGHT INDUCED KINETIC EFFECTS ON ATOMS. IONS AND MOLECULES

eds. Mol L., Gozzlnl S.. Gabbanlnl C. Aiimondo E., Strumla F.. 1991, ETTS Exlitrlce.Plsa 291

SINGLE ION OPTICAL SPECTROSCOPY*

J.C. Bergqulst, Wayne M. Itano, FYank Elsnert, M.G. Ralzen

and D.J. Wlneland

Time and Frequency Division

National Institute of Standards and Technology

Boulder. Colorado 80303, U.SA.

ABSTRACT

A single ^^^Hg""" ion is confined in a miniature rf trap and is lasercooled to nearly 0.001 K. We have studied the spectrum of the narrow

5d^°6s ^Sjj - 5d^6s^ ^^5/2 electric-quadrupole-allowed transition near 282 nm and obtain a linewidth below 80 Hz (FWHM) . The measured lifetime of the

metastable ^05^2 state gives a natural linewidth limit for this transition

of 1.8 Hz. The narrow linewidth, the ability to detect transitions with

unit probability, and the small perturbations of a single laser-cooled ion

make it an attractive candidate for an optical frequency standard. Our

present resolution is limited by the spectral purity of the frequencydoubled dye laser at 563 nm. Optical heterodyne measurements between two laser beams locked to independent, high-finesse cavities give a beat note

of less than 40 Hz (FWHM at 553 nm) that is dominated by noise in the

frequency range from to 10 Hz. This is caused by the insufficient

isolation of the cavities from mechanical vibrations in this frequency

range. Better isolation methods intended to improve the laser linewidth to about 1 Hz or less are being investigated. A linear Paul trap, in which it

would be possible to trap and cool many ions unperturbed by rf

micromotion, is being tested.

EXPERIMENT

The ion trapping and laser-cooling have been described elsewhere^ ^ '

. A ^^^Hg atom is ionized and trapped in the harmonic pseudopotential well

created by an rf potential applied between the electrodes of a miniature

Paul trap. The separation between the endcap electrodes (2zo) is about 650

^m. The frequency of the rf potential is about 21 MHz. Its amplitude can

be varied up to 1.2 kV resulting in a secular frequency of up to 4 MHz.

The ion is laser cooled to a few millikelvins by a few microwatts of

radiation from two 194 nm sources. One source induces transitions from the

5di°6s 2s^(F=l) to the 5d^°6p 2Pi,(F=0) level. This is essentially a two

TN-70
^92

level system suitable for laser-cooling, except for weak of f -resonance pumping into the ^S. (F=0) state. The second 194 run source, tuned to the

^Sj^(F=0) to ^P^(F=1) transition, returns the ion to the ground state F=l

hyperfine level. The frequency separation between the two radiation

sources is equal to the sum of the ground and excited state hyperfine

splittings (about 47 GHz). The complication of two- laser cooling is

dictated by the hyperfine structure of ^^^Hg"*". An isotope with hyperfine

structure is required in order to have a first-order field- independent

clock transition near B - 0. The two 194 run beams are made collinear and

irradiate the ion at an angle of 55° with respect to the symmetry (z) axis

of the trap. In this way, all motional degrees of freedom are cooled to near the Doppler cooling limit of 1.7 mK. 194 nm fluorescence from the

ion, collected in a solid angle of about 5 x 10"^ x ^n sr, is detected with

an efficiency of 10% to give a peak count rate on resonance of about

25 000 s"^

.

The 282 nm radiation that drives the narrow 5d^°6s ^Sj^ - Sd^Ss^ ^^5/z

transition is obtained by frequency-doubling the radiation from a cw dye

laser that is stabilized to a Fabry-Perot cavity with finesse'^'

F = 60 000. Prior to being frequency doubled, the 563 nm radiation is

passed through an acousto-optic modulator so that its frequency can be

tuned through the S-D resonance. Since the Zerodur'' ' reference cavity

contracts at a nearly constant rate of 3.3 Hz/s^ '

, the acousto-optic

modulator is driven by a frequency obtained by summing the rf output of two

synthesizers. The frequency of one synthesizer sweeps opposite to the

cavity drift; the frequency of the second synthesizer is stepped back and

forth through the S-D resonance. The 282 nm laser and the two- frequency

194 nm source are turned on and off sequentially using shutters and the

acousto-optic modulator. This prevents any broadening of the narrow S-D

transition due to the 194 nm radiation. Optical-optical double

resonance^ • ^

' (electron shelving)'' is used to detect each transition made

to the metastable D state as a function of the frequency of the 282 nm

laser. At the beginning of each cycle, both 194 nm lasers irradiate the

ion. The fluorescence counts in a 10 ms period must exceed a minimum

threshold (typically 20 counts) before the interrogation sequence can continue. The 194 nm beams irradiate the ion for sequential 10 ms periods

until the threshold is met. Then the 194 nm radiation tuned to the

^S;^(F=0) - 2p,|j(F=l) transition is chopped off for 5 ms . During this time

the 194 nm radiation tuned to the 2s,,(F=l) - ^Pj,(F=0) transition optically

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293

pumps the Ion into the ^S^(F=0) ground state. Then this 194 nm source is

turned off. One millisecond later, the 282 nm radiation, tuned to a frequency resonant or nearly resonant with the ^S^(F=0, rap=0) - ^05/2(^=2,

mp=0) transition irradiates the ion for an interrogation period that was varied up to as long as 15 ms . At the end of this period, the 282 nm

radiation was turned off and both 194 nm sources were turned on. Another

10 ms detection period was used to determine whether a transition to the D state had been made (fluorescence counts > threshold, no; fluorescence

counts < threshold, yes). The result was recorded as a 1 or (no or yes)

and averaged with the previous results at this frequency. Then the

frequency of the 282 nm radiation was stepped and the measurement cycle

repeated.

Since the frequency drift of the 282 nm laser depended not only on the

reference cavity contraction rate, but also on small pressure and

temperature changes, on laser power variations, etc.®'^', we locked the

frequency of the laser to the narrow S-D transition to remove any long term

frequency drifts. To do this, we modified the measurement cycle to include

a lock cycle. We began each measurement cycle by stepping the frequency of

the 282 nm radiation to near the half maximum on each side of the resonance

N times (N varied from 8 to 32). At each step, we probed for 5 ms , and

then looked for any transition with the electron-shelving technique. We

averaged the N results from each side of the resonance line, took the

difference and corrected the frequency of the synthesizer drifting against

the cavity. The gain of this lock needed to be properly adjusted to avoid

adding frequency noise to the laser. In this way, variations in the

frequency of the 282 nm laser for time periods exceeding a few seconds were

reduced.

In Fig. 1, we show the spectrum obtained by scanning in this driftfree way through the Doppler-free resonance of the ^S^(F=0, mp=0) - ^05/2(^=2, mp=0) transition. The lineshape shown is the result of 138

consecutive scans. The probe period was 15 ms and the step size was 15 Hz

at 563 nm (30 Hz at 282 nm) . The resonance shows a clearly resolved

triplet with the linewidth of each component less than 40 Hz (< 80 Hz at

282 nm) . We first thought that the triplet structure might be due to 60 Hz

modulation of the frequency of the 563 nm laser either due to grounding

problems, line pickup or inadequate servo gain. However, when the

radiation from two independently stabilized laser beams were heterodyned

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294

together (see next section), the 60 Hz modulation index was far too small

to account for the sideband structure observed on the S-D resonance. In addition, the frequency separation of the peaks is nearer to 50 Hz, not

60 Hz. We now think that, most likely, the triplet structure is caused by

Rabi power broadening. The 282 nm radiation is focussed to a spot size of

about 25 ^m; therefore, on resonance, fewer than 10^ photons/s (< 1 picowatt) will saturate the transition. Below the data is a theoretical

lineshape calculated for an ion at rest, for no broadening due to collisions or laser bandwidth, for a pulse length of 15 ms and for

sufficient power at resonance to give a 3.5 jr-pulse. Qualitatively the

figures compare well. The fluctuations from measurement cycle to measurement cycle in the quantum-occupation-nuinber of the ion in the

harmonic well of the trap cause variations in the transition probability of

the ion. This, and the finite laser linewidth, likely cause the general

broadening and weakening of the signal. We will investigate the lineshape

and the effects of power broadening in more detail in future experiments.

1 00

5d'°6p 'p.,

5d'°6s ^v.

080

-360 -240 -120 120 240 360

•F =

040

-360 -240 -120 120 240 360 Laser freq. offset (Hz)

Figure 1. On the left is a simplified energy-level diagram for ^^^Hg* at zero field. Shown in the upper figure on the right is the power-broadened

lineshape obtained by scanning through the Doppler-free resonance of the

^S;^(F=0, mp=0) - ^T)^

/ 2^^-'^ < nip=0) transition in a single laser cooled

^^^Hg"^ ion. A 563 nm laser that is stabilized to a high finesse reference

cavity, which in turn is long-terra stabilized to the ion, is frequency

doubled and stepped through the resonance for 138 consecutive sweeps. The

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295

step size is 15 Hz at 563 nm (30 Hz at 282 nm) . The lower-right figure

shows the lineshape calculated for conditions similar to the experimental

conditions for the upper figure, except that the ion is assumed to have

zero temperature and the laser is assumed to have zero linewidth.

REFERENCE CAVITY STABILITY

To simplify the study of the performance of a laser locked to a reference cavity against external perturbations, we constructed a second

Zerodur reference cavity. This study was carried out by heterodyning the

light from laser beams locked to the two independent cavities and analyzing

the fluctuations in the power spectrum of the beat frequency. Each Laser

beam was locked to its reference cavity using the reflection sideband

technique discussed in detail by Drever et ai.^' In attempting to narrow

the 563 nm laser to the order of 1 Hz or less, great care must be taken in

the electrical and optical set-up. Many of the techniques and subtle

difficulties are treated in earlier papers by Hough et al.^' and by Salomon

et al.^°' The optical and electrical problems limit the laser linewidth in our work at or below the 1 Hz level; the dominant contribution to the laser

linewidth is the length stability of the reference cavities . ^° " ^^

' Some

aspects of our stabilized laser systems and efforts made to isolate the

cavities from external perturbations will be summarized here . ^ ''

Each cavity is constructed with a Zerodur*' cylindrical spacer that

has a diameter of 10.2 cm with a 1 cm holed bored down the axis. The

mirrors are highly polished Zerodur'' ' substrates that are coated to give

high finesse and good efficiency and then optically contacted to the

polished ends of the spacer. The cavities are suspended by two thin

molybdenum wires inside an aluminum vacuum vessel that has an inner

diameter of 26.7 cm. The wires were slung around the spacers at the nodal

positions for the lowest order bending mode (about L/5 from each end where

L is the cavity length) and attached to the walls of the vacuum vessel by

small clamps. The two wires open into a symmetric "V" at each end; with

opening angles of about 10° at one end and about 15° at the other end.

This allowed free movement of the cavities along the direction of their

axis. The damping of this motion was primarily into the aluminum housing

and from there into the table and padding used for isolation. The aluminum

vacuum vessels were thermally insulated and temperature controlled to the

order of a few mK. The temperature coefficient of our Zerodur'"' spacers

was approximately 6 x 10"^/ C near room temperature. The thermal time

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296

constant from the walls of the evacuated aluminum housing to the spacers

was on the order of 1 day. Low pressure was maintained in each vacuum

vessel by an ion pump that was directly attached to the vessel. The

pressure in one vessel was about 8 x 10"^ Torr (1.1 x 10^^ Pa) but only 2 x 10"^ Torr (2.7 x 10"^ Pa) in the second system. This is marginally

adequate, since 10% pressure fluctuations in the second system would cause frequency fluctuations of a few Hz to the laser stabilized to its cavity

from changes in the index of refraction.

The length of one Zerodur'' ' rod results in a free spectral range of

562 MHz while the other gives 622 MHz. The finesse of the longer cavity is

60 000 and the transmission efficiency on resonance exceeds 23%. The

shorter cavity has a finesse of 90 000 and an efficiency of about 27%.

(Two Fabry-Perot cavities have been constructed from ULE'' ' spacers that are about 10 cm long by about 7.6 cm in diameter. Both cavities have a measured finesse that exceeds 130 000. The frequencies of the mechanical

resonances of these shorter, stiffer bars will be about a factor of 3 higher than the Zerodur bars. In general, if the frequencies of the

mechanical vibrations are high enough, the clock transition can be probed

by the "unperturbed" carrier of the laser spectrum.) The aluminum vacuum

vessels rested on Viton* ' rubber strips attached to V-blocks made of

aluminum. The V-blocks were secured to a rigid plexiglass*' plate. Each

reference cavity system was mounted on separate optical tables that

(initially) consisted of surface plates damped into sand. The sand box sat on soft rubber pads and cinder block legs in one case and on low-pressure

inner tubes and cinder block legs in the second. Noise vibrations on the

floor and on the table tops were monitored with sensitive seismometers.

Some isolation from mechanical vibrations at frequencies above 5 Hz was achieved for both table tops.

For heterodyning purposes a small fraction of the frequency stabilized

light from each cavity was combined on a beam splitter and detected with a fast diode. The heterodyned signal was amplified and analyzed with two

spectrum analyzers used in tandem. This allowed us not only to look

directly at the beat note but also to Fourier analyze the noise terras that

contribute to its linewidth. The first analyzer could be used to observe

the beat signal or as a frequency discriminator. As a discriminator, the scan was stopped and the analyzer was used as a tunable rf receiver with a variable bandwidth. The center frequency was shifted so that the

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297

heterodyne signal lay at the 3 dB point of the response curve. The

bandwidth was adjusted so that the frequency excursions of the beat signal

were much less than the bandwidth. This produced a one-to-one map of

frequency excursions to voltage excursions whose Fourier power spectrum

could be analyzed In the second, low-frequency (0-100 kHz) spectrum

analyzer. This allowed us to study the nature and origin of the

vibrational noise that contributes to the llnewidth of the stabilized

laser

.

The width of the heterodyne signal between the two stabilized lasers

was less than 40 Hz. The noise power spectrum revealed that low frequency

fluctuations in the range from near to 30 Hz dominated this llnewidth.

The vibrational noise spectrum measured by the seismometers on the table

tops matched the largest noise components of the beat note. The

frequencies of the pendulum motion of the suspended cavities were about 1.4

Hz and 1.48 Hz which gave FM at these frequencies. There were also bright

features In both the laser and the seismometer power spectra that came from

floor motion at 14.6 Hz, 18.9 Hz and at 29.2 Hz. These all had enough

power to contribute to the beat note llnewidth. When the pendulum motions

of the bars were quiet, the integral of the nearly featureless noise power

spectrum from ~ Hz to 10 Hz most strongly contaminated the spectral

purity of the lasers. Some, If not all of this noise was mechanical in

origin, but it was not clear how it coupled to the suspended cavity. To

help elucidate the connection, we drove the table tops in either the

horizontal or vertical direction with a small loudspeaker connected to an audio signal generator. The motion of the speaker diaphragm was coupled to the table by a rod glued to the diaphragm and gently loaded against the

table. The table could be driven at frequencies from a few Hz to about 500

Hz with enough power to be 40 dB above any background noise term. When the

loudspeaker drove the table in the horizontal plane in a direction parallel

to the axis of the cavity, the isolation of the suspended cavity was sufficiently good that the beat signal showed no evidence of the

perturbation even at the high drive levels. However when the drive was applied vertically at a level barely perceptible above the vertical

background noise, the heterodyne signal revealed added noise power at the

drive frequency. The stiff support in the vertical direction strongly

coupled vertical motion into effective cavity length changes. The

sensitivity of the Fabry-Perot cavity to vertical motion was orders of

magnitude higher than for horizontal motion parallel to the cavity axis.

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298

For practical reasons, we were unable to drive the table in the horizontal

plane in a direction perpendicular to the cavity axis.

In order to improve the vertical isolation, one table was suspended

just above the floor with latex-rubber tubing attached to the ceiling. The resonance frequencies for both the vertical motion and the horizontal

pendulum motion of the suspended table were near 0.33 Hz. These were damped to the floor with two small dabs of grease. The linewidth of the

heterodyne signal between the laser radiation stabilized to the cavity

supported on this table and the laser radiation stabilized to the cavity

supported on the best sandbox table dropped from 40 Hz to less than 20 Hz. The Fourier noise power spectrum from 0-10 Hz is shown in Fig. 2. The

beatnote linewidth obtained by integrating the power spectrum is about

15 Hz. We suspect that the laser stabilized to the cavity on the sandbox

table is the dominant contributor to the width of the heterodyne signal

since the vibrational noise measured on this table is greater. Current

efforts are devoted to measuring the narrow S-D transition in ^^^Hg'^ using

the laser stabilized to the cavity on the suspended table. A linear rf

Paul trap i*"!^' has been constructed and is also being tested. With this

trap it is possible to store and laser-cool many ions which gives a better

signal-to-noise ratio (thereby better stability) ^ but it is still possible

to have a small second-order Doppler shift.

5 10 FREQUENCY (Hz) Figure 2. Fourier noise-power spectrum of laser heterodyne signal.

We acknowledge the support of the Air Force Office of Scientific Research

and of the Office of Naval Research.

* Contribution of the US Government, not subject to copyright.

TN-77
299

^Present address: UniversitSt Hamburg, D-2000 Hamburg, Federal Republic of

Germany

.

REFERENCES

1. Bergquist, J. C, Itano, W. M. and Wineland, D. J., Phys . Rev. A36

.

428 (1987).

2. Bergquist, J. C, Wineland, D. J., Itano, W. M. , Hemraati, H., Daniel,

H. -U. and Leuchs , G., Phys. Rev. Lett. 51, 1567 (1985).

3. Bergquist, J. C., Diedrich, F. , Itano, W. M. and Wineland, D. J., in

Laser Spectroscopy IX . ed. by M. S. Feld, J. E. Thomas, and A. Mooradian (Academic Press, San Diego, 1989) p. 274.

4. Mention of specific trade names is for purposes of technical

communication alone and does not imply endorsement by N.I.S.T.

5. Bayer-Helms, F., Darnedde , H. and Exner, G., Metrologia 21, 49 (1985).

6. Wineland, D. J., Bergquist, J. C., Itano, W. M. and Drullinger, R. E.,

Opt. Lett. 5, 245 (1980); Wineland, D. J. and Itano, W. M. , Phys.

Lett. 82A, 75 (1981).

7. Dehmelt, H., Bull Am. Phys. Soc . 20 60 (1975); Dehmelt, H., J. Phys.

(Paris) CoUoq. 42, 08-299 (1981).

8. Drever, R. W. P., Hall, J. L. , Kowalski, F. V., Hough, J., Ford,

G. M. , Munley, A. J. and Ward, H., Appl. Phys. BH, 97 (1983).

9. Hough, J., Hils, D., Rayman, M. D., L. -S., Ma, HoUberg, L. and Hall,

J. L. , Appl. Phys. B33, 179 (1984).

10. Salomon, Ch., Hils, D. and Hall, J. L. , J. Opt. Soc. Am. B5, 1576

(1988) .

11. Bergquist, J. C., Diedrich, F. , Itano, W. M. and Wineland, D. J., in

Frequency Standards and Metrology , ed. by A. DeMarchi (SpringerVerlag, Berlin, Heidelberg, 1989) p. 287.

12. Hils, D. and Hall, J. L. , ibid, p. 162.

13. Bergquist, J. C., Eisner, F. , Raizen, M. G., Itano, W. M. and

Wineland, D. J., in preparation.

14. Prestage, J., Dick, G. J. and Maleki , L. , J. Appl. Phys. 66, 1013

(1989) .

15. Dehmelt, H. G., in Frequency Standards and Metrology , ed. by A. DeMarchi (Springer-Verlag, Berlin, Heidelberg, 1989) p. 286.

16. Wineland, D. J., Bergquist, J. C., Bollinger, J. J., Itano, W. M. , Heinzen, D. J., Gilbert, S. L. , Manney, C. H. and Raizen, M. G., to be

published in I.E.E.E. Trans, on Ultrasonics, Ferroelectrics , and

Frequency Control.

17. Walther, H., in these proceedings.

TN-78
JOURNAL OF MODERN OPTICS, 1992, VOL. 39, NO. 2, 233242

Linear trap for high-accuracy spectroscopy

of stored ionsf

M. G. RAIZENt J. M. GILLIGAN,

J. C. BERGQUIST, W. M. ITANO and

D. J. WINELAND

Time and Frequency Division, National Institute of

Standards and Technology, Boulder, Colorado 80303, USA

(Received 3 July 1991 ; revision received 23 September 1991)

Abstract. In a linear r.f. Paul trap, 'crystallized' structures of laser-cooled '^^Hg"^ ions are observed. The ground-state hyperfine transition at 405 GHz is observed in microwave—optical double-resonance spectroscopy. Future prospects are also discussed.

1. Introduction

Trapping and cooling of large numbers of ions, each confined to much less than a wavelength of an atomic transition (the Lamb-Dicke regime), is of great interest for high-accuracy spectroscopy, improved frequency standards, and atom-radiation

field experiments. In a quadrupole r.f. Paul trap, the kinetic energy of a single

trapped ion is on the order of the secular energy, and confinment of a single ion to the

Lamb-Dicke regime for an optical transition has been successfully demonstrated

[1, 2]. For a single '^^Hg^ ion laser-cooled to the Doppler limit [3], the fractional

second-order Doppler shift is "(Avo2/Vo)= — 23 x 10~ '^ [4]. However, two or more

ions in the trap are pushed by their mutual Coulomb repulsion from the centre of the

trap, which leads to relatively large micromotion associated with the non-zero r.f. field and limits laser cooling and the achievable confinement [5-7].

The linear trap design is descended from a race-track-configuration r.f. trap, first used by Drees and Paul for short-term confinement of an electron-ion plasma [8] and

later by Church for longer confinement of atomic ions [9] . Race-track traps consist of

quadrupole r.f. electrodes, similar to those used in mass filters, bent to form a closed

path. Charged particles feel a force toward the axis of the quadrupole and follow

trajectories that carry them around the path formed by the trap electrodes. Dehmelt

first suggested using a string of ions in a linear trap to suppress the second-order

Doppler shift [10]. In race-track traps, the ions are usually free to move along the axis of the trap, but to satisfy the Lamb-Dicke criterion it is necessary to fix the position

of each ion in this direction as well as in the radial direction. The linear trap described

here uses a static electric field to confine the ions in the axial direction. Using a static

field allows us to fix the axial positions of the ions, but sacrifices the race-track trap's

ability to confine both positive and negative charges simultaneously. The static field also weakens the radial confinement, as will be discussed below.

t Work of the U.S. Government. Not Subject to U.S. copyright.

X Present address: Department of Physics, University of Texas at Austin, Austin, Texas

78712, U.S.A.

0950-0340/92 $300 © 1992 Taylor & Francis Ltd.

TN-79
234 M. G. Raizen et al.

Prestage et al. [11] have trapped a cloud of ^^^Hg"*" ions elongated along the axis of a linear trap and have demonstrated a ^^^Hg"*" frequency standard with excellent frequency stability. Crystallized strings of laser-cooled ^*Mg^ ions have been

observed in a race-track trap at the Max Planck Institute for Quantum Optics [12]. In our group at the National Institute of Standards and Technology (NIST), we

have constructed a linear r.f. trap and have observed simple 'crystallized' linear structuresof up to tensof ^^^Hg"*" ions [13]. By varying the axial well depth, we have

also observed more complex structures. In the next section we describe the principle of the linear trap and present the

results of numerical simulations of ion configurations. We then describe our trap and

present images of trapped 'crystallized' structures of ions. We next present

preliminary observations of the 405 GHz ground-state hyperfine transition of

'^^Hg"*" using microwave-optical double-resonance spectroscopy with a trapped

string of ions and discuss the potential for a microwave frequency standard. We

conclude with a discussion of future experiments and prospects.

2. Linear trap, theory

Consider the trap shown in figure 1 . The r.f. electric fields are transverse to the

trap axis for the entire axial extent of the centre segment of the trap. We assume that near the trap axis the time varying potential is

<P =

^^2

cosQt, (1)

where R is approximately equal to the distance from the axis to the surface of the trap

electrodes. For sufficiently high drive frequency Q this results in a harmonic

pseudopotential well in the radial direction of the form

where (Oj^qVQJiyl2mQR^) is the oscillation frequency in the radial direction. The

confinement in the radial direction is identical to that of a quadrupole mass analyser

[14].

To provide confinement along the axis of the trap, a static voltage Uq is applied to the end sections. Determining the potential in the trap due to this static potential on

Z r^

^ ( r

Vo cos lit r^=Tr ( r'=w

Figure 1. Linear trap configuration. An alternating r.f. voltage VqCosQi is applied to diagonally opposite electrodes as shown. The segments of each rod are capacitively

coupled so that each one can be biased at a different static potential, yet remain at the same r.f. potential. We assume that the end segments of the electrodes are long enough

so that in the centre section of the trap the r.f. electric fields are parallel to the xy plane.

To trap ions along z we assume the centre four electrodes are at static ground potential

and the two sets of four electrodes on either end are at a static potential f/g (^o'*^ ^^ trap positive ions).

TN-80
Linear trap for spectroscopy 235

the electrodes would require a full three-dimensional calculation, which is beyond

the scope of the present work. However, near the centre of the trap, the static potential can be approximated by

2q

(3)

where co^^ilkqUolmy^ is the axial oscillation frequency in the trap and ^ is a geometric factor. The potential well in the radial direction is weakened by the

addition of the static potential, and is given by

m m

(/), = - i(oj-w'j2){x^+y')^- ico'^ix^+y'),

2q Iq

(4)

where coj. is the radial oscillation frequency in the presence of the static potential and

pseudopotential

.

The kinetic energy in the micromotion of a string of ions trapped along the z axis can be about equal to the kinetic energy in the secular motion. At the Doppler cooling

limit, the fractional second order Doppler shift can then be as low as —2-3 x 10" ^®

for all the Hg^ ions in the string [4]. We have performed numerical simulations to find ion configurations in a linear trap as a function of the number of ions and the

radial and axial secular frequencies [15]. This calculation found the equilibrium

positions of the ions by minimizing the potential energy, assuming harmonic

potentials in the radial and axial directions and Coulomb repulsion between the ions. The simplest configuration of ions is a linear string, which is obtained when the

radial well is much deeper than the axial well. A computer simulation of eight ions in a string is shown in figure 2. The ions can be pushed into planar or three-dimensional

configurations by increasing the axial well depth relative to that of the radial well. In

figure 3 a planar zig-zag structure of 11 ions is shown. Such configurations are

closely related to structures predicted for cold ions in storage rings [16].

3. Linear trap, experiment

The linear trap constructed in our group at NIST follows a geometry similar to the outline of figure 1 . This design ensures that the r.f. fields are transverse to the trap

axis. The trap dimensions are: rod diameters 16 mm, distance of the rod centres from the praxis of the trap ISSmm, and the axial centre trapping section 25 mm. The

Figure 2. Numerical simulation of a crystallized string of eight "'Hg* ions for secular frequencies a)r/2Jt = 435kHz, (0^/27: = 41 -7 kHz. These secular frequencies were obtained by scaling secular frequencies measured under different conditions to the

potentials applied to the trap when figure 5 was acquired.

TN-81
236 M. G. Raizen et al.

• • • •

• • • • •

50 ^m

Figure 3. Numerical simulation of a crystallized zig-zag structure of 1 1 '^^Hg"*" ions. An

asymmetry in the two radial directions {x,y) was assumed, pinning the structure in the weaker of the two wells, and the projection is at 45° from the plane of the structure. The

secular frequencies are oi'J2n = n\ii\z, (y^/27t = 92-4kHz, and (0^/271 = 31 kHz. These

secular frequencies were obtained by scaling measured frequencies to the potentials

applied when figure 6 was acquired and increasing (dy by 20% ad hoc to make the results agree with figure 6. F = \ F =

laser 1

F= 1

F =

'P^ /2

laser 2

'S, /2

Figure 4. Energy level structure for the ^Si^j and ^Pi/2 levels of ^^^Hg"^ near zero magnetic field. The ^Si/2{F=0) — {F=l) hyperfine splitting is 405 GHz, and the ^Pi/2(^=0)~(^= 1) hyperfine splitting is 6-9GHz. The intervals are not drawn to

scale. Lasers 1 and 2 are used for cooling, optical pumping, and detection as described in the text.

radial size of the trap was chosen to permit Hg"*^ ions to be confined near the LambDicke regime for an optical transition by a 12-7 MHz r.f. drive with a voltage

amplitude Vq of approximately 700 V. In the work so far, amplitudes Vq of up to 350 V were used. Axial confinement is achieved with a static voltage of typically 1 V

or less. '^^Hg^ ions are loaded into the trap by leaking neutral ^^^Hg (isotopic purity

91%) into the vacuum chamber to a pressure of approximately 10"^ Pa and then

ionizing some of these atoms in the trap with electrons from a field emission point.

After the trap is loaded, the chamber is evacuated to a pressure of approximately

10~^Pa. The ions are laser-cooled by a few microwatts of c.w. laser radiation at 194nm [2].

TN-82
Linear trap for spectroscopy Til

Due to the hyperfine structure of ' ^^Hg ^ , optical pumping must be considered in the laser cooling scheme. In figure 4 we show the hyperfine levels for the ^Sj/2 ^'^d ^Pi/2 states of ^^^Hg"*^ in zero magnetic field. Two lasers at 194 nm are required. The

ions are both cooled and detected using a laser (laser 1 in figure 4) that is tuned

slightly below the resonant frequency of the optical transition ^Si/2

(F= l)->^Pi/2(^=0). This is a 'cycling transition' since the ions that are excited to the ^Pi/2(^— 0) level can decay only to the ^81/2(^=1) level according to dipole

selection rules. The ions are detected by collecting photons scattered in this transition. Laser 1 can also weakly excite the ^Si/2(^— l)^^Pi/2(^= 1) optical

transition (it is detuned 69 GHz from this resonance, whose natural linewidth is y/271 = 70 MHz). From the '^V^i-^{F=\) level the ions can then decay to the '^^^^^{F={S)

level. By itself, laser 1 would optically pump the ions into the ^Si/2(^'=0) level,

which would spoil the laser cooling. To prevent this, we have a second laser (laser 2 in figure 4), that is offset 47-4 GHz to the blue of laser 1 in order to repump the ions from

the ^Si/2(^==0) level back to the ^Si/2(^==1) level. In addition to the above

considerations, we must prevent optical pumping into the ^^-^i^iF—X, »Zf=±l)

levels. We do this by applying a magnetic field of about 5xlO~*T (5 G) at approximately 45° to the electric field vector of the 194nm radiation.

To cool both the radial and axial motion of the ions, we direct the laser beam

through the trap at an angle of approximately 9° to the trap axis. Fluorescence

emitted perpendicular to the axis of the trap is collected by a fast lens system, which

images it with a magnification of 22 onto an imaging photon detector. The secular

frequencies of the trap are measured by applying a drive to the trap electrodes and

observing a drop in the fluorescence (due to strong excitation of the secular motion of

the ions) as the drive sweeps through resonance [17]. These measurements permit a quantitative comparison between the observed ion configurations and the numerical

simulations described earlier. The secular frequencies were measured for several

different r.f. and static voltages and had the correct functional dependence on the

voltages. The unperturbed radial secular frequency (M^ was proportional to the r.f. voltage and the axial secular frequency cd^ was proportional to the square root of the

static voltage. For very small axial voltage (less than lOOmV) the effects of local contact potentials can come into play, making it difficult to characterize the axial well

depth. By changing the loading conditions (that is, by changing the neutral Hg

background pressure and the duration and intensity of the field emission), we can

trap different numbers of ^^^Hg^ ions and have resolved as many as 33 ions in a string and as few as a single ion. To crystallize and resolve significantly longer

strings, higher laser power and improved imaging optics will probably be required.

Another limitation may be our ability to cancel the effects of local patch potentials over the entire axial extent of the string. In the present trap, static offsets can be

independently applied to three of the four trapping rods to compensate for such

potentials, but as the string of ions approaches the radial size of the trap (750 (im),

these potentials may vary over the length of the string. In this case, we expect the

static compensation to be less effective. In addition, we can apply a different static voltage to each end to translate the entire structure along the axis. Two representative pictures of crystallized structures are shown in figures 5 and

6. In figure 5, eight ions in a string are clearly resolved. Figure 6 shows a structure of

11 ions (10 ^^^Hg"^ ions and one impurity ion). This structure was obtained by

lowering the r.f. voltage on the trap for a fixed static voltage. Both ion configurations

are in close agreement with the numerical simulations shown in figure 2 and in figure

TN-83
238 M. G. Raizen et al.

Figure 5. Image of crystallized string of eight '^^Hg"^ ions. The potentials applied to the

trap when this image was acquired were used as input for the numerical simulation

shown in figure 2.

Figure 6. Image of a crystallized structure of ten ^^^Hg"*" ions and one impurity ion. The

potentials applied to the trap when this image was acquired were used as input for the numerical simulation shown in figure 3.

3, which used secular frequencies that corresponded to the trap conditions for figures

5 and 6, respectively. These values were obtained by scaling secular frequencies

measured under diflferent trap conditions to the potentials applied to the trap when

figures 5 and 6 were acquired. In order for the numerical simulation shown in figure

3 to agree with the ion configuration of figure 6 it was necessary to assume an

azimuthal asymmetry in the radial potential sufficient to force the ions into a plane.

The source of this asymmetry is not known, but it may be due to diflferences in contact potentials or static electric charges on the electrodes. For a large number of

ions and a weak axial well, more complicated two and three-dimensional structures

are seen. Such structures are closely related to the structures predicted for cold ions

in storage rings [16]. A systematic and quantitative study of these two and threedimensional crystallized structures will require eliminating the azimuthal asymmetry or characterizing it more completely.

4. Microwave—optical double-resonance

A string of cold ions is of great interest for high-resolution spectroscopy and

improved frequency standards. With imaging techniques, each ion can be treated as an independent atomic clock, which is detected with 100% efficiency (using

Dehmelt's 'electron shelving' technique [2, 18, 19]). We assume that the Ramsey

separated field method is used in the time domain to interrogate the clock transition

at frequency cDq (in rad s~ '). In this method, excitation of the clock transition is by

TN-84
Linear trap for spectroscopy 239

two phase-coherent pulses of radiation, each of duration ATr, separated by a time

Tr. We assume that ATr« Tr. By probing the clock transition on each side of the

Ramsey peak, we can obtain an error signal to drive the average frequency to cOq. The

frequency stability for the locked oscillator as characterized by the two-sample Allan

variance [20] is a,(T)-(TiVTX)-i/2, {x>T^), (5)

where t is the averaging time and N is the number of atoms. This expression shows

that there is an advantage in using large N, (Oq, Tr, and T. The trap environment can

enable long interrogation times Tr. To achieve high accuracy, we must minimize and

account for external perturbations due to electric, magnetic, and gravitational fields. These include ion-trap and ion-ion interactions, collisions with neutral background

atoms, external magnetic and electric fields, and gravitational red shifts. For a microwave transition on a cold string of ^^^Hg"*" ions, the uncertainty could

eventually be as small as the uncertainty of the second-order Doppler shift. At the

Doppler limit of laser cooling, the fractional second order Doppler shift of '^^Hg"*" is — 2-3 X 10"^^ [4]. To realize this small Doppler shift it will be necessary that the

equilibrium radial position of the ions coincide exactly with the line along which the

r.f. trapping fields vanish. Otherwise, the micromotion will cause the second-order

Doppler shifts to be much larger. The contributions of other effects, such as Stark

shifts due to neighbouring ions, are discussed in [21,22].

This potential for very high accuracy has led us to investigate the possibility of a microwave frequency standard based on the 405 GHz ground-state hyperfine

splitting of ^^'Hg"*" with a trapped and laser-cooled string of ions. This transition is independent of the magnetic field to first order at zero field. For a Ramsey

interrogation time of Tr= 100 s and cOo/27t = 40-5 GHz, the frequency stability of a clock 'ensemble' of N=50 ions would be (7y{T) = 5-5 x

10"''* t"^'^. As a first step toward this goal we have recently detected the 405 GHz groundstate hyperfine transition of a string of '^^Hg^ ions by microwave-optical doubleresonance. For this preliminary measurement the total fluorescence of the whole

string of ions was measured. This meant that the measurement was sensitive to noise

arising from intensity and frequency fluctuations in the cooling/detecting laser, so the stability figure given by equation (5) is not applicable.

To describe the measurement sequence we rely on the discussion of laser cooling

and optical pumping given in the previous section and on figure 4. As described

above, when the lasers are on, it is necessary to apply a magnetic field to prevent

optical pumping into the (F= 1, mp= + 1) levels of the ground-state. However, to avoid second-order Zeeman shifts of the (F=l) — (F=0) ground-state hyperfine

interval, we wanted to have zero magnetic field during the measurement. To this end, we used two sets of Helmholtz coils. One set cancelled the ambient field and the

secondset applied a field of about 5 x 10"'* T (5 G) to prevent optical pumping. With

the second set of coils switched off, the residual magnetic field near the trap was measured to be approximately l-6xl0"^T (016 G). The sequence of the measure- ment is as follows:

(a) Lasers 1 and 2 and the magnetic field are all initially on to laser-cool the ions.

(b) Laser 2 is turned off and the ions are optically pumped by laser 1 into the

F=0 ground-state.

TN-85
240 M. G. Raizen et al.

(f) Both lasers are turned off, the second set of Helmholtz coils are switched off, bringing the magnetic field near zero. The ^Si/2(^=^0)^^Si/2(^= 1) hyperfine transition near 405 GHz is driven using the Ramsey method with two

microwave pulses, each of duration A T^ and separated by a time Tr. AT^and

the microwave power are adjusted to be close to values that give a Jt/2 pulse at the resonant frequency.

{d) The magnetic field is switched back to the value at step (a), laser 1 is turned

on, and we measure flurorescence from the ions. The fluorescence is proportional to the number of ions (driven by the microwaves) that have

made the transition to the ^Si/2(^= 1) state.

At the end of this sequence, the frequency of the microwave source is stepped,

and the sequence is repeated. The results of many such scans are averaged together.

The results of such a measurement, made with eight ions and T^= IBs, is shown in figure 7. The line centre and linewidth were determined from a least-squares fit of a cosine function to the central lobe of the observed Ramsey spectrum. For the signalto-noise ratio of this spectrum, the difference between a cosine function and an exact Ramsey profile is insignificant. The measured linewidth is 25 1 + 6 mHz, which gives

(0C

o

o

c

o

-M

o

.c

Cl.

Freq. - 40 507 348 013.3 Hz

Figure 7. Microwave-optical double-resonance of the ^S^|^{F=Q)-^'?>^|2{F= 1) hyperfine

splitting of '^^Hg^ at 405 GHz. Shown is the central peak of a Ramsey resonance with

the measurement sequence as described in the text, taken with a linear string of eight

ions. The circles represent data; the solid line is a cosine function fit to the data using the

least-squares criterion. For this measurement Tf^= 1-8 s, AT'r = 130ms, and each point

is the average of 27 measurements. Including time for cooling, pumping and detection, each measurement cycle takes about 3 s.

TN-86
Linear trap for spectroscopy 241

a fractional resolution of 6-2x10"^^. The theoretical linewidth for the pulse

duration and free precession times used in this measurement is 254 mHz, in good

agreement with the data. For future experiments, in which the individual ions will be monitored

separately, the fluorescence detection efficiency must be high enough that when laser

1 is turned on, enough photons will be detected that we can be sure whether each ion

has or has not made the microwave transition. The number of photons scattered by

an ion before it is pumped into the ^Si/2(^=0) level is approximately equal to the

square of the ratio of 69 GHz (the detuning of laser 1 from the ^Siy2(i^= l)~^^Pi/2

(F= 1) transition) to 35 MHz (half the radiative linewidth). This gives about 40 000

photons if laser 1 is resonant with the ^Si|2(^= l)~*^Pi/2(^— 0) transition. Therefore, the detection efficiency must be significantly better than the inverse of

this number which is 3 x 10~'. If this condition is met, it should be possible to detect

the microwave transitions in each ion with nearly 100% efficiency. We have

measured the detection efficiency of the present apparatus to be near 10~*, which is marginally acceptable.

5. Future prospects

To reach the very high accuracy which should be possible with such a clock, it will be advantageous to use longer interrogation times. To reduce heating caused by

collisions with background neutral atoms during the interrogation time, cryogenic

pumping or sympathetic cooling [23] will most likely be required. Although in our

preliminary measurements the total fluorescence from the entire string was

measured, it appears possible to measure the fluorescence from each ion individually. It is then a data handling problem to route the counts from each ion to the correct 'bin' in the servo electronics.

Beyond the applications to high-accuracy spectroscopy, confining a string of ions

so that each is in the Lamb-Dicke regime should make it possible to perform

interesting experiments in fundamental physics such as studies of interference,

cavity QED, collective behavior, and the fundamental quantum noise in the

measurement of a transition in a single atom.

Acknow^ledgments

We gratefully acknowledge the support of the U.S. Office of Naval Research and

and the Air Force Office of Scientific Research. We thank F. Moore and H. Robinson

for helpful suggestions on the manuscript.

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TN-87
242 Linear trap for spectroscopy

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Germany, edited by R. W. Hasse, I. Hofmann and D. Liese (GSI report GSI-89-10,

ISSN 0170-4546), p. 33; Habs, D., Ibid., p. 43.

[17] Wineland, D. J., Bollinger, J. J., and Itano, W. M., 1983, Phys. Rev. Lett., 50, 628.

[18] Nagourney, W., Sandberg, J., and Dehmelt, H., 1986, Phys. Rev. Lett., 56, 2797.

[19] Sauter, Th., Neuhauser, W., Blatt, R., and Toschek, P. E., 1986, Phys. Rev. Lett. ,57, 1696; Sauter, Th., Blatt, R., Neuhauser, W., and Toschek, P. E., 1986, Optics

Commun., 60, 287.

[20] Barnes, J. A., Chi, A. R., Cutler, L. S., Healey, D. J., Leeson, D. B., McGunigal, T.

E., Mullen, J. A., Jr., Smith, W. L., Snydor, R. L., Vessot, R. F. C, and Winkler,

G. M. R., 1971, I.E.E.E. Trans. Instrum. Meas., 20, 105.

[21] Itano, W. M., Lewis, L. L., and Wineland, D. J., 1982, Phys. Rev. A, 25, 1233.

[22] Wineland, D. J., Itano, W. M., Bergquist, J. C, Bollinger, J. J., and Prestage, J. D., 1985, Ann. Phys. (Paris) 10, 737.

[23] Larson, D. J., Bergquist, J. C., Bollinger, J. J., Itano, W. M., and Wineland, D. J., 1986, Phys. Rev. Lett., 57, 70.

TN-88
PHYSICAL REVIEW

LETTERS

Volume 63 4 SEPTEMBER 1989 Number 10

Test of the Linearity of Qaantmn Mechanics by rf Spectroscopy of the 'Be '*' Ground State

J. J. Bollinger, D. J. Heinzen, Wayne M. Itano, S. L. Gilbert, and D. J. Wineland

National Institute ofStandards and Technology, Boulder, Colorado 80303

(Received IS May 1989)

A hyperfine transition in the ground state of 'Be'*' was used to test a nonlinear generalization of quan- tum mechanics recently formulated by Weinberg. We searched for a dependence of the frequency of a coherent superposition of two hyperfine states on the populations of the states. We are able to set a limit of 4 X 10 ~^^ on the fraction of binding energy per nucleon of the 'Be'*' nucleus that could be due to nonlinear corrections to quantum mechanics.

PACS numbers: 03.6S.Bz, 06.30.Ft, 32.30.Bv

Since the 1920s, quantum mechanics has passed numerous tests as illustrated by the agreement between

the predictions of specific quantum-mechanical theories

and experimental measurements. For example, the measured energy levels of the hydrogen atom are in excellent agreement with the predictions of quantum-mechanical

theory. However, this could be regarded as a test of the

accuracy of the Hamiltonian rather than a test of quantum mechanics itself. It should be possible to test the

basic framework of quantum mechanics independently of

and more precisely than any particular quantum-mechanical theory. Recently, Weinberg''^ has formulated a

general framework for introducing nonlinear corrections

to quantum mechanics which enables such a test. He

has suggested that a sensitive way to search for possible

nonlinearities is to look for a change in a transition fre- quency as the wave function of a system changes be- tween initial and final states. Monochromatic radiation

used to drive the transition would therefore not stay in resonance throughout the entire transition. If the transition can be driven experimentally in time T, then the

maximum frequency shift due to the nonlinearity must

be on the order of l/T. Because nuclear magnetic reso- nance transitions in ^Be"** have been observed with T as long as 1 s, Weinberg sets a limit of ~10~'* eV on the

magnitude of any such nonlinear corrections to the ener- gy of the 'Be nucleus. In this Letter we report an experiment which improves this limit by 5 orders of magnitude. In the formalism developed in Refs. 1 and 2, the equation which describes the time evolution of the wave function yfU) is nonlinear and derivable from a Hamiltonian

function h(yf,\tf*)- For a discrete system, it takes the

form

^^dn _ a/>(y,y*)

(1)

where \ifk is the amplitude of state k. In general, h is not a bilinear function of v^ and y^* as in ordinary quantum

mechanics, but the property of homogeneity [/i(X^,yr*)

A(v^,A,y*)—XA(v^,V'*) for any complex X] is retained. Homogeneity guarantees that if iff (t) is & solution of Eq.

(1) then XyfU) is also a solution representing the same

physical state. Homogeneity ensures the proper treat- ment of physically separated systems and distinguishes

this formalism from previous nonlinear generalizations^-^

and tests ^'^ of quantum mechanics.

Consider a two-level system which in the absence of

nonlinear corrections has eigenvalues Ek, k^\,2. Be- cause any nonlinear corrections to quantum mechanics

are expected to be small, it is reasonable to write the

Hamiltonian function as the sum of the bilinear term

hoiyff¥*^''^k~i.2Ek¥k¥k of ordinary quantum mcWork of the U. S. Government

Not subject to U. S. copyright 1031

TN-89
Volume 63, Number 10 PHYSICAL REVIEW LETTERS 4 September 1989

chanics and a term h„\(\ff,if/*) that is not bilinear and

contains the small nonlinear corrections. A form of Ani appropriate for the work discussed here is hpi^nhia),

where « - I V'l I ^+ I v^2 M, fl = | v^2 1 V/i, and A is a real function.''^ The nonlinear time-dependent Schrodinger

equation then takes the form

ff/\ = hQ)\(a)y/] ,

ih

d\V2

dt

E2 + h-\-{\-a) dh. da

y/2 = h (02(a) y/2.

which has solutions

/.\ — i<uj(a)»

, A:-l,2, (2)

where a and the c^'s can be parametrized by c\

"sin(0/2) and C2~a''^^~cos(0/2). The relative phase

of the two components of the wave function (specifically, the time dependence of the coherence v'lV* ) evolves with a frequency

oip = (o\ia) — (02(a) ~(oo~ (dh/da)/h ,

where q)o''(E\ —E2)/h is the atomic transition frequen- cy in the absence of nonlinearities. A two-level system is mathematically equivalent to a spin-j system in an

external, uniform magnetic field, where 6 is the angle by

which the spin is tipped with respect to the magnetic

field and cop is the precession frequency of the spin about

the magnetic field. ^ In the language of the equivalent

spin-y system, the effect of the nonlinear correction

dh/da is to create a dependence of the precession fre- quency (Op on the tipping angle 6 between the spin and

the magnetic field. We searched for a 6 dependence of the precession

frequency of the (mi,mj )~(— j,+ t)—(— i,+ y)

hyperfine transition at --303 MHz in the ground state of

'Be"^ (see Fig. 1). At a magnetic field B of 0.8194 T,

this transition, referred to as the clock transition, depends only quadratically on magnetic field fluctuations. With v^i = V'( - 2 . + } ) and v':- ( - 2 , + 2 ) the simclock

transition

(mi, mj)

'i::S- (-1/2, 1 F=1

F=2

1/2) 1/2)

(1/2, 1/2)

(3/2, 1/2)

FIG. I. Hyperfine energy levels (not drawn to scale) of the 'Be* Is ^S\n ground state as a function of magnetic field. At

fl— 0.8194 T the 303-MHz clock transition is independent of magnetic field to first order.

1032

plest nonbilinear addition to the Hamiltonian function of

the free 'Be"*" nucleus for the two states is''^ /i(a)-2ea2 (3)

where f is a measure of the strength of the nonlinear

correction. This gives rise to a dependence of (Op on 9 of

ojp^(Oo-4(€/h)cosHe/2). (4)

This discussion assumes that the 'Be''" nuclear spin is decoupled from the valence electron spin and therefore

the (— T.y) and (— f,y) states are pure (m/,my)

states. At a magnetic field of 0.8194 T these states have

a 0.02 to 0.03 amplitude admixture of my " — y states. This creates small corrections to Eqs. (3) and (4) which we neglect.

Between 5000 and 10000 'Be''" ions and 50000 to 150000 ^^Mg"*" ions were simultaneously stored in a cylindrical Penning trap^ with 5=0.8194 T under conditions of high vacuum (;$10"^ Pa). To minimize

second-order Doppler shifts of the clock transition, the

'Be''" ions were cooled to less than 250 mK. The ^^Mg"'"

ions were directly laser cooled and compressed by a narrow-band (~1 MHz) radiation source at 280 nm.'

The 'Be"*" ions were then sympathetically cooled"' by

their Coulomb interaction with the cold Mg"*" ions. A

narrow-band 313-nm radiation source was used to optically pump and detect the 'Be''" ions."'^ With the

313-nm source tuned to the 2^ ^S\/2(mi " y ,mj " y ) to 2p^P3/2(2>2) transition, 94% of the 'Be"*" ions were

optically pumped into the 2j^5i/2(y, 2" ) ground

state. "'^ The 313-nm source was then turned off to avoid optical pumping and ac Stark shifts. The sympathetic cooling of the 'Be ''" ions by the Mg "*" ions pro- vided a steady cooling source independent of the 3 1 3-nm

radiation and therefore permitted the use of long transi- tion times.

The clock transition was detected by the following

method. After the 313-nm source was turned off, the

ions in the ( y , y ) state were transferred to the (

2"

, y ) state and then to the ( — y , y ) state by two successive rf

Tt pulses. Each pulse was 0.2 s long and resonant with

the appropriate transition frequency (around 321 and

311 MHz, respectively). The clock transition was then

driven by Ramsey's method of separated oscillatory

fields ' ^ with rf pulses of about 1 -s duration and a free- precession time on the order of 100 s. This transferred some of the ions from the ( ~ y , 2^ ) state to the

( — y , y ) state. Those ions remaining in the ( — y , y ) state were then transferred back to the ( y , y ) state by

reversing the order of the two rf Jt pulses. The 313-nm

source was then turned back on, and the population of

ions in the ( ~ y , 2" ) state was registered as a decrease i in the 'Be"*" fluorescence, relative to the steady-state ' fluorescence, during the first second that the 313-nm

source was on. [The optical repumping time of the ions ' from the ( — y , y ) state to the ( i , I ) state was an or-

'

TN-90
Volume 63, Number 10 PHYSICAL REVIEW LETTERS 4 September 1989

der of magnitude longer than this.] For the test of nonlinearities, Ramsey's method with

unequal rf pulses was used to drive the clock transition and measure (Op for different values of 9. First an rf 6

pulse of duration re was applied. This prepared the ions

into a coherent superposition of the (—2.1) and

( ~ } . 2" ) states given by Eq. (2) for a particular value

of 9. The value of 9 was determined from 0""(Te,r,);r,

where t, was the length of time to drive a k pulse at the same rf power. After the rf 9 pulse, the ions freely pre- cessed for a time T. This was followed by an rf n/2 pulse

coherent with the first pulse which completed the Ramsey excitation. In the limit that r» Te,T,/2, the Ramsey

line shape [specifically, the number of ions remaining in the ( — y , T ) state as a function of the rf frequency (o in the Ramsey excitation] is proportional to

1 —s\n9coi{[oi — (Op{9)]T}

,

where (t)p{9) is given by Eq. (4). The center frequency

of the Ramsey line shape is the precession frequency

o)p{9). Figure 2 shows a Ramsey signal obtained with

r- 150 s and 0-;r/2.

The Ramsey signal was used to steer the frequency of a synthesized rf source.

' ' Ramsey-signal measurements

were taken near both of the fuU-width-at-half-maximum

frequencies Q)-\.=2nv+ and (o-=lnv-, where v+ and

V- are indicated in Fig. 2. The difference in the measured signal strengths on either side of the line center was used to electronically steer the average frequency of

the synthesizer to Q}p(,9). Eight pairs of measurements

were taken with an angle 9a " 1 .02 rad followed by eight

pairs of measurements with an angle 0b "2.12 rad. This

pattern was repeated for the length of an entire run as indicated in Fig. 3. The average frequency of the syn15000

(0c

s§

I.o

3

-0.005 0.000 0.005

frequency - 303 016 377.265

(Hz) FIG. 2. Ramsey signal of the clock transition with T— 150 s and 0—;r/2. The data are the result of one sweep (that is, one measurement per point). The sweep width is 9 mHz and the frequency interval between points is 0.750 mHz. The dots are experimental and the curve is a least-squares fit. The signal- to-noise ratio is limited by the frequency stability of the refer- ence oscillator. The fuU-width-at-half-maximum frequencies

are indicated by v+ and v-.

thesizer for 9^ 9a was then subtracted from the average

frequency of the synthesizer for 9^ 9b. Run lengths

varied between 4 and 10 h. The uncertainty was due to the frequency instability of the reference oscillator used

with the synthesizer. Most runs were taken with a commercial cesium beam clock [fractional frequency stabili- ty'^ <t^(t)~6x10~'^t~'''^ for measurement time t in seconds] as the reference oscillator. For a limited time we had access to a passive hydrogen maser'^ [ct^(t)

~(2-3)xlO~'^T~'^^] and a few runs were taken with

the passive hydrogen maser as the reference oscillator. Runs were taken with free-precession periods of T

"30, 60, and 100 s and rf pulse lengths of xa "0.65t,/2,

Tfl~1.35T,/2 with T,/2'"0.5, 1, and 2 s. A weighted

average of the synthesizer frequency differences for 9—9a and 9 — 9b from 25 runs is 2.7(6.0) /iHz. The uncertainty (in parentheses) is the external error calculated

from the scatter of the 25 measurements from the weighted average and is in good agreement with the

internal error of 5.7 /iHz calculated from the uncertainties of each of the 25 runs. The time constant of the ser- vo would have decreased the apparent size of a real fre- quency difference by 28%. This results in a possible

dependence of the precession frequency on 9 of

[ft)p(0fi)-a)p(0/i)]/2;r-3.8(8.3) //Hz and from Eq. (4) a value for the parameter e of

6/2ffft -1.8(4.0) /xHz. (5)

The error is a 1 standard deviation uncertainty. A few

runs were also taken with 9^n/2. The frequency (Op{n/

l)l2n was compared with the frequencies (Op{9A)l2n

and aip{9B)l2n for runs taken within a few days of each

other. The standard deviation of the frequencies from

their average was 6 /iHz, consistent with the 7-/iHz uncertainty of the frequencies. The 9^nl2 runs do not improve the limit of Eq. (5) on a possible correction to (Op linear in a, but in general can be used to help place limits

X3

csl

Q. 3

200

100

I -100

Q. 3 -200

IxlC^ 2x10'^ 3x10^ 4x10^^ time (s)

FIG. 3. 'Be"*" precession frequency £Up(r), referred to a passive hydrogen maser, as a function of time for a single run with r— 100 s. The periods A (B) during which the initial Rabi

pulse created a mixed state with angle 9a — 1.02 rad (0a— 2.12 rad) are indicated.

TN-91 1033
Volume 63, Number 10 PHYSICAL REVIEW LETTERS 4 September 1989

on a more complicated form for hia).

Equation (5) sets an upper limit of |f | < 2.4x10"^°

eV (5.8 /iHz) for a nonlinear contribution to the 'Be^

nuclear Hamiltonian. This is less than 4 parts in 10^^ of

the binding energy per nucleon of the 'Be ''' nucleus and

improves the limit set in Ref. 1 by roughly 5 orders of

magnitude. The limit on 1^1 is also 5 orders of magnitude smaller than experimental limits placed by neutron

interferometry'"^ on \b\, where b is the coefficient of a logarithmic addition —b\if(x)ln\\i/ix)\^ to the oneparticle Schrodinger equation. However, this nonlinearity does not satisfy the property of homogeneity and

therefore these experiments^'^ test for a nonlinearity

which does not satisfy the requirements of the framework developed by Weinberg. Our experimental result is limited by statistics due to the frequency instability of

the reference oscillator. The largest known systematic

error of our measurement of (OpiO) is the second-order

Doppler (time dilation) frequency shift due to the temperature and ExB rotation of the ions in the trap." Its size is less than 3 /iHz (1 x 10 ~ '*). We believe it can be

held constant to significantly better than 10% over the

time required to make a frequency difference measurement. With a better reference oscillator or a second

'Be**" clock, it should therefore be possible to improve

our limit on | ^ | by more than an order of magnitude.

Improvements on this measurement may also be possible

using nuclear magnetic resonance techniques on neutral

atoms.''*"

We gratefully acknowledge the support of the U.S.

Air Force Office of Scientific Research and the U.S.

Office of Naval Research. We thank S. Weinberg for a number of helpful discussions clarifying the theory, and

J. C. Bergquist and S. R. Lundeen for their comments on

the manuscript.

'S. Weinberg, Phys. Rev. Utt. 62, 485 (1989).

2S. Weinberg, Ann. Phys. (N.Y.) 194, 336 (1989).

'I. Bialynicki-Birula and J. Mycielski, Ann. Phys. (N.Y.)

100,62(1976).

'*A. Shimony, Phys. Rev. A 20, 394 (1979).

5C. G. Shull, D. K. Atwood, J. Arthur, and M. A. Home,

Phys. Rev. Lett. 44, 765 (1980). 6R. Gahler, A. G. Klein, and A. Zeilinger, Phys. Rev. A 23, 1611 (1981). 'R. P. Feynman, F. L. Vernon, Jr., and R. W. HcUwarth, J. Appl. Phys. 28, 49 (1957).

*S. L. Gilbert, J. J. Bollinger, and D. J. Wineland, Phys. Rev. Lett. 60, 2022 (1988).

'W. M. Itano and D. J. Wineland, Phys. Rev. A 24, 1364

(1981); W. M. Itano, L. R. Brewer, D. J. Larson, and D. J. Wineland, Phys. Rev. A 38, 5698 (1988).

'^'D. J. Larson, J. C. Bergquist, J. J. Bollinger. W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 57, 70 (1986). "J. J. Bollinger, J. D. Prestage, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 54, 1000 (1985).

'^L. R. Brewer, J. D. Prestage, J. J. Bollinger, W. M. Itano, D. J. Larson, and D. J. Wineland, Phys. Rev. A 38, 859

(1988).

'^N. F. Ramsey, Molecular Beams (Oxford Univ. Press, London, 1956).

^^<j}(T)-{[f,+,U)-f,U)V/2p), where //(r) is the ith measurement of the frequency over a measurement time inter- val r, the brackets denote an average over all measurements,

and 7~(//). See J. A. Barnes et ai, IEEE Trans. Instrum. Meas. 20, 105 (1971).

'5f. L. Walls, IEEE Trans. Instrum. Meas. 36, 596 (1987).

'^S. K. Lamoreaux, J. P. Jacobs, B. R. Heckel, F. J. Raab,

and E. N. Fortson, Phys. Rev. A 39, 1082 (1989).

"T. E. Chupp, E. R. Oteiza, J. M. Richardson, and T. R.

White, Phys. Rev. A 38, 3998 (1988).

1034

TN-92

.Ji»V - S^.K....\J,K
QUANTUM OPTICS OF SINGLE, TRAPPED IONS"

Wayne M. Itano, J.C. Bergquist, F. Diedrich,^ and D.J. Wineland

Time and Frequency Division

National Institute of Standards and Technology

Boulder, CO 80303

INTRODUCTION

Single ions in ion traps can be localized in small volumes and held for long periods of time. This makes it easier to observe certain nonclassical properties of the

electromagnetic field, such as photon antibunching and sub-Poissonian photon statistics, which are reduced when large numbers of atoms are present. Such properties can

be observed in atomic beams so dilute that the probability of having more than one atom in the observation volume is low.^'^ Trapped ions can be studied for much longer

times. This makes it possible, for example, to observe repeated quantum jumps of the same atom. When there are only a few (or one) ions in the trap, their number can

be known and kept fixed. Thus, sub-Poissonian photon statistics can be observed^'' without the time gating which is necessary with atomic beams. ^ Another advantage of trapped ions is that, thanks to their isolation from collisions and other perturbations,

they can be Iciser-cooled to low temperatures and studied spectroscopically with great

precision. A single ion has even been cooled to the ground energy level of the harmonic

weU of the trap, so its motion must be treated quantum mechanically.^

EXPERIMENT

Hg"^ ions were confined in a Paul trap that consists of a toroidal ring electrode,

with an inner diameter slightly less than 1 mm, and two endcap electrodes, which are placed symmetrically on opposite sides of the hole through the ring.^ A combination

of static and rf potentials applied between the electrodes effectively creates a three- dimensional harmonic well. The classical motion consists of a small-amplitude oscillation, at the frequency of the applied rf potential, superimposed on a large-amplitude,

harmonic motion, called the secular motion. The frequencies of the secular motion were

1-4 MHz.

A cw, tunable 194 nm radiation source'° with a power of about 5 y.\N wcis used

to excite the ^5i/2 to ^Fi/2 first resonance line of Hg"*". In order to cool the ions, the frequency of the 194 nm source was tuned slightly lower than the atomic resonance. The radiation pressure force increeises when the ion's velocity is opposed to the direction

of 194 nm propagation, because the Doppler shift brings the light frequency closer to resonance. This form of laser cooling is called Doppler cooling. The 194 nm fluorescence

'Work of the NIST. Not subject to U.S. copyright. ^Present address: Gsanger Optoelektronik GmbH, Planegg, Federal Republic of Germany.

Coherence and Quantum Optics \ I 539

Edited by J. H. Eberlv et al. Plenum Press, Ne« York, 1990

TN-93
11 fum

194 nm

excitation

Figure 1 The lowest four energy levels of Hg"*".

from a single, cooled ion was easily observed. The photon detection efficiency was

about 5 X 10"'*. Approximately 50 000 photons/s from a single ion were detected with

a photomultiplier tube. QUANTUM JUMPS

Quantum jumps of single, trapped ions have been observed in Ba"'',^''' Hg"*",^ and

Mg"*".^ In this context, the term "quantum jump" refers to a sudden change in the fluorescence intensity when the atom makes a transition to or from a metastable level. We have observed photon antibunching and sub-Poissonian statistics in the system

shown in Fig. 1. Measurements of the radiative decay rates were reported previously.^

The 194 nm source drives the Hg"*" ion from the ground ^5'i/2 level to the ^Pi/2 level, which decays at a rate of 4 x 10* s~^. Usually, the ion decays back to the ground state, emitting a 194 nm photon. However, about once in 10^ times, it decays instead to the metastable '^1)3/2 level, and emits an 11 /im photon. The ^Dj,i2 level decays with a

total rate of 71 = 109 ± 5 s~^. It decays directly to the ground state with probability

/i — 0.491 ±0.015 and to the metastable ^0^12 level with probability /j = 1 - /i. The

^ D^j2 level decays to the ground state at a rate 72 — 11.6 ± 0.4 s~^

Figure 2 shows the 194 nm fluorescence from a single Hg"*" ion as a function of time. When the ion makes transitions between the ^5i/2 level and the ^Pi/2 level, the fluorescence has a high, steady level. When the ion makes a transition (quantum jump)

to the ^ D3/2 level and emits an 11 /zm photon, the fluorescence drops to the background

level. When the ion returns to the ground state, the fluorescence goes back to the high

level. We call the sudden drops in fluorescence "on-to-ofF" quantum jumps and the sudden increases "off'-to-on" quantum jumps.

500 1000 time (ms)

1500

Figure 2. 194 nm fluorescence of a single Hg"*" ion as a function of time.

540

TN-94
Table 1. Table of observed and calculated values of Np{n) and Q. Values of Np{n)

and Q for Poissonian distributions having the same (n) are shown for comparison.

Pin) Q

n = 1 2 3 4 5 6 7 Obs. 350 785 572 222 58 12 1 -0.253 ±0.025

1-ion Calc. 360.3 771.5 568.3 228.2 59.2 10.9 1.5 0.2 -0.243

Poiss. 470.5 680.9 492.6 237.6 86.0 24.9 6.0 1.2 0.000

Obs. 796 1503 1118 445 99 35 2 2 -0.194 ±0.019

2-ion Calc. 775.0 1558.1 1077.1 436.8 122.3 25.8 4.3 0.6 -0.198

Poiss. 969.0 1373.8 973.9 460.2 163.1 46.3 10.9 2.2 0.000

Let 5*'^'(t) be the normalized intensity correlation function of the 11 fim field generated by the radiative decay from the ^Pi/2 level to the ^0^/2 level. Reference 7 used a solution of the density-matrix equations of the four-level atomic system, valid

for times long compared to the lifetime of the ^Pi/2 level, to show that

^(^'(r) = 1 - C+e-^+" - C_e-^-\ (1)

where

7± = 2{(')'o + 71 + 72) ± [(70 + 7i + 72)^ - 4/27071 - 47172 - 47072]^}, (2)

C± = ±7t(/i7± - 72)/[72(7+ - 7-)]- (3) Here, 70 is the transition rate from the "on" to the "off" state. It depends on the

intensity and frequency detuning of the 194 nm source. Since g^^\T) —> as t —» 0, the

11 ^m light is antibunched. The calculated and measured g^^^r) are in good agreement

[see Fig. 2(a) of Ref. 7]. Photon antibunching was also observed with two Hg"*" ions separated by about 3 fim7 The observed g^^^r) was consistent with the assumption

that the two ions were independent.

Given 5'^^'(t), we can compute the complete photon counting distribution. Let p{m) be the probability that m, 11 /xm photons are emitted in an interval of length T.

It can be shown that

1 ^(-1)'

pi^") = zr\Yl

Am+T)

ml ), (4) r=0 where the rth factorial moment of n is defined for r = 1, 2, 3 . .

. , by

00

(n'"")) = {nin - 1) . . . (n - r + 1)) = ^ n(n - 1) . . . (n - r + l)p(n) (5) n=0

and (n*°^) = 1.^^ With the aissumption that the ion loses memory of its previous history

after each 11 ^m photon emission, the factorial moments are given by

(n(^)) = {iyr\ r dU... r dU g^'HU - ir-i) .

. g^'\t2 -U), r = 2, 3, . . .

,

Jo Jo

(n(')) = (n) = {i)T, nn = 1, (6)

where (/) is the average number of 11 /xm photons emitted per unit of time. The

normalized second factorial moment Q is given by Eq. (11a) of Ref. 13. This parameter

measures the departure of the variance of a distribution from that of a Poissonian

distribution. Negative values of Q indicate sub-Poissonian statistics.

541

TN-95
1000 • Data Calculation

Poissonian

1 2 3 4 5 6 Number of photons in 200 nns

Figure 3. Observed and calculated photon count distributions. The uncertainties are

less than or approximately equal to the radii of the dots. A Poissonian distribution is shown for comparison.

Observed and calculated values of Np{n) and Q for one and two ions are shown

in Table 1. Here, Np{n) is the number of intervals in which n photons were detected,

where N is the total number of intervals. For the one-ion data, A'^ = 2000, T = 200 ms, 7o = 12.55 s'K For the two-ion data, N = 4000, T = 100 ms, 70 = 12.1 s'^ The

observed and calculated values are in good agreement with each other and clearly differ from the values for a Poissonian distribution. The statistical uncertainties of Np{n)

are approximately the square roots of the measured values. The values of Q depend on

the value of T. For example, Q was measured to be —0.242 ± 0.025 for T = 200 ms.'^ Values of p{n) were calculated by computing (n*""') for r = 0, . .

. , 11 from Eq. (6) and

using Eq. (4). Figure 3 shows the observed and calculated values of Np{n) for one ion. The dots represent the experimental data. The solid line is a smooth curve connecting

the calculated values.

If the atom loses all memory after each quantum jump, then the lengths of suc- cessive intervals between jumps should show no correlation. Let Ti, T2, . .

. , T^ be the

successive intervals between off-to-on or on-to-off quantum jumps. A nonrandom pat- tern in the scatter plot of T„ vs T^+i would be an indication that the seemingly random

sequence of quantum jumps was actually governed by a low-dimensional chaotic attractor. Such patterns have been observed, for example, in the time sequence of drips from

a leaky faucet.^'' Plots of r„ vs T„+i for both the off-to-on and the on-to-off quantum

jumps are shown in Fig. 4. No nonrandom structure is apparent.

500

+

c

• - f • \

if ft .".• *••

500

+

c

Tp (ms)

500

Tp (ms)

500

Figure 4. Scatter plot of Tn+i vs T„, where r„ is the nth interval between successive

(a) on-to-off quantum jumps and (b) off-to-on quantum jumps.

542

TN-96
QUANTUM HARMONIC OSCILLATOR

The wavefunctions of a single charged particle in a Paul trap have been calculated

in Refs. 15 and 16. Since the Hamiltonian is time dependent, stationary states do not

exist. However, quasi-stationary states, obtained by solving for the eigenvalues of the

Floquet operator, do exist. They correspond in some sense to the stationary states of the secular harmonic oscillator potential.

The absorption spectrum of a narrow optical resonance of an ion in a Paul trap

consists of an unshifted carrier, surrounded by discrete motional sidebands. The side- bands are spaced by multiples and combinations of the secular frequencies. We have

observed these sidebands in the absorption spectrum of the ^Si/2-io- Ds/2 transition of

a single trapped Hg''' ion.^ The carrier results from transitions in which the vibrational

quasi-energy is unchanged. The upper and lower sidebands correspond to transitions

which increase or decrease the quasi-energy.

Recently, we have cooled a single Hg"^ ion almost to the lowest (n„ = 0) vi- brational quasi-energy state, by a method called sideband cooling. First, the ion was

Doppler cooled to about 2 mK with the ^Si/2-io-^Pi/2 transition, so that the mean value

of n^ was about 12 at a secular frequency of 2.96 MHz. Laser radiation tuned to the

first lower sideband of the narrow ^Si/2-to-^Ds/2 transition was then applied to the ion,

lowering n„ by one for each photon absorbed. After the sideband cooling period, laser radiation of saturating intensity Wcis applied at the lower sideband frequency. Absence

of absorption, detected by optical double resonance, indicated that the ion was in the

Uy — state. The ion was found to be in the (n^ = 0) state about 95% of the time.

Starting from the n^ = state, other quantum harmonic oscillator states could

be prepared by manipulation of the electric potentials applied to the trap. A coherent

state could be prepared by quickly shifting the static potential of one endcap relative

to the other, thus shifting the center of the potential well. A squeezed state could be

prepared by shifting the static potential on both of the endcaps relative to the ring.

Acknowledgments

We wish to acknowledge the generous support of the Air Force Office of Scientific Research and the Office of Naval Research.

REFERENCES

1. H. J. Kimble, M. Dagenais, and L. Mandel, Phys. Rev. Lett. 39, 691 (1977); M.

Dagenais and L. Mandel, Phys. Rev. A 18, 2217 (1978).

2. R. Short and L. Mandel, Phys. Rev. Lett. 51, 384 (1983).

3. W. Nagourney, J. Sandberg, and H. Dehmelt, Phys. Rev. Lett. 56, 2797 (1986).

4. Th. Sauter, W. Neuhauser, R. Blatt, and P. E. Toschek, Phys. Rev. Lett. 57, 1696

(1986).

5. J. C. Bergquist, R. G. Hulet, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 57, 1699 (1986).

6. F. Diedrich and H. Walther, Phys. Rev. Lett. 58, 203 (1987).

7. W. M. Itano, J. C. Bergquist, and D. J. Wineland, Phys. Rev. A 38, 559 (1988).

8. F. Diedrich, J. C. Bergquist, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 62, 403 (1989).

9. J. C. Bergquist, W. M. Itano, and D. J. Wineland, Phys. Rev. A 36, 428 (1987).

10. H. Hemmati, J. C. Bergquist, and W. M. Itano, Opt. Lett. 8, 73 (1983).

11. R. G. Hulet, D. J. Wineland, J. C. Bergquist, and W. M. Itano, Phys. Rev. A 37,

4544 (1988).

12. W. M. Itano, J. C. Bergquist, R. G. Hulet, and D. J. Wineland, Phys. Rev. Lett. 59, 2732 (1987).

13. L. Mandel, Opt. Lett. 4, 205 (1979).

14. P. Martien, S. C. Pope, P. L. Scott, and R. S. Shaw, Phys. Lett. IIOA, 399 (1985).

15. R. J. Cook, D. G. Shankland, and A. L. Wells, Phys. Rev. A 31, 564 (1985).

16. M. Combescure, Ann. Inst. Henri Poincare 44, 293 (1986).

543

TN-97
PHYSICAL REVIEW A VOLUME 41, NUMBER 5 1 MARCH 1990

Quantum Zeno effect

Wayne M. Itano, D. J. Heinzen, J. J. Bollinger, and D. J. VVineland

Time and Frequency Division, National Institute of Standards and Technology, Boulder, Colorado 80303

(Received 12 October 1989)

The quantum Zeno effect is the inhibition of transitions between quantum states by frequent measurements of the state. The inliibition arises because the measurement causes a collaj^se

(reduction) of the wave function. If the time between measurements is short enough, the wave function usually collapses back to the initial state. We have observed this effect in an

rf transition between two Be"*" ground-state hyperfine levels. The ions were confined in a Penning trap and laser cooled. Short pulses of light, applied at the same time as the rf field, made the measurements. If an ion was in one state, it scattered a few photons; if it was in tiie other, it scattered no photons. In the latter case the wave-function collapse was due to a nidi measurement. Good agreement was found with calculations.

I. INTRODUCTION

The quantum Zeno effect (or paradox) is the inhibi- tion of transitions between quantum states by frequent

measurements.^"^ Misra and Sudarshan^ were the first theoretically, to call the effect by that name, but closely related work

was done much earlier.^ Consider the decay of an unstable state, such as an

unstable particle. An observation that the state has not decayed causes a collapse (reduction) of the wave function to the undecayed state. The probability that the

state decays after this collapse grows quadratically with

time, for short enough times. Suppose n measurements,

spaced in time by T/n, are made. The probability that

the state will survive for a time T goes to 1 in the limit

75 —f oo. Hence, Misra and Sudarshan argued, a continuously observed state can never decay. -^ This effect is diffi- cult to observe in spontaneous decay because the interval during which the probability grows quadratically is very

short compared to the time required to make a measurement. Ghirardi et al.^ have shown, by general arguments

based on the time-energy uncertainty relations, that the

dependence of the hfetime on the frequency of measurements, although present in principle, would be extremely

difficult to observe. Deviations from an exponential de- cay law, expected theoretically for very short and very

long times, have not yet been observed experimentally.^

The term "quantum Zeno effect" is applied also to the inhibition of induced transitions by frequent measurements. This effect can easily be observed experimentally,

in contrast to the inhibition of spontaneous transitions. Consider a system made up of two levels, labeled 1 and

2. Assume that the system can be driven from level 1 to

level 2 by applying a resonant perturbation for a given

length of time. Assume that it is possible to make measurements of the state of the system, which project the

system into one of the two levels, and which take a negligible amount of time. If the system is initially in level

1, and we make n equally spaced measurements while

the perturbation is applied, the probability of finding the

system in level 2 at the end of the period decreases as n

increases. Various cases of this type have been examined

10-12

II. THEORY

Cook^^ proposed an experiment on a single, trapped

ion to demonstrate the quantum Zeno eflfect on an in- duced transition. Ti-apped ions provide very clean sys- tems for testing calculations of the dynamics of quantum

transitions. They can be observed for long periods, free from perturbations and relaxations. Their levels can be

manipulated easily with rf and optical fields. In Cook's proposed experiment, the ion was assumed

to have the level structure shown in Fig. 1. Level 1

is the ground state. Level 2 is an excited metastable

state. Spontaneous decay from level 2 to level 1 is Jis- sumed to be negligible. If the ion is in level 1 at time r =0, and a perturbation having the resonance frequency

(£"2 — Ex)/h is applied, a coherent superposition state

is created. Let Pi and P2 be the probabilities for the

ion to be in levels 1 and 2. Then P2{''') = sin^(Qr/2)

2

41

FIG. 1. Energy-level diagram for Cook's proposed demonstration of the quantum Zeno effect. TN-98

2295
2296 ITANO, HEINZEN, BOLLINGER, AND WINELAND 41

and Pi(r) = cos^(Qr/2), where Q, the Rabi frequency,

is proportional to the amphtude of the apphed field. If a measurement of the state of the ion is made after a

short time, such that fir <C 1, tlien Pi(r) ^ 1 and

Pni^) ~ ^fi'T^ <C 1- If, instead, the ion starts out in level 2, the situation is reversed, so that Po{t) rs 1 and

A(r)«ifiV2.

Assume that level 3 is connected by a strongly allowed

transition to level 1 and that it can decay onhj to level

1. The state measurement is carried out by driving the

1 —"^ 3 transition with an optical pulse. This pulse causes a collapse of the wave function. The wave function of the

ion is projected by the measurement into level 1 or 2 with

probabilities equal to the squares of the wave-function

amplitudes for being in level 1 or 2. If the ion is projected

into level 1 at the beginning of the pulse, it cycles between

level 1 and level 3, and emits a series of photons until the

pulse is turned off. If it is projected into level 2, it scatters no photons. The latter case is an example of what

Porrati and Putterman have called a wave-function col- lapse due to a null measurement.^"' That is, the absence

of scattered photons when the optical pulse is applied is enough to cause a collapse of the wave function to level

2. The pulse must be long enough so that an ion in level

1 would have time to scatter a few photons. It does not matter whether aswitched-on detector capable of detect- ing the photons is actually present. The state of the ion is recorded in the electromagnetic Field. The measurement

(and the subsequent wave-function collapse) takes place

after the field has interacted with the ion for a sufficient amount of time.^^''^'^ If a measurement finds the ion to be in level 1, the ion returns to level 1 after the end of

the measurement, within a time approximately equal to the lifetime of level 3. If a measurement finds the ion to be in level 2, the ion never leaves that level during the

measurement. If this measurement is followed immediately by a second one, the result will almost always be the same. Thus the optical pulses make nearly ideal, nondestructive measurements. However, since a finite time is required to make a measurement, the wave function can

evolve between measurements. Therefore, it is possible

for the result of the second measurement to differ from

that of the first. Cook's proposed experiment was to drive the 1-^2

transition with an on-resonance tt pulse (a square pulse

of duration T = 7r/fi) while simultaneously applying a

series of short measurement pulses. The duration of a measurement pulse was assumed to be much less than

the time between pulses. Suppose the ion is in level 1 at time r =: 0. The it pulse is then applied. Without

the nfieasurement pulses, the probability P2{T) to be in level 2 at r = T is 1. Let n measurement pulses be applied at times r = kT/n = kw/^nQ), where k — I, . . . ,n. The level populations at the end of the n pulse are easily

calculated with the use of the vector representation of a two-level system. ^^ The equations simplify if we trans- form to a coordinate system in which the rotating component of the rf perturbation is stationary. (We ignore the

counter-rotating component.) The system is described

by a vector R= {Ri, R2, R3), whose components can be

expres.sed in terms of the density matrix p:

Ri = pi2 +P21,

R2 = KPn - P21),

R3 = P22 - Pn = P2 P, The equation of motion for R is

(lYl/dt =r a> X R,

(1)

(2)

where w = (fi,0,0). The geometrical interpretation of Eq. (2) is that R precesses about uj with fixed magnitude

and angular velocity \u)\ = fi. At r =: 0, R = (0,0,-1).

Just before the first measurement pulse at r = •n-/(nfi),

R=[0,sin(fir),-cos(fir)] = [0,sin(7r/7z), — cos{Tr/n] (3)

The measurement pulse projects an ion into level 1 or

2. Its effect on the density matrix, which corresponds to an ensemble average, is to set the coherences (pio and

P21) to zero, while leaving the populations (p22 and pn)

unchanged. Hence, Ri and R2 are set to zero, while R3

remains unchanged:

R=: [0,0, -cos(7r/?i)]. (4)

Thus, at r = 7r/(?ifi), just after the first measurement

pulse, R is the same as it was at r = 0, except that its magnitude has been decreased by a factor of cos(7^/7^). After the second measurement [t = 27r/(?ifi)], |R| is de- creased by another factor of cos(7r/?i). This follows from

the fact that Eq. (2) is linear with respect to R. After n

measurements, (r = 7r/fi), R(r) = [0,0,-cos"(7r/n)].

We use Eq. (1) to express P2 in terms of R3:

(5)

P2 = R3 + Pl = R3 + il = ^{l + R3).

Pi)

(6)

In deriving Eq. (6), we used the conservation of probability for a closed two-level system: Pi-\- P2 — 1- Substituting the value of /?3(T) from Eq. (5) into Eq. (6), we

h ave

Po(T) = ^[l-cos"(7r/n)]. (7)

It can be shown from Eq. (7) that P2{T) decreases monotonically toward zero as n goes to infinity. For large ?j,'-

P2(r)«^[l-exp(-i7rVn)] (8)

Equation (8) was derived from Eq. (7) by expanding

cos(;r/);) in a power series and using

iim (1 -x/n)" = c- TN-99 (9)
41 QUANTUM ZENO EFFECT 2297

III. EXPERIMENT

Our experiment is very similar to that proposed by

Cook. Levels 1 and 2 are the (m/, mj) = (|,-^) and (^,-^) hyperfine sublevels in the ground 25^5i/2 state of ^Be"*" (see Fig. 2). These levels are separated by 320.7 MHz at the magnetic field used in the experiment {B w 0.8194

T). Level 3 is the (m/ = |, mj = |) sublevel of the

2p^P3/2 state, which decays only to level I. Spontaneous

decay from level 2 to level 1 is negligible.

The experimental apparatus has been described

previously.^^"^'* About 5000 ^Be"*" ions were stored in a cylindrical Penning trap. The pressure in the trap was

about 10~* Pa. The storage time of the ions in the trap was several hours. A frequency-doubled cw dye laser gen- erated 313-nm radiation to drive the 1 —> 3 transition in order to optically pump, detect, and laser cool the ^Be"*" ions. The 313-nm fluorescence from the ions was de- tected by an imaging photon-counting detector. ^^ About

100 000 ^^Mg"*" ions were confined together with the ^Be"*" ions. The ^^Mg"*" ions were laser-cooled by 280-nm radi- ation from a frequency-doubled cw dye laser. The ^Be+

were kept cold (^250 mK) by long-range Coulomb col- lisions with the ^^Mg"*" ions even when they were not

directly laser-cooled by the 313-nm source.

^^ The polarization of the 313-nm beam was perpendicular to the magnetic field. When the 313-nm radiation was nearly resonant with the 1—^3 transition and no

rf field was applied, about jy of the population was optically pumped to the (|,i) ground-state sublevel (level 1).^°""^ The remaining population was in the (|, —

^)

sublevel. When the 313-nm radiation was on continuously, the populations approached the steady state with

a time constant of about 1 s. The measurement sequence for the 1 —» 2 transition was as follows: The 313-nm radiation was left on for

2p^B

'1/2

RF transition ,,---7,-- 2

2s ^S,

B

FIG. 2. Diagram of the energy levels of ^Be"*" in a magnetic field B. The states labeled 1, 2, and 3 correspond to those in Fig. 1 .

about 5 s to prepare most of the ^Be"*" ions in level 1 and

to empty level 2. The 313-nm radiation was then turned

off. The 320.7-MHz rf field was turned on for T = 256 ms.

Its frequency and amplitude were adjusted to make this an on-resonance n pulse. During the rf pulse, n pulses of

length Tp = 2.4 ms and wavelength 313 nm were applied,

where n was 1, 2, 4, 8, 16, 32, or 64. The pulses were

long enough to collapse each ion's wave function without

causing significant optical pumping. The delay from the

beginning of the rf pulse to the beginning of the first 313-nm pulse was {T/n— 1.3) ms. The time between the

beginning of one 313-nm pulse and the beginning of the next one was T/n.

After the end of the rf n pulse, the 313-nm radiation was turned on and left on to prepare the state. The number of photons counted in the first 100 ms was recorded.

This signal was roughly proportional to the number of

ions remaining in level 1. However, background counts, counter deadtime, and optical pumping during the 100 ms cause deviations from this proportionality. In order

to calibrate the signal, known level populations were cre- ated by applying rf pulses of lengths t — 0, 32, 64, . . .

, 544

ms. The population of level 1 (per ion in the subsystem

made up of levels 1 and 2) was then given by cos^(f2r/2).

From these data, the precise value of Q and also the cal- ibration of the signal as a function of the population of

level 1 were obtained. The deviations of the calibration

points from a smooth curve gave an indication of the

measurement errors. The 313-nm radiation was turned on and off with an

electromechanical shutter that had a rise or fall time of

about 0.2 ms. The 320.7-MHz rf field was turned on and off" with a semiconductor diode switch, which had a switching time of about 150 ns and an on-to-off ratio of

about 75 dB. The 280-nm beam was left on continuously.

The measurement sequence was the same for the 2^1

transition, except that first an rf n pulse, free from 313-

nm pulses, transferred the level 1 population to level 2 immediately after the 313-nm radiation was shut off. The 1 —* 2 transition frequency decreases by 22 Hz

for a l-/iT increase in 5. If the transition frequency

shifts by more than a small fraction of l/T w 4 Hz, the

observed transition probabilities will deviate significantly

from the calculated ones. The center frequency of the

1 —> 2 resonance was measured before and after each

run. If the frequency shifted by more than about 0.1 Hz, the data from that run were not used. Drift of the

magnetic field was the main obstacle to increasing T. Relaxations of the ground-state hyperfine levels are very weak when the 313-nm radiation is not applied. This was shown in studies of the {mj , mj)=:(— 2 ,2) to (—2 '2) transition. The derivative of this transition frequency

with respect to B goes to zero at B « 0.8194 T. At this value of B, weak relaxation processes due, for example,

to collisions or to the 280-nm radiation can be studied.

This transition showed no sign of population or coherence

relaxations for times up to 550 s.^^ The average number of photons scattered by an ion in

TN-100
2298 ITANO, HEINZEN, BOLLINGER, AND WINELAND 41

TABLE I. Predicted and observed values of the 1 —+ 2 and 2 — 1 transition probabilities for different values of the number of measurement pulses n. The uncertainties of the observed transition probabilities are about 0.02. The second column shows the transition probabilities that result from

a simplified calculation, in which the measurement pulses are assumed to have zero duration and

in which optical pumping is neglected.

i[l -cos"(7r/n)]

1 — 2 transition Predicted Observed2 —* 1 transition Predicted Observed

124

81632

64

1.0000 0.5000

0.3750

0.2346 0.1334

0.0716

0.0371

0.995 0.995 0.999 0.998

0.497 0.500 0.501 0.496

0.351 0.335 0.365 0.363

0.201 0.194 0.217 0.209

0.095 0.103 0.118 0.106

0.034 0.013 0.073 0.061 0.006 -0.006 0.080 0.075

level 1 during a single 313-nm pulse of length Tp is approxinnately TpRc/{eciN), where Re is the observed steadystate photon count rate, e^ is the probability of detecting

a scattered photon, and TV is the total number of ^Be"*" ions. The photon detection efficiency e^ was estimated

from the solid angle of the lens system and the quantum efficiency of the detector to be about 2 x 10"''. For

typical experimental values Tp = 2.4 ms, Re = 30 000

s~^, and N = 5000, the number of scattered photons per

ion per pulse was therefore about 72, more than enough

to cause the collapse of the wave function. We emphasize that it is the number of scattered photons which is important, not the number that can be detected by the

apparatus. The number of photons detected per ion per

pulse is much less than 1. As a further check that the

pulses were long enough, some runs were taken with Tp decreased to 1.4 ms (the shortest that the shutter could

make). The results indicated that these pulses were still long enough to collapse the wave functions. These data

are not reported here because the pulse shapes were not

the same for different pulse repetition rates. This made

quantitative interpretation of the data difficult. With a faster optical shutter, such as an acoustooptic modulator, the 313-nm pulses could be decreased in length and still be long enough to collapse the wave functions. However, the minimum time required for a measurement pulse depends not only on the average photon

scattering rate, but also on the time required to ensure

that every ion will pass through the 313-nm beam. The

^Be"*" ions occupied a cylindrical volume with a height

of about 1000 nm and a radius of about 350 fim. The

313-nm beam was focused to a radius of about 50 /im. It propagated perpendicular to the axis of the cylinder and

intersected the ^Be"*" ions near the center of the volume.

If the 313-nm beam were expanded radially and directed

along the axis, so that it intersected the entire volume,

this problem could be avoided.

IV. RESULTS

Table I shows the calculated and observed values of the

probabilities of making the 1 —» 2 and 2 ^^ 1 transitions

for values of n = 1 , 2, . .

. , 64. The predicted and observed

values agree within the measurement error of about 0.02 estimated from the scatter of the signal calibration data. The general decrease of the probabilities with n demonstrates the quantum Zeno effect. Probabilities must take

values from to 1. However, with our method of deter- mining the transition probability, random fluctuations in the photon count rate can lead to an apparent transition

probability which is less than zero or greater than 1. The

value of —0.006 for the observed 1 —>• 2 transition probability for n = 64 just means that the number of photons

detected was slightly higher than the number expected

for a transition probability of 0. Figures 3 and 4 show

the probabilities for the 1-^2 and 2 —> 1 transitions,

respectively.

The assumptions made in the calculations are as fol- lows: When the rf field is present and the 313-nm radi- ation is not, the transition between levels 1 and 2 is as- sumed to proceed without relaxations. During the 313-

O

o

cd

FIG. 3. Graph of the experimental and calculated 1 — 2 transition probabilities as a function of the number of measurement pulses n. The decrease of the transition probabilities with incrccising n demonstrates the quantum Zeno effect. TN-101
41 QUANTUM ZENO EFFECT 2299

>.

cdo

o

FIG. 4. Graph of the experimental and calculated 2 — 1 transition probabilities as a function of the number of measurement pulses n. The transition probabilities for n — 32 and

n — 64 are higher than the corresponding ones for the 1 —» 2 transition because of an optical pumping effect discussed in the text.

nm pulses, the coherence between levels 1 and 2 is as- sumed to be destroyed so quickly that the rf field can be

ignored. The 313-nm radiation causes optical pumping

from level 2 to level 1 with a time constant of about 1 s. This rate weis measured by using an rf tt pulse to prepare

the ions in level 2, turning on the 313-nm radiation, and

recording the 313-nm fluorescence as a function of time.

Optical pumping by the 313-nm pulses can have an observable effect on the measured transition probabilities,

especially for large n. This optical pumping causes a de- crease in the 1 —+ 2 transition probability and an increase

in the 2 —+ 1 transition probability. The decrease in the

1 —<• 2 transition probability is not noticeable in our data,

since the transition probability is already small for large

n. The increase in the 2 —* 1 transition probability is noticeable for n = 32 and 64. The predicted transition

probabilities shown in Table 1 and plotted in Figs. 3 and

4 take into account optical pumping, the finite 313-nm

pulse durations, and the measured value of Q. The measured value of Q, differed by less than 3% from the ideal value Q = n/T. For comparison, ^[1 — cos"(7r/n)], the

transition probability calculated if these effects are not

included, is also shown in Table I. For the larger values of n, the sum of the measurement

periods, uTp, is not negligible compared to T. Therefore,

part of the decrease in the transition probability is due

merely to the decrease in the time during which rf field can act. For the most extreme case (n = 64) iiTp is 60% of T. However, even for this case, the decrease in the transition probability is much more than could be

accounted for by the decrease in time. For n = 16, riTp is only 15% of T, but the transition probability is decreased

by about 90%.

V. DISCUSSION

Cook^^ originally proposed an experiment with a sin- gle ion. Such an experiment should be feasible, since

single ions have been observed in Penning traps^^'^^ and

in Paul traps. ^'^'^^ The ensemble average, which is needed

for comparison with calculations of the density matrix,

would be obtained by repeating the experiment many

times. In the present experiment, each measurement is an average over an ensemble of about 5000 independent

ions, since their mutual interactions can be neglected. An

experiment might be done with a single Hg+ ion. Level 1 would be the ground 5d^°6s^5i/2 state, level 2 would be

the metastable b(P6s^ "£^5/2 state, and level 3 would be

the 5c/^°6p2Pi/2 state. The 1 -^ 2 transition at 282 nm

and the 1 —* 3 transition at 194 nm have already been

observed in a single Hg"*" ion.^^

It might be argued that previous observations of ef- fects such as collisional relaxation of rf transitions al- ready contain the quantum Zeno effect. This is possible,

but we know of no experiment in which the effect has

been demonstrated simply and unambiguously. Colli- sions cannot usually be interpreted as quantum measurements. That is, they do not necessarily project a single

quantum system, such as an atom, into one state or the

other. Rather, in many cases, they perturb the phase of

the coherence, represented by Ri and R2 in the vector model, for each system. For the ensemble average, Ri

and i?2 are driven to zero, just as in our experiment, but

the underlying physics is different. Collisions may also perturb R3, which is not desirable for a demonstration of

the quantum Zeno effect. Some experiments that involve continuous measure- ments have, in a sense, already demonstrated the quantum Zeno effect. Although the measurements are contin- uous, it takes a finite amount of time to make a measure- ment with a given degree of uncertainty. Thus a contin- uous measurement might be regarded as a series of measurements, one after the other. In such experiments, the

measurement times are not separated from the free evolution periods, so the analysis is more difficult than for the

present experiment. One example of such an experiment

is the spin flip resonance of a single, trapped electron. '^'^ The spin state is detected by coupling the electron to an

electronic circuit. As the time required to make a measurement is decreased by increasing the degree of coupling, noise broadens the spin flip resonance. This broadening decreases the rate of transitions induced by a weak

microwave field. Another example is the three-level Hg''' system described previously.^^ Radiation resonant with

the bd^°6s^Si/2 to 5c/'°6p2Pi/2 (1^3) transition per- forms a continuous state measurement. Photons are not

scattered if the ion is in the 5rf^6s^^£)5/2 state (level 2); otherwise they are. We have observed a decrease of the

rate of the 1 — 2 transition, induced by a narrow-band

laser, when radiation at the 1 —> 3 transition frequency

is applied.

Normally, the probability for spontaneous emission of

TN-102
2300 ITANO, HEINZEN, BOLLINGER, AND WINELAND 41

a photon by an atom grows quadratically only for ex- tremely short times, approximately the inverse of the

frequency of the emitted photon. However, it might be

possible to increase this time by placing the atom inside a resonant cavity. Jaynes and Cummings^^ showed that a two-level atom coupled to a single mode of a resonant,

lossless cavity oscillates between the ground and excited

states. Suppose an excited atom is placed in a cavity

that initially contains no photons. Then the probability

for the atom to be in the excited state is cos^(gr), where

g is the coupling constant (the vacuum field Rabi fre- quency) and r is the time that the atom has been in the

cavity. The probability for the atom to be in the ground

state is proportional to r^ for r short compared to g~^.

If the atom is coupled to a single damped cavity mode,

the probability to be in the ground state is still pro- portional to r^, provided that r is short compared to both g~^ and 77 \ where jc is the dissipation rate of the

cavity.^^ It might be possible to demonstrate the quantum Zeno effect on the decay of an atom in a cavity. This

would not contradict the results of Ghirardi ei a/.,^ since

they did not consider such a system.

ACKNOWLEDGMENTS

This research was supported in part by the U. S. Air Force Office of Scientific Research and the Office of Naval

Research. We thank Dr. S. L. Gilbert and Dr. M. G. Raizen for reading the manuscript.

^B. Misra and E. C. G. Sudarshan, J. Math. Phys. 18, 756

(1977). ^C. B. Chiu, E. C. G. Sudarshan, and B. Misra, Phys. Rev. D 16, 520 (1977).

''G. C. Ghirardi, C. Omero, T. Weber, and A. Rimini, Nuovo

Cimento 52A, 421 (1979). *A. Peres, Am. J. Phys. 48, 931 (1980). ^E. Joos, Phys. Rev. D 29, 1626 (1984). ^K. Kraus, Found. Phys. 11, 547 (1981). ^D. Home and M. A. B. Whitaker, J. Phys. A 19, 1847

(1986).

*L. A. Khalfin, Zh. Eksp. Teor. Fiz. 33, 1371 (1957) [Sov. Phys.—JETP 6, 1053 (1958)]; R. G. Winter, Phys. Rev. 123, 1503 (1961); W. Yourgrau, in Problems in the Phi- losophy of Science, edited by I. Lakatos and A. Musgrave

(North-Holland, Amsterdam, 1968), pp. 191 and 192. E. B. Norman, S. B. Gazes, S. G. Crane, and D. A. Bennett,

Phys. Rev. Lett. 60, 2246 (1988). ^°H. Dehmelt, Proc. Natl. Acad. Sci. U.S.A. 83, 2291 (1986);

83, 3074 (1986).

"G. J. Milburn, J. Opt. Soc. Am. B 5, 1317 (1988). ^^R. J. Cook, Phys. Scr. T21, 49 (1988).

"M. Porrati and S. Putterman, Phys. Rev. A 36, 929 (1987).

^''H. J. Carmichael, S. Singh, R. Vyas, and P. R. Rice, Phys. Rev. A 39, 1200 (1989).

^^I. I. Rabi, N. F. Ramsey, and J. Schwinger, Rev. Mod.

Phys. 26, 167 (1954); R. P. Feynman, F. L. Vernon, Jr. , and R. W. Hellwarth, J. Appl. Phys. 28, 49 (1957).

^^S. L. Gilbert, J. J. Bollinger, and D. J. Wineland, Phys. Rev. Lett. 60, 2022 (1988); S. L. Gilbert, J. C. Bergquist,

J. J. Bollinger, W. M. Itano, and D. J. Wineland, in Atomic

Physics 11, Proceedings of the International Conference,

Paris, 1988, edited by S. Haroche, J. C. Gay, and G. Grynberg (World Scientific, Singapore, 1989), p. 261. ^M. J. Bollmger, S. L. Gilbert, W. M. Itano, and D. J. Wineland, in Frequency Standards and Metrology, Proceedings of the Fourth Symposium, Ancona, Italy, 1988, edited by A. Be Marchi (Springer- Verlag, Berlin, 1989), p. 319.

^*J. J. Bollinger, D. J. Heinzen, W. M. Itano, S. L. Gilbert,

and D. J. Wineland, Phys. Rev. Lett. 63, 1031 (1989). ^^D. J. Larson, J. C. Bergquist, J. J. Bollinger, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 57, 70 (1986). 2°D. J. Wineland, J. C. Bergquist, W. M. Itano, and R. E. Drullinger, Opt. Lett. 5, 245 (1980). ^^R. G. Hulet, D. J. Wineland, J. C. Bergquist, and W. M.

Itano, Phys. Rev. A 37, 4544 (1988).

^^L. R. Brewer, J. D. Prestage, J. J. Bollinger, W. M. Itano, D. J. Larson, and D. J. Wineland, Phys. Rev. A 38, 859

(1988). ^^D. J. Wineland and W. M. Itano, Phys. Lett. 82A, 75

(1981).

^^W. Neuhauser, M. Hohenstatt, and P. E Toschek, Phys. Rev. A 22, 1137 (1980); W. Nagourney, G. Janik, and H. Dehmelt, Proc. Natl. Acad. Sci. U.S.A. 80, 643 (1983); F. Diedrich and H. Walther, Phys. Rev. Lett. 58, 203 (1987); R. G. DeVoe, J. Hoffnagle, and R. G. Brewer, Phys. Rev. A 39, 4362 (1989); J. D. Sankey and A. A. Madej, Appl.

Phys. B 49, 69 (1989).

"J. C. Bergquist, R. G. Hulet, W. M. Itano, and D.

J. Wineland, Phys. Rev. Lett. 57, 1699 (1986); J. C. Bergquist, W. M. Itano, and D. J. Wineland, Phys. Rev. A 36, 428 (1987). 2«E. T. Jaynes and F. W. Cummings, Proc. IEEE 51, 89

(1963).

2^S. Sachdev, Phys. Rev. A 29, 2627 (1984).

TN-103
PHYSICAL REVIEW A VOLUME 43, NUMBER 9 1 MAY 1991

Reply to "Comment on 'Quantum Zeno effect' "

Wayne M. Itano, D. J. Heinzen, J. J. Bollinger, and D. J. Wineland

Time and Frequency Division, National Institute of Standards and Technology, Boulder, Colorado 80303

(Received 12 November 1990)

Various interpretations of quantum mechanics are valid insofar as they predict the same ex- perimental results. Some invoke "wave-function collapse" and some do not. An interpretation

based on the collapse postulate provides a simple explanation for a recent experimental demonstration of the quantum Zeno effect [Itano et al., Phys. Rev. A 41, 2295 (1990)], but other

interpretations are also valid.

The theory of quantum mechanics has the unusual

property that, while there is general agreement on the

experimental consequences, there is endless debate on

matters of interpretation. The preceding Comment by

Ballentine^ on our article" illustrates this point.

The choice among interpretations that yield identical predictions, including the standard Copenhagen interpretation, the "statistical" or "ensemble" interpretation fa- vored by Ballentine,"' or even Everett's "relative-states" or "many-worlds" interpretation,^ would seem to be a matter of personal taste. Ballentine states that "collapse

of the wave function" is not necessary to quantum mechanics. We agree. However, we feel that the explanation

given in our article, which invokes von Neumann's "col- lapse" postulate, is useful for giving a simple explanation

of our experiment. Other situations may require a more

elaborate treatment. One such case is the limit in which

the optical "measurement" pulses are too weak to completely "collapse" the wave function. Peres and Ron^

have carried out a calculation for this case. We agree that the optical pulses perturb the atom, and we do not state otherwise in our article. However, we

disagree with Ballentine's statement that the inhibition

of the excitation is due to a strong perturbation rather

than to a collapse of the wave function caused by measurement. We regard both statements as valid, under

different interpretations of quantum mechanics.

Ballentine criticizes the phrase "wave-function collapse

due to a null measurement," as being misleading. We

used this term in the sense in which it was used by Porrati and Putterman,^ to mean that the nonemission of a photon during (or immediately subsequent to) the optical

pulse is correlated with the atom being in the metastable

state. We did not mean to imply that the optical pulses

did not perturb the atom.

Ballentine objects to our use of the word "measure- ment" and quotes the Oxford English Dictionary to sup- port his case. Dictionaries contain definitions of words

like "energy," "force," or "work" that do not correspond

to their meanings in physics. Therefore, one has to look

at the way in which the words are used by physicists. It

is true that the "measurement" pulses did not actually

yield a detectable signal. Our usage of the word "measurement" in this case seems to be the same as that of

Carmichael ei al7 in their discussion of the theory of the

detection of photons from resonance fluorescence. (This

situation is very similar to that of an atom subjected to a measurement pulse.)

"The detector is not the agent causing state reduction.

The atom collapses to its ground state due to its irre- versible decay into the vacuum; the collapses proceed at an average rate given by the inverse atomic lifetime, quite

indifferent to the successful or unsuccessful recording of

the emitted photons." (Ref. 7, p. 1202.)

". . . the source is an open system that loses energy

irreversibly to the vacuum. The irreversibility effectively

performs a continuous quantum measurement, without

the need for a conscious observer to record the emitted

photons." (Ref. 7, p. 1215.)

Ballentine carries out a simplified interpretation of our experiment. He calculates that the probability that

each of a sequence of n optical "measurement" pulses

results in the emission of a photon, leaving an atom in the ground state after the nth pulse, is [cos(7r/2n)]^".

This calculation is correct, but is not particularly useful for comparison with our observable. Our experiment measures the total probability that an atom is left in the ground state after the nth pulse, regardless of

whether or not each optical pulse resulted in the emission of a photon. This probability is, in our notation,

Pi(r) = l-PoiT) = |[l-l-cos"(7r/n)]. This result comes

from Eq. (7) of Ref. 2. (Here we neglect the duration of

the optical pulses, as does Ballentine.) The two probabilities are not identical, because the rf field can drive the

atom to the metastable state in one interval and return

it to the ground state in a later interval. [Ballentine^ states in his Ref. 10 that his method will also yield the

proper value of Pi(T), although he does not show this

explicitly.] In summary, quantum mechanics can be interpreted

in different ways. In this ca.se, interpretations with and

without the collapse postulate correctly predict the ex- perimental data. Therefore, the experiment neither verifies nor falsifies the notion of "wave-function collapse."

Work of the U. S. Government

43^ 5168 Not subject to U. S. copyright

TN-104
43 COMMENTS 5169

^L. E. Ballentine, preceding Comment, Phys. Rev. 43, 5165 ^H. Everett, Rev. Mod. Phys. 29, 454 (1957).

(1991). ^A. Peres and A. Ron, Phys. Rev. A 42, 5720 (1990).

^W. M. Itano, D. J. Heinzen, J. J. Bollinger, and D. J. ^M. Porrati and S. Putterman, Phys. Rev. A 3G, 929 (1987).

Wineland, Phys. Rev. A 41, 2295 (1990). ''H. J. Carmichael, S. S. Singh, R. Vyas, and P. R. Rice,

^L. E. Ballentine, Rev. Mod. Phys. 42, 358 (1970). Phys. Rev. A 39, 1200 (1989).

TN-105
Volume 67, Number 13 PHYSICAL REVIEW LETTERS 23 September 1991

Search for Anomalous Spin-Dependent Forces Using Stored-Ion Spectroscopy

D. J. Wineland, J. J. Bollinger, D. J. Heinzen,*"' W. M. Itano, and M. G. Raizen

Time and Frequency Division. National Institute of Standards and Technology, Boulder, Colorado 80303

(Received 10 June 1991)

Resonances in atomic ions can be used to search for new, weak, spin-dependent interactions. Upper

limits on anomalous dipole-monopole and dipole-dipole couplmgs for the neutron and electron are deter- mined by examining hyperfine resonances in stored 'Be"*" ions. These experiments also place strict limits on anomalous weights of spinning gyroscopes.

PACS numbers; 42.50.Wm, 04.90.-l-e, l4.80.Gl, 32.80.Pj

The existence of weakly interacting bosons has been suggested previously [l-lO]. Laboratory experiments might

detect scalar or pseudoscalar couplings of such particles to photons [11-15] or matter [6,7,16-30]. In the latter case, new spin-dependent forces would occur. Here, we report the use of stored atomic ion spectroscopy to search for anomalous potentials having a dipole-monopole or dipole-dipole character. The first is expected to include terms like [7,9,16,31]

V% = h-DSA-H\/X^r+\/r-)Q\p(-r/X^), (l)

where the spin S .) (in units of ft ) of particle A couples to particle B, r is the distance between particles, X^ is the range

of the force, and D is a coupling constant with units of (mass)

~

' [see Fig. 1 (a)]. In 1968, observation of an interaction

like Eq. (1) was reported [22] where particles A were proton spins and particles B comprised the Earth. Subsequent

measurements [23,24] contradicted this measurement and found null results (see Table I). A dipole-dipole interaction [Fig. 1 (b)] would be expected to include terms like [7,16,31]

^'l8=(ftVc)rexp(-rA«)[(lA,r--l-l/r^)S,.rSfl-(lA> + 3/V" + 3/r^)(S,4 f)(Sfl-f)], (2)

(a) "AB

Vab i

(b)

w

M M M

M M Pb ^- M M I M M M

2R

where the spin of particle A interacts with that of particle

B. T has units of (mass)"" and characterizes the strength of the interaction [see Fig. Kb)]. This type of

interaction is sought in acceleration [7,16,19,20], reso- nance [16,22-27,29], and induced magnetism [17,18] ex- periments.

Such forces mediated by axions [4] have received the most attention because the axion emerges in schemes at- tempting to resolve the strong CP problem [4,10]. The

mass and coupling of axions and related particles to matter can be severely constrained by arguments based on observed energy-loss rates of stellar objects [32]. As

discussed below, these constraints on axions appear to be

much stronger than those derived from current laboratory

experiments. Nevertheless, the laboratory experiments

are still useful because they can search for interactions outside the scope of the axion-type models.

TABLE \. Values of the coupling strength D [Eq. (1)1 evalu- ated from laboratory experiments. Here, we assume X.*»/?

FIG. 1. Experimental configurations sensitive to (a) Vab and

(b) K'fl (refer to Eqs. (I) and (2) of the text). In both parts, we assume the size of sample A is small compared to d, /?, and

/. In (a), B is assumed to be spherical with density of B particles pfl. In (b), B is assumed to be a cylinder of radius R and

height / and spin density pg. Experimentally, y% and Vab are sensed as an anomalous change in energy when Scz is changed. In the experiments reported here, Sri for the elec- tron and neutron arc changed by driving a hyperfine transition m the ground stale of atomic ''Be"*^ ions.

Experiment Measured quantity Result (kg"')

Ref. [22] D(proton) -0.02

Ref. [23] /)(proton) <2xlO-''

Ref. [24] /)(deuteron) <6.7x|0-»

Ref. [28) ^(electron) < 3.7x10'

This work D(^Be) <9.0xl0-'

This work £>(electron) <4.5xI0~'

This work /)(neutron) <2.7x|0-*

Ref. 130] D(-»'Hg)-D( ""Hg) <3xlO""'

Work of the U. S. Government

Not subject to U. S. copyright TN-106 1735
Volume 67, Number 13 PHYSICAL REVIEW LETTERS 23 September 1991

For the assumed configurations shown in Fig. 1, either V% or Vab could be observed as an anomalous dependence of energy on S^rZ- In the experiments reported

here, we looked for such a dependence in the energy

InhvQ of the atomic "'Be"'' ~S\i2 ground-state (F = \,mf

=0)—' (F = I,mf = — 1 ) hyperfine transition [33] in a magnetic field fio— 0.8194 T (transition frequency wo = 303 MHz). Here, mr is the sum of the electron spin J (=y) and nuclear spin I (=y) projections along the magnetic-field axis, taken to be along z. Measurements

of vo were made on about 5000 ''Be"'' ions stored in a Penning ion trap [34]. The ''Be"'' ions are trapped and

laser cooled (to reduce Doppler shifts), then optically

pumped into the (F = 1 ,mF —0) level using a combination

of laser and radio-frequency (rf) radiation. Additional rf radiation is then used to drive the ion population from the (F'=l,Wf=0) to the (F = l,mf=-1) state. Changes

in the population of the two states are sensed by looking

for changes in laser light scattered by the ions [34]. The

frequency vq is the frequency of radiation which drives the above transition with maximum probability. The

reference for vq is an ensemble of hydrogen maser and

cesium atomic clocks. Particles A are assumed to be ei- ther the ^Be nucleus or the unpaired outer electron in

'*'. For the above hyperfine transition, the changes in are given by |A(S,rz)|-U5',(rl — ' or 1^5,4.-1

"Be S^ z

= 2x10^'* for the ''Be nuclear spin and electron spin, re- spectively. (We assume that V^tg affects either the elec- tron or ''Be nucleus but not both at the same time.)

In the search for Vab, particles B were taken to be the nucleons in the Earth. We looked for a change in vo be- tween the cases where Bo was parallel or antiparallel to the vertical direction in the laboratory. (Since the hyperfine transition is a |AAn/^| = l transition, we had to subtract the effects of the Earth's rotation.) In the search

for Kifl, particles B were taken to be the electron spins in the iron pole faces of an electromagnet. We compared vq when Bq was created by this electromagnet with vo when

Bo was created by a superconducting solenoid (Sg spins

absent). The value of Bo was chosen such that bvJbBo

=0. Bq was set by measuring a magnetic-field-dependent

transition in ''Be"'" [34]. (The inaccuracy in our measure- ment of Bq due to Vab is negligible on this field- dependent transition.) For 9vo/S5o=0, we could rule out any positive results which might arise from small changes

in Bq between the different configurations. The largest

systematic error in the experiment was due to a background-gas pressure shift of vq [34]. For the geometries of Figs. 1 (a) and 1 (b) we integrate over the volumes containing the B particles to find the anomalous energy f.) of a single A particle. Assuming

S,4 =S i,z and Ss =5b,z, we find [Fig. I (a)]

(4)

E5-C^LD=C%(/? + ^)"M[/?(/? + ^)+M2/? + ^)'+?^«]exp[-(2/? + ^)AJ + [/?(/?+^)-M-^«]exp(-c/A,)}

^-IC'^R^HR + d)- (for l,:^R,d) (3)

and [Fig. I (b)]

£l-C^^r"C^{(^/r,)exp(-riA*)-[(</ + /)/r2]exp(-r2/?i«)-exp(-^A«) + exp[-(rf + /)AJ}

"C'^id/rt-id + D/rj} (for X^»R,d,l),

where C'^'^lnh^DpeSA-, C'^ ='2n{hVc)TpBSA-.SBz, r\

^{R^+d^y'\ ri = \R^-\-U + iyV'\ and ps is the number density of particles B (assumed here to be nucleons for Vab and electron spins for Vab)-

In our experiment, we found the shift Avq of vq due to Vab (where the magnetic field is upward) to be Avq

--6.4±2.9±6.4/iHz ^ot d-\.5 m [35]. The first er- ror is the random uncertainty, the second, an estimate of

the pressure shift variation between field-up and -down

measurements. The estimate of the pressure shift variation was made by measuring the fluctuations in vq over a period of many months with the magnetic field in one

direction. Adding these errors in quadrature, we find IAvoI < 13.4 //Hz (<5.5xlO"^"eV). Therefore we find for the coupling constant defined in Eq. (1), DCBc)

<3.8/Z,Dkg"' or/)(f ")< 1.9xl0VZ,D kg "', where Z-o [Eq. (3)] is in centimeters. If we assume X^»/?,J and

/?-/?Earih-6.36x!0'* cm [pB=p(Earth) = 3. 33x10--'

nucleons/cm^] we can make the estimates of D shown in Table I. To obtain /^(neutron) we argue as follows: In the simplest version of the shell model, the spin of the Be

nucleus is due to the odd neutron, which is in the \piii

shell. In this model, the expectation value of a component of the neutron spin is j of the expectation value

of the corresponding component of the total nuclear spin.

From this we obtain D(neutron) <2.7xlO~'* kg"'— 4.8 xlO"^'' GeV~'. This constrains one model proposed in Ref. [9] where D might be as large as 10 "^•' GeV "'. We also found A vo = — 1 3 ± 30 ± 1 70 /v Hz between vq measured in an electromagnet [36] and a superconducting magnet system. The first error is the uncertainty in the reference oscillator for the time between measure- ments ( = 5 yr), the second, relatively large error is due to the uncertainty in an estimate of the pressure shift [34] in different apparatuses. Adding the errors in quadrature, we obtain JavqI < 186 /jHz (< 7.7x10"''' eV). For the electromagnet, d = l.l cm, /=33 cm, R = \5 cm {V^b is enhanced by 2 because of the two pole faces of the magnet). We neglect the magnet yoke which will increase the

effects of Vab slightly. From Av'o and Eq. (4), we can put an upper limit on T [Eq. (2)] for assumed values of X,. Here, for brevity, we will assume that \^':^d,LR in which case V^b has the same form as normal magnetic

coupling. In Table II, we list the ratio a of Ea [Eq. (4))

to the normal magnetic interaction for various cxperi1736 TN-107
Volume 67, Number 13 PHYSICAL REVIEW LETTERS 23 September 1991

TABLE II. Experimental determinations of a{A,B), the ratio of £j [Eq. (4)] to normal

magnetic coupling for particles A and B as indicated in Fig. 1. Here, e~ denotes electron, p

denotes proton, n denotes neutron, and d denotes deuteron.

Experiment a(A,B) Method

Ref. [25] a(p,p)<5y.\0~- Molecular spectra

Ref. [26] a(-"'Hg,e-)<10-"'-10-" NMR

Ref. [26] a(''^Hg,f-)<10-'°-10-" NMR

Ref. [27] a{p,e-)-a(d,e-) <2xlO-'° NMR

Ref. [20] a(e-,e-)<3x\0-'° Acceleration

Ref. [19] a(e~,e~)<0.9xl0-" Acceleration

Ref. [17] a(e",e~)<0.005xlO-" Induced magnetism

Ref. [18] a(e",e-)<0.00085xlO-" Induced magnetism

This work a(''Be,e")<3.8xlO~" NMR

This work a(n,e")<2.3xlO~" NMR

This work a(e~,e~)<4.1xl0-" NMR

ments. From ReL [32], A.«= 10 cm is within a "window"

on the axion which has not been ruled out by stellar energy-loss rate arguments. If we take T=g,4gB/\6n

xMaMb, where g4 and ge are coupling strengths and

Ma and Mb are the masses of the elementary particles

responsible for V^b, then from Refs. [7] and [32] we esti- mate theoretically for axions a(A,B) ^ 10~'^ for X^ = 10 cm, considerably smaller than the sensitivity of the current laboratory experiments.

So far we have assumed that Sa is due to intrinsic spin.

From Eq. (1), if S^ were orbital angular momentum

rather than intrinsic spin, we would expect spinning gyroscopes to have a weight dependent on S.4"g/|gL where g

is the acceleration of gravity. Although an effect like this (for one sign of S,4g) has been reported [37], more ac- curate subsequent experiments [38-40] found a null re- sult (an earlier experiment on the intrinsic spin of the neutron was reported in Ref. [21]). The authors of Ref.

[40] found Am < 9x 10"** kg for a 0.143 kg rotor whose

angular momentum was approximately equal to 7.2 X \0^^h. If we assume a gyroscopic weight is due to Vab,

and >.«»/? Earth, then the force derived from Eq. (3) gives

Z)< 4.0x10"^ kg"' for the experiment of Ref. [40]. In our experiment, if we consider the orbital motion of the unpaired neutron in ^Be to act like a gyroscope with orbital angular momentum L, we obtain Z)<5xlO~'

kg"', about 10^ times more stringent a limit on an S^-g/|g| effect for angular momentum [4]. From Refs. [37-40], for our experiment, we might expect to replace

S^ in Eq. (1) by wneutron'' eqW — L/req, whcrc Teq is defined

in Ref. [37] (taken to be 10"'^ cm here). If we assume

A.#»/?,f/ in Eq. (3), then the resonance test reported here

is approximately lO'" times more sensitive than the ac- celeration test of Nitschke and Wilworth [40,41].

It is interesting to compare the sensitivity of resonance experiments (such as the type of experiment reported

here) with acceleration tests. From Eqs. (3) and (4), we

calculate the accelerations ao = — Of^/d^)/^^ and ar = — idEA/dd)/mA, where m^ is the unit of mass associ- ated with each spin in A. If we use parameters of the

current resonance and acceleration [16] experiments, the

limits placed on C^ and C^ are roughly equal when \^ is on the order of the size of laboratory experiments. Reso- nance (acceleration) tests are more sensitive for larger

(smaller) values of X^. Significant improvements in the sensitivity of the 'Be"*" experiment could be expected. For 100% detection

efficiency [34], a resonance time Tn (time taken to drive

the vo transition), A^ stored ions, and total measurement

time T, the statistical uncertainty in a frequency measurement is Sv^(47r^NT/tT) "'^^ [34]. In future experiments A' = 10^, T/? = 100 s, and t ~30 days do not seem

unreasonable, whence 5vo=3xlO~^ Hz. In the 'Be"*" experiments, the pressure shift can be made negligible by

using cryogenic pumping. By operating on other field- independent transitions in 'Be"*" or other ions, we could

make |Acts.. | == I for the electron and be much more sensitive to anomalous couplings to electron spin. The characteristic dimensions of the ion experiments can also be

made very small and therefore can investigate relatively short ranges (A.« < 1 mm).

The authors gratefully acknowledge the support of

ONR and AFOSR. We thank D. Bartlett and J. Gilligan

for helpful suggestions on the manuscript.

^^^Present address: Department of Physics, University of Texas, Austin, TX 78712.

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(1991)1.

Il9l R. C. Ritter el al., Phys. Rev. D 42, 977 (1990).

[20] D. Graham and R. Newman, in Proceedings of the Elevenlh International Conference on General Relativity

and Gravitation, Stockholm, Sweden. 1986, edited by M.

A. M. MacCullum (Cambridge Univ. Press, New York,

1987), p. 614.

[21] J. W. T. Dabbs, J. A. Marvey, D. Paya, and M. Horst- mann, Phys. Rev. 139. B756 (1965).

[22] G. E. Velyukhov. Zh. Eksp. Teor. Fiz. Pis'ma Red. 8. 372

(1968) [Sov. Phys. JETP Lett. 8. 229 (1968)1.

[23] B. A. Young, Phys. Rev. Lett. 22, 1445 (1969); B. V.

Vasil'ev, Zh. Eksp. Teor. Fiz. Pis'ma Red. 9, 299 (1969)

[Sov. Phys. JETP Lett. 9, 175 (1969)1.

[24] D. J. Wineland and N. F. Ramsey, Phys. Rev. A 5, 821

(1972).

[25] N. F. Ramsey, Physica (Amsterdam) 96A. 285 (1979).

[26] E. B. Aleksandrov, A. A. Ansel'm, Yu. V. Pavlov, and R. M. Umarkhodzhaev, Zh. Eksp. Teor. Fiz. 85, 1899

(1983) ISov. Phys. JETP 58, 1 103 (1983)1.

[27] A. A. Ansel'm and Yu. I. Neronov. Zh. Eksp. Teor. Fiz. 88, 1946 (1985) [Sov. Phys. JETP 61, 1154 (1985)1.

[281 C.-M. Hsieh et ai. Mod. Phys. Lett. A 4, 1597 (1989); Y.

Chou. W.-T. Ni. and S.-L. Wang, ibid. 5, 2297 (1990).

From the first paper we estimate /n-i— 800 u (see discussion in text about acceleration tests). [29] Preliminary reports of our work have been given in Proceedings of the Twelfth International Conference on General Relativity and Gravitation, Boulder, Colorado, 1989 (to be published) and in Proceedings of the Elevenlh Moriond Workshop on Tests of Fundamental Laws in Physics, Les Arcs, Savoie, France, 1991 (to be published).

[30] B. Meckel et ai, in Proceedings of the Eleventh Moriond

Workshop on Tests of Fundamental Laws in Physics

(Ref. [29]) (final results of this work should appear at the same time as our report). See also S. K. Lamoreaux, J. P. Jacobs, B. R. Meckel, F. J. Raab, and E. N. Fortson. Phys. Rev. Lett. 57, 3125 (1986).

[31] Moody and Wilczek's derivation (Ref. [7]) of the dipolemonopole and dipole-dipole potentials assumed that parti- cles A and B have spin f . Mowever, by use of the Wigner-Eckart theorem, we can show that for the conditions discussed in our paper Eqs. (1) and (2) are also valid for systems with higher spins, such as the spin- \ ''Be nucleus.

[32] See recent reviews: M. S. Turner, Phys. Rep. 197. 67

(1990); G. G. Raffelt. ibid. 198. I (1990).

[33] We designate the states using the low- (magnetic-) field notation where the total atomic angular momentum F

and projection ntr along the B-field axis are good quan- tum numbers. At higher fields, mr is still a good quan- tum number.

[34] J. J. Bollinger, D. J. Meinzen, W. M. Itano, S. L. Gilbert,

and D. J. Wineland, IEEE Trans. Instrum. Meas. 40, 126

(1991); D. J. Wineland. J. C. Bergquist. J. J. Bollinger,

W. M. Itano. D. J. Meinzen. S. L. Gilbert. C. M. Manney.

and M. G. Raizen. IEEE Trans. Ultrasonics, Ferroelectrics, and Frequency Control 37, 515 (1990).

[35] The 'Be"*" ions are located 1.5 m above a concrete foundation which rests on the ground. To a good approximation

the experimental apparatus closer than 1.5 m to the ions

is placed symmetrically around them. If a positive result was observed, the effects of the surrounding geometry

would have to be studied in detail. [36] J. J. Bollinger. J. D. Prestage, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 54. 1000 (1985).

[37] H. Mayasaka and S. Takeuchi, Phys. Rev. Lett. 63. 2701

(1989).

[38] J. E. Faller, W. J. Mollander. P. G. Nelson, and M. P. McMugh, Phys. Rev. Lett. 64, 825 (1990).

[39] T. J. Quinn and A. Picard. Nature (London) 343, 732

(1990).

[40] J. M. Nitschke and P. A. Wilmarth. Phys. Rev. Lett. 64, 2115 (1990).

[4ll One might assume other forms of interaction such as (for A.,— oo) |/,fl«Srf/r In this case also, the resonance

tests are more stringent.

1738

TN-109
ATOMIC PHYSICS TESTS OF NONLINEAR QUANTUM MECHANICS*

J. J. Bollinger, D. J. Heinzen, Wayne M. Itano,

S. L. Gilbert, and D. J. Wineland

National Institute of Standards and Technology,

Boulder, Colorado 80303

ABSTRACT

Atomic physics experiments which test a nonlinear generalization

of quantum mechanics recently formulated by Weinberg are described.

The experiments search for a dependence of hyperfine transition

frequencies or nuclear spin precession frequencies on the relative

populations of the hyperfine or nuclear spin states. The experiments

set limits less than 10 /iHz on the size of the possible nonlinear

contributions to these frequencies. In some cases this can be

interpreted as a limit of less than ~ 10"^^ on the fraction of

binding energy per nucleon that could be due to a nonlinear

correction to a nuclear Hamiltonian. The possibility that a nonlinear addition to quantum mechanics violates causality is

discussed.

INTRODUCTION

Quantum mechanics has survived numerous tests since its inception

in the 1920' s. Many of these tests can be regarded as tests of a specific quantum-mechanical theory as well as tests of the basic

formalism of quantum mechanics. For example, the fact that the

measured energy levels of the hydrogen atom are in agreement with the

predictions of quantum-mechanical theory is a test of both the

formalism of quantum mechanics and the accuracy of the Hamiltonian

for the hydrogen atom. However, this only tests quantum mechanics at

the level we know the Hamiltonian is accurate. Because of its

importance in modern physics, quantum mechanics should be tested

independently of and, if possible, more precisely than any particular

quantum-mechanical theory. Surprisingly few such tests exist. Some

examples are the experiments '^' which have conclusively ruled out

hidden variable theories in favor of quantum mechanics. These

experiments, however, do not provide a test of quantum mechanics

itself to much less than 1%.

The lack of precise, independent tests of quantum mechanics is at least partially due to the absence of generalized versions of quantum

mechanics with which to plan or interpret an experimental test. Generalized versions of quantum mechanics which have a logically

consistent interpretation have been difficult to construct.

Recently, however, Weinberg has developed a formalism for introducing

nonlinear corrections to quantum mechanics.'^'''' The size of the

nonlinear corrections is determined by a parameter c . In the limit

that e = 0, Weinberg's nonlinear generalization reduces to ordinary

Contribution of the U.S. Government; not subject to copyright.

461

in Atomic Physics 12 . (proc. of the 12th International Conference on Atomic Physics), ed. by Jens C.

Zorn and Robert R. Lewis, (Amer. Inst, of Physics press. New York, 1991), p. 461.

TN-110
462 Tests of Nonlinear Quantum Mechanics

linear quantum mechanics. This paper includes a discussion of atomic

physics experiments'''"^' which search for nonlinear corrections

permitted by Weinberg's formalism. The experiments completed to date

have found null results which put upper bounds on the parameter €

.

In some cases these bounds are 26 orders of magnitude smaller than

the size of the ordinary, linear quantum-mechanical terms.

The following simple discussion gives an idea of how the atomic

physics tests work. Consider a two- level system with energy

eigenstates |k>, k = 1,2 and energies fii^ , with w^ > 1^2 "^^^ time

evolution of these eigenstates is given by e ^ l^^- Now suppose at time t = 0, the state

|V'(0)> = (l-a)^|l> + a'«|2> (1)

is formed from a superposition of the two energy eigenstates. The

linearity of quantum mechanics (specifically, the linearity of the

Hamiltonian) requires that the time evolution of ]V'(0)> be given by

|V(t)> = e'^'^i^[(l-a)^|l> + a'»e^^'^i""2)t|2>]. (2)

Except for an overall phase factor, the time evolution of the

coherent superposition is determined by the frequency w = Wj^ ' ^2 < which is independent of the parameter a, the probability of finding

the system in the eigenstate |2>. This result can be stated

graphically by using the mathematical equivalence between a two- level

system and a system with a spin-4 particle in a magnetic field.'®'

The eigenstates | 1> and | 2> correspond to spin states antiparallel

and parallel to the magnetic field. The state |V>(0)> corresponds to

a state where the spin is tipped by an angle d with respect to the

magnetic field, where cos(5/2) = a^ . The frequency w is then the

precession frequency of the spin about the magnetic field (or z axis). In the language of the equivalent spin-4 picture, the

linearity of quantum mechanics requires that the precession frequency

Wp be independent of the tipping angle 6 of the spin.

A test for a nonlinear addition to quantum mechanics can therefore

be made by searching in a coherent superposition of states for a dependence of the frequency u> on the state probabilities (the

parameter a). After a brief discussion of Weinberg's formalism, we will describe the different experimental tests. They include an ion

trap experiment'*' which uses a ground-state hyperfine transition in

^Be* , two different experiments'^''' which use optical pumping

techniques to polarize Ne and Hg nuclei in gas cells, and an experiment'^' which uses a hydrogen maser. The first three

experiments may be viewed as tests of the linearity of quantum

mechanics in a nuclear system. The hydrogen maser experiment may be

viewed as a test of the linearity of quantum mechanics in an atomic

system. In addition to the experimental tests, Weinberg's work has

stimulated theoretical discussions'^"^*' on the possibility of a nonlinear addition to quantum mechanics. In particular, a nonlinear

addition to quantum mechanics may violate causality and make it

TN-111
J. J. Bollinger et al. 463

possible, at least in principle, to have arbitrarily fast

communications .' ^ ^ ^^ ' ^*' We summarize this discussion and give an example of how a nonlinear correction to quantum mechanics in the Be*

experiment could be used to communicate faster than the speed of

light.

WEINBERG'S NONLINEAR QUANTUM MECHANICS

In order to help motivate tests of nonrelativistic quantum

mechanics, Weinberg has developed a formalism' ^ ^

' which generalizes

quantum mechanics by permitting possible nonlinear terms. In this

formalism, observables are represented as operators on a Hilbert

space of states just as in ordinary quantum mechanics. The

difference with ordinary quantum mechanics is that the operators no longer need be linear. In order to have a sensible physical

interpretation, Weinberg requires that any nonlinear additions do not

violate Galilean invariance. He shows that there is a possible class

of such additions to the Hamiltonian describing the internal degrees

of freedom of a particle with spin. This means that the time

evolution of the wave function ^(t) is no longer given by a linear

Schrodinger equation. Instead it takes the form

dt dip^ ^

-^

where \j\ = <k|\6> is the amplitude of state |k> (k = -I, -I+l,..,

I-l, I for a particle of spin I) and the Hamiltonian function h(^,V'*)

is a homogeneous function of V> and ^* [h(AV>,V'*) = h(V',A^*) = Ah(V',V'*)

for any complex X] . Homogeneity guarantees that if V'(t) is a solution of Eq. (3) then AVi(t) is also a solution representing the same physical state. This ensures the proper treatment of physically

separated and uncorrelated systems.''" In general, h(V',V'*) is not

bilinear as it would be in ordinary quantum mechanics. A possible

form of h(V),V'*) which Weinberg has investigated is

h(^,\t*) - I

r

= 1 v^H,,v>, + ^ I ^v-:g,,„„0„^„

ki " kimn

(4)

where the first term is the linear term which describes ordinary

quantum mechanics, the second term is the lowest order nonlinear

addition, and n = ^i^* ^ Galilean invariance puts restrictions on

the coefficients Hjj ^ , Gj,,„,„ If we ignore translational degrees

of freedom, Galilean invariance reduces to rotational invariance.

The Hamiltonian function must transform as a scalar under rotations.

The only scalars that can be formed from the two components of a spin-h wave function are functions of the norm n. Therefore, in the

absence of external fields, the potential nonlinear contributions of

Eq . (4) to the Hamiltonian function of a spin-4 particle are functions of the norm and have no effect other than to set the zero

TN-112
464 Tests of Nonlinear Quantum Mechanics

of energy for the spin. Weinberg shows that in the presence of

external fields (where the G^,„n transform according to the rotation

properties of a tensor), a spin-4 particle can have a first-order

quadrupole interaction. Particles with spin I > 1 can have a nonlinear contribution to the Hamiltonian function even in the

absence of external fields. The lowest order nonlinear addition

h^^(s h^/n) for a spin I = 3/2 can be written as'^'

Ki =

T: (17603/2^--^ - y2V^V^|2 + |3i^3/20.3/2 - ^^^|^.^ 2 n

+ \J(>i>-3,2^k - 720-%^--^ I'). (5)

For a nucleus with I = 3/2, Eq. (5) gives a possible nonlinear

addition to the nuclear Hamiltonian. The parameter e gives the

strength of the nonlinear term. Weinberg does not make a prediction

for a value of e ; he only constructs a theory in which e ?= is

allowed.

As a simple example, consider the two-level system consisting of

the mj = -3/2 and -H states of a spin I = 3/2 nucleus. Let | 1> =

|mj = -3/2> and 1 2> = |mj = -h> and hQ(i/),!/i*) = ^w^V^V^* + ftw2V'2V'2* denote the bilinear Hamiltonian function for the system. (Suppose

the degeneracy of the levels is split by a magnetic field and that no population exists in the +h and +3/2 sublevels.) The Hamiltonian

function for the lowest order nonlinear contribution given by Eq. (5)

reduces to

h„, = n2ea2, (6)

where a = |V>2l^/n is the probability of finding the system in 1 2> and

is identical to the parameter a in Eqs . (1) and (2). From Eq . (3)

the time evolution of the superposition state |i/)(0)> from Eq. (1) is

|V.(t)> = e'^'^i^^)" [(l-a)Ml> + a^e^^"^^^^-"2(a))t|2>]_ (7

where w^ (a) = Uy-2(e/ft)a^ and (jo^ia) = <ji^-l{(./fi)a^ + 4(£/fi)a. The

coherence frequency cj = Wj^ (a) - Wj (a) now depends on the state

probabilities according to

'^p = '^0 -^(e/^)a

(8) = Wo-4(€/fi)cos2(fl/2),

where Wg s w^ - u>^ is the transition frequency in the absence of

nonlinearities . In the language of the equivalent spin-h system, the

precession frequency w of the spin now depends on the tipping angle

Q between the spin and the magnetic field. The equation of motion

for a two-level system with the nonlinear addition given by Eq . (6)

is formally identical to that of Jaynes' neoclassical theory of the

interaction of the electromagnetic field with atoms .' ^ * ^^

' However,

the origin of the nonlinearity is quite different. In Jaynes' theory

TN-113
J. J. Bollinger et al. 465

the nonlinearity is due to a postulated radiation reaction of the

atomic dipole on itself.

The frequency w is the "instantaneous" transition frequency of

the two-level system. That is, if an external field of frequency w

is used to induce transitions between the two states, this transition

rate is maximized for w = u> . Suppose all the population is

initially in state |1>. Application of an external field with

frequency w = Wg = w (tt) starts driving population from state 1 1> to state |2>. If the Rabi frequency Q due to the applied field is much

smaller than the nonlinear correction e [U « (e/ft)], then the

population can never be completely inverted. This is because the

transition frequency given by w is chirped as the state of the

system evolves. Consequently, unless the frequency of the applied

field is similarly chirped, a transfer of all the population to state

I 2> never occurs. Weinberg used this fact, along with the

experimental observation that nuclear magnetic resonance transitions

in ®Be"^ have been driven in times as long as 1 s to set a limit of

- 10"^^ eV on the magnitude of a nonlinear correction to the energy

of the ^Be nucleus.'^'

Very few proposals for nonlinear terms in quantum mechanics have

been made and experimentally tested. One proposal that has been

experimentally tested is the addition of a logarithmic term

-bV'(x)in| ViCx) 1^ to the one-particle Schrodinger equation discussed by

Bialynicki-Birula and Mycielski . ' ^ '

^ This nonlinear addition can be

derived from a Hamiltonian function, but the Hamiltonian function is

not homogeneous. Physically, it predicts that the time evolution of

a wave function depends on its normalization, in contrast to Weinberg's theory. Nevertheless it gives a correct treatment of

physically separated systems. Shimony'^®' suggested an experimental

test based on neutron interferometry for this nonlinear term. Experiments' ^^ ^° ' have put a bound of 3 x 10"^^ eV on the magnitude

of the parameter b.

EXPERIMENTAL TESTS

Weinberg's generalization of quantum mechanics has motivated

recent experimental tests which improve upon the limit set by

Weinberg by more than 5 orders of magnitude. The first experiment to be described here was done at the National Institute of Standards and

Technology and used ion storage techniques to measure the frequency

of a hyperfine transition in ®Be* to high precision.'''' Trapped ions

provide very clean systems for testing calculations of the dynamics

of quantum transitions. The ions can be observed for long periods,

relatively free from perturbations and relaxations. Their levels can

be manipulated easily with rf and optical fields. This has led to their use in observing quantum effects such as quantum jumps, photon

antibunching, sub-Poissonian statistics ,' ^^

' and the quantum Zeno

effect' ^^' . In addition, trapped ions have been used to test the

isotropy of space' ^•'^ and search for a spin-dependent fifth

force. '^'' For a test of nonlinear quantum mechanics, we searched

TN-114
466 Tests of Nonlinear Quantum Mechanics

for a state population dependence (a 9 dependence) of the (eDj , nij ) = (-1/2, +1/2) - (-3/2, +1/2) hyperfine transition at - 303 MHz in the

ground state of ^Be* (see Fig. 1). At a magnetic field B of

0.8194 T, this transition, called the clock transition, depends only

quadratically on magnetic field fluctuations. With | 1> =

1-3/2, +l/2> and | 2> = |-l/2, +l/2>, the lowest order nonbilinear

addition to the Hamiltonian function of the free ^Be* nucleus for the two states is given by Eq. (6). Equation (8) then gives the

dependence of the precession frequency c; on the tipping angle 6, which describes the relative amplitude admixture of the two states.

A search for a nonlinear addition to quantum mechanics was made by

measuring cjp at two different tipping angles.

Clock <""• '"J^ transition

1/2)

1/2) (1/2, 1/2) F=1

^^..^^'^^ (3/2, 1/2)

~3^(-3/2, 1/ -iri^(-1/2, 1/

F=2

(3/2, -1/2)

a/2, -1/2)

(-1/2, -1/2)

(-3/2, -1/2)

Fig. 1. Hyperfine energy levels (not drawn to scale) of the ^Be*

2s ^Sjj ground state as a function of magnetic field. At B = 0.8194 T

the 303-MHz clock transition is independent of magnetic field to first order. (From Ref. 4)

Between 5000 and 10 000 ^Be* ions and 50 000 to 150 000 ^^Mg* ions were simultaneously stored in a cylindrical Penning trap'^^' with B = 0.8194 T under conditions of high vacuum (- 10"® Pa). To minimize

second-order Doppler shifts of the clock transition, the ^Be* ions were cooled to less than 250 mK by the following method. The ^^Mg*

ions were directly laser cooled' ^^' and the ^Be* ions were then

sympathetically cooled'^'' by their Coulomb interaction with the cold

Mg* ions. Narrow-band 313-nm radiation was used to optically pump

and detect the ^Be* ions. With the 313-nm source tuned to the

2s 2Sj,(3/2, 1/2) to 2p 2p3^2(3/2, 3/2) transition, 94% of the ^Be*

ions were optically pumped into the 2s ^S^(3/2, 1/2) ground

state. ' ^® ^^ ' The 313-nm source was then turned off to avoid optical

pumping and ac Stark shifts.

The clock transition was detected by the following method. After

the 313-nm source was turned off, the ions in the (3/2, 1/2) state were transferred to the (1/2, 1/2) state and then to the (-1/2, 1/2)

state by two successive rf ?r pulses. The clock transition was then

TN-115
J. J. Bollinger et al. 467

driven by Ramsey's method of separated oscillatory fields' ^°' with rf

pulses of about 1-s duration and a free-precession time on the order

of 100 s. This transferred some of the ions from the (-1/2, 1/2)

state to the (-3/2, 1/2) state. Those ions remaining in the (-1/2,

1/2) state were then transferred back to the (3/2, 1/2) state by

reversing the order of the two rf n pulses. The 313-nm source was then turned back on, and the population of ions in the (-3/2, 1/2)

state was registered as a decrease in the ^Be* fluorescence, relative

to the steady-state fluorescence, during the first second that the

313-nm source was on. [The optical repumping time of the ions from

the (-3/2, 1/2) state to the (3/2, 1/2) state was an order of

magnitude longer than this.]

For the test of nonlinearities , Ramsey's method with unequal rf

pulses was used to drive the clock transition and measure w for

different values of . First an rf 6 pulse of duration Tg was applied. This prepared the ions in a coherent superposition of

states I 1> and | 2> of the clock transition given by Eq. (7) for a particular value of a = cos^ (6/2) . After the rf 9 pulse, the ions

freely precessed for a time T. This was followed by an rf n/l pulse,

coherent with the first pulse, which completed the Ramsey excitation.

In the limit that T » r^ , T » t„

^ 2 i the Ramsey line shape

(specifically, the number of ions remaining in the state | 2> as a function of the rf frequency w in the Ramsey excitation) is

15000

COc

0).

(0

So

CO. o

3

(J

-0.005 0.000

frequency - 303 016 377.265

(Hz)

0.005

Fig. 2. Ramsey signal of the ^Be* clock transition with T = 150 s and 6 = n/2. The data are the result of one sweep (that is, one measurement per point). The sweep width is 9 mHz . The dots are experimental and the curve is a least-squares fit. The signal-tonoise ratio is limited by the frequency stability of the reference

oscillator. The full-width-at-half -minimum frequencies are indicated

by i/+ and u. . (From Ref . 4)

TN-116
468 Tests of Nonlinear Quantum Mechanics

proportional to

1 -sin^cosl [w-ojp (d) ]T) , (9)

where <^p(0) is given by Eq . (8). The center frequency of the Ramsey

line shape is the precession frequency w (5). Figure 2 shows a Ramsey signal obtained with T = 150 s and 8 = n/2.

Ramsey-signal measurements were taken near both of the fullwidth-at-half -minimum frequencies u>^ = 2i^v^ and w. h 2-ku. , where t/^. and 1/. are indicated in Fig. 2. The difference in the measured

signal strengths on either side of the line center was used to electronically steer' ^®' the average frequency of a synthesized rf source to w (5). Eight pairs of measurements were taken with an angle 8f^ = 1.02 rad followed by eight pairs of measurements with an angle 8^ = 2.12 rad. This pattern was repeated for the length of an entire run as indicated in Fig. 3. The average frequency of the

synthesizer for 8 = 8 p^ was then subtracted from the average frequency

of the synthesizer for 8-8^. Runs were taken with free-precession

periods of T = 30, 60, and 100 s. A weighted average of the

synthesizer frequency differences for 8=8^ and 8=8^ from about

110 h of data gave a possible dependence of the precession frequency

on 8 of [WpC^s) -'*'p(^A)]/2'f = 3.8 ± 8.3 /iHz . From Eq . (8) a value

for the parameter e of i/{lnh) = 1.8 ± 4.0 /iHz is obtained. The

uncertainty is expressed as a 1 standard deviation.

200

-200

1x10"^ 2x10"^ 3x10^

time (s)

4.x 10"^

Fig. 3. ^Be'^ precession frequency Wp(t) as a function of time for a single run with T = 100 s. The periods A(B) during which the initial

Rabi pulse created a mixed state with angle 5^ = 1.02 rad {B^ =2.12

rad) are indicated. (From Ref . 4)

TN-117
J. J. Bollinger et al 469

This sets an upper limit of |e| < 2.4 x lO'^o gV (5.8 ^Hz) for a nonlinear contribution to the ^Be* nuclear Hamiltonian. This is less

than 4 parts in 10^^ of the binding energy per nucleon of the ^Be*

nucleus. Our experimental result is limited by statistical

fluctuations due to the frequency instability of the reference

oscillator used with the synthesizer (typically a commercial Cs atomic clock) . The largest known systematic errors of our measurement of Wp(S) are an apparent, surprisingly large pressure

shift' ''^' of ~ 30 mHz and a second-order Doppler shift of ~ 3 ^Hz due

to the ExB rotation of the ion cloud in the trap.'^^^ We believe

both shifts are constant to better than 1 ^Hz over the time required

to make a frequency difference measurement (about 40 min)

.

Experiments'^''' with ^^Ne and ^°^Hg nuclei have produced

comparable results to the ^Be* experiment described above. All three

experiments use spin- 3/2 nuclei to test the linearity of quantum

mechanics. In the ^^Ne and ^°^Hg experiments, optical pumping

techniques are used to polarize the nuclear spins of an atomic vapor

along a weak magnetic field. Let fiw^ , k = +3/2, +1/2, -1/2, -3/2

denote the energies of the nuclear spin levels. In a coordinate

system with the z axis along the polarization axis, the density

matrix p describing the spin- 3/2 state populations is diagonal and

can be expressed' ^ ^' in terms of irreducible spherical tensors T^q

3 according to p s S Plo'^lo- '^^® multipole polarizations

Plo - Tr(pTLQ), L = 0,1,2,3 (the monopole , dipole, quadrupole, and

octupole atomic polarizations) are given by

Poo = ^(<IV'3/2l'> + <l^^%l'> + <|V'-1.|'> + <|V'-3/2l'>).

PlO - 275 '3<l^3/2l'> + <IV'%l'> - <IV>-^|'> -3<|V-3/2l'>).

(10)

P20 = ^(<IV'3/2l'> - <IV'J'> - <\^-^\^> + <|V'-3/2l'>).

^30 = YJl <<l'/'3/2l'> - 3<|V'j.|'> + 3<|V.J2> - <|\/'-3/2l'>'.

where < > denotes the ensemble average over the spin- 3/2 state

populations. A coherent superposition of all of the nuclear spin

levels is created by tipping the polarized nuclear spins with respect

to the magnetic field by an angle 9. The transverse polarizations of

the nuclear spins then precess about the magnetic field at frequencies determined by the Zeeman frequency splittings.

Measurement of these free precession frequencies as a function of the

state populations provides a test of the linearity of quantum

mechanics. The Zeeman frequencies co^ , i = 1,2,3 are'^^'

wi = W3/2 - w^ = 27r(D + Q + Y2 0).

W2 = w^ - w.^ = 27r(D + J2 °^' ^^^^

TN-118
470 Tests of Nonlinear Quantum Mechanics

13

W3 H w.^ - w.3/2 = 27r(D - Q + Y2 °) >

where D, Q, and are the dipole, quadrupole, and octupoLe

contributions to the splitting of the Zeeman levels. If the Zeeman

frequencies are resolved (for example, if Q' ^ is less than the

relaxation time of the coherent superposition), Weinberg's nonlinear

corrections to these frequencies from Eq. (5) can be expressed in

terms of PiQ(d), the multipole polarizations along the precession

axis, where Pi^q(6) = Pi^QF^(cosd) and P^^icosd) is the L'-'' -order

Legendre polynomial, according to'^^

«wi = f

[- 75 PioC^'.t) + P2o(9,t) + -j^ P3o((?,t)J

^"^2 = J [

- 7I PioC^.t) - yf P3o(«.t) ) (12)

*'^3 = f [

- 75 Pl0(<'.t) - P20(S,t) + yl P3o((?,t)j.

The monopole component of the polarization has been omitted because

it is unimportant to the discussion of these experiments. The

variable t explicitly denotes the dependence of the multipole

polarizations on time due to relaxation.

The precession frequencies in these experiments are relatively low

(< 1 kHz) compared to the - 300 MHz frequency in the ^Be* experiment.

In addition, the number of atoms used in these experiments is very

large (10^'' - 10^^ atoms) compared with the number of ions (- 5000)

used in the ^Be* experiments. As a result, the second order Doppler

shifts in these experiments are negligible and the signal- to-noise

ratio is typically high enough to measure the precession frequencies

to much better than 1 ^Hz . Time -dependent , systematic frequency

shifts, however, typically limit the precision of the precession

frequency measurements. In addition to the test of quantum mechanics

discussed here, experiments like these have been used to test the

isotropy of space' '2" •^ '' and to search for T-vioiating

interactions .

' ^ ^

'

In the 2°^Hg experiment done at the University of Washington,'^'

circularly polarized light from a ^^®Hg discharge lamp is used to optically pump about 10^^ ^^^Hg atoms contained in a quartz cell and

polarize the ^^^Hg nuclei along a few milligauss (1 mG = 10"^ T)

magnetic field collinear with the light. A polarization of 45% of

the value for a maximally polarized four- level system (all atoms in

the mj = +3/2 or -3/2) was obtained. After optical pumping, the

polarized nuclear spins are adiabatically rotated by an angle 4> by

slowing rotating the magnetic field. The magnetic field is then

suddenly changed to a direction perpendicular to the incident light

and at an angle 8 = 90° - 4> with respect to the polarized spins. By

changing the angle 8 the spin polarization moments given in Eq . (12)

could be varied. The 2 Hz precession of the spins about the magnetic

TN-119
J. J. Bollinger et al. All

10 |-n—I—r-1—r—'

^~~^—i

I i i—i

i I ' '

'—""1—"""^—'—' I

'

'—'

i I

' ' ' ^

/_\ _io I '

I I I I I 'I I I I 'I I I I I I I I

'

I I I I I—I I I I I I I I ^ ' •" 20 30 40 50 60 70 80 TIME (sec) 10

6E5

4E5

z3

2ES

(b)

T 1 1 r

L....1....1....1. 1.6

T 1 1 1 1 1 1 1

1 1 1 r

{u^_+u^/Z-n

(tJi+Uo)/2"

((^t-u),)/2Tr

U,

+ 3/2

All.

-1/2

-3/2

1.8 FREQUENCY (Hz)1.9

Fig. 4. (a) Typical 2°^Hg free-precession signal for ff = 90°. The

signal-to-noise is photon shot-noise limited (roughly 500 to 1 here),

(b) A fast Fourier transform power spectrum of the data in (a) . Also

shown are the relevant ^°^Hg Zeeman levels and the parametrization of

the three coherence frequencies used in Ref. 7. (From Ref. 7)

TN-120
472 Tests of Nonlinear Quantum Mechanics

field is measured from a modulation of the light transmitted through

the cell for about 240 s which is roughly 2.5 coherence lifetimes.

The three precession frequencies are resolvable due to a Q = 50 mHz quadrupole interaction between the ^°^Hg spins and the cell wall. A

typical free precession signal and its fast Fourier transform are shown in Figs. 4a and 4b. In this experiment, the difference in the

center frequency and the average of the outer two frequencies, u>2 - (wi + aj3)/2, was fitted for a possible dependence on the dipole and

octupole polarization given by Eq. (12). This combination of

frequencies is sensitive only to the octupole contribution [0 in Eq

.

(11)] of the Zeeman splitting. This provides good isolation from

possible systematic shifts; the only known octupole shift is due to a misalignment of the cell axis with the magnetic field and was measured to be small [0 = 2.5 ± 0.5 mHz]. Furthermore, this octupole

shift is independent of atomic polarization as well as precession

angle 8. The result of several thousand 240-s runs taken at different precession angles 9 between 15° and 45° is t/{lifh) = 1.1 ±

2.7 ^Hz . Thus the fraction of the binding energy per ^°^Hg nucleon

which could be due to nonlinear quantum mechanics is less than

2 X 10-27 (3 8 ^Hz).

In the 2^Ne experiment done at Harvard University,'^' ^^Ne and ^He

(about 2 X 10^^ atoms of each) are contained in a sealed glass cell

and polarized along a static . 3 mT magnetic field by spin exchange

with optically pumped Rb. The ^^Ne spins are then tipped by ^ = 20°

with respect to the static magnetic field axis by a pulse of

oscillating magnetic field at the precession frequency. The free

precession of the ^^Ne spins is then measured over a period of 4.5 h

by monitoring the transverse magnetization with the voltage induced

in a pickup coil. The ^^Ne spins remain coherent over the entire

measurement. The - 995 Hz ^^Ne precession frequency is mixed with

the ^He precession frequency (- 9600 Hz) divided by 9.649 to produce

a - 1/60 Hz carrier frequency which is relatively free of the effects

of magnetic field fluctuations. Figure 5 shows the ^^Ne precession

data for one 4.5-h run. The three precession frequencies are resolvable due to the interaction of the electric quadrupole moment

of the ^^Ne nucleus with the glass cell wall. The quadrupole

contribution to the Zeeman splitting is Q - 240 /iHz . Because the

dipole component of the frequency splitting is sensitive to spinexchange shifts, originally the quadrupole component of the Zeeman

splitting was used to test the linearity of quantum mechanics.'^'

Specifically the difference frequency w^ - Wj was extracted from the

precession frequency data and fit for a possible dependence on the

quadrupole polarization. The quadrupole polarization would decay

during a run due to relaxation. The result of five 4.5 h runs is

£/(2jrA) = 12 ± 19 mHz. The upper limit on \e/i2ifh)\ of 31 ^l)^z is 1.6 X 10"^^ of the binding energy per nucleon of the ^^Ne nucleus. This

measurement is limited at least in part by the small quadrupole

polarization (pjo ^ 0.025). The quadrupole polarization is small

since it arises only from quadrupole relaxation in the presence of

spin exchange, where spin exchange is a dipole interaction that

produces dominantly dipole polarization. Recently, these ^^Ne data

TN-121
J. J. Bollinger et al. 473

B -

2 4 6

time (hf)

2 -I

w OH

ff -" H;=::l!l!i!!.,,

'^^iiiiiiiiii''^''"''

n 1 1 1—

8 10 1 ^ 1,4 time (hr)

2 -I

ra

2-'-

2 -I

re

1 r 16 18 2 time (hr)r

2 2

2-1

-V.

2 4 —1 r~

2 6 2.8

time (hr) —

r

3

2 -I

re

2 -I

^0^fmm^fmi0^ 78

~r"

3 2

1 r—

3 4 3 6

time (hr)

^^nmmm'^

3 8 —1 1 4 4 2 time (hr)

4 4

Fig. 5. Data for coherent free precession of ^^Ne over 4.5 b. Each

panel represents a 45-min measurement of the beat frequency between

the ^^Ne signals and a reference frequency derived from freely

precessing 'He. •'He is not coherent from panel to panel; ^^Ne is.

(From Ref. 5)

TN-122
474 Tests of Nonlinear Quantum Mechanics

have been reanalyzed'-'^' by searching for a dipole and octopole

polarization-dependent shift in the frequency component 0, similar to the technique developed by the University of Washington group in

their ^°^Hg experiment. The new limit on a possible nonlinear

correction to the ^^Ne nuclear Hamiltonian of \€\/(2nfi) < 1.5 /iHz is

about a factor of 20 improvement over the previous limit.

In addition to these two experiments, a similar experiment'^®' is

planned which uses radioactive spin-3/2 ''^Ar nuclei (ti^ = 35 days)

polarized by spin exchange with optically pumped Rb . The

polarization of the ^^Ar is detected by observing the asymmetry in

the emission of the internal bremsstrahlung accompanying the electron

capture decay of '^Ar to •'^Cl.

Hydrogen masers at the Smithsonian Astrophysical Observatory have

also been used to search for a possible nonlinear correction to quantum mechanics.'®' The (F,mp) = (1,0) -» (0,0) clock transition in

a hydrogen maser (see Fig. 6) is a transition between

states of two coupled spin-4 particles. As noted earlier, a spin-H

particle cannot have a nonlinear correction to the Hamiltonian

describing its internal degrees of freedom. However, the interaction

between two spin-h particles can. Walsworth and Silvera have

determined the lowest order allowed nonlinear Hamiltonian function

for the H ground-state hyperfine structure.'-'^' The resulting

nonlinear contribution to the clock transition frequency oig of the maser is

5-0 = [^^') d^il' - l^ol^). (13)

where Vi and Vo ^^e the amplitudes for the atom to be in the F = 1, mp = and F = 0, rap =0 states respectively, the nonlinearity

parameters e^' and €g ' include a possible dependence on the atom's

state, and n is the norm. A test for a nonlinear correction to

F=l

Maser Clock Transition (1420 MHz)

F=0 :J!zi

Fig. 6. Atomic hydrogen ground state hyperfine energy levels as a function of magnetic field, for fields < 0.02 T. Energy level shifts

due to possible nonlinear effects are shown. (From Ref. 6)

TN-123
J. J. Bollinger et al. Alb

quantum mechanics was made by searching for a dependence of the maser

frequency on the relative populations of the clock states as shown in

Fig. 7. The relative population of the two states was varied by

changing the input H flux to the maser, and it was measured from the

dependence of Q^ , the line Q of the maser signal, on the

populations.'''''' Increasing the H flux decreases both Q^ and the

population difference between the states. The result of eight -50 h

runs searching for a dependence of the maser frequency on the state

populations was (e^' + (. ^

' ) / {lirh) = 1.5 ± 7.4 pHz . This limit is

comparable in magnitude to the limits placed on nonlinear effects in

a nuclear system described in three previous experiments. It sets an upper limit of 3 x 10"^^ on the fraction of the 13.5 eV electronic

binding energy that could be due to a nonlinear contribution to the

atomic Hamiltonian of the hydrogen atom.

630

10 20 30

time (hours)

Fig. 7. The raw beat frequency data from a single run (#6). Six

different H flux levels were used in this run. An overall linear

frequency drift of - 6 /iHz/day is evident in the data, due to the

settling of the materials in the resonant cavity. (From Ref . 6)

Table I summarizes the experimental results. In the absence of a model which gives a specific prediction for £, it is important to have experimental tests in several different systems. Limits have now been set at - 10 ^Hz or less in four different systems. This is

5 orders of magnitude smaller than experimental limits placed by

neutron interferometry' ^ ^ ^° ' on the size of a logarithmic addition

to the one-particle Schrodinger equation. Three of the experiments

can be interpreted as setting a fractional limit of better than 10'^^ on a nonlinear addition to a nuclear Hamiltonian. The hydrogen maser experiment sets a fractional limit of 3 x 10"^^ on a possible

nonlinear contribution to an atomic Hamiltonian. All of the

experiments are first attempts at measuring a nonlinear component of

quantum mechanics and have the potential for an order of magnitude

improvement in sensitivity.

TN-124
476 Tests of Nonlinear Quantum Mechanics

Experiment

Limit on

\e\/anft) Interpretation Reference

9 Be 5.8 /iHz

^°^Hg 3.8 /iHz

2iNe 31 mHz

1.5 mHz

8.9 mHz

nuclear

nuclear

nuclear

atomic

4

7

5 37

TABLE I. Summary of experimental tests for a nonlinear addition to quantum mechanics permitted by Weinberg's formalism. The

interpretation column gives the type of Hamiltonian which the

experiment is interpreted as testing. The limits are one standard

deviation.

THEORETICAL DISCUSSIONS

In addition to motivating experimental tests, Weinberg's formalism

for introducing nonlinear corrections to quantum mechanics has

stimulated theoretical discussions.'^"^'"' These discussions center

on whether there is any theoretical reason why quantum mechanics has

to be linear. While the discussions do not rule out the

possibility of a nonlinear component to quantum mechanics, they show

the difficulty in constructing such a theory.

A. Peres has reported'^' that nonlinear variants of Schrodinger '

s

equation violate the second law of thermodynamics, Weinberg's

response 1 10 ) was that this may be true if entropy is defined as in

ordinary quantum mechanics, but that is not an appropriate definition

of entropy in a generalized version of quantum mechanics. An

interesting question is how to generalize the concept of entropy in a generalized version of quantxim mechanics.

Recently, A. Valentini has shown' ^^' that small nonlinearities of

the SchrSdlnger equation lead to a violation of the wave-particle

complementarity in quantum mechanics. Specifically, if the

Schr6dinger equation was nonlinear, a double slit experiment could be

done where both an interference pattern and the slit the particle

went through could be measured. Valentini points out that this does

not necessarily rule out nonlinear theories, but that the wave- particle aspect of the physical interpretation of such theories must

differ from that of standard quantum mechanics.

An interesting connection between nonlinear time evolution in

quantum mechanics and causality has been discussed by

I

TN-125
J. J. Bollinger et al All

N. Gisin. ' ^ ^ ' ^^

' Gisin shows, by the example of an EinsteinPodolsky-Rosen (EPR) experiment consisting of pairs of correlated

spin-h particles traveling in opposite directions, that a nonlinear

coupling between a spin-H particle and an electric quadrupole field

(permitted in Weinberg's formalism) would enable arbitrarily fast

communications.'^^' Gisin argues that the same conclusion follows

for any nonlinear time evolution. ' ^^

' Gisin points out that there

may be ways out of this dilemma by changing the usual interpretation

of the EPR experiments. In addition, Weinberg's formalism

generalizes nonrelativistic quantum mechanics only. As a nonrelativistic theory it may not necessarily give a correct

description of causality. However, as Gisin points out, it does

appear to create a problem where before there was none.

Following Gisin' s example, we can show how a nonlinear

contribution to the ^Be* clock experiment could result in

communication faster than the speed of light. As before, let | 1> and

I 2> denote the two states of the ^Be* clock transition. Let |+H>2

and I -H>j denote the two states of a spin-h particle, say an electron, along the same quantization axis (the z axis) as the ^Be"^ system. Suppose a source of correlated states,

y2(|l> |-H>, - |2> |+H>J, (14)

is placed between two regions A and B in space. Assume the source continuously emits the correlated states with the electrons moving

toward region A along the y axis and the ^Be"^ clock (the ^Be* ion in an 0.8194 T magnetic field) moving in the opposite direction toward

region B. (If desired the ^Be"^ clock can be at rest and only the

electron moves.) In region A an observer measures the spin of the

electron in a Stern-Gerlach apparatus. In region B a different

observer drives the clock transition and then measures the population

of state |2>. Specifically, a weak rf pulse (duration r » {2-Kh) / 1 , Rabi frequency fi « e/ft) of frequency w = w {6 = n) = Wq is applied in

region B. Due to the nonlinear contribution to the clock transition

frequency, this applied field will start to drive a transition from a pure I 1> state to state |2>, but not from | 2> to |1>. After the

applied rf pulse, the population of state | 2> is measured as described in the previous section.

Suppose the observer in region A orients the Stern-Gerlach

apparatus so that the z component of the electron spin is measured.

This prepares the ^Be* clock in either the | 1> state or the | 2>

state. The rf pulse takes the | 1> state and puts it into a mixture

of I 1> and I 2> but leaves the | 2> state unchanged. Observer B will

then measure on the average that state | 2> has a population greater

than H. Now suppose the observer in region A rotates the SternGerlach apparatus and measures the x component of the electron spin.

The correlated state given by Eq. (14) can be written in terms of

I +W>j( and \-H>^, the electron spin states along the x axis, as

TN-126
478 Tests of Nonlinear Quantum Mechanics

^(|1> - |2»|-H>, - ^(|1> + |2»|+4>,. (15)

According to the standard interpretation of the EPR experiments,

when obseirver A measures the x component of the electron spin the

^Be* clock will be prepared in either the (

| 1> - \2>)/j2 state or the

( 1 1> + \2>)/j2 state. Both of these states are described by 5 = ir/2 and will not be affected by the rf pulse with frequency w = w (tt) ^

Wp(n-/2). In this case, observer B will measure that state 1 2> has a population equal to h. When observer A rotates the Stern-Gerlach

apparatus, observer B instantly sees a change in the state | 2>

population. It will take observer B a time greater than {2i^h) / i to measure this change, but since regions A and B are arbitrarily far

apart, it appears that communication at arbitrarily fast speeds is

possible

.

Recently, Polchinski' ^ *

' has investigated whether nonlinear

quantum mechanics necessarily violates causality by determining the

constraints imposed on observables by the requirement that

communication does not occur in EPR experiments. He shows that this

leads to a different treatment of separated systems than that

originally proposed by Weinberg, but that it is possible for quantum

mechanics to be nonlinear without violating causality. However, he

shows that this necessarily leads to another type of unusual

communication: the communication between different branches of the wave funct ion. '* ^

' Polchinski points out that, in effect, this means

the wavefunction is never reduced and that this may in turn lead to a dilution of any nonlinear effects.

Clearly a number of theoretical questions about nonlinear theories

of quantum mechanics remain to be fully resolved. These questions at present may not prove that quantum mechanics must be linear, but they

do point out some problems with the physical interpretation of a nonlinear theory. Experimental work to date has shown that a nonlinear component to quantum mechanics must be very small. In some

cases, the nonlinear terms must be 26 orders of magnitude smaller

than the linear terms. The theoretical discussion and the

experimental tests given here, involve nonlinear generalizations of

the Hamiltonian. A nonlinear version of quantum mechanics could

possibly take a different form. For example, Weinberg's formalism

allows the possibility that operators other than the Hamiltonian

could be nonlinear.

The NIST work is supported by the U.S. Air Force Office of

Scientific Research and the U.S. Office of Naval Research. We thank

M. Raizen for carefully reading the manuscript and P. K. Majumder, S. K. Lamoreaux, R. L. Walsworth, and T. E. Chupp for their useful

comments and help in understanding their experimental work.

TN-127
J. J. Bollinger et al 479

REFERENCES

1. A. Aspect, P. Grangier, and G. Roger, Phys . Rev. Lett. 49, 91 (1982), and references therein.

2. S. Weinberg, Phys. Rev. Lett. 62, 485 (1989).

3. S. Weinberg, Ann. Phys. (N.Y.) 194, 336 (1989).

4. J. J. Bollinger, D. J. Heinzen, W. M. Itano, S. L. Gilbert

and D. J. Wineland, Phys. Rev. Lett. 63, 1031 (1989).

5. T. E. Chupp and R. J. Hoare , Phys. Rev. Lett. 64, 2261

(1990).

6. R. L. Walsworth, I. F. Silvera, E. M. Mattison, and R. F. C. Vessot, Phys. Rev. Lett. 64. 2599 (1990).

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and E. N. Fortson, submitted for publication.

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12. N. Gisin, Helv. Phys. Acta 62, 363 (1989).

13. A. Valentini, Phys. Rev. A 42, 639 (1990).

14. J. Polchinski, submitted for publication.

15. M. 0. Scully and K. Wodkiewicz , submitted for publication.

16. See for example C. R. Stroud, Jr. and E. T. Jaynes, Phys. Rev. A 1, 106 (1970).

17. I. Bialynicki-Birula and J . Mycielski, Ann. Phys. (N.Y.) 100 . 62 (1976).

18. A. Shiraony, Phys. Rev. A 20, 394 (1979).

19. C. G. Shull, D. K. Atwood, J. Arthur, and M. A. Home, Phys. Rev. Lett. 44, 765 (1980)

.

20. R. Gahler, A. G. Klein, and A. Zeilinger, Phys. Rev. A 23,

1611 (1981).

21. W. Nagourney, J. Sandberg, and H. Dehmelt, Phys. Rev. Lett

56, 2797 (1986); Th. Sauter, W. Neuhauser, R. Blatt, and P. E. Toschek, ibid. 57, 1696 (1986); J. C. Bergquist, R. G. Hulet, W. M. Itano, and D. J. Wineland, ibid. 57., 1699

(1986); F. Diedrich and H. Walther, ibid. 58, 203 (1987); W. M. Itano, J. C. Bergquist and D. J. Wineland, Phys. Rev. A 38, 559 (1988); W. M. Itano, J. C. Bergquist, F. Diedrich,

and D. J. Wineland, in Proceedings of the Sixth Rochester

Conference on Coherence and Quantum Optics (Plenum Publishing

Corp., N. Y.) in press.

22. W. M. Itano, D. J. Heinzen, J. J. Bollinger, and D. J. Wineland, Phy s. Rev. A 41, 2295 (1990).

23. J. D. Prestage, J. J. Bollinger, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 54, 2387 (1985).

24. J. J. Bollinger, D. J. Heinzen, W. M. Itano, S. L. Gilbert,

and D. J. Wineland, in Abstracts of the 12th International

Conference on General Relativity and Gravitation . Boulder,

CO, 1989, p. 500.

25. S. L. Gilbert, J.J. Bollinger, and D. J. Wineland, Phys. Rev. Lett. 60, 2022 (1988).

TN-128
480 Tests of Nonlinear Quantum Mechanics

26. W. M. Itano and D. J. Wineland, Phys . Rev. A 24, 1364 (1981);

W. M. Itano, L. R. Brewer, D. J. Larson, and D. J. Wineland,

Phys. Rev. AM. 5698 (1988).

27. D.J. Larson, J. C. Bergquist, J. J. Bollinger, W. M. Itano,

and D. J. Wineland, Phys. Rev. Lett. 57, 70 (1986).

28. J.J. Bollinger, J. D. Prestage, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 54, 1000 (1985).

29. L. R. Brewer, J. D. Prestage, J. J. Bollinger, W. M. Itano,

D. J. Larson, and D. J. Wineland, Phys. Rev. A 38, 859

(1988).

30. N. F. Ramsey, Molecular Beams (Oxford Univ. Press, London,

1956).

31. D. J. Wineland, J. C. Bergquist, J. J. Bollinger, W. M. Itano, D. J. Heinzen, S. L. Gilbert, C. H. Manney, and M. G. Raizen, IEEE Trans, on Ultrasonics, Ferroelectrics , and

Frequency Control, to be published.

32. S. K. Lamoreaux, J. P. Jacobs, B. R. Meckel, F. J. Raab , and

E. N. Fortson, Phys. Rev. A 39, 1082 (1989).

33. T. E. Chupp, R. J. Hoare , R. A. Loveman, E. R. Oteiza, J. M. Richardson, M. E. Wagshul , and A. K. Thompson, Phys. Rev. Lett. 63, 1541 (1989).

34. T. E. Chupp, E. R. Oteiza, J. M. Richardson, and T. R. White,

Phys. Rev. A38, 3998 (1988).

35. S. K. Lamoreaux, J. P. Jacobs, B. R. Heckel, F. J. Raab, and

E. N. Fortson, Phys. Rev. Lett. 57, 3125 (1986).

36. S. K. Lamoreaux, J. P. Jacobs, B. R. Heckel, F. J. Raab, and

E. N. Fortson, Phys. Rev. Lett. 59, 2275 (1987).

37. T. E. Chupp and R. J. Hoare, to be published.

38. M. L. Pitt and F. P. Calaprice, Phys. Dept., Princeton Univ.,

private communication.

39. R. L. Walsworth and I. F. Silvera, Phys. Rev. A 42, 63

(1990).

40. S. B. Crarapton, Phys. Rev. 158, 57 (1967).

41. The Many-Worlds Interpretation of Quantum Mechanics , eds B.

S. DeWitt and N. Graham (Princeton University Press,

Princeton, 1973).

TN-129
in: Proc. of the Workshop on Crystalline Ion Beams, Wertheim, W. Germany, Ed. by R W Hasse

I. Hofmann, D. Liese (GSI report GSI-89-10, ISSN 0170-4546), p. 241.

'

Coulomb Clusters of Ions in a Paul Trap

Wayne M. Itano J. C. Bergquist D. J. Wineland

Time and Frequency Division

National Institute of Standards and Technology

(formerly National Bureau of Standards)

Boulder, Colorado 80303, U.S.A.

Abstract

Ordered structures of as many as 16 laser-cooled Hg"^ ions, confined in a Paul trap, have been

observed. These structures, called Coulomb clusters, match those calculated by minimizing

the effective potential energy of the system. The bd}'^6s ^S\/2 to ^(fQs^ ^ Dsi2 transition in

Hg"^ has been observed by optical-optical double resonance. The resolution was high enough

that Doppler-induced sidebands, due to the harmonic motion of a single ion in the trap, were

clearly resolved. Additional sidebands, due to the relative vibration of two ions forming a pseudomolecule, have also been observed.

I. Introduction

A group of a few ions in a Paul (radiofrequency) trap is a system that can be modelled

with simple calculations, but which still exhibits phenomena, such as spatial ordering and

phase transitions, which are associated with condensed matter. Stimulated in part by re- cent theoretical predictions that systems of laser-cooled ions would form spatially ordered

structures, several experimental groups have produced and studied such structures."""

We have created systems of a few Hg"^ ions, which we call Coulomb clusters, in which the

positions of the ions are approximately fixed relative to each other. These relative positions are determined by a balance between the confining forces of the trap and the mutual Coulomb

repulsion of the ions. We have also observed a modification of the optical absorption spectrum

of the ions by the harmonic vibration of two ions relative to each other. This vibration is an indication that the ions have formed a spatially ordered state. Direct evidence of spatial

ordering has been obtained by two-dimensional imaging of the laser-induced fluorescence

from the ions. Images of up to 16 ions have been recorded and compared with calculations.

These clusters are a novel form of matter, in which the spacing between atoms is larger by

about three orders of magnitude than in an ordinary crystal or molecule. Observations of

similar structures of ions in a Penning trap have been made recently. Those structures

'Work of the NIST. Not subject to U.S. copyright.

TN-130
11 jum

Figure 1: The lowest energy levels of Hg"*".

contained more ions than those described here (as many as 15 000). However, individual

ions were not resolved in the Penning trap, as they were in the Paul trap. The smallest

clusters studied in the Penning trap had about 20 ions, so that the two experiments taken

together cover a range of ion numbers from one to 15 000. We have also observed Coulomb

clusters of charged particles with diameters of several micrometers in a large Paul trap. Such

clusters of mascroscopic particles are more easily produced and observed than ion clusters.

However, the charge-to-mass ratio cannot be made as uniform as for ions. In Thomson's

model of the atom, the potential energy function for the electrons was essentially the same

as for the ions in a Paul trap. As a result, some of the cluster shapes that we observe are the same as those Thomson calculated for atomic electrons. Preliminary reports of this work

have appeared elsewhere. '

II. Hg+ levels

The lowest energy states of the mercury ion are shown in Fig. 1. The ground state is ^5i/2, as in an alkali atom. The strong ^5i/2 to ^Pi/2 electric dipole transition at 194 nm is used

for the laser cooling and fluorescence detection. The lifetime of the ^P\/2 state is about 2

ns. Usually, it decays directly back to the ground ^5i/2 state. About once in 10^ times, it decays instead to the metastable ^0^/2 state, which has a lifetime of about 9 ms. If this

happens the 194 nm fluorescence abruptly turns off. The ^Da/? state decays about half the

time directly to the ground state and the rest of the time to the metastable ^Ds/2 state,

which has a lifetime of about 86 ms. After the decay to the ground state the fluorescence

turns on again. Thus, the 194 nm fluorescence from a single ion is bistable and switches

randomly between zero and a steady level. The statistical properties of this bistable signal

I TN-131
PHOTOMULTIPLIER TUBE

LENS

SYSTEM

Figure 2: Schematic view of the apparatus. The separation between the endcap electrodes is approximately 625 fim. The inner diameter of the ring electrode is approximately 890 //m.

have been used to measure the decay rates of the metastable states. "^

III. Apparatus

The experimental apparatus is shown schematically in Fig. 2. The electrodes of the Paul trap

are shown in cross section. Radiofrequency and static voltages are applied between the electrodes to control the effective potential well for the ions. The 194 nm laser beam passes along

a diagonal between the electrodes. The 194 nm source is required both for detecting the ions

and for cooling them (by resonant radiation pressure). The 194 nm fluorescence is detected

along another diagonal. For some experiments, the 194 nm detector is a photomultiplier

tube. In other cases, a resistive-anode photomultiplier tube that provides two-dimensional

positional information from single photoelectrons is used. For optical-optical double reso- nance experiments, 282 nm light from a frequency- doubled dye laser is focused on the ions.

A. Paul trap

The electric potential inside an ideal Paul trap is given by the expression,

(j){x,y,z) =

^o-K Vbcos(fiO

A2 {x'^y'-2z'). (1)

The potential V applied between the ring electrode and the two endcap electrodes is the

sum of a static and a radiofrequency part: V = Uq + Vocos{Qt). The parameter A depends

TN-132
on the geometry of the trap electrodes and has the dimensions of area. The radiofrequency

voltage Vq gives rise to an effective potential which confines in all directions. This potential

is four times as strong in the axial (z) direction as in the radial (r) direction. The ratio

of the axial to radial confining force can be changed by varying the static voltage Uq. The

effective potential energy which governs the average motion, called the secular motion, is

= ^(^' + y') + -^^'-

(3)

We call the harmonic frequencies for the secular motion along the axial and radial directions

cOz and ojr- The ion also oscillates at the frequency Q, of the applied field. This motion is called the micromotion. Under normal operating conditions, is much greater than lj^ or

ujr. If there is a slight deviation from axial symmetry, we can describe the effective potential

in terms of secular frequencies lOx and Uy, provided that we align the coordinate axes along

the principal axes of the potential. The effective potential is qM^, y, z) = -^x" + -^y" + ^^^

(4)

where lo^ '^ ^y

The trap used in these experiments has been described previously. The electrodes were made of molybdenum. The design of the electrodes, which were machined with simple

straight cuts, has been discussed in detail elsewhere. The inside diameter of the ring

electrode was less than 1 mm. This allowed very strong electric field gradients to be applied.

The maximum value of Vo was about 1 kV. The frequency 0/27r was about 23 MHz. The

ions were created inside the trap by electron impact ionization of isotopically purified ^^^Hg.

B. Lasers

The cw, tunable 194 nm source has been described previously. Radiation at 257 nm was

generated by frequency doubling the output of a 515 nm single-mode cw argon ion laser in an ammonium dihydrogen phosphate crystal. This radiation was mixed with the output of a 792 nm cw dye laser in a potassium pentaborate crystal to generate the sum frequency at 194 nm. The efficiencies of the frequency doubling and mixing processes were enhanced by

using ring buildup cavities to increase the powers of the input beams circulating through the

crystals. About 5 /uW were generated at 194 nm. The bandwidth was about 2 MHz.

The narrowband 282 nm source was obtained from a dye laser that was frequency doubled

in a deuterated ammonium dihydrogen phosphate crystal. The bandwidth of the dye laser,

which was stabilized to a Fabry-Perot cavity, was about 15 kHz for these experiments.

Recently, the linewidth of this laser has been decreased further by stabilizing it to a higherfinesse cavity.

C. Photon imaging detection

A three-stage lens system projected an image of the ion fluorescence onto the photocathode

of the position-sensitive photon counter with a magnification of about ISO. The first stage

4 TN-133
was an aberration-corrected multi-element lens with an /-number of 4.5. The positional

information from single detected photons was available in either analog or digital form.

Images could be observed in real time on an oscilloscope screen, using the analog pulses.

The digital information went to a computer, in order to make time exposures. An image of a single ion in the trap, recorded with this apparatus, has been published previously.

IV. High resolution optical spectroscopy

A. Optical-optical double resonance

We have observed the 282 nm ^S'1/2 to ^D^i2 electric quadrupole transition with high res- olution. The metastable ^.05/2 state has a lifetime of about 86 ms, so the transition has a

natural linewidth of about 2 Hz. It would be very difficult to observe this transition in a

single ion by detecting the emitted 282 nm photons, since there would be at most about 11

photons per second. To increase the efficiency of observation, we use optical-optical double

resonance with quantum amplification. '' The method works in the following way. The

194 nm source is turned off and 282 nm source is turned on. Suppose the ion absorbs a 282

nm photon, and is put into the metastable state. When the 194 nm light is then turned on, no fluorescence is observed. If, on the other hand, the ion is still in the ground state when

the 194 nm light is turned on, fluorescence is observed at the normal intensity (typically

thousands of photons per second detected). After a few milliseconds, the state of the ion can be determined with almost complete certainty. This technique was developed with the

goal of making an optical frequency standard. Here it is used to yield information about the

temperature and mutual interactions of the trapped ions.

B. Single-ion motional sidebands

The Doppler broadening of an absorption resonance of an ion bound in a harmonic well is modified by the confinement.'" Instead of a single, broadened resonance line, the spectrum

takes the form of a series of discrete resonances, each having a width equal to the natural

width of the resonance, which are separated by the frequency of harmonic motion.

Consider an ion moving in the x direction in a harmonic well with frequency u^ and

amplitude Xq, so that

X = Xq COs{(jJj:t). (5)

Let it be irradiated by a monochromatic laser beam of frequency u propagating along the x

axis. The electric field E{x^ t) of the laser beam is

E{x,t)=^KeEoe'^''''-'''\ (6)

where the symbol Re denotes the real part of a complex expression. In the frame of the

ion, there is a frequency modulation of the laser field. The electric field in the ion's frame is obtained by substituting the expression for x given by Eq. (5) into Eq. (6):

+00 = Re^o E ^V„(A:Xo)e~'^""""^^^ (7)

TN-134
In addition to the carrier at to, the ion sees sidebands, called motional sidebands, spaced by

the harmonic oscillation frequency lOx- The amplitudes of the sidebands are proportional to

the Bessel functions J„. The ion absorbs light when the laser frequency is equal to one of

its resonance frequencies and also when one of the motional sidebands matches a resonance

frequency.

The intensities of the sidebands can be used to measure the temperature of the ion.

A quantum mechanical treatment, in which a thermal-state-probability distribution was

assumed, has been given previously. According to Eq. (44) of Ref. 21, the intensity of the

nth sideband o-{n), is

a{n) = croexp It^^ - ^'^i^'^)] ^n fexp f ^^(fcxo)^K)

j j

. (8)

In this notation, the intensity of the carrier is cr(0), /„ is a modified Bessel function, and

{x"^) is the mean squared displacement of the harmonic oscillator, given by

{x') = (K) + i) = 2xl{{n,) + i), (9)

mcOx

where m is the mass of the ion and (n^) is the mean occupation number of the oscillator.

The temperature T is related to {ux) by

Equation 8 can easily be generalized to cases where there are two or more distinct frequencies

of motion and where the laser beam is not directed along one of the principal axes of the

trap. We have measured the temperature of a single, laser-cooled Hg"*" ion from the motional

sidebands of the ^Si/2 to ^0^/2 transition. The measured temperature was about 2 mK,

91 which agrees with theoretical predictions.

Figure 3(a) shows the absorption spectrum of a single Hg"*" ion, observed by the opticaloptical double resonance. The carrier (at optical frequency Vq) and the motional sidebands

above and below Uq are clearly resolved. The potentials Uq and Vq were adjusted so that

Ur ~ 2u2 « 473 kHz, where Ur = i^r/^Tr and u^ = u^JIt:. This condition helped to simplify the

sideband spectrum. The frequencies v^ and Vr were determined experimentally by applying

a radiofrequency voltage between the electrodes. When the frequency of the applied voltage

matched one of the secular frequencies, the fluorescence decreased, due to heating of the

ion. In our notation for the sidebands, (32r,r -f z), for example, denotes the two overlapping

sidebands at the frequencies Uq -\- du^ and Uq + Ur -\- VzC. Two-ion motional sidebands

If Uz < Ur, two ions have the lowest possible potential energy when they are located on the

z axis, at 2 = i^min, where z^in = (a^/4ma;^)^/^. The normal-mode frequencies of vibration

about the minimum of the potential have been calculated previously.^' '"^ The center of

mass of the two ions behaves like a particle of mass 2m and charge 2q and thus has the same

fi TN-135
0.6

-1500 -1000 -500 500 1000

Frequency detuning (kHz)

1500

Figure 3: Absorption spectra of the ^Si/2{mj = 1/2) to ^D^j2{rnj = —1/2) transition for

(a) one Hg"*" ion and for (b) two Hg"'' ions. The carrier at optical frequency Uq (defined as zero frequency detuning) and the motional sidebands are clearly resolved. The sidebands

are labeled in a notation that is explained in the text. The additional sidebands in (b) are due to the vibration of the two ions with respect to each other at frequency v'^.

frequencies of motion as a single ion. The frequency at which two ions vibrate with respect to

each other along the z axis is u'^ = yZv^ . This mode is like the stretch vibration of a diatomic

molecule. The other vibrational frequency is {u^ — u^Y^^, which, for Ur = 2//^, is also equal

to \/3i/j. This vibration is a torsional motion in the x-z or y-z plane. Figure 3(b) shows an

absorption spectrum taken with 2 ions in the trap. The trap potentials were the same as for

the data of Fig. 3(a). The electronics of the position-sensitive photon detector were adjusted

so that the fluorescence from only one of the two ions was detected. The motion of one ion

is a superposition of harmonic motions at all of the normal-mode frequencies of the system.

Therefore, all of the sidebands in Fig. 3(a) are also present in Fig. 3(b). The additional lines are due to the additional vibrational modes, at frequency v'^, of the 2-ion pseudomolecule.

This kind of vibrational mode of two trapped ions has been observed by another method

described in Ref. 9. Those workers applied a radiofrequency electric field between the trap

electrodes, thus directly exciting the vibrational mode. When the frequency of the applied

field matched the vibrational resonance, the fluorescence signal decreased.

TN-136
V. Multi-ion cluster shapes

A. Calculations

The effective potential energy V^ff for a system of A'^ ions of mass m and charge q is given by

the expression,

Kff({^.,y.-,-a) = ?i;("k'+"j!/?+"k')+5Lr?^- (11)

" V ' -^ ^ ' effective trap potential Coulomb repulsion

Here, r^ = [xi,yi,Zi) is the position of the zth ion. The first term in Eq. (11) is the effective

potential of the ions due to the trap fields. The second is the potential energy due to

the Coulomb repulsion between the ions. The equihbrium configurations were calculated

by finding the set of 3A^ ion coordinates that minimized the potential energy. A variable

metric, or quasi-Newton, method was used to find the minimum, as in Ref. 3. We used

the FORTRAN subroutine DFPMIN from Ref. 23. For a cylindrically symmetric Paul trap,

there is an infinite set of solutions, since Kff is unchanged by an arbitrary rotation about

the z axis. To avoid possible convergence problems, and also to simulate the asymmetry of a real trap, oj^; and LOy were assumed to differ by 0.2 %. Because of the remaining symmetry

of the trap, there were in some cases several solutions, which were related by rotations and

reflections. The algorithm sometimes converged to a minimum which was not the global

minimum. In order to detect such false minima, at least 10 random initial configurations were used for each case. The solution with the lowest final value of V^n was assumed to be

a global minimum.

B. Observations

The ions were observed at an angle of approximately 54.7° with respect to the z axis of

the trap. This allowed the three-dimensional structure of the clusters to be seen. In the

apparatus of Ref. 6, the ions were observed along the z axis, so only the projection on the

x-y plane could be seen. Figures 4-10 show the observed ion images and the corresponding

calculated configurations. The magnification was the same for all of the experimental images. The scale of the experimental and calculated images is the same. The scale for the

experimental images was determined from the separation between two ions, which is easily

calculated. An approximate scale can be determined from the caption to Fig. 5. Experimental images of two, three, and four ions, obtained with less optical magnification, were

published previously.

Figure 4(a) shows five ions for u, = 0.308 MHz and i/, = 0.376 MHz. Two of the ions

lie along the z axis, symmetrically above and below the x-y plane. The other three ions

circulate in the x-y plane about the z axis. They are not fixed in position, because the

trap's effective potential is nearly cylindrically symmetric. Thus, the small, unintentional

torque that the 194 nm beam applies to the system is enough to cause it to rotate. In this case, we determined the number of ions by reducing v^, relative to u^, until they were clearly

separated in a chain along the z axis. Figure 4(b) shows the calculated ion positions for these

^ TN-137
trap parameters. The circle in the calculated figure is obtained by rotating the configuration

about the z axis.

(a) (b)

Figure 4: Nonplanar configuration of five ions for v^ = 0.308 MHz and Vr = 0.376 MHz. (a)

Experimental data, (b) Calculation.

Figure 5 shows six ions for u^ = 0.497 MHz and Vr = 0.256 MHz. One is located

at the origin. The other five circulate in the x-y plane. For these trap parameters, the

configurations calculated for six, seven, and eight ions all have one ion at the origin and the

rest in a ring in the x-y plane. In this case, we assume that there were six ions, because that

number gives the best agreement between the observed and calculated ring sizes.

(a) _ (b)

m

Figure 5: Planar configuration of six ions for u^ = 0.497 MHz and i/^ = 0.256 MHz. The

diameter of the ring is approximately 16 ^m. (a) Experimental data, (b) Calculation.

Figure 6 shows six ions for Uz = 0.780 MHz and Ur = 0.347 MHz. One is at the origin,

and four others lie at the vertices of a pentagon in the x-y plane. The fifth vertex of the

pentagon appears to be vacant, but we assume that it is occupied by an ion which does not

fluoresce. It may be a heavier isotope of mercury, such as ^^^Hg"*", or a molecular ion, such

as HgOH"*", neither of which would fluoresce. If it had a higher charge-to-mass ratio than

^^^Hg"*", it would be more tightly bound to the trap, and would go the center. The additional

asymmetry, due to the odd ion, keeps the configuration from rotating.

Figures 7 and 8 show two different configurations of the same nine ions. The secular

frequencies for Fig. 7 were u^ = 0.497 MHz and Ur = 0.256 MHz; for Fig. 8, they were

9

TN-138
(a) (b)

•

Figure 6: Planar configuration of six ions for u^ = 0.780 MHz and Ur = 0.347 MHz. Five

ions are presumed to lie at the vertices of a pentagon in the x-y plane, but one does not

fluoresce. Another one lies at the origin, (a) Experimental data, (b) Calculation.

//^ = 0.439 MHz and i^r = 0.304 MHz. The ions in Fig. 7 lie in the x-y plane. This

configuration is the same as the calculated one shown in Fig. 1 of Ref. 3. The outer ring

of the experimental image appears to be incomplete in this case because of nonuniform

illumination of the cluster by the tightly focused 194 nm beam. In Fig. 8, two ions lie along

the z axis, symmetrically above and below the x-y plane. The others form a ring in the

x-y plane. The number of ions in Figs. 7 and 8 was determined uniquely from the observed

configurations for the two different sets of conditions.

(a) . (b)

Figure 7: Planar configuration of nine ions for u^ = 0.497 MHz and u^ = 0.256 MHz. (a)

Experimental data, (b) Calculation.

j Figure 9 shows 15 ions for u^ = 0.780 MHz and z/r = 0.347 MHz. All of them lie in

the x-y plane. Five form a pentagon, and the other ten form the outer ring. The observed

pattern is only consistent with the one calculated for 15 ions.

Figure 10 shows 16 ions, for u^ = 0.626 MHz and Ur = 0.479 MHz. The large ring of

eight ions lies nearly in the x-y plane. Two smaller rings of four ions each are displaced

along the z axis, above and below the x-y plane. If another ion is added, at the same trap

voltages, then, according to calculations, one ion goes to the center. By using data taken

at another set of trap voltages, we could determine the number of ions. The 16 ions lie close to the surface of a spheroid. For larger numbers of ions, one would expect to see a

10

TN-139
(a)

Q)

(b)

Figure 8: Nonplanar configuration of nine ions for v^ = 0.439 MHz and v^ = 0.304 MHz.

Seven circulate in the x-y plane. The other two are displaced along the z axis, (a) Experimental data, (b) Calculation.

(a) (b)

Figure 9: Planar configuration of 15 ions for u^ = 0.780 MHz and Ur = 0.347 MHz. (a)

Experimental data, (b) Calculation.

(a) (b)

Figure 10: Nonplanar configuration of 16 ions for v^ = 0.626 MHz and Ur = 0.479 MHz. (a)

Experimental data, (b) Calculation.

11

TN-140
series of concentric spheroids like those which have been predicted and observed in Penning

traps. ^2,25

VI. Related physical systems

A. Macroscopic particle traps

Perhaps the earliest studies of spatially ordered systems of charged particles in a Paul trap were made by Wuerker, Shelton, and Langmuir about 30 years ago. In these studies,

aluminum particles that had diameters of a few micrometers were suspended in a large Paul

trap and cooled by collisions with gas molecules.

A similar trap has been built in our laboratory. The inner diameter of the ring was

about 2.5 cm. Typical trap parameters were Vq ^ 350 V, C/q ~ V, and Q/27r « 60 Hz.

Aluminum oxide particles of approximate mass 10~^ g and charge approximately 10^ times

the proton charge were trapped and observed to form ordered structures.

B. The Thomson model of the atom

In an early model for the atom, first suggested by Kelvin and later studied in detail b

Thomson, electrons were assumed to be embedded in a uniform sphere of positive charge.

The potential energy of an electron inside the sphere due to the positive charge is proportional

to x"^ -{- y"^ -\- z^. Thus, the stable spatial configurations of the electrons in such a model

are the same as for ions in a Paul trap with a spherically symmetric effective potential.

Thomson's calculations of the minimum-energy configurations, which actually were done in

two dimensions, showed patterns of concentric rings. He tabulated the configurations for

up to 100 electrons. The 100-electron configuration contains seven rings. For v^ > ^r^ the

three-dimensional configurations for small numbers of particles are the same as for Thomson's

two-dimensional calculations. For example, Thomson's 15-electron configuration has five in

the inner ring and ten in the outer ring, like the data of Fig. 9. Thomson's calculations are perhaps the earliest predictions of layered structures in a bound Coulomb system. Such

layered structures are predicted to occur in one-, two-, and three-dimensional systems and

have been observed in the Penning trap, a three-dimensional system.

Thomson discussed experiments with floating magnets, which he used to verify his calculations of the two-dimensional configurations. For these experiments, equally magnetized

needles, oriented in the same way, were pushed through corks and floated on water. An

attractive force was produced by a large magnet placed above the surface of the water, the

lower pole having the opposite sign to that of the upper poles of the floating magnets.

VII. Discussion

The equiHbrium shapes of clusters that contain 16 ions or fewer and the intra-cluster \ibrational frequencies can be calculated from the effective potential approximation. Other

properties, however, such as the Doppler spectrum or the order-chaos transition, arc affected

by the micromotion, and require a more complete theoretical treatment.

12 TN-141
Experiments with charged micrometer-sized particles in Paul traps might be useful in

testing calculations, since they are relatively simple to perform, compared to experiments

with ions. Large numbers of particles (on the order of 100) can be cooled and optically

observed. The melting and freezing phase transitions have been observed previously. Perhaps further experimental work could reveal details of the transition to chaotic behavior."'"

Other kinds of traps, with linear or "racetrack" geometries ' might be useful in simulating

ordered structures in ion storage rings. Such structures have been predicted, but not yet

observed."

Acknowledgments

We gratefully acknowledge support from the Air Force Office of Scientific Research and

the Office of Naval Research.

References

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2. E. V. Baklanov and V. P. Chebotayev, Appl. Phys. B 39, 179 (1986).

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6. F. Diedrich, E. Peik, J. M. Chen, W. Quint, and H. Walther, Phys. Rev. Lett. 59, 2931

(1987).

7. D. J. Wineland, J. C. Bergquist, W. M. Itano, J. J. Bollinger, and C. H. Manney, Phys.

Rev. Lett. 59, 2935 (1987).

8. J. Hoffnagle, R. G. DeVoe, L. Reyna, and R. G. Brewer, Phys. Rev. Lett. 61, 255 (1988).

9. R. Blumel, J. M. Chen, E. Peik, W. Quint, W. Schleich, Y. R. Shen, and H. Walther,

Nature 334, 309 (1988).

10. D. J. Wineland, W. M. Itano, J. C. Bergquist, S. L. Gilbert, J. J. Bollinger, and F. Ascarrunz, in Non-neutral Plasma Physics, edited by C. W. Roberson and C. F. Driscoll,

A. I. P. Conf. Proc. 175, (Am. Inst. Phys., New York, 1988), p. 93.

11. P. Toschek et al., these Proceedings.

12. S. L. Gilbert, .J. J. BoUinger, and D. J. Wineland, Phys. Rev. Lett. 60, 2022 (1988).

13. W. M. Itano, .1. C. Bergquist, R. G. Hulet, and D. J. Wineland, Phys. Rev. Lett. 59,

2732 (1987).

14. J. C. Bergquist, D. J. Wineland, W. M. Itano, H. Hemmati, H.-U. Daniel, and G.

Lcuchs, Phys. Rev. Lett. 55, 1567 (1985).

13 TN-142
15. E. C. Beaty J. Appl. Phys. 61, 2118 (1987).

16. H. Hemmati, J. C. Bergquist, and W. M. Itano, Opt. Lett. 8, 7.3 (198.3).

17. W. M. Itano, J. C. Bergquist, and D. J. Wineland, Science 237, 612 (1987); D. .J. Wineland and W. M. Itano, Phys. Today 40 No. 6, 34 (1987).

18. H. G. Dehmelt, Bull. Am. Phys. Soc. 20, 60 (1975).

19. D. J. Wineland and W. M. Itano, Phys. Lett. 82A, 75 (1981).

20. R. H. Dicke, Phys. Rev. 89, 472 (1953).

21. D. J. Wineland and W. M. Itano, Phys. Rev. A 20, 1521 (1979).

22. J. C. Bergquist, W. M. Itano, and D. J. Wineland, Phys. Rev. A 36, 428 (1987).

23. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes

(Cambridge Univ. Press, London, 1986).

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25. A. Rahman and J. P. Schiffer, Phys. Rev. Lett. 57, 1133 (1986); D. H. E. Dubin and

T. M. O'Neil, ihid. 60, 511 (1988); J. P. Schiffer, ihid. 61, 1843 (1988).

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Sons, New York, 1907).

27. D. A. Church, J. AppL Phys. 40, 3127 (1969).

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Europhys. Lett. 1, 55 (1986); D. Habs, in Lecture Notes in Physics 296, edited by M.

Month and S. Turner (Springer-Verlag, Berhn, 1988), p. 310.

,, TN-143

14
OBSERVATION OF CORRELATIONS IN FINITE, STRONGLY COUPLED ION PLASMAS*

J.J. BOLLINGER, S.L. GILBERT, D.J. HEINZEN, W.M. ITANO, and D.J. WINELAND

National Institute of Standards and Technology, 325 Broadway, Boulder, CO

80303

1 . INTRODUCTION

We have observed spatial correlations with up to 15 000 Be ions in a Penning trap with a coupling (defined below) of F > 100. These correlations

are strongly affected by the boundary conditions and take the form of

2-4

concentric shells as predicted by computer simulations. In this paper we briefly describe the experimental confinement geometry and the method of

producing low temperature ions. The relatively large spacings between the

ions (~ 20 ;im) permit the shells to be directly viewed by imaging the Be

laser- induced fluorescence onto a photon- counting camera. Diagnostic

techniques capable of measuring the ion diffusion are then discussed.

Qualitative observations of the ion diffusion are compared with theoretical

predictions

.

2. CONFINEMENT GEOMETRY

The Penning trap uses a static, uniform magnetic field and a static,

axially symmetric electric field for the confinement of charged particles.

The magnetic field, which is directed along the z axis of the trap, provides

confinement in the radial direction. The ions are prevented from leaving the

trap along the z axis by the electric field. In the work described here, the

electric field was provided by three cylindrical electrodes as shown in Fig.

1. The dimensions of the trap electrodes were chosen so that the first

anharmonic term (i.e. fourth order term) in the expansion of the trapping

potential was zero. Over the region near the trap center, the potential can

2 2 be expressed (in cylindrical coordinates) as $ = AV (2z -r ) where A = 0.146

-0 -8 -10 cm A background pressure of 10 Pa(= 10 Torr) was maintained by a triode sputter- ion pump. The confinement geometry is similar to that used by

the group of the University of California at San Diego (UCSD) with the

exception that our trap is smaller than the UCSD traps.

in Strongly Coupled Plasma Physics , ed. by S. Ichimaru, (Elsevier Science Publishers B.V. / Yamada

Science Foundation, 1990) p. 177.

TN-144
LENS

SYSTEM

PROBE

BEAM

PERPENDICULAR

COOLING BEAM

DIAGONAL

COOLING BEAM

FIGURE 1 Schematic drawing of the trap electrodes, laser beams, and imaging system (not

to scale). The overall length of the trap is 10.2 cm. The trap consists of

two end cylinders and two electrically connected central cylinders with 2.5 cm inner diameters. Ion clouds are typically less than 1 mm in both diameter and

axial length. The diagonal cooling beam crosses the cloud at an angle of 51°

with respect to the z axis In the experiments, B = 1.92 T or 0.82 T and V

ranged between 20 V and 200 V.

°

The stored ions can be characterized by a thermal distribution where the

"parallel" (to the z axis) temperature Ty is approximately equal to the

"perpendicular" temperature T . This thermal distribution is superimposed on

6-9

a uniform rotation of the cloud at frequency w which, at the low

-* -* -* temperatures of this experiment, is due to the ExB drift, where E is the

electric field due to the trap voltage and the space charge of the ions. In a frame of reference rotating with the ions, the static thermodynamic properties

of an ion cloud confined in a Penning trap are identical to those of a one- component plasma (OCP) . An OCP consists of a single species of charge

embedded in a uniform-density background of opposite charge. For the system

of ions in a Penning trap, the trapping fields play the role of the

neutralizing background charge. An OCP is characterized by the Coulomb

7,10 coupling constant,

r.qVca^k^T).

which is a measure of the nearest-neighbor Coulomb energy divided by the

thermal energy of a particle. The quantities q and T are the ion charge and

3 temperature. The Wigner-Seitz radius a is defined by Una n /3=1 , where -qn

is the charge density of the neutralizing background. In the Penning trap the

TN-145
background density n- depends on the rotation frequency w and the cyclotron

frequency fl and is given by

n^ = mw(n-c.,)/(2rrq^). (1)

3. LASER COOLING AND COMPRESSION

The ion density that can be achieved in a Penning trap is limited by the

magnetic field strength that is available in the laboratory. Consequently to

obtain large values of F and therefore strong couplings, a technique to obtain

low ion temperatures is necessary. In our work, radiation pressure from

lasers is used to reduce the temperature of the stored ions to less than 10

11-13

mK. This technique, known as laser cooling, uses the resonant scattering

of laser light by atomic particles. The laser is tuned to the red, or lowfrequency side of the atomic "cooling transition" (typically an electric

dipole transition like the D lines in sodium) . Ions with a velocity component

opposite to the laser beam propagation (k«v < 0) will be Doppler shifted into

resonance and absorb photons at a relatively high rate. Here, k is the photon

wave vector (|k|=2?r/A, where A is the wavelength of the cooling radiation).

For the opposite case (k'V > 0) , the ions will be Doppler shifted away from

the resonance and the absorption rate will decrease. When an ion absorbs a photon, its velocity is changed by an amount Av = ftk/m due to momentum

conservation. Here Av is the change in the ion's velocity, m is the mass of

the ion, and IttH is Planck's constant. The ion spontaneously reemits the

photon symmetrically. In particular, when averaged over many scattering

events, the reemission does not change the momentum of the ion. The net

effect is that for each photon scattering event, the ion's average velocity is

reduced by ftk/m. To cool an atom from 300 K to millike Ivin temperatures takes

4 8 typically 10 scattering events but, since scatter rates can be 10 /s , the

cooling can be rapid. + 2 2 In our work with Be , the 2s S. .-

"* 2p P- .^ "D." transition was used as the cooling transition as indicated in Fig. 2. Cooling laser beams were

directed both perpendicularly and at an angle with respect to the magnetic

field as indicated in Fig. 1. This enabled us to control the cloud size and

14 15

obtain the lowest possible temperatures. The 313 run radiation required

to drive this transition was obtained by frequency doubling the output of a continuous wave, narrow band (3 MHz) dye laser. The 313 nm power was typically 50 /jW. The theoretical cooling limit, due to photon recoil

11-13

effects, is given by a temperature equal to ^7/(2k ) where 7 is the

o radiative linewidth of the atomic transition in angular frequency units. For

the Be cooling transition (7 = 27r x 19.4 MHz), the theoretical minimum

temperature is 0.5 raK. TN-146
P3

2

,

Probe Pump

Si

2

3/2

1/2

-1/2

-3/2

1/2

-1/2

mj

FIGURE 2

Energy level 9+2

structure of the Be S ground state and the first excited

P_ .- state. The magnetic field splits each state into its m sublevels. The

laser cooling (pump) and depopulation (probe) transitions are shown.

Laser scattering can also be used to change the angular momentum and

8 9

compress the stored ion plasma. ' The z component of the canonical angular

momentum for an individual ion in the plasma is

2

i. = mv.r +

z B

qBr

2c

(2)

The two terms in Eq. (2) are the ion's mechanical angular momentum and the

field angular momentum. The total z component of the angular momentum of the

plasma is

L = m(n/2-w)N<r >. (3)

Here N is the total number of ions and <r > is the mean- squared radius of the

plasma. For most of the work described in this paper cj « n and

mi^N

L «

z <r > > 0, (4)

Therefore the total angular momentum is dominated by the field angular

momentum. Suppose the cooling laser beam is directed normal to the z axis but

at the side of the plasma which is receding from the laser beam due to the

plasma rotation. Because the rotation of the positive ions is in the -Q

direction, the torque of the laser on the ions will also be negative.

Consequently, angular momentum is removed from the plasma and according to Eq

.

(4) the radius of the plasma must decrease. In general, the plasma is

compressed until the torque due to the cooling laser is balanced by another

TN-147
external torque. As the radius decreases, the density of the plasma

increases

.

Even in the absence of external torques, there is a limit to how far the

plasma can be compressed. From Eq. (1), the maximum density, known as the

Brillouin density, occurs when the rotation frequency to = 0/2. The Brillouin

density is given by

2 mn

nmax o 2 OTrq

We have recently been able to achieve densities at or near the Brillouin

limit. In fact we have also been able to achieve rotation frequencies w > 0/2

where according to Eq. (1) the ion density decreases. In these experiments,

the temperature was not determined. At the magnetic field of 1.92 T used in

9 -3 some of the work discussed here, the Brillouin density is 1.1x10 cm . This

density with the theoretical minimum 0.5 mK temperature, results in a coupling

r ~ 5500. For the work reported here, we have been able to obtain ion

temperatures in the 1-10 mK range with densities 5-10 times less than the

Brillouin density. This results in couplings F of a few hundred.

We measured ' ' the ion density and temperature by using a second

laser, called the probe laser, to drive the "depopulation" transition as indicated in Fig. 2. The cooling laser optically pumps the ions into the

2 8 S. .- m^=+l/2 state. The resonance fluorescence (i.e. laser light scattered

by the ions) from this transition is used as a measure of the ion population

in the m =+1/2 state. The probe laser drives some of the ion population from

2 2 the S. _ m =+1/2 state to the P^ ,« m =-1/2 state where the ions decay with

1/^ J f. -5/^ J 2/3 probability to the S, .^ m =-1/2 state. This causes a decrease in the

1/^ J 2 observed ion fluorescence because the S, .„ m =-1/2 state is a "dark" state a (state which does not fluoresce in the cooling laser) . The ion temperature is

obtained from the Doppler broadening of the resonance lineshape when the probe

laser is scanned through the depopulation transition. The ion rotation

frequency is measured from the shift in the depopulation transition frequency

as the probe laser is moved from the side of the plasma rotating into the

laser beam to the side of the plasma rotating with the laser beam. From the

measured rotation frequency, the density is calculated from Eq. (1). The

measured density and temperature is used to calculate the coupling F.

4. OBSERVED CORRELATIONS

With measured couplings F > 100, we anticipate that the ions will exhibit

correlated behavior. If the number of stored ions is large enough for

TN-148
approximating spheroids. Independent theoretical investigations ' of the

infinite voliime behavior, the ions may be forming a bcc lattice. Until now

we have cooled and looked for spatial correlations with up to 15 000 Be ions

stored in the Penning trap of Fig. 1. A currently unanswered question is how

many stored ions are required for infinite volume behavior, i.e. the

appearance of a bcc lattice for T > 178. For a finite plasma consisting of a hundred to a few thousand ions, the boundary conditions are predicted to have

a significant effect on the plasma state. Simulations involving these numbers

of ions in a spherical trap potential predict that the ion cloud will separate

2-4 into concentric spherical shells. Instead of a sharp phase transition, the

system is expected to evolve gradually from a liquid state characterized by

short-range order and diffusion in all directions, to a state where there is

diffusion within a shell but no diffusion between the shells (liquid within a shell, solid-like in the radial direction), and ultimately to an overall

4 solid- like state. These conclusions should apply to a nonspherical trap

potential as well if the spherical shells are replaced with shells

approximating spheroids. Independent thee

nonspherical case support this conjecture.

9 + We have observed shell structures with Be ions stored in a Penning trap

by imaging the laser induced fluorescence from the cooling transition. This

technique is sensitive enough to observe the structures formed with only a few

18-21

ions in a trap. About 0.04% of the 313-nm fluorescence from the decay of

2

the Fo /9 state was focused by f/10 optics onto the photocathode of a resistive -anode photon-counting imaging tube (see Fig. 1). The imager was located along the z axis, about 1 m from the ions. The imaging optics was composed of a three-stage lens system with overall magnification of 27 and a resolution (FWHM) of about 5 /im (specifically, the image of a point source when referred to the position of the ions was approximately 5 /im in diameter) . Counting rates ranged from 2 to 15 kHz. Positions of the photons arriving at

the imager were displayed in real time on an oscilloscope while being

integrated by a computer. The probe laser could be tuned to the same transition as the cooling laser and was directed through the cloud

perpendicularly to the magnetic field. With the probe laser turned on continuously, the cooling laser could be chopped at 2 kHz (50% duty cycle) and

the image signal integrated only when the cooling laser was off. Different

portions of the cloud could then be imaged by the translation of the probe

beam, in a calibrated fashion, either parallel or perpendicular to the z axis.

Images were also obtained from the ion fluorescence of all three laser beams.

TN-149
o

(a)

>lff»ti»t«lJi«*M«IIM»MMHfM

200 /im

(b)

\

i£>. ^

HMlMMwMflf MUM

DIAGONAL

COOLING

PROBE

(c)

PERPENDICULAR

~ COOLING

FIGURE 3 Images of shell structures obtained with B = 1.92 T. (a) A single shell in a cloud containing approximately 20 ions. Trap voltage V^ = 14 V and cloud

aspect ratio a (axial length/diameter) = 6.5. This image was obtained from

the ion fluorescence of the perpendicular and diagonal cooling beams. (b)

Sixteen shells (probe-beam ion fluorescence only) in a cloud containing about

15 000 ions with Vg = 100 V and a - 0.8. (c) Eleven shells plus a center

column in the same cloud as (b) , with V_ = 28 V and a - 2.4. This image

shows the ion fluorescence from all three laser beams. Integration times were about 100 s for all images.

We have observed shell structure in clouds containing as few as 20 ions

(one shell) and as many as 15 000 ions (sixteen shells) . Images covering this

range are shown in Fig. 3. Even with 15 000 ions in the trap there is no evidence for infinite volume behavior. We measured the coupling constant F

for several clouds containing about 1000 ions. Drift in the system parameters

was checked by verifying that the same images were obtained before and after

the cloud rotation frequency and ion temperatures were measured. Figure 4

shows examples of shell structures at two different values of F. The first

image is an example of high coupling (F ~ 180) and shows very good shell

definition in an intensity plot across the cloud. The second image is an example of lower coupling (F = 50) and was obtained with cooling only

perpendicular to the magnetic field. Variations in peak intensities

equidistant from the z axis are due to signal-to-noise limitations and

imperfect alignment between the imager x axis and the probe beam.

TN-150
• MtMliM

(a)

-100 100 -100 100

XOim)

FIGURE 4

Intensity plots along the imager x axis (parallel to the probe beam) through

the center of the ion cloud with corresponding images (above). (a) r = 180!f° (1 = 6:!:^ mK, n^ = 7x10^ ions cm"^). Cloud aspect ratio a =3.5.

(b) r = 5011° (T = 33111 niK. "o = 2x10^ ions cm"^), a = 5. The clouds

contained about 1000 ions and B = 1.92 T in both cases.

We obtained three-dimensional information on the shell structure by taking

probe images at different z positions; two types of shell structure were

present under different circumstances. The first type showed shell curvature

near the ends of the cloud, indicating that the shells may have been closed

spheroids. Shell closure was difficult to verify because of a lack of sharp

images near the ends of the cloud where the curvature was greatest. This may

have been due to the averaging of the shells over the axial width of the probe

beam. In the other type of shell structure, it was clear that the shells were

concentric cylinders with progressively longer cylinders near the center. An

example of these data is shown in Fig. 5. Other evidence for cylindrical

shells was obtained from the observation that shells in the diagonal-beam

images occurred at the same cylindrical radii as those from the perpendicular

beams. This can be seen in the three -beam images such as that shown in Fig.

3(c). Systematic causes of these two different shell configurations have not

yet been identified.

One comparison which can be made between the theoretical calculations and

our experimental results is the relationship between the number of shells and

the number of ions, N, in a cloud. For a spherical cloud, two independent

2 22 1/3 1/3 approaches ' estimate (N/4) ^ and (3N/47r) shells. For the nearly

spherical cloud of Fig. 3(b) (N = 15 000), these formulae predict 15.5 and

TN-151
Zp (jjtm)

-15

-40

-65

-90

-115

-140

-152

-165

-178

FIGURE 5 Data showing evidence for concentric cylindrical shells. On the right is a series of images obtained with the probe beam for different z positions z of

the probe beam (lower half of the cloud only). Intensity plots for z =

^ -40 /im and z = -178 /im are shown on the left. The cloud aspect ratio a was about 1.9 anH B = 1.92 T. ^

15.3 shells and we measure 16. At present, it is difficult to make further

quantitative comparisons between our data and the theoretical calculations.

For example, there is substantial uncertainty in our measurement of T due to

uncertainty in the temperature measurement. Our data do agree qualitatively

with the simulations with the exception, in some cases, of the presence of an open-cylinder shell structure as opposed to the predicted closed spheroids.

Shear (that is, different rotation frequencies) between the shells could

possibly account for this discrepancy. In our experiment, shear could be

g caused by differential laser torque or the presence of impurity ions. For

the data here, we have determined that the rotation frequency does not vary by

more than 30% across the cloud.

5. ION DIFFUSION

The probe laser can be used to optically tag ions and observe the ion

14

diffusion. With the probe laser tuned to the depopulation transition (see

Fig. 2), ions in the path of the probe laser beam are put into the "dark" m =

-1/2 ground state. These ions will not fluoresce when they pass through the

cooling laser until they are optically pumped back into the m =+1/2 ground

state. This repvunping time is typically on the order of 1.0 s. By pulsing

the probe laser on and measuring the length of time it takes the dark ions to

TN-152
diffuse from the probe beam to the cooling Laser beam, it should be possible

to measure the ion diffusion. By directing the probe laser beam to the radial

edge of the plasma so that only the outer shell is intersected by the probe

beam, it should be possible to observe the diffusion of ions in the radial

direction or between shells. By directing the probe beam to the axial edge of

the cloud it should be possible to observe the diffusion of the ions in the

axial direction or within a shell. According to the simulations of Dubin and

4 O'Neil, for intermediate values of the coupling (F ~ 100) we expect to

observe that the diffusion between shells is much slower than the ion

diffusion within a shell (solid- like behavior between shells, liquid- like

within a shell). As the temperature is lowered and the coupling T increases,

the diffusion within a shell should smoothly slow down. At high enough

couplings (i.e. T > 400) the diffusion should be very slow both between and

within shells, indicative of solid- like behavior.

We have qualitatively observed the ion diffusion at intermediate values

of r (r ~ 100-200) . We observed that the diffusion of ions between shells is

slow compared to the optical repumping time (~ 1 s) but that the diffusion

within a shell (i.e. from the axial end of a shell to the z = plane) is fast

compared to this repumping time. We have also observed states with higher

couplings (the couplings F were not measured) where the diffusion of ions both

between and within a shell was slow compared to the optical repumping time.

In the future we plan to make quantitative measurements of the ion diffusion.

ACKNOWLEDGEMENT

We gratefully acknowledge the support of the U.S. Office of Naval Research

and the Air Force Office of Scientific Research. We thank M. Raizen and F. Moore for carefully reading the manuscript.

REFERENCES

1) S.L. Gilbert, J.J. Bollinger, and D.J. Wineland, Phys . Rev. Lett. 60

(1988) 2022.

2) A. Rahman and J. P. Schiffer, Phys. Rev. Lett. 57 (1986) 1133; J. P. Schiffer, Phys. Rev. Lett. 6i (1988) 1843.

3) H. Totsuji, in Strongly Coupled Plasma Physics, eds . F.J. Rogers and H.E

DeWitt (Plenum, New York, 1987) pp. 19-33.

4) D. Dubin and T. O'Neil, Phys. Rev. Lett. 60 (1988) 511.

5) C.F. Driscoll, J.H. Malmberg, and K.S. Fine, Phys. Rev. Lett. 60 (1988)

1290; J.H. Malmberg and J . S . deGrassie, Phys. Rev. Lett. 35, 577 (1975).

6) T.M. O'Neil, in Non-Neutral Plasma Physics, eds. C.W. Roberson and C.F.

Driscoll (American Institute of Physics, New York, 1988) pp. 1-25.

TN-153
7) J.H. Malmberg and T.M. O'Neil, Phys . Rev. Lett. 39 (1977) 1333.

8) L.R. Brewer, J.D. Prestage, J.J. Bollinger, W.M. Itano, D.J. Larson, and

D.J. Wineland, Phys. Rev. A38 (1988) 859.

9) D.J. Wineland, J.J. Bollinger, W.M. Itano, and J.D. Prestage J. Opt. Soc

.

Am. B2 (1985) 1721.

10) S. Ichimaru, H. lyetomi, and S. Tanaka, Phys. Rep. 149 (1987) 91 and

references therein.

11) D.J. Wineland and W.M. Itano, Phys. Rev. A20 (1979) 1521.

12) W.M. Itano and D.J. Wineland, Phys. Rev. A25 (1982) 35.

13) D.J. Wineland and W.M. Itano, Phys. Today 40(6) (1987) 34; S. Stenholm,

Rev. Mod. Phys. 58 (1986) 699.

14) L.R. Brewer, J.D. Prestage, J.J. Bollinger, and D.J. Wineland, in Ref.

3), pp. 53-64.

15) J.J. Bollinger and D.J. Wineland, Phys. Rev. Lett. 53 (1984) 348.

16) J. P. Schiffer, Argonne Natl. Lab., Argonne, IL, private communication.

17) D.H.E. Dubin, Dept. of Physics, UCSD, La Jolla, CA, private

communication.

18) F. Diedrich, E. Peik, J.M. Chen, W. Quint, and H. Walther, Phys. Rev.

Lett. 59 (1987) 2931.

19) D.J. Wineland, J.C. Bergquist, W.M. Itano, J.J. Bollinger, and C.H.

Manney, Phys. Rev. Lett. 59 (1987) 2935; D.J. Wineland, W.M. Itano, J.C.

Bergquist, S.L. Gilbert, J.J. Bollinger, and F. Ascarrunz , in Ref. 6),

pp. 93-108.

20) J. Hoffnagle, R.G. DeVoe , L. Reyna, R.G. Brewer, Phys. Rev. Lett. 61

(1988) 255.

21) Th. Sauter, H. Gilhaus, I. Siemers, R. Blatt, W. Neuhauser, P.E. Toschek,

Z. Phys. DIO (1988) 153.

22) D.H.E. Dubin, Phys. Rev. A40 (1988) 1140.

TN-154
Microplasmas

Two or more atoms—stripped of their outer electrons, trapped by

electromagnetic fields and cooled to temperatures near absolute zero—

array themselves in structures that behave like both liquids and solids

by John J. Bollinger and David J. Wineland

In

1973 a container whose "walls" were built from electric and magnetic fields trapped a single elec- tron. Then in 1980 a similar device

confined a single atom. The technol- ogy enabled physicists to measure

the properties of electrons and atoms

in unprecedented detail. The workers

who initiated these experiments, Hans

G. Dehmelt of the University of Washington and Wolfgang Paul of the University of Bonn, shared the 1989 Nobel

prize in physics. Employing the same

control over the temperature and position of atoms, we and our colleagues

are investigating fundamental theories of atomic structure by trapping as many as 15,000 ions (atoms stripped

of one or more of their electrons). The

result is called a microplasma, by ex- tension from the large groups of ions

and electrons known as plasmas.

A microplasma is made by first ap- plying electromagnetic fields to confine the ions to a specified region of

space. A technique called laser coohng

can then cool the trapped ions to temperatures of less than a hundredth of a kelvin. Because microplasmas can be built up practically one ion at a

time, they provide an excellent opportunity to explore mesoscopic systems.

JOHN J. BOLLINGER and DAVID J. WINELAND are physicists at the National Institute of Standards and Technol- ogy in Boulder, Colo. Their work on

microplasmas grew out of efforts to develop techniques for high-resolution

spectroscopy of stored ions. Bollinger received a B.A. in physics from Cornell University and a Ph.D. from Har- vard University in 1981. He enjoys play- ing volleyball and hiking in the Rocky

Mountains. Wineland earned a B.A. in physics from the University of Cahfornia, Berkeley, and a Ph.D. from Harvard

University in 1970. He was a research

associate at the University of Washington and joined the Institute in 1975. An

avid bicyclist, Wineland "plans to race in the Tour de France in his next life."

that is, collections of ions too small

to behave like a familiar, macroscopic system and yet too complex to be

identified with the behavior of a single

ion. Furthermore, microplasmas can

serve as models for the dense plasmas

in stellar objects.

Like the atoms in liquids, the ions in some cold microplasmas can diffuse through a somewhat ordered state. In other cases, the ions can resemble the atoms in solids, diffusing very slowly through a crystal lattice. Yet the nature of microplasmas is quite dif- ferent from that of conventional liq- uids and sohds. Whereas common liq- uids and sohds have densities of about

lO^^atomsper cubic centimeter, microplasmas have concentrations of about

10* ions per cubic centimeter. Consequently, the average distance that separates ions in a microplasma is about

100,000 times greater than the dis- tance between atoms in common liq- uids or solids. Furthermore, whereas

internal attractive forces between the atoms hold a conventional liquid or

solid together, external electric and

magnetic fields hold the trapped ion

microplasmas together, hideed, the

ions, which all have the same charge,

actually repel each other and tend to disperse the microplasma.

The first investigations of these

cold plasmas began more than

a decade ago. In 1977 John H. Malmberg and Thomas M. O'Neil of

the University of California at San Die- go suggested that a collection of elec- trons or ions in an electromagnetic

trap would resemble a type of matter known as a one-component plasma. In such a plasma, a rigid, uniform back- ground of charge confines mobile,

identical particles of opposite charge.

The specific heat, melting point and

other thermodynamic properties of a one-component plasma depend greatly on the density and the temperature

of the mobile particles.

A single dimensionless parameter

called the coupling, which can be

derived from particle densiry and

temperature, describes the thermodynamic properties of a one-component

plasma by providing a measure of how strongly neighboring ions interact. The coupling is defined as the Coulomb potential energy between

nearest neighboring ions divided by

the kinetic energy of the ions. The

Coulomb potential energy depends on

both the average distance betv\'een the ions (a function of density) and the charge of the ion species. The kinetic energy is simply the temperature multiplied by a physical constant known

as the Boltzmann constant. When the Coulomb potential energy

is less than the kinetic energy—that is, when the couphng is less than one—

the one-component plasma should

have no obvious structure and should

behave like a gas. But a one-component plasma whose coupling is greater

than one should show some spatial order. In such strongly coupled one- component plasmas, the ions should

stay away from each other because the repulsive Coulomb forces are greater

than the thermal forces. At couplings

of two or more, a plasma should exhibit liquid behavior. At couplings near 180, a one-component plasma

should change from a Hquid to a solid phase, in which the ions are arranged

in a body-centered cubic crystal.

These theoretical predictions for one-component plasmas pertain to

"infinite" systems, ones whose macroscopic properties do not change w hen

a large number of ions are added

or subtracted. In addition, the predic- tions are valid as long as the ions in the plasma behave classicallv, that

is, as long as the effects of quantum

mechanics can be neglected. Under

conditions of high density and low- temperature, quantum mechanics can be important, as Eugene P. Wigner first investigated in \9^A.

Lxamples of stronglv coupled one- component plasmas can be found in

124 Scientific American January 1990 TN-155
MICROPLASMAS composed of six, nine and 16 ions of mercury grams are based on the predictions by Wayne Itano of the Na-

{top, middle and bottom, respectively) are held in a Paul trap. A tional Institute of Standards and Technology. Although the ions

structural diagram is shown next to each photograph. The dia- keep the same relative positions, they orbit around the z axis.

TN-156
the universe, especially in dense stel- lar objects. The outer crust of a neutron star (the collapsed remnant of a

large star that has exploded as a su- pernova) is expected to contain from

10^^ to 10^^ iron atoms per cubic centi- meter—a density at least 1,000 times

greater than anything on or within the

earth. How does a strongly coupled

one-component plasma form in this environment? The tremendous pres- sure in the star's crust breaks down

the iron atoms into iron nuclei and

free electrons. The iron nuclei behave

classically: one positively charged nucleus simply repels its identical neighbors. On the other hand, the free electrons obey the laws of quantum

mechanics, specifically the exclusion

principle: every electron must occupy

a different energy state. Because of the high density of electrons in a neutron

star, the electrons are forced into very high energy states. The electrons are therefore unaffected by the motion

of the much lower-energy iron nuclei. Hence, they form a uniform density

background of negative charge. The

mobile nuclei in the electron background form a one-component plasma whose coupling is estimated to range from 10 to 1,000.

To learn more about such natural plasmas, workers have attempted to generate a strongly coupled one-com- ponent plasma in the laboratory. Be- cause the thermodynamic properties

of a one-component plasma depend

only on the coupling, a one-com- ponent plasma that is cool and dif- fuse can have the same properties

as a one-component plasma that is hot and dense. Yet until recently one- component plasmas in the laboratory

have not been dense enough or cool enough to become strongly coupled.

In the outer crust of a neutron star, for example, the density of iron nuclei is roughly 20 orders of magnitude greater than the typical density of ions in the trap. To create a one-component

plasma whose coupling matches that of a neutron star, workers must cool

trapped, charged ions to a temperature that is roughly nine orders of magnitude less than that of the star. Attaining a high enough coupling de- mands temperatures well below one

kelvin. In addition to having a coupUng

equal to that of the natural system,

the laboratory one-component plasma

must include enough ions to reveal the behavior that is characteristic of

the many ions in the natural system.

Physicists have recently taken the first step and are now working on achiev- ing the second.

Both efforts involve the technology

of electromagnetic traps. The technol- ogy is roughly 30 years old: in 1959

Ralph F. Wuerker, Haywood Shelton

and Robert V. Langmuir in the labora- tory at Thompson Ramo-Wooldridge,

Inc., in California tested an electro- magnetic trap that confines charged

metallic particles. This beautiful ex- periment demonstrates the effects caused by the strong coupling of the

particles. At about the same time,

workers began trapping electrons and

ions [see "The Isolated Electron," by

Philip Ekstrom and David Wineland;

Scientific American, August, 1980].

In principle an electromagnetic trap can be nothing more than a sphere

uniformly filled with negative charge.

When a positively charged particle is re- leased within the sphere, it is pulled

toward the sphere's center by the uni- formly distributed negative charge.

Overshooting the center, the positive

particle experiences a "restoring" force that is proportional to its dis- tance from the center and eventually

pulls it back toward the center again.

As long as the particle is free to move

through the sphere, it will oscillate about the center. If the positive particle is cooled gradually, however—that

is, if its kinetic energy is decreased—it will oscillate over an ever smaller dis- tance until it settles in the center.

Practical devices can approximate

this ideal trap. Two types of elec- tromagnetic traps in particular—the Paul trap and the Penning trap—can produce strongly coupled one-com- ponent plasmas.

The Paul trap consists of three

metallic electrodes: a ring and

two end caps [see illustration on

opposite page]. The ring is wired to a generator whose voltage fluctuates si- nusoidally at the so-called driving fre- quency. The electrodes are supported

within a vacuum chamber to ensure

that air molecules do not collide with

the ions. How does the Paul trap confine

ions? The sinusoidal ring voltage pro- duces a time-varying electric field be- tween the ring and the end caps. The

direction of the field alternates: half the time the ion is pushed toward

the ring and pulled away from the end caps, and half the time the ion is pulled away from the ring and pushed

toward the end caps. The ion wiggles

sinusoidally at the dri\ing frequency. Because the driving force varies si- nusoidally over time, one might ex- pect the net force on the ion to be

zero. In other words, one might think

that the ion would move back and

forth under the influence of the oscil- lating electric field but that the center

of its oscillation would remain fixed over time. What, then, drives the ion

toward the center of the trap?

The answer stems from the fact that because of the shape of the electrodes, the electric field is weaker at the center of the trap than it is near the

electrodes. By considering a special

case, one can gain a basic idea of how

the spatial variation in the field affects an ion. Suppose that for a time the forced sinusoidal motion of an ion is centered closer to the top end cap than to the bottom end cap. When the ion is at the top of its oscillation, it encounters a strong force toward the center of the trap. When it is at the bottom of its oscillation, the force is toward the top end cap but is some- what weaker. The ion therefore experi- ences a net force toward the center of

the trap—the so-called ponderomotive force. In the same way, the ion will ex- perience a ponderomotive force that tends to drive it toward the center of

the trap if it oscillates below the center or to either side. In the vertical direction the ponderomotive force is called the axial force; in the lateral direction it is called the radial force. At all times, then, an ion in the trap

experiences a driving force and a pon- deromotive force. It turns out that the

axial ponderomotive force is greater

than the radial ponderomotive force. An ion therefore oscillates with three

characteristic frequencies: driving, ra- dial and axial. The trap is usually de- signed so that the radial and axial ponderomotive frequencies are about

10 times lower than the driving fre- quency. Hence, the motion caused by

the driving force is a small, fast wiggle

superposed on a large, slow oscilla- tion about the center of the trap that is caused by the ponderomotive force.

If one disregards the smaller, faster

oscillation caused by the driving force, an ion in the Paul trap moves in the same way it would inside a negativ ely

charged sphere. Because the ponderomotive force differs in the axial direction, however, it is more appropriate

to think of the Paul trap as a spheroid that is oblate (pancake-shaped) or prolate (cigar-shaped). To control the shape of the spheroid and the combined forces on the ions, one can apply an additional constant voltage be- tween the ring and the end caps. When

the overall radial force is greater than

the axial force, the trap acts as a prolate spheroid. Conversely, when the

radial force is less than the a\ial force, then the trap acts as an oblate spher126 Scientific American January 1990

TN-157
END-CAP ELECTRODE

PAUL TRAP creates a time-varying electric field (red lines) be- tween the electrodes. As a generator applies a changing volt- age to the ring, the electric field changes strength and direction, but its shape stays constant. The resulting ponderomotive forces (black arrows) confine charged particles. A laser beam directed at the trap's center cools and probes the ions.

old. Thus, the Paul trap can both confine the ions in the center and orient the collection in the axial or radial

direction. In

1987, working at the National

Institute of Standards and Technology with James C. Bergquist,

Wayne M. Itano and Charles H. Manney, we used a Paul trap to observe strongly coupled microplasmas

of mercury ions. At the same time,

groups led by Herbert Walther and

Frank Diedrich of the Max Planck In- stitute for Quantum Optics in Garching, by Peter E. Toschek of the University of Hamburg and by Richard G. Brewer of the IBM Almaden Research

Center in San Jose, Cahf., were con- ducting similar experiments on vari- ous ion species. At the start of our experiment, a small amount of mercury vapor was

allowed to leak into the vacuum sys- tem containing the Paul trap. As the mercury atoms passed through the

trap, they were bombarded with a beam of electrons. The electrons in the beam had just enough energy to knock a single electron out of any

mercury atom they struck, thereby

ionizing the atom. To cool the ions, we relied on the technique of laser cooling [see "Cooling and Trapping Atoms," by William

D. Phillips and Harold J. Metcalf ; Sci- entific American, March, 1987]. We

generated a laser beam at a frequency

slightly below a frequency that the ions could easily absorb. The ions

traveling toward the laser source, however, "saw" the frequency as be- ing slightly increased because of the Doppler effect. These ions absorbed

the light strongly and slowed down.

The ions traveling away from the source encountered the light at a low- ered frequency; as a result, they ab- sorbed the light weakly and did not speed up much. Overall the average motion of the ions was reduced: the ions were cooled. To observe the individual mercury ions and their spatial structures, we illuminated them with ultraviolet light, which mercury ions scatter

strongly. An ultraviolet video camera

recorded up to 100,000 photons of

ultraviolet light per second and there- by generated a motion picture of the trapped ions. The camera could re- solve details as small as one micron

(one millionth of a meter). At first we worked with only one mercury ion in the trap and cooled it to millikelvin temperatures. The ponderomotive force confined the ion to the center of the trap. When we allowed two ions into the trap, we dis- covered two possible configurations.

If the radial ponderomotive force was

stronger than the axial force, the two

ions lined up in the axial direction, so that they were equidistant from

the trap center. Conversely, if the ra- dial ponderomotive force was weaker

than the axial force, the ions lined up

in the radial plane equidistant from

the trap center. As we added more

ions to the trap, our intuition about

the locations of the ions became questionable. But, with the assistance of a computer to keep track of the many

forces and ions, Itano simulated vari- ous conditions in the trap and accurately predicted the resulting configurations of ions. By

confining ions, we confirmed

that a Paul trap could support

a one-component plasma. Al- though the ions were strongly coupled, as the observed spatial struc- tures demonstrated, we needed to calculate the coupling from measure- ments of density and temperature. We

could easily determine the density of

the ions from images, but we had to measure temperature indirectly. We

observed (as have many other groups

in other experiments) that the motion

of the ions modified the absorption

TN-158
spectrum associated with the ions. (The absorption spectrum of an atom

or molecule reveals the frequencies of

radiation absorbed most strongly by

the atom or molecule.)

A collection of absolutely stationary

ions would have a very sharp absorption spectrum, indicating absorption

only at well-defined frequencies. On

the other hand, if the ions moved

around to some degree, the absorption spectrum would be blurred. The

blurring results from the motion of

the ions toward or away from the

radiation source. From the perspective

of the ions, however, it is the source

that is moving toward or away from

them, and so the frequency of the light

is shifted by the Doppler effect. Thus,

ions moving toward the source will be able to absorb radiation of slightly lower frequencies more effectively

than motionless ions can, and ions

moving away from the source will be

able to absorb radiation of slightly

higher frequencies. The combination

of many ions moving in many direc- tions has the effect of "smearing out"

the spectrum, and the amount of smearing discloses the temperature

of the ions. This technique proved that the temperature of the ions in our trap

approached 10 millikelvins. The couplings, then, were as large as 500. These measurements of the couplings, which indicated that the Paul

trap could support a strongly coupled

one-component plasma, were done

with fewer than about 25 ions in the

trap. We found it difficult to create

similar solid states with more ions. The difficulty arose from the effects of

the oscillation induced in the ions by

the driving force. As an ion oscillates

at the driving frequency, the repulsion

between the ion and its neighbors en- ables it to influence the oscillations of the other ions. This additional perturbation can cause a plasma's structure

to heat to a breaking point under cer- tain operating conditions of the trap. The effect is known as radio-frequen- cy heating, because the driving fre- quency for atomic ions is about the same as the frequency of radio waves. Radio-frequency heating was first ob- served in the experiments of Wuerker,

Shelton and Langmuir.

More recently Reinhold Bliimel and

co-workers at the Max Planck Institute and John A. Hoffnagle and colleagues

at the IBM Almaden Center have studied radio-frequency heating of atomic ions in great detail. These studies

show that a small change in the sys- tem parameters of the Paul trap can cause a sudden transition between

cold, crystalline states and hot, gaseous states. For example, the rate at which the ions are cooled is very sensitive to the frequency of the laser

light employed for cooling. If the laser

is tuned far below the optimal cool- ing frequency, radio-frequency heat- ing transforms the ion plasma into a

hot, disordered state. If the laser fre- quency is increased, the rate of laser cooling increases, until at a critical frequency the laser cools the ion plas- ma quickly enough, so that some or- der starts to appear. At that point the radio-frequency heating diminishes

drastically, and the ion plasma suddenly freezes into an ordered state. This frozen state is quite stable and

will persist even if the laser frequency is again decreased somewhat. The

irregular nature of radio-frequenc>

heating makes it difficult to study the hquid-to-solid phase transitions pre- dicted for one-component plasmas.

The problem of radio-frequency

heating grows as the number of ions

in the trap increases. When many ions

are in the trap, some are pushed to- ward the electrodes, where the dri\ ing

force is stronger. Those ions then oscillate at the driving frequenc> with a large amplitude and thereby increase

the effects of radio-frequenc> heating.

These considerations ha\e limited the number of ions that can be cooled at one time in a Paul trap to about JOO. If

the difficulties associated u1th radio- frequency heating can be overcome,

the Paul trap should allow workers to study the liquid-to-solid phase transi- tion as well as other properties of the

"infinite" one-component plasma.

At present the Penning trap pro- /\ vides a more hospitable en\iron-

±. jL ment for experimenting with

large strongly coupled one-component plasmas than the Paul trap does. Unlike the time-\ar\ing electric fields of the Paul trap, the electric and magnetic fields that confine charged particles in the Penning trap are static. In 1988, with Sarah L. Gilbert, we constructed a Penning trap to confine be- ryllium ions. Malmberg and colleagues

have also empkned a Penning-Upe

trap to confine strongly coupled plas- mas of electrons. Our trap consisted of four c\lindrical electrodes arranged end to end

along a common axis {sac illustration on page 130]. A positi\e \oltage was

applied to the two outer c\lindrical electrodes. This \oltage generated an

electric field between each inner elec- trode and the adjacent outer elec- trode. These fields trapped the ions in the axial direction, near a plane be- tween the inner electrodes. A powerful magnet placed around

the electrodes created a uniform mag128 Scientific American January 1990 TN-159
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SHELL STRUCTURE of a microplasma consisting of about 1,000

beryllium ions is revealed in a photograph (leffi made by shin- ing a laser beam through the microplasma and recording the

resulting fluorescence. The illustrations depict the complete

shell structure as well as the diffusion of the beryllium ions from one moment (center) to the next (right). The microplasma

behaves like both a solid and a liquid: the ions (colored dots) travel around within the shells but not between the shells.

netic field directed along the axis of

the cylinders. The magnetic field pre- vented the ions from leaving the trap

in the radial direction. The radial force

of the electric field near the center of the trap is directed away from the trap

center. This force combines with the

axial magnetic field to cause the ions

to orbit about the trap axis. As the

orbiting ions pass through the magnetic field, they experience a Lorentz

force that is directed radially inward.

The Lorentz force is what confines the ions radially.

Except for this uniform rotation of the microplasma, the confining forces of the Penning trap are equivalent to the confining forces in the uniformly

charged spheroid. Hence, even though

ions in a Penning trap rotate, they

behave like ions in a one-component

plasma—in particular, they have the same thermodynamic properties.

To begin the experiment, we pro- duced beryllium ions by a method

similar to that described for making

mercury ions in the Paul trap. The ions were cooled by two intersecting la- ser beams. A third laser beam, called the probe, was employed to measure

the temperature of the ions. The light scattered by the ions was collected to make an absorption spectrum. As in the mercury-ion experiment, the temperature of the ions could be deduced

from the blurring of certain features

of the spectrum. This technique re- vealed that the ions had been cooled

to below 10 millikelvins. The rotation frequency of the beryllium microplasma could also be de- duced from the spectrum. Beryllium

ions circulated inside the trap at a rate of from 20,000 to 200,000 rotations per second. Because the rotation fre- quency is directly related to the radial

electric fields, which are in turn relat- ed to the ion density, we were able to calculate that the ion density ranged

from 50 to 300 million ions per cubic

centimeter. From ultraviolet images of

the plasmas we could determine the volume occupied by the trapped ions

and therefore the number of trapped

ions. The temperature and density

measurements yielded couplings as

large as 200 to 400 for less than about

15,000 ions in the trap. We expect a phase transition from a liquidhke to a sohdlike

state to occur at a coupling of 180 for a one-component plasma with an infinite number of ions. Our measured values for the coupling indicate

that the trapped ions should form a crystaUine ion sohd, if the trap contains enough ions. Should a system of 15,000 trapped ions, though, behave

hke an infinite system?

Recently some computer simulations have elucidated this question.

The late Aneesur Rahman of the University of Minnesota at Minneapolis,

John P. Schiffer of the Argonne National Laboratory, Hiroo Totsuji of Okayama University and Daniel H. E. Dubin

and O'Neil of the University of Califor- nia at San Diego have created simulations of a trapped plasma containing

as many as several thousand ions. The

simulations reveal several remarkable

features. When the couplings exceed

one, the ions are concentrated in concentric shells, which are spaced evenly. For couplings around 10, the shells are in a liquid state characterized by

short-range order and diffusion in all directions. As the coupling increases, the shells become more clearly defined; the ions diffuse quickly within

the shells and slowly between the

shells. For high couplings (above 200), the diffusion of the ions within a shell slows down and the ions form a solidlike state. Instead of showing a sharp

phase transition, the plasma evolves gradually from a liquidlike to a solidlike state. Experiments have confirmed these

predictions. Even though the plasma

of beryllium ions rotates around the

axis of the Penning trap, the shell structure is preserved in the radial

direction. The scattered light from

TN-160
a laser beam cutting across the plasma shows alternating bright and dark

bands corresponding to the shells. We

looked for shell structure in plasmas

that contained as few as 20 ions and

as many as 15,000 ions. In a plasma of 20 ions, a single shell was clearly observed. In a plasma of 15,000 ions, we

could distinguish 16 shells. So far we

have not detected any distinct struc- ture within a shell because of the rota- tion of the plasma in the trap. We were able to test predictions that

for couplings around 100, the plasma

will act like a liquid within a shell but

like a sohd between the shells. In par-

.'/

COOLING BEAM

COOLING BEAM

PROBE BEAM

LENS- SYSTEM

IMAGER

PENNING TRAP generates electric fields {red lines) and magnetic fields (green lines) to produce forces (black arrows) that confine charged particles. The electric and magnetic fields also cause the microplasma to rotate. Two laser beams cool the particles; a third serves as a probe for various experiments. The magnetic field arises from

an electric current flowing through a solenoid (not shown) that encircles the trap.

ticular, the ions should diffuse fast- er within shells than between shells. To demonstrate the effect, we optically "tagged" the ions by tuning the probe laser to a specific frequency.

The probe laser suppresses the emission of light from the ions it strikes by

placing them in a "dark" energy state

in which they do not scatter light from

the cooUng laser beams.

First we darkened an outer shell of

the plasma and measured the time

required for the dark ions in the outer

shell to move to the inner shells. Then we darkened part of the plasma across

several shells and measured the time

required for dark ions in one part of

a shell to move to other parts of the same shell. These measurements veri- fied that for moderate couplings the

diffusion of ions between shells is more than 10 times slower than the

diffusion of ions within a shell. Many questions about microplasmas are still unresolved. At what point will the behavior that is characteristic of an infinite system start to appear? How many

ions are required for the system to exhibit a sharp phase transition? At what stage will the soUd state become

a body-centered cubic lattice rather than a collection of shells? At present these questions are diffi- cult to answer even in theory. Dubin

predicts, however, that perhaps as many as 50 to 60 shells may be re- quired before a body-centered cubic

lattice would become an energetically favorable configuration. That would

require about a million ions, more

than 50 times the number of ions in the largest, strongly coupled microplasmas created so far. Current technology' should be able to confine cold plasmas of this size. If coupUngs of

200 or more can be maintained for a plasma of a million ions, we may be

able to visit the surface of a neutron

star in a laboratory on the earth.

FURTHER READING

Non-neutral Plasnu Pm sics. Edited by

C. W. Roberson and C. F. Driscoll. .MP Conference Proceedings 175, American

Institute of Physics, March, 1988. Proceedings of the Eleventh lNTt:RNA- TiONAL Conference on Atomic Pfhsics, July 4-8, 1988, in Atomic Phvsics

11. Edited by S. Haroche, J. C. Gay and

G. Gr>Tibcrg. World Scientific l^iblish- ingCo., 1989. Proceedings of the Yam-\da Confer- ence on Strongly Coupled Plas- ma Physics, August 29-Septembf.r 2, 1989. Edited by Setsuo Ichimaru. Else- vier Science Publishers, in press.

130 Scientific American January 1990 TN-161
Volume 66, Number 16 PHYSICAL REVIEW LETTERS 22 April 1991

Rotational Equilibria and Low-Order Modes of a Non-Neutral Ion Plasma

D. J. Heinzen,**^ J. J. Bollinger, F. L. Moore, Wayne M. Itano, and D. J. Wineland

Time and Frequency Division, National Institute of Standards and Technology, Boulder, Colorado 80303

(Received 4 October 1990)

We study rotational equilibria and low-order electrostatic modes of a magnetically confined, nonneutral ion plasma. The plasma rotation rate is controlled with radiation pressure from a laser beam and

is continuously varied over the entire allowed range, including Brillouin flow. Excitation of an asymmetric plasma mode by a static field asymmetry is observed. The symmetric quadrupole mode is also studied; its behavior is characteristic of a strongly magnetized plasma at low density, and of an unmagnetized plasma at Brillouin flow.

PACS numbers: 52.25. Wz, 32.80.Pj, 52.35.Fp

Recently, much attention has focused on the properties of plasmas containing charges with only one sign.

Such non-neutral plasmas are unusually simple both

theoretically and experimentally, and are unique in that steady-state thermal equilibria are possible. Nevertheless, they exhibit a wide range of collective phenomena,

such as Debye shielding and collective oscillations. Typically, non-neutral plasmas are confined radially by a uni- form axial magnetic field B=Bz, and axially by an electrostatic potential.^ Such a plasma may be characterized by a self-field parameter^ S = 2(0p/il^, where (Op "{.^nnq^/m)^'^ is the plasma frequency, with n, q, and

m are the particle density, charge, and mass, respectively, and ft ~qB/mc is the cyclotron frequency. Of partic- ular theoretical'"^ and practical** interest is the condition

5 — 1, referred to as "Brillouin flow."^ In this state the

plasma rotates rapidly about the z axis at a frequency

ft/2 and is compressed to its maximum possible density

of nB~mil^l%7iq^, the "Brillouin density." Near Bril- louin flow, the plasma behaves in many ways like an un- magnetized plasma.^ Values of S near unity have been

obtained in only a few experiments^'* where the plasma

confinement times were not long enough to reach ther- mal equilibrium.

This paper describes the use of laser-induced torques on a magnetically confined non-neutral ^Be"^ plasma to access the entire range of allowed rotational thermal

equilibria including Brillouin flow. Geometrical errors in the trapping fields tended to limit the range of accessible

rotational equilibria. A static, azimuthally asymmetric

field error was observed to excite an asymmetric collective resonance and heat the plasma. When this asymmetry was eliminated, the techniques described here were used to study a symmetric quadrupole mode of the plasma over a wide range of conditions including Bril- louin flow. The experimental apparatus (Fig. I) has been described previously.^ Electroformed copper cylinders pro- duce a trap with accurate cylindrical symmetry. A potential Vj of between 10 and 300 V, applied between the end and central electrodes, confines the ions axially. The

potential near the trap center is

H,,.)-^ m(02

2q

.2-£_ (I)

where ^7^ = 1.85 cm and (Oz is the frequency at which a

single trapped ion oscillates along the z axis. The trap

is placed in a uniform magnetic field B—Bz, with B

=0.8194 T, which confines the ions radially. The 'Be"*" cyclotron frequency is ft/2;r = l.40 MHz. Background

gas pressure is approximately 10 ~* Pa. For this work,

between 1000 and 5000 'Be"*" ions are confined in the trap for many hours. The plasma dimensions are typically less than 0.1 cm. We remove contaminant ions with a mass-to-charge ratio greater than 'Be "*" by raising

Vt to -550 V (a)z/ft~0.67 for 'Be"^). Agreement be- tween the predicted and observed mode frequencies is obtained only when the contaminant ions are removed.

The ion plasma evolves into a near-thermal equilibrium state characterized by a uniform rotation of the plasma about the z axis at frequency (o. Under the lowtemperature conditions of this experiment, the density is

Microwave Horn

Torque Cooling

\ Beam I Beam

Plasma

Imager Lens System

FIG. 1. Schematic drawing of the experimental apparatus. For clarity, the electrodes are shown in a cross-sectional view,

and the diagonal laser beam is omitted. The size of the plasma

is exaggerated.

2080

Work of the U. S. Government

Not subject to U. S. copyright

TN-162
Volume 66, Number 16 PHYSICAL REVIEW LETTERS 22 April 1991

uniform over length scales large compared to the inter- particle spacing and the plasma frequency is given

by co^=2ft>(ft — ft))." The density falls to zero at the boundary of the plasma in a distance on the order of the Debye length,^ which in our experiment is much less than the plasma dimensions. With the confining potential of Eq. (1), the plasma boundary is spheroidal with an aspect ratio a^zoAo determined by co from the ex- pression 0)} =(o^Q^(a/(a^ - \)^'^)/ia^ - \)^ Here 2zo

and 2ro are the maximum axial and radial extent of the plasma and QP is the associated Legendre function of

the second kind. We neglect the effect of image charges

because the plasma dimensions are much smaller than

the trap dimensions.

The plasma density and temperature are controlled with radiation pressure from two cw laser beams nearly resonant with the 2s^S\/2imi = + j ,mj = + j) — 2p^Py2(+ i,+ I ) transition of ^Be"*" at frequency

ft^o (A. « 3 1 3 nm), where m/ and mj are the nuclear- and

electron-spin projections along z. One of these, the "cooling beam," is tuned 10-50 MHz below coq, has a power of 50-500 juW, and is directed through the plasma center in a direction perpendicular to the z axis. The

second beam, the "torque beam," is directed perpendicularly to the z axis through the side of the plasma that re- cedes from the laser beam (Fig. 1). Its frequency (Ot is between and 1500 MHz above (oq and its power is 1-3

/iW. For some of the measurements, a third laser beam

is directed diagonally through the plasma with a component along z, to provide additional laser cooling.^

Each beam diameter is 25-50 nm, an order of magnitude less than the typical plasma diameter. The torque

beam supplies energy to the plasma and provides a torque which changes the plasma angular momentum

and increases (o. Equilibrium is maintained because the energy input by the torque beam is removed by the cool- ing beam, '° and because the laser-beam torques and

torques from static field asymmetries sum to zero. ""'^ Temperatures typically below 250 mK are obtained.

Light scattered from the ions is imaged by a lens sys- tem onto a photon-counting imaging tube.^ A real-time

display of the image is used to monitor the plasma kinetic energy; a hot plasma has a more diffuse boundary and

less ion fluorescence. With this simple diagnostic we did

not, in general, distinguish between an increase in the random thermal energy of the ions and the excitation of coherent modal motion of the ions. In order to measure

the rotation frequency of the plasma, we drive the (-l-y,-l-y)—>-(-hy,— y) electron spin-flip transition of the 'Be''" ground state at ioj2n=21 GHz. The transition is observed as a decrease in the ion fluorescence.''' Absorption at both the "carrier" frequency ft), and the motional "sideband" frequencies ft;^ ± ft) are observed.

These sidebands occur because the ions see the mi- crowave field as being phase and amplitude modulated at frequency (o, due to the coherent rotational motion of the

ions." The rotation frequency (i)/2n is determined to an accuracy of approximately 5 kHz.

With the two-laser-beam technique, we are able to es- tablish a stable rotation frequency at any value in the allowed range from ftj^ to H— &;„,, where (Or„ = il/2 — (nV4 — ftJrV2)' is the single-ion magnetron frequency." Initially, with w small, the torque laser frequency

(Ot is tuned near (oq, and the torque beam induces very

little fluorescence due to the rotation-induced Doppler

shift. Then, as cot is slowly increased, the ion fluores- cence and torque increase, and the plasma compresses

and rotates faster. For co < ft/2, the plasma radius de- creases and the aspect ratio a increases with increasing

ft). A typical curve of rotation frequency versus torque

laser tuning is shown in Fig. 2. At high rotation fre- quencies hysteresis occurs. On the lower branch (increasing coj), the plasma becomes much hotter and ro- tates at a nearly constant frequency, until an abrupt

transition to a cold state of faster rotation occurs. With

further increases in cut, the rotation frequency can be

smoothly varied through Brillouin flow (ft) = ft/2) to fre- quencies slightly less than Cl—com- For ft) > ft/2, the plasma radius increases and the aspect ratio decreases

with increasing cd. On the upper branch of the hysteresis (decreasing wt), the rotation frequency decreases

smoothly, and the plasma remains cold until some heating is again observed, at a rotation frequency slightly

higher than that at which heating first occurred on the lower branch. The size of the hysteresis depended sensitively on the angle 9q between the trap symmetry axis and the magnetic field. For 0o>O. 1°, the plasma rotation frequency could not be increased beyond the point at which heating first occurred. We could make Oq < 0.01°

by searching for an alignment which gave no apparent

heating or hysteresis.

1000

N

C\2

800

600

400

200

— Brillouin flow

Heating resonance

limits —1

FIG

200 400 600 800 1000

Torque laser detuning {u> -u^)/2n (MHz)

2. Plasma rotation frequency as a function of torque

laser frequency. Arrows indicate the direction of the frequency

sweep. Insets A and B illustrate the appearance of hot and

cold plasmas, respectively. Image A was recorded on the lower branch of the hysteresis, and image B on the upper. Bright

fluorescence from the cooling and diagonal beams, and weak

fluorescence from the torque beam, is visible.

TN-163

2081
Volume 66, Number 16 PHYSICAL REVIEW LETTERS 22 April 1991

The rotation frequency co at which heating occurs for 00 —0.02° is shown in Fig. 3 as a function of co,. The

lower bounds indicate where heating first occurs with in- creasing rotation frequency and the upper bounds indi- cate where heating first occurs with decreasing rotation

frequency. We have identified this heating resonance as an excitation of a collective (/,m)=(2, 1) plasma mode

by the static field asymmetry associated with the trapmagnetic-field misalignment. [The indices / and m refer

to a description of the modes in spheroidal coordinates,

with m describing the azimuthal (<p) variation, and / the

variation along a spheroidal surface in the direction per- pendicular to ^.] This mode is similar to an m = l diocotron mode in a cylindrical plasma column with a wavelength equal to the plasma length.'^ In this mode

the plasma density remains constant and the shape

spheroidal, but the plasma symmetry axis is tilted with

respect to the z axis and precesses backwards relative to the plasma rotation at frequency w. That is, the mode

frequency ft>2i°'^ in the rotating frame is negative.

Therefore when (02°^^=— (o, the mode has frequency

CO21 =0 in the laboratory frame and may be excited by

static field asymmetries.""'^ We show as a solid line in Fig. 3 the calculated rotation frequency (o at which

ft)2i =0. Excellent agreement between the predicted and

observed rotation frequencies is obtained. We calculate''' 0)21 by considering small axial and radial displacements of the ions from their equilibrium positions consistent with a tilt of the plasma symmetry axis. A self- consistent calculation of the axial and radial restoring

forces to first order gives two linear differential equations

describing the motion of the axial and radial displace- ments. The requirement that the eigenfrequencies for these two equations be equal gives a cubic equation for

(02\.A static field error cannot transfer energy to the plasma. However, it can convert potential energy to thermal

energy by producing a radial expansion of the plasma.

In our work, the laser beams are sources of strong

torques and energy input as well as cooling. We have

not attempted to model all of the energy input and

torques and obtain a quantitative description of the radial transport and heating produced by the static (2,1)

mode.

We have also measured the (2,0) axially symmetric

quadrupole mode frequency of the plasma. This mode is similar to an m —0 plasma mode in a cylindrical plasma

column with a wavelength equal to the plasma length. In this mode the plasma density remains uniform and the shape spheroidal, but the plasma aspect ratio a oscillates

in time. The mode is excited with an oscillating potential applied between the trap's end and center electrodes and resonance is observed by a change in the ion fluores- cence. Results are shown in Fig. 4 for two different trap

potentials. The solid line shows the mode frequency (O20 calculated in a manner similar to the (2,1) mode and

given by'^

2(U2^o = n,2-^ft>^-{(n2-fi,^)2-4n.?((y,^-4/3)}'/2.

(2X

Here n,. = ft— 2w is the vortex frequency, nu=(fi,?

+ (0p)^^^ is the upper hybrid frequency, and (OQ=Q)p

X 3aQHa/(a^- 1 ) ^^^)/ia^- 1 ) ^^l The frequency coq is the (2,0) mode frequency in the absence of a magnetic

field (i.e., a plasma confined by a uniform background of

opposite charge). Again, good agreement between the

predicted and observed mode frequencies is obtained

with no adjustable parameters. Independently, a more

general calculation of spheroidal plasma mode frequencies has been recently carried out by Dubin. '* Figure 4 shows two additional calculations. In the

FIG. 3. Rotation frequency (o at which heating was observed as a function of the single-particle axial frequency Wz. Both frequencies are expressed in units of the cyclotron fre- quency n. The error bars on the measurements are explained

in the text. The solid line indicates the calculated rotation fre- quency ft) at which W21 =0. This is a universal curve involving no adjustable parameters.

0.40

0.35

^ 0.30 o

a' 0.25 \

0.20

0.15

0.0

'T:....A.. r,

* u^/n = 0.256

• ^2/0 = 0.151

.r^^W*^

0.1 0.2 0.3 0.4 0.5

FIG. 4. Plasma quadrupole frequency 020 as a function of

rotation frequency co for ftj.Vft =0.151 (Fr=28 V) and cojil

=0.256 (f^r "80 V). All frequencies are expressed in units of

the cyclotron frequency fl. The circles and triangles give the experimental data. The solid lines give the cold-fluid model

predictions for 6)20- The dashed and dotted lines give the highand low-magnetic-field calculations for coio, respectively.

2082

(Fig. 4 has been corrected as per erratum: Phys. Rev. Lett. 66, 3087 (1991).

TN-164
Volume 66, Number 16 PHYSICAL REVIEW LETTERS 22 April 1991

first, shown as dashed lines, the magnetic field is as- sumed to be effectively infinite; that is, the ions are not allowed to move radially, and the mode frequency is cal- culated assuming a simple axial stretch of the charged

spheroid. In the second, shown as dotted lines, the magnetic field is assumed to be effectively zero; that is, the curve shows o)q. The quadrupole mode frequency varies

smoothly from the high-magnetic-field calculation at low

rotation frequencies to the zero-magnetic-field calculation at the Brillouin limit. We also studied plasmas containing about 40000

'Be"*" ions at a magnetic field fi-6 T («fl-l.lxlO'°

cm"^) and trap voltage Kr -500- 1500 V. At this

field, we were also able to obtain rotation frequencies

throughout the allowed range. The (2,1) heating reso- nance was much stronger and more sensitive to the

magnetic-field alignment. In addition, weaker heating

resonances, which were also sensitive to the magneticfield alignment, were observed at lower rotation frequencies than the (2,1) heating resonance. We were unable

to remove contaminant ions from the plasma at this high

magnetic field. In conclusion, we have shown that the entire range of

allowed rotational equilibria can be realized in a nonneutral ion plasma using laser-induced torques and by

careful control of trap asymmetries. This should permit

detailed investigations of non-neutral plasmas near Bril- louin flow. Here we studied the excitation of an asymmetric tilt mode of the plasma by a static field asymmetry. This mode heated the plasma and tended to limit the plasma density. Observation of this mode allows the

magnetic- and electric-field axes to be carefully aligned.

We also studied a symmetric quadrupole mode and

showed that the plasma exhibits unmagnetized behavior near Brillouin flow. Measurement of this mode frequency, perhaps through induced image currents in the trap

electrodes,'^ could provide useful information on plasma

rotation frequency, shape, and density when the plasma

cannot be directly imaged.

We gratefully acknowledge the support of the Office of

Naval Research. We also thank Dan Dubin and Fred

DriscoU for stimulating conversations, and Sarah Gilbert

and Jim Bergquist for their comments on the manuscript.

'''^Current address: Department of Physics, University of Texas, Austin, TX 78712. ^Non-Neutral Plasma Physics, Proceedings of the Symposi- um of Non-Neutral Plasma Physics, Washington, D.C., 1988,

edited by C. W. Roberson and C. F. Driscoll, AIP Conf. Proc. No. 175 (AIP, New York, 1988).

2j. H. Malmberg and J. S. deGrassie, Phys. Rev. Lett. 35, 577 (1975).

'R. C. Davidson, Physics of Nonneutral Plasmas (AddisonWesley, Reading, MA, 1990); R. C. Davidson, in Non-Neutral

Plasma Physics (Ref. 1).

"* Wayne Vernon, in Proceedings of the Workshop in Intense Positron Beams, Idaho Falls. Idaho. 1987, edited by E. H. Ottewitte and W. Kells (World Scientific, Singapore, 1988), p. 135.^A. J. Theiss, R. A. Mahaffey, and A. W. Triveipiece, Phys. Rev. Lett. 35, 1436 (1975).

^Guy Dimonte, Phys. Rev. Lett. 46, 26 (1981).

^S. L. Gilbert. J. J. Bollinger, and D. J. Wineland, Phys. Rev. Lett. 60, 2022 (1988).

*L. R. Brewer, J. D. Prestage, J. J. Bollinger, W. M. Itano, D. J. Larson, and D. J. Wineland, Phys. Rev. A 38, 859

(1988). 'C. F. Driscoll, J. H. Malmberg, and K. S. Fine, Phys. Rev.

Lett. 60, 1290(1988).

'°W. M. Itano, L. R. Brewer, D. J. Larson, and D. J. Wineland, Phys. Rev. A 38, 5698 (1988).

"Rhon Keinigs, Phys. Fluids 24, 860 (1981); 27, 1427

(1984).

'^C. F. Driscoll and J. H. Malmberg, Phys. Rev. Lett. 50, 167

(1983).

'^D. L. Eggleston, T M. O'Neil, and J. H. Malmberg, Phys. Rev. Lett. 53, 982(1984).

"D. J. Wineland, J. C. Bergquist, W. M. Itano, and R. E. Drullinger, Opt. Lett. 5, 245 (1980).

'^H. S. Lakkaraju and H. A. Schuessler, J. Appl. Phys. 53, 3967 (1982).

'^S. A. Prasad and T M. O'Neil, Phys. Fluids 26, 665

(1983).

'^J. J. Bollinger et al. (to be published).

'*D. Dubin, preceding Letter, Phys. Rev. Lett. 66, 2076

(1991).

"D. J. Wineland and H. Dehmelt, Int. J. Mass Spectrom. Ion Phys. 16, 338 (1975); 19, 251 (19/6).

2083

TN-165
Cooling methods in ion traps^

Wayne M. Itano, J. C. Bergquist, J. J. Bollinger, D. J. Wineland

Time and Frequency Division

National Institute of Standards and Technology

Boulder, Colorado 80303

I. Introduction

Trapping of ions is used to make measurements that are

difficult or impossible to perform by other techniques, be- cause the ions can be held for long periods in a well- controlled environment. Cooling of the ions has several

beneficial effects for such experiments. First, it increases

the time that the ions remain in the trap. Second, it can

lead to increased precision for measurements of masses, magnetic moments, and optical or microwave spectra.

Third, some phenomena, such as the formation of spatially ordered structures of ions, can be observed only at low temperatures.

We use the term "cooling" to mean a reduction in the velocities of the ions. Use of the term does not imply

that the ions are in thermal equilibrium with each other nor that they have a well-defined temperature. The motional modes that are cooled may be random or coherent.

The secular motion in a Paul (rf quadrupole) trap and the

axial motion in a Penning trap are examples of random

modes. The micromotion in a Paul trap and the magnetron rotation in a Penning trap are examples of coherent

modes. Laser cooling is very efi^ective and has been used

to obtain temperatures much less than 1 K.^^^'^ ' However,

since laser cooling depends on the energy level structure

of the ions and on the availability of narrowband radiation sources matched to the level spacings, it has been

applied only to a few kinds of ions. Other cooling methods are more generally applicable.^ ''^^' Resistive cooling,

active feedback cooling, or rf sideband cooling can be applied to all ions, since they depend only on the charge and mass of the ions. Collisional cooling is also widely applicable. Sympathetic cooling is the cooling of one ion species

through Coulomb collisions with another, laser-cooled ion

species. It heis been demonstrated with ions in Penning

traps^ ''^ ' and for (a few) ions in Paul traps. ^

' In spectroscopy the main reeison for cooling ions is to reduce Doppler shifts. If the transition is broadened

by the first-order Doppler shift, which is linear in velocity, cooling the random motion will reduce the observed

linewidth. Methods other than cooling can eliminate the

first-order Doppler shift. Dicke narrowing' ' and two- photon absorption' ' are two examples. Reduction of the

.second-order (time dilation) Doppler shift, however, re- quires cooling.

'Work of the N.I.S.T. Not subject to U.S. copyright.

Dicke narrowing has been applied to many ion trap

experiments. It occurs when the atoms or ions are confined to spatial dimensions less than about half the wavelength of the radiation that excites the transition. The

lineshape contains a sharp component, which is free of the

first-order Doppler eflfect. In addition, the lineshape contains a broad pedestal or a discrete series of sidebands,

depending on whether the motion of the ions is random or

periodic. The pedestal or sidebands are reduced in intensity as the spatial confinement is improved.

For radiofrequency or microwave transitions, this ef- fect can be seen without any special cooling techniques.

As long £is the ions are confined in the trap, they are also confined to dimensions less than the wavelength. Hyperfine resonances a few hertzes wide (Q's of 10^ to 10^") were observed in early work with uncooled ^He"^ ions' ' and with ^^^Hg"*" ions.' ' For optical transitions, which

have much shorter wavelengths, cooling to temperatures

less than 1 K is required to obtain Dicke narrowing. A

Dicke-narrowed optical spectrum has been observed in laser-cooled, trapped Ba"*" ions' ^ and in Hg"*" ions.' ' Precise mectsurements of masses and ^^-factors also benefit from cooling.' ^~' ' Reduction of the velocities lowers uncertainties due to the relativistic mass increase

and due to anharmonicities of the electric potentials. Cooling the magnetron motion in a Penning trap confines ions

to regions of smaller spatial extent. Over a small region,

the magnetic field can be more uniform and the electric potential more nearly quadratic.

Cooling of trapped ions is also useful in studies of plasmas and ordered structures. With sufficient cooling so that the Coulomb potential energy between nearest neighbors is much greater than their average kinetic energy, the

ions can form ordered structures. Ordered, crystal-like structures of highly-charged micrometer particles of alu- minum in a Paul trap were observed when the particles were cooled with background gas.' ' Similar structures of a few laser-cooled atomic ions have recently been observed

in Paul traps.' '' ^~' ^ A concentric shell structure of several thousand laser-cooled Be"'' ions hais been observed in a Penning trap.' '

II. Resistive cooling

Resistive cooling involves connecting the trap electrodes to an external circuit, so that the ions dissipate their energy

by inducing electric currents in the circuit. If there are

TN-166
no sources of heating besides the coupling to the external

circuit, the ions come to equilibrium at the temperature

of the external circuit. The method is, in principle, applicable to all ions. The cooling time can be estimated from a simple

model.^

^ Consider a single ion of mass m and charge q. The ion is bound by a harmonic restoring force to the in- terior of a parallel plate capacitor. The separation between

the plates is 2zo. The ion oscillates in the direction per- pendicular to the plates (the z direction) at a frequency

u)z. The plates of the capacitor are connected by a resistor R. The capacitance C of the plates is assumed to be small,

so that R <C l/{uizC). If the ion moves with a velocity Vz, electric charges are induced on the plates, and a current

i = qVz/2zo flows through RP^^ Let Wz = rn{v^) be the

total (kinetic plus potential) energy of the ion. The brackets denote a time average. The average rate at which the

moving ion does work on the external circuit is dWz

dt (i'R) =

q^RWz

Amzl

The energy is damped with a time constant equal to

tQ =

AmzQ

q^R

'

(1)

(2)

This model can be applied to the axial (z) motion

in either a Penning or a Paul trap. In those cases, the

external circuit is connected between the two endcap elec- trodes. It could also be applied to the x or y mode in a Paul trap or the cyclotron mode in a Penning trap. In such

a case, the ring electrode must be split, and the external

circuit connected between the two halves. This method

cannot be applied to the magnetron mode in a Penning

trap, because the energy in the magnetron mode, which is mostly potential, decreases as the amplitude of the mode

increases. Dissipation of energy leads to an increase in the

diameter of the magnetron orbit and an increase in the

magnetron velocity. As will be discussed in Section III, active feedback can be used to reverse this process. In a real trap, as opposed to an infinite parallel

plate capacitor, the expression for the induced current

is i = BiqVz/{2zo), where Bi depends on the geometry

of the electrodes. The cooling time to is proportional to 5j~ . Methods for calculating Bi have appeared in the

literature. ^^'^^'^^^^ For typical trap designs, Bi « 0.8. Equation (2) shows that resistive cooling is most eff'ec- tive for ions with high charge-to-mass ratios. For a given mass and charge, the cooling time can be reduced by de- creasing zo or increcising R. In practice, the high value

of R is usually obtained by using a parallel inductance L

to cancel the reactance due to the capacitance C of the

trap electrodes. The effective value of i? is then Q/{uzC),

where Q is the quality factor of the tuned circuit. Typically, Q s» 1000. However, Cornell et al}^ ' have used

a superconducting tank circuit with Q — 2b 000. This

reduces the axial cooling time to 6 s for N^ ions. Consider A'^ independent ions in a trap. Let i =

Ylj-i b

^^ ^^^ total current induced by the ions in the

external circuit, where ij is the current induced by the

jth ion. Since the ions are assumed to be independent,

(ijik) = ^jjt^o, where the brackets denote a time average

and where I'o is the rms current induced by a single ion. Thus, the total mean-squared induced current, and hence

the rate of energy loss, is N times that of a single ion: N N N

^") = EE(b--o-EoJ) = ^^-o- (3)

i = l Jb = l j=l

The average energy of N ions is N times the average energy

of a single ion. Thus, the time constant for damping the

energy of N independent ions is the same as that for a

single ion, provided that the relaxation of energy between

the center-of-mass mode and the internal modes is fast. This is not necessarily the case, particularly if the trap

potentials are very nearly harmonic.

Resistive cooling of the axial motion of a gas of elec- trons in a Penning trap was observed by Dehmelt and

Walls. ^

' The cooling time was about 0.1 s. Since there was very little heating from other sources, the temperature

of the electrons could be brought down to the temperature

of the external circuit. This system was studied in more

detail by Wineland and Dehmelt .^^^ Resistive cooling of protons in a Paul trap was demonstrated by Church and Dehmelt.^ ' The temperature with- out resistive cooling was about 12 000 K. Coupling to an

external circuit reduced the temperature to 900 K and in- creased the storage time from 100 s to 3000 s. The cooling

time was a few seconds. Heating caused by the rf trap

fields ("rf heating") prevented cooling to the temperature

of the external circuit. Resistive destabilization (heating) of the magnetron

motion of electrons in a Penning trap was observed exper- imentally by White, Malmberg, and DriscoU.'^^^ A resis- tance was connected between two sections of the cylindrical ring electrode. The magnetron mode is a displacement

of the center of the electron plasma from the central axis of the trap. The magnetron mode is also called the / = 1, kz — diocotron mode, since it is an excitation that has an e''^ azimuthal dependence, where / = 1, and no dependence on 2. The displaced plasma rotates around the axis

due to the drift caused by crossed electric and magnetic

fields {E X B drift). This displacement grew exponentially

with time. The observed variation of the growth rate with

resistance was in agreement with theory.

III. Active-feedback cooling

Cooling trapped ions by negative electrical feedback has

been discussed theoretically and demonstrated experimentally, though it has not yet been widely applied. Like re- sistive cooling, it can be applied to all ions. Dehmelt ei al} ' have treated theoretically the case

in which the feedback is applied continuously, to a single,

harmonically bound ion. A displacement of the ion from

its equilibrium position induces a signal on, for example, one of the endcap electrodes. Let this signal be amplified

TN-I67
by a factor g and fed back with reversed phase to the

other endcap electrode. The fed-back signal forces the ion

back toward the equilibrium position. The cooling time of

the ion is (1 + g) times shorter than for ordinary resistive cooling. However, there is a trade-off between the cooling

rate and the final temperature. The thermal noise voltage

of the external circuit resistance is also amplified, so the

final temperature is (1 -l-^f) times higher than for resistive cooling. One potential problem with this method is that

the correction signal can be picked up by the detector and

interfere with the signal from the ion. This effect could be

reduced by coupling part of the correction signal, with a phase reversal, to the detector. If there is more than one

ion, the cooling is not as straightforward. To the extent

that the electric field from the electrodes is uniform over

the ion cloud, only the center of mass of the ions can be

detected and corrected, not the deviations of the individual

ions from the center of mass. Continuous feedback has

been used to cool ions in Penning traps, but details have

not been published. (See footnote 26 of Ref. [4].) Resistive destabilization of the magnetron motion of an electron plasma in a Penning trap was described in Section II. The magnetron mode can be damped with active feedback similar to that described in the preceding

paragraph. A signal is picked up on one section of the ring

electrode and fed back on another section, with the proper

gain and phase shift. Cooling of the magnetron mode by

this method has been demonstrated experimentally.^ ''^ ' The magnetron mode is coherent; that is, the displacements of the individual ions have the same phase. Thus,

the cooling of a plasma is similar to that for a single ion. Stochastic cooling is a kind of negative feedback cool- ing which has successfully been applied to antiprotons in storage rings. ^ ' In stochastic cooling, a signal proportional to the average displacement of the particles is picked

up on one section of the storage ring. It is amplified and

fed back to a set of electrodes, called a kicker, at another

section of the ring. The timing is adjusted so that the

signal arrives at the electrodes at the same time as the

particles. If there were only a single particle in the ring,

its displacement from the desired orbit could be brought

to zero by proper adjustment of the gain and phase of

the amplifier. In practice, there are many particles, whose

signals are not resolved. At best, only the center-of-mass

deviation of a sample of particles can be brought to zero

in a single application of feedback. Stochastic cooling de- pends on having a spread in the energies of the particles, so that, after a period of time, called the mixing time, the

center-of-mass deviation again becomes nonzero and more

energy can be extracted. This mixing time is analogous

to the time required for coupling of energy between the

center-of-mass and internal modes in an ion trap.

Stocheistic cooling has been extended to traps by Beverini ei al} '~^ Mn a trap, the pick-up and kicker elec- trodes cannot be separated physically, as they can be in a storage ring. In order to avoid feed-through from the

kicker to the pick-up, the detection and correction are sep- arated in time from each other. The amplitude and phase

of the center-of-mass motion is measured. Then a short

electric pulse is applied, with the proper amplitude and

phase to stop the center-of-mass motion. The process is then repeated.

To show that cooling can be obtained by this method, we consider a simplified model.^ ' Consider N noninteracting ions, each oscillating at a fixed frequency Uk

{k = 1,2, ... ,N). In the real case, there is a spread in frequencies due to space charge and the anharmonicity of

the trap potential, but the frequency of a given ion is not

fixed. The axial position of the kth ion is

Zk = Ak sin(wjtf -I- <^k)- The position of the center of mass is

(4)

Zb =

Xlifc^fcsin(cjfcf -|-0fc) = ylBsin(wB< + <;iB), (5)

where Ab is the instantaneous center-of-mass amplitude.

We assume that the frequencies and amplitudes of

the individual ions are distributed randomly. The time

average of the square of the center-of-mass amplitude is then

kA\) = {A^)IN, (6)

where (^4^) is the ensemble average of the squared amplitude of a single ion. The instantaneous value of A^

fluctuates around the mean value given by Eq. (6) sis the

phases of the ions change with respect to each other. Let

(T^^ be the standard deviation of the wjfc 's from their mean

value ijjQ. The time required for A?^ to change significantly

(the mixing time) is on the order of r^ z=. a~^

.

Suppose a short electric field pulse of amplitude E is applied to the ions at time V . The pulse duration r is assumed to be much shorter than the oscillation period of

the ions. Before the pulse, the position of the fcth ion is

Zk =yljtsin(a;fc< + 0fc).

After the pulse, it is

Z; = A'fcsin(w,<-F<?i'fc).

The change in the squared amplitudes is

(7)

(8)

^k ^k mujkj

+ 2^ AkCos{ukt* -\-<t>k), (9)

mujk

where m and q are the mass and charge of an ion. The

first term on the right side of Eq. (9) is positive and causes

heating. The second term on the right side of Eq. (9) can be positive or negative, depending on the argument of the

cosine.Suppose that the amplitude of the applied field is E = -g

mABU)Q

qr

(10)

where g is the gain. The average change in the squared

amplitudes of the particles is

|2\ _ {A")-{A') = {9'-2g)A'B- (11)

TN-168
The ions are cooled if < g < 2, which makes the right

side of Eq. (11) negative. The optimum cooling is obtained

for J = 1. For this case, we take a statistical average and

use Eq. (11) to obtain

{A'-')-{A') = -{A%) = -{A')IN. (12)

The fractional change in {A^) is \/N for one feedback

pulse. Another pulse can be applied after a period of

about Tm- Thus, the optimum damping rate is approximately l/{NTm)-

A more detailed calculation^ ' shows that the energy

per ion decreases to a limiting value enm which depends on the electrical noise and on the gain g. The cooling time

Tcool also depends on g. There is no advantage to increasing g above about 0.5, since e\\m increases rapidly, while

Tcool decreases slowly. For g < 0.5, there is a trade-off between cnm and Tcool) such that their product is approximately independent of ^. If the electrical noise is due to thermal noise in the circuit.

ksTto, (13)

where ks is Boltzmann's constant, T is the electrical noise

temperature, and to is the time constant for resistive cool- ing given by Eq. (2). Beverini et al. have carried out an experimental

demonstration of stochastic cooling of ions in a Penning

trap.^^^^ The temperature was measured from the noise current induced on the trap electrodes by the ions. They

measured a time constant for stocheistic cooling of 55 ± 10

s, in good agreement with their calculation. The time constant for resistive cooling was about 160 s. The effective- ness of the cooling was verified by varying the pheise of the

feedback signal. For a 180° phase shift, heating occurred.

Stochastic cooling is most useful when the ions are

initially at high energies. At low energies, the signal-tonoise ratio suffers, and stochastic cooling is less effective.

IV. Collisional cooling

Trapped ions can be cooled by collisions with charged or neutral particles which have lower temperatures. The colli- sions may lead to loss of the ions through charge-exchange

or other reactions, or they may cause perturbations of the

energy levels. However, in some Ccises the benefits of the

cooling outweigh these disadvantages. One advantage of

collisional cooling over resistive cooling is that it works

with ions having a wide range of energies and oscillation frequencies. Because of the required high-Q tuned circuit, resistive cooling works for only a narrow range of oscil- lation frequencies. Two applications of collisional cooling

will be given here: buffer gas cooling of ions in a Paul trap

and electron cooling of antiprotons in a Penning trap. The

latter is a form of sympathetic cooling (the cooling of one

ion species by another). The case where one of the ion

species is laser cooled is discussed in Section VI. Consider collisions of ions in a Paul trap with neutral

molecules which have lighter masses than the ions. Major

and Dehmelt^ ^ showed that such collisions should damp

the secular motion. The micromotion should not be much

disrupted by a collision, but only slightly modified in amplitude and phase, so the trapping should not be affected. Other theoretical work on buffer gas cooling of ions in a Paul trap has been done by Blatt et alJ-^^' and Vedel et

The final ion temperature results from a balance

between heating and collisional cooling. Usually, rf heating^ ' is the dominant source of heating. The mechanism of rf heating has recently been clarified. Computer

simulations show that rf heating results from the existence

of a chaotic regime for a system of many interacting ions

in a Paul trap.^ ' The chaos does not require stochastic

forces. It results from the deterministic, classical dynamics of the coupled, nonlinear system. There are also crystalline or quasiperiodic states, in which the ions do not absorb energy from the rf trapping field. Chaos results

in a continuous, rather than a discrete, power spectrum

of the motion of the ions. This allows the ions to absorb

energy from the rf field. If the density of the ions becomes

low enough, their motions become nearly independent and

are not chaotic. The rf heating then stops. This is why

ions can be confined for long times even without cooling. In an early experiment, Wuerker et al} ' confined

charged micrometer particles of aluminum in a Paul trap.

The particle motions could be damped by background gas

at a pressures up to 1.3 Pa (10~^ Torr). When the pres- sure was high enough, the particles crystallized into regular patterns. These patterns corresponded to the spatial

configurations which minimized the effective potential en- ergy of the system. The time required for the particles to

crystallize, starting from a disordered state, increased as the pressure was reduced. An increase in the confinement

time of heavy ions (Hg"*") by the introduction of a light buffer gas (Ne) was observed by Dawson and Whetton.'^ ' This increase in confinement time was interpreted as an

effect of collisional cooling. However, the ion temperature was not measured directly.

Schaaf et aly ' measured the temperature of Ba"*" ions

confined in a Paul trap in the presence of He buffer gas. They measured the spatial density distribution of the ions

by scanning a laser beam across the trap and observing the

fluorescence. (Knight and Prior^^^^ had previously used a similar method to measure the density distribution of

Li"*" ions in a Paul trap.) Schaaf et al. found the density

distribution to be well described by a Gaussian in both

the radial and axial directions. Since the strength of the

confining potential was known, the ion temperature could

be inferred from the width of the spatial distribution. The

temperature of the ions decreeised as the He pressure was

increased. At a pressure of 5 x 10"'' Pa, the temperature was reduced by a factor of 3. At higher pressures, the ions were rapidly lost from the trap.

Cutler et a/.f'"'^''''^^ have collisionally cooled ^^Ug+

ions with He for use in a microwave frequency standard.

The frequency standard is based on measurements of the

ground-state hyperfine separation, which is approximately

TN-169
40.5 GHz. The ions are prepared in the (F = 0) groundstate hyperfine level by optical pumping with 194 nm ra- diation from a ^°^Hg resonance lamp. This radiation is nearly resonant with the transition from the {F = 1) level of the ground state to the 6p "P1/2 state. If 40.5 GHz mi- crowave radiation repopulates the F = 1 level, the trapped

ions absorb and scatter radiation from the lamp, and the

scattered radiation is detected by a photomultiplier tube.

The frequency of the hyperfine resonance increased

rapidly, by about 5 parts in 10^^, as the He pressure was

increased from to about 1 x 10~^ Pa.^ ^ Cutler et at. interpreted this as a reduction of the second-order Doppler

shift, due to collisional cooling of the secular motion. A

slow, linear increase in frequency with pressure remained,

which they interpreted as pressure shift of the hyperfine resonance. The fractional pressure shift was 4 x 10~^^

Pa~^. The maximum He pressure was approximately 10~^

Pa. The addition of He increased the ion storage time,

increased the ion density, and stabilized the second-order

Doppler shift. The storage time was 2.5 x 10^ s. The amplitude of the micromotion depends only on

the intensity of the rf electric fields at the positions of the

ions. The fields go to zero at the center of the trap and

increase in all directions outward from the center. Hence,

the second-order Doppler shift due to the micromotion de- pends on the spatial distribution of the ions in the trap.

The spatial distribution depends on the number of ions,

the temperature of the secular motion, and the trap parameters. The He collisions do not cool the micromotion

directly, but by cooling the secular motion, they increase

the ion density, so the ions are closer to the center of the

trap. This reduces the micromotion second-order Doppler

shift, since the rf electric fields sampled by the ions are reduced.

Cutler et al^'^^ measured the second-order Doppler

shift of the ^^^Hg"^ resonance. For 10® ions and typical

trap parameters, it was —1.2 x 10~^^ times the resonance

frequency. An analysis of the sidebands of the Dickenarrowed spectrum of the hyperfine resonance showed that

the temperature of the secular motion was reduced to ap- proximately 500 K by the He.' ' Without cooling the ki- netic energy of the ions was about 2 eV, which corresponds

to a temperature for the secular motion of 8000 K.

Neutral buffer gas cooling is not used in Penning

traps, since such collisions would quickly drive the ions out of the trap. Unlike the Paul trap, which is stable in

all directions, the Penning trap is unstable in the radial

(i-y) direction. Although the particle orbits are stable

in the absence of collisions, there is no restoring force in the radial direction. Rather, there is a balance between

an electrostatic force directed away from the axis and a magnetic {v x B) force directed toward the axis. However, cooling of one ion species by another charged

species is feasible in some cases in a Penning trap. The

Penning trap can simultaneously hold ions of different charge-to-mass ratios. The stability of the trapping is not

destroyed by collisions between the diff"erent kinds of ions. There must be some means of removing heat from the second species.

Electron cooling is used in storage rings to cool pro- tons or other particles. ^^^^ In this method, a beam of elec- trons is sent through a straight section of the storage ring,

with a velocity that matches the average velocity of the cir- culating particles. In the frame moving with the average

velocity of the ions, the ions lose energy to the electrons. Since the electrons pass through once and are dumped,

they do not have to be cooled. Like stochastic cooling,

electron cooling was first demonstrated in storage rings

and has recently been adapted to ion traps.

Gabrielse et alS ^ have demonstrated cooling of trapped antiprotons by collisions with trapped electrons. Cold antiprotons can be applied to tests of CPT invari- ance (through measurements of their inertial mass) and to measurements of the gravitational force on antimatter.

In the experiment of Gabrielse et al., pulses containing

approximately 3 x 10* 5.9 MeV antiprotons were extracted

from a storage ring and directed toward the trap. After passing through a titanium window and an aluminum

plate, some of the antiprotons were slowed down to 3 keV

or less and were captured in a Penning trap. A maximum

of about 60 000 antiprotons were captured. The Penning

trap was made up of several cylinders, which can be held

at diff"erent electric potentials. The magnetic field was 6 T. The longitudinal energy distribution was measured by

lowering the confining potential at one end of the trap, so that antiprotons that had an energy greater than the potential could escape. The antiprotons were annihilated

when they struck the vacuum enclosure. The charged pi- ous that were created were detected by scintillators. Soon

after being trapped, the antiprotons had a broad energy

distribution, about 2 keV wide. This distribution was al- most unchanged after 2.7 d (in the absence of electron

cooling).

Cooling of the antiprotons was observed when elec- trons were introduced to the trap. To improve the control and monitoring of the electron cooling, a small harmonic

well was created inside the trap to hold the electrons. This was done by adjusting the potentials of five of the cylindrical electrodes to approximate a quadrupole potential over a short region. Approximately 10^ electrons were loaded

into the trap from a field-emission point. Since the elec- trons had low energies, they collected at the minimum of the harmonic well. The electrons cooled, by emitting syn- chrotron radiation, to the temperature of the surrounding apparatus (4.2 K) with a time constant of 0.1 s. The

electrons induced a noise voltage across a parallel resonant

circuit, which was connected between one of the electrodes and ground potential. This signal was used to determine

the number of trapped electrons.'- •'

If antiprotons were introduced after the electrons had

been loaded and cooled, the voltage across the resonant

circuit increased, due to heating of the electrons by the antiprotons. The time constant for cooling of the antiprotons was approximately 10 s. The energy distribution of the antiprotons was measured after cooling. Energy widths

TN-170
Time

1 1 1 1 1 1 r 100 _ —

«/) -

1 ms • c

o 80 - - ^^

ro

'Z.c

c 60 - -

r«c

o

L— 9 meV -

"o 40 h- -

c_ L^ o. - " .^ - c

< 20 " 1 . 1 50 meV.

n t

-^^^

u

1 '

1 w ell depfh

Figure 1: Graph of the number of antiprotons detected

as a function of the well depth, which is reduced linearly

in time. The width of this distribution reflects the energy

distribution of the trapped antiprotons, which are cooled

by electrons. (From Ref. [47].)

as small as 9 meV were observed (see Fig. 1). Part of

this energy spread may have been due to conversion of the

space charge potential energy to kinetic energy, as the antiprotons escaped from the trap. Thus, the actual energy

spread of the kinetic energies of the antiprotons in the trap

may have been even lower.

V. Sideband cooling

Rf sideband cooling and laser cooling (also known as optical sideband cooling) were developed for very different types of measurements. Rf sideband cooling was developed

to improve measurements of elementary particle ^r-factors and of ion masses, by confining the particles to smaller

spatial volumes.^ ' Laser cooling was developed to reduce

Doppler shifts in atomic spectroscopy.^ ''^ ^ However, the

two methods are fundamentally the same. Both can be de- scribed in terms of mode coupling by a parametric drive. Figure 2 is a generalized diagram of the mode struc- ture of the cooled system. The system contains two modes,

represented by the subscripts a and 6. We assume that

'^a < wj, SO the energy levels separate into manifolds

as shown. For simplicity, we assume the level separations in each manifold are the same, so that the energy

is h{na(Ma -f-njo;;,), but this condition is not essential. The

{OP)

Figure 2: Energy level structure for a particle which is suitable for sideband cooling, having two modes, labeled a and h.

cooling relies on (1) parametric coupling of the two modes

and (2) relaxation of the h mode by spontaneous decay at a rate 7j. Spontaneous decay of the a mode is assumed to be negligible. Here, the a mode is a mode of oscillation of an ion, which is to be cooled. Figure 2 shows one part of

the cooling process. The particle is initially in the state {ua = 1, nj = 0). It absorbs a photon of energy h.{u)i —u)a)

from the externally applied field and is driven to the state

{jia = 0, nj = 1). The particle then spontaneously emits a photon of energy hu)i, and makes a transition to the state [ua =: 0, nj = 0). In this process, system's energy is re- duced by Tiuia- This is the basic idea of sideband cooling.

Heating occurs if the coupling causes absorption of photons of energy h{un, -f- Wa)-

In general the process is more complicated. Other ab- sorption processes of the form (na,nj) —* (hq — l,nj,-|- 1) can be driven by the same applied coupling. The entire distribution of ria and nj must be taken into account. Also, even when the coupling is tuned to resonance, other non- resonant transitions can occur, since the resonances are broadened by spontaneous emission. For rf sideband cooling, wj and Wa are rf frequencies. For example, the b mode might be the axial motion and

the a mode might be the magnetron motion, in which case the a mode energy level structure is inverted. Cooling then

occurs when the applied coupling is at frequency (u-'(,-|->v'a). For laser cooling, wj, is an optical frequency. In this case, the b mode is just a two-level system (nj, = or

TN-171
1). The recoil of the cooled particle upon the spontaneous

emission of a photon of frequency u>b leads to heating. Let (Ana) be the average change in Ua per spontaneous emission. Then,

nu>a{Ana) = R = {huif /2mc'^ (14)

where R is the recoil energy (the kinetic energy of an ini- tially stationary particle of mass m after emission of a photon of frequency wj). We assume that the photon is emitted into free space. That is, the emitted radiation is not influenced by cavity effects. Two cases of laser cooling

will be treated separately: u>a •C 76 (sidebands unresolved)

and Ua >> 7j (sidebands resolved).

Sideband cooling is closely related to methods of dynamic nuclear orientation. '^ ^ The method of polarization

by forbidden transitions is directly analogous to sideband

cooling. In this method, an applied rf field drives transi- tions between states which differ in both the nuclear and

electronic spin quantum numbers. This corresponds, for example, to the {ria = l,ni = 0) — [ua = 0,nb = 1) transition in Fig. 2. The electronic spin relaxes quickly

[{ua = 0,nb — 1) —> (ua = 0,nb = 0)]. The nuclear (a)

mode reaches the same polarization as the electronic (6) mode.

Vyatchanin'- ' has discussed, in general terms, the

cooling of a quantum oscillator (or two-level system) by

parametric coupling to another quantum oscillator (or twolevel system) which is more strongly damped. This treat- ment is directly applicable to sideband cooling.

A. Rf sideband cooling

Rf sideband cooling has been used with great success, first on electrons stored in Penning traps^^^^ and more recently on atomic or molecular ions stored in Penning traps. ^^^^''^ '

It has been especially helpful in reducing the radii of the

magnetron orbits of a few particles. Cooling of the magnetron motion is especially important, since most external

perturbations, such as collisions with neutral gas, cause

the magnetron radius to grow. Without cooling, the ions

would eventually strike the ring electrode and be lost. With rf sideband cooling, the ions can be held in a Penning

trap for as long as desired.

The case of rf sideband cooling which has received the

most attention is when the a mode is the magnetron mode

{ua = u>m) and the b mode is the axial mode (wj = u^).

This is similar to other cases. Since the a energy levels are inverted, the frequency of the applied drive is tuned

to cause transitions of the form {ria, rib) < * ("a -|- 1, nj -|-

1). Spontaneous emission occurs at frequency Ub, through

coupling of the axial motion to a tuned circuit. We assume

that damping of the magnetron motion, in the absence of

sideband cooling, can be neglected. Finally, we assume

that huj^ <C ksTz, where T^ is the temperature of the

axial motion, and that the axial mode is nearly in thermal

equilibrium with the external circuit. The average axial

oscillation quantum number (n^) is much greater than 1.

We wish to calculate the magnetron cooling rate. Let

the ion be subjected to a parametric coupling in the form

of a time-varying potential.^^^^'^^^^'f^^^

V =

Vd

-r^XZCOs{uJdt), (15) t

where d has the dimensions of a length, and x and z are coordinates of the ion. Consider the case where

i^d = ^z+f^m- The basic mechanism for cooling in this case

is as follows: Suppose the x component of the magnetron

motion is initially of the form x = Vm cos{uimt). Substituting this expression for x into Eq. (15), we see that there

is a 2 component of electric field {—dV'/dz) at frequency

Uz, which can excite the axial motion. The axial motion

is then damped by the tuned circuit, at a rate 7^. By

this parametric process, the magnetron motion is trans- ferred to the axial motion, which is then removed from

the ion by the tuned circuit. We can show from Eq. (15)

that the magnetron-induced axial motion gives rise to an

electric field in the x direction, and that this field has fre- quency Um and the correct phase to damp the magnetron

motion.t^^.t^^J

If saturation can be neglected, that is, if the axial damping is fast compared to the rate of energy transfer between the axial and magnetron modes, then the magnetron amplitude decays with a rate,^^^'^^^^

7m

I'V}

4m2u;^74w<; -2ujm)'

(16)

Brown and Gabrielse^ ' have considered a more general

case, where saturation is allowed and where w^ is not necessarily equal to ui^ + Um

Figure 3 shows rf sideband heating and cooling of a

single electron in a Penning trap. The magnetron radius

Vm, plotted on the vertical axis, increases when the drive

is applied at angular frequency u>d equal to Uz — u)m =

2n{uz — I'm)- It shrinks when u>d = (-^z + WmIf damping of both the axial and magnetron motions

can be neglected, this type of parametric coupling leads

to a sinusoidal exchange of the energies in the two modes.

Parametric coupling of modes of ions in a Penning trap

has been studied theoretically, for this case, by Cornell et

The exchange of excitation between modes in a Pen- ning trap, by coupling with a parametric drive, has been

used to cool the cyclotron motion of ions in a recent ex- periment at MIT.^ ' In this experiment, the ratio of the masses of CO"^ and N^ was measured to an accuracy of 4 X 10~^°. This is the most accurate mass comparison to date. A short rf pulse of the appropriate amplitude and

duration exchanges the phases and actions of the cyclotron

and axial modes. The action of the rf pulse is mathematically like that of a tt pulse applied to a two-level system.

Since the axial motion is kept cold by resistive cooling,

application of the n pulse cools the cyclotron motion. The

axial motion is heated immediately after the pulse, but

is cooled again resistively. Ordinary rf sideband cooling,

TN-172
^ SB -drive ^

itognetron

orbit

radius

(mn

•

—

to

off 1 ivf-vjon -H off |(i^, + i»,)on _

fi, (min) s« 0.014mm / |

405

1

- u| '-7-- IMJi^ U2 4

Time (minutes)

Figure 3: Rf sideband heating and cooling of the magnetron mode of a single electron in a Penning trap. The

vertical scale is the axial oscillation frequency. This is re- lated to the magnetron radius because of a deliberately applied magnetic inhomogeneity "bottle." (From Ref. [15].)

with cw excitation at frequency w^ -f- u>m was used to cool

the magnetron motion.

For both cw and pulsed parametric coupling, the minimum achievable temperature is determined by heating of

the axial motion by thermal noise. When the axial motion

is coupled to the tuned circuit, it has an amplitude due

to random thermal noise. This causes a random excitation of the magnetron motion, given by Eq. (15), through

the parametric coupling between the modes. The result- ing noise excitation of the magnetron motion (or cyclotron

motion in in the case of the MIT group) gives the limit for sideband cooling. This theoretical limit can be calculated

from the equations of motion,^ ''^ ^ by thermodynamic

arguments,^ ' or by detailed-balance arguments.'^ ' The

result, for the case where ksTi ;^» huii, is

UJ„ |Ta(min)|=^|rt|. (17)

Cooling is routinely observed in several experiments, but

the temperatures achieved are somewhat higher than the

limit given by Eq. (17).

B. Optical sideband cooling (resolved sidebands)

In optical sideband cooling, the b mode is an internal optical transition in a trapped particle. Generally, the transition is an electronic transition (at an optical frequency)

which relaxes radiatively by spontaneous emission at the

rate jb- The quantum number nj takes only the values

and 1. Here we consider the resolved sideband case

(Tj "C Wa), where uia is the oscillation frequency of the

cooled mode of the bound particle. (The unresolved side- band ccise will be discussed in Section V. C.) Optical side- band cooling of trapped ions and neutral atoms has been

an active field of research.^ ''^ ' New methods of laser cooling, which are distinct from optical sideband cooling, have

recently been developed for neutral atoms. ^ '~'- ' These

methods have not yet been applied to trapped ions.

The parametric coupling from the ground state vibrational manifold (nj — 0) to the excited state vibrational

manifold (nj = 1) is done with narrowband radiation. The

width Awft of the radiation should be less than or on the

order of 74 for optimal cooling. It should be resonant with

the first lower sideband at frequency wj — Wg. If the recoil energy R [defined in Eq. (14)] is less than the vibrational

energy hua, then sideband cooling is possible. (Otherwise

the cooling which results from the photon absorption is undone by the recoil from the photon emission.) An atom

absorbs photons of energy ^(wj — lOa) and reemits photons of average energy huit. Hence, on the average, each

scattered photon reduces the atom's vibrational energy by

hua and the atom's vibrational quantum number Ua by

1. In this way, it is possible to make the mean quantum

number {na) much less than 1, provided that R >C huiaIf {ria) <C 1 for all motional degrees of freedom, then the

atom resides in the ground state level of the confining potential most of the time.

Optical sideband cooling of the thermal degrees of

freedom of many simultaneously trapped particles is possible in principle. However, the kinetic energy in coherent

degrees of freedom usually limits the minimum attainable

kinetic energy. The higher energies of the coherent motions in both the Penning trap and in the Paul trap are consequences of Coulomb repulsion. In a Penning trap, any ion that does not lie on the

symmetry axis of the trap rotates about the axis, due to crossed electric and magnetic fields {E x B drift). This ro- tation gives rise to a kinetic energy that depends quadratically on the radial distance of the ion from the trap cixis.

It might be possible for a few ions to lie along the symmetry axis. However, if there are many ions, the combined

effects of the trapping fields and the mutual Coulomb re- pulsion of the ions will force some of the ions to lie at a

finite distance from the axis. In a Paul trap, the Coulomb repulsion between ions is balanced by a force from the trapping potential, directed

toward the center of the trap. Only one ion can occupy

the center of the trap, where the applied trap fields go to

zero. Any offset of an ion from the trap center leads to micromotion of the particle at the frequency of the applied rf

field. The kinetic energy in this nonthermal motion can- not be reduced by sideband cooling. So, although it is possible to reduce the kinetic energy of the secular motion

for more than one ion, the kinetic energy in the micromotion remains undiminished and can be substantial. This

limitation could be overcome in a Paul trap with linear geometry. ^^°'"'^^' In such a trap, the magnitude of the rf

field approaches zero on a line, rather than at a point.

Even for a single ion, optical sideband cooling in a Penning trap is difficult. The magnetron, cyclotron, and

axial frequencies cannot be degenerate for stable trapping.

This leads to a complicated sideband spectrum. ^^ ' The

spectrum contains sidebands not only at the fundamental

motional frequencies (for example, uii, iu^m), but also all intermodulation products (ujj, -t- j^j'^ + k^. + <"--•„,, where

j,k,£ are any integers). At high temperatures this specTN-173
trum is nearly continuous. Therefore, precooling to near

the Dicke limit (xrms "C A/2) is required before sideband

cooling is done. Here, Xrms is the rms value of the x coor- dinate of the ion, and A = 2'Kc/uih)

Attaining the lowest vibrational energies (the minimum {ua)) for all degrees of freedom would require cool- ing with a laser tuned to an individual sideband for each

of the three motional degrees of freedom. Finally, the condition that the recoil energy R be much less than any of

the motional energies is more readily satisfied in the Paul

trap than in the Penning trap. For both Paul and Penning traps, the sideband cool- ing limit for a single ion is

{na) = \U+\]{lhlu^a)\ (18)

where a depends on the angular distribution for photon

emission and is of order 1. Equation (18) is valid when

the intensity of the cooling radiation is below saturation

and when 7t/wa "C 1- The limit results from a balance

between cooling and heating. The most important cooling process is on-resonance absorption of a photon on the lower

sideband [ua —* n^ — 1), followed by emission on the carrier {ua —»• Tia)- (In the Dicke limit, almost all emission

is on the carrier.) The most important heating processes

are (1) off-resonance Ua —> "a + l transitions, followed by

emission on the carrier, and (2) off-resonance n,, —> Ua

transitions, followed by emission on the lower sideband

("a —* "a + !)• This result was derived by Neuhauser et

al}-^^^ and others.^^^'f^^^ Wineland ei a/.^^^^ considered the

effect of finite laser bandwidth and also explicitly treated

resolved optical sideband cooling in a Penning trap.

Lindberg'^ ' and Javanainen et alS ' have derived the

steady-state energy of a single two-level atom, confined

in a one-dimensional harmonic well of frequency Wa, for arbitrary laser frequency, laser intensity, and 'ihjua'-

{Ea) = hUa (^M + (19)

-^[u;2(a2 + jI + 6k2) + 4t|(A2 -f yl +

2«2)]-i

x{a[a;2(A2 + 57|+4K2-u;2)2

+47|(A2+7|-h2/c2-2a;2)2]

+uliA'' + 7| + 2k2)(a2 + 57I + 8k2 + ul)

+47|[(A2 + 72 + 2k2)2 + 2k\A' - 3jI - Zul)]}.

This result holds in the limit that R/{hu>a) <^ 1- Here, k

is the Rabi frequency, which is proportional to the square

root of the laser intensity, and 72 = 76/2. The detuning A

is defined as A = w — wj , where to is the laser frequency.

It must be negative (Iciser frequency below resonance), or

else there is no steady state. Equation (20) reduces to Eq. (18) in the limit k —» 0, 72/wa —* 0, and A -^ —Wa- The value of (ria) can be determined experimentally

from the relative transition strengths for on-resonance absorption on the first lower and upper sidebands.^ ''^ ' When {ria) <C 1, the strength of resonant absorption on

398 nm

.5nm

Figure 4: A simplified energy level diagram of Hg+,

showing the transitions used for optical sideband cooling.

(From Ref. [68].)

the lower sideband (wj— Wa), which is proportional to (ua),

approaches zero. When the ion is in the lowest kinetic en- ergy state {ua = 0), it is no longer possible to extract vibrational quanta from the ion. The strength of the up- per sideband (wj +Ua) is proportional to (ua) -f- 1. Thus,

if the lower and upper sidebands are probed with saturat- ing power, the ratio of their absorption strengths becomes

independent of power and directly gives {ria).

A single ^^^Hg"'' ion, confined in a small Paul trap, has

been laser-cooled, in the resolved sideband limit, to near

the zero-point energy of motion. '^ ^ The electronic energy

levels of Hg"*" are shown in Fig. 4. The procedure was

as follows: First, the ion was laser-cooled in the Dopplercooling limit (discussed in Section V. C. 1.), by radiation

scattered from the strongly allowed 194 nm first resonance

line. This reduced the temperature to about 1.7 mK and

the mean vibrational quantum number {ua) to about 12. The secular frequency {ua/2'K) was about 3 MHz. Laser

radiation, tuned to the first lower vibrational sideband of

the narrow 282 nm bd^°Qs '^Sx/2-to-5d^6s'^ ^^5/2 transition, was then applied to the ions. In order to speed the

cooling process, 398 nm laser radiation was used to drive

ions from the "^0^/2 state to the 5c/^°6p ^^3/2 state, which

decays rapidly to the ground 2^1/2 state. This allowed

the absorption of 282 nm photons to proceed faster than

the 90 ms natural lifetime of the ^Z?5/2 state would have

allowed. After scattering enough photons to remove 12 vi- brational quanta from the ion for each degree of freedom,

laser radiation of saturating intensity probed the 282 nm

sideband spectrum.

The results are shown in Fig. 5. The fact that the

intensity of the lower sideband {Sl) is much less intense

than the upper sideband {Su) indicates that {ua) <C 1- From these data, it was deduced that the ion was in the lowest vibrational quantum state, for the y and z degrees

of freedom, 95% of the time. This corresponds to a temperature of 47 ± 3 /iK. Since the 282 nm probe beam was

TN-174
1. Paul traps

100

t 50 o</> Xi<

->—I—'—

r

->L Su —»4 \

I

I L -3.0 -2.9 2.9 3.0 Frequency Detuning (MHz)

Figure 5: Absorption spectrum of the ^5i/2-to-^Z)5/2

electric quadrupole transition. The vertical scale is the

probability that the ion makes a

"^ 81/2-^0-"^ D5/2 transi- tion when the probing 282 nm radiation is applied. The

spectrum in the small inset weis taken before the sideband

cooling was applied. Sl is the first lower sideband (fre- quency ujb—u>a)', Su is the first upper sideband (frequency

Wft + Wo). The asymmetry between Su and 5^ indicates

that the average quantum number for the oscillation of the

ion is much less than 1. (From Ref. [68].)

nearly orthogonal to the motion in the x direction, the

measurement was not very sensitive to the energy in this degree of freedom. For this reason, a precise measurement

of the X vibrational energy could not be made in this ex- periment. Cooling of all degrees of freedom could be done

by orienting the 282 nm beam at a different angle with

respect to the trap axes.

C. Optical sideband cooling (unresolved sidebands)

In most experiments on optical sideband cooling, a

strongly allowed optical transition, which heis a natural

linewidth 75 of approximately 20 MHz or more, is used.

This is much greater than the motional frequencies of the

trapped ion (typically 3 MHz or less). The sideband struc- ture of the optical resonance is not resolved. It is then

natural to treat the motion of the ion classically. The

scattering of laser light by the ion leads to a force on the

ion. The scattering takes place in a time of about 7^ , much less than the orbital period, so it can be cLssumed

to give an instantaneous impulse to the ion. In the low

intensity limit, the average force on an atom is the product of the average momentum transfer per scattering (the

photon momentum hk) and the scattering rate. The size of the force is maximized when the laser frequency, in the

frame moving with the ion, is in resonance with the optical

transition. Fluctuations in the force arise from (1) the ran- domness of the direction of the scattered photon and (2)

the random time distribution of the scattering events. As

an alternative to this semiclassical approach, the cooling can also be explained with the general concepts of side- band cooling. ^^^^'^^^•' It is necessary, in this case, to include a large number of sidebands in the analysis.

An ion in a Paul trap can be treated, to a good approximation, as a particle in a three-dimensional harmonic potential, undergoing simple harmonic motion at the secular

frequencies. The micromotion can be added by numerical

simulation, f^°^'f'^°^'t^^^ but is hard to treat analytically.

For simplicity, consider an ion moving in a harmonic

well in the x direction, interacting with a laser beam prop- agating in the +x direction. If the ion has a velocity v and

the laser detuning is A, the average force is

lo-Q ll

huj [(A - kvY -f 72]

hk

I<^0f2 MA2+7I)

Fq — mVv,

1 +

2Aibt; A2+7IJ

hk

(20)

where / is the laser intensity, ctq is the resonant scattering

cross-section, and m is the mass of the ion. The approximation holds for low intensities [IcrQ/{h.u) <C 72] and low

velocities {kv <C 72)- In the last line of Eq. (20), Fq is a velocityindependent force, which slightly displaces the equilibrium position of the ion, and mTv is a damping force (for A <C 0), which leads to cooling. The fluctuations in the

force, which are due to the discreteness of the photon scat- terings, have properties similar to electronic shot noise. ^

' The equation of motion for the ion position is the same as that of a series RCL (resistance-capacitance-inductance)

circuit driven by a fluctuating emf. The steady-state en- ergy of the ion can be calculated by methods analogous to those used to calculate noise in electronic circuits.^ ^^ The steady-state energy is

{Ea) (l-fa)%| + A2)

-4A

(2i;

This agrees with Eq. (20) in the limit where k —> and

Wa/72 -^ 0. The lowest energy is obtained when A = —72:

{E,) = ksT (1 -I- Q;)^72

(22)

For a ss 1, this yields the Doppler-cooling limit, ksT «

^72 = hji,/2, which also holds, under suitable conditions,

for free two-level atoms.^ ^'^ ''^ "' For typical cases, T % 1 mK.

Optical sideband cooling in a Paul trap was first demonstrated experimentally by Neuhauser ei al}*^ ' They

cooled Ba"*" ions, using the 493 nm, 6s ^Si/2-to-6p ^Pi/2

transition. One complication in the experiment was that

the 6p ^Pi/2 state can decay to the metastable bd 'D3/2

state. A 650 nm laser beam drove the ions from the bd ^0^12 state back to the 6p "P1/2 state, so that the cooling could continue. The eff'ect of cooling was to in- crease the storage time of the ions in the trap. Figure 6 shows the eff'ect of laser cooling when the 493 nm laser is tuned below the 6s "^Siin-^o-^p 'Pif2 transition. Later,

TN-175
1 1 1 1 1 1 1 TIME— y^A^n^vvyY

1 1 1

\ -1 klOsec

'if jT -

1

-0 vT A-saW-V^

aj = cuo (J =cuo-A,/2 Ci> = tUo*^l/2

Figure 6: The first optical sideband cooling of ions in a Paul trap. The Ba"*" fluorescence intensity is plotted

cLS a function of time. The trap is initially empty when

the laser frequency w is equal to the atomic resonance

Wo- The fluorescence increases when u is tuned below the

atomic resonance to wq — ^xfi- It decreases rapidly when w = Wo + Ai/2- Here, A1/2 is the resonance half-width in the discharge tube used as a reference. (From Ref. [64].)

Neuhauser ti al. were able to estimate the temperature

of a single ion to be less than 36 mK, from the size of its photographed image. ^

' Bergquist et al}^^' cooled a single Hg"*" ion, confined in a Paul trap, using the 194 nm 5d^°6s ^'S'1/2 to 5d^°6p ^Pi/2 transition. The narrow 282 nm bd^^Gs %i/2

to 5(i^6s^ ^D5/2 transition was used to determine the temperature. The relative intensities (transition probabilities)

at the frequencies of the carrier (ujf,) and at the first side- bands (wj ± Wa) were measured. Equation (44) of Ref.

[55] relates the relative intensities of the carrier and side- bands to the temperature. The measured temperature was

1.6 ±0.5 mK, in good agreement with the Doppler-cooling

limit of 1.7 mK. In later experiments with a single ^^^Hg"*"

ion, the 282 nm carrier was observed with a linewidth of

under 200 Hz (Q « 5 x 10^^).^ '^^ A typical resonance curve

is shown in Fig. 7. Similar experiments have been done

with a single Ba"*" ion by Nagourney et al} ' They observed the 1.8 fim 6s ^5i/2-to-5rf ^£^5/2 transition with a

linewidth of about 40 kHz.

Optical sideband cooling of ions in Paul traps has

been applied to other areas of physics. Radiative life- times have been measured in single, laser-cooled ions of

Mg+ (Ref. [76]), Ba+ (Refs. [77] and [78]), and Hg+ (Ref.

[79]). Rates for quenching of a metcistable state by vari- ous gases have been measured in single, laser-cooled Ba"*"

ions.'^ ' The fluorescence from a single cooled ion can be

easily observed. This has made it possible to observe cer- tain inherently quantum properties of the electromagnetic

field, such as photon antibunching^*°^~^ ' and quantum

[77] [81] [83] jumps.^ ''' ''^ '

If several ions in a Paul trap are cooled to a low

enough temperature, they arrange themselves into spatial

configurations which minimize the potential energy. Such

ordered patterns have been observed with Iciser cooled

1.00( ««*«-J

• •»%« J*"***

\

1/

•

>

-375 -250 -125 125 250

Laser detuning (Hz)

375

Figure 7: Absorption resonance for a single Zee- man-hyperfine component of the 5<i^°6s ^5i/2-to-5(i^6s^

^D^/2 transition in a single ^^^Hg+ ion. The probability

that the ion did not make a transition to the ^0^/2 state is plotted on the vertical axis. The relative frequency of the

563 nm laser, which was frequency doubled to 282 nm, is plotted on the horizontal axis. The curve is a least-squares

fit to a Lorentzian. The width of the Lorentzian is about

90 Hz. (From Ref. [74].)

Mg+ (Ref. [19]), Hg+ (Ref [7]), and Ba+ (Refs. [20] and

[21]). Phase transitions of these ions between ordered and

disordered states have been studied, especially with regard

to classical chaos.[^^]-[20],[40]

2. Penning traps

In some respects, optical sideband cooling of ions in Pen- ning traps and Paul traps are similar. The eixial and cy- clotron modes are cooled if the laser is detuned below res- onance. The differences arise from the E x B rotation of

the ions around the trap axis. The rotation frequency of a single ion is the magnetron frequency Um- The rota- tion frequency of many ions is increased by electric fields caused by space charge.

Cooling of the rotation requires control of Lz, the

z component of the canonical angular momentum of the

ions.^*^^ For a single ion of charge q and mass m, the canonical angular momentum is £ = f x p, where r and p are the position and the canonical momentum of the ion. The

canonical momentum is defined as p = mv -}- {q/c)A{r),

where v is the velocity of the ion and A{r) is the vector potential at r. For convenience, we choose the symmetric

gauge, where A{r) = |5 x r and B is the magnetic field. The z component of i is

£z = mvgp +

q_Bl

2c

(23)

where vg and p = (x^-j-y^)^/^ are the azimuthal component

of the velocity and the cylindrical radius of the ion. For N identical ions of charge q and mass m,

N

- 5Z I ^^9'Pi +

» =1

qBp]

2c

(24)

TN-176
LASER ON

IONS REMOVEDFigure 8: The first optical sideband cooling of ions in a Penning trap. The vertical scale is the axial temperature

of the Mg"^ ions. The ions were heated by the laser before

the beginning of the data. The laser was turned on with

a frequency below the resonance frequency, cooling the

ions to below 40 K. After the laser is turned off, the ions

thermalize to the ambient temperature of 350 K. (From

Ref. [87].)

where vg . and p,- are the azimuthal component of the ve- locity and the cylindrical radius of the ith ion. If the trap

has perfect axial symmetry and there are no interactions

with the outside world, L^ is conserved. For the typical

Ccise, where the rotation frequency is much less than the

cyclotron frequency qB/{mc), the second term of Eq. (24)

dominates. Conservation of Lz is then equivalent to con- servation of the mean squared radius of the ion plasma.

A more general proof of the fact that conservation of Lj, leads to radial confinement has been given by O'Neil.^*^^ There are confined thermal equilibria, whose properties are determined by the three conserved quantities:

N, Lz, and the energy.'®^^ One property of these thermal

equilibria is uniform rotation. That is, the velocity distri- bution, with respect to a rotating frame, is Maxwellian. In a continuous fluid model, the zero-temperature equilibria are uniformly rotating spheroids of uniform density.

^^'^^ Asymmetries of the trap fields apply torques to the

ions, so that Lz is not perfectly conserved. This leads to radial expansion. The light-scattering force from a laser beam can be used to apply a torque to the ions. This

torque can be used to cancel the torques from the trap

fields and establish radial confinement. Such a torque

could also be applied with a neutral atomic beam.

The first experimental demonstration of optical side- band cooling in a Penning trap was made by Wineland et alS ' Approximately 5 x lO'* Mg"*" ions were cooled by a 280 nm laser beam tuned slightly lower in frequency than

one Zeeman component of the 3s ^5i/2-to-3p ^P3/2 transition. The temperature of the ions was determined from

noise current induced on the electrodes. The ions were

cooled below 40 K. This was the measurement limit set by the electronic noise. Figure 8 shows the decrease in temperature when the laser is turned on, with a frequency

below the resonance frequency. Collisions with neutral gas molecules cause the ions to rethermalize after the laser is turned off. Optical sideband cooling of Mg"*" ions in Penning traps has also been observed by Plumelle et al} ' and

by Thompson et al}^^^

Drullinger et al} ' laser-cooled Mg"*" ions to approximately 0.5 K. The temperature was determined from the

Doppler width of the transition. They also demonstrated

the connection between the plasma radius and the canonical angular momentum. Placing the laser beam on one side or the other of the axis exerted a torque on the ions, which

caused radial expansion or contraction of the plasma.

If the cooling laser beam is perpendicular to the rota- tion axis, the steady state temperature can be calculated

from the rotation frequency and the laser frequency and

intensity profile.^ ' The temperature is calculated from the

requirement that the work done on the ions by the laser beam, averaged over the velocity distribution, be zero. The temperature increases with the rotation frequency,

the distance between the axis and the laser beam, and the

laser intensity. Itano et al^^' obtained good agreement

between the calculated and observed temperatures of Be"^

ions, over a wide range of experimental parameters.

In general, lower temperatures are obtained when the

cooling laser beam is not perpendicular to the magnetic

field. Brewer et alS ' observed temperatures of Be"^ ions

as low as 2 mK, when cooling beams were directed simultaneously at 90° and at 55° with respect to the magnetic

field. They also confirmed experimentally the relationship

between the rotation frequency and the shape of the ion

plasma that was predicted in Ref. [84]. The reduction of Doppler shifts and the spatial con- finement which results from the cooling has been useful for several experiments. Some examples are in mass

spectroscopy^ ' and in the observation of single-ion quantum jumps.' ' High-resolution optical spectra of Mg"*"

(Ref. [5]) and Be"*" (Ref. [94]), have yielded hyperfine constants, fine structure separations, and isotope shifts. Rfoptical double resonance experiments have yielded groundstate hyperfine constants and ^-factors of ^^Mg"*" (Ref.

[95]) and ^Be+ (Ref. [92]). A frequency standard, based on a 303 MHz groundstate hyperfine transition in laser cooled ^Be"*" ions, has

been demonstrated.^ ' The estimated error of this stan- dard, about 1 part in 10^^, is comparable to that of primary cesium atomic beam standards. A search was made

for any dependence of the frequency on the orientation of

the magnetic field, with null results.^ ' This is one of the

most precise tests of local Lorentz invariance. Ion plasmas which have high enough densities and low

enough temperatures are called strongly coupled. In such

plasmas, the positions of neighboring ions are correlated

with each other, cis in a liquid or a solid. Strongly coupled

plasmas have been produced, by optical sideband cooling

of ions in Penning traps. ^

' Spatial ordering, in the form of

concentric shells, has been directly observed by imaging of

the resonance fluorescence.^ ' Numerical simulations predict the existence of such shells. ^^ ^'^^ °^

TN-177
VI. Sympathetic laser cooling

Because laser cooling depends on having a favorable en- ergy level structure, it has been demonstrated on only a handful of different ion species (Be"*", Mg"*", Ba"*", Hg"*", and Yb"*"). However, the technique of sympathetic laser cooling^ ^'^ ' can be used to cool any ion species to subkelvin temperatures. In sympathetic laser cooling, two

different ion species are loaded into a trap. One of the ion

species is laser cooled. The other species is sympathetically cooled by its Coulomb interaction with the Iciser- cooled species. In a Penning trap, the sympathetically

cooled species must have the same sign of charge as the

laser-cooled ion species. Sympathetic laser cooling may

enable the formation of cold, strongly coupled plasmas of

positrons.^ ' In principle, sympathetic laser cooling can be done

in either a Paul or a Penning trap. In a Paul trap, or- dered, crystal-like structures in which at least one of the

"lattice" sites was occupied by an impurity (perhaps an- other isotope) that was not directly laser-cooled have been

observed. '^ ''' '•* ' These ions did not fluoresce, but their presence was detected by their effect on the positions of

the other ions. This shows that at least a small number

of ions can be sympathetically laser-cooled to sub-kelvin

temperatures in a Paul trap. (If the nonfluorescing ions

had not been cold, the ordered structures would not have

formed.) In general, rf heating and the available cooling

laser power limits the number of ions that can be sympathetically Iciser-cooled in a Paul trap. In the Penning

trap, which is free of rf heating, experiments have demonstrated sympathetic laser cooling on much larger numbers

of ions.^ ^'^ '<^ ' Sympathetic laser cooling is a form of

collisional cooling between charged particles. However, the

low temperatures that can be achieved with laser cooling

produce some new features. For simplicity, consider two ion species with like charges but difl"erent masses confined in the same Penning

trap. The rotation of the ions about the magnetic field axis of the trap produces a centrifugal separation of the

ion species.^ ^ For typical experimental conditions, where

the rotation frequency is much less than the cyclotron fre- quency, the rotation frequency is nearly independent of ion mass. This is because the rotation is a circular Ex B drift. The velocity for linear E x B drift is independent of mass. However, a more exact calculation shows that the higher mass ions rotate at a higher frequency than the lower mass

ions, if they occupy they occupy the same volume in the

trap, so that they are subjected to the same electric fields. The equal and opposite momentum transfers due to the diff"erent rotation frequencies will cause the lighter ions

to move in (toward the trap axis) and the heavier ions to move out. This separation of the ions continues until both

species are rotating at the same frequency, as a result of the

change in the space charge induced electric fields. Uniform

rotation is one characteristic of thermal equilibrium of a nonncutral plasma.^ ' In the low temperature limit, the

separation is complete, and a gap forms between the ion

species. ^^^'f^°^^'f^°^^ The higher mass ions form a "doughnut" around the lower mass ions. Although the separation

of the ion species limits the thermal coupling between the

species, sympathetic laser cooling has been demonstrated

experimentally in a number of cases.

^^^'^^^'^^°^^ The first temperature and density measurements on

sympathetically cooled ions were done with Hg"*" ions, sympathetically cooled by laser-cooled Be+ ions.^^^ The ions were confined in a Penning trap. Up to 12 000 Be"*" ions,

with temperatures less than 0.2 K, sympathetically cooled a larger number of Hg"*" ions to temperatures less than

1.8 K. As expected, Hg"*" ions were observed at larger dis- tances from the trap axis than the Be"*". However, since the laser beams which probed the spatial distribution of the ions were perpendicular to the trap axis, not along the

axis, complete separation could not be experimentally verified. The sympathetic cooling also prevented the radial diffusion of the Hg"^ ions. The Be+ cooling laser applied a torque to the Be"^ ions, and this torque was transmitted

by the Coulomb interaction to the Hg"*" ions. If the Be"*" cooling laser was blocked or tuned off resonance, the Hg"*" ions left the trap in several minutes. With sympathetic

cooling, the Hg"*" ions were confined indefinitely. In another Penning trap experiment, laser-cooled Mg"^ ions were used to sympathetically cool and radially confine Be"*" ions.^^°^^''^^°^^ The sympathetic cooling was used to improve the performance of the ^Be"^ fre- quency standard discussed in Section V. C. 2. In order

to avoid ac Stark shifts, the laser used to cool and detect the Be"*" ions was turned off when the hyperfine transition was driven. With this laser off, the Be"*" plasma

expanded radially, due to torques originating from trap

asymmetries. The radial expansion releases electrostatic potential energy, leading to heating of the ions.'^ ' The ex- pansion and subsequent heating were eliminated by using

approximately 100 000 Mg"*" ions to sympathetically cool about 5000 Be+ ions to temperatures less than 0.25 K. The

ac Stark shift on the Be"^ hyperfine transition, due to the

Mg"*" cooling laser, was estimated to be less than 1 part

in 10^^. This could be neglected relative to other systematic shifts. The Mg"*" ions were observed, with an imaging

photomultiplier tube, to form a doughnut around the Be"*" ions. The size of the gap between the two species could not be determined, because the optical system could not be focused at both wavelengths at the same time. In this way a cold, steady-state cloud of Be"*" ions was obtained,

independent of the Be"*" cooling laser. The sympathetic

cooling enabled long measurement periods (up to 550 s), free from the perturbing effects of the Be"*" cooling laser. The observed resonance linewidth is inversely proportional

to the measurement time. For a measurement time of 550

s, 900 /iHz linewidth was observed on the 303 MHz hyperfine transition.^ ^ A typical resonance curve is shown in Fig. 9.The frequency standard based on this hyperfine

transition was used to test the linearity of quantum

mechanics.' ' A nonlinearity would have resulted in a dependence of the resonance frequency on the degree of

TN-178
15000

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 frequency (mHz) - 303 016 377 265

Figure 9: Resonance signal of the 303 MHz hyperfine

transition of ^Be"*" at a magnetic field of 0.8194 T. A de- crease in the fluorescence intensity corresponds to an in- crease in the transition probability. The width of the res- onance is 900 /iHz. (From Ref. [103].)

excitation of the transition, analogous to the frequency

shift of an anharmonic oscillator with amplitude. An upper limit of 6 /iHz was placed on a nonlinear correction to the Hamiltonian of the Be"*" nucleus.

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TN-181
RECENT EXPERIMENTS ON TRAPPED IONS AT

THE NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY

D. J. Wineland, J. C. Bergquist, J. J. Bollinger, W. M. Itano, F. L. Moore, J. M. Gilligan, M. G. Raizen, D. J. Heinzen,

C. S. Weimer, and C. H. Manney

Time and Frequency Division

National Institute of Standards and Technology

Boulder, Colorado 80303

INTRODUCTION

In these notes, we discuss recent experiments conducted by the ion storage group of the

National Institute of Standards and Technology in Boulder, Colorado. The overall goal of this work

has been the development of techniques for high resolution spectroscopy using stored ions. These

techniques have also been applied to problems of practical and fundamental interest. In the following

we summarize the work for the year preceding July, 1991.

(1) Be^ Hyperfme pressure shift In this experiment, an oscillator has been locked to the (m, = -1/2, m, = 1/2) ••— (-3/2, 1/2)

nuclear spin-flip, hyperfme, clock transition (oijlir = 303 MHz) in the ground state of 'Be'' (Fig. 1). The details of this experiment have been described previously [1].

clock ^f"'' "ij) transition '~3^(-3/2.

-irl— r-1/2

1/2)

(-1/2, 1/2) F-1 —-

^^-^^^^J^ n'z, 1/2) ^~^ ^^ ^-"^ (3/2, 1/2)

F=2.

(3/2, -1/2)

(1/2, -1/2)

(-1/2, -1/2)

'(-3/2, -1/2)

Fig. 1. Hyperfme energy levels (not drawn to scale) of the 'Be'^ 2s ^S.^^ ground state as a function of

magnetic field (horizontal axis). At B = 0.8194 T the 303 MHz clock transition is independent of

magnetic field to first order.

A shift of the clock transition frequency, with an unexpectedly large value, was discovered

when the background gas pressure was increased [1]. The background gas pressure could be

increased by moving the magnet of the sputter ion pump (which evacuated the trap region) so that it overlapped fewer pumping cells and reduced the pumping speed. We checked to make sure the

magnetic field at the site of the ions was not disturbed.

The composition of the gas was not known since the pressure was measured with a BayardProc. of the Enrico Fermi Summer School on "Laser manipulation of atoms and ions,"

ed. by E. Arimondo and W. D. Phillips, July, 1991, Varenna, Italy.

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Alpert gauge. It was difficult to install a residual gas analyzer on the system without a thorough

reprocessing of the vacuum system. Therefore we opted to leak in various gases which are known to be present in high vacuum systems [2] that are similar to our ^e^ apparatus. We monitored the

frequency of the oscillator that was locked to the clock transition as an increase in the background gas

pressure was separately provided by the following gases: Hj, Nj, Oj, HjO, CO, He, CH4, and COj.

The maximum pressure for any gas introduced into the system was around 4 x lO"^ Pa (3 x 10"^ Torr), and the base pressure of the system (as measured by a Bayard-Alpert gauge calibrated for Nj) was 1.3 X 10"* Pa (1 x 10'° Torr). (Below, we use pressure readings of the Bayard-Alpert gauge

which was calibrated for Nj; the quoted results must be corrected for changes in sensitivity for the

various gases.) The gases Hj, CO, HjO, He, O2 and N2, gave no observable pressure shift. Specifically, the gases Hj, CO, and Nj caused fractional frequency shifts of the ^e^ clock transition

less than 1.5 x lOVPa (2 x lO-'VlO"'" Torr). Similarly, the gases H2O, He, and O2 caused

fractional frequency shifts less than 3.8 x 10''/Pa (5 x 10"'VlO'° Torr). The gas CH4 gave a pressure shift of (-1.7 ± 0.4) x laVPa ((-2.2 + 0.5) x la^VlO"'" Torr). When CO2 was admitted

to the system the hyperfme frequency slowly changed (over about one hour) as though CO2 slowly

reacted with some consituent inside the vacuum system and released a gas which caused a pressure

shift of the clock transition.

The large difference between our data for CH4 and the other gases we measured, and the

difference between our data for CH4 and ion hyperfine pressure shifts induced by inert gases which were measured by other groups [3,4], is not understood at this time. One possible explanation is suggested by studies [5] of radiative association of C^ with H2 to form CH2 . In the models of this process, it is assumed that the Hj can stick to the C"^ for a long enough time to allow the C^-Hj

complex to radiatively stabilize. This sticking is possible because the collision energy can be taken up

by the internal degrees of freedom in the H2 molecule or the H2 - C^ complex. The sticking time can be orders of magnitude longer than the interaction time during a simple elastic collision. If a similar

sticking mechanism is active in CH4 + Be* collisions, it may account for the apparent large pressure

shift. This pressure shift might affect all precision measurements of hyperfine structures of ions

similarly. That is, we might expect the sticking time to depend only on the charge of the ion. In that case, it may be necessary to achieve significantly better vacuums by using cryopumping.

(2) Linear Paul Trap

The main advantage of using a single ion in a Paul quadrupole trap is that the kinetic energy

of micromotion can be on the order of the secular motion energy. For a single ''^g* ion cooled to the Doppler-cooling limit, the second-order Doppler shift would be [6,7] <^Vq2IVo> = - 2.3 x lO"

'*. In a quadrupole ion trap, two or more ions in the trap are pushed from the center of the trap by

their mutual Coulomb repulsion. Therefore, the second-order Doppler shift is higher due to increased

micromotion.

Consider the trap shown in Fig. 2. In this trap, the rf electric fields are transverse to the trap

axis for the entire z extent of the trap. If a single string of ions is trapped along the z axis, the

kinetic energy of micromotion is about equal to the kinetic energy in the secular motion. This kind of

trap was first demonstrated for atomic ions by Drees and Paul [8]; lower temperatures were later obtained by Church [9]. Prestage et al. [10] have demonstrated a '^Hg* microwave clock with

excellent short term stability {Oy{T) = 1.6 x 10"^' t'^) using a cloud of ions elongated in the z direction in a linear Paul trap. Dehmelt first suggested using a string of ions to suppress the secondorder Doppler shift in such a trap [11]. Strings of atomic ions have been observed at Garching [12]

and NIST [13]. In Fig. 3a, we show a string of '''Hg* ions confined in a linear trap that has rod diameters

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1.60 mm and distance of the rod centers from the z axis of the trap equal to 1.55 mm. The spacing

of the ions is approximately 20 /im.

S S

^ VoCOsHt C^ S!

-2zo

Fig. 2. Linear trap configuration. The alternating rf voltage V^cosOt is applied to diagonally

opposing electrodes as shown. We assume the end portions of the electrodes are long enough that the

resulting rf potential at the position of the ions is independent of z, so that the rf electric fields are

parallel to the x-y plane. To trap ions along z, the center four electrodes are held at static ground

potential and the two sets of four electrodes on either end are held at a static potential U„ (Uo > to trap positive ions). The average position of the ions could be made to coincide with the rf electric

field null by applying slightly different static potentials to the four central rods to correct for offsets from contact potentials or static charge on the electrode surfaces. This geometry allows laser beams

to be directed along the z axis.

When the linear density of ions is increased by increasing the static voltage applied to the end sections

of the rods, or when the x-y potential is weakened, zig-zag structures like those shown in Fig. 3b

result. These structures are the lowest-energy configurations expected for a certain range of the ratio of z confining potential and x-y potential. They have been predicted to be the lowest energy

configurations in ion storage rings [14].

Strings of ions should be useful for high resolution spectroscopy. If we use imaging

techniques, each ion can be treated as an independent atomic clock where the clock transition could be

detected with 100% efficiency (using Dehmelt's "electron shelving" scheme [6,15]). If we use 50

ions, a resonance frequency uiJl-K = 40.5 GHz (the ground state hyperfine transition for ''llg^), and

if we use the Ramsey method of interrogation where the free precession time is 100 s, the frequency

stability of this clock "ensemble" should be Oyir) = 5.5 x 10''*t'''. One or more ions whose

positions are localized to less than X/lir are also interesting for experiments investigating interference,

superradiance and subradiance, and cavity-QED.

(a) «««^f«## m

(b) « t > * « « ^ # if » ^

Fig. 3. Images taken of crystallized structures of '^Hg* ions in a linear rf trap like the one of Fig.

2. In (a), 10 ions form into a string with length 220 /xm. In (b), the the lowest energy configuration

for 13 ions is a zig-zag structure. In (a) and (b), the trap conditions are different but the

magnification is the same. (Ref. 13.)

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The 40.5 GHz hyperfme resonance has been observed on a string of ions and also on a single

ion using electron shelving. Currently, we are investigating the possibility of using cryogenic

pumping to minimize ion loss due to Hg"^' + Hg ^ dimer formation and to minimize possible

pressure shifts in the hyperfme resonance.

(3) Penning trap density limitations

We have discovered and investigated a particular limitation to the number and density of

charged particles that can be stored in a Penning trap. The energy and angular momentum of a cloud

of 'Be^ ions were controlled with radiation pressure from two different lasers. One of the laser beams was directed through the radial edge of the ion cloud and supplied a radiation pressure torque

which increased the ^e"^ rotation frequency and density. We discovered that a slight misalignment of

the trap synmietry axis with the magnetic field would heat the plasma when we attempted to increase

the rotation frequency (or density) beyond a certain value [16]. With about 2000 ^e"^ ions in the

trap, we could use this heating to align the trap symmetry axis with the magnetic field to better than

O.Or. Larger ion clouds were even more sensitive to this misalignment. In fact, in this trap, all other known methods of alignment were less sensitive.

By measuring the rotation frequency where the heating occurred as a function of the trap

voltage, we were able to associate this heating with the excitation of a collective mode of the plasma

by the static field asymmetry produced by a misalignment of the trap electric field axis from the

direction of the magnetic field [16]. The mode is called an (£,m) = (2,1) mode. In a (2,1) mode the

plasma tilts with respect to the magnetic field and precesses about the magnetic field axis. In a frame

of reference rotating with the plasma, there is a particular (2,1) mode which precesses in a direction

opposite to the rotation. For a particular value of the rotation frequency, this becomes a static mode

(no time dependence in the lab frame) that can be excited by static asymmetries. It is this static (2,1)

mode which we observed to limit the plasma density. While this is the lowest-order and therefore the

strongest static mode that can be excited, other weaker heating resonances which tended to limit the

plasma density to even lower values were observed. These weaker heating resonances are also due to the excitation of coherent modes by static field asymmetries. These static modes have important

implications for experiments where storage of large number of charged particles is important. For

example, they may provide a limit to the number of antiprotons or positrons which can be stored in a

trap. The plasma modes also play an important role in the asymmetry-induced transport which occurs

in a Penning trap even at low densities. An understanding of the modes may therefore improve the

long term confinement of many ions in a Penning trap.

(4) Search for anomalous spin-dependent forces using ^Be"" hyperfme

spectroscopy

The existence of weakly interacting bosons (such as axions) has been suggested previously.

Laboratory experiments might detect scalar or pseudoscalar couplings of such particles to matter in the form of new spin-dependent forces. We have used spectroscopy of ^e^ to search for anomalous

potentials having a dipole-monopole or dipole-dipole character [17]. The first is expected to include

terms like

V^ = 'h''D S^ f (1/X^r -H l/r2)exp( -r/X^) , (1)

where the spin S ^ (in units of fi) of particle A couples to particle B, r is the distance between

particles, X^ is the range of the force, and D is a coupling constant with units of (mass)"'. A dipoleTN-185
dipole interaction would be expected to include terms like

(2)

where the spin of particle A interacts with that of particle B. T has units of (mass)'^ and characterizes

the strength of the interaction.

We have been able to place experimental limits on D and T by examining the ^e^ hyperfme

"clock" transition frequency Vq under various conditions. Particles A are assumed to be either the ^e

nucleus or the unpaired outer electron in ^e"^. In the search for Vab, particles B were taken to be

the nucleons in the earth. We looked for a change in Pq between the cases where B „ was parallel or

antiparallel to the vertical direction in the lab. In the search for VJb, particles B were taken to be the

electron spins in the iron pole faces of an electromagnet. We compared Vq when^B,, was created by

this electromagnet with Uq when B„ was created by a superconducting solenoid (S b spins absent).

From these data, upper limits on D and T for the electron and neutron have been established and

compared to the results of others [17].

(5) Theory of Sisyphus cooling for a bound atom (ion)

Cooling that results from optical dipole forces has been considered for a bound atom (or

trapped ion) [18]. Through optical pumping, the atom can be made to feel decelerating optical dipole

forces more strongly than the accelerating optical dipole forces. This effect, which has previously

been realized for free atoms, is called Sisyphus cooling [19]. A simple model for a bound atom is examined in order to reveal the basic aspects of cooling and heating when the atom is confined in the

Lamb-Dicke regime. Results of semiclassical and quantum treatments show that the minimum energy

achieved is near the zero-point energy and can be much lower than the Doppler cooling limit. Sisyphus cooling of trapped Mg^ and Hg* have been examined theoretically.

(6) Observation of "atomic projection" noise

In spectroscopy, "technical noise," such as laser amplitude fluctuations caused by an unstable

power supply, often dominates the noise. These sources of noise can be eliminated by careful

engineering. Two examples of more fundamental noise sources are: (1) the detection shot noise in a

laser absorption spectroscopy experiment and (2) the fluctuations in signal caused by the fluctuations

in the number of atoms in an atomic beam experiment. These sources of noise can also be eliminated or significantly reduced. For example, laser shot noise can, in principle, be reduced by use of

squeezed light [20]. In a stored-ion experiment, the number of ions can be held constant thereby reducing the

atomic number fluctuations to zero. Also, when absorption is detected using electron shelving [6,15],

the detection noise approaches zero since 100% detection efficiency is possible. Because of this immunity from some sources of noise, we have been able to observe, in a clear way, what might be

called "quantum projection noise." Basically, this source of noise is caused by the statistical fluctuations in the number of atomic absorbers (ions in our case) which are observed to make the

transition in an absorption spectroscopy experiment [21].

For simplicity, assume that the number N of atoms is fixed and the efficiency of detection of

absorption by each atom is 100%. We also assume the resonance in question is driven using the

Ramsey method in the time domain where the pulse times are much less than the free precession time

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T and the lifetimes of both states involved in the transition are much greater than T. For optimum

power, where the transition probability on resonance is unity, the probability of driving the absorber

from one state to the other is given by[1,21]

p - ( 1 + cos(a) - o)q) T)I 2

,

(3)

where Wq is the resonance frequency, and co is the applied frequency. Upon detection, the average

number of atoms that are observed to make the transition is pN and the rms fluctuations in the

number of ions that are observed to make the transition is given by the dispersion of the binomial

distribution

^N = ^Np(l-p) (4)

We call this the atomic projection noise because it arises from the fluctuations in the state the atom is projected into when a measurement is made. When no excitation occurs (p = 0), or when the atom

is totally transferred into the other state (p = 1), this noise is zero. This is to be expected when the

atom is in an eigenstate, since a measurement should always find the atom in that state. The noise is maximum for p = '/^ which, for Ramsey excitation, is at the point of maximum slope of the resonance curve. Interestingly, we can show that, if this is the only source of noise, the sensitivity to frequency fluctuations is independent of where measurements are taken on the Ramsey curve (Eq.

(3)). This is because the slope of the resonance curve, dp/du, which gives frequency sensitivity (the

signal), is proportional to AN as can be verified from Eqs. (3) and (4). Therefore the ratio of signal

to noise is independent of w. However, if there are any added sources of noise, it is more

advantageous to measure at the points of maximum slope (the "half-power" points) for maximum

frequency sensitivity. In Fig. 4, we show the noise observed on a Ramsey resonance taken on the clock transition on approximately 20 'Be'^ ions. The method for measuring the curve is described in Ref. 1. The

qualitative evidence for the projection noise is the fact that the observed noise is larger on the sides of

the Ramsey resonance than on the peaks and valleys of the curve.

2000

g> 1000

(/I

500

A-

\

\

\

\

\

//

1 1

t

w

\\

\ r

1 H''

-0.10 -0.05 0.00 0.05 0.10 Freq. - 303016377.265 Hz Fig. 4. A Ramsey resonance taken on the "clock" resonance of 'Be'^ ions in a Penning trap (refer to Fig. 1). The number of ions (approximately 20) was held fixed. At each rf frequency (horizontal

scale), 30 measurements were made. The vertical scale is the number of detected photoelectron

counts from the ions' fluorescence observed in a 1 second integration time [1]. The dots indicate the means of the measurements. The error bars indicate the standard deviations. We determined, by

independent measurements, that the frequency fluctuations of the oscillator driving the clock transition

caused a negligible contribution to the observed signal fluctuations. The fact that the standard

deviations are higher on the sides of the resonance than at the peaks and valleys is a result of "atomic

projection noise."

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(7) Precision measurement of the g, factor of Mg"^

Penning traps are well suited to precision measurements of g-factors of ions. The required

magnetic field is already present. When the magnetic field is generated by a superconducting

solenoid designed to have high homogeneity and stability, there is a potential for very high accuracy.

Systematic errors can be well controlled because the ions occupy a small volume, over which the field can be very uniform. Further, the ion sample can be moved by adjusting the trap's electric potentials

in order to map out the magnetic field variations.

Previously, we measured the gj factor of the 'Be'^ ion [22]. This was done in a Penning trap

that used a nonsuperconducting electromagnet. The frequencies of a magnetic-field-dependent

hyperfine transition were measured at the same time as the ion cyclotron resonance and other

motional resonances of the ions. From these measurements, and using the ratio of the ion mass to the

electron mass, which was known from other experiments, we obtained gj(^Be^) to an accuracy of

about 1.6 parts in 10^. This measurement could be improved by the use of a superconducting

magnet.

This year, we measured the ratio of the gj factor of the ^Mg* ion to that of ^e*. The two

types of ions were trapped and cooled simultaneously in nearly the same volume. (The higher-mass

^Mg"^ ions form a ring around the ^e"^ ions [23].) A magnetic-field-dependent hyperfine transition

in 'Be* and the electron spin-flip resonance in ^Mg+ were measured, one after the other, by

microwave-optical double resonance. From these measurements, we obtained the ratio

gj(^Mg*)/gj(^e'^) to an accuracy of 1.3 parts in 10^ Combined with our previous measurement of

gj^Be*), we obtain gjC^Mg"^) to an accuracy of about 1.6 parts in 10^. This agrees with a Hartree- Fock calculation [24] to about 3 parts in 10*. The deviation of gjf^Mg"^) from the g factor of the free

electron, which is the physically interesting quantity, is about 3 parts in 10^. Our measurements

already determine this deviation to about 0.5 %. The g, factors of the neutral alkali atoms (Li,

Na,...) are well known. Comparisons with the gj factors of the isoelectronic positive ions (Be*,

Mg*,...) may reveal systematic trends and lead to a better understanding of calculational methods.

(8) Subharmonic excitation of a single electron

A single electron in a magnetic field can have its cyclotron motion excited by an oscillating

electric field. Similarly, a single electron in a Penning trap can have its cyclotron, magnetron, or

axial motion excited by an oscillating electric field. Usually, the excitation of these motions is made

at a frequency which coincides with the resonance frequency Wq of one of the motions of the electron,

but excitation at a multiple of the motional frequency is also possible. This excitation of a single

electron (whose orbit can be well controlled) is interesting because it is one of the simplest systems in

which to study parametric and subharmonic excitation and because it should be possible to study these

excitations in both the quantum mechanical and classical regimes in the same system.

We have recently studied excitation of the cyclotron and magnetron motions of a single

trapped electron [27] by an electric field which we assume to be of die form

00

n=0 Because the field is spatially nonuniform (the x" factors), excitation at some harmonic of the

fundamental motional frequency is possible; that is, where co = ncoo- This is called subharmonic

excitation. One example of this process is subharmonic excitation of the magnetron motion of an

electron. The magnetron orbit is first excited to a certain radius by a potential, oscillating at a frequency near coq, which is applied to one electrode of a split ring electrode of the Penning trap.

Since the damping of the magnetron motion is extremely small, the orbit size is nearly constant after

TN-188
this first drive is removed. The orbit size is observed as an anharmonic frequency shift of the axial resonance frequency [28]. Application of an electric field of frequency oj = ncoo causes an excitation near Wq through the C„., and C„+i terms in Eq. (5). In our experiment, this excitation was difficult to observe on the cyclotron motion because the stability of the magnet was poor and because the

damping time of the cyclotron motion allowed initial excitation only due to thermal noise.

Nevertheless, it was possible to observe excitation for n = 3 on the cyclotron motion (ooq = 2.82

GHz) and for n = 9 on the magnetron motion (coq = 47 kHz).

One feature of this kind of system may be of practical importance. For example, if the

subharmonic excitation can be detected by observing the motion of the electron at its fundamental

frequency, the resulting device acts as a frequency divider [25,26]. That is, we inject a certain

frequency into the device and measure a submultiple of the input frequency. For the electron, this parametric down-conversion process can, in principle, be extended to very high orders; that is, it should be possible to inject a laser frequency and read out a microwave frequency (the cyclotron

frequency).

(9) Observation of time varying radiation pressure forces

This work arose out of attempts to observe the effects of optical dipole forces on trapped ions.

One manifestation of dipole forces would be Sisyphus cooling (Section (5)). It should also be

possible to observe the dipole force on ions without the effects of optical pumping (required for

Sisyphus cooling). One way the dipole force could be observed is to modulate it at a frequency near

a motional resonance frequency of the ions and observe the excitation as a change in fluorescence

[22], or a change in the orbit size observed on an imaging device. Although not immediately

apparent, this excitation has many similarities with the nonlinear excitation of a single electron

(Section (8)). One configuration for observing time-varying optical dipole forces is the following: Suppose

we have a single ion localized near the center of the trap. We focus a Gaussian laser beam so that its waist coincides with the center of the trap along the direction of the beam, but we position the beam

transversely so that the center of the trap coincides with the point of half intensity on the side of the

beam. Specifically, assume the orgin of coordinates is at^the trap center and the z axis is the trap

symmetry axis. Assume the beam is incident along the y direction, and its center is in the z =

plane, but is located at a position x = (ln2/2)''^Wo, where Wq is the Gaussian beam waist. The laser frequency co is assumed to be detuned by an amount 5 from a particular resonance transition

frequency Wq of the ion (6 = co - Wo). For 6 > the dipole force [29] is in the minus-x direction; for

6 < 0, it is in the plus-x direction. If we now modulate the intensity of the laser we modulate the

force in the x direction and can observe the corresponding excitation of the ion. A similar excjtation occurs when we use two overlapping laser beams whose incident directions (taken to be along k i and

kj) are, in general, different. Assume the beams are plane waves. If the frequencies of the beams

are the same,_the intensity as a ftinctjpn of position goes as cos[(k 1 - k2)-r ]. If we choose k , - kj along x , then the intensity is spatially modulated in the x

direction. If we now let the frequencies of the lasers Wi and coj be different, then the dipole force in the X direction is modulated in time at the frequency Wi - Wj. When coi - coj is adjusted near the

motional resonance frequency of the ion, the orbit size will increase. This case is closely analogous

to the excitation of a single electron by two laser beams [26].

For a traveling wave laser beam, the ion also feels the force along the direction of the beam

due to the spontaneous scattering force. If the intensity of the laser is modulated, the spontaneous

force is modulated and motional excitation will occur.

In order to see the dipole forces clearly, the effects of the spontaneous force (laser cooling

and heating) must be suppressed. This requires that the laser intensity and detuning be sufficiently

8

TN-189
large. In our preliminary experiments, our laser intensity was small enough that the spontaneous

forces dominated. We were able to observe the axial and magnetron excitation of a small number of

'Be* ions in a Penning trap due to the modulated spontaneous force. A steady state excitation was

observed because the spontanous force was balanced by laser cooling.

(10) Hg"" optical spectroscopy

The basic idea for this experiment has been discussed previously [6]. The goal of the

experiment is to lock a laser local oscillator to the 282 nm quadrupole transition in Hg"^. To obtain

the best performance, it will be necessary to observe the linewidth of the quadrupole transition with a resolution near the natural linewidth of 2 Hz. The main problem in our experiment, as well as all other current optical frequency standard experiments, is to have the laser linewidth be narrower than

the inverse of the time it takes to lock the laser to the resonance line. By using a single, laser-cooled ''^g'^ ion as an absolute frequency spectrum analyzer, we

have measured a linewidth below 40 Hz for a ring dye laser (oscillating at — 5.3 x 10'"* Hz) that was

stabilized to an ultra-quiet reference cavity of high finesse [31]. This is the best absolute stability that has been demonstrated for an optical laser (as well as the highest atomic or molecular Q reported to

date). If we try to improve the laser performance by using the spectrum of the single ion, we are

limited by the low signal-to-noise ratio and by the lifetime of the ion in the trap.

To more easily study the performance of the cavity-stabilized laser, a second cavity was

constructed on an independent optical table to which the frequency of a laser could be locked. A

measurement of the noise fluctuations in the electronic error signals indicated that the laser frequency

could be stabilized to each cavity to less than 30 mHz. In other words, the frequency of the

stabilized laser tracked the resonance frequency of the reference cavity to within 30 mHz. Some

authors claim this number to be their laser linewidth, but this is not the relevant linewidth for

metrology. The dominating contribution to the laser linewidth comes from the length instability of

the reference cavities. The study was carried out by heterodyning the light from the two independently stabilized

laser beams and analyzing the fluctuations in the power spectrum of the beat frequency. We have

attempted to isolate the cavities thermally, barometrically, and mechanically. We begin with a cavity

constructed from a cylindrical spacer made from an ultra-low expansion

/• /^

c /\ o

(0 / \ >

A

/

1 b \

CO f

1 \

CO J \

f \

llj \

25 Hz / Division Fig. 5. Spectrum of the beat frequency between the two independent cavity-stabilized lasers discussed

in the text. A linear, relative cavity drift is removed by mixing the beat note with the frequency from

a synthesizer which is swept in time. The resolution bandwidth is 30 Hz. Total integration time for

this data is 70 s. The 14.6 Hz sidebands are due to a floor vibration (indepedently detected with a seismometer). The apparent bandwidth is about 30 Hz; the bandwidth of the most stable laser of the

two is at least ^2 narrower.

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material. The cavity is suspended by two thin wires in an aluminum vacuum chamber that is evacuated to about 10* Pa. (= 10"* Torr). The walls of the vacuum vessel are temperature-regulated

to better than 10 mK. The vacuum vessel is supported by, but thermally isolated from, a stiff surface

plate. We have attempted several passive schemes to isolate the surface plates from seismic noise.

The best isolation was achieved by supporting the surface plate from the ceiling with latex tubing.

The vertical and horizontal resonant frequencies for the latex-support system were 0.33 Hz which is lower by more than a factor of 10 a brick, rubber pad, and sandbox support. The linewidth of the

heterodyne signal between the laser radiation stabilized to the cavity supported by the latex tubing and

the cavity supported on the sandbox table was approximately 30 Hz or less (Fig. 5). Presumably the

width of the heterodyne signal is dominated by the stability of the cavity on the sandbox table. Currently, we are working to verify this and working to build better cavities (higher finesse,

increased stiffness and lower sensitivity to temperature and aging).

ACKNOWLEDGEMENTS

We gratefully acknowledge the support of the U.S. Office of Naval Research and the U.S.

Air Force Office of Scientific Research. We thank A. DeMarchi, R. DruUinger, H. Robinson and

M. Young for helpful comments on the manuscript.

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Bergquist, J. J. Bollinger, W. M. Itano, D. J. Heinzen, S. L. Gilbert, C. H. Marmey, and M.

G. Raizen, IEEE Trans, on Ultrasonics, Ferroelectrics, and Frequency Control 37, 515

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(Chapman and Hall, London, 1968).

3. J. Vetter, M. Stuke, and E. W. Weber, Z. Phys. A273, 129 (1975).

4. L. S. Cutler, R. P. Giffard, and M. D. McGuire, Proc. 37th Ann. Symp. Freq. Control,

1983 (copies available from IEEE, 445 Hoes Lane, Piscataway, NJ 08854), p. 32.

5. E. Herbst, Astrophys. J. 252, 810 (1982), and references therein.

6. J. C. Bergquist, W. M. Itano, and D. J. Wineland, Phys. Rev. A36, 428 (1987).

7. D. J. Wineland, W. M. Itano, J. C. Bergquist, J. J. Bollinger, F. Diedrich, and S. L.

Gilbert, in Frequency Standards and Metrology. Proc. 4th Symposium, ed. by A. DeMarchi,

Ancona, Italy (Springer-Verlag, Berlin, Heidelberg, 1989) p. 71.

8. J. Drees and W. Paul, Z. Physik 180, 340 (1964).

9. D. A. Church, J. Appl. Phys. 40, 3127 (1969).

10. J. Prestage, G. J. Dick and L. Maleki, J. Applied Physics, 66, 1013 (1989); J. D. Prestage,

G. J. Dick, and L. Maleki, IEEE Trans. Instrum. and Meas. 40, 132 (1991).

11. H. G. Dehmelt, in Frequency Standards and Metrology. Proc. 4th Symposium, ed. by A.

DeMarchi, Ancona, Italy (Springer-Verlag, Berlin, Heidelberg) p. 286.

12. H. Walther in Light Induced Kinetic Effects on Atoms. Ions and Molecules , ed. by L. Moi,

S. Gozzini, C. Gabbanini, E. Arimondo, and F. Strumia, (ETS Editrice, Pisa, 1991), p. 261;

10

TN-191
G. Birkl, S. Kassner, W. Quint, H. Walther, private communication.

13. M. G. Raizen, J. C. Bergquist, J. M. Gilligan, W. M. Itano, and D. J. Wineland, J. Modern

Optics, special issue on the Physics of Trapped Ions, ed. by R. Blatt, P. Gill, and R. C.

Thompson, to be published.

14. See for example: R. W. Hasse and J. P. Schiffer, Proc. Workshop on Crystalline Ion Beams,

Wertheim, W. Germany, ed. by R. W. Hasse, I. Hofinann, D. Liese (GSI report GSI-89-10,

ISSN 0170-4546), p. 33; D. Habs, ibid, p. 43; R. W. Hasse and J. P. Schiffer, Ann. Phys.

203, 419 (1990).

15. W. Nagourney, J. Sandberg, and H. Dehmelt, Phys. Rev. Lett. 56, 2797 (1986).

16. D. J. Heinzen, J. J. Bollinger, F. L. Moore, W. M. Itano, and D. J. Wineland, Phys. Rev.

Lett. 66, 2080 (1991).

17. D. J. Wineland, J. J. Bollinger, D. J. Heinzen, W. M. Itano, and M. G. Raizen, Phys. Rev.

Lett. 67, 1735 (1991).

18. D. J. Wineland, J. Dalibard, C. Cohen-Tannoudji, J. Opt. Soc. Am. B9, 32 (1992).

19. C. Cohen-Tannoudji and W. D. Phillips, Physics Today, vol. 43, no. 10, 1990, p. 33.

20. B. Yurke and E. A. Whittaker, Optics Lett. 12, 236 (1987); A. Heidmann, R. J. Horowicz,

S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, Phys. Rev. Lett. 59, 2555 (1987).

21. D. J. Wineland, W. M. Itano, J. C. Bergquist, and F. L. Walls, Proc. 35th Ann. Symp.

Freq. Control, Philadelphia, PA, May 1981, p. 602-611. (Copies available from: Annual

Frequency Control Symposium, c/o Electronic Industries Assoc, 2001 Eye St., Washington,

D. C, 20006.)

22. D.J. Wineland, J.J. Bollinger, and W.M. Itano, Phys. Rev. Lett. 50, 628 (1983).

23. D. J. Larson, J. C. Bergquist, J. J. Bollinger, W. M. Itano, and D. J. Wineland, Phys. Rev.

Lett. 57, 70 (1986).

24. L. Veseth, Univ. of Oslo, unpublished calculations.

25. D. J. Wineland, J. Appl. Phys. 50, 2528 (1979).

26. Y. J. Ding and A. E. Kaplan J. Opt. Soc. Am. B6, 1299, (1989) and references therein.

27. C. Weimer, et al., in preparation.

28. F. L. Moore, D. L. Farnham, P. B. Schwinberg, and R. S. Van Dyck Jr., Nuclear

Instruments and Methods in Physics Research B43, 425 (1989).

29. see for example: J. Dalibard and C. Cohen-Tannoudji, J. Opt. Soc. Am. B2, 1707 (1985).

30. D. J. Heinzen and D. J. Wineland, Phys. Rev. A42, 2977 (1990).

31. J. C. Bergquist, F. Diedrich, W. M. Itano, and D. J. Wineland, in Laser Spectroscopy IX. ed. by M. S. Feld, J. E. Thomas, and A. Mooradian (Academic, San Diego, 1989), p. 274;

J. C. Bergquist, W. M. Itano, F. Eisner, M. G. Raizen, and D. J. Wineland, in Light

Induced Kinetic Effects on Atoms. Ions and Molecules , ed. by L. Moi, S. Gozzini, C.

Gabbanini, E. Arimondo, and F. Strumia, (ETS Editrice, Pisa, 1991), p. 291.

11

TN-192
NIST-114A

(REV. 3-90) U.S. DEPARTMENT OF COMMERCE

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY

BIBLIOGRAPHIC DATA SHEET

1. PUBUCATION OR REPORT NUMBER

TN 1353

2. PERFORMINQ ORGANIZATION REPORT NUMBER

B92-0134

3. PUBUCATION DATE April 1992

4. TITLE AND SUBTITLE Trapped Ions and Laser Cooling III

5. AUTHOR(S)

J.C. Bergquist, J.J. Bollinger, W.M. Itano, D.J. Wineland

6. PERFORMINQ ORQANIZATION (IF JOINT OR OTHER THAN NIST, SEE INSTRUCTIONS)

U.S. DEPARTMENT OF COMMERCE

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY

GAITHERSBURG, MD 20899

7. CONTRACT/GRANT NUMBER

e. TYPE OF REPORT AND PERIOD COVERED

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (STREET, CITY. STATE, ZIP)

10. SUPPLEMENTARY NOTES

ions, ions-cooling, ions-micro, ions-opt, ions-nonn, ions-tests

11. ABSTRACT (A 200-WORD OR LESS FACTUAL SUMMARY OF MOST SIGNIFICANT INFORMATION. IF DOCUMENT INCLUDES A SIGNIFICANT BIBUOQRAPHY OR

LITERATURE SURVEY, MENTION IT HERE.)

This collection of papers represents the work of the Ion Storage Group,

Time and Frequency Division, National Institute of Standards and Technology from

October, 1988 to February, 1992. It follows the collections of papers contained in

NBS Technical Note 1086 "Trapped Ions and Laser Cooling" (June 1985) and NIST

Technical Note 1324 "Trapped Ions and Laser Cooling II" (September 1988). Although

the primary goal of this work has been the development of techniques necessary

for achieving high resolution spectroscopy, we have also been able to investigate

related areas of research.

12. KEY WORDS (6 TO 12 ENTRIES; ALPHABETICAL ORDER; CAPITAUZE ONLY PROPER NAMES; AND SEPARATE KEY WORDS BY SEMICOLONS)

atomic clocks; atomic spectroscopy; frequency standards; ion storage; laser cooling;

non neutral plasma; quantum measurement

13. AVAILABIUTY UNUMITED

FOR OFFICIAL DISTRIBUTION. DO NOT RELEASE TO NATIONAL TECHNICAL INFORMATION SERVICE (NTIS). X

ORDER FROM SUPERINTENDENT OF DOCUMENTS, U.S. GOVERNMENT PRINTING OFFICE, WASHINGTON, DC 20402. ORDER FROM NATIONAL TECHNICAL INFORMATION SERVICE (NTIS). SPRINGFIELD, VA 22161.

14. NUMBER OF PRINTED PAGES

204

15. PRICE

ELECTRONIC FORM

«U S GOVERNMENT PRINTING OFFICE: 19a^e7S^)7«/ 65066

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