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PUBUCATIONS

United States Department of Commerce

Technology Administration

National Institute of Standards and Technology

NIST Technical Note 1380

Technical Note 1380

Trapped Ions and Laser Cooling IV

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

Edited by

James C. Bergquist

John J. Bollinger

Wayne M. Itano

Christopher R. Monroe

David J. Wineland

Time and Frequency Division

Physics Laboratory

National Institute of Standards and Technology

325 Broadway

Boulder, Colorado 80303

Supported in part by

U.S. Office of Naval Research

800 North Quincy

Arlington, Virginia 22217

U.S. Army Research Office

P.O. Box 12211

Research Triangle Park, North Carolina 27709

January 1996

, .

.—

^

U.S. DEPARTMENT OF COMMERCE, Ronald H. Brown, Secretary

TECHNOLOGY ADMINISTRATION, Mary L. Good, Under Secretary for Technology

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY, AratI Prabhakar, Director
National Institute of Standards and Technology Technical Note

Natl. Inst. Stand. Technol., Tech. Note 1380, 204 pages (January 1996)

CODEN:NTNOEF

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1996

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 May 1992 to January 1996. It follows the collections of papers contained in NBS Technical Note 1086,

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

Laser Cooling II (September 1988), and NIST Technical Note 1353, Trapped Ions and Laser

Cooling III (April 1992). 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 viii were published or prepared during the period May 1992 to January 1996 but are not included here. Copies 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 contributions of many colleagues to this collection. In

particular, we thank Amy Barton, Ed Bell, Dana Berkeland, Flavio Cruz, Dan Dubin, Ulli

Eichmann, Jon Gilligan, Phil Gould, Dan Heinzen, Steve Jefferts, Brana Jelenkovid, Brian

King, Dawn Meekhof, John Miller, Fred Moore, Martin Poitzsch, Mark Raizen, Norman

Ramsey, Max Rauner, Andy Steinbach, Carl Weimer, and Joseph Tan. We gratefully

acknowledge the continued support of the U.S. Office of Naval Research and the U.S. Army

Research Office. We thank Wendy Ortega Henderson for assembling this collection.

James C. Bergquist

John J. Bollinger

Wayne M. Itano

Christopher R. Monroe

David J. Wineland

Boulder, Colorado

February 1996

m

CONTENTS

LIST OF ADDITIONAL PUBLICATIONS viii

FREQUENCY STANDARDS AND METROLOGY

1. "Laser stabilization to a single ion," J. C. Bergquist, W. M. Itano, TN-1

and D. J. Wineland, in Frontiers in Laser Spectroscopy, proc.

International School of Physics < < Enrico Fermi > > , Course

CXX, ed. by T. W. Hansch and M. Inguscio (North Holland,

Amsterdam, 1994), pp. 359-376.

2. "Ionic crystals in a linear Paul trap," M. G. Raizen, J. M. Gilligan, TN-19

J. C. Bergquist, W. M. Itano, and D. J. Wineland, Phys. Rev.

A41, 6493-6501 (1992).

3. "Cryogenic linear ion trap for accurate spectroscopy," M. E. TN-28

Poitzsch, J. C. Bergquist, W. M. Itano, and D. J. Wineland, Rev.

Sci. Instrum. 67, 129-134 (1996).

4. "Minimizing the time-dilation shift in Penning-trap atomic clocks," TN-34

J. N. Tan, J, J. Bollinger, and D. J. Wineland, Proc., Conf. on

Precision Electromagnetic Meas., Boulder, CO, June, 1994; IEEE

Trans, on Instrum. Meas. 44, 144-147 (1995).

5. "High-magnetic-field corrections to cesium hyperfme structure," TN-38

W. M. Itano, Proc. , Fifth Symp. Freq. Standards and Metrology,

Woods Hole, MA, Oct. 1995 (Worid Scientific).

6. "Application of laser-cooled ions to frequency standards and TN-40

metrology," D. J. Wineland, J. C. Bergquist, D. J. Berkeland, J. J. Bollinger, F. C. Cruz, W. M. Itano, B. M. Jelenkovid, B. E. King,

D. M. Meekhof, J. D. Miller, C. Monroe, and J. N. Tan, ibid.

7. "Accurate measurement of time," W. M. Itano and N. F. Ramsey, TN-49

Sci. Am. 269(1), 56-59 and 62-65 (July 1993).

LASER COOLING AND QUANTUM STATE CONTROL

1. "Resolved-sideband Raman cooling of a bound atom to the 3-D TN-57

zero-point energy," C. Monroe, D. M. Meekhof, B. E. King, S. R.

Jefferts, W. M. Itano, D. J. Wineland, and P. Gould, Phys. Rev.

Lett. 75, 4011-4014 (1995).
2. "Quantum projection noise: population fluctuations in two-level TN-61

systems," W. M. Itano, J. C. Bergquist, J. J. Bollinger, J. M.

Gilligan, D. J. Heinzen, F. L. Moore, M. G. Raizen, and D. J. Wineland, Phys. Rev. AiZ, 3554-3570 (1993).

3. "Spin squeezing and reduced quantum noise in spectroscopy," D. J. TN-78

Wineland, J. J. Bollinger, W. M. Itano, F. L. Moore, and D. J. Heinzen, Phys. Rev. A4^, R6797-R6800 (1992).

4. "Squeezed atomic states and projection noise in spectrosopy," TN-82

D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heinzen,

Phys. Rev. A5Q, 67-88 (1994).

5. "Spin squeezing applied to frequency standards," J. J. Bollinger, TN-104

D. J. Wineland, W. M. Itano, and D. J. Heinzen, Proc. Fifth

Symp. Freq. Standards and Metrology, Woods Hole, MA, Oct.

1995 (World Scientific).

6. "Demonstration of a fundamental quantum logic gate," C. Monroe, TN-112

D. M. Meekhof, B. E. King, W. M. Itano, and D. J. Wineland,

Phys. Rev. Lett. 7^, 4714-4717 (1995).

INTERFERENCE

1. "Young's interference experiment with light scattered from two TN-116

atoms," U. Eichmann, J. C. Bergquist, J. J. Bollinger, J. M.

Gilligan, W. M. Itano, D. J. Wineland, and M. G. Raizen, Phys.

Rev. Lett. 7Q, 2359-2362 (1993).

2. "Light scattered from two atoms," W. M. Itano, U. Eichmann, TN-120

J. C. Bergquist, J. J. Bollinger, J. M. Gilligan, M. G. Raizen, and

D. J. Wineland, Proc. Lasers '93, Lake Tahoe, NV, edited by V.

J. Corcoran and T. A. Goldman (STS Press, McLean, VA, 1994),

pp. 412-419.

NON-NEUTRAL PLASMAS AND SINGLE ELECTRON STUDIES

1. "Electrostatic modes of ion-trap plasmas," J. J. Bollinger, D. J. TN-128

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

Dubin, Phys. Rev. A48, 525-545 (1993).

2. "Electrostatic modes as a diagnostic in Penning-trap experiments,TN-149

C. S. Weimer, J. J. Bollinger, F. L. Moore, and D. J. Wineland,

Phys. Rev. A49, 3842-3853 (1994).

vi
3. "Long-range order in laser-cooled, atomic-ion Wigner crystals TN-161

observed by Bragg scattering," J. N. Tan, J. J. Bollinger,

B. Jelenkovid, and D. J. Wineland, Phys. Rev. Lett. 7i, 4198-4201

(1995).

4. "Observation of long-range order in trapped ion plasmas by Bragg TN-165

scattering," J. N. Tan, J. J. Bollinger, B. Jelenkovid, W. M. Itano,

and D. J. Wineland, Proc., Intl. Conf. on the Physics of Strongly

Coupled Plasmas, Binz, Germany, Sept. 11-15, 1995, (World

Scientific).

5. "Laser-cooled positron source," D. J. Wineland, C. S. Weimer, and TN-175

J. J. Bollinger, Proc., Anti-Hydrogen workshop, Munich, July SCSI, 1992, ed. by J. Eades, in Hypeifine Interactions 7^, 115-125

(1993).

6. "High-order multipole excitation of a bound electron," TN-186

C. S. Weimer, F. L. Moore, and D. J. Wineland, Phys. Rev. Lett. 70, 2553-2556 (1993).

APPARATUS

1. "Coaxial-resonator driven rf (Paul) trap for strong confinement," TN-190

S. R. Jefferts, C. Monroe, E. W. Bell, and D. J. Wineland, Phys.

Rev. A^, 3112-3116(1995).

vu
ADDITIONAL PUBLICATIONS

The following publications were published or prepared during the period between May 1992

and January 1996, but are not included in this technical note.

1. "Experimental results on normal modes in cold, pure ion plasmas," J. J. Bollinger,

D. J. Heinzen, F. L. Moore, C. S. Weimer, W. M. Itano, and D. J. Wineland, in

Strongly Coupled Plasma Physics, ed. by Van Horn and Ichimaru (University of

Rochester Press, Rochester, NY, 1993), pp. 393-398.

2. "Quantum measurements of trapped ions," W. M. Itano, J. C. Bergquist, J. J. Bollinger, J. M. Gilligan, D. J. Heinzen, F. L. Moore, M. G. Raizen, and D. J. Wineland, Proc., Inti. Symp. on Quantum Physics and the Universe, Waseda Univ.,

Tokyo, Japan, Aug. 19-22, 1992, ed. by M. Namiki, K. Maeda, and I. Ohba, Vistas

in Astronomy 37, 169-183 (1993).

3. "Precise spectroscopy for fundamental physics," W. M. Itano, J. C. Bergquist, J. J. Bollinger, J. M. Gilligan, D. J. Heinzen, F. L. Moore, M. G. Raizen, and D. J. Wineland, Hyperfme Interactions Tg, 211-220 (1993).

4. "Non-neutral ion plasmas and crystals, laser cooling, and atomic clocks," J. J. Bollinger, D. J. Wineland, and D. H. E. Dubin, Phys. Plasmas 1, 1403-1414 (1994).

5. "Interference in the resonance fluorescence of two trapped atoms," U. Eichmann,

J. C. Bergquist, J. J. Bollinger, J. M. Gilligan, W. M. Itano, M. G. Raizen, and

D. J. Wineland, Laser Spectroscopy, Proc., Xlth Inti. Conf,, ed. by L. Bloomfield,

T. Gallagher, and D. J. Larson (AIP Conf. Proc. 290, New York, NY, 1994), pp.

43-48.

6. "Paul trap for optical frequency standards," S. R. Jefferts, C. Monroe, A. S. Barton,

and D. J. Wineland, IEEE Trans. Instrum. Meas. 44, 148-150 (1995).

7. "Progress on a cryogenic linear trap for ^^g"*^ ions," M. E. Poitzsch, J. C.

Bergquist, W. M. Itano, and D. J. Wineland, Proc., 1994 IEEE Inti. Frequency

Control Symp., pp. 744-746, June 1994.

8. "Nonneutral ion plasmas and crystals in Penning traps," J. J. Bollinger, J. N. Tan,

W. M. Itano, D. J. Wineland, and D. H. E. Dubin, Proc. Nobel Symp. on Trapped

Charged Particles and Related Fundamental Physics, Lysekil, Sweden, August 1994,

Physica Scripta T59, 352-359 (1995).

9. "Cooling methods in ion traps," W. M. Itano, J. C. Bergquist, J. J. Bollinger, and

D. J. Wineland, ibid, pp. 106-120.

10. "Quantum effects in measurements on trapped ions," D. J. Wineland, J. C. Bergquist,

J. J. Bollinger, and W. M. Itano, ibid, pp. 286-293.

viii
11. "Quantum mechanics with one, two, and many atoms," W. M. Itano, J. C. Bergquist,

J. J. Bollinger, and D. J. Wineland, in Group Theoretical Methods in Physics, Proc.,

XX Intl. Colloquium (Yamada Conference XL, "Quantum mechanics widi one, two,

and many atoms,"), Toyonaka, Japan, 1994, ed. by A. Arima, T. Eguchi, and

N. Nakanishi (World Scientific, Singapore, 1995), pp. 10-18.

12. "Laser-cooled trapped-ion experiments at NIST," J. N. Tan, J. J. Bollinger, A. S.

Barton, and D. J. Wineland, in Non-neutral Plasma Physics II, ed. by J. Fajans and

D. H. E. Dubin, AIP Conf. Proc. 331 (AIP Press, New York, NY, 1995), pp. 215-

228.

13. "A cryogenic linear ion trap for ^^g* frequency standards," J. Miller, M. E.

Poitzsch, F. C. Cruz, D. J. Berkeland, J. C. Bergquist, W. M. Itano. and D. J. Wineland, Proc., 1995 IEEE Inti. Frequency Control Symp., June 1995, pp. 110-

112.

14. "Quantum mechanically correlated states and atomic clocks," C. Monroe, D. M.

Meekhof, B. E. King, W. M. Itano, J. J. Bollinger, and D. J. Wineland, Proc. 1995

Moriond Workshop (Editions Frontieres, Gif-sur-Yvette, France, 1996).

15. "Experiments at NIST with trapped ions: 3-D zero-point cooling, quantum gates,

Bragg scattering, and atomic clocks," C. Monroe, A.S. Barton, J.C. Bergquist,

D. J. Berkeland, J.J. Bollinger, F. Cruz, W.M. Itano, S.R. Jefferts, B.M.

Jelenkovid, B.E. King, D.M. Meekhof, J.D. Miller, M.E. Poitzsch, J.N. Tan, and

D.J. Wineland, Proc., 12th Inti. Conf. on Laser Spectroscopy, Capri, Italy, June

1995 (World Scientific), to be published.

16. "Minimizing time dilation in Penning trap ion clocks," J. N. Tan, J. J. Bollinger,

W. M. Itano, and D. J. Wineland, Proc. 5th Symp. on Frequency Standards and

Metrology, Woods Hole, MA, Oct. 1995 (World Scientific) to be published.

17. "Primary atomic frequency standards: New developments," R. E. DruUinger, S. L.

Rolston, and W. M. Itano, in Review ofRadio Science, in press.

18. "CW second harmonic generation with elliptical Gaussian beams," A. Steinbach, M.

Rauner, F. C. Cruz, and J. C. Bergquist, Opt. Commun., in press.

19. "Generation of non-classical motional states of a trapped atom," D. M. Meekhof,

C. Monroe, B. King, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett., in press.

20. "A 'Schrodinger Cat' superposition state of an atom," C. Monroe, D. M. Meekhof,

B. E. King, and D. J. Wineland, submitted to Science.

21. "Optimal frequency measurements with maximally correlated states," J. J. Bollinger,

W. M. Itano, D. J. Wineland, and D. J. Heinzen, submitted to Phys. Rev. Lett.

IX

Laser Stabilization to a Single Ion.

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

Time and Frequency Division, National Institute of Standards and Technology

Boulder, CO 80303

I. - Spectrally narrow Optical Oscillators.

I.l. - Frequency references.

An unperturbed optical resonance in an atom or molecule provides a good

absolute frequency reference, but practically there are limitations. The signalto-noise ratio is limited by the number of atoms in the interrogation region and

by saturation of the resonance. If the probe laser is spectrally broad, then the

measured atomic line is broadened, which degrades the stability. Furthermore,

atoms recognize interactions and collisions with neighboring atoms, usually

with shifts to the internal energy level structure. The motion of the atoms also produces Doppler shifts and broadening. A single laser-cooled atom removes many of these problems but only with a severe penalty in the signal-to-noise ratio [1]. Even so, by detecting each transition in the single atom, it should be

possible to stabilize the frequency of a laser oscillating in the visible to better

than 10"^^ t"^/^ with an accuracy approaching 10"^* [2], if the laser were suffi- ciently stable (or spectrally narrow) for times that are comparable to the inter- rogation time of the transition in the atom.

A two-step approach might then be appropriate; spectrally narrow the laser by some scheme that offers good short- to medium-term stability {e.g., 1ms to 10 s), then stabilize the frequency of the laser for longer times to a narrow reso- nance in a single atom. The reference for the short- to medium-term stabiliza- tion of the laser not only needs good stability in this time frame, but also low

phase and frequency noise. If a Fabry-Perot cavity is used as the frequency dis- criminator, then the response of the frequency discriminator can be nearly lin- ear as a function of power and the signal-to-noise ratio can be high [3]. Frequency shifts and fluctuations arise due to thermal distortion in the mirror coat- ings caused by absorption of Light in a small volume in the dielectric stack and

due to photochemical processes at the surfaces of the coatings. Also, practical

359

TN-1
360 J. C. BERGQUIST, W. M. ITANO and D. J. WINELAND

limits are reached at powers that saturate the detector or at powers (circulating

in the cavity) that cause radiation pressure noise, but the signal-to-noise ratio

from a cavity can be many orders of magnitude larger than that obtained with

atoms. Although there are other types of optical frequency references, the most

widely used is the Fabry-Perot interferometer, principally because the fre- quency excursions to error-voltage can be extremely large in a high-finesse cavity. We will spend time in the next few sections discussing some of the details and limits of a suitably stable reference cavity, and then turn our attention to the single atom.

1.2. - Reference cavity limitations.

We can begin with a brief look at the demands that a spectral purity of 1 Hz

places on the physical stability of the reference cavity. If the cavity has a length

of 30 cm and the optical wavelength is 500 nm, then the optical-path length between the mirrors must not change by more than 10 "^^m, the approximate size of the nucleus of any of the constituent atoms in the dielectric coatings. Researchers who study parity-nonconserving interactions in atomic systems

sometimes use the analogy that a human hair added to the radius of the Earth is equivalent to the distortion in an atom caused by the parity-nonconserving part

of the Hamiltonian. By the same taken, if the spacer for an optical cavity were

the Earth, a human hair added to the diameter would cause a frequency shift of

about 300 Hz! In the first part of this section, we investigate some of the fundamental limits to the attainment of an average spacing between two mirrors that

is stable to better than 1 • 10"^^ and to the achievement of a laser that is spectrally narrowed to better than 1 Hz. We will also address some of the limitations

imposed by various environmental factors. We will see that, although fundamental limits come from quantum mechanics and thermodynamics, important

practical limitations come from mechanics and gravitational coupling to the

noisy terrestrial environment. In particular, at low Fourier frequencies

( < 100 Hz), seismic noise and pendulum motion dominate the noise budget.

At the quantum level, the measurement of the length of the cavity to which

the laser is locked brings about the inevitable competition between the measurement precision and the perturbation of the measurement to the system.

The measurement precision can be improved by a factor of 1/\N by increasing

the number A'^ of (signal) photons in the measurement interval, whereas the

shot noise of the radiation pressure on the mirrors increases as yN. The optimum flux of signal photons, or input power (assuming 100% of the light is cou- pled into the cavity on resonance and that the detection efficiency is unity), is given when both effects are equal in magnitude [4]. For the laser interferometers used in gravity wave detection, the mirrors are suspended as pendulums

and the optimum power is calculated for frequencies cj/Ztz well above the reso- nance frequency (i}()/2k of the pendulum support. As a function of frequency,

TN-2
LASER STABILIZATION TO A SINGLE ION 361

this limit length uncertainty is given by [5]

(1) Aa;«(4^A//m)i/Vw.

where m is the mass of suspended mirrors and A/ is the measurement bandwidth. Interestingly, this is the same measurement precision allowed by the

standard quantum limit [4-7]. In our case the resonator is composed of a single

bar, or spacer, to which the mirrors are rigidly attached. The resonator is then

suspended, often by small-diameter wires, inside a temperature-regulated,

evacuated housing. The resonator bar can also be treated as a harmonic system,

but now the frequency of the lowest mechanical resonance frequency a)Q/2n is typically greater than a few kHz. The noise spectrum of length fluctuations that

is of particular importance to us is at Fourier frequencies that are below the

lowest resonance frequency of the bar. At frequencies lower than coo/27z and for optimum power, the measurement precision limit is independent of Fourier

frequency

(2) Aa: - (4^A//my/Va>o .

For cjq/2k = 10 kHz, a cavity finesse of 100000 at A = 500 nm and a near optimum input power of about 3 W, this limit corresponds to a fractional length uncertainty of less than (5 10"^ w/VHz)(A/)^''^ fo^ ^ typical bar mass of 4kg.

Thus, with as little as 100 [aW incident on the cavity, the quantum fluctuations

in the radiation pressure acting on the mirrors are neghgible, yet there is suffi- cient signal-to-noise ratio that the frequency of a laser can be made to track the resonance of the cavity to well below 1 Hz [8] (if only limited by the shot noise of

the detected signal).

The technical fluctuations in the laser light that is coupled into the cavity

must also be considered. If the finesse is 100 000 and the corresponding power

enhancement as high as 30 000, then 100 jj.W of input power translates into 3 W

of circulating power when the laser is resonant with the cavity. This in turn

gives a force W on each mirror of about 2 • 10"* N. If the mirrors are treated as clamped disks of thickness t, the deflection of each mirror, S, is given

by [9]

(3) S = SWrHl-p^)/47:Et\

where the light force is assumed to act uniformly over a concentric area much

less than the area of the mirror. The radius r of the mirror is measured from the

center to the clamped edge, p is Poisson's ratio and E is Young's modulus of

elasticity. For a mirror substrate with a material composition similar to fused

silica, p is about 0.17 and E is about 7 • 10^" N/m^. If t is 5 mm and r is 10 mm,

then the deflection at the center of the mirror is about 1.7 • 10"^^ m for a radiation mode size (wq ) of 200 |j.m. For a cavity that is 30 cm long, the corresponding

fi-actional length change (2S/L) is about 1 • 10"^^, or about 0.5 Hz. The dimenTN-3
362 J. C. BERGQUIST, W. M. ITANO and D. J. WINELAND

sions and physical properties of the resonator used in this example are typical.

Cavities have been constructed that have been shorter, that have used thinner

mirrors, and that have coupled in more Ught; all of these conspire to degrade

the frequency stability through fluctuations of the intracavity light intensity. In our example, if 1 mW of power is used to stabilize the laser to the cavity,

then the amplitude of the circulating light must be constant to about 10% to achieve a laser stabihty better than 1 Hz. The radiation pressure also works to stretch the entire bar. The strain, or fractional length change, induced by a

force acting normal to the end of a cylindrical bar of cross-sectional area A is given by [9]

(4) AL/L = F/AE

.

The fractional length change for 3 W of circulating power is about 2.5 • 10~".

This is smaller than the elastic distortion of our typical mirror, and even power

fluctuations as large as 10% would cause only millihertz frequency fluctuations

through this term. The clear indication is that the fluctuations in the circulating

power must be controlled if the laser frequency is to be stabilized to much bet- ter than 1 Hz. Cavity power fluctuations are caused by intensity fluctuations of

the laser light external to the cavity and by variations in the amount of hght

that is coupled into the cavity. The latter fluctuations are caused, for example,

by any motion of the resonator with respect to the input beam.

Another problem is the local heating of the dielectric mirror coating from

the light circulating in the cavity. With high-finesse low-loss mirror coatings, one might assume that this would not be an important concern. However, as we

have seen, even with as little as 100 [xW of power coupled into the cavity that

has a finesse of 100000, the circulating power inside the cavity can exceed 3W.

If the absorption losses are as little as a few p.p.m., (5 ^ 10) jj.W are absorbed in the coating. Most of this power is absorbed in the first few layers of the dielectric stack where the light intensity is the highest. When the light amplitude

fluctuates, there is a transient response followed by relaxation to a steady-state

condition. For a radiation mode size of 200 [j.m, we have measured a 2 Hz/(jlW

shift of the cavity resonance to higher frequencies with increased power. The

magnitude has been measured to be as much as 20 Hz per ijlW change in the

power of the input coupled light [10]. Both the thermal distortion of the mirror

and the light pressure problem could be reduced by adjusting the mirror radii and the cavity length so that the mode size is larger at the mirrors (for example,

by using a near-spherical resonator).

The thermal noise in the bar or spacer must also be considered. We can think

of this as the weak coupling of the fundamental mode of the spacer to its environment, e.g., the residual background gas, the wire suspension, radiation from

the walls of the vacuum vessel, etc. If the bar is thermalized to this background

or thermal bath at some physical temperature, T, then the weak coupling to the

thermal bath causes the mode's amplitude to execute a random walk in the doTN-4
LASER STABILIZATION TO A SINGLE ION 363

main | Ax| ^ Xr.m.s. • Xr.m.s. is the average deviation of one end of the spacer from

the equilibrium position assuming the other end is fixed. The magnitude of this deviation can be found by equating the energy in the harmonic motion of the

fundamental mode to k^T. After rearranging, this gives

(5) Xr.^.. = {^hT/M,^coiy^\

where ^b is Boltzmann's constant and M^n is the effective mass of the spacer. The fluctuation-dissipation theorem states that the time scale on which this ran- dom walk produces changes of order Xr.m.s. is the same as the time scale given by

the decay time of the fundamental mode [11]. The decay time tq is related to the

quality factor of the fundamental mode by tq = 2Q/ojq. Q is inversely proportional to the fractional energy loss per cycle. For times t shorter than tq, the

mean-square change in the mode's amplitude is XT.m.s. reduced by the ratio t/tq, ^^ - Zr.m.s. '^/'Z'o[12]. Physically, this expresses the fact that a harmonic oscilla- tor is a tuned system that responds to noise and other perturbations in a narrow

range of frequencies. So, whUe the noise is proportional to temperature, the

fluctuating forces cannot appreciably change the mode amplitude in times short

compared to the decay time.

To reach an appreciation of the size of the length fluctuations due to thermal

noise, we can calculate the frequency of the lowest mode of the spacer and solve

for >^r.m.s. • Alternatively, since M^ff is only estimated, we can equate the work

necessary to stretch or compress a cylindrical bar by Xr.m.s. to the energy k^ T. The force necessary to elastically stretch a bar by a small amount x is given by

eq. (4),

(6) F = EAx/L.

F diX is the incremental work done by this force in going fi:'om a: to a: + dx. Inte- grating from X = to X = Xr.m.s. > the work done is

(7) W = EA{xr.^,,f /2L . When this is equated to k-^ T, Xr.m.s. is simply related to the temperature of the

spacer and to its physical properties,

(8) Xr.m.s. = {^Lk^T/EAY^'.

Recall that, for a spacer made from a material comparable in its properties to fused sihca, E is about 10^^ N/m^ . If the spacer is 30 cm long and 10 cm in diameter, then Xr.m.s. ~ 1.8 • 10"^^m at a thermal-bath temperature of 300 K. The fre- quency of a laser in the mid-optical locked to this cavity would move by about

3 Hz for a length change of this magnitude. However, this excursion occurs dominantly at the vibrational frequency of the lowest fundamental (mechanical)

mode of the spacer. For a fused-sUica spacer of this size, the resonance fre- quency of its lowest mode is about 10 kHz. Hence, there is little power in the

TN-5
364 J. C. BERGQUIST, W. M. ITANO and D. J. WINELAND

thermally induced sidebands at ± 10 kHz since the modulation index is so small [13] (== 3 10"^). It is worth noting that the resonant Q in a fused-sUica

bar at room temperature is only about 10^ [14]. Therefore, the decay time for the

lowest mode is about 10 s, and, unlike our colleagues looking for gravity waves

with high-Q resonant-bar detectors, we are sensitive to the full change in the

mode amplitude on time scales of critical interest to us. However, a few hertz at 10 kHz driven by the Brownian motion of the cavity is not a limitation to the

frequency stability nor spectral purity of a 1 Hz laser. The temperature sensitivity of low-expansion materials suitable for spacers can be better than 10~^/K, which implies tjK control at the cavity in order to achieve stabilities of a few hertz. However, if the cavity is suspended in a thermally massive, evacuated chamber, then the thermal coupling to the environment is primarily radiative. The time constant can exceed a day. Therefore, if the temperature fluctuations of the walls of the vacuum vessel never exceed

10 mK, then the frequency fluctuation rate of the laser will be less than 1 Hz/s,

independent of the time rate of charge of the wall temperature.

Pressure fluctuations in the gas between the mirrors produce density variations which in turn cause refractive-index changes. This causes an effective optical-length change to the cavity, nL, where n is the index of refraction for air. Near room temperature, the index of refraction of dry air is linearly related to

its pressure, P, by

(9) w-l-310-^P,

where P is in Pa (133 Pa = 1 Torr). Thus, even if the cavity is evacuated to 10~^

Pa, the absolute shift in the cavity resonance for optical frequencies near

X «= 500 nm is about 15 Hz (from that of Pa). 10% fluctuations in this pressure

cause frequency excursions of the laser of approximately 1.5 Hz. The air pres- sure at 10"^ Pa also causes a strain in the bar but the length compression is negligible. The pressure at 10"^ Pa is less than 10"^ N/m^, which produces an axial strain in the spacer of about 1 • 10"'^. The fractional length change is not fully this magnitude since the pressure-induced stress in the axial direction is some- what compensated by the radial stress. Consequently, in a reasonably stiff spacer, fluctuations in the cavity length due to fluctuations in the pressure are dominated by index-of-refraction changes.

Additional limitations to the stability of a laser locked to a Fabry-Perot res- onator come from technical problems such as optical feedback, intensity fluctuations, beam pointing stability, etc. [15], but seismic noise or ambient vibrations

that cause changes in the cavity length are the most important practical problems limiting the frequency stability for times longer than a few ms. Generally,

there are two distinct effects: high-frequency vibrations, which may excite fundamental mechanical resonances of the bar, and low-frequency vibrations,

which tend to produce nonresonant distortions of the bar. The first effect typiTN-6
LASER STABILIZATION TO A SINGLE ION 365

cally occurs at frequencies in the range of (100 -^ 1000) Hz and often can be ef- fectively eliminated, for example, by mounting the system on alternating lay- ers of shock-absorbing material and passive masses, such as thin rubber and

cinder block. Low-frequency vibrations ((0.1 ^ 100) Hz), which are typically

driven by ground noise and building vibrations, are much harder to eliminate.

Some sophisticated vibration isolation systems, both active and passive, have

been developed to reduce noise in this frequency interval [16]. While an active

vibration isolation system may ultimately be necessary for sufficient attenuation of seismic noise to reach laser spectral purity below 1 Hz, we, for the moment, isolate only with passive systems. The simplest method of passive vibration isolation consists of mounting the device to be isolated on a resilient support, such as a pendulum or spring. A pendulum isolates in the horizontal plane,

using gravity as its spring constant (note that this spring can be nearly lossless). A spring can isolate horizontally and vertically simultaneously. It is relatively simple to show that the attenuation in the amplitude of motion between

the support and the bar increases as a»^ for motion at frequencies higher than

the resonance frequency wq (wq = k/m, spring; a>o = g/l, pendulum). Damping

must be included to limit the ampUtude of motion, aj, at resonance. The re- sponse of a damped system is

(10) tti = a2(l + iroj)/{l - (w/wo)^ + iroj),

where aj is the amplitude of the motion at the bar, 02 is the amplitude of motion

of the support at frequency w, and F is the coefficient of damping. The amplitude ttj is complex (phase shifted) and everywhere bounded. In practice the

choice of damping is a compromise between low resonant amplitude and suffi- cient high-frequency isolation. There are more complex passive systems that of- fer better high-frequency isolation while at the same time reduce the resonance peaking to a factor of 2 or less [16]. Calculations of the vibration isolation pro- duced by various mechanical suspensions have been driven by the work done on

gravity wave detectors; a good treatment is that of Veitch [17]. The most important function of the isolation system is to reduce fractional

length changes of the reference cavity to below (ideally, well below) 1 • 10"^^. If the cavity is suspended with its axis horizontal by wires that act as vertically

stiff pendulums, then it has been demonstrated that the isolation from horizontal vibrations in the direction of the cavity axis can be adequate to attain a stability approaching 1 Hz [18]. However, a suspended bar is subjected to a dis- tributed load resulting from the pull of gravity, which produces considerable

stress to the support structure and to the bar. In addition, the bar is not perfectly rigid and must distort at some level under its weight and this causes additional stress. All these stresses can produce sudden acoustical emission [19] at the rate of up to several per second and at frequencies that may or may not coincide with the eigenfrequencies of the bar. Also, since the wires are essentially

TN-7
366 J. C. BERGQUIST, W. M. ITANO and D. J. WINELAND

stiff to vertical vibrations, these perturbations can be coupled into the bar. If, though, the wires were connected to the nodal points of the bar, then the external force would be unable to excite that mode (or modes) through that nodal position. Unfortunately, there are many bending modes and vibrational modes

with disparate nodal positions, so, although it is possible to reduce some of the

cavity sensitivity to vertical perturbations, single suspension points are not

sufficient to eliminate excitation of all modes. Further improvement can be

achieved if the cavity is isolated vertically as well. We now turn our attention to some of the experimental studies pursued over the past few years in the ion

storage group at NIST.

1.3. - Experimental results: cavity comparisons.

In some of our studies, a direct heterodyne comparison of two independently

frequency-stabilized light beams was made. The linewidths and frequency stabilities were analyzed in various ways. The beat note was recorded in an r.f. or microwave spectrum analyzer, thereby directly displaying the combined

linewidth of the two sources. The signal-to-noise ratio was improved and the

long-term relative frequency drift was studied by averaging many successive scans of the beat note. It was also possible to use two spectrum analyzers in tan- dem to identify the specific frequency noise components that contribute to the

laser linewidth. This was particularly useful toward unraveling the noise sources that cause fluctuations to the length of the cavities. For instance, if the

seismic noise was independently studied with a seismometer, a correspondence

between the seismic-noise terms and the dominant contributors to the laser

linewidth could be made. If the length fluctuations of the two cavities were similar in frequency and amplitude, but otherwise independent, the linewidth of ei- ther frequency-stabilized system is smaller than the recorded beat note by \/2.

If the effective length stability of one cavity had been worse than that of the

other, the measured linewidth would have been be dominated by the frequency

fluctuations of the laser locked to the noisier cavity.

Either the transmitted light beam or the reflected beam that interferes with

the light re-emerging from the cavity can be used to stabilize the frequency of

the laser. In our experiments, we used the beam reflected from the cavity. The

error signal can be derived either near zero frequency or at some higher fre- quency. Since technical-noise terms are present at low frequencies, it is better

to modulate and detect at a frequency at which the signal-to-noise ratio is limit- ed by the shot noise in the detected light beam. This is the Pound-Drever-Hall

reflected-sideband technique that has been treated in detail elsewhere [8, 20].

Attention to the optical layout and to the electronic-noise terms is critical to achieving a spectral linewidth that is smaller than a few hertz. This also has

been discussed in papers by HouGH et al [21] and by Salomon et al. [15]. By several separate measurements, the electrical problems in our work were deterTN-8
LASER STABILIZATION TO A SINGLE ION 367

mined to be unimportant to the attainment of a laser linewidth below 1 Hz; the

dominant contribution to the laser linewidth were mechanical (and perhaps optical) perturbations that caused length changes in the reference cavity.

The frequency-stabilized light beams were derived from a home-built ring

dye laser oscillating at 563 nm. The wavelength of the laser was chosen because

its frequency would eventually be doubled into the ultraviolet to probe a nar- row transition in ^^^Hg"^ . Historically, the dye laser was locked in a two-step

process. The laser was pre-stabilized to the order of a few hundred Hz by lock- ing the laser to a lower-finesse (about 800) cavity. The cavity resonance was

probed by the reflected-sideband technique using a modulation frequency of

about 10 MHz. Rapid frequency fluctuations of the laser were removed by a fast (bandwidth > 2 MHz) servo driving an intracavity E/0 modulator, and the

lower-frequency fluctuations were corrected by a second-order servo-loop driving an intracavity PZT-mounted mirror. The frequency-stabilized light was

then transmitted through optical fibers to the high-finesse cavities. Since the

laser linewidth was less than 1 kHz, a smaller-bandwidth (100 kHz), lowernoise servo could be used to strip the remaining noise from the laser beams.

Again, the reflected-sideband technique was used to probe the resonance of the

high-finesse cavities. In the second stage, the frequency-correcting element was an acousto-optic modulator through which the laser beam was singly or doubly passed. The fi^equency corrections were simply written onto the acousto-optically shifted beam by the servo. Long-term corrections were fed back to the low-finesse cavity to maintain frequency alignment of the laser with one of

the high-finesse cavities. The laser power was also stabilized by adjusting the

r.f. power applied to the acousto-optic modulator. The overall intensity regulation was better than 0.1%, but this was applied to the beam before it entered

the cavity. From the arguments made in sect. 1.2, intensity variations in the

Ught circulating in the cavity cause length fluctuations by light pressure

changes and by heat variations in the mirror coatings. Therefore, although we

have not done so yet, it may be better to stabilize the intensity of the light cir- culating in the resonator by using the light transmitted from the cavity. This

should give a first-order insensitivity of the frequency of the laser to power

fluctuations caused by relative motion between the cavity and the injected light

beam.

The cavities were constructed from a cylindrical, Zerodur[22] rod that was

cut and rough ground to a diameter of about 10 cm and a length of about 27 cm. A 1 cm round hole was bored along the axis of the spacers, and a smaller one was

bored through the center of the rod midway from the ends and normal to its axis. The smaller bore permitted evacuation of the space between the mirrors.

The length of the cavities and the mirror radii of curvature were chosen so that

the cavities were highly nondegenerate for the spatially transverse modes. In

particular, the frequency of the TEMqi and TEMjo modes are separated from

the lowest-order TEMqo mode by approximately 30% of the free spectral range.

TN-9
368 J. C. BERGQUIST, W. M. ITANO and D. J. WINELAND

Even at the 12th transverse order, the frequencies are still separated from the

lowest-order fundamental mode (modulo c/2L) by several percent of the free spectral range. This gives good immunity to any line pulling if any light power

is coupled into the higher-order modes. Even so, great care is taken to mode

match into the cavities. Note that this high immunity to line pulling would be

somewhat compromised by going toward a near spherical resonator as suggested in sect. 1.2. Each cavity is suspended by two thin (250 (xm) molybdenum wires inside a

thick-walled (19.1mm) aluminum vacuum vessel that has an inner diameter of

26.7 cm. The wires are placed as slings under either end of the spacer, about 1/5

the cavity length from each end, in an attempt to support at the nodal positions

for the lowest-order bending mode. The ends of each wire are attached to the in- ner wall of the vacuum vessel by small clamps. Each wire either travelled verti- cally upward from either side of the spacer to the wall («U» shaped), or opened

slightly away from the bar («V» shaped). This allowed free movement of the

cavity along the direction of its axis and restricted, but not rigidly, the motion

perpendicular to the axis. Damping of the pendulum motion was weak and dominantly into the table and its padding through the aluminum housing. The alu- minum vessels are thermally insulated and temperature regulated to the order

of a few mK. The temperature coefficient of the spacers is approximately

6-10"^K"^at7' = 300 K. The thermal time constant from the walls of the evac- uated aluminum housing to the spacer is on the order of one day, giving adequate isolation to small temperature fluctuations at the vessel walls. Since the

variations in the wall temperature were controlled to less than 10 mK for any

time period, the maximum rate of change in the temperature of the bar did not exceed lOOpK/s. This corresponds to a frequency fluctuation rate of about

0.3 Hz /s and a maximum frequency change of about 50 kHz. A pressure of

1 • 10"^ Pa (8 • 10"^ Torr) is maintained in each vacuum vessel by an ion pump

that is rigidly attached to the vessel. This is adequate to give frequency fluctuations of less than 1 Hz for pressiu-e fluctuations of 10%.

The length of the longer rod corresponds to a free spectral range of 622 MHz

and the shorter rod has a free spectral range of 562 MHz. The mirrors are highly

pohshed Zerodur substrates that are coated to give high fimesse and good effi- ciency and then optically contacted to the polished ends of the spacer. The fi- nesses of the cavities are about 60000 and 90000, respectively, and, for both

cavities, the transmission on resonance exceeded 30%. The high finesse F, or

the long optical storage time, translates into a narrow fringe whose HWHM is given by c/2LF. Consequently, the ratio of error voltages to frequency excursions can be very high, even for short cavities. Shorter cavities have been constructed from ULE[22] that have measured finesses that exceed 130000 (and,

recently, finesses exceeding 10^ have been reported [23]). The frequencies of

the mechanical resonances of these shorter, stiffer bars wQl be about a factor of

3 higher than those of the Zerodur bars. As long as the frequencies of the meTN-10
LASER STABILIZATION TO A SINGLE ION 369

chanical vibrations are high enough, a narrow optical resonance can be probed

with high resolution by the unperturbed carrier of the laser spectrum.

The aluminum vacuum vessels rested on Viton [22] rubber strips attached to v-blocks made of aluminum. The v-blocks were secured to a rigid acrylic plastic

plate. Each reference cavity system was mounted on a separate optical table

that (initially) consisted of surface plates that were deadened by damping their

internal vibrations into sand. The sandbox sat on soft rubber pads and cinder

blocks in one case and on low-pressure inner tubes and cinder block legs in the

second. Noise vibrations from the floor and on the table tops were monitored

with moving-coil seismometers that had a sensitivity of 629V-s/m from ap- proximately (1 -^ 100) Hz. Measurements of the floor motion in our laboratory

revealed bright resonances at 14.6 Hz, 18.9 Hz and 29.2 Hz on top of a broad

pedestal from about 1 to 40 Hz. The average amplitude of the resonant motion was greater than 10 ~^ m/s and the pedestal peak was about 10"^ m/s. Isolation

from mechanical vibrations began at frequencies above about 5 Hz for both ta- ble tops. By 100 Hz the isolation from floor noise for either table had improved

by a factor of 40 or better. For heterodyning, 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 series. This allowed us to look directly at the beat note and also Fourieranalyze the noise terms that contributed to its linewidth. The first analyzer

could be used to observe the beat signal, or as a frequency discriminator. As a frequency discriminator, the scan was stopped and the analyzer was used as a tunable r.f. filter /receiver with a variable bandwidth. The center frequency of

the analyzer was then shifted so that the heterodyned signal lay at the half- power point of the response curve. The bandwidth was adjusted so that the fre- quency 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. The noise power spectrum in the second analyzer helped to reveal the nature and origin of the vibrational noise that contributes to the linewidth of the stabilized lasers. The width of the heterodyne signal between the two stabilized lasers was

less than 50 Hz. The noise power spectrum disclosed that low-frequency fluctu- ations in the range from near zero to 30 Hz dominated this linewidth. 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 the laser power spectra

that came from the floor motion at 14.6 Hz, 18.9 Hz and 29.2 Hz. The modulation indices of the latter three noise components were about one, so they all had

enough power to contribute to the beat note linewidth. When the pendulum mo24 - Rendiconti S.I.F. - CXX

IN- 11
370 J. C. BERGQUIST, W. M. ITANO and D. J. WINELAND

tions of the bars were quiet, their FM contributed Httle to the laser linewidths.

However, the integral of the nearly featureless noise power spectrum from

about Hz to 10 Hz contributed about 15 Hz to the combined spectral purity of

the lasers. Some, if not all, of this noise was mechanical, but it was not clear how it coupled to the suspended cavity. To help elucidate the connection, we

drove one of 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 hertz to about 100 Hz with enough power to be 40 dB above background noise. 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 little 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 showed 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

r\/^

/\

c

rV

/

o \

'co

r> /

\ \

1 \

/ \

/ \

\

25 Hz/division

Fig. 1. - Spectrum of the beat frequency between the two independent, cavity-stabilized lasers discussed in the text. The resolution bandwidth is 30 Hz. Total integration time for these data is about 70 s (the relative linear drift between the two cavities is removed by

mixing the beat note with the frequency from a synthesizer that is swept in time). The partially resolved sidebands at 14.6 Hz are due to a resonant floor vibration. The apparent

hnewidth is about 30 Hz; the linewidth of the better-stabilized laser is at least ^/2 narrower.

TN-12
LASER STABILIZATION TO A SINGLE ION 371

cavity axis. (For practical reasons, we did not 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, but the damping did not significantly change

the isolation afforded by the latex tubing at higher frequencies. The isolation from vibrational noise above 1 Hz was more than an order of magnitude

better than that of the quietest sandbox table. This was partially reflected

in 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; it dropped from about

50 Hz to less than 30 Hz. In fig. 1 the spectrum of the beat frequency

is shown. Zerodur is known to temporally contract with a time constant

of years [8, 24]. The creep rate for both cavities corresponds to a linear

frequency drift of (3 ^ 5) Hz/s, but the rates are not identical. The linear,

relative cavity drift is removed by mixing the beat frequency with the

frequency from a synthesizer that was swept in time. The spectrum of

fig. 1 represents an integration time of 70s. The noise vibrational sidebands

at 14.6 Hz are partially resolved. Note that the linewidth of the best stabilized

laser is at least V^ narrower. The Fourier noise power spectrum from

to 10 Hz is shown in fig. 2. The beat note 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. Perhaps the linewidth of the laser stabilized to the cavity on

220-

0.22-

5 10 frequency (Hz)

Fig. 2. - Fourier noise power spectrum of the laser heterodyne signal (shown in fig. 1) from to 10 Hz.

IN- 13
372 J. C. BERGQUIST, W. M. ITANO and D. J. WINELAND

the table suspended by the latex tubing is below 1 Hz. We are working

to verify this and to build better cavities.

II. - Single-atom Spectroscopy.

II.l. - Single-ion results.

The ion trapping and laser cooling relevant to our experiments have been

described elsewhere [25, 26]. A ^^^Hg atom is ionized and trapped in the harmonic pseudopotential well created by an r.f. potential applied between the

electrodes of a miniature Paul trap. The separation between the endcap elec- trodes (2^0 ) is about 650 [j.m. The frequency of the r.f. potential is about

21 MHz. Its amplitude can be varied up to 1.2 kV; at the maximum r.f. amplitude, the quadratic pseudopotential is characterized by a secular frequency of

nearly 4 MHz. The ion is laser-cooled to a few millikelvin by a few microwatt of

radiation from two 194 nm sources. One source drives transitions from the

5d^%s ^Sy2(F = 1) to the 5d^°6p ^PyziF = 0) level (see fig. 3). This is essen5d'°6p'p,

0.40 -360 -240 -120 120 240

laser frequency offset (Hz)

360

Fig. 3. - 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 ^Si/2iF = 0, mF = 0)-^D^f2(F = 2, m,p = 0) transition in a single laser-cooled '^^Hg^ ion. A 563 nm laser that is stabilized to a high-fi- nesse reference cavity, which in turn is long-term stabilized to the ion, is frequency-doubled and stepped through the resonance for 138 consecutive sweeps. The 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.

TN-14
LASER STABILIZATION TO A SINGLE ION 373

tially a two-level system suitable for laser cooling, except for weak off-reso- nance pumping into the ^Si/^^F = 0) state. The second 194 nm source, tuned to the ^Si/2{F = 0) to ^Pi/2(F = 1) transition, returns the ion to the ground-state

F = 1 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 two 194 nm beams propagate coUinearly 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 • 10"^ 47i: sr, is detected with an efficiency of 10% to give a peak count rate on

resonance of about 25000/s. The complication of laser cooling with two lasers is brought about by the hyperfine structure of ^^^Hg^ . Only an isotope with nonzero nuclear spin can have first-order, field-independent transitions, which

give great immunity to magnetic-field fluctuations. In ^^^Hg^ , the nuclear spin

is 1/2. Near B = 0, the narrow 5d^%s ^Si/2-5d^6s2 ^D^/z transition at 282 nm is first-order field-independent. The decay rate of the metastable ^Dr,/2 state cor- responds to an optical linewidth of less than 2 Hz—certainly, a suitably chal- lenging test for the stabilized dye laser. The 282 nm radiation is obtained by frequency doubling the radiation from

the dye laser that is stabilized to the Fabry-Perot cavity on the sandbox table. (The cavity comparisons were done subsequent to the single-ion studies and we

had not yet suspended a table with latex tubing.) Prior to being frequency-doubled, the 563 nm radiation (beam 1) is passed through an acousto-optic modulator (A/0-1) so that its frequency can be tuned through the S-D resonance. We

also used A/0-1 to suppress the linear drift of the cavity and the frequency

fluctuations caused by relative motion between the cavity and the ion trap

(which are supported on different tables separated by 3 m). These Doppler ef- fects can be removed in a fashion similar to that used by Vessot to remove

Doppler frequency shifts between a ground-based microwave source and a rocket-borne microwave oscillator [27]. Another acousto-optic modulator (A/0-

2) is placed in an auxiliary laser beam (beam 2) near the ion trap. The frequency

of beam 2 need not be stabilized. A/0-2 generates a frequency-shifted beam

(beam 3) that is sent to the cavity table and returned on a path very close

( < 2 cm) to that followed by beam 1. The light paths need not be overlapping in order to reach a frequency stability of 1 Hz. Beam 3 is recombined with its carri- er to produce a beat note at the r.f. frequency of A/0-2. However, because the

shifted beam traveled over to the cavity and back to the trap, the frequency

fluctuations caused by relative motion between the tables and atmospheric tur- bulence are impressed on the beat note. Dividing the beat frequency by 2 gives

the one-way path noise information carried at half the radiofrequency of A/0-2.

If this frequency is summed with the right quadrature to the frequency that

sweeps the stabilized laser through the S-D resonance, then path noise is elimi- nated. This is equivalent to bringing the cavity and trapped ion together. StepTN-15
374 J. C. BERGQUIST, W. M. ITANO and D. J. WINELAND

ping the frequency of the stabilized laser through the S-D resonance and re- moving the linear cavity drift are accomplished vdth an r.f. drive frequency to A/0-1 obtained by summing the output of two synthesizers. The frequency of one synthesizer sweeps opposite to the cavity drift and the frequency of the sec- ond synthesizer is stepped back and forth, sweeping the frequency of the stabilized laser through the narrow atomic resonance. The 282 nm radiation and the two-frequency 194 nm source are turned on and off sequentially using shutters and the acousto-optic modulator. This pre- vents any broadening of the narrow S-D transition due to the 194 nm radiation.

Electron shelving [25, 28] 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. The 194 nm radiation

tuned to the ^Si/2iF = 0)-^Pi/2(F = 1) transition is chopped off for 5ms. During

this time, the 194 nm radiation tuned to the ^Si/2{F = iy^Pi/2iF = 0) transition

optically pumps the ion into the ^Si/2iF = 0) ground state. Then this 194 source

is turned off. One milHsecond later, the 282 nm radiation, tuned to a frequency

resonant or nearly resonant with the ^Si/2iF = 0, vtf = 0)-^D^/2(F = 2, nif = 0)

transition, irradiates the ion for an interrogation period that is varied up to 25 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 refer- ence cavity contraction rate, but also on small pressure and temperature

changes, on laser power variations, and so on (as discussed in sect. 1,2), we

locked the frequency of the laser to the narrow S-D transition to remove longterm frequency drifts. To do this, we modified the measurement cycle to in- clude a locking cycle. We began each measurement cycle by stepping the fre- quency of the 282 nm radiation to near the half maximum on each side of the res- onance A^ 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 aver- aged the A^ results from each side of the resonance line, took the difference and

corrected the frequency of the synthesizer used to compensate the cavity drift. 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. 3, we show the spectrum obtained by scanning in this drift-free way

IN- 16
LASER STABILIZATION TO A SINGLE ION 375

through the Doppler-free resonance of the ^51/2 (F = 0, m^ = 0)-^1)5/2 (-?" =

= 2, thf = 0) transition. The Uneshape shown is the result of 138 consecutive

scans, each of which included a locking cycle. 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 hnewidth 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 was heterodyned together, 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 to 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 ij.m; therefore, on resonance, fewer than 10^ photons/s ( < 1 pW) will saturate the transition. Below, the data is a theoretical

lineshape calculated for an ion a 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 TT pulse (which roughly corresponds to the power used).

Qualitatively, the figures compare well. The fluctuations from measurement

cycle to measurement cycle in the quantum occupation number 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. Current efforts are devoted to measuring the narrow S-D transition using the laser stabilized to the cavity on the suspended table. A

cryogenic, linear r.f. Paul trap has been constructed and will soon be tested. With this trap, it should be possible to laser-cool many ions and to store them

without attrition for days. The increased numbers of trapped ions wUl give a

better signal-to-noise ratio (thereby better stability), but it will still be possible

to have a small second-order Doppler shift. We also plan to investigate the line- shape and the effects of power broadening in more detail in future experiments.

The authors gratefully acknowledge the contributions of the colleagues who

participated in the work reported here: F. Diedrich, F. Elsner and M. Raizen.

We also acknowledge the support of the Office of Naval Research.

REFERENCES

[1] D. J. WiNELAND, W. M. Itano, J. C. Bergquist and F. L. Wells: in Proceedings ofthe 35th Annual Symposium on Frequency Control, Philadelphia, Pa., May 1981

(copies available from Electronic Industries Assoc, 2001 Eye St., Washington, DC,

20006), p. 602.

TN-17
376 J. C. BERGQUIST, W. M. ITANO and D. J. WINELAND

[2] 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., Ferroelectr. Fre- quency Control, 37, 515 (1990).

[3] MiAO Zhu and J. L. Hall: J. Opt. Soc. Am. B., 10, 802 (1993). ,

[4] C. M. Caves: Phys. Rev. Lett, 45, 75 (1980).

[5] W. A. Edelstein, J. Hough, J. R. Pugh and W. Martin: J. Phys. E, 11, 710

(1980).

[6] C. M. Caves: Phys. Rev. Lett, 54, 2465 (1985).

[7] R. Loudon: Phys. Rev. Lett., 47, 815 (1981).

[8] D. HiLS and J. L. Hall: in Frequency Standards and Metrology, edited by A. DeMarchi (Springer-Verlag, Berlin, 1989), p. 162.

[9] R. J. Reark and W. C. Young: in Formulas for Stress and Strain (McGraw-Hill, New

York, N.Y., 1975).

[10] N. Sampras, Stanford University: private communication.

[11] H. B. Callen and R. F. Greene: Phys. Rev., 86, 702 (1952); H. B. Callen and T. A. Welton: Phys. Rev., 83, 34 (1951).

[12] G. W. Gibbons and S. W. Hawking: Phys. Rev. D, 4, 2191 (1971).

[13] P. F. Panter: Modulation, Noise and Spectral Analysis (McGraw-Hill, New York,

N.Y., 1965).

[14] V. B. Braginsky, V. P. Mitrofanov and V. I. Panov: Systems with Small Dissipation

(University of Chicago Press, Chicago, 111., 1985).

[15] Ch. Salomon, D. Hils and J. L. Hall: /. Opt Soc. Am. B, 5, 1576 (1988).

[16] D. L. Platus: SPIE Proceedings, 1619, 44 (1991); P. R. Saulson: Gravitational Astronomy; Instrument Design and Astrophysical Prospects, edited by D. E. McClelland and H.-A. Bachor (World Scientific, Singapore, 1991), p. 248.

[17] P. J. Veitch: Rev. Sci Instrum., 62, 140 (1991).

[18] J. C. BERGQUIST, W. M. Itano, F. Elsner, M. G. Raizen and D. J. Wineland: in Light

Induced Kinetic Effects on Atoms, Ions, and Molecules, edited by L. Moi, S. GozziNl,

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

[19] F. Larsen: Acoustic Emission (IFI/Plenum, New York, N.Y., 1979).

[20] R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley

and H. Ward: Appl. Phys. B, 31, 97 (1983).

[21] J. Hough, D. Hils, M. D. Rayman, L.-S. Ma, L. Hollberg and J. L. Hall: Appl.

Phys. B, 33, 179 (1984).

[22] Mention of a commercial product is for technical communication only.

[23] G. Rempe, R. J. Thompson, H. J. Kimble and R. Lalezari: Opt Lett., 17, 363

(1992).

[24] F. Bayer-Helms, H. Darnedde and G. Exner: Metrologia, 21, 49 (1985).

[25] J. C. BERGQUIST, W. M. Itano and D. J. Wineland: Phys. Rev. A, 36, 428

(1987).

[26] J. C. BERGQUIST, D. J. Wineland, W. M. Itano, H. Hemmati, H.-U. Daniel and

G. Leuchs: Phys. Rev. Lett, 55, 1567 (1985).

[27] R. F. C. Vessot, M. E. Levine, E. M. Mattison, E. L. Blomberg, T. E. Hoffman,

G. U. Nystrom, B. F. Farrel, R. Decher, P. B. Eby, C. R. Baugher, J. W. Watts,

D. L. Teuber and F. D. Wills: Phys. Rev. Lett, 45, 2081 (1980).

[28] H. Dehmelt: Bull. Am. Phys. Soc, 20, 60 (1975); J. Phys. (Paris) Colloq., 42, C8-299

(1981).

TN-18
PHYSICAL REVIEW A VOLUME 45, NUMBER 9 1 MAY 1992

Ionic crystals in a linear Paul trap

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

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

(Received 31 October 1991)

We describe a configuration for a linear Paul rf ion trap. This trap can store a long string of ions with

a small second-order Doppler shift, comparable to that achieved with a single ion in a quadrupole Paul

trap. Crystallized strings of trapped ions, as well as more complicated structures, have been observed in the trap. We report an observation of the 40.5-GHz ground-state hyperfine interval of "'Hg^ by

microwave-optical double-resonance spectroscopy and discuss prospects for a microwave frequency

standard based on a trapped string of ions. PACS number(s): 32.80.Pj, 32.30.Bv, 95.55.Wk, 36.40. + d

I. INTRODUCTION

Experiments with trapped and laser-cooled ions have

been motivated by the possibility for high-accuracy spec- troscopy, improved frequency standards, and experiments in fundamental physics. For a large range of spec- troscopic experiments, the goal for ion confinement is the

Lamb-Dicke regime, in which the extent of the motion of

each ion is much less than the wavelength of an atomic

transition. For an optical transition this requirement is severe. In a quadrupole rf Paul trap, the total kinetic en- ergy of a single trapped ion can be on the order of the

secular kinetic energy, and confinement of a single laser- cooled ion to the Lamb-Dicke regime for an optical transition has been verified spectroscopically in Ba"*" [1,2] and

Hg"^ [3]. Laser cooling can significantly reduce the

second-order Doppler shift due to the ion motion, which

can otherwise limit the accuracy of a spectroscopic measurement. As an example, for a single ^^^Hg"*" ion laser- cooled to the Doppler limit [4], the fractional secondorder Doppler shift is

<Av/vo> = -2.3X10" 18

(Ref. [5]). In many experiments it is advantageous or even necessary to work with a larger number of ions. However, in a quadrupole rf Paul trap, two or more ions are pushed by

their mutual Coulomb repulsion from the center of the

trap to regions where the rf field is stronger. This leads

to increased ampHtude of the micromotion [6-8], Hmits

the achievable confinement, and may inhibit laser cooling

[7,8]. One way to circumvent this problem is to design a trap in which the rf field vanishes along a line instead of

at a single point. This goal has led to the development of the linear trap.

The linear-trap design descends from a "racetrack"

configuration rf quadrupole trap first used by Drees and

Paul for the short-term confinement of an electron-ion

plasma [9] and later used by Church to trap atomic ions

for longer times [10]. The racetrack trap resembles a quadrupole mass filter bent into a closed path. Charged

particles are confined to trajectories along the closed path

formed by the electrodes. Dehmelt first suggested using a

string of ions in a linear trap to suppress the second-order

Doppler shift [11]. In addition to confining the ions radially, it is desirable to fix the axial positions of the ions. In racetrack traps, axial localization can arise from patch

effects on the trap electrodes, which pin ions in one re- gion of the trap. Coulomb repulsion among the ions then

localizes the remaining ions. The traps described here

and in Ref. [12] provide axial confinement with a pur- posely applied static electric field. This allows the trap to confine the ions more strongly to their axial positions

than racetrack traps do and gives more control over the

axial positions. The static field weakens the radial confinement, as will be discussed below. Also, racetrack

traps can confine both positively and negatively charged

particles simultaneously, while traps such as ours, which

use static confining potentials, cannot.

Prestage, Dick, and Maleki have trapped a cloud of

i99jjg+

JQjjg elongated along the axis of a Hnear trap and

have demonstrated a '^^Hg^ microwave frequency stan- dard with excellent frequency stability [12]. Crystallized

strings of laser-cooled ^^Mg"*" ions have been observed in a racetrack-type trap at the Max Planck Institute for Quantum Optics [13]. We have constructed a Hnear rf trap and observed simple crystallized linear structures of

up to 33 '^^Hg"*^ ions [14]. By varying the strength of the

axial confining potential, we have also observed more

complex structures. In Sec. II we discuss the theory of the linear trap and

provide details of our trap. We then present images of

trapped crystallized structures of ions and compare these observations with the results of numerical simulations.

We next describe an observation of the 40.5-GHz

ground-state hyperfine transition using microwaveoptical double-resonance spectroscopy with a trapped

string of '^^Hg^ ions and discuss the potential for a mi- crowave frequency standard based on this transition. We

conclude with a discussion of future experiments and

prospects.

II. THEORY AND DESIGN OF THE LINEAR TRAP

The starting point in the discussion of our linear trap is the quadrupole mass analyzer [15,16]. This device con- Work of the U. S. Government

45 6493 Not Subiect to U. S. r.onvHtrht TN-19
6494 RAIZEN, GILLIGAN, BERGQUIST, ITANO, AND WINELAND 45

sists of four parallel rods. Typically, each rod has circular cross section in the plane perpendicular to the axis of

the trap, as shown in Fig. 1 . A time-varying rf potential

Focoslir is applied to two opposing rods. The remaining

two rods are held at rf ground. The rf electric fields are transverse to the axis of the mass analyzer. The timevarying potential near the axis can be approximated by

^=^ 1 + - R

cosQ,t (1)

where R is the distance from the axis to the surface of the

electrodes. For sufficiently high drive frequency O, a particle of mass m and charge q moves in an effective pseudopotential

<I) =

^vl

4m ft^/?^

(x

Iq

where

o),=qVQ/(V2m^R

(2)

(3)

is the angular frequency of oscillations in the radial direc- tion [12,15,16].

To reduce the anharmonicity of our trap, we chose the

rods to have radius /?' = 1.03/?. The anharmonicity is minimized with a ratio of 1.146 [17], but is not dramatically worse for our design. A numerical multipole expansion of the field for the geometry of our trap shows that near the axis the harmonic term in the pseudopotential is 1% smaller than that predicted by Eq. (2). The total contribution of anharmonic terms in the pseudopotential, up

to order {r /R )"^, is less than 0.1% of the quadratic term

for (r /R)<Q.l, where r is the distance from the trap

axis. While electrodes with hyperbolic cross sections

would give a more nearly harmonic pseudopotential, the use of circular rods facilitates construction and is a good

compromise since we are primarily interested in confining

the ions near the axis. Such a geometry has found wide

use as a mass analyzer, but it cannot function as a trap

Vn cos Vtt y\j

V

K3^0

FIG. 1. Cross section of a linear quadrupole trap. An alter- nating rf voltage Focosflr is applied to a pair of diagonally opposite rods. The other pair of rods is maintained at rf ground.

In the actual trap, the radius R' of the rods is 0.794 mm, 1.03 times the distance R (0.769 mm) from the trap axis to the sur- face of the rods.

since particles can move freely along the axis. To confine the ions axially, a static potential can be applied from opposing sides along the axis, creating a re- gion in the center in which particles can be trapped. Fig- ure 2 shows how this is accomplished in our trap. Each

of the circular trap rods is divided into two sections of

unequal length. The static potential of the longer "central" sections is maintained at ground and a static bias

voltage Uq is applied to the shorter "end" sections. The

segmented design divides the trap into three regions

along the axis: two end regions where the end segments

of two rods overlap the central segments of the other two

and a central region where all four central segments overlap. As a function of axial position, the static potential

has a minimum in the central region of the trap. The

static potential thus provides axial confinement, but is ex- pected to add far greater anharmonicity to the trap potential than does the use of circular rods. A schematic diagram of the actual trap is shown in Fig. 3. Four circular rods, 12.6 mm long and 1.588 mm

in diameter, are aligned with their axes parallel at a radial distance of 1.563 mm from a common axis (the z axis). Each rod is made of two segments of beryllium copper: an end segment 5 mm long and a central segment 7.5 mm

long. The central region of the trap, where the four central segments overlap, is 2.5 mm long. The two segments

are electrically isolated from each other using spacers of

alumina and machinable ceramic, as shown in Fig. 4. This configuration allows independent static potentials to be applied to the segments of the rods. The insulating spacers are hidden from the outside of the rods, as shown

in Fig. 4, to avoid perturbations caused by surface charge on the insulators. The ends of the rods are held in two

machinable ceramic fixtures (not shown), which are themselves held together by four outer support rods (also not shown). The fixtures were coated with titanium, ex- cept where the trap electrodes are held, and grounded in OO

oo

(a)

i\ u

I'o

(b)

FIG. 2. (a) End view and (b) side view of the linear trap, showing how the axial confining potential is applied. For simplicity, the rf potentials are not indicated on this figure. Each

trap rod is divided into two sections that are rf common and

held at different static potentials. The dashed lines show the

divisions between the sections of the rear rods (which are hid- den behind the front ones). The short segments of the rods are held at a positive static potential Uq and the long segments at static ground, as indicated for the front rods. With this ar- rangement, we create an axially confining potential for positive ions in the region where the four longer segments overlap. In our trap, this region has length 2zo = 2.5 mm
45 IONIC CRYSTALS IN A LINEAR PAUL TRAP 6495

VQCOsflt + AUi

Vq cos fit + Uq

VQCOsQt + AU2

A^3

Uq + At/5

Ac/4

Off

\^

FIG. 3. Linear-trap configuration. An rf voltage Vocosilt is applied to diagonally opposite electrodes as shown. The seg- ments of each rod are rf common so that each central segment

can be biased at a different static potential A U, yet remain at the same rf potential. The end segments of the electrodes are 5 mm

long, much longer than the 0.769-mm radial distance from the

axis of the trap to surface of the rods, so the rf electric fields in the center section of the trap are nearly parallel to the xy plane.

The region in the center of the trap in which the four central segments of the electrodes overlap is 2.5 mm long. To trap positive ions along the z axis, the four central segments are held near static ground potential and a positive static potential Uq is placed on the end segments. Small potentials AU^-^U^ can be

applied to the central segments to compensate for contact and

patch potentials on the electrodes. The axial position of the

ions can be varied by applying a bias A C/5 to one set of end seg- ments. The ions are detected by an imaging photomultiplier tube that looks along an axis normal to the z axis and 45° to the X and y axes.

order to minimize effects due to charging of exposed insulators. An rf potential is applied to two opposing rods, while

the two other rods are kept at rf ground. The two seg- ments that make up each rod are rf common. The end

segment of each rod is biased at a static voltage Uq > 0,

Beryllium copper

Insuloting spocer

•2.5 mm

.588 mm

12.6 mm

FIG. 4. Section through two trap rods (the front rods in Fig.

2). The end segment of each trap rod is a sleeve that fits over an

insulating spacer, which in turn fits over an extension of the

central segment. The insulating spacer is completely hidden

from the outside of the rod to prevent surface charge on the in- sulators from perturbing the trap potential.

and the central segment is nominally held at zero. This

arrangement confines positive ions within the central re- gion of the trap. To compensate for contact and patch

potentials on the trap electrodes, there are provisions for applying small static bias potentials (AC/i -^U^ in Fig. 3) to the central segments of the individual rods and to one

pair of end segments (A C/5 ). The rf voltage is supplied by a helical resonator, which

acts as a resonant transformer with a 40: 1 turns ratio. To

allow the different segments of the rods to have different

static potentials while remaining at the same rf potential,

the secondary winding of the helical resonator is a trifilar coil, whose three filaments supply the voltages

FoCosflr + At/j, Vocosilt + Uq, and Focosftr +AC/2, as shown in Fig. 3. The rf-grounded rods are capacitively

coupled to one another. The resonator produces up to about 700 V ampHtude at its 12.7 MHz resonant frequen- cy when driven with about 5W. The high-voltage output

is monitored with a capacitive voltage divider, which has

been calibrated with 10% accuracy. Because there is so httle symmetry to the trap, describing the static potential in detail would require a full three-dimensional numerical solution. However, the re- gion of interest is near the midpoint of the trap axis, a saddle point of the static potential. In this region the

static potential can be approximated by the harmonic potential

<f>s m

[z^-Ux^+y^)]

=-.5(.- [x^+y^] (4)

where Zq is half the length of the central region of the

trap (zo = 1.25 mm for our trap), /c is a geometric factor

(/fwO. 31 for our trap; see below), and

(0^ = {2KqUQ/mzQ)2 \l/2

(5)

is the angular frequency of axial oscillations in the trap.

The factor k depends on the geometry of the trap, and

hence will vary with Zq, so the secular frequency does not

scale simply as I/zq. The pseudopotential well in the ra- dial direction is weakened by the addition of the static potential and is given by

m

<!>, = —{coi-^coiKx'+y

2q

= -^{co:)Hx'+y')

2q

where — ^,,2_ l^,2^1/2

(6)

(7)

is the angular frequency of radial oscillations in the pres- ence of the static potential.

The radial size of the trap was chosen to allow

confinement of Hg^ ions near the Lamb-Dicke regime

for a 194-nm optical transition. We define the LambDicke regime by the criterion that the rms motion of

each ion be less than X/l-rr, where A is the waveleneth of

TN-21
6496 RAIZEN, GILLIGAN, BERGQUIST, ITANO, AND WINELAND 45

the transition. Lamb-Dicke confinement in the radial direction is expected for a 12.7-MHz rf drive with an amplitude Fq of 1.1 kV applied to the trap rods, assuming

that the ions have been cooled to the 1.7-mK Doppler

limit (this limit applies when motion along each axis is cooled equally [4]). Axial confinement is typically

achieved with a static voltage [/q of 1 V or less. Extrapolating from measured axial secular frequencies, we expect

that a static potential of about 400 V would confine the

axial motion of a single ion to the Lamb-Dicke regime at the Doppler limit of laser coohng.

III. ION CRYSTALS IN THE LINEAR TRAP

The trap is located in a vacuum chamber maintained at a base pressure of about 10~^ Pa (lPaw7.5 mTorr) by a

sputter ion pump. To load '^^Hg"*" ions into the trap, the

pump is turned off and neutral '^^Hg vapor (isotopic purity 91%) is bled into the vacuum chamber to give a pres- sure of approximately 10~^ Pa. Neutral atoms in the

trap are ionized by electron bombardment from a field emission point. After the trap is loaded, the pump is turned on to return the chamber to near its base pressure. The ions are laser-cooled using the 65 ^5,/2—>-6/) ^Pi/2

transition [3]. A few microwatts of 194-nm laser radiation is sufficient to cool the ions. Because optical pumping between hyperfine levels of

the '^^Hg'*' ion can inhibit laser cooling, two lasers at 194

nm are required. Figure 5 shows the hyperfine levels for the ^S,/2 and '^P\/2 states of '^^Hg"*" in the absence of a magnetic field. The ions are cooled and detected using a

laser (laser 1 in Fig. 5) that is tuned slightly below reso- nance on the

F = l F =

laser 1

F = l

F =

1/2

laser 2

>l/2

FIG. 5. Energy-level structure for the 6s ^8^/2 ground state and 6p^Pi/2 state of '^'Hg"'' near zero magnetic field. The

^S,/2(F=0)-(F=1) hyperfine splitting is 40.5 GHz, and the ^Pi/2(F=0)-(F=1) hyperfine splitting is 6.9 GHz. The inter- vals are not drawn to scale. Lasers 1 and 2 are used for cooling,

optical pumping, and detection, as described in the text.

transition. This is a "cycling transition," since selection

rules require that the ions in the ^P 1/2(^ = 0) level decay

only to the ^5,/2(F=l) level. Ions are detected by col- lecting photons scattered as the ions cycle on this transition. Laser 1 can also weakly excite the

'1/2 :F=i)^2p,/2(F=i)

transition (the laser frequency is detuned 6.9 GHz from

this resonance, whose natural linewidth is 7/277 = 70

MHz). From the ^Pi/2(jF= 1 ) level, the ions can decay to the ^Si/2(F =0) level. This decay path acts to optically

pump the ions into the ^Si/2(F=0) level, and ions in this

level are not cooled by laser 1. To prevent optical pumping into the ^5,/2(F =0) level, a second laser (laser 2 in Fig. 5) is used to repump ions from the ^Si/2(F =0) level into the ^51/2(^=1) level via the ^Pi/2(F=1) state. Laser 2 is offset about 47.4 GHz to the blue of laser 1, putting its frequency near the

2S,/2(F =0)- 'Pwiif- 1)

resonance. In addition to the above considerations, it is necessary to prevent pumping into the

^5i/2(F= l,W;r = + l) levels. To this end, we apply a magnetic field of approximately 5X lO"'^ (5G) at about

45° to the electric-field vector of the laser radiation.

The laser is directed through the trap at an angle of ap- proximately 9° from the trap axis in order to cool both ra- dial and axial motion. At this angle, the theoretical

Doppler limit of laser cooling is about 1.1 mK for the axial motion and 24 mK for the radial motion [4]. Fluores- cence is collected by a fast lens system that produces an

image, magnified 22 times, on the front surface of an imaging photomultiplier tube. The spatial resolution of the

lens and detector system, defined as the full width at half maximum of the image of a single localized ion, corre- sponds to 6 fim at the center of the trap. The imaging

system looks along an axis normal to the trap axis and at 45° to the X and y axes. At the Doppler limit of laser cooling described above, the rms secular motion along

the z axis is calculated to be 1.4 fim for 6;^/27r=25 kHz,

the smallest secular frequency used in these experiments,

so the extent of the secular motion of the ions is less than

the spatial resolution of the detector. As the secular fre- quency is increased, the ions are pushed closer together

along the trap axis. For sufficiently high secular frequencies, the ion spacing will decrease below the 6-/im spatial

resolution of the imaging system and we will be unable to resolve the individual ions. With two ions in the trap,

this should occur at a secular frequency of about

6),/277-=400kHz.

The temperature of the ions was not measured spectroscopically [3], but we can place an upper bound on the

temperature by noting that the images of the ions were

about 6 fj.m full width at half-maximum. We think that

this width is due largely to the spatial resolution of the

imaging system, but even if the width were entirely due to motion of the ions, we would have z^^^^i/xm, from

which F^ w 5 mK for co, = 25 kHz. If the sizes nf the imTN-22
45 IONIC CRYSTALS IN A LINEAR PAUL TRAP 6497

ages of the ions were primarily due to thermal motion,

the images would be elliptical, since the secular frequencies in the radial and axial directions are different. The

lack of eccentricity in the images supports our supposition that the widths are due to the imaging system and,

hence, that the axial temperature is significantly less than

5 mK for ft), =25 kHz.

The secular frequencies of the ions are measured by ap- plying a drive to the trap electrodes and observing a drop

in the fluorescence as the drive sweeps through resonance

[18]. The drop in fluorescence when the amphtude of the

secular motion increases is due to the combined effects of

the ion's motion taking it out of the laser beam and of

Doppler shifts detuning the ion resonance. These measurements permit a quantitative comparison between the

observed ion configurations and numerical simulations.

The secular frequencies were measured at several

different rf and static voltages and showed the expected

functional dependence on the voltages. The measured

values for the axial secular frequency co^ were proportional to the square root of the static voltage. The constant k

in Eq. (5) was determined from the measured secular fre- quencies to be 0.31 with an uncertainty of about 10%.

The unperturbed radial secular frequency co^ was deter- mined from the measured secular frequencies co'^ and co^ using Eq. (7). The values for co^ were proportional to the

rf voltage and were consistently 17% ± 10% lower than

predicted by Eq. (3), where the uncertainty is dominated

by a systematic error in measuring the rf voltage applied

to the trap. For very small axial voltages (less than about

100 mV), the effects of local static fields, due to surface

charge or patch potentials on the electrodes, make it difficult to characterize the axial potential.

By varying the neutral Hg background pressure and

the duration and intensity of the electron bombardment

when loading the trap, we can capture different numbers

of '^^Hg"*" ions. We have observed single ions and crystalline structures of up to 33 ions, as well as much larger

clouds that did not crystallize. The simplest crystalline

configuration of ions is a linear string, which is obtained

when a)'^»(o^. We have observed crystalhzed linear

strings of 2 to 33 ions. Figure 6 shows images of linear crystals containing 1

5

and 33 ions. The number of ions we can observe in such

crystals is limited by our ability to image the entire string

while resolving the individual ions. Large strings, such as the string in Fig. 6(b), are longer than the approximately

200-jUm-long region in which the laser intersects the trap

axis, and it is not possible to illuminate the whole string

at once. To obtain this picture, it was necessary to sweep

the position of the laser beam back and forth to il- luminate the whole string. A more powerful laser could

illuminate a larger area with sufficient intensity to observe the ions easily. Alternatively, more ions could be

confined within the illuminated region by increasing co^, but the imaging optics cannot resolve ions less than about

6 /xm apart, so this approach would require improved imaging optics. We have formed crystallized strings with

enough ions that when the spacing between the ions was

sufficient for the imaging system to resolve them, the

strings were longer than the field of view of the imaging

(a) #

100 Jim

(b)

0% #•

100|im

FIG. 6. (a) A picture of a linear crystal of 15 ions, (b) A pic- ture of a linear crystal of 33 ions. This string was longer than

the region of the trap axis illuminated by the cooling laser. To

obtain an image of the entire string, it was necessary to sweep

the laser beam back and forth.

system (about 300/im). The fact that only part of the

string was illuminated by the coohng laser did not seem

to inhibit coohng and crystallization, although when

much larger clouds of ions were trapped, crystallization was not observed. A limitation to forming extremely

long strings may be our inability to compensate for the

effects of local perturbations to the static potential due to patch fields or surface charge. In the present trap, static potentials can be independently applied to three of the four trapping rods to compensate for such perturbations.

This technique works well for short strings, but as the

length of the string approaches the radial dimension of

the trap (about 750 /im), the patch fields may vary over

the length of the string. In this case, we expect the static compensation to be less effective. Figure 7(a) shows the measured positions of eight ions

in a string and the positions calculated by a numerical

simulation. The simulation, performed with no adjust- able parameters, determines equilibrium positions for the

ions by minimizing the potential energy of a given number of particles in harmonic radial and axial wells, with

Coulomb repulsion between the particles. The secular frequencies used for the simulation were determined by

scaling previously measured secular frequencies by the

static and rf voltages applied to the trap. The hnear

configurations of ions are quite insensitive to the radial secular frequency, because the ions are confined to the

axis of the trap.

The ions can be pushed into planar or three- dimensional configurations by increasing co^ relative to

0)'^. The simplest structure beyond a linear string is a pla- nar zigzag. Figure 8 shows such a structure composed of

1 1 ions (10 '^^Hg"*" ions and 1 impurity ion, which did not

fluoresce). We think that the impurity ion is another iso- tope of mercury, which would be consistent with the

91% isotopic purity of the mercury vapor used to load

the trap. This structure was obtained by lowering the rf voltage on the trap. Figure 7(b) compares the measured

positions of the ions to the predictions of a calculation.

To obtain agreement between the zigzag configuration

TN-23
6498 RAIZEN, GILLIGAN, BERGQUIST, ITANO, AND WINELAND 45

and a numerical simulation with 11 '^^Hg"*" ions, it was

necessary to introduce into the calculations an ad hoc

asymmetry of about 20% between the secular frequencies

in the x and y directions. This azimuthal asymmetry

forces the ions to He in a plane; without it, the calculation

predicts a quasihelical structure that disagrees

significantly with the observed ion positions. The axial secular frequency co^ and the smaller radial secular fre- quency co'y were determined by scaling previously measured secular frequencies by the voltages present on the

trap when this crystal was observed. The larger radial secular frequency co'^ was set to l.lco'y. The source of the azimuthal asymmetry in the potential well is not known. One possible cause is unequal contact or patch potentials on the trap rods. Another possi- ble cause is the azimuthal asymmetry of the static trapping field. At the geometric center of the trap the static

field should be the same in the x and y directions. However, the position of the ions along the axis was not necessarily at the center and varied from load to load. We

suspect that this was due to surface charge deposited during loading. When the ions are not at the midpoint of the

trap axis, there is a quadrupole component to the radial

static field, which breaks the azimuthal symmetry of the

20

: (a)

'

1

-^ •f + + +

1

1 1 1 T

•

i 1 1

100 -50 100

z (/xm) FIG. 7. Observed positions of ions in the trap (crosses) and

positions predicted by a numerical simulation (circles). The

arms of the crosses extend to the half-maximum intensity points of the images (about 4 /im radius). The coordinates are as in Fig. 3: z represents the displacement along the trap axis and _v, represents the projection of the radial displacement onto the im- age plane, (a) A linear crystal of 8 ions. The secular frequencies used for the calculation are co'r/2TT = 435 kHz and CL)./2n = 4\.l

kHz. These frequencies were determined by scaling secular fre- quencies measured under different conditions by the voltages

present on the trap when this image was acquired, (b) A nonlinear crystal of 1 1 ions. One site in the crystal was occupied by

an impurity, which did not fluoresce. An asymmetry between

the X and y directions was introduced ad hoc to make the re- sults of the calculation agree with the data. This asymmetry

makes o)'^ = l.lco'y and confines the structure to the yz plane.

The secular frequencies used for the calculation are w;/27r = 92.4 kHz, co',/2Tr=ll kHz, and «,/2i7=31 kHz. The

frequencies cj'y and w^ were determined by scaling frequencies

measured under different conditions by the voltages present on

the trap when this image was acquired.

."A •••»

50|xm

FIG. 8. A picture of a crystal of 11 ions in a zigzag

configuration. The crystal consists of 10 '^'Hg^ ions and 1 impurity ion, which did not fluoresce. This is the same crystal

shown in Fig. 7 (b).

effective radial potential given by Eq. (6). In the case of

Fig. 7(b), the axial secular frequency is approximately

40% of the mean radial secular frequency, and it seems

plausible that the quadrupole asymmetry could produce

the observed planar structure. However, further work is required to characterize the asymmetry more completely.

The observed crystalline structures are closely related

to the structures predicted for cold ions in storage rings

[19]. With many ions and small co'^, we have observed

complex crystalhzed structures which appear to be nonplanar and require further study. To conduct a systematic and quantitative study of two- and three-dimensional

crystallized structures, we will have to ehminate the az- imuthal asymmetry from our trap or characterize it more

completely.

IV. MICROWAVE-OPTICAL DOUBLE RESONANCE

A string of cold ions is of great interest for highresolution spectroscopy and improved frequency stan- dards. With imaging techniques, each ion can be treated

as an independent atomic clock. Using Dehmelt's "elec- tron shelving" technique, we should be able to detect the clock transition with 100% efficiency in each ion

[1,3,20,21]. This would make the signal nearly immune

to fluctuations arising from, for example, fluctuations in the laser intensity.

Consider a clock that uses the Ramsey separated-field

method in the time domain to interrogate the clock transition at angular frequency coq. The clock transition is ex- cited by two phase-coherent pulses of radiation, each of

duration A 7"^, separated by a time T^ (AT^ «T]i). By

probing the clock transition on each side of the Ramsey

peak, we can obtain an error signal to correct the average

frequency of an oscillator to match the atomic resonance

coq [16,22]. The fractional frequency stabihty of such a locked oscillator, as characterized by the two-sample Allan variance [23], is a (t) = [tNTj^coI)

-1/2

(8)

where r is the averaging time (t> Tj^) and A'^ is the number of atoms. Thus, it is advantageous to use large A'^, (Oq,

Tjf , and t. The trap environment can make long interro- gation times Tj^ possible (r;j=550 s has been achieved

with ^Be"*" ions in a Penning trap [22]). To achieve high accuracy, we must minimize and ac- count for external perturbations due to electric, magnetTN-24
45 IONIC CRYSTALS IN A LINEAR PAUL TRAP 6499

ic, 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 [5,6,11,24,25]. For a microwave transi- tion in a cold string of '^^Hg"'" ions, it should be possible

to reduce these effects to the point where the uncertainty

in the transition frequency is dominated by the uncertainty in the second-order Doppler shift. When the secular motion is cooled to the Doppler limit, the fractional

second-order Doppler shift can be as low as — 2X10~'*

[16]. This hmit can be realized if the ions he exactly

along the axis of the trap, where the rf trapping fields ap- proach zero. In this case, the kinetic energy in the rf micromotion is about equal to that in the secular motion,

and the two make approximately equal contributions to the second-order Doppler shift. If contact or patch potentials on the trap electrodes push the ions away from

the region of minimum rf field, the kinetic energy in the

micromotion, and hence the second-order Doppler shift, can be significantly greater. Thus, in a clock based on

this trap, it will be necessary to adjust the static shimming potentials At/i-At/4 on the trap rods to compensate for contact and patch potentials.

This potential for very high accuracy has led us to in- vestigate the possibility of a microwave frequency stan- dard based on the 40.5-GHz ground-state hyperfine splitting of '^^Hg'*' with a trapped and laser-cooled string of

ions. This transition is, to first order, independent of

magnetic field at zero field. For a Ramsey interrogation

time of Tj^ = 100 s and 6)o/27r=40. 5 GHz, the frequency

stability of a clock "ensemble" of A'^ = 50 ions would be

(a) Initially, lasers 1 and 2 and both sets of Helmholtz

coils are on. This allows the ions to be laser-cooled.

(b) Laser 2 is turned off and the ions are optically

pumped by laser 1 into the F = ground state.

(c) Both lasers are turned off. The second set of

Helmholtz coils is switched off, reducing the magnetic

field to approximately 1.6X 10~^ T. The

25,/2(F =0)->^S,/2(F=l)

ground-state hyperfine transition is driven using the

Ramsey method with two microwave pulses, each of

duration AT^ and separated by a time T^. The pulse

duration AT^ is adjusted to be close to the value that

gives a it/2 pulse at the resonant frequency. At the mi- crowave intensities used here, this was obtained for AT^j -0.13 s. (d) the second set of Helmholtz coils is switched back

on, laser 1 is turned on, and the number of photons

counted by the detector during a gate time is recorded.

Only those ions that made the microwave transition are

in the (F~l) hyperfine level of the ground state, so only

those ions fluoresce. With laser 2 off", these ions are eventually optically pumped into the (F = 0) level and cease

fluorescing. The gate time is chosen to be shorter than

the optical pumping time. In these experiments, gate

times of 30-40 ms were used (at the laser intensity used

here, the time constant for optical pumping was about 50

ms).

cr„(T) = 5.5XlO" •14, 1/2

where r is expressed in seconds.

As a first step toward this goal, we have recently observed the 40.5-GHz ground-state hyperfine transition of

a string of '^^Hg"*" ions by microwave-optical double reso- nance. In this preliminary observation the total fluores- cence of the entire string of ions was detected. Fluctuations in the fluorescence from the ion cloud due to amplitude and frequency noise in the cooling and detection

laser were too large for us to detect transitions with

100% efficiency. To describe the measurement sequence, we rely on the discussion of laser coohng and optical

pumping given in Sec. Ill and on the level diagram

shown in Fig. 5. To prevent optical pumping into the

^S,/2(F= l,myr = ±l) levels of the ground state, it was

necessary to apply a magnetic field while the cooling

lasers were on. However, we also wanted to observe the

microwave transition in near-zero magnetic field, so we

used two sets of Helmholtz coils. One set was used to nearly cancel the ambient magnetic field in the trap. No

magnetic shielding was used. The residual field near the

trap was measured to be approximately 1.6X10"^ T

(0.16 G). A second set of coils produced the field used to prevent optical pumping when laser 1 was on. The

current in the second set of coils was switched off and on

to switch the field between 1.6X10^^ T and approximately 5X lO"'^ T, respectively. The sequence of the ex- periment is the following.

c

o

o

coo

0.

-0.4 -0.2 0.0 0.2

f - 40 507 348 013.3 Hz

0.4

FIG. 9. Microwave-optical double-resonance spectrum of the

ground-state hyperfine transition in "^Hg"*" at 40.5 GHz. This

figure shows the central peak of a Ramsey resonance, obtained

using the measurement sequence described in the text, with a

linear string of eight ions in the trap. The circles represent the measured fluorescence, and the solid line is a cosine function fit to the data. For this measurement 7;, = 1.8 s, AT^ =0.130 s, and each point is the average of 27 measurements. Including

time for cooling, pumping, and detection, each measurement cycle takes about 3 s. TN-25
6500 RAIZEN, GILLIGAN, BERGQUIST, ITANO, AND WINELAND 45

At the end of this sequence, the frequency of the mi- crowave source is stepped, and the sequence is repeated.

The results of many scans of the microwave frequency

are averaged together. Several sets of data were acquired,

with Tjf ranging from 0.32 to 1.8 s. The result of one observation, using a string of eight ions, is shown in Fig. 9. The linewidth, determined by a least-squares fit of a cosine function to the data, is 251 ±6 mHz, giving a frac- tional resolution of 6.2X 10~'^, and showing good agree- ment with the 254-mHz theoretical Hnewidth for the

0.1 3-s pulse duration and 1.8-s free precession time used

in this measurement. Because we did not take advantage

of the nearly 100% detection efficiency that would be

possible if we counted the fluorescence from each ion in- dividually, the signal-to-noise ratio is limited by fluctua- tions in the ion fluorescence caused by intensity and fre- quency noise in laser 1. With this amount of noise, the

diff'erence between a pure cosine and the central lobe of a Ramsey profile is insignificant. The amount of technical

noise from laser 1 also means that the stability figure

given by Eq. (8) is not applicable.

V. FUTURE PROSPECTS

Future experiments will measure the fluorescence from

each ion individually. It should be possible to detect the

microwave transition with 100% efficiency if the fluores- cence detection efficiency is high enough that when laser

1 is turned on, many photons are detected from each ion

in the (F= 1 ) level before it is optically pumped into the

(F = 0) level. If laser 1 is detuned by half a hnewidth

from the

transition, the number of photons scattered by an ion in the ^51/2(^=1) level before it is pumped into the (F =0)

level is approximately the square of the ratio of 6.9 GHz

[the detuning of laser 1 from the

2S,/2(F-l)^2p^^^(^=l)

transition] to 35 MHz (half the radiative hnewidth). This

gives about 4X10'^ photons. Therefore, the detection

efficiency must be significantly greater than the inverse of

this number, 2.5X10"^. If this condition is met, it should be possible to detect the microwave transitions

with nearly 100% efficiency. By observing the rate of

quantum jumps by a single ion in the trap, we determined

the rate at which the ion fluoresced [26], and by comparing this rate to the count rate from the detector, we measured the overall detection efficiency of our present ap- paratus to be about 1 X 10"'*, which is marginally acceptable. 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 the effects of collisions with neutral background gas atoms and molecules

during long interrogation intervals, we plan to use cryogenic pumping. Although in our preliminary measure- ments the fluorescence was collected from the entire

string, it should be possible to treat each ion as an in- dependent clock, since our imaging system resolves the

individual ions in the string. Beyond the applications to high-accuracy spectroscopy, the possibility of confining a

string of ions so that each is in the Lamb-Dicke regime

should permit experiments in fundamental physics, such

as studies of interference, cavity QED, and collective be- havior.

ACKNOWLEDGMENTS

We thank Sarah Gilbert and Fred Moore for helpful

comments on this paper. This work was supported by

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

Office of Scientific Research.

*Present address: Department of Physics, University of Texas at Austin, Austin, TX 78712.

[1] G. Janik, W. Nagourney, and H. Dehmelt, J. Opt. Soc. Am. B 2, 1251 (1985); W. Nagourney, N. Yu, and H.

Dehmelt, Opt. Commun. 79, 176 (1990).

[2] I. Siemers, M. Schubert, R. Blatt, W. Neuhauser, and P. Toschek (unpublished).

[3] J. C. Bergquist, W. M. Itano, and D. J. Wineland, Phys. Rev. A 36, 428 (1987); F. Diedrich, J. C. Bergquist, W. M.

Itano, and D. J. Wineland, Phys. Rev. Lett. 62, 403 (1989).

[4] See, for example, W. M. Itano and D. J. Wineland, Phys. Rev. A 25, 35 (1982); D. J. Wineland and W. M. Itano, Phys. Today 40 (6), 34(1987).

[5] D. J. Wineland, W. M. Itano, J. C. Bergquist, and R. G.

Hulet, Phys. Rev. A 36, 2220 (1987); D. J. Wineland, W.

M. Itano, J. C. Bergquist, J. J. Bollinger, F. Diedrich, and

S. Gilbert, in Frequency Standards and Metrology,

Proceedings of the Fourth Symposium, Ancona, Italy, 1988, edited by A. DeMarchi (Springer, Berlin, 1989), pp.

71-77.

[6] D. J. Wineland, in Precision Measurement and Fundamental Constants II, edited by B. N. Taylor and W. D. Phillips, Natl. Bur. Stand. (U.S.) Spec. Publ. No. 617 (U.S. GPO, Washington, DC, 1984), p. 83.

[7] R. G. DeVoe, J. Hoffnagle, and R. G. Brewer, Phys. Rev. A 39, 4362 (1989).

[8] R. Bliimel, C. Kappler, W. Quint, and H. Walther, Phys. Rev. A 40, 808(1989).

[9] J. Drees and W. Paul, Z. Phys. 180, 340 (1964).

[10] D. A. Church, J. Appl. Phys. 40, 3127 (1969).

[11] H. G. Dehmelt, in Frequency Standards and Metrology

(Ref. [5]), p. 286.

[12] J. D. Prestage, G. J. Dick, and L. Maleki, J. Appl. Phys.

66, 1013 (1989); IEEE Trans. Instrum. Meas. 40, 132

(1991).

[13] H. Walther, in Proceedings of the Workshop on Light In- duced Kinetic Effects on Atoms, Ions, and Molecules, edit- ed by L. Moi et al. (ETS Editrice, Pisa, Italy, 1991), p. TN-26
45 IONIC CRYSTALS IN A LINEAR PAUL TRAP 6501

261; G. Birkl, S. Kassner, W. Quint, and H. Walther

(private communication).

[14] M. G. Raizen, J. C. Bergquist, W. M. Itano, and D. J. Wineland, in Quantum Electronics Laser Science, 1991 Technical Digest Series (Optical Society of America,

Washington, DC, 1991), Vol. 11, p. 170; M. G. Raizen, J. M. Gilligan, J. C. Bergquist, W. M. Itano, and D. J. Wineland, J. Mod. Opt. (to be published).

[15] W. Paul, H. P. Reinhard, and U. von Zahn, Z. Phys. 152, 143(1958).

[16] 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. Ferroelectr. Freq. Control 37, 515 (1990).

[17] D. R. Denison, J. Vac. Sci. Technol. 8, 266 (1971).

[18] D. J. Wineland, J. J. Bollinger, and W. M. Itano, Phys. Rev. Lett. 50, 628(1983).

[19] R. W. Hasse and J. P. Schiffer, Ann. Phys. (N.Y.) 203, 419

(1990): R. W. Hasse and J. P. Schiffer, in Proceedings of

the Workshop on Crystalline Ion Beams, Wertheim, Germany [Gesellschaft fiir Schwerionenforschung Report No.

GSI-89-10, 1989 (unpublished)]; D. Habs, ibid. [20] W. Nagourney, J. Sandberg, and H. Dehmelt, Phys. Rev.

Lett. 56, 2797(1986).

[21] T. Sauter, W. Neuhauser, R. Blatt, and P. E. Toschek,

Phys. Rev. Lett. 57, 1696 (1986); T. Sauter, R. Blatt, W.

Neuhauser, and P. E. Toschek, Opt. Commun. 60, 287

(1986).

[22] J. J. Bollinger, D. J. Heinzen, W. M. Itano, S. L. Gilbert,

and D. J. Wineland, IEEE Trans. Instrum. Meas. 40, 126

(1991).

[23] J. A. Barnes et al., IEEE Trans. Instrum. Meas. IM-20,

105(1971).

[24] W. M. Itano, L. L. Lewis, and D. J. Wineland, Phys. Rev. A 25, 1233(1982).

[25] D. J. Wineland, W. M. Itano, J. C. Bergquist, J. J. Bol- Hnger, and J. D. Prestage, Ann. Phys. (Paris) 10, 737

(1985).

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

TN-27
Cryogenic linear ion trap for accurate spectroscopy

M. E. Poitzsch,^' J. C. Bergquist, W. M. Itano, and D. J. Wineland

Time and Frequency Division, National Institute of Standards and Technology,

Boulder, Colorado 80303

(Received 15 May 1995; accepted for publication 11 October 1995)

We have observed linear "crystals" of up to tens of laser-cooled '^^Hg"*" ions in a linear rf ion trap. The trap operates at liquid-He temperature and is designed for use as a prototype 40.5 GHz

frequency standard with high accuracy and stability. © 7996 American Institute of Physics.

[80034-6748(96)05001-1]

I. INTRODUCTION

The 40.5 GHz ground-state hyperfine transition of the '^Hg"^ ion provides the basis for a high-performance micro- wave frequency standard.'"^ Our work on '^Hg"^ has been

devoted to obtaining a system that will provide high accuracy as well as high stability.^ To help achieve this goal, we

have incorporated laser cooling to suppress the second-order

Doppler shift. In this paper, we report preliminary results using a rf ion

trap in a cryogenic (~4 K) environment. This should yield

high vacuum, thereby reduce ion loss and frequency shifts due to background gas collisions, should provide the basis

for superconducting magnetic shielding, and will suppress

shifts due to blackbody radiation.^ Figure 1 shows the energy levels of interest in the '^^Hg^ ion. The ^Sy2{F=l,M = 0) to ^5,/2(F = 0,M = 0)

ground-state hyperfine splitting is ~40.5 GHz. It has no first- order Zeeman shift near zero field and is used as the clock

transition.' (F=l) to 2Pi/2(F = 0) ^ The upper-state ^P 1/2 hyperfine splitting is ~7 GHz. Laser cooling and fluores- cence detection are accomplished using the 194 nm electric dipole cycling transition from the ground ^Si/2iF= 1) state

to the excited ^PyjiP^^) state. ^'^ The natural linewidth of

this strongly allowed transition is 70 MHz, which is more

than an order of magnitude larger than the width of the 194 nm laser used for cooling and detection. To prevent optical

pumping of the ion into the F=0 ground-state sublevel by

off-resonant driving of the ^5i/2(F= 1) to ^PmiF— 1) transition and subsequent decay, a second (collinear) laser at 194 nm is present during the laser cooling and tuned about 47.4

GHz to the blue of the main cooling laser

Previously, a rf-trap '^^Hg"^ ion frequency standard (us- ing helium buffer gas cooling) was shown to have high fre- quency stability.^ It contained yV«=2X10^ ions and had a

fractional second-order Doppler shift of approximately

-2X10"'^ More recently, a short-term fractional frequency

instability of <7X lO"'"* t"''^ has been demonstrated in a

linear rf trap (also using helium buffer gas cooling), which

operated with A^^2.5X10^ ions, and a fractional secondorder Doppler shift of approximately -4X10"'^ was

inferred."* In comparison, the fractional second-order Doppler

shift of a single '^^Hg^ ion laser cooled to the Doppler limit

is about -2X10~'^ (see Ref 8). The fractional frequency

shift of the 40.5 GHz clock transition with magnetic field is 0.24B^, where B is expressed in teslas. Thus, a '^^Hg"*" ion

confined in an ion trap at near-zero magnetic field and laser cooled to the Doppler limit should constitute a highly accurate 40.5 GHz microwave frequency standard. To improve

the signal-to-noise ratio (and hence the fractional frequency

stability), it will, however, be desirable to have many '^^Hg"^

ions, all with equally low Doppler shifts.

II. LINEAR RF ION TRAP

The linear rf quadrupole trap, which uses four rf rods for

radial confinement and a static axial potential for longitudi- nal confinement, was developed as a way of confining multiple ions with low Doppler shifts due to rf micromotion."*'^'^"" In this scheme, the four rods are configured as in a rf mass analyzer, with a zero-field node along the center line instead of at a single point as in a spherical Paul

quadrupole rf trap.'^ Axial confinement is achieved by applying static potentials at the ends of the trap, using positively

biased rings, pins, or split sections in the trap rods. Figure 2 shows the linear rf trap used in the present cryogenic experi- ment. The four rf rods are 7?' =0.20 mm in radius, centered on a radius of /? + 7?'=0.64 mm from the trap axis (about

half the size of our previous linear rf trap^). Axial confine- ment is achieved by positively biasing rings (electrodes C

and R in Fig. 2), separated by 4 mm, at either end of the four-rod quadrupole.

Two (diagonally opposite) rf rods of the trap are driven

with a rf potential Vq cos(nr), while the odier two rf rods are maintained at rf ground. Near the trap axis, the time-varying

potential is approximately a saddle-point potential in x and

J.

^0 / x--y^ , (1)

"Present address: Schlumberger-Anadrill, Sugar Land, TX llllS.

where x and y are the transverse coordinates. The centers of

the two rf-driven rods intersect the x axis, one at ~{R + R')

and the other at R + R' . The approximation assumes that the rods of circular cross section give the same potential as rods whose surfaces are equipotentials of Eq. (1). If the drive frequency H is high enough (that is, much higher than the

ion's radial secular oscillation frequency), the ion of mass m

and charge q will experience an approximately harmonic

pseudopotential that confines in both ;c and ^'i

Rev. Sci. Instrum. 67 (1), January 1996 0034-6748/96/67(1 )/1 29/6/$6.00 1996 American Institute of Physics 129 TN-28
6p ^P^a 6.9 GHz

i99Hg* ^o +47.4 GHz

6s ^S.

FIG. 1. Diagram of the hyperfine energy levels of the "'Hg'^ ground 65^5 1/2 state and the 6p^Py2 state. Here vq corresponds to a wavelength of 194.2 nm. The 40.5 GHz ground-state hyperfine transition is intended for use as a reference for a frequency standard. m

^{x,y)'^—a)l{x^ + y^). (2)

where (o^ is the radial angular oscillation frequency (secular

frequency), which is given by

qVo

0) =

yllmCLR^

(3)

Axial confinement (along z) results from biasing the

rings at either end of the trap with a positive potential Uq . Near the center of the trap, an ion experiences the static saddle-point potential

m

cl>,{x,y,z)^—<ol[z'-Hx' + y')l (4)

where the axial angular oscillation frequency o)^ is given by

(O, mzo

(5)

The separation of the rings is 2zo=4.0 mm, and /c is a geometrical factor that is ~0.004 in this trap (as determined

from measurements of w^). As Eq. (4) indicates, the radial pseudopotential <t> is weakened by the addition of the static axial potential. Thus, the effective radial angular oscillation frequency is

0)1 = {0)1 -0)1/2) 1/2 (6) Under typical operating conditions in this trap, we expect

ojr/27r«=350 kHz and a»,/27r«=25 kHz, assuming CI/2tt=\3

MHz, Vo=100 V, and Uo = 25 V.

Endcap

^Laser Beam Path

(a)

R

Linear RF

Quadrupole Electrodes

FIG. 2. Schematic diagram of the linear rf (Paul) trap electrodes, (a) Side view (normal to the trap z axis). The experimental trap is located between "endcap" electrodes C and R. Trapping normal to c is provided by pon- deromotive forces resulting from rf voltage applied between diagonally opposite quadrupole electrodes [Eq. (1)]. Trapping along the c direction is provided by applying a static positive potential f/g between the endcap

electrodes and quadrupole rods [Eqs. (4) and (5)]. The region between elec- trodes L and C is intended for loading ions which are then transferred to the experimental trap. The paths for two of the laser beams are shown in black. A third laser beam (not shown) passes through the center of the trap and normal to the figure, (b) View along the z axis showing the endcap elec- trodes with the four rf-quadrupole electrodes superimposed. The shaded

area indicates relief for passage of the laser beams shown in (a).

130 Rev. Sci. Instrum., Vol. 67, No. 1, January 1996 Linear ion trap TN-29
As shown in Fig. 2, the trap's axial-confinement "ring"

electrodes are in fact, bored-out conical disks of beryllium

copper, appropriately drilled to allow laser access to the trap

center. The four rf rods are beryllium-copper wires. The di- mensions of the quadrupole rods were chosen to allow //I

light collection from the center of the trap. Also, the choice

of the small R dimension makes the trap radially unstable,

due to the Mathieu instability, for unwanted contaminant

ions (created during the loading procedure) with (u^>Cl/2 or, equivalently

w^ 10.6m X

'0 100 V

(7)

where u is the atomic mass unit. The four outer support rods

for the trap structure are alumina (AI2O3) rods sheathed in thin beryllium-copper tubing of an outer diameter of 1.28 mm. These support rods can each receive independent bias- ing potentials to "shim" the ions back onto the trap's geometrical center line, in the event that patch charges on the rod electrodes should cause the trap's geometrical and dc

electrical centers to not coincide. (Without this compensation, the ions could experience large rf electric fields and

undergo substantial rf micromotion.) The ceramic insulating

endpieces that define the positions of the trap rods are highpurity alumina slabs 2.0 mm in thickness, with appropriately

placed laser beam access holes. Figure 2(a) shows that the electrodes form two linear

traps in tandem. The leftmost trap, between electrodes L and

C, is intended for capturing ions in the initial loading process. Ions would then be pushed through electrode C into the experimental trap (between electrodes C and R). This design was implemented with the idea of gaining immunity from

contact potentials or electric charge buildup on the electrodes near where the ions were loaded. Appropriately placed mirrors allow laser cooling to be

performed along any one of three noncoplanar directions at the center of the trap, as illustrated in Fig. 2(a): in the plane

of the page at ±20° to the trap's z axis or normal to the plane

of the page (using a small mirror located directly below the

trap). We require only one of the beams shown in Fig. 2(a) for cooling, optical pumping, and detection. The three beam

paths are required for probing the velocity distribution in three dimensions. The kinetic temperatures achievable in these cooling geometries are derived using Ref. 13. The temperatures at the Doppler-cooling limit with either of the ±20° beams correspond to rradiai^5.6 mK and T^^^^^^l.l mK, under the assumption that the radiation is emitted iso- tropically. These temperatures imply a fractional secondorder Doppler shift of the 40.5 GHz hyperfine transition fre- quency that is approximately -5.5X10"'^.

Previously, a slightly larger linear rf trap, in which the rods were segmented in such a way that dc potential differ- ences could be applied to the different segments, was used, thus providing axial confinement.^ In that apparatus, operating at room temperature and at a pressure of about 10"^ Pa, several tens of '^^Hg"*^ ions were crystallized at fixed positions in a single row along the trap's nodal center line. Such

a geometry is optimal for the present frequency standard application, since the ions can be imaged independently for

improved signal-to-noise ratio, and all have approximately

the same low second-order Doppler shift as a single ion in a quadrupole trap. The major limitation of this previous apparatus was the background gas pressure in the vacuum chamber, which was high enough that ions would be lost in several minutes due to chemical reactions. For a frequency

standard application, it is desirable to be able to interrogate a

particular ensemble of ions, located at relatively fixed positions, for periods of many hours, several days, or longer.

Also, even at 10~* Pa, shifts of the transition frequency with

changes in the background gas pressure could limit the

accuracy.'"*

III. CRYOGENIC LINEAR RF ION TRAP

Our solution to the background gas pressure problem is to maintain the trap and vacuum vessel at liquid-He temperature (~4 K). At this low temperature, most gases cryopump

to the walls of the chamber, giving a very low background

pressure. In a similar sealed vacuum can, lowered to 4 K,

Gabrielse et al.^^ report background pressures below 10"'"*

Pa. Thus by lowering the pressure by several orders of magnitude, we should be able to store trapped ions for at least several days, interrogate them with Ramsey free-precession

times as long as tens or hundreds of seconds, and eliminate

or greatly reduce pressure shifts of the 40.5 GHz clock fre- quency. In addition, the 4 K temperature should allow us to operate a superconducting shield around the ion trap region

to help in shielding changes in the magnetic field. We have constructed and have made initial tests of a prototype apparatus based on these concepts. The trap described in Sec. II and related components are mounted in an

In-sealed OFHC copper vacuum can (see Fig. 3), inside a nested liquid-He/liquid-N2 Dewar, heat sunk to the outside

bottom of the liquid-He reservoir and surrounded by radiation shields at 4 and 77 K. Figure 4 shows a schematic view

of the small vacuum can mounted inside its nested

liquid-He/liquid-N2 Dewar, which has a liquid-He hold time

of about four days. Optical access to the trap region is through windows around the base of the Dewar, aligned with

baffled holes in the radiation shields and windows in the

sides of the can. The laser beams are introduced into the vacuum apparatus through In-sealed fused silica windows.'^

Electrical and microwave access is through heat-sunk ca- bling leading down from connectors on the Dewar's top vacuum flange. The cables are routed along the liquid-N2 and

liquid-He reservoirs, into the radiation-shielded 4 K space around the experimental vacuum can, to cryogenic vacuum

feedthroughs on the sides of the can. The 13 MHz rf drive

for the trap (see below) and the dc currents for the field coils surrounding the trap are routed down the neck of the Dewar,

through the liquid-He, and into the can by cryogenic

feedthroughs in its top plate (which is also the bottom plate for the liquid-He reservoir). The superconducting shield consists of a 5 fim coating of lead, electroplated onto the inside of the copper vacuum vessel. The rf potential Vq is applied to the trap electrodes by a resonant step-up transformer. In the frequency range of inter- est here, the most convenient type of resonator consists of a

helical "secondary" coil that is effectively K/4 in length inRev. Sci. Instrum., Vol. 67, No. 1, January 1996 Linear ion trap 131 TN-30
HelmhottzColl (One Pair Shown) Indium-Sealed Quartz Window

Microwave DC Lines Feed Through

Objective Lens

Collected Ruorescence

Light Path

FIG. 3. Top view of pillbox vacuum assembly. The trap is shown at the center of the diagram in the same orientation as Fig. 2(a). Two of the laser beam paths are shown at 20° with respect to the trap z axis. The top of this pillbox asssembly forms the bottom of the liquid-He reservoir shown in Fig. 4.

side a shielding cylindrical can,'^ with an inductive-coupling

input "primary" loop at the end of the input coaxial line. In our cryogenic apparatus, it is desirable to dissipate as little rf power as possible into the liquid helium. Thus, we constructed a superconducting lead helical resonator with

(2 '^ 3000 when attached to the trap apparatus. The losses are mainly in the connecting leads and not in the resonator. (The

resonator by itself had an unloaded Q in excess of 200 000.)

It was determined that 4 mW of rf at 13 MHz produced an amplitude Vo^^l^O V at the trap rods. The superconducting

resonator sits at the bottom of the Dewar reservoir, immersed

in liquid He, with the high-potential end of its secondary coil attached to a copper cryogenic feedthrough leading through

the top plate of the experimental vacuum chamber and con- nected to the two rf-driven rods of the trap. The input coupling loop is a small off-axis coil, attached to a stainless-steel coaxial cable leading out through the top of the Dewar

through an O-ring compression seal. This coaxial cable can be rotated from outside the Dewar so as to orient the coupling loop properly for optimized power coupling into the

resonator. The trap is surrounded by orthogonal pairs of small

coils, each consisting of 350-600 turns of 0.076 nun diam

Nb wire, in order to produce the desired magnetic field components inside the superconducting shield. The coils allow

arbitrary x, y, and z components of magnetic induction to be

produced at the trap center with efficiencies of 5-10 /iT/mA

(in each coil). To ensure 100%-efficient "electron shelving" detection, we need to collect a large fraction of the ions'

132 Rev. Sci. Instrum., Vol. 67, No. 1, January 1996

Thermal

Contact Point

Resonant RF

Transformer

Copper Plate

Trap Pill-Box

FIG. 4. Side view of cryostat assembly (not to scale). The pillbox containing

the trap and related hardware is bolted to a copper plate which forms the bottom of the liquid-He reservoir. The resonant rf transformer is shown on

top of this copper plate. The high-voltage output lead is connected to a vacuum feedthrough which is then connected to the trap quadrupole elec- trodes inside the trap pillbox.

fluorescence.''^ For this reason, the imaging objective lens has a large numerical aperture. Also, it is advantageous to be

able to resolve the fluorescence of the individual ions so as to be able to detect them independently using different portions

of the photocathode of the imaging UV detector. The lens used in the experiment therefore also has resolving power at 194 nm sufficient to resolve point objects separated by less than 4 /u,m. The field of view over which the lens can image

with aberrations insignificant at this level is about 250 /xm.

The / number is //I, with the lens's front surface sitting 8.1 mm away from the center of the trap. In order for the lens to be capable of surviving temperature cycles over a range of

about 370 K ("bakeout" temperature during vacuum pro- cessing) to 4 K (operational temperature in the experiment),

the housing for the five-element lens was constructed of the same UV-grade fused silica as the elements themselves, and

the assembly was performed without any bonding agent or cushioning material. Loading the trap with '^^Hg"^ ions is accomplished by

decomposing a sample of isotopically enriched (92%)

'^^HgO powder in a small ceramic oven tube wound with

heater wire, located below the trap. A fraction of the diffuse beam of '^^Hg atoms is ionized at the center of the trap by

electrons emitted by a field-emitter point. When a sample of

ions has been captured, the oven and electron beam are turned off, and the vacuum chamber returns to a uniform 4 K

temperature from whatever local warming has occurred. We first tried loading ions between electrodes L and C

[see Fig. 2(a)] and transferring them to the experimental trap. This was unsuccessful. If the ions were loaded between elec- trodes L and C, we were unable to push them through elec- trode C into the experimental trap by biasing electrode L

Linear ion trap TN-31
* 4» ^ ^ # « m

FIG. 5. A linear crystal of laser-cooled '"Hg* ions, illuminated by 194 nm radiation. The gaps are due to the presence of other ions, possibly other mercury

isotopes, which do not fluoresce at the same wavelength as '"Hg"^. The spacing between adjacent ions is —lOfim.

with a positive potential with respect to electrode C (up to about 250 V). This result may have been caused by shielding

from the quadrupole rf electrodes and by the non-negligible

thickness of the central conical electrode. Subsequent to these experiments, the HgO source and electron source were moved near the experimental trap and the ions were created and captured directiy into this trap. We load and optically resolve individual cold ions, coalesced into linear crystals with interion spacings of 10-30

fim. We have seen crystals ranging in number from one to several tens of ions, very similar in appearance to those re- ported in Ref. 5. An example is shown in Fig. 5. With laser cooling, these crystals are stable over periods of at least 10 h. One rough measure of the background gas pressure is the

rate at which trapped "impurity" ions (which show up as readily identifiable nonfluorescing spots in the crystal) ex- change places with their '^^Hg"*" ion neighbors. In the roomtemperature apparatus at 10~^ Pa, these interchanges oc- curred every few minutes, but they do not occur over periods

of several hours or more in our cryogenic vacuum. We have observed all three ground-state hyperfine mi- crowave transitions (AM=0,±1) by the method discussed in Refs. 5 and 7.

IV. PROSPECTS

We plan to study the microwave transitions in detail after making technical improvements to the magnetic shielding,

the nulling of the residual rf micromotion, and the micro- wave local oscillator frequency synthesis. The scheme for operating this apparatus as a microwave clock will initially

be that described for the room-temperature experiment.^ The

ions will be imaged onto separate portions of the detector's photocathode and detected individually as an ensemble of

independent atomic clocks. Dehmelt's technique of electron shelving can be used to detect the clock transition in each ion

with nearly 100% efficiency.^''* Under these assumptions,

the fractional frequency stability is given by*

o-yi t)

1

(Oq ^NT^r

(8)

where Wo/2 77=40.5 GHz, A'^ is the number of ions in the

linear crystal ensemble, T/j is the free-precession time be- tween the two phase-coherent rf pulses (in Ramsey's interro- gation scheme), and T>Tjf^ is the averaging time. Assuming a 100 s Ramsey interrogation time, the short-term fractional frequency stability of an ensemble of 30 ions would be

—7X10"''' r""'^, and longer free-precession times with

linger samples of ions seem possible. Tne fractional secondorder Doppler shift to the frequency should be no greater than a few parts in lO'* with ions on the trap's nodal center

line, cooled to the Doppler limit. "Pie second-order Zeeman

shift, as a fraction of 40.5 GHz, is equal to O.IAB^, where B

is in teslas. Control of this shift may require conventional

external magnetic shielding as well as the internal superconducting magnetic shield surrounding the trap. With suffi- cientiy cold ion and small field fluctuations, a fractional in- accuracy of <1X10 of the clock frequency appears

attainable. In addition, this apparatus contains features (the super- conducting coil pairs) that should allow us to investigate new

effects based on motional Zeeman coherences. These include

a novel cooling scheme (proposed by Harde'^) using optical

pumping in conjunction with a motional magnetic coupling

between the spin orientation and the harmonic oscillator state of the ions in the trap potential, as well as a scheme for "squeezing" the total ensemble spin,^° which could improve

the signal-to-noise ratio in frequency standards where the dominant noise contribution is projection noise.'

ACKNOWLEDGMENTS

We thank John Miller for making improvements in the

light-collection optics used to acquire the image shown in Fig. 5. We thank Dale Thoele, Doug Gallagher, Richard

McLaughlin, and David Kelley for their fine technical support in the construction of the apparatus as well as Donald

McDonald and Ronald Ono for their advice on cryogenic

matters. We thank John Miller and Joseph Tan for reading

the manuscript. This work was supported by the U. S. Office

of Naval Research.

'F. G. Major and G. Werth, Phys. Rev. Lett. 30, 1155 (1973). ^M. D. McGuire, R. Petsch, and G. Werth, Phys. Rev. A 17, 1999 (1978). ^L. S. Cutler, R. R Giffard, R J. Wheeler, and G. M. R. Winkler, in Proceedings of the 41st Annual Frequency Control Symposium. Philadelphia, 1987 (IEEE, Piscataway, NJ, 1987), pp. 12-19.

''R. L. Tjoelker, J. D. Prestage, G. J. Dick, and L. Maleki, in Proceedings of

the 1993 IEEE International Frequency Control Symposium, Salt Lake

City. UT. 1993 (IEEE, Piscataway, NJ, 1993), pp. 132-138. ^M. G. Raizen, J. M. Gilligan, J. C. Bergquist, W. M. Itano, and D. J. Wineland, Phys. Rev. A 45, 6493 (1992). *W. M. Itano, L. L. Lewis, and D. J. Wineland, Phys. Rev. A 25, 1233

(1982). ^W. M. Itano, J. C. Bergquist, J. J. Bollinger, J. M. Gilligan, D. J. Heinzen,

F. L. Moore, M. G. Raizen, and D. J. Wineland, Phys. Rev. A 47, 3554

(1993). *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.

Ferroelectr. Freq. Control 37, 515 (1990). 'P T. H. Fisk, M. A. Lawn, and C. Coles, Appl. Phys. B 57, 287 (1993).

'°J. D. Prestage, G. J. Dick, and L. Maleki, J. Appl. Phys. 66, 1013 (1989). " H. G. Dehmelt, in Frequency Standards and Metrology. Proceedings of the 4th Symposium. Ancona. Italy. 1988, edited by A. DeMarchi (Springer- Verlag, Berlin, 1989), p. 286. '^W Paul, O. Osberghaus, and E. Fischer, Frosch. Berichte des Wirtschaft- mmistenums rsordrnem-Westtaien, Mr. 413 (lyssj. '^W. M. Itano and D. J. Wineland, Phys. Rev. A 25, 35 (1982). '"D. J. Wineland, J. C. Bergquist, J. J. Bolhnger, W. M. Itano, F. L. Moore,

J. M. Gilligan, M. G. Raizen, D. J. Heinzen, C. S. Weimer, and C. H. Manney, in Laser Manipulation of Atoms and Ions, Proceedings of the International School ofPhysics "Enrico Fermi," Course CXVIII, Varenna,

Rev. Sci. Instrum., Vol. 67, No. 1, January 1996 Linear ion trap 133 TN-32
Italy, edited by E. Arimondo, W. D. Phillips, and F. Strumia (North- '«H. G. Dehmelt, IEEE Trans. Instrum Meas IM-31 83 (1982)

_ Holland. Amsterdam, 1992), pp. 553-567. i^H. Harde, in International Conference on Quantum Electronics Technical G. Gabnelse X. Fei, W. Jhe, L. A. Orozco, J. Tan, R. L. Tjoelker, J. Haas, Digest, Series 1990 (Optical Society of America, Washington. DC. 1990) H. Kalmowsky, T. A. Trainor, and W. Kells, Am. Inst. Phys. Conf. Ser. Vol. 8, p. 278. le^^n^t^ ^'o^^^- . , „ ,„„ „„„,,

'°D- J- Wineland, J. J. Bollinger. W. M. Itano, F L. Moore, and D. J. nS; „,.?' ^^^- '^"'- ^ ' ^°^ ^^^^^^- "^'"^^"> Phys. Rev. A 46. R6797 (1992); D. J. Wineland. J. J. Bollinger. W. W. Macalpme and R. O. Schildknecht. Proc. IRE 47, 2099 (1959). W. M. Itano, and D. J. Heinzen. ibid. 50 67 (1994)

134 Rev. Sci. Instrum., Vol. 67, No. I.January 1996 i i„^ . Linear ion trap TN-33
144 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT. VOL. 44. NO. 2. APRIL 1995

Minimizing the Time-Dilation Shift

in Penning Trap Atomic Clocks

Joseph N. Tan, J. J. Bollinger, and D. J. Wineland

Abstract—If environmental perturbations are made negligible, the time-dilation shift is expected to give one of the largest

systematic uncertainties in a stored ion clock. In general, this shift increases with the number of trapped ions. Fluctuations in the time dilation shift therefore could limit the frequency stability of an ion clock. We show that in a Penning trap, relativistic time di- lation can be minimized if the laser-cooled ions are prepared in a

special spheroidal state. In addition, a modest stabilization of the spheroid near the minimum-shift configuration can significantly reduce fluctuations in the time dilation. The results obtained for a single-species ion clock also provide a good approximation for a sympathetically cooled system.

I. Introduction

ONE of the goals of the Time and Frequency Division

at NIST-Boulder is to realize an ion frequency standard with absolute uncertainty better than 1 part in 10^^. In recent

years, experiments at NIST with ^Be^ ions stored in a Penning

trap have demonstrated that an RF oscillator can be steered by a nuclear spin-flip "clock" transition (~ 303 MHz) with

ay{T) '^3y.\0~^- / y/r fractional frequency stability [l]-[4].

If collisional shifts [3], [4] are negligible, the expected perfor- mance of a stored ion clock using the Ramsey method [5] of

interrogation is limited by the "projection-noise" [6] frequency

instability

1 2'kUo^/NTrt

(1)

and the uncertainty in the time-dilation shift (second-order

Doppler shift)

2 6-2 (2)

The frequency stability (1), depends on the clock transition frequency Uo, the number of ions TV, the Ramsey interrogation

period Tr, and the averaging time r :$> Tr. The possibility

of using "squeezed" atomic states to obtain higher frequency

stability than (1) is being explored [7]. The time-dilation shift,

(2). depends on the average squared velocity {v') of the ions. A systematic error of 5x 10"^"^ due to time dilation has been

determined for a stored ^Be"^ ion clock [l]-[4]. In this paper, we give a detailed treatment of the time-dilation effects in Penning traps to identify the important factors for optimized

performance [8].

Manuscript received July 1, 1994; revised October 15, 1994. This work

was supported in part by the Office of Naval Research. The authors are with Time and Frequency Division, NIST, Boulder, CO

80303 USA.

IEEE Log Number 9408707.

II. Laser-Cooled, Single-Species Ion Clock

The simplest stored-ion clock consists of one trapped ion,

with charge q and mass M. In an ideal Penning trap, the ion is confined by a pure quadrupole electrostatic potential superimposed on a uniform magnetic field. The motions of the trapped

ion consist of a harmonic oscillation (angular frequency tUz) along the trap's symmetry axis, a high-frequency cyclotron

orbit (a.'^) about the magnetic field, and a low-frequency

magnetron orbit (tUm = '^;/2u;^) about the trap 2-axis that generates a gv x B force for radial confinement [9]. A suitable

internal transition (e.g., between hyperfine levels) which is field-independent at a particular field Bo, can be used in a frequency standard. If the ion motions are cooled (or heated

in the case of the magnetron orbit) to their ground states, the time-dilation shift in the selected "clock transition" is very

small. The potentially high accuracy obtainable with a single ion (N — 1), however, exacts a cost in low signal-to-noise

ratio and long averaging times. The signal-to-noise ratio and frequency stability can be

improved significantly by using large numbers of ions—see

(1). However, the time-dilation shift also increases with the number of ions. This is because, in general, a larger ion "cloud" extends farther from the trap axis, and because, for

stable trapping, an ion cloud or plasma [10], [11] must rotate (spin) about the trap's symmetry axis (taken as 2:-axis). This

rotation frequency tUr lies in the range

< UJr < IJ^n (3)

where u;^ = cjc — Wm, and ujc is the cyclotron frequency Uc = qBo/M. The accompanying thermal motions of the ions can be reduced greatly by Doppler laser cooling (typically

to < 1 K, with a cooling limit of ~ 1 mK for ^Be"^). We

can assume that the time dilation due to thermal motions is negligible, and hereon study the T = K limit. In a nearly

ideal Penning trap [9], the laser-cooled ions form a uniform

density spheroid bounded by

+ -1 =1 (4)

with an aspect ratio a = Zg/vg which varies in a known

way with the rotation frequency ujr [10], [11]. When the gv X B radial restoring force is weak (uJr -^ i^m) or when

the centrifugal force is very strong {ur — o;^), the spheroid

is stretched into a thin circular disk (a -^ 0). Hence, near the upper and lower bounds of Ur, the magnitude of the timedilation shift {Au/uo — —ulrljbc^) for the thin disk of ions

0018-9456/95$04.00 © 1995 IEEE TN-34
TAN et ai: MINIMIZING THE TIME-DILATION SHIFT IN PENNING TRAP ATOMIC CLOCKS 145

is very large. We now show that the magnitude of the timedilation shift goes to a minimum at a particular (low) value of

uir which brings the ions closer to the trap axis. For a single ion species, the time-dilation shift due to the plasma rotation may be written as

/\u = 2i^2 -SiX'Y (5)

We use a dimensionless spheroid radius X = rg/b scaled by

the radius 6 of a fictitious sphere enclosing A^ ions with the minimum cold fluid (T — K) density ue = CoMujl/q^.

This mass-independent radius h may be written as 6 =

yZNh^^JuJl where r = q~ /(ArreoMc-). The dimensionless

rotation frequency Y is given by y = {ujr/ujm){i^c/^c)- The

scaling factor Si > is defined by

Si =

1 UJ2 tj-rl 3iV-^

2U yjc_ c

--12/3

(6)

The radius and rotation frequency of the spheroidal ion cloud

are related through the conditions [1 1, (2.1) and (2.6)] required

for rotational equilibrium. For our purpose, these conditions

are recast in the form

X^a 2 V LOr Y\

^^"^"^-^ = ^<7^)

(7)

(8)

where <5i(x) is an associated Legendre polynomial of the second kind in x- The first of these coupled equations, (7)

is derived by taking the divergenece of the Lorentz force law

and eliminating the electric field with Gauss' law. It has a

centrifugal term oc y^ in a frame corotating with the ions. In the cold fluid model, (8) represents the solution to Maxwell's

equations for a uniformly charged spheroid rotating in an ideal Penning trap. It uses the electrostatic appoximation, wherein

fields generated by the ion motions are negligible. Interaction with image charges induced in the trap electrodes is also negligible since the ions are assumed to be distributed in a

central volume much smaller than the trap size [10], [11]. The dependence of the time-dilation shift on system parameters comes almost entirely from the mass-independent scaling

factors Si, ujm, and 6 because (7) depends weakly on {uj^/uJcY

for small values of u,/uc, as illustrated in Fig. 1. Convergence

is particularly rapid for low rotation frequencies (i.e., Y of

order unity). In the limit u^/lUc —> 0, the product X^Y'^ yields a universal curve. This universal (w^/wc —> 0) limit is of

experimental interest since it gives the smallest time-dilation

shift and provides a good approximation for the experimental

regime uJz/uJc < 0.1. For convenience, subsequent discussions

employ this limit. The minimum shift occurs at X = Xo ~

1.06 (see Fig. 1) and is attained by preparing the ions in a spheroid with aspect ratio a = Zg/rg ~ 0.460 rotating at Ur c^ l.S2LJrn{<^c/i^'c) « 1.82ujm- To the cxtent that this limit

is a good approximation, the optimal shape of the spheroid is independent of system parameters, and the fractional frequency

CO

><

Fig. 1. Time-dilation shift versus spheroid radius, for various values of

€ = uJ:/u>c. Solid curve gives the universal limit as e — 0.

Fig. 2. Root-mean-squared frequency jitter in a single-species ion clock as a function of mean spheroid radius A", for various values of the Gaussian

standard deviation 6 in the radius fluctuations.

shift near the minimum is

3.721+3.4

Ar,

(9)

This is expressed in terms of the deviation Ar^ from the optimal spheroid radius r^o = 1 .06 fc because the spheroid

radius is typically monitored in real-time with an imaging

device and thus provides an observable for locking schemes to stabilize the spheroid. By stabilizing the spheroid radius near

the optimal value rgo, it should be possible to reduce the jitter in the time-dilation shift. If the spheroid radius fluctuates about

X with Gaussian standard deviation 6. the frequency jitters by

jitter dX' —7=exp

{X' -X

2(52

i21

u{X') - {u)

(10)

As shown in Fig. 2, near the minimum-shift configuration, even a modest stabilization of 5 < 0.1 (i.e., (Ars)rms/'"so ~

0.1) can reduce the frequency jitter to less than 1% of the time-dilation shift, i.e., t^Vrmsl^ S 4 x lO^^Si.

In the ijJzl^c —*0 limit, the minimum time-dilation shift {/^Urnin/j^o — —3.721 Si) is independent of the ion mass. Thus, ions with clock transitions which are field-independent

at high magnetic fields are desirable since the scaling factor Si

TN-35
146 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 44, NO. 2. APRIL 1995

TABLE I Expected Performance for a Single-Species Ion Clock with A' =10^, Tr = 100 s, f> = 0.46, AND rso = 4.2 mm. Equation (1) is Used to Evaluate (jy. Values of i/o and Bo are Taken from [12], [13]

Ion t/o (GHz) So(T) W^/Uc <7,(1 s) xlO^s

minimum

xl0i5

^Be+ 0.303016 0.8194 0.0989 53 -241

«Mg+ 0.291996 1.2398 0.1089 55 -105

6^Zn+ ~1 -8.0 0.0276 16 -2.5

199Hg+ 20.9 43.9 0.0087 0.76 -0.084

201 Hg+ 7.73 3.91 0.0979 2.1 -11

is inversely proportional to the square of the magnetic field. The expected performance for several systems with N =10^,

Tji = 100 s, and fixed spheroid dimensions (a = 0.46,

rso — 4.2 mm) is given in Table I. The gain in frequency

stability with large numbers of ions will allow measurements

of the time-dilation shift in reasonably short integration time. Since Si oc Vo (K is the trapping voltage), it should be

possible, by measuring the minimally shifted clock frequency

for various 14, to extrapolate down to the Vq = limit for the unshifted clock frequency to an accuracy limited by Oy{T).

The slope of the extrapolation would provide an independent

determination of the number of ions.

III. Sympathetically Cooled Ion Clock

In a frequency standard, it is desirable to interrogate the clock transition without the perturbations (a.c. Stark shifts) caused by the cooling laser. If the heating by the envi- ronment is sufficiently weak, then the cooling laser can be

turned off during the interrogation cycles. However, when

the interrogation time is rather long or heating is significant,

continuous cooling of the clock ions is required. This cooling can be provided by using a second, simultaneously-stored

laser-cooled ion species, which sympathetically cools the clock

ions through the Coulomb coupling between species [14]-[16].

When the second ion species, the "coolant" ions, have smaller charge-to-mass ratio than the clock ions, the two species of

ions centrifugally separate [14]-[16] with the coolant ions occupying an annular region surrounding a central core of

clock ions. Ramsey interrogation times as long as Tr ~

550 s have been attained using laser-cooled ^^IVIg"*" ions to sympathetically cool ^Be"^ clock ions [3], [4]. Moreover, the

time-dilation shift is smaller for sympathetically cooled clocks

since the clock ions are distributed closer to the axis than in a single-species configuration. For low rotation frequency and u>z/uJc < 1, the surface enclosing the two species may be approximated by the spheroidal surface for a single species. This is because, for a

single species in this limit, the centrifugal term in (7) becomes

negligible and the ion distribution becomes independent of ion

Fig. 3. Cross-sectional diagram of a two-species ion plasma for an ion clock. Coolant ions occupy the annular region surrounding the clock ions (shown

darker) in the cylindrical core.

N/N,

Fig. 4. Ratio of the time-dilation shift in a sympathetically cooled clock to that in a single-species clock. The number of clock ions is fixed.

mass to order {uj,/uc)'^. Therefore, if a cylindrical core is re- moved from a minimum-shift configuration and is substituted by an equal number of clock ions, as illustrated in Fig. 3, the spheroidal boundary is well preserved. Neglecting the small gap [14]-[16] between the two species, (5) and (9) derived for

the time-dilation shift in a single species, are still applicable to the cylindrical core of clock ions provided the scaling factor Si is replaced by

Ss = S'lA (11)

where S{ is Si evaluated assuming N = Ni + N2 {Ni is the number of clock ions and A^2 is the number of coolant ions)

and A is defined by

2/3]

/ ; vo \ A = 1 1 3iV2

2A^ N1+N2 (12)

Fig. 4 plots S2/S1 as a function of N2/N1, comparing the time-dilation shift in a two-species clock to that in a singlespecies clock with the same number of clock ions A^i. In the regime N2/N1 > 1, we have S2/S1 ~ {5/6) i/N^/N^- For

a typical N2/N1 ~ 10, the time-dilation shift is about 60%

smaller than in a single-species clock. The design and realization of an ion frequency standard, of

course, must take into account other considerations [12]. For

instance, preliminary studies showed that the ^Be"*" hyperfine

clock transition (303 MHz) has an unexpectedly large pressure

shift { (-1.7±0.4)xl0-^ Pa-^ for CH4} [3], [4]. For some

ions, a cryogenic environment may be required to suppress this

effect. Significant improvement in both frequency stability and

accuracy also requires the ability to trap and laser-cool large

numbers of ions, as well as to characterize and control the ion

distribution for minimizing the time-dilation shift. Experiments

TN-36
TAN etai: MINIMIZING THE TIME-DILATION SHIFT IN PENNING TRAP ATOMIC CLOCKS 147

are underway in a new Penning trap [17] designed for these and other studies.

Acknowledgment

The authors thank A. Barton and S. Gilbert for carefully

reading the manuscript.

References

[1] J. J. Bollinger, J. D. Prestage, W. M. Itano, and D. J. Wineland,

"Laser-cooled-atomic frequency standard," Phys. Rev. Lett., vol. 54, pp. 1000-1003, 1985.

[2] J. J. Bollinger, D. J. Heinzen, W. M. Itano, S. L. Gilbert, and D. J. Wineland, "Test of the linearity of quantum mechanics by rf spec- troscopy of the ^Be"*" ground state." Phys. Rev. Lett., vol. 63, pp. 1031-1034, 1989.

[3] , "A 303-MHz frequency standard based on trapped Be+ ions," IEEE Trans. Instrum. Meas.. vol. 40, pp. 126-128, Apr. 1991.

[4] D. J. Wineland, etal., "Reent experiments on trapped ions at the National

Institute of Standards and Technology," in E. Arimondo, W. D. Phillips, and F. Strumia, Ed. Laser Manipulation ofAtoms and Ions. New York:

North-Holland, 1992, pp. 553-567.

[5] N. F. Ramsey, Molecular Beams. London: Oxford Univ. Press, 1956,

p. 124.

[6] W. M. Itano, J. C. Bergquist, J. J. Bollinger, J. M. Gilligan, D. J. Heinzen

et ai, "Quantum projection noise: Population fluctuations in two-level

systems," Phys. Rev. A. vol. 47, pp. 3554-3570, 1993.

[7] D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heinzen,

"Squeezed atomic states and projection noise in spectroscopy," Phys.

Rev. A, vol. 50, pp. -67-88, 1994.

[8] J. N. Tan, J. J. Bollinger, and D. J. Wineland, "Experiments with large numbers of laser-cooled, trapped ions," Bull. APS, vol. 39, p. 1218,

1994.

[9] L. S. Brown and G. Gabrielse, "Geonium theory: Physics of an electron or ion in a Penning trap," Rev. Mod. Phys., vol. 58, no. 1, pp. 233-31 1, 1986.

[10] L. R. Brewer, J. D. Prestage, J. J. Bollinger, W. M. Itano, D. J. Larson, and D. J. Wineland, "Static properties of a non-neutral ^Be^-ion

plasma," Phys. Rev. A, vol. 38, pp. 859-873, 1988.

[11] J. J. Bollinger, D. J. Heinzen, F. L. Moore, W. M. Itano, D. J. Wineland,

and D. H. E. Dubin, "Electrostatic modes of ion-trap plasmas," Phys.

Rev. A. vol. 48, pp. 525-545, 1993.

[12] D. J. Wineland, J. C. Bergquist, J. J. Bollinger. W. M. Itano, D. J. Heinzen et al. "Progress at NIST toward absolute frequency standards using stored ions," IEEE Trans. Ult. Per. Freq. Control, vol. 37, no. 6, pp. 515-523, 1990.

[13] W. M. Itano and D. J. Wineland, "Frequency and time standards

utilizing laser-cooled ions," Bull. APS, vol. 24, p. 11 85, 1979; "Precision measurement of the ground-state hyperfine constant of -^Mg'^ ," Phys.

Rev. A. vol. 24, p. 1364, 1981.

[14] D. J. Larson, J. C. Bergquist, J. J. Bollinger, W. M. Itano. and D. J. Wineland, "Sympathetic cooling of trapped ions: A laser-cooled twospecies nonneutral ion plasma," Phys. Rev. Lett., vol. 57, pp. 70-74,

1986.

[15] T. M. O'Neil, "Centrifugal separation of a multispecies pure ion plasma," Phys. Fluids, vol. 24, pp. 1447-1451, 1981.

[16] J. J. Bollinger, et al, "Ion trapping techniques: Laser cooling and

sympathetic cooling," in E. H. Ottewitte and W. Kells, Ed., Intense Positron Beams. Singapore: World Scientific, 1988, pp. 63-73.

[17] J. N. Tan, J. J. Bollinger, and D. J. Wineland, "A new Penning trap for laser-cooling large ion numbers," Bull. APS, vol. 38, p. 1972, 1993.

TN-37
HIGH-MAGNETIC-FIELD CORRECTIONS TO

CESIUM HYPERFINE STRUCTURE *

WAYNE M. ITANO

Time and Frequency Division, National Institute of Standards and Technology,

Boulder, Colorado 80303, USA

Corrections to the Breit-Rabi formula for the ground state of ^^^Cs aire calculated. For a recently

proposed high-magnetic-field frequency standard, the corrections amount to a few peirts in 10^^

De Marchi has proposed a cesium atomic-beam frequency standard based on the {Mp =

—1, AMp — 0) transition, at the magnetic field (jB w 82 mT) where the derivative of the

frequency with respect to B is zero,^ and has carried out preliminary experiments? In order

for this to be a primary frequency standard, it is necessary to relate the measured frequency

to that of the zero-field transition frequency z/Q) which defines the SI unit of time. The BreitRabi formula predicts the frequency to be y/15/16fo = 8 900 727438.257 Hz.^ For higher

accuracy, additional terms must be taken into account.

Three corrections to the Breit-Rabi formula have been measured for the ground-state

hyperfine structure of rubidium by Larson and coworkers.^"^ These are the dipole diamagnetic shift, the quadrupole diamagnetic shift, and the hyperfine-assisted Zeeman shift. These terms have not yet been measured for cesium.

The dipole diamagnetic shift results from a cross term between the contact hyperfine

interaction and the atomic diamagnetic interaction in second-order perturbation theory.

This leads to a shift, proportional to B^ , in the dipole hyperfine constant A. I evaluated

the perturbation sum over states, including the continuum, by solving an inhomogeneous

differential equation for the perturbed wavefunction. I used an empirical potential for the

valence electron derived by Klapisch.^ The result is 6A/A = 5.46 x 10~^° B^, where B is expressed in teslas. This leads to an increase in the {Mf — —1) field-independent transition

frequency of 33 mHz. Previous calculations have been reported by Bende/ and by Ray et alf In order to check the method, I used Klapisch's potential to calculate 6A/A for rubidium.

The result is 10% below the experimental value. I estimate an accuracy of around 20% for

the present calculation in cesium.

The quadrupole diamagnetic shift results from a cross term between the nuclear quadrupole hyperfine interaction and the electronic diamagnetic interaction in second-order perturbation theory. The interaction is diagonal in Mj and Mj. The perturbation sum over

states is the same one that appears in the calculation of the quadrupole shielding factor

7oo-'* Using the value ToolCs"*") = —86.8,^ I obtain the result

(M7,Mj|ffQD|Mz,Mj) = -4.6 X iq-b^^ [3M|^-^/(/ + 1)]

^ ^^^

The result is in hertz if B is expressed in teslas. A similar calculation for rubidium yields a

coefficient in good agreement with experiment."* For cesium, the estimate of the coefficient

'Work of the US government. Not subject to US copyright.

TN-38
for the shift is probably accurate within around 30%. However, for the special case of the

{Mp = — 1) field-independent transition, the shift vanishes. The two states involved in the

transition are (| - 1/2, -1/2) ± | - 3/2, l/2))/\/5, in the \Mi, Mj) basis, where + refers to the higher-energy state and — to the lower-energy state. Hence, the states are shifted by

the same amount, since they have the same M/-content.

The hyperfine- assisted Zeeman shift was explained by Fortson as a third-order perturbation, in which the contact hyperfine interaction acts twice and the electronic Zeeman

interaction acts once.^° Fortson gave diagonal matrix elements in the |M/,Mj) basis. For

the (Mp = — 1) field-independent transition, it is necessary to extend Fortson's calculation to include off-diagonal matrix elements. In the \[IJ)F^Mp) basis, the nonzero matrix

elements for ^^^Cs are

(3,MF|iyHAz|4,Mir) = AhpB^Jw - M^. (2)

Fortson used quantum-defect theory to make a semi-empirical estimate for (5 in rubidium;

it agreed with experiment within 10%. The same method, applied to cesium, yields /3 = 88

mHz/T. For the [Mp = —1) field-independent transition, the shift at 5 = 82 mT is — 2v'T5/?5 = —56 mHz. This estimate of the shift is probably accurate within around 30%.

To achieve a theoretical error of 10~^* in the total frequency, which is the projected

accuracy of the high-field frequency standard,^ the larger shifts must be known within less than 1%. This could possibly be done by ah initio calculations or by experiments carried

out at much higher magnetic field, similar to those of Larson and coworkers in rubidium.

Acknowledgments

This work was supported by the Office of Naval Research and the Army Research Office.

References

1. A. De Marchi, in Proc. 7th European Frequency and Time Forum, Neuchatel, Switzerland, 1993, p. 541.

2. G. Costanzo, E. Rubiola, A. De Marchi, in Proc. 9th European Frequency and Time

Forum, Besangon, France 1995 (in press).

3. N. P. Economou, S. J. Lipson, and D. J. Larson, Phys. Rev. Lett. 38, 1394 (1977).

4. S. J. Lipson, G. D. Fletcher, and D. J. Larson, Phys. Rev. Lett. 57, 567 (1986).

5. G. D. Fletcher, S. J. Lipson, and D. J. Larson, Phys. Rev. Lett. 58, 2535 (1987).

6. M. Klapisch, C. R. Acad. Sci. Paris 265, 914 (1967).

7. P. L. Bender, in Quantum Electronics, Proceedings of the Third International

Congress, Paris, ed. P. Grivet and N. Bloembergen (Columbia Univ. Press, New

York, 1964), p. 263.

8. S. N. Ray, M. Vajed-Samii, and T. P. Das, Bull. Am. Phys. Soc. 24, 477 (1979).

9. H. M. Foley, R. M. Sternheimer, and D. Tycko, Phys. Rev. 93, 734 (1954).

10. N. Fortson, Phys. Rev. Lett. 59, 988 (1987).

TN-39
APPLICATION OF LASER-COOLED IONS

TO FREQUENCY STANDARDS AND METROLOGY*

D. J. Wineland, J. C. Bergquist, D. Berkeland, J. J. Bollinger, F. C. Cruz,

W. M. Itano, B. M. Jel^ikovid, B. E. King, D. M. Me^hof,

J. D. Miller, C. Monroe, M. Rauner, and J. N. Tan

National Institute of Standards and Technology, Time and Frequency Division

Boulder, CO, 80303

E-mail: dwineland@nist.gov

ABSTRACT

With the first experiments on laser cooling, it was realized that a principal application

of the technique would be to reduce Doppler shifts in frequoicy standards. Laser cooling is

still thought to be essential to achieve uncertainty significantly smaller than 1 part in 10'^; however, a number of other problems must be solved to achieve this goal with stored atomic

ions. For both RF/microwave and optical frequmcy standards based on trapped ions, some of

these problems are discussed. Laser cooling of tnq)ped ions also spears to be important for other reasons in metrology. For example, for frequency standards, cooling to the zero-point of

motion should enable the creation of quantum mechanically correlated states for improved

signal-to-noise ratio in spectroscopy, or provide accurate measuremoit of Stark shifts. Cooling

to the zero-point of motion also enables the creation of nonclassical states of motion or

correlated states which may be applicable to sensitive detection, quantum computation, or to

test quantum measurement principles. Techniques to achieve laser cooling to the zero-point of

motion are briefly described.

1. Introduction

The first laser cooling experiments^'^ were accomplished on trapped ions in

1978. It was apparent then that one important application for laser cooling would be

the reduction of the time-dilation (2nd-order-Doppler) shift in frequency standards. In

fact by 1980, temperatures achieved from laser cooling on ions were low enough that

the corresponding time-dilation shift was significantly below 1 part in 10^*. This is illustrated in Fig. 1, where we plot time-dilation shifts (shown as a fraction of the

transition frequency) corresponding to temperatures reported in some laser cooling

experiments. Since these shifts are significantly smaller than the smallest reported

inaccuracies for atomic clocks, we can ask why the overall accuracies reported do not

approach these numbers.

One source of the discrepancy is that, for trapped ions, laser cooling affects

only the thermal or random modes of motion. However, ion motion in both Paul and

Penning traps^* in part consists of unavoidable coherent motion; this motion can result

in large ion velocities and accompanying Doppler shifts. In a Penning trap, a cloud

of ions must rotate about the_ symmetry axis of the trap. The velocity associated with

this rotation gives rise to a v x B Lorentz force which provides trapping normal to the symmetry axis. For large clouds, the velocity in this rotation motion can be

significantly larger than the thermal velocity in the axial and cyclotron motions (which

can be laser-cooled), thereby limiting the degree to which the time-dilation shift can

* Contribution of the U.S. Government; not subject to U.S. copyright.

TN-40
o<

o

1980 1990

Time, Years

Fig. 1. The circles show magnitudes of calculated (fractional) time-dilation shifts | Ai^/^o | based on r^>orted temperatures achieved in selected trapped-ion, laser cooling experiments.

These calculations omit the contributions from the coherent modes of motion (see text). The

squares show (fractional) imcertainties in trapped-ion, laser-cooled atomic frequency standard

experiments in which overall errors were estimated. The nimibers beside the symbols indicate the corresponding reference numbers.

be suppressed.* In a Paul trap, ions are forced to oscillate at the driving field

frequency (the micromotion). Since the driving fields are spatially inhomogeneous,

the force on an ion, averaged over one cycle of the driving field, gives rise to trapping (the ponderomotive force). For ions near the outside of a large ion cloud

this force must be large to balance the Coulomb repulsion from the ions near the

center of the trap. This implies that the velocity of this coherently driven motion is large; it is often much larger* than the velocity of the ions associated with their

"secular" motion in the ponderomotive well (which can be laser-cooled).

In 1973, Dehmelt suggested storing a single ion in a Paul trap.^ This system

has the advantage that the kinetic energy in the ion micromotion is approximately

equal to the kinetic energy in the secular motion. Therefore, if the secular motion is laser-cooled, the time-dilation shift from secular motion and micromotion can be very

small. On the other hand, a single-ion frequency standard may not give sufficient

short-term frequency stability. For N ions, the frequency stability (two-sample Allan

variance) is limited to*'*

a/x) =

1 (^Jtn\

(1)

where cOo is the clock transition frequency, T is the transition interrogation time (the

Ramsey method is assumed, where the time of the two Ramsey pulses is much less

than the free-precession period T), t is the averaging time, and we assume 100%

detection efficiency. For a single-ion clock using RF or microwave transitions, this

may imply a long averaging time to reach a measurement precision better than one

part in 10*'*. For example, for an ion with v^ = ujlir = 10 GHz, and T = 10 s,

TN-41
Eq. (1) implies (7y(r) = 5.0x10'^^ t"'*. Therefore, in general, for both Penning and

Paul trap RF/microwave frequency standards, where we desire large N to make 0^(7)

sufficiently small, we require methods to minimize the velocity in the coherent modes

of motion. Some strategies are discussed below.

If the velocity in the coherent motion can be suppressed sufficiently, Doppler

cooling' will suffice (at least for the next generation of frequency standards) to provide small enough time-dilation shifts for frequency standards. If we neglect the

velocity in the coherent modes of motion, minimum temperatures achieved from

Doppler cooling give rise to a time-dilation shift of Avjyi/v^ = -3.3x10'^*(7/2t)/M

where 7/2T is the laser-cooling transition linewidth in megahertz and M is the ion mass in relative atomic mass units. For example, for ^''Hg'^ ions (y/lr = 70 MHz,

M = 199 u), we could expect Avoif^o = - 1.2x10".

However, there now appear to be other reasons for achieving better laser

cooling - in particular for reaching the zero-point energy state of motion. New

applications are emerging for this extreme form of laser cooling, both for frequency

standards and other forms of metrology. Some of these applications and methods to reach the zero-point energy through laser cooling are discussed below. Reference 10

provides a more comprehensive review of cooling methods in ion traps.

2. Reductioii of the coherent motion (micromotion) in a Paul trap

One way to suppress the velocity in the micromotion and still achieve a large

number of ions for good signal-to-noise ratio is to use an elongated "linear" trap

geometry. In this type of trap, shown schematically in Fig. 2, the RF electric fields

vanish along the axis of the trap. Therefore, if ions are confined near the trap axis in an elongated ion cloud geometry, the micromotion can be suppressed. This idea was

first developed and used by Prestage et al." In part, because of the relatively small

time-dilation shift, excellent stabilities have been achieved in clocks where the ions'

secular motion in a linear trap is cooled by buffer gas (see the contributions at this

conference by Tjoelker et al. and Fisk et al.).

Dehmelt suggested applying this idea to frequency standards in its limiting

form,^^ that is, where a single string of ions is confined to the trap axis like "beads on

a string." Here, as for a single ion in a spherical quadrupole trap, the velocity of

micromotion can be approximately the same as for the secular motion. With laser

cooling, the time-dilation shifts can therefore be very small. At NIST, this idea is being pursued for application to a frequency standard of high accuracy. Strings of

laser cooled ^'^g'^ ions have been obtained in a room temperature apparatus" and in a cryogenic (4 K) apparatus^"* whose good vacuum suppresses collisional frequency

shifts and ion loss from chemical reactions. The bottom part of Fig. 2 shows an image of a string of laser cooled '''Hg'^ ions taken in this cryogenic apparatus. The

Munich group has also achieved strings of laser-cooled (Mg"^) ions in a race-tracktype trap*^; this type of apparatus could also be used for frequency standard

applications.

Another approach to suppress micromotion is to use arrays of conventional

(spherical quadrupole) ion traps."" Here, single laser-cooled ions could be stored in

each trap thereby ensuring small time-dilation shifts. Important technical problems

appear to be the loading, addressing, and detecting of the individual ions.

TN-42
r VoCOsQt

Fig. 2. The i^)per part of the figure shows a schematic diagram of the electrode configuration

for a linear Paul-RF trap." Typical parameters for trapping '*^g"^ ions are: rod-electrode

separation of (proximately 1 mm, V^ =: 400 V, and C/2x ss. 10 MHz. The lower part of the

figure shows an image of a string of ^^g*^ ions, illiuninated with 194 imi radiation, Xskesa. with a UV-s^isitive, photon counting imaging tube. The spacing betwe^i adjac^it ions is approximately 10 iim. The "g^s" in the string are occupied by inqiurity ions, most likely

other isotopes of Hg"^, which do not fluoresce because the frequ^icies of their resonant

transitions do not coincide with the 194 nm ^.^ -» ^.^ transition of ^'^g'*'.

3. Reduction of the coherent (rotation) motion in a Penning trap

Following the example of the linear or racetrack Paul traps, the coherent

motion in a Penning trap could be suppressed by storing ions near the axis of the

trap - ideally as a single string along the axis. A more modest approach might be

the following. For a cloud of laser-cooled ions in a Penning trap, as the radius of the

cloud is varied, the time-dilation shift goes through a minimum (see Ref. 18 and Tan,

et al., this conference). Therefore if the cloud of ions is stabilized at this extremum

point, the time-dilation shift and its fluctuations can be made very small.

Any configuration of ions in a Penning trap is in an unstajble equilibrium (as

opposed to a sample of ions in a Paul trap). Therefore, once created, the

configuration would not be preserved if, for example, collisions are present. Torques

from laser scattering could restore the configuration but at the expense of possible

perturbing effects from the cooling laser.

4. Sympathetic laser cooling

One solution to the problems associated with Penning trap cloud stability is sympathetic laser cooling.^' As an example, in the laser-cooled frequency standard

experiment reported in Ref. 20, a ^^

ion sample was cooled sympathetically,

through Coulomb coupling, by a simultaneously stored, surrounding cloud of Mg"^

ions which were directly laser-cooled. This method has the advantage that the laser

cooling (and laser torques) can be applied continuously to the Mg* ions, thereby

maintaining the geometry and temperature of the 'Be* ions. Because the frequency of

TN-43
the Mg"^ optical cooling transition is significantly different than the optical transitions

in 'Be'^, the 'Be* ions are not appreciably perturbed by the Mg* cooling laser beam.

It should be possible to apply this technique to sympathetically cool a string of ions in a Penning trap or a cloud that is configured to give the minimum time-dilation shift.

^* Sympathetic cooling might also be employed to advantage in a (linear) Paul

trap. When long Ramsey interrogation times T are desired, the ions may heat from,

for example, RF heating. However, in a long string of ions, cooling could be applied

to ions on one end which then cool the "clock" ions sympathetically. In principle, the

"cooling ions" could be of the same species if the string is long enough or, if the

string is bent around a (light-baffled) comer or, if the clock ions are shielded by a

coupling electrode^^ to minimize the perturbing effects of scattered light from the

cooling ions.

5. Laser-cooled, trapped-ion, optical frequency standards

Most researchers agree that, in the future, the most accurate and stable clocks

will be made using optical or higher frequency transitions. From Eq. (1), if the clock

transition frequency is high enough, then a single ion would give good short term

stability. For example for the ^Svi -* ^Dsa 282 nm transition in ^''Hg*, assuming N =

1, T = 0.01 s (the radiative lifetime of the '^D^n level is approximately 0.1 s), Eq. (1)

implies <7y(r) = 1.5x10"^* r'^. As discussed above, the time-dilation shift could be

very small. In addition, shorter interrogation times will mitigate the effects of various sources of ion heating, since clock cycles could be alternated with relatively frequent

laser cooling. Work on optical frequency standards is represented by several groups

at this conference; see, for example, the stored-ion contributions by Barwood, et al.

,

Fermigier, et al.. Gill, et al., Knoop, et al., Madej, et al., Nagoumey, et al., Peik, et

al. , and Tamm and Engelke.

Currently, laser local oscillators are not as stable as desired. A representation

of the state of the art for trapped-ion optical frequency standards is given in Ref. 22,

where a laser with frequency bandwidth of less dian 25 Hz (for 60 s averaging times)

was locked to the 282 nm quadrupole transition in a single laser-cooled ^''Hg'^ ion.

Finally, an additional technical difficulty for all optical frequency standards is that of comparing laser local oscillators to other oscillators (particularly oscillators at low frequency). Several approaches are discussed in these proceedings.

6. Present and future laser-cooled ion frequency standards

So far, there has been only one report of an ion-based, laser-cooled atomic

frequency standard whose accuracy has been evaluated. In this experiment, lasercooled ^Bq* ions in Penning trap were used.^ A second evaluation^" of this system

uncovered an unexpected background gas pressure shift. This shift was apparently

caused, in part, by background methane gas, which has a strong perturbing effect on

'Be"^ ion hyperfine structure.^ (Interestingly, methane does not have nearly as large a

perturbing effect in the ^^^Yb"^ experiments reported by Bauch and Tamm at this

conference.) At present, the inaccuracy of the laser-cooled 'Be'^ ion clock is limited

to about one part in 10" from this pressure shift.^®

Further work is required to minimize possible pressure shifts and a number of

other typical systematic effects (for example, magnetic field shifts) present in all

TN-44
frequency standards. However, we can be optimistic tiiat frequency standards based

on laser-cooled ions will eventually have inaccuracies considerably smaller than 1 part

in 10^*, perhaps as small as 1 part in 10" or less.^'"*^

7. Applications of trapped ions which are laser-cooled to the zero-point of motion

Although Doppler cooling should yield sufficientiy small time-dilation shifts

for frequency standards, there now appear to be other reasons for achieving better

laser cooling. At very low kinetic energies, the quantum nature of the motion

becomes apparent; this quantized motion, particularly the zero-point state, can be used

for various purposes.

For example, when inaccuracies of optical clocks become smaller than one

part in 10^^, Stark shifts may become significant.^ As Dehmelt has pointed out", the

intercombination transitions in group niA ions are superior in this regard, although

for various technical reasons, optical quadrupole S - D transitions in other ions can be easier to implement. The quadratic Stark shifts on these latter transitions depend

on the kinetic energy of the ions. Since the secular motion is quantized, the quadratic

Stark shift from the secular motion becomes quantized. This could be used to identify

the zero-point energy states of the ion^ or, conversely, the Stark shift could be

calibrated from its discrete spectrum caused by quantized motion.

Another possible frequency standard application of zero-point cooling is to improve the quantum-limited signal-to-noise ratio in the detection process (see Ref. 28

and Bollinger, et al., this conference). If spectroscopy is performed on atoms which

are first prepared in particular quantum mechanically correlated, or entangled, states,

the frequency stability would be given by ay(r) = 1/(woN(Tt)'*) rather than that given

by Eq. (1). In this case, the time to reach a desired measurement precision would be

reduced in proportion to 1/N.

Outside the realm of frequency standards, there may be other metrological

applications of ions which are laser-cooled to the zero-point of motion. Cirac and

Zoller have recentiy suggested a very interesting scheme to perform quantum

computation using an array of ions confined in a linear trap.^' Quantum computers

have received attention lately because of their ability to crack the most common form

of public-key encryption. The first quantum logic gates using prepared input "qubits"

have now been realized using trapped ions which are first cooled to the zero-point

energy. ^° A second possible application might be to the generation of multiparticle

"EPR"-type correlated states which would be interesting from the standpoint of

quantum measurement theory. ^^ Finally, if a single ion or single mode of oscillation

is cooled to the zero-point state, it is possible to generate nonclassical states of motion

such as squeezed states^^-^*'^^'^' (see also Blatt et al., this conference). Such states may be useful for sensitive detection, for example, mass spectroscopy at the quantum

leveP*. In all of these applications, efficient cooling to the zero-point of motion is a

crucial prerequisite.

8. Methods for zero-point laser cooling

With the use of Doppler cooling,' minimum energies for a bound atom in one

dimension are approximately equal to ^7/2, where 2t^ is Planck's constant and 7 is the natural linewidth of the cooling transition. For an atom bound in a harmonic well

TN-45
with "vibrational" frequency Wv, this energy can be expressed as fiQi^(<n^> +V^)

where nv is the harmonic oscillator quantum number. Since Doppler cooling is valid

in the regime where 7 > co^, limiting kinetic energies for Doppler cooling necessarily

imply <ny> > 1. When 6)v > 7, cooling can be achieved in the "resolved sideband" regime.**

Consider a two-level atom characterized by resonant transition frequency wq and

radiative linewidth 7. If a laser beam (frequency wj is incident along the direction of

the atomic motion, the absorption spectrum is composed of a "carrier" at frequency

Wo and resolved frequency-modulation sidebands spaced by Wv which are generated

from the Doppler effect. Cooling occurs if the laser is tuned to a lower sideband, for

example, at Wl = wq-Wv ^ this case, photons of energy ^(cob-cov) are absorbed and

spontaneously emitted photons of average energy ^coq return tiie atom to its ground

internal state (assuming that fiojy is much greater than the photon recoil energy).

Therefore, the atom's kinetic energy is reduced by ^co^ per scattering event. Cooling

proceeds until the atom's mean vibrational quantum number in the harmonic well is given by^' (n^),^ = (y/2(a^)^ < 1. Two experiments on single ions have reported

laser cooling to the zero-point of motion.^'"'** In Ref. 36, resolved-sideband stimulated

Raman transitions replaced single photon transitions; cooling to the zero-point of

motion was achieved 98% of the time in 1-D and 92% of the time in 3-D. Before

correlated states between ions can be obtained, zero-point cooling must be extended to certain modes (such as the center-of-mass oscillation) on collections of N (^ 2) ions.

Dramatic advances in laser cooling of neutral atoms were made witii the

introduction of polarization-gradient and Sisyphus cooling.'^ Because of this success,

it was interesting to investigate whether or not these methods would yield < n^ > =

for bound atoms. Theoretical investigations have shown that particular applications of

Sisyphus cooling or polarization-gradient cooling^*'^''"' to trapped atoms appear to give

cooling near <n> = 1, but not <n> -» 0. More recentiy, it has been suggested

that ions in the n = state could be selected from a distribution by null detection.^^

Other discussions of laser cooling appear in the contributions to this conference by

Knoop et al. , Sugiyama and Yoda, and Alekseev and Krylova.

9. Summary

Laser cooling, which can suppress time-dilation shifts, appears to be essential

to achieve high accuracy in atomic frequency standards. However, before the benefits

of laser cooling can be realized with trapped ions, the velocity in the coherent or nonthermal modes of motion must be suppressed. We have discussed several

approaches which address this problem. It also appears that laser cooling may have

other benefits in metrology. We have pointed out the application of zero-point laser

cooling to the generation of correlated states for improved signal-to-noise ratio in

spectroscopy, fundamental quantum measurements and quantum computation.

10. Acknowledgements

We gratefully acknowledge the support of the U.S. Office of Naval Research

and the U.S. Army Research Office. We thank D. Howe, D. Lee, J. Kitching, and

M. Young for helpful comments on the manuscript.

TN-46
11. References:

1. D.J. Wineland, R.E. Drullinger, and F.L. Walls, Phys. Rev. Lett. 40, 1639

(1978).

2. Neuhauser, M. Hohenstatt, P. Toschek, and H. Dehmelt, Phys. Rev. Lett. 41, 233

(1978).

3. H.G. Dehmelt, Adv. At. Mol. Phys. 1, 53 (1967) and 5, 109 (1969).

4. D.J. Wineland, Wayne M. Itano, and R.S. VanDyck, Jr., Adv. At. Mol. Phys. 12,

135 (1983).

5. R. C. Thompson, Adv. At. Mol. Phys. M, 253 (1993),

6. 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 22, 515 (1990).

7. H.G. Dehmelt, Bull. Am. Phys. Soc. IS, 1521 (1973).

8. W. M. Itano, J. C. Bergquist, J. J. Bollinger, J. M. Gilligan, D. J. Heinzen, F. L.

Moore, M. G. Raizen, and D. J. Wineland, Phys. Rev. A47, 3554 (1993).

9. see, for example: D.J. Wineland and W.M. Itano, Physics Today, vol. 40, no. 6,

p. 34 (1987).

10. W. M. Itano, J. C. Bergquist, J. J. Bollinger, and D. J. Wineland, Proceedings

of the Nobel Symposium on Trapped Charged Particles and Related Fundamental

Physics, Lyseldl, Sweden, August, '94, to be published in Physica Scripta.

11. J. Prestage, G. J. Dick and L. Maleki, J. Applied Physics, ^, 1013 (1989); J. D. Prestage, G. J. Dick, and L. Maleki, IEEE Trans. Instnim. and Meas. 4Q, 132

(1991).

12. H. G. Dehmelt, in Frequency Standards and Metrology, Proc. 4th Symposium,

ed. by A. DeMarchi, Ancona, Italy (Springer-Verlag, Berlin, Heidelberg, 1989) p. 286.

13. M. G. Raizen, J. M. Gilligan, J. C. Bergquist, W. M. Itano, and D. J. Wineland, Phys. Rev. A45, 6493 (1992).

14. M. E. Poitzsch, J. C. Bergquist, W. M. Itano, and D. J. Wineland, Proc. 1994

IEEE International Frequency Control Symposium, p. 744, June, 1994; J. Miller, M.

E. Poitzsch, F. C. Cruz, D. J. Berkeland, J. C. Bergquist, W. M. Itano. and D. J. Wineland, 1995 IEEE International Frequency Control Symposium, June, 1995.

15. G. Birkl, S. Kassner, and H. Walther, Nature 251, 310 (1992).

16. F. Major, J. Phys. Lett. 2S, L221 (1977).

17. R. Brewer, R. G. DeVoe, and R. Kallenbach, Phys. Rev. A46, R6781 (1992).

18. J. N. Tan, J. J. Bollinger, and D. J. Wineland, Proc. Conf. Prec.

Electromagnetic Meas., Boulder, CO, June, 1994; IEEE Transactions on

Instrumentation and Measurement 44, 144 (1995).

19. D.J. Larson, J.C. Bergquist, J.J. Bollinger, W.M. Itano, and D.J. Wineland,

Phys. Rev. Lett. 52, 70 (1986).

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

IEEE Trans, on Instrum. and Measurement 4Q, 126 (1991).

21. D. J. Heinzen and D. J. Wineland, Phys. Rev. A42, 2977 (1990).

22. J. C. Bergquist, W. M. Itano, and D. J. Wineland, in Frontiers in Laser

Spectroscopyy proc. International School of Physics < < Enrico Fermi > > , Course

TN-47
CXX, ed. by T. W. Hansch and M. Inguscio, (North Holland, Amsterdam, 1994),

pp. 359 - 376.

23. J. J. Bollinger, J.D. Prestage, W.M. Itano, and D.J. Wineland, Phys. Rev. Lett. 54, 1000-1003 (1985).

24. 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, in

Laser Manipulation ofAtoms and Ions, ed. by E. Arimondo, W. D. Phillips, and F.

Strumia (North-Holland, Amsterdam, 1992), pp. 553-567.

25. H. Dehmelt, in Frequency Standards and Metrology, Proc. 4th Symposium, ed.

by A. DeMarchi, Ancona, Italy (Springer-Verlag, Berlin, Heidelberg) p. 15.

26. D.J. Wineland, W.M. Itano, J.C. Bergquist and R.G. Hulet, Phys. Rev. A26,

2220 (1987).

27. N. Yu, X. Zhao, H. Dehmelt, and W. Nagoumey, Phys. Rev. A5Q, 2738 (1994).

28. D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heinzen, Phys. Rev.

A5Q, 67 (1994).

29. J. I. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995).

30. C. Monroe, D. M. Meekhof, B. E. King, W. M. Itano, and D. J. Wineland,

Phys. Rev. Lett. 75, 4714-4717 (1995).

31. D. M. Greenberger, M. A. Home, and A. Zeilinger, Phys. Today 4^(8), 22

(1993).

32. J. I. Cirac, A. S. Parkins, R. Blatt, and P. Zoller, Phys. Rev. Lett. 7Q, 556

(1993); J. I. Cirac, R. Blatt, A. S. Parkins, and P. Zoller, Phys. Rev. Lett. 7Q, 762

(1993).

33. D. M. Meekhof, C. Monroe, B. King, W. M. Itano, D. J. Wineland, submitted.

34. D. J. Wineland and H. G. Dehmelt, BuU. Am. Phys. Soc. 2Q, 637 (1975).

35. F. Diedrich, J.C. Bergquist, W. M. Itano, and D.J. Wineland, Phys. Rev. Lett. 62, 403 (1989).

36. C. Monroe, D. M. Meekhof, B. E. King, S. R. Jefferts, W. M. Itano, D. J. Wineland, and P. Gould, Phys. Rev. Lett. 75, 4011-4104 (1995).

37. see, for example, C. Cohen-Tannoudji and W. D. Phillips, Phys. Today 4^(10),

33 (1990).

38. D. J. Wineland, J. Dalibard, and C. Cohen-Tannoudji, J. Opt. Soc. Am. B2, 32

(1992).

39. J. I. Cirac, R. Blatt, A. S. Parkins, and P. Zoller, Phys. Rev. A4S, 1434 (1993).

40. S. M. Yoo and J. Javanainen, Phys. Rev. A48, R30 (1993); J. Javanainen and S.

M. Yoo, Phys. Rev. A48, 3776 (1993).

41. J. Eschner, B. Appasamy, and P. E. Toschek, Phys. Rev. Lett. 74, 2435 (1995).

42. W. Neuhauser, M. Hohenstatt, P. E. Toschek, and H. Dehmelt, Phys. Rev. A22,

1137 (1980).

43. J.C. Bergquist, W.M. Itano and D.J. Wineland, Phys. Rev. A2^, 428 (1987).

TN-48
Accurate Measurement of Time

Increasingly accurate clocks—now losing no more than

a second over millions ofyears—are leading to such advances

as refined tests of relativity and improved navigation systems

by Wayne M. Itano and Norman F. Ramsey

Fewpeople complain about the ac- curacy of modern clocks, even If they appear to run more quickly

than the harried among us would like. The common and inexpensive quartzcrystal watches lose or gain about a sec- ond a week—making them more than

sufficient for everyday living. Even a spring-woimd watch can get us to the church on time. More rigorous applications, such as communications with in- terplanetary spacecraft or the tracking of ships and airplanes from sateUites, rely on atomic clocks, which lose no more

than a second over one million years.

WAYNE M. ITANO and NORMAN F. RAMSEY have collaborated many times

before writing this article: Itano earned

his Ph.D. at Harvard University under the

direction of Ramsey. Itano, a physicist at the Time and Frequency Division of the National Institute of Standards and Tech- nology in Boulder, Colo., concentrates on the laser trapping and cooling of ions and conducts novel experiments in quan- tum mechanics. He is also an amateur

paleontologist and fossil collector. Ramsey, a professor of physics at Harvard, earned his Ph.D. from Columbia University. He has also received degrees from

the University of Oxford and the University of Cambridge, as well as several honorary degrees. A recipient of numerous

awards and prizes, Ramsey achieved the highest honor in 1989, when he shared

the Nobel Prize in Physics for his work

on the separated oscillatory field method and on the atomic hydrogen maser.

There might not seem to be much

room for the improvement of clocks or even a need for more accurate ones. Yet many applications in science and

technology demand all the precision

that the best clocks can muster, and

sometimes more. For instance, some

pulsars (stars that emit electromagnetic radiation in periodic bursts) may in certain respects be more stable than

current clocks. Such objects may not be accurately tuned. Meticulous tests of relativity and other fimdamental

concepts may need even more accurate

clocks. Such clocks will probably be- come available. New technologies, relying on the trapping and cooling of at- oms and ions, offer every reason to be- Ueve that clocks can be 1,000 times more precise than existing ones. If his- tory is any guide, these future clocks may show that what is thought to be

constant and Immutable may on finer scales be dynamic and changing. The

sundials, water clocks and pendulvmi

clocks of the past, for example, were

sufficiently accurate to divide the day

mto hours, minutes and seconds, but

they could not detect the variations In the earth's rotation and revolution. A

clock's accuracy depends on the

regularity of some kind of pe- L riodic motion. A grandfather

clock relies on the sweeping oscillation of its pendulum. The arm is coupled to a device called an escapement, which

strikes the teeth of a gear m such a way

that the gear moves In only one direction. This gear, usually through a series of additional gears, transfers the motion

to the hands of the clock. Efforts to Im- prove clocks are directed for the most

part toward finding systems m which

the oscillations are highly stable. The three most important gauges of frequency standards are stability, re- producibility and accuracy. StabUity is a measure of how well the frequency

remains constant. It depends on the length of an observed interval. The

change In frequency of a given stan- dard might be a mere one part per 100

billion from one second to the next, but it may be larger—say, one part per 10 billion—from one year to the next. Reproducibility refers to the ability of Independent devices of the same de- sign to produce the same value. Accuracy is a measure of the degree to which the clock replicates a defined In- terval of time, such as one second.

Until the early 20th century, the most

accurate clocks were based on the reg- ularity of pendulum motions. Gahleo

had noted this property of the pen- dulum after he observed how the period of oscUlation was approximately In- dependent of the amplitude. In other

words, a pendulmn completes one cycle In about the same amount of time, no matter how big each sweep is. Pen- dulimi clocks became possible only

after the mid- 1600s, when the Dutch

scientist Christiaan Huygens Invented an escapement to keep the pendulum

swinging. Later chronometers used the

oscillations of balance wheels attached

56 Scientific American 7u/y J 993

TN-49
TRAPPED MERCURY IONS, separated by about 10 microns, flu- oresce under illumination by ultraviolet light (photograph).

The ions are held by oscillating electric fields generated by

electrodes (cutaway diagram). Static electric potentials (not

shown) prevent the ions from escaping through the ends of the trap. Strings of trapped ions may lead to new timing de- vices more stable than conventional atomic clocks.

to springs. These devices had the ad- vantage of being portable.

Considerable ingenuity went Into improving the precision of pendulum

and balance-wheel clocks. Clockmakers would compensate for temperature

changes by combining materials with

different rates of thermal expansion.

A more radical approach came in the 1920s, when William H. Shortt, a Brit- ish engineer, devised a clock in which a

"slave pendulum" was synchronized to a "free pendulum." The free pendulum

oscillates in a low-pressure environment

and does not have to operate any clock mechanism. Instead it actuates an electrical switch that helps to keep the

slave pendulum synchronized. As a result, the period of the Shortt clock is extremely stable. These clocks had an

error of a few seconds In a year (about one part per 10 million) and became

the reference used in laboratories. The next major advance in timekeeping was based on the development

of quartz-crystal electronic oscillators. The frequency of such devices depends on the period of the elastic vibration of

a carefully cut quartz crystal. The vi- brations are electronically maintained

through a property of such crystals called piezoelectricity. A mechanical

strain on the crystal produces a low

electric voltage; inversely, a voltage in- duces a small strain. The quartz vibrates at a frequency

that depends on the shape and dimensions of the crystal. In some wrist- watches, it is cut into the shape of a

tioning fork a few millimeters long. In other timepieces, it is a flat wafer. The

quartz is connected to an electric cir- aiit that produces an alternating current. The electrical feedback from the quartz causes the frequency of the cir- cuit to match the frequency at which

the crystal naturally vibrates (usually

32,768 hertz). The alternating current from the circuit goes to a frequency di- vider, a digital electronic device that generates one output pulse for a fixed number of input pulses. The divider

also actuates either a mechanical or

digital electronic display. In the late 1920s Joseph W. Horton

and Warren A. Marrison, then at Bell Laboratories, made the first clock based on a quartz-crystcil oscillator. In the 1940s quartz-crystal clocks replaced

Shortt pendulimi clocks as primary lab- oratory standards. These clocks were

stable to about 0.1 millisecond per day

(about one part per biUion). Relatively

inexpensive, quartz clocks continue to be extensively used. The timekeeping el- ements of common quartz watches and

clocks are simplified and miniaturized

versions of quartz frequency standards. Quartz wristwatches became common

once the abihty emerged to cut the quartz into thin, tuning-fork shapes reli- ably and to manufacture miniature, low- power digital electronic components.

Yet quartz-crystal clocks prove inad- equate for many scientific applications,

such as tests of relativity. According

to Albert Einstein's calculations, gravity

distorts both space and time. The difference in gravitational potential causes

time to pass more quickly high in the at- mosphere than it does on the surface. The difference is slight. Time runs about

30 nullionths of a second per year faster

at the top of Movmt Everest than it does

at sea level. Only atomic frequency stan- dards achieve the requisite precision.

The quantized energy levels in at- oms and molecules provide the

physical basis for atomic frequen- cy standards. The laws of quantum mechanics dictate that the energies of a boimd system, such as an atom, have

certain discrete values. An electromagnetic field can boost an atom from one

energy level to a higher one. The pro- cess can also work in reverse. If the atom is in a high energy level, it can drop to a lower level by emitting elec- tromagnetic energy. The maximtmi amoimt of energy is absorbed or emitted at a definite fre- quency—the resonance frequency, or

the difference between the two energy

levels divided by Planck's constant. This value is sometimes called the Bohr

frequency. Such frequencies make ideal time standards because they are ex- tremely stable. Time can be kept by ob- serving the frequencies at which elec- tromagnetic energy is emitted or ab- sorbed by the atoms. In essence, the atom serves as the master pendulimi

whose oscillations are counted to mark

the passage of time. Although we have described generad

quantum properties, the effects exploitScEENTiFic American July 1993 57

TN-50
ed in atomic docks are slightly more

complicated. In most atomic clocks the energy that atoms absorb or release ac- tually results from transitions between

so-called hyperfine energy levels. These

levels exist because of an intrinsic property of particles known as the magnetic

moment. Electrons and the nuclei of

most atoms spin about their axes as if they were tops. In addition, they are magnetized, like compass needles ori- ented along their axes of rotatioiL These

axes can have different orientations with respect to one another, and the

energies of the orientations may differ.

These positions correspond to the hyperfine levels. The nomenclature comes

about because the levels were first ob- served in spectroscopy as small split- tings of spectral lines. On paper, standcirds based on atomic processes are ideal. In practice, perfection is elusive. Atoms do not absorb

or emit energy precisely at the reso- nance frequency. Some energy is spread

over a small interval surrounding the frequency—a smearing of frequencies, so to speak. All else being equal, the precision to which the resonance fre- quency can be measured is inversely proportional to this smearing. The

greater the spread, the less precise the measurement. The spread is often ex- pressed in terms of the quality factor, or Q, which is equal to the resonance

frequency divided by the frequency

spread. In many cases, the higher the resonance frequency, the higher the Q.

Furthermore, smearing is often inversely proportional to the time the atom is in the apparatus. In those situations, the Q of the resonance, and hence the

precision of the measurement, increas- es as the measuring time increases. The motions of the atoms also intro- duce uncertainty by causing apparent

shifts in the resonance frequencies.

Such changes appear because of the Doppler effect. The phenomenon can be divided into first- and second-order

shifts if the atoms are moving much

slower than the speed of light. The

first-order Doppler shift is an apparent

change in the frequency of the appUed

electromagnetic wave as seen by a moving atom. The amount of the shift is proportional to the velocity of the atom.

If the atom moves in the same direc- tion as the wave does, the shift is to a lower frequency. If the atom's motion

is opposed to that of the wave, the

shift is to a higher frequency. If the di- rections are perpendicular, the first-or- der shift is zero. The second-order Doppler shift comes about as a consequence of time

dilation. According to relativity, time

slows down for objects in motion; a moving atom "sees" a slightly different frequency than does a stationary counterpart. The effect on the resonance fre- quency is usually much smellier than

the first-order shift. The second-order

shift is proportional to the square of the atomic velocity and does not depend

on the relative directions of the atomMASTER PENDULUM of this 1920s Shortt clock oscillates in cin evacuated enclosure. It actuates an electriccd switch to synchronize a slave pendulum, which

drives the clock mechanism.

ic motion and the electromagnetic wave.

Several other factors affect the quali- ty of the information. Atoms in the sys- tem may coUide with one another; the impacts add noise to the signal. The

svtrroimding environment can perturb

the resonance frequencies. Defects in the

electronic equipment, stray electromagnetic fields and the ever present thermal

radiation all introduce errors. Therefore, a good atomic frequency standard not

only must establish a steady, periodic

signal but also must minimize these

potential errors. Oneof the earliest and now widely used methods to sidestep

many of these difficulties is called atomic beam resonance, pioneered by 1. 1. Rabi and his colleagues

at Columbia University in the 1930s. The atoms emerge from a small chamber, exit through a narrow aperture and

then travel as a beam. The entire in- strument can be shielded from stray

magnetic eind electric fields and insu- lated from external sources of heat. Perhaps more important, coUisions of

atoms are virtually eliminated, because

the entire device is housed in a long,

evacuated chamber. The pressure in the chcimber is so low that the atoms

are vmlikely to strike anything before

reaching the other end.

In simplified form, atomic beam reso- nance involves three steps. The first is to select only those atoms in the appropriate energy level. This selection is ac- complished by using a specially shaped

magnetic field, which acts as a kind of

filter. It allows atoms in one energy level to pass and blocks aU others by bending the beam. Only atoms in the correct energy level are bent the correct amount

to reach and pass through the aperture

that serves as the entrance to the cavity.

The second and crucial step is to send the selected atoms into another energy level. The task is accompUshed

by passing the atoms through an oscil- lating microwave field inside a cavity.

The atoms will go to another energy

level only if the frequency of the applied oscillating microwaves matches

their Bohr frequency.

The third step is to detect those at- oms that have chcinged energy levels. At this point, the beam of atoms passes through another magnetic field fil- ter, which allows only atoms in the correct energy level to strike a detector that records the atoms as current flow. An

abimdance of such atoms will exist if the frequency of the applied oscillating

microwaves precisely matches their natural frequency. If the frequency of

the applied microwave field is off the mark, fewer atoms change their energy

58 Scientific American July J 993

TN-51
levels, and so fewer will strike the de

tector. One knows, therefore, that the applied microwaves match the natural frequency of the atoms if the niunber

of atoms striking the detector is maximal. An electronic feedback mechanism, called a servo loop, keeps this value constant. If it finds that the cur- rent from the detector is falling off, it changes the frequency of the applied

field imtU the current reaches a maximum again.

By keeping the current from the detector at a maximvun, the servo loop

maintains the frequency of the applied

microwave field at the natural frequen- cy of the atoms. To measure time, one

couples the applied field to a frequen- cy divider, which generates timing pulses. By analogy, the atoms represent the quartz crystcil in a watch or the master pendulum in a Shortt clock. The applied microwave field is the oscillating

circuit or the slave pendulum, which

actucdly drives the clock mechanism.

Minor variations of the atomic beam

standard exist. For example, in some de

vices the atoms that undergo a change

in energy level are made to miss, rath- er than strike, the detector. Not much

difference in accuracy exists, however.

Rather all the versions to some extent represent trade-offs in terms of size, cost and complexity.

A more important modification of

the atomic beam came in 1949, when

one of us (Ramsey) invented the so- called separated oscillatory field method. Instead of Irradiating the atoms

with a single applied field, this tech- nique relies on two fields, separated by some distance along the beam path. Applying the oscillating field in two steps

has many benefits, including a narrowing of the resonance and the elimination of the first-order Doppler shift. Jerrold

R. Zacharias of the Massachusetts Insti- tute of Technology and Louis Essen and

John V. L. Parry of the Natioucd Physical Laboratory in Teddington, England,

adapted this method to working fre- quency standards in the mid-1950s.

Currently the separated oscillatory

field method provides the most repro- ducible clocks. The best ones are locat- ed at a few national laboratories, al- though smaller and less accurate ver- sions are commercially available. The

docks rely on cesixim, which has several advantages over other elements. It has a

relatively high resonance frequency—

about 9,192 megahertz—and low reso- nance width, which lead to an excellent Q. Cesium can also be detected readily

and efiidentiy; all that is needed is a hot

meted filament. When a cesivmi atom

strikes the filament, it ionizes and be- comes observable as electric current.

Resonance Frequency

Atomic frequency standards deL pend on the quantization of

the internal energies of atoms or molecules. A pair of such energy

levels, shown here as levels f, and

£2, is associated with an atomic resonance. The resonance fre- quency f, at which it absorbs or emits electromagnetic radiation, is f= (fj- f|)//7, where h is Planck's constant. The radiation, however,

is not precisely f but instead is spread over a range near f, called

A^. The precision to which fcan

be measured is proportional to the quality factor, Q, defined by

Q = f/Af. The higher the Q, the more stable the clock.

>-o

UJ

The Qs of these standards are about

100 miUion, exceeding the Qof quartz

wristwatches by a factor of several thou- sand. The greatest reproducibilities are about a part per 10^''. The best cesium

frequency standards cire so much more

reprodudble than the rate of rotation

and revolution of the earth that in 1967 the second was defined as 9,192,-

631,770 periods of the resonance fre- quency of the cesium 133 atom. One of the most promising improvements in cesivmi atomicbeam standards is the use of

optical piunping to select the atomic

states. Beginning in the 1950s opticalpumping techniques were devdoped by

Frands Bitter of M.I.T., Alfred Kastler and Jean Brossd of the Ecole Normale

Superieure ctnd others. In this method,

light, rather than a magnetic Add, se

lects atoms in the desired states. Before

the atoms are subjected to the micro- wave fidd, radiation from a laser is used

to drive (or pump) the atoms from one

energy levd into another. In fact, one

can control the number of atoms in en- ergy levds by tuning the frequency of

the light. After the atoms have been irradiat- ed by the microwave fidd, they pass through a second light beam. Only at- oms occupying the correct energy levd

absorb this light, which they quickly reemit. A Ught-sensitive detector records

the reemissions and converts them

into a measurable current. As in atomic

beam resonance that relies on magnetic sdection, one knows that the applied

microwave fidd matches the natural frequency of the atoms if the current from the detector is at a maximum.

Using light instead of magnets has

many advantages. Perhaps the most cru- dal is that, with the right optical-pumping techniques, all the atoms in the beam can be put into the desired energy

levd. Magnetic sdection merdy filters out those that are in the other energy

levels. Hence, the signal strength from

optical pumping is much higher than it is from magnetic sdection. Researchers

at various laboratories are devdoping

optically pimiped cesivun atomic-beam

docks. One such dock, at the National

Institute of Standards and Technology

(mist) in Boulder, Colo., has recentiy be- come the primary frequency standard

for the U.S. Designated nist-7, it has an

expected error of one second in about one million years, making it many times more stable than its predecessor.

There is an optically pumped atomic dock that is available commerdally. Such a dock is based on the 6,835-

megahertz, hyperfine resonance of ru- bidium 87. Rather than moving through

the apparatus as a beam, the rubidium

atoms are contained in a glass cell. The

ceU also houses a mixture of gases that prevents the rubidium atoms from col- liding with the ceU walls. A discharge

lamp containing rubidiiun vapor, rath- er than a laser, irradiates the atoms. A

photovoltaic sensor on the opposite

side of the cell detects changes in the amount of Ught absorbed by the at- oms. The atoms are prepared, the mi- crowaves applied and the Ught detected

in one cell. As a result, rubidium docks

can be made to fit in a cube about 10

centimeters on a side. In contrast, cesi- um beam docks can extend from about

50 centimeters to more than five meters. Rubidium docks are also much

less expensive than are cesiimi ones. The drawback is that the rubidium

devices are generally less accurate and

less reprodudble. The Q of rubidium

standards is about 10 million, a factor of 10 less than the cesivmi beam's quality factor; their reproducibility is only

about a part per 10'". Shifts in the reso- nance frequency mostly account for the poor reprodudbiUty. The frequent

collisions of the rubidium atoms with

other gas molecules cause the shifts. But the rubidium standards' short-term

stabilities are good—in fact, better than

those of some cesium atomic beams.

The atomic docks described thus far work in a rather roundabout way—by

SoDENTinc American July 1993 59

TN-52
detecting a change in some signal, such

as the number of atoms striking a detector, as the frequency of the applied oscillatory field shifts. One way to make

use of the radiation emitted by the at- oms more directly relies on the principle of the maser (an acronym for mi- crowave amplification by stimulated

emission of radiation). In 1953 Charles H. Townes and his associates at Co- lumbia invented the first maser, which was based on ammonia. Beguining in 1960, Ramsey, Daniel Kleppner, now at

M.I.T., H. Mark Goldenberg, then at Harvard University, cind Robert F. C. Vessot, now at the Harvard-Smithsoni- an Center for Astrophysics, developed

the atomic hydrogen maser, the only

type that has been used extensively as an atomic clock. In this instrument, a radio frequen- cy discharge first splits hydrogen molecules held in a high-pressure bottle

into their constituent atoms. The at- oms emerge from a small opening in the bottle, forming a beam. Those in the higher energy level are focused by magnetic fields and enter a specially coated

storage bulb surrovmded by a tuned, resonant cavity. In the bulb, some of these atoms will drop to a lower energy level, releasing

photons of microwave frequency. The

photons will stimulate other atoms to

fall to a lower energy level, which in turn releases additional microwave photons. In this manner, a self-sustaining

microwave field builds up in the bulbthus the name "maser." The tuned cavity arovmd the bulb helps to redirect photons back into the system to maintain the stimulated emission process. The maser oscillation persists as long

as the hydrogen is fed into the system.

A loop of wire in the cavity can detect the osdllation. The microwave field inDETECTOR

MAGNET B MAGNET A

FREQUENCY

DIVIDER

DIGITAL COUNTER

AND DISPLAY

ATOMIC-BEAM frequency standards provide the most accurate, long-term timekeeping. Conventional atomic clocks rely on magnets {a). Atoms in the correct energy level are deflected by magnet A through the microwave cavity. Microwave fields oscil- lating at the resonance frequency of the atoms drive some of them into a second en- ergy level. These atoms are deflected by magnet B so as to strike a detector. The

servo mechanism monitors the detector and maintains the frequency of the applied

microwaves at the resonance frequency. To keep time, some of the microwaves are

duces a current in the wire, which leads out of the cavity to a series of circuits. The circuits convert the induced cur- rent to a lower frequency signal suit- able for generating timing pulses.

The resonance frequency in the hydrogen maser is about 1,420 megahertz, which is much lower than the resonance frequency of cesium. But be- cause the hydrogen atoms reside in the bulb much longer than cesium atoms do

in a beam, the maser's resonance width

is much narrower. Consequently, the Q

of a hydrogen maser standard is about

10^, exceeding the Q of the cesium

atomic clock by an order of magnitude.

In addition, a hydrogen maser has the

TUNED CAVITY MICROWAVE

PHOTONS

MAGNET

SOURCE ^H

Ol

DIGITAL COUNTER

AND DISPLAY

highest stabiUty of any frequency stan- dard, better than one part per 10 '^ Unfortunately, the maser's superior

attributes last just for a few days. Be- yond that, its performance falls below

that of cesium beams. The stabihty de- creases because of changes in the cavity's resonant frequency. CoUisions be- tween the atoms and the bulb shift the frequency by about one part per 10".

One way to overcome the problem is to operate the hydrogen maser at low

temperatures. This condition allows more atoms to be stored (thus resulting in a stronger signal) and reduces elec- tronic noise. Coating the inside of the bulb with superfluid liquid heUimi also enhances performance. This substance

acts as a good surface against which the hydrogen atoms can boimce. More effec- tive magnets, better coating substances

and servo loop techniques that keep the

cavity resonance centered on the atomic resonance cire other approaches now

being taken to improve maser stability.

FREQUENCY

DIVIDER

Altl A

STORAGE BULB ATOMIC HYDROGEN MASER relies on a self-sustaining microwave field to serve as a frequency standard. Hydrogen atoms in the correct energy level are deflected by a magnet into a storage bulb. Some atoms will drop to a lower level, releasing a mi- crowave photon. The photon stimulates other atoms to drop to a lower level, which

produces more photons. The process quickly builds up a microwave field in the bulb. The field induces an alternating current in a wire placed in the cavity. The tuned cavity helps to redirect the photons back into the bulb to maintain the process.

Ithough the cesium atomic-beam

frequency standard is the most

accurate, long-term standard we have, several breakthroughs have

indicated that it is possible to fabricate even more precise clocks. One of the most promising depends on the reso- nance frequency of trapped, electrically

charged ions. Trapped ions can be sus- pended in a vacuum so that they are almost perfectly isolated from disturb- ing influences. The ions themselves

stay well separated from one another

62 SciENTinc American July 1993

TN-53
'A MICROWAVE

CAVITY

SOURCE

LASER A I

MICROWAVES

SERVO

MECHANISM

DIGITAL COUNTER

AND DISPLAY FREQUENCY

DIVIDER

directed to a device that divides the frequency into usable timing pulses. Optically

pumped standards ([?) use light rather than magnets to select atoms. Laser A pimips

the atoms into the right energy level, preparing them to be excited by the micro- waves. Only atoms placed in the correct energy level by the microwaves absorb light

from laser B. They quickly reemit that energy, which is sensed by a photodetector.

An optically pumjied clock using cesiimi atoms at the National bistitute of Standards

and Technology, called Nisr-7, now keeps time for the U.S. (photograph).

because they have the same electric charge. Hence, they do not suffer col- lisions with other particles or with

the walls of the chamber. Ions can be

trapped for long periods, sometimes

for days.

Two different types of traps are used. In a Penning trap, a combination

of static, nonuniform electric fields and

a static, uniform magnetic field holds

the ions. In a radio frequency trap (of- ten called a Paul trap), an oscillating,

nonuniform electric field does the job.

Each type of trap has its own characteristic shortcoming. The strong magnetic fields of Penning traps can alter the resonance frequency. The electric field in Paul traps Ccin create heating effects that cause Doppler shifts. The kind of

trap chosen depends on its suitability for a particvilar experimental setup.

Workers at Hewlett-Packard, the Jet Propulsion Laboratory in Pasadena,

CaM., and elsewhere have fabricated ex- perimental standard devices using Paul

traps. The particles trapped were mercury 199 ions. This ion was selected because it has the highest hyperfine

frequency—40.5 gigahertz—of aU the atoms that are appropriate for the trapping technique. A few million such

ions are caught between the electric

fields generated by electrodes. Then

the ions are optically pumped by ultra- violet radiation from a lamp. Subsequent operation resembles that of the

optically pumped standards, but the maximum Qs of trapped-ion standards

exceed 10'^ This value is 10,000 times

greater than that for current cesiiun

beam clocks. Their short-term stabili- ties are also extremely good, cilthough

they do not yet reach those of hydrogen masers. The second-order Doppler

shift limits the reproducibility to about one part per lO'l

The Doppler shifts can be greatly re- duced by laser cooling. In 1975 David J. Wineland, now at mist, Hans G. Dehmelt of the University of Washington,

Theodor W. Hansch, now at the University of Munich, and Arthur L. Schawlow

of Stanford University first proposed

such a technique. In essence, a beam

of laser light is used to reduce the velocities of the ions. Particles directed

against the laser beam absorb some of

the laser photon's momentum. As a result, the particles slow down. To compensate for the Doppler shifting as the

particle moves against the laser, one

tunes the beam to a frequency slightly

lower than that produced by a strongly

allowed resonance transition. Many laboratories are developing fre- quency standards based on laser-cooled ions in traps. A standard based on be- ryllium 9 ions, laser-cooled in a Penning

trap, has been constructed. Its reproducibility is about one part per 10'^

Umited as it is by coUisions of the ions

with neutral molecules. Improvements

in the quality of the vacuiun should

significantly increase the reproducibility because the imcertainty of the sec- ond-order Doppler shift is only about

five parts per 10^^

During flie past few years, there have

been spectacular developments in trap- ping and coohng neutral atoms, which

had been more difficult to achieve than

trapping ions. Particularly effective la- ser cooling results from the use of three

pairs of oppositely directed laser-cool- ing beams along three mutually perpendicular paths. A moving atom is then

slowed down in whatever direction it moves. This effect gives rise to the designation "optical molasses." Several inves- tigators have contributed to this break- through, including William D. Phillips of NiST in Gaithersburg, Md., Claude

Cohen-Tannoudji and Jean DaUbard of

the Ecole Normale Superieure and Ste- ven Chu of Stanford [see "Laser Trapping of Neutral Particles," by Steven Oiu;

SciENTinc American, February 1992].

Neutral-atom traps can store higher

densities of atoms than can ion traps, because ions, being electrically charged,

are kept apart by their mutual repulsion. Other things being equal, a larger

nvunber of atoms results in a higher

signal-to-noise ratio. The main hurdle in using neutral atoms as frequency standards is that the resonances of atoms in a trap are strongly affected by the laser fields. A

device called the atomic fotmtain sur- mounts the difficulty. The traps capture

and cool a sample of atoms that are then given a Uft upward so that they move into a region free of laser light. The atoms then faU back down under

the influence of gravity. On the way up

SciENTinc American July 1993 63

TN-54
and again on the way down, the atoms

pass through an oscillatory field. In this way, resonance transitions are induced,

just as they are in the separated oscil- latory field beam apparatus.

The Q of such a device can be higher

than that of an atomic beaim because

the time between the two passages can be longer. Experiments on atomic fountains have been done by Chu and his co-workers at Stanford and by Andre

Clairon of the Primary Laboratory of Time and Frequency in Paris and Chris- tophe Salomon of the Ecole Normale

Superieure and their co-workers. Much current research is directed

toward laser-cooled ions in traps that resonate in the optical realm, where fre- quencies are many thousands of giga- hertz. Such standards provide a promising basis for accurate clocks because

of their high Q. Investigators at nist

have observed a Q of 10" in the ultra- violet resonance of a single laser-cooled,

trapped ion. This value is the highest Q

that has ever been seen in an optical or microwave atomic resonance. Because

LASER

LASER %

MICROWAVE

CAVITY

of technical difficulties, none of the optlccd frequency clocks constructed so

far, however, has operated over extend- ed periods.

The variety of high-performance

frequency standards that exist today might seem to obviate the need for future devices of even greater

performance. After £iU, current atomic

clocks are so accvirate that they have

redefined some of our basic units. As

mentioned ecirUer, the second is now

based on the resonance frequency of

the cesium atom. Also by intemationeil agreement, the meter is defined as the

distance light travels in 1/299,792,458

of a second. The voltage imit is maintained by the characteristic frequency

associated with a voltage that appears

in a so-called Josephson jimction in a superconducting circuit. There are, however, applications that tax the capacity of modern clocks. Radio astronomy is a good example. As- tronomers often use several telescopes

spaced thousands of kilometers apart

MICROWAVES

LASER

LASER

ATOMIC FOUNTAIN uses atoms that have been cooled and trapped by sbc laser beams (J). The vertical beams then briefly impart an upward velocity

to the atoms. The atoms rise, passing through the microwave cavity on the way up (2) and again on the way down (3). The rest of the process resembles optically pumped atomic-bccun standards: the atoms pass through an- other laser beam (4), and their fluorescence is recorded by a photodetector

(5). Servo loops and frequency dividers {not shown) generate timing pulses.

64 Scientific American July 1993

TN-55
to study a stellar object, a technique

that dramatically increases the resolu- tion [see "Radio Astronomy by VeryLong-Baseline Interferometry," by Anthony C. S. Redhead; SciENTinc American, June 1982]. Two radio telescopes

spaced 10,000 kilometers apart have an

effective angular resolution more than one million times better than either tele- scope alone. But to combine the data

from each telescope appropriately, in- vestigators need to know precisely

when each telescope received the signal.

Present-day hydrogen masers have the

stability required for such observations. More stable clocks may be needed for space-borne radio telescopes.

Highly stable clocks are essential for the best tests of relativity. Timing measurements of millisecond pulseirs, some

of which are as stable as the best atomic

docks, offer evidence for gravity waves.

In 1978 Joseph H. Taylor, Jr., and his as- sociates at Princeton University found

that the period of a binary-pulsar sys- tem has been slowly varying by just the amount that would be expected for the

loss of energy by gravitational radiation, as predicted by general relativity. Greater precision can be achieved

if measurements are taken over many

years, so clocks with better long-term

stability would be useful.

In other tests of relativity, Vessot and his colleagues confirmed the pre- dicted increase in clock rates at high al- titudes. They sent on board a rocket a hydrogen maser and measured the

small, relativistic clock shift to within an accuracy of 0.007 percent at an alti- tude of 10,000 kilometers. Highly sta- ble clocks have also been used by Irwin

I. Shapiro, now at the Harvard-Smithsonian Center for Astrophysics, to ob- serve the relativistic delay of a Ught sig- nal passing by the sun. Ultraprecise timekeeping has more

practical applications as well—most

notably, for navigation. The location of Voyager 2 as it sped by Neptune was

determined by its distance from each

of three widely separated radar tele- scopes. Each of these distances in turn was obtained from accurate measure- ments of the eight hours it took for

light to travel from each telescope to the spacecraft and return. Navigation is, of course, also important on the earth. One of the latest applications of precise clocks is the satel- lite-based assemblage called the Global

Positioning System, or GPS. This system

relies on atomic clocks on board orbit- ing satellites. The GPS enables anyone

with a suitable radio receiver and computer to determine his or her position to approximately 10 meters and the cor- rect time to better than 10'' second.

Coordinating Time Scales

In the article, we discuss the measurement of an interval of time, such as a second or a minute. This process requires only a good clock. But to be

able to state that an event happened at a particular time, say, 22 seconds af- ter 12:31 P.M. on July 5, 1993, requires synchronization with a clock that is, by mutual agreement, the standard. The world's "standard clock" exists on

paper as an average of the best clocks in the world. The International Bureau

of Weights and Measures in Sevres, France, is responsible for coordinating

international time. This coordinated time scale is called International Atomic

Time, orTAI. Many users require a time scale that keeps pace with the rotation of the

earth. That is, averaged over a year, the sun should be at its zenith in Greenwich, England, at noon. The day as determined by the apparent position of

the sun is irregular but on the average longer than the 24 hours as defined

by TAI. To compensate, another time scale, called Coordinated Universal Time, or UTC, is specified by occasionally adding or subtracting a whole

number of seconds from TAI. These seconds, or leap seconds, are inserted or deleted, usually on December 31 or June 30, to keep UTC within 0.9 sec- ond of the time as defined by the rotation of the earth. The record of leap

seconds must be consulted to determine the exact interval between two

stated times.

Two observers monitoring the same

sateUite can synchronize their docks to within a few nanoseconds.

It is expected that the GPS will have

widespread practical applications, such

as pinpointing the positions of ships,

airplanes and even private automobiles.

The GPS was used during the 1991 Per- sian Gulf War to enable troops to de- termine their positions on the desert. Commercial receivers can be purchased

for less than $1,000, although these

civilian versions are limited to an accuracy of about 100 meters because of deliberate scrambling of the signals trans- mitted from the satellites. A fuU complement of 24 satellites would give 24- hour, worldwide coverage. The system

is nearly complete.

These and other apphcations show

the importance of time and frequency standards. The anticipated improvements in standards will increase the ef- fectiveness of the current uses and open

the way for new functions. Only time

will teU what these uses will be.

FURTHER READING

From Sundials to Atomic Clocks: Un- derstanding Time and Frequency. J. Jespersen and J. Fitz-Randolph. Dover,

1982. History of Atomic Clocks. N. F. Ramsey in Journal of Research of the National Bureau of Standards, Vol. 88, No. 5,

pages 301-320; September/October 1983. Precise Measurement of Time. N. F. Ramsey in American Scientist, Vol. 76, No. I, pages 42-49; January/February 1988. Time and Frequency. Special issue of Pro- ceedings of the IEEE. Vol. 79, No. 7; July 1991.

Scientific American July 1993 65

TN-56
Volume 75, Number 22 PHYSICAL REVIEW LETTERS 27 November 199>

Resolved-Sideband Raman Cooling of a Bound Atom to the 3D Zero-Point Energy

C. Monroe. D. M. Meekhof, B. E. King, S. R. Jefferts. W. M. Itano, and D. J. Wineland

Time and Frequency Division. National Institute of Standards and Technology. Boulder. Colorado 80303

P. Gould

Department of Phvsics, University of Connecticut. Storrs. Connecticut 06269

(Received 19 December 1994)

We report laser cooling of a single ^Be* ion held in a rf (Paul) ion trap to where it occupies the quantum-mechanical ground state of motion. With the use of resolved-sideband stimulated Raman

cooling, the zero point of motion is achieved 98% of the time in ID and 92*7^ of the time in 3D. Cooling to the zero-point energy appears to be a crucial prerequisite for future experiments such as the

realization of simple quantum logic gates applicable to quantum computation.

PACS numbers: 32.80.Pj, 42.50.Vk. 42.65.Dr

Dramatic progress in the field of atomic laser cooling

has provided cooling of free or weakly bound atoms

to temperatures near the Doppler cooling limit [1], near

the photon recoil limit [2], and, more recently, below

the photon recoil limit [3,4]. For a tightly bound atom,

a more natural energy scale is given by the quantized

vibrational level n, where the energy is £ = ha)y{n + 5) for an atom confined in a harmonic potential of frequency

a>„. In this case, the fundamental cooling limit is the « = zero-point energy of the binding potential. In this Letter, we demonstrate a new technique for laser cooling a trapped atom to the 3D zero-point energy. Attainment of the 3D ground state is significant for two

primary reasons: (i) it appears to be a goal of intrinsic interest as it is the fundamental limit of cooling for a bound atom and approaches the ideal of an isolated

particle at rest and (ii) it will be important in future planned experiments. For example, once the ion is cooled

to the « = state, it should be possible to realize the Jaynes-Cummings interaction [5] in the regime of strong

coupling and generate other nonclassical states of motion

such as squeezed states [6-8]. If the collective motion

of two or more trapped ions can be cooled to the zero

point, it may be possible to transfer correlation from the

external motional state to the internal spin state of the

ions. Generating "EPR"-like atomic spin states would not only be interesting from the point of view of quantum

measurements [9], but may also allow a reduction of

quantum noise in spectroscopy [6,7]. Zero-point cooling

combined with long coherence times may make it possible

to construct a quantum computer [10]. Cirac and Zoller have proposed a quantum computer based on a system of

trapped ions, in which information is stored in the spin

and motional states of the ions [11]. The fundamental

switching action in this implementation of a quantum

computer is a coherent exchange between the spin state of

an individual ion and a collective vibrational state of all the ions. Cooling to the zero-point energy and realizing the Jaynes-Cummings coupling is critical to this scheme.

We cool a single beryllium ion bound in a rf (Paul) ion

trap to near the zero-point energy using resolved-sideband

laser cooling with stimulated Raman transitions along

the lines suggested in Ref. [6]. The idea of resolved- sideband laser cooling with a single-photon transition is as follows [12]. Consider a two-level atom characterized by

resonant transition frequency &>o and radiative linewidth

y. We assume the atom is confined by a ID harmonic

well of vibration frequency to^ » y. If a laser beam

(frequency ojl) is incident along the direction of the atomic motion, the absorption spectrum is composed

of a "carrier" at frequency &>o and resolved frequencymodulation sidebands spaced by Wy, which are generated

from the Doppler effect. Cooling occurs if the laser is tuned to a lower sideband, for example, at uiL = wq —

(Oy. In this case, photons of energy ^(&>o ~ (^v) are absorbed and spontaneously emitted photons of average energy flcDQ return the atom to its initial internal state thereby reducing the atom's kinetic energy by ficoy per

scattering evenir( assuming Hcoy is much greater than the photon recoil energy). Cooling proceeds until the atom's mean vibrational quantum number in the harmonic well is given by (n)n,in — (y/2<w„)^ «: 1 [1,13]. The interaction with the laser is significantly reduced once in the « =

state; thus the zero-point state satisfies the operational

definition of a dark state [14] for y/a>y sufficiently small. Resolved-sideband cooling in 2D was previously achieved on a '^^Hg"*" ion using a narrow single-photon optical

quadrupole transition [15]. For laser cooling with stimulated Raman transitions [4,6,16,17], the single-photon transition is replaced by

a stimulated Raman transition between metastable levels

(e.g., hyperfine or Zeeman electronic ground stated, and

spontaneous Raman transitions irreversibly recycle the in- ternal state of the atom. Stimulated Raman cooling offers the important practical advantages that the cooling

linewidth can be varied experimentally, and the effective

laser linewidth can be made very narrow by the use of

optical frequency modulators. These features of stimu0031-9007/95/75(22)/4011(4)$06.00 © 1995 The American Physical Society 4011

TN-57
Volume 75, Number 22 PHYSICAL REVIEW LETTERS 27 November 1995

lated Raman cooling have already been used to achieve very low temperatures for free or weakly bound neutral atoms in the unresolved-sideband limit [4]. Since narrow

single-photon transitions are not required, the resolved- sideband Raman cooling technique described here can be

generalized to many ion species and may also be applied

to strongly bound neutral atoms held in dipole traps [18]

and optical lattices [19]. The experiment is conducted as follows. We first achieve (ny) — 1 in 3D (u = x,y,z) by performing

Doppler cooling on an allowed electric dipole transition (y large) and making the trap strong enough that (Oy — 7- This Doppler "precooling" places the ion into the LambDicke regime (Ajc <c A/Itt, where Ax is the rms spread

of the ion position and A is the wavelength of the dipole

transition). We reduce (riy) further in 3D by employing

a second stage of cooling on narrower Raman transitions between hyperfine ground states (yRam "^ <Wu)- Finally we extract (riy) by measuring the asymmetry of the resolved motional sidebands in the Raman absorption

spectrum [15]. A single ^Be"^ ion is stored in a coaxial-resonator-based

rf (Paul) ion trap (ro = 170 /mm, zo — 130 /im) described

in Ref. [20]. A potential Vq cosCHoO is applied to the ring

(Vo - 600 V, Clo/ln - 231 MHz), yielding ^Be^ pseudopotential oscillation frequencies of (oi^, Qjy,oj.)/27T —

(11.2, 18.2, 29.8) MHz along the principal axes of the trap

[21]. Once a ^Be"*" ion is loaded in the trap, its lifetime is about 6 h (background pressure <10"^ Pa). The geometry and polarizations of the various laser beams as well as the relevant energy levels in ^Be"*" are summarized in Fig. 1. The quantization axis is defined

by an applied magnetic field |B| ==0.18 mT. Laser

radiation (beam D2, a'^ polarized) detuned slightly to the red of the ~S\/2{F = 2) —» ^Pa/T transitions (A =

313 nm, y/ln — 19.4 MHz, ~P3/2 hyperfine structure — 1 MHz) is directed at oblique angles to all the principal axes of the trap, providing Doppler precooling in all trap dimensions. A second 313 nm source (beam Dl, cr'^ polarized) tuned to the ^S\/2iF = 1) —* ^Ps/i transition prevents optical pumping to the F = 1 ground state. A pair of Raman beams is also directed into the trap (beams Rl and R2) to drive a much narrower transition between the 'S\/2 \F = 2) and \F = 1) hyperfine ground

states of ''Be^ through the virtual ^P\/2 state. The

Raman beams are detuned —12 GHz to the red of

the "5 1/2 — ~P\/2 transition with a difference frequency

very near the ~5i/2 hyperfine splitting of coq/Itt ~

1.250 GHz. A third 313 nm source (beam D3, cr'^ polarized) is tuned to the "5i/2(F = 2) — 'P\/2{F = 2) transition and depletes the \F.mf) = |2. 1) ground state. Beams Dl. D2, and D3 are derived from two frequencydoubled dye lasers, producing 5-10 /xW of power in each beam, enough to saturate the ion near resonance. Beams R\ and R2 are derived from a third frequencydoubled dye laser, providing a few mW of power in each

(a) (b)

FIG. 1. (a) Laser beam geometry. The trap ring electrode

(in the x-y plane) is shown rotated 45° into the page

(the endcap electrodes along the z axis are not shown; see Ref. [20]). A magnetic field B defines a quantization axis along x/V2 + y/2 -I- z/2, and laser beam polarizations are indicated. Fluorescence light is collected along the direction perpendicular to the page, (b) Relevant ^Be^ energy levels (not to scale), indicated by F, rriF, quantum numbers in the ^Si/2 ground state OP fine-structure splitUng is =197 GHz,

^S]/2 hyperfine splitting is (oq/Itt — 1.250 GHz, and the ^Pj/z

hyperfine and Zeeman structure is not resolved). All optical transitions are near A = 313 nm. Dl and D2: Doppler

precooling and detection beams; D3: |2, 1) deplefion beam; R]

and R2: Raman beams. The detuning of R 1 and R2 from the '5,/2 'Pi/2 U-ansition is A/lv — 12 GHz.

beam. The difference frequency of the Raman beams

is tunable over the range 1200-1300 MHz with the use of a double-pass acousto-optic modulator (AOM). The

counterpropagating Raman beams are also at oblique

angles to the trap's principal axes and are therefore

sensitive to motion in all dimensions. All beams are shuttered with AOMs. The 313 nm fluorescence from the trapped ion is imaged through f/2 optics onto a positionsensitive photomultiplier tube, resulting in a photon count

rate as high as —10 kHz (quantum efficiency —2 X

lO"'*). The background count rate is —100 Hz. Doppler precooling, Raman cooling, and measurement

of the Raman absorption spectrum are accomplished

by following the sequence outlined in Table I. After precooling (beams Dl, D2, and D3), the ion is prepared

in the ^Si/2|2, 2) electronic ground state by turning off beam D2. The relative tuning of the Raman beams is set to a first lower (red) sideband near a>o " w„, driving a

stimulated Raman transition from the |2, 2)|n„) state to the |l,l)|«i, - 1) state. The ion is then recycled with

nearly resonant beams Dl and D3, inducing spontaneous

Raman transitions predominantly to the |2, 2)|«y — 1)

state. This cycling of stimulated and spontaneous Raman

transitions (steps 3 and 4 of Table I) is repeated as desired on any or all of the three dimensions. The relative tuning of the Raman beams is then set near a»o + 5pr, and a stimulated Raman "probe" transition is driven

between the states, where An^j = 0, ±1. The probability of driving the probe transition is then measured by

4012

TN-58
Volume 75, Number 22 PHYSICAL REVIEW LETTERS 27 November 1995

driving a stimulated Raman "exchange tt pulse" from

|2,2)|«„) <- |l,l)|/jy), followed by driving the cycling

^5i/2|2,2) -* 2/'3/2|3,3) transition with beam D2 and

gating and collecting the fluorescence. (The exchange tt pulse in step 6 of the table reduces the fluorescence noise

when {riy) — 0.) The ion scatters thousands of photons on the cycling transition before decaying into the 1 1 , 1

)

electronic state due to imperfect circular polarization of

beam D2, resulting in a net quantum efficiency near 1. As this sequence is repeated at —A kHz, 5pr is slowly

swept, yielding the Raman absorption spectrum.

The Raman carrier represents the transition |2, 2) |/iy) ^

|l,l)|«i,) and occurs at 5pr = (compensating for sta- ble Zeeman and ac Stark shifts of —2 MHz). In the Lamb-Dicke regime, the carrier feature has strength Ic =

sin^(nTpr), where 11 = g\g2/^ is the carrier Rabi flopping frequency, Xpr is the exposure time of the atom to the probe Raman beams, g \ and g2 are the resonant Rabi

frequencies of Raman beams Rl and ^2, and A is the detuning of the Raman beams from the excited state (we

assume A » y,g],g2) [6]. In each dimension, blue and

red sidebands occurring at 5pr = ±a»„ represent the transitions |2,2) \ny) •^ |1, l)\ny ± 1). The strengths of the blue and red sidebands in the Lamb-Dicke regime are given by /„^'"« = {sin^[aTprVvin, + D'^^]) and Il^<^ =

(sin^[nTprr7y«y^]), where the average is performed over

the distribution of «„. The Lamb-Dicke parameters are given by rj^ = Sk^r^ = (0.21, 0.12, 0.09), where Sk^ =

Iky is the component of the difference in the counterpropagating Raman beam wave vectors in the uth dimension,

and r„ = (^/2/na»„)'/^ is the spread of the n^ = Q wave

function. The sideband Rabi flopping frequencies Clriv are typically a few hundred kilohertz, thus the absorption

features are well resolved and spontaneous emission during all stimulated Raman processes is negligible.

A Raman absorption spectrum of the first blue and red sidebandsof the X direction (5pr = ±w) is shown in Fig. 2

(solid points) for Doppler precooling only (omitting steps 3 and 4 in Table I). Similar features appear at 5pr = ±coy

and ±a>;. If n^ is thermally distributed, then it is straightforward to show that /;"V/^^'"^ = («i.)/(l + {riv)), inde- pendent of fl Tpr 7)v . By recording several spectra varying

Tpr, we find the ratio of red to blue sideband strength re- mains approximately constant, indicating a nearly thermal

distribution of «i,. (We ensure that fl is the same on the blue and red sidebands.) We measure {riy) for a variety

of red detunings of Doppler precooling beams D 1 and D2

and obtain values as low as {riy) — (0.47, 0.30, 0.18) in the

three dimensions at a detuning of about -30 MHz (we es- timate — 10% uncertainties in the measurements). At other

detunings, we measure values as high as (n^) — 7. These

values and their behavior with detuning are consistent with

the theoretical limit of Doppler cooling [17]. Figure 2 includes a Raman absorption spectrum following additional sideband Raman cooling (hollow points).

Five Raman cooling cycles in the x dimensions are em-

-11 11 Raman detuning (MHz)

FIG. 2. Raman absoqjtion spectrum of a single '^Be" ion

after Doppler precooling (solid points) and after five cycles of additional resolved sideband stimulated Raman cooling on

the X dimension (hollow points). The observed count rate

is normalized to the probability P{|2, 2)} of the ion being

in the |2, 2) state. The first blue and red sidebands of the X dimension are shown at S^r/lir — ±11.2 MHz. Similar sidebands are found near ±18.2 MHz (y dimension) and at ±29.8 MHz (z dimension). For precooling, the asymmetry in the sidebands indicates a thermal average vibrational occupation

number of (/?,) = 0.47(5). For Raman cooling, the reduction of the red sideband and growth of the blue sideband implies

further Raman cooling in the .v dimension to («,) = 0.014(10). The widths of the features are consistent with the 2.5 yiis Raman probe time. Each point represents an average of 400

measurements, corresponding to =5 min of integration time for the entire data set. The lines connect the data points.

ployed (steps 3 and 4 of Table I with co^ = (o^). The

exposure time XRam of the last Raman pulse is set so

that nTRam''7jr — 7r/2 or TRam — 2.5 /iS, corresponding

to nearly a tt pulse from the rix = 1 state to the n, =

state. The durations of the earlier pulses are set shorter

to optimize the cooling. The recycle time between each

stimulated Raman transition is about 7 ^ts, ample time

for the ion to scatter the expected average of —3 photons

from beams Dl and D3 and ultimately get recycled to the |2, 2) state. The suppression of the red sideband

(and growth of the blue sideband) indicates the extent of the additional cooling, from {n^) — 0.47 to («f) =

0.014(10) (the n,, = state is occupied =98% of the

time). We observe no further cooling by increasing

the number of Raman cooling cycles beyond about five and see little sensitivity to the details of the Raman

cooling pulse durations. As discussed above, we verify

that the distribution oi n^ is nearly thermal after Raman

cooling. We achieve resolved-sideband Raman cooling

in 3D by sequentially driving on all three red sidebands.

Five cycles of cooling are applied to each dimension

(order xyzxyz . . .) by alternating the tuning of the Raman

cooling beams between the three red sidebands. From

the measured asymmetry of each pair of sidebands, we

infer 3D Raman cooling to («„) = (0.033, 0.022, 0.029), or the Hx = riy = n~ — 3D ground state being occupied

=^92% of the time. The limit of Raman cooling is determined by heating

from off-resonant stimulated-Raman transitions [expected

to result in a limit of (n;c)stRam =" {g\g2)^/iiOv^)^ =

4013

TN-59
Volume 75, Number 22 PHYSICAL REVIEW LETTERS 27 November 1995

TABLE I. Timing sequence for Doppler precooling, resolved-sideband stimulated Raman cooling and Raman detection of («„). As the sequence is repeated through steps 1 repeated as desired within the sequence.

-7, 5pr is slowly swept across absorption features. Raman cooling steps (3 and 4) are

Duration Beams Raman

Step i/MS) D1,D3 D2 Rl, R2 tunnting Function

1 =50 On On Off

2 =7 On Off Off

3 1-3 Off Off On a»o ~ w„

4 -7 On Off Off

5 1-3 Off Off On COo + 5pr 6 = 1 Off Off On Wo

7 =200 Off On Off

Doppler precool

Prepare in |2,2) state Stimulated Raman transition |2, 2)|«y) —' |1, l)|ni, — 1) Spontaneous Raman recycle |1,1)|«„ — 1) — |2,2)|m„ - 1) Probe (rt„) with stimulated Raman transitions |2, 2)|«„) -* |1, \)\n'^}

Exchange tt pulse: |2, 2)|ni,) —» |1, 1)|«„) Detect transition in step 5: cycle on |2, 2) — |3, 3); collect fluorescence

10~^] and heating from off-resonant spontaneous emission from the Raman beams [a limit of (rtjc)spRam —

ig\g2/^^)Vxy'''Ram — 10~^]. We believe the minimum

measured value of («^) = 0.02 may be due to anomalous

heating of the ion, which we measure to be d{n)/dt —

+ l/msec by inserting various amounts of time between

cooling probing. We are currently investigating the source of this heating.

This work is supported by the Office of Naval Research

and the Army Research Office. We acknowledge considerable contributions from J. C. Bergquist. We thank N. R. Newbury, J. D. Miller, and M. Young for comments on

the manuscript.

[1] D.J. Wineland and W. M. Itano, Phys. Today 40. No. 6, 34(1987).

[2] C. Cohen-Tannoudji and W. D. Phillips, Phys. Today 43, No. 10,33 (1990).

[3] A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and

C. Cohen-Tannoudji, Phys. Rev. Lett. 6L 826 (1988); J. Lawall, F. Bardou, B. Saubamea, M. Leduc, A. Aspect, and C. Cohen-Tannoudji, Phys. Rev. Lett. 73, 1915

(1994).

[4] M. Kasevich and S. Chu, Phys. Rev. Lett. 69, 1741 (1992); H. Lee, C. Adams, N. Davidson, B. Young, M. Weitz, M.

Kasevich, and S. Chu, in Atomic Physics XIV. edited by

D.J. Wineland, C.E. Wieman, and S.J. Smith (AIP Press, New York, 1995), p. 258.

[5] E.T. Jaynes and C.W. Cummings, Proc. IEEE 5L 89

(1963).

[6] D.J. Heinzen and D.J. Wineland, Phys. Rev. A 42. 2977

(1990).

[7] D.J. Wineland. J.J. Bollinger. W. M. Itano, and D.J. Heinzen, Phys. Rev. A 50. 67 (1994).

[8] J. 1. Cirac. A. S. Parkins. R. Blatt. and P. Zoller, Phys. Rev.

Lett. 70. 556 (1993); J.l. Cirac, R. Blatt, A.S. Parkins, and P. Zoller, Phys. Rev. Lett. 70, 762 (1993).

[9] D. M. Greenberger, M. A. Home, and A. Zeilinger, Phys.

Today 46, No. 8. 22(1993).

[10] A. Ekert. in Atomic Physics XIV, edited by D.J. Wineland, C.E. Wieman, and S.J. Smith (AIP Press, New York.

1995), p. 450, and references therein. [11] J.I. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091

(1995).

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

[13] W. Neuhauser, M. Hohenstatt, P. Toschek, and H. Dehmelt. Phys. Rev. Lett. 41, 233 (1978).

[14] R. Dum. P. Marte, T. Pellizari, and P. Zoller, Phys. Rev.

Lett. 73, 2829(1994).

[15] F. Diedrich, J.C. Bergquist, W.M. Itano, and D.J. Wineland. Phys. Rev. Lett. 62, 403 (1989).

[16] P.E. Toschek, Phys. (Paris) 10, 761 (1985); H. Dehmelt,

G. Janik, and W. Nagoumey, Bull. Am. Phys. Soc. 30, 111 (1985); B. Appasamy, I. Siemers, Y. Stalgies, J. Eschner,

R. Blatt, W. Neuhauser, and P. E. Toschek, Appl. Phys. B

60,473 (1995).

[17] M. Lindberg and J. Javanainen, J. Opt. Soc. Am. B 3, 1008 (1986).

[18] S. Chu, J.E. Bjorkholm, A. Ashkin, and A. Cable, Phys. Rev. Lett. 57, 314 (1986); W.D. Phillips, in Laser

Manipalation of Atoms and Ions, Proceedings of the International School of Physics "Enrico Fermi," Varenna,

1991, edited by E. Arimondo and W.D. Phillips (NorthHolland, Amsterdam, 1992), p. 325; J.D. Miller, R.A.

Cline, and D.J. Heinzen, Phys. Rev. A 47, R4567 (1993).

[19] P. Verkerk, B. Lounis, C. Salomon, and C. CohenTannoudji, Phys. Rev. Lett. 68, 3861 (1992); P. Jessen, C. Gerz, P. Lett, W. Phillips, S. Rolston, R. Spreeuw, and C. Westbrook. Phys. Rev. Lett. 69. 49 (1992); A. Hemmerich

and T. Hansch, Phys. Rev. Lett. 70, 410 (1993).

[20] S.R. Jefferts. C. Monroe, E. Bell, and D.J. Wineland,

Phys. Rev. A 51, 3112 (1995).

[21] The drive frequency Ho = 231 MHz is much larger than the pseudopotential vibration frequencies a»t,, so the contribution of the driven "micromotion" at Hq to the overall amplitude of the ion motion is very small. This

includes the micromotion induced §5i>m static background

fields, which are minimized with compensation electrodes

(see Ref. [20]). Moreover the small FM micromotion

sidebands at ±flo are well resolved (D.q » y) and far from the region of interest in the absorption spectrum.

Thus the trap can be accurately approximated as a simple

harmonic oscillator with frequencies a»i,.

4014

TN-60
PHYSICAL REVIEW A VOLUME 47, NUMBER 5 MAY 1993

Quantum projection noise: Population fluctuations in two-level systems

W. M. Itano, J. C. Bergquist, J. J. Bollinger, J. M. Gilligan, D. J. Heinzen,* F. L. Moore,*

M. G. Raizen,* and D. J. Wineland

Time and Frequency Division, National Institute of Standards and Technology, Boulder, Colorado 8030S

(Received 9 December 1992)

Measurements of internal energy states of atomic ions confined in traps can be used to illustrate fundamental properties of quantum systems, because long relaocation times and observation times

are available. In the experiments described here, a single ion or a few identical ions were prepared in well-defined superpositions of two internal energy eigenstates. The populations of the energy levels were then measured. For an individual ion, the outcome of the measurement is uncertain, unless the amplitude for one of the two eigenstates is zero, and is completely uncertain when the magnitudes

of the two ampUtudes are equal. In one experiment, a single ^^^Hg"*" ion, confined in a linear rf trap, was prepared in various superpositions of two hyperfine states. In another experiment, groups

of ^Be"^ ions, ranging in size from about 5 to about 400 ions, were confined in a Penning trap and

prepared in various superposition states. The measured population fluctuations were greater when

the state amplitudes were equal than when one of the amplitudes was nearly zero, in agreement

with the predictions of quantum mechanics. These fluctuations, which we call quantum projection

noise, are the fundamental source of noise for population measurements with a fixed number of atoms. These fluctuations are of practical importance, since they contribute to the errors of atomic

fi"equency standards. PACS number(s): 03.65.Bz, 32.80.Pj, 32.30.Bv

I. INTRODUCTION

Quantum mechanics is not a deterministic theory, even though the time development of the quantum-state vec- tor is governed by the Schrodinger equation, which is deterministic. That is, quantum mechanics does not, in general, predict the result of an experiment. Rather,

it provides a prescription for predicting the probability

of observing a given result. The relationship of the

quantum-state vector to the physical system that it describes is central to the interpretation of quantum mechanics. There are at least two distinct interpretations

of the quantum state [1]. According to the Copenhagen interpretation, the state vector provides a complete description of an individual

system (a single atom, for example). This is not the only

definition of the Copenhagen interpretation, but it is the one that we adopt here. According to this interpretation,

the state vector of a system develops in time according

to the Schrodinger equation until a measurement causes

it to be projected into an eigenstate of the dynamical

variable that is being measured. The assumption that

the state vector "collapses" in this manner is considered

unattractive by some, because of its ad hoc nature. According to another interpretation, sometimes called the statistical-ensemble interpretation, the state vector is merely a mathematical construct which describes an en- semble of similarly prepared systems [2-4] . One common

misconception is that this interpretation is not capable of

describing an experiment on a single atom. In this case, the state vector describes a conceptual ensemble (a Gibbs

ensemble) of similarly prepared atoms. The single atom

in the experiment is a member of that ensemble. Experimentally, an ensemble is generated by repeatedly preparing the state of the atom and then making a measurement. The state vector, in this interpretation, is analogous to a statistical distribution function of the kind that

appears in classical statistical mechanics. The difference

is that, in quantum mechanics, there is no underlying

microscopic theory which can predict the behavior of a

single system, even in principle. The statistical-ensemble

interpretation has the virtue of avoiding the necessity of "reducing" or "collapsing" the state vector. In spite of occasional claims to the contrary [5] , it ap- pears that the Copenhagen and statistical-ensemble in- terpretations do not differ in their predictions of exper- iments when properly applied [6]. In practice, either in- terpretation may motivate a particular calculation. For example, some problems in quantum optics have been

solved by simulating the behavior of the wave function

of a single atom, explicitly including the reduction of the wave function at random times [7-10]. Such methods follow naturally from the Copenhagen interpretation

(which is not to say that the practitioners of these methods would necessarily advocate the Copenhagen inter- pretation as opposed to the statistical-ensemble interpretation). On the other hand, the conventional method

of solving the density-matrix equations follows naturally

from the statistical-ensemble interpretation. The re- sults of averaging many wave-function simulations are the same as those of solving the density-matrix equations. The experiments described here can be interpreted

within either framework.

Perhaps the simplest example of the indeterminism of

quantum mechanics is the behavior of a two-level sys- tem prepared in a superposition ji/^) = c^|.<4) -\- cb\B) of the two states \A) and \B) and subjected to a measure- Work of the U. S. Government

47 3554 Not subject to U. S. copyright

TN-61
47 QUANTUM PROJECTION NOISE: POPULATION . . 3555

ment. The measurement yields one indication or "pointer

reading" for a system in |^) and another for a system

in \B). Except when either ca or Cb is zero, the out- come of the meeisurement cannot be predicted with cer- tainty. Provided that the state vector is properly nor- malized (|cAp + |cb|^ = 1), |caP = Pa, and |cbP = Pb

are the probabilities of finding the system in \A) or \B).

The indeterminacy is present no matter how accurately

the state has been prepared. It is an inherent feature of

quantum mechanics. We will call this source of measure- ment fluctuations "quantum projection noise," since it can be interpreted as arising from the random projection

of the state vector into one of the states compatible with

the measurement process. In some experiments we have a sample of A'' identical

systems that are effectively independent. If we carry out

the same kind of state preparation and measurement as that just described for a single system, then we should

get the same result as by repeating the experiment N

times. That is, the sum over all N atoms of the measured

quantity should have the same mean and fluctuations as the sum of A'' independent measurements on one system.

The internal states of a set of A'' ions in an ion trap constitute a system of this type. The ions are well separated

from each other by their mutual Coulomb repulsion, so, to a very good approximation, the state of one ion has no effect on that of another ion. If all of the ions are subjected to the same optical and radiofrequency fields, they can all be described by the same state vector for their internal degrees of freedom. In an ion trap, unlike an atomic beam, for example, we can repeatedly prepare

and observe the same set of N atoms. To illustrate the main ideas, we will first describe a

simplified version of the experiment. The actual experiments, carried out with single ^^^Hg"^ ions in a linear rf trap and with numbers of ^Be"*" ions ranging from about

5 to 400 in a Penning trap, will be explained in detail in later sections. Consider a single atom, or several identical atoms, with

three energy levels \A), \B), and \C) (see Fig. 1). The

FIG. 1. A simplified energy-level diagram of an atom of the type used in the experiments. The atom or atoms are prepared in a superposition of \A) and \B) by a combination

of laser optical pumping and rf excitation. The state is measured by applying laser radiation resonant with the |B) —> \C)

transition. If the atom is in \A), it does not absorb any photons from the laser beam and remains in \A). If the atom

is in \B), the laser radiation drives it to \C). The atom then

decays back to \B) and emits a photon. It can then be excited

to \C) again.

lower two levels \A) and \B) are stable. They are sepa- rated by a radiofrequency photon energy. Level \C) decays, by emission of an optical photon, to \B) only. The

state vector is prepared in a superposition of \A) and

\B) hy first preparing it in \A) and then applying an rf pulse near the \A) —> \B) transition frequency. Any desired superposition of \A) and \B) can be obtained by

controlling the frequency, amplitude, and duration of the

rf pulse. The number of atoms in \B) is then measured.

The measurement is made by applying a laser beam that

is resonant with the transition from \B) to \C) and de- tecting the photons emitted in the decay from \C) to \B).

If an atom is found in \B) when the laser pulse is applied,

it is excited to \C). It quickly decays to |5) and emits a photon. It can then be excited bcick to \C) by the laser and emits another photon. Thus, an atom in \B) emits a

series of many photons, while an atom in \A) emits none. The rate at which photons are emitted by the entire sample of atoms is proportional to the number of atoms in \B) when the laser beam is applied. If there is only one atom in the sample volume, the detection is particularly

simple: If some photons are emitted, the atom was in \B); otherwise it was in \A). Quantum-amplification de- tection of this type, sometimes called electron shelving, was first proposed by Dehmelt [11] and, to the best of our knowledge, was first used by Wineland et al. [12]. In the preceding discussion, we assumed that each

atom either emits a burst of photons or does not. According to the Copenhagen interpretation, the wave function

of each atom is a superposition of \A) and \B) before

the measurement and collapses to one state or the other

when the detection laser beam is applied. According to the statistical-ensemble interpretation, the atom is found

to be in \A) or \B) when the measurement is made, with

probabilities pa and pb, respectively. There is no point

in asking which state the atom was in just before the

measurement, since this is not a question that can be

answered by the experiment.

Quantum projection noise in the measured populations

of states prepared in superpositions may be of some practical interest. This point was discussed previously, in the context of atomic frequency standards, by Wineland et

al. in Appendix A of Ref. [13]. In some atomic frequency

standards, such as cesium atomic beams, a signal which

is proportional to the population of a particular quantum

state is measured as a function of the frequency of an applied rf field. The signal is a maximum (or a minimum,

depending on the detection method used) when the fre- quency matches the transition frequency ujq between two

atomic states and decreases when the frequency increases or decreases from this value (see Fig. 2). The frequency

to of an oscillator is matched to the resonance frequency

ujo by measuring the signal at two frequencies to — Auj

and u! + Au! and making a correction to u by an amount

proportional to the difference between the two signals. In the example shown in Fig. 2, the signal at a; -I- Au) is less than the signal at oj — Acj, which indicates that u

must be decreased in order to match uq. If the noise is independent of the position on the resonance line shape,

measurements at the points of maximum slope would give

the greatest sensitivity for determining uiq. However, the

TN-62
3556 W. M. ITANO et al. 47

O

c

0)

/^ ' ^

0) \

Ci)—Acj i/

\/ I—

i cj+Ao)

V/

frequency

FIG. 2. A graph of the mean detected signed as a function

of the frequency of the applied field for a resonance like that used in an atomic clock. The signal is proportional to the population of a particular energy level after the field has been

applied. The maximum signal occurs at firequency cjo- If the

fine shape is sjrmmetric, uq, can be determined by measuring

the signaJ at two different frequencies u — Aui and w + Aw and

Vcirying u until the signal intensities aie equal. The frequency

u> determined in this way is an estimate of ujq.

noise will in general vary with the signal level, due to quantum projection noise and also to other causes. As- sume we have the extreme case where the only noise is quantum projection noise, where the maximum and the

minima of the line shape correspond to pure energy eigenstates \A) and \B), and where the half-intensity points

correspond to equal superpositions of |.i4) and \B). In that case, it is not so obvious what the optimum value

of Aa; should be. The noise goes to zero at the maxima

and minima of the Une shape, where the sensitivity to frequency deviations also goes to zero. The noise is the

greatest at the half-intensity points, which, for a typical line shape, is where the frequency sensitivity is ap- proximately the greatest. It will be shown later that the

precision with which the line center can be determined

is independent of Ao;, if the line shape has a cosinusoidal form, often a good approximation when Ramsey's

separated-oscillatory-field resonance method [14] is used.

II. THEORY

A. Single atom

Suppose a single system, such as a single atom, is pre- pared in a well-defined superposition of two stable or metastable states \A) and \B), which have energies Hlua and hujB- This could be done by preparing the atom in

I A), by optical pumping, for example, and then applying

a neaurly resonant rf field of well-controlled fi-equency, amplitude, and duration. Suppose that the system is in \A) at time t = 0. We then apply an oscillatory perturbation

V which has matrix elements:

{A\V\A) = {B\V\B)=0,

{A\V\B) = {B\V\A)* = Kbe'^K

The state at a later time f > is [14]

(2.1)

(2.2)

m))=CA{t)\A)+CB{t)\B) sm n

.26 V2

nt -^^sml-|exp

Sit

+ cos I -— exp

--(a; -uja -(jJB)t

-{u -uja -u>B)t

\B),

\A)

(2.3)

(2.4)

where

Uq = ub — l-^A, n= V(u;o-w)2 + (26)2.

(2.5)

(2.6)

In deriving Eq. (2.4), we assumed that V does not in- duce transitions to states other than \A) and \B) and that

spontaneous decay can be neglected. Also, all other per- turbations, such as those due to collisions, are assumed

to be negligible. For the levels studied experimentally

in this work, which are hyperfine Zeeman sublevels of

ground electronic states of atoms, spontaneous decay can

be neglected. Such states decay primarily by magnetic

dipole radiation. The rate for magnetic dipole decay from

a higher sublevel |e) to a lower level \g) is

7(e -^ g) =

4u;3eg Shc^ \{e\M\' (2.7)

where hueg is the energy difference between je) and \g) and n is the magnetic dipole operator. For typical val- ues of the parameters for transitions between hyperfineZeeman sublevels, the mean lifetime for spontaneous de- cay is many years. For example, ifugg = 27r x 30 GHz and

|(e|/i|5)| = fJ-B, where hb is the Bohr magneton, then the

decay rate given by Eq. (2.7) is 2.7 x 10"^^ s~^; that is, the lifetime is 1200 years. In order to prepare a state with a given value of pb =

|cB(Op! W6 can adjust b, u, and the time t during which

V is applied. For example, if a; = ujq, then

Pb = sin^

f ir ) = sin^(6t). (2.8)

Any value of ps from to 1 can be obtained by adjusting

the value of the product bt. Alternatively, it may be more

convenient to vary w, keeping 6 and t fixed at values so

TN-63
47 QUANTUM PROJECTION NOISE: POPULATION 3557

that bt — TT. (This is what is done in order to observe

the resonance hne shape, for example.) This induces a complete inversion (ps = 1) at exact resonance (cj =

loq). Any value of pb down to can be obtained by

varying co between ujq and wq — ^iry/S/t or between uq

and ujQ + 27r\/3/t.

Ramsey's method of separated oscillatory fields [14]

is another way of creating a given value of pb- In

this method, the perturbation is applied in two phasecoherent pulses of duration r with a delay T between

them. If the strength of the perturbation b is adjusted

so that br = 7r/2, then ps = \cb{T + 2r)p can be made

to take any value between and 1 by varying u. If, in addition, t <^T and \u> — u)q\t <C 1, the probability ps

to be in \B) at time T + 2t is

PB^\{l+cos[{u-uo)T]}. (2.9)

An expression valid for arbitrary values of the parameters

has been given by Ramsey [14]. It is useful to define a vector operator r on the subspace

spanned by j^l) and \B). The components of r are defined

as

r, = \{\A){B\^-\B){A\\

r2^\{\A){B\-\B){A\\

r^^\{\B){B\-\A){A\).

(2.10)

The operator r is equivalent to a spin-^ angular mo- mentum operator, since it operates on a two-dimensional

complex vector space and since the commutators satisfy

the same algebra:

\r^,rA =ieiikrk, (2.11)

where €ijk is the Levi-Civita symbol. For a general pure

state of the form of Eq. (2.3), the expectation values of these operators are

(ri) = 1{c*aCb + c*bCa) = I sine cos0,

(r2) = ^{c*aCb - c*bCa) = ^

sini9 sin0,

(r3) = i(|cB|'-|c^|^) = i

(2.12) cos 9,

where we have used the notation (ri) = {ip\ri\ijj), etc. The expectation values can be represented geometrically

by a three-dimensional vector (r ) of length | . The spherical polar angles (^,0) define the orientation of (r). As

pointed out by Feynman, Vernon, and Hellwarth [15], this representation is sometimes useful because of the

way in which the evolution of the quantum state can be

visualized as a rotation of a vector. The quantities {2(ri)

,

2(r2), 2(r3)} are equivalent to the quantities {ri, r2, r^}

defined by Feynman, Vernon, and Hellwarth. The reason

for using the definition of r given by Eq. (2.10), rather

than one differing by a factor of 2, is that r then corre- sponds precisely to a spin-| angular momentum operator. The vector representation can be generalized to deal

with mixed states (statistical ensembles of pure states),

but that will not be necessary here. The third component

of r is proportional to the internal energy operator. The

eigenstates of r^ with eigenvalues m = —^ and m = +^

correspond to \A) and \B), respectively.

The variance of the measurement of the state {\A) or

\B)) of a single atom is particularly simple to calculate. We define a projection operator Pb = \B){B\. The ex- pectation value of Pb is |cbP = Pb, the probability of a measurement finding the atom in \B). The variance of

the measurement is

(APb)^ = {{Pb - (Pb)?) = {pI-2{Pb)Pb + {Pb?) (2.13) = (Pb) - (Pb)' = {Pb) - {Pb}' (2.14) = {Pb){1-{Pb))=Pb{1-Pb). (2.15)

In Eq. (2.14), we have used the fact that

P'b = {\B){B\){\B){B\)^\B){B\=Pb. (2.16)

Equation (2.15) shows that the uncertainty is zero when

Pb is or 1 and has its maximum value when Pb = 2-

B. N atoms

If cooperative effects can be neglected over the time

of the experiments, we may consider the atoms to be

independent. For the systems used in the experiments,

the decay rate of an individual atom is extremely small.

It is possible for a suitably prepared collection of N atoms

to exhibit a maximum spontaneous-decay rate per atom

of about A'' times the usual rate given by Eq. (2.7), due

to cooperative effects [16]. Even if the rate is enhanced

by a factor of A'', though, it can still be ignored for the

small values of N that were used in these experiments.

One natural way to calculate the fluctuations of the

measured populations is to consider the A'' atoms to be

independent and to combine the probabilities according

to the binomial distribution [17]. Let N^ and Nb be the

numbers of atoms found to be in \A) and \B), respectively

{Na + Nb = N). Then, according to this model, the

probability of measuring a given value of A^b is

P{Nb,N,pb) =

Nl

NbI{N - Nb)1 {pBf^l-pB)^''-''^\

(2.17)

where pb = |cbP is the probability for a single atom to be in \B) and (1 — Pb) = Pa ^ |ca|^ is the probability

to be in |^). The variance of the binomial distribution is

[17]

a^ =Npb{1-Pb)- (2.18)

The variance is zero when ps = or ps = 1 and has

its maximum value of A''/4 when Pb = ^- For N = 1, Eq. (2.18) agrees with Eq. (2.15). Figure 3 shows plots of probability distributions calculated from Eq. (2.17), for A" = 20 and PS =0,0.1,...,!.

In general, it would be better to use a formalism that

treats the N atoms as a combined quantum system. This

should give a correct description both when cooperative

TN-64
3558 W. M. ITANO et al. 47

FIG. 3. A plot of the probability distributions of Nb, the number of atoms in | B) , for ps = 0,0.1,...,!. The total number of atoms N is 20. As pB increases from 0, the distribution broadens and reaches its maximum variance at pb = 0.5. As

PB increases beyond 0.5 the distribution narrows, eventually

becoming a spike at Nb = 20 for ps = 1.

effects are important and when they are negligible. Each

two-level atom can be described by a spin-^ operator Tj,

defined as in Eq. (2.10), where i = 1,...,N labels the

atom. The eigenstates of ri^ with eigenvalues rrii = —

^ or +1 correspond, respectively, to the \A) and \B) states of the ith atom, which we denote by \Ai) and \Bi). One

basis for the 2^-dimensional Hilbert space of A'' atoms is provided by the direct-product states of the form

N

\mi, 1712,..., tun) = Yll'^i)- (2.19)

1=1 A collective-angular-momentum operator for the N

atoms can be defined by

N

R-E (2.20)

t=i

It is implicitly assumed that each term in the summation

is multiplied by the identity operators for all of the other atoms. Another basis for N atoms is given by states of

the form 1^, R, M), which are linear combinations of the

product states defined by Eq. (2.19) [16]. Here, R{R+l)

2 is the eigenvalue of R , M is the eigenvalue of R3, and g

is an index that may be required to provide a complete

set of labels. The eigenvalue M is related to the variables

previously defined by

M=^{Nb-Na), (2.21)

(2.22)

Thus, fluctuations in M correspond directly to fluctua- tions in Nb, although M and Nb differ by a constant. The state in which all of the atoms are in \A) has

R = ^N and M = -^N and is nondegenerate. This

state is denoted by

N N

\R=^N,M = -^N) = ll\m,^

t=i

(2.23)

t=i

We assume the atoms are far enough apart that they

do not overlap spatially, so that symmetrization or an- tisymmetrization of the total wave function is unnecessary. Suppose that we apply the same perturbation to

all of the atoms, so that the state of the ith atom goes

to Cyi|Ai) + cslBi) for each i. That is, the new state is N

]l{CA\Ai)+CB\Bi)). (2.24)

t=i We can visualize this as a rotation of the collective state vector from the negative Rz axis {9 = it) to new values of

9 and given in terms of c^i and cb by Eq. (2.12). Equivalently, it can be described as a rotation of the coordinate

system by tt — ^ about the original R2 axis, followed by a

rotation of —^ about the new R3 axis. The properties of

states of this kind were discussed by Arecchi et al. [18], who called them atomic coherent states. They also called them Bloch states, because they resemble the spin states studied in nuclear magnetic resonance. The new state given by Eq. (2.24) is a linear combination of the states \R = ^N, M), for M = -^N, -^N + 1,. . . ,^N, since it

is generated from Eq. (2.23) by a rotation. This state is M=N/2

\R^^^N,9,<I>)= Yl D\^%^{0,n-9,-<l>)

M=-N/2

x\R=^N,M), (2.25)

where Dj^ -n/2 ^^ ^ matrix element of the rotation operator as defined in Eq. (4.1.10) of Ref. [19]. If the system

is prepared in the state given by Eq. (2.25), the probability of measuring a given value of M is the absolute

square of the coefficient oi \R= ^N, M), which is

ftl„ch(M, N, 0) = |Di^{.%j(0,T - e, -4>)\^ = [<%.(-«)]'

m

~ {\N + M)\{\N - M)\ cosQ

JV+2M sm N-2M

(2.26)

(2.27)

(2.28)

(2.29)

TN-65
47 QUANTUM PROJECTION NOISE: POPULATION 3559

N\

Nb\{N - NbVNb\{N - Nb)\

m

NbKN - NbVcos^ r 1 f sin^ - 5j

(N-Nb)

[i(l +cos^)]^«[i(l -cos^)](^-^«)

(2.30)

(2.31)

(2.32)

The final result is the same as Eq. (2.17), which was

obtained by another method. Arecchi et al. obtained an equivalent result from the algebraic properties of the

angular momentum operators, without explicitly making

use of the rotation-operator matrix elements [18]. The

quantity rfi5^-%2 *" ^^- (2-27) is defined in Eq. (4.1.12)

of Ref. [19]. Equation (2.28) follows from Eq. (2.27)

by making use of Eqs. (4.2.4)-(4.2.6) of Ref. [19]. The

expUcit form for d^j^ ^^',^{6) given in Eq. (2.29) is firom Eq. (4.1.27) of Ref. [19]. Equation (2.30) results from

using Eq. (2.22) to express M in terms of Nb and N,

and Eq. (2.31) follows by making use of trigonometric

identities. The final expression [Eq. (2.32)] results from

using Eq. (2.12) to express cos^ in terms of ps = |cs|^-

The variance of the measured value of M can be cal- culated by the standard formula

(Ai?3)' = {RD - {Rs)' (2.33)

For a state with the form of Eq. (2.24), in which the CAi 's and cb;'s are the same for all z, {R3) and (R^) can be

easily be evaluated:

A'' (^3) = j{Pb-Pa), (2.34)

and

^\[k'"))

(2.35)

N N

^JUl^^'^^^h) (2.36)

i=l j=l

N

i=l i,j

(2.37)

N N(N-l), ,2 = 4+4 ^^^ ^^^ • (2.38)

In obtaining Eq. (2.38), we used the fact that the first sum in Eq. (2.37) contains A^ terms, all equal to

ic*AM^\+c*B,{B,\)rl{cAMi) + CBABi))

\CA,\

+

|CBil = 4, (2.39)

and the second sum contains N{N — 1) terms, all equal

to

icAAAi\+c*BABi\)icX{A2\+c*B,{B2\)rur2,

x{caMi)+cb^\Bi)){caM2)+cb^\B2))

JCBil \caA'

2 2 = j{PB -PAfcb. \<^A,2 J

(2.40)

Substituting the values of {R3) and (R^) from Eqs.

(2.34) and (2.38) into Eq. (2.33), we obtain

2_N , iV(iV-l),^_ _,2 N\__ _ ,2 N,

(Pb-Pa) --^{pb-PaY

= j[i-{PB-PAf] = j{1-\Pb-{1-Pb)?} = NpBil~PB)- (2.41)

This result agrees with Eq. (2.18), which was based on the

properties of the binomial distribution. Equation (2.41) was derived for the special case of an uncorrelated TV- atom system in which all of the atoms have the same

state vector. For more general states, in which the state vectors of different atoms are correlated with each other,

it may be possible for the variance to be either larger or smaller than this value [20-23].

The error of an atomic frequency standard depends on

the ratio of the noise in the signal to the firequency derivative of the signal. Aside firom an additive constant, the

signal of an idealized frequency standard with N atoms

is proportional to {R3). The quantum projection noise

is proportional to -^/(Aila)^. If the Ramsey method is used, so that pb is given by Eq. (2.9), the ratio of the

quantum projection noise to the frequency derivative is independent of the frequency u at which the measure- ment is made:

djRz) \NT sin[{u - uJo)T],

1 ^(^3) TyfN'

(2.42)

(2.43)

(2.44)

The error is proportional to 1/\/N, which is sometimes

called the shot-noise limit. If noise from other sources

is significant, the ratio is not constant, and it is best to TN-66
3560 W. M. ITANO et al. 47

measure the signal near the points of maximum slope. If the Ramsey method is not used, the exact trade-oflF of

Eq. (2.44) does not hold, but the situation is not greatly

different. A simple graphicad representation of a Bloch state can be used to provide an estimate of (AiJa)^ which

agrees qualitatively with the result of Eq. (2.41). We

can represent the Bloch state \R = ^N,M = —^N) =

\R = ^N,9 = tt) by the set of all vectors of length

FIG. 4. Graphical representations of Bloch states \R = \NM = -iiV) = \R = \N,6 = tt),

(a)

. . . (b) \R = \N,e = f

,</. = 0), and (c) \R = \N,e = 0). The

imcerteiinty in the number of atoms in \B) (or \A)) is rep- resented by the spread in the third {R3) component for the

points on the circle surrounding the base of the cone. The

uncertainty goes to zero for (a) and (c) and has its largest value for (b).

^/R{R^\^ having their third {Rz) component equal to —R. This set forms a cone whose base has a radius

equal to y/R, as shown in Fig. 4(a). The feict that

{^Rzf = for this state is represented by the fact that

all of the points on the circle surrounding the base of the cone have the same third component. The Bloch state \R = \N,e = ^7r,0 = 0) is represented by Fig. 4(b).

The points on the base of the cone have third components which vary over a range of ±\/R, while the rms

deviation is approximately y/R/2. Thus, we obtain the

estimate (Aila)^ w R/2. For a Bloch state with an ar- bitrary value of 6, we cam use the same method to make

the estimate

(Ail3)2 « |sin2 = NpBil -Pb). (2.45)

This agrees with the actual value, which is given by

Eq. (2.41). The Bloch state \R = \N,d = 0) is rep- resented by Fig. 4(c). For this state, (Aila)^ = 0.

III. SINGLE-ATOM EXPERIMENT

Quantum-state preparation and detection experiments were carried out with ^^^Hg"*" ions confined in a Unear rf trap. Detailed observations were made of single ions,

although some experiments were also carried out with

several simultaneously trapped ions.

A. ^®*Hg+ energy levels

Figure 5 shows the energy levels of ^^^Hg"*" which were

important for the experiments. The ground electronic state has the configuration 5d^°6s ^Si/2- The first electric dipole transition, at 194 nm, is to the 5d^°6p^Pi/2 state. The metastable 5d^65^ ^^3/2 and 5c^6s^ ^^^5/2 states (not

shown in Fig. 5) lie below the ^Pi/2 state. The ^^^Hg nucleus has spin ^, so both the ^Si/2 and the ^Pi/2 states are

split by hyperfine interactions into states with total angular momentum F = and F = 1. For both the ^5i/2 and

^Pi/2 states, the F = \ hyperfine state is higher in energy

than the F = hyperfine state. The ^Si/2 hyperfine split-

'^'""^ tl

/

laser 1

,

/

F=l y

/

65^5

1/2 F=0

FIG. 5. Energy levels of ^^^Hg"*". The transition between

the ^Si/2 and ^P\/2 states is at 194 nm. The hyperfine split- tings in the ^Si/2 and ^Pi/2 states are 40.5 and 6.9 GHz, re- spectively. The ^Si/2 (F = 1) state is detected by exciting the ^Si/2 (F = 1) -> ^Pi/2 (F = 0) transition with laser 1. TN-67
47 QUANTUM PROJECTION NOISE: POPULATION . . . 3561

ting has been measured by microwave resonance methods

to be 40.507 347 996 9(3) GHz [24]. The '^Pi/2 hyperfine

splitting has been measured by Fabry-Perot interferometry to be 6.955(90) GHz [25]. The natural linewidth of

the 2Pi/2 state is about 70 MHz [26,27].

B. Apparatus

The ^^^Hg"*" ions were confined in a linear rf trap that

consisted of four parallel cylindrical electrodes of radius

0.794 mm arranged symmetrically around a central axis

[28]. The distance from the central axis to the inner sur- face of each electrode was 0.769 mm. The rf potentials on any two adjacent electrodes were 180° out of phase.

The amplitude of the rf potential difference between two

adjacent electrodes was about 500 V, and its frequency was 12.7 MHz. The electric fields produced by these elec- trodes created a force which pushed the ions to the central axis. The electrodes were divided into sections, to which static electric potentials of 1 V or less were applied

to keep the ions from escaping along the axis. Ions were produced by electron-impact ionization of

neutral atoms inside the trap volume. A sample of ^^^Hg

of isotopic purity 91% was used. Typically, the pressure was about 10~^ Pa (1 Pa « 7.5 mTorr), except when the

ions were being created, when it weis raised to a higher

level. The ions were confined to a region of a few hundred micrometers extent around the center of the trap.

After being laser cooled, an individual ion was localized

in position to about 1 jjira or less. Narrow-band cw radiation at 194 nm was required for laser cooUng and optical detection of the ^^^Hg"*" ions. This was generated by a combination of second-harmonic

generation and sum-frequency mixing, starting with cw

lasers [29]. About 5 ^W of 194-nm radiation were avail- able. In order to laser cool and continuously observe the

ions, 194-nm radiation near both the '^Si/2{F = 1) ^

2Pi/2(F = 0) and the 251/2 (F = 0) -^ ^Pi/2{F = 1) transition wavelengths was required [28]. We call these

two sources laser 1 and laser 2. Otherwise, with only

laser 1 or laser 2 on, the ions were optically pumped to a hyperfine state which could not absorb the 194-nm ra- diation. Also, in order to prevent trapping of the ions in Zeeman sublevels of the ^Si/2{F = 1) state, we had to apply a magnetic field of approximately 5 x 10~^ T at an

angle of approximately 45° with respect to the electric

field vector of the radiation from laser 1 (see Appendix)

The magnitude and orientation of the magnetic field were

controlled by several pairs of Helmholtz coils. The 194-

nm beams were directed through the trap at an angle of

9° with respect to the trap axis, so that both the axial

and radial degrees of freedom were laser cooled. For this geometry, the theoretical Doppler cooling limit was 1.1 mK for the axial motion and 24 mK for the radial motion

[30].Some of the 194-nm radiation emitted by the ions

perpendicular to the trap axis was focused by a multielement lens onto a two-dimensional imaging photoncounting tube. The probability of a photon emitted by an

ion being detected was about 10~^. Individual ions could

be resolved with this apparatus. Some images showing

several clearly resolved ions have been published previously [28]. The electronics could be adjusted so that the

photons from any rectangular region of the image, for ex- ample, a region including only one ion, could be counted

separately. To a good approximation, the ions in the trap

do not interact and are independent.

A stable source of 40.5-GHz radiation was required for exciting the '^Si/^iF = 0) -^ (F = 1) hyperfine transition. The output of a 10.125-GHz frequency synthesizer was amplified and then frequency multiplied by four. The

output of the frequency multiplier was directed at the

ions with a horn antenna. The data from the photon-counting tube were recorded

by a computer. The computer also controlled the light

shutters, a microwave switch, the firequency of the mi- crowave frequency synthesizer, and the magnitude and

orientation of the magnetic field.

C. State preparation and detection

Optical pumping can be used to prepare the ions in either the {F = 1) or the {F = 0) hyperfine level of the ^51/2 ground state. In order to prepare them in the

(F = 1) state, both laser 1 and laser 2 are left on. If an ion in the ground (F = 1) state is excited to the

2Pi/2(F = 0) state, it is forbidden by electric dipole se- lection rules from decaying to the ground {F — 0) state, so it must return to the (F = 1) state. There is a weaJc

transition rate from the ground (F = 1) state to the

ground (F = 0) state, via the "^Pi/^iF — 1) state. This

rate is approximately 3 x 10~^ times the rate of leaving

the ground (F = 1) state and returning to the same state, via the '^P\/2 (F = 0) state, since laser 1 is far from reso- nance with the '^Si/2{F = 1) —> ^Pi/2{F = 1) transition.

If an ion does malce a transition to the ground (F = 0) state, laser 2 quickly drives it back to the '^Pi/2{F — 1) state, which decays, with probability |, to the ground

(F = 1) state. If laser 1 and laser 2 are both blocked at the same time, the ion will be in the ground (F = 1) state with high probability, after a few multiples of the ^Pi/2 state lifetime (2.3 ns). This method does not select out

a particular Mp Zeeman sublevel of the ground (F = 1) state. If desired, a single Mp state could be selected by

switching the polarization of laser 1 (to right or left cir- cular polarization with propagation along the magnetic

field for Mp = ±1 or linear polarization perpendicular to the magnetic field for Mp = 0). However, this was not done in this work.

An ion can be prepared in the ^Si/2{F = 0) hyperfine

level by blocking laser 2 while leaving laser 1 unblocked. For a typical intensity of laser 1, the ion is pumped to the

ground (F — 0) state in about 10 ms. In contrast to the

previous method, the ion is prepared in a single Mp state. Laser 1 can also pump an ion from the ground (F = 0) state to the ground (F = 1) state. However, the rate for (F = 0) ^ (F = 1) is less than that for (F = 1) ^ (F =

0), mainly because laser 1 is farther from resonance for this process. In the steady state, the probability of being

in the (F = 0) state is about 94% (see Appendix). A

TN-68
3562 W. M. ITANO et al. 47

possible way to improve the state selection, for a single

ion, is to observe the 194-nm photons emitted and to block laser 1 when the fluorescence drops to zero. There is a probability of about 10"^ that an ion in the ^Pi/2 state will decay to the ^r>3/2 metastable state rather than to the ^5i/2 state [27]. The ^Dz/2 state has a lifetime of 9.2 ms and decays, with about equal probability, to the ^5i/2 state or to the metastable ^D^/2 state, which has a Ufetime of 86 ms [27,31]. This process is a

potential problem for the state-selection methods, since

the metastable states decay randomly to both ground hyperfine states. The effect was kept small by reducing the

intensity of laser 1 so that the transition rate to the ^1)3/2 state was less than about 1 s~^. After the ion was prepared in the ground {F = 0)

state, any desired superposition with the ground {F =

1,Mf = 0) state could be created by applying rf fields of well-controlled frequency, amplitude, and duration.

These two states were used, because the magnetic shift of

the transition frequency is queidratic, rather than linear, in the field. Thus, the transition frequency was stable

enough to create the superposition states consistently. State detection was carried out by counting the 194-

nm photons emitted by the ions for a period, typically

15 ms, with laser 1 unblocked and laser 2 blocked. The mean number of photons detected was proportional to the number of ions in the (F = 1) state. However, the

number is subject to statistical fluctuations, since the

detection is essentially a Poissonian process. Also, laser intensity fluctuations lead to fluctuations in the overall counting rate. Quantitative studies were made with only a single ion. In this case, the ion was presumed to be in the (F = 1) state if the number of photons detected was greater than

a given threshold value and in the {F = 0) state if it was

not. This threshold was set empirically, and was usually

either or 1. There was some possibility of error with

this detection method. Dark counts from the phototube

or photons scattered from some surface and then detected

could lead to a false (F = 1) signal. Also, since the mean number of photons detected from an (F = 1) ion was small (about 5 in 15 ms), it was possible that not enough of them would be observed, thus leading to a

false (F = 0) signal. If the detection efficiency could be

improved, both of these problems could be reduced by

using a higher threshold number of photons to distinguish

between (F = 1) and (F = 0).

D. Results

The ability to prepare an ion in either the (F = 0) or (F = 1) ground hyperfine state is shown in Fig. 6. Each vertical line denotes the detection of a single photon at a particular time. The detection electronics were

adjusted so that photons firom only a single ion were detected. Before recording the data shown in Fig. 6(a), the

ion was prepared in the ground (F = 1) state by leaving

both laser 1 and laser 2 unblocked for about 0.1 s and

then blocking them both. After a short delay, laser 1 was unblocked, and for 0.1 s, the photons were counted.

0.00 0.02 0.04 0.06 0.08 0.10 Time (s)

0.00 0.02 0.04 0.06 0.08 0.10 Time (s)

FIG. 6. Typical record of the detected photons from a sin- gle ^^^Hg"*" ion prepared in the ground (a) (F = 1) state and

(b) (F = 0) state. The horizontad axis represents the time

after laser 1, the detection laser, is unblocked. Eax;h vertical

line represents the detection of a single photon. In (a), eight

photons were detected (two photons were too close in time to resolve on the graph). In (b), no photons were detected.

Their detection times were recorded with a resolution of

100 /is. In the example shown in Fig. 6(a), eight photons were recorded, but two of them were too nearly simultaneous to be resolved on the graph. Before recording

the data shown in Fig. 6(b), the ion was prepared in the

ground (F = 0) state by leaving laser 1 unblocked and

laser 2 blocked for 0.05 s. Laser 1 was then blocked. After a short delay, laser 1 was unblocked again, and the computer was set to record photons, as for Fig. 6(a). None

was recorded, so, with high probability, the ion was in the (F == 0) state. Figure 7 shows the results of 100 successive single-ion state preparation and detection cycles of the type shown

in Fig. 6. Preparation of the (F = 1) state [Fig. 7(a)] was

alternated with that of the (F = 0) state [Fig. 7(b)], so that slow drifts in parameters such as the laser intensity

would affect both plots in the same way. Each point rep)- resents the mean photon-count rate during a given 0.5-ms

period after laser 1 was unblocked. The data firom the

first 1.5 ms, which was the time required for the light

shutter to open fully, are not shown. The solid curve in Fig. 7(a) is a least-squares fit of the data to a decreasing

exponential plus a constant base line. The fitted time

constant of the exponential is 8.66 ±0.41 ms. The fitted amplitude of the exponential is 769 ± 22 counts/s. The

fitted base line is 47.3 ± 4.3 counts/s. The solid line in Fig. 7(b) corresponds to 47.3 counts/s, the base fine determined firom Fig. 7(a). This value does not differ significantly from the mean of all of the points in Fig. 7(b),

which is 50.05 counts/s. The background count rate due

to all sources other than photons emitted by the ion was

measured by detuning laser 1 far ft-om resonance. This

TN-69
47 QUANTUM PROJECTION NOISE: POPULATION 3563

c

oo

\J\J<J 1 (a)

800 r

-

600

V*

-

400

• \ •

\* *

-

200

•V*

• \

* \ *

•

•

•

• • •

•

•

• •

• •• • •

• • • • • • • • • ••• •• • •• •

-

n

«•• • •(• • • • • •_, 0.00 0.02 0.04 0.06 0.08 0.10 Time (s)

CO \(n

*-*c

13O

o

200

• (b)

~w« mwm—> — via— • «H—•»— —

wa w 0.00 0.02 0.04 0.06 0.08 0.10 Time (s)

FIG. 7. Plot of the mean photon count rate as a function

of the time after unblocking laser 1 for (a) a single ion pre- pared in the {F — 1) state and (b) the same ion prepared

in the (F = 0) state. These plots were generated from 100 measurements like those in Fig. 6. The solid curve in (a) is a

least-squares fit of the data to a decreasing exponential plus a basehne. The solid line in (b) is the base hne determined

from (a). The origin of the time axis is 1.5 ms after the laser

1 shutter is activated. After 1.5 ms this shutter is fully open.

rate was 4 ± 2 counts/s, so the actual base line for light

scattered from the ion was 43 ± 5 counts/s. The ratio of

the base line to the sum of the base line and the amplitude of the exponential is 0.053 ± 0.006. This represents

the steady-state probability that an ion is in the ground

(F = 1) state when laser 1 alone is applied. This compares well with a calculated value of approximately 0.06

(see Appendix).

The Rabi resonance line shape shown in Fig. 8 was

measured by repeatedly preparing the {F = 0) state, applying a microwave pulse at a given frequency, and

determining whether the ion was in (F = 0) or (F = 1). At each frequency, 19 measurements were made, and the

results were averaged [(F = 0) corresponding to and

(F = 1) to 1]. For this run, the sequence of operations

for each measurement was as follows.

(1) Both laser 1 and laser 2 were left on for 200 ms to

laser cool the ion. The magnetic field was set to about

5 X 10~^ T to avoid optical pumping.

(2) The fluorescence from the ion was measured for 5 ms. This step was repeated if the number of photons

counted was 50 or less. This was done to avoid proceeding

while the ion was trapped in one of the metastable D

states.

(3) Laser 2 was blocked to allow the ion to be pumped

to the ground (F — 0) state. The fluorescence was measured for 10 ms. If any photons were detected during this

1.0

>> 0.8

5 0.6 o

1 ' ,TTT 1 T— / • \

/ •\

- /• \ -

/

/

\

\

/ \

"

1 \ /• •\

• /

\

\

• .

> • / \ i • p \ • •

^ ^ / •\ V

.ii. 1 —

0.4

0.0-40 -20 20 40

l/(Hz) - 40 507 348 017.3

FIG. 8. Microwave resonance of a single ion. Each dot

represents the average of 19 measurements. The curve is a

calculated Rabi Une shape. The calculation does not taJce into account the imperfect state preparation.

period, another 10-ms measurement was made. This was

to ensure that the ion was in the ground (F — 0) state before proceeding.

(4) Laser 1 was blocked, and the magnetic field was

decreased to approximately 5 x 10~^ T in order to reduce

the effect of magnetic field fluctuations on the resonance

frequency. A 30-ms rf pulse near the 40.5-GHz resonance was applied. The magnetic field was then increased to

its previous value.

(5) Laser 1 wets unblocked, and the fluorescence was

measured for 15 ms. If or 1 photons were detected, the

ion was assumed to be in (F = 0). If more photons were

detected, it was assumed to be in (F = 1). The dashed line is a theoretical Rabi line shape for optimum power and a 30-ms rf pulse duration, calculated

from Eq. (2.4). The deviation between the data and the

calculation is most obvious at the minima of the line shape. The deviations result from the combined effects of

imperfect (F = 0) state preparation and imperfect state

detection. The combination of these effects results in a

signal that goes from about 0.1 to 0.95 instead of from

to 1 . The frequency instability of the microwave source is known, from separate measurements, to be small enough

as to have a negligible effect on the line shape [28]. We can also compute the variance of the signal at various positions on the resonance curve. At the point

corresponding to the maximum signal, it is 0.053. At

the two points corresponding approximately to the half- maximum points, it is 0.26. At the two points corre- sponding to the minimum signal on both sides of the

resonance, it is 0.051. We have published a microwave resonance curve in which the total fluorescence from several ions was measured [28]. Such a signal is more sensitive to noise from

the intensity fluctuations of the 194-nm sources, background scattered light, and other sources. In order to re- duce this noise, it should be possible to count the photons

from each ion separately, in order to clearly discriminate

between ions in the ground (F = 0) and (F = 1) states, TN-70
3564 W. M. ITANO et al. 47

Num

5 10

ber of counts

5

(c)

-

5

n ll.l.lll 1

5 10 Number of counts

15 5 10 Number of counts15

FIG. 9. Photon count distributions for a

single ^^^Hg"*" ion prepared in (a) the (F=0,

Mf=0) state, (b) the (F=l, Mf=0) state, and (c) an equal superposition of these two

states. The bars represent the number of

cases in which a given number of fluores- cence photons was detected when the detection laser was appUed. The number of measurements was 38 for (a), 19 for (b), and 38

for (c).

as was done here with a single ion. Figure 9 shows the distributions of the numbers of photons detected at various points on the resonance curve of

Fig. 8. Figure 9(a) shows the distribution at the points

corresponding to the minima on both sides of the reso- nance. In most cases, no photons were observed. However, in a few cases one or more photons were observed,

presumably due to a combination of background scat- tered light and imperfect state preparation. Figure 9(b)

shows the distribution at the maximum of the resonance. There is a broad distribution of numbers of photons detected, with a mean of about 5.5. In one measurement,

no photons were measured. This may have been due

to imperfect state preparation or to the possibility that no photons were detected, even though some were emitted. Figure 9(c) shows the distribution at the two half- maximum points on the resonance. The distribution is a superposition of those at the maximum and the minima.

This bimodal distribution is the signature of the quantum projection noise for the single-atom case. That is, for a superposition state with equal amplitudes of the two

components, the measurement finds the ion randomly in one state or the other.

bers 1-8, as shown in Fig. 10, which correspond, in order

of decreasing energy, to |-§,+|), |-^,+^), 1 + 5,+^),

|j_3 ,i\ 1,3 _lv 1,1 _U |_1 _1\ andl-^ -^\ i'2''2''l'2' 2''l~2' 2''l 2' 2'' I 2 ' 2

'

"

The 313-nm transition to the 2p'^Pzj2 electronic state was

used for state selection and detection.

B. Apparatus

The experimental apparatus and techniques have been

described previously [32-35]. The Penning trap was made of cylindrical electrodes, to which static electric potentials were applied. It was inserted into the bore of a superconducting solenoid magnet, which generated a uniform magnetic field Bq of approximately 0.82 T. The

combination of the electrostatic potentials and uniform

magnetic field trapped the ions in three dimensions. The

pressure in the trap was approximately 10~* Pa. The

ions were created by electron-impact ionization of neutral atoms. The state preparation experiments required that the

313-nm light be blocked for some periods in order to avoid

IV. iV-ATOM EXPERIMENT

A^'-atom quantum-state preparation and detection ex- periments were carried out with ^Be"'" ions in a Penning

trap. The Penning trap was used, rather than the rf trap, because the number of atoms that could be stably

trapped could be varied from a few to several thousand.

However, loading and detecting single ions was difiicult with this apparatus.

A. ^Be+ energy levels

Figure 10 shows the energy levels of ^Be"*" which were

important for the experiments. The ground electronic state has the configuration 2s ^S1/2- The ^Be nucleus

has spin |, so the ground state has eight hyperfine Zee- man sublevels. In a high magnetic field, as is present in the Penning trap, the energy eigenstates are approximate

eigenstates of I^ and J^ , the z components of the nuclear

and electronic angular momenta. The static magneticfield direction defines the z axis. These eigenstates will be referred to by their main components in the |M/, Mj)

basis. For brevity, they will also be referred to by num25^5

1/2

FIG. 10. Level diagram for ^Be"*". The grotmd-state Zee- man hyperfine states are labeled by nvunbers 1-8. Their quan- tum numbers axe given in the text. Laser radiation at 313 nm

optically pumps most of the ions into state 4. These atoms are then transferred to state 3 and then to state 2 by rf tt pulses.

Other rf pulses then create a superposition of states 1 and 2. Atoms which remain in state 2 are transferred bax;k to state

4. The laser reuiiation is applied again, and the number of atoms in state 4 is determined from the fluorescence intensity

in the first second after the laser radiation is reappUed.

TN-71
47 QUANTUM PROJECTION NOISE: POPULATION 3565

perturbations of the energy levels. In order to keep the

^Be"*" ion plasma from heating and increasing its spatial extent when the 313-nm light was not available for laser cooling, approximately 1000 ^^Mg"'" ions were trapped

and laser cooled at the same time. This cooled the ^Be"*" ions by long-range Coulomb collisions [36]. The 280-nm

3s'^5i/2 —> 'ip^Pz/2 transition was used for laser cooling

the 2^Mg+ ions. The 313-nm and 280-nm beams required for state selec- tion and detection of the ^Be"*" ions and for laser cooling

the ^^Mg"*" were generated by frequency doubling the out- puts of cw dye lasers in nonlinear crystals. Fluorescence

from the ions was focused by a multielement lens onto the

photocathode of an imaging photon-counting tube. The

overall detection efficiency was approximately 2 x 10""*. To generate the rf magnetic fields required for the state preparation and detection, the amplified output of a fre- quency synthesizer was coupled into a wire loop antenna. The light shutters, rf switches, and frequency synthesizer were controlled by a computer, which also recorded the

photon counts.

C. Ion-number measurement

In the ^^^Hg"'" experiments, the number of ions could

be determined directly from the image. This could not be

done for the ^Be"*" ions in the Penning trap, since they

rotate rapidly around the z axis. Therefore, the number was determined indirectly. Calculations based on a charged fiuid model relate the density of the ion plasma

to its shape, for given external fields [36-38]. The shapes

and sizes of the nonneutral, ion plasmas were determined

by moving the laser beam and observing the imaged fluo- rescence. The product of the density and volume yielded

the number of ions with an uncertainty estimated to be

about 30%.

D. State preparation and detection

In the ^Be"*" experiments, coherent superpositions of

two internal states (states 1 and 2) were created and then

subjected to measurements. These states were chosen because, for a value of Bq near 0.8194 T, the first derivative

of the transition frequency with respect to Bq goes to

zero. The resulting insensitivity to magnetic-field fluctu- ations makes it easier to generate coherent superposition

states reproducibly.

The state preparation began by subjecting the ions to 313-nm radiation, polarized perpendicular to the magnetic field, for approximately 15 s. The frequency of the

313-nm radiation was slightly below the 2s'^Si/2 {Mj =

+ |,Mj = +\) -> 2p2p3/2 {Mj = +|,M/ = +1) transi- tion frequency. This is a cycling transition, since electric dipole selection rules require that the ion return to the

ground-state sublevel that it started from. Spontaneous

Raman transitions induced by the 313-nm radiation es- tablished a steady state in which approximately y| of the

ions were in state 4 and the remaining ^ were in state

5. This optical pumping has been discussed previously

[12,39] and studied experimentally [40]. There are fluctu- ations about these average values, since any given ion is continually making transitions between states. The ions

could have been completely optically pumped into state 4 by circularly polarized light propagating along the z

axis, but this was not convenient experimentally.

The 313-nm beam was blocked to stop the optical

pumping and to prevent perturbations to the ^Be"*" en- ergy levels. Next, the ions in state 4 were transferred to

state 3 and then to state 2 by 0.2 s resonant rf pulses.

These were tt pulses; that is, the products of the rf magnetic fields and the pulse durations were adjusted so that

6t = TT in Eq. (2.8). The frequencies were 320 712 280

Hz for the (4 -^ 3) transition and 311 493 688 Hz for the

(3 —> 2) transition. The Ramsey method was then used

to create various superpositions of states 1 and 2. The

two Ramsey rf pulses were 0.5 s long and were separated

by 5 s; the frequency was 303 016 377.265 Hz. Then a measurement was made of the number of ions

in state 2. First, the ions in state 2 were transferred to

state 3 and then to state 4 by applying the tt pulses in the

opposite order. Then the 313-nm beam was unblocked,

and the fluorescence photons were counted for 1 s. The ions which were left in state 5 at the time that the

313-nm beam was blocked (about ^ of the total number) contributed to the fluorescence signal. This is so, because the time constant for exchanging population between states 4 and 5 by spontaneous Raman transitions was about 0.1 s, which was much less than the 1-s observation time. The time constant for ions in state 1 to be

optically pumped to states 4 and 5 was approximately

10 s. Therefore, the number of ions pumped from state

1 back to states 4 and 5 could be neglected.

E. Results

Ion plasmas containing numbers of ^Be"*" ions ranging

from a few to a few hundred were studied. Figures 11(a)-

11(d) show the results from plasmas containing approximately 5, 21, 72, and 385 ions. The rf power for the

(2 —> 1) Ramsey resonance was adjusted so as to give a minimum fluorescence signal at the line center. Measurements were made at rf frequencies corresponding to the

transition maximum (minimum fluorescence), the first upper and lower transition minima (maximum fluores- cence), and the points halfway between the transition

maximum and the upper and lower transition minima.

The measured signal is the number of photons detected

in the first second after the laser is unblocked. This is S = B + K{Ni + N^), (4.1)

where B is the background signal, A^4 and N^ are the

numbers of ions in states 4 and 5, and K is a, constant which must be calibrated for a given set of conditions.

At the line center, the signal is B + {KN/17), where N

is the total number of ^Be"*" ions, while at the points of maximum fluorescence, it is B + KN. For the N = 5, 21,

72, and 385 data shown in Figs. ll(a)-ll(d), K = 87.3,

TN-72
5000

c

3oo

-T

1 r N=72 ^.

\ i

I

-0.1

\ / \ /

0.0 0.1 ly(Hz) - 303 016 377.265

W. M. ITANO et al.

2000

V)3

oo

1

1 1 'X N=21 5

\.,/"

-0.1

(b) 0.0 0.1 30000

3

Oo

1 r N=385

\

\

5000

-0.1

(d) /

_] L. 0.0 0.1 Z/(Hz) - 303 016 377.265

47

FIG. 11. Plots of the fluorescence detected from ^Be"*" ions confined in a Penning trap as a function of the frequency of the apphed rf radiation for (a) 5 ions, (b) 21 ions, (c) 72

ions, and (d) 385 ions. The dots are the ex- perimental mean signals, and the error bars show the experimental standard deviations. On the sides of the resonances, the standard

deviations are mainly due to quantum projection noise. The dashed hues show the calcu- lated line shapes, fitted to the experimental

minimum and maxima.

73.9, 58.0, and 48.6 counts/ion, respectively. The fact that X ^ 1 for all of these cases means that quantum

projection noise should be more important than photon

shot noise, except when the vanishing of one of the state amplitudes causes the quantum projection noise to go to

zero. This is a good example of electron-shelving detection. In Figs. ll(a)-ll(d), the dots are the experimental mean signals:

1 "

(4.2)

where 5i, 52, . .

.

, 5„ is the sequence of n measurements

made under the same conditions. The error bars corre- spond to ±a, where a was calculated from

n-l

Equation (4.3) was used, rather than the usual formula,

1 ""

1=1 Sf, (4.4)

for the standard deviation, because it is less sensitive to slow drifts of the signal, such as those caused by varia- tions in the laser intensity or other experimental parameters [41]. Table I summarizes the data shown graphically in Figs. ll(a)-ll(d). For each value of N, the data from

the point of minimum fluorescence is labeled "dip," the

data from the two points of maximum fluorescence are combined and labeled "peaks," and the data from the

two points halfway between the minimum and the maxima are combined and labeled "sides." Four contributions to cr^ are listed in Table I: cr?proji

pump

'

^shot' and a^ They are assumed to be independent, so they are added in quadrature to yield <7^aic'tech'

_2 2 2 2 2 = ^proj "T" ^pump "I" ^shot + ^tech* (4.5)

The quantum projection noise iTproj is approximately

zero at the peaks and at the dip and is ^{^N)^/'^K on

the sides [see Eq. (2.41)]. The factor of y| appears in this expression because, on the average, jj of the ions are left in state 5 by the optical pumping that precedes

the rf pulses. The fluctuations in the number of ions

left in state 5 are the source of <t,pump- This has the

greatest effect at the dip, where the only contribution to the fluorescence is from the ions in state 5. At the dip.

<7pump=[N{^){^)Y^'K=^VNK. (4.6)

This is derived from the expression for the variance of a binomial distribution [see Eq. (2.18)]. At the peaks,

'pump is approximately zero, because all of the ions are

either in state 4 or state 5 and contribute to the signal. At

the sides, Cpump is half as large as at the dip. The shot

noise cTshot is equal to (5)^^^ and results from Poisson

statistics in the photon detection. All other contributions to a, such as those due to in- tensity fluctuations of the laser, are called technical noise

(Ttech- Laser-intensity fluctuations will lead to a contri- bution to (Ttech proportional to S. Fluctuations in the

shape and temperature of the ion plasma may make a

large contribution to (Ttech) approximately proportional

to S. Such fluctuations have been observed in other laser- cooled ion plasmas in Penning traps [42,43], but are not

well understood. Small fluctuations in the frequency of

the microwave source, or in the resonance frequency of

the ions, would cause an increase in (Ttech on the sides of

the line shape but not at the dips or peaks. However,

from auxiliary measurements, including the observation

TN-73
47 QUANTUM PROJECTION NOISE: POPULATION 3567

TABLE I. Mean signals S and variances cr^ for TV = 5, 21, 72, and 385 ^Be''' ions. The terms

"dip," "peaks," and "sides" refer to the points of minimum fluorescence, the points of maximum

fluorescence, and the points halfway between the minima and the maxima, respectively. The number

of measurements is n. The measured varieince is cTexpt- The calculated variance a^a,\c is equeil to

<^proj + <^pump + Cshot + <''?ech • The VEilue of (T^xpt at the peaks was used to empirically determine

CTtech- Hence cr^^ic is not hsted at the peaks, since it is necessarily equal to CTexpt- However, the agreement between (Texpt and CTcaic at the sides and at the dip is a test of the theory. The units for N Position n S

(counts)

_2

C'expt

2 "proj

_2

'^pump C^shot ^2

^tech _2^calc (10^ counts^)

5 dip 19 89 1.0 «0 2.1 0.1 0.1 2.3 5 sides 38 275 6.6 9.0 0.5 0.3 0.7 10.5 5 peaks 38 500 2.8 kO ^0 0.5 2.3

21 dip 30 232 5.5 %o 6.3 0.2 0.1 6.6 21 sides 60 810 37.8 27.0 1.6 0.8 1.3 30.7

21 peaks 60 1693 7.6 wO «0 1.7 5.9

72 dip 30 498 11.5 «o 13.4 0.5 0.3 14.2 72 sides 60 2432 61.0 57.0 3.4 2.4 6.3 69.1 72 peaks 60 4429 25.2 «0 wO 4.4 20.8

385 dip 30 6642 69 «o 50 7 26 83

385 sides 60 16108 600 214 13 16 151 394

385 peaks 60 24 253 367 kO wO 24 343

of lineshapes as narrow as 900 fxHz [34] , we determined

that this potential source of noise was negligible. Since

known contributions to cTtech are roughly proportional to

S, cTtech was determined empirically from crgxpt at the

peaks, where the only other contribution to a is CTshoti which is small. For A^ = 5, 21, 72, and 385, atech/S at the peaks was 9.6%, 4.5%, 3.2%, and 2.4%, respectively.

The values of crtech at the sides and the dips were esti- mated by assuming that, for a given set of experimental

conditions, atech was proportional to S. The entries on Table I show that different types of noise

dominate at each of the three positions on the line: <7prQj on the sides, cTp^rnp ^^ ^he dip, and crtech ^^ ^^^ peaks.

Shot noise is not a large contribution for any of the cases shown in Table I. Considering the uncertainties in the

experimental parameters, particularly in N, the agree- ment between cTg^pt and u^^ij. is quite good. Figure 12 shows that the noise on the sides of the lines is mainly

attributable to quantum projection noise. The quantity

plotted is a/AS and is defined as

o- _ CTexpt (sides)

AS 5(peaks) - 5(dip)

(4.7)

The dashed line is the theoretical prediction for quantum

projection noise alone: yVf _ 0.515

AS " 8//V ~ \/iV

'proj (4.8)

The deviation of the experiment from theory for large A^ is presumably due to technical noise. As noted previously, CTtech /-S" decreases as N increases, apparently be- cause the ion plasma becomes more stable in shape and

temperature. However, it should not continue to decrease

indefinitely with N, since the contribution from laser in- tensity fluctuations, for example, should approach a constant value for large N. Hence a

j

AS should approach

a constant value for large N rather than continuing to decrease as 1/VN. For A^ = 10 000, (Tproi/S is approximately 1% on the sides, while we might, from the present

results, expect atech/S to be about 2%. This is consis- tent with previous experimental results from a firequency

<

0.1

0.01

-1—I I I I I II] 1—I I M I iri

I ' ' ' I I 10 100

N

1000

FIG. 12. Plot of the normalized standard deviations as a

function of N, the number of ^Be"*" ions. The quantity a/AS

is the ratio of the experimental standard deviation on the

sides of the resonance to the difference in the signal between

the peaks and the dip. The dashed line is the theoretical prediction for the contribution from quantum projection noise

alone. „^ ^ _ TN-74

I
3568 W. M. ITANO et al. 47

standard based on 5000-10 000 ^Be+ ions [34]. However,

there is no fundamental reason that the technical noise

cannot be greatly reduced.

V. CONCLUSION

We have prepared one or many ions in well-defined su- perpositions of two states and then measured the population of one of these states. The population measurements

display quantum fluctuations which are greatest when

the amplitudes of the two states are equal. In the case

of a single ion, where photon scattering is used to distin- guish between the two states, the fluctuations take the

form of random switching between two distinguishable

conditions — the observation of zero photons (or background) or of some photons. There is good agreement

between the observations and theoretical predictions for both one and many ions. Population fluctuations of the type observed here

(quantum projection noise) are present and could be observed in other kinds of experiments, for example, those

using atomic or neutron beams. Quantum projection

noise might be observed in neutron interferometry [44]. In this case, the two neutron beams which emerge from

the interferometer correspond to the two energy states

\A) and \B). Introducing a phase shift in one arm of

the interferometer allows one to create coherent superposition states. Ranch et al. [44] show counting-rate his- tograms for phase shifts corresponding to the maximum

and minimum counting rates in one beam. However, they

do not show the histograms for a phase shift halfway between a maximum and a minimum, which might show an

increased variance due to quantum projection noise. In the preceding discussions, the single-atom and Natom experiments were treated from different perspectives. This was because of the different experimental

methods which were used for the ^^^Hg+ and ^Be+. However, if the N atoms are uncorrelated, there is not a fun- damental difference between the two cases. We could

simulate the results of an A'^-atom measurement by making N successive measurements on a single atom and

summing the results. In many experiments, quantum projection noise would

be obscured by other forms of noise, such as fluctuations

in the beam intensity or the statistics of the detection

process. In the experiments described here, trapping

nearly eliminates fluctuations in the number of atoms,

and the electron-shelving detection method nearly eliminates statistical fluctuations in the detection, thereby

making quantum projection noise the dominant source

of noise. In spectroscopic measurements on fixed numbers of atoms, where transitions are detected by monitoring changes in population, this is the fundamental noise

limit.

ACKNOWLEDGMENTS

We acknowledge financial support from the Office of

Naval Research. We thank Jean Dalibard for pointing out the simple calculation of the variance for one atom

[Eq. (2.15)].

APPENDIX: OPTICAL PUMPING IN i»»Hg+

For low enough laser intensities, optical pumping

among the Zeeman hyperfine sublevels of the ground

state of ^^^Hg+ can be described by population rate equations. The following set of equations applies to the populations of the Zeeman hyperfine sublevels of the 6s ^5i/2

and &p^P\/2 states, when a linearly polarized optical field of the form E(r, t) = iEo cos(k r — cj^f ), nearly resonant with the ^Si/2{F = 1) applied:

^Pi/2{F — 0) transition, is

^ = -^sm^eR^L{A)P, - 1(1 + cos2^) (^) R^Pi + ^{P, + P5 + Pe),

^ = -isin2eiloL(A)P3 - i(l + cos2^) {^^ R0P3 + |(P4 + Pe + ^7),

^ = i sin^ ^i?oi^(A)(Pi + P3) + cos'' e R^L{A)P2 - 7P4,

dPs 1.2 7/2

A5 + Ap

Ii^Po + cos^e{^^ RoPi + lsin'eC^] R^P2-^Ps,

dPe — 2a( 7/2 — = cos 6 -—-

dt VA5 + ^"J RQP^ + \sm^d(^^ RoiPi + P3) - yPe,

dPi 1.2

dt

^^ =isin^e(— 2 V As

^-1- Aj

PoPo + ^sin2^(^) P0P2 + cos2 ^ ( ^] R^Ps-jPj. 7/2

Ap

(ill \Ap

(Al)

(A2)

(A3)

(A4)

(A5)

(A6)

(A7)

(A8)

TN-75
47 QUANTUM PROJECTION NOISE: POPULATION 3569

Figure 13 shows the correspondence between the nu- merical state labels used in Eqs. (Al)-(A8) and the quantum numbers. The labels 0-3 correspond to the 6s ^51/2

{F^O,Mf = 0), {F = 1,Mf^-1), {F^1,Mf^0),

and {F = 1,Mf = 1) states, respectively. The labels 4-7 correspond to the 6p^Pi/2{F = 0,Mf = 0), {F =

1,Mf = -1), {F = 1,Mf = 0), and (F = 1,Mf = 1) states, respectively. The quantity 7 is the natural decay

rate of the ^Pi/2 state, HAs and hAp are the ^5i/2 and

^Pi/2 hyperfine energy splittings, and 9 is the angle between the z axis, defined by the magnetic field, and e. The quantity Rq, which has dimensions of inverse time,

is

Rq

67/12

(A9)

where {Qp^Pi/2 II D^^^ \\ 65^5^/2) is a reduced matrix

element of the electric-dipole-moment operator, defined

according to the conventions of Edmonds [19]. The quantity L{A) is a Lorentzian resonance line shape, defined

by

L(A). (^/^)^

A2 + (7/2)2'

(AlO)

where A = Wfe — cjq is the laser detuning from the

251/2 (F = 1) ^ ^Pi/2{F = 0) resonance frequency

ujQ. We consider the low-intensity limit, defined by

RqL{A) <^ 7. a typical experimental value for RqL{A)

was 10'^ s~^, while 7 is 4.3 x 10^ s~^. The Zeeman shifts are not included in A, because they are assumed to be

small compared to 7. For a typical magnetic field of

5 X 10~^ T, states 1 and 3 are separated from state 2 by

± 7 MHz, while 7, in the same units, is 70 MHz.

When a second laser near the ^5i/2(F = 0) —

'^Pi/2{F = 1) resonance is present, the atom spends

nearly all of its time in the ^5i/2(F = 1) and ^Pi/2(F = 0)

states. The following reduced set of rate equations then

describes the system:

HP^ = -^{smH)RoL{A)P, + ^7^4,

-^ = -(cos^ e)RoL{A)P2 + ^7^4,

dP2 ^

dt^ 2

dt dP

= -Usm^e)RoL{A)P3 + hP^,

dP3

^ = Usin^9)RoL{A){P, + Ps)

+(cos2 9)R^L{A)P2 - 7F4.

(All)

(A12)

(A13)

(A14)

The mean populations of the states are obtained by set- ting the time derivatives in Eqs. (All)-(A14) to zero. The mean populations are

(A15)

(A16)

P3

2 cos2 9 "

l + 3cos2 9' sin^e ~

l + 3cos2 9'

2 cos2 9

l + 3cos2e'

(A17)

6p2p

1/2 F-1

F=0

6s2S

1/2 F-0

Mp— 1 Mp-0 Mp-1

FIG. 13. Hyperfine Zeeman sublevels of the 6s^Si/2 and

6p ^Pi/2 states of ^^^Hg+. The labels 0-7 for the sublevels are used in the discussion of optical pumping in the Appendix.

3sin^gcos2g

1 -h 3 cos2 9 RoL{A)

7

<1. (A18)

The dependence of P4 on 9 is of interest, because the

number of fluorescence photons emitted per ion is 7P4.

Prom Eqs. (A18)-(A15), it can be seen that the fluores- cence goes to zero for ^ = or tt, when the population

is pumped into states 1 and 2. The fluorescence also goes to zero for 9 = tt/2, when the population is pumped

into state 2. The fluorescence has its maximum value for cos 9 = 3~^/2 (^ ~ 54.7°), when states 1, 2, and 3 are equally populated.

Another quantity of interest is the mean population

of the '^Si/2{F — 1) state when only one laser beam is present. This quantity is equal to P1+P2 +P3 and can be

determined from the steady-state solutions to Eqs. (Al)-

(A8). In the low-intensity limit,

Pi + P2 + P3

As + Aj

2(l + 3cos2g)

TTTcos^y

(A19)

The quantity inside the square brackets varies from 2 to 3, depending on 9. It reaches its maximum value

for cos^ — 3~^/2^ the same value for which Eq. (A18)

predicts the maximum fluorescence when both lasers are

present. Since 9 was set empirically so as to maximize the

fluorescence, it was probably close to this value in the ex- periment. Prom Eq. (A19), and using P0 +P1+P2 +P3 «^

1, we find that Pi -I- P2 + P3 lies between 0.041 and

0.06. This compares well with the experimental value of

0.053 ± 0.06 determined from the data shown in Fig. 7. In order for the rate-equation approach to be valid, the

laser intensity must not be too large. A density-matrix

calculation including the coherences between states 1, 2, 3, and 4 shows that if the magnetic field is not large

enough compared to the optical electric field, the atom

is optically pumped into a nonfiuorescing superposition

state [45]. While the density-matrix approach is valid over a larger range of parameters, the rate-equation ap- proach is used here because it is simpler and is still a good approximation for the low laser intensities which

were used in the experiment. ^ „^ TN-76
3570 W. M. ITANO et al. 47

* Present address: Department of Physics, University of [23 Texas, Austin, TX 78712.

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[6] D. Home and M. A. B. Whitaker, Phys. Lett. A 160, 325

(1991). [27

[7] J. Dalibard, Y. Castin, and K. M0lmer, Phys. Rev. Lett. 68, 580 (1992). [28

[8] H. J. Carmichael, R. J. Brecha, and P. R. Rice, Opt.

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[9] G. C. Hegerfeldt and T. S. Wilsner, in Quantum [29 Systems—Foundations and Symmetries: Proceedings of

the Second International Wigner Symposium, Goslar, [30 1991, edited by W. Scherer, H. D. Doebner, and F. Schroeck, Jr. (World Scientific, Singapore, 1992).

[10] R. Dum, P. Zoller, and H. Ritsch, Phys. Rev. A 45, 4879 [31 (1992).

[11] H. G. Dehmelt, Bull. Am. Phys. Soc. 20, 60 (1975). [32

[12] D. J. Wineland, J. C. Bergquist, W. M. Itano, and R. E. DruUinger, Opt. Lett. 5, 245 (1980). [33;

[13] D. J. Wineland, W. M. Itano, J. C. Bergquist, and F. L. Walls, in Proceedings of the 35th Annual Frequency Con- [34

trol Symposium, Philadelphia, 1981, NTIS Doc. No. ADAl 10870 (Electronic Industries Association, Washington,

DC, 1981), pp. 602-611 [reprinted in Trapped Ions and [35 Laser Cooling, edited by D. J. Wineland, W. M. Itano, J. C. Bergquist, and J. J. BoUinger, Natl. Bur. Stand. (U.S.) [36 Technical Note 1086 (U.S. GPO, Washington, DC, 1985), pp. TN-47-TN-56].

[14] N. F. Ramsey, Molecular Beams (Oxford University [37^ Press, London, 1956).

[15] R. P. Feynman, F. L. Vernon, Jr., and R. W. Hellwarth, [38

J. Appl. Phys. 28, 49 (1957).

[16] R. H. Dicke, Phys. Rev. 93, 99 (1954).

[17] P. R. Bevington, Data Reduction and Error Analysis for [39 the Physical Sciences (McGraw-Hill, New York, 1969).

[18] F. T. Arecchi, E. Courten, R. Gilmore, and H. Thomas, [40

Phys. Rev. A 6, 2211 (1972).

[19] A. R. Edmonds, Angular Momentum in Quantum Me- [41 chanics (Princeton University Press, Princeton, NJ,

1974). [42

[20] B. Yurke, Phys. Rev. Lett. 56, 1515 (1986); B. Yurke,

S. L. McCall, and J. R. Klauder, Phys. Rev. A 33, 4033 [43 (1986).

[21] M. Kitagawa and M. Ueda, Phys. Rev. Lett. 67, 1852

(1991); in Noise in Physical Systems and 1/f Fluctuations, Proceedings of the International Conference, Kyoto, Japan, 1991, edited by T. Musha, S. Sato, and M. [44

Yamamoto (lOS, Burke, VA, 1992), p. 355.

[22] G. S. Agarwal and R. R. Puri, Phys. Rev. A 41, 3782 [45 (1990).

D. J. Wineland, J. J. Bollinger, W. M. Itano, F. L. Moore,

and D. J. Heinzen, Phys. Rev. A 46, 6797 (1992).

L. S. Cutler, R. P. Giffard, and M. D. McGuire, in Proceedings of the 13th Annual Precise Time and Time Interval Applications and Planning Meeting, edited by L.

J. Rueger, NASA Conf. Publ. 2220 (NASA, Washington,

DC, 1982), pp. 563-578. Y. Guern, A. Bideau-Mehu, R. Abjean, and A. JohanninGilles, Phys. Scr. 14, 273 (1977).

P. Eriksen and O. Poulsen, J. Quant. Spectrosc. Radiat. Transfer 23, 599 (1980).

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

M. G. Raizen, J. M. Gilligan, J. C. Bergquist, W. M.

Itano, and D. J. Wineland, J. Mod. Opt. 39, 233 (1992);

Phys. Rev. A 45, 6493 (1992). H. Hemmati, J. C. Bergquist, and W. M. Itano, Opt.

Lett. 8, 73 (1983).

See, for example, W. M. Itano and D. J. Wineland, Phys. Rev. A 25, 35 (1982); D. J. Wineland and W. M. Itano, Phys. Today 40 (6), 34 (1987). A. G. Calamai and C. E. Johnson, Phys. Rev. A 42, 5425

(1990).

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

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

S. L. Gilbert, J. J. Bollinger, and D. J. Wineland, Phys. Rev. Lett. 60, 2022 (1988).

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

and D. J. Wineland, IEEE Trans. Instrum. Meas. 40, 126

(1991).

W. M. Itano, D. J. Heinzen, J. J. Bollinger, and D. J. Wineland, Phys. Rev. A 41, 2295 (1990). D. J. Larson, J. C. Bergquist, J. J. Bollinger, W. M.

Itano, and D. J. Wineland, Phys. Rev. Lett. 57, 70

(1986). D. J. Wineland, J. J. Bollinger, W. M. Itano, and J. D.

Prestage, J. Opt. Soc. Am. B 2, 1721 (1985).

L. R. Brewer, J. D. Prestage, J. J. Bolhnger, W. M. Itano, D. J. Larson, and D. J. Wineland, Phys. Rev. A 38, 859

(1988). R. G. Hulet and D. J. Wineland, Phys. Rev. A 36, 2758

(1987). R. G. Hulet, D. J. Wineland, J. C. Bergquist, and W. M.

Itano, Phys. Rev. A 37, 4544 (1988). H. Mark and J. Workman, Statistics in Spectroscopy

(Academic, San Diego, 1991), pp. 57-68. R. C. Thompson, G. P. Barwood, and P. Gill, Appl. Phys.

B 46, 87 (1988).

W. M. Itano, L. R. Brewer, D. J. Larson, J. J. BoUinger,

S. L. Gilbert, and D. J. Wineland, in Frequency Standards

and Metrology, Proceedings of the Fourth Symposium,

Ancona, Italy, edited by A. De Marchi (Springer-Verlag,

Berlin, 1989), p. 447. H. Rauch, J. Summhammer, M. Zawisky, and E. Jericha,

Phys. Rev. A 42, 3726 (1990).

W. M. Itano (unpublished calculations).

TN-77
PHYSICAL REVIEW A VOLUME 46, NUMBER 1

1 DECEMBER 1992

Spin squeezing and reduced quantum noise in spectroscopy

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

Time and Frequency Division, National Institute of Standards and Technology, Boulder, Colorado 80303

D. J. Heinzen

Physics Department. University of Texas, Austin, Texas 78712

(Received 25 March 1992)

We investigate the quantum-mechanical noise in spectroscopic experiments on ensembles of N twolevel (or spin-i) systems where transitions are detected by measuring changes in state population. By

preparing correlated states, here called squeezed spin states, we can increase the signal-to-nose ratio in spectroscopy (by approximately /V '^^ in certain cases) over that found in experiments using uncorrelat- ed states. Possible experimental demonstrations of this enhancement are discussed. PACS number(s): 42.50.Lc, 03.65. Bz, 42.50.Dv

Squeezed states of the electromagnetic field have re- ceived much attention for over a decade [I] and have now

been applied to fundamental metrology such as the reduction of noise in optical interferometers below the standard

quantum limit [2]. Related to this are theoretical investi- gations of noise reduction in fermion interferometers with

correlated input states [3,4). In this paper we investigate an application of squeezing to the reduction of quantummechanical noise in spectroscopic experiments on ensembles of two-level systems. We also suggest how this squeezing might be realized in an experiment.

We will be specifically interested in experiments where

transitions are detected by measuring the changes in population of one of the two levels. This is to be contrasted

with experiments which detect the radiation transmitted

through an absorber; these experiments benefit from

squeezed radiation [5]. We first show that preparation of

correlated input states (squeezed spin states) improves, in a fundamental way, the signal-to-noise ratio (SNR) in any spectroscopic experiment which is limited by the quantum-mechanical uncertainty in the measurement of

the level populations. We then discuss, as an example, a

possible experimental realization of this improvement us- ing laser-cooled stored ions, where the SNR is currently

limited by quantum fluctuations in measured populations

[6].Since the dynamics of a two-level system interacting

with radiation are the same as the dynamics of a spinj

particle in a magnetic field [7] we will discuss only the

latter case. We begin by assuming an ensemble of A' spin

S=j systems each with magnetic moment /i =/ioS and

whose direct dipole-dipole interaction can be neglected.

Each spin interacts with an externally applied magnetic

field Boi (flo=const) through the Hamiltonian Hq = — fi- (Bqz) = h(OoSz where (Oq is the precession fre- quency of each spin about the z axis. We assume A' is fixed and the relaxation negligible [6]. Suppose all the

spins are initially prepared at time / =0 in the I"})

state. In this case, the spin wave function for the ensemble can be represented by the Dicke state \j,M = — J)

where J =S/S, (S, is the spin of the /th particle), J =N/2,

and M={J,} [8]. For this state, AA(/ =0) =Ay^(0)

= (y/2) '''^ and AJ^iO) =0, where AA is the square root of

the variance of operator /I, {AA)^ = {A'^} — {A)^. In Fig.

I (a), the lower cone shows, pictorially, a more general initial state with (J) =(7^)2, iJz) <0, and AJyKAJ^ in a reference frame which corotates about the z axis with the

spin precession frequency ojq. In this rotating frame, the

field Bq transforms to zero and the state remains station- ary [9]. We perform spectroscopy (here, essentially nuclear

magnetic or electron spin resonance) using the Ramsey

method of separated oscillatory fields [9]. Description in terms of the Ramsey method leads to mathematical

simplification, but the improvement of SNR is quite general. The Ramsey method consists of applying an oscillat-

<^(^*2f^,,)>

4) = (u -cojr

FIG. 1. Pictorial representation of the Ramsey method of spectroscopy in a frame corotating with the spin precession fre- quency. In (a), the lower cone represents the initial spin state

for the Ramsey spectroscopy described in the text. After appli- cation of the second n/l pulse [upper cone in (b)], we detect the number of spins in the | + j ) state. M R6797 Work of the U. S. Government

Not subject to U. S. copyright

TN-78
R6798 WINELAND, BOLLINGER, ITANO, MOORE, AND HEINZEN 46

ing classical field yB\(t)cos((ot) where a> = (oo. This field can be decomposed into two fields which rotate about the z axis; one of these fields B, (|B;.| =B\/D rotates in the same direction as the spin precession and the other can

usually be neglected as is assumed here [9]. We will as- sume that B\ is nonzero and constant with value 5io from

time r =0 to t=t„ii such that ilRt„i2 = n/2 and

l<w~<0ok;r/2«l, where nR=noB\o/i2h) is the Rabi fre- quency. At / =0 in the rotating frame, B^ lies along the y

axis as shown in Fig. 1 (a). The lower cone of Fig. 1 (a) ro- tates about B,- and preserves its size and shape so that

after time t„/2, the cone lies along the x axis. At / =t„/2,

B\ is reduced to zero and the cone remains stationary

along X. After a time T i^t„/2), B\ is made equal to B\q

again for a time t„/2. If (o^(oq, B^ is now at an angle

^ = {a) — (OQ)T with respect to its original direction and

the cone precesses about B;- as shown in Fig. 1 (b). At

time tf=T + 2t„/2, the number of particles in the |+ y)

state is measured. The expected number of particles (A' + )

in the |+y) state, where N+=J + J:, is given by the Ramsey resonance curve

(yv + ) =7 - (^z (O))cos(w - 0)0)7^ • (1)

The discussions of correlated-particle interferometer ex- periments [3,4] parallel the description of the Ramsey

method.

With A' constant we eliminate noise due to fluctuations

in A' as found in an atomic beam experiment. However,

successive measurements of A'+ for a particular value of

co — coo fluctuate by AA'+ due to fluctuations in the number of spins measured to be in the | + y ) state. This

produces apparent fluctuations in the center frequency

of the Ramsey curve, given by \Aco\=AN +/\BiN + }/d(o\

=AJz(tf)/\d{J:(tf)}/dco\. It is these fluctuations we

desire to make small. If the initial state is the |y, —J)

Dicke state, we find \Aa)\os~UTN^^^. The somewhat

surprising independence of IAwIds on ^ occurs because

the quantum noise is proportional to the slope of (A' + ). In the presence of added noise it is advantageous to operate

at frequencies co where \d{N + }/da)\ is maximized in which case (A^ "*") =7 =A'/2. This condition on (o is as- sumed in the remaining discussion. The frequency noise

|Ac(j|ds has been observed in experiments on trapped ions

[6]. By use of suitably prepared "squeezed" initial spin

states, which show correlations between the individual

spin-y particles, it should be possible to achieve |A(u| < IAo^Ids- Hence, we define squeezing in Ramsey spec- troscopy as ^R < \ where

^R = \Aco\/\Aa)\os'=[2J]'^^AJyiO)/\{JAO)}\ (2)

Spin squeezing has been defined in other ways. From

the commutation relations for angular momentum, the corresponding uncertainty relation can be given by

AJxAJy> |<yzV2| and the expressions which follow from

cyclic permutations. From this expression, it is natural to define squeezed states [10] as those where AJi/\{Jj)/

2|'''^<1 for some i^j&{x,y,z}. This squeezing is present in Bloch states [8], which are obtained by rotating the |y, ±7) Dicke states [10]. For example, during the

first Ramsey pulse [Fig. 1(a)], AJj\U,)l2y'^-icosey'^

< 1. However, these rotated |7, ±7) Dicke states have

^R = I and do not improve the spectroscopy we describe

here. Spin squeezing might also be defined as follows [4]: Let A7< denote the smallest uncertainty of a spin component perpendicular to mean spin vector (J). The spin is squeezed if ^spin < 1 where

^spin=A7</|a)/2|'/^ (3)

For the Bloch states, i^spin'^^l- A squeezed spin state with

i^spin < 1 can be rotated so that (J)=(7z)z and A7< =A7^

and can then be used in Ramsey spectroscopy with

^/,=(7/|a)|)'%in.

We now examine possible methods to prepare the initial squeezed spin state. For a single particle (7=y),

^spin,*?/?^ 1- For 7 > 1. states with ^^p\„,^R< 1 can be

prepared using an interaction proportional to the square

of angular momentum operators [4]. Because we were

unable to find a physical interaction of this type for our problem, we have investigated an interaction of the form — HoJxzB' cosicomtp). Here, B' has units of a field gradient (dBx/dz), 2 =zo(o +a) is the quantum-mechanical

amplitude of a harmonic oscillator of frequency ftj^, zq is

its zero-point amplitude [zq—Az (coherent state)], a^ and

a are raising and lowering operators, tp is the time during

the preparation period, and we assume (Oz^oo- For a suitable choice of com, this interaction gives rise to a reso- nant coupling between J and z. In the interaction picture

and in the rotating-wave approximation it has the form

Hi = -hilU + a+J-a^) , (4a)

for cOm =coo~ (Oz and

H2=^-hnU+a^ + J-a) , (4b)

for ci)r„=coo + o)2 where J + =Jx + iJy, J-=iJ+) , and

n=/io5'zo/4/i.

The Heisenberg equations of motion for a and 7- can

be solved analytically for Eqs. (4) when dJ:/dtp =0. If the initial spin state is the |7, — 7) Dicke state, this amounts to the small-angle approximation 7^ = — 71

where I is the identity operator [11]. For the interaction

//i, we find

2(, =1 <^spin=<?^=COS^fi;vrp + ^z('A'=0)sin^a^rp, (5)

where n^^An^ and ^^(/p =0) = Az(rp =0)/Az (coherent

State) is the initial squeezing of the harmonic oscillator. Therefore if t,^ (/^ =0) < 1 , this squeezing is transferred to the spins (in a time k/2D.n) like wave-function exchange

between coupled harmonic oscillators [12]. Because of

the limited validity of the small-angle approximation, we

have numerically evaluated Schrodinger's equation to compute i^spin and £,r for two special cases shown in Figs. 2 and 3 assuming an initial |7, — 7) Dicke state. For our implementation of squeezing [Eqs. (4)], these figures

show the improvement in SNR which can be obtained

over the quantum limit obtained with uncorrelated input

states. This squeezing can, in principle, be generated by the in- teraction of an ensemble of atoms with a single-mode cavity field [1,11,13]. It can be viewed as the complement of

TN-79
^ SPIN SQUEEZING AND REDUCED QUANTUM NOISE IN . . . R6799

(a) 10

0.5

^R •

^sp,n A

J=l

. small an

theory

.\

\-' <n^(0)> O

I

<— "°^n o <> 0.0 0.5

P (0)

^ 7

2.5

2.0

1.5

1.0

0.5

1.0

(a) 10

0.9

0.8 1

<

^R •

^spin

5 10 <n,(t =0)>

(b)

1-0

0.5

0.0

1 . 1 1 . 1 1

1

1 1 1

o f^-~^.

i^spin

e^(o) o ^8

(b) 10.

10

J

100

FIG. 2. Squeezed spin state preparation assuming the in- teraction of Eq. (4a) and assuming the harmonic oscillator is initially in the squeezed vacuum state where (z(?p=0)>

=(z(/p=0)>=0, ^zUp=0) < !. In (a), we plot the first minima

(as a function of preparation time tp) of <^spin, the resulting value of ^R, and the corresponding value of {nzilp=0))

i = {aHO)alo)}) vs ^Aip=0) for the case 7 = 1. As expected,

the small-angle theory is valid for ^j(/p =0)— 1, ({nAO))^0).

In (b), we show the first minimum values (as a function of preparation time tp) of <^spin which have also been minimized

with respect to ^z(tp =0), the resulting value of ^r, and the cor- responding values of ^2 itp =0) vs J.

the squeezing of a single-mode radiation field by coupling

to an ensemble of spins [11,1 3]. It might also be generated in an ensemble of trapped ions where the harmonic oscillator corresponds to the center-of-mass (cm.) oscilla- tion in a particular (z) direction. For simplicity, we consider A' ions trapped along the axis of a linear rf trap [14]. The two-level system is the Zeeman doublet for an unpaired electron on each ion. B' is a gradient field dB^/dz

generated by current /^y in two wires (which could double

as trap electrodes) situated as the positions z = ±zt relative to the ions. We assume the Coulomb interaction be- tween the ions is large enough that the frequencies of the

ions' internal modes are significantly different from w,. The cm. mode is first laser cooled to the zero-point state

[15]. A "squeezed vacuum" state of the cm. mode (appropriate for Fig. 2) could be obtained by suddenly changing the ions' well depth or driving the z oscillation parametrically at Ico^ [12]. A coherent state of nonzero

amplitude (appropriate for Fig. 3) could be created by

suddenly changing the center position of the well or driv- ing the zero-point state with a classical resonant excitation [12]. When po is equal to two Bohr magnetons, we

find il/27r=2Iyzf^ -^{Moijln)'^'^ where ly, zt, M,

and cDz/ln are expressed in amperes, centimeters, atomic mass units, and megahertz, respectively. For 7^=0.1 A,

zr=O.OI cm, Af =24u (^''Mg + ), and w,/2;r = I MHz, we

0.5

0.0

' '

' ' '11111.

1 fR •

f ^^o : ^~^

1

-^ <n^(0)> o|-

^^•^

• 1

q 100

A

10

10

J 100

FIG. 3. Squeezed spin state preparation assuming the in- teraction of Eq. (4b) and assuming the harmonic oscillator is initially prepared in a coherent state [^2(/p=0) = l] with

izitp =0))?iO and (zitp =0)) =0. In (a) we show the first mini- ma of i^spin (as a function of tp), and the resulting values of ^r vs

(nz Up =0)) for 7 = 1 . In (b), we show the first minimum values

(as a function of preparation time tp) of i^spin which have also been minimized with respect to {rizUp =0)), the resulting values

of ^R, and the corresponding values of irizUp =0)> vs J. We find the first minima of i^spin (as a function of tp) occur after Uz) has reached its maximum value and the spin vector is moving back

toward the negative z axis. In this case the squeezed state must

be rotated back to the negative z axis before applying the Ramsey fields.

find n/2;r— 400 Hz. The interesting cooling method pro- posed by Harde [16] uses the same parametric coupling

described here. For Ramsey spectroscopy, the relative phase between the squeezing and Ramsey fields must be

reproducible from measurement to measurement.

When A' is even, there exist correlated states where

^R={N/l+\)~^''^ (see also Ref. [3]) in which case the uncertainty in the measurement of coq will be approximately proportional to TV ~' rather than the usual A'

~'''^. For example, for A' =2 (7 = 1), the |7 = l,A/=0) Dicke

state is maximally squeezed ((J) =0 for this state, but

^R

1/2 as |(J)| —* 0). It can be used in Ramsey spec- troscopy by detecting j} rather than 7^ [17]. For this state the correlations between spins is clear. If one spin is measured to be in the + j

( — y ) state, the other spin

will be found in the — y (+ y ) state. Starting from the |«z=0)|7 = l, A/ = — 1) state, we can prepare this squeezed state by making w^ anharmonic to break the de- generacy of the 5/Jz = 1 transitions. With a classical field we drive to the 1 1 )| I , — 1 ) state and then turn the anharmonicity off. We then apply the interaction of Eq. (4a) to drive to the |0>| 1,0) squeezed state. In summary, it should be possible to improve the

signal-to-noise ratio in spectroscopic experiments which

TN-80
R6800 WINELAND, BOLLINGER, ITANO, MOORE, AND HEINZEN 46

detect the changes of state population over the case where

uncorrelated particles are used. It might also be possible

to demonstrate spin squeezing in individual atoms or ions with / > 1 such as in the Zeeman sublevels of a particular

hyperfine state [17]. The relatively large value of il suggests that cavity-QED problems in the strong-coupling

(weak relaxation) regime might be studied in the ion sys- tem. The correlated states described here could be used in Bell's inequality experiments. A possibility is the creation

of multiparticle correlated states which strongly violate

Bell's inequality [18]. The individual particle states could

be detected with nearly 100% efficiency [6].

We gratefully acknowledge the support of ONR. We

thank M. Kitagawa for sending us copies of the work of

Ref. [4]. D.J.H. acknowledges support from the Sloan

Foundation. We also thank J. Bergquist, J. Kimble, M.

Poitzsch, C. Weimer, and B. Yurke for comments on the manuscript.

[I] See for example the feature issue edited by H. J. Kimble

and D. F. Walls, J. Opt. Soc. Am. B 4, 1450 (1987); special issue edited by R. Loudon and P. L. Knight, J. Mod.

Opt. 34, Nos. 6/7 (1987); H. J. Kimble, in Fundamental

Systems in Quantum Optics, edited by J. Dalibard, J. M.

Raimond, and J. Zinn-Justin (Elsevier, New York, 1991).

[2] M. Xiao, L.-A. Wu, and H. J. Kimble, Phys. Rev. Lett. 59, 278 (1987); P. Grangier, R. E. Slusher, B. Yurke, and

A. LaPorta, ibid. 59, 2153 (1987).

[3] B. Yurke, Phys. Rev. Lett. 56, 1515 (1986); B. Yurke, S.

L. McCall, and J. R. Klauder, Phys. Rev. A 33, 4033

(1986).

[4] M. Kitagawa and M. Ueda, Phys. Rev. Lett. 67, 1852

(1991); in Noise in Physical Systems and Iff Fluctuations, edited by T. Musha, S. Sato, and M. Yamamoto

(Ohmsha, Ltd., Tokyo, 1 99 1 ), p. 355; and (unpublished).

[5] C. K. Hong, S. R. Friberg, and L. Mandel, Appl. Opt. 24, 3877 (1985); B. Yurke and E. A. Whittaker, Opt. Lett.

12, 236 (1987); A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, Phys. Rev.

Lett. 59, 2555 (1987); M. Xiao, L.-A. Wu, and H. J. Kimble, Opt. Lett. 13, 476 (1988); C. D. Nabors and R. M. Shelby, Phys. Rev. A 42, 556 (1990); P. R. Tapster, S.

F. Seward, and J. G. Rarity, ibid. 44, 3266 (1991).

[6] W. M. hano et al. (unpublished).

[7] R. P. Feynman, F. L. Vernon Jr., and R. W. Hellwarth, J. Appl. Phys. 28, 49 (1957).

[8] See for example, F. T. Arecchi, E. Courtens, R. Gilmore,

and H. Thomas, Phys. Rev. A 6, 221 1 (1972).

[9] N. F. Ramsey, Molecular Beams (Oxford, London,

1963), p. 124.

[10] D. F. Walls and P. Zoller, Phys. Rev. Lett. 47, 709

(1981); K. Wodkiewicz and J. Eberly, J. Opt. Soc. Am. B

2, 458 (1985); J. D. Macomber and R. Lynch, J. Chem.

Phys. 83, 6514 (1985).

[ll] See, for example, S. Haroche and J. M. Raimond, Adv.

At. Mol. Phys. 20, 347 (1985); S. Haroche, M. Brune, and

J. M. Raimond, in Atomic Physics 12, edited by J. C. Zorn and R. R. Lewis (AIP, New York, 1991), p. 204; G. Rempe, M. O. Scully, and H. Walther, ibid., p. 219.

[12] D. J. Heinzen and D. J. Wineland, Phys. Rev. A 42, 2977

(1990).

[13] A. Heidmann, J. M. Raimond, and S. Reynaud, Phys. Rev. Lett. 54, 326 (1985); S. M. Barnett and M.-A. Dupertuis, J. Opt. Soc. Am. B 4, 505 (1987); M.-A. Dupertuis, S. M. Barnett, and S. Stenholm, ibid. 4, 1 124 (1987);

M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, and H.

J. Kimble, Phys. Rev. Lett. 59, 198 (1987).

[14] M. G. Raizen et ai, Phys. Rev. A 45, 6493 (1992).

[15] F. Diedrich, J. C. Bergquist, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 62, 403 (1989).

[16] H. Harde, in International Conference on Quantum Electronics Technical Digest, Series 1990, Vol. 8 (Optical Society of America, Washington, DC, 1990), p. 278.

[17] K. Abdullah, C. Carlberg, E. D. Commins, H. Gould, and

S. B. Ross, Phys. Rev. Lett. 65, 2347 (1990).

[18] D. M. Greenberger, M. Home, and A. Zeilinger, in Bell's Theorem, Quantum Theory, and Conceptions of the Universe, edited by M. Kafatos (Kluwer Academic, Dordrecht, The Netherlands, 1989), p. 69.

TN-81
PHYSICAL REVIEW A VOLUME 50, NUMBER 1 JULY 1994

Squeezed atomic states and projection noise in spectroscopy

D. J. Wineland, J. J. Bollinger, and W. M. Itano Time and Frequency Division, National Institute ofStandards and Technology, Boulder, Colorado 80303

D. J. Heinzen

Physics Department, University of Texas, Austin, Texas 78712

(Received 11 January 1994)

We investigate the properties of angular-momentum states which yield high sensitivity to rotation. We discuss the application of these "squeezed-spin" or correlated-particle states to spectroscopy. Transitions in an ensemble of A'^ two-level (or, equivalently, spin-y) particles are assumed to be detected by

observing changes in the state populations of the particles (population spectroscopy). When the particles' states are detected with 100% efficiency, the fundamental limiting noise is projection noise, the noise as- sociated with the quantum fluctuations in the measured populations. If the particles are first prepared in particular quantum-mechanically correlated states, we find that the signal-to-noise ratio can be improved

over the case of initially uncorrected particles. We have investigated spectroscopy for a particular case of Ramsey's separated oscillatory method where the radiation pulse lengths are short compared to the time between pulses. We introduce a squeezing parameter ^^ which is the ratio of the statistical uncertainty in the determination of the resonance frequency when using correlated states vs that when using

uncorrected states. More generally, this squeezing parameter quantifies the sensitivity of an angularmomentum state to rotation. Other squeezing parameters which are relevant for use in other contexts can be defined. We discuss certain states which exhibit squeezing parameters ^j(^N~^^^. We investigate

possible experimental schemes for generation of squeezed-spin states which might be applied to the spec- troscopy of trapped atomic ions. We find that applying a Jaynes-Cummings-type coupling between the ensemble of two-level systems and a suitably prepared harmonic oscillator results in correlated states with^^<l.

PACS number(s): 03.65.Bz, 32.80.Pj, 06.30.Ft

I. INTRODUCTION

Intriguing features of quantum-mechanically correlat- ed particles have become well known through the cele- brated gedanken experiment of Einstein, Poldolsky, and

Rosen (EPR) [1]. The most important of these have been

demonstrated in "EPR-type" experiments which use

correlated photons [2,3]. Optical fields have also provided the basis for numerous studies devoted to nonclassical

harmonic-oscillator states such as squeezed states [4-7].

Demonstrated applications of squeezed optical states in- clude the improvement of the signal-to-noise ratio in in- terferometers [8,9] and absorption spectroscopy [10]. By

contrast, analogous studies of correlated or squeezed

states of material particles are much less common, al- though they may lead to interesting new phenomena and

applications.

First, it should be possible to generate squeezed states of harmonic oscillators which are not associated with the

electromagnetic field. For example, as a result of the in- vestigations devoted to the detection of gravitational waves with macroscopic antennas [11,12], it should be

possible to generate squeezing in a material harmonic oscillator. In atomic physics, it has recently become possible to generate quantized states of motion for trapped

atoms [13-16]. Therefore, it may also be possible to generate states of squeezed position or momentum for these atomic particles [15,17-21]. In addition, interesting studies have been devoted to the (phase) correlations

between the different internal states within the same

atomic particle (as opposed to correlations between

different atomic particles). As a recent example, atten- tion has been focused on atomic states which yield dispersion without absorption and "lasing without inversion"

[22]. In this paper, we examine methods for the generation

of quantum-mechanical correlations between the internal

states of different atomic particles and investigate the use of these correlated atoms in spectroscopy. For example,

consider atoms which have two internal levels with corre- sponding wave functions denoted 1 1 ) and 1 2 ) . For the case of two such atoms, a and b, we will be interested in finding ways to generate correlated or "entangled" states of the form

i/' = [2cosh(20)]"'/2

X(e-^|2)j2),+e''*e^|l)Jl),), (1)

where the subscripts refer to the atoms. (Here, the entan- glement of the particle wave functions is evident because

the total wave function cannot be written as the product

of wave functions of the separate atoms). These states

(for 0^-0, (f)-^TT) are the correlated two-particle states discussed in Bohm's version of the Einstein, Podolsky,

and Rosen experiment [23]. We will be interested in finding ways to generate these states and the analogous

1050-2947/94/50( 1 )/67(22)/$06.00 50 67 ©1994 The American Physical Society

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68 WINELAND, BOLLINGER, ITANO, AND HEINZEN 50

correlated states for much larger numbers of particles.

If we make the connection between spin-y particles

and two-level systems [24], such states can also be called "squeezed-spin" states, where squeezing is defined in analogy to the squeezing for the electromagnetic field. Various aspects of spin squeezing have been considered

previously [25-38]. For example, in the work of Ref.

[26], the relationship between squeezing in a two-level

system and squeezing of the radiation which is emitted by

the two-level system is discussed. Reference [33] considers the generation of correlated or squeezed-spin states, in ensembles of two-level systems, by broadband squeezed vacuum radiation. The use of correlated states in inter- ferometers has been discussed in Refs. [30] and [35]; these

states correspond formally to the squeezing of angularmomentum operators.

We have found it useful to investigate spin squeezing in the context of spectroscopy. Part of the reason for this is that we were led to consider it as a natural outgrowth of our experiments on the spectroscopy of stored atomic

ions [39,40]. Also, the squeezed-spin states that are useful in one physical context or application are not necessarily the ones of interest in another. For example, the

form of spin squeezing discussed in Refs. [25-28,33]

does not, as we describe below, appear to be relevant for the spectroscopy we describe here. However, the method

of spectroscopy we assume is formally equivalent to the

description of particle interferometry discussed earlier

[30,35]. Therefore, the same form of squeezed states is of

interest in both contexts even though the states describe

quite different physical systems and will require different interactions to be produced. The states of interest in spectroscopy and interferometry can be put in a more

general context. In any system which can be represented

by a net angular momentum J, the states we are interest- ed in are those which give the highest angular resolution

of ( J ) about a particular axis. In our experiments on stored ions [39], we detect atomic transitions by observing changes in atomic state population. Typically, we first localize an ensemble of N identical atoms in a trap, where N remains fixed throughout

the experiments. We initially prepare each of the atoms

in the same internal eigenstate, which we take to be state

1 1 ) . We then apply (classical) radiation, which we will

call the clock radiation, to the atoms. The clock radiation has a frequency that drives the atoms from state 1 1

)

to another state, designated state 1 2 ) . After application

of this radiation, an atom is, in general, in a coherent su- perposition state Ci|l)+C2|2), where |c, P+ |c2l'^= 1. We assume that relaxation of states 1 and 2 is negligible;

this is usually a good approximation for stored atomic

ions. We then detect the number of atoms in state 1 1 ) (or

|2 )). In the detection process, we will find the atom to be

in either state 1 or state 2; that is, the measurement can

be thought of as projecting the atom into one of these

states [40]. If we perform this preparation, irradiation,

and detection many times, on average, we will detect

A'^i = |cil^iV atoms to be in state 1. However, unless

|cil = l or 0, the number of atoms found in state 1 will fluctuate from measurement to measurement. We call these fluctuations "projection noise" [40]. They are given

by

A7V, = [Ar|c,P(l-|c,P)]2\ll/2

Recently, we have reduced all other sources of noise in the experiments so that the signal-to-noise ratio is limited

by projection noise [40]. In those experiments, AiVj was

given by the expression above. The ability to see the projection noise clearly was enabled by detecting a fixed number of atomic ions with high efficiency.

Projection noise is the fundamental limiting noise in spectroscopic experiments which detect transitions by

monitoring changes in population on a fixed number of

particles. In Ref. [38], we have investigated, theoretically, ways to increase the signal-to-noise ratio over what

has been observed in Ref. [40], that is, when all of the

atoms are initially prepared in the same internal eigenstate. This can be accomplished if we can initially

prepare the atoms in particular correlated states. The generation and detection of correlated atomic particle states by the methods described below is interesting

because it would allow the investigation of squeezed

states outside the domain of the electromagnetic field. Also, the increase in signal-to-noise ratio possible in spec- troscopy with correlated states may be of practical in- terest. A dramatic example is the following: In atomic

clocks which use uncorrelated atoms and are limited by

projection noise, the signal-to-noise ratio is equal to K(Nt)^^'^, where A'^ is the number of atoms, r is the total measurement time, and K is a constant. However, if the

atoms are initially prepared in particular correlated

states, the signal-to-noise ratio would be equal to KNt^ '^. Therefore an atomic clock using lO"^ correlated atoms

would yield the same precision in 1 s as an atomic clock

using lO"' uncorrelated atoms taking 300 years. On a less ambitious scale, the attainment of modest squeezing

should also be of practical significance. This is particularly true in measurements on atomic clocks where precision measurements take many days, weeks, or even

years of averaging time. Therefore, for example, even if a squeezing of only ^r~\ is obtained ( ^^ defined in Sec. V A), a four-day measurement of a certain precision is re- duced to a one-day measurement.

The purpose of the present paper is to extend the dis- cussion of Ref. [38]. As in that paper, we will be interest- ed in two things: (1) identifying states which reduce projection noise in the context of spectroscopy, and (2) how

these states might be generated in an experiment. We

will concentrate on squeezing in systems of trapped

atoms because it is relevant to our experimental work and

might be where such effects will first be demonstrated.

However, the results are quite general, and the increase

in signal-to-noise ratio will, in principle, apply to spec- troscopy on any ensemble of two-level systems.

In Sec. II, we discuss the signal-to-noise ratio, or

transition-frequency measurement uncertainty, in spec- troscopic experiments which detect transitions by observing changes in level populations. As is often done, we use

the analogy between spectroscopy on two-level systems

and spectroscopy on spin-y systems. This analogy will lead to a useful pictorial representation of the noise and

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50 SQUEEZED ATOMIC STATES AND PROJECTION NOISE IN 69

will make obvious what is desired of the squeezed-spin

states. In Sec. Ill, we apply this analysis to a special case of spectroscopy which uses Ramsey's method of separated oscillatory fields [41]. Here, we discuss the connection

between Ramsey's method and interferometry. In Sec. IV, we examine the signal-to-noise ratio when the particles are initially prepared in particular uncorrected pure

states. This is essentially a summary of the results of Ref.

[40]. In Sec. V, we show that some correlated particle

states can be used in spectroscopy to increase the signalto-noise ratio over that found in experiments which use uncorrelated particles. We define a squeezing parameter

^;j which is relevant for these experiments and also show

that ^n expresses the measurement sensitivity of an angular-momentum state to rotations. In Sec. VI, we dis- cuss alternate definitions of spin squeezing. In Sees. VII

and VIII, we investigate possible ways to generate the

correlated states useful for spectroscopy. The difficulty will be in finding a practical scheme; we discuss some

possibilities in Sec. VIII. Although the discussion will apply to an arbitrary number N of correlated particles, in many cases we will find it useful to illustrate various as- pects of the squeezing for two particles.

II. POPULATION SPECTROSCOPY

We are interested in the spectroscopy of an ensemble of

A'^ identical two-level particles. We assume that transi- tions are detected by observing changes in the populations of the two levels after application of classical radiation. It will be convenient to use the fact that the dynamics of an individual two-level system interacting with ra- diation is the same as the dynamics of a spin-y particle in a magnetic field [24]. This spin representation provides a simple pictorial way to follow the evolution of operators

(or quantum states) under the influence of radiation. For

an ensemble of particles, the spin representation also provides a simple way to visualize the noise in the measure- ment of populations, which will make obvious what is desired of the correlated states discussed in Sec. V.

Therefore, we begin by assuming that we have an en- semble of N identical particles, where N is fixed. Each

particle has spin S where S = ~. Associated with each

spin is a magnetic moment fi=[IqS. We assume that the

particles are far enough apart that they do not overlap

spatially, so that antisymmetrization (or symmetrization

for integral-spin two-level particles) of the total wave

function is unnecessary. We will also assume the particles are far enough apart that the direct dipole-dipole

coupling or other direct interactions between particles can be neglected. Relaxation is assumed to be negligible.

These are good approximations for many spectroscopic

experiments on trapped atomic ions where the storage

time is long, the background gas pressure is low, and the

Coulomb repulsion between ions typically restricts particle separations to greater than 1 /zm.

We apply a uniform external magnetic field BqZ to the

ensemble of spins, so the Hamiltonian for each particle is given by

where coq^ —^oBq/H. The eigenstates of this Hamiltoni- an are the \m) = \+j) and I ~ y ) states where

S^\m)—m\m). The Heisenberg equation for S is aS/8r=6)oXS , (3)

H()~-fi-Bo-ii(Of;S, , (2)

where 6>o — ^o^- ^f the particles are electrons, Ho—gjfJ-B

where gj{^—2) is the electron g factor and /ig is the

Bohr magneton. In this case, a»o>0 and S precesses in the +(f) direction defined by Eq. (3). The upper and lower

energy levels of this two-level system correspond to the

I + Y

) and I ~y ) states, respectively. In spectroscopy, we are interested in experimentally

measuring coq. In our spin-y model, the spectroscopy is essentially equivalent to NMR (nuclear magnetic reso- nance) on a spin-y nucleus or ESR (electron spin reso- nance). In general, we might be interested in the case where the particles are first prepared in a mixed state. In

this case, we would employ a density matrix treatment to describe the evolution of the system. However, since we

will be concerned with maximizing the signal-to-noise ratio, we will assume that the particles are initially

prepared in pure states with an average value of S which,

for most of our discussion, we take to be aligned along

the z axis and denote (S^). {(A) denotes the mean

value of operator A.) We then apply (classical) radiation,

which we call the spectroscopic or "clock" radiation, at a frequency a> (near coq), which changes (S^). The measurement apparatus is configured to detect this change in (S^). We assume the amplitude and duration of the ra- diation is adjusted so that (S^) reverses sign when

An alternative method of performing spectroscopy is to observe the radiation transmitted through the sample

of particles. If the intensity of the radiation is chosen appropriately, the maximum change in the transmitted in- tensity corresponds to the condition o) =coq. In some

cases, for example, when A'^ is large and the clock radiation weak, the fluctuations of the detected radiation

(which limit the signal-to-noise ratio) are dominated by

the fundamental quantum fluctuations of this radiation. In these experiments, the signal-to-noise ratio can be improved by using squeezed radiation [10,42-46]. Here, we have assumed that the radiation used for the spectroscopy is classical: that is, it is in a coherent state, and the

number of photons in the radiation source is large

enough that the quantum fluctuations in this radiation can be neglected. We assume all sources of technical

noise (for example, fluctuations in the amplitude of the

clock radiation caused by an unstable power supply) are

also negligible. Since N is assumed constant, fluctuations

in N, such as those which occur in an atomic-beam exper- iment, are absent. In the measurement of S^ , we assume that each particle

is projected into either the l+y) or |— y) eigenstate

[40]. We will assume that we can detect which state the

particle is measured to be in with 100% efficiency. With

these assumptions, the noise in the measurements of S^

(for the same value of co) from measurement to measure- ment is due to fluctuations in finding particles in either the I + y ) or I — y ) eigenstate.

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70 WINELAND, BOLLINGER, ITANO, AND HEINZEN 50

It will be useful to use a formalism which treats the N

particles as a combined quantum system. This has the

advantage that, for one choice of basis states, it will provide a simple way to visualize the measurement noise

when the particles are either uncorrected or correlated. One possible choice of basis states is the set of direct- product states

Wi./n 1'"'2' N

/ = 1 mi), (4)

where |m, ) is an eigenstate for the rth particle. However, we will use another common representation in which we define a collective angular-momentum operator

N

J=2 (5)

where S, is the spin operator for the /th particle. The set of basis states we use are of the familiar form \J,M),

which are linear combinations of the states in Eq. (4) [47,48]. For these states, J^\J,M)=-J(J +l)\J,M) and

JJJ,M)=M\J,M).

Heisenberg picture

In the presence of the field Bg, Eqs. (3) and (5) yield the

Heisenberg equation for J (see Fig. 1): aj/af=<yoXJ • (6) The clock radiation is assumed to be a classical field, which rotates about the z axis, of the form

B,=5, xcos{(ot + 6) + -.—rysm(cot + 0) (7)

where co^coq and the factor Wq/I'^oI insures that Bi ro- tates in the same sense as 6)o. In general, 5, is also a function of time. We assume that B i is switched rapidly

FIG. 1. Pictorial representation of the precession of the an- gular momentum J about an applied magnetic field B. The an- gular momentum is assumed to have a net magnetic moment

fi=HoJ, where /io>0 for the figure. The figure represents the motion of ( J ) or the operator J in the Heisenberg representation [Eq. (6)].

between and some constant value. In practice, the

clock radiation may be an oscillating (magnetic) field which is perpendicular to Bg. This oscillating magnetic

field can be decomposed into two rotating fields: one which rotates at frequency (o [given by Eq. (7)], and the

other rotating in the opposite sense. Usually the effects of the oppositely rotating field can be neglected [41] (rotating-wave approximation); we assume this to be the case here. To solve for the system evolution, it will be convenient

to transform to a frame which rotates at frequency o) [41,49]. In this frame of reference, the angular momentum J^ will interact with the effective field B = z5,+5i(xcos0+ysin0) , where

B^=BQ + -hai/^Q =BQ((x)Q~o))/a)Q .

Without loss of generaUty, we will assume = v/2, in which case B = z5, + y5, . (8)

In this rotating frame, the Hamiltonian is given by

(9)

where (o' = a)^z. +u>^, (o^ = —fiQB^/ii = (OQ — co, C0^ = —^qB^ /h, and J is now the angular momentum in the

rotating frame. We have omitted an explicit subscript or

superscript on J since only J^ will be measured and J^ is the same in both frames. The Heisenberg equation for J

in the rotating frame is

aj/3r = 6)'XJ (10)

In Fig. 1 we represent, pictorially, the time evolution of J

(or (J)) when jIq, coq —ci>>Q, \o}qi~cl)\^\(o-^\, and

<J(f=0)) = -z|<y,(0))|.

As discussed by Ramsey [41], we could now transform

to a second frame which rotates about B at a frequency —jUgB /^. In this frame, the effective magnetic field is and therefore wave functions (in the Schrodinger picture) or operators (in the Heisenberg picture) remain constant. In this case, the system evolution is obtained by applying

the appropriate rotation operators to the initial (laboratory) wave function or operators. In this paper, we will describe the system evolution in the first rotating frame cor- responding to Eq. (10). We will assume that, after application of the clock ra- diation, the detector measures the number of particles

Nj^ in the \+\) state. In the Heisenberg picture, this is given by the operator

N+{tf)=J,{tf) +JI (11)

where / is the identity operator and tj- is the "final" time

corresponding to the time just after the clock radiation is applied. For a particular value of co, we denote a single

measurement of N^ by {N^)^, and the average of M

measurements of A'^+ by (N+)j^f. By making measure- ments of {N+(tf))nf for various values of co, we obtain a resonance curve as a function of co. To find ojn. we could

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50 SQUEEZED ATOMIC STATES AND PROJECTION NOISE IN . 71

fit this resonance curve to a particular function. The uncertainty in the determination of cOq would result from

the noise in our measurements of {N+(tj-))nf at each

value of (o. In many cases of experimental interest, we can assume

the resonance curve is symmetric about coq] this is true

for the spin-y example discussed here when

<J(f =0)) = -z|<7^(0))|. In this case, rather than

fitting to the entire resonance curve, it will, in general, be

advantageous to measure iV+ at particular values of co which minimize the uncertainty in the apparent position

(as a function of (o) of the curve. For a particular value of

CO, the deviation of the apparent position of the curve from the true curve ( A''_,_ ) is given by

dcoM = [(N+ )M-(N+)]/{d{N+)/d(o) , (12)

where the subscript M on da denotes the results for M

measurements. The magnitude of the rms fluctuations of

8a> for repeated measurements of (A'^_|_ )i at a particular

value of CO is given by

\^co\=^N+{tJ)/\^{N+)/^co\

=u,{tf)/\^{j,(tj))/M , (13)

where, as usual, for operator A, ^A is the square root of

the variance, {AA)^={ A'^) — { aV. If we make M

measurements at a particular value of co, the uncertainty

in the value of coq is reduced to | Aa)|j|^ = | A6)|(M)~'^^.

Given A^ and tj-, our task in spectroscopy is to minimize

|A6l)|. Equivalently, since the signal-to-noise ratio can be

defined as being proportional to |A(y|^', we therefore

want to maximize signal-to-noise ratio. If the exact form of the resonance curve were known, it would be necessary to measure the curve at only one fre- quency to determine the best value of coq. This is impractical since, for example, the height of the resonance curve (a|{j^(0))|) depends on the exact value of 5, and this may not be precisely determined. However, when the resonance curve is symmetric about coq, this difficulty is overcome, experimentally, by measuring two points on

the resonance curve, at frequencies a)^>coQ and cog <coq where co ^—coQ^coQ —cog. If (N _^_{co =co^)) = {N^{co =cog)), then (co ^+cob)/2 =coq. In practice, we approximate this condition with a servo mechanism

[50].

III. POPULATION SPECTROSCOPY

USING THE RAMSEY METHOD

In order to illustrate the basic ideas behind the improvement in signal-to-noise ratio using correlated states, we will examine | Awl for a special case of the resonance method due to Ramsey [41]. This is an important case to analyze because, experimentally, for a given time tj- in

which to apply the clock radiation, the narrowest

linewidths are obtained with this method. The Ramsey

method [41] breaks the resonance period {t=0- 7

mto

three parts. In the special case discussed here, during the

first part of the resonance period, B j is nonzero and constant with value B,^'10 from time f=0 to t = t„n such that 7r/2 n^f^/2 = 7r/2 and D.^ »\co —coq\, where fl^ =/iQ5,o/^ is usually called the Rabi frequency. At time f^/2, Bi is re- duced to zero. After an additional time T, Bi is again

made equal to B^q for a time r^/j and then reduced to zero. We will assume that 5, is switched between zero and 5 10 in a time much less than r^/2 in which case

fy- =2?^/2 + r. We make the further assumption that

With these assumptions, and assuming (J(0)) = —z.\{J^(0))\, during the first part of the Ramsey

period, that is, during the first "Ramsey pulse,"

B =5^z+5,oy — 5,oy. In the rotating frame, from Eq.

(10), J (or (J)) precesses around the y axis until it lies along the x axis at which point 5, is reduced to [Fig.

2(a)]. From time t = t„/2 to ?^/2 + r, J precesses around

B =5^z so that J( ? ^/2 + T ) lies at an angle (coq~co)T with

respect to x but still in the x-y plane [Fig. 2(b)]. In the

third part of the Ramsey period, from time f ^/2 + T to 2t„/2 + T = tj-, J again precesses around Bi^y so that at time tj- it lies in the y-z plane [Fig. 2(c)]. At this point,

A'^+ or, equivalently, J^ is measured [Eq. (11)]. We have

{N+(tf)) =J - {J,{0))cos(coo-co)T , (14)

which is the Ramsey resonance curve for T»t^/2 [51]. In general, we can choose B'^Q = +(n+\)Trfi/{yiQt^/2)

(n =0,1,2,3, ... ) with the same result [52]. Other

choices of B^q will result in smaller values of

1 3 { J^ ( ?y ) ) /bco I . For brevity we will assume « =0. Application of the Ramsey fields corresponds, in the

Heisenberg picture in the rotating frame, to the transfor- mations

Jy{tf)=J^(0)smco,T + Jy(0)cosco,T

(15)

(16)

and

J^(tf)= -J,(0)coso),T+Jy{0)smco,T , (17) where co^=coq—co. Therefore, in general, from Eqs. (11) and(15)-(17),

<A^+(r^))=J-</,(0))cos6>,r+<7^(0))sin&),r

2il/2 =j-[{j,(o)y+{jy{o)y]

Xcos[&),r + tan"'[(^.(0))/(J,(0))]]

(18)

and

A«r=|A7,(0)'cosV7^ + ^-^/0) sin^r

+ [{j(0)){j^{0))-^{j^{0)JJ0)+JJ0)J,{0))]sm2co,T][T({J^(0))smco,T+{J (0))cosco,T)]

-2

(19) TN-86
72 WINELAND, BOLLINGER, ITANO, AND HEINZEN 50

We see from Eq. (18) that the resonance curve is not symmetric about 6^0 unless (J^(0))=0. Even when

{ 7^,(0) ) =0, for initial states where

{J,{0)JA0) +JA0)J^{0))^0 ,

(a) t = - t,,j

(b) t = t,,2 -* t„2 + T

(Wo - ^)T

(c) t = t,„ + T -» 2t,„ + T = t,

FIG. 2. Pictorial representation of the three parts of the Ramsey separated-field method. The figures apply to a frame

which rotates with the applied time-varying field given by Eq.

(7) of the text. Part (a) shows the precession of J during the first "7r/2 pulse." In (b), J precesses about the residual field B^. In

(c), J rotates into the y-z plane under the application of the second 7r/2 pulse. We assume |Bj « |B,| and t„/2 « T, so the

total "interrogation time" tf = t^/2 + T + t^/2=^T. After the second -rr/2 pulse, J^itj-) is measured; this yields the signal used

in spectroscopy [Eqs. (11) and (14)].

the quantum noise on the Ramsey curve AiV+(ry)

[ =AJ^(ry )] is not symmetric about coq. This is not necessarily a problem for finding the best value of coq since the

contribution to the noise in a measurement of

N+{co^) —N+(cog) from this term drops out. If we as- sume

{J,{0}JJ0)+JJ0)J,(0)) = {JA0))=0,

then

I Awl

T^{J,m)h\n^ai,T

(20)

In practice, some additional noise will be present in the

measurement of A'^^. Assuming that this noise is uncorrelated with the projection noise and with o), the noise

A.N+(tj-) in the measurement of A'^+ must be replaced by

[AA^+(r^)HAiV^dd]'^^ where AiV^d^ is the rms value of

the added noise. From Eq. (13), Eq. (20) becomes

\Ao}\

T^{JAQ))hin^co,T

(21)

Equation (21) shows that with Ay^(0)^0 and/or

AA'^add'^0, |A(y| is minimized for the choices (Dq — o) = ±(«+l)7r/r(«=0,l,2,3, . . .) [53]. Independent of the values of A7^(0) and ^N^^^, | Aft;| is minimized for «=0 or equivalently for coq — co = ±Tr/{2T). These values of o)q — (o will be assumed

throughout the remainder of the paper except where noted. In other words, we will measure the resonance curve

at frequencies co ^=(o-\-tt/(1T) and 0)^=0) — tt /(IT)

where |a; —cjqI is assumed to be much less than v/{2T).

In terms of Eq. (14), this means that we measure N^ at two frequencies corresponding to the half-intensity points on the central lobe of the Ramsey resonance curve. With

this assumption on co^ and (Og, |A(y| is independent of

(y,(0)//0) +/^(0)J,(0)) and <J^(0)).

In the remainder, we will assume AA^^^^^j «AJj,(0), in which case we end up with a simple expression for lAwl

[38]: |Aa;|=AJJO)/[rK/,(0))|] (22) We can represent /!^N+{t/ = Ay^(?y)] pictorially.

More generally, we can represent AJ^, AJ^, and AJ^ by

an "error spheroid" (or error ellipsoid) [28] which lies at the end of the vector J and whose dimensions are given

by A/^, AJy, and AJ^. This is shown schematically in Fig. 3 for the case (j) = -|<J)|z. (In Refs. [29], [36],

[38], and [40], AJ^ and AT^, are represented as semimajor

axes of an ellipse placed at the end of, and perpendicular

to, (J).) The pictorial representation shown in Fig. 3

will be useful for the method of spectroscopy described

above, but is somewhat hmited. For example, certain

states with (J) = -|<J)|z and {J,Jy+JyJ,)^0 might

have, in a (primed) coordinate system rotated with

respect to the spheroid shown in Fig. 3, a value of

A.J^<A.J^,A.Jy,A,J^. This state might be more appropriately represented with an error spheroid which is tilted TN-87
50 SQUEEZED ATOMIC STATES AND PROJECTION NOISE IN . . . 73

with respect to the one shown in Fig. 3. Also, when

(J) = — KJ)|z, and \{J)\ ¥^J, the distribution of measured values of J^ is not symmetric about {J^(0)).

Therefore the error spheroid should be distorted along z. A more accurate representation is given by plotting the

probability distributions for N+{tf) which are, in general, asymmetric about (N^itf)} (see, for example. Fig.

3 of Ref. [40]). Other pictorial methods can be used to describe the probability distribution for measurements of

7V+ or J. For example, Ref. [37] provides a way of

visualizing correlated or squeezed-spin states through the use of Wigner functions for angular momentum.

Since we have chosen Q)Q —co = ±7r/{2T), the evolution

of the error spheroid at various times during the Ramsey

period has a simple representation, as shown in Fig. 4. Since the three parts of the Ramsey period correspond to 7r/2 rotations about -Bjoy, B^% and 5,oy, respectively,

the evolution of the variances is particularly easy to fol- low. In Fig. 4(d), we plot {N+(tj^)) as a function of o)

FIG. 3. Pictorial representation of the fluctuations in the measured values ofJ^, Jy, and J^. In this figure < J) is taken to point along the negative z axis. The dimensions of the axes of

the spheroid are AJ^, A7^, and AJ^ where A/,^= <//) — < J, )^ and i=x,y,z.

(a) t = -* tw/2

= AJ,(0)

(b) t = Kn - Kn + T

AL(t,,2 + T)> = AT(0)

(c) t = t,,2 + T - 2t,;2 + T ^ tf

AJ,(tf) = AJ rO)

(d) <N^(g>

N

J + |<J(0)>|

AJ^(t,)=AN^(t,)

J-|<J(0)>|.

FIG. 4. Pictorial representation of the evolution of < J ) and the fluctuations at various parts of the Ramsey method for the condition coo-co=Tr/2T. From these figures we see that the fluctuations in 7V+(f/) are given by AJ^f) or AJ^(O). In part (d) of the

figure, we show the point on the Ramsey curve corresponding to (Oq—(o=tt/2T. AiV+ represents the rms fluctuations for repeated measurements of iV+ for coq-co^tt/IT. For part (d), we have assumed that the initial state is such that < J ) is aligned with the negative z axis but | { J ) | < 7. TN-88
74 WINELAND, BOLLINGER, ITANO, AND HEINZEN 50

[Eq. (14)], the particular point on the resonance curve corresponding to coQ —co = Tr/(2T), and the corresponding

value of the uncertainty in the measurement AN+itj-)

AJyiO).

A. Schrodinger picture

In some cases, it will be useful to represent the system

dynamics in the Schrodinger picture. As before, we as- sume we are in the rotating frame in which Bj is stationary [Eq. (8)]. In this frame, operators will now be as- sumed to be time independent, and the wave function

evolves according to

*(r)=C/(f,0)*(0) (23)

where the evolution operator U is the solution to the

equation i-fidU /dt=H,U [54]. In the case of two particles (7=1), for example, fol- lowing the application of the Ramsey fields, we have

U(tf,0)

(coso)^T—\) —iVlsinco^T (cosd^^T+l)

i'^lsino^T —Icosco^T iV'2sinco^T

icosco^T+l) —iVlsincOi-T (cosco^T—l)

(24)

where we use the notation

V'=a|l,l)+fe|l,0>+c|l,-l) (25)

B. Comparison of Ramsey spectroscopy to interferometry

Yurke and co-workers [29-31] and others [55] have

discussed the connection between a linear lossless passive

device with two input and output ports and the SU(2)

symmetry group. They construct abstract operators

[56] J^=(a\a^+alaO/2,

)"i

32+02^1 )/2, Jy = —i(a\a2

[2^2)' ^^^ N = (a\ai— 0202),

J^ = —i(a\a2—ala] )/2,

where a- and Oj (/ = 1,2) are the creation and annihilation operators for particles (bosons or fermions) entering ports 1 or

2 of the device. For example, the system might be photons injected into a Mach-Zehnder interferometer. These

abstract operators 7, {i=x,y,z) have the same

mathematical properties as the angular-momentum

operators and therefore the interferometer can be de- scribed in terms of this abstract spin space. A'^ corre- sponds to the total number of particles entering both

ports of the interferometer. Beam splitters and

diflFerential phase shifters between the two arms of the in- terferometer behave like rotations of the net spin operator. If we compare our case to this formalism for inter- ferometers, the Ramsey spectroscopy we discuss is for- mally equivalent to a Mach-Zehnder interferometer

where the two Tr/2 Ramsey pulses are identified with 50- 50 beam spHtters in the interferometer, and the phase

shift incurred between the spin precession and the applied field in Ramsey spectroscopy corresponds to the

differential phase shift between the interferometer arms.

If we define the phase sensitivity in Ramsey spectroscopy

as A(f)jf =LcoT, then the sensitivity of the Ramsey method

is equivalent to the phase sensitivity of a Mach-Zehnder

interferometer [see Eq. (13) of this paper and Eq. (3.16) of

Ref. [29]]. Because of this connection, we wish to create the same form of correlated input states as desired for in- terferometers.

IV. SPECTROSCOPY OF UNCORRELATED PARTICLES

In this section, we consider the particles to be initially

uncorrelated and independent. This case is treated in detail in Ref. [40]. Because the particles are assumed to be

uncorrelated, the initial wave function can be written as a

direct product

rl){Q)-- n '^/(o)

'

(26)

1 = 1

where 1/^,(0) is the initial wave function for particle /. By optical pumping techniques we can prepare atoms

in energy eigenstates. Therefore, in our spin-y model, we

will be particularly interested in the case where all of the

particles are initially prepared in, for example, the

j/',(0)=|— y) state. In this case, the initial state is represented by the wave function

xl,(Q)=\J =N/2,M=-N/2) = Y[ l-j),

/ = i

(27)

which is a particular Dicke state [47]. For this state, </,(0)) =

and

J, A/J0) =0, AyJ0) = AJ,(0) = (//2)'''^

{N +(tf))=\N[\+co%(co- 0)^)7] .

Equation (20) is identical to Eq. (22) for all values of co^ since A.J^(0) =0. In this case, the best value of Wq can be

determined independent of where we measure on the res- onance curve. This somewhat surprising result occurs because the quantum noise ANj^{tf) is proportional to the signal slope ^{N+ )/9a) for all values of co. Therefore, from Eq. (13), |Aft>| is independent of our choice of

CO. The noise is equal to zero on the peaks and valleys of

the Ramsey resonance curve because, there, we detect

eigenstates of 7^ [Fig. 5(a)]. Unfortunately, the sensitivity to frequency shifts is also equal to on the peaks and val- leys since the slope of the Ramsey curve is equal to 0. As

discussed in the last section, if AN^^^ is not negligible, we

TN-89
50 SQUEEZED ATOMIC STATES AND PROJECTION NOISE IN . . . 75

minimize |A&)| if we operate at frequencies coq—co = ±(n -\-\)tt/T (n =0,1,2,3, • • •), and, for simplicity, we

have restricted ourselves io (j}Q —co = +\Tr/T.

It is instructive to consider the evolution of the wave

function in the Schrodinger picture. For example, for J = l, our initial wave function is |l, — l) [a=b=Q,

c =1 in Eq. (25)]. After appHcation of the Ramsey fields we have from Eq. (24)

^lj{tf) = \(co%o)J+\)\\,\)^-^%mo),T\\,Q) + \{co%oiJ-\)\\,-\) = (cos6)^r/2| + )] + / smo^T /2\ — ),)(cos&)^r/2| + )2 + / ^inco^T /l\ — )2)

(28)

where, for simplicity, we write | + 1 ) = | + ) and

I — y ) = I — ) . The evolution operator for particle j

satisfies i-fidUj/dt=H^Uj and ^j(tj) is the wave function of particle j after being acted upon by the Ramsey

fields. Therefore, the final wave function is the product

of the final wave functions of the individual particles.

The particles remain uncorrelated (or unentangled) after application of the Ramsey fields. This is true for the application of any form of the (classical) clock radiation and

for any number of particles since the initial wave function

is given by Eq. (26) and U{tj) = X[Uj{tf).

This case (when all particles are initially prepared in eigenstates) serves as a benchmark and has been realized

experimentally. In Ref. [40], we reported spectroscopic

experiments on ^Be"*^ and '^^Hg"*" ions where the signalto-noise ratio was Hmited by the projection noise A7V+ in the measurements. In these experiments, the ions were

prepared in eigenstates corresponding to the initial Dicke

state given by Eq. (27) in our spin-y model. For this case

I A«| is given by Eq. (22), which we denote

1 1 AdJ DS T(2J il/2 TN 1/2 (29)

This value of |A6;| is the minimum that can be obtained

using uncorrelated states in the Ramsey spectroscopy de- scribed above. It is Hmited by the fundamental quantum

fluctuations in the measurement. This limitation on |A<y|

observed in the experiments led us to consider the possibility of using correlated states for spectroscopy where

even smaller values of | A&)| might be obtained.

V. SPECTROSCOPY OF CORRELATED PARTICLES

A. Squeezing parameter for spectroscopy

Before we examine specific states which can reduce

I A(y| below lAcyl^s, we introduce a parameter which indi- cates the level of improvement. As described in the last section, |Aa)|i5s provides a useful benchmark. Therefore we will define a "squeezing" parameter

= {1J)'^^UA0)/\{JA0))\ , (30)

where |Aw|, U.itf), {Jyitf)), AJ^(O), and {J,m) refer to the new states to be considered and the subscript R

denotes that this is the relevant squeezing parameter for the particular form of the Ramsey method of spectroscopy considered. The goal of the paper is to examine ways

to make ^^ less than 1 and as small as possible.

More generally, consider a state with arbitrary (J). Let e denote a direction perpendicular to (J) ((J)-e=0)

such that A(J-e) is a minimum. This state can always be

rotated (by, for example, application of a classical field)

<N^(g>

<N^(g>--

FIG. 5. Plot oi {N+(tf)) and LN+(tf) for three values of a

corresponding to cjq— w=0, v/2T, and tt/T, for 7V=2 (7=1).

For both (a) and (b), ^m= (e-^\\,\) +e^\\,-\))/

[2cosh(20)]'^^. In (a), we assume 0^>-oo, that is, the initial state is the |l, — l) Dicke state. For coq — (o = and v/T, the noise ^N+{tf) = since the corresponding final states

^(fy^) = il,l) and |l, — l) are eigenstates. However, the sensitivity to frequency fluctuations 8< iV+ ( ?/ ) ) /8ai also goes to zero

at these frequencies, so the signal-to-noise ratio is not improved.

In general, for ^(0)=|/, —/) (or \J,J)), the error Aco in our

determination of a)o [Eqs. (13) and (22)] is independent of <y. In

(b), = 0.25. This state leads to a reduced value of Ata for coQ-(o = rr/2T(^K=0.11%). However, for |wo-a>|^Oand tt/T, A<a increases substantially over case (a). TN-90
76 WINELAND, BOLLINGER, ITANO, AND HEINZEN 50

so that e and ( J ) point along the y and negative z axes,

respectively. This state can then be used in Ramsey spec- troscopy with ^^ given by Eq. (30). If we define

A7j^ = A(J-e), a more general definition of ^^ is given by

^;j=(2J)'/2AJ,/KJ)| . (31)

B. Squeezing parameter to indicate sensitivity to rotation We may view the squeezing parameter ^j^ as having a more general geometrical interpretation which indicates

the degree to which we can sense rotations of angular

momentum states. This generalization can then be applied to specific cases such as interferometry [29] or spec- troscopy [38]. To see this, first consider a state which has

{J) = — z|(7^)| (Fig. 3). Suppose we are interested in measuring a small rotation of J about the x axis by the

angle 6. A way to do this is to measure Jy . We define 6

through the relation {Jy)=sind\(J)\. The meansquared noise in is given by (A0)^ =

(d(Jy ) /dd)^. Therefore the precision of the angle measurement is given by Ad = ^Jy/(cosd\{J)\), which is minimized for 6 = ±mr (« =0, 1,2, . . . ). More generally,

for a state with arbitrary (j), consider rotations (pirj) of

(j) about an axis ij where (J)-^ = 0. We can measure

these rotations by measuring J^=a-J where

a-{l)=a-TJ =0. In this case the uncertainty in our measurement of ^(^) is given by

A(^ = AJi/KJ)| (32)

If we compare this angle uncertainty to the value Ac^^g

obtained using the Bloch states (states obtained from the

\J, —J) Dicke states by a rotation [48]), we can define a squeezing parameter indicating sensitivity to rotation as Ai^/A^Dg. We have

A<^/A<^ds = (27)'/2^J^/Kj)|=|^ (33)

Therefore the subscript R can also be used to signify rota- tion and 1"^ the improved sensitivity to rotations using

squeezed-spin states. From this definition of squeezing, we gain a pictorial

representation of what is desired in this kind of

squeezed-spin state. Referring to Fig. 3 and Eq. (30),

(31), or (33), we desire a state where \{J)\ is as long as possible and the error spheroid is compressed as much as possible in a direction perpendicular to ( J ) [along y in Fig. 3(a)], but, at the same time, minimizing AJj^/KJ)|.

As discussed in Sec. V C and in Refs. [30] and [34], some

states show the best squeezing when |(J)|^0. The pictorial representation of such a state is a spheroid nearly

centered on the origin with My(Q) «A7^(0), Ay^(O).

C. A simple example (squeezed states for N =2)

Consider two particles which are initially prepared in the state

V'(0) = [2cosh(2a)]~'/^

X(e-V)il + >2 + eV)i|->2) • (34)

This is a correlated or entangled state of particles 1 and 2

since the wave function cannot be written as a direct product (except for 0^ + oo). Many properties of this kind of state have been considered by Rashid [25], who

investigated states of the form

*'(/,M,0) = C^exp(-07Jexp(-/7r/^/2)|J,M ), where C^ is a normalization constant. The state in Eq.

(34) is, to an overall phase factor, the Rashid state ^'( 1,0,0). It has also been considered in Ref. [32] in the context of the radiation emitted by the two particles. It would be produced by coupling the two particles to a broadband squeezed vacuum [33]. This state has the

properties that {J^{0)) = {JJO))=0, <J,(0)) = -tanh20, Ay^'(0) = i-( 1 +sech20), A//(0) = i-(l -sech20), AJ,(O) = sech20, and ^^ =sech0[(cosh20)/

2]'''^. This state also satisfies the condition

AJ^{0)AJy(0) = -^\{j^{0))\ and is therefore a minimumuncertainty state [25]. When 6 is large, i/'(0)^ 1 1, — 1

)

and ^ji^l; this is just the initial state considered in the

example in Sec. IV. When 6 is small,

|^^(l+0V2)/2'^. Therefore, as 6^0,

^,i^\/V2=l/VN. We can also show that, for / = 1, the state of Eq. (34) is the state which minimizes ^j^

.

In Fig. 5, we plot the Ramsey resonance curve ({N^(tj-)) vs (o) when using this initial state for two

cases, 0-^00 ( |-^ = 1) and = 0. 25 (^;j= 0.728). We also

plot the fluctuations AN+it^-) for values of co given by

((Oq — co)T = 0, tt/I, and w. For (coq — co)T = and n, we

see that plotting AN+itj-) is some what misleading be- cause, for example, values of A'^_|_(ry^) outside the range of

0-2 are not allowed. However, at the primary values of

interest, (ci)Q~co)T = ±Tr/2, AN_^.{tf) gives a reasonable

representation of the fluctuations between measurements.

From Fig. 5(b), the key point is that, although the amplitude of the Ramsey curve becomes smaller, the noise

at the probe frequencies (coQ —co)T — +Tr/2 becomes

smaller more rapidly. Therefore the signal-to-noise ratio

( oc |A&)|^') is increased at these frequencies. It is also clear that the signal-to-noise ratio approaches near the

peaks and valleys of the curve; hence it is important to probe near the half-intensity points.

A physical explanation for the noise reduction is as fol- lows. In the Schrodinger picture, for 0^0, the final wave function (to an overall phase factor) is given from

Eqs. (24) and (34) as

Xl>(tf l,0)=(|+),|-)2+|-)il+)2)/^2

This state has the property that, if J^ is measured for ei- ther particle 1 or 2, we find {J^)=0; that is, it is equally

likely to find the particle in the | + ) or the | — ) state. On the other hand, when we measure J^ sequentially for each particle, we find opposite values of J^ for each particle. For example, if when measuring particle 1 we find it to be in the I + ) i state, then xl)( tj- ) has been projected

into the l + )il~)2 state. Therefore, when 7^ is measured for particle 2, we find it to be in the I — )2 state. This property of correlated states is at the heart of the

correlations observed in two-particle EPR experiments

TN-91
50 SQUEEZED ATOMIC STATES AND PROJECTION NOISE IN 77

[1,23]. The correlations exist even if the particles do not

interact. Letting d-^0 in Eq. (34) clearly shows the correlations

between particles but gives rise to practical problems

since the signal ( « |<y^(0) ) | =tanh20) also approaches 0. More generally, for 6—>-0, <J)-*0 and the signal goes to zero for any form of the (classical) resonance radiation,

since any excitation, Ramsey or otherwise, is equivalent

to a rotation. Therefore, for example, in the presence of

added noise, we want to choose a value of 6 which minimizes |A6j| given by Eq. (21).

D. Squeezed states for other values of N

We first consider N=\ (7 = { ). We would not expect a way to make |;^ < 1 because there is only one particle and

the issue of correlations does not arise. For / = {, the

most general initial pure state can be written in the form

iA(O) =cos0/2e ~"*^2| + ) +sme/2e'^^^\- )

in which case 1^ = ( 1 + tan^^ cos^<t> )

' ''^ > 1

For A^ > 2, states which give rise to minimum values of

^f^ are not immediately apparent. For example, for

A'^ = 3, as the parameters of the wave function are varied,

local minima can be found. Using a minimization pro- gram we find locally minimum states ^(0)^0.935||,|>+0.354|j,-l)

and V(0) = *'(|,i,0), the Rashid state for 6-^0. These

states give locally minimum values of ^j^ with values

^n ^0.76 and_0.66, respectively. Neither of these states has ^^ = 1 /a/AT =0.577.

Finding states which minimize ^j^ becomes more complicated for larger N. NeverthelesSj^ we can guess at states which approximate ^,^ = \/Vn . When N is odd,

consider the relatively simple ^«a/ states (considered pre- viously by Yurke [30] for interferometers)

rfjUf) = —^{\j,\)^\J,-{)) (AT odd (35)

The appropriate initial states can be obtained through ro- tations (for example for reversing the order of the Ramsey pulses)^ These states have the property that ^R={\/VN)[l/(\ + \/N)]. When N is even consider

the final states

1 ^(tf)= —^sina|y, 1 ) -H/ cosa|y,0)

7=^sina|y, — 1 ) (A^ even) V2

-1/2

(36)

For these states, ^^ =( 1 +A^/2)"'^Vcosa. For N = l, this state is equivalent to the Rashid state of Eq. (34). We

see that ^^ is minimized for a^-0, but as noted in Sec. V C, the amplitude of the Ramsey curve also goes to zero

as a^^Q. Agarwal and Puri [34] have shown that the

states of Rashid [25]

vl''(7,M,0) = Cjvexp( -07, )exp( -inJ^ /1)\J,M )

can be used in Ramsey spectroscopy and give minimum

values of ^r=[\^-U^-M^)/J]-^^^ for 0^0. Therefore, for A^ even, ^n=(\+N /l)'^^'^ with M=0. For A^ odd, ^«=[1+(A^^-1)/2A^]''^^ with M= ±\. This

gives a value of ^j^ somewhat smaller than the states of

Eq. (35).

E. Minimum value of ^^ Although the particular Rashid states considered by

Agarwal and Puri [34] give the best value of squeezing of

the states considered above, it is useful to consider the

minimum possible value of ^^ . We find that ^^ ^N~^''^

from the inequalities

>(7/2)'/VA7 >(2J)-'/2 = A^-'^2 _ (37) The first inequality follows from the uncertainty principle

applied to angular momentum, AJy > | ( 7, ) | /2^J^ (see Sec. VIA). The second inequality follows from A,J^ 17.

This last relation can be seen by noting that, if we quantize the angular momentum along the x direction, it is straightforward to show that the states with the largest

value of A7^ are of the form

^,=2-'/2(|7,7),+e"^|7,-7)j,

in which case ^J^ =7. In the states considered above, the

only state which shows this maximum degree of squeezing is the 7 = 1 state considered in Eq. (34).

VI. ALTERNATE DEFINITIONS OF SPIN SQUEEZING

A. Squeezing parameter based on angular-momentum commutation relations

Different definitions of spin squeezing can be used de- pending on the context in which squeezing is considered.

From the commutation relations for angular momentum,

the uncertainty relations between different components of

the angular momentum are given by

A7,A7 >K7,)/2| (38)

and the expressions which follow from cyclic permutations of indices. From these uncertainty relations, it is natural to define squeezed-spin states as states where

AV,<K7,)/2| for i^j. Hence a squeezing parameter

for this "natural" definition of squeezing might be writ- ten as^„=A7,/|(7,.)/2r^ i^jG(x,y,z) (39)

Squeezing in the context of this definition is discussed in Refs. [26-28] and [33]. For example. Walls and Zoller

[26] illustrate the relation between spin squeezing defined

in this way and the squeezing in resonance fluorescence

light from two-level atoms. This squeezing is exhibited

by Bloch states which are states derived from the

|7, —7 ) Dicke states by a rotation. We can see this using

Eq. (10). During the first Ramsey pulse where a'^coiy

(corresponding to a rotation about y), the angularmomentum operators transform according to TN-92
78 WINELAND, BOLLINGER, ITANO, AND HEINZEN 50

JAt)=JAO)cosco,t—J^{0)sinco,t erated by interaction of the spins through nonlinear

and

JAt)=JJO)cosco,t+JAO)sma),t

Therefore, if we define squeezing as ^„=A/^/

|(yJ/2^/^ we find ^„{t ) =cos^ ^^co ^t < I when

^(0)=|/,-/>.

B. Squeezing parameter to indicate tlie degree of correlation As pointed out by Kitagawa and Ueda [36] the

definition of squeezing discussed in Sec. VI A does not

necessarily reflect the correlations between particles. For

example, the Bloch states remain uncorrelated under ro- tation [cf. Eq. (28)] even though they show ^„ < 1 . Kitagawa and Ueda therefore regard the spin to be squeezed

only if the variance of one spin component J^ normal to the mean spin vector is smaller than the variance for a Bloch state (=J/2) [36]. In this view, a spin-squeezing

parameter might be defined as

^5 = A/^/(//2)'/2 ^ (40)

where the i subscript refers to an axis perpendicular to (j) where the minimum value of AJ is obtained. A

squeezed-spin state ( l'^ < 1 ) can be rotated so that

( J ) = ( J^ ) z and A7| = A/^, and could be used in Ramsey

spectroscopy (or interferometry) with ^r=(J /\{Jz)\)isIn Ref. [38], we used a somewhat different definition of l;^ which we called I'spjnVII. GENERATION OF SQUEEZED-SPIN STATES

As discussed in the last section, states for which ^„<l

can be generated by rotations of the \J,±J) Dicke states. Since |^/j = ^s = 1 for these states, we do not consider

them further. The specific correlated or squeezed states considered in Refs. [29-31] and in Eqs. (35) and (36) above were constructed to emphasize the benefits of spin

squeezing in the context of various experiments, but generators for these states were not given.

Rashid [25] investigated the class of angularmomentum states which satisfy the equality in Eq. (38). These minimum-uncertainty or "intelligent" states can be

formed from the \J,M) Dicke states by the transformation of the form

^'{J,M,d} = Cf^exp{-ej,)exp{-iTrJ^/2)\J,M) , where C/y is a normalization constant. These kinds of

states show ^j^^N'^^'^ for 0^-0 as discussed in Sec. VD. Agarwal and Puri have shown [33] that the

^'(J,0,d) Rashid states for integral / can be created by

the interaction of an ensemble of two-level or spin-y particles with broadband squeezed vacuum radiation. Barnett and Dupertuis [32] considered the correlated

states between pairs of particles (1 and 2) generated by a Hamiltonian of the form H = i(g*J^ + J2+ —gJi-Jj-)^

where J +=J^ +iJy and J _—J^ ~^'^y ^^^ ^^^ usual rais- ing and lowering operators. Kitagawa and Ueda [35,36]

considered the squeezed-spin states, for arbitrary A'^, genHamiltonians of the form H =fixJ^ and

H =fix(J\ —J^- )/2i. Both of these interactions lead to

useful squeezing. They have shown that the Coulomb in- teraction between electrons in the two arms of an elec- tron interferometer corresponds to a Hamiltonian of the

form H — fixJ2 and might be used to generate the

squeezed states in that system.

A. Coupling to a harmonic oscillator

In Ref. [38], we considered the spin squeezing pro- duced when an ensemble of two-level particles was coupled to a (single) harmonic oscillator. One reason we

considered this coupling was the prior theoretical demonstration of the complementary eff"ect —harmonicoscillator squeezing through the same coupling to twolevel systems [57] (see also Ref. [21]). The hope was that, for certain initial conditions, the same type of system

might produce spin squeezing. As discussed below, it may be possible to realize coupling to a suitable harmonic

oscillator, whereas we have not thought of a practical

way to realize the couplings discussed by Kitagawa and

Ueda [35,36] for ensembles of two-level atoms.

Therefore, in the laboratory frame, the Hamiltonian we

consider is given by

Hi = -2fi[l{a^ + a)(J++J_)cosco^t ,

(41)

where the first term is given by Eqs. (2) and (5), a and a

are the creation and annihilation operators for the harmonic oscillator of frequency o^ and amplitude

z=ZQ[a-\-a^), and Cl represents the strength of the coupling, which we assume, in general, is sinusoidally modulated at frequency co^. It will be useful to transform to the interaction picture where, for n =0, the operators (or wave functions) are constant in time. In this interaction

picture, the new operators are given by a=a cxpiico^t),

J_=J_ exp(icoQt), and the adjoint expressions. The

operators J are the operators in the rotating frame intro- duced in the subsection of Sec. II. In this interaction pic- ture, the Hamiltonian representing the coupUng becomes

+ aJ+e

Hco^—Oq)!

(42)

where, for convenience, we have dropped the tilde symbols ( ~ ). Resonant interactions occur for two values of

0)^. For oj^^coq and co^ = ±(coq —co^), the Hamiltonian

is Hi=-—nmaJ++a^J_) , (43)

plus high-frequency terms which we neglect (rotating- wave approximation). This is the Hamiltonian of the

Jaynes-Cummings model [58] which is of considerable in- terest in quantum optics. For = ±(a)o +cJz ), the Hamiltonian is

o),^con and «.

TN-93
50 SQUEEZED ATOMIC STATES AND PROJECTION NOISE IN 79

H,= -fin{a^J++aJ. (44) ^l{t) = ^l{0 )cos^ft^ t +sin^ft^ t

plus high-frequency terms which we neglect. When and

o}q=co2, n should be replaced by 2D, in the expressions

for Hi and Hj. For brevity, we will assume co^^^coq. H^

and H2 are essentially equivalent since H^ becomes H2 if we switch the roles of the I + j ) and I — y ) states for each particle.

We now consider the preparation of the spins into

correlated states through the interactions H^ or Hj- The

states which are prepared this way could then be used in Ramsey spectroscopy. For simplicity of notation, we will consider this preparation phase to start at r— 0. However, this time should not be confused with the time when

the first Ramsey pulse is applied.

B. Approximate solution One way squeezing can be imparted to the spins is by

first squeezing the harmonic oscillator and transferring

this squeezing to the spins through fT, [38]. To get a

feeling for how this works without making the problem

mathematically complicated, consider the following special case. The spin is assumed to be initially prepared in the \J,—J) Dicke state and the harmonic oscillator is prepared in a squeezed vacuum state characterized

by (z(r=0)) = <z(0))=0 and Az(0) < Az(coherent

state)=Zo- W^ ^^^ characterize the squeezing of the harmonic oscillator by the parameters ^^(t)^Az(t)/zQ or

^y(f) = Az{f)/(«^Zo). The condition ^^ < 1 indicates amplitude squeezing and ^^ < 1 indicates velocity or momentum squeezing. When Hy applies, the Heisenberg equations of motion are

da /dt = iilJ_ , dJ_ /dt liDaJ, (45)

and the adjoint expressions. We now make the assumption that the initial mean number of quanta {«(0)) = (a (O)a(O)) in the harmonic oscillator is small enough

and/or A'^ is large enough that the value of ( J^ ) does not change appreciably during the time //, is applied. In this case, we make the approximation J^(t)= —JI, where / is the identity operator. With this approximation, the

Heisenberg equations of motion can be solved analytically to give

J ^{t)=J _{0)cosn!^t + iN(n/CLf^)a(0)smnf^t ,

a{t) = a(0)cosi:if^t+iJ_(0)N~^^hmn;^t , and the adjoint expressions, where ft^^iVH^.

these equations, we find

(46)

(47) From

(48)

(49)

^l(t) = f,{0)cos^aj^t+sm^aj^t

(50)

(51)

\I/2/ In Eq. (49), we make the definition ^^^ =(27)'^^A/^/

I (7^ ) |. If a state with ^r^x <^ is produced, it could be

used in Ramsey spectroscopy by first rotating the state by

tt/2 about the z axis so that ^^ ^ ~^^R,y —^r • From Eqs.

(48) and (49), we can create squeezed-spin states by first squeezing the harmonic oscillator (|;^(0) < 1 or ^^^(O) < 1) and transferring the squeezing to the spins. Correspondingly, from Eqs. (50) and (51) we see that the squeezing is "drained away" from the harmonic oscillators as it is transferred to the spins. In this small-angle approximation, squeezing is transferred to the spins like wavefunction exchange between harmonic oscillators [18]. The squeezing is sinusoidally transferred back and forth between the harmonic oscillator and the spins; however,

in a more precise treatment, where we no longer assume

(J^) to be constant, this will no longer be true. We find the same expressions for ^R(t), ^R^x^t), ^^(t), and ^^(t)

when o)^=±(coq +co^) {Hj apphes), t/'(0)= |J, +/), and

if we make the small-angle approximation J^ — +JI.

For Hy, if we instead assume ij;(0)=\J,+J) and make

the small-angle approximation /^(r)= +//, we find

(52)

(53)

(54)

(55) The same expressions are found for Hj if we assume

i/'(0)=|y, —7) and make the small-angle approximation

J^(t)= —JI. In these cases, for short times, no squeezing

is transferred to the spins and the harmonic-oscillator

squeezing degrades.

C. Numerical solution

Because of the limited validity of the small-angle ap- proximation, we have integrated Schrodinger's equation

for Hy and H2 for some special cases. We write the wave

function for the combined harmonic-oscillator -spin en- semble as *c(')=2c„,m(^)I«)I^> (56) «,M

where | n ) are the harmonic-oscillator eigenstates and we

use the shorthand notation \M) = \J ,M) since / is a constant of the motion for i/, and H2. From Schrodinger's

equation, we find for H^

11/2/ rfC,,^/dx=/(« + l)'^^[y(y + l)-M(M-l)]'/X+,,^_, + /«'^^[;(7 + l)-M(M+l)]'^X_,,M + , , (57)

where x = Ctt. We have integrated this equation and the corresponding one for H2 using a fourth-order Runge-Kutta

method [59]. For the initial wave function, we assume the atoms and harmonic oscillator are uncorrelated so that the wave function can be written as a direct product TMj q

,
80 WINELAND, BOLLINGER, ITANO, AND HEINZEN 50

*J0) = ^c„m\n) V'(O) , (58)

where c„(0) are the initial harmonic-oscillator wavefunction coefficients [60] and we will assume xplO) = \J, —J ) or \J, +J). Of course, we must truncate the

basis of harmonic-oscillator states; an adequate number

of states is determined by increasing the basis until the re- sult remains unchanged.

For / = 1, we also solved Schrodinger's equation for

Hi and H2 with a different numerical approach. For

J = l, the Hamiltonians i/j or H2 can be written as blocks of 3 X 3 matrices on the diagonal and zeros else- where. Explicitly, Hi only couples states where « +M is a constant and H2 only couples the states | « — 1 ) | — 1 )

,

|«)|0), and |n + l)| + l). The 3X3 matrices corre- sponding to the Hamiltonian of the coupled states were

diagonalized and the wave function for the combined

harmonic-oscillator -spin ensemble written as a sum over

the eigenstates. The time evolution of the wave function

(q)

2.5

2.0

in 1.5

•My

1.0

0.5

0.0

^R •

^S <n(0)>

-small— angle

theory —

»

- 0.5

o—60.0

2.5

2.0

1.5

1.0

A

cV

0.0

(b)

1.0

•My

0.5

•My

I 1

1 1

1 1 r

1 1 1

' "1

, , ,1

1 1 1 1 1 1 1

1

^R •

1

1 1 1 1 1 1

1

0.5 1.0 1 P (0)

(c)

11 '^ >

1.0<

My

00 \ • /

>

0.3

> A

0.8

a: My

/

\ \ / ; 0.2

) cV

0.7

0.6

0.5

°\ \. y -°

0.1

- ^R • ^0—0-^ - <n>

1

10J

100

1 nt

FIG. 6. Plots of ^R [Eq. (30)] and ^s [Fq- (^)] assuming the squeezed-spin states are prepared by coupling a single harmonic oscil- lator to the spins via the interaction H^ [Eq. (43)]. We assume i/'(0)= | J, —J ) and the harmonic oscillator to be initially prepared in the squeezed vacuum state where (z(0)) = <z(0))=0 and |:^ ( ) < 1 . In (a), J = 1 . We plot the values of the first minima of ^^ and ^5

as a function of time after H^ is turned on [refer to (c)] vs the initial value of ^^(0). We also plot the value of («(0)) = (a^c2 ) vs

^^(0). In addition, we plot the prediction of the small-angle theory, ^« =^5 = |;^(0), as a dashed line. As expected, the small-angle

theory is valid only for very small values of ^^(0) (or (n )). In (b), we plot the first minimum values of ^^ and ^5 less than 1, after /f

,

is turned on, vs J. For each point, i^^iO) has been adjusted to give the smallest value of ^^ and ^5. In (c), we plot ^k, ^S' ^nd ( « ) as a function of time for y = 1 , and |;,(0)=0.60. At ilt^ 1.35, ^b reaches its minimum value of =^0.84. ^ TN-95
50 SQUEEZED ATOMIC STATES AND PROJECTION NOISE IN . .

is then obtained from the time evolution of the eigenstates. With this method a larger number of harmonicoscillator states could be included in the calculation. We

obtained results that agreed with the numerical approach

described above. This method can be generalized for J>1 with the complication that (2/ + l)X(27 + l) matrices would need to be diagonalized. In Fig. 6, we plot 1"^ and ^^ ^^^ ^\ assuming

rp{0)—\J,—J) and assuming the oscillator is initially in an amplitude-squeezed vacuum state where (z(0)) = ( z ( ) ) = and ^^{0)<\. This figure indicates the hmited validity of the small-angle approximation, which is shown as a dashed line. We see that there are states for which l^s < 1 while |^^ > 1. This emphasizes the need to

define squeezing in the context of a particular problem.

From Fig. 6(a), we also see that the minimum values of

^5 and ^ji are obtained for diiferent initial values of

^^(0). In Fig. 6(c) we plot the evolution of ^^, l'^, and

(n ) as functions of time for 7 = 1 and |;^(0) =0.60. Initially, ^ji becomes greater than 1 but eventually reaches a minimum value of 0.840 corresponding to that shown in Fig. 6(a). In Fig. 7, we plot ^j^^ and ^^ for Hj assuming

xp{0)= \J, —J ) and assuming the oscillator is initially in a coherent state where <z(0))^0, (z(0))=0, and ^^(0) = |^y(0)=l. Before using the resulting states in Ramsey

spectroscopy, we want to first rotate { J ) so that it points

along the negative z axis and then rotate the state about

the z axis by 7r/2 so that the squeezing in the x direction

is transferred to the y direction (^R^x~^^R,y—^R ) The

harmonic oscillator, in combination with Hj, drives the

spin ( J) to lie in the y-z plane. This is indicated in Fig.

7(c) where we plot, for 7= 1, ^r^' is,x> and (n ) vs time

and (J(/)) as viewed in the — x direction, assuming the

value of («(0)) which minimizes ^nx- Spin squeezing

along these lines may be of practical interest since we re- quire the initial harmonic-oscillator state to be only a coherent state, not a squeezed state. A squeezed state of the harmonic oscillator can be generated by parametric pumping of the oscillator at fre- quency 2co^. In the laboratory frame this pumping in- teraction can be represented by

Hp{lah)=- mO-p ia+a^ )hm2co, t «

hinlo), t .

If we add this interaction to i^j, the total Hamiltonian in the interaction frame becomes

H-^ = -'nf^{aJ++a^J_) + ifiVLp(a^-{a^9) (59)

plus high-frequency terms which we neglect. In general, we could let fip vary in time. The case considered in Fig.

6 is equivalent to having ftp nonzero and large for a very

short time after r=0 and then allowing the system to evolve under the influence of //[. In Fig. 8, we consider a

special case where ftp and ft are constant in time and applied together. We show a plot of ^p and ^^ ^^ time for J = \, i/'(0) = |l, — l), the harmonic oscillator initially in the vacuum state [<z(0)) = <z(0)) =0, and

^,(0) =|:„(0) = 1], andftp/ft=0.23. This value of ftp /ft

yields a (local) minimum in ^p for these initial conditions.

<n(t=0)>

(c)

2.0

1.5

0.5

0.0

^R •

^S o

y

<n> o -o—

8.0

6.0

A

c

4.0^

2.0

1 2 nt

FIG. 7. Plots of ^^.^ [or ^^, Eq. (31)] and ^s [Eq. (40)] as- suming the squeezed-spin states are prepared by coupling a har- monic oscillator to the spins via the interaction Hi [Eq. (44)]. We assume ;/'(0)= |/, —/) and the harmonic oscillator is ini- tially prepared in a coherent state where (z(0))=5^0, {i(0))=0,

and |:,(0) = ^„(0)=1. In (a), J = \. We plot the values of the

first minima of ^g^ and |;s, as a function of time after H^ is turned on [refer to (c)] vs <«(0)). In (b), we plot the values of

the first minima of ^^ and ^s less than 1 after Hi is turned on vs

J. For each point, (z(0) ) has been adjusted to give the smallest value of |;« and ^s- The harmonic oscillator, in combination

with Hi, drives the spin ( J ) to he in the y-z plane. Before using

the resulting states in Ramsey spectroscopy, we want to first rotate ( J ) so that it points along the negative z axis and then rotate the state about the z axis by tt/I so that the squeezing in the X direction is transferred to the y direction

(^«,x -^^R.y=iR )• In (c), we plot, for 7 = 1, ^r^^, ^s,x, and (n)^ vs time and (J{t)), shown as an arrow, as viewed in the — x

direction, assuming the value of ( «(0) ) which minimizes ^r,;. TN-96
82 WINELAND, BOLLINGER, ITANO, AND HEINZEN 50

1.0

- 0.5 -

en

0.0

-^

o

^/•^*.

-•—•-

-o-o-~.

^R •

1 1 2 Qt

4

^O-o-o

FIG. 8. For / = 1, we plot ^« [Eq. (31)] and ^s [Eq. (40)] vs time assuming the squeezed-spin state is prepared by coupling a parametrically pumped harmonic oscillator to the spins via the

interaction H^ [Eq. (59)]. We assume V'(0)=|l, — 1) and the harmonic oscillator is initially prepared in the vacuum state. The curves use the value ilp/il=0.23 which minimizes (locally) the value of ^r at ilt ^2.5. At certain times, the values of both

^R and ^5 are reduced below those found in Fig. 6(a).

The values of both |^^ and ^^ ^""^ ^^^^ than in the case of

Fig. 6(a).

VIII. POSSIBLE EXPERIMENTAL METHODS

As shown by Agarwal and Puri [33,34], a way to pro- duce good squeezing is to couple the spins to a broadband

squeezed vacuum. However, this may be difficult in practice, because all spatial modes of the field must be

squeezed [61]. These modes are then properly phased

only at a particular spatial location and the atoms must

all be locahzed about this point to within a small fraction

of a wavelength. For optical transitions with stored ions

this is difficult because the Coulomb repulsion typically

results in ion-ion separations of more than 1 jum. Neutral

atoms could be confined with spacings less than X, but

then the direct dipole-dipole coupling must be included.

An alternative strategy might be to confine the atoms in a cavity that is driven by a squeezed vacuum field [61]. From the previous section, another way to produce

spin squeezing is to first prepare a harmonic oscillator in a squeezed or coherent state and then couple the oscilla- tor to the spins through a Jaynes-Cummings-type interac- tion [Eqs. (43) or (44)]. Using the extensive work on

"cavity QED" as a guide, this could be accomplished by

the interaction of an ensemble of atoms with a suitably

prepared electromagnetic field of a single cavity mode

[4-7,62-64]. In these types of experiments, the pres- ence of thermal noise and/or cavity and atomic relaxation would reduce the degree of spin squeezing that could

be achieved. Although we can expect these problems to be overcome in the future, it may be useful to consider al- ternate systems. Our experiments have led us to consider

coupling of the internal levels of atoms to a different harmonic oscillator —that associated with the oscillation of

atoms in a trap. In this paper, we will concentrate on the use of trapped atomic ions, but many of the considerations apply to trapped neutral atoms as well.

A. Trapped-atomic-ion oscillator The interest in a trapped-ion oscillator is due to its potential immunity from relaxation and thermal noise excitation. To a good approximation, a single ion confined in an ion trap [65,66] can be modeled as a charged harmonic

oscillator. For an ensemble of ions localized near the center of the trap this model is also valid for the center- of-mass (cm.) motion of the ensemble [66]. In what fol- lows, we will assume all internal mode frequencies are

shifted away from the cm. frequency by the ions' Coulomb interaction. The cm. charged harmonic oscil- lator is subject to radiative decay and heating from the

environment. It can be considered to be confined in a

cavity formed by the trap electrodes. Typically, the

wavelength of the harmonic-oscillator radiation (corresponding to oscillation frequencies of a few megahertz or

less) is much larger than the dimensions of the trap elec- trodes. A useful representation of this situation is to model the cm. motion of the harmonically bound ion (in one direction) as a series LC circuit which is shunted by

the capacitance of the trap electrodes as shown in Fig. 9

[66]. The resistance r is due to losses in the electrodes

and conductors which connect the electrodes. The

equivalent inductance of the ions is given [66] by

l^md^/NiZq)^ where m is the ion mass, d the characteristic internal dimension of the ion-trap electrodes, N

the ion number, Zq the charge of a single ion (^ =charge

of the proton), and we neglect geometrical factors on the

order of 1. The resistance r both damps and imparts

thermal energy to the ions with time constant I /r. To

characterize the damping and thermal heating, we calculate the time t* for heating the ion's cm. motion from

the « =0 to the n = 1 state,

t* = lfi(o,/{rksT)^2.9{co^/2Tr)d^M(u)/(NrTZ^) ,

(60)

where kg is Boltzmann's constant, T the temperature of

A

C,:^

IONS

FIG. 9. Equivalent circuit representation of one component

(z) of the trapped ions' center-of-mass motion coupled to the surrounding trap electrodes (from Ref. [66]). We have

l^md^/NiZq)^ where m is the ion mass, d the characteristic

internal dimension of the ion-trap electrodes, A'^ the ion number,

Zq the charge of a single ion (9=charge of proton), and we

neglect geometrical factors on the order of unity. The resis- tance r is due to losses in the electrodes and conductors which

connect the electrodes. The equivalent capacitance of the ion, c = 1 /cojl, is typically much less than Cj- which is usually on the order of a few picofarads. As discussed in the text, damping

and thermal heating from r can be neglected in many cases.TN-97
50 SQUEEZED ATOMIC STATES AND PROJECTION NOISE IN 83

resistor r, co^ is the oscillation frequency, and the ion mass M is expressed in atomic mass units. For co^ /27r=

MHz, d = 2 mm, M=100 u, N =2, r =0. IH, r = 300 K,

and Z = 1, we obtain f * ^58 s. In this case, and in general when Hci)^ «kg T, the decay time from n = 1 to « =

is much longer. In the experiment reported in Ref. [13], (co,/2-rr^5 MHz, d ^0.1 mm, M^200 u, N=\, r^300

K, and Z = 1), t* was measured to be 0.17 s even though no care was taken to make r small or to reduce external sources of noise. If we can make Cl large enough

{»l/t*), it should be possible to avoid the effects of

damping and thermal heating on the ion harmonic oscillator. Similarly, it should be possible to avoid radiative relax- ation in the two-level system. This is particularly true if the levels are separated by rf or microwave frequencies

where radiative decay rates can be extremely small [41]. Even at optical frequencies, many forbidden transitions

have radiative decay times on the order of seconds or longer and may be viable as candidates for spin squeezing.

B. Squeezed or coherent states of ion motion

A single trapped ion can be laser cooled to its quantum

ground state of motion [13]. It should also be possible to extend this technique to cool the cm. motion of an en- semble of ions to the ground state. In analogy with

methods used in quantum optics, a squeezed vacuum

state of the cm. mode (appropriate for Fig. 6) could be

obtained from the ground state by suddenly changing the

ion's well depth or parametrically modulating the well depth at 2ci>^ as indicated by the second term in Eq. (59) [17,18]. A coherent state of nonzero amplitude (appropriate for Fig. 7) could be obtained from the vacuum

state by suddenly shifting the center position of the ions'

well, or by driving the oscillator with a classical resonant

excitation [17,18].

Cirac et al. [19] have shown that the ion could be laser cooled and squeezed at the same time by superimposing

the nodes of two standing-wave laser beams at the mean

position of the ions and tuning the laser frequencies to the first lower and upper sidebands of the two-level transition frequency, that is, to ajg — ^j^ and coq +o)^. Independently of the way the harmonic-oscillator state is pro- duced, the phase of the oscillator and/or squeezing must

be referenced to the phase of the clock radiation in a pre- dictable and reproducible way.

C. Possible realizations of the Jaynes-Cummings-model coupling

After the harmonic-oscillator state is prepared, squeezing could be imparted to the spins (or two-level systems)

by application of the Jaynes-Cummings interaction given

by Eqs. (43) or (44). This form of coupling has already

been realized to couple the spin and cyclotron motion of

a single electron in the classic g-2 experiments of

Dehmelt and his collaborators [67]. In Ref. [38], we considered one possible realization of the Jaynes-Cummings

model for atoms. We considered A'^ ions having an unpaired outer electron which are trapped along the axis of

a linear rf trap [68] where, here, we take the axis of the

trap to be in the y direction. The ions are subjected to a homogeneous magnetic field which quantizes the spins

according to Eq. (2). We then superimpose an inhomogeneous field gradient B ' = dB^ /dz whose value, averaged

over the ions' orbits, is zero. However, as the ions oscillate, they experience a motional oscillating magnetic field which tends to flip the spin and reduce or increase the

ions' cm. harmonic oscillator as in Eq. (43) for H^ [69]. The B' field could be generated by a current /y in two

wires (which could double as trap electrodes) situated at the positions z = ±Zj- relative to the ions. We find n/2Tr^2I zfHMco^/2Tr) -1/2 where /,,, Zj, M, and

cl)^/2tt are expressed in amperes, centimeters, atomic mass units, and megahertz, respectively. For 7^=0. 1 A,

Zj = Q>.0\ cm, M= 24 u(^^Mg + ), and 6J, 7277=1 MHz, we

find n 7277 ^400 Hz.

Blockley, Walls, and Risken [70] have shown that the

Jaynes-Cummings model is realized for a harmonically

bound atom or ion which interacts with a traveling-wave

laser tuned near the transition frequency of the atom. They assume the atom is confined to the Lamb-Dicke

limit and the resulting coupling strength is much smaller

than the oscillator frequency [£L«(o^ in our Eq. (41)]. If the laser is tuned to the first upper or lower motional

sideband of the atomic transition, and spontaneous emission from the excited state can be neglected, Rabi oscilla- tions occur between ground and excited states accompanied by oscillations between adjacent harmonicoscillator states. Cirac et al. [71] have shown that the

Jaynes-Cummings model is realized for the harmonic

motion of a two-level ion confined to the Lamb-Dicke

limit whose mean position is located at the node of a standing-wave laser field tuned near the ion's transition

frequency. In both cases, if a suitably narrow optical level (one with weak relaxation) can be used, one might hope

to impart squeezing to optical levels. These schemes

could, in principle, also be applied to transitions of much

lower frequency (infrared or microwave frequencies).

The coupling Cl would be reduced for the same value of

co^ (reduced Lamb-Dicke parameter) but the radiative de- cay could be expected to be considerably reduced.

References [12-1A] and [18] have theoretically considered the use of stimulated Raman transitions to achieve laser cooling of trapped ions to the zero-point energy. (Stimulated Raman transitions have recently been

used to cool free atoms to a kinetic energy less than that corresponding to the recoil of one photon [75].) The sys- tems used to achieve this cooling can also be used to real- ize the Jaynes-Cummings model. Under certain conditions, stimulated Raman transitions produce a Rabi oscil- lation [18] which can be described by an effective JaynesCummings interaction. For convenience, we use the notation of Ref. [18]. Reference [18] considered stimulated Raman laser cooling to proceed by repeated applications of a sequential, two-step process. Figure 10 illustrates the first step

of this process, where we have restricted our attention to harmonic motion (of frequency o^) along one direction

taken to be the z direction. The eigenstates of the system

TN-98
84 WINELAND, BOLLINGER, ITANO, AND HEINZEN 50

FIG. 10. Schematic representation of the energy levels for stimulated Raman transitions between internal (electronic) levels 1 and 2 of the trapped ion. The states are designated by the notation \n,j ) where n denotes the ion harmonic-oscillator level and 7 = 0,1,2 denotes the electronic state, k) and k2 are the wave vectors for the traveling-wave fields which couple elec- tronic level to levels 1 and 2. We assume A»w^. In the

figure, we schematically show the |2,1 )<-*|l,2) stimulated Ra- man transition. Under the assumptions discussed in the text, the stimulated Raman transitions are equivalent to a JaynesCummings-type coupling between electronic levels 1 and 2 and

the harmonic (center-of-mass) motion of the ions.

are denoted by \n,q) = \n)\q)^ where n is the occupation number of the harmonic motion and q =0,1,2

denotes a particular electronic level. Decay from levels |l)g and |2)^ is assumed to be negligible; level lO)^ de- cays at rate y. The ions are irradiated by two laser beams

with classical field amplitudes Ej=Rc\EQjexp[i(kj-r —cojt)]] (7 = 1,2) which couple level \0)^ to levels |l)^

and |2)^ with Rabi frequencies gjQ^\fijQ-EQj\/2ii where

fiQj is the dipole matrix element between states |0)g and

ly/g. The frequencies of the beams are assumed to be

equal to a>^=coQ^ — A —co2+b and co2 =coq2

is the transition frequency between states |rt,0)

\n,j). We assume 0)01,0)02 »A »ct)^ »8. The ions are

initially prepared in the electronic state 1 1 )g. When the

ions are confined to the Lamb-Dicke limit (amplitude of

motion « 1 /kj^ where kj^ = k^ -z), a reduction in the

harmonic-oscillator energy by Hco^ occurs by stimulated

Raman transitions |«,l)^|«,0)^|n — 1,2) and

|n, 1 )->!« — l,0)->|n — 1,2) which reduce n by 1. In the second step of the cooling process, a laser tuned near

o)o2 causes spontaneous Raman transitions from level |2)^ back to |l)^. In the Lamb-Dicke limit, transitions

of the form |« — l,2)^-|n — l,0)-^|n — 1,1) are dominant in this second step. Therefore, after the two steps

A where coqj and

of the cooling process, ( « ) is reduced by 1 . After repeated applications of this two-step process, the ions are eventually cooled until (n ) « 1 [18]. The first step of the cooling process realizes a JaynesCummings coupling between the ions' harmonic oscilla- tor and the internal levels 1 1 )g and |2 )^. If we ensure the condition A»TrY /(2\bkJzQn^^^), where 6/c^ = ^2z ~^iz'

then the ions undergo stimulated Raman transitions with

negligible probability of spontaneous decay from level |0)^ [18]. Under these conditions, the first step of the

cooling process is adequately described by amplitude

equations derived from Schrodinger equation. If we

write the wave function for the system as

*(r)=2 2 ^""^

n g=0

'1,9" ,q) (61)

then Schrodinger's equation leads to the amplitude equations [18]

cfa„,,/ir = -^2 2 ««.,^ "'"''' K,,,n' (62)

n' q' = where fico n,q,n

is the energy of state \n,q) minus the

energy of state ]n',q') and F„^ „.^. is the matrix element

of the perturbation —/i-(E, + E2) between states \n,q)

and \n',q'). If

0;, O) 20 »A»7,g,o,g20.«z

we can adiabatically eliminate the excited states from the

equations. With the additional assumption that

0)^ »gj/J^, and neglecting terms in the equations of

motion that vary with frequency co^ or higher, we arrive

at the amplitude equations

\/2„i&t da„j/dt = iAsia„j-n*n ^^^e a. 1,2 and

da,_i^2/dt = i^s2^n-\,2 + ^*n^^^e -ibK

'",1

(63)

(64)

where O.* =g^Qg2Qbk^z^/^, A^i^gjo/A, and A<^52

=g2o/A. The first terms in these equations are due to the ac Stark shifts of levels 1 1 )^ and |2 )^ by lasers 1 and

2, respectively. If A »|c[;,o — ojjol, we have

Asi = (gio+.?i2)/A, and ^s2 = '<s\o+sl\)/^^ where

g,2 = lAtio'Eo2l/2^ and gj, = |/i2o'Eoil/2^- We can now

make the transformations a„^ = iA„^t\^p{il^s\t) and

a„2^^«,2exp(/A52?)- If we let 8 = A^-, — A52, the equations of motion become

dA„,/dt=iQ.*n'''A '«,! 1,2

and

dA n-1,2 /dt = ia*n'^'A n,\

(65)

(66)

The choice of 8 is that which compensates for the

differential ac Stark shifts in levels ll)^ and \2) ^ and

makes (0^ — 0)2 resonant with the first lower motional side- band of the stimulated Raman transition. These equations are identical to Eqs. (57) (for J = j) if we make the

identification A n,r >C„ 1/2 and A nX

we identify the lO^, and |2)^

'«, + 1/2' that is, if states of Fig. 10 with the

TN-99
50 SQUEEZED ATOMIC STATES AND PROJECTION NOISE IN . 85

I — ) and I + ) states of the spin-| model. Hence, when

spontaneous Raman transitions can be neglected, the sys- tem of Fig. 10 gives rise to the Jaynes-Cummings model

coupling between the two-level system consisting of states |l)^ and |2)^, and the z cm. harmonic motion of the

ions. This causes Rabi oscillations between states | n , 1

)

and \n — 1,2) at frequency ft*« '^^. To indicate possible experimental parameters, we consider the ^Si/2(F=l)^^Pi/2-^^5,/2(F = 2) stimulated

Raman transition in ^Be"*" (1 = 313 nm, (ii2Q —(o-^Q=\.25

GHz). If we assume \bk^\ = \k^\, co^/lv — S MHz,

S\Q^S20 = 1^^ MHz, A/27r = 20 GHz, we find |8A:^ZoN0.21, ft*/27r^l50 kHz, and the probability of

spontaneous emission from the excited state during the

time for complete transfer from the F = \ to F = 1 ground state [18] to be approximately 0.01. If we assume

that |Uio=/i2o~9(0-5 X 10~^ cm) where q is the electron

charge and that the ions are at the center of Gaussian

laser beams with waist u;o = 20 yum, then we require ap- proximately 325 /iW in each beam.

Realizations of the Jaynes-Cummings model using optical transitions are potentially interesting because the

coupling frequencies fl may be much higher than that provided by inhomogeneous magnetic fields acting on

electron spins. In addition, lasers may allow spin squeezing to be applied to atomic levels which are of more in- terest for atomic clocks, such as hyperfine and optical

transitions.

IX. SUMMARY AND DISCUSSION

In this paper, we have discussed the application of

correlated particle states, or squeezed-spin states, to spec- troscopy. Spin squeezing in other contexts has been dis- cussed elsewhere [25-37]. This work extends that of a previous paper which introduced some of the ideas [38]. Transitions are assumed to be excited by classical radiation and detected by observing changes in the state populations of the particles (population spectroscopy). In this case, the fundamental limiting noise is projection noise

[40], the noise associated with the quantum fluctuations

in the measurement of populations. We find that the

signal-to-noise ratio can be improved over the case of ini- tially uncorrelated particles if the particles are first prepared in particular quantum-mechanically correlated or squeezed-spin states. We have considered a particular case of Ramsey's separated-oscillatory-field method of

spectroscopy [41] since it gives the narrowest linewidth

for a given interaction time. We introduce a squeezing

parameter |;^ which is the ratio of the uncertainty in the

determination of the resonance frequency when using

correlated states vs that when using uncorrelated states. This squeezing parameter has more general applicability, in that it gives a measure of the sensitivity of angularmomentum states to rotation. Since one description of

interferometers is formally equivalent to Ramsey spec- troscopy, the squeezing parameter might also be used in that context. Other squeezing parameters which are relevant in other contexts can be defined. We discussed

certain states which exhibit a squeezing |;^ ^iV^'^^. Be- cause of our experimental background in the spectroscopy of stored atomic ions, we have investigated possible

experimental schemes which might yield ^^^ < 1 in this system.

The investigation and demonstration of squeezed-spin

states is interesting for various reasons. These studies ex- tend the realm of squeezing beyond the electromagnetic

field. The ideas apply, in principle, to particle inter- ferometry [29-31,35,36] and spectroscopy. From the

practical side, since some trapped-ion spectroscopy ex- periments are currently limited by projection noise [40], the use of squeezed-spin states would yield more precise

measurements. As a byproduct of these investigations we

have discussed some schemes where it might be possible

to realize the Jaynes-Cummings model [58] by coupling

atomic internal levels to harmonic particle motion. In

practice, this might allow the study of this fundamental

quantum system and related cavity-QED experiments in the regime of weak relaxation and high detection

efficiency. Also, it may be possible to generate correlated

states which would be useful in multiparticle EPR experiments [2,38]. In spectroscopy, if we assume that the state preparation and detection time are small compared to the Ramsey interrogation time T, and if the measurement noise is limited by projection noise, the measurement uncertainty

of the particle's transition frequency Oq, expressed fractionally, is [76]

Aft) h

(OoirNT) 1/2 (67)

where t»T is the total measurement time and ^j^ ac- counts for the use of correlated states. In a particular application, if we require a certain fractional frequencymeasurement precision and if A'^ and T are fixed, then the

time r required to reach this measurement precision is proportional to |;| . The reduction in measurement time

due to |;^ < 1 would be particularly important in many

applications using atomic clocks, where, to reach the

highest measurement precision, the frequency is averaged

over very long times, perhaps years. For states with

^j^=N~^^'^, the required measurement time would be re- duced by A'^. We emphasize the importance of setting and maintaining the phase relationship between the fields which

prepare the squeezed-spin states and the Ramsey fields. In the case where the squeezed-spin state is prepared by

couphng to a harmonic oscillator, we must maintain the

correct phase relationship between the oscillation of the

harmonic oscillator, the coupling D. at frequency co^ , and

the Ramsey fields. When we start from a squeezed- vacuum state of the harmonic oscillator, slight phase er- rors will increase the value of ^^ that is obtained, but

should not affect the accuracy of the measurement since

the correlated state that is created should have

{Jy{0))=0 [see Eq. (18)]. However, in the case where

the squeezed-spin state is created from a coherent harmonic oscillator, it may be difficult to ensure that {Jy{0))=0. The resulting off"set in the Ramsey curve would affect the accuracy (that is, it would give a sys- tematic offset in our determination of Oq), but if the oflFset IN- 100
86 WINELAND, BOLLINGER, ITANO, AND HEINZEN 50

can be maintained constant it will not affect the relative measurement precision attained. After the particles are put in squeezed-spin states, the

correlations exist even though the particles do not in- teract. During the application of the squeezing Hamiltonian, the particles interact through their mutual coupling to the quantized harmonic oscillator(s). After this preparation stage, the particles do not interact even though the correlations remain. When the final particle

states are measured, these correlations between particles are manifested even though, as in the classic EPR experiment, the particles have no way of "communicating" be- tween one another. In Sec. V C, we noted that it may be necessary to work

with states that are not maximally squeezed since, when

^j^ reaches its minimum value, the signal also approaches

[for example, 6-^0 in Eq. (34)]. However, we can re- cover the advantages of squeezing without losing the sig- nal by using a different measurement strategy. We can accomphsh this by measuring higher moments of J^. As

an example, we consider measuring J^ rather than J^ (or iV+) [38,77]. We illustrate the idea for Af = 2 (/=1). Let

V'(0)=|1,0). Using Eq. (24) we obtain

{jHtf)}=sm^(o^T

and

MJ^(tf))- [sinlco^T

(68)

(69) As a function of co (or co^), (J^itj-)) oscillates twice as

fast as (J.itf)) [in Eq. (14)]. In analogy with Eq. (13), the frequency imprecision for a single measurement is given by

\Aco\=MJ^(tf))/\d{J^{t^))/Qco\ = \/2T , (70)

which is_ independent of co. Therefore, effectively,

^R = 1 /y'2 ( = 1 /Vn ) is obtained. For states of higher

J, the maximum sensitivity may be given by measuring

higher moments or combinations of higher moments.

Although we are most interested in states which exhibit quantum correlations between different particles, some

of the interesting features and spectroscopic advantages

of squeezed states can be studied in a different model sys- tem. In particular, consider an individual atom with

7 > y. As a concrete example assume 7=1, as in the

^5i/2 (F= 1 ) ground hyperfine state of '^^Hg'*'. Suppose we are interested in measuring the Zeeman transition fre- quency for AMf = ±1 transitions when the ion is placed

in a weak magnetic field. By weak, we mean a field where

the Zeeman energy is much smaller than the ground-state

hyperfine frequency, so the Zeeman sublevels are split

equally. Suppose the ion is prepared in the \F—\,

M= 0) state. This state is equivalent to the squeezed

state of Eq. (34) for 0^-0. We then apply radiation

which drives the ion from the |l,0) state to the |l,l)

and 1 1, — 1 ) states. We can measure Jj, after application

of the Zeeman radiation, by measuring the probability

that the ion is detected to be in the |l,0) state by

methods similar to those of Refs. [40] and [68]. As outlined in the last paragraph, this effectively gives

^;j=2~'^^. One possible apphcation of spin squeezing

within a Zeeman level is for improved signal-to-noise ratio in electric-dipole-moment (EDM) experiments [77]. Several issues need further investigation. For example, we have not proved that, for a given measurement time

T, the minimum value of Ao) is obtained for the particular method of spectroscopy we have assumed. Even for

this method, other states may more closely approach the

lower hmit of ^r—N~^^'^ than the particular Rashid

states investigated by Agarwal and Puri [34]. In considering the parametrically pumped oscillator [Eq. (59)], we

have not optimized the squeezing for initial conditions or form of flp(t). In considering the application of H^ (or Hj), we have not considered all possible initial conditions, interaction times, or forms of ft(f). In all that we have discussed, we have assumed wave

functions constructed with \J,M ) basis states where

J =N/2. This is because, prior to preparation of the

spin-squeezed states, we assume initial states of the form

\J = N/2,M), and the generators for the squeezed states preserve J. For completeness, we should consider similar

basis states with J <N/2. Since, in general, ^^ is larger

for smaller values of J, we suspect that the smallest values of ^^ are obtained for states with J = N/2. In the spectroscopy we have described, we have as- sumed that, once the initial states are prepared, transi- tions are driven by classical fields. It will be interesting to investigate the signal-to-noise ratio when the spectroscopy is performed with quantized fields. In this case, we

could consider both the case where transitions are detect- ed by observing changes in state population and when

they are detected by looking at changes in the transmitted or scattered radiation. From the experimental side, we are hopeful that simpler practical ways can be found to generate spin

squeezing than the schemes we have discussed. Nevertheless, even if states with only modest squeezing could

be produced they would be very important in spectroscopy and perhaps other applications.

ACKNOWLEDGMENTS

We gratefully acknowledge the support of ONR. We

thank G. S. Agarwal for sending a prepubhcation copy of

Ref [34]. We thank C. Monroe, J. Tan, M. Young, and

P. Zoller for critical comments on the manuscript.

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88 WINELAND, BOLLINGER, ITANO, AND HEINZEN 50

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not dominated by projection noise.

IN- 103
SPIN SQUEEZING APPLIED TO FREQUENCY STANDARDS'

J.J. Bollinger, D.J. Wineland, and Wayne M. Itano Time and Frequency Div. , NIST, 325 Broadway, Boulder, CO 80303

E-mail: jjb@enh.nist.gov

and

D.J. Heinzen

Physics Dept. , University of Texas, Austin, TX 78712

ABSTRACT

In experiments on N atoms initially prepared in uncorrected states, the precision of a

transition frequency measurement (limited only by the quantum fluctuations in the measurement) improves as N"''^. We show that the precision of a frequency measurement

can improve as N"', if the atoms are prepared in particular quantum mechanically

correlated states. These correlated states can be interpreted as squeezed spin states in analogy with the squeezing that has been discussed with reference to the electromagnetic

field. We discuss states which provide the maximum possible precision in a transition frequency measurement and briefly describe possible methods for generating correlated

states.

1. Spin Squeezing and Ramsey Spectroscopy

In this manuscript we consider tlie spectroscopy of N two-level atoms, where

transitions are driven by classical radiation and detected by measuring changes in the

state populations. We assume that the relaxation of the atomic states can be neglected.

The use of correlated states to improve the signal-to-noise ratio in this type of

spectroscopy has been discussed recently under the subject of spin squeezing. ^"^ In

particular, Refs. 1 and 2 discuss spin squeezing for the Ramsey technique of separated

oscillatory fields (SOF). After reviewing this discussion below, we discuss in Sec. 2 the

best squeezed spin states for N=2 atoms. In Sec. 3 we give a state that can be used in

Ramsey spectroscopy to provide a frequency imprecision of (NT)"' where T is the time

of a single measurement. From a time-energy uncertainty relation, we show that this is the best precision possible. Finally in the last section we briefly describe possible

methods for generating correlated or squeezed spin states of ions in a trap.

Let I e) denote the excited state of a two-level atom and | g) the ground state, and

consider the case where N atoms are uncorrelated and each atom is initially prepared in

its ground state |g). Because any two-level system interacting with classical radiation

*Work of the US government; not subject to US copyright.

TN-104
is equivalent to a spin-V2 magnetic dipole in a magnetic field/ the initial state of the

composite system for this case is equivalent to the | J=N/2, Mj=-N/2) state of a J=N/2

spin. This initial (t=0) state has (J,>o=-N/2, (JJo=(Jy)o=0, and AJ,(0)=AJy(0)=N'''V2.

(Here (A)t denotes the expectation value of an operator A at time t and AA(t)=(AA^)t'^%

where AA^=A^-(A)'^.) The Hamiltonian for the equivalent spin system is H= — /i-B

where /i=/XoJ is the magnetic moment of the composite system and B is the applied

field. Here B=BoZ + Bi, where B^ is a time independent field set by the frequency of

the transition {oiQ=-iJi^Jh, where we assume iUo<0), and Bj is an applied oscillating

field used to perform spectroscopy on the atoms. We assume Bj is perpendicular to, and

is rotating about the z axis according to Bi=Bi[-xsinwt + ycoscot]. In the Ramsey SOF

technique,^ Bj is applied (that is, nonzero) for two periods of length t^/2=ir/(2fiR) where

Q^=\ix^x\lh is the Rabi frequency, separated by a period of length T during which

Bi=0. Throughout this manuscript we assume 1]r > > |cOo— ^^1 and T > > t^/2 which

results in a simple expression for the Ramsey lineshape. In a frame of reference rotating

with Bi, the first pulse rotates the spin vector about Bj (the y axis in the rotating frame)

by 90°. The spin vector then precesses about the z axis during the field free period,

acquiring an angle (cOo-ca)T relative to the x axis in the rotating frame. This angle could

be read out by measuring, for example, J^ in the rotating frame. Experimentally this is done by the second 7r/2 pulse, which rotates the spin vector by 90° about the Bi axis,

and then measuring the number of atoms in |e), or equivalently J^. We obtain

(7^),^ = (M2)cos(co,-co)r (1)

where tf=2t^/2+T.

Measurements of J^ along with Eq. (1) can be used to estimate the frequency Wo. Because of the statistical nature of quantum mechanics, the number of particles which

make a transition to |e) from measurement to measurement will fluctuate by ^i^{\.f).^

This produces an uncertainty in the estimate of co^, of | Aco| =^^J^\.f)l\d{^^Jb(j3\. We

obtain

|Aco| = A7/0)/[r|(7^>J] = AT-'/^-i (2)

independent of co. (Experimental measurements are usually made with co — C0o±7r/(2T)

where Eq. (1) has its steepest slope. This minimizes the effect of any added technical

noise.) The fundamental noise limit of Eq. (2) has been achieved with small numbers

of ions in a trap.^ Qualitatively, the dependence of | Aco| on AJy(O) in Eq. (2) can be

understood from the picture that the Ramsey SOF technique measures the accumulated

angle (cjo-w)T of a spin in the x-y plane which is precessing about the z axis. In

particular, spectroscopy with the Ramsey SOF technique is equivalent to determining the

rotation of a spin state (after the first 7r/2 pulse) about the z axis. The precision of this

TN-105
measurement is limited by the uncertainty of the spin normal to the mean spin vector and

in the x-y plane, which here is AJy(O). The idea of spin squeezing is to start with a state {J)=z{J^)q such that AJy(O) < N'^V2, Such a state can be used to improve the frequency

resolution in Ramsey spectroscopy if the squeezing parameter ^r = (2J)'^AJy(0)/ 1 (Jjo I < 1.^ The parameter ^r is a measure of the improvement in Ramsey spectroscopy over

the case where the initial state consists of uncorrected atoms. States with ^r < 1 are

in general entangled; they cannot be written as a product of individual atomic wave

functions. These states are multiparticle versions of the correlated two-particle states discussed in the Einstein-Podolsky-Rosen experiments. Spin squeezing has also been

discussed in the context of interferometry.^"" The discussion there parallels the

discussion here because there is a one-to-one correspondence between the Ramsey SOF

technique and a Mach-Zehnder interferometer.

2. Two-Atom Squeezed States

For simplicity, consider N=2 (J = l) and let the mean spin vector be parallel to the z-axis, (J )=(Jjz. In this case it is straightforward to show that the state with the

best squeezing (lowest value of ^r) is lO =(2cosh(2e)r^{e-'\J=l,m,=l} +^^1,-1)}, (3)

where 6 is determined by (Jj=-tanh(2^). As discussed above, except for the 0-»±oo

limits, this is a correlated state of the two atoms; it cannot be written as the product of

individual atomic states. In addition, AJy^ = y2(l-sech(20)), AJx^ = V2(l + sech(2^)), and

^R^v/2sech^/(cosh(2^))'''\ The smallest (that is, best) values of ^r are obtained for small

6. As ^-*0, ^r-*1A/2. However, as d-M), the mean spin vector (J )=(Jz)z and the SOF

signal also approach 0. The next section indicates how this problem can be solved by

detecting an operator different than J^.

3. Optimal Frequency Measurements

Some squeezed states for N>2 are discussed in Refs. 2 and 3. Reference 3

shows that certain "intelligent" states, discussed in detail by Rashid,'^ provide a squeezing ^^->V2N''^' as N-^oo. By detecting an operator different than J^,, we can fmd

a state with a squeezing ^r=N"'''\ In order to motivate the choice of this state, recall that

the angle (cOo-w)T which is measured in the Ramsey method is, in the rotating frame, just

the phase factor e'^^o"^"^ that the excited state | e) acquires relative to the ground state | g)

during the free precession period T. Consequently it may be possible to improve the

precision of Ramsey spectroscopy by making a coherent superposition (after the first x/2

pulse) of two energy eigenstates whose energies differ by more than ^(coo-w). For N

TN-106
two-level atoms the eigenstates | N/2, -N/2) and |N/2, N/2) provide the largest energy

difference, with an accumulated phase difference of N(a)o-a))T over the free precession

period. It appears that the state

1 1^^) =

{

I NI2,NI2) + I Nil, -NI2)]lV2 (4)

may therefore provide a frequency imprecision of (NT)'\ which is N''^ smaller than that

provided by N independent atoms. However, because (i/'^l J I^Am) = 0» some operator

other than J^ with higher-order tensor components must be detected.

Consider the operator O =.n a^. which is a product of the z-component Pauli spin

matrices for the atoms. For J=N/2 this operator is diagonal in the |J, Mj) basis with

eigenvalues ( — 1/"'^-'. (This follows by expressing | J,Mj) as a sum over all states which

are products of individual spin up or spin down states with J-Mj spins down.) Quantum

jump detection can be used to measure O (or JJ without added noise. "^ The result of

measuring x atoms in the spin up state is assigned the value (-1)^". Suppose at the end

of the first 7r/2 pulse the state 1 0^) is created. After the second -^12 pulse we detect {O)^^ = {i/f\0\\pf) where

• '> r \^.) =e ^'.-'^"•-"^^'H^^)

(5)

Explicit computation gives (0)t^=(-l)^cos[N(cOo-w)T] and, because 0^ = 1,

(A(y)t^=sin^[N(cOo-a))T]. By detecting the operator O, we can therefore use the state

I ^^ in spectroscopy with a frequency imprecision

|Aco| = AO(rp/|a(0),/da)| = {NT)-' (6)

independent of w. The Ramsey method measures cOo by measuring the phase cOoT (we now use the

laboratory frame of reference) which state | e) acquires relative to | g) during the free

precession period; that is, cOq is measured by observing the free time evolution of the

system. With the state 1 1/'^,) a frequency imprecision of (NT)"^ is obtained. We show

that this is in fact the best precision that can be obtained on N identical two-level atoms

undergoing free time evolution. This follows from an application of the time-energy

uncertainty principle

bt'' (AH^y > fiV4 , C^)

where ( A//^) is the variance of the Hamiltonian and 5f- is the variance in estimating time,

derived from measurements of an operator of the system. (In many texts dt is interpreted

TN-107
as a characteristic evolution time of the system and Eq. (7) then relates the uncertainty

in the system's energy to this characteristic evolution time. However dt can also be

interpreted as the uncertainty in determining time from measurements of an operator of

the system. For example, measurements of an operator A can be used to determine time

with an uncertainty AA/|d(A)/dt| . See Ref. 13 for a simple proof of the timeuncertainty relation and Refs. 14-16 for a more rigorous discussion.) For the system of

N identical two-level particles, Eq. (7) can be expressed in terms of dimensionless

quantities

6</)2 <M2> >l/4 (8)

where <l>=03j and

h=Y:{V2\eUe\-V2\gUg\}. (9)

We are here considering the full I'^-dimensional Hilbert space and not just the J=N/2

subspace discussed earlier. We can establish an upper limit (A/z^)<lSP/4 from

{Ah^)={h^)—{hy < {h^) and (h^) < N^/4. The last inequality follows because the maximum

eigenvalue of h^ is lSP/4. From these inequalities and Eq. (8) we have

5</)2> i >J_. (10)

4<A/2 2> N^

For a free time evolution of duration T, Eq. (10) implies that

5a;>J-, (11)

where Sco^ is the uncertainty in determining o)^ from measurements at the end of the

evolution period. Therefore the optimum precision in measuring co^, can be obtained

using Ramsey spectroscopy with 1 1/-^) and detecting the operator O as described above.

We can apply the above results to particle interferometry. In particular, for a Mach-Zehnder interferometer employing n input particles for each measurement, the

desired state corresponding to Eq. (4) (that is, the state after the first beam splitter) is

1 0) = {

I n>a 1 0)b+ 1 0)a 1 n)b}A/2 where a and b denote the two modes after the first beam

splitter. ^'^ In this case, the operator O can be detected by measuring the number of

particles n(b') detected in the b' output mode of the second beam splitter. The result of

such a measurement is assigned the value (-1)°^'\ Equation (6) implies that the phase

sensitivity of the interferometer is given by A0=n'\ and Eq. (11) shows that this is the

maximum sensitivity possible.

TN-108
4. Methods for Making Squeezed States

In this section we briefly discuss several ideas for making squeezed spin states with ions in a trap. (There are also ideas for making squeezed spin states with neutral

atoms. One possibility replaces the center-of-mass mode of the ion cloud discussed

below with an electromagnetic cavity mode.^^) Ion traps provide an isolated environment

for the internal energy states of trapped ions. At first sight, this isolation appears to make correlating the internal states of the trapped ions difficult. However, with cooling,

the translational states of the trapped ions can be strongly interacting. (For example,

with laser cooling, ions in a linear rf trap freeze into a 1-D string.'*-'^) In the schemes

discussed so far for generating correlated states of trapped ions, the internal states of the

ions are coupled to a particular (shared) motional degree of freedom. In Refs. 1 and 2,

quasi-static inhomogeneous fields and stimulated Raman transitions were discussed as possible methods for coupling ground state hyperfine levels of an ion with the center-of- mass (COM) mode of a trapped ion cloud. The ions were assumed to be prepared in

either one of the | N/2, +N/2) states and all ions were assumed to see the same coupling

with the COM mode. (For the stimulated Raman interaction, this will be satisfied if the

laser beam waists and confocal parameters are large compared to the ion cloud.) In this case, in the interaction picture and the rotating wave approximation, the interaction Hi

between the internal states and the COM mode can take the form of a Jaynes-Cummingstype interaction

H^ = -m(J^a+J_a') (12)

where a'^ (a) is the raising (lowering) operator for the COM mode, J+ (J_) is the raising

(lowering) operator for the J=N/2 ladder of atomic states, and Q is the strength of the

interaction. Two different schemes were investigated. In one, the COM mode is

initially assumed to be a coherent state and in the other a squeezed state. The interaction

of Eq. (12) was then applied for a specific time and the resulting squeezing of the spins

calculated. In general, this method results in a mixed state of the ions' internal states and the COM mode, and a squeezing less than the optimal value discussed in the last section. By preparing appropriate (probably nonclassical) states of the COM mode, we

may be able to prepare to prepare the state | i/'m). For example, with N=2 the coupling

in Eq. (12) can be used to evolve the |n = l)|J = l, Mj=-1) state into the |n=0)|J = l,

Mj=0) state. For N=2, the internal state differs from \\p^) by a rotation.

More recently, Cirac and Zoller ^° have discussed the use of a string of N ions

in a linear rf trap for quantum computation and described a method for preparing general

quantum states of this system. Their method i«^ similar to that discussed in the previous

paragraph but differs in that it uses a well focussed laser beam to couple the internal

states of individual ions with a mode of the ion string. The mode is initially prepared

TN-109
in |n=0). The laser beam is then used to address an individual ion and correlate, or

entangle, the internal state of the ion with the mode. The laser beam can then be used

to address a different ion and change the correlation between the ion and the mode to a

correlation between two different ions. Reference 20 lays out the steps needed to prepare

the state Ji/'m)- Here we discuss the simple case of N=2. We assume lasers can be

focussed and used to drive transitions on an individual ion without perturbing the other

ion. (If |g) and |e) are hyperfme levels of an atomic ground state, lasers can be used

to drive a stimulated Raman transition between the states.'^') Suppose that the 2 ions are prepared in the initial state |g)j|g)2|n=0) where the indices refer to the internal states of ions 1 and 2 and | n=0) is the n=0 state of a mode of the 2-ion string. First a 1^12- pulse on the blue sideband of ion 1 takes the |g)i|g)2|n=0) state into the

|g)2{|g)ili^=0)+|e)i|n = l)}A/2 state. This is then followed by a tt pulse on the red

sideband of ion 2. This leaves the |g)i|g)2|n=0) state unperturbed, but takes the

I £)i 1 2)2 1 n = 1) state into the | e)i | e)2 1 n =0) state. In this manner 1 1/'^) can be prepared

for N=2.

5. Acknowledgements

We gratefully acknowledge the support of the Office of Naval Research and the

Army Research Office. We thank Joseph Tan, John Kitching, and Matt Young for their

comments on the manuscript.

6. References

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

Phys. Rev. A46 (1992) R6797.

2. D.J. Wineland, J.J. Bollinger, W.M. Itano, and D. J. Heinzen, Phys. Rev. A50

(1994) 67.

3. G.S. Agarwal and R.R. Puri, Phys. Rev. A49 (1994) 4968.

4. R.P. Feynman, F.L. Vernon Jr., and R.W. Hellwarth, J. Appl. Phys. 28 (1957)

49.

5. N.F. Ramsey, Molecular Beams (Oxford, London, 1963), p. 124.

6. W.M. Itano, J.C. Bergquist, J.J. Bollinger, J.M. Gilligan, D.J. Heinzen, F.L.

Moore, M.G. Raizen, and D.J. Wineland, Phys. Rev. A47 (1993) 3554.

7. As discussed in Refs. 1 and 2, to achieve the full benefit of a squeezed state, the

oscillating field in the x/2 pulses must have the correct phase relative to the

squeezed state.

8. B. Yurke, S.L. McCall, and J.R. Klauder, Phys. Rev. A33 (1986) 4033.

9. B. Yurke, Phys. Rev. Lett. 56 (1986) 1515.

TN-UO
10. M. Kitagawa and M. Ueda, Phys. Rev. Lett. 67 (1991) 1852.

11. M. Kitagawa and M. Ueda, Phys. Rev. A47 (1993) 5138.

12. M.A. Rashid, J. Math. Phys. 19 (1978) 1391; 19 (1978) 1397.

13. A. Messiah, Quantum Mechanics, Vol. I (North-Holland, Amsterdam, 1961) pp.

319-320.

14. C.W. Helstrom, Quantum Detection and Estimation Theory (Academic, New

York, 1976), Chaps. VIII.2.C and VIII.4.C.

15. A.S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (NorthHolland, Amsterdam, 1982), Chaps. III.2, III. 8, IV. 7, and VI. 3. 16. S.L. Braunstein and CM. Caves, Phys. Rev. Lett. 72 (1994) 3439.

17. H.J. Kimble, to appear in Laser Spectroscopy XII, edited by M. Ignuscio, (World

Scientific); M, Brune, F. Schmidt-Kaler, J. Dreyer, A. Maali, and J.M.

Raimond, to appear in Laser Spectroscopy XII, edited by M. Ignuscio, (World

Scientific).

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

Phys. Rev. A45 (1992) 6493.

19. G. Birkl, S. Kassner, and H. Walther, Nature 357 (1992) 310.

20. J.I. Cirac and P. ZoUer, Phys. Rev. Lett. 74 (1995) 4091.

21. C. Monroe, D.M. Meekhof, B.E. King, S.R.Jefferts, W.M. Itano, and D.J.

Wineland, "Resolved-sideband Raman cooling of a bound atom to the zero-point

energy," to be published in Phys. Rev. Lett..

TN-ii;
Volume 75, Number 25 PHYSICAL REVIEW LETTERS 18 December 1995

Demonstration of a Fundamental Quantum Logic Gate

C. Monroe, D. M. Meekhof, B. E. King, W. M. Itano, and D. J. Wineland

National Institute of Standards and Technology. Boulder, Colorado 80303

(Received 14 July 1995)

We demonstrate the operation of a two-bit "controlled-NOT" quantum logic gate, which, in conjunction with simple single-bit operations, forms a universal quantum logic gate for quantum

computation. The two quantum bits are stored in the internal and external degrees of freedom of a single

trapped atom, which is first laser cooled to the zero-point energy. Decoherence effects are identified for the operation, and the possibility of extending the system to more qubits appears promising.

PACS numbers: 89.80.+h, 03.65.-w, 32.80.Pj We report the first demonstration of a fundamental

quantum logic gate that operates on prepared quantum

states. Following the scheme proposed by Cirac and

Zoller [1], we demonstrate a controlled-NOT gate on a

pair of quantum bits (qubits). The two qubits comprise

two internal (hyperfine) states and two external (quantized

motional harmonic oscillator) states of a single trapped atom. Although this minimal system consists of only two

qubits, it illustrates the basic operations necessary for, and

the problems associated with, constructing a large scale quantum computer.

The distinctive feature of a quantum computer is its ability to store and process superpositions of numbers

[2]. This potential for parallel computing has led to the discovery that certain problems are more efficiently

solved on a quantum computer than on a classical computer [3]. The most dramatic example is an algorithm

presented by Shor [4] showing that a quantum computer

should be able to factor large numbers very efficiently.

This appears to be of considerable interest, since the

security of many data encryption schemes [5] relies on the

inability of classical computers to factor large numbers.

A quantum computer hosts a register of qubits, each of

which behaves as quantum mechanical two-level systems

and can store arbitrary superposition states of and 1.

It has been shown that any computation on a register

of qubits can be broken up into a series of two-bit

operations [6], for example, a series of two-bit

"controlled-NOT" (CN) quantum logic gates, accompanied by simple rotations on single qubits [7,8]. The CN

gate transforms the state of two qubits €\ and ei from

|ei)|e2) to ki)|6] © 62), where the © operation is addition modulo 2. Reminiscent of the classical exclusive-OR

(XOR) gate, the CN gate represents a computation at the most fundamental level: the "target" qubit If:) is flipped

depending on the state of the "control" qubit \€\). Experimental realization of a quantum computer re- quires isolated quantum systems that act as the qubits, and

the presence of controlled unitary interactions between the qubits that allow construction of the CN gate. As pointed

out by many authors [6,9,10], if the qubits are not suffi- ciently isolated from outside influences, decoherences can

destroy the quantum interferences that form the computation. Several proposed experimental schemes for quantum

4714

computers and CN gates involving a dipole-dipole inter- action between quantum dots or atomic nuclei [6,7,1 1,12]

may suffer from decoherence efforts. The light shifts on atoms located inside electromagnetic cavities have been

shown to be large enough [13,14] that one could construct a quantum gate where a single photon prepared in the cavity acts as the control qubit [7,15] for the atomic state. However, extension to large quantum registers may be difficult. Cirac and Zoller [1] have proposed a very attrac- tive quantum computer architecture based on laser-cooled trapped ions in which the qubits are associated with in- ternal states of the ions, and information is transferred between qubits through a shared motional degree of free- dom. The highlights of their proposal are that (i) decoherence can be small, (ii) extension to large registers is relatively straightforward, and (iii) the qubit readout can have nearly unit efficiency. In our implementation of a quantum CN logic gate, the

target qubit |5) is spanned by two ^5i/2 hyperfine ground

states of a single ^Be ion (the \f = 2,mf = 2) and

|F = \,mf = 1) states, abbreviated by the equivalent

spin- 1/2 states | ]) and | |)) separated in frequency by

(oq/Itt — 1.250 GHz. The control qubit \n) is spanned

by the first two quantized harmonic oscillator states of the trapped in (|0) and |1)), separated in frequency

by the vibrational frequency (jj^jItt — 11 MHz of the harmonically trapped ion. Figure 1 displays the relevant ^Be^ energy levels. Manipulation between the four basis eigenstates spanning the two-qubit register (|«)|5) =

10)1 i), |0)| T), 1 1)1 i), 1 1)1 D) is achieved by applying a

pair of off-resonant laser beams to the ion, which drives stimulated Raman transitions between basis states. When

the difference frequency 5 of the beams is set near 8 —

a>o (the carrier), transitions are coherently driven between

internal states |5) while preserving \n). Likewise, for 5 —

ctiQ — u>r^ (the red sideband), transitions are coherently

driven between |1)| 1) and |0)| |), and for 5 — wq + f^x (the blue sideband), transitions are coherently driven

between |0)| 1) and |1)| ])• Note that when 5 is tuned to either sideband, the stimulated Raman transitions entangle

|5) with |«), a crucial part of the trapped-ion quantum CN

gate.We realize the controlled-NOT gate by sequentially

applying three pulses of the Raman beams to the ion

TN-112
Volume 75, Number 25 PHYSICAL REVIEW LETTERS 18 December 1995

Detection

(c+)

l0>|aux>

FIG. 1. ^Be^ energy levels. The levels indicated with

thick lines form the basis of the quantum register: internal levels are |5) = \ i) and | T) CSi/jlF = 2,mf = 2) and

^5i/2|F = l,m/r = 1) levels, respectively, separated by

(Oq/Iv = 1.250 GHz), and |aux) = ^SipjF = 2,w/r = 0) (separated from | 1) by —2.5 MHz); external vibrational levels are \n) = |0) and |1) (separated by coJItt - 11.2 MHz).

Stimulated Raman transitions between ^Si/j hyperfine states are driven through the virtual ^Pi/2 level (A — 50 GHz) with

a pair of =313 nm laser beams. Measurement of 5 is accompHshed by driving the cycling | [} —» ^P3/2\F = 3,mf = 3)

transition with cr ^ -polarized light and detecting the resulting ion fluorescence.

according to the following format:

(a) A 7r/2 pulse is applied on the carrier transition. The effect is described by the operator V'^^(77-/2)

in the notation of Ref. [1].

(b) A In pulse is applied on the blue sideband

transition between | t) and an auxiliary atomic (1) level |aux) (see Fig. 1).

(c) A 77-/2 pulse is applied on the carrier transition,

with a 77- phase shift relative to (a), leading to the operator V^^H-ir/l) of Ref. [1]. The 7r/2 pulses in steps (a) and (c) cause the spin |5) to undergo + 1 /4 and - 1 /4 of a complete Rabi cycle,

respectively, while leaving |n) unchanged. The auxiliary

transition in step (b) simply reverses the sign of any

component of the register in the 1 1)| |) state by inducing a complete Rabi cycle from |1)| |) — |0)|aux) — -|1)| ]}. The auxiliary level |aux) is the ^S\/2 \F = 2, m/r = 0)

ground state, split from the | i) state by virtue of a Zeeman shift of —2.5 MHz resulting from a 0.18 mT

applied magnetic field (see Fig. 1). Any component of

the quantum register in the |«) = |0) state is unaffected by the blue sideband transition of step (b), and the effects of the two Ramsey 7t/2 pulses cancel. On the other hand,

any component of the quantum register in the |1)| t) state acquires a sign change in step (b), and the two Ramsey

pulses add constructively, effectively "flipping" the target qubit by tt radians. The truth table of the CN operation

is as follows:

Input state — Output state 10)11)- 10)11)

10)1 T)- 10)1 T) (2) IDID- IDIT)

IDIT)- 11)11).

The experiment apparatus is described elsewhere

[16,17]. A single ^Be"^ ion is stored in a coaxial- resonator rf-ion trap [17], which provides pseudopotential

oscillation frequencies of (tt)^, a»v. a<-)/2 77 — (11.2, 18.2, 29.8) MHz along the principal axes of the trap. We cool

the ion so that the rix = vibrational ground state is occupied ===95% of the time by employing resolved-sideband

stimulated Raman cooling in the x dimension, exactly

as in Ref. [16]. The two Raman beams each contain = 1 mW of power at ==313 nm and are detuned —50 GHz

red of the ^P\/2 excited state. The Raman beams are applied to the ion in directions such that their wave-vector

difference 5 k points nearly along the x axis of the

trap; thus the Raman transitions are highly insensitive to motion in the other two dimensions. The Lamb-Dicke

parameter is rj;^ = Skxo — 0.2, where xq — 7nm is the spread of the n^ = wave function. The carrier

(l«)l 1) — \n)\ T)) Rabi frequency is Cl(/l7r - 140 kHz,

the red (|1)|1)- |0)| T)) and blue (|0)| 1) - |1)| !)) sideband Rabi frequencies are rj^^lo/lir — 30 kHz, and

the auxiliary transition (|1)| t) —* |0)| 1)) Rabi frequency

is 77;ci^aux/27r — 12 kHz. The difference frequency of

the Raman beams is tunable from 1200 to 1300 MHz

with the use of a double pass acousto-optic modulator

(AOM), and the Raman pulse durations are controlled with additional switching AOMs. Since the Raman beams

are generated from a single laser and an AOM, broadening

of the Raman transitions due to a finite laser linewidth is negligible [18]. Following Raman cooling to the |0)| 1) state, but before

application of the CN operation, we apply appropriately

tuned and timed Raman pulses to the ion, which can

prepare an arbitrary state of the two-qubit register. For

instance, to prepare a |1)| 1) eigenstate, we apply a n

pulse on the blue sideband followed by a tt pulse on

the carrier (|0)| 1) -^ |1)| T) — 1 1)1 1))- We perform two

measurements to detect the population of the register

after an arbitrary sequence of operations. First, we

measure the probability P{S =1} that the target qubit

\S) is in the | 1) state by collecting the ion fluorescence when cr"*" -polarized laser radiation is applied resonant with the cycling | 1) —» ^P:i/2\F = 3,m/r = 3) transition (radiative linewidth y/lir — 19.4 MHz at A = 313 nm;

see Fig. 1). Since this radiation does not appreciably

couple to the | f) state (relative excitation probability:

—5 X 10"^), the fluorescence reading is proportional to P{S =1}. For 5 = 1, we collect on average ==1 photon

per measurement cycle [16]. Once 5 is measured, we

4715

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Volume 75, Number 25 PHYSICAL REVIEW LETTERS 18 December 1995

perform a second independent measurement that provides

the probability P{n = 1} that the control bit \n) is in the

|1) state: After the same operation sequence is repeated, an appropriate Raman pulse is added just prior to the detection of 5. This "detection preparation" pulse maps

|/j) onto \S}. For instance, if we first measure S to be i, we repeat the experiment with the addition of a "tt pulse" on the red sideband. Subsequent detection of S resulting

in the presence (absence) of fluorescence indicates that n = (1). Likewise, if we first measure S to be t, we

repeat the experiment with the addition of a 'V pulse" on the blue sideband. Subsequent detection of S resulting

in the presence (absence) of fluorescence indicates that n = 1 (0). In the above measurement scheme, we do not obtain

phase information about the quantum state of the register

and therefore do not measure the complete transformation

matrix associated with the CN operation. The phase

information could be obtained by performing additional operations (similar to those described above) prior to the measurement of S. Here, we demonstrate the key features of the CN gate by (i) verifying that the populations

of the register follow the truth table given in (2), and

(ii) demonstrating the conditional quantum dynamics

associated with the CN operation.

To verify the CN truth table, we separately prepare each

of the four eigenstates spanning the register (|«)|5) =

|0)| i), |0)| T), 1 1)1 i),|l)l T)), then apply the CN operation

given in ( 1 ). We measure the resulting register population

as described above after operation of the CN gate, as shown in Fig. 2. When the control qubit is prepared

in the \n) = |0) state, the measurements show that the gate preserves 5 with high probability, whereas when the

initial control qubit is prepared in the \n) = |1) state, the CN gate flips the value of S with high probability. In

contrast, the gate preserves the population n of the control qubit \n) with high probability, verifying that the register

populations follow the CN truth table expressed in (2). The fact that the measured probabilities are not exactly zero or one is primarily due to imperfect laser-cooling,

imperfect state preparation and detection preparation, and

decoherence effects. To illustrate the conditional dynamics of a quantum

logic gate, we desire to perform a unitary transformation on one physical system conditioned upon the quan- tum state of another subsystem [19]. To see this in the present experiment, it is useful to view steps (a) and (c) of the CN operations given in (1) as Ramsey radiation pulses [20], which drive the \n)\ i) —» \n)\ ]} transition

with the addition of the perturbation (b) inserted between

the pulses. If we now vary the frequency of the Ramsey

pulses, we obtain the typical sinusoidal Ramsey interfer- ence pattern indicative of the coherent evolution between

states |5) = | i) and | t). However, the final population

5 depends on the status of the control qubit \n). This

is illustrated in Fig. 3 where we plot the measured probability P{S = 1} as a function of detuning of the Ram4716

1= Prob.( I S = Prob.(|n=l>)

|0>lt> I |I)|i>

Initial State

|l>lt>

FIG. 2. Controlled-NOT (CN) truth table measurements for

eigenstates. The two horizontal rows give measured final values of n and S with and without operation of the CN gate, expressed in terms of the probabilities P{n = 1} and P{S =1}.

The measurements are grouped according to the initial prepared

eigenstate of the quantum register (|0)| 1), |0)| T), 1 1)1 1), or

1 1)1 t)). Even without CN operations, the probabilities are not

exactly or 1 due to imperfect laser-cooling, state preparation and detection preparation, and decoherence effects. However,

with high probability, the CN operation preserves the value of

the control qubit n, and flips the value of the target qubit 5

only if n = 1.

sey pulses. For initial state |0)| i), we obtain the normal

Ramsey spectrum since step (b) is inactive. For initial state [1)1 i), the Ramsey spectrum is inverted indicating

the conditional control (by quantum bit \n)) of the dynamics of the Ramsey pulses. Appropriate Ramsey curves are also obtained for initial states |0)| t) and |1)| f)- The switching speed of the CN gate is approximately

20 kHz, limited mainly by the auxiliary Itt pulse in step

(b) given in (1). This rate could be increased with more Raman beam laser power, although a fundamental limit in switching speed appears to be the frequency separation of

the control qubit vibrational energy levels, which can be

as high as 50 MHz in our experiment [17]. We measure a decoherence rate of a few kHz in the experiment, adequate for a single CN gate operating at a speed of —20 kHz, but certainly not acceptable for a more extended computation. We identify several sources

responsible for decoherence, including instabilities in the

laser beam power and the relative position of the ion with

respect to the beams, fluctuating external magnetic fields (which can modulate the qubit phases), and instabilities

in the rf-ion trap drive frequency and voltage amplitude.

Substantial reduction of these sources of decoherence can

be expected. Other sources of decoherence that may

become important in the future include external heating

and dissipation of the ion motion [16,21], and spontaneous

emission caused by off-resonant transitions. We note that decoherence rates of under 0.001 Hz have been achieved

for internal-state ion qubits [22]. The single-ion quantum register in the experiment comprises only two qubits and is therefore not useful for computation. However, if the techniques described here are applied to a collection of many ions cooled to the n =

state of collective motion, it should be possible to impleTN-114
Volume 75, Number 25 PHYSICAL REVIEW LETTERS 18 December 1995

1 0)1 4') initial state

1 1)1 4-) initial state

II

Ramsey detuning (kHz)

FIG. 3. Ramsey spectra of the controlled-NOT (CN) gate. The detuning of the Ramsey Tr/2 pulses of the CN gate [steps

(a) and (c)] is swept, and 5 is measured, expressed in terms of

the probability P{S = [}. The solid points correspond to initial preparation in the \n)\S) = |0)| j) state, and the hollow points

correspond to preparation in the \n)\S) = |1)| i) state. The

resulting patterns are shifted in phase by tt rad. This flipping of \S) depending on the state of the control qubit indicates the conditional dynamics of the gate. Similar curves are obtained when the \n)\S) = |0)| t) and |1)| t) states are prepared. The

lines are fits by a sinusoid, and the width of the Ramsey fringes

is consistent with the —50 /isec duration of the CN operation.

ment computations on larger quantum registers. For ex- ample, the CN gate between two ions (m and n) might be

realized by mapping the internal state of the mth ion onto

the collective vibrational state of all ions, applying the single-ion CN operation demonstrated in this work to the nth ion, then returning the vibrational state back to the in- ternal state of the mth ion. (This mapping may be achieved

by simply driving a tt pulse on the red of blue sideband of

the mth ion.) This approach is equivalent to the scheme

proposed by Cirac and Zoller [1,23]. An arbitrary computation may then be broken into a number of such operations on different pairs of ions, accompanied by single qubit ro- tations on each ion (carrier transitions) [6-8].

We are currently devoting effort into the multiplexing

of the register to many ions. Several technical issues remain to be explored in this scaling, including laser- cooling efficiency, the coupling of internal vibrational modes due to trap imperfections, and the unique addressing of each ion with laser beams. Although we can trap

and cool a few ions in the current apparatus, other ge- ometries such as the linear rf-ion trap [24] or an array of ion traps each confining a single ion [25] might be

considered.

This work is supported by the U.S. Office of Naval

Research and the U.S. Army Research Office. We

acknowledge useful contributions from J. C. Bergquist

and J.J. Bollinger. We thank Robert Peterson, Dana

Berkeland, and Chris Myatt for helpful suggestions on the manuscript.

[1] J.l. Cirac and P. Zoller, Phys. Rev. Lett. 74. 4091 (1995).

[2] R.P. Feynman. Int. J. Theor. Phys. 21. 467 (1982); Opt. News 11, 11 (1985).

[3] D. Deutsch, Proc. R. Soc. London A 425, 73 (1989); D. Deutsch and R. Jozsa, Proc. Soc. London A 439, 554

(1992).

[4] P. Shor, in Proceedings of the 35th Annual Symposium

on the Foundations of Computer Science (IEEE Computer

Society Press, New York, 1994), p. 124.

[5] R. L. Rivest, A. Shamir, and L. Adelman, Comm. ACM,

28, 120(1978).

[6] D.P. DiVincenzo, Phys. Rev. A 51, 1015 (1995).

[7] A. Barenco, D. Deutsch, A. Ekert, and R. Jozsa, Phys. Rev. Lett. 74,4083 (1995).

[8] A. Barenco, C.H. Bennett, R. Cleve, D.P. DiVincenzo,

N. Margolus, P. Shor, T. Sleator, J. Smolin, and H. Weinfurter, Phys. Rev. A 52, 3457 (1995).

[9] R. Landauer, in Proceedings of the Drexel-4 Symposium

on Quantum Nonintegrability-Quantum Classical Corre- spondence, edited by D. H. Feng and B-L. Hu (International Press, Boston, to be published).

[10] W.G. Unruh, Phys. Rev. A 51, 992 (1995).

[11] K. Obermeyer, W.G. Teich, and G. Mahler, Phys. Rev. B

37,8111 (1988).

[12] S. Lloyd, Science 261, 1569 (1993).

[13] M. Brune, P. Nussenzveig, F. Schmidt-Kaler, F. Bemardot, A. Maali, J.M. Raimond, and S. Haroche, Phys. Rev.

Lett. 72, 3339 (1994); M. Brune, F. Schmidt-Kaler, J. Deyer, A. Maali, and J. M. Raimond, "Laser Spectroscopy

XII, " edited by M. Inguscio (World Scientific, Singapore,

to be published).

[14] H.J. Kimble, in "Laser Spectroscopy XHI" Ref. [13]. [15] T. Sleator and H. Weinfurter, Phys. Rev. Lett. 74, 4087

(1995).

[16] C. Monroe, D.M. Meekhof, B.E. King, S.R. Jefferts, W.M. Itano, D.J. Wineland, and P. Gould, Phys. Rev.

Lett. 75, 4011 (1995).

[17] S.R. Jefferts, C. Monroe, E.W. Bell, and D.J. Wineland,

Phys. Rev. A 51, 3112 (1995).

[18] J. E. Thomas, P. R. Hemmer, S. Ezekiel, C. C. Leiby, R. H.

Picard, and C. R. Willis, Phys. Rev. Lett. 48, 867 (1982).

[19] A. Ekert and R. Jozsa, Rev. Mod. Phys. (to be published).

[20] N. F. Ramsey, Molecular Beams (Oxford University Press, London, 1956).

[21] F. Diedrich, J.C. Bergquist, W.M. Itano, and D.J. Wineland, Phys. Rev. Lett. 62, 403 (1989).

[22] J.J. Bollinger, D.J. Heinzen, W.M. Itano, S. L. Gilbert, and D.J. Wineland, IEEE Trans. Instrum. Meas. 40, 126

(1991).

[23] The controlled-NOT operator proposed in Ref. [1]

is Vy-(7T/2)Ul;°U;-Wl;°VyH-Tr/2), adopting their notation. This is equivalent to the controlled-NOT

operator proposed here between ions m and «, U!„'>vy-{TT/2)U;'vyH-TT/2)Ul;'', since V„ and

Urn commute.

[24] M.G. Raizen, J.M. Gilhgan, J.C. Bergquist, W.M. Itano, and D.J. Wineland, Phys. Rev. A 45, 6493 (1992).

[25] F. Major, J. Phys. (Pans) Lett. 38, L221 (1977); R. Brewer, R. G. DeVoe, and R. Kallenbach, Phys. Rev. A

46, R6781 (1992).

4717

TN-115
PHYSICAL REVIEW

LETTERS

Volume 70 19 APRIL 1993 Number 16

Young's Interference Experiment with Light Scattered from Two Atoms

U. Eichmann/^) J. C. Bergquist, J. J. Bollinger, J. M. Gilligan, W. M. Itano,

and D. J. Wineland

National Institute of Standards and Technology, Boulder, Colorado 8030S

M. G. Raizen

Department of Physics, University of Texas, Austin, Texas 78712

(Received 18 December 1992)

We report the first observation of interference effects in the light scattered from two trapped atoms. The visibility of the fringes can be explained in the framework of Bragg scattering by a harmonic

crystal and simple "which path" considerations of the scattered photons. If the light scattered by

the atoms is detected in a polarization-sensitive way, then it is possible to selectively demonstrate

either the particle nature or the wave nature of the scattered light.

PACS numbers: 03.65.Bz, 32.50.+d, 32.80.Pj, 42.50.-p

Young's two-slit experiment [1], in the context of waveparticle duality, is often regarded as a paradigm for quantum phenomena. To some it "has in it the heart of quantum mechanics. In reality it contains the only mystery"

[2]. In this Letter we report, for the first time, a version of

Young's experiment where we detect the interference of

weak laser light scattered from two localized atoms which

act as two slits. The visibility of the interference fringes

has a simple interpretation in terms of Bragg scattering

and "which way" arguments based on the changes of the atoms' internal states. Interference of light scattered from atoms has received

recent attention because it has provided information on

the degree of localization of laser-cooled neutral atoms in optical wells [3,4]. Optical interference has also been observed in the time-resolved atomic fluorescence following

the photodissociation of a molecule [5]. Although fixed

numbers of atomic ions have been localized in ion traps

[6-9] , interference has not been previously reported due,

in part, to inadequate localization. In the experiments

described here, we observe interference of light scattered

from two ^^®Hg+ ions localized in a linear Paul trap [10]. The experiment is shown schematically in Fig. 1. Using established procedures, we trapped two ^^®Hg+ ions

along the axis of a linear trap [10]. The ions were irradiated by a linearly polarized, traveling wave laser beam tuned below the resonance frequency of the ^^^Hg"*"

65 ^5i/2-6p ^Pi/2 transition at 194 nm. The beam serves two functions in the experiment. It reduces the ions' kinetic energy by preferentially imparting photon mo- mentum to them in a direction that opposes their motion (Doppler laser cooling) [11]. The low temperatures

strongly localize the ions in our trap. The beam also

FIG. 1. Schematic diagram of the experiment. Scattered

light from two ions (represented by small circles near the center of the trap) was imaged onto detector D2 via the collecting

lens L, aperture A, and an optional polarizer P. The polarizer was a UV-absorbing glass plate oriented at Brewster's angle

(see text). Detector Di served as a monitor of ion number.

© 1993 The American Physical Society 2359

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Volume 70, Number 16 PHYSICAL REVIEW LETTERS 19 APRIL 1993

acts as the light source for the interference experiment.

It was directed through the center of the trap at an angle ^=62° with respect to the trap axis (z axis). The

beam waist was about 50 /xm and the laser power could

be varied to a maximum of about 50 /zW. A 7.77 MHz

rf drive with an amplitude Vq — 1 kV confined the ions

radially while a static voltage Uq applied to both ends of

the trap provided axial confinement. Scattered light was

observed with two position-sensitive imaging detectors

Di and D2 (Fig. 1). On one side, a lens system pro- duced a real image of the ions on detector Di [10]. This

allowed for continuous monitoring to make sure that ex- actly two ions had been trapped. On the opposite side,

detector D2 was set up to measure the intensity of the

light as a function of the scattering angle. This scat- tered light was collected with an //I lens and imaged

with a magnification of 4.7 onto an aperture 300 ^m in diameter. The light that passed through the aperture was directed onto the position-sensitive photocathode of

detector D2 about 0.1 m from the aperture. Although

the interference fringes could, in principle, be detected

using D2 alone, the lens and aperture suppressed background stray light firom reaching the detector. With this configuration, detector D2 covered an in-plane detection

angle (p = 10° to 45°, where (p is defined with respect

to the 194 nm beam direction and the plane is defined

by the traps z axis and the 194 nm beam. The out-of- plane detection angle $ ranged from —15° to 15°. The

ion separation d could be adjusted by varying the axial

voltage Uq- The separation can be calculated with our

trap parameters using the formula d — 19.4Uq '

, where

d is in //m and Uq is in volts [8]. In the experiment,

Uq was varied between 10 V and 200 V, which produced

spacings from 9 fim to 3.3 /xm, respectively. At these separations, the dipole-dipole interaction between ions can be neglected.

Figure 2 shows examples of the experimental results with polarization-insensitive detection. The fringes were

recorded at three different ion separations: 5.4 /xm,

4.3 /xm, and 3.7 /xm. The fringe spacing increases with

decreasing ion separation, as expected. Additionally, the

firinge contrast is highest close to the forward scattering

direction and deterioriates with increasing 0. At higher

angles (f), the scattered light loses its coherence due to the residual ion motion.

We have performed calculations that quantitatively describe the fringe contrast and spacing. These calculations

proceed along the same lines as those for Bragg scattering

by a harmonic crystal [12] and use Fermi's "golden rule" to determine the scattering rate from an initial state to a final state. Since our detector does not diflFerentiate between elastically and inelastically scattered light, both

processes contribute to the detected interference signal.

We have calculated the loss of contrast caused by the

residual ion velocity at our ion temperatures and find it to be negligible. Also, loss of fringe contrast due to the

resulting variations in the energy of the scattered light is 2360

$

FIG. 2. Interference pattern for three different ion separ

rations: (a) 5.4 ^m, (b) 4.3 ^m, and (c) 3.7 /um. The two

white spots are caused by stray reflections of the laser beam.

negligible. We obtain the scattered light intensity as a

function of momentum transfer ^q = ^(kout — kjn):

7(q) = 27o(l + cos(g,d)exp{-([q- (ui - U2)]2)/2}),

(1) where kout and kjn are the scattered and incident photon wave vectors and Iq represents the scattered intensity of a single ion (assumed to be equal for both ions).

The brackets () denote an ensemble average and Uj de- notes the displacement of the ith ion from its equilibrium position. We assume that the positions of the ions are characterized by a thermal distribution of the nor- mal modes of the two ions in the trap. These modes are the stretch mode along the trap axis, two rocking or tilt modes normal to the trap axis, and the center-of-mass

modes. Excitation of these modes during the scattering process gives rise to inelastic light scattering similar to phonon excitation processes in Bragg scattering by crystals. If we could measure only the elastically scattered

contribution, it would have 100% fringe contrast independent of the momentum transfer, but the intensity would

be weighted by an overall exponential factor (the DebyeWaller factor) as it is in the case of elastic Bragg scat- tering by crystals. The fringe contrast expressed by the

exponential factor in Eq. (1) is due to the contribution of

inelEistic scattering processes in the interference of light

scattered from two atoms. For two ions with fixed posiTN-117
Volume 70, Number 16 PHYSICAL REVIEW LETTERS 19 April 1993

tions, Eq. (1) recalls the result obtained by Heitler in his

classic treatment of resonance fluorescence [13] giving a

fringe contrast equal to 1. The center-of-mass modes are absent from the exponential factor. These modes do not

contribute to loss of fringe contrast, because they do not

affect the relative phase of the scattered light.

We can calculate ([q • (ui — U2)]^) from the residual

kinetic energy (temperature) in the normal modes of the

two ions given by kBTj/2 = mjUj'j{Xj)/2, where ks is the Boltzmann constant, j denotes the mode, ruj is the

effective ion mass for mode j (for the stretch and tilt modes rrij equals one-half of the single ion mass), and

{Xj) is the mean squared amplitude of the modes. At the

Doppler-cooling Umit, the kinetic energy of the stretch mode is {1 + [3 cos^(^)]~^}^7/8 and the kinetic energy of

the rocking mode is {1 + [3sin^(^)]-^};i7/8 [11], where

7 is the natural linewidth. Since the laser detuning and

the degree of saturation were not precisely determined, we might expect somewhat higher kinetic energies. The

frequencies of the normal modes are conveniently ex- pressed in terms of the single-ion secular frequencies,

which can be easily measured. We have Wgtretch = V^ujz

and wtiit = (oj'^ — ^l)^^'^, where the single-ion axial and

radial secular frequencies at the maximum applied volt- ages were about Uz/2t:=1 MHz and u'j./2t:—1.5 MHz,

respectively.

To "compare our data with theory, we normalized the

interference data with respect to the detected fluores- cence light of a single ion. This method compensates

for net efficiency variations across the detector. Further- more, we need to include a constant background intensity

lb. This background is partly due to stray light, partly

due to incoherent fluorescence radiation (see below), and

partly due to quantum jumps from the 6p '^P\/2 state into

the metastable level 5(i^6s ^Dz/2 [14], which leave only one ion fluorescing. The background due to single-ion

fluorescence caused by quantum jumps depends on the mean population of the 6p ^Pi/2 state and the lifetime of the metastable state. We fltted Eq. (1) (including h) to the data and found

excellent agreement for all measured interference pat- terns with a temperature of about twice the 1.7 mK

Doppler-cooling limit. Radial confinement was about

30 nm, and the residual amplitude in the axial direction was about 60 nm to 300 nm, depending on Uq. Figure 3

shows an example where C/o=193 V, corresponding to an

ion separation of 3.4 jim.. Our experiment resembles the classic Young's two-slit arrangement with the slits replaced by the two atoms.

In discussions of Young's type interference experiments,

the position-momentum uncertainty relation is often used

to show that it is impossible to determine through which

slit the photon or particle passes without interacting with

the photon or particle strongly enough to destroy the in- terference pattern. The position-momentum uncertainty

relation need not be invoked; the destruction of the in- terference can arise due to correlation between the meaFIG. 3. Shown are the interference fringes for d=3.35 /zm

together with a fit to the data. For these data, the temperature of the ions was determined to be 1.5 times higher than

the temperature at the Doppler-cooling limit.

surement apparatus and the measured system [15]. The

present experiment offers the possibility to obtain "which

path" information by exploiting the internal level struc- ture of the atom. The ^^^Hg"*" ground state 6s ^5i/2 and

the excited state 6p ^Pi/2 are twofold degenerate with re- spect to the magnetic quantum number rrij. The effect of this level structure is that scattering linearly polarized light off the two ions can result in either tt- or a- polarized scattered light. Assume that only one photon

is scattered at a time. In the case of 7r-polarized scattered

light (Amj = transition) the ions' final states are the same as the initial states. Which atom scattered the pho- ton cannot be determined. Quantum mechanics therefore

predicts that interference must be present in the light

scattered from the two ions. On the other hand, obser- vation of the (T-polarized scattered light (|Amj|= 1 transition) indicates that the final state of one atom differs from its initial state. This allows us, at least in principle, to distinguish the scattering atom from the "spectator"

atom, and hence to determine which path the photon

traveled. Consequently, there is no interference in the

light scattered from the ions. In this context, the exis- tence of interference fringes indicates wavelike behavior,

while the absence of fringes, consistent with a single pho- ton trajectory, which begins at the source, intersects one

of the atoms, and continues to the detector, indicates a

particlelike behavior. Thus, polarization-sensitive detec- tion of the scattered photons can serve as a switch to extract either the wavelike or the particlelike character

of the scattered photon. These two complementary pic- tures are essentially classical. Both types of behavior are contained in the quantum description. The interference

then occurs between Fejniman path amplitudes, and the

presence or absence of fringes depends on whether or not

there are two or one possible paths from the initial state to the final state. Our simple "which path" considerations are confirmed

2361

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Volume 70, Number 16 PHYSICAL REVIEW LETTERS 19 APRIL 1993

FIG. 4. Polarization-sensitive detection of the scattered

light (unnormalized) : (a) 7r-polarized scattered light, showing interference; (b) cr-polarized scattered light, showing no

interference pattern as explained in the text.

by a detailed theoretical study of the coherence properties of the resonance fluorescence from a single atom with

the same level structure as ^^^Hg"*" [16]. It reveals that

for a weak, linearly polarized field, only the 7r-polarized

scattered light is coherent, while the cr-polarized components are incoherent. In our experiment, polarization-sensitive detection of the 194 nm scattered light was accomplished with a glass

plate oriented at Brewster's angle. It was mounted be- tween the aperture and the detector. Light linearly polarized normal to the plane of incidence of the Brewster

plate was reflected toward the detector with about 15%

efficiency. Light polarized in the plane of incidence was

transmitted into the glass, where it was absorbed. Consequently, with the laser beam polarized normal to the

plane of incidence, 7r-scattered light was detected. Conversely, when the laser beam was polarized in the plane of

incidence, the 7r-scattered light was absorbed, and only

(7-scattered light reached the detector. Figure 4 shows

the results (unnormalized) of the polarization-sensitive

detection. Figure 4(a) displays the interference pattern

£is expected for the case of 7r-scattered light. When

cr-scattered light was detected, no interference pattern

could be observed [Fig. 4(b)], in agreement with the

quantum mechanical predictions. In summary, we have reported the observation of inter- ference fringes in the light scattered from two localized

atoms driven by a weak leiser fleld. The measured fringe

pattern and contrast can be explained in the firamework

of Bragg scattering by a harmonic crystal. These results show that interference measurements provide another

method to determine ion temperatures and separations in traps. By exploiting the atom's internal level structure, we showed, without invoking the position-momentum uncertainty relation, that the possibility of determining the

path of the scattered photon destroyed the interference

fringes.

Future prospects include measurements of the dependence of the fringe contrast on the saturation of the

atomic transition, since the coherent component decreases in a predictable way as the light field intensity

is increased [17]. We plan to study the interference pat- tern of more than two ions and have already been able

to observe the interference pattern of three ions. Finally,

heterodyne measurements could distinguish the different contributions of the elastic and inelastic scattering processes to the interference pattern.

We gratefully acknowledge the support of ONR. U.E.

gratefully acknowledges financial support of the Deutsche

Forschungsgemeinschaft. We thank M. E. Poitzsch and

J. Tan for their suggestions on the manuscript.

^^^ Permanent address: Universitat Freiburg, 7800 Freiburg,

Germany.

See, e.g., Great Experiments in Physics, edited by M.H.

Shamos (Henry Holt and Company, New York, 1959), p. 93. R. Feynman, R. Leighton, and M. Sands, The Feynman

Lectures on Physics (Addison-Wesley, Reading, 1965),

Vol. ni, p. 1-1. P. Verkerk, B. Lounis, C. Salomon, C. Cohen-Tannoudji,

J.-Y. Courtois, and G. Grynberg, Phys. Rev. Lett. 68,

3861 (1992).

P.S. Jessen, C. Gerz, P.D. Lett, W.D. Phillips, S.L. Rolston, R.J.C. Spreeuw, and C.I. Westbrook, Phys. Rev.

Lett. 69, 49 (1992).

P. Grangier, A. Aspect, and J. Vigue, Phys. Rev. Lett. 54, 418 (1985).

W. Neuhauser, M. Hohenstatt, P.E. Toschek, and H. Dehmelt, Phys. Rev. A 22, 1137 (1980); Th. Sauter, H.

GiUiaus, I. Siemers, R. Blatt, W. Neuhauser, and P.E. Toschek, Z. Phys. D 10, 153 (1988).

F. Diedrich, E. Peik, J.M. Chen, W. Quint, and H. Walther, Phys. Rev. Lett. 59, 2931 (1987). D.J. Wineland, J.C. Bergquist, W.M. Itano, J.J. Bollinger, and C.H. Manney, Phys. Rev. Lett. 59, 2935

(1987). R.G. Brewer, J. Hoffnagle, R.G. DeVoe, L. Reyna, and

W. Henshaw, Nature (London) 344, 305 (1990).

M.G. Raizen, J.M. Gilhgan, J.C. Bergquist, W.M. Itano, and D.J. Wineland, Phys. Rev. A 45, 6493 (1992).

The trap asymmetry reported in this reference has been

greatly reduced. W.M. Itano and D.J. Wineland, Phys. Rev. A 25, 35

(1982).

N.W. Ashcroft Eind N.D. Mermin, Solid State Physics

(Saunders College, Philadelphia, 1976), Appendix N. W. Heitler, The Quantum Theory of Radiation (Dover,

New York, 1984), p. 194. W.M. Itano, J.C. Bergquist, R.G. Hulet, and D.J. Wineland, Phys. Rev. Lett. 59, 2732 (1987).

M.O. Scully, B.-G. Englert, and H. Walther, Nature

(London) 351, 111 (1991). D. Polder and M. F. H. Schuurmans, Phys. Rev. A 14,

1468 (1976).

L. Mandel, Phys. Rev. A 28, 929 (1983); C.K. Hong, Z.Y. Ou, and L. Mandel, Phys. Rev. A 37, 3006 (1988).

[1

[2:

[3:

[4

[5]

[6]

[7]

[8]

[9

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11

12

13;

14

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17

2362

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LIGHT SCATTERED FROM TWO ATOMS*

W. M. Itano, U. Eichmann," J. C. Bergquist, J. J. Bollinger,

J. M. Gilligan.t M. G. Raizen.tt and D. J. Wineland

National Institute of Standards and Technology

325 Broadway Street Boulder, Colorado 80303

Abstract

We have observed interference fringes, like those in Young's clcissic experiment, in the laser light scattered

by two trapped atoms. The interference fringes are present only in one polarization of the scattered light.

The polarization dependence is related to the complementarity principle, which forbids the simultaneous

observation of wave-like and particle-like aspects of light. The interference fringes are due to processes

in which a single photon scatters from two atoms. We describe methods which might be used to observe

other interference effects due to two photons scattering from two atoms.

Introduction

We have observed Young's interference fringes resulting from the light scattered from two atoms. In Young's original

experiments, sunlight, passing through two slits, produced a pattern of closely-spaced fringes on another screen. Those

observations established the wave nature of light and resulted in the first accurate determination of the wavelength of

light.i Interference experiments of this form have played an important part in the conceptual development of quantum

mechanics. Either matter (electrons, for example) or light can display interference (a wave-like property). However, if it

is possible to infer which slit the matter or light passed through, thereby extracting a particle-like property, interference is not observed. Wave-particle duality is a particular example of a more general principle, the principle of complementarity,

which states, "For each degree of freedom, the dynamical variables are a pair of complementary variables."^ Variables

are complementary if a precise determination of one implies an inability to predict the other. For example, the position

and momentum of a particle are complementary variables, since, according to Heisenberg's uncertainty principle, the

product of their uncertainties must be greater than h/2. In our experiments, in contrast to various gedanken experiments,

complementarity is enforced in a manner which does not require the position-momentum uncertainty relations.

Experimental Apparatus

194 nm

light beam 9.

»bt

Figure 1: Diagram of the experimental apparatus.

Figure 1 shows the experimental apparatus, which has been described previously.^ A linear rf trap"* was used to confine

i98jjg+ iQijg ju i98jjg+^

^Yie lowest-frequency electric-dipole transition is from the ground 6s^5i/2 level to the Qp^Pi/2

level at 194 nm. The ions were laser-cooled to temperatures of a few millikelvins with a beam of linearly polarized,

'Work of the National Institute of Standards and Technology. Not subject to U.S. copyright. "Present address: Max Bom Institut, Berlin, Germany.

^Present address: Environmental Research Laboratory, National Oceanic and Atmospheric Administration, Boulder, Colorado. 'Present address: Department of Physics, University of Texas, Austin, Texas.

TN-120
continuous-wave light. Cooling in the trap resulted in strong localization of the ions, which is essential for observation of

interference fringes. The trap potentials were arranged so that a pair of ions would be oriented along the symmetry (z)

axis of the trap. The laser beam intersected the ions at an angle 6 of 62* with respect to the trap axis. Light emitted by

the ions was focused by a lens L through an aperture A and then directed to the surface of an imaging detector D2. This

detector was used to observe the fringes. The cixis of the detection optics was in the plane defined by the laser beam

and the trap axis. The in-plane detection angle <i> varied from 15* to 45* with respect to the laser beam direction. The

out-of-plane detection angle varied from -15* to -f-15*. Optionally, a linear polarizer P was inserted before D2. Another

lens system formed a real image of the ions on another imaging detector Di. This image was used to determine when

there were precisely two ions in the trap.

One-Photon Two-Atom Scattering

Our version of Young's interference experiment, in which the

two slits are replaced by two atoms, is an example of a case

in which each photon interferes only with itself. (We call this second-order interference, i.e., second-order in the fields, in order to distinguish it from other forms of interference to be discussed later.) From the point of view of quantum

mechanics, we expect interference whenever there are two

or more possible paths from a given initial state to the same

final state. The absolute square of the sum of the complex

amplitudes assigned to these paths yields the probability for the final state. The two paths which yield the interference are represented by the two parts of Fig. 2. In each path, the

photon, represented by the wavy arrow, is scattered by one

of the two atoms, represented by circles, and reaches the

detector, represented by the half circle. After the photon

is detected, there is no way to tell which atom scattered it,

i.e., which of the two paths the system took.

An example of interference data is shown in Fig. 3. The

two ions were separated by 3.4 /im. The angular separation

of successive fringes is about 3* . The angle <f) between the

incident and scattered photon directions increaises to the

right. Interference fringes were obtained for ion separations varying from 9 ^m to 3.3 /im. The angular separation of

the fringes increased as the distance between the ions was decreased. The fringe visibility was observed to decrease with

increasing 0. The dependence of the fringe spacing on the ion separation and the variation in the fringe visibility with angle are contained in the following expression for the intensity I of the scattered light as a function of q, where hq — ft(^out — ^in).

and ^,n and ^out are the wavevectors of the incoming and outgoing photons:

+

Figure 2: The two paths which contribute to the Young's

interference fringes. The paths correspond to the photon

being scattered by one atom or the other and then being

detected.

/(g) = 2/0 [1 + cos{q,d) exp {- {[q- {uy - u,)]') /2}] . (1)

In this expression, 7o is the intensity due to scattering by a single ion, d is the ion separation, and ui and U2 are the

displacements of the ions 1 and 2 from their equilibrium positions. The angular brackets denote an ensemble average. Equation (1) was derived by methods similar to those used for Bragg scattering by a harmonic crystal.^' ^ The fringe

separation is contained in the factor coslq^d) and is therefore inversely proportional to d. The fringe visibility depends on

the quantity (^[q- {ui — tl2)]^). The visibility is greatest in the forward scattering direction, where g = 0, and decreases

with increasing scattering angle and with increasing ion temperature. The temperature of the ions has been determined

from the rate of decrease in fringe visibility with angle and is approximately equal to the theoretically calculated value.^ The data shown in Fig. 3 were obtained with polarization-insensitive detection. When a linear polarizer was placed in front of the detector, the fringes were present only in the light having polarization in the plane defined by the polarization

of the incident light and ^out (7r-polarization) and not in the light having the orthogonal polarization ((X-polarization).

Figure 4 shows interference patterns observed in (a) 7r-polarized and (b) tr-polarized light. The 7r-polarization shows high

fringe visibility, while the c-polarization shows no fringes, only the slow variation with scattering angle of the detection

sensitivity.

TN-121
Figure 3: Interference fringes observed for an ion separation of 3.4 /im. The deviation from the forward scattering

direction increases to the right.

C/3

Figure 4: Intensity of the scattered light from two ions as a function of the scattering angle 4>- (a) 7r-polarized light, (b)

CT-polarized light. Only the 7r-polarized light shows interference fringes. The data are not normalized for the detection

efficiency, which varies with 4>-

TN-122
6p2Pi/2 -<I

A =194 nm

65^51/2

TT

-<1

W

mj = +1/2

mj = -1/2

mj = +1/2

mj = -1/2

Figure 5: Zeeman sublevels involved in the 194 nm, 6s^5i/2-to-6p^Pi/2 transition of ^^^Hg+. The allowed x and a- transitions are labeled. The Zeeman splitting of the levels is exaggerated.

TT

TT

TT

TT

(a)

/^^

7^

^^^(b)^^^

Figure 6: Each box represents the combined state of the two atoms. The ordering of energy levels is the same as in

Fig. 5. In (a), a tt- transition is made from the ground state to the excited state and is followed by a ir-transition back

to the ground state. There are two possible paths, both of which lead to the same final state, so that interference is possible. In (b), the 7r-transition from the ground to excited state is followed by a (T-transition back to the ground state. The two paths do not lead to the same final state, so there is no interference.

TN-123
The source of the polarization dependence of the fringe visibility lies in the internal level structure of the ^^^Hg'*' ion

and in basic principles of quantum mechanics. Figure 5 shows the magnetic sublevels involved in the 6s'^5i/2-to-6p'^Pi/2

transition. The static magnetic field is small enough that we are free to define the quantization axis of the ions to be

along the electric polarization vector of the incident light. If the static magnetic field is along some other direction, then

the Zeeman sublevels defined according to the electric polarization vector are not stationary states. This does not change

the analysis as long as the Zeeman precession frequency is much less than the inverse of the scattering time, which is approximately equal to the 6p^Pi/2-state lifetime (2.3 ns). In the experiments described here, the magnetic field was

small enough that this was always the case. The TT-transitions (Amy = 0) and tr-transitions (Amy = ±1) are labelled in Fig. 5. The incident laser light drives

only TT-transitions from the ground to the excited state. However, the decay from the excited state to the ground state can be either a 7r-transition or a (r-transition. If the decay is a 7r-transition, the atom returns to the same mj sublevel it occupied before it absorbed a photon. Hence, it is not possible to tell, by examining the atoms, which one scattered the

photon. On the other hand, if the decay is a cr-transition, the atom returns to a different mj sublevel and it would be

possible in principle to examine the atoms after the scattering and tell which one scattered the photon. In the context of

the principle of complementarity, a particle-like property is present, and interference, a wave-like property, must vanish.

From a more modern point of view, we say that interference occurs when there is more than one transition amplitude

connecting the initial and final states.^ Figure 6 illustrates this point of view. There are four possible initial states of the

combined system of two atoms, since each atom can be in either of two mj sublevels. We consider a particular initial state, in which both atoms are in the mj = -1-1/2 sublevel. All four choices of initial states contribute to the interference

fringes in the same way. In Fig. 6(a), a 7r-polarized photon is absorbed and a 7r-polarized photon is emitted. There are two paths for this process. Since they lead to the same final state, interference can occur. In Fig. 6(b), a ;r-polarized

photon is absorbed and a cr-polarized photon is emitted. There are two paths for this process. However, they lead to

different final states, so there is no interference.

Polder and Schuurmans'^ calculated the spectrum of resonance fluorescence from a (.7 = l/2)-to-(J = 1/2) transition. They found that, for low intensities, the 7r-polarized scattered light is coherent with the incident light, while the <r- polarized scattered light is not. Thus, it is reasonable that the ir-polarized light should show interference fringes while

the cr-polarized light should not. The fringe visibility for the 7r-polarized light should decrease with intensity, since, as

is the case for two-level atoms,®' ^ the ratio of incoherent scattering to coherent scattering increases with intensity. The

intensity dependence of the visibility has not yet been examined experimentally.

Two-Photon Two-Atom Scattering

Next, we consider the interference process represented by Fig. 7. Two photons are scattered by two atoms and are each

detected at different spatial positions. However, it is not possible to tell which atom scatters which photon, so there is interference between the two paths. This is a kind of fourth-order interference (i.e., fourth-order in the fields), which can

persist even when there is no second-order interference. The interference effects that we have labeled fourth-order and

second-order are called second-order and first-order, respectively, by others,^ since they are second-order and first-order in the intensities. Interference by independent quantum sources has been discussed in detail by various authors. ^°' ^^ If the sources

are independent, there is no stationary interference pattern. However, both classical and quantum calculations predict

intensity correlations between two detectors. In particular, if the sources are two single atoms, there will be a finite coincidence rate if the detectors are separated by n interference fringes (n=0, 1,2,.. .) and no coincidences if the detectors

are separated by n -f- 1/2 fringes.^" In this case, the quantum calculation predicts a fringe visibility of 1, while the

corresponding classical calculation predicts a visibility less than or equal to 1/2. The difference arises from the fact that a single atom cannot emit two photons simultaneously. After emitting one photon, it must be excited to the upper state again before it can emit another. In the classical calculation, the simultaneous detection of two photons emitted by the same atom is allowed, and this gives rise to a background signal that reduces the fringe visibility. The observation of a

fringe visibility greater than 1/2 for independently phased atoms would thus be an example of a quantum phenomenon

having no classical analog.

Consider a simple case originally treated by Dicke.^^ Two two-level atoms are both in the excited state and are separated by much more than the wavelength of the light that they emit (see Fig. 8). This state can be written as |e)i|e)2,

where e denotes the excited state of an atom and 1 and 2 label the two atoms. The first photon can be emitted in any

direction consistent with the dipole radiation pattern of a single atom. The state of the atoms immediately after the first photon has been emitted can be symmetric, i.e., (|e)i|g)2 + |g)i|e)2)/\/2, or antisymmetric, i.e., (|e)i|g)2 — |g)i|e)2)/>/2,

where g denotes the ground state of an atom. In general, the state can be a linear combination of these two cases.

TN-124
+

Figure 7: The two paths which contribute to fourth-order interference fringes. The paths correspond to each of the two

photons being scattered by one of the two atoms and then being detected by one of the two detectors.

Symmetric

Antisymmetric • Atoms

Figure 8: Radiation patterns for two atoms (indicated by circles) separated by 5 wavelengths and initially both in the

excited state. The solid curve is for the case in which, after the emission of one photon, the atoms are in a symmetric

state. The dcished curve is for the case in which they are in an antisymmetric state.

TN-125
Like the radiation pattern of pair of classical dipoles, the probability distribution for the emission of the second

photon is made up of many interference fringes. A pair of classical dipoles has a pattern which shifts according to their relative phases. Here, the pattern shifts according to the relative phases of the wavefunctions of the two atoms. Figure 8

shows an example, for a separation of five wavelengths. If the wavefunction is symmetric with respect to the two atoms,

the radiation pattern is given by the solid curve. If it is antisymmetric, it is given by the dashed curve. If it is a linear combination of symmetric and antisymmetric wavefunctions, it is intermediate between the two curves. The direction of the second photon is constrained by the direction of the first. It cannot be in a direction differing

by n + 1/2 fringes and has the greatest probability of differing by n fringes. The emission of the first photon fixes the

symmetry of the wavefunction and hence the probability distribution for the emission of the second photon. The symmetry

must be such that the direction of the first photon is along a fringe (lobe) of the radiation pattern. It is important to remember that each photon is emitted by both atoms. If, after the emission of the first photon, a measurement is made

to determine which of the atoms is in the excited state, the coherent superposition is totally destroyed, and the second

photon can be emitted in any direction. Consider the following modification of the experiment. Wait until the first photon is detected. Its direction of emission

fixes the symmetry of the wavefunction. Then, before the second photon is emitted, change the separation between the

atoms to a different value. In an ion trap, this can be done by changing the electric potentials on the electrodes. This

changes the radiation pattern, adding or subtracting fringes. The second photon will be emitted into a fringe of the new

radiation pattern. Thus, the second photon can be aimed toward certain directions and away from others by controlling

the distance between the atoms.

Estimated Signals

Experimentally, a higher data- collection rate could be achieved by exciting the atoms continuously and using coincidence

detection, rather than by sequentially exciting the atoms and then detecting the emitted photons. It should be possible

to observe fourth-order interference with a slight modification of the present experimental apparatus. Two imaging

detectors would be used, together with fast coincidence electronics. In order to avoid the intensity variations due to the second-order interference [as in Fig. 4(a)], only the <7-polarized light should be detected. For each coincidence, the

position of both photons would be recorded. Let iJcoinc he the detected coincidence rate. Then,

-Rcoinc = 2(7;ilscatt) <5r, (2)

where t] is the fraction of scattered photons that are detected, ilscatt is the rate at which photons are scattered by one atom, and 6t is the coincidence time window. A reasonable value for t} is 10"^, assuming a detection solid angle of 1%

and a detector quantum efficiency, including the polarizer efficiency, of 10%. If the incident light intensity is high enough

to saturate the transition, Racatt approaches 7/2, where 7 is the decay rate of the upper state. The coincidence time

window 6t should be less than the coherence time for the fourth-order interference, which is less than or approximately

equal to 7"^. If we let St « 0.257~^, then

Hcoinc w 1.25 X 10-'^7- (3)

For Hg"^, where 7 w 4.3 x 10® s~^, iJcoinc '^ 54 s~^. Thus, it should be possible to observe a fringe pattern with a few

minutes of observation time.

Other ions might be better suited than Hg"^ for this experiment. For example, Mg"*" hcis a smaller mass, thus allowing

better spatial localization, and a longer resonance wavelength, thus allowing the use of more efficient detectors and

polarizers. On the other hand, it has a smaller value of 7 than Hg"*" (7 s» 2.7 x 10® s~^).

It would also be of interest to observe the coincidences in the 7r-polarized light, particularly in the high-intensity

limit. In this limit, the second-order interference disappears, but the fourth-order interference remains.^ The visibility

of the fourth-order interference fringes is 1 for arbitrary intensity, but the time window during which it can be observed

decreases with intensity.

Quantum Erasers The term "quantum eraser" was used by Scully and Driihl^^ to describe a gedanken experiment involving the observation

of interference fringes in the light scattered by two atoms. Other forms of quantum erasers have been described. ^'*' ^^ The common element is that interference is destroyed when it is possible to tell, in principle, which path the system

has taken, and is recovered when that information is erased or hidden. In Scully and Driihl's original proposal, two

atoms are excited by a laser pulse. In the case of two-level atoms, interference fringes are observed, as in the case of

TN-126
TT-polarized emission discussed in the previous Section. In the case of three-level atoms, interference is not observed when

the light emission from the atoms is accompanied by a transition to a level different than the one occupied before the

laser pulse. This is because it is possible to tell which atom scattered the photon, as in the previous case of cr-polarized

emission. This information is erased by exciting the atoms to a fourth level, from which they decay back to the initial

level. Fringes are recovered, but only in a coincidence measurement, which is in principle similar to the fourth-order

interference experiment that we have proposed. The experiment can be operated in a "delayed choice" mode. That is, the interference fringes can be made to appear or not, depending on whether or not a shutter is operated, even though

this is done after the photon is on its way to the detector. The fourth-order interference experiment we have proposed is a kind of quantum ereiser, though, in common with some

other quantum erasers,^'*' ^^ it lacks the "delayed-choice" feature. If we detect only the <T-polarized light, no interference

fringes are detected in a non-coincidence experiment. In principle the states of the atoms could have been mecisured

before and after the photon scattering, and it would be possible to tell which atom scattered the photon. In a coincidence

detection, both atoms have scattered a photon and have changed their states. However, it is impossible to tell which

atom scattered which photon. That is, it is impossible to tell which of the paths shown in Fig. 7 the system took. This

is what makes it possible to observe interference fringes.

Conclusion

We have observed Young's interference fringes in the light scattered from two localized atoms. The interference patterns can be used to infer the separation and temperature of the two atoms. The fact that the interference appears only

for one polarization of the light has a simple explanation based on the fact that only paths leading to the same final state can interfere. Fourth-order interference effects might be observed by detecting coincidences in two photon detectors.

This work was supported by the Office of Naval Research.

References

[1

[2:

[3:

[4:

[6;

[8:

[9

[10

[11

[12

[13;

[14;

[15;

In Great Experiments in Physics, edited by M. H. Shamos (Henry Holt and Company, New York, 1959), p. 93. M. 0. Scully, B.-G. Englert, and H. Walther, Nature 351, 111 (1991). U. Eichmann, J. C. Bergquist, J. J. Bollinger, J. M. Gilligan, W. M. Itano, D. J. Wineland, and M. G. Raizen, Phys. Rev. Lett. 70, 2359 (1993).

M. G. Raizen, J. M. Gilligan, J. C. Bergquist, W. M. Itano, and D. J. Wineland, Phys. Rev. A 45, 6493 (1992). N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College, Philadelphia, 1976), Appendix N.

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms (Wiley, New York, 1989), pp. 204-215. D. Polder and M. F. H. Schuurmans, Phys. Rev. A 14, 1468 (1976). B. R. Mollow, Phys. Rev. 188, 1969 (1969).

Th. Richter, Opt. Commun. 80, 285 (1991).

L. Mandel, Phys. Rev. A 28, 929 (1983). H. Paul, Rev. Mod. Phys. 58, 209 (1986). R. H. Dicke, in Quantum Electronics, proceedings of the Third International Conference, Paris, edited by P. Grivet

and N. Bloembergen (Columbia University Press, New York, 1964) p. 35. M. 0. Scully and K. Driihl, Phys. Rev. A 25, 2208 (1982).

P. G. Kwiat, A. M. Steinberg, and R. Y. Chiao, Phys. Rev. A 45, 7729 (1992). A. G. Zajonc, L. J. Wang, X. Y. Zou, and L. Mandel, Nature 353, 507 (1991).

IN- 127
PHYSICAL REVIEW A VOLUME 48, NUMBER 1 JULY 1993

Electrostatic modes of ion-trap plasmas

J. J. Bollinger, D. J. Heinzen,* F. L. Moore,* Wayne M. Itano, and D. J. Wineland

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

Daniel H. E. Dubin

Department ofPhysics, University of California at San Diego, La Jolla, California 92093

(Received 4 January 1993)

The electrostatic modes of a non-neutral plasma confined in a Penning or Paul (rf) trap are discussed

in the limit that the Debye length is small compared to the plasma dimensions and the plasma dimensions are small compared to the trap dimensions. In this limit the plasma shape is spheroidal and analytic solutions exist for all of the modes. The solutions for the modes of a Paul-trap plasma are a special case of the modes of a Penning-trap plasma. A simple derivation of some of the low-order quadrupole

modes is given. Experimental measurements of these mode frequencies on plasmas of laser-cooled Be^

ions in a Penning trap agree well with the calculations. A general discussion of the higher-order modes

is given. The modes provide a nondestructive method for obtaining information on the plasma density

and shape. In addition, they may provide a practical limit to the density and number of charged particles that can be stored in a Penning trap. PACS number(s): 32.80.Pj, 32.90. +a, 52.25.Wz, 52.35.Fp

I. INTRODUCTION

Penning traps [1,2] typically use a uniform magnetic

field superimposed along the axis of azimuthally symmetric electrodes to confine charged particles. Radial

confinement is provided by the axial magnetic field and

axial confinement is provided by electrostatic potentials

applied to the trap electrodes (see Fig. 1). Paul (rf) traps

[1] use an electrode structure similar to Penning traps,

but there is no magnetic field, and a combination of rf and electrostatic potentials are applied to trap electrodes. Charged particles are confined by the inhomogeneous rf

fields (the pondermotive force) to a region of minimum rf

field strength near the trap center. Penning and Paul

traps are used in a number of studies on charged atomic

particles such as mass spectroscopy [3-8], high-precision

magnetic-moment measurements [1,9,10], high-resolution

spectroscopy and frequency standards [1,11-14],

charge-transfer studies [15], non-neutral-plasma studies

[2,16-21], and antimatter storage [7,22-24]. In many of

these experiments more than one charged atomic particle

(a "cloud") are stored in the trap and cooled to low temperatures. A cloud of charged particles in a Penning trap can be considered a plasma, in particular a non-neutral

plasma, when the Debye length is less than the cloud di- mensions. This paper discusses the electrostatic modes of

these plasmas in the limit that the Debye length is much

less than the plasma dimensions and when the plasma di- mensions are small compared to the trap dimensions. As

discussed below, the modes of a Paul-trap plasma are de- scribed by a special case of the Penning-trap plasma

modes. Therefore in this paper we concentrate our dis- cussion on the modes of Penning-trap plasmas and show

how solutions for the modes of an rf-trap plasma follow

from the Penning-trap results. Measurement of the plasma mode frequencies may

have applications for ion trap experiments. For example,

mass spectroscopy experiments done with a cloud or plasma of ions in a Penning trap can have systematic shifts associated with the distances of the ions from the trap center and the small anharmonicities of the trapping poImager

Lens _^[ ^,_—-.^ I System y^"^ ^^

Microwave Horn

Torque

Beam

Cooling

Beam

9Be+ Plasma

FIG. 1. Sketch of a Penning trap used to make mode measurements on a plasma of 'Be'*' ions. The size of the plasma is exaggerated. The trap electrodes (shown in cross section) are

right circular cylinders with inner radius po=\.n cm. They

provide a quadratic potential near the trap center with

^2~0-236. Some other Penning traps use hyperboloids of re- volution as trap electrodes. The laser beams, microwave horn,

and imaging system are used in the measurement of the plasma

modes as described in Sec. IV.

1050-2947/93/48( 1 )/525(2 1 )/$06.00 48 525 Work of the U. S. Government

Not subject *" TT c

.^„„^-j,*^^^
526 J. J. BOLLINGER et al. 48

tential [6,25,26]. In high-resolution atomic spectroscopy

of ions in a Penning trap, one of the largest systematic

shifts is the second-order Doppler shift associated with

the EXB rotation of the ions about the magnetic field axis of the trap [12]. The larger the radius of the plasma,

the larger the second-order Doppler shift. A third appHcation is illustrated by experiments which measure

electron-ion recombination by passing an electron beam

through a sample of ions trapped in a Penning trap [27]. In these studies, it would be desirable to have a direct knowledge of the geometrical overlap of the electron

beam with the ion sample. In all of the above examples,

the plasma shape and density, along with the number of

trapped ions, determine the radial and axial positions of

the ions and therefore the size of the required corrections. Similar considerations apply to experiments on

ions in a Paul trap. In some cases the plasma shape and

density can be determined with a laser [20,28]. When

this or other techniques are not available, detection of the

plasma modes, perhaps through induced image currents

in the trap electrodes, could provide this information.

Recent experiments have trapped positron [23] and antiproton [7,24] plasmas in Penning traps. The current techniques for obtaining information about these plasmas

involves ejecting the plasma from the trap. Detection of

the plasma modes should provide a nondestructive diagnosis for obtaining information on the antimatter plasma

density, temperature, and shape. A goal of the antimatter work is the high-density storage of large numbers of positrons and antiprotons for transport to

different laboratories around the world. Field errors in the trapping potential can excite plasma modes and

enhance radial transport. Excitation of the plasma

modes may therefore set a practical hmit on the density

and number of antiparticles that can be stored in a Penning trap. In addition to having potential applications to current experiments, the electrostatic modes of a Penning-trap

plasma may also provide a tool for studying the dynamics

of non-neutral and strongly coupled plasmas. A simple,

analytic form has been obtained for all of the electrostatic modes of a low-temperature Penning-trap plasma [29]. This is the only finite length geometry for which exact mode eigenfrequencies and eigenfunctions have been cal- culated. Therefore the Penning trap may provide an in- teresting geometry for the study of plasma modes and

their importance to the dynamical behavior of nonneutral plasmas. For example, it should be possible to study how errors in the trapping fields couple to the plasma modes and enhance radial transport [19]. Measurement of the damping of the modes should provide infor- mation on the plasma's viscosity. This measurement

could presumably be done with a strongly correlated

plasma over a range of magnetic field strengths where

very little information is available. In Sec. II we review the static properties of a cold,

Penning-trap plasma in thermal equilibrium. We assume

the trapped particle number iV » 1 . For sufficiently low

temperature or small Debye length, the plasma has constant density. If the plasma dimensions are small compared to the trap electrode dimensions, then the confining

potential may be assumed to be quadratic and the eflFect of image charges neglected. In this case, the plasma has a spheroidal boundary with an aspect ratio determined by

the plasma density and the axial restoring force of the

trap. This is the starting point for describing and calculating the modal excitations of a plasma in thermal equilibrium. In Sec. Ill we describe a simple but exact calculation for some of the quadrupole modes of a spheroidal

Penning-trap plasma. We show how these modes could

be used to measure the density and shape of a Penningtrap plasma. Section IV compares the results of the

quadrupole mode calculations with measurements [21]

done on about 2000 laser-cooled ^Be"*" ions in a Penning

trap. In that section we show how a zero-frequency

quadrupole mode was excited by a misalignment of the

magnetic field axis with respect to the electrostatic symmetry axis of the trap. Excitation of this mode tended to

limit the plasma density. Section V discusses the higherorder modes of a Penning-trap plasma [29]. Modes that can be excited by static field errors are identified.

II. STATIC PROPERTIES

The Penning trap shown in Fig. 1 consists of four cy- hndrical electrodes. The outer cylinders are called the

"end-cap" electrodes in analogy with the end caps of a hyperboHc Penning trap [1]. The inner cyHnders are electrically shorted and together called the "ring" electrode. With a positive potential Vj apphed to the end-cap elec- trodes with respect to the ring electrode, positively

charged particles (ions) can be electrostatically confined

in the direction of the trap axis. A static, uniform magnetic field B=Bz parallel to the trap's symmetry axis confines the ions in the radial direction. Near the center of the trap, where the ions are confined, the radial component of the trap electric field is directed outward. This

field produces an E X B circular drift of the ions about the

symmetry axis of the trap. As the ions rotate through the

magnetic field, they experience a Lorentz force directed

radially inward.

With sufficiently long confinement, the ions evolve to a

state of thermal equiUbrium characterized by a uniform

"rigid" rotation of the ions at a frequency «, [18,28,30].

Specifically, the rotation frequency o^ is independent of

radius. In the Umit of zero temperature, the plasma density Hq and plasma frequency cOp are constant in the plasma interior and drop abruptly to at the plasma edge.

The density depends on the rotation frequency according

to (Systeme International units are used throughout)

nn =

2€Qmco^{fl —cOr) (2.1)

where Q. =qB /m is the ion cyclotron frequency, q and m

are the charge and mass of the ion, and €q is the permittivity of the vacuum. The plasma frequency is therefore

related to the rotation frequency through the equation

COr Com — 2(0^01—0)/, (2.2)

In this paper we use the convention that the symbols o)^ IN- 129
48 ELECTROSTATIC MODES OF ION-TRAP PLASMAS 527

and ft denote positive quantities. However, for positive

ions, the sense of the rotation and the sense of the cyclo- tron motion with respect to B are actually negative.

Specifically, when viewed from above the x-y plane, the

ions move in clockwise orbits. For nonzero temperatures, the density drops to at the plasma edge in a dis- tance on the order of a Debye length A.^ [18,30], where

1/2

(2.3) €okT

"09

k is Boltzmann's constant, and T is the ion temperature.

For kj) « (plasma dimensions), the plasma therefore has

a uniform density, given by Eq. (2.1), with sharp boundaries. For very low temperatures, there are correlations

in the ion positions [20,31,32], and the ion density is not

constant over length scales small compared to the in- terion spacing (~n^'^^). However, as long as the in- terion spacing is small compared to the plasma dimensions (that is, as long as the number of ions A'^ »1 ) and

small compared to the wavelength of the plasma modes,

the plasma can be treated as a constant-density plasma

even in the presence of spatial correlations.

The plasma boundary has a simple shape in the hmit

that the plasma dimensions are small compared to the

trap dimensions [28,33]. Near its center, the electrostatic potential of the trap, relative to the potential at the trap

center, can be written as

<l>j'(r,z)-- mco'

Aq

-Hz' (2.4)

where r and z are cylindrical coordinates, and co^ is the

frequency at which a single trapped ion (or the center-of- mass of a cloud of ions) oscillates along the z axis. For

the cylindrical trap of Fig. 1,

0), 4qVjA2

mpl

1/2

(2.5)

where p^ is the inner radius of the trap electrodes and A 2 is a dimensionless parameter that depends on the

geometry of the trap design. For the trap of Fig. 1, ^2~0-236. In general, the total electrostatic potential is the sum of the trap potential, the space-charge potential

of the ions, and a potential due to the induced image

charges on the trap electrodes. If the plasma dimensions

are much less than the trap dimensions, the trap potential over the region of the plasma is given by Eq. (2.4), and

the effect of the induced image charges can be neglected. In this case the shape of the plasma boundary is a spheroid (an ellipsoid of revolution) as shown in Fig. 2

[28,33]. Let Ivq and 2zo denote the diameter and the axial extent of the plasma as shown in Fig. 2. The plasma

aspect ratio a=Zo//'o is related [28] to the plasma fre- quency (Op and the trap axial frequency a^ by

(litp (a^-1 1I/2 /

(a^-i: (2.6)

where QP is the associated Legendre function of the

second kind [34]. When the plasma is a prolate spheroid

FIG. 2. Spheroidal shape of a Penning-trap plasma. This

shape is obtained under the conditions of thermal equilibrium

and kj) «(plasma dimensions) «(trap dimensions).

(a > 1), Eq. (2.6) can be written as

,2 Oi,

Oin

1 a^-1

^in "p + 1 Up-\

1 "P ^TM2_iU/2 ia'-lY

(2.7) When the plasma is an oblate spheroid (a< 1), Eq. (2.6) can be written as

(o.

co„

1 a^-l

Wq tan

-1 "0

-1 Un =

a

(l-a2)'/2 •

(2.8)

Figure 3 shows a graph of o)] /(oj versus the plasma as- pect ratio a. Experimental measurements of £t)j.,a)^, and a

discussed in Ref. [28] are in good agreement with the

CV2

C\2

FIG. 3. Relationship between the plasma aspect ratio a and

(ol/coj for spheroidal-shaped plasmas in a Penning trap. The

solid Hne is a theoretical curve from Eq. (2.6) with no adjustable

parameters. The experimental measurements, described in Ref.

[28], were taken with two different traps at three different axial frequencies between co, /fl =0. 07 1 and 0.121. IN- 130
528 J. J. BOLLINGER et al. 48

theoretical calculation of Eq. (2.6). In the next section the potential inside a uniformly

charged spheroid (the space-charge potential of a cold

Penning-trap plasma) will be used to calculate some of

the quadrupole mode frequencies. For r,z inside the plasma, the space-charge potential <f>j can be written

[28,33,35] as

(f)j(r,z)= —

mor.

6q

-[a{a)r'^^-b(a)z^] , (2.9)

where a is the plasma aspect ratio, cOp is the plasma fre- quency given by Eq. (2.2), b(a) =3(2?(a/(a2- 1)'/^)/

(a^— 1), and Poisson's equation requires

2a(a) + bia) = 3. A spherical plasma has a =b = 1. At

the limit b =3, the aspect ratio a=0, and the plasma is an infinitely thin disk. (Because we require Zq »Xj), here

infinitely thin means Zq «rQ.) With a=j, the aspect ratio a is infinite, and the plasma is an infinitely long cylin- drical column (zo»A-o). In Eq. (2.9) we choose the potential at the center of the spheroid to be 0. If the potential is chosen to be at oo , then Eq. (2.9) neglects a term

which depends on the plasma aspect ratio a but is in- dependent of r and z. It is instructive to consider the plasma equilibrium as a function of rotation frequency o^ for fixed trapping conditions (fixed co^, ft, and AO. Constant density equilibria

exist for 6)^ < ft /V2 and co^ <co^<D,—co^, where

fi)„=ft/2-(ftV4-£yV2)\W2(2.10)

is a single-ion magnetron frequency [1]. For (o^ slightly

larger than co^ , col /o)^ ^ 1 , and the plasma is shaped like a pancake (an oblate spheroid). In the limit that (o^^-co^, the plasma's aspect ratio a—>0, and the

plasma's radius Tq ^ oo . As co^ increases, co] /co^ de- creases and the plasma's aspect ratio a increases by de- creasing Kq and increasing Zq. At a)^ = ft/2 the plasma

attains its maximum aspect ratio (smallest r^ and largest

Zq) and maximum density ng=€QmD,^/2q^. The condition aj^ = ft/2 is often called Brillouin flow [36]. In a frame of reference rotating with the plasma, the motion

of an individual ion within the non-neutral plasma consists of circular gyrations (perturbed cyclotron orbits) at the frequency Cl — 2co^. At Brillouin flow, these gyrating

orbits become free streaming (straight-line trajectories),

and the plasma behaves in many ways like an unmagnetized plasma [36]. Therefore at 6)^ = ft/2, a Penning-trap

plasma behaves dynamically like a plasma confined in an

rf (Paul) trap (neglecting the rf micromotion). As co^ in- creases beyond ft/2, the plasma's aspect ratio a and density Mq decrease. Because «o is an even function of co^ about (0^ = 0. /2 [see Eq. (2.1)], the plasma's aspect ratio,

radius, and axial extent are even functions of co^ about

co^ = ft /2. Figure 4 shows a graph of the radius of a plasma of ^Be"^ ions as a function of rotation frequency. The

plasma's rotation frequency and radius were measured

with techniques described in Sec. IV. Good agreement was obtained between the observed and predicted dependence of the plasma's radius on rotation frequency.

cj /n

FIG. 4. Radius Kq of a plasma of ==2000 'Be"*^ ions as a function of rotation frequency cOr- The radius is plotted in units of

Tj, , the plasma radius at the Brillouin limit, and the rotation fre- quency is plotted in units of the cyclotron frequency n. The

solid line is a theoretical curve involving no adjustable parameters. The data were taken as described in Sec. IV with

il/2Tr= 1.4 MHz and co,/il=0. 151.

With the assumption of thermal equilibrium and max

(Xp, nQ^^^)« (plasma dimensions) « (trap dimensions), a Penning-trap plasma has uniform density with a spheroidal boundary. Modal excitations on this equilibrium can be conveniently described with spheroidal coordi- nates by two integers {l,m) with / > 1 and m >0 [29]. (Negative integral values of m are allowed, but do not

give rise to new modes.) The index m denotes an azimuthal dependence e"""^ of the plasma mode potential. The

index / describes the variation along a spheroidal surface

(for example, the plasma boundary) in a direction perpendicular to <f>. In this paper. Sees. Ill and IV give a simple,

detailed discussion of some of the / =2 modes and Sec. V

gives a general discussion of higher-order modes of a Penning-trap plasma. Excitation of an / = 2 mode pro- duces a quadrupole deformation of the plasma shape.

The / = 1 modes are the famiUar center-of-mass modes.

Here the ion plasma's shape remains unchanged, but the

center-of-mass of the plasma executes one of the three

motions of a single ion in a Penning trap. For example,

the (1,0) mode is the axial center-of-mass mode at fre- quency CO2- There are two (1,1) modes which correspond

to the perturbed cyclotron and magnetron center-of-mass

modes at frequencies ft — co„ and to^ . (In experiments on

long columns of electrons where the induced image

charges cannot be neglected, the analog of the magnetron

center-of-mass mode is the / = 1 diocotron mode [37,38].)

These center-of-mass frequencies can, in general, be measured or calculated very precisely. Figure 3 gives the

(1,0) mode frequency or co^ in units of the plasma fre- quency as a function of the plasma aspect ratio or shape.

Because cOp is typically unknown, a measurement of co^ does not provide information on the plasma aspect ratio. However, measurement of an / = 2 mode frequency along

with the (1,0) mode frequency will determine the plasma's

aspect ratio and density.
48 ELECTROSTATIC MODES OF ION-TRAP PLASMAS 529

III. QUADRUPOLE PLASMA MODES

In this section we calculate two diflFerent types of / =2

Penning-trap plasma modes. This is done by making a

guess for the plasma eigenmode, then verifying that this guess is a mode of the plasma and calculating the eigenfrequency. In particular, we consider small axial and radial displacements of the ions from their equilibrium positions consistent with the assumed eigenmode. A self- consistent calculation of the axial and radial restoring

forces gives two linear diflFerential equations describing

the motion of the axial and radial displacements. We

then require that the eigenfrequencies from these two

equations be equal.

A. (2,0) modes

We first consider an azimuthally symmetric quadrupole mode. We assume that in this mode the plasma al- ways stays spheroidal, but the aspect ratio of the plasma

oscillates in time as shown in Fig. 5. This mode turns out

to be a (2,0) mode [29]. Let p=x+iy and z denote the

equilibrium radial and axial position of an ion (or group

of ions) in a plasma with an equilibrium density Kq and

aspect ratio a. The displacements from equilibrium

p^p[\+l{t)].

(3.1)

parametrized by 6 and b = bj^+ibj, give another

spheroidal plasma but with a density and aspect ratio given by

a-

(l+e)|l+6|2

1+e

(3.2)

|1+5|

where |1+5| = [(l+5;j )^ + 8^]'^^ Here e, 8^, and 5/

characterize axial, radial, and azimuthal displacements.

For an eigenmode the axial and radial displacements

should have the same time dependence. We assume

€{t)

b„{t) V (3.3)

where 7], the mode parameter, is independent of time.

FIG. 5. Sketch of one cycle of the (2,0) mode at time inter- vals r/4 where T=2Tr/co2o is the period of the mode. The plas- ma always stays spheroidal with a uniform density, but the as- pect ratio oscillates in time.

Equations (2.4) and (2.9) can be used to calculate the electric field at the displaced ion positions with the results that

Ei = - mco„

3q

36)z ^ a[a(l+€)\l+8\'^]

20)1 (l+6)|l+8p

p(l+S)

_ fn(Op

39

39

mcol,

A(d,€)p(\+8) ,

3^z ^ 6[a(l+6)|l+6rM

CO' (l + e)|l+6p

39

-B{b,e)z{l+€)

(3.4)

z(l+6)

(3.5)

Here E^ and E^ are the components of the electric field perpendicular and parallel to the z axis and cOp is the

equilibrium plasma frequency. The equations describing

the radial and axial motions of the displaced ions are

^[p(i+5)j+/n-|-{p(i+6)j

coz —^^(6,e)p(l + 6)=0,

CO

dt

{z{l+e)}-^B(d,€)z(\+€)=0

(3.6)

(3.7)

After transforming to a frame rotating at the equilibrium rotation frequency co^ and keeping terms no higher

than first order in 8 and 6, we obtain linear differential equations for 8 and e. The equations are

d^ ~ d ~ ^l

-7Y8 +/(ft-2a),)^8 {-{r]A,+ A^)5,i=0, (3.8) dt dt 3

d^ ^^2-'-^{B,+B,/'n}e^0. (3.9)

Here A^ = dA{0,0)/d6R and ^, = a^(0,0)/a6, with

similar definitions for B^ and ^g. There are no first-order bj terms in the expansion of A and B, because A and B

are even functions of 8/. The real and imaginary parts of

Eq. (3.8) can be solved simultaneously. The solution is oscillatory with eigenfrequency cojo given by

,2

CO

2 —

20

COn (n-2co,y —^(77^,+^g) (3.10)

The eigenfrequency of Eq. (3.9) is given by

.2 ^20 ~ , {B, +B,/ri\ (3.11)

The requirement that the two eigenfrequencies be equal

gives a quadratic equation for r]. Substitution of the solu- tions for 7] back into either Eq. (3.10) or (3.11) gives the

(2,0) mode frequency CO20. After a number of algebraic

manipulations we find IN- 132
530 J. J. BOLLINGER et al. 48

+ \2 2(6)^)

2(a)

' = al+(o'n

+

20' ml

-4n2(c^2. -4/3)]'/2 (3.12)

Here ll^ = fi — 2<y, is the vortex frequency (the cyclotron

frequency as seen in the rotating frame),

ft„ =(fti^+a)p )'^^ is the upper hybrid frequency, and

2 2 3a

ia^-l)3/2 Q'2

(a^-1) 1/2 (3.13)

The lower-frequency mode [denoted by the negative sign

in Eq. (3.12)] is called a plasma mode and is characterized

by 17 < and a mode frequency cojo 5 fi/V3. The higherfrequency mode [denoted by the positive sign in Eq.

(3.12)] is called an upper hybrid mode and is characterized by r]>0 and a mode frequency cojq - ft/^3. Plasma

and upper hybrid modes are discussed in Sec. V. The fre- quency coq is the (2,0) plasma mode frequency in the ab- sence of a magnetic field (that is, ft^^-0 or a plasma

confined by a uniform background of opposite charge). It

is a mode frequency of a cold cloud of ions in an rf trap.

Most Penning-trap experiments are done with eo^KKil.

For simplicity we therefore discuss the two modes of Eq.

(3.12) for this experimentally interesting case. Figure 6

shows a graph of cojo and ^20 ^s a function of rotation fre- quency for a)^ /n = 0. 1 5 1 . In the limit of low rotation

frequencies (co^-^o^) the plasma mode frequency ap- proaches cOp. At <y^=<y^, the mode frequency

a)^=6) =co^. In this limit the mode parameter 17 for the

plasma mode frequency is large in magnitude and negative. The plasma mode looks predominantly Hke an axial stretch mode with only small radial excursions which are

1.0

0.8

a 0.6

o

CM 3 0.4

0.2

0.0

V

1 T 1 —r-

- V"^^,

/

.ny" \ r "30/" S

0.0 0.2 0.4 0.6

cj /n

0.8 1.0

FIG. 6. Graph of the plasma CO20 and upper hybrid CO20 mode

frequencies [from Eq. (3.12)] as a function of rotation frequency

(Or for <U2 /ft = 0.151. Also shown are the vortex frequency

il^—Q, — 2cOr (dotted line) and the plasma frequency cOp (dashed

line) in units of the cyclotron frequency.

180° out of phase with the axial excursions. This is be- cause for (o^^co^ «ri the magnetic field in the rotating

frame ( «ft —2w^) is large and constrains the radial ex- cursions of the plasma. At the Brillouin limit (n^,=0) or

in a cold plasma confined in an rf trap, the plasma mode

frequency equals coq and the mode parameter rj=—2.

This means the volume and therefore the density of the

plasma stay constant during the excitation of this mode.

At the Brillouin limit, the plasma mode therefore consists

of incompressible deformations of the plasma shape (a surface mode). The (2,0) plasma mode of a spheroidal

plasma is similar to an m =0 plasma mode of a cylindrical plasma column with a wavelength equal to the plasma

length [37,38].

The upper hybrid mode frequency approaches

fl^=ft — 2w^ for low rotation frequencies and is equal to n — 2co^ at (0^=0)^. In this hmit the mode parameter tj for the upper hybrid mode is small and positive (radial

excursions » axial excursions). For low rotation fre- quencies we can show that this mode looks like a coherent excitation of the perturbed cyclotron orbits of

the ions. For two ions on opposite sides of the trap axis,

the perturbed cyclotron orbits are 1 80° out of phase. At

the Brillouin limit (or for ions confined in an rf _trap), the

upper hybrid mode is equal to o)p = il/V2. Here

7] = a~^, which implies that the plasma oscillates (or

"breathes") from a small spheroid to a larger confocal

spheroid and back again. The electric field outside the

plasma does not change under such an oscillation. This can be called a bulk mode because it is confined to the

plasma interior and is difficult to couple to with external

fields. Both of the (2,0) modes are symmetric (even functions) about the Brillouin condition (6i)^ = ft/2), as can be

seen in Fig. 6. If either one of the (2,0) mode frequencies can be measured, it can be used together with the (1,0) mode fre- quency 0}^ and the cyclotron frequency ft to obtain the

plasma rotation frequency co^. The plasma rotation fre- quency can then be used to determine the plasma's density Wq ^nd aspect ratio a from Eqs. (2.1), (2.2), and (2.6). Figure 7(a) shows points of constant 6)20 (the upper hybrid frequency) and Fig. 7(b) shows points of constant w^

(the plasma mode frequency) on graphs of rotation fre- quency versus the trap axial frequency. All frequencies

are normahzed by ft. The axial frequency co^ [equivalently the (1,0) mode frequency] depends only on the trap

design and the potential difference between the trap end

cap and ring electrodes. Typically it can be measured or

calculated very precisely. Similarly the cyclotron fre- quency can be measured or calculated very precisely and

the operating point for co^ /ft along the x axis of Fig. 7

precisely determined. A measurement of a (2,0) mode

frequency can then be used with Fig. 7 to determine the

rotation frequency. Figure 7(b) shows that the plasma

mode frequency CO20 gives accurate information on (o^ /ft

mainly for slow rotation frequencies sUghtly greater than

co^ . However, the upper hybrid frequency gives accurate information on 6)^ /ft over a wide range of conditions,

especially over the experimentally interesting range

&)2/ft«l. Section IV describes some (2,0) mode mea- TN-133
48 ELECTROSTATIC MODES OF ION-TRAP PLASMAS 531

surements on plasmas of ^Be"*" ions. The modes were ex- cited by applying an oscillatory potential between the

ring electrode and the trap end caps which was resonant with the mode frequencies.

Figure 7 assumes a spheroidal plasma with A,^,, Hq^''^ « (plasma dimensions) as discussed in Sec. XL

However, even for finite temperatures, the (2,0) modes

should provide useful information on the plasma rotation

frequency. For example, in the limit that the ion space- charge fields can be neglected, the upper hybrid mode fre- quency is ft — 26)^. The diflFerence between ft— 2a)„ and

the measured upper hybrid mode frequency is therefore a measure of the importance of the space-charge fields. Also, an oscillatory drive on the ring electrode at

il — 2co^ can parametrically excite the perturbed cyclo0.5

FIG. 7. Sensitivity of the (2,0) mode frequencies for deter- mining the plasma rotation frequency. Points of constant (a) upper hybrid frequency CO20 or (b) plasma mode frequency coio are plotted on a graph of rotation frequency cOr vs the trap axial frequency o)^ (in units of the cyclotron frequency CI). A measurement of co^, il, and CO20 or «m can be used with these graphs

to determine the rotation frequency. The cojo plots are symmetric about <y,/fl=0.5. The boundary defined by the low-Or end of the curves denotes the condition (Or—o)„. Ion confinement requires (o^^co^.

tron motion of the plasma center of mass [the highfrequency (1,1) mode] whose frequency is ft —o)^ in the

laboratory frame. This parametric excitation requires an

initial, nonzero mode ampHtude.

B. (2,1) modes

We now consider an azimuthally asymmetric quadrupole mode. As in the previous section, we let p=x+iy

and z denote the equihbrium radial and axial position of an ion (or group of ions) in a plasma with an equilibrium

density Kq and aspect ratio a. We guess that there is a mode described by displacements from equilibrium of the

form

X —i-X +8z cos(co2it) ,

z ^z + e[x cos( co2it)+y sin( co2i't ) ] .

(3.14)

As the subscripts in Eq. (3.14) indicate, this turns out to be a (2,1) mode. The superscript lab indicates that the

description is in the laboratory frame of reference. [This

labehng was not required with the azimuthally symmetric

(2,0) mode because its frequency is the same in either the

laboratory or rotating frame.] Equation (3.14) can be

rewritten using complex notation as p-^p + 8(t)z , z^z + Re{?(r)p*j ,

(3.15)

where the time dependence is now included in 5(f) and

?(f), the asterisk denotes the complex conjugate. Re

denotes the real part, and

?(?)

6(f)

V (3.16)

where the mode parameter 77 is independent of time and

real. Equation (3.15) transforms a uniform density

spheroid into a uniform density elhpsoid. The elhpsoid is rotated with respect to the original, equilibrium spheroid

by an angle \f\, where, to first order in ? and 5,

1 (3.17) a

The modulus | f | denotes the magnitude of the rotation

angle. The phase of f denotes the plane in which the ro- tation takes place. Specifically, the rotation occurs about

the unit vector ( Imf /|f |, —Rey/\y\,0) or, equivalently, the p= —if axis. To first order in ? and S, the tilted ellipsoid is a spheroid with the same aspect ratio as the

equilibrium spheroid. Therefore to calculate the electric

field at the displaced ion positions to first order in ? and

5, we have only to calculate the electric field inside the

equilibrium spheroid rotated by an angle \f\ about the

p= —if axis. We obtain

mcol _ mcol E^^^^{p + bz) + —^(a -b)fz

E,^

2q

tncOr Iq

3q

-(a-b)Re{fp*}

(3.18)

(3.19) TN-134
532 J. J. BOLLINGER et al. 48

Here ojp is the equilibrium plasma frequency and a {a)

and bia) are evaluated at the equilibrium aspect ratio a. After transforming to a frame rotating at the equilibrium rotation frequency, the following diflFerential equations are obtained for ^^^cxp{io}^t]5 and

?^=exp{/6),f I?:

,2 4T8,+/av4s.--^/(a)(aHr7)S,=0

dt^ at 3

d' ^ "^P f, ^ + 1 ?^=0

(3.20)

(3.21)

Here

f(a) = [a{a)-b(a)]/{l-a^) = {(\-3col/(ol)/(l-a^).

With the convention that the time dependence of S^ and

?^ is given by exp(ico2\t), Eqs. (3.20) and (3.21) give the

following two equations for co2\, the (2,1) mode eigenfrequency in the rotating frame. We obtain

COr -<y^i-ft^2i —r^/(a)(a'+7/)=0 ,

1 ^p + 1 =0

(3.22)

(3.23)

When Eqs. (3.22) and (3.23) are solved simultaneously, a cubic equation for -q is obtained.

illv + -^fia)(a^ + V^(V-'^)^=0 . (3.24)

Solutions for 77 from Eq. (3.24) can be plugged back into

either Eq. (3.22) or (3.23) to obtain values of tuji for three

diflFerent (2,1) modes. The laboratory (2,1) mode frequency co2\ 'S obtained from 6>2^i^=(y2i ~<yr- With the convention used here, a positive 6)21 (^2*1'') denotes a counter- clockwise precession when viewed from above the x-y plane in the rotating (laboratory) frame. This agrees with

the usual right-hand rule for the sign of an angular fre- quency.

Figure 8 shows a graph of the three different (2,1) mode frequencies (in the laboratory) versus rotation fre- quency for 6)^ /n=0.151. As discussed in the previous

two paragraphs, all three modes correspond to a tilt of

the equilibrium spheroid with respect to the z axis. This

tilted spheroid then precesses about the trap symmetry

axis at the frequency a>2i' as shown in Fig. 9. Because the

density remains constant during the mode excitation, all three (2,1) modes are surface modes. The modes with the

highest and lowest values of |<y2^i''| are characterized by

77 >0 (axial and radial excursions in phase). The third

mode with intermediate frequency is characterized by

T/<0 (axial and radial excursions 180° out of phase). For

the mode with the highest value of |<y2^i'| in Fig. 8, the

tilted spheroid precesses rapidly at a frequency near fl. For co^^co^ «fl, the mode frequency approaches

co^f^ ^—(fl — co^) and the mode parameter satisfies r/ « 1

.

Equations (3.15) and (3.16) show that the axial excursions

are small compared to the radial excursions and the pre- cession can be thought of as being due to a coherent (al- though asymmetric in z) excitation of individual ion cy0.2

0.0

V

1

-0.2 - ^ -0.4

-^

3 -0.6

-

-0.8 - \

-1.0

0.0 0.2 0.4 0.6 0.8 1. cj /n

FIG. 8. Graph of the three (2,1) mode frequencies coix as a function of the plasma rotation frequency cOr for 6;j/Q=0. 151. The mode frequencies in the laboratory frame are plotted. The

mode with the lowest value of Itu^i*"! has a frequency equal to foro,/n=0.219.

clotron orbits. For low rotation frequencies, this mode is classified as an upper hybrid mode (see Sec. V). For the

mode with the lowest \co2\\ i^ Fig- 8, the tilted spheroid

precesses at a low frequency. For co^

ft and

6), ^6t)^«ft, the mode parameter satisfies 7/»l and

a)2\^a>2—o)r. Here this mode can be thought of as a coherent (and azimuthally asymmetric) excitation of the

axial motion of individual ions. For low rotation fre- quencies, this mode is classified as a magnetized plasma

mode (see Sec. V). At the Brillouin hmit, the highestand lowest-frequency modes both have r/=l, and their frequencies are equally spaced about — ft/2. Stated

FIG. 9. Sketch of the (2,1) mode in the laboratory frame. The plasma spheroid is tilted with respect to the z axis and

precesses about the z axis at co2t'- In this figure the sense of the precession is positive. The picture in the rotating frame is the same except the precession occurs at the frequency

IN- 135
48 ELECTROSTATIC MODES OF ION-TRAP PLASMAS 533

diflferently, in the rotating frame the two modes have

equal and opposite frequencies at the Brillouin Umit.

Near the Brillouin limit, these two modes are classified as evanescent modes (see Sec. V). The intermediate frequency mode has rj<0. For co^KKil and lo^^a^ «ft, the

intermediate mode has |7;|»1 and (o^2\—~^z~'^rTherefore this mode has an interpretation similar to the

lowest-frequency mode. However, the behavior of this mode is quite different for a>^»oi^. At the Brillouin

limit 7}=— a}, which means that the plasma shape does

not change (that is, the plasma spheroid does not tilt), and this mode frequency in the rotating frame is 0. Therefore this mode vanishes at the Brillouin hmit and is called a magnetized plasma mode (see Sec. V).

Figure 8 shows that the (2,1) mode with the lowest

magnitude of frequency in the laboratory frame is at

ft)^ /ft =0.2 19 for 0)^ /ft =0.151. This zero-frequency

mode can be excited by a static field error [19,21,39].

From the ion's point of view (the rotating frame) the static field error looks hke a rotating field with a frequency

co^. When 0)21=6)^ or (o^2\~^2\~(^r~^ the (2,1) mode is excited. Section IV describes the excitation of the static

(2.1) mode by a tilt of the electrode symmetry axis of the

trap relative to the trap magnetic field axis. If the rota- tion frequency (o^ was increased from small values by an

external torque, excitation of this mode tended to limit

(o^ to values less than the rotation frequency where

4f=0. In addition to the (2,0) and (2,1) modes, there are two

(2.2) modes. The (2,2) modes are discussed with the other

l—m modes in Sec. V. In a (2,2) mode, the plasma forms an ellipsoid with unequal principal axes in the x-y plane.

The ellipsoid rotates about the z axis at the (2,2) mode

frequency. A measurement of one of the (2,1) or (2,2) mode frequencies can be used to determine the plasma ro- tation frequency and aspect ratio in the same manner as described for the (2,0) modes (see Fig. 7).

IV, EXPERIMENTAL RESULTS

In this section we compare experimental measurements

of some quadrupole modes of a Penning-trap plasma with

the calculations of Sec. III. Much of the experimental

work has been described previously in Ref. [21]. Typically between 1000 and 5000 ^Be"*" ions were stored in a cy- hndrical Penning trap shown schematically in Fig. 1. Most of the experimental work was done with 5 =0.82 T

where ft(^Be"^)/27r= 1400 kHz. Some singly charged,

heavier ions were created when the ^Be"*" ions were creat- ed or loaded into the trap. In addition, singly charged,

heavier ions slowly formed after loading ^Be"*", presumably due to ion-molecule reactions involving the ^Be"*" ions. Their presence appeared to shift the observed mode

frequencies. We could eliminate them from the trap by

momentarily raising Vj so that 6>^ /ft > 1 /a/2 for the

heavier ions, making them unstable. In practice we

would raise Vj to satisfy 0)^/0.^0.61 for ^Be"*". We then

obtained repeatable and consistent measurements for the

mode frequencies.

Radiation pressure from a laser was used to cool the

^Be"*" plasma [20,21,28]. A 313-nm laser (power

==^100)tiW) was tuned 10-50 MHz below the rest frequency (Oq of the 2s ^5i/2(mj = +y, mj = H-y)

-^2p ^P3/2( j,y) transition in ^Be"*". In addition to cool- ing the ions, this laser optically pumped the ions into the

(y, j) ground state [28]. The ions were detected by imaging the laser-induced ion fluorescence onto the photocathode of a photon-counting imaging tube (see Fig. 1). A real-time display of the image was used to monitor

qualitatively the plasma's kinetic energy; a hot plasma

had a more diff"use boundary and less ion fluorescence. With this simple diagnostic we were unable to distinguish

between an increase in the random thermal energy of the

ions and the excitation of a coherent modal motion of the

plasma. The cooling laser was split into two beams. One

beam (shown in Fig. 1) was directed perpendicularly to the 2 axis near the center of the trap. The second beam

(not shown in Fig. 1) was used along with the first beam

when the lowest ion temperatures were desired

[20,21,28]. It was directed at a 51° angle with respect to the z axis. (The projection of this beam along the z axis cooled the ion's axial motion directly.) With laser cooling, the ^Be"*" ions quickly evolved into a near thermal

equilibrium distribution and could be confined for many

hours [28]. The temperature of the ions could be measured from the Doppler broadening of an optical transi- tion [28]. Typical temperatures ranged from 5 to 200

mK. (Here temperature refers to the MaxwellBoltzmann velocity distribution which occurs in thermal

equihbrium in the rotating frame.)

The plasma density was determined from Eq. (2.1) by

measuring the plasma rotation frequency. We measured

the rotation frequency by driving the ( m^ = + y, my = +y)—»-(+y, — y) clcctron spin-flip transition at fre- quency oi^ in the Is ^S^/2 ^^^^ ground state. The transi- tion frequency o)^ is approximately equal to 22 GHz at 5 =0.82 T. The transition was observed as a decrease in the ion fluorescence when the frequency of the appHed

microwave field is resonant with 0^ [40]. In addition, a decrease in the ion fluorescence was observed at the side- band frequencies 6jj±<y^. This is because from the ion's point of view the phase and amplitude of the microwave

field are modulated due to the rotation of the plasma.

This produces sidebands in the microwave spectrum observed by the ions at (o^+cOr [41]. Measurement of these sidebands enabled co^/Itt to be determined to about 5 kHz. At 5 =0.82 T, the ^Be"*" ion densities usually were measured to be greater than lOVcm^. This density with

temperatures less than 200 mK results in Debye lengths

less than 10 jim. With 1000-5000 ions in the trap, the

typical plasma dimensions were between 100 and 1000

jum, which is one or two orders of magnitude larger than

A^). The radius of the cylindrical trap electrodes (1.27 cm) was another order of magnitude larger. Consequently the

laser-cooled ^Be"*" plasmas satisfied the condition

Xj) «(plasma dimensions) «(trap dimensions) and

formed constant density, spheroidal plasmas. The axial frequency co^ was measured by applying an oscillating potential to one of the end caps [42]. The cyclotron fre- quency ft was calculated from the magnetic field. The

magnetic field was determined accurately from measure- TN-136
534 J. J. BOLLINGER et ah 48

ments of the electron spin-flip or nuclear spin-flip transi- tion frequencies in the ^Be"*" ground state. The angular momentum and therefore the rotation fre- quency (0^ of the ^Be"*" plasmas were controlled by a torque from the radiation pressure of a second, weak

laser beam (power ^2 ^W) whose frequency coj was

tuned above the cooling transition frequency coq. This

torque beam (see Fig. 1) was directed perpendicularly to the z axis but displaced from the z axis through the side of the plasma which recedes from the laser beam due to the plasma rotation. This beam imparted to the plasma a torque which tended to increase the plasma rotation fre- quency co^. Initially the plasma was cooled by the cooling

laser beam with the result that 6>^ «n. The torque laser beam was then directed through the side of the plasma

with a frequency coj- tuned slightly below (Oq. As coj- was

increased above coq,coj- became resonant with the frequency of the Doppler-shifted ions in the torque beam. This

produced ion fluorescence and a torque on the plasma

which increased the rotation frequency. As shown in Fig. 4, an increase in co^ produced a decrease in the plasma radius Tq for ct), <ft/2 and an increase in Tq for

co^ > n/2. The plasma radius was measured with the imaging tube. Accurate measurements of the plasma's axial extent were more difficult to obtain. However, we could

qualitatively check that the plasma obtained its max- imum axial extent (and therefore maximum aspect ratio) at the Brillouin limit by scanning the position of one of

the laser beams along the axial direction. By appropriate tuning of the torque laser frequency

and with some adjustment of the positions of the coohng

and torque lasers, we could obtain a steady-state, equilibrium plasma with a rotation frequency anywhere in the

allowed range from co„ to ft— «„. In addition to supplying a torque, the torque laser also supplied energy to the

plasma when C[)7'>6}o [^3]. Equilibrium occurred when

the energy input from the torque laser was removed by

the cooling laser and when the laser beam torques and

torques from static field asymmetries summed to 0. We

were also able to increase co^ to values greater than ft/2

by using just the cooling laser. An increase in the cooling

laser frequency increased co^- In this case, however, the

rotation frequency sensitively depended on the cooling

laser frequency and a steady-state condition was more

difficult to achieve.

We discovered [21] that as the plasma rotation fre- quency was increased, there was a range of rotation fre- quencies where the plasma acquired a diflFuse boundary

and a low level of ion fluorescence characteristic of a hot

plasma. The range over which this apparent heating oc- curred depended sensitively on the alignment of the

trap's symmetry axis with the magnetic field axis. Let ^o denote the angle between these two axes. If we assume

that the trap is aligned when the apparent heating reso- nance is minimized, then we could adjust 6q<0.0V by

searching for an alignment which gave no apparent heating. For df^yOA" the plasma rotation frequency could

not be increased beyond the point at which heating first occurred. This heating resonance appeared to get

stronger with an increase in the number of ions. We have

been able to identify this heating resonance as an excita- tion of a collective (2,1) plasma mode by the static field asymmetry associated with the misahgnment of the trap

symmetry axis with the magnetic field. Figure 10 shows

the measured rotation frequencies where heating oc- curred with 00 — 0-02° for different trap axial frequencies.

Also shown (solid hne) is the calculated rotation frequen- cy at which £02^/' =0, that is, co,=co2i as calculated from

Eqs. (3.22)-(3.24). Excellent agreement is obtained be- tween the predicted and measured rotation frequencies

where heating is observed.

We did not attempt to understand the source of energy

in the heating resonance. However, two possibihties are

listed below. A static asymmetry cannot change the total energy of the plasma, but it can convert potential to thermal energy by producing an expansion of the plasma.

Because the ratio of potential to thermal energy is large

in our plasmas, a small expansion can produce a large in- crease in the ion thermal energy. This energy source in- creases with the potential energy per ion of the plasma.

For example, it increases with the ion number or plasma

density. With the lasers operating continuously, the

time-averaged plasma radius does not change. However,

the plasma radius may fluctuate. For example, there

could be a small plasma expansion between photon

scattering events, balanced by a plasma contraction when

a photon is scattered. Heat could be generated by expansion during plasma fluctuations. Another source of ener- gy is the torque laser. The energy input from this laser could increase when the static (2,1) resonance is excited if there is an increase in the scatter rate from this laser. If the density is increased from low values corresponding to co^ «ft/2, excitation of the static (2,1) mode may

provide a practical hmit to the density and number of

charged particles that can be stored in a Penning trap.

For large ion numbers or Oq > 0. 1° we were unable to obtain densities greater than nQ(cOr=co2i). For

6)^ /ft > 0.53, Fig. 10 shows that 6)^=021 occurs for

G

FIG. 10. Rotation frequency co^ at which heating was ob- served as a function of the single-particle axial frequency co^. Both frequencies are expressed in units of the cyclotron frequen- cy il. The experimental data were obtained with 1000-5000

'Be"*" ions stored in the trap of Fig. 1. The soHd hne indicates the calculated rotation frequency &>, at which &>2i=0. This is a

universal curve involving no adjustable parameters.

TN-137
48 ELECTROSTATIC MODES OF ION-TRAP PLASMAS 535

6}^ /ft > 0.5 and therefore the static (2,1) mode will not

prevent reaching the Brillouin hmit in this case. However, for a high magnetic field and a trap with large dimensions, the condition (o^ /ft > 0.53 may be difficult to obtain because of practical limitations on the voltage that can be applied to the trap electrodes. In addition,

(y^ /ft =0.5 with 6J^ /ft =0.53 occurs for an aspect ratio a =0.44. For storage of a large number of charged particles, this would require a trap with a large radius. However, a large volume trap is more easily constructed by

extending it axially while keeping the radius fixed [44]. For storage of large ion numbers, the plasma dimensions

may not be small compared to the trap dimensions. This

changes the calculation of the modes because the trap potential will not, in general, be quadratic and the image

charges in the trap electrodes cannot be neglected. In

spite of the difficulty of increasing co^ (and the density)

through the point where the heating resonance occurs, there is still the potential to "jump through" this condition by sudden switching of co^ or creating the plasma

with 0)^ > 1 6)21 1

Laser torques were also used to increase co^ in ^Be"*" plasmas consisting of 40 (XX) ^Be"*" ions at 5 =6 T. We

were unable to remove possible contaminant ions at this high magnetic field because of the high voltages required on the trap electrodes. In addition, we did not have a mi- crowave source (=^160 GHz at B=6 T) to drive the

ground-state electron spin-flip transition and measure the

plasma rotation frequency. However, a rough measure of

cOr could be obtained from the plasma aspect ratio. We

were able to obtain rotation frequencies cOr>Cl/2. Fig- ure 11 shows an image of a rapidly rotating Be"*" plasma

with <y^/27r=;ft/27r=10 MHz. The propeller shape of

the ion fluorescence is due to the fast rotation of the plas- ma. The ions get excited in the laser beam, but decay

outside the laser beam because the 8.2-ns lifetime of the

excited P state is longer than the typical transit time

FIG. 1 1 . Image of the ion fluorescence of a rapidly rotating 'Be+ plasma at 5=6 T with &),/27r^ft/2ir= 10 MHz.

Fluorescence due to the perpendicular and diagonal cooling

beams is observed. The plasma diameter in the z =0 plane is approximately 1 mm. The ion kinetic energy due to rotation at the radial edge of the plasma is approximately 50 eV.

through the beam. The (2,1) heating resonance was much

stronger and more sensitive to the magnetic field alignment at B =6 T than in our work at 0.8 T. This may

have been due to the larger number of trapped ions at 5 = 6 T. We also observed additional heating resonances

at lower rotation frequencies than the (2,1) heating reso- nance. Section V discusses other static modes that are

potential candidates for these additional heating reso- nances. We note that many of these resonances tend to hmit the plasma density to even lower values than the

static (2,1) resonance. Density hmits on trapped ion plasmas have also been

reported in other Penning-trap experiments. In Ref. [45] the experimentally observed ion fluorescence in a Penning trap was interpreted in terms of limits to the ion

density imposed by single-particle resonant transport

[46]. In single-particle resonant transport, enhanced

transport (and lower ion density) may occur when the axial bounce frequency of an individual ion is commensurate with the ion's rotation frequency. In this process an

individual ion resonantly interacts with an external field error. This is different from transport due to the excita- tion of a zero-frequency mode where a collective plasma

mode resonantly interacts with an external field error. In our work we observe no evidence for single-particle reso- nant transport as an important mechanism for limiting

the ion density. An important difference between our work and that of Ref. [45] is the ion temperature. In our work the ions are sufficiently cold that the ion-ion collision time is short compared to the axial bounce time.

The experimental measurements of Ref. [45] were done

on relatively hot ion clouds (^5000 K) where the ions

may bounce faster than they coUide. In addition, the Debye length in this work was Ukely comparable to or larger

than the cloud dimensions. In this case the ion density is not constant, and calculation of the space-charge-shifted

axial bounce frequency of an individual ion requires a model diffierent from the cold thermal equilibrium dis- cussed in Sec. II. The theoretical analysis of Ref. [45] ^

would be more convincing with a self-consistent calculation of the space-charge shifts. Such a self-consistent cal- culation is described in Ref. [47]. When space charge is important, the potential variation in the axial direction is no longer harmonic, so the axial bounce frequency de- pends on the ion temperature and axial dimension of the

ion cloud. Both of these dependencies are neglected in Ref. [45]. We also measured the (2,0) plasma and upper hybrid

modes. These modes were excited by applying a

sinusoidal potential between the two central and two

outer electrodes in Fig. 1. They were detected by a change in the ion fluorescence when the frequency of the

applied rf was resonant with the mode frequency. This

change was presumably caused by a change in the ions' Doppler width and possibly by a change in the laser beam -plasma overlap. Measurements of the plasma

mode frequency as a function of the plasma rotation fre- quency are shown in Fig. 12 for two different trap axial frequencies. The soUd Unes show the plasma mode fre- quency calculated from Eqs. (3.12), (3.13), and (2.6). Again, good agreement between the predicted and obTN-138
536 J. J. BOLLINGER et al. 48

0.40

0.35

.^ 0.30

o

I w

3 0.25

0.20

0.15

_ - - ^i>^-»*- TfTTTT..

: /,••' A cj^/n = 0.256-

• a;yn = 0.151

- — — ~ ~ _ J ^p-^-^* . ^. . . • w ^ w

r . 1 0.0 0.1 0.2 0.3 0.4 0.5

FIG. 12. Plasma mode frequency 6J^ as a function of the ro- tation frequency cOr for two different axial frequencies co^. All

frequencies are expressed in units of the cyclotron frequency ft. The circles and triangles give the experimental data. The solid

lines give the cold-fluid model predictions for cojo- The dashed

and dotted lines give the high- and low-magnetic-field calculations for cl>2o, respectively. The dotted line corresponds to the

oj^ frequency for an rf-trap plasma with the same density and

aspect ratio.

served modes is obtained with no adjustable parameters.

Figure 12 shows two additional calculations. In the first, shown as dashed lines, the magnetic field is assumed to be

eflFectively infinite; that is, the ions are not allowed to move radially, and the mode frequency is calculated as- suming a simple axial stretch of the charged spheroid.

The mode frequency in this case is calculated from Eq.

(3.11) by setting the 55/77 term equal to 0. In the second

calculation, shown as dotted lines, the magnetic field is assumed to be effectively 0; that is, the curve shows coq

[see Eq. (3.13)]. From this figure, the (2,0) plasma mode

behaves like a mode of a strongly magnetized plasma at low rotation frequencies and an unmagnetized plasma

near the Brillouin limit. We attempted to excite the

upper hybrid (2,0) mode with a drive frequency near the

upper hybrid frequency. A resonance (a change in the

ion fluorescence) was observed as the frequency of the applied rf was swept. The amplitude of the drive was de- creased until the resonance signal was barely detectable. Figure 13 shows a plot of the measured resonance fre- quency along with a calculation of the upper hybrid (2,0) mode from Eqs. (3.12), (3.13), and (2.6). Agreement is good, indicating that the observed resonance was likely

the excitation of the upper hybrid (2,0) mode. As dis- cussed in Sec. V, our drive could also potentially excite an upper hybrid (4,0) mode. For the conditions of this experiment, the two mode frequencies would be difficult to distinguish. However, for sufficiently weak drive

strength, the (2,0) mode should be excited more strongly

than the (4,0) mode. We could not excite this upper hybrid mode near the Brillouin limit because, as discussed

in Sees. Ill and V, the upper hybrid modes become

difficult to couple to with external fields there. The agreement obtained here with the zero- temperature calculations requires that Xj)

(plasma

dimensions) «(trap dimensions). This condition is usualFIG. 13. Upper hybrid mode frequency CO20 as a function of

the rotation frequency co^ for co^ /ft = 0. 1 5 1 . All frequencies are expressed in units of the cyclotron frequency ft. The circles give the experimental data. The uncertainty for measurements

done at low rotation frequencies is approximately the size of the

circles. At high rotation frequencies the uncertainties increase because the mode becomes diflBcult to excite. The solid line gives the cold-fluid model predictions.

ly easy to satisfy with a laser-cooled ion plasma and, in

fact, may not be difficult to satisfy with a cryogenic

{T=4 K) plasma. For example, a plasma with

tiQ^lOi^ /cm^ at r =4 K has a Debye length A.^, =44 /zm.

Consequently if the dimensions of this plasma are greater

than 1 mm, the inequaUty Xj) «(plasma dimensions) is satisfied. A plasma with Zq = Tq =^ 1 mm and

n Q ~ 1 0^ /cm^ requires loading A-irr qM q /3 =4 X 1 0"* charged particles within 1 mm of the trap axis. The

charged particles must then be cooled and evolve into a thermal equilibrium state before the plasma radius ex- pands [18,46] due to radial transport produced by trap

asymmetries. For electrons this appears to be possible

because at large magnetic fields (5^6 T) the cyclotron

radiation time is short ( < 1 s) and the time over which

the plasma spreads due to trap asymmetries can be long

[16] (many hours). In addition, localized sources such as

field emission points can be used to load many electrons

close to the trap axis. For ions which cannot be laser cooled the inequahty for a zero-temperature plasma may

be more difficult to achieve. This is because radiative

cooUng times are typically longer. In addition, the radial transport in Penning traps due to trap asymmetries is likely faster for ions than electrons because of the larger

ion mass. However, with care in loading the plasma, or with the development of a technique to reduce the plasma

radius and increase the ion density, it should be possible

to satisfy the conditions for a zero-temperature plasma

with ions at 4 K. Ions in a Paul trap can be rapidly

cooled to the temperature of a background buffer gas. In

this case it should be possible to obtain a plasma where

the Debye length is small compared to the plasma dimensions. Some experiments in Paul traps approximate this regime [47].

V. GENERAL LINEAR MODES

The quadratic oscillations discussed in Sees. Ill and IV

are important special cases of an infinite set of hnear nor- TN-139
48 ELECTROSTATIC MODES OF ION-TRAP PLASMAS 537

mal modes. Although we have, up to now, emphasized

the behavior of the quadratic modes, there are several reasons to consider the general modes here. First, al- though the general higher-order modes have not yet been

experimentally measured, we will see that it should be

possible to observe at least a few of them using the same

experimental techniques as were described in Sec. IV. Second, a deeper understanding of the quadratic modes

may follow from consideration of their place in the

hierarchy of general oscillations. Finally, a simple exact

analytic solution for all of these modes has been found

[29]. In Sec. V A we briefly review some features of this general solution. We then discuss a simple form of the general dispersion relation for the modes which allows one to obtain numerical solutions of the mode frequencies using

a polynomial-root-finding algorithm. We also consider some examples, including some of the modes which can

be resonantly driven by applying an oscillating potential

between the two end electrodes and the center electrodes. In Sec. V B, we focus on modes which can be resonantly

driven by static field errors. We find that such reso- nances occur in profusion, becoming dense when the

cloud rotation frequency is small compared to the cyclo- tron frequency.

A. Description of the normal modes

In order to obtain a general solution for the normal

modes several assumptions must be made. We assume

the cloud is near thermal equilibrium, and we assume the

oscillations around this equilibrium are small so that we

can linearize the equations of motion. The temperature is assumed to be sufficiently small that pressure effects on

the fluid dynamics are negligible, and correlation effects are also neglected; these are good approximations provided that both the Debye length and interparticle spacing

are small compared to both the size of the cloud and the

wavelength of the mode. Electromagnetic eff'ects are neglected (that is, VXE= VXH=0 in Maxwell's equations) since the cloud is small and the mode frequency is relatively low, and the effect on the dynamics of image

charges in the electrodes is neglected.

These approximations are identical to those used in the

earlier sections and apply well to present experiments on

small cold ion clouds. However, unlike in the previous

analysis, we do not assume a particular form for the density perturbations. Rather, a general linear perturbation

is employed. The dynamics are described in a frame ro- tating at the constant rotation frequency co^ of the equilibrium plasma. In this frame the plasma is stationary

and the density, fluid velocity, and the potential are per- turbed from their equilibrium values:

n(x,t) = nQ{ii) +bn(\,t) ,

v(x,?)=0+5v(x,r) ,

^(x,^)--<^o(x) + V'(x,r) ,

(5.1a)

(5.1b)

(5.1c)

where bn, 6v, and i/* are the perturbed density, fluid velocity, and potential, respectively. The equilibrium density /2o(*) is uniform within the plasma spheroid and zero

outside, and <{>q is the total equilibrium potential in a ro- tating frame, equal to [28]

2q

The first two terms in (f>Q are the external trap and space- charge potentials, given by Eqs. (2.4) and (2.9), respectively, and the last term is a pseudopotential due to rota- tion through the magnetic field. We substitute Eqs. (5.1) into the cold-fluid equations

[48] describing conservation of momentum and particle

number, as well as into Poisson's equation. In the rotat- ing frame, the cold-fluid equations have the same form as

in an inertial frame (for example, the laboratory frame),

except that the electrostatic potential is changed as indi- cated above and the cyclotron frequency is changed to the vortex frequency 0,^—0,-20)^. When equilibrium

force balance is taken into account and the equations are

linearized in the small perturbations, we obtain the

linearized continuity, momentum, and Poisson equations

of cold-fluid theory:

dbn

+ V-(«o6v)=0,

-^ + -^VV'-6vXft^=0,

at m"

2,/.= _-2- V'V bn

(5.2a)

(5.2b)

(5.2c)

where z is a unit vector along the z axis. Equation (5.2b)

and further equations implicitly assume q >0. For

q <0, fly-^ — 0,y. Using the assumption that the per- turbed quantities have a time dependence of the form

e~"^' in the rotating frame (so that co is the mode fre- quency as seen in this frame), a differential equation for xp follows from standard algebraic manipulations of Eqs.

(5.2):

V-e-Vt/'=0 , (5.3a)

where e is the cold-plasma dielectric tensor. In Cartesian

coordinates

e= le-,

-l€2

^3

(5.3b)

where 6, = 1 ill), €2 = n^p /[coico^- 0.1)],

and 63 = 1— (i)^/&)^. Equation (5.3a) is just Maxwell's

equation V-D — for a medium with a linear frequencydependent anisotropic dielectric tensor e. After a solu- tion to Eqs. (5.2) for the perturbed potential ^ is found,

Eqs. (5.2) can be used to calculate the perturbed density

bn and perturbed fluid velocity Sv. The normal mode problem requires a solution to Eq.

(5.3) subject to the boundary condition that ^-^0 at

infinity, which is a problem in the theory of electrostatics. Outside the plasma e=l and rp satisfies Laplace's equation, V^i/'°"' =0. Inside the plasma the dielectric tensor is anisotropic and the solution of Eq. (5.3a) is more compliTN-140
538 J. J. BOLLINGER et al. 48

cated. The inner and outer solutions must be matched

across the plasma-vacuum boundary according to

,/,in — ,/,out I W V I boundary '

n-e-Vi/''"=n-VV'°"' I boundary

{5.4a)

(5.4b)

where n is a unit vector normal to the plasma-vacuum

boundary.

Although the formulation of the problem as one in the

theory of electrostatics is a step forward, simple solutions are generally available only in one of the standard

geometries for which a separable solution exists, and this

is not a standard geometry. The surface of the plasma is spheroidal, while the dielectric tensor is anisotropic with

a different, cylindrical, symmetry. However, Ref. [29] showed that Eq. (5.3) does in fact have a separable solu- tion in an unusual frequency-dependent coordinate sys- tem. This is the only known exact analytic solution for normal modes in a magnetized plasma of finite size. The

solution for the mode potential is MCr(|'i/^)^r(^2)e""'^""" (outside plasma) , "^^

J5P/'"(fi/^)^r(^2)e"'"^""'* (inside plasma) , ^^"^^

where A and B are constants and Ql" and P/" are associ- ated Legendre functions. Outside the plasma, the solu- tion is expressed in terms of spheroidal coordinates [49] (^1,^2'^^ defined by the relations

x = [{fi-d^){\-^l)y^^cos<f> ,

The coordinate l^i is a generalized distance coordinate

taking the value I", e[zo,oo) outside the cloud, ^2 is a

generalized latitude in the range [—1,1] and <f> is the usual

azimuthal angle. Surfaces of constant ^^ are confocal

spheroids with the surface of the cloud defined by ^^^ =Zq,

and surfaces of constant ^2 ^""^ confocal hyperboloids.

The foci are a distance 2\d\ apart, where d^ =ZQ — rl. The coordinates {^i,^2'^^ become the usual spherical

coordinates ( r, cos0, (f)

) when Zq = rg

.

Inside the cloud the coordinates (|^i,|'2,<^) are employed in order to obtain a separable solution. These

coordinates are transformed spheroidal coordinates

defined by the equations

X = [(|?-rf')(l-|^)]>/2coS(^,

y = [(f,-d'){\-il)]''^sm<l>, (5.6)

2(6,/63)'/^ = ^,^2

where d^=z^ — r\ and ZQ^Zo^^i/^a)'^'^- These coordi- nates are frequency dependent; the different possible to- pologies of the coordinate surfaces are discussed in Ref.

[29]. Returning to Eq. (5.5), we note that different normal

modes are enumerated by the integers / and m, where

/ > and \m\<l. In fact, values of m < do not give rise to new modes if negative frequencies are allowed. Therefore m > is assumed throughout, and negative frequencies are allowed. For m¥'0, positive and negative fre- quency modes rotate about the z axis in opposite directions. As discussed in Sec. Ill, we use the convention

that modes with positive frequency rotate counterclockwise when viewed from above the x-y plane (so that

d<f>/dt >0 for (o>0). The two directions of rotation are not equivalent due to the apphed magnetic field. The

mode frequencies for positively and negatively charged

particles diflFer by a minus sign.

For a given pair (l,m) the mode potential outside the

cloud decays away like s*~'~*' at large distances s from

the cloud center [because Qi"(x)-^x -(/ + i) for large x

and ^x^>-s for large s\. The modes can also be

diflFerentiated by the number of oscillations in ^. For ex- ample, there are I —m zeros in the potential as one moves

in ^2 along a given spheroid outside the plasma from one

pole to the other [that is, from |;2 = 1 to — 1 on a constant

{^\,^) curve]. This is because Pf^ix) has / — m zeros in the range [—1,1].

The variation of the potential outside the cloud is in- dependent of the mode frequency, up to the overall constant A. However, inside the cloud, the frequency dependence of the coordinates (through their dependence on ej

and 63) impHes that the behavior of the mode potential

varies depending on the mode frequency, the plasma fre- quency and vortex frequency [except for two exceptional

cases described in Eqs. (5.9) and (5.10) below]. This

behavior can be understood qualitatively from the spatial

Fourier transform of Eq. (5.3a), e,kl+€,k^=0 , (5.7)

where k^ and k^ are the components of the wave vector perpendicular and parallel to the magnetic field, respectively. When co/cOp and fl/Wp are such that ei/e3<0, a

solution of this equation exists with both kj^ and k^ real, which is a propagating mode. However, when e, /e^ > 0, Eq. (5.7) implies that either kj^ or k^ must be imaginary

and the mode is evanescent. The frequency dependence

of the 6's imphes that the propagating mode relation

ei/e3<0 is satisfied by frequencies in the ranges

0< |6j| <min[(y , In^l] and max[(y^, |0^,| ] < |w| <ft„,

where ft„ =(ft}J+ft^)'^^ is the upper hybrid frequency.

Modes in the first frequency range are called magnetized

plasma oscillations, and modes in the second frequency

range are called upper hybrid modes [50]. As we will see presently, evanescent modes occur only when |ft^,| <o)p,

and such modes have frequencies in the range

|ft^| <\oi\<a}p. These regimes are shown in Fig. 14 as a function of the rotation frequency. (Here, remember that

both the vortex frequency and the plasma frequency are functions of co^.) Modes with frequencies faUing in areas

labeled A are magnetized plasma modes, modes falling in areas B are upper hybrid modes, and modes falling in C are evanescent. Modes which are evanescent have a different potential

variation within the cloud than modes which are propagating. Propagating modes reflect off the interior surface

of the spheroid and set up a standing-wave pattern inside

the cloud, whereas evanescent modes propagate along the

cloud surface but decay with distance from the surface

TN-141
48 ELECTROSTATIC MODES OF ION-TRAP PLASMAS 539

0.4 0.6

FIG. 14. Sketch showing the areas of allowed mode frequencies w as a function of the rotation frequency a>, . Modes in re- gions A, B, and C are, respectively, magnetized plasma, upper

hybrid, and evanescent modes.

into the plasma. (Because of the curvature of the cloud's

surface, this decay is algebraic rather than the exponential decay we would observe for an evanescent mode at a

planar interface.)

For example, the zeros of the (2,0) mode potential

within the cloud are shown in Fig. 15 for the case of a

spherical plasma with ily/cOp =0.5. As discussed in Sees.

Ill and IV, there are two possible forms for the potential

corresponding to a high-frequency upper hybrid mode

[labeled (a)] and a lower-frequency mode [labeled (b)]. The upper hybrid mode is propagating, but for this value

of ^y/ci)p the lower-frequency mode is evanescent, decaying with distance into the plasma. This behavior is best observed in Figs. 15(c) and 15(d), which show the potential variation of the modes along z =0 and r =0, respectively.

Furthermore, either Fig. 15(a) or Figs. 15(c) and 15(d)

show that the potential of the upper hybrid mode nearly

vanishes along the cloud surface, so there is almost no

potential variation exterior to the cloud. This is because

as fly/cOp —>0 (the Brillouin limit in a Penning trap or the

condition for ions in a Paul trap) the upper hybrid modes

have frequency co —cOp and, as shown below, become bulk

plasma oscillations with ^°"'=0. If ft^/cj. ^0 and

oi-^o)p, then 6=0 and Eq. (5.4) implies V^°"'-n = along

the plasma boundary. The only solution which satisfies both the boundary condition at the plasma and ^°"'—>-0 at infinity is ^°"*=0. Furthermore in the Cl^/cOp ^-0 hmit

the magnetized plasma modes disappear as their frequency a>^>-0. Thus, only the evanescent modes with 61/63 > may be observable near the 11^/0)^^0 limit if only elec- trostatic detection of the modes is employed. In this limit the evanescent modes satisfy V^i/''" =0, so from Eq.

(5.2) there is no density perturbation except at the surface

of the cloud. In this hmit the evanescent modes induce

incompressible deformations of the cloud's shape, and for

this reason they are often called surface modes.

As a second example, the interior potential variation of

the four possible (4,0) normal mode potentials are shown

in Fig. 16, again for a spherical cloud with ft^/a)p=0.5.

Like the (2,0) modes, these modes could also be driven by

in-phase oscillation of the end-cap potentials. There are now four zeros in the exterior potential as we move from

pole to pole along the spheroid, as can be observed in Figs. 16(a)- 16(d). Figures 16(a) and 16(b) correspond to upper hybrid modes, whereas Fig. 16(d) is a magnetized

plasma mode and Fig. 16(c) is an evanescent mode. The

decay of the evanescent mode with distance into the plasFIG. 15. (a) and (b) show the zeros of the potential for the two 1=2, m=0 modes in a spherical plasma with Q,^/cOp=0.5.

(a) is the upper hybrid mode and (b) is the plasma mode which is evanescent for these conditions. In (c) the variation of the mode

potential (normalized to the potential at the plasma center) is shown as a function of cylindrical radius r in the z =0 plane,

and in (d) the potential is shown as a function of z along r =0.

TN-142
540 J. J. BOLLINGER et al. 48

(a)

1 /

\

(c) (d)

ma is most easily observed in Figs. 16(e) and 16(f). Just as with the (2,0) modes, for this relatively low value of

^y/(Op the upper hybrid modes induce almost no potential variation outside the cloud, as opposed to the magnetized plasma mode or the evanescent mode.

Finally, there are two exceptional cases for which the

form of the mode potential is independent of co, (Op , and

fl^. When / = m or / = m + 1, we can substitute into Eq.

(5.5) the general form of the Legendre function P/",

(/-m)/2

7=0

2\m/2_ -.l — m—lj

(5.8)

where the /p^'s are given numbers, and use Eq. (5.6) to show that

t(m,m) = Ar'"e'^'"'^~'^'^

and

V'i(m +l,m) =Br'"ze mHm4>— {ot)

(5.9)

(5.10)

(e)

> 0.5

(f)

No

c

FIG. 16. (a) -(d) show the zeros of the potential for the four /=4,m=0 modes in a spherical plasma cloud with

ily/cOp =0. 5, in order of highest frequency (a) to lowest frequency (d). In (e) the variation of the mode potential (normalized to the potential at the plasma center) is shown as a function of cylindrical radius r in the z =0 plane, and in (f) the potential is shown as a function of z along r=0. The labels (a) -(d) corre- spond to Fig. 15(a)- 15(d).

where A and B are constants. (In fact, for all / and m, V*'" can be expressed as a finite-order multinomial in x, y, and

z.) In the cases of Eqs. (5.9) and (5.10) the form of the

mode potential is independent of co, oi„, and fl^, because

the mode satisfies 8^i^'"/3z^=0 and V^V'"~0 separately,

and so Eq. (5.3a) is satisfied for any 6] and €y Further- more, these particular modes satisfy V^^'" =0, so they cause incompressible deformations of the cloud for all cOp and fl^,; that is, they are always surface modes. For example, for the case of the (2,1) mode, t/-;^ ,,=^e''^"'"'Vz,

which is the potential inside a tilted cloud precessing at frequency co, in agreement with the analysis of Sec. III. The (1,0) and (1,1) modes are also examples of incompressible cloud deformations, which correspond to the well- known axial center-of-mass and magnetron modes.

The (2,2) mode is an incompressible distortion of the

cloud into a triaxial ellipsoid, leaving the length of the

cloud fixed at 2zo. In general, the (/,/) modes are finite length extensions of the z-independent diocotron and

upper hybrid surface modes of cylindrical non-neutral

plasmas [51]. Although the I =m and / = m + 1 surface modes ap- pear to be fundamentally different from other magnetized

plasma, upper hybrid, and evanescent modes, in fact they

display many characteristics which are similar to these

modes. For example, when co^ «£l two of the three (2,1) oscillations can be thought of as finite length versions of

magnetized plasma modes in a cylindrical column. The

modes (indicated by the upper two curves in Fig. 8) have

m =\ and a half wavelength potential variation over the

length of the column; they rotate in opposite directions

around the column. Similarly, the other (2,1) mode is a

finite length version of an upper hybrid oscillation which

also has a half wavelength variation over the column

length. Modes with larger values of / simply have more

wavelengths fitted into the column length, and so are not fundamentally different. Indeed, we will soon see that the

frequencies of these I =m and / = m + 1 modes behave in a qualitatively similar fashion as those of the other propagating and evanescent modes. When discussing the genTN-143
48 ELECTROSTATIC MODES OF ION-TRAP PLASMAS 541

eral frequency dependence of the modes, we therefore do

not need to make a distinction between these modes and

modes with other values of / and m.

Turning now to the normal mode frequencies, Ref. [29] showed that substitution of Eq. (5.5) into Eq. (5.4) leads

to two homogeneous linear equations for A and B which

have a nontrivial solution only if e^Pr+ma ' ^^ a'-

1/2M ^2

€^/€,

a^-l

1/2

Q!

(5.11)

Here P!;"=PJ"(a/{a^-€^/€^)^^^), Qr = Qr(aAa^ — 1)*^^), a=ZQ/rQ, and the primes denote diflFerentiation with respect to the entire argument.

The general behavior of the solutions to this equation was considered in Ref. [29]. Here, we discuss a

simplification of Eq. (5.11) which aids in the determination of the solutions, and we consider some examples.

Analysis of the roots of this nonlinear equation is aided

by the fact that it can be expressed as a polynomial in the

frequency co. This polynomial can be derived by substitu- tion of Eq. (5.8) into Eq. (5.11), which leads, after some

algebra, to the expression

1-x

(/ 2\m/2

-m)/2

j=0

ma^{€i + e2) + (l —m —2j)e^ — a

(a2_i)l/2 = (5.12)

r

where x =0/(0^—63/61)'^^. The factor before the sum

is nonzero and can be discarded. Furthermore, since

x~^-'=( 1—63/0^61)^, and €], €2, and €3 are rational functions of CO, the sum itself may be expressed as a polynomial in CO. For example, e^/€x = ((ip- — £l\)(ap-—ci?p)/

[(o\oi^-al], and e^ + €2^(o)^-^oiil^-(ol)/[oi{ai + a^)].

Substitution of these results into Eq. (5.12) leads, after

some further reduction, to the following polynomial

equation:

(/-m)/2

(5.13)

where

a, =pi (a) + ftJ (/--m -IjW-co])-

(a^- 1)1/2 gm^

b=aW{(o^-£ll)-{(o^-col){oi^-al) ,

+ ma^(o{co^—(0^p +6)0^)

and

The function int[(/— m)/2] denotes the largest integer

less than or equal to ( / — m ) /2.

This form of the dispersion relation (the dependence of

CO on a for a given l,m) is considerably more simple to solve numerically than Eq. (5.12), using any polynomialroot-finding algorithm. Furthermore, the equation leads

to some simple analytic results. For example, we can

count the number of normal modes by determining the

order of the polynomial. The order is 3+4int[(/ — m)/2]; however, we must be careful to ex- clude any spurious roots generated in the derivation of

Eq. (5.13) through multiplication by resonant denominators of 6], €2, or €3. When m —0 and / is odd, there is a

single spurious root at <y= — ft^ [due to the (co + il^) term

in Oj]. If m =0 and / is even there are three spurious

roots at (y2 = and co= — fl^,. Subtracting out these roots from the total, we find for m =0 there are 2/ normal

modes. However, when m =0, Eq. (5.13) is a polynomial

of order / in co^. The roots then come in / pairs at ±co, and the pairs do not really correspond to two separate

modes; indeed, Eq. (5.5) shows that the mode potential is identical for both ±co when m =0.

When m¥^0 and when / — m is even, there is a single

spurious root at co =0, while when / — m is odd, there are no spurious roots. Thus, when /— m is even there are [2(/— m) + 2] modes, and when /— m is odd there are [2(/— m)+l] modes. The roots no longer come in ±a)

pairs because, for mv^O, modes with positive and negative frequencies rotate in opposite directions around the z

axis, and these directions are not equivalent because of

the magnetic field. Some simple analytic results are also possible for the

mode frequencies when /=Anor/=m+l. In this case only the j —0 term survives from Eq. (5.13) and the

modes are determined by the equation 0^ =0. When

I =m this is a quadratic equation with roots given by

co,j = -a,/2±V^lM+coj/[l-{rQ/lad)Ql'/QJ] .

(5.14)

For n^>0 the low-frequency mode (upper sign) corre- sponds to the diocotron mode of a cylindrical non-neutral

plasma [51] and the high-frequency mode (lower sign) is a

IN- 144
542 J. J. BOLLINGER et al. 48

surface mode in the upper hybrid frequency regime.

When / =m +1, Eq. (5.13) becomes

(a)

(a,+Pi)coHo) + n^)-a,(ocol-n^l=0 , (5.15)

where a/ = l+a^(/-l) and l3, = -[a/{a^

-l)'^^]e/"'7e/~'- There are three solutions for the

mode frequency which, when / = 2, are the same as the

solutions of Eqs. (3.22)-(3.24) for the (2,1) mode. When

/ = 1, one of the roots is spurious and the other two ap- proach the single-particle axial bounce frequency

(o^=(Op/{ H-/3, )'''^ in agreement with Eq. (2.6); this is the

axial center-of-mass mode.

Results for the frequencies when I =m +2,

m +3, ... require the solution of even higher-order po- lynomial equations. We have found the solutions numerically, and some results are displayed in Figs. 17(a)- 17(c) as functions of the rotation frequency co^ . It is important

to remember here that both H^ and Op are functions of

the CO/, in fact, in the rotating frame the plasma becomes

unmagnetized at the Brillouin limit 11^=0.

Examination of the mode frequencies at the Brillouin

limit (equivalent to the case of ions in a Paul trap) shows

that for given / and m there are two modes which become

unmagnetized surface plasma oscillations. These surface

mode frequencies are described by the particularly simple

limiting form of Eq. (5.11), ei =PrQr' /Pr'QP [here the

argument of the P,'"'s simplifies to a/{a}—W^\ Thus,

when n^=0 there are a pair of surface modes with fre- quencies of opposite sign. For n^=?^0 but |ft^| <(i)p and

/ ^ 00 , these modes are evanescent, approachmg the

magnetized surface plasma frequencies +fl„/A/2. This

result is independent of the shape of the cloud. However,

for \£i^>(i)p the behavior of these two modes depends on

m. When m =0 the pair of modes remains in the magnetized plasma regime (oKojp, as shown in Figs. 17(b)

and 17(c). When m^O, however, only one mode remains

in the magnetized plasma regime, while the other enters the upper hybrid range, as shown in Fig. 17(a). The rest of the modes always remain in either the magnetized plasma or upper hybrid regime. Their numbers

are as follows; when m 7^0 there are / — m magnetized

plasma modes and 2int[(/ — m)/2] upper hybrid modes;

when m =0 there are int[(/ — 1 )/2] pairs of magnetized

plasma modes, and int[//2] pairs of upper hybrid modes.

[These numbers can be determined by analysis of the

solutions of Eq. (5.13) in the hmits Cl^^- oo and ft^^-0.]

Finally, we compare the frequencies of (2,0) and (4,0) modes, which, like the (2,0) modes, can be excited using

the technique described in Sec. IV. The frequency of the evanescent (4,0) mode, shown in Fig. 17(b) for the experimental value co^/Cl =0.l5l, is quite different from the evanescent (2,0) frequency, which was measured experimentally. Evanescent branches with higher / are even

further^ removed from the / =2 root, approaching

fty /V2. This provides further evidence that the measured mode shown in Fig. 12 is in fact a (2,0) mode exci- tation and not a higher-order oscillation. However, it should be possible to excite other m =0 even / modes us- ing the same experimental technique. For larger values

of w, /ft, however, the evanescent (2,0) mode becomes

a

3

a

3

a

1 V

'y/-

0.8 ^»-^ ^^^..xjy °

'

•

'

.V

0.6 •y ^•

/

•;°° ^

8»'

S° ^ 0.4

/oooooo-°::-*

••.. 1 ftOOOt 0.2 7 ...•• 0"°°°° 8* "::::::!>..:^iSS^«i?ssssssss-»'' 0.2 0.4 0.6 0.8 1 CO /Q.

(b) 1

0.6

0.4 I- ;

0.2

' _oOOOO°°°°ooooooooooooooooooooooooooooooo \ .o

(c) 1

0.4 0.6 (o/Q

r FIG. 17. Mode frequencies co in the rotating frame vs rota- tion frequency oi^ for different values of (l,m) and co^ (From

Sec. VB, co^^'° = co — m(Or-) For aKO the diagrams should be

reflected through the point («;.,w) = (ft/2,0). Also shown for comparison are the upper hybrid frequency Hu^cOr) (solid curve), the vortex frequency £iJ,(o,) (dotted curve), the plasma

frequency copico,) (dot-dashed curvj), and in (c) the surface upper hybrid frequency D,^{o}r)/^2 (dashed curve). All fre- quencies are expressed in units of the cyclotron frequency CI. (a) 6;,/a=0.151; (/,m) = (2,l), filled circles; (/,m) = (5,l), open

circles. The (2,1) modes for this condition are also shown in Fig. 8 in the laboratory frame, (b) w,/n = 0.151; (/,m) = (2,0),

filled circles; (/,/n) = (4,0), open circles. The (2,0) modes for

this condition are also shown in Fig. 6. They are the same in the laboratory or rotating frame, (c) 0)^/11=0.5; (/,m) = (2,0),

filled circles; (/,m) = (4,0), open circles. TN-145
48 ELECTROSTATIC MODES OF ION-TRAP PLASMAS 543

nearly degenerate with the evanescent (4,0) mode [see Fig. 17(c)], making it more difficult to distinguish be- tween modes with different values of / merely from fre- quency measurements.

B. Resonances between modes and static field errors

Now that the linear modes have been enumerated, we

turn to the problem of modes driven by an external

source. In particular, as we saw in Sec. IV, a static field error can drive to large amplitude a (2,1) mode, when the

mode satisfies the linear resonance condition that it has

zero frequency in the laboratory frame. In this section we consider the resonance between a static error and

modes with general ( /, m ). In the rotating frame, modes have a variation proportional to exp{im<f> — io)t). The azimuthal angle (f> as viewed in the rotating frame is related to the angle (^^ as viewed in the laboratory frame through the GaHlean

transformation (f>=(f>i+ci)^t. Therefore, the condition

that a mode has zero frequency in the laboratory frame is

(o—meOr (5.16)

Solutions of this equation can be found graphically in Fig. 17 by finding the intersections of the line given by

Eq. (5.16) and the curves of mode frequency versus rota- tion frequency. In general, for a given / and m there are

/ — m intersections for m^O and no intersections when

m —0. [Equation (5.16) implicitly assumes m>0 and

positively charged, trapped particles as discussed earher. With the sign convention used here where co^ is always

positive, the condition for negatively charged particles is co= —mo)^.]

For a given co^/D, the solutions of Eq. (5.16) provide

specific values of co^/fl at which resonances can occur between a static error and a mode with given (l,m).

These values trace out curves as co^/fl is varied; the curves are plotted for m = 1 and m =2 in Fig. 18. Since

modes with / — m odd are odd in z, these modes will be

excited (in linear theory) only by errors which are also odd in z, such as the error induced by a tilt of the elec- trodes with respect to the magnetic field. Similarly, er- rors which are even in z excite modes with I —m even. Furthermore, only modes with the same value of m as the

perturbation can be excited (in Hnear theory). This implies that the tilt field error, which (for small tilt angles) is

(a) 1

0.8

0.6

0.4

0.2

"^

1 '""^^^7^-^ 4,1 --6,1

\.

y/>'\ -

- ^-'^^^^y-

'^S^^

(b)

0.25 ._ 0.5 0.75

a

3"

1 "^

^S. "»*

I I --3,1 —5,1

0.8 ^^^

s^U

0.6

- \

0.4

-

0.2 /^'v^^

I

V

1 1 0.25 0.5 (o/Q

0.75

(c) 0.5 (d) 0.5

0.25 ,^ 0.5

CO /Q

0.75

a

3'

0.25 0.5 o)/Q

0.75

FIG. 18. Values of a,/^ and a^ /U for which different m = 1 and m =2 modes become zero frequency in the laboratory frame. The limiting points U and P, which fall on the confinement boundary, are discussed in the text. The confinement boundary, defined by (i),=a)„ and W;. = fi— <u„, where a)„ is given by Eq. (2.10), is also drawn. The m = 1 modes are shown in (a) and (b), and the m =2

modes are shown in (c) and (d). Modes odd in z are shown in (a) and (c) and modes even in z are shown in (b) and (d). The curve for the zero-frequency (2,1) mode in (a) is the curve in Fig. 10. IN- 146
544 J. J. BOLLINGER et al. 48

an A72 = 1 perturbation that is odd in z, can linearly resonate only with the (2,1), (4,1), (6,1), .. . modes.

The (2,1) mode is most strongly excited since it has the

slowest spatial variation and hence has the largest "over- lap" with the slowly varying tilt error. Furthermore, the

(4,1) and (6,1) resonance curves have quite diflFerent forms

from the (2,1) curve, which indicates that the excitation

observed in the experiments is in fact a (2,1) mode, not a higher-order excitation. However, the other resonances may also be measurable.

Turning now to the hmiting behavior of the resonance

curves, we observe that in the rotating-frame mode fre- quencies are always less than the upper hybrid frequency

ft„ =(6;^+ft^)^^^. Then Eq. (5.16) impUes that resonant

rotation frequencies must satisfy the relation

m(o,<V0)1 + 0.1 = V2a),{a-(D,) + m-lco,

which can be solved to give

<u,/n<(Vm2-l-l)/(m2-2)=l, 0.366,0.261,. . . .

This implies that as m increases the resonances become

packed into a smaller and smaller region of the experimental operating regime. Since most experiments

operate in the regime 6j^/ft«l, we may have cause to worry that these many resonances could have a deleteri- ous effect on the plasma confinement. However, the

mode resonances with smaller values of / and m are most

dangerous since they are more easily excited by perturbations with slow spatial variation, and larger / and m

modes may be heavily damped due to viscous eflFects or

coUisionless Landau damping [52]. Other limiting values of the resonance curves can also be found. For example, consider the limit in which the

plasma becomes extremely oblate, that is, a thin disk or pancake. This occurs along the edge of the confinement

region defined by the equations oir = co^ and co^ = Cl — (o^,

where w^ is given by Eq. (2.10). In this limit, the mode

frequencies approach either ft^ or (Op. In the former case resonances occur when fl^=ma)^, or a),/n=l/(m+2) = l,l,l,... .

This limit is labeled by the point P in Fig. 18. In the

latter case resonances occur when cOp=ma)^, which can

be solved for co^ to yield

6),/ft = 2/(m2 + 2) = |,l,^,... .

This limit is labeled by the point 17. Resonance curves which connect to the point P are due to magnetized plasma modes, whereas curves connecting to point U are due

to resonances with upper hybrid modes.

Finally, we briefly discuss how the strength of these resonances should behave as the size or density of the

plasma increases. Since the plasma is generally small

compared to the distance to the electrodes, external field errors have slow spatial variation compared to the plasma size, and hence only the lowest-order / and m reso- nances should be observed. However, as the plasma in- creases in size higher-order modes have a larger "overlap"' with the field error. Furthermore, as either plasma

size or density is increased more electrostatic energy is h- berated by radial expansion of the plasma due to the

torque of the error acting on the cloud. Thus, we expect

heating due to resonances to increase as more ions are trapped, and more resonances with higher values of / and

m should be observable. This qualitative picture appears

to agree with observations of higher density ion clouds. However, a quantitative analysis of this important question involves effects which are beyond the scope of this work, such as the effect of viscosity. Landau damping,

laser light pressure, and nonlinearities on the saturation

level of the mode amplitude. These issues need to be considered in a future paper.

ACKNOWLEDGMENTS

We thank John Schiffer for stimulating discussions on

this subject and Steve Jefferts and Steve Barlow for their comments and careful reading of the manuscript. We

gratefully acknowledge support by the U.S. Office of

Naval Research Contract No. N00014-92-F-0038.

D.H.E.D. acknowledges the support of the U.S. Office of

Naval Research Contract No. N00014-89-J-1714 as well

as National Science Foundation Grants No. PHY87-

06358 and No. PHY9 1-20240.

'Current address: Department of Physics, University of

Texas, Austin, TX 78712.

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TN-148
PHYSICAL REVIEW A VOLUME 49, NUMBER 5 MAY 1994

Electrostatic modes as a diagnostic in Penning-trap experiments

Carl S. Weimer, J. J. Bollinger, F. L. Moore,* and D. J. Wineland

Time and Frequency Division, National Institute ofStandards & Technology, 325 Broadway, Boulder, Colorado 80303

(Received 4 October 1993)

A subset of the electrostatic modes of a cold cloud of electrons, a non-neutral electron plasma, trapped

in a Penning trap has been observed and identified using a recent theoretical model. The detection of

these modes is accomplished using electronic techniques which could apply to any ion species. The

modes are observed in the low-density, low-rotation limit of the cloud where the cloud approaches a two-dimensional charged disk. We observe both axially symmetric and asymmetric drumhead modes.

The shape, rotation frequency, and density of the cloud are found in a real-time nondestructive manner

by measuring the frequency of these modes. In addition, it is found that radio-frequency sideband cool- ing compresses the cloud, increasing its density. The ability to measure and control the density of a trapped ion cloud might be useful for experiments on low-temperature ion -neutral-atom collisions, recombination rates, and studies of the confinement properties of non-neutral plasmas.

PACS number(s): 32.80.Pj, 35.80. + s, 52.25.Wz, 52.35.Fp

Penning ion traps have become an important tool in studies involving low-energy charged particles [1]. Recently, a theory of the electrostatic fluid modes of a plasma confined in a Penning trap (and Paul rf trap) was

developed [2-4]. A number of these modes were observed experimentally in a plasma of Be"*" ions using opti- cal fluorescence techniques [3,4]. By measuring the

eigenfrequencies of two or more of these modes, plasma

characteristics such as density, rotation frequency, and

shape can be found nondestructively. Several types of ex- periment would benefit from this information. For example [1,4], experiments on recombination rates, ion -neutral-atom and ion-ion collisions, and strongly

coupled systems depend on the ion sample density, size, and shape. These modes can also be used to study the

confinement properties of the trap. However, a general

method for detecting these modes is desired. Optical

detection of the modes, as demonstrated in Ref. [3], is practical only for a small number of ion species. The

method of characterizing the plasma by ejecting the plasma from the trap onto a set of charge collectors [5] has

the disadvantage (for some experiments) of being a destructive measurement.

The object of the work reported here was to investigate

detection techniques which would allow these mode fre- quencies to be measured nondestructively for any type of

'Present address: Dept. of Physics, University of Texas, Austin, TX 78712.

ion species. To do this, we used electronic methods for detecting the modes and electron plasmas to test the

methods. Related observations with trapped ions have

been made by Barlow, JeflFries, and Dunn [6], and with

electrons and positrons by Tinkle, Greaves, and Surko

[7]. In the low-density, low-rotation limit of the electron plasmas we were able to detect a number of modes. By

comparing the frequency dependence of the detected

modes with the theoretical model by Dubin [2,4], an

identification of the modes was made. This allowed us to calculate the density and aspect ratio of the electron plas- mas and study their evolution. By combining this infor- mation with a measurement of the number of electrons,

the plasma size could then be determined.

A second goal of this work was to find a way to manipulate and increase the plasma density. For trapped atomic ions, lasers can be used to apply a torque to the plasma, compressing it and increasing its density [3,4,8]. In

addition, lasers can be used to cool the atomic ion plasma, reducing its temperature to below 10 mK [8]. The

possibility of compression of a non-neutral plasma by wave excitation has also been discussed [9]. In the work

reported here, we found that the parametric coupHng

technique called magnetron sideband cooling [10,11] in- creased the electron plamsa density. This technique uses only radio-frequency fields, is independent of the internal structure of the ion, and therefore can be used for any

type of ion. The ability to increase a trapped plasma's

density and decrease its temperature is important for a number of experiments including proposals to produce

antihydrogen [12]. A prominent candidate for antihydrogen formation is a three-body collision (e'^,e'^,p) whose

cross section scales as nlT~'^^^, where «o is the density

and Tihe temperature [12,13].

1050-2947/94/49(5)/3842(12)/$06.00 49 3842 ©1994 The American Physical Society

IN- 149
49 ELECTROSTATIC MODES AS A DIAGNOSTIC IN PENNING- 3843

I. THEORY

A Penning trap [1,5,10,11] confines charged particles

by combining static magnetic and electric fields. Referring to Fig. 1, a static voltage Vq applied between the

endcaps and ring forms a potential (in spherical coordinates).

<P(r,9,(l>)=Vo 2 C2„(r/rfrP2„(cos6») , (1) n=0

where Pjn ^rc the Legendre polynomials and C2„ are constants. Here d =r, /a/2 is a characteristic trap size, with

r, the ring's internal radius (for our trap, rf =3.54 mm).

The electrode shapes are designed so that the dominant

term in the electrostatic expansion is that labeled by C2. This results in a potential VQC2[z^-ix^+y^)/2]/2d^

which confines the electrons axially in a harmonic well. In this case, the axial center-of-mass (cm.) motion of the

electrons is the same as that of a single electron and has a frequency (o^ — ilcjqVQ/md^)^^^ where q/m is the

charge-to-mass ratio. A homogeneous axial magnetic

field Bq is superimposed on this electric potential ensuring radial confinement. In the radial plane the center-of- mass motion is composed of two superimposed circular

motions: the EXB (or magnetron) motion at frequency

0)^, and the modified cyclotron motion at frequency

a)[=a)c—o)^, where co^=qBQ/m (SI units). In general.

rco: -o)„co'^, and for our experiments, (o„ KKco^ «(o[.

Cryogenic cooling used in this work was capable of re- ducing the electron temperature to close to 4 K. For the

magnetic fields used here, and with 4 K temperatures, the

cyclotron radius of an electron was small compared with

the distance of closest approach between two electrons. Under this condition the plasma is strongly magnetized

and the coupling between motion parallel and perpendicular to the magnetic field is weak [14,15]. This may have

resulted in two different temperatures for motion parallel

and perpendicular to the magnetic field. Except for a

possible difference in these temperatures, we assume the

electrons had sufficient time to evolve (due to electron/electron collisions) to a state of thermal equilibrium. In an experiment [16] with electrons with eV energies, global equilibration times were measured to be

about 1 s at 5o=0.01 T. This time should increase with

larger magnetic fields. It may scale as 5 g or Bq depending on the conditions of the plasma [16-18]. For our ex- periment the non-center-of-mass modes were not observed until times greater than 100 s. Over long times, trapped plasmas have been observed

to undergo a radial expansion resulting in a decrease in their aspect ratio and density with time and eventual gra- dual particle loss from the trap as the particles strike the

ring electrode. In the absence of external torques, the total canonical angular momentum of the plasma is conserved [19], which places a limit on the radial expansion

and the number of electrons that can be lost from the

trap. Therefore, some external torque is responsible for

this radial transport of particles and energy [20]. For ex- periments done in high vacuum apparatus (such as used

here), the most likely candidates for this external torque

are azimuthally asymmetric components in the trapping

electric or magnetic fields [14]. Resonant enhancement

of this transport rate can occur if the static field asymmetries couple to collective modes [3]. In thermal equilibrium, a cold electron (or ion) plasma

with dimensions much less than those of the Penningtrap electrodes can be modeled as a uniform density

spheroid which rigidly rotates about z at the rotation fre- quency co^, with (o^ <(Or<o)'c [3,4,21-23]. The density is nQ = 2EQmcOr(co^—cOr)/q^ in the plasma interior and falls off at the plasma boundary over a distance approximately

equal to the Debye length X^ = (zQkT /n^q^)^^^, where k

is Boltzmann's constant, T the temperature, and Zq the

permittivity of free space. For a comparison of experiment with theory, we will assume that the plasma is sufficiently cold that the Debye length is much smaller

than the cloud dimensions. The plasma frequency (o., defined by (oj=q^nQ/mEQ, is related to the rotation fre- quency by C0p—2o}^(o)^—a)^). The maximum density occurs for (0^=0)^/2, which is called the Brillouin limit. The aspect ratio of the plasma, defined by a=Zo/ro

with 2zq the cloud length and 2ro its diameter, is a function of the rotation frequency and therefore the density

[23]. As (0^-^10^/2 (Brillouin limit) the aspect ratio in- creases to its maximum value. If, in addition, (o^ «cOc'

the cloud is a long cigar shape (prolate spheroid) along z. For either co^^>-co^ or co^—>-co'^, a decreases and the cloud

becomes a flat pancake shape (oblate spheroid). An exact expression relating a and co^ is given in Refs. [4] and [23]. For a « 1, this expression can be expanded to first order

in a with the result (Op^co^il +ira/4).

We have neglected the effects of correlations in the

plasma due to the electron/electron Coulomb interaction

[24]. A measure of these correlations is the coupling constant r = q^/{4TreQa^kT), where a^ is the Wigner-Seitz ra- dius given by 47ra/no/3= 1. For F « 1, the plasma acts as a weakly interacting gas, so correlations can be ignored. For r> 1, the plasma correlation increases and

the plasma becomes liquidlike. As F further increases,

the plasma crystalUzes into a rotating lattice. In a Penning trap, the onset of crystallization has been directly

observed using laser-cooled Be^ ions [25]. A similar

crystallization has also been reported in other systems,

for example, in Paul traps [26], electrons on Hquid helium

[27], and, in the high-density limit, electrons in GaAs

junctions [28]. For the work here, r^O.2 was measured

and therefore correlations can be ignored. In general, even in the presence of spatial correlations, if the

electron/electron spacing is short compared to the wavelength of the modes, the plasma can be treated as a

constant-density plasma.

The theory of the electrostatic fluid modes of a cold,

constant-density, spheroidal plasma confined by a harmonic well was solved analytically for some simple cases

in Refs. [3] and [4], and was solved for the general case by Dubin [2,4]. The theory, appHed at the Brillouin limit, also applies to ion plasmas in the rf or Paul trap [4]. Dubin assumes that the plasma is in thermal equilibrium

and its temperature sufficiently low that pressure effects can be ignored. In addition he assumes that a^ and X^

TN-150
3844 WEIMER, BOLLINGER, MOORE, AND WINELAND 49

are much less than the plasma dimensions and the wavelength of the modes. By combining the continuity,

momentum, and Poisson's equations, the differential equation for the electric potential was solved using scaled

spheroidal coordinates. The solutions are a set of mode

potentials expressed as products of associated Legendre

functions P^ and Ql" and a factor e'('"^-"", where (f) is the azimuthal angle about the trap axis and a> is the mode

frequency. Here / and m are integers which label the

modes, with l>\m\. Without loss of generality, m >0

may be assumed and negative frequencies allowed.

As examples of these modes [2-4] the (/,m) = (l,0)

mode is just the axial center-of-mass mode. There are two

(1,1) modes which are just the magnetron and cyclotron

center-of-mass modes, so all the 1 = 1 modes are center- of-mass modes. The / = 2 modes correspond to quadrupole deformations of the plasma shape. For example,

there are two (2,0) modes. Here the plasma shape

remains spheroidal, but the aspect ratio of the spheroid

oscillates in time. The low-frequency (2,0) mode is a quadrupole oscillation where the axial and radial extents of the plasma oscillate out of phase with each other. The

high-frequency (2,0) mode is similar, except that the axial and radial extents oscillate in phase as if the plasma were

breathing. The three (2,1) modes are azimuthally asymmetric modes. For these modes, for small amplitude, the

z axis of the plasma tips and precesses relative to the

magnetic axis. In general, since the mode potential is proportional to e'""^, modes with m =0 are azimuthally

symmetric while those with m > are azimuthally asymmetric. The (1,0) and (1,1) modes (center-of-mass modes)

are commonly observed both with electronic and optical

detection. The (2,0) and (2,1) modes have been observed

using optical detection, and excitation of a (2,1) mode by

a static field asymmetry was observed to heat the cloud

[3].A polynomial equation for the mode eigenfrequencies

in a frame rotating at frequency a>^ is given in Eq. (5.13) of Ref. [4]. The mode frequencies in the laboratory frame

are related to those in the rotating frame by

ft)/^ ( lab ) = £l)/^( rotating ) — m£t)^, where the term mco^ is due to the Doppler shift. For co^ «(o^ «co^, the modes

fall into three frequency ranges: high-frequency modes

with co~o}[, intermediate-frequency modes with o)~co^

(or cOp), and low-frequency modes with co^co^ (or oi^). Because of our detection technique (described below), the

modes which we detected were the intermediatefrequency modes. For co^, to^ KKco^ (the case here), they

are magnetized plasma modes [4]. Here we summarize

the w =0 and m = 1 intermediate mode frequencies with

/ — w < 7 in the a « 1 (low-density) limit. This was obtained by solving Eq. (5.13) of Ref. [4] for each value of / and m, to first order in a in the Hmit co^cOp «co^ [29]. In the laboratory frame

K[^~^^«]

\co

k[l-M7ra]

^^s.ol =

l^p

'

<y7,oi =

\co„ (2)

'^" [0)p±C0, ,

CO,6,ll \(0p±(0, ,

\^l,l\-(^p^(^r .

I'^z^co.-^^Traco,

(Og

\C0 ±0), ,

3,0l

\Ct)n

and in addition, for completeness,

and to first order in a

i^p^^zi^ + j-n-a] ,

co, = {(ol/2o},)[l + \7ra] .

The braces and ± signs indicate when there are more

than one intermediate mode frequencies to first order in a. Notice that for all values of / and m (except for

I —m = 1 ) there are mode frequencies which decrease

with decreasing a. In general, these frequencies occur for more than one mode; they are degenerate to first order in

a. However, when / — m is odd, there are also mode fre- quencies that increase with decreasing a. Each of these frequencies occurs for a single, nondegenerate mode. The

center-of-mass modes (1,0) and (1,1) are independent of a

and this the density of the cloud. However, (1) anharmonic terms in the trapping potential [n > 1 in Eq. (1)], and (2) image charge shifts, which are neglected in these

expressions, cause the cm. frequencies to depend slightly on their amplitudes of oscillation as well as the spatial

distribution of the plasma in the trap. As a result of

these effects, as the plasma expands due to radial trans- port the cm. mode frequencies will shift slightly.

By knowing co^ and identifying and measuring the fre- quency of any non-center-of-mass mode (or alternatively

by measuring the frequencies of any two non-center-ofTN-15I
49 ELECTROSTATIC MODES AS A DIAGNOSTIC IN PENNING- . 3845

mass modes), the plasma's aspect ratio and therefore its rotation frequency and density can be calculated. We

note that if modes that are azimuthally asymmetric

(m >0) can be resonantly excited, then it might be possi- ble to change the rotation frequency and increase or de- crease the density. This is because the m > modes carry angular momentum. Therefore, by driving them it might

be possible to apply to the plasma a torque which in turn could affect the plasma density.

The linewidths of the modes will be set by some combination of damping and dephasing processes. Mode

damping could occur through resonant or nonresonant

coupling to resistive elements making up the trap and

support structure. Careful trap design can decrease (or

increase) this loss mechanism. Couphng between electrostatic modes could also cause energy transfer out of any

one mode, damping its amplitude. Landau damping of

the modes has also been reported [7]. Dephasing of the

modes, leading to hne broadening, could occur because of

small anharmonic terms in the trapping or mode potentials. In addition, electron/electron collisions could

cause this dephasing.

II. EXPERIMENTAL APPARATUS

The trap used in this study was previously used to in- vestigate the nonlinear excitation of a single trapped elec- tron and the center-of-mass modes of electron clouds

[30]. The trap electrodes (Fig. 1) were machined from

oxygen-free copper and had an inner ring radius of 0.50

cm, resulting in d =0.35 cm. When a voltage Vq was applied between the endcaps and the ring, the resulting

electric potential near the trap's center closely approximated a harmonic potential along the z direction (with

^2 =0.487) [31]. The cylindrical symmetry of the elec- trodes minimized azimuthally asymmetric electric perturbations. The leading order perturbation to the potential was the term labeled by c^ in Eq. (1). However, by applying a voltage Vg to the additional guard electrodes [32] and observing the axial cm. mode hne shape, this perturbation could be reduced to |c4l<5X10~^. For a precision hyperbolic trap without guard electrodes, typically

|c4l>10~^ [11]. The ring electrode was split into three

sectors (Fig. 1). Typical operating parameters for this trap were Fo = - 10.45 V, ^q =0.1 10 T, co,/2Tr=6l.54

MHz, o)„ /2ir=615.0 kHz, and co'^ /27r=3.079 GHz.

The magnetic field Bq was produced with a multifilament Nb-Ti superconducting magnet. The field homogeneity near the trap center was specified as A5 /Bq < 2 X 10~^ for a 1-cm-diam volume. The magnetic field's long term drift was measured to be within

lX10"5/h-7X10"^/h for 5o=0.1 T. The trap was mounted at the center of the magnet in a sealed copper vacuum container. The magnet and vacuum container

shared a common liquid-helium bath. The large cryo- sorption rate of the liquid-helium cooled walls maintained the vacuum inside the trap. The background pres- sure in a similar system was measured to be less than

7X10"'^ Pa (5X10~'^ Torr) [33]. An additional set of

Helmholtz coils whose axis was perpendicular to Bq and

whose center was common with the trap center was mounted external to the magnet dewar. These coils were

used to minimize the angle Qq between the trap magnetic

and electric axes when Bq^OAT.

Electrons were loaded into the trap by using a field- emission point (FEP) ahgned along the f axis and mounted in one endcap. By applying a negative potential of 600

V to the FEP an electron beam which passed into the

trap and back out through matching holes in the endcaps was formed. Inside the trap the electron beam ionized

background gas, which in turn supphed the low-energy

secondary electrons which were trapped. By varying the

beam current, the loading rate was varied from 0.01 to 100s"'.

The electronic system (Fig. 2) used to detect the signal

induced by the electron's motion was similar to that developed at the University of Washington [10,11,34,35].

At the heart of the system is a helical resonator tuned cir- cuit (TC) with frequency cojq. The resonator is attached

to one endcap and its frequency set such that cojq^co^.

The current induced in this endcap by the electrons' axial motion produces a voltage drop across the tuned circuit which was then amplified with a dual gate GaAs FET

preamplifier. Both the tuned circuit and preamplifier were immersed in the liquid helium. The real part of the

tuned circuit impedance sets the damping rate and

FIG. 1. A schematic diagram of our Penning trap. The electrodes have been separated

along the axis to show details. The inner ring

diameter is 1 cm.

Endcap Endcap

IN- 152
3846 WEIMER, BOLLINGER, MOORE, AND WINELAND 49

linewidth of the axial cm. motion [10,11,36]. The axial cm. Unewidth Aw^ for our experiment was A&j^ =N X 19 s"' for (02=a)jQ, where N was the number of electrons. Therefore, an estimate for the number of electrons

trapped was made by measuring Aco^. The thermal amplitude of the axial cm. motion (and the cyclotron cm.

motion via synchrotron radiation) had a characteristic

temperature set by the liquid-helium bath, T =4 K.

Different detection methods were used in this study.

The following is a brief description of each.

1. Noise power. In this detection method the noise in- duced in the endcap by the electrons' thermal motion was

detected using a spectrum analyzer which acted as a square-law power detector [34,36]. When this spectrum

analyzer was used to detect noise near the axial cm. fre- quency, a noise decrease (or "dip") was observed for co^^cojQ as the electrons short circuited the tuned

circuit's noise voltage [36]. A noise peak occurred for values of co^ outside of the tuned circuit resonance [36]. The advantage of this detection technique was that it was

passive, requiring no oscillating drive voltages to be applied to the trap.

A related detection method is sometimes called the

"bolometric" method [6,34,36,37]. In this method, the

spectrum analyzer's local oscillatory frequency is fixed so

that only noise power around co^ is observed. A parametric drive at 2co^ can be apphed to heat the cloud increasing coUisions and coupUng between the modes. An oscil- lating drive voltage at frequency coj^co^ can simultaneously be appHed to any of the trap electrodes. If the elec- trons absorb energy at coj, and if some of this energy is

Mixer

FIG. 2. A schematic diagram of the electronic detection sys- tem used in detecting the electrostatic modes of the electron plasma. The primary detection method was to look at the noise induced in one endcap using the spectrum analyzer, with the

voltages at oj and oj^od set to zero amplitude. In addition, the response of the plasma to the coherent drives coj and co^od was

also detected and recorded by the X- Y recorder. A dynamic sig- nal analyzer which recorded the fast Fourier transform (FFT) of

the response was used as a diagnostic tool. V^„ is the ser- vomechanism correction voltage.

coupled through some mechanism to the axial cm.

motion at (o^, then an increase in the noise is observed.

2. Coherent. In this method, typically, the electrostatic well is modulated at a frequency (Of^^/2ir=l MHz

producing frequency modulation sidebands on o)^. A

second drive of frequency coj is then applied to an endcap

with a)J^(02+o)JJ^^. As co^j is swept through resonance

with the sideband (6>^=<y2+a)^od), the electrons' axial cm. is excited, and this excitation is turn induces a signal

in the other endcap near co^. This signal is then amplified

and phase-sensitively (linearly) detected with a doubleheterodyne receiver [10,11,35]. The modulation allows

the drive to be applied at a frequency different than (Oiq,

thus preventing saturation of the preampHfier. This

method is very sensitive to small coherent changes in the

axial cm. ampHtude.

3. Axial frequency shift. Because of imperfections in the static trap fields (c2„^0 for « > 1, and Bq not homogeneous) and because of relativistic effects, the axial cm.

motion is sHghtly anharmonic and weakly coupled to the

other degrees of freedom [10,30,38]. As an example, the

axial cm. frequency co^ becomes dependent on the ampHtude of oscillation of the cm. magnetron motion. If the

magnetron motion is excited, the axial frequency shifts because of the residual c^ term in the trap potential. For

a change of r^ =0—*r^o i" the magnetron amplitude, the

axial frequency shifts by do)2/co2^3c^r^Q/(2c2d^). To

observe the axial frequency shift, an electronic ser- vomechanism is used [10,38]. An axial drive is used to weakly excite the axial cm. motion. The response to this drive is then detected and the output of the coherent

phase detector is integrated and added (fedback) to Vq

with the correct sign. This effectively locks the axial fre- quency, holding it fixed relative to the (synthesized) drive

and modulation frequencies. Any effiect which would

otherwise change the axial cm. frequency is observable

by monitoring the correction signal of the servo. An additional experimental technique which was inves- tigated was a parametric mode-coupling technique called magnetron sideband cooling [10,11,39], which is analogous to laser cooling [40]. It was first reported in experiments involving single electrons at the University of

Washington [10]. The technique is used to reduce the

metastable magnetron cm. ampUtude by parametrically

coupUng this motion to the axial cm. motion which is damped by its coupHng to the tuned circuit (a)2=6)Tc)-

This parametric coupling is accomplished by applying a

spatially inhomogeneous electric potential of the form

<f>(x,y,z)=VpXZ cos(co2+o)m)t. The magnetron amplitude can be damped to a theoretical Umit given by

r^=2[(co„/(o,){r^)]^^^, where (r^)^^^ is the thermalized axial amplitude. Similarly, the magnetron motion

can be damped through coupling to the cyclotron

motion. Theoretical models of the cooling have not in- cluded plasma effects [11,39,41], although plasma models

for parametric couphngs between plasma modes might

apply [42]. Evidence that sideband cooling affects the ra- dial transport rate of a plasma has been found previously,

in that, with the cooling on, cloud lifetimes are extremely

long [33]. However, the effect of sideband cooling on the

IN- 153
49 ELECTROSTATIC MODES AS A DIAGNOSTIC IN PENNING- 3847

aspect ratio and density of a plasma has not been studied

before. For our trap the sideband coohng drive was ap- phed to the ring sector A. The necessary axial dependence to the drive arose from the different impedances at- tached to the endcaps.

III. EXPERIMENTAL RESULTS

A. Observation of the modes from the noise power

Detection of the noise power of the induced currents in one endcap (method 1 above) gave the most definitive results, so we focused our attention there. After a cloud of

electrons was loaded and the field emitter (and all drives)

turned off, the axial cm. motion with frequency o)^ =6^1 q could be seen in the noise spectrum (see inset in Fig. 3). Initially, the frequency of the cm. motion decreased

slightly with time. At a later point in time, noise peaks

began to appear in the noise spectrum (Fig. 3). The fre- quencies of these noise peaks increased in time, with some crossing the axial cm. frequency and others ap- proaching it asymptotically (Fig. 4). If q)^=cojq, so the

axial cm. signal was a noise dip, then as the peaks ap- proached (o^ they too would change to noise dips, then

back to peaks after the crossing. After the last crossing,

the frequency of the axial cm. mode would begin to in- crease slightly. Eventually, one by one, the amplitudes of

the peaks decreased until they were no long observable,

with the axial cm. mode typically the last to be seen. Surprisingly, it was found that the larger the value of IC4I the smaller the absolute frequency shift of the axial center of mass. The number of noise peaks which were observed de10

i-D

O

Q.

0)n

B -

W1.0 t = 150 s B„ = 0.11 T N = 43 000 electrons

t = 1720 s

Tuned Circuit

Frequency Marker

60. B 61.0 61.2 61.4 61.6 61.8 62.0 62.2 62.4 Frequency (MHz)

FIG. 3. The noise spectrum induced in one endcap as seen by the spectrum analyzer (resolution bandwidth is 10 kHz). The

broad peak centered at 61.6 MHz is the axial tuned circuit reso- nance excited by thermal (^4 K) noise. The inset shows the

noise spectrum of a newly loaded cloud (r = 150 s after loading)

in which only the axial center-of-mass mode and tuned circuit resonance are seen, and with co^ slightly detuned from cojc- In the main figure the signals from six electrostatic modes includ- ing the axial center of mass mode can be seen at a later time

(f = 1720 s after loading). Here Bq =0. 1 1 T.

62.4

62.0

61.6 o

c

<0 61.2

CTV

U- 60.8

60.4

-

. ...rrr V (2.1) ,.•••••'

,•••**

...... (4.1) •;;.... (6.1)

(1.0) :-•• '.'.:•.::'.• (8.1) (1.0)

(3.0)

1

(5.0)

1

(7,0)

1 .1. 1 1 500 1000 1500 2000 2500 3000 3500 Time (seconds)

FIG. 4. A plot of the frequencies of the modes vs time after the cloud was loaded. The modes are labeled by their (/,/n) values. These data were taken using a spectrum analyzer and

were from the same data set as in Fig. 3. For these data the sideband cooling was off and the guard electrode voltage de- tuned so that |c4l=;4X10~\ The measurement uncertainty in the frequency is less than the symbol size used in the plot and is the same for all points.

pended most strongly on the number of electrons in a

cloud. For clouds of 7000 electrons only the cm. signal

and one other peak were observed. For clouds of 70 000

electrons as many as ten peaks besides the axial cm. peak were observed. The time at which the first additional

noise peak appeared was found to be strongly dependent on the magnetic-field strength Bq. For Bq =0.11, 0.25,

and 1.4 T (and for all A?^ between 7000 and 70000) the first peaks (other than the co^ q = co2 mode, which was seen immediately) appeared at the respective average times 520,

2000, and 15 800 s after loading. The modes were observed as the clouds' aspect ratio decreased below a ~0.02. It was likely that the magnetic-field dependence was partly due to a slower radial expansion with larger

magnetic fields. Also, the larger magnetic fields should

have produced a more tightly confined electron beam

that was used to load the electrons. This probably result- ed in an initial cloud of smaller radial dimension.

Figure 4 shows a plot of the frequency of the noise

peaks, including the axial center-of-mass peak, as a function of the elapsed time after an electron cloud was load- ed into the trap. As discussed above, the frequency of the

center-of-mass peak decreased slightly for early times

after loading and then increased slightly. The figure

shows two sets of noise peaks: those which asymptotically approach the axial center-of-mass frequency co^ and

those which asymptotically approach co^+co^. These

peaks can be identified with some of the electrostatic mode frequencies of Eq. (2) for a pancake shaped (strongly oblate) plasma (a«l). The peaks which asymptotically approach co^ can only be m =0 modes. The peaks

which asymptotically approach co^+co^ can only be

m =1 modes. In addition to the noise peaks shown in Fig. 4, some peaks were observed at early times and ap- peared to be m =2 modes. These modes were only

detected for a brief time which made their identification uncertain.

IN- 154
3848 WEIMER, BOLLINGER, MOORE, AND WINELAND 49

As discussed previously, azimuthal asymmetries in the

trap cause a radial expansion of the plasma. The radial expansion should produce a decrease in the plasma's as- pect ratio with increasing time. Therefore the mode fre- quencies detected in Fig. 4 apparently increase with de- creasing aspect ratio (except the center-of-mass mode,

whose frequency remains relatively constant). From Eqs.

(2), only the frequencies of a subset of the modes with

l—m odd increase, or stay the same, for decreasing as- pect ratio. For modes in this subset with m —0, the (1,0) mode has the highest frequency, the (3,0) mode has the next highest, the (5,0) the next, etc. For modes in this subset with m =1, the (2,1) mode has the highest fre- quency, the (4,1) the next highest, etc. In this manner, we

were able to make the probable identification shown in Fig. 4 of the noise peaks.

The modes can be used to obtain information on the

aspect ratio as a function of time. Following Dubin's

cold fluid theory, we let co^q be an estimate of co^. (This means our estimate of co^ varies shghtly as a function of

time which disagrees with the ideal cold fluid theory.)

Then, from Eqs. (2), every non-cm. mode which we

detected provides an estimate for the plasma aspect ratio a. Figure 5 shows the estimated aspect ratio as a function of time from the data of Fig. 4, using the measured

value 6j^/27r=3080 MHz. Fair agreement between the

different estimates is obtained. Excluding the CO21 mode

estimate, all the other estimates agree to better than

20%. It appears that the differences between the esti- mates are due to a systematic effect not included in the

theory. In Fig. 5(b) it is seen that the estimated aspect ratio depends on l—m. As l—m increases, the estimated

aspect ratio decreases. For example, the (2,1) mode gives

the highest estimated aspect ratio. The aspect ratio esti- mates from the (4,1) and (3,0) modes agree, but are less than the (2,1) estimate, and they are in turn greater than

the (6,1) and (5,0) estimates. If the systematic dependence of the aspect ratio estimates on / — m could be

corrected, it appears that the mode frequencies could be

used to calculate the plasma aspect ratio to better than

5%. We also note that the noise fluctuations in the esti- mates for the plasma aspect ratio decrease with increasing l—m. This is a result of the different sensitivities of a to the different mode frequencies [see Eqs. (2)]. The systematic shifts in the mode frequencies can also be seen in Fig. 4. From Eqs. (2), the intervals in Fig. 4

between neighboring m =0 noise peaks and also between

neighboring m=l noise peaks should equal co^a to within a few percent. A visual inspection of Fig. 4 shows

that the differences in these intervals are not equal within

a few percent. In general, the intervals decrease in fre- quency as / increases. The smallest frequency intervals are 25-30% less than the largest frequency intervals. Similar results are found for |c4l<5Xl0~^. The

discrepancy between the detailed predictions of Eqs. (2) and the observed noise peak frequencies of Fig. 4 may be

due to (1) couphng of the modes with each other causing

relative frequency shifts, (2) shifts due to image charges

because the plasma radius is approaching the ring radius,

and (3) the small thickness of the cloud. Because the

detected modes appear to be two dimensional (discussed

0.016

0.012

D

ct:o

0) <

0.008

0.004

0.000

1000 2000 3000

Time (seconds)

4000

(a)

0.004

o

a:o

Q.

CO <

0.003

0.002

1 1

• '

'• A /(2.1)

V....' •-. .-, i \^>(4.1)\/\/\ .

•

- \ "> ^^ '•_ (5:0) \ \-....^ M?'°) \

(7^^

' 1 v^H;

2000 2400 2800

(b)

Time (seconds)

FIG. 5. (a) Estimates of the cloud aspect ratio a as a function

of the time since the cloud was loaded, for the data of Fig. 4. For each time, the (1,0) mode is used to estimate co,. Each (/,w ) mode is then used to provide an estimate for a. The calculated

values for a have been connected by straight lines for each mode. The values of a from m =1 (m =0) modes are connect- ed by dashed (solid) lines, (b) An expanded view of (a) from

1800 to 3000 s showing the systematic dependence of the es- timated aspect ratios on / and m. IN- 155
49 ELECTROSTATIC MODES AS A DIAGNOSTIC IN PENNING-

3849

below) the small thickness of the cloud should be less important. Evidence for the mode coupling is discussed

below. The frequency variation of the (1,0) mode is possibly due to a combination of mode coupUng, anharmonic

eflFects, and image charge effects. If the size of the fre- quency variation of the (1,0) mode is used as an estimate

of potential systematic shifts, then these shifts are

sufficient to explain the apparent differences in the inter- vals between the mode frequencies.

As an example of what the mode frequencies infer about the plasmas, from Fig. 5 at time t = 1720 s, we find that a=0.0039±0.0005. The uncertainty here is equal to the scatter between the average aspect ratio estimate and

the individual estimates. From the hnewidth of the co^q

mode for 6)],o~^tc when first loaded, we determined

N =43 000. The calculated electron density is then

4.7X 10^ cm~^, which impUes an interparticle spacing of

17.2 jum and cloud dimensions of Zq = \4.9 /xm and

ro = 3830 fitn. Assuming the plasma was thermaUzed to T=4 K, the Debye length was kj=20.l /xm and the

coupling strength was r=0.24.

The evolution of the mode frequencies was studied

while varying different trap parameters. As discussed

above and below, the two parameters which had the most

profound effect on the plasma in this low-density limit were the magnetic-field strength and the application of a cooling drive. The appearance time of the mode peaks

depended strongly on ^q (the trap axial frequency was

kept constant for all ^q). For all ^q, the signals from the

modes appeared when the plasmas' aspect ratios were ap- proximately the same, a ^0.02. This means the mode

frequencies for different Bq were also about the same. However, the rate at which the frequencies of these peaks

changed in time due to radial expansion was found to be

fairly insensitive to Bq (see Fig. 6). This needs to be studied in more detail with better control over all other ex- perimental parameters (such as Gq). Increasing the numo

-Mo

Cl

(0 <

0.016

I 1

^\ 0.11 T

-

0.012

T

\

T

\

\ \

0.008

0.004

T

\

\T

\

\ 0.25 T

" \ -

n nnn

T

X

T

1 1 \

1 1000 2000 3000

Time (seconds)

FIG. 6. The evolution of cloud aspect ratio as calculated from the frequencies of the co^o mode and co^ for two diflFerent magnetic fields. For Bq=OA T, TV =43 000 electrons and

Go<0.03°; and for 5o=0-25 T, iV = 57000 electrons and

00-0. r.

ber of trapped electrons from 7000 to 70000 had little effect on the mode evolution. Changing the trap angle

from 00 < 0.030° to ==0.7° (increasing the azimuthal

asymmetry), distorting the electrostatic well by adding to

0.5 V to ring sector A (again, increasing the azimuthal

asymmetry), or decreasing the resistance to ground of

sector A from 105 to 16 ft (which might change the radial damping) also had Uttle or no effect on the mode evolution. Changing the guard voltage so that the trap was

less harmonic (larger IC4I) allowed the modes to be observed for a longer time after they appeared. It also slowed the rate of change of the aspect ratio, but by only

a factor of 2.5 for a change of IC4I < 5 X 10~' to 4X 10~^

Having the cooUng drive on or off during the initial loading had no effect on the mode evolution, nor did having

6)2 tuned or detuned from cojq. We also tried to change

the plasma's density by applying a drive to ring sector A

resonant with the azimuthally asymmetric (m >0) modes

whose frequencies were near co^ . No change in the mode

frequencies was observed. Of these different control pa- rameters, the only one which had any dramatic effect on

the signal size, or number of observed modes, was the

number of electrons in the plasma. We found that (as noted above) the larger the number of electrons, the

larger the number of modes that were observed.

As mentioned previously, we detected only a subset of

all of the possible modes with frequency near oy^. Why

we observed these modes and not others involves a number of factors. We found that by inspecting the mode potential [4] that the modes detected in Fig. 4 look hke

drumhead modes of a two-dimensional disk. At a given

radius and azimuthal coordinate, all the electrons of the

plasma oscillate axially in phase. Parts of the plasma

with different radial positions and azimuthal coordinates

oscillate with different phases. These modes are two di- mensional and will exist in the limit that the thickness of

the electron plasma is much less than the mean interparticle spacing. Modes which we did not detect have struc- ture in the axial direction (the phase of the motion de- pends on z) which can be characterized by an effective wavelength which is less than, or on the order of, the

plasma's axial extent. The continuous, fluid description

of these modes breaks down when the interparticle spacing is comparable to or larger than this wavelength. In

particular, when the plasma is very thin it is not clear how to think about these modes, and their calculation

from the formulas of Refs. [2] and [4] is not correct. In order to understand some of the other factors that determined which modes we observed, we require a better understanding of the detection mechanisms. There are at

least two mechanisms by which the modes could couple

to the endcap and be detected. The first is that, like the

axial center of mass, the modes couple directly to the

endcap by electrostatic coupUng [6,43]. In general, the

efficiency of inducing axial currents in the endcap de- pends on the radius of the plasma. Therefore, modes

whose radial extent is on the order of r, might induce a

detectable current in the endcap even though the net axial velocity of the plasma is zero. For detection of an az- imuthally asymmetric (m >0) mode this requires that the cylindrical symmetry of the trap be broken by, for ex- TN-156
3850 WEIMER, BOLLINGER, MOORE, AND WINELAND 49

ample, a misalignment of the trap's electric and magnetic

axes [44]. A second coupling mechanism for detection would be

that the modes couple directly to the axial cm. mode,

which in turn couples to the endcap. This coupling could

be by field asymmetries or mediated by image currents in- duced in the electrodes. For an ideal trap, small plasma

size, and low mode excitation, all the modes are independent and therefore there would be no such coupling be- tween the modes. However, Fig. 7 shows a narrow fre- quency span as the frequencies of two of the modes inter- sect that of the axial cm. The (2,1) mode forms an anticrossing (or avoided crossing) with the axial center-of- mass mode; the mode frequencies never overlap. This is a

general characteristic of two coupled oscillators [45]. Further evidence of the mode coupling was observed in the mode linewidths. Modes with frequencies diflFerent from co^ by more than 150 kHz had full width at half

maximum (FWHM) hnewidths less than 10 kHz. These

linewidths were observed to increase as the mode fre- 200

Frequency (MHz)

FIG. 7. A series of noise spectra (as in Fig. 3) of a cloud of 20000 electrons (with IC4I <5X10~^ 5o=0.11 T) showing an

avoided crossing between the axial center-of-mass mode and the

CO21 mode (plus an avoided crossing of an unidentified mode at f«380 s). As the modes approach each other they couple to- gether resulting in the mode frequencies never becoming equal.

The modes were identified by observing their behavior after the cloud was first loaded and also their asymptotic behavior for long times. The resolution bandwidth is 10 kHz.

quencies approached co^, as would be expected since the

axial center-of-mass linewidth was larger.

These two detection mechanisms are not completely in- dependent, making it difficult for us to differentiate be- tween them. In addition, we might observe the modes

through a combination of the two mechanisms. Additional evidence comes from the fact that we did not

detect the m = 1 subset of modes whose frequencies went

as (o^—a>^~k(l)a, where k(l) is a constant which de- pends on / [see Eq. (2)]. This was true for both

(0^—0)^ ^cojQ and co^ ~oi^ <oiic- These modes have the same mode potential as the subset of modes whose fre- quencies went as co^+co^—kil)a, which we did detect. However, this subset of modes which we did detect crossed or had a small detuning from co^, while the others

did not. Another important factor was that, when we

purposely increased the trap azimuthal asymmetry (as discussed above), the signal size and number of observed

m —\ modes did not noticeably increase. Also, when we

increased IC4I we were able to observe all the modes for a longer time. Together, these observations imply that coupling to the axial cm. mode might have been the more

important detection mechanism for us.

B. Effect of sideband cooling

The eflFects of sideband cooling were studied by first loading a cloud and allowing it to evolve to the point

where some of the other modes besides the axial cm. mode were observed. Then a strong sideband cooling

drive was applied with frequency co^+co„ (for co^^cojq).

After a few seconds, all the modes other than the cm. mode disappeared. In addition, the cm. mode frequency was back to its initial loading value. If the drive was then

turned off, the other modes reappeared in approximately

the same time, and in the same order, as a newly loaded

cloud. This process was found to be very repeatable.

Each time after the strong cooling drive was applied, the

mode frequencies evolved at the same time and rate as that of a newly loaded cloud, except that the time was

measured from when the drive was turned oflF. Thus,

after each time the cooling was applied, the plasma density was reset back to the same value. Electrons were lost only when the cloud was allowed to expand radially until they struck the ring electrode before the cooling drive was applied. The point at which electrons began to be

lost was consistent with the ring electrode radius being

equal to the cloud radius as calculated from the mode fre- quencies. A typical drive strength to observe the eflFect of

the sideband cooling drive was F^ /Fg ^ 10""* at ring sec- tor A for 10000 electrons and 5o=0. 1 T, where V^ is the voltage on A. Similar results were found for the

diflFerent magnetic fields used, as well as with the cooling

drive applied at frequency coj—co^+lto^ with a larger

drive amplitude. With a weak coohng drive on continuously, additional modes were observed in the noise spec- trum but they were not positively identified. From these observations we conclude that the sideband

cooling not only can stop radial expansion, but can re- verse it by increasing the rotation frequency, and thus the

aspect ratio and density, of the cloud. The rotation fre- quency was increased to approximately the same value as IN- 157
49 ELECTROSTATIC MODES AS A DIAGNOSTIC IN PENNING- . . . 3851

the loading value. Applying the drive after the cloud ex- panded increased the aspect ratio by more than a factor of 20 from a < 0. 00 1 to a > 0. 02. For our conditions this impUes an increase in the density of >3%. The mechanism for this compression of the cloud and what deter- mines the Umit to that compression are not known. One

possibility is that the parametric coupling directly

reduces the magnetron center-of-mass ampUtude just as

for a single electron. Then, anharmonic terms in the

trapping potential might couple the magnetron cm.

mode to other non-cm. modes [36] resulting in a compression. A second possible mechanism is that the

parametric drive directly couples some other azimuthally

asymmetric mode whose frequency is close to <y^ (Uke

the lower frequency coj^i mode) to the axial center-of- mass mode. The coohng would then proceed until the az- imuthally asymmetric mode frequency had shifted away

from the resonance condition as the plasma density in- creased. Both of these mechanisms are possible in that we observed: (1) apparent electrostatic coupling between cm. and azimuthally asymmetric non-cm. modes (see Fig. 7) and (2) parametric coupling between a cm. and an

azimuthally asymmetric non-cm. mode (discussed

below).

C. Alternate detection methods

Detection of non-cm. modes was also possible using

the other techniques discussed in Sec. II. The

"bolometric" technique (described in method 1 above)

did not detect any modes, but it was not thoroughly test- ed. However, by using coherent detection (method 2

above) some of the modes were detected. Figure 8 shows

the output of the phase-sensitive detector when the drive

frequency was swept through (d^+cOj^^^. Two modes besides the axial cm. mode are seen in the figure. This

demonstrates that not only can we detect these modes

coherently, but also that we can excite them with exter- nal drives. Figure 9 shows the servo correction voltage in the axial frequency shift detection method (method 3 of above).

Here the axial cm. frequency was kept fixed by the ser- vomechanism while a drive was applied to ring sector A. For 5o=0. 1 T, a resonance appeared when the drive fre- quency was between ~ 1(X) and 500 kHz, and the reso- nance had a dispersive line shape. As time increased, this resonant frequency increased, always staying below oi^. The resonance was very repeatable from cloud to cloud

and was observable on clouds with as few as 300 elec- trons. As the drive strength was increased, the line shape

became hysteretic, depending on the direction that the

drive frequency was swept and the amplitude of the drive. Evidence for the origin of this resonance was found

when we simultaneously detected both the noise spec- trum (method 1) and the axial frequency shift (method 3). We found that the resonant frequency of the dispersive resonance occurred at exactly the measured diflFerence frequency between fu^^^i.o ^"*^ ^2,1 for ^^^ the time the resonance could be observed. However, this dispersive resonance could be observed only after the (02^ , mode had

crossed over a>^. Apparently the drive was capable of

parametrically coupling together the (2,1) mode and the

J I I I I L 3 61.4 61.5 61.6 61.7 61.8 61.9

Frequency (MHz)

FIG. 8. Data taken using coherent detection (method 2 in the

text) with (o^^o^xc and a detection bandwidth of 10 Hz. The

axial potential well was modulated at £iJn,od/27r= 1 MHz. A

coherent drive applied to one endcap was swept across

o>2 + (o^ad 2nd the response detected phase sensitively in the absorptive mode. These data were taken for Bq—0.\\ T, A'^=26000 electrons, and IC4I <5X 10~'. The signals from the

axial center-of-mass plus two other modes can be seen.

axial cm. mode, similar to magnetron sideband cooling.

This was done either directly by the parametric electric

field or indirectly through the field's modulating the

coherent axial motion necessary for the detection. This

mode coupling might also explain the dispersive Une

shape since, as the modes become more strongly coupled

, i

3 -

2 - N

X^ 1

•4-* H-Ic

(/)

VHB -1

X

< -2

-3 -

-4

540 550 560 570 580

Frequency (kHz)

FIG. 9. The detection of the coi^ \ mode by monitoring the correction voltage to the locked axial center-of-mass frequency

when a spatially inhomogeneous drive was applied through ring sector A and its frequency swept through coi^^—co^ (method 3 in the text). These data were taken for 5o =0. 1 1 T, A^ = 3000 elec- trons, IC4I < 5 X 10~^ and (o„ /2it=615 kHz. The arrows show

the direction of the sweep. The hysteresis in this dispersive reso- nance can be seen. For sufficiently hard drives, the axial fre- quency could be shifted by much more than its linewidth. TN-158

I
3852 WEIMER, BOLLINGER, MOORE, AND WINELAND 49

together by the field, the modes would form an avoided

crossing [45]. Our detecting just the frequency of the axial cm. mode, while varying the coupling by sweeping the

drive frequency, would result in a dispersive line shape.

This line shape would become more hysteretic the

stronger the coupling strength.

IV. DISCUSSION AND CONCLUSIONS

We have shown that it is possible to detect the electrostatic modes of a pure electron plasma in a Penning trap

using nondestructive electronic methods. We detected

several azimuthally symmetric and asymmetric modes in the low-density limit where the plasma approaches a two-dimensional charged disk. We observed these modes

in the low-density limit because the noise power detection

technique which we primarily used is facihtated by the

mode frequencies being close to the axial center-of-mass

frequency. This occurs for the intermediate frequency

modes when the plasma density is low. We found that the frequencies of these modes fit Dubin's theoretical

model [2,4] reasonably well. Also, we found that we

could excite these modes directly with external drives,

and in addition, by parametrically coupling modes to- gether. The observation of these modes allowed the rota- tion frequency, shape, and density of the plasma to be

determined. From auxiliary measurements of the elec- tron number, we could then determine the plasma size. The detection techniques demonstrated here should work

for any type of non-neutral trapped plasma, and should

be applicable to the detection of modes other than the in- termediate frequency modes. Other modes might also be

detectable with a different geometry of electrodes to act as induced current detectors. The modes observed here

may correspond to some of the features reported by Bar- low [6], but a comparison is difficult because of

differences in the detection. In studying these modes we also observed behavior

which departed from Dubin's model. This included a time dependence to the axial center-of-mass mode fre- quency due in part to the plasma radial expansion in an anharmonic trap. Also, we observed coupling between

the axial center-of-mass mode and other modes evi- denced, primarily, by avoided crossing behavior as their frequencies became degenerate. It is likely that this coupling is one cause for the deviations between observed

and predicted mode frequencies, and will have to be studied further in order to get a more complete picture of the

plasmas' characteristics. In addition, in comparing our

data with Dubin's theory we found there to be a sys- tematic shift in our measured aspect ratios as a function

of / — m where ( /, m ) is the mode used to determine the

aspect ratio. To be able to measure plasma aspect ratios to much better than 20% will require this systematic

effect to be understood and corrected for. We also demonstrated that the parametric coupling

technique called magnetron sideband cooling, which couples the magnetron and axial center-of-mass motions, not only stops the radial expansion of the plasma, but actually compresses the plasma increasing its density. The

mechanism for this compression needs to be found in hopes of achieving higher densities with this technique.

An interesting test would be to look for compression by

sideband cooling when the coupling is between the magnetron and cyclotron motions. The ability to control the

density without requiring optical methods might open new avenues for the study of cold non-neutral plasmas in Penning traps.

This study leaves a number of issues to be investigated.

While the origin for the coupling between modes might

be related to static electric field asymmetries, this needs

to be studied further. Also, the mechanisms by which the

modes are detected need not be positively identified. A

detailed study of the signal amplitudes and linewidths of

the modes would be a helpful first step towards solving

these problems. A detailed study of the hnewidths might

also yield information on electron/electron collisions in trapped plasmas. If trapped, strongly coupled electron plasmas could be achieved, then these mode linewidths

might be a measure of spatial correlations in the plasma.

For example, in experiments on electrons localized near

the surface of liquid heHum, it has been demonstrated

that the linewidth of shear wave resonances depended on r [46]. Similar results have been predicted for the elec- trostatic mode linewidths discussed here [47]. One important plasma parameter which we could not

easily control was the plasma temperature T. Other

groups have controlled their ion and electron plasmas'

temperatures by introducing a buffer gas (e.g., Refs. [7], [43], and [48]). This has the disadvantages of applying an

external torque on the cloud causing it to expand, being

limited to temperatures greater than 4 K, and introducing what can be undesirable ion/neutral collisions affecting the physics to be studied. The problem of radial expansion due to the torque apphed by the buffer gas can

be eliminated by applying simultaneously a sideband

cooling drive [48]. However, as noted in the introduction, there is also interesting physics to be studied for temperatures less than 4 K. Two methods which allow

temperature control of the plasma for T < 4 K are using

a dilution refrigerator, or sympathetic coohng by laser- cooled ions [49]. While both these methods are practical,

they suffer from being expensive and difficult to imple- ment. However, with advances in space-borne ^He

liquefiers, a liquefier well suited for cryogenic ion trap ex- periments has been developed [50]. These liquefiers act

as both refrigerator and thermal isolator allowing the

temperature to be varied within the range 0.3 to 30 K.

While such a device would not give the lowest achievable

temperature, it would give a useful control parameter.

ACKNOWLEDGMENTS

We gratefully acknowledge the support of ONR for

this work. We would like to thank Dan Dubin for his help on the mode theory. We would also like to thank

Steve Jefferts, Joseph Tan, and Matt Young for their comments on the manuscript.

IN- 159
49 ELECTROSTATIC MODES AS A DIAGNOSTIC IN PENNING- . . . 3853

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Spectrom. Ion Proc. 74,97(1986).

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Ph.D. thesis, University of California at San Diego, 1993

(unpublished).

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1990), Vol. 35.

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Volume 75, Number 23 PHYSICAL REVIEW LETTERS 4 December 1995

Long-Range Order in Laser-Cooled, Atomic-Ion Wigner Crystals Observed by

Bragg Scattering

Joseph N. Tan, J.J. Bollinger, B. Jelenkovic,* and D.J. Wineland

Time and Frequency Division, National Institute of Standards and Technology, Boulder, Colorado 80303

(Received 26 June 1995)

We report the first observation of Bragg scattering from atomic ions confined in an electromagnetic

trap. The results reveal long-range order and give evidence for bulk behavior in a strongly coupled

collection of laser-cooled ^Be"^ ions in a Penning trap. Long-range order emerges in approximately

spherical clouds with as few as 5 X 10"* ions (cloud radius ro = 37a where a = Wigner-Seitz radius). Bulk behavior is evident with 2.7 X 10^ trapped ions (ro = 65a), with Bragg scattering patterns

characteristic of a body-centered cubic lattice.

PACS numbers: 32.80.Pj, 52.25.Wz

Systems of crystallized charged particles have gained

wide interest, with implications for the pure Coulomb sys- tems of one-component plasmas [1], trapped atomic ions [1-3], ion storage rings [4], and dense astrophysical matter

[5], and for the shielded Coulomb systems of colloidal sus- pensions [6] and dusty plasmas [7]. For example, Paul rf traps have been used to study crystallization and melting of

small "Coulomb clusters" [8]. In both the rf and Penning

traps, cold ions form shell structures when the dimensions

of the system exceed a few interparticle spacings [9,10].

Such transitions of Coulomb fluids into crystal-like struc- tures are examples of Wigner crystallization in the classical regime (i.e., the wave functions of the particles do

not overlap). However, the pure Coulomb systems real- ized so far are not large enough to neglect surface effects and hence do not allow observation of infinite-volume or bulk behavior [11]. For example, the "micromotion" of

ions in the rf trap, which generates the trapping potential, can cause heating which limits the smallest dimension of

laser-cooled ion collections to a few interparticle spacings

[3]. The drawback of the Penning trap (which uses static trapping fields) vis-a-vis the rf trap is that the ions rotate about the trap magnetic field —this has hindered imaging

of the ion crystals as seen in Paul traps [3,8,10]. However,

the Penning trap, which has no micromotion heating, can

potentially store very large systems of laser-cooled ions and hence was used in the studies here.

In related works, optical lattices formed by the interfer- ence of intersecting laser beams have been used to localize and cool neutral atoms [12,13]. Unlike the ion Wigner

crystals, optical lattices are not formed by the interaction of the confined atoms; rather, they conform to the periodic

confinement wells created by the laser standing waves.

Also, trapped ions differ from systems of colloidal suspensions and dusty plasmas in that the ions interact through

a pure Coulomb (rather than a Yukawa-type) potential,

and the charges (q) and masses (m) of the particles are

identical. A system of ions confined in a Penning trap (hereafter

called an "ion cloud") is an example of a one-component

plasma (OCP) [1]. A OCP is a system of identical charged particles that is neutralized by a uniform-density

background of opposite charge. The thermodynamic

properties of an ion OCP are fully characterized by the coupling parameter

^2 r s

1 (1) 477 60 aksT

'

the ratio of the nearest-neighbor Coulomb interaction energy to the thermal energy (ksT) of an ion. The

Wigner-Seitz radius a, a measure of the interparticle

spacing, is defined by 47ra^no/3 = 1, where «o is the density of the neutralizing background (which is formally

equivalent to the confinement fields in ion traps), and is equal to the ion density at low temperatures [1]. An

unbounded OCP is predicted to exhibit liquidlike behavior

for r > 2 and undergo a liquid-solid phase transition to a bcc lattice at T = 172 [14]. The OCP has become one

of the most carefully studied models in statistical physics

because the interaction is simple and because of its importance in understanding dense astrophysical plasmas

[5]. In contrast to the growing number of theoretical works, there are few laboratory systems to test the models.

Until now, only shell structures have been observed with

A'^ < 1.5 X 10"^ ions in a Penning trap [9], and in a race- track rf trap [3,10]. This shell phase can be viewed as arising from the strong influence of the curved surface of a small cloud [11]. Simulation studies show that the ions in each shell form distorted (2D) hexagonal lattices, exhibiting short-range order with little correlation between

shells [4,15].

We report here the first observation of Bragg scatter- ing from laser-cooled ions in a trap [16]. In ion clouds with all dimensions much larger than the Wigner-Seitz

radius, we find interesting departures from the earlier observed shell phase [9]. Shell structures are still ob- served with N < 2 X 10* ions (or radii ro < 27a for spherical clouds). Long-range order emerges with as few

as // ~ 5 X 10"* ions (ro = 37a). We show that for N ^ 2.1 X 10^ ions (ro = 65a) the Bragg scattering pat4198

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Volume 75, Number 23 PHYSICAL REVIEW LETTERS 4 December 1995

tern is consistent with a bcc lattice. The ion crystals are estimated to be >150 /xm (~20a) long on a side. Theoretical estimates indicate that /-q S 50a is required

for bulk behavior [11,17]. We have built a large Penning

trap wherein clouds with N < 2.1 X 10^ ions (ro ^ 65a)

and r values as high as 600 have been attained. The

trap is formed by a 127 mm long stack of cylindrical electrodes with 40.6 mm inner diameter, in a 10~^ Pa

vacuum. A NMR superconductive magnet provides a uniform magnetic field (Bq = 4.5 T) parallel to the trap axis (z axis), confining the ions in orbits around this axis (cyclotron frequency of (1/277- = qBo/m = 7.55 MHz).

A static electric field, generated by applying Vq = 1 kV

between the end and central electrodes, confines the ions along the z axis near the center (z = 0) of the trap (a single ion oscillates at o),/2tt = 795 kHz).

The trapped ^Be"*" ions are cooled by a laser beam

propagating along the z axis (Fig. 1) and are optically

pumped into the 25^5i/2(A// = -^.My = —2) state by

tuning the laser frequency (A = 313 nm) slightly below

the 2s^S\/2{—2>2)~^^P^P^/2{~2^~7) resonance fre- quency [2]. A laser-cooled ion cloud first condenses into a bounded fluid state (a liquid drop). In a Penning trap with quadratic potential and negligible image charge ef- fects, a laser-cooled ion cloud forms a uniform density

spheroid bounded by {x^ + y^)/rj + z^/z] — 1 (for a

spherical cloud, Ts "= Zs — ^0) [18]. The residual ther- mal motions are superimposed upon a rigid rotation of the ion cloud about the z axis. The aspect ratio a = Zs/rs

of the spheroid varies with its rigid rotation frequency Wr

[18,19]. A laser beam directed normal to the z axis (not

shown in Fig. 1 and turned off during the Bragg scatter- ing) can exert a torque and change ojr within 40.8 kHz <

<jjr/2TT < 7.51 MHz, thus controlling the shape and density of the cloud [18]. An //5 imaging system along the

Polarizer 2 ^i

1^^ photocathode 'Lens 2

aperture A

photocathode

A

»LeDsl

'beam deflector \h

Polarizer 1 -* \

laser beam

1 Pemiing trap

vacuum envelope

FIG. 1. Schematic diagram (not to scale) for detection of Bragg scattering from an ion cloud, shown as a small prolate

spheroid, inside a Penning trap. An //5 imaging system (left) monitors the shape of the cloud, thereby giving the ion density.

The diffraction pattern is relayed to the upper photocathode,

filtered by a small aperture (A) and crossed polarizers, as described in the text.

X axis, with a laser beam along the z axis (beam waist ~ 0.4 mm S Tj, power ~200 /xW), gives a side-view image of the ion cloud, thus monitoring Zs and Tj which, in turn, yields a, a, (Or, no, and A^ [18]. Bragg scattering is used to study the spatial correlations because the rigid rotation of the ion cloud makes it difficult to image a crystal lattice. The photons resonantly

scattered by the ions interfere to form Bragg-scattering

peaks at certain angles determined by the crystal lattice. Because A/a ~ 0.04, the diffraction pattern occurs very

close to the laser beam direction (Fig. 1), within about

0. 1 rad. Since the ions scatter far fewer photons than the vacuum windows, special care is needed to minimize this potentially strong background.

As shown in Fig. 1 (not to scale), the laser beam (k, =

27rz/A) propagates first through a linear polarizer 1, and then into the vacuum chamber. Upon exiting the

ion trap, the laser beam is diverted away from the detector by a set of mirrors. The photons scattered by

the ions (k^ = 27rkj/A) are collected by lens 1 (/ =

19 cm, z = 25.5 cm from the ions), forming an image

of the ion cloud at a small aperture (A) to reduce the background. The diffraction pattern is then relayed by

lens 2 (/ = 24 cm) to the photocathode of a photoncounting imager (z = 160 cm). The linear polarizer 2

is inserted after the aperture. The polarization axes

of polarizers 1 and 2 are crossed to attenuate, with

extinction ratio ^ 6 X 10^, the stray light that leaks through the small aperture and has the same polarization

as polarizer 1. The photons scattered from the ions,

however, are attenuated only by a factor of 2 since they

are mainly circularly polarized.

The Bragg scattering from ion clouds with A^ S 5 X

lO'* ions (ro S 37a) exhibits long-range order characteristic of a crystal. Because of the ion cloud rotation

{cor /27r ~ 10^ Hz), the dots in a crystalline Laue pat- tern are rotated into rings. (A ring pattern will also be

produced by a crystalline powder or fluid.) Figure 2(a)

gives an example of a Bragg scattering pattern obtained

with N = 2J X 10^ trapped ions with long-range order. The circular intensity maxima are Bragg peaks, with radii inversely proportional to the Wigner-Seitz radius a. To

facilitate analysis, a differential scattering cross section

is generated from each diffraction pattern by averaging

the photon counts azimuthally about the z axis, as il- lustrated in Fig. 2(b). A density-independent plot is obtained by using the dimensionless parameter A^a, where

Ak is the length of the vector Ak = k, — k^. This can be compared with the interference function [20] or the

static structure factor SiAka) calculated for various sys- tems. For incoherent scattering, 5(A/:a) = 1. The short

correlation lengths (~a) of the fluid and shell phases

are reflected in their SiAka), shown in Fig. 2(c) [21],

which have only one strong, narrow peak. In contrast. Fig. 2(b) has four strong, narrow peaks indicating the presence of a crystal. Similar diffraction patterns for

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Volume 75, Number 23 PHYSICAL REVIEW LETTERS 4 December 1995

solid = supercooled fluid f = 1000

dotted = shell structure F > 500

15.0

FIG. 2(color). Observed Bragg scattering from N = 2.1 X

10^ ions (a) and (b) compared with the S(^ka) calculated for a supercooled fluid and an TV = 5 X 10^ ion cloud with shells

(c) [21]. The diffraction pattern ions (a) is partially blocked,

as indicated by a rectangular shadow due to the laser beam

deflector and a square shadow due to a wire mech. The color

scale is logarithmic. The total flux reaching the detector is = 1 X 10* photons/s but is attenuated to avoid saturating the photocathode. The field of view subtends a 5.4° scattering

angle from the z axis. The differential cross section (b) is obtained by azimuthally averaging (a) about the z axis.

N > 5 X 10** ions (ro > 37a) have been obtained with up to nine narrow Bragg peaks. Such long-range order

is not observed every time we cooled an ion cloud. In

this case, if a crystal was formed, the reason it was not observed may be that some crystal orientations did not produce Bragg peaks.

The crystal structure may be determined from the positions of the Bragg peaks [22]. For an unbounded

OCP the minimum-energy structure is a body-centered

cubic (bcc) lattice [14]. However, there are metastable

structures with slightly higher energies, such as the face- centered cubic (fee) and the hexagonal close-packed

(hep) lattices [11.23]. In a bounded ion cloud, residual surface effects may determine which structure dominates.

An analysis of data from a cloud with N Si 2.7 x

10"' ions (ro == 65a) shows that the diffraction patterns

are consistent with a bcc lattice. Figure 3(a), the evidence

of bulk behavior, shows a histogram of the number of

observed Bragg peaks (not intensity) as a function of

\ka. constructed with no adjustable parameters from

14 diffraction patterns obtained from the same ion cloud

(/V = 2.7 X lO"" ions) with various Wigner-Seitz radii (20 < a/ A < 27). Typically the ion cloud was heated

and recooled several times between each recorded pattern.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 best-fit Aka

FIG. 3. Evidence of bulk behavior in an ion OCP. Histogram

(a) counts the number of observed peaks (not intensity) from

various diffraction patterns of the same cloud (A' = 2.7 x

10^ ions), with no adjustable parameters (a is determined from

the observed a). By fitting to the bcc lattice for the WignerSeitz radius a, even the high Aka peaks are observed to be

consistent with a bcc lattice (b). The ticks indicate the location of Bragg peaks for various lattice types.

The scatter of the peak positions in Fig. 3(a) is consistent with the ~5% uncertainty in the determination of the Wigner-Seitz radius a from the observed value of the aspect ratio a. This uncertainty also makes comparison

with known lattices difficult for Aka > 10, where the density of Bragg peaks is higher. However, for Aka <

10, the bcc lattice is the best match to the observed

patterns. Having identified the lattice, we can also determine the Wigner-Seitz radius a by fitting each

pattern to the calculated bcc peaks. With these fitted values of a, the histogram Fig. 3(b) matches the bcc

lattice very well even as far as Aka ~ 14. The minimum size of the ion crystals can be esti- mated from the intensity widths of the Bragg peaks [see

Fig. 2(b)]. For a perfect crystal of characteristic length

L. the finite-size broadening of the Bragg peaks is esti- mated by the formula L == A/A, where A is the angular

FWHM of the Bragg peak [24]. We find that, on average, A ~ 2 mrad. This gives L ~ 150 /im (~20a). With the

typical density («o ~ 5 X 10^/cm^), this corresponds to a few thousand ions per crystal. This is a lower limit since

the observed widths can be instrumentally broadened, as suggested by the small change in A with A^. The relative intensity of the Bragg peaks indicates

that the ions do not freeze into a "powdered" sample of

randomly oriented crystals. For example, in some cases, peaks expected in such a powder pattern are missing.

The crystallized region may be surrounded by at least =20 shells on the cloud surface (based on a simulation

with 2 X 10"* ions) [25]. For a cloud with 2.7 x 10^ ions these boundary shells contain about 2.3 X 10^ ions.

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Volume 75, Number 23 PHYSICAL REVIEW LETTERS 4 December 1995

The scattering from the shells may contribute to the size of the first Bragg peaks, but will produce a fairly flat background [SiAka) ~ 1] for the other peaks. As shown

in Fig. 2(b), the Bragg peaks are superimposed upon a broad background. Many of the observed Bragg peaks

are often as large as, if not a few times larger than, the background. If these are Bragg peaks from a single crystal, the crystal must contain A^ ~ 10"* ions (for L ~ 240 /xm

or 35a on a side) in order to produce peak intensities greater than the background due to 2.3 X 10^ ions. In the future, we hope to remove the averaging due to the rigid rotation by stroboscopic detection of the Bragg

scattering. This could be useful in determining whether

the ions form more than one crystal. It may reveal other types of crystals in the ion OCP. Quasicrystals, for example, would produce a fivefold azimuthal symmetry in the

diffraction pattern [26]. The uncertainty in Wigner-Seitz

radius a can be reduced by measuring (Or directly [18]. This work is supported by the Office of Naval Research. We acknowledge useful and stimulating discus- sions with D. H. E. Dubin, J. P. Schiffer, H. E. DeWitt, and

S. Ichimaru. We thank J. P. Schiffer for providing simulation data of shell structure, D. H. E. Dubin for providing

calculations of SiAka) from shells and OCP fluids, and S. Ichimaru for permission to reproduce the fluidlike S{Aka)

[plotted in Fig. 2(c)]. We thank S. Gilbert, S. Jefferts, H. J. M. Hanley and M. Young for useful comments on

the manuscript.

*0n leave from the Institute of Physics, University of

Belgrade, Yugoslavia.

[1] J.H. Malmberg and T.M. O'Neil, Phys. Rev. Lett. 39, 1333 (1977).

[2] J. J. Bollinger and D. J. Wineland, Phys. Rev. Lett. 53, 348

(1993).

[3] H. Walther, Adv. At. Opt. Phys. 31, 137 (1993).

[4] A. Rahman and J. P. Schiffer, Phys. Rev. Lett. 57, 1133

(1986); J. P. Schiffer, Phys. Rev. Lett. 61, 1843 (1988); D. Habs, Max-Planck Institut fur Kemphysik, Heidelberg,

Report No. MPI H-1987-V10, 1987 (unpublished).

[5] S. Ogata and S, Ichimaru, Astrophys. J. 361, 511 (1990);

T. Strohmayer et ai, Astrophys. J. 375, 679 (1991).

[6] N. A. Clark, A.J. Hurd, and B.J. Ackerson, Nature (London) 281, 57 (1979); C.A. Murray, W.O. Sprenger, and R.A. Weak, Phys. Rev. B 42, 688 (1990); L.B. Chen

et ai, Phys. Rev. Lett. 69, 688 (1992).

[7] J.H. Chu and Lin I. Phys. Rev. Leu. 72. 4009 (1994); H. Thomas, G. E. Morfill, V. Demmel, and J. Goree, Phys. Rev. Lett. 73, 652 (1994).

[8] R.F. Wuerker et ai, J. Appl. Phys. 30, 342 (1959);

F. Diedrich et al, Phys. Rev. Lett. 59, 2931 (1987); D.J. Wineland et al. Phys. Rev. Le«. 59, 2935 (1987).

[9] S.L. Gilbert, J.J. Bollinger, and D.J. Wineland, Phys. Rev. Lett. 60, 2022(1988).

[10] G. Birkl, S. Kassner, and H. Walther, Nature (London)

357,310(1992).

[11] D. H. E. Dubin. Phys. Rev. A 40, 1 140 (1989).

[12] P. Verkerk et al, Phys. Rev. Leu. 68, 3861 (1992); P.S. Jessen et ai. Phys. Rev. LeU. 69, 49 (1992); A. Hemmerich and T. W. Hansch, Phys. Rev. Lett. 70, 410(1993).

[13] D.R. Meacher et ai, Phys. Rev, Lett. 74, 1958 (1995). and references therein. [14] W.L. Slattery, G.D. Doolen, and H.E. DeWitt, Phys.

' Rev. A 26, 2255 (1982); D.H.E. Dubin. Phys. Rev. A

42, 4972 (1990).

[15] D.H.E. Dubin and T. M. O'Neil, Phys. Rev. Lett. 60. 51

(1988); R.W. Hasse and J. P. Schiffer, Ann. Phys. (N.Y.)

203,419(1990).

[16] Some preliminary results are in J.N. Tan et ai. Nonneutral Plasma Physics II, edited by J. Fajans and D. H. E. Dubin (AIP Press, New York, 1995), p. 215.

[17] R.W. Hasse and V.V. Avilov, Phys. Rev. A 44, 4506

(1991).

[18] J.J. Bollinger et at., Phys. Rev. A 54, 525 (1993).

[19] L.R. Brewer et ai, Phys. Rev. A 38, 859 (1988).

[20] A. Guinier, X-Ray Diffraction in Crystals, Imperfect

Crystals, and Amorphous Bodies (Dover Publications,

Inc., New York, 1994), p. 29. [21] The supercooled fluid S(Aka), calculated for an un- bounded system with F = 1000. is from Fig. 6(b) in

S. Ichimaru et al, Phys. Rep. 149, 91 (1987). The shell phase S(Aka) is calculated from simulation data for an N = 5000 spherical cloud and F > 500, received from

J. P. Schiffer. Similar shells S(Aka) have been calculated by D. H. E. Dubin.

[22] N.W. Ashcroft and N.D. Mermin, Solid State Physics

(Holt, Reinhart, and Winston, Philadelphia, 1976), pp. 104-107.

[23] S. Ogata. Phys. Rev. A 45, 1122 (1992).

[24] Guinier (Ref. [20]), pp. 125-130.

[25] J. P. Schiffer, in Non-neutral Plasma II, edited by J. Fajans and D.H.E. Dubin (AIP. New York, 1995), p. 191.

[26] M. V. Jaric, Phys. Rev. B 34, 4685 (1986); S. Ogata and

S. Ichimaru, Phys. Rev. A 39, 1333 (1989); H.C. Jeong and P.J. Steinhardt, Phys. Rev. Lett. 73, 1943 (1994).

4201

TN-164
OBSERVATION OF LONG-RANGE ORDER IN TRAPPED ION

PLASMAS BY BRAGG SCATTERING^

JOSEPH N. TAN, JJ. BOLLINGER, B. JELENKOVIC*, WAYNE

M. ITANO, and D.J. WINELAND

National Institute of Standards and Technology

325 Broadway, Boulder, CO 80303

ABSTRACT

We have detected long-range order (crystal lattices) in atomic ions confined in a Penning trap. The crystals were observed by the Bragg scattering of nearly resonant

laser light at small (1-5°) scattering angles. Long-range order was observed in approximately spherical plasmas with as few as 5x10'' ions (plasma radius ro«37aws

where aws = Wigner-Seitz radius). With 2.7x10^ trapped ions (ro«65aws), Bragg

scattering patterns were obtained that were consistent with a body-centered cubic lattice.

1. Introduction

Charged particles in a trap form a realistic model of a classical one-component

plasma (OCP). For example, in a Penning trap under conditions of thermal equilibrium

the trapping fields provide a uniform background of opposite charge. ' With Doppler

laser-cooling on trapped ions, temperatures of a few millikelvins can be obtained with

densities greater than 10* cm'^ which results in couplings T = q^/(4xeoawskBT) > 250.^'^

Here q is the ion charge, e^ is the permittivity of the vacuum, aws is the Wigner-Seitz

radius, kg is Boltzmann's constant, and T is the ion temperature. These couplings are

larger than the predicted liquid-solid phase transition (F— 172)'* for an infinite OCP and

enable trapped ion plasmas to be used to study strongly coupled one-component plasmas.

However, in the trapped ion experiments to date, the boundary and relatively small size

of the ion plasmas have strongly affected the observed spatial correlations. In plasmas

where the smallest dimension is only a few interparticle spacings, the ions form curved

shells which approximately conform with the boundary of the plasma. ^"^ These shell structures have been observed with imaging techniques in both the Penning and rf

traps.*"^°In this manuscript we describe some recent experiments on approximately

spherical plasmas of up to 4.7x10^ Be"*" ions whose dimensions are large compared to aws- Through Bragg scattering we occasionally observe the presence of long-range order

or crystal lattices in the ion plasmas. Crystal lattices have been observed in the shielded

Coulomb systems of colloidal suspensions" and dusty plasmas'^. However, we believe

this is the first observation'^ of crystal lattices in a pure Coulomb strongly coupled

^ork of the US Government; not subject to US copyright 'On leave from the Institute of Physics, University of Belgrade, Yugoslavia

TN-165
plasma. We use the Penning trap in this work because it permits dimensionally large

plasmas to be laser-cooled. The Penning trap'"* uses a uniform, static magnetic field to confine ions in directions perpendicular to the magnetic field. A static electric field is used to confine ions in the direction parallel to the magnetic field. Due to the axial

magnetic field and the radial electric field in a Penning trap, the ion plasma rotates about

the magnetic field axis of the trap. The plasma rotation makes it difficult to directly

image the ions in a crystal. However, Bragg scattering can still be used to observe the

crystals. In the final section of this manuscript we also discuss a time-resolved

experiment where the Bragg scattering signal is used to measure the plasma rotation

frequency.

In an infinite, homogeneous one-component plasma, the body-centered cubic

(bcc), face-centered cubic (fee), and hexagonal close-packed (hep) lattices have the same

Madelung energy within 1 part in 10'', with the bcc lattice the energetically favored

configuration.^^ For a finite system, the surface energy and details of the boundary also

determine the energetically favored configuration. Dubin^^ considered a zero-temperature

model which was infinite and homogeneous in two dimensions but bounded and confined

in the third direction by a quadratic potential. He found that if the system was greater

than ~ 60 lattice planes (in the bounded direction), a bcc-like structure was the minimum

energy configuration. However, with less than 60 lattice planes the minimum energy

configuration depended sensitively on the exact number of lattice planes. This calculation

indicates that a trapped ion plasma may need to be larger than 60 lattice spacings along

its smallest dimension to exhibit behavior which is not strongly influenced by the plasma

boundary. A different analytical method developed by Hasse gives a comparable

prediction.'^ With a spacing of — LSa^s between typical low order lattice planes,'^ a

spherical plasma with a diameter of 60xl.5aws consists of ^9x10"* ions. We have

observed long-range order (crystals) in approximately spherical plasmas with greater than

5x10"* ions. With 2.7x10^ ions, the observed Bragg scattering pattern was

predominantly consistent with a bcc lattice, the expected infinite volume structure.

2. Experimental Set-up

Figure 1 shows a sketch of our experimental set-up. The Penning trap is formed

by a 127 mm stack of cylindrical electrodes with 40.6 mm inner diameter, in a 10* Pa

vacuum (133 Pa = 1 Torr). A superconductive magnet provides a uniform magnetic

field (B„=4.5 T) parallel to the symmetry axis (^Be"^ cyclotron frequency of

fi/27r=qBo/m=7.55 MHz, where m is the ion mass). The static electric field is generated by applying Vo=l kV between the end and central electrodes of the trap (a

single ion oscillates at 00^/2 7r= 795 kHz).

The trapped ^Be"^ ions are laser-cooled and optically pumped into the

2s^S,^(Mi=-3/2, Mj = -l/2) state by tuning the laser frequency (X«313 nm) slightly

below the 2s2S,^(-3/2,-l/2) -* 2p%2(-3/2, -3/2) resonance frequency.^''* A lasercooled ion plasma in thermal equilibrium forms a uniform density plasma (number

density nj with sharp boundaries. In a Penning trap with quadratic potential ($ =

TN-166
polarizer 2 >

aperture A

photocathode

lens 1 'beam deflector

I

polarizer 1

laser beam

Pemiing trap

vacuum envelope

Figure 1. Schematic diagram (not to scale) of the experimental set-up for detection of the Bragg scattering as described in the text.

m(j}^[2z^-T^y[2q]) and in which

image charge effects from the trap

electrodes are negligible, the

boundary is a spheroid given by

(x^+y^)/T,^ + z^lz^ = 1 (for a

spherical plasma r, = z, = rJ . The

residual thermal motions of the

ions are superimposed upon a

rigid rotation of the plasma about

the magnetic field axis (z-axis) at a frequency oj^. The aspect ratio

0L = zJi^ of the spheroid varies

with cOf.^^''^ A laser beam directed

normal to the z-axis (not shown in

Fig. 1 and turned off during the

Bragg scattering) can exert a torque that changes co^ within 40.8

kHz < co,/(2x) < 7.51 MHz,

thus controlling the shape and

density of the plasma.'^ An f/5 imaging system along an axis perpendicular to the z-axis,

with a laser beam along the z-axis (beam waist — 0.4 mm, power — 200 jwW), gives a

side-view image of the ion plasma, thus monitoring z^ and r^, which, in turn, yield a., w^,

"oj ^wsj and the number N of trapped ions.^^ Typical values for these parameters in the

work described here are a—l, Wr/(27r)~125 kHz, no~'4xl0^ aws~8.4 ^m. (The

measurement of w^ and therefore most of these parameters is significantly improved by

an autocorrelation measurement of the Bragg-scattered light described in the last section.)

The laser beam along the z-axis in Fig. 1 is used both to laser-cool the ions and

also to perform the Bragg scattering. Because X/aws ^ 0.04, the diffraction pattern occurs at small angles (a few degrees) relative to the initial beam direction. The main

purpose of the Bragg scattering apparatus in Fig. 1 is to detect the light scattered at small

angles by the ions while hiding from the forward-scattered light from the vacuum

windows. The incident laser beam (ki=2'7rz/X) first passes through linear polarizer 1,

and then into the vacuum chamber. Upon exiting the ion trap, the laser beam is diverted

away from the detector by a set of mirrors. The photons scattered by the ions

(k3 = (2x/X)(i^) are collected by lens 1 (f=19 cm, z=25.5 cm from the ions), forming an image of the ions at a small aperture (A) to reduce the background. The diffraction

pattern is then relayed by lens 2 (f«24 cm) to the photocathode of a photon-counting

imaging tube (z~160 cm). Linear polarizer 2 is inserted after the aperture. The

polarization axes of polarizers 1 and 2 are crossed to attenuate, with extinction ratio >

10^, stray light which leaks through the small aperture and has the same polarization as polarizer 1. The photons from the ions are attenuated by only a factor of 2 since they

are mainly circularly polarized. The angular acceptance of the system is limited by lens

1 to be +5.4°. For plasmas of a few hundred thousand ions, the ion fluorescence is

TN-167
strong enough that it was necessary to insert a 10-20 dB attenuator in front of the photoncounting imaging tube.

3. Bragg Scattering Results

The following procedure was typically used in obtaining a Bragg scattering

pattern. First, the perpendicular (to the z-axis) laser beam was used to set the density

and aspect ratio of the cloud. Typical densities were n^- 4 x 10^ cm'^ with aspect ratios

close to spherical (between 0.5 and 2). The perpendicular beam was then blocked and

the parallel beam unblocked with its frequency well below atomic resonance where little scattering takes place. The frequency of the parallel beam was then increased to near resonance and a Bragg scattering pattern was recorded. The laser frequency was then

detuned and increased again (or perhaps the laser was blocked and then unblocked) and

another pattern was recorded. The expansion of the plasma was slow enough that this

process could be repeated many times before the perpendicular beam was used to reset the plasma density and aspect ratio. On some occasions we observed Bragg scattering

patterns that consisted of several sharp rings such as shown in Fig. 2a. Patterns like this

with up to 9 narrow rings were observed with as few as N = 6x10'^ trapped ions. The

circular intensity maxima in Fig. 2a are Bragg peaks. The radius of a Bragg peak is inversely proportional to aws, a fact which was verified by changing the Wigner-Seitz

radius using laser torque. Figure 2b is a differential scattering cross section generated

from Fig. 2a by averaging the photon counts azimuthally about the z-axis (the laser beam

axis which is normal to the figure). Figure 2b is qualitatively different from the static structure factor S(q) in Fig. 2c for the quenched OCP fluid and shell structure, ^° both of

which exhibit short range order characteristic of a fluid. Here q= | k-k^ | =(4'7r/X)sin(0/2)

where 6 is the scattering angle. The fluid-like patterns have only one strong narrow

peak. In contrast. Fig. 2b has 4 narrow peaks which indicate the formation of a crystal

lattice. As mentioned earlier, such long-range order was not observed every time we

cooled an ion plasma. We have not determined whether this was because a crystal had

not formed or a crystal had formed but was not observed (because, for example, the

crystal orientation did not produce Bragg peaks).

In order to compare the Bragg peak positions with the calculated positions for the

bcc, fee and hep lattices, the radii of the Bragg rings must be calibrated in terms of the

scattering angle. This was done for a set of 14 diffraction patterns which were obtained on a plasma of 2.7x10-^ ions. For each pattern, aws was determined with about 5%

uncertainty from the measured a. Figure 3a shows a histrogram of the number of

observed Bragg peaks as a function of qa^s- For qa^s < 10, the histogram shows 5

groups of peaks. These groups are consistent with a bcc lattice but not an fee lattice. (They are also nominally consistent with a subset of hep peaks, but if many hep crystals were forming randomly, we would expect to see more groups or at least broader groups

of peaks.) Comparison of the groups of peaks with different calculated Bragg peaks for

qav/s > 10 is limited by the uncertainty in the determination of aws- However, if aws is determined by fitting each diffraction pattern used in Fig. 3a to the calculated bcc peaks.

TN-168
we see from Fig. 3b that with 2.7 x 10^ trapped ions the Bragg peaks are consistent with

a bcc lattice even as far as qaws— 14.

A lower limit on the size of the ion crystals can be obtained from the widths of

the Bragg peaks as observed in Fig. 2b. A crystal with dimension L has a Bragg-peak

width of A - X/L, where A is the angular FWHM of the Bragg peak. We find that, on

; (c)

CO

» I II—r—I—I—[-

-I—I—I r solid = supercooled fluid r = 1000

dotted = shell structure F > 500

0.0 2.5 5.0 7.5 10.0 12.5 15.0

ws

Figure 2. (a) Bragg scattering pattern observed from 2.7 x 10' ions showing the presence of a crystal(s).

The diffraction pattern is partially blocked by a rectangular shadow due to the laser beam deflector and a square shadow due to a wire mesh, (b) Differential cross-section obtained from an azimuthal average of

(a) about the z-axis. (c) Fluid-like S(q) calculated for a super-cooled OCP and an N= 5xlO^ ion cloud

with shells.^

TN-169
average, A — 2 mrad, which gives L— 150 /xm (~20aws) and a few thousand ions per

crystal. This is a lower limit, since the observed widths could be instrumentally

broadened. The negligible change measured in A with N supports such broadening. In

addition, the intensity in the Bragg peaks increased with N, which indicates that the

crystal size is growing with the number of trapped ions. The crystallized region is probably surrounded by at least -20 shells (based on a simulation with 2x10^ ions).^^

For a plasma with 2.7x10^ ions, this corresponds to about 2.3x10^ ions or a large

fraction of the total plasma in these boundary shell layers. The positions of these ions

are correlated only over a few interparticle spacings and may be the source of the

background signal in Fig. 2b. If the peaks in Fig. 2b are due to Bragg scattering off a

single crystal, we estimate that the crystal must contain — IC ions in order to produce

peak intensities greater than the uncorrelated background due to 2.3x10^ ions.

14 15

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

best-fit qaw

Figure 3. The number of observed peaks (not intensity) from 14 diffraction patterns with N= 2.7xl0^

ions. In (a) there are no adjustable parameters (a^s is determined from the observed a). In (b), a^s is determined by fitting the diffraction patterns used in (a) to a bcc lattice. The ticks indicate the location of

Bragg peaks for the simple cubic, hep, fee, and bcc lattices.

TN-1701
4. Time-Resolved Experiments

The Bragg scattering pattern from a single, stationary crystal consists of dots (a

Laue pattern). In our experiment the crystal rotates (o)J(It) — 125 kHz) about the

magnetic field axis (which is aligned with the laser beam axis to within 0.5°). This

rotation turns the Laue pattern of dots into rings. At any given point on a ring, the light

intensity should be modulated at the rotation frequency or a multiple of the rotation

frequency if the ring consists of more than one dot. The experimental set-up in Fig. 1 was modified to look for a time dependence of the Bragg-scattered light. The camera

in Fig. 1 was replaced with a mirror which made a 45° angle with respect to the laser

beam axis (the z-axis). The mirror had a small hole drilled parallel to the z-axis which

could be positioned on one of the Bragg rings. The light which passed through this hole

was detected by a photomultiplier tube. A pulse from the photomultiplier triggered a

multichannel scaler. The multichannel scaler then measured the arrival times of

photomultiplier pulses for —0.5 ms after the trigger. The Bragg scattering pattern

reflected from the 45° mirror was re-imaged onto the camera.

Figure 4a shows an example of an autocorrelation pattern obtained on the first ring from a plasma of 4.7 x 10^ trapped ions. The pattern indicates that the ring consists

of 4 equally spaced dots of unequal intensity. From the side-view image of the plasma,

27r/cOr is determined within — 15%, which unambiguously identifies the period of the

autocorrelation pattern. However, a strong autocorrelation signal can be used to improve

the measurement of w, to —0.1% with less than 10 s of averaging. The output of the

phototube was also used to "trigger" the camera recording the Bragg-scattered light. A

pulse from the phototube would enable the position of a photon to be recorded if it arrived within a window (duration much less than 2x/cOr) that was a fixed delay after the

photomultiplier pulse. Figure 4b shows the gated image that was recorded

simultaneously with the autocorrelation pattern in Fig. 4a. The gated Laue pattern

provides more information for determining the crystal type and the crystal orientation

relative to the laser beam. So far most, but not all, of the Laue patterns we have

observed are consistent with a single crystal.

The data in Fig. 4 can be understood as Bragg scattering off a single bcc or fee

crystal as follows. First, Bragg scattering is observed at small (1-5°) scattering angles.

This means that the planes which give rise to Bragg-scattered light are nearly vertical

(parallel to the z-axis). Second, a Bragg ring with momentum transfer q will persist over

an angular change in crystal orientation of — (27r/L)/q. For low-order planes, q — (2'7r/a),

where a is the lattice constant (a[bcc] =2.03aws; a[fcc] =2.56aws)- For our small crystals,

this means the crystal can be tilted with respect to the z-axis over a range a/L — 0.1 rad

or 5° and still give rise to the same Bragg peak. This means a low order, nearly vertical

plane will produce two Laue dots separated by 180°. The four equally spaced dots

making up the first Bragg ring in Fig. 4 are therefore produced by two vertical planes

which are 90°from each other. This rules out the hep lattice as a possibility; however

the planes could be bcc 110 planes or fee 200 planes where the laser beam is approximately aligned with a 4-fold symmetry axis of the cubic cell. The rest of the

TN-171
c

0)c

(a)

1200

900

600

300

10 20 30

Time (^s)

. '<{W{'-t " :.

,

40

- • -^ »i;>- '>4^'S{;,.' -.k^-i^. .

- ,. '*^

i'V

4 •

^ • " i'C

(b) • •= :s-''- '• "

Figure 4. (a) Autocorrelation pattern and (b) gated image simultaneously obtained on 4.7x10^ trapped

ions. The photomultiplier tube was sampling counts from the first ring.

Laue pattern in Fig. 4b is consistent with this interpretation. To distinguish between bcc

and fee in this case, an absolute angular calibration of the rings must be done. We are doing this with the time resolved set-up as well as analyzing other Laue patterns. This

will enable us to check with increased confidence that the lattice most frequently formed

with a few hundred thousand trapped ions is bcc.

5. Acknowledgements

We gratefully acknowledge the support of the US Office of Naval Research. We

acknowledge stimulating discussions with D.H.E. Dubin, J. P. Schiffer, H.E. DeWitt,

TN-172
and S. Ichimaru. We thank J, P. Schiffer for providing simulation data of shell structure,

D.H.E. Dubin for providing calculations of S(q) from shells and OCP fluids, and S. Ichimaru for permission to reproduce S(q) for the supercooled OCP fluid (Fig. 2c). We

also thank D. Berkeland, F. Cruz, and M. Young for carefully reading the manuscript.

6. References

1. J.H. Malmberg and T.M. O'Neil, Phys. Rev. Lett. 39 (1977) 1333.

2. J.J. Bollinger and D.J. Wineland, Phys. Rev. Lett. 53 (1984) 348.

3. J.J. Bollinger, D.J. Wineland, and D.H.E. Dubin, Phys. Plasmas 1 (1994) 1403.

4. E.L. Pollock and J.P. Hansen, Phys. Rev. A8 (1973) 3110; W.L. Slatterly, G.D.

Doolen, and H.E. DeWitt, Phys. Rev. A26 (1982) 2255; S. Ogata and S. Ichimaru,

Phys. Rev. A36 (1987) 5451; D.H.E. Dubin, Phys. Rev. A42 (1990) 4972.

5. A. Rahman and J.P. Schiffer, Phys. Rev. Lett. 57 (1986) 1133.

6. H. Totsuji, in Strongly Coupled Plasma Physics, ed. F.J. Rogers and H.E. Dewitt

(Plenum, New York, 1987) p. 19.

7. D.H.E. Dubin and T.M. O'Neil, Phys. Rev. Lett. 60 (1988) 511.

8. S.L. Gilbert, J.J. Bollinger, and D.J. Wineland, Phys. Rev. Lett. 60 (1988) 2022.

9. G. Birkl, S. Kassner, and H. Walther, Nature 357 (1992) 310.

10. R.F. Wuerker, H. Shelton, and R.V. Langmuir, J. Appl. Phys. 30 (1959) 342; F.

Diedrich, E. Peik, J.M. Chen, W. Quint, and H. Walther, Phys. Rev. Lett. 59

(1987) 2931; D.J. Wineland, J.C. Bergquist, W.M. Itano, J.J. Bollinger, and C.H.

Manney, Phys. Rev. Lett. 59 (1987) 2935; Th. Sauter, H. Gilhaus, W. Neuhauser,

R. Blatt, and P. Toschek, Europhys. Lett. 7 (1988) 317.

11. N.A. Clark, A.J. Hurd, and B.J. Ackerson, Nature 281 (1979) 57; C.A. Murray

and D.G. Grier, Amer. Sci. 83 (1995) 238; L.B. Chen, C.F. Zukoski, B.J.

Ackerson, H.J.M. Hanley, G.C. Straty, J. Barker and C.J. Glinka, Phys. Rev. Lett. 69 (1992) 688.

12. J.H. Chu and Lin I, Phys. Rev. Lett. 72 (1994) 4009; H. Thomas, G.E. Morfill, V.

Demmel, and J. Goree, Phys. Rev. Lett. 73 (1994) 652.

13. Preliminary results are in J.N. Tan, J.J. Bollinger, A.S. Barton, and D.J. Wineland,

in Non-Neutral Plasma Physics II, ed. J. Fajans and D.H.E. Dubin (AIP Press,

New York, 1995) p. 215.

14. H.G. Dehmelt, Adv. At. Mol. Phys. 3 (1967) 53; 5 (1969) 109.

15. S.G. Brush, H.L. Sahlin, and E. Teller, J. Chem. Phys. 45 (1966) 2102.

16. D.H.E. Dubin, Phys. Rev. A40 (1989) 1140.

17. R.W. Hasse and V.V. Avilov, Phys. Rev. A44 (1991) 4506.

18. L.R. Brewer, J.D. Prestage, J.J. Bollinger, W.M. Itano, D.J. Larson and D.J.

Wineland, Phys. Rev. A38 (1988) 859.

19. J.J. Bollinger, D.J. Heinzen, F.L. Moore, W.M. Itano, D.J. Wineland, and D.H.E.

Dubin, Phys. Rev. A48, (1993) 525.

20. The supercooled S(q), calculated for an unbounded OCP with T = 1000, is from Fig.

6b in S. Ichimaru, H. lyetomi, and S. Tanaka, Phys. Rep. 149 (1987) 91. The shell

TN-173
21.

structure S(q) is calculated from the simulation data, received from J. P. Schiffer,

for an N=5000 spherical plasma with r>500. Similar shell S(q) have been

calculated by D.H.E. Dubin.

J. P. Schiffer, in Non-Neutral Plasma Physics II, ed. J. Fajans and D.H.E. Dubin

(AIP Press, New York, 1995) p. 191.

TN-174
Hyperfine Interactions 76 (1993) 1 15-125 115

Laser-cooled positron source

DJ. Wineland, C.S. Weimer and J.J. Bollinger

Time andFrequency Division, NIST. Boulder. CO80303. USA

We examine, theoretically, the feasibility of producing a sample of cold (<4 K), high-density (« lO'^/cm^) positrons in a Penning trap. We assume 'Be"*" ions are first loaded into the trap and laser-cooled to approximately 10 mK where they form a uniform density column centered on the trap axis. Positrons from a moderator are then injected into the trap along the

direction of the magnetic field through an aperture in one endcap of the trap so that they inter- sect the 'Be"*" column. Positron/'Be"*" Coulomb collisions extract axial energy from the posi- trons and prevent them from escaping back out the entrance aperture. Coohng provided by

cyclotron radiation and sympathetic cooling with the laser-cooled 'Be^ ions causes the posi- trons to eventually coalesce into a cold coliunn along the trap axis. We present estimates of the

efficiency for capture of the positrons and estimates ofdensities and temperatures of the result- ing positron column. Positrons trapped in this way may be interesting as a source for antihydro- gen production, as an example of a quantum plasma, and as a possible means to produce a

bright beam ofpositrons by leaking them out along the axis ofthe trap.

1. Introduction

1.1. MOTIVATION AND BACKGROUND

For over a decade, slow positron beams using moderators [1-3] have found use

in diverse areas such as the formation of positronium, measurement ofatomic scat- tering cross sections, and surface studies. In addition to the importance of slow

positron beams, it would be useful to have an accimiulator and reservoir of cold

positrons for various experiments. An important application of such a reservoir

might be for forming antihydrogen by passing cold antiprotons through the positron reservoir and relying on three-body recombination [4]. Since the recombination rate is expected to scale as n^T"^/^, where n and T are the positron density and

temperature [4,5], it is important to achieve as high a density and as low a temperature as possible in the reservoir.

A cold positron reservoir has other potential applications. Surko et al. [6] point

out possible applications such as the study ofelectron /positron recombination, the

Contribution of the National Institute of Standards and Technology; not subject to US

copyright.

©J.C. Baltzer AG, Science Publishers

IN- 175
1 16 D.J. Winelandet al. / Laser-cooledpositron source

Study of resonances in electron /positron scattering [6,7], a positron source for accelerators, a diagnostic for fusion plasmas, and studies of basic plasma physics.

With the techniques described here, the resulting positron plasma may reach the

conditions for a quantum plasma [8] in the sense that hu^ >k^T where a;p is the plasma frequency and k^ is the Boltzmann constant. The techniques described here

may also be useful for providing a reservoir of cold high-Z ions or other ions which

are difficult to cool by other means.

If cold positrons are extracted from the reservoir, a beam source with high

brightness might be obtained. Such a source might be useful in atomic scattering

experiments which seek to observe positron /atom scattering resonances, diffraction from atoms, and high angular resolution [3]. Several other groups have already investigated and demonstrated trapping of

positrons in Penning traps. Schwinberg, Van Dyck, and Dehmelt have captured

small numbers of positrons in a Penning trap by injecting positrons from a ^^Na

source through an aperture in one endcap of the trap and extracting the axial en- ergy by resistive damping of the currents induced in the electrodes [9]. Gabrielse

and Brown have discussed a similar idea with potential increased loading efficiency

by use of a moderated positron source [10]. Surko et al. have achieved a nonneutral

positron plasma stored in a Penning trap [6]. In this work, positrons from a moderator are injected into the trap and their energy reduced by collisions with background gas. Conti, Ghaffari, and Steiger have trapped positrons in a Penning trap

by injecting positrons from a moderator through one endcap while ramping the voltage of the trap [11]. Mills has discussed a scheme for positron accumulation in a magnetic bottle [12].

In this paper, we investigate the feasibility of making a positron accumulator

and reservoir by injecting positrons from a moderator through an aperture in one endcap of a Penning trap and extracting axial energy from them through collisions

with a simultaneously trapped laser-cooled ion plasma.

1.2. BASIC IDEA

The basic idea of the scheme is illustrated in fig. 1 . We assume that 'Be'*' ions are first loaded into a Penning trap and laser-cooled by established techniques [13].

With only ^Be"*" ions in the trap, the ions form a uniform density nonneutral ion

plasma (temperature Ti^^o^) « 10 mK) which rotates about the trap axis at fre- quency (jj. We will assume the Debye length of the plasma is small compared to its dimensions. Therefore, the potential inside the plasma is independent of the axial

coordinate. We assume that positrons from a moderator are injected through an

off-axis aperture in one endcap (endcap 1) with just enough energy to pass through

the center of the aperture. The potential at the center of the aperture is less than

the potential V\ of endcap 1 because of the proximity of the tube (held at potential

Vt<V\) between the moderator and endcap 1. The diameter of the 'Be"*" plasma

is assumed to be large enough that the incoming positrons pass through the ^Be"^

TN-176
D.J. Winelandetal. / Laser-cooledpositron source 117

ENDCAP 2

POSITRON

MODERATOR

SOURCE T APERTURE ^^

Fig. 1. Schematic diagram of a possible Pemiing trap apparatus for capturing and cooling positrons

injected into the trap from a moderator. The trap magnetic field (not shown) is assumed parallel to the

axis of the electrodes. The 'Be"*" ions damp the motion ofthe positrons along the B field direction causing them to be trapped. As the positrons are cooled, they are forced inside the 'Be'*' ions where they

eventually form a cold column centered on the trap axis.

ion plasma. The other trap endcap (endcap 2) is biased at a potential V2 which is sufficient to reflect the positrons back through the ^Be"*" plasma. Positrons inside

the ^Be''' plasma also rotate about the trap axis at approximately frequency a;. If a;

is large enough and/ or the ^Be"*" plasma is sufficiently long, the positrons will

have been displaced azimuthually when they approach endcap 1 from inside the

trap and are reflected by the potential V\ back through the 'Be''" plasma. As the positrons pass through the 'Be'*' ions they undergo Coulomb collisions which extract

axial energy from them. At the low energies we consider, positron annihilation by

collisions with the 'Be"*" ions is suppressed by Coulomb repulsion. Initially, the positrons' axial energy is primarily transferred into their cyclotron motion energy since

the recoil energy of the 'Be"*" ions is small. If these collisions extract enough axial

energy before the positrons encounter the entrance aperture again, they will be

trapped. If the entrance aperture is mounted off axis, the positrons can make many

oscillations along the trap axis (and through the 'Be"*" ions) before encountering

the entrance aperture again. This is the same idea proposed by Dehmelt et al. [14]

and used successfully by Schwinberg et al. [9] except we assume the axial energy is extracted by t^/^"Rt^ collisions rather than resistive damping.

Once trapped, the positrons cool by cyclotron radiation and sympathetic cooling with the cold 'Be"*" ions. When sufficiently cooled, the positrons coalesce to a uniform density plasma column along the trap axis which co-rotates with the 'Be"*" plasma (both at frequency oJ) as shown in fig. 1 [8,15,16]. At low temperatures, the

TN-177
118 D.J. Winelandetal. / Laser-cooledpositron source

^Be"*" plasma separates from the positron plasma forming an annular region outside the positron plasma [8,15,16]. Quahtatively, the separation occurs because of

the larger outward centrifugal force on the ^Be^ ions. The positron cyclotron motion is strongly coupled to the ambient temperature To (4 K assumed here) by cyclotron radiation. At temperatures below about 10 K, the positron axial motion

decouples from the cyclotron motion [17] and we expect the positron axial or "parallel" motion temperature T\\ to be reduced to less than Tq by the Coulomb interaction with the cold 'Be"^ ions. Therefore we expect the parallel temperature T\\ to be lower than the cyclotron or "perpendicular" temperature Tj.. For a;< /2/2, the

positron and ^Be"^ plasma densities will be approximately equal [8] and limited by

the maximimi attainable ^Be"*" plasma density (Brillouin density). Here we will

make the approximation «(e+) w n{ Be"*") = n. Since the Brillouin density for an

ion species ofmass Mscales as B^/M, where B is the magnitude of the Penning trap

magnetic field, we want B as large as possible and M as small as possible. This is the reason for choosing ^Be"*" ions; other ions would work in the scheme described

here, but 'Be"*" has the smallest mass of any of the positive ions which can be easily

laser cooled therefore giving the highest density {n « 10^° cm~^ for 5 = 6 T).

1 .3. USEFUL FEATURES OF A LASER-COOLED POSITRON SOURCE

If this reservoir of cold positrons can be achieved, it should have some interesting features compared to those of other schemes. (1) Under the assumptions

described below, the efficiency of capturing moderated positrons can be very high,

approaching unity. (2) Relatively direct visual information is available on the positron plasma. By use of imaging techniques, the ^Be"*" plasma can be observed in

real time [13,18]. The positron plasma is then observed by the "hole" it creates in

the image ofthe 'Be"*" plasma. In this way we obtain fairly direct information on the

size and shape of the positron plasma. (3) If there is sufficient Coulomb coupling

between the positrons and 'Be"*" ions, the density and temperature of both species can be determined by spectroscopic techniques [1 3,1 8]. (4) The density ofboth plasmas can be varied by controlling the angular momentum imparted to the 'Be"*" plasma by the laser [13,18]. (5) The apparatus can be operated at high vacuum,

thereby reducing positron loss due to annihilation or positron formation with neutral background gas.

1 .4. APPROXIMATIONS

The calculations contained here are, admittedly, based on approximations. In

addition, the availability of a moderated source with certain characteristics (in

particular a small energy spread, A£i = 0.1 eV) is freely assumed. The purpose of

these notes is to give an estimate of what might be achievable with the proposed

technique.

IN- 178
D.J. Winelandetal. / Laser-cooledpositron source 119

2. Positron capture

2.1 . POSITRON INJECTION

We assume positrons are emitted from a moderator (fig. 1) with axial energy

spread and cyclotron energy spread both equal to A^",. They enter the Penning trap

along magnetic field lines through a tube held at potential Fj. We assume they enter the trap through a circular aperture in endcap 1 which is offset from the trap

axis by distance R^. Near the center of the aperture the potential is between V\ and

Ft; we consider an area A\ (= Tir^) centered on the aperture which is smaller than

the geometrical area ofthe aperture. The area A\ is such that V\ minus the potential

within A\ is larger than A^i. We make the approximation that over v4i the potential

is constant and equal to Fa, where Fi > Fa> Ft and V\ - FA>A£i. Outside A\,

we assume the potential is equal to V\ . We assume the potential Fg is adjusted so

that the positrons have just enough axial energy to spill over the potential Fa and

enter the trap. The positrons then are accelerated to an energy E\ by the potential

difference between Fa and the ^Be"^ plasma potential (f)p{Rx) at radius Rp^; we have

Ei = q[V\ — (j)^{R\)) + Eq, where q is the positron charge and Eq (« A^i) is the

mean axial kinetic energy of the positrons at the aperture. When the positrons enter

the plasma they drift through it with a velocity {2E\/m) '

, where m is the positron mass. Positrons exit the far side of the plasma and are then reflected back by endcap 2. They then traverse the plasma in the opposite direction and approach endcap

1 . If the length of the plasma is long compared to the spacings between the plasma

and the endcaps, the positrons spend most of their time in the plasma. Therefore,

upon returning to endcap 1 for the first time, they rotate an azimuthal angle

Ad « 7wh{lEilmy^'^ = 0.n{uj/{n/2))hBEr^/^, (1)

where A^ is in radians, uj is the plasma rotation frequency, and in the last expression the plasma length h is expressed in cm, B in T, Ei in eV and Q is the ^Be"*" cyclotron frequency. We have Q/2n = 1 .70 x B[Y] MHz when B is expressed in T. If A^

is larger than the angle subtended by the effective entrance aperture

^a(~ 2rA/i?A), then positrons are again reflected back into the plasma where they

have a greater chance to lose axial energy [14]. As long as A^ > 6a,, the positrons typically take 2{ln/6\) passes through the ^Be"*" plasma before encountering the en- trance aperture again. If A0<^a, only a fraction of the positrons which enter the

trap make multiple bounces [19]. In order to have all of the positrons enter the trap

we want ^j > A^*!. We would like E^ to be as small as possible to satisfy the condition M> 6\ and to make the positrons scatter through the largest angles.

Ifthe length ofthe positron/'Be"^ plasma is much longer than its diameter, the potential inside the plasma as a function of radius is approximately equal to that of an

infinitely long uniformly charged cylinder inside of a concentric grounded cylinder.

Therefore the potential at radius r inside the plasma is approximately equal to

TN-179
n =

120 D.J. Winelandetal. / Laser-cooledpositron source

4,p{r) « ngnr^ll ln(r,/rp) + 1 - {r/rp)% (2)

where r^ is the plasma radius, and r^ is the radius of the grounded outer cyHndrical

"wall" electrode. Since the potential inside the plasma is independent of the axial

coordinate, to keep the plasma confined along the axis, we require Vq, V2 > <f>p{0),

where Vq is the potential on the central portion of endcap 1 as shown in fig. 1 . We

adjust Vo and Vi (and Vj and Fj) so that Ei is kept small. This implies Vq > V\ and

that the end of the plasma bulges out towards endcap 1 at the radius R\. The 'Be'*' plasma density n is related to the rotation frequency u by [20]

where M is the ^Be"^ mass and in the last expression B is expressed in T. The maximum density, called the Brillouin density, is obtained when u = Q/2.

2.2. POSITRON DAMPING

We assume that the positrons initially lose axial kinetic energy by Coulomb scat- tering off the 'Be"*" ions. This transfers some of the initial positron axial kinetic en- ergy Ei into cyclotron motion energy. We assume that the cyclotron motion

energy is predominantly extracted by cyclotron radiation in the first stages ofcooling. In the first stages of the cooling, a good approximation is to assume that the

'Be"*" ions are stationary and infinitely heavy; cooling of the positrons by the recoil

of the ^Be"*" ions is negligible. We assume that the positrons scatter through a net

angle ^s (relative to the magnetic field direction) before encountering the entrance

aperture from inside the plasma. The scattering angle ^s results primarily from multiple small angle scattering of the positrons in the ^Be"*" plasma. The mean scatter- ing angle <^s> = 0. The mean-squared scattering angle is nonzero and given by

[21]

(A0s)' = <^>=^^ln(!»„,ax/^min), (4)

where L is the effective length of the plasma. When A9>dA, L is equal to

2h{2n/9/Cl- For a weakly magnetized plasma ftmax/^min is given by Vc/b, where r^ is the positron cyclotron radius and b is the distance of closest approach

(b = q^/k^T) [22]. For the plasma to be weakly magnetized, we require rc/b»l.

We assume that \mfiir^ w AEi, where fi^ is the positron cyclotron frequency.

Therefore, we can write

r,/b = 131Q^^^^, (5)

where the energies are expressed in eV and the field in T. Therefore, when A0>6a,

we can write eq. (4) in the form

TN-180
D.J. Winelandetal. / Laser-cooledpositron source 121

A^s = 1.28 X \Q-%n{r^/b)nh/eK]^"'/E;, (6)

where the angles are expressed in radians, h in cm, n in cm"^, and E\ in eV.

From simple geometric considerations, the axial energy lost before re-encountering the entrance aperture (assuming the initial cyclotron energy is much less than

EOis

AEs=EiSm^9s. (7)

We will make the approximation that the positrons are captured if A£'s>A£i.

Therefore from eq. (7), positrons are captured if they scatter through angles greater

than 6c where

ec = sm-\AEi/Eif\ (8)

If we assume that the distribution of angles ^s is Gaussian, the fraction of positrons

which enter the trap and are captured is given by

r7c = l-erf(^c/(2'/'A^s)). (9)

In table 1 , we list values ofrjc for various positron and ^Be"*" plasma parameters.

3. Beam source

In addition to using this configuration as a cold positron reservoir, it might

also be possible to use it as a positron beam source by leaking positrons out a hole in

endcap 2, which is centered on the trap axis. If the positrons are injected into the

trap at an arbitrarily slow rate, they can be extracted as a beam with an internal

temperature governed by the final equilibrium conditions achieved for the reser- voir discussed in the previous section. For a beam source we would like as high a throughput as possible. Therefore it is useful to estimate the temperature of the exTable 1 Calculated values of n/2n, 6\, A9, n, 0p(O), rc/b, rjc, and T - To for various values of input parameters. We assume A = 10 cm and Afj = 0.1 eV for all entries. The last column for (T - To) applies

only for the positron beam source described in the text (for all rows we assume dN/dt = 10' s~'). The

first row gives a high value of r/c but requires a very high trap potential (>(^p(0)). The last two rows assume the entrance aperture for positrons is centered on the trap axis. B Ei ta Ra rp /•w 2w/Q n/2n 9a M n </)p(0) Tclb r/c {T - To)

(T) (eV) (cm) (cm) (cm) (cm) (MHz) (rad) (rad) (lO'o (V)

cm-3)

(K)

10 1 0.05 0.5 0.7 2.0 0.5 17 0.2 9.0 2.2 15200 73 0.91 0.013

5 1 0.05 0.25 0.4 2.0 0.1 8.5 0.4 0.9 0.14 427 147 0.55 2.52

5 10 0.025 0.25 0.3 1.0 0.1 8.5 0.2 0.28 0.14 194 1470 0.27 44.8 10 10 - - 0.1 1.0 0.5 17 2n 2.2 560 734 0.11 6.38 10 10 - - 0.1 0.5 0.1 17 2n 0.56 107 734 0.001 25.2

TN-181
122 D.J. Winelandetal. / Laser-cooledpositron source

tracted beam versus throughput. As a conservative estimate we will assume that

the positron energy is extracted only by cyclotron radiation. In steady state, energy

balance is therefore described by the expression (d£'/df)i^ ~ {dE/di)^ + {dE/dt\,

where {dE/dt):^^ is the energy input from the incoming positron beam, {dE/dt)^ is the energy extracted by cyclotron radiation, and {dE/dt\ is the energy removed by

the extracted beam. The cyclotron energy decays at the rate 7c ^ 0.395^ [T] s~^

This rate is lower than the equilibration rate between T\\^ and Tj, for temperatures

down to about 10 K (for n « 10^, 5 w 6 T, see fig. 3 of ref. [23]), therefore above

this temperature T\\ = T± = T. In this case, the expression for energy balance can

be written

{dN/dt)Ei = \ioNk^{T - To) + lkBT{dN/dt), (10)

where (dN/dt) is the positron flux, A'^ is the total number of trapped positrons,

and To is the ambient temperature. This equation yields the solution

r-7-„.4.43><,0^(^^. (U)

where Ej is given in eV and B in T. Examples are included in table 1 . Similar considerations must be addressed to ensure that the laser beams can remove the (canonical) angular momentum and energy input to the ^Be"^ ions due to the incoming

positrons. With care these requirements can be satisfied. For small values of

dN/dt, the parallel and perpendicular motions become decoupled [22] and we

expect Ty « Tj, ~ 4 K. In this case the positron plasma is expected to be strongly

coupled. Under these conditions, it may be possible to extract a "string" of positrons which is located on the axis of the trap [18]. For these positrons, the angular

momentum is zero, so when the positrons are extracted from the B field, the trans- verse energy can be quite small. In the extraction to zero magnetic field, the cyclotron energy is converted to axial energy spread. At ^ « 6 T and Tl ~ 4 K, the

cyclotron energy is concentrated in the n = and 1 quantum levels, so the axial

energy of the extracted positrons is concentrated in two energy peaks.

4. Experiments

Cooling positrons with 'Be"*" ions has not been tried yet. However, some important aspects of the scheme have been demonstrated by using one laser-cooled ion

species to sympathetically cool another ion species. In fig. 2 we show an image ta- ken by viewing along the z axis of a Penning trap [13,18]. In this (UV) photograph,

a column of 'Be"*" ions, centered on the trap axis, is sympathetically cooled by la- ser-cooled Mg"*" ions at a magnetic field B w 0.82 T [24]. We have also demonstrated the use oflaser torque to control the shape and density of a ^Be"*" ion plasma,

reaching densities very near the Brillouin density (n « 10*°cm~^for5 « 6T)[13].

TN-182
D.J. Winelandetal. / Laser-cooledpositron source 123

1 mm

'Be^

26A/f«+ Mg^

1 mm 'I

Fig. 2. Ultraviolet photograph showing the Hght scattered from laser beams which intersect a

'Be"'^/^*Mg"'^ plasma confined in a Penning trap {B = 0.82 T). The view is along the direction of the trap axis (and B field) as described in refs. [13,18]. The laser beam for the Mg"*" ions intersects only a

portion of the Mg"*" plasma whose boundary is indicated by the dashed lines. The separation of the

species is probably exaggerated because the photograph required a double exposure. The optics were

refocused between the 313 nm ('Be"*^) and 280 nm (Mg"*") images. This probably resulted in different magnifications of the two wavelengths. The separation between plasmas is expected to be approximately equal to the interparticle spacing for low rotation frequencies (a> « /?/2) [ 1 5 , 1 6]

.

5. Discussion

If the configuration above can be achieved, antihydrogen formation from

three-body recombination with cold antiprotons passed through the positron plasma should be very efficient. If the positron plasma is strongly magnetized, the re- combination rate should depend on T\\ as rP'T'^ '

. In this case the probability of

antihydrogen formation might be unity for one pass of cold antiprotons through

the positron plasma [4,5]. Similarly, for the positron plasma to be quantized we re- quire ^p>/:b7'||.

On the theoretical side, more exact calculations are required to achieve better es- timates of r)c and other parameters. We have omitted any details of the positron

cooling for T'< 10 K. For example, for u!«Q, the separation between plasmas is approximately the interparticle spacing. As u> approaches Q, the separation be- comes large compared to the interparticle spacing and the Debye length [8,15]. In

this case, we expect the thermal coupling between the positron axial motion and the

^Be"*" ions to become weak. If the coupling between the positron axial motion and

the ^Be"*" ions remains relatively weak compared to the coupling between parallel

and perpendicular positron motions for T « 4 K, then T\\ will not be significantly

below T±. In principle, we could decouple the positrons' cyclotron motion from the

ambient temperature by avoiding the resonances of the cavity formed by the trap

electrodes [25]. However, this decoupling will be more difficult to achieve for plasmas whose dimensions are large compared to the cyclotron wavelength.

IN- 183
124 D.J. Winelandetal. / Laser-cooledpositron source On the experimental side, we have assumed narrow energy distributions A£i

from the moderator, which we have implicitly assumed is in the trap magnetic field. These values of A£i must be demonstrated experimentally. Use of a transmission

moderator would be desirable; the high energy particles from the moderator could

perhaps be rejected with an .E x 5filter. The long ^Be"*" plasmas assumed here (// = 10 cm) have not been demonstrated

yet. This is an important question which must be resolved experimentally; in general, longer plasmas are more prone to mode excitation from static field azimuthal

asymmetry which Hmits their density [13]. Currently we are assembling a new apparatus which should allow us to investigate the storage of large plasmas. If the desired long plasmas are difficult to achieve, it should be possible to stack shorter

plasmas with traps stacked along the axis of the system. This configuration may

also have the advantage that between the trap sections, E x B azimuthal drift sec- tion could be configured to increase A^ for the incident positrons.

The ideas presented here should apply to the trapping of other ions such as multiply charged ions which are difficult to cool by other means. In principle, these ideas

would apply to coohng of antiprotons or other negative ions. This would require use of simultaneously stored laser-cooled negative ions. Unfortunately, laser cooling of negative ions has not been achieved so far because of the apparent lack of a

suitable optical transition in a negative ion. Sympathetic laser cooling of negative

ions might be achieved using a coupled trap geometry [26].

Acknowledgement

We thank the US Office of Naval Research for support. We thank S. Gilbert, S. Jefferts, T. Steiger, D. Sullivan and M. Young for helpful comments on the manuscript. We acknowledge helpful discussion with B.R. Beck, R.S. Conti, G. Gabrielse, M.E. Glinsky, A.P. Mills, T.M. O'Neil, T.D. Steiger, T.S. Stein and CM.

Surko.

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D.J. Winelandetal. / Laser-cooledpositron source 125

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TN-185
Volume 70, Number 17 PHYSICAL REVIEW LETTERS 26 April 1993

High-Order Multipole Excitation of a Bound Electron

Carl S. Weimer/"' F. L. Moore/''^ and D. J. Wineland

Time and Frequency Division, National Institute of Standards and Technology, Boulder, Colorado 80303

(Received 9 November 1992)

The nonlinear resonant response of a bound electron to a time-varying spatially inhomogeneous electric field was studied experimentally. By use of the artificial atom "geonium" (an electron bound in a Penning trap), we observed up to ninth-order multipole (pentacosiododecapole) coherent excitation of

the electron's magnetron motion, and up to third-order (octupole) excitation of the cyclotron motion.

Also, by applying two fields simultaneously, we have observed coherent stimulated Raman excitation of

the electron's motion.

PACS numbers: 42.65. Ky, 36.90.+r, 42.79.Nv

The interaction of an electromagnetic field with a medium is, in general, nonlinear. There are three com- mon origins of the nonlinearity [1-3]: (l) the interaction

of the charges with the spatial inhomogeneity of the exci- tation fields, (2) the anharmonic binding of the medium's

charges, and (3) the interaction of the medium with the magnetic component of the field. A number of resonant

nonlinear effects have been studied, for example, harmonic generation, multiwave mixing, and multiphoton absorption. These nonlinear effects have almost exclusively relied on electric dipole transitions (with notable exceptions

[4]), and the origin of the nonlinearities falls primarily

into the second category. The first nonlinear mechanism,

which corresponds to excitation of higher-order multipole

transitions, is less common. Although multipole excitation is observed in some atomic and nuclear physics ex- periments, the excitation and detection are usually in- coherent. We have studied this nonlinear mechanism by

using an artificial atom [5]: an electron bound in a Penning trap [6-8]. This has provided for the first time a means to excite and detect, coherently, high-order multipole transitions, and observe high-order subharmonic

generation. We have also observed several degenerate

and nondegenerate nonlinear parametric processes which

have been discussed by Kaplan and Ding [2,3]. Our observations complement the body of work on coherent har- monic excitation and multiwave mixing in atoms. An electron bound in a Penning trap can be thought of

as the artificial atom "geonium" [6-8]. The "nucleus" of

geonium (the trap) is created with the electrodes (and

fields) shown schematically in Fig. 1. A static voltage Vq

applied between the end caps and ring forms a potential

(in spherical coordinates)

<D(r,0,0) = KoZ C2n(r/d)^"P2n(cose) , /i=0(1)

where the Pzn are the Legendre polynomials and the Cjn

are constants. Here ^=/-o/V2 =3.54 mm is a characteristic trap size, with /-q the ring internal radius. The

term labeled by C2 forms a harmonic well along the z

axis. Our trap is designed to have C2 =0.477, and

C4 = C6 — [9]. Applying a voltage Vg to two additional

guard electrodes [10] allows C4 to be varied; in practice.

|C4|<5xl0~ . Although IC4I was small, it remains the dominant electrostatic perturbation to our harmonic potential term. The leading perturbation to an applied

homogeneous magnetic field Bqz is a magnetic quadrupole bottle [6-8] of order 52/^0 = 7x10"^ cm~^ due to magnet imperfections and the copper electrode diamagnetism. Bo was measured to have a minimum long term

drift rate of \(dBo/dt)/Bo\ =7x 10"^ h

"', and for times on the order of 1 min, field fluctuations of ABo/Bo — 2 xlO"^for5o=0.1 T. The motion of a nonrelativistic electron bound in an

ideal Penning trap (C„ =0 for n>2, Bq homogeneous) is composed of three independent modes [6-8]. The axial

(z) motion is harmonic with amplitude r^ and frequency

(0^={.2C2qyQ/md^)^'^ where q/m is the electron's charge-to-mass ratio. In the radial {x-y) plane the motion is two superimposed circular motions: the magnetron motion of radius r^ and frequency cOm corresponding

to the ExB drift of the electron, and the cyclotron

motion of radius r^ and frequency a)'c—a)c~com where

(Oc =qBo/mc. For Kq = - 1 0.45 V and 5o =0. 1 T we obtain (o,/27c = 6\.5 MHz, W;„/2;r = 615 kHz, cOc/2k = 3.0S GHz, and in general we maintain (o'c^coz^com- The axial energy is damped [F^ (measured) — 20 s~'] by connecting a resonant tuned circuit to one of the end cap

electrodes (analogous to a two-level atom in a resonant

optical cavity) [6-8,11]. The magnetron motion is metaEndcap

FIG. I. A schematic diagram of the Penning trap used in the experiment. The electrodes have been separated along the z

axis to show details. © 1993 The American Physical Society TN-186

2553
Volume 70, Number 17 PHYSICAL REVIEW LETTERS 26 APRIL 1993

stable. However, its amplitude can be damped at a vari- able rate of 0< F^w < y H to a theoretical limit of r^ = 2{{o)J(o,){r^)V'^ [where {r})^'^ is the thermalized

(7=4 K) axial amplitude] by the use of sideband cooling

[6-8]. The cyclotron energy is radiatively damped at the

free space rate modified by the presence of the trap and

secondary electrodes [12] [rf(measured)=0. 1 s~' (1 s~') for Bq—0.\ T (1.4 T)]. Three main factors lead to deviations from the ideal motion: (I) anharmonic terms

in the electrostatic potential {C„^0,n>2), (2) relativistic effects, and (3) inhomogeneity of the magnetic field [6-8,13]. These cause the electron's modes of motion to be slightly anharmonic and weakly coupled together.

A single electron is loaded into the trap using established techniques [6-8,11]. The frequencies cD^MmyCOc

are measured by applying oscillating drive potentials to

different electrodes and observing a resonant response of

the electron's motion. For the axial motion, the electron's response is observed by measuring the image currents in- duced in one end cap using a phase sensitive heterodyne

detector (analogous to phase sensitive detection of atomic

resonance fluorescence) [6-8,1 1]. We chose to detect the magnetron or cyclotron excitations via their coupling to the axial motion [6-8]. Specifically, the observed axial frequency depended upon the magnetron amplitude due

to the electrostatic C4 term, and it depended on the cyclotron amplitude due to the slight magnetic field inhomogeneity 82- For example, for a magnetron excitation

(r„=0^r^o), A(oJ(o,= -3C4(r„,oy/2C2d^. The shift

*' 1

L^ -i

r^ ^' ^ ^oh\AnAPi ,

' .» c

V ^\l\ \

' w

V 1 3 \

*> o- 1 K*^' = -

,? is beat 1( . notes ^ f\

K) J

' L"

"5 h ^

.

**c« \

, 11 \}%v 1 >

. A A/ J lAll>JA/)AilA/\) 3 r•(IfffW^MX 0.5 H2

or

tiJ

-. i ., M

m

Drive Frequency cj^ FIG. 2. The subharmonic response of order wd/wm — ^ of the center-of-mass magnetron motion of a cloud of approximately

seventy electrons. Baselines for the two traces have been offset

for clarity. The arrows show the direction the drive frequency was swept (at a rate of 0.1 Hz/s). The axial frequency shifts due to the magnetron excitation (the vertical steps) and the beat note between the free oscillation at com and the subharmonically excited oscillation at cod/S are observed. For this figure the sideband cooling was off and the magnetron amplitude had been initialized in a large orbit. The inset shows a representation of this coherent process for particular quantum

levels.

A(Oz is effectively observed by monitoring the correction

voltage of an electronic servomechanism that locks the

axial frequency to an external reference [6-8]. If, in addition to the driven oscillation (at frequency <o</) there is a free (or thermally excited) oscillation at (o^, then an

interference beat note at wj — ci}„ will be observable in the correction voltage [7,14]. If a second coherent drive

field is applied with frequency coq, then three beat notes can be observed with frequencies (Od — (Om, (Oo — CDm, and

(Od — (oq. The magnetron and cyclotron motions are excited by applying a drive voltage to sector A of the ring electrode (Fig. I). To estimate the (x-y) drive field the

electrode geometry is approximated by a box with sides of

length (IxJyJz) =2d(-/2,y/2, 1 ). One side of the box per- pendicular to X (representing sector A) is assumed to be

at a potential Vj with respect to the other sides. This

model gives the resulting potential (in the x-y plane)

<t>Ax,y,0) = Vdj;,J,a„,„(x/d)'"(y/d)' (2)

with am.n constants [15]. A quantum description of the nonlinear excitation in- volves calculating amplitudes for processes like those in the insets in Figs. 2 and 3. However, in our experiment

the mean occupation numbers of the electron's quantized

motion were much larger than I, so a classical model

suffices [2,3]. When excited by a homogeneous field, the

magnetron and cyclotron motions can be treated as one40

m

•o

30

-g 20

Z3

E<

10

1

1 1 1 S 10 IS Frequency (Hz) FIG. 3. The Fourier transform of the response of the magnetron motion of a single electron to drive fields at frequencies oo,

(0\, and o>2. The magnetron free oscillation is at frequency (Om, and the driven oscillations are at fi)o •" (Um ~ 2 ;r( 7 Hz) and

(02 — (oi = (tim "^ 2n(4 Hz), with (U| =ft)m + 2;r(97 Hz). Three in- terference beat notes are observed. For short averaging times,

r < 10 s, the amplitudes and widths of the three beat notes were

comparable. For longer averaging times (r = IOO s is shown

here) the beat note responses between the driven and free oscil- lations (left two peaks) broaden and decay because of the magnetic field noise affecting the free oscillation. The beat between two driven oscillations (right peak) is unaffected (other than an

amplitude modulation noise pedestal) and its width remains

limited by the resolution of the dynamic signal analyzer (FFT).

When measured with a phase detector, this peak showed that the nonlinear response was phase coherent with the drives. The

inset shows a representation of this stimulated Raman response

at <U2~ft)|.

2554

TN-187
Volume 70, Number 17 PHYSICAL REVIEW LETTERS 26 April 1993

dimensional harmonic oscillators. For an inhomogeneous

drive field this is no longer true, but the main effect of the

field inhomogeneity is to modify somewhat the strength

and phase of the force. We therefore assume that the x

components of the magnetron and cyclotron motions obey

the approximate equations of motion (ignoring mode coupling) x + rjx + (ojxl\+^U)+p2(x/dy+p4(x/d)*+ ]

'^ajFd[aQ-¥a\(xld)-\-ai{x/d)^-\- hosicojt) , (3)

where Fd^gYd/md, ao-0.19, ai -0.35, 02-0.29, and

03=0.13 [from Eq. (2)], and j refers to either the cyclotron or magnetron motion. This equation neglects the y

and z dependence of the force. However, since the a, are only approximate, this further simplification should not

significantly affect the results. For the magnetron motion

(Oj^Om, aj''Oimlioim-o)'c)—-(oJ(Oc, and Pi^^Ci+i- For the cyclotron motion (oj —<Uc, aj — (o'cl{oi'c — oim) — \, and the ^, arise from the relativistic mass shift [13l. The

term ^(r) is a multiplicative noise term tl6] which, in our

experiment, is caused primarily by the magnetic field fluctuations. The magnetic component of the drive fields [category (3) of the introduction] can be shown to be

negligible here. Equation (3) holds also for the center- of-mass motion of small clouds of electrons (cloud size «(/); experimentally, both small clouds and single elec- trons were used. Equation (3) is solved by assuming solutions of the form xU)''A{t)co%[o}t+<l)U)] where y4(r) and tpit) are slowly varying. Nearly resonant excitation (o — toy) of

the motion can occur when cod—ncoj, n an integer, in which case 0) = {l/n)(0d. This corresponds to subharmonic excitation (i.e., frequency division). It can be

caused by (I) the electron's motion in the inhomogeneous

drive field [a, in Eq. (3)], and/or (2) the anharmonic

motion of the electron [Pi in Eq. (3)], as noted in the in- troduction. The first is due to electric multipole transi- tions of frequency nwj. The second is due to dipole transitions of frequency ncoj which arise from mixing of

the quantum oscillator levels due to the anharmonicity.

Since p„-\<^n^a„-\/aQ for our experiment, Eq. (3) can

be used to show that the first mechanism dominates.

Parametric excitation (« =2) has been observed previously in traps for the axial motion of a single electron [II]

and clouds of electrons [17-19].

If we neglect Pi with / > 2, Eq. (3) reduces to Duffing's

equation [3] driven by a nonlinear force. Subharmonic

response of order n exhibits both hysteresis and n stable phases of response relative to the phase of the drive [2,3]. In addition, from Eq. (3), a steady state response requires

[for|(r)-0] [3,20]

ajOn-iFd

(A/d) n-l

(4)

l"-^(Ojd

Therefore Fd and A must exceed certain threshold values

Fdt and Af. The origin of these thresholds is that the en- ergy supplied by the drive at Wd to the oscillation at (OdIn

is dependent upon the amplitude A of that oscillation and

must overcome the damping. No such threshold exists

for harmonic excitation (frequency multiplication), (Od — (Ojim. The initial amplitude At necessary for subharmonic response can be generated by a resonant homogeneous drive ((Od — (Oj), a thermal drive, or a free oscil- lation. When frequency noise is included [^it)^0], Tj in Eq. (4) is replaced by an effective damping rate which

depends upon the form of the noise [16], and the frequen- cy sweep rate of the drive. However, the ratios of Eq. (4)

for different orders of n should be insensitive to Tj.

Subharmonic excitation of the magnetron motion was

observed for « — 1 -9 on samples of up to about seventy

electrons when the magnetron amplitude was initialized

in a large orbit. Both the frequency shift and modulation

of the axial frequency (by the beat notes) were observed

as the magnetron amplitude resonantly increased. This is seen in Fig. 2 for « —5, where the beat note is due to the

interference between the magnetron free oscillation and

the subharmonically excited driven oscillation. With a single electron the threshold values for subharmonic response for n=2, 3, and 4 were measured with

the initial magnetron amplitude cooled to its minimum

value (set by the sideband cooling). Independently, we

determined the magnetron amplitude at its sideband cool- ing limit by varying C4 and measuring the resulting

change in com [8,14]. This gave r^— 220± 140 /im, or Tm— (55±35)rtheo, where rtheo is the theoretical cooling

limit. This disagreement with theory is consistent with

measurements made by other groups, and its origin is not known [7,8,14]. We measured the threshold values

Vdin =2) =6 /iV, Vdin =3) =300 /iV, and Vdin =4) =6

mV, for ft>^/2;r=615.0 kHz (i9o'=0.1 T, sideband cooling

off, sweep rate 0.1 Hz/s). From these measurements

and Eq. (4), we calculated 01/02 = 1. 6 ± 1.0 and 01/03 = 1.0-^;^. These agree with the values given by the electrostatic model [Eq. (2)] within the measurement uncertainty. Similar results were found for C0m/2n =41.34 kHz

(fio=1.4T).

The origin of the subharmonic response at « = 3 was

experimentally investigated. This response could arise from the excitation of the octupole moment of the magnetron orbit, or from a weakly allowed dipole transition due

to the C4 anharmonic term in the trap electrostatic potential. Changing the guard voltage allowed C4 to be varied

in small steps from C4<5xl0~^ to >4xl0~^ No

change in the required threshold drive strength was observed. If the subharmonic response had been due to the anharmonic component of the magnetron motion, this threshold drive strength would be proportional to |C4~' I- The more general resonance condition [21] cod^in/

m)(Om, rn,n^\ was also investigated. We observed reso- nant excitation when o}d~\(Om (for Vd = 'i mV, (Oml'^Jt =615.0 kHz, and the same conditions as above), which

corresponds to three-photon excitation of the quadrupole

\

IN- 188

2555
Volume 70, Number 17 PHYSICAL REVIEW LETTERS 26 APRIL 1993

resonance. For two drive fields we observed stimulated Raman excitation of the magnetron motion when (O2=qo)\=ro)m

with (^ = 1, /• = 1,2,3) and ((7 = 1,2,3, r=\). These reso- nances could be observed either with the sideband cooling

(damping) on or off, and typical detunings A = ft>i— w^ = 2;r(l to 150 Hz). Resonant enhancement of the re- sponse was observed as A was decreased. The beating be- tween these excitations and the free magnetron oscillation was observed as well as the anharmonic pulling of the magnetron frequency. When another driven oscillation near too — fi>m was generated, three beat notes were observed, two between the driven and free oscillations, and

one between the two driven oscillations near oim (see Fig.

3). The observation of these beat notes allows the magnetron responses (linear or nonlinear) to be measured in a continuous phase sensitive manner [20]. We observed subharmonic excitation of order n = 2 and

3 for a single electron's cyclotron motion (o)c/2k = 3 GHz). Since co'c^com, magnetic field drift and noise

caused significantly more dephasing of the cyclotron

motion than that of the magnetron motion; this resulted

in poor quantitative agreement between the cyclotron re- sults and the simple theory. When two drive fields were

applied with frequencies coj and a)\, stimulated Raman

excitation of the cyclotron motion of a single electron was

observed for a}2~ o)\^=(j)'c with a)\=oi'c + di. Resonant

enhancement of the response was observed as the detuning A was decreased; typically A/2;r=IOO to 600 kHz.

Magnetic field noise prevented us from observing beat

notes between the different cyclotron excitations.

In summary, we have observed, coherently, high-order

nonlinear resonant excitation of a single bound electron's motion as well as that of the center-of-mass motion of a cloud of electrons. These effects were observed in the

limit where the nonlinearity was dominated by the multipole interaction between the electron and the drive field. Initial comparisons with a simple one-dimensional theory

show quantitative agreement for excitation of the magnetron motion. Detailed quantitative comparisons between

experiment and theory requires better magnetic field stability, better estimates for the shape of the drive fields (the a,) and orbit amplitudes, as well as the extension of

Eq. (3) to three dimensions. In addition, the effects of mode coupling and noise-induced fluctuations of the mode frequencies must be included in the nonlinear

analysis, especially because they can inhibit very highorder excitation. Experimentally, it should be possible to greatly reduce the magnetic field noise, which in turn should facilitate high-order subharmonic excitation of

the cyclotron motion. If very high-order subharmonic

response can be observed a)=a)dln with n — \Q^, the sys- tem could have applications in frequency metrology

[3,22]. In general, this system might be useful in studies of the effect of noise on nonequilibrium phase transitions

[16], as well as chaotic behavior in the transition from

classical to quantum dynamics.

We thank J. Bergquist, J. Bollinger, R. Fox, S. Jefferts, 2556

D. Sullivan, and M. Young for their comments on the manuscript. We gratefully acknowledge the support of

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

Office of Scientific Research.

^"^Also at Colorado State University, Fort Collins, CO

80523.

^'''Current address: Physics Department, University of Texas at Austin, Austin, TX 78712.

[1] N. Bloembergen, Nonlinear Optics (Benjamin, New

York, 1965).

[2] A. E. Kaplan, Phys. Rev. Lett. 56, 456 (1986).

[3] A. E. Kaplan, Opt. Lett. 12, 489 (1987); Alexander E. Kaplan and Yu J. Ding, IEEE J. Quantum Electron. 24, 1470 (1988), and references therein.

[4] D. C. Hanna, M. A. Yuratich, and D. Cotter, Nonlinear

Optics of Free Atoms and Molecules (Springer-VerJag,

Berlin, 1979), and references therein.

[5] Marc A. Kastner, Phys. Today 46, No. 1, 24 (1993).

[6] Robert S. Van Dyck, Jr., Paul B. Schwinberg, and Hans

G. Dehmelt, Phys. Rev. D 34, 722 (1986).

[7] Robert S. Van Dyck, Jr., Paul B. Schwinberg, and Hans

G. Dehmelt, in New Frontiers in High-Energy Physics,

edited by Behram Kursunoglu, Arnold Perimutter, and

Linda F. Scott (Plenum, New York, 1978).

[8] Lowell S. Brown and Gerald Gabrielse, Rev. Mod. Phys.

58,233 (1986).

[9] Ear! C. Beaty, J. Appl. Phys. 61, 2118 (1987), and unpublished calculations.

[10] R. S. Van Dyck, Jr., D. J. Wineland, P. A. Ekstrom, and

H. G. Dehmelt, Appl. Phys. Lett. 28, 446 (1976).

[II] D. Wineland, P. Ekstrom, and H. Dehmeh, Phys. Rev.

Lett. 31, 1279 (1973).

[12] R. S. Van Dyck, Jr., P. B. Schwinberg, and H. G. Dehmelt, in Atomic Physics 9, edited by R. S. Van Dyck, Jr. and E. N. Fortson (World Scientific, Singapore, 1984);

Gerald Gabrielse and Hans Dehmelt, Phys. Rev. Lett. 55, 67 (1985).

[13] Gerald Gabrielse, Hans Dehmelt, and William Kells, Phys. Rev. Lett. 54, 537 (1985).

[14] F. L. Moore, L. S. Brown, D. L. Farnham, S. Jeon, P. B. Schwinberg, and R. S. Van Dyck, Jr., Phys. Rev. A 46, 2653 (1992).

[15] J. D. Jackson, Classical Electrodynamics (Wiley, New

York, 1975), 2nd ed. [16] R. L. Stratonovich, Topics in the Theory of Random

Noise (Gordon and Breach, New York, 1967), Vol. II; A. Schenzle and H. Brand, Phys. Rev. A 20, 1628 (1979); H. Haken, Synergetics (Springer-Verlag, Berlin, 1978).

[17] H. G. Dehmelt and F. L. Walls, Phys. Rev. Lett. 21, 127

(1968).

[18] R. S. Van Dyck, Jr., F. L. Moore, D. L. Farnham, P. B. Schwinberg, and H. G. Dehmelt, Phys. Rev. A 36, 3455

(1987).

[19] J. Tan and G. Gabrielse, Phys. Rev. Lett. 67, 3090

(1991).

[20] Carl S. Weimer, Ph.D. thesis, Colorado State University,

1992.

[21] N. N. Bogoliubov and Y. A. Mitropolsky, Asymptotic

Methods in the Theory of Non-Linear Oscillations (Hindustan, Delhi, India, 1961).

[22] D. J. Wineland, J. Appl. Phys. 50, 2528 (1979).

TN-189
PHYSICAL REVIEW A VOLUME 51, NUMBER 4 APRIL 1995

Coaxial-resonator-driyen rf (Paul) trap for strong confinement

S. R. Jefferts, C. Monroe, E. W. Bell,* and D. J. Wineland

National Institute ofStandards and Technology, Boulder, Colorado 80303

(Received 22 August 1994)

We describe a variant of the quadrupole ri" (Paul) ion trap capable of localization of a trapped ion to much less than an optical wavelength (Lamb-Dicke regime). The trapping potentials are generated by a high-Q, vacuum-compatible, quarter-wave resonator driven at about 240 MHz. The binding strength of

the trap has been characterized. The trap contains compensation electrodes which allow the cancellation of stray static electric fields within the trap. Secular frequencies of tens of megahertz have been

achieved for trapped magnesium and beryllium ions. PACS number(s): 32.80.Pj

I. INTRODUCTION

In this paper we describe an ion trap apparatus capable

of very strong confinement. Strong confinement is important in several applications. In optical atomic spectroscopy, strong confinement allows the attainment of the

Lamb-Dicke condition whereby the extent of the atom's motion is less than X/2ir, where k is the wavelength of

the exciting radiation [1]. Attainment of the LambDicke condition is important because it suppresses broadening due to the first-order Doppler effect, which,

in harmonically bound atoms, takes the form of motioninduced sidebands [2-5]. It also suppresses the fluctua- tions of the "carrier" from measurement to measurement

[6]. More recently, strong confinement has been suggested as a way to observe the effects of super and subradi- ance for two closely spaced atoms under well-controlled

conditions [7]. Achievement of the Lamb-Dicke limit also suggests experiments on single-atom cavity QED and

nonclassical atomic fluorescence features under controlled conditions [8,9]. With strong confinement, we should be able to achieve

resolved sideband cooling [10,11] using allowed electric- dipole transitions. Previously, resolved sideband cooling

has been achieved using weakly allowed optical transi- tions [11]. For resolved sideband cooling, we require that the trap is made deep enough that the atom oscillation frequencies (o),, / =x,y,z) satisfy 6), »7, where y is the

linewidth of the transition. Use of resolved sidebands on

allowed electric-dipole transitions has the potential advantage that cooling to the ground state of motion can be

achieved very rapidly. Resolved sidebands should also

facilitate precise measurement of the lifetimes of rapidly

decaying levels by measuring the linewidth of the

resolved carrier. In a Penning trap [12], single-ion confinement is limit- ed by the strength of binding along the magnetic-field direction, which we can characterize by the (harmonic)

binding frequency co^. For stable trapping in the radial direction, we must have o)^ < co^ /v^, where co^ is the ion

cyclotron frequency. This impHes Inv^^o^

Kl'^^^QB/M, where Q is the ion charge, B is the

magnetic-field strength, and M is the ion mass. For

5 = 10 T, Q = le, and M =9 u (e.g., ^Be+ ), v^ < 12 MHz.

In this paper, we discuss only the Paul trap because, for practical field strengths, the binding should be stronger.

For a single ion bound in a quadrupole rf Paul trap

[12], the axial or z frequency o}^ must satisfy o}^-^/^,

where ft is the rf drive frequency. If we assume that the

static potential apphed between the electrodes is zero, then, at the Mathieu stability limit (9^ =0.908) [12,13] in an axially symmetric trap, 6>2(max) = ft/2 and the radial secular frequency is given by <y^(max)^0. 35a)^(max).

The maximum secular frequencies can then be characterized by the expression for V2(max) [12,13]

1/2

V;,(max) = ft 7277 1 Indr

QVo

0.908M

(1)

'Present address: National Institute of Standards and Technology, Building 221, Gaithersburg, MD 20899.

where Q is the ion charge, Vq is the peak rf potential applied between the ring and end caps, M is the ion mass, and do is the characteristic trap dimension. For a trap

with hyperbolic electrodes, 2dl = rl+2zl, where Tq is the

inner radius of the ring electrode and 2zq is the iimer

end-cap to end-cap distance. Using Q = le, M =9 u, Fo = 2 kV, and c?o =0. 15 mm, we find v^(max)^164

MHz and v^(max)^58 MHz. These conditions would al- low resolved sidebands on the S—>-P transition in ^Be"*", where y /2'n= 19.4 MHz. If we assume v^ = 58 MHz and

the ion is cooled to the ground sate of motion, the LambDicke parameter is 27rXo A=0.063, where Xq is the

spread of the zero-point wave function.

From Eq. (1), to achieve strong confinement, we re- quire large values of Kg and small values of dg- Implicitly, we assume that ft is adjusted to give stable trapping,

that is, £i>2(o^. Large values of Fq can be limited by

electric-field breakdown or arcing. Also, if the high volt- age is generated external to the required vacuum enclo- sure for the trap, the voltage at the trap is reduced by the

capacitance of the vacuum feedthroughs. This problem

becomes more difficult for the required high values of ft. As the trap dimensions are reduced, it becomes more

51 3112

TN-190
51 COAXIAL-RESONATOR-DRIVEN rf (PAUL) TRAP FOR 3113

difficult to machine the desired electrode shapes. This

problem can be minimized by using simpler electrode

shapes where the leading anharmonic terms in the trap

potential can be nulled for appropriate electrode shapes.

For example, nominally quadrupolar electrodes with simple conical surfaces can be used [14]. "Planar" traps

formed with holes in parallel plates [15,16] or with con- ducting rings [16] simplify construction even further. These electrodes have the advantage that they can be

made small by using lithographic techniques [16]. As multielectrode traps (such as the conventional

quadrupole trap) become smaller, the relative electrode

positions also become more difficult to maintain. One

way to eliminate this problem is to keep either the ring or end-cap electrodes small and let the complementary

electrode(s) recede to large distances. The Paul-Straubel

or "ring" trap [17] is essentially a ring electrode with the

end caps at large distances. This trap has been analyzed

[16,18,19] and demonstrated for Ba"*" ions [18] and In"*" ions [19]. For this kind of trap, an efficiency e can be

defined as the ratio of the rf voltage required in a quadrupole trap with hyperbolic electrodes to the voltage re- quired in the ring trap (of the same internal ring diameter) which gives the same secular frequency. References

[16,18,19] find £ in the range of 0.1-0.2. In a similar

spirit, Janik, Prestage, and Maleki [20] have analyzed the

case of a linear trap with two and three electrodes rather

than four electrodes as for the quadrupole hnear trap.

For the two-electrode trap they find e^^^O. 25. The "end-cap" trap is complementary to the PaulStraubel trap in that it has closely spaced end cap elec- trodes with the ring receding to infinity. Schrama et al. [19] analyzed and demonstrated a version of this trap us- ing Mg"*" ions. They find an efficiency e in the range of

0.2-0.5. Both the Paul-Straubel or ring trap and the

end-cap trap additionally provide a more open optical access for collecting ion fluorescence hght than the conventional quadrupole trap. In the work described below [21], we discuss a strategy

for achieving large values of Vq and ft and small values

of do in a trap with nominal quadrupolar geometry where

II. DESCRIPTION

The trap described here is very simple; it consists of

two thin sheets of metal: the first, the ring, has a hole,

the second, perpendicular to the first, has a slit and acts

as the end caps. The arrangement is illustrated in Fig. 1. One trap constructed with this geometry is used for trapping ^'*Mg'^ ions and has a ring inner diameter Ir^ of 0.43 mm. The end-cap slit width 2zq is 0.32 mm. The

molybdenum sheet from which the trap was constructed was 0.13 mm thick. As shown in Fig. 1, the relative posi- tion of the trap electrodes is maintained by ceramic positioning rods. The trap is mounted at the end of an ultrahighvacuum-compatible coaxial quarter-wave resonant

transmission line. This oxygen-free high-conductivity

copper quarter-wave line is installed inside a vacuum en- closure which has fused silica windows to allow the passage of uv laser beams for cooling and probing the atom

and resonance fluorescence light from the ions. The

transmission line is resonant at a frequency of approximately 240 MHz and has an unloaded quahty factor Q„

on the order of 2000. The transmission line is excited

with a small loop antenna at the base of the transmission

Hne. The area of the loop is adjusted so that, on reso- nance, the antenna plus resonator presents a 50-ft load to the driving amplifier. With some care adjusting the cou- pHng, voltage standing-wave ratios of less than 1 .05 can

be achieved. With 1 W of incident rf power, the calculat- ed rf trapping voltage Vq at the trap is about 700 V peak,

neglecting the trap capacitance and end effects of the

resonator. From the modeled trap parameters (Sec. Ill) and the measured secular frequencies (below), we calculate the actual voltage between the ring and end caps to be Vq =; 600 V for 1 W of incident power. We have been

unable to apply voltage much beyond 1 kV due to a vacu- um discharge problem which we are attempting to solve. One advantage of this type of design results from the

fact that the rf voltage at the vacuum feedthrough is small, thus minimizing problems associated with highvoltage breakdown at or before the feedthrough as well as reduction of the resonator Q by the vacuum feedthrough.

An extension of the design would be to mount the "ring"

electrode at the antinode of a half-wave transmission line resonator with the end caps attached to the outer wall of

the transmission line. This geometry would allow a

direct current to be passed through the ring electrode to evaporate any contaminants on the trap ring [18]. Uncontrolled static electric fields arising from contact

potentials or charged patches can cause significant problems in miniature rf traps. These fields displace ions from

the zero of the rf trapping fields, thereby leading to undesirable increases in the "micromotion" amplitudes [18]. A common scheme for dealing with these problems is the

introduction of "compensation electrodes," which are used to cancel the stray electric fields in the trapping re- gions. We have included compensation electrodes, similar to those used by Nagourney [22], in the manner

shown in Fig. 1. Static potentials appHed to various combinations of the compensation electrodes allow for cancellation of static electric fields in an arbitrary direction

in the trapping region.

With zero static potential between the electrodes and

in the pseudopotential approximation, the secular (motional) frequencies of a trapped ion can be expressed

as

«,=«,

^2QVq

mdlQ,

(2)

where a, is a numerical constant for ion motion along the

/th axis (/ =x,y,z; see Fig. 1). In the case of a quadrupolar hyperbolic ion trap, the constant a, is j for the x and

y directions and 1 for the z direction. The trap described

here breaks the symmetry about the z direction, causing

oi^<cOy. In the pseudopotential approximation, with

zero static electric field between the trap electrodes,

co^+o)y=co^. Measured ion motional frequencies for this trap fall in the ratios of co^/o^—OAO and co^/co, =0.60.

lN-191
3114 JEFFERTS, MOhfROE, BELL, AND WINELAND 51

50Q Feedthrough

;- Vacuum Flange Compensation Eleclrodes(4ea) -

Optical Access Holes

Fine Thread Adjuster Serein/

Ceramic Adjuster Rod i of 4-

Center Conduibctor of Transmission Lme^

Detail - Adjuster Rod:

FIG. 1 . Approximate scale drawing of the coaxial resonator and rf trap electrodes (shown in an expanded view in the lower right of the figure). The ring electrode is attached to the inner conductor at the position of the antinode of the quarter-wave hne. The end

cap electrode is held near the rf ground by the short section of center conductor at the right-hand side of the resonator. The resonator is driven by a coaxial line external to the vacuum system shown at the left. The area of the loop antenna is adjusted so the resonator (on resonance) presents a matched load (50 ft) to the input line. The resonator is surrounded by a vacuum envelope (not shown in the figure) which has quartz windows to allow the passage of uv laser beams and fluorescence light. The position of the ring electrode

relative to the end cap electrode is maintained by four adjustable ceramic rods (alumina) shown in the detail in the lower left of the

figure. Static potentials are applied to four compensation electrodes to compensate for stray electric fields due to contact potentials

and charge buildup on the trap electrodes. For a proper selection of compensation potentials, the average position of a single trapped

ion coincides with the position of a zero rf electric field. The two compensation electrodes in front of the end cap electrode are shown in the expanded view in the lower right of the figure. Two more compensation electrodes (behind the end cap electrode) are not shown in the figure.

The secular frequencies of trapped ^^Mg"*" ions were measured by monitoring the intensity of scattered laser- cooling light (A. ^280 nm) as an rf drive was applied to one of the compensation electrodes and swept over the

frequency range which included the secular motion fre- quencies. Data taken with a single trapped ^"^Mg"*" ion

and Fo=^450 V at frequency ft are shown in Fig. 2. The

measurements of ion secular frequencies at a given rf voltage in ft yield the efficiency of this trap relative to the

normal hyperbolic rf trap. From these measurements, we

can assign a trap efficiency factor e of about 0.9 for the

axial (z) direction of this trap. This compares favorably

with the efficiency factor of about 0. 1 3 for the single-ring

trap [18] and between 0.2 and 0.5 for the multiring trap

[16], or end cap trap [19]. We have constructed a similar

but slightly smaller trap (a-q^O. 15 mm, Zo=0. 10 mm),

for ^Be"*" ions; this trap has demonstrated confinement

with secular axial frequencies in excess of 50 MHz

(ft 7277 =230 MHz), the largest secular frequency reported to date to our knowledge. Previously, secular frequeno ^

Is

Q_

m tK**f * II w^tr'iiii <"< i^itMiipti^'itijL

.5 MHz 3.5 MH^ 5.5 MHz

Frequencency

FIG. 2. Response of a single ^"^Mg"*" ion to an oscillating potential applied to one of the compensation electrodes shown in Fig. 1. The cooling laser for ^'*Mg"'" ions is tuned several linewidths to the red of the ^Si/2—>-'^Pi/2 transition. When the ion is excited by the oscillating electric field, the Doppler shift caused by its increased velocity leads to an increase in fluores- cence scattering. The three resonances shown correspond to ex- citation of the three secular frequencies co^yW^, and co^- IN- 192
51 COAXIAL-RESONATOR-DRIVEN rf (PAUL) TRAP FOR 3115

cies of approximately 10 MHz have been reported for

Ba"*" ions [16,18]. In this second trap the secular frequen- cy was inferred by measuring the ion-ion spacing at low

rf-drive levels, which calibrates the "spring" constant of

the trap, and then scaling the spring constant to higher

rf-trapping voltages. This measurement, coupled with

the modeled electric potential in the trap (below) and the

results from the first trap using trapped ^'*Mg''", allows

the assignment of the secular frequencies at arbitrary

values of the rf-trapping voltage. The secular frequencies

will be limited by the requirement that co^ <D,/2 for

stable trapping (Mathieu stability hmit). To increase 6)^, one must increase O,; one way this can be accomplished

with no change to the apparatus is by driving the

transmission line at the first overtone, thereby raising ft, and hence the limit on co^, by a factor of 3.

III. MODELING

This trap design does not lend itself well to analytic

methods of solution for the electric potential in the trapping region, so the trap potential must be modeled using

a computer program which can describe the full three- dimensional extent of the trap electrodes. We have per- formed modeling of the three-dimensional trap structure using the MAFIA [23] computer program.

The ring and end-cap electrodes of the ^'*Mg''' trap were discretized on a cubical mesh whose overall side was

1.02 mm on a side with a distance between individual

mesh points of 0.013 mm. The mesh axes lay along the same coordinate system as the trap coordinate system

shown in Fig. 1, with the origin centered on the trap.

The compensation electrodes were not included in the

model. The mesh spacing was chosen so there were a

sufficient number of mesh points to describe the smallest

feature of the trap. The electrostatic field was then calculated and the secular frequencies determined from the

field. The calculated motional frequencies are in the ratios 0)^ /(o^ =0.44 and o)y /co^ =0. 63, in reasonable agree- ment with the measured ratios of co^/co2=0A0 and

o)y/iL>^=0.60. The relatively coarse grid used in the trap

modehng limits the accuracy of the modeled trap potential thus causing the calculated co^ and cOy to not sum exactly to 0)^. Higher-order terms in the potential are also of importance. These have also been estimated by fitting the

MAFIA results to the expansion (in spherical coordinates)

^(r,d,<f)]

n=0

InPi^icosO) (3)

As a result of the reflection symmetry in % the odd order terms in the expansion have been eliminated. The constant Cj is the coefficient of the quadrupolar trapping potential and is of order 1 . The C^ term in the expansion,

the lowest-order correction to the described quadrupole

potential, is less than 0.3. Higher-order even terms in the

expansion have not been calculated. Due to the small size of the trap, significant misalignment of the trap electrodes is a possibihty. To determine

the sensitivity of the trapping potential to this effect, possible misalignments of the components of the trap were

modeled. Several cases where the end caps were shifted up to 0.13 mm relative to the ring in the y and z direc- tions were computed. This distance corresponds to about

one-third of the ring diameter and represents a "worst"

case misalignment. The predicted motional frequencies

for maximal misahgnment differed by less than 5% from

the frequencies for the aligned case. This calculated shift

is approximately the uncertainty introduced due to discretizing the trap on the finite mesh used. Therefore,

it seems that small misalignments of the trap electrodes

do not significantly alter the characteristics of the trap.

IV. CONCLUSIONS

This trap has demonstrated the highest secular fre- quencies for trapped ions reported to date and can be ex- tended to even higher frequencies. Unlike some other

designs, it retains the trapping efficiency of conventional

hyperbolic quadrupolar designs, thereby allowing larger

traps to be built to achieve a given secular frequency and

thus relaxing dimensional tolerance requirements. Use of

a coaxial-resonator-based design minimizes problems as- sociated with high-voltage breakdown and facilitates the

attainment of high values of the drive frequency fl. ACKNOWLEDGMENTS

Support for this project has been provided by the

ONR. S.R.J, and CM. received financial support from

the National Research Council. We thank J. C. Bergquist for help in setting up the coohng laser used in these experiments. A. S. Barton provided significant help

in the early stages of this work. We thank the AT-7 code

group at Los Alamos National Laboratories for the use of

the MAFIA modehng program. We thank R. Blatt, J. Miller, and M. Young for helpful comments on the

manuscript.

[1] R. H. Dicke, Phys. Rev. 89, 472 (1953).

[2] D. J. Wineland and W. M. Itano, Phys. Rev. A 20, 1521

(1979); J. C. Bergquist, W. M. Itano, and D. J. Wineland,

ibid. 36,428(1987).

[3] W. Neuhauser, M. Hohenstatt, P. E. Toschek, and H. G.

Dehmelt, Phys. Rev. A 22, 1137 (1980).

[4] G. Janik, W. Nagoumey, and H. Dehmelt, J. Opt. Soc. Am. 2, 1251 (1985).

[5] A. A. Madej, K. J. Siemsen, J. D. Sankey, R. F. Clark, and

J. Vanier, IEEE Trans. Instrum. Meas. 42, 234 (1993). TN-193
3116 JEFFERTS, MONROE, BELL, AND WINELAND 51

[6] H. Dehmelt, Bull. Am. Phys. Soc. 18, 1521 (1973).

[7] R. G. Devoe and R. G. Brewer, Bull. Am. Phys. Soc. 38, 1140(1993).

[8] Fundamental Systems in Quantum Optics, Proceedings of

the Les Houches Summer School of Theoretical Physics, Session LIII, Les Houches, France, 1990, edited by J. Dalibard, J. M. Raimond, and J. Zinn-Justin (Elsevier, Amsterdam, 1992).

[9] W. Vogel, Phys. Rev. Lett. 67, 2450 (1991); W. Vogel and

R. Blatt, Phys. Rev. A 45, 3319 (1992).

[10] D. J. Wineland and H. DehmeU, Bull. Am. Phys. Soc. 20, 637(1975).

[1 1] F. Diedrich, J. C. Bergquist, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 62, 403 (1989).

[12] H. G. Dehmelt, Adv. At. Mol. Phys. 3, 53 (1967); 5, 109

(1969); D. J. Wineland, W. M. Itano, and R. S. Van Dyck,

Jr., ibid. 19, 135 (1983); R. C. Thompson, ibid. 31, 63

(1993).

[13] R. F. Wuerker, H. Shelton, and R. V. Langmuir, J. Appl.

Phys. 30,342(1959).

[14] E. C. Beaty, J. Appl. Phys. 61, 21 18 (1987).

[15] M. H. Prior and H. A. Shugart, Phys. Rev. Lett. 27, 902

(1971).

[16] R. G. Brewer, R. G. DeVoe, and R. Kallenbach, Phys. Rev. A46,R6781(1992).

[17] H. Straubel, Naturwissenschaften 18, 506 (1955).

[18] N. Yu, W. Nagourney, and H. Dehmelt, J. Appl. Phys. 69, 3779(1991).

[19] C. A. Schrama, E. Peik, W. W. Smith, and H. Walther,

Opt. Commun. 101, 32(1993).

[20] G. R. Janik, J. D. Prestage, and L. Maleki, J. Appl. Phys.

67,6050(1990).

[21] S. R. JeflFerts, C. Monroe, A. S. Barton, and D. J. Wineland, Bull. Am. Phys. Soc. 39, 1167 (1994).

[22] W. Nagourney (private communication).

[23] T. Weiland, mafia (Release 3.1), October 1991, CST,

Ohlystrasse 69, D-6100 Darmstadt, Germany.

IN- 194
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