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WIneland, David J/Trapped Ions and laser QC100 .U5753 N0.1324 1988 V198 C.I NISTNIST TECHNICAL NOTE 1324

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

Nxyrn. ac 100 .usjs's i^o. 024

Trapped Ions and Laser Cooling II

Selected publications of the Ion Storage Group of thie Time and Frequency Division, NIST, Boulder, Colorado

Edited by

David J. Wineland

Wayne IVI. Itano

James C. Bergquist

John J. Bollinger

Time and Frequency Division

Center for Basic Standards

National IVleasurement Laboratory

National Institute of Standards and Technology

(fornnerly National Bureau of Standards)

Boulder, Colorado 80303-3328

Supported in part by

U.S. Office of Naval Research)

800 North Quincy

Arlington, VA 22217

and

U.S. Air Force Office of Scientific Research

Boiling Air Force Base, DC 20332

U.S. DEPARTMENT OF COMMERCE, C. William Verity, Secretary

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY, Ernest Ambler, Director

Issued October 1988 -^^^

x'^*^'^'v.>iJ^'
National Institute of Standards and Technology Technical Note 1324

Natl. Inst. Stand. Technol., Tech Note 1324, 200 pages (Oct. 1988)

CODEN:NTNOEF

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1988

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

This collection of papers represents the work on laser cooled, stored

atomic ions by the Time and Frequency Division of the National Institute of

Standards and Technology (formerly the National Bureau of Standards) from

July, 1985 to September, 1988. It follows the collection of papers, contained

in NBS Technical Note 1086, "Trapped Ions and Laser Cooling" (June 1985).

Although the primary goal of this research 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 with the prefix A are not included here but can be obtained

on request. We intend to update this publication periodically to include new work not contained here. We hope this collection of papers will be useful to our colleagues in this and related fields.

We gratefully acknowledge the important contributions of several

colleagues, post-doctoral and guest researchers, and students, including

Charles Manney, Earl Beaty, Larry Brewer, Sarah Gilbert, Hamid Hemmati, Randy

Hulet, John Prestage, Ule Daniel, Frank Diedrich, Dan Larson, Gerd Leuchs

,

Shao Zhong Xing, and Franklin Ascarrunz. We also acknowledge the continued

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

Scientific Research.

David J. Wineland

Wayne M. Itano

James C. Bergquist

John J. Bollinger

Boulder, Colorado

September, 1988

111

CONTENTS Page

Contents of previous NBS Technical Note 1086 ix

List of Additional Publications xi

SPECTROSCOPY AND FREQUENCY STANDARDS

1. "Energy and Radiative Lifetime of the 5d^ 6s^ ^^5/2 State in Hgll

by Doppler-Free Two-Photon Laser Spectroscopy,"

J.C. Bergquist, D.J. Wineland, Wayne M. Itano, Hamid Hemmati,

H . -U . Daniel , and G . Leuchs TN- 1

2. "Measurements of the gj - Factors of the 6s ^S^ and 6p ^ P^ States

in i9 8Hg+,..

"W.M. Itano, J.C. Bergquist, and D.J. Wineland TN-5

3. "Absorption Spectroscopy at the Limit: Detection of a Single Atom,"

D.J. Wineland, W.M. Itano, and J . C . Bergquist TN-8.

4. "Laser-Cooling Limits and Single-Ion Spectroscopy,"

D.J. Wineland, W.M. Itano, J.C. Bergquist and R.G. Hulet TN-11

5. "Recoilless Optical Absorption and Doppler Sidebands of a Single

Trapped Ion , " TN- 24

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

6. "Hg"'' Single Ion Spectroscopy,"

J.C. Bergquist, F. Diedrich, W.M. Itano, and D.J. Wineland TN-27

7. "High Accuracy Spectroscopy of Stored ions,"

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

F. Diedrich, and S.L. Gilbert TN-33

8. "Frequency Standards Utilizing Penning Traps,"

J.J. Bollinger, S.L. Gilbert, W.M. Itano, and D.J. Wineland TN-39

QUANTUM JUMPS

1. "Observation of Quantum Jumps in a Single Atom,"

J.C. Bergquist, R.G. Hulet, W.M. Itano, and D.J. Wineland TN-46

2. "Quantum Jumps via Spontaneous Raman Scattering,"

R.G. Hulet and D.J. Wineland TN-50

3. "Precise Test of Quantum Jump Theory,"

R.G. Hulet, D.J. Wineland, J.C. Bergquist, and W.M. Itano TN-55

4. "Radiative Decay Rates in Hg"*" from Observations of Quantum Jumps

in a Single Ion,

W.M. Itano, J.C. Bergquist, R.G. Hulet, and D.J. Wineland TN-59

5. "Photon Antibunching and Sub-Poissonian Statistics from Quantum

Jumps in One and Two Atoms ,

W.M. Itano, J.C. Bergquist, and D.J. Wineland TN-63

V
NONNEUTRAL PLASMA STUDIES

1. "Angular Momentum of Trapped Atomic Particles,"

D.J. Wineland, J.J. Bollinger, W.M. Itano, andJ.D. Prestage TN-67

2. "Sympathetic Cooling of Trapped Ions: A Laser-Cooled TwoSpecies Nonneutral Ion Plasma,"

D.J. Larson, J.C. Bergquist, J.J. Bollinger, W.M. Itano,

and D.J. Wineland TN-76

3. "Ion Trapping Techniques: Laser Cooling and Sympathetic Cooling,"

J.J. Bollinger, L.R. Brewer, J.C. Bergquist, W.M. Itano,

D.J. Larson, S.L. Gilbert, and D.J. Wineland TN-80

4. "Static Properties of a Nonneutral ^Be"*" Ion Plasma,"

L.R. Brewer, J.D. Prestage, J.J. Bollinger, W.M. Itano,

D.J. Larson, and D.J. Wineland TN-91

5. "Atomic Ion Coulomb Clusters in an Ion Trap,"

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

and C.H. Manney TN-106

6. "Shell-Structure Phase of Magnetically Confined Strongly Coupled

Plasmas ,

S.L. Gilbert, J.J. Bollinger, and D.J. Wineland TN-110

7. "Liquid and Solid Ion Plasmas,"

D.J. Wineland, W.M. Itano, J.C, Bergquist, S.L. Gilbert,

J.J. Bollinger, and F. Ascarrunz TN-114

8. "Quantitative Study of Laser Cooling in a Penning Trap,"

W.M. Itano, L.R. Brewer, D.J. Larson, J.J. Bollinger, S.L. Gilbert,

and D. J . Wineland TN-130

GENERAL ARTICLES

1. "Optical Pumping of Stored Atomic Ions,"

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

and J.D. Prestage TN- 1 32

2. "Laser Spectroscopy of Trapped Atomic Ions,"

W.M. Itano, J.C. Bergquist, and D.J. Wineland TN-144

3. "Precise Optical Spectroscopy with Ion Traps,"

W.M. Itano, J.C. Bergquist, R.G. Hulet, and D.J. Wineland TN-150

4. "Laser Cooling,"

D. J . Wineland and W.M. Itano TN-156

vi
APPARATUS

1. "Simple Electrodes for Quadrupole Ion Traps,"

E.G. Beaty TN-163

2. "Ion Traps for Large Storage Capacity,"

D.J. Wlneland TN-168

3. "Thermal Shifts of the Spectral Lines in the '"'P^ii ^° '''^11/2

Manifold of an Nd: YAG Laser,"

Shao Zhong Xing and J.C. Bergquist TN-182

vii

CONTENTS OF NBS TECHNICAL NOTE 1086

"Trapped Ions and Laser Cooling,"

(Issued June 1985)

Page

1. "Radiation-Pressure Cooling of Bound Resonant Absorbers,"

D. J. Wineland, R. E. Drullinger, and F. L. Walls TN-1

2. "Laser Cooling of Atoms,"

D. J. Wineland and W. M. Itano TN-5

3. "Laser-to-Microwave Frequency Division Using Synchrotron

Radiation,"

D. J. Wineland TN-25

4. "Laser to Microwave Frequency Division Using Synchrotron

Radiation II,"

J. C. Bergquist and D. J. Wineland TN-30

5. "Frequency and Time Standards Utilizing Laser Cooled Ions,"

W. M. Itano and D. J. Wineland TN-34

6. "Laser Induced Magnetron Compression (Expansion) of Ions

Stored in a Penning Trap,"

D.J. Wineland, R.E. Drullinger, J.C. Bergquist, and W.M- Itano.. TN-35

7. "High- Resolution Optical Spectra of Laser Cooled Ions,"

R. E. Drullinger, D. J. Wineland, and J. C. Bergquist TN-36

8. "Double-Resonance and Optical- Pumping Experiments on Electromagnetically Confined, Laser Cooled Ions,"

D.J. Wineland, J.C. Bergquist, W.M. Itano, and R.E. Drullinger.. TN-40

9. "Spectroscopy of a Single Mg Ion,"

D. J. Wineland and W. M. Itano TN-43

10. "Proposed Stored kg+ Ion Frequency Standards,"

D. J. Wineland, W. M. Itano, J. C. Bergquist, and F. L. Walls... TN-47

11. "Precision Measurement of the Ground-State Hyperfine Constant

of 25Mg+,"

W. M. Itano and D. J. Wineland TN-5

7

12. "Laser Cooling of Ions Stored in Harmonic and Penning Traps,"

W. M. Itano and D. J. Wineland TN-67

13. "Shift of ^S , Hyperfine Splittings Due to Blackbody Radiation,"

W. M. Itano, 'l. L. Lewis, and D. J. Wineland TN-87

14. "Laser Cooled, Stored Ion Experiments at NBS and Possible

Applications to Microwave and Optical Frequency Standards,"

D. J. Wineland, J. C. Bergquist, R. E. Drullinger, H. Hemmati,

W. M. Itano, and F. L. Walls TN-90

ix
Page

15. "Spectroscopy of Stored Ions,"

D. J. Wineland TN-97

16. "Magnetic Field Dependence of (s,£) Electron Configurations,"

D. J. Wineland and W. M. Itano TN-107

17. "Laser Cooling and Double Resonance Spectroscopy of Stored

Ions,"

W. M. Itano and D. J. Wineland TN-108

18. "Generation of Continuous-Wave 194-nm Radiation by SumFrequency Mixing in an External Ring Cavity,"

H. Hemmati, J. C. Bergquist, and W. M. Itano TN-117

19. "Laser Fluorescence Mass Spectroscopy,"

D. J. Wineland, J. J. Bollinger, and W. M. Itano TN-120

20. "Time and Frequency Standards Based on Charged Particle

Trapping,"

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

J. J. Bollinger TN-125

21. "Frequency Standard Research Using Stored Ions,"

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

and H. Hemmati TN-128

22. "Precision Measurements of Laser Cooled 9Be+ Ions,"

J. J. Bollinger, D. J. Wineland, W. M. Itano, and J. S. Wells... TN-132

23. "Hyperfine Structure of the 2p P ,„ State in ^Be+,"

J. J. Bollinger, J. S. Wells, D. J. Wineland, and W. M. Itano... TN-137

24. "Trapped Ions, Laser Cooling, and Better Clocks,"

D. J. Wineland TN-141

25. "Spectroscopy of Stored Atomic Ions,"

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

and J. D. Prestage TN-147

26. "Strongly Coupled Nonneutral Ion Plasma,"

J. J. Bollinger and D. J. Wineland TN-172

27. "Laser-Cooled-Atoraic Frequency Standard,"

J. J. Bollinger, J. D. Prestage, W. M. Itano, and

D. J. Wineland TN-176

28. "Limits for Spatial Anisotropy by Use of Nuclear- SpinPolarized "Be+ Ions,"

J. D. Prestage, J. J. Bollinger, W. M. Itano, and

D. J . Wineland TN-180
Additional publications of the Time and Frequency

Division Ion Storage Group not Included in this technical note

or NBS Technical Note 1086

Al . "New Possibilities for Frequency Standards Using Laser Cooling and Detection

of Stored Ions," F.L. Walls, D.J. Wineland, and R.E. Drullinger. Proc . 32nd

Annual Symp . on Frequency Control, June 1978, p. 453-459. (Copies available

from: Annual Frequency Control Symposium, c/o Electronic Industries Assoc, 2001

Eye St., Washington, DC, 20006).

A2 . "Laser Cooling of Ions Bound to a Penning Trap," R.E. Dullinger and D.J.

Wineland, in: Laser Spectroscopy IV . Eds. H. Walther and K.W. Rothe, (Springer,

Berlin, 1979) p. 66-72. Same title by D.J. Wineland and R.E. Drullinger, Proc.

6th Vavilov Conf. on Non-Linear Optics, Novosibirsk, Springer, June 1979.

A3. "The Isolated Electron," Philip Ekstrom and David Wineland, Sci. American

243, no. 2, p. 105-121, August 1980.

A4. "Shift of ^S^ Hyperfine Splittings Due to Blackbody Radiation and Its

Influence on Frequency Standards," Wayne M. Itano, L.L. Lewis, and D.J. Wineland,

J. de Physique, Colloque C8, suppl. to no. 12, vol. 42, Dec. 1981, p. C8-283-287.

A5 . "Prospects for Stored Ion Frequency Standards," D.J. Wineland, Proc. 13th

Annual Precise Time and Time Interval (PTTI) Applications and Planning Meeting,

Naval Research Laboratory, Washington, DC, Dec. 1981. NASA Conf. Publ. 2220., p.

579-592.

A6 . "Prospects Toward a Stored Ion Frequency Standard at NBS," Wayne M. Itano,

D.J. Wineland, J.C. Bergquist, and F.L. Walls. Proc. Conf. on Precision

Measurements and Fundamental Constants; Gaithersburg, MD June 1981; Precision

Measurement and Fundamental Constants II, B.N. Taylor and W.D. Phillips, eds

.

National Bureau of Standards (US) Spec. Publ. 617 (1984), p. 93-97.

A7 . "High Power Second Harmonic Generation of 257 nm Radiation in an External

Ring Cavity," J.C. Bergquist, H. Hemmati, and W.M. Itano, Opt. Commun. 43, 437-

442 (1982).

A8. "High Resolution Spectroscopy of Stored Ions," D.J. Wineland, Wayne M. Itano

and R.S. Van Dyck Jr., in Advances in Atomic and Molecular Physics . Vol. 19, ed.

by Bederson & Bates, (Academic Press, Aug. 1983), p. 135-186.

A9. "Laser Cooled ^Be"^ Accurate Clock," J.J. Bollinger, Wayne. M. Itano, and

D.J. Wineland, Proc. 37th Annual Symp. Freq . Control, 1983, p. 37-41 (copies

available from Systematics General Corp., Brinley Plaza, Rt . 38, Wall Township,

NJ 07719.)

AlO. "Spectroscopy of Stored Ions Using Fluorescence Techniques," D.J. Wineland,

Wayne M. Itano, J.C. Bergquist, and H. Hemmati, S.P.I.E., Vol. 426- Laser Based

Ultrasensitive Spectroscopy and Detection V . Society of Photo-Optical

Instrumentation Engineers, 1983, p. 65-70.

XI
All. "Frequency and Time Standards Based on Stored Ions," J.J. Bollinger, D.J.

Wineland, W.M. Itano, J.C. Bergquist, and J.D. Prestage, Proc. 16th Ann. Precise

Time and Time Interval (PTTI) Applications and Planning Meeting, Naval Research

Laboratory, Washington, D.C., Dec. 1984, p. 48.

A12. "Doppler-Free Two-Photon Laser Spectroscopy of Hgll," J.C. Bergquist, D.J.

Wineland, Wayne M. Itano, Hamid Hemmati , H.-U, Daniel, and G. Leuchs , Proc, 39th

Ann. Symp . on Frequency Control, 1985.

A13. "Two-Photon Optical Spectroscopy of Trapped Hgll," J.C. Bergquist, D.J.

Wineland, W.M. Itano, H. Hemmati, H.-U. Daniel, G. Leuchs, Laser Spectroscopy

VII . T.W. Hansch and Y.R. Shen eds . , (Springer Verlag, Berlin, Heidleberg, 1985).

p. 6-9.

A14. "Frequency Standards Based on Stored Ions," D.J. Wineland, IEEE Special

Issue on Radio Measurements & Standards: Proc. IEEE 74, 147 (1986).

A15. "A High-r, Strongly-Coupled, Non-neutral Ion Plasma," L.R. Brewer, J.D.

Prestage, J.J. Bollinger and D.J. Wineland, In Strongly Coupled Plasma Physics . F.J. Rogers andH.E. Dewitt, Eds., (Plenum, New York, 1986), p. 53.

A16. "The Observation of Quantum Jumps in Hg"*" , " W.M. Itano, J.C. Bergquist, R.G.

Hulet, and D.J. Wineland, Laser Spectroscopy VIII . W. Persson and S. Svanberg

eds., (Springer Verlag, Berlin, Heidelberg, 1987) p. 117.

A17. "Cooling in Traps," R. Blatt, G. Lafyatis , W.D. Phillips, S. Stenholm, and

D.J. Wineland, Physica Scripta T22, 216 (1988).

A18. "Frequency Standards in the Optical Spectrum," D.J. Wineland, J.C.

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

F. Bassani, T.W. Hansch, and M. Inguscio, (Springer Verlag, Heidelberg, 1988) to

be published.

A19. "The Digitized Atom and Optical Pumping," D.J. Wineland, W.M. Itano, J.C.

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

Grynberg, (World Scientific Press, Singapore, 1988) to be published.

A20. "Liquid and Solid Phases of Laser Cooled Ions," S.L. Gilbert, J.C.

Bergquist, J.J. Bollinger, W.M. Itano, and D.J. Wineland, ibid.

A21. "Laser Cooling to the Zero Point Energy of Motion," F. Diedrich, J.C.

Bergquist, W.M. Itano, and D.J. Wineland, submitted for publication.

A22. "Perpendicular Laser Cooling of Ion Plasmas in a Penning Trap," Wayne M.

Itano, L.R. Brewer, D.J. Larson, and D.J. Wineland, Phys . Rev. A., to be

published.

xii
Volume 55, Number 15 PHYSICAL REVIEW LETTERS 7 October 1985

Energy and Radiative Lifetime of the 5d^6s^ ^^5/2 State in Hg 11 by

Doppler-Free Two-Photon Laser Spectroscopy

J. C. Bergquist, D. J. Wineland, Wayne M. Itano, Hamid Hemmati,*^' H.-U. Daniel/"^

and G. Leuchs''-* Time and Frequency Division, National Bureau ofStandards, Boulder, Colorado 80303

(Received 22 July 1985) The Doppler-free, two-photon 5

<^ '"65 ^S 1/2 -5d'^6s^^Ds/2 transition in singly ionized Hg, attractive as an optical-frequency standard, has been observed for the first time on a small number of

i98j_jg+

jpj^j confined in a radio-frequency trap. The radiative lifetime of the ^Ds/2 state and the absolute wave number of the two-photon transition were measured to be 0.090(15) s and

17757.152(3) cm"', respectively. Optical amplitude-modulation sidebands, induced by the secular (thermal) motion of the harmonically bound ions, were observed also for the first time.

PACS numbers: 32.70.Fw, 32.30.Jc

Microwave or optical transitions of laser-cooled ions

that are confined in electromagnetic traps offer the

basis for frequency standards of high stability and accuracy.'"^ The advantages of such devices are numerous:

Very long interrogation times and, therefore, high

transition-line ^'s can be achieved; fractional frequency perturbations that are due to the trapping fields can

be held below 10"'^; collisions with background gas and cell walls can be largely avoided; Doppler shifts are directly reduced by trapping and cooling; and finally, nearly unit detection efficiency of transitions to metastable states is possible so that the signal-to-noise

ratio need be limited only by the statistical fluctuations

in the number of ions that make the transition.^ Details of ion traps and laser cooling have been published

elsewhere.'"^ A particularly attractive candidate for an optical-frequency standard is the 5d/'°6s^5'i/2-

5d^6s^^Ds/2ii&^ transition driven either by two photons with a wavelength near 563 nm'* or by one photon at half this wavelength.^ The lifetime of the ^Ds/2

state, which decays by the emission of electric quadrupole radiation at 281.5 nm, is calculated to be of order

0.1 s.^ This gives a potential optical-line Q of about

7x 10'"*. In this Letter we describe the first results of our investigation of the two-photon transition in i98j^g+ (^hich is free of hyperfine structure) stored in a miniature rf trap. Our trap is similar to the small radio-frequency traps

used in the ion-cooling experiments that were con- ducted at Heidelberg University on Ba"*",^ and at the

University of Washington on Mg"*" and Ba"*".* A cross

section of the trap electrodes is shown in Fig. 1. It is interesting to note that, although the inner surfaces of our trap electrodes were machined with simple conical

cuts, the trap dimensions were chosen to make the

fourth- and sixth-order anharmonic contributions to the potential vanish.^ The rf drive frequency was 21 MHz with a voltage amplitude Kq^ 1 kV. The partial pressure of the background gas, with the exception of

deliberately added mercury and helium, was ^ 10"^

Pa (133 Pa = 1 Torr). After loading of 50-200 mercury ions the mercury vapor was frozen out in a liquid-nitrogen cold trap and the vacuum vessel was

backfilled with He to the order of 10~^-10"^ Pa. This was sufficient to cool the trapped Hg"*" coUisionally to near room temperature as verified by the Doppler

width of the ^Si/2-^P\/2 resonance line near 194 nm.

Optical double resonance'^" was used to detect the

two-photon transition. About 5 /i,W of narrow-band cw sum-frequency-generated radiation near 194 nm'^

was tuned to the 6s ^Si/2 to 6p ^P\/2 first-resonance

transition and was directed diagonally through the trap

(between the ring electrode and the end caps). The

fluorescence light scattered by the ions was detected at right angles to the 194-nm beam with an overall detection efficiency of about 10"'*. Our signal level depended on the ion number and temperature and,

typically, was (2-10) x 10^ counts/s. The signal-tobackground ratio was better than 10/1. When the ions

Later b«ain path

FIG. 1. Schematic showing a cross-section view of the

trap electrodes. The electrodes are figures of revolution about the z axis and are made from molybdenum.

Work of the U. S. Government

Not subject to U. S. copyright TN-1 1567
Volume 55, Number 15 PHYSICAL REVIEW LETTERS 7 October 1985

were driven by the radiation from a 563-nm cw ring

dye laser out of the '^S^j ground state into the ^05/2

metastable state, there was a decrease in the 194-nm

fluorescence corresponding to the number of ions in the D state. The dye-laser beam was also directed diagonally

through the trap; the axes of the dye-laser beam, the

194-nm beam, and the collection optics were mutually

perpendicular. A near-concentric standing-wave cavity was placed around the trap in order to enhance the

power of the 563-nm radiation and better to ensure

a nearly equal intensity of the counterpropagating

beams. The cavity was positioned so that its waist

( ^0 = 25-30 /u.m) was located near the center of the

cloud of trapped ions. The power-buildup factor was

approximately 50, giving nearly 5 W of circulating

power for typical input power levels of 100 mW. The

frequency of the laser was offset locked and precisely

scanned with respect to a second dye laser locked to a hyperfine component in the Doppler-free, saturatedabsorption spectrum of '^^l2. The linewidth of the

swept laser was determined to be of the order of 300

kHz by spectrum analysis of the beat note between the

two lasers. A typical resonance curve and simplified energyievel diagram is shown in Fig. 2. The full scan width is 4 MHz. The electric field vector of the 563-nm laser radiation is nearly parallel to a small applied magnetic

field of approximately 11.6x10"'* T (11.6 G) which

differentially Zeeman splits the ground and excited

states. The selection rule for the two-photon transi- tion for this polarization is Awy = 0, and, thus, only

two components are observed, separated by approximately 13 MHz (approximately 6.5 MHz at the dyelaser frequency). In Fig. 2, we scan over only one of

these Zeeman components (my = — -7 — wy = — 7

)

but see sideband structure. This structure is due to amplitude modulation of the 563-nm laser intensity

due to the secular motion of the ions in the rf trap. Any frequency modulation of the laser caused by the

motion of the ions'^ is strongly suppressed if the cavity

is well aligned so that the k vectors of the counterrunning beams are antiparallel. By a change in the well depth of the trap the harmonic frequency of the secular motion is changed, thereby shifting the sideband

components of the two-photon signal. To our knowledge, this is the first observation of secular- motion sidebands at optical frequencies. In principle,

the ion temperature could be derived from the relative amplitude of the sidebands.'^ The depth of the central feature in Fig. 2 is nearly 25% of full scale, implying

that we have nearly saturated the two-photon transition. For the data of Fig. 2, the 194-nm laser irradiates the ions continuously, and, since the Zeeman splitting

is unresolved for the first resonance transition, it mixes the ground state, maintaining equal populations

in the two electron spin states. The linewidth of the

two-photon resonance is about 420 kHz, and is determined in nearly equal parts by the 563-nm laser linewidth of about 320 kHz and by the nearly 270-kHz

excitation rate of the ^S^/2 state by the 194-nm radiation. When the 194-nm radiation is chopped, the

two-photon linewidth drops to approximately 320 kHz,

consistent with the dye-laser linewidth. The pressure

broadening of the resonance in Fig. 2 was estimated to be less than 50 kHz by observation of the resonance at

5 times higher He pressure. We have experimentally measured the radiative life- time of the 5d%s^^Ds/2 state to be 0.090(15) s (one

standard deviation) in agreement with the calculated

lifetime of 0.105 s.^ Again, in this measurement the

ground-state population was monitored by measure-

\-

3300

3100

O 2000 -

%-^

194 nm

_li -

554 555 556 557 563 nm Laser frequency offset (MHz) FIG. 2. Two-photon ^Si/2 -'^Ds/2 transition in '"^"Hg"^. Amplitude modulation by the harmonic secular motion of the ions is visible in this scan. The frequency scan is 4 MHz at the fundamental laser frequency (A =563 nm). The depth of the central component is about 25% of full scale. The integration time is 2 s/point. In the inset is a simplified energy-level diagram of "^Hg"^ , depicting the levels of interest.

568

TN-2
Volume 55, Number 15 PHYSICAL REVIEW LETTERS 7 October 1985

merit of the laser-induced fluorescence of the 194-nm

transition. The radiation from the dye laser near 563

nm was tuned to resonance with the two-photon transition and chopped on and off. During the time that the laser radiation was on, it drove (10-20)% of the

ion population into the D state. The time constant for the atomic system to relax during the radiation-off

period could be determined from the exponential re- turn of the 194-nm fluorescence to steady state. An

example is shown in Fig. 3. The relaxation rate was measured over a range of He pressures differing by a factor of 4. The reported radiative lifetime is the

result of an extrapolation to zero pressure of a linear

least-squares fit to the data. The pressure-induced decay rate was determined poorly, but amounted to only

about 25% of the radiative decay rate at the highest

pressure (about 6x 10~^ Pa).

We have also measured the absolute wave number

of the ^Si/2 -^D^u two-photon transition by measuring

the frequency difference between the two-photon res- onance and the / hyperfine component of the nearby

/?(33) hne of the 21-1 band in '^^Ij (line No. 1220 in the iodine atlas). '"^ At zero magnetic field, the two- photon transition in '^^Hg"^ lies 551(2) MHz to the

red of this component. To a fairly good approximation,

the center of gravity for the ^(33) line is situated at a point \ the distance between the t and b hyperfine

components.'^ From this approximation, we find that the two-photon signal is 998(5) MHz to the red of the center of gravity of this iodine line. Thus, the meas8000

O 7250

7000

0.3 0.4 0.5 Time (s) FIG. 3. Measurement of the radiative lifetime of the 5d^6s^^Ds/2 state in Hg"^. The dots are the experimental

points indicating the return of the atomic system to steady

state after chopping off of the radiation from the 563-nm

dye laser that drives the ions from the ground state to the metastable D state by two-photon absorption. The solid line

is a least-squares fit of an exponential to the experimental

data. The vertical axis is the counts per second obtained by

measurement of the laser-induced fluorescence from the continuously driven ^P\n-^Si/2 transition. The horizontal

axis shows time after the turning off of the 563-nm laser ra- diation.

ured wave number is 17 757.1522(20) cm"' as determined relative to the Gerstenkorn and Luc value of

17 757.1855(20) cm"' for the center of gravity of the

i?(33) 21-1 line in iodine. '"* '"' Tellinghuisen finds a consistent discrepancy of the order of 0.002-0.003

cm"' between his predictions for the line center of

gravity and those of Gerstenkorn and Luc for lines with odd J values in this spectral region (there is good

agreement for lines with even J values).'^ Tellinghuisen calculates a value for the line center of

gravity of 17 757.182(2) cm"'. Using this value, we

find the two-photon transition to be at 17 757.149(2)

cm"'. These values are in good agreement with the

direct wavelength measurement of the electric quadrupole emission of the ^D^/2-^^\/2 transition obtained

from a mild electrodeless discharge in a ''*Hg cell backfilled with about 700 Pa of neon and from a standard '^^Hg cell backfilled with 33 Pa of argon. '^ The

wavelength measured in standard air is 2 814.9389(5)

A, which corresponds to a vacuum wave number of

17 757.146(3) cm"' for the two-photon transition. In

this Letter, we report the wave number of the two- photon transition to be 17 757.152(3) cm"'. The

quoted uncertainty allows for the discrepancy in the

determinations of the iodine-line center of gravity. In the near future, we anticipate narrowing the 563-

nm laser linewidth to the order of a few kilohertz and

studying various systematic effects including pressure

broadening and shifts, power broadening, and light

shifts. Ultimately, we would like to narrow the laser linewidth to a value near that imposed by the natural

lifetime of the D state, and to drive this two-photon,

or single-photon, electric quadrupole transition on a single, laser-cooled ion. Single-photon excitation of

the ^^5/2 state is particularly attractive because light shifts are eliminated. Light shifts are expected to contribute the largest uncertainty to the resonance fre- quency for two-photon excitation.^

We gratefully acknowledge the support of the U. S. Air Force Office of Scientific Research and the U. S. Office of Naval Research. We wish to thank R. Blatt (University of Hamburg) for technical assistance in the construction of the rf trap and H. Layer (National

Bureau of Standards) for supplying the '^^Hg. We take

particular pleasure in acknowledging the help of

W. Martin, J. Reader, and C. Sansonetti (National

Bureau of Standards) who provided the correct wavelength for the ^Ds/2 -^•S'1/2 transition. We also wish to thank D. Huestis (SRI) and our colleague

R. Drullinger for many useful conversations and in- sights. One of us (G. L.) is a Heisenberg Fellow of

the Deutsche Forschungsgemeinschaft.

(a)present address: Allied Bendix Aerospace Corporation,

TN-3 1569
Volume 55, Number 15 PHYSICAL REVIEW LETTERS 7 October 1985

Columbia, Md.

"'^Present address: Springer-Verlag, Heidelberg, West

Germany.

'"^^Present address: Max-Planck-Institute fiir Quantenoptik, Garching, West Germany.

'H. G. Dehmelt, Adv. At. Mol. Phys. 3, 53 (1967), and

5, 109 (1969), and IEEE Trans. Instrum. Meas. 31, 83 (1982). ^D. J. Wineland, W. M. Itano, and R. S. Van Dyck, Jr., Adv. At. Mol. Phys. 19, 135 (1983).

3d. J. Wineland, W. M. Itano, J. C. Bergquist, J. J. Bollinger, and J. D. Prestage, in Atomic Physics 9, edited by R. S. Van Dyck, Jr. and E. N. Fortson (World Scientific, Singapore, 1984), p. 3. "P. L. Bender, J. L. Hall, R. H. Garstang, F. M. Pichanick,

W. W. Smith, R. L. Barger, and J. B. West, Bull. Am. Phys.

Soc. 21,599 (1976). 5D. J. Wineland, J. C. Bergquist, R. E. DruUinger,

H. Hemmati, W. M. Itano, and F. L. Walls, J. Phys. (Paris),

Colloq. 42, C8-307 (1981); D. J. Wineland, W. M. Itano,

J. C. Bergquist, and F. L. Walls, in Proceedings of the Thirty-Fifth Annual Symposium on Frequency Control, 1981 (unpublished), p. 602 (copies available from Electronic

Industries Assoc, Washington, D.C.).

6R. H. Garstang, J. Res. Nat. Bur. Stand. Sect. A 68, 61 (1964).

''W. Neuhauser, M. Hohenstatt, P. Toschek, and H. G.

Dehmelt, Phys. Rev. Lett. 41, 233 (1978).

^W. Nagourney, G. Janik, and H. G. Dehmelt, Proc. Nat. Acad. Sci. U. S. A. 80, 643 (1983); G. Janik, W. Nagourney,

and H. Dehmelt, J. Opt. Soc. Am. 2, 1251 (1985).

'E. C. Beaty, to be published.

'OD. J. Wineland, J. C. Bergquist, W. M. Itano, and R. E. DruUinger, Opt. Lett. 5, 245 (1980).

"H. G. Dehmelt, in Advances in Laser Spectroscopy, edited by F. T. Arecchi, F. Strumia, and H. Walther (Plenum, New

York, 1983), p. 153. '^H. Hemmati, J. C. Bergquist, and W. M. Itano, Opt. Lett. 8,73 (1983).

•^H. A. Schuessler, Appl. Phys. Lett. 18, 117 (1971); F. G. Major and J. L. Duchene, J. Phys. (Paris) 36, 953 (1975); H. S. Lakkaraju and H. A. Schuessler, J. Appl. Phys. 53,

3967 (1982); M. Jardino, F. Plumelle, and M. Desaintfuscien, in Laser Spectroscopy VI , edited by H. P. Weber and

W. Luthy (Springer-Verlag, Berlin, 1983), p. 173; L. S. Cutler, R. P. Giffard, and M. D. McGuire, Appl. Phys. 36,

137 (1985).

•''S. Gerstenkorn and P. Luc, Atlas du Spectre d'Absorption de la Molecule de I'lode Entre 14 800-20000 cm'^ (Editions

duC. N. R.S., Paris, 1978).

•^S. Gerstenkorn and P. Luc, Rev. Phys. Appl. 14, 791

(1979).

'^J. Tellinghuisen, private communication.

'^J. Reader and C. J. Sansonetti, to be published.

570

TN-4
Reprinted from Journal of the Optical Society of America B, Vol. 2, page 1392, September, 1985 Copyright © 1985 by the Optical Society of America and reprinted by permission of the copyright owner. Measurements of the gj factors of the 65 ^Si/2 and 6p ^Pi/2

states in i98jjg+

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

Time and Frequency Division, National Bureau of Standards, BouJder, Colorado &0303

Received February 22, 1985; accepted April 19, 1985 Measurements of ^^^Hg^ gj factors by two methods are reported. The first method was based on optical wavelength measurements of the Zeeman components of the 6s ^i\/2 (ground state) to 6p ^Pi/2 transition at 194 nm. The lines were observed by the absorption of tunable 194-nm radiation by Hg* ions created in a rf discharge. The

results were gj (6s ^Si/2) = 2.0036(20) and gj(6p 2P1/2) = 0.6652(20). The second method was based on microwaveoptical double resonance of ions confined in a Penning trap. They were optically pumped by the 194-nm source, which was tuned to a particular Zeeman component. An increase in the resonance-fluorescence intensity was ob- served when the microwave frequency was tuned to the ground-state Zeeman resonance. The result is g,y (6s ^Si/2) = 2.003 174 5(74).

INTRODUCTION

Few measurements of gj factors in Hg+ have been made.

Measurements made with optical precision (typically a few

parts in 10'^) are useful for determining the purity of the

configurations and coupling schemes. Van Kleef and Fred^

measured thegj factors of the 7s "^81/2, 6p ^P3/2, and bd%s'^

2D5/2 states by optical spectroscopy. The values agree with

those calculated for LS coupling to within the accuracy of the measurements. No measurements of the 6p '^Pi/2 state have

been published previously. The present results for this state agree with the LS coupling value. Measurements made with higher precision, usually by microwave-resonance techniques, are sensitive to relativistic and

many-body effects and test calculational methods. Schuessler and Hoverson^ observed a ground-state magnetic resonance

in i^^Hg+ by spin exchange with optically pumped Rb and

derived a value for the ground-state gj factor. The results

of the present study are in considerable disagreement with this value but are in agreement with a recent calculation by Dzuba

etal.^

EXPERIMENTAL METHOD

Two different methods of measuring gj factors were used in this study. The first relied on optical wavelength measure- ments of the Zeeman components of an absorption line. The

ions in this case were created in a rf discharge. The second

method used microwave-optical double resonance. This

method had much higher resolution than the first but could

be applied only to the ground state. The ions in this case were

contained in a Penning ion trap.^

A narrow-band, tunable, cw 194-nm radiation source was

used in both methods. This source was described previously.^

The 194-nm radiation was generated by sum-frequency

mixing 257- and 792-nm radiation in a potassium pentaborate

crystal. The 257-nm radiation was generated by frequency

doubling the output of a single-mode cw 514.5-nm Ar"*" laser

in an ammonium dihydrogen phosphate crystal. The 792-nm

radiation was generated by a single-mode cw dye laser, which

provided the tunability. Ring cavities were used to increase the circulating powers at the input frequencies at both the frequency-doubling and frequency-mixing stages. The

linewidth of the 194-nm radiation was about 2 MHz, and the

output power was a few microwatts. The Ar"*" laser was fre- quency stabilized to a saturated-absorption feature of known

wavelength in the ^^'l2 spectrum. The wavelength of the dye

laser was measured with an interferometric wavemeter of the lambdameter type.^ For the optical gj factor measurements, the Hg""" ions were

created in a cell with fused-silica windows containing Ne at a pressure of about 800 Pa (6 Torr) and a droplet of ^^^Hg. A

discharge was excited by applying a few watts of power at a frequency of approximately 150 MHz to a coil surrounding the

cell. The cell was placed in the gap of an electromagnet. The

magnetic field of approximately 1.3 T was measured with a nuclear magnetic resonance (NMR) probe.

The intensity (after attenuation) of the 194-nm radiation

transmitted through the discharge was measured as a function

of the dye-laser frequency with a photomultipHer tube (PMT).

The detected signal was normalized to the incident power in order to reduce noise and to eliminate sloping baselines owing

to variations of the power as a function of frequency or time. The results of a typical laser scan are shown in Fig. 1. The

width of the scan, approximately 6.5 GHz, was calibrated with

a Fabry-Perot interferometer of 250-MHz free-spectral range. The wavelength of the dye laser was measured at a point near

the resonance center with the lambdameter. For the microwave-optical double-resonance measurements, the Hg+ ions were confined in a Penning trap. The

trap apparatus and the optics for introducing the light and

detecting the fluorescence were similar to those described

previously for use with Mg+ ions'' except that the trap elec- trodes were made from Mo rather than Cu. The characteristic inner dimensions were ro = 0.417 cm and 20 = 0.254 cm. The magnetic field was approximately 1.3 T, and the potential

between the ring electrode and the end-cap electrodes was

approximately 1 V for trapping Hg"*" and about 6 V for trapping electrons. Ions were created inside the trap by

ionization of neutral '^^Hg, introduced through a leak valve,

0740-3224/85/091392-03$02.00 1985 Optical Society of America TN-5
Itano et al. Vol. 2, No. 9/September 1985/J. Opt. Soc. Am. B 1393

'----N^ A^Mo^ni ^^^ ^

\ / >. •. / • * m

c

^

% s

T3 \ <D . F

CO / \ /

c

CO

2 GHz '

Laser frequency

Fig. 1. Graph of the intensity of the 194-nm radiation transmitted through the rf discharge as a function of laser frequency, showing the absorption that is due to the Mj = —1/2 to Mj = +1/2 component of the i98Hg+ 6s 2Si/2 to 6p ^Pu2 transition.

36 kHz

1—

1

0)

Z3CO

<Da

F

a>ro a

.' . m • .". . ...•'

..;. v v- V •...:.•

'

a

35 963.364 MHz

Microwave frequency

Fig. 2. Graph of the temperature of electrons in the Penning trap

as a function of the applied microwave frequency, showing the cy- clotron resonance.

a

a

a

m / s //.

. . y. .

280 kHz

£

c

0)o

c

9

U

0)a1

a

.'. X ' "\

/ \ a \

a \ m \ a

a " a —

1 a

36 020.334 MHz

Microwave frequency

Fig. 3. Graph of the 194-nm resonance-fluorescence intensity of

i98{-jg+ j(jris in the Penning trap as a function of the applied micro- wave frequency, showing the ground-state Zeeman resonance. The

curve is a least-squares fit to a Gaussian.

with an electron beam directed along the trap axis. The

storage time was typically a few minutes, limited by neutral '^^Hg pressure. The 194-nm radiation was introduced

through a hole in the ring electrode along a direction per- pendicular to the trap axis. Photons scattered by the ions back through the hole in the ring electrode were counted by

a PMT. The net detection efficiency (number of photoelectrons counted per photon emitted by the ions) was of the order

of 0.5 X lO""*. Microwave radiation was transmitted through

a window in the vacuum envelope using a pair of horns (one

as a transmitter outside the vacuum apparatus and one as a

receiver inside the vacuum apparatus) and introduced into the trap through the gap between the ring and one of the end

caps. The number of ions trapped was estimated by driving the axial motion of the ion cloud in the wings of the resonance with a rf electric field and measuring the induced currents. The maximum number was about 8 X lO"*. The magnetic field was stabilized by means of a NMR probe. The magnetic field inside the trap was calibrated by detecting the cyclotron res- onance of a cloud of electrons, using the bolometric method.^

In this method the translational motions of the electrons are detected by the currents induced in an external circuit coupled

to the trap electrodes. The microwave source for the elec- tron-cyclotron excitation was a frequency tripler driven by

a frequency synthesizer. A typical cyclotron-resonance curve

is shown in Fig. 2. To observe the Hg"*" ground-state Zeeman resonance, the 194-nm source was tuned to the 6s '^Sii2(Mj = —1/2) to 6p

2Pi/2(Mj = +1/2) transition. The ions were pumped to the Mj = +1/2 ground-state sublevel and ceased scattering

photons, since they were no longer in resonance. When the frequency of the applied microwave radiation matched the ground-state Zeeman resonance, the Mj = —1/2 sublevel was

repopulated, and an increase in the resonance fluorescence was detected. Here, the microwave source was a klystron,

phase-locked to a harmonic of a frequency synthesizer. Up

to 100 mW of power was available at the end of the horn. A

typical resonance curve is shown in Fig. 3. The linewidth was

larger than for the electron-cyclotron resonance. This was

probably due to magnetic-field inhomogeneities and a larger

spatial extent of the Hg+-ion cloud compared with that of the

electron cloud.

RESULTS

Two or three optical measurements were made of each of the

four Zeeman components. The centers of the absorption

curves were determined by least-squares fitting to a function

of the form, transmitted intensity = Ci + C2 exp(— C3

expj- [(u — C4)/C5]2|), where v was the dye-laser frequency and

C1-C5 were constants. This line shape included the effect of saturation on the Gaussian Doppler profile, which tended to broaden the lines. All the line shapes were fitted with a Doppler-width parameter (C5) corresponding to a temperature of approximately 400 K. The temperature of the rf dis- charge was not measured directly. The reproducibility of the

line centers was ±20 MHz. The uncertainties in absolute

frequencies of up to several hundred megahertz, which were due mostly to the uncertainty of the frequency of the reference

laser of the lambdameter, were unimportant since only fre- quency differences were needed to determine the gj factors. Similarly, perturbations that shifted all magnetic sublevels

TN-6
1394 J. Opt. Soc. Am. B/Vol. 2, No. 9/September 1985 Itano et al.

of a state equally, such as pressure shifts, could be neglected.

From the four frequency measurements, there were two in- dependent ways to obtain the S-state Zeeman splitting and

two independent ways to obtain the P-state splitting. The

average values were 36 033(30) MHz for the S state and

11 964(30) for the P state. The measured-proton NMR fre- quency in a mineral-oil sample was 54.708 MHz, corre- sponding to a magnetic field of 1.284 95 T. The resulting values for the gj factor were 2.0036(20) for the S state and

0.6652(20) for the P state. The gj factor of the ground S state was obtained from the Hg+ microwave-resonance frequency u(Hg+) and the unper- turbed electron cyclotron frequency Vc by the equation gj =

2u(Hg+)/uc. The measured cyclotron frequency Vc' differed from Uf because of the electric fields of the Penning trap.*

The correction was calculated from the axial resonance fre- quency Uz, measured by the bolometric method, according to

Vc ^ v/ + Vz'^/2v/. At a particular value of the magnetic

field, u(Hg+) was measured to be 36 020.477(48) MHz, whereas

v/ was measured to be 35 963.345(24) MHz. The line centers were determined by least-squares fitting to Gaussians. The

value of Vz was 59.25 MHz. The result was gj{6s ^Si/2) =

2.003 174 5(74). The statistical error was only 1.5 parts in 10^ (ppm), but a possible systematic error may have arisen be- cause the Hg+ ions and the electrons may not have sampled

the same average magnetic field. We estimated this error to be no more than the half-width of the Hg"*" resonance (3.4 ppm).

DISCUSSION

The measured gj factor of the 6p ^Pi/2 state is equal to the

value calculated for LS coupling (0.665 89) within the ex- perimental error. This is the first reported measurement of

this quantity.

The values of the ground-state gj factor measured by the two methods are in agreement. They are both in serious disagreement with the measurement of Schuessler and Hoverson.^ The present measurement is, however, in good

agreement with a recent calculation based on the relativistic Hartree-Fock equations and many-body perturbation theory.-^ It is probably more instructive to express these results

in terms of the small deviation, Ag = gj{6s ^Si/2) — ge, of the gj factor from the free-electron g factor {gg = 2.002 319 30)

than in terms of the gj factor itself. The present experimental

result is Ag = 0.8552(74) X 10"^^ the experimental result of Schuessler and Hoverson^ is Ag = 14.7(1.0) X 10"'^, and the

theoretical result is Ag = 1.02 X 10"^. An empirically cor- rected result of 0.85 X 10"^ is also given in Ref. 3. The cor- rection is based on the difference between the experimental

and theoretical values for isoelectronic Au.

In light atoms, deviations of the gj factor of a ^Si/2 state are due to the relativistic and diamagnetic interactions given by

Abragam and Van Vleck.^ For the alkali atoms up to K, cal- culations based on these interactions are in good agreement

with experiment. ^'^ For these atoms Ag is negative. However, for the heavier alkali atoms and for Au and Hg"*", which

also have •^Si/2 ground states, A^ is positive. This is primarily

due to a perturbation that is of the second order in the spinorbit interaction and of the first order in the residual Coulomb

interaction. Calculations that take this effect into account

agree well with experiment.'^'^^'^^

ACKNOWLEDGMENTS

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

Office of Scientific Research and the U.S. Office of Naval

Research. We wish to thank Howard Layer (National Bureau

of Standards) for supplying the ^^^Hg and the authors of Ref.

3 for making their results available to us before publication.

REFERENCES

1. T. A. M. Van Kleef and M. Fred, "Zeeman effect measurements

in neutral and singly ionized mercury," Physica (Utrecht) 29, 389-404(1963).

2. H. A. Schuessler and S. J. Hoverson, "Measurement of the hyperfine structure Zeeman splitting of the Hg"*" ion," Phys. Lett. 43A, 25-26 (1973); S. J. Hoverson, "Measurement of the hyperfine

structure Zeeman splitting of the ground state of the mercury

ion," Ph.D. dissertation (Texas A&M University, College Station, Tex., 1974).

3. V. A. Dzuba, V. V. Flambaum, P. G. Silvestrov, and 0. P. Sushkov,

"Anomalies of ^-factor in heavy atoms," Phys. Scr. 31, 275-280

(1985).

4. H. G. Dehmelt, "Radio frequency spectroscopy of stored ions. 1; Storage," Adv. At. Mol. Phys. 3, 53-72 (1967).

5. J. C. Bergquist, H. Hemmati, and W. M. Itano, "High power

second harmonic generation of 257 nm radiation in an external

ring cavity," Opt. Commun. 43, 437-442 (1982); H. Hemmati, J. C. Bergquist, and W. M. Itano, "Generation of continuous-wave

194-nm radiation by sum-frequency mixing in an external ring

cavity," Opt. Lett. 8, 73-75 (1983); D. J. Wineland, W. M. Itano,

J. C. Bergquist, J. J. Bollinger, and J. D. Prestage, "Spectroscopoy

of stored atomic ions," in Atomic Physics 9, R. S. Van Dyck, Jr., and E. N. Fortson, eds. (World Scientific, Singapore, 1984), pp.

8-9.

6. J. L. Hall and S. A. Lee, "Interferometric real-time display of cw

dye laser wavelength with sub-Doppler accuracy," Appl. Phys.

Lett. 29,367-369(1976).

7. D. J. Wineland, R. E. DruUinger, and F. L. Walls, "Radiationpressure cooling of bound resonant absorbers," Phys. Rev. Lett.

40, 1639-1642 (1978).

8. H. G. Dehmelt and F. L. Walls, "Bolometric technique for the rf spectroscopy of stored ions," Phys. Rev. Lett. 21, 127-131

(1968).

9. A. Abragam and J. H. Van Vleck, "Theory of the microwave Zeeman effect in atomic oxygen," Phys. Rev. 92, 1448-1455

(1953).

10. L. Veseth, "Many-body calculations of atomic properties: I. gj-factors," J. Phys. B 16, 2891-2912 (1983).

11. 0. P. Sushkov, V. V. Flambaum, and I. B. Khriplovich, "The

nature of the strongly forbidden Ml transitions and the g-factor anomalies in heavy atoms," Zh. Eksp. Teor. Fiz. 75, 75-81 (1978)

[Sov. Phys. JETP 48, 37-40 (1978)].

12. G. V. Anikin and I. L. Zhogin, "Calculation of corrections to g

factors of alkali-metal atoms," Opt. Spectrosk. 51, 549-550 (1981) [Opt. Spectrosc. (USSR) 51, 303-304 (1981)].

TN-7
Reprinted from Optics Letters, Vol. 12, page 389, June 1987. Copyright © 1987 by the Optical Society of America and reprinted by permission of the copyright owner.

Absorption spectroscopy at the limit: detection of a single atom

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

Time and Frequency Division, National Bureau of Standards, Boulder, CoJorado 80303

Received January 21, 1987; accepted February 26, 1987 We investigate the sensitivity limit of absorption spectroscopy. An experiment is described in which the decrease

in transmitted light intensity that is due to absorption by a single, electromagnetically confined atomic ion is observed.

Absorption spectroscopy dates back to the early 19th

century when Fraunhofer observed characteristic

Hnes in the solar spectrum (Fraunhofer lines) and to the mid-19th century when Kirchhoff and Bunsen

identified specific absorption lines with certain ele- ments.^ Since that time, absorption spectroscopy has

played an important role in the development of atomic

and molecular theory and continues to be an important anal3^ical tool in several fields of science. As

with most experimental measurements, signal-tonoise ratio and resolution are of primary concern, and

various advances have led to increased sensitivity.

The laser led to a renewed interest in spectroscopic

measurements. Not only can source brightness and

resolution be greatly increased but the high spectral

intensities have made it possible to observe a host of

nonlinear effects that have greatly improved sensitivity and resolution.2 It is always interesting to ask

what the ultimate limits to a technique are. In this Letter we investigate the sensitivity limit of absorption spectroscopy when absorption is detected as a decrease in the radiation transmitted through the absorber.

There are many different kinds of sensitivity limits

that could be established given the constraints of a

particular experimental configuration. Here we in- vestigate one such limit by asking the question, "What

is the smallest number of atoms or molecules that one

can detect by using absorption spectroscopy?" The

fundamental limit is one atom or molecule, and in this Letter we describe a particular realization of this limit. Previously, the lower limit on number sensitivity was

found to be about 65 atoms^ when a four-wave mixing

technique^ was used.

We can estimate the signal-to-noise ratio in an absorption experiment in the following way. We assume

that we have a collimated radiation source of power P

incident upon a collection of N atoms or molecules.

For simplicity, we assume that the intensity of the

source at the position of the atoms is uniform with

cross section A. We write 'the fraction of power absorbed by the atoms as rjP and assume that ?? « 1. We

further assume that the detector converts the incident

photons into electric current with quantum efficiency

/3. Therefore the total detected current is given by Id = P(l — r])e0/{hp), where e is the electron charge, h is

Planck's constant, and v is the frequency of the inci- dent photons. We define the signal Is as the change in

detector current due to absorption. Therefore we

have

/, = vPemhv).

The rms noise current can be written as

in = Hd + h + W)2\l/2

(1)

(2) In Eq. (2), we have separated the statistically independent contributions from the detector noise iu (e.g., Johnson noise), radiation-source technical noise ir

(e.g., intensity fluctuations in the radiation from a

laser source), and shot noise in the detection process

ish- The shot-noise contribution is simply written as

I'sh^ = 2e/£)Az^, where Lv is the detection bandwidth.

The contributions from in and ir will depend on the

specific experimental configuration, but we desire in, It « ish for maximum signal-to-noise ratio.

If absorption is due to a particular atomic transition

with resonance frequency vq, then, below saturation,

the absorption coefficient tj is given by the expression

7]P = NaP/A, where a = oqK. oq is the resonant cross section for the particular transition observed, oq is of

the order of \'^/{2-k), where X is the wavelength at resonance.^ K is a factor {K < 1) that expresses the

reduction of the cross section from the resonant value

by (1) radiative decay into atomic states other than the

lower state of the transition under investigation, (2)

Doppler broadening, (3) coUisional broadening, (4) detuning of the source away from vq, or (5) finite bandwidth of the radiation source. We can write the signal-to-noise ratio, S/N, as

S/N = 7A =

NcoKelSP/iAhvo)

[ifl2 + i^2 + 2e^^PAp/{hvo)] 1/2 (2)

Our experimental configuration is shown in Fig. 1. Since we are interested in the case when N = 1, a

single, laser-cooled ^^^Hg''" ion was confined in a miniature rf trap^ with inside endcap-to-endcap spacing 2zo = 0.064 cm and inner ring diameter 2ro = 0.091 cm. With a ring-to-endcap voltage Vt — (10 + 400

cos ^t)V, where 9,/2-k = 21.07 MHz, the ion was confined to a space with linear dimensions <0.25 ixm.^ A

narrow-band (width < 5 MHz) radiation source at 194

0146-9592/87/060389-03$2.00/0 © 1987, Optical Society of America TN-8
390 OPTICS LETTERS / Vol. 12, No. 6 / June 1987

ENDCAP Hg+ ION /VACUUM

INCIDENT ,v'

TMTM^'°°^

RADIATION. '^^Ai\\ 2r

RING'

+90°-, n

-90°

-H2s^ UP/DOWN

COUNTER

DISPLAY

Fig. 1. Schematic diagram of the apparatus. Absorption

from a single ^^^Hg+ ion is detected as a change in current in the vacuum photodiode. This absorption is modulated at frequency fi by applying a voltage VTM to one endcap elec- trode of the trap. To avoid a false signal from stray radia- tion picked up by the detector, the sign of V^ is alternated at frequency w. To avoid drifts at the output of mixer M2, a

third modulation at 0.5 Hz of the phase of w and the sign of the signal at M2 is converted to dc by the up/down counter. Vm, (Pa, and </)j^2 are adjusted to maximize the display output.

nm (P !^ 3 mW) was tuned near the 6s '^Sy2 ~^ 6p '^Pi/2 first resonance transition in ^^^Hg"*" and focused

through the trap and onto the ion. This radiation source was derived from sum-frequency mixing a dye

laser (790 nm) and a frequency-doubled Ar"*" laser (257

nm)J The detector was a vacuum photodiode with a Cs-Te photocathode. We had fi ^ 0.055 (10% tube

efficiency; 55% transmission from trap to detector). In order to suppress ir from the 194-nm source, it is desirable to modulate the absorption at a high fre- quency.^ This was accomplished by applying a volt- age Vm to one trap endcap. This displaced the ion

from its equilibrium position in the rf trap and caused

its velocity, which has a component along the direction

of the 194-nm beam, to be modulated at the trap rf

Q. That is, the static electric force was compensated

for by the pseudopotential force that gave rise to ion

micromotion at frequency ft.^ Owing to the first- order Doppler shift, the resonance frequency of the ion was modulated at fi. Such frequency-modulation

schemes were first used in nuclear magnetic reso- nance^ '^ and microwave spectroscopy.^^ This scheme

of velocity modulation is essentially identical with

that developed by Saykally and co-workers. ^^ As

is indicated in Fig. 1, the modulated-absorption signal

current was detected by a tuned circuit whose equivalent parallel resistance, Rd ==: 300 kfi, was derived from

the measured resonance Q and parallel capacitance

( =^ 3 pF owing to tube and stray capacitance) . Unfortunately, the detector also picked up radiation from

the unshielded trap electrodes, which gave a false signal that was unstable. To avoid this problem, we

switched the phase of the absorption signal by 180°

by switching the sign of the voltage Vm applied to the

endcap at the frequency a)/27r = 2 kHz. Therefore,

after the first mixer (Ml), the absorption signal

appeared at frequency w, which we then converted to a dc voltage in M2. To avoid drifts at the output of

M2, we switched the phase, <p^i, of Vm by 180°

every second and used a voltage-to-frequency converter (V/F) followed by an up/down counter to display

the signal. The values of Vm and the phases (/)n and 0c^2 were adjusted to give maximum absorption

signal.

From Eq. (3), in order to maximize S/N, we want to make P as large as possible until saturation is reached.

When the incident radiation exceeds saturation intensity, the numerator of Eq. (3) reaches a constant value

given by aoKP/iAhvo) r^i 7/2, where 7"^ is the radiative

lifetime of the upper state. Increasing P further only

increases I'sh, thereby reducing S/N. If we assume that

the incident radiation is just at saturation intensity,

and if we neglect iVin Eq. (3), Eq. (3) becomes (for AT =

1)

S/N = (tt^)"

e/37/(2Aj'i/2)

(4) [4kBT/Ro + e^0yA/(aoK)f^

The factor {t^)~^ accounts for the modulation

scheme used but makes the simplifying assumption

that the atomic absorption changes linearly between

zero and its maximum value as the atomic-resonance

frequency is modulated through its full extent. We

have made the identification io"^ = 4kBTAv/RD, where

ks is Boltzmann's constant and T is the effective temperature of the input circuit. For the first resonance

transition in Hg"*", 7"^ = 2.3 nsec, and co = 6.0 X 10~^^

cm^. The ion was at the center of our focused Gaussian laser beam [wq ^ 5 nm). K ^1, which was the key

to this experiment. Therefore, we calculate that the

second term in the denominator of Eq. (4) is negligible.

This was verified by a direct measurement of i'shAd- We determined that the detector-noise temperature was T !^ 600 K (i>;3-dB noise figure) by measuring

photon shot noise on the detector and assuming that ir « Id at 21.07 MHz. Making the identification Aj^ =

(27)"^ where t is the integration time per point out of

the up/down counter, we calculate that S/N « 12.6 (77 « 2.5 X 10-5) for T = 50 sec from Eq. (4). This compares reasonably well with the data of Fig. 2, where we

measured S/N c^i 5.7. For the maximum signal, the

laser is tuned halfway down the resonance line and the

absorption of the single ion is modulated between a few percent and nearly 100% of the maximum absorption at frequency Q. The disagreement between theory and experiment may be due to two causes. First,

the simple theory assumes that the absorption signal occurs only at frequency fi and neglects signal content

at harmonics of fl. Second, we have neglected loss of

signal due to radiative decay from the ~Pi/2 state to the

metastable 2D3/2 level. Data similar to those of Ref. 6

indicate this may account for reduction in signal by a

factor of about 2 in the present experiment.

TN-9
June 1987 / Vol. 12, No. 6 / OPTICS LETTERS 391

2500

-2500

Single Ion Absorption and Fluorescence

-e -5000

& -7500-

g -10000

30000 ,^

20000

10000

-360 -180 180

Relative Laser Frequency (MHz)

Fig. 2. Absorption signal observed as the 194-nm source is tuned through vq. Lower trace shows the simultaneously

observed fluorescence scattering; the flat-topped appear- ance of this curve is due to the frequency modulation of the

ion resonance. Integration times per point are 50 and 10 sec

in the upper and lower traces, respectively.

preaches ao, the absorption of the input beam by the

ion would approach several percent. In this limit, the

detector could be a photomultiplier tube operated in

the photon-counting mode, io would be replaced by

tube dark counts, and S/N would approach the square

root of the number of detected counts. If we can

neglect io and ir, the choice of absorption or fluores- cence detection is straightforward. If the fraction e of

fluorescence photons striking the detector is greater

than 7], fluorescence detection gives the greater S/N.

The reverse is true for 77 > e. We have assumed that jS is the same for the absorption and fluorescence detectors. We gratefully acknowledge the support of the U.S.

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

of Scientific Research. We thank R. G. Hulet for assistance in several experiments that led to the ex- periment reported here. We also thank R. G. Hulet,

A. De Marchi, L. Hollberg, C. Wieman, C. Howard,

and P. Hammer for helpful suggestions on the manuscript.

We expect the signal to show a dispersion shape as the frequency of the 194-nm source is tuned through

the resonance frequency vq- Only half of the dispersion feature (Fig. 2) was observed because the ion

rapidly heated when the frequency of the 194-nm

source was tuned above the resonance frequency vq.^^ This caused a significant decrease in scattering owing

to increased Doppler broadening and decreased spatial overlap of the laser beam with the ion; i.e., K

became very small. If we simultaneously observed the

fluorescence scattering (lower trace in Fig. 2), the absorption signal first reached its maximum amplitude

(minimum of upper curve in Fig. 2) and then returned

to zero at the resonance frequency, as indicated by the

sudden drop in fluorescence in the lower trace. This

verified that the signal was mostly due to frequency

modulation rather than to amplitude modulation

(achieved by moving the ion in and out of the laser beam) since the maximum signal for amplitude modulation would be achieved at i^o- The frequency modulation also accounts for the flat-topped shape of the

fluorescence curve in Fig. 2. The optimum value of

Vm was independently verified by observing the intensity modulation of the fluorescence signal at 21.07

MHz by measuring time correlations of the photons

scattered by the single ion. S/N in the present experiment could be increased by

a factor of about 4 if io could be reduced so that the

noise was dominated by igh- When tgh dominates, Eq.

(4) tells us that we can improve S/N by decreasing the

laser spot size A. Stronger focusing in this experiment was prevented by the size of the input windows

to the trap. In order just to maintain saturation we

must reduce P accordingly. In the limit where A apReferences

1. A concise historical review is given by R. A. Sawyer,

Experimental Spectroscopy, 3rd ed. (Dover, New York,

1963), pp. 1-17.

2. See, for example, K. Shimoda, ed., High Resolution Laser Spectroscopy (Springer- Verlag, Berlin, 1976); R. A.

Keller, ed., Laser-based Ultrasensitive Spectroscopy

and Detection V, Proc. Soc. Photo-Opt. Instrum. Eng.

426 (1983); M. D. Levenson, Nonlinear Laser Spectroscopy (Academic, New York, 1982).

3. J. Bialas, R. Blatt, W. Neuhauser, and P. E. Toschek,

Opt. Commun. 59, 27 (1986).

4. S. Le Boiteux, D. Bloch, M. Ducloy, A. Pesnelle, S. Runge, and M. Perdrix, Opt. Commun. 52, 274 (1984).

5. See, for example, R. Loudon, The Quantum Theory of Light (Oxford U. Press, London, 1973), p. 287.

6. J. C. Bergquist, R. G. Hulet, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 57, 1699 (1986).

7. H. Hemmati, J. C. Bergquist, and W. M. Itano, Opt.

Lett. 8, 73 (1983).

8. J. J. Snyder, R. K. Raj, D. Bloch, and M. Ducloy, Opt.

Lett. 5, 163 (1980); L. Hollberg, Ma Long-Sheng, M.

Hohenstatt, and J. L. Hall, Proc. Soc. Photo-Opt. In- strum. Eng. 426, 91 (1983); E. A. Whittaker and G. C. Bjorklund, Proc. Soc. Photo-Opt. Instrum. Eng. 426, 81

(1983); M. Ducloy and J. J. Synder, Proc. Soc. PhotoOpt. Instrum. Eng. 426, 87 (1983). 9 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., Adv. At. Mol. Phys. 19, 135 (1983).

10. N. Bloembergen, E. M. Purcell, and R. V. Pound, Phys.

Rev. 73, 679 (1948).

11. R. H. Hughes and E. B. Wilson, Jr., Phys. Rev. 71, 562L

(1947).

12. C. S. Gudeman, M. H. Begemann, J. Pfaff, and R. J. Saykally, Phys. Rev. Lett. 50, 727 (1983).

13. R. E. Drullinger, D. J. Wineland, and J. C. Bergquist,

Appl. Phys. 22,365(1980).

TN-10
PHYSICAL REVIEW A VOLUME 36, NUMBER 5 SEPTEMBER 1, 1987

Laser-cooling limits and single-ion spectroscopy

D. J. Wineland, Wayne M. Itano, J. C. Bergquist, and Randall G. Hulet

Time and Frequency Division, National Bureau ofStandards, Boulder, Colorado 80803

(Received 9 January 1987)

The limitations to the achievement of low kinetic energies for laser cooling of single ions confined

in electromagnetic traps are discussed. Sideband cooling of an ion in an rf (Paul) trap is reexamined

including the effects of finite laser bandwidth and the energy of the rf micromotion. The micromotion is the oscillatory motion of the ion at the same frequency as the rf voltage applied to the trap

electrodes. Sideband cooling of ions in a Penning trap is examined for the first time. In both cases, cooling to the zero-point energy of the ion in the trap should be possible and a method for verifying

this condition is suggested. The implications for high-resolution, high-accuracy spectroscopy are in- vestigated. Under certain conditions, the uncertainty in the second-order Doppler shift may be

significantly less than 1 part in 10'*.

I. INTRODUCTION

Over the past few years, the number of experimental

and theoretical investigations of laser cooling has in- creased dramatically.'"'* Much of this initial work has

concentrated on characterizing the cooling process and

understanding the lower limits to cooling. This is partly

because the attainment of very low velocities will be important in applications such as high-resolution spectroscopy (reduction of Doppler shifts) and low-energy studies of

atom-atom and atom-surface interactions. Minimum ki- netic energies have often been expressed in terms of temperature. For free or weakly bound atomic particles, we

make the identification m { u,^ ) /2 = /c^ T, /2 where

m{vf) /I is the kinetic energy in the rth degree of free- dom, r, is the temperature for that degree of freedom,

and kg is Boltzmann's constant. For atomic particles

bound in a harmonic well, r^ defined by the above rela- tionship is therefore a measure of the average kinetic ener- gy in the x direction. When the above relationship between kinetic energy

and temperature holds, and when laser cooling is accomplished by driving a single photon transition, the minimum achievable temperature is given by the "Doppler" cooling limit (or "heavy particle" limit"*).

r^=^/2/cs (1)

In Eq. (1), 7~' is the radiative lifetime of the laser-cooling

transition, iTrfi is Planck's constant, and we have assumed

that the incident radiation is well below saturation intensity. For atoms bound in a harmonic well with characteristic "vibration" or oscillation frequency co,,, the Doppler

cooling limit applies when y »«„. The free-particle case

is for the hmit cj^^O, and thus Eq. (1) still applies. T^ is typically in the range of 1 mK for strongly allowed electric dipole transitions. For the case of stored ions, temperatures less than about 10 mK have been achieved.^ ~^ The uncertainties in the measurements have often been

consistent with the temperature being at the theoretical

limit given by Eq. (1). The lowest measured temperatures

36

have recently been achieved by Chu and co-workers using

neutral sodium atoms.^ They demonstrated a temperature (r =240160^ /iK) equal to the Doppler cooling limit using three intersecting and mutually perpendicular,

Standing-wave laser beams which were tuned to give max- imum cooling.

For trapped ions or atoms, cooling in the "sideband"

limit (when y •5C<aj„) is achieved by tuning the incident ra- diation to one of the lower motional sidebands of the atomic absorption. For example, in Fig. 1, the laser is as- sumed to be tuned to coq — co^, where coq is the transition frequency for the atom at rest. In this case, the minimum

kinetic energy achieved is often given in terms of the mean occupation number {«„ ) of the harmonic oscillator

state for the atom or ion in the well. The brackets ( ) denote the average over time or the average over an eno<

'fy±

C0o-2c0v OJq-^m

Frequency of laser for cooling

A —=T>- cuo-hco^ Wq + 2(xi^

Laser frequency (ojj

FIG. 1. Absorption spectrum of a single ion bound in a har- monic well, e.g., the absorption spectrum for the secular motion

in an rf trap. It is assumed that w,, (oscillation frequency of the

ion in the harmonic well) »y (radiative hnewidth of the cooling

transition). The Lamb-Dicke limit is assumed, i.e., ion excursions are less than k/ln=c /coq. Therefore, the intensity of the sidebands is small compared to the carrier at wo- Maximum

cooling is achieved for a narrow-band laser (bandwidth less than

y) tuned to Wq — '^i.- ork of the U. S. Government

2220 TN- 1

1

4ot subject to U. S. copyright
36 LASER-COOLING LIMITS AND SINGLE-ION SPECTROSCOPY 2221

semble of identically prepared systems, assumed to be the same. In the sideband limit we have"*''"'"

{n„), = C,(Y/co„: (2)

In Eq. (2), Cj is a constant on the order of 1 which de- pends on the atomic selection rules for the absorption and

reemission. We have assumed that the incident radiation

is well below saturation intensity. So far, sideband cool- ing has not been accomplished for trapped atoms or atomic ions. It has been realized in the special case of cooling

the magnetron motion of trapped electrons,'^ but in this case, the minimum achievable temperature is limited by

thermal excitation."''^ Since laser cooling gives rise to a thermal distribution of

occupation numbers,'^ we can write

< «j, ) = [expdko^, /kg D — 1

]

(3) Combining Eqs. (2) and (3), we achieve the minimum

temperature in the sideband cooling limit,

=fko,/[ksln(col/C,y^)] (4)

When fiio^ «kgT, Eq. (3) reduces to the expression

m{v}) /2 = kBTi/l, which applies to the Doppler cooling

limit of Eq. (1). Equations (1) and (4) show that lower temperatures

could be achieved by using weakly allowed transitions (smaller values of y). In addition, lower temperatures

could be achieved by using combinations of evaporative

and adiabatic cooling and/or novel cooling schemes using more than one atomic transition or nonadiabatic manipulation of trap parameters.''*"'* For example, several re- cent papers

'^~^^ have considered cooling by using stimulated Raman transitions. Here the cooling limits of Eqs.

( 1 ) and (4) apply but y is now the linewidth of the stimulated Raman transition, which can be quite narrow. Other authors speculate about schemes to achieve temperatures as low as 10~'° K for free atoms. '^

In this paper, we investigate the limits of laser cooling

for single trapped ions. We also consider the implications

of this cooling for accuracy in high-resolution spectroscopy. Some of the conclusions will also apply to the case of

trapped neutral atoms but here we concentrate on the case of trapped ions. As we will try to demonstrate, describing

the cooling limits in terms of temperature may not be particularly relevant for spectroscopic purposes. This may

also be true for free neutral atoms. For example, if a sin- gle atom [mass of 100 u (atomic mass units)] starts from

rest, the time it takes it to reach a velocity t; corresponding to an effective temperature T^ = 10" "* K (via the rela- tionship jmv^ — \kBT^) in the earth's gravitational field is about 10 /iS. If we desired to maintain an effective temperature below lO"^ K, the interrogation time between

applications of cooling radiation would be quite short. For trapped ions or atoms, gravity need not play an important role; it merely shifts the origin of the trap. Therefore, barring other heating effects, interrogation times can

be extremely long. Shifts in spectra caused by velocitychanging effects due to recoil can be made negligible. For

example, the maximum of the recoil-free "carrier" (at fre- quency (Do in Fig. 1) is shifted slightly due to the overlapping effects of the sidebands at coQ —oi^ and <Uo +6^u which

are different in amplitude." (See Sec. V.) These pulling

effects can be made extremely small. Perhaps the most interesting case is when the ion or atom is cooled to near the zeroth quantum level,

( «(,)«!. Independently of the method used, if («u)«l can be achieved, the accuracies for highresolution spectroscopy need not be limited by motional

effects. Since this condition can be achieved using laser cooling on single photon transitions, we investigate only

this relatively simple case here. At the outset, unless

specifically stated, we consider only a single ion in an rf or Penning trap.^"*'^^ The reason is simply that for two or more ions in the trap, the limiting kinetic energies are considerably higher than those given by Eqs. (l)-(4). For

example, for more than one ion in an rf trap, in order to overcome the Coulomb repulsion between ions, there must be a force on each ion from the trapping potential

towards the center of the trap. This implies a certain ki- netic energy of micromotion for each ion which can far exceed the kinetic energy of the secular motion. The micromotion^'*' ^^ in an rf trap is the oscillatory motion of the

ion(s) at the frequency of the imposed rf voltage applied to the electrodes. The secular motion is the additional,

lower-frequency motion of the ion(s) due to the resulting

pseudopotential well established by the spatially inhomogenous rf fields. It is this secular motion, at frequency

(o^,, which is cooled by laser cooling. The kinetic energy

in the micromotion is directly related to the kinetic energy

in the secular motion (Refs. 24 and 25 and Sec. II B) but

is not directly cooled by the laser. For more than one ion

in the trap, even though very low temperatures of the sec- ular motion can be achieved through laser cooling, the ki- netic energy would be dominated by the rf motion.^"* For

a single ion (Sec. II), this effect is considerably reduced.

Similarly, for more than one ion in the Penning trap, the

velocity in the rotation of the ion sample*'^'* gives rise to a

kinetic energy which can be greater than the limits of Eqs.

(l)-(4). For a single ion this effect can be very small (Sec.

III).In Sec. II we reexamine the sideband cooling limit for ions in an rf trap where we include the effects of finite laser bandwidth and the importance of micromotion. In

Sec. Ill, we examine sideband cooling in a Penning trap;

this has not been done previously. In Sec. IV, we suggest

a possible practical scheme for achieving {n^) «\, and

in Sec. V, we discuss a method for verifying that the zeroth quantum level is achieved. In Sec. VI we discuss the implications of these results for spectroscopy.

II. SIDEBAND COOLING FOR A SINGLE ION

IN AN rf (PAUL) TRAP

A. Laser cooling of secular motion in an rf trap We assume that the secular motion of the ion in the rf trap is harmonic and characterized by the frequencies co^,

cOy, and w^ for the x, y, and z directions, respectively.

TN-12
2222 WINELAND, ITANO, BERGQUIST, AND HULET 36

This case, or more generally the cooling of atoms trapped

in a harmonic potential well, has been treated extensively

in the literature. The most complete treatment appears to be that of Lindberg, Javanainen, and Stenholm, who in- clude the effects of saturation. ''^^ However, lowest

minimum temperatures are achieved with laser intensities below saturation, and simple perturbation methods'"'"

can be used to find the minimum kinetic energies. Here, we use the formalism of Refs. 11 and 27, which treats each photon absorption and subsequent reemission as a

scattering event —spontaneous Raman scattering. This

process is spontaneous scattering because we assume low

intensity and therefore the ion returns to its ground state

via spontaneous emission. It is Raman scattering because

when the internal atomic states and vibration states (oscil- lation states in the trap) are treated together, the ion can change the vibration quantum number n^, in a scattering

event. Let tii^ denote the nth vibration state of the ion in the X direction. The subscript / means that this is the initial (/, lower) vibration state of the ion before scattering. Let Pinjx) be the probability that the ion is in the initial state with quantum number /i/^, where "^n Pini^)—l.

Assume that the ion is irradiated with a narrow-band

laser (bandwidth Ad)/, «/) directed along the x direction. The average rate of energy change in the /th direction due

to this laser in the x direction is given by

Ix' JX

I

I

This is a straightforward generalization of Eq. (48) of Ref.

11. Nx=Ix<^ox/^o is the scattering rate for the laser tuned to resonance with the ion at rest, coq is the rest fre- quency of the ion, R ={fik)^/2m is the recoil energy

where k is the photon wave vector (k —kx^ky^k^),m is the ion mass, Hk^/f^j is the rms value of reemitted photon momentum along the /th direction, ^^ Eini^) ='fuOx(nix+j) where (o^ is the harmonic well frequency

in the x direction, di^ is the Kronecker 5, and coj^^ is the

laser frequency. The sum is over all possible initial vibrational states (with quantum numbers ni^) and intermediate vibrational states j. The intermediate states are the

states with the ion in the excited electronic state and with

vibrational quantum number rij^- Similar expressions

hold for laser beams directed along the y and z directions. Equation (5) holds for arbitrary ion kinetic energy

(neglecting relativistic corrections). The most interesting

case, however, is that for maximum laser cooling; that is, when the laser is tuned to the first lower sideband fre- quency coq —cOx- For the final stages of cooling, we antici- pate that the ion will be in the Lamb-Dicke regime.^*

This condition is given by the relationship (/c^(x^))'^^ = /cxo(2«^ + l>'^2«l where <«;c > =2«;^-P(«/x )n/;c- Qualitatively, the Lamb-Dicke regime implies that the amplitude of the ion's motion is less than k/ln.

XQ — {fi/2moix)^^^ is the zero-point amplitude so that

/cxo = (/? /fioix )^^^=PY^. Hence, for the Lamb-Dicke limit to apply we must have R «fux)x, or equivalently,

oix»u>n where (On—R/fi is the recoil frequency. For

A, = 300 nm and m = 100 u, w/j/2ir~22 kHz. For these conditions and for oix/'2.-n-=\ MHz, then Xo^O.007 /zm

and /3;f =0.022. If the Lamb-Dicke limit applies, then

e''^'^\-\-ikx, and only three excited states j contribute to the sum in Eq. (5). Therefore (see Fig. 1), <£„>^iVJ<«,>iS,(/?/,,-8„feJF,+/?/„rV(4.jl)

+ {Rf,i+b,x-lvjOx )(<«;,) + 1 )Pxr^/( \tcol )] .

(6)

I <«> \e' "''\n,x)\^

In Eq. (6), we have now included the factor F,= J

I^ico) ?

1 +

r

icOQ —Oix —(o)

2 ' (7)

which allows for the fact that the laser spectral width may

be larger than y. In practice, it may be desirable to make

the laser bandwidth much greater than y to allow for anharmonic contributions to or instabilities in (o^. For

simplicity, we assume that this width is still much less than cOx- Ixio)) is a normalized intensity such that f

Ix(co)dci}=\. Therefore, the laser spectral intensity is given by Ixixio)) where I^ is the total intensity. The aver- age scatter (or absorption) rate (iV^) is related to Eq. (6) as

{N,x>=Nx[(nx)l3xFx+y'/(4col)

+ ({nx) + \)Pxr^A\6col)] . (8)

If we also have laser beams directed along the y and z axes, equations analogous to Eqs. (5)-(8) hold for (Eiy),

{N,y}, <£^),and (N^).

Case 1. Suppose we have only one laser beam (say,

along the x axis) but that the x, y, and z degrees of free- dom are thermalized to the same temperature. This

would correspond to the case of more than one ion in the trap where Coulomb coupling provides the thermalization, but where we must assume that the individual ion

oscillation frequencies are not significantly perturbed.

That is, we neglect the effects of space-charge repulsion.

The total rate of energy change is given by Eq. (6) as {E)= 2 <£„)

i{=x,y,z)

=.iVJ<n^)[-/?F^-|-/?yV(16a)^)]-h5/?)'V(16a)l)) ,

(9) where terms of higher order in /?, have been neglected.

TN-13
36 LASER-COOLING LIMITS AND SINGLE-ION SPECTROSCOPY 2223

In steady state ((£) =0), we have

{n,)=5r'{l6col[F,-r'/{l6col)]}-' . We will assume throughout that the spectral density of

the laser is fairly uniform with width Aco^ . If the spectral

width of the laser is larger than y, we can get an estimate

of F by assuming that the laser spectrum is flat with

width Aa)£, that is, I^ = l/A,coi. For Acoi»y, Eq. (7) gives Fx=Try/2Aa>i^. Since we have assumed Aco^ «o)^,

then F»y^/col and we can neglect the y^/{\6co\) term

in the previous expression and obtain

{n^)=5y^/{l6(olF^) (10)

If the laser linewidth is much less than y, F^ = \ yielding

{n^ ) = 5y^/\6co\, which is the result of Ref. 11. Equation (10) is valid in the Lamb-Dicke limit,

R /fio)^ « 1 . To find corrections to this expression which

are first order in P^—R /-fkox we must include matrix ele- ments (n^—2\e''^\nx} and in^+lle''^ {n^} in Eq. (5) and take e '^ = 1 + ikx —k^x^/2. In this case we obtain

in,)

16 (U,

l + i^J/^. (11)

Case 2. Suppose we have three laser beams directed

along the x, y, and z axes which are tuned to the appropriate first lower sidebands for each direction. We must

write an equation for (E^), the rate of energy change in the X direction, using Eq. (6), and the analogous equations

which give contributions to (E^) from the y and z beams.

Similarly, we must write equations for (Ey ) and {E^}. In steady state we can solve these three equations for in^},

(ny), and (n^). These expressions are somewhat complicated but reflect the basic idea that if one of the scatter

rates is large, say N^ »N ,N,, and if 7,;^ -fsy -fsz and

o^c^oiy^o)^, then {uy) ,(n^) »{nx) due to the recoil heating in the y and z directions from the laser directed

along the x direction. If, however, we examine the special case where Nj^Nj, cOi^coj, FjC^Fj, and /^,~/y~|

{i,j =^x,y,z), then in^) c^iuy) ^(n,) where (n^) is given by Eq. (10). In this case, laser cooling and recoil heating are approximately equal in the x, y, and z directions. Case 3. Suppose we have one laser beam (intensity /) at some oblique angle with respect to the x, y, and z axes. If

f^x c^fsy ~/52 , then it is natural to choose k parallel to x + y + z in order to avoid excessive recoil heating along

any axis. This is also a natural choice for a typical ion

trap where the inner ring diameter is equal to ^2 times

the endcap-to-endcap separation. Here, the x + y + z

direction splits the gap between ring and endcaps. Thus

the x + y + z direction is a practical choice for introducing

laser beams into the trap. In this case, N^^ from Eqs. (5) and (6) is equal to iV/3 where N=IaQ/fitOQ is equal to the

scatter rate for a narrow-band (Aa)^ «y) laser tuned to resonance {(X)i=(j)q] for the ion at rest. Assume that

a),f ~&)^~6j;,~(y„ and that the laser bandwidth is broad

enough to cover all three of the first lower sidebands, i.e.,

must have

Fy^F2=F. To achieve the best laser cooling, we

(Oj—cOj I k, RNF/iko^, ii^j). The reason for

this can be seen by first assuming C0x=c0y. In this case

the choice of the x and y axes_is arbitrary. If we chose new axes where y' = (x — y)/V2 so that y'-k = 0; this implies recoil heating without bound in the y' direction."

For co^'^^'j^y, again assume k is parallel to x + y + z, and

choose fsx'^fsy—fsz^f—i (condition for isotropic

scattering). Recoil heating occurs at a rate RfN^ in the y' direction (i.e., perpendicular to the laser beam) where N^

is the total scattering rate from the ion. For kinetic energies near the sideband cooling limit, the dominant scatter- ing occurs in the wings of the carrier (at frequency coq) of

the ion absorption spectrum so that Ns^Ny^/Aoil [see Eq. (8)]. The closer co^ is to cOy, the higher is the energy

in the y' direction. To estimate this energy, we consider

the following. In the absence of laser scattering, if at some time t =0 all of the ion's energy (above the zero- point energy) is in the y' direction, then after a time r such that \cL>^—a>y \t = tt, all of the energy is in the

x' = (x + y)/v'2 direction. This situation is like the case of two coupled oscillators (here, in the x' and y' directions) which exchange energy in time t. Therefore, the decay of energy from the y' direction [denoted E(y')] is approximately equal to E{y')/T. In steady state, we have

RfN,^[E(y')-E{x')]/T. If we desire to keep

[E{y')—E{x')] <{n,. )fico^ due to the recoil heating in the

y' direction, then we require RfN^r < (n^ )^„ or approximately

co^—co^ I ^NRF/iuo,, (12) From the above arguments, it is clear that we must also make the same restriction on Ico^—co^l and {cOy—co^ \

.

The condition in Eq. (12) can also be obtained by assuming that we want | coj—coj \ to be larger than the inverse

of the time constant for laser cooling.^ In an axially symmetric rf trap, ojj^=cOy. Therefore, in order to achieve the condition given by Eq. (12), the axial symmetry must in- tentionally be destroyed.^ Thus, if the condition of Eq.

(12) holds, Eq. (6) and the analogous equations for (Ejy ) and { Ei^ ) in steady state give

(«,) =

12/„ + l 16F

iy/co. (13)

Equation (13) reduces to Eq. (10) for/j,=|. If the laser spectrum is uniform with bandwidth Acoi{I'=l/Ao)i),

and Acoj^ »y,

<«,) =

12/,, + 1

877 yAcoL /(o; (14)

The right-hand side of Eq. (14) is larger than Eq. (2) by a

factor of approximately Aco^/y. As an example, assume

co,/2tt=\ MHz, A6;i/27r~100 kHz, r/27r=l kHz, and

/„=|. Then <«,)^2X10-^

The meaning of { «, ) « 1 is the following. After most

photon scattering events, the ion's energy in the well is equal to 3fko^ /2, the zero-point energy. That is, most of

the time the ion scatters a photon without recoil energy

being imparted to the oscillation in the well. Infrequently, after some scattering events, the ion's energy in the

well is equal to (3/2+l)iko,, that is, the ion recoils into the first harmonic oscillator state, for some direction. TN-14
2224 WINELAND, ITANO, BERGQUIST, AND HULET 36

Once in the first harmonic oscillator state, it is pumped by

the sideband cooling mechanism back to the zero-pointenergy state. That the ion does not assume some inter- mediate energy state in the well is reflected in the discrete spectrum of fluorescence. ' ' We also note that when the

ion spontaneously emits a photon, its energy, when aver- aged over many scattering events, increases by R per emission event," even though after any given scattering

event, the vibrational energy can only change by zero or some multiple of %)„. Therefore, on the average, approximately HcOij/R scattering events occur before the ion

recoils into the first harmonic oscillator state. With very high probability, the next scattering event returns the ion

to the zero-point energy state. Therefore, most of the time, the ion is in the zero-point

energy state and photons are being scattered at a rate Ns=Ny'^/4-col (scattering in the wings of the carrier). In order to minimize the relative time the ion spends in the

rty = 1 state and therefore minimize the ion's kinetic ener- gy [i.e., make Eq. (14) valid], we must avoid saturation of

the first lower sideband when the ion is in the «„ = 1 state. From Eq. (8), we require NPF/3 <y. This implies that the scattering in the wings of the carrier N^ is less than

3Y^/4Fo)yCt)ji. For the conditions of our previous example («„/27r=l MHz, 7/277=1 kHz, «/j/27r=22 kHz),

N^ S 0.2/F, which is small for F near 1.

B. Limits of micromotion energy

In Sec. II A, we have discussed the limits to laser cool- ing for an ion or atom in a purely harmonic well, e.g., for the secular motion of an ion in an rf trap. When an ion is confined in an rf trap, one must also consider the energy in the micromotion.^"*' ^^ If we treat the motion classically, one can show that the average kinetic energy in the micromotion is equal to the average kinetic energy in the secular

motion for two trap configurations of practical interest:^'*' ^^ (1) when there is no static potential applied be- tween the ring and end caps {co^ =ci)y=co^/2) or (2) when

the appropriate static potential is applied between ring

and end caps to give a nominally spherical trap

{cOx =cOy ^co^). For other cases, this equality of micromotion and secular energies does not strictly hold, but the

key point is that they are of the same order. Since the very low kinetic energies predicted by Eqs.

(2), (10), and (13) require a quantum-mechanical treat- ment of the ion motion, it is useful to address the question

of micromotion energy again, but now treating the ion

motion in an rf trap quantum mechanically. Here, we ex- amine only the case of zero static potential applied to the

trap electrodes. For the quantum treatment. Cook et al}'^

have examined the solution to Schrodinger's equation for the rf trap in the form

w(x,/)=exp[— /i;(x)sin(nr)/^n] , [17)

8

^^ i-n^-\b=--—vV+y(x)cos(ftr)i/' , at 2m

(15)

where i;(x)cos(nr) is the potential inside the trap. By

writing the wave function in the form

Cook et al}^ show that we can identify g with the secular motion. If we take the expectation value of the kinetic energy, Ef^ — —ff'V^ /2m, and average over one period of

the drive frequency, we obtain

{^P\Ek\,P),,= —^ ^ d'xVvVvg*g+ ^ d\E^g*g . AmW *' •'

(18) The first term on the right-hand side of Eq. (18) is the ex- pectation value of the eff"ective potential (pseudopotential), calculated with the wave function g. The second term is the expectation value of the kinetic energy calculated with

the wave function g. The net result is the same as in the

classical case. That is, the average value of the effective potential energy ((Vy)^)/4mft^ (pseudopotential energy)

is equal to the kinetic energy of the micromotion. In the simple case of a quadratic pseudopotential, this contribution is equal to the kinetic energy of the secular motion.

In an experimental situation, the kinetic energy in the micromotion is typically larger than indicated above.

Sometimes there can be contact potential diff'erences, or

differences in stray charge buildup, between the elec- trodes. In the experiments using Ba"*", Mg"*", or Be"*", contact potential differences on the electrodes might result from nonuniform deposition on the trap electrodes from

the source of these atoms. ^° For example, suppose one endcap has a surface which has a higher contact potential

than the other two electrodes. This implies that in addition to the usual rf trapping forces, there is an additional

steady force along the z axis which shifts the center of the

trap. The new center of the trap is given by the condition

that the total average force on the ion is zero. Thus, the steady force due to the contact potential must be balanced

by the pseudopotential force; hence, there is a resultant micromotion at the new center of the trap where the secular motion amplitude is zero (i.e., the position of the ion

when it is cold). To estimate classically the size of this

effect, we write the potential inside the trap due to applied

voltages in the form^'*'^^

(f){r,z)

{7o+f^ocos(n?)

rl + 2zl

r^-2z' (19)

where Uq and Vq are the static and time-varying voltages

applied between the ring and end caps and where 2rQ and

2zq are the internal ring diameter and endcap-to-endcap

spacing as indicated in Fig. 2. (Effects of truncating the

electrodes are neglected.) This gives rise to a pseudopotential of the form^^

(l)nir,z) =

qVl Ua

+

mnHrl + 2zl)^

^qvl

+

n+2z

7^

2C/n

m£l\rl^2zl)^ '"0 + 225

z^

^(x,r)=g(x,Ow(x,r) , where

116) mco]. mcol . T +-Z—z^

Iq 2q TN-15

(20)
36 LASER-COOLING LIMITS AND SINGLE-ION SPECTROSCOPY 2225

B^BqZ (required for Penning trap)

Endcap

A=V/(rg+2z^)

V= Uo + Vocos m

(Vo= for Penning trap)

FIG. 2. Electrode configuration for an rf (Paul) or Penning

trap. The inner surfaces of the electrodes are assumed to be

equipotenitals of </>, and the effect of truncating the electrodes is neglected, to is the inner radius of the ring electrode and 2zo is the endcap-to-endcap spacing.

where C0r=o)x—(0y and 7 and z are the radial and axial positions of the ion averaged over one cycle of the rf

drive, that is, over a time Itt/H. The z force towards the

center of the trap due to the pseudopotential is given by

F^^ —qd(l)p/dz. The force away from the center of the

trap due to a contact potential on one endcap is given by

the following argument. Suppose we have a voltage AV

on one endcap. This can be written as the sum of symmetric and antisymmetric terms as indicated in Fig. 3(a). The z electric field due to the antisymmetric term can be

approximated^' by a parallel-plate capacitor of separation

Izq/u as shown in Fig. 3(b). The factor a has been numerically estimated^^'^^ to be about 0.8 for a trap where

T-Q = 2zo. If we assume the case of an rf trap with zero in- tentionally applied voltage Uq, then the center of the trap

is given by the condition F2 = —qd(f>p/dz—qaAV/2z() = 0, or

z = —

ImcolzQ

(21)

From this expression, we can find the kinetic energy in

(a) =

(b) —

+

AV/2

t, 2zja

AV/2 FIG. 3. Part (a) indicates how a potential A F applied to one endcap electrode can be viewed as the sum of voltages applied

symmetrically and antisymmetrically to the electrodes. For ions near the center of the trap, the lowest-order contribution to the

electric field from the antisymmetric part can be viewed as aris- ing from a voltage A V applied between a parallel plate capacitor of separation Iz^/a. This is indicated in part (b) of the figure. From Refs. 32 and 33, a~0. 8.

the micromotion Ej^^ by recalling that it is equal to the pseudopotential energy, that is, Ej(^ =q(l)p. Therefore, for the z motion we have

EKfxiz) =

qa^V

COZq (22)

To get an idea of the size of this effect, we examine the conditions of the miniature rf trap used to store single

Hg ions. There, a^/lvc^l.^ MHz, Zo=::320 /xm.

Vq^ASO iim. If AF = 1 V, £;j^^(z):^7xlO=fe,. Thus,

offsets due to contact potential variations or similar effects are extremely important if it is desired to reach kinetic energies approaching those given by Eqs. (2), (10), and (13). Such problems can be alleviated by applying external volt- ages to the electrodes to compensate for any of the above

type of asymmetries. This is easily accomplished along

the z axis by applying different static voltages to the two

endcaps. In the x and y directions, separate compensating

electrodes could be added or, for example, the ring could

be split into quadrants.^'* Different static voltages could

then be applied to the quadrants in order to compensate

for any asymmetric contact potential difference on the

ring.

Next, we consider the effects of rf voltages which may

be asymmetrically applied to the electrodes. This situa- tion might occur, for example, because of different impedances from either endcap to ground. If the amplitude

of the rf voltage on the two endcaps is different, this amounts to shifting the origin of the coordinate system for the rf voltage. In this new coordinate system, a voltage

Uq applied symmetrically to the endcap electrodes or to the ring electrode causes a static force which displaces the

ion from the origin for the rf fields and yields a resultant micromotion which can be deduced from the previous ar- guments. A new problem to consider, however, is that the phase of the rf voltages on the electrodes might be

different. To estimate the effects of this, assume that the

voltage FoCos(ftr + 0/2) is applied to one endcap and the voltage VqCOsHIi —6/2) is applied to the other endcap.

The ring electrode is assumed to be grounded. If 0« 1, we can consider the ion motion to be the sum of a solution

for the condition when Focos(ftf) is appHed to both endcaps and a perturbation to this solution for the additional

voltages (Fo^/2)sin(ar) and -( Fo0/2)sin(nf) applied to the two endcaps, respectively. Classically, we can easily

solve for the kinetic energy E^q due to the perturbed

motion (averaged over Itt/^I ). We have

(o,{rl + 2zl)a6

^^

"Ke- m

2 8Zn

(23)

For our example of the '^^Hg"*" ion,^ we obtain

£'^e~2.4x 10^0^^^. Thus, very precise control of the phase of the rf voltage is required to reach micromotion

kinetic energies near the zero-point energy of the secular motion. This can be accomplished by injecting a portion

of the drive signal (with the proper phase) onto one endcap.Because of the relatively strong effect of asymmetrically

applied potentials, it is natural to ask whether imperfections in the electrode shapes can lead to residual rf motion

TN-16
2226 WINELAND, ITANO, BERGQUIST, AND HULET 36

even when the ion reaches the position given by ( F ) = 0. To estimate the effects of electrode distortion, we must

modify Eq. (19). About a suitable origin we can write

(f){r,z) = [Uo + VoCOs(nt)]

'-2z'

+ A2(x^-yn

rl+lzl

(24)

where Laplace's equation is satisfied for each order. To

estimate the effects of the higher-order terms, we examine the term UqC->,x^z = UqC'i,x^z /zq in Eq. (24). Unless

the electrodes are severely distorted, we have C\«\.

Near the origin, this term gives rise to a z component of

electric field E'^ = —( Uq /zq K'^ix^) /zq. If we can cool

to near the zero-point energy ({«;,.)< 1), then {x^} ~Xo.

For the conditions of a Hg"*" ion in a miniature rf trap,^

(Ox/2Tr=\.5 MHz, Zo = 320 jum, Xo=^4.1XlO~^ fim,

C/q ~ 10 V (conditions for an approximately spherical

trap). For C3~0. 1, the resulting field E^ would be

equivalent to applying a voltage AF~4x 10 '° V to one

endcap. From Eq. (22), Ef:^(z)^\XlO~ fico^ for this value of A F which is negligible. Therefore, this effect by

itself can be small. However, electrode distortion may

give rise to unbalanced rf phases on the endcaps. Other

terms in Eq. (24) appear to be of this order or smaller.

and magnetron motions are both in the x-y plane, the only way this can be done is to use a spatially inhomogeneous laser beam. This case has been treated previously."

In the sideband cooling limit (y kkco^jCo'^co^), we are

able to use spatially homogeneous, plane-wave laser beams for the following reason. In Fig. 4, we show the absorption spectrum in the x-y direction averaged over many scattering events for an ion that has been cooled

within the Lamb-Dicke hmit (k^x^) ^k^rlin^+n^

+ 1)/2«1, where" rQ = [lfi/m(co',-co„)\'^^. In this

limit, only the first-order sidebands are of appreciable

size. The schematic representation of Fig. 4 may not be

to scale, but the expressions for the relative intensities above the indicated sidebands should be valid. For this figure, we have assumed o'^ = ico^ /I in order to simplify the drawing. This diagram and the remarks above show

that simultaneous sideband cooling of the magnetron and

cyclotron degrees of freedom can be achieved by having

two plane-wave laser beams directed normal to the z axis. One laser beam can be tuned to coq — co'^ and the other

beam tuned to c^o + '^m • I" principle, it should be possible

to achieve cooling by having one laser tuned to the side- band at frequency (OQ — a'^+co^, but, in the Lamb-Dicke

limit, the absorption intensity of this sideband is considerably reduced. Below, we examine the limits to sideband

cooling for the Penning trap. However, we first note that sideband cooling will work only at relatively low temperatures in the Penning trap where particular sidebands are

well resolved as shown in Fig. 4. At much higher temperIll. SIDEBAND COOLING

FOR A SINGLE ION IN A PENNING TRAP

Sideband cooling for a single ion in a Penning trap re- quires special attention because of the peculiar properties

of the magnetron motion. ^^ Cooling for the cyclotron and

axial degrees of freedom is essentially the same as for the

secular motion in the rf trap. However, for the magnetron motion, we will define cooling to be the reduction of

the kinetic energy in this degree of freedom. This implies

that the total energy of this degree of freedom increases, since most of the energy in the magnetron motion degree

of freedom is due to electrical potential energy from the

radially outward trap force. For positive charges, this potential energy is given by Eq. (19) with Vq=Q) and Uq <0.

In the axial direction, the ion is bound in a harmonic

well, and in the radial direction the ion sits on a potential

hill whose maximum is at the center of the trap where we

choose <^=0. First, consider the Doppler cooling limit

iy »a>^,o)'c,(02 where a>^, (o[, and co^ are the magnetron,

cyclotron, and axial oscillation frequencies in the Penning

trap^^'^^). To achieve cooling, we require that the ion

preferentially absorb photons when it moves toward the

laser due to cyclotron motion and simultaneously preferentially absorb photons when it moves away from the

laser due to the magnetron motion. Since the cyclotron

+

E

C2.E

E ii- c 1- 6 ^

S +

3+

CDi

FIG. 4. Amplitudes of the lowest-order sidebands in the ab- sorption spectrum for a single ion confined to the Lamb-Dicke

region in a Penning trap. (co[ =5w„ /2 is assumed.) The radiation source (laser) is assumed to be directed normal to the z axis. The carrier amplitude (at frequency coq) is normalized to 1. k =^coo/c,ro^[2^/m(a'c-(a,„ )]"'-,)3p = (/cro)V4and «<; and n„, are the cyclotron and magnetron quantum numbers, respectively. Simultaneous cooling of the magneton and cyclotron degrees

of freedom will occur for two laser beams tuned to wo + 'Jm and

&jo — a)c, respectively. TN-17
36 LASER-COOLING LIMITS AND SINGLE-ION SPECTROSCOPY 2227

atures, the magnetron and cyclotron sideband distribu- tions become nearly continuous, and it is impossible to simultaneously cool the kinetic energies for both degrees

of freedom using plane waves. This implies the necessity

of some form of precooling as discussed in Sec. IV. The basic formahsm for cooling of ions in Penning traps

has already been established.^^ In the quantum treatment

of the ion motion, the total energies of the axial, cyclotron, and magnetron degrees of freedom can be

written as E^=(n2+j)iico2, £^={17^+^)^10)'^, and

E^ = —{n^ +j)iia)„, respectively. The mean amplitudes

of the motion we desire to reduce are given by (z^)

=z^<2«, + l), {r^)=rl{n,+\), and {rl)=rl{n^

+ y), where Zo = (^/2mw2 )'^^ as in Sec. II. The axial cooling is identical to that described for the secular

motion in the rf trap in Sec. II. Now, however, we are entirely free of the energy in the micromotion since the fields are static in the Penning trap. From Ref. 27 we can write

a general expression for the average rate of change of the

axial, cyclotron, and magnetron motion quantum numbers for the |;th laser beam (total intensity I^) as

rf<«,>

dt = 2 Pi\n'})r(\n'}^{n^])(nf-n!) . (25)

In Eq. (25), {«'j = {«/,«/, «4 j is the set of initial (/= lower) quantum numbers, jn-'^j is the set of final (/= final) quantum numbers, and i =z, c, or m. P is the

initial probability distribution for quantum numbers {«'j

[P[n'] takes the role of /•(«/,) from Sec. II]. T is the

transition rate given by

T({n']^\n^]) = {I^/-ho)Q)a*[\n'\-^[n^\) . (26)

In Eq. (26), a* is the cross section given by

a*(|«'l->(/t^)) = (ro(r/2)'/

{\n^ — ik -r \n'\){\n}\ \e'^-'\

coQ — co — iY/2-\-(x){[n^\) — co({n']

PAK)dil (27)

In Eq. (27), k and k, are the wave vectors of the incident and scattered light. P,{k, )dCl is the probabiHty that a photon

is emitted into a solid angle dCL in the k^ direction (kj = k^ / | k, | ), and fi[co({nJ])-o)i{n'])]isthe difference in motional

energy between the initial (/) and intermediate (_/) states. We have

/„= /p,(k,)(kjVn.

where k^, is the component of kj in the /th direction. In Eq. (27), we have assumed Acoj^ «y. Equations (25) -(27) reduce

to Eq. (5) for the case of harmonic oscillator cooling or cooling the secular motion in an rf trap.

If we use Eqs. (25)-(27) and the operator relations"'^' for [Ni,e'^'^], where Nj=aiaj is the operator whose eigenvalue

is «,, we obtain the following relationship for the cooling due to the ^th laser beam:

din;}

dt =N^fr^aj 2 ^(("'D

l«'l.l"^l

WJs:+[nAJ)-n;{l)]]\{\n^\e

ikf-r

1 + -[coo-(o + co(\nJ])-(o{\n'\)] r

(28)

In this expression, N^=I^ao^/fkoo and we have now al- lowed the laser spectral distribution to have some width

as in Sec. II, such that f

r^{(o)dco—l. Also,

Ukro)^/4 (i=c or ml

fs, \fsx+fsy (' =C or

\fsz (i=z).

(29) m,

Equation (28) looks very similar to Eq. (5) except that the

recoil energy (/?,/^, term) divides between the cyclotron

and magnetron degrees of freedom in such a way that the

average change of {n,„ ) or (n^. ) is the same upon photon

reemission (i.e., the incremental changes in r^ and r„ are the same).

In the Lamb-Dicke limit, Eq. (28) simplifies considerably because e ^ ~l+k^-r and only matrix elements for transitions where An,=0, ±1 need be considered. This is the same situation as for the harmonic oscillator case of

Sec. II. However, for the Penning-trap case, both the cy- clotron and magnetron cooling beams are assumed to be

incident in the x-;; plane. Therefore, both beams cause

transitions where A/7,„=0,±l and ^n^=0,±l, and some

extra terms are encountered. For example, the contribution of dirii) /dt due to the cyclotron cooling beam

(tuned to wq — co'^ and assumed to be directed along the >' axis) is given by TN-18
2228 WINELAND, ITANO, BERGQUIST, AND HULET 36

iPifsi-K)\(n,-l\e''y\n,)\'F,+PJ,^

+ ()8,/.,+S,c)-

+ (/5//5,+S,m)

{n,\e''y\n,)\'

[2{co',-co^)/rY

+ (/?,/., -S,m )

-

[2(co',+co^)/rY

(30)

As in Sec. U, F, = f

I^{co)\\ + [2{a}o-co',-(o)/yf]-^dco,

and we have assumed A.col «co^,ci}2,o)'c for simplicity.

We can find the other two equations which are analogous to Eq. (30) for the magnetron cooling beam and axial cooling beam. By summing these equations, we arrive at expressions for the total rates of change of average quantum numbers for the three degrees of freedom d{ni)/dt

(i = c, m, or z). These expressions take the form

a =c, m, or z) (3))

where the coefficients C,y and c?, come directly from Eq.

(30) and the analogs for the cyclotron and magnetron

beams. For steady state, (dirij) /dt =0) and we find that the ( «, ) reach certain minimum values for given N^^ and

F^ (^ =c,m,z). These solutions can in general be quite

complicated. To get some estimate for the minimum

values for {«,), we have solved these equations for the following simple conditions. We assume f^^ =fsy —fsz = \, (o„ «(02 «o)'c, and require a solution where

{n,) = (n^) = {n,) = {n). For F,=F^=F,=F, this implies the condition iV^ =2iV^ = 8iV^ /ll on the laser beam intensities and we obtain

<«) = (ll/48F)(r/6>„ (32)

This has the same form as Eqs. (2), (10), and (13). For

these solutions to hold, we have required /? « 1 or

(Oji «co^ and co^ «co2 «(o'^ <oi^. These conditions

may only be approximated in an experiment. To see this, we note that we can write coR/cOc = \.'i (A/ 100

nm)~^jB~'Z~', where B is in tesla, Z is the ion charge (in units of the proton charge), w^ is the unshifted cyclotron

frequency, co^=qB /mc, and X is the wavelength of the cooling laser radiation. To satisfy the above conditions, we require a large magnetic field and a relatively long

wavelength cooling transition. A possible case might be

5+ [X('5o^^Pi) = 268 nm] at 15 T. Here, (Or/o, = 0.012, and (x),„ and <y^ can be adjusted so that Eq. (32) should be reasonably accurate. In practice, it is usually easier to make (i)x,oiy, and co^ in the rf trap be much larger than a^ can be made in the Penning trap. Therefore, the conditions for sideband

cooling are more easily satisfied in the rf trap than in the

Penning trap. The limit on how high we can make co^ is a

limit on how large a magnetic field we can achieve, since we must always have" «,„ <ftj^/V2 <&;^ < w^. =^5/mc

(conditions for stable trapping). From a comparison of

Eqs. (10) or (13) and Eq. (32), we might conclude that much lower values of the kinetic energy (above the zero- point energy) can be obtained with the rf trap than with

the Penning trap. This may indeed be the case for the secular motion of the rf trap, but, as we have seen at the end of Sec. II, the kinetic energy in the micromotion of

the rf trap may be much more difficult to suppress. For

the Penning trap, there is not an analogous problem to the micromotion since the imposed fields are static. Therefore, it may, in practice, be possible to obtain smaller

values of the kinetic energy in the Penning trap.

IV. TWO-STAGE LASER COOLING

To reach the cooling limits provided by sideband cool- ing [Eqs. (10), (13), or (32)], some sort of two-stage cool- ing may be necessary. This has already been referred to

in Sec. Ill for the Penning trap, where simultaneous side- band cooling of the magnetron and cyclotron motions can

only be effective at low enough temperatures where the Lamb-Dicke limit is approached. Therefore, to approach

the Lamb-Dicke limit, it may first be necessary to reach a

kinetic energy provided by the Doppler cooling limit on a strongly allowed electric dipole transition. For the Penning trap, this also implies the use of a spatially inhomogeneous cooling beam.^'

The same restriction does not apply for the rf trap. In

principle, it should be possible to tune to the first lower

sideband (for y «cl)^) and wait a sufficient time to reach

the steady state given by Eq. (10) or (13). Initial cooling

could be enhanced by tuning to the /th lower sideband

where Ico,, is approximately equal to one-half of the Doppler linewidth at the initial temperature." The max- imum cooling rate is given by dE /dtc=—l-fki)^,y /2. To

get an idea of the initial cooling rate, we can estimate the time Ar it takes to reduce the ion temperature from

r,=300 K to r,/2. We have At ^3kBT^A-fiIa},y)

~3.8xl0Vr, where we assume lo)^,=2Tr(0.5) GHz. If the cooling transition is very weak (e.g., y ~ 100 s~'), then

Af ~380 s. Even under ideal conditions, this time is very

long. Moreover, spurious heating mechanisms, such as

collisional heating, may prevent any initial cooling at all. Thus, in the case of the rf trap as well as the Penning

trap, there may be some motivation for using a precooling

stage [e.g., cooling to the Doppler limit of Eq. (1) where ]'

is due to a strongly allowed electric dipole transition].

To get some idea of how such two-stage cooling might

work, we examine the case of cooling in the rf (or har- monic) trap. Assume that the initial cooling is performed

TN-I9
36 LASER-COOLING LIMITS AND SINGLE-ION SPECTROSCOPY 2229

TABLE I. Examples of k{{x^})^^^, («„ ), and epi for a few ions of recent experimental interest. We have assumed that the ions are confined in a harmonic well with oscillation frequency W;, /In and are cooled to the Doppler cooling limit [Eq. (1)]. y /In and k are the

radiative linewidth and wavelength corresponding to the cooling transition, M is the ion mass in atomic mass units, k —1tt/\, ( «„ ) is the mean occupation number of the ion in the harmonic well [from Eq. (33)], dsvdi ^={kx^)^ =R /{fuii^). ie^i is the magnitude of the

fractional second-order Doppler shift for the x,y, and z directions combined [three times the value in Eq. (35)]. If the ions are confined

in an rf trap, the magnitude of the second-order Doppler shift is valid only for the secular motion contribution.

Ion Y/2tt

(MHz)

X

(nm) M{u)

co^/2n

(MHz) kiix'))' in,} P i€.

2''Mg+ 43 280 24 1 2.14 21 0.106 6X10-'^

(7>s^S,n--ip^P^n)

^*Mg+ 43 280 24 5 0.43 3.8 0.021 6X10-'"

Os^Sui^ip'Pui)

.98Hg + 70 194 198 \.. 0.91 22.8 0.018 1.2xl0-'«

(es^Su2~*(>p'P^n)

"^Hg+ 70 194 198 15 0.091 1.8 0.0018 1.2X10-'*

{(>s^Su^^^P^P^n)

•38Ba+ 21 493 138 1.5 0.24 6.5 0.004 5.1X10-"

(6s^5,/2->6p^P,/2)

"3ln+ 0.3 231 113 0.1 1.0 1 0.33 9.0X10-^'

(5s^'5o-*55 5/>^Pi)

on a strong transition (transition 1) where 7i»a)„. In

the Doppler cooling limit, the mean oscillation quantum

number can be derived from {{n^) -\-\)fuji^ =^7i/2 or

{n,)=[r^/co,-\)/l . (33)

In the Doppler cooling limit, we can also write

A:(<x^))'/^ = 27r(^,/2m)'/V(Xi&),) . (34)

In Table I, we have calculated <«„ ) and k{{x^))^^^ from

Eqs. (33) and (34) for some of the favored ions in recent experiments. This table shows that the Lamb-Dicke limit [A:((x^))'^^«l] is closely approached in the Doppler 2 cooling limit. If the Doppler cooling limit is first reached, the next

step might be to drive a much weaker transition in the same ion (linewidth YjKKco^) where the sideband cooling

limit applies. As an example, consider the case of '^^Hg"*"

whose relevant electronic structure is shown in Fig. 5. Initial laser cooling can be achieved^ by coohng on the

'1/2" 1/2 transition at A =194 nm. Sideband cooling

^1)3/2 quadrupole

^2)5/2 quadrupole

could be achieved by driving the S^/jtransition at A =198 nm or the ^Si/2-

transition at 1 = 281 nm. We could also drive the two- photon transition to accomplish this; in this case the

effective k vector of the transition is k' = k, + k2, where kj

and k2 are the wave vectors of the two photons required

for the transition, so that if k'||x, then ^^ = (|k,+k2|Xo)'/'.

Suppose we tune the second cooling laser to the first lower sideband at frequency [oiQ2 — o}^). Equation (8) shows that the ion scatter rate on this sideband is equal to Nn^PF/3, where we assume the geometry of Sec. II

A

(case 3). For the minimum time to reach the sideband

cooling limit, we require Nn^lSF /Zc^y2/2 for n^ = \. If much higher power is used, the scattering in the wings of

the carrier increases while the cooling rate saturates. This

would give a higher limit than that of Eq. (10) or (13).

Since each scattering event removes an energy fico^ from

the harmonic oscillator state, a minimum time

~27^'(«y) is required to reach the sideband cooling

limit. Here, ( «„ ) is the value obtained from precooling.

For the conditions of the second ^^^Hg"*" example in Table

I, we estimate a minimum time of approximately 80 ms

for the ^'S'i/2->^I>3/2 transition and 400 ms for the

^Si/2 ^^1)5/2 transition. If the goal is to perform very- high-resolution spectroscopy on the ^Sxn^D^/i

"clock" transition, we would Hke to perform the sideband

coohng using the ^Sx/2^D^/2 transition because of the

shorter cooling time required. (We designate the

Si/2-> ^2)5/2 transition as the clock transition because its narrow linewidth suggests that it could be used as a fre- quency standard or clock.) This still is not an optimum

198Hg +

5d^°6p

3/2

1/2 •T = 2.3 ns

FIG. 5. Simplified energy-level diagram for '^*Hg+. The

281.5-nm quadrupole "clock" transition can be observed by

monitoring the 194-nm fluorescence. If the ion has made a transition from the ^51/2 to the ^D^n level the 194-nm fluorescence

disappears. From Ref 7. TN-20
2230 WINELAND, ITANO, BERGQUIST, AND HULET 36

-*^D^/2 transithis spectrum

situation because we would like the cooling time to be

much less than the interrogation time ( — 7^') for the

clock transition in order to achieve maximum data rate and therefore best signal-to-noise ratio.

V. MEASURING THE ZEROTH QUANTUM LEVEL

We can verify that the sideband cooling limit given by

Eqs. (10), (13), or (32) has been realized by measuring the spectrum of the sideband cooling transition. This corre- sponds to either the ^•S' 1/2-^^/) 3/2 or ^5'i/2 —

tion in the Hg+ example. To measure

(after cooling) we can use a simple double-resonance

scheme.^'^^'^^ For the Hg"*" example, we can measure ab- sorption on the ^Si/2—>-^D^/2 clock transition by monitoring the changes in fluorescence from the 194-nm cooling

transition.^ In general, it will be necessary to drive the clock transition with the 194-nm laser switched off in or- der to avoid ac Stark shifts of the clock-transition spec- trum. After the clock radiation is turned off, the fluores- cence from the 194-nm cooling radiation will be absent if the clock transition has been made. If the ion remains in the ^51/2 state after application of the 281-nm radiation,

then the fluorescence will immediately appear after admitting the 194-nm radiation. After each such detection cycle, the ion must be recooled by sideband cooling.

First, assume we measure the absorption spectrum

along one of the principal axes (x, y, or z) while staying

below saturation intensity on the carrier. In the LambDicke Hmit, the strength of the lower sideband (assuming

the carrier strength is equal to 1) is ( «„ )P and that of the

upper sideband is ((n„) + l))8. Here we assume that

n^ = ni and P—Pi for i=x, y, or z. If («„)«!, the lower sideband will be very small and the upper sideband

will have amplitude P, as indicated qualitatively in Fig.

6(a). For the '^^Hg"^ examples of Table I, P is so small

that the upper sideband may not be discernable above the

CO.

c

o

CO <

(a)

1 Y

^+

: ^ ca.> ^

c

I

f <^0-^v COr

i;1

(b)

f^O + ^v ^Liil

t^O-^v COr 0^0 + t^V

FIG. 6. Absorption spectrum of an ion in a harmonic potential for {n^, ) « 1 and P=R /fuo^ « 1. In (a) relative amplitudes

are given for the unsaturated spectrum. In (b) the radiation in- tensity is adjusted so that the upper sideband is just saturated. The amplitude of the weak lower sideband is a direct measure of

background. If, however, we increase the power so that we are just saturating the upper sideband, then the carrier and upper sideband have amplitude ~ 1 and the lower

sideband has amplitude in^), as indicated qualitatively in

VI. IMPLICATIONS FOR SPECTROSCOPY

One of the original motivations for laser cooling was

the reduction of Doppler shifts and broadening in very- high-resolution spectroscopy. ^^~^^ Achieving the LambDicke limit essentially eliminates first-order Doppler

broadening effects since the intensity in the sidebands is very small. This result is independent of temperature, but

is easier to achieve at low temperatures. Historically, the uncertainty in the second-order Doppler frequency shift has been the primary limiting systematic effect in highresolution experiments on stored ions. This was true in the first high-resolution experiments on ^He"*" ions'^ (ac- curacy approximately equal to 1 part in 10^) and the most

accurate experiments on ions to date, those on ^Be"''

ions'*' (accuracy approximately equal to 1 part in lO'^). If the limits provided by sideband cooling can be realized, the second-order Doppler shift may no longer be the limiting systematic effect. First, we note that even in the Doppler cooling limit for an allowed transition, the second-order Doppler shift would be extremely small. From Eq. (1), the limiting mean kinetic energy is given by fr/ /A for each degree of

freedom. The Penning trap has one axial degree of free- dom and two cyclotron degrees of freedom which contribute; therefore, the limiting kinetic energy is 3^/4. For

the rf trap, the kinetic energy is at least twice this value

due to the micromotion contribution (Sec. II B). The

magnitude of the resulting minimum second-order

Doppler shift per kinetic-energy degree of freedom e^^ is given by

^Di'- ^^Di Jry l.lXlO~'^[y/27r(MHz)]

vo AmC M

(35) where the right-hand side denotes the Doppler cooling

limit. In Eq. (35), 7 /27r is expressed in MHz and M is the ion mass in u (atomic mass units). Values of €j)2 for a few kinds of ions confined in a harmonic trap are given

in Table I. eD2 gives the mean value of the second-order

Doppler shift. The uncertainty in ej)2 can be less depending on actual experimental conditions. In spite of

the smallness of these numbers, it may be desirable to suppress the second-order Doppler shift further since

other systematic effects may be controllable''^ to levels below one part in lO'*. In the sideband cooling limit, since (w^, )«1, the second-order Doppler shift is simply due to the kinetic en- ergy in the zero-point oscillations fuo^JA for each degree

TN-21
36 LASER-COOLING LIMITS AND SINGLE-ION SPECTROSCOPY 2231

of freedom. Therefore,

^VD2

.37

^Dl=-

Vo (2(«„> + l) Amc'

l.lXlO-'*K/27r(MHz)]

M

(36)

where the right-hand side denotes the sideband cooHng

limit. 6j)2 is at least twice this value for the rf trap due to micromotion. Although these values of 6^)2 can be very

small, the uncertainty in these values can be even lower."

For example, the uncertainty in 6^2 given by Eq. (36) might likely be due to the uncertainty of fluctuations in

03^. As an example, assume we perform sideband cooling

of '^^Hg"*" ions in an rf trap where Wi,/27r=10 MHz. If the uncertainty in this oscillation frequency is 100 Hz and

(«y)«10"~^, the uncertainty in the second-order

Doppler shift is 3.4 X 10" ^'*! It is not difficult to think

about even much lower limits. If we go back to the ex- pression for the ion temperature [Eq. (4)], we see that for

fixed 6)y, the temperature for the sideband cooling limit is not a strong function of 7 as it is for the Doppler cooling

limit [due to the ln((ny)~') factor]. In fact, for fixed y, the Doppler cooling limit gives a lower temperature than

the sideband cooling limit. However, as in the example

above, the uncertainty in 602 may be due to the uncertainty in oi^, and not directly a function of temperature.

Thus, it would appear that temperature may not be a very

useful parameter to describe the limits to high-resolution

spectroscopy on bound atoms or ions. With the potentially very low values of uncertainty in the second-order Doppler-shift correction, it continues to be an open question as to what will be the ultimate limit on accuracy in single-ion spectroscopy. The various

eff'ects of multipole electric and magnetic field interactions have to be considered;'*^ in principle, these eff'ects could be

measured and controlled to much less than 1 part in lO'*. Certainly, practical issues now play an important role. At

present, the required tunable narrow-band lasers do not

exist, but recent experimental advances'*^ may provide a

solution. If we assume that the spectral purity and amplitude stability of the lasers will not be a problem, we must

still consider the fundamental limits to the signal-to-noise

ratio. The high quantum multiplications achievable in double-resonance spectroscopy allow 100% detection

efficiency." This has been demonstrated in recent experiments on single ions.'"'*'*''*^ In this case, the noise in the ex- periments is dominated by the shot noise in the clocktransition measurement. That is, the noise in the measure- ment process is governed by the fluctuations in whether or not the ion has undergone the clock transition after application of the clock radiation. If we assume 100% detection efficiency in the double-resonance experiment de- scribed in Sec. V (Fig. 6), the maximum signal-to-noise ratio for determining hne center of the clock transition is given by measuring the transition probability at the half- power points on both sides of the clock resonance and

forcing the average frequency of the interrogating radiation to line center. If we make the assumption that the

time-domain Ramsey method is used to interrogate the clock transition, then the frequency stability (two-sample

Allan variance ) is given as

C7,(T) = [<(<6>,>,-<6;, + ,),)2>,/26>g]'/2

=(2wgr^T)-'/2

.

(37)

In this expression, {coi^)^. is the kth measurement of the frequency of the locked oscillator averaged over time t,

( )^. denotes an average over many measurements, coq is the nominal frequency of the clock transition, and Tj^ is the interrogation time between pulses in the Ramsey

method. Thus ayir) is a measurement of the rms fluctua- tions of the frequency for measurement times r. Equation

(37) assumes that optical state preparation (cooling and

optical pumping) and double-resonance detection (from

the 194-nm fluorescence signal in the example of Fig. 6) takes a time much less than Tj^

.

The accuracy of the measurement is limited to the value ayir) for a measurement time r. If, for example,

Tji =0.5 s [linewidth approximately equal to (2r^)~' = l Hz] and Wo/27r=10'^ Hz (^2 = 300 nm), then

r~2.5x lO'* s is required to make a measurement with an

imprecision cr (t) of 1 part in lO'^. To reach a precision

of about 10 ~ would require a time equal to the age of

the universe. Therefore, even though the intrinsic accura- cy may be quite high, it may take a long averaging time to reach a comparable measurement precision. In other

words, measurement precisions beyond 1 part in lO'* may

require even higher Q transitions (through smaller values of 7) and narrower linewidth lasers.

VII. CONCLUSIONS

If the theoretical limits to laser cooling can be reached on a single trapped ion, the ion will nearly be held in the zeroth quantum level of its bound motion. This would be

the closest possible realization of the ideal of a single iso- lated atom at rest. When cooling to near the zeroth quantum level is achieved, the temperatures achieved appear to lose their relevance; rather, we might be interested in the

fluctuations of the zero-point energy due, for example, to fluctuations in the well depth of the trap. In addition to the intellectual interest of achieving

confinement to the zero-point energy state, the implications for accuracy in high-resolution spectroscopy are re- markable. For example, uncertainties in the second-order

Doppler-shift correction might be 10 "^"^ or lower. The

possibility of these extremely low numbers clearly emphasizes the importance of many practical problems. For

the rf trap, minimizing the kinetic energy in the micromotion may be a very difficult problem (Sec. II). For the Penning trap, no analog for the micromotion is apparent

although it will be more difficult to obtain conditions

where the sideband cooling limit applies (Sec. III). In addition, there is the fundamental problem of the limiting

signal-to-noise ratio on a single ion; that is, we must ask,

"Can we reach the required measurement precision in a reasonable length of time (Sec. VI)?" We must also consider the problems of laser spectral purity and the measurement of laser frequencies. Equally important, we TN-22
2232 WINELAND, ITANO, BERGQUIST, AND HULET 36

must address the problems of controlling the various per- turbations to the internal structure of the ions produced

by external electric and magnetic fields. Nevertheless, in the end, it does not seem unreasonable to think that accuracies and measurement precisions at or beyond 1 part in lO'^ will be achieved.

ACKNOWLEDGMENTS

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

Force Office of Scientific Research (AFOSR) and the

Office of Naval Research (ONR). We also thank J. Cooper and L. Hollberg for helpful comments on the paper.

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edited by B. N. Taylor and W. D. Phillips, (U.S. GPO,

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3, 53 (1967) and 5, 109 (1969); D. J. Wineland, W. M. Itano, and R. S. Van Dyck, Jr., ibid. 19, 135 (1983). 26m. Lindberg, J. Phys. B 17, 2129 (1984); J. Javanainen, M.

Lindberg, and S. Stenholm, J. Opt. Soc. Am. B 1, 111 (1984). "W. M. Itano and D. J. Wineland, Phys. Rev. A 25, 35 (1982). 28R. H. Dicke, Phys. Rev. 89, 472 (1953). 29r. J. Cook, D. G. Shankland, and A. L. Wells, Phys. Rev. A

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PHYSICAL REVIEW A VOLUME 36, NUMBER 1 JULY 1, 1987

Recoilless optical absorption and Doppler sidebands of a single trapped ion

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

Time and Frequency Division, National Bureau of Standards, Boulder, Colorado 80303

(Received 13 April 1987)

Spectroscopic measurements of the electric-quadrupole-allowed 5d^%s^S\/2 to 5d^f)S^^Dsn

transition near 282 nm on a single, laser-cooled Hg"*" ion give a recoil-free absorption line (carrier) and well-resolved motional sidebands. From the intensity ratio of the sidebands to the carrier, the effective temperature of the Hg *

ion was determined to be near the theoretical minimum

of 1.7 mK. A fractional resolution of better than 3xlO~" for this ultraviolet transition is achieved.

Following the success of recent experiments on single, laser-cooled ions, most notably those experiments showing

1-3 quantum jumps, '"^ photon antibunching, • and absorption by a single ion,^ several groups are now at the threshold of atomic spectroscopy that promises fractional resolution and accuracy exceeding 1 part in 10'^. In the work

reported here, we experimentally demonstrate the spectroscopic resolution and signal-to-noise ratio that are attain- able with a single atom nearly at rest in space. We have

resolved the recoilless optical resonance^'^ and motional

sidebands of the 5J'°65^5'i/2 to 5d^6s^^Ds/2 transition

(A. = 282 nm) on a laser cooled ''^Hg"*" ion confined in an

rf trap. Since we can detect each transition to the metastable ^Ds/2 state with nearly 100% efficiency,^ there is essentially no instrumental noise. The noise results only

from the quantum statistical fluctuations in the transition probability of the single ion.* The unshifted resonance and sidebands shown in Fig. 1 are a convincing demonstration in the optical region of Dicke's original prediction

of the spectral features of an atom whose spatial excursions are constrained to the order of A.. ^ From the side- band structure we are able to make a determination of the amplitude of the motion of the ion and its eff"ective temperature. ^''° We anticipate that by further cooling with

narrowband laser radiation that is tuned to the first lower

sideband of the ^S\/2-^Ds/2 transition, the Hg"*" ion can be

made to reside at the zero-point energy of the trap's harmonic well most of the time.^'"-'^ The experimental setup is essentially the same as that described in our earlier studies on the two-photon spec- troscopy of Hg"'",'-' quantum jumps,^ and single-ion absorption.^ A mercury atom that is ionized by a weak elec- tron beam is captured in a miniature rf (Paul) trap that has internal dimensions of tq —455 /xm and zq — 320 pm.

The rf trapping frequency was 21.07 MHz with a peak

voltage amplitude of about 730 V. The ion is laser cooled

by a few microwatts of cw laser radiation that is frequency tuned below the 6s^S\/2-6p^P\/2 first resonance line near 194 nm.^'" When the Hg"*" ion is cold and the 194- nm radiation has sufficient intensity to saturate the

strongly allowed .S'-/' transition, 2x10** photons/s are scattered. With our collection efficiency, this corresponds to an observed peak count rate of about 10^ s~' against a background of less than 50 s ~

'. The 282-nm radiation that drives the ^S 1/2-^0^/2 transition is obtained by frequency doubling the radiation from

a cw ring dye laser that has been spectrally narrowed and

stabilized in long term to less than 15 kHz. For this ex- periment, the dye laser is stabilized to a high-finesse optical cavity'^ to give good short-term stability. In long

term, the laser is stabilized by FM optical heterodyne

spectroscopy to a saturated-absorption hyperfine component in '^'l2. '^ The pressure in the iodine cell is not

controlled against temperature variations, and this gave

rise to pressure-induced frequency fluctuations that dominated the stability of the laser for times greater than

about 1 s. The frequency of the laser is scanned by tuning the frequency applied to an acousto-optic modulator

through which the laser beam is passed. We eliminated

angle variation in the beam that was frequency shifted

1.00

0.95

2^ 0.90

T3

(DNC

0)3

a

0.85

0.80

1 I I I r I II I

I I I I I (Ap^ MfrVl

0.75-4

J I L J \ \ L J I L -2 2 Frequency Detuning (in MHz)

FIG. 1. Quantized sigml showing the electric-quadrupole- allowed 5d^%s^Si/2(mj = -\)-5d%s^^Di/2(mj = j) transi- tion in a single, laser-cooled "^Hg"*" ion. On the horizontal axis

is plotted the relative detuning from line center in frequency

units at 282 nm. On the vertical axis is plotted the probability

that the fluorescence from the 6s^S\/2-6p^P\/2 first resonance

transition, excited by laser radiation at 194 nm, is on. The

electric-quadrupole-allowed S-D transition and the first- resonance S-P transition are probed sequentially in order to avoid light shifts and broadening of the narrow S-D transition. Clearly resolved are the recoilless absorption resonance (carrier) and the Doppler sidebands due to the residual secular motion of

the laser-cooled ion. The integration time per point is about 16 s (230 measurement cycles).

M 428 TN-24 Work of the U.S. Government

Not Subject to U.S. Copyright
RAPID COMMUNICATIONS

M RECOILLESS OPTICAL ABSORPTION AND DOPPLER . 429

and scanned by using the twice-frequency-shifted beam

obtained by retroreflecting the first-order frequencyshifted beam back through the crystal. A computer controls the frequency and amplitude of the synthesizer that

drives the acousto-optic modulator. Up to a few microwatts of 282-nm radiation could be focused onto the

ion in a direction either orthogonal to, or counterpropagating with the 194-nm light beam. A magnetic field of

approximately 1.2 mT (12 G) was applied parallel to the

electric field vector of the linearly polarized 282-nm radiation to give well-resolved Zeeman components. This

configuration gives a selection rule of | Amy | = 1 for the

electric-quadrupole-allowed transitions to the various

Zeeman states. Optical-optical double-resonance utilizing quantum

amplification '^''^"'^ was used to detect transitions driven

by the 282-nm laser to the metastable '^D^/i state. Our

version of this method makes use of the fact that the 194-

nm fluorescence intensity level is bistable; high when the

ion is cycling between the S and P states (the "on" state)

and nearly zero when it is in a metastable D state (the

"off^' state). '"^ The fluorescence intensity in the on state

is high enough that the state of the atom can be deter- mined in a few milliseconds with nearly 100% efficiency.

The full measurement cycle was as follows: A series of

measurements of the 194-nm fluorescence was made, us- ing a counter with a 10-ms integration period. As soon as the counter reading per measurement period was high

enough to indicate that the ion was in the on state, the

194-nm radiation was shut off" and the 282-nm radiation was pulsed on for 20 ms. Then, the 194-nm radiation was

turned on again, and the counter was read. If the reading was low enough to indicate that the ion had made a transi- tion to the ^Dsn state (the off" state), the signal was

defined to be 0. Otherwise, it was defined to be 1. The

282-nm laser frequency was then stepped, and the cycle was repeated. As the laser frequency is swept back and

forth through the resonance, the quantized measurement

of the fluorescence signal at each frequency step is aver- aged with the previous measurements made at that same

frequency. Since we could detect the state of the ion with

nearly 100% efficiency, there was essentially no instru- mental noise in the measurement process. Occasionally,

while the 194-nm radiation was on, the ion decayed from

the ^P\i2 state to the metastable ^Z)3/2 state rather than

directly to the ground state. ^ This process led to a background rate of false transitions which was minimized by

the quantized data-collecting method described above and

by decreasing the 194-nm ffuorescence level (thereby de- creasing the '^Pxii-'^Dyi decay rate) until it was just high

enough for the quantized detection method to work. The

quantized measurement scheme also removes any contri- bution to the signal base line due to intensity variations in the 194-nm source. The 282- and 194-nm radiation were chopped so that they were never on at the same time. This

eliminated shifts and broadening of the narrow 282-nm

resonance due to the 194-nm radiation, '^^o.^' Figure 1 shows the fluorescence signal obtained from an 8-MHz scan of the 282-nm laser through the

5d ^%s ^S^ijimj = - i- ) -- 5d%s 2 ^Dsn^mj = '^ )

Zeeman component of the laser-cooled Hg"*" ion. The

recoilless absorption resonance (carrier) ^'^ and the motional sidebands due to the secular motion of the ion in the harmonic well of the rf trap are completely resolved. The

inhomogeneous rf electric field produces a pseudopotential

harmonic well with a radial frequency of approximately

1.46 MHz and an axial frequency of approximately twice

the radial frequency, or about 2.9 MHz. An ion trapped

in this pseudopotential well will execute nearly independent harmonic motions in the radial and axial directions. These harmonic motions frequency modulate the laser ra- diation, as seen by the trapped ion and produce sidebands on the carrier frequency at the radial and axial frequencies and their harmonics.' Thus, the number and amplitude of sidebands are a direct measure of the amplitude of the ion's motion and its effective temperature. In Fig. 1, only two pairs of sidebands are obtained. In this figure the 282-nm intensity is adjusted to be close to saturation on the carrier in order to enhance the relative strength of the sidebands to the carrier. Also, the laser linewidth was

broadened to about 80 kHz (at 563 nm) by modulating

the frequency of the acousto-optic modulator in order to reduce the number of data points required for the sweep. A careful comparison of the sideband intensities [Ref. 9, Eq. (44)], including the effects of saturation, gives an ion temperature for the secular motion of about 5.8 mK. This

is near the theoretical laser-cooling limit for '^^Hg"*" on

the first resonance line at 194 nm given by'

Tmm = h yllks — 1 .7 mK.

The spectrum in Fig. 1 was taken with the 194-nm

beam counterpropagating to the 282-nm beam. This ar- rangement gives optimum cooling to the ion motion in the

direction that is probed by the 282-nm beam. We have

also probed in a direction perpendicular to the propagation direction of the 194-nm beam. In this case, if there is perfect symmetry in the radial plane of the trap, then the

ion will heat without bound in the radial direction perpendicular to the cooling beam.''" Trap asymmetry in the

radial plane permits cooling in all directions provided the k vector of the cooling beam is not in the radial plane nor

collinear with the trap axis.''"''^ Even though cooling in

all directions is possible, the limiting temperature will be

higher in the direction perpendicular to the cooling beam

due to recoil heating in that direction. From our data, we

find this temperature to be only a few mK higher, indicat- ing a fairly rapid energy transfer between the ion's motional degrees of freedom.

Stray static electric fields in the trap can produce resid- ual micromotion at the rf drive frequency of 21.07 MHz.

This happens because the static field shifts the ion's equilibrium point to a region where the force from the static

field is compensated for by the force from the pseudopotential. In this case, there is residual micromotion at the bottom of the well even for a cold ion. The micromotion

will produce sidebands on the carrier at harmonics of the micromotion frequency analogous to the secular motion

sidebands. The appearance of rf sidebands in our initial scans indicated the presence of a small stray static field perhaps caused by contact potential variations on one or more of the electrodes. It was possible to null out the rf sidebands by applying a compensating dc voltage of about

TN-2.S
RAPID COMMUNICATION'

430 J. C. BERGQUIST, WAYNE M. ITANO, AND D. J. WINELAND 26.

0.3 V to one of the end caps. To null the rf sidebands in

all directions would likely require extra compensation

electrodes.

In Fig. 2, we show high-resolution scans through the Doppler-free resonance at line center and across the first upper sideband. The average integration time per point is about 10 s (160 measurement cycles). The linewidth at half maximum is approximately 30 kHz at 282 nm (or 15 kHz at 563 nm), giving a fractional resolution of about

3xlO~". If we can obtain a resolution that approaches

the natural linewidth limit of 1.6 Hz, the required integration time per point for the same signal-to-noise ratio of

Fig. 2 will increase by a factor of less than 2. For the data

of Fig. 2, an effective temperature of 1 .6 ± 0.5 mK and an rms amplitude of 28 ± 5 nm for the secular motion was

obtained. However, over several sets of data similar to that of Fig. 2, we obtained temperatures that ranged from

1.6 to 6.7 mK. The discrepancies between the measured

temperatures and the theoretical minimum might be due

to the uncertainty in the tuning of the 194-nm cooling

laser to the half-power point on the S-P transition. If the

frequency of the laser were detuned to a longer wavelength by a natural linewidth, the temperature obtained

would be about twice the minimum. Also, some heating

may have been caused by collisions with the background

gas. Finally, if the frequency of the 282-nm laser is tuned

to the first lower secular sideband, it would be possible to cool the ion until it occupied the zero point or («,) =0

(/ =x,y,z) energy state of the harmonic well most of the

tij^e 9.11,12,18

In summary, we have laser cooled a single ion and

probed with high resolution the ^S\/2-^Ds/2 quadrupole

resonance at 282 nm. The ability to detect each transition with almost no ambiguity eliminated nearly all instrumental noise. We have resolved the recoilless resonance line and the Doppler-generated motional sidebands. From the

relative strength of the sidebands, a direct measurement

of the ion temperature and its spatial excursion was made.

1.00

0.95

03cw

01o

c

a>

£ 0.90 -

O

0.85

0.80 1340 1400 1460 1520 1580

Frequency Detuning (in kHz)

FIG. 2. High-resolution scan through the recoilless absorption resonance and first (upper) sideband of the ^Su2-^Ds/2

(mj = — j-mj^ j) transition in a single laser-cooled "^Hg"*"

ion. The full width at half maximum of the recoilless absorption resonance and of the upper sideband is about 30 kHz at A, = 282 nm (15 kHz at A. =563 nm). A comparison of the carrier-to- sideband intensity gives a temperature of 1.6 ±0.5 mK, in agreement with the theoretical minimum of 1.7 mK. The in- tegration time per point is about 10 s (160 measurement cycles).

We anticipate improved spectral purity of the probe laser and better magnetic shielding of the Hg"^ ion in the future

to give increased spectral resolution. We acknowledge the support of the Air Force Office of

Scientific Research and the Office of Naval Research. We

thank L. Hollberg, R. Hulet, and M. Young for suggestions and comments on the manuscript.

'W. Nagourney, J. Sandberg, and H. Dehmelt, Phys. Rev. Lett. 56,2797 (1986).

2j. C. Bergquist, R. G. Hulet, W. M. Itano, and D. J. Wineland,

Phys. Rev. Lett. 57, 1699 (1986).

^T. Sauter, W. Neuhauser, R. Blatt, and P. E. Toschek, Phys. Rev. Lett. 57, 1696 (1986).

F. Diedrich and H. Waither, Phys. Rev. Lett. 58, 203 (1987).

^D. J. Wineland, W. M. Itano, and J. C. Bergquist, Opt. Lett. 12,389 (1987). 6R. H. Dicke, Phys. Rev. 89, 472 (1953).

^See, for example, Hans Frauenfelder, The Mdssbauer Effect (Benjamin, New York, 1962). «D. J. Wineland, W. M. Itano, J. C. Bergquist, and F. L. Walls,

in Proceedings of the 35th Annual Symposium Frequency

Control, 1981, p. 602 (copies available from Electronic Industries Association, 2001 Eye Street, N.W., Washington, DC

20006).

'D. .1. Wineland and W. M. Itano, Phys. Rev. A 20, 1521

(1979).

'°G. Janik, W. Nagourney, and H. Dehmelt, J. Opt. Soc. Am. B

2, 125! (1985).

"D. J. Wineland. W. M. llano, J. C. Bergquist, and R. G. Hulet, Phys. Rev. A (to be published). '^W. Neuhauser, M. Hohenstatt, P. Toschek, and H. Dehmelt,

Phys. Rev. Lett. 41, 233 (1978).

'3j. C. Bergquist, D. J. Wineland, W. M. Itano, H. Hemmati,

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

(1985).

'*H. Hemmati, J. C. Bergquist, and W. M. Itano, Opt. Lett. 8, 73 (1983).

'5j. C. Bergquist, R. L. Barger, and D. J. Glaze, in Laser Spectroscopy IV, edited by H. Waither and K. W. Rothe, Springer

Series in Optical Science, Vol. 30 (Springer-Verlag, New

York, 1979), p. 120.

'^J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, Appl.

Phys. Lett. 39, 680(1981).

'^D. J. Wineland, J. C. Bergquist, W. M. Itano, and R. E. Drullinger. Opt. Lett. 5, 245 (1980); D. J. Wineland and W. M.

Itano, Phys. Lett. 82A, 75 (1981).

i^H. Dehmelt, J. Phys. (Paris) Colloq. 42, C8-299 (1981).

'^H. G. Dehmelt, Bull. Am. Phys. Soc. 20, 60 (1975).

20W. M. Itano and D. J. Wineland, Phys. Rev. A 24, 1364

(1981).

2' W. Happer, Rev. Mod. Phys. 44, 169 (1972).

TN-26
From: Proc. 4th Symposium on Frequency

Standards and Metrology, Ancona, Italy, Sept.

1988; A. Demarchi, ed. (Springer Verlag,

Heidelberg)

Hg"*" Single Ion Spectroscopy*

J.C. Bergquist, F. Diedrich,''' Wayne M. Itano, and D.J. Wineland

Time and Frequency Div., National Institute of Standards and Technology

(formerly the National Bureau of Standards)

Boulder, CO 80303

Abstract

A single Hg'*' ion that is confined in an rf (Paul) trap can be laser cooled so that

the amplitude of its motion is much less than a wavelength (the Dicke limit) for

optical transitions. Recently, we have used the technique of optical sideband

cooling to reach the zero point of motion. This realizes for the first time the

fundamental limit of laser cooling for a bound atom and the ideal of an isolated

atomic particle at rest in space to within the quantum mechanical limits imposed by

the surrounding apparatus. In both limits, Doppler effects become negligible to

all orders, the interrogation time is long and the fundamental shot noise detection

limit of a single atom is readily attained.

Introduction

Laser cooling of a single atom has led to a number of important experiments in

fundamental physics [1-5] and may allow optical spectroscopy and metrology with a resolution and accuracy that approach 1x10"^° [6-9]. One of the candidates for an optical frequency standard that approaches this accuracy is a single, laser-cooled

^^'Hg"*" ion confined in a miniature rf trap [8]. At NIST we have begun with an investigation of a single ^'^Hg"*" ion, laser cooled into the Dicke limit for optical

transitions [8] and, in a second experiment, laser cooled in the sideband limit to

the zero-point energy of motion [10,12]. The ^^^Hg"*" ion offers the experimental

advantage of zero nuclear spin; hence, no hyperfine structure and a simpler optical

configuration for cooling. However, all the narrow optical transitions have a magnetic field sensitivity of the order of 10 MHz/mT (1 MHz/G), whereas the ^^^Hg"*"

ion has several first-order magnetic field- independent transitions.

In the work siumnarized here we begin to show the spectroscopic resolution and

signal-to-noise ratio that are possible with a single atom nearly at rest in space.

We have resolved the recoilless optical resonance and the motional sidebands of the

5d^°6s ^S^Cmj— i) - 5d^6s^ ^D5/2(mj«i) electric quadrupole transition (X5282 nm) on

a laser-cooled ^^^Hg"*" ion [8]. Since we can detect each transition to the

*Contribution of the NIST, not subject to copyright in the U.S. TN-27
metastable ^©5/2 state (y/2"it « 1.7 Hz) with nearly lOOZ efficiency, there is

essentially no technical noise. The noise results only from the quantum

statistical fluctuations in the transition probability of the single ion [6]. From

the sideband structure we are able to make a determination of the amplitude of the

motion of the ion and its effective temperature [10,11,13]. The sidebands at ojq ±

ntu^ (n an integer) are generated by the oscillatory motion at frequency w^ of the

ion in its confining harmonic well. If the ion is driven with narrowband radiation

tuned to the first lower sideband, the atom will absorb photons of energy h(ajQ-ai^)

and in most cases re-emit photons of energy I^ujq. This reduces the atom's

vibrational quantum nxunber by 1 for each scattered photon at frequency ojq. In this

way we can obtain <a^> « 1, which means that the atom spends most of its time in

the ground state level of its confining potential [10,12-14]. To the extent that

the atom is in the ground state of its confining potential the fundamental limit of

laser cooling for a confined particle has been reached [12].

Experiment

The mercury ion is confined in a miniature rf trap that has internal dimensions of

rQ 3 466 urn and zg » 330 um [8,15]. The amplitude of the trapping field (frequency

Q/2tt 5 21-23 MHz) could be varied to a peak of 1.2 kV. The ion is laser cooled to

a few millikelvins by a few microwatts of cw laser radiation [16] that is frequency

tuned below the ^Si - ^Pi first resonance line near 194 nm [2,8]. In order to cool

all motional degrees of freedom to near the Doppler cooling limit (T h>r/2kg s 1.7

mK) the 194 nm radiation irradiates the ion from 2 orthogonal directions, both of

which are at an angle of 55° with respect to the symmetry (z) axis of the trap.

The 282 nm radiation that drives the narrow ^Si - ^05/2 transition is obtained by

frequency-doubling the radiation from a narrowband cw ring dye laser. In the long

term, the laser is stabilized by FM optical heterodyne spectroscopy to a saturated

absorption hyperfine component in ^^^l2 [17]. The frequency of the laser is

scanned by an acousto-optic modulator that is driven by a computer controlled

synthesizer. Up to a few microwatts of 282 nm radiation could be focussed onto the

ion in a direction coxinterpropagating with one of the 194 nm light beams.

Optical-optical double resonance (electron shelving) [1-3,8,15,18-20] with a net

quantum amplification (including detection efficiency) in excess of 100 at 10 ms was used to detect transitions driven by the 282 nm laser to the metastable ^^^/2

state. The fluorescence rate from the laser-cooled ion is high when the ion is

cycling between the ^Si and ^Pi states and nearly zero when it is in the metastable

^05/2 state [1-3,8]. Thus the ^Si - D5/2 resonance spectrum was obtained by

probing the S-D transition at a particular frequency for the 282 nm radiation for

20 ms, then turning off the 282 nm radiation and turning on the 194 nm radiation to

look for the presence or absence of scattered photons at 194 nm (the two radiation

fields are alternately applied to avoid light shifts and broadening of the narrow

TN-28
S-D transition). If there was no fluorescence at 194 nm, a transition into the

metastable D state had occurred; the presence of 194 nm fluorescence indicated that

the ion was in the ground state and no transition was recorded for this frequency

of the 282 nm laser. The frequency was then stepped and the measurement cycle

repeated. As the frequency was stepped back and forth each new result at a particular frequency of the 282 nm radiation was averaged with the previous

measurements at that frequency. Normalization (or quantization) of the signal was obtained by assigning a 1 to each measurement of high fluorescence and a to each

measurement of no fluorescence. The high fluorescence level made it possible to

determine the state of the atom with almost no ambiguity in a few milliseconds.

Thus, it is easy to reach the shot noise limit imposed by the single atomic

absorber [8].

The quantized fluorescence signal obtained from an 8-MHz scan of the 282 nm

laser through the ^Si(mj"-i) -* ^D5/2(n»j"i) Zeeman component of the electric

quadrupole transition is shown in Fig. 1 [8]. The recoilless-absorption resonance

(carrier) [21,22] and the motional sidebands due to the secular motion in the

harmonic well of the rf trap [10] are completely resolved. The number and

amplitudes of the sidebands are a direct measure of the ion's motion and its

effective temperature. A careful comparison of the intensities of the sidebands to

that of the carrier [9] for several sets of data similar to that shown in Fig. 1,

including the effects of saturation, gave temperatures that ranged from 1.6 to 6.7

mK. This is consistent with the theoretical laser-cooling limit for ^^^Hg"*" on the

first resonance line at 194 nm given by T^^^j^ fiy/lk-^ a 1.7 mK [10].

(Sc

ai vi«

ue

«

uM

«

3

a

iZ

e«

aO

100

095

vMl /*t^|^P^ MM Nf^AjTij/^J^.

0.8S 1

1

0.80

rt 7«

"

Freautncy Detuning (in MHz)

10, on, On

Fig. 1. Quantized signal showing the electric-quadrupole-allowed 5d^"^6s

^Si(mj— i)-5d^6s^ ^D5/2(mj-i) transition in a single, laser-cooled ^^^Hg"*" i the horizontal axis is plotted the relative detiining from line center in frequency

units at 282 nm. On the vertical axis is plotted the probability that the

fluorescence from the 6s ^Si-6p ^Pi first resonance transition, excited by laser

radiation at 194 nm, is on. The electric-qxiadrupole-allowed S-D transition and the

first-resonance S-P transition are probed sequentially in order to avoid light

shifts and broadening of the narrow S-D transition. Clearly resolved are the

recoilless absorption resonance (carrier) and the Doppler sidebands due to the

residual secular motion of the laser-cooled ion. The integration time per point is

about 16 s (230 measurement cycles).

TN-29
In Fig. 2 we show a high resolution scan through the Doppler-free (recoilless) resonance at line center The full width at half maximum (FWHM) is approximately

1.7 kHz at 563 nm (3.4 kHz at 282 nm). This corresponds to a fractional resolution

of about 3.4x10"^* (Q*3»<10^M. For this trace the laser was spectrally narrowed

by locking to a mechanically, acoustically and thermally quiet reference cavity

that had a finesse of about 60,000. The frequency of the laser fluctuates less

than 1 Hz relative to this reference cavity. This is determined from the

measurement of the residual noise in the error signal; the actual frequency

fluctuations of the laser are then governed by the stability of the well isolated

cavity [23]. The natural linewidth for the ^Si - D5/2 transition is approximately

1.7 Hz [15]. The measured linewidth in Fig. 2 is dominated by fluctuations in the

first-order magnetically-sensitive electric quadrupole transition in the ^^^g"*"

ion. The field sensitivity of the ^Si(mj—i) to 2D5/2(n»j"i) transition is on the

order of 22.4 MHz/mT .(2.24 MHz/G). In our laboratory there are field fluctuations

of a few milligausses with a frequency-noise spectrum that is dominated by 60 Hz

and its harmonics. Shielding would help but is difficult in the present

experiment. The better approach is to switch to the ^'^g"*" isotope. This isotope

has first-order, field- independent transitions near zero magnetic field that should

permit us to reach the 1.7 Hz natural linewidth. In this case, the ion will

provide a good test of the spectral purity of the 282-nm laser.

|1.0

</1 J • . ••.,•... • •

'' • S 0,9

5 0.8 -

•

•

•

•

"S 0.7 a|o.6

~

• till,

•

-6 -4-2 2 Frequtncy Detuning (in kHz)

Fig. 2 High resolution scan through the

recoilles absorption resonance of

the ^S^(mj— i)-^D5/2(mj»l) transition in a single laser cooled

^^^g"*" ion. The full width at half

maximum is about 1.7 kHz at X-563 nm

(3.4 kHz at X-282 nm).

The possibility of cooling the Hg"*" ion further, to the zero point energy of

motion, is intriguing for several reasons. First, cooling so that the average

occupational number <n^> for the motional energy of the bound atom is zero is a fundamental limit to laser-cooling for any bound particle [8]. This limit is

imposed by the position-momentum Heisenberg uncertainty principle where the spatial

confinement (Ax) is provided by the physical surroundings. Driving a single atom

in a macroscopic trap to the zero point energy of motion also exploits the benign

environment near the center of an rf trap - certainly a system worthy of

consideration for the high resolution and accuracy in optical spectroscopy,

metrology and frequency standards. Second, with an ion prepared in the lowest

TN-30
vibrational state (most of the time), experiments such as squeezing the atom's

position and momentum can be demonstrated.

Cooling to the zero point energy of motion is possible for a bound atom in the

resolved sideband limit where the linewidth of the cooling transition (^Si - ^05/0

in our case) is less than motional oscillation frequency of the atom [10,12,13].

When the ion is in the lowest or ground vibrational state (n^=0) it can no longer

resonantly absorb a photon at uq-u»^; at this point the resonant feature at -uq-jj^

disappears. Experimentally, a measurement of the amplitude of the resonant feature

at ojQ-w^ (for saturating intensity) is a direct measurement of <n^> [13]. The

result from ref. 12 is shown in Fig. 3.

-30 -2.9 2.9 3.0 Frequency Detuning (MHz)

Fig. 3 First lower (S^) and upper (S^)

motional sidebands of the ^Si-^D5/2 slecric quadrupole transition of a lasersideband-cooled ^'^Hg"*" ion measured 10 ms after cooling. Data points are averages

of 41 sweeps.

From these data we deduce that the y and z degrees of freedom are in the zero point energy state (0^=0) 95^ of the time. Because the 282 nm probe beam is nearly

orthogonal to the x-axis, we were not sensitive to the x-motion. The uncertainty

in the second-order Doppler shift in this example is dominated by the uncertainty

in <n^> and amounts to Av/v « 10"^'^ [13]. It could be made even lower by

adiabatically lowering the potential well depth (thereby lowering u^) after the ion

is cooled into the ground state.

With this system, the absorption of a single quantum of energy at a frequency in

the megahertz range would raise n^ by one unit and this could be detected with an efficiency of nearly 100%. Also, it should be possible to produce squeezed states

[24] of the atom's motion from the zero point energy state by a sudden, non- adiabatic weakening (and, in general, shifting) of the trap potential or by driving

the atomic motion parametrically at 2 bi^. If after some time the atom could then

be returned to the ny.»0 state by reversing the above procedures, the zero point

energy state could be unambiguously detected by the absence of the lower sideband.

This would convincingly demonstrate the squeezed state of the single atom in the rf

trap.

We acknowledge the support of the U.S. Air Force Office of Scientific Research,

the U.S. Office of Naval Research and the continued support of the NIST. F. Diedrich thanks the Deutsche Forschungsgemeinschaft for financial support.

"''Permanent address: Max Planck Institut fiir Quantenoptik, Garching, West Germany.

TN-3I
References

1. W. Nagoumey, J. Sandberg, and H. Dehmelt, Phys. Rev. Lett. 56, 2797 (1986)

2. J.C. Bergquist, R.G. Hulet, W.M. Itano, and D.J. Wineland, Phys. Rev. Lett.

57, 1699 (1986)

3. T. Sauter, W. Neuhauser, R. Blatt, and P.E. Toschek, Phys. Rev. Lett. 57_, 1696 (1986)

^. F. Diedrich and H. Walther, Phys. Rev. Lett. 58, 203 (1987); W.M. Itano, J.C.

Bergquist, D.J. Wineland, Phys. Rev. A38, 559 (1988)

5. D.J. Wineland, W.M. Itano and J.C. Bergquist, Opt. Lett. 12, 389 (1987)

6. D.J. Wineland, W.M. Itano, J.C. Bergquist, and F.L. Walls, in Proc. of the

35th Ann. Symp. Freq. Control, 1981, p. 602 (copies available from Electronic

Industries Association, 2001 Eye Street, N.W., Washington, DC 20006)

7. H. Dehmelt, IEEE Trans. Instrum. Meas. IM-31 . 83 (1982); D.J. Wineland,

Science 226, 395 (1984).

8. J.C. Bergquist, W.M. Itano, and D.J. Wineland, Phys. Rev. A36, 428 (1988)

9. See articles by D.J. Wineland et. al.; Dehmelt, et. al.; W. Nagoumey, et.

al.; R. Blatt, et al.; P. Gill et. al. in this volume

10. D.J. Wineland and W.M. Itano, Phys. Rev. A20, 1521 (1979)

11. G. Janik, W. Nagoumey, and H. Dehmelt, J. Opt. Soc. Am. B2, 1251 (1985)

12. F. Diedrich, J.C. Bergquist, W.M. Itano, and D.J. Wineland, submitted for

publication

13. D.J. Wineland, W.M. Itano, J.C. Bergquist, and R.G. Hulet, Phys. Rev. A36,

2220 (1987)

14. W. Neuhauser, M. Hohenstatt, P. Toschek, and H. Dehmelt, Phys. Rev. Lett. 41_,

233 (1978)

15. J.C. Bergquist, D.J. Wineland, W.M. Itano, H. Hemmati, H.-U. Daniel, and G.

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

16. H. Hemmati, J.C. Bergquist, and W.M. Itano, Opt. Lett. 8, 73 (1983)

17. J.L. Hall, L. Hollberg, T. Baer, and H.G. Robinson, Appl. Phys. Lett. 39, 680

(1981)

18. D.J. Wineland, J.C. Bergquist, W.M. Itano, and R.E. Drullinger, Opt. Lett. 5,

245 (1980); D.J. Wineland and W.M. Itano. Phys. Lett. 82A, 75 (1981)

19. H. Dehmelt, J. Phys. (Paris) Colloq. 42, C8-299 (1981)

20. H. Dehmelt, Bull. Am. Phys. Soc. 20, 60 (1975)

21. R.H. Dicke, Phys. Rev. 89, 472 (1953)

22. See, for example, Hans Frauenfelder. The Mossbauer Effect (Benjamin, New

York, 1962)

23. J. Hough, D. Hils, M.D. Rayman, Ma L.-S., L. Hollberg, and J.L. Hall, Appl.

Phys. B33, 179 (1984)

24. See for example, J. Janszky and Y.Y. Yushin, Opt. Coramun. 59, 151 (1986); Fan

Hong-Yi and H.R. Zaidi, Phys. Rev. A37, 2985 (1988)

TN-32
From: Proc. 4th Symposium on Freq. Standards and Metrology,

Ancona, Italy, Sept. 1988; A. Demarchi, ed. (Springer Verlag,

Heidelberg)

High Accuracy Spectroscopy of Stored Ions~

D.J. Wineland, Wayne M. Itano, J.C. Bergquist, J.J. Bollinger,

F. Diedrich,"^ and S.L. Gilbert

Time and Frequency Div. , National Institute of Standards and Technology

Boulder, Colorado 80303

ABSTRACT: Physical limitations to the achievement of accurate, high resolution

spectroscopy on stored ions are briefly discussed. For experiments on ion

clouds, a compromise between frequency stability and second order Doppler shift

uncertainty must be made. Expected performance for specific examples using

single ions and ion clouds is discussed.

1. Introduction

In this paper we discuss some of the basic physical limitations to the

achievement of accurate, high resolution spectroscopy of electromagnetically

confined ions. For brevity, we do not address many practical problems, such as local oscillator spectral purity. These problems, although important, will

probably not cause the ultimate limit in accuracy and resolution.

This topic naturally divides between experiments on many ions and single

trapped ions. We anticipate that for experiments using more than one stored ion,

the uncertainty in the second order Doppler (time dilation) shift will be the

accuracy- limiting systematic effect[l-3]. This has been the case in the past.

For single ions, the uncertainty in the second order Doppler shift can be made

extremely small. The limiting uncertainty for single ions may depend on the

experiment [ A-6 ] , but inaccuracies should eventually be less than 1 part in 10-'^°. Use of single ions however might be precluded if the signal-to-noise ratio (S/N)

is not high enough to achieve the desired measurement precision in a reasonable

length of time. Therefore, we must consider a trade-off between second order

Doppler shift and S/N, both of which increase as the ion number increases.

As in any spectroscopic experiment, we must account for the perturbations due

to static and time varying multipole interactions for electric, magnetic, and

gravitational fields. These include atom-trap field interactions, collisions,

shifts due to uncontrolled electric and magnetic fields, gravitational red

shifts, etc. (see refs 1-9 and other contributions in these proceedings).

2. Optical-Pumping, Double-Resonance Experiments on Stored Ions

We will assume ion spectra are observed by optical-pumping, double-resonance

methods in which optical fluorescence changes are monitored. In Fig. 1, we

"Contribution of the U.S. Government, not subject to U.S. copyright.

TN-33
assume that level 3 is at an optical energy above the ground state 1. The

transition of interest for high resolution spectroscopy is between levels 1 and 2 (frequency ooq) which we designate the "clock" transition. A2 and A3 are radiative decay rates and R is the excitation rate between levels 1 and 3. R' and R" denote pumping rates between the clock levels due to optical pumping

sources (independent of the clock radiation) applied simultaneously with R.

Fig. 1 Schematic diagram for optical

pumping, double resonance experiments.

2.1. "Depletion" pumping . Assume A2, R' , R" « R < PA3, and a = ^. When R is

applied, atoms are pumped from level 1 to level 2 through level 3 (level 1 is

depleted). If radiation (not shown in Fig. 1) is applied near the clock

frequency ojq = i'K2~^i)l^, some of the atoms will be transferred back to level 1. This causes an increase in the detected fluorescence. [7 ,8]

2. 2. "Electron Shelving.

[4] We assume that the clock radiation and pumping

radiation are applied sequentially to avoid light shifts on the clock transition.

Assume a = 0, R' » R" , and A3 » (A2+R'-R"). In steady state with R, R' and R"

applied, the ions spend most of the time in level 1. If ions are in level 2 after

the clock radiation is applied and if R s A3, we can observe the absence of about

^d ~ nA3(A2+R' -R")~ detected (3^1) photons for each photon absorbed on the clock

transition (n is the net photon detection efficiency). When N^j » 1 and sources

of technical noise are absent, the fluctuations in detected coxonts are caused not

by the photon shot noise but by the fluctuations in the number of ions which have

made the transition from level 1 to level 2 (Ref. 1). This is the fundamental

noise limit. In Ref. 5, a = 10"^, R' , R" = 0, A3 = 4.3x10^ s~^, A2 = 12 s"^,

n 3 5x10'^, N(i = 20000. In Ref. 9, a = 0, A2 = 0, N^ = 1 , and R » R' » R" (R,

R' , and R" result from the same radiation source).

3. Frequency Stability

The signal is the response of the fluorescence detector to changes in frequency

of the oscillator which drives the clock transition. As a specific example, we assume use of the Ramsey separated field method in time domain. That is,

excitation of the clock transition is by two phase coherent pulses of radiation

of duration ATj^ separated by Tj^.[9] For mathematical simplicity we assume ATj^ «

Tj^ « A2"^ and R'-R" >> A2 (In Ref. 5, R' , R" = 0). The electron shelving method

of detection is assumed in the case where H^ » 1 in a time much less than (A2 + R'-R")"^ « T{^; that is, the detection time is less than the repumping time, which

TN-34
is less than T^^. If we probe the clock transition on each side of the central

Ramsey peak, we can develop an error signal which steers the average frequency to

ojQ, Under the above assumptions, the measurement precision as characterized by

the two-sample Allan variance[10] for the locked oscillator is[L,ll]

Oy(x) = (xNTrojq^)'' (i)

where x is the measurement time and N is the ion number.

4. Second Order Doppler Shift in Ion Clouds

From Eq. 1, if ojq 5r 1-^ is too small, it may be necessary to use large N to

obtain the desired measurement precision in a reasonable length of time. We

anticipate that the largest systematic effect in an experiment on many stored

ions will be the uncertainty in the second order Doppler shift, Avj|)2.

Fractionally, this shift is equal to <Av^2/^0'* " -<v^>/2c^ where vq = 'jjq/Z-tt, \? is

the ion velocity, c is the speed of light, and < > denotes an average over the N ions and over time. We would like to minimize the second order Doppler shift

while simultaneously making the number of ions as large as possible to increase

stability. These two requirements are contradictory and so a compromise between

accuracy and stability must be made.

4.1. rf Trap

We assume that the pseudopotential approximation [ 12] is valid and the kinetic

energy of the secular motion (the motion associated with the pseudopotential) is

reduced to a value where the Debye length of the ion cloud is less than the cloud

dimensions [ 13] . If the cloud is large enough, this can be accomplished by

bringing the secular motion nearly into thermal equilibritun with a background gas

(e.g.. He) through ion-neutral collisions [7, 14] . In principle it can also be

accomplished by the method of laser cooling but so far this has been achieved

with only a few ions in an rf trap because of rf heating.

Under the above conditions, the average kinetic energy per ion is dominated by

the rf driven micromotion [3,14]. For a spheroidal cloud of radius r^y and

height 2Zj,]_ we calculate <L\>-q2^^Q^ ~ "^^(^^cl "*" ^cO^-*^^ where a = q~VQ^/(mQ^(ro +2zq^)^). Here, q and m are the ion mass and charge, Vg and Q. are

the applied rf potential and frequency and vq and zq are characteristic trap

dimensions [ 12] . From this expression and the ion density[ 12, 13] , we have for a spherical cloud [3,14]

<Avj)2/vo> = -3ujg^r^j_/l0c2 = -3q2N/ 10rj,2.nic^ = -3(u)gNq2/ra)^/^/ LOc- (2)

where ojg is the secular frequency for a single ion in the trap. From Eq

derive a relation between N and <t^VY^2^^Q^

1

TN-35
N = -2.16 X lO^^r^^M <A\^2/^0>/2^ (3)

where Z is the ion charge (in units of the proton charge), M is the ion mass in

atomic mass units, and r^-^ is the radius of the cloud in centimeters. Equation 3 shows that for an assumed value of <Avp2/'^0'^> ^® would like to work with as large

a cloud (low density) and ion mass as possible.

4.2. Penning Trap

We will assume that the internal (cyclotron and axial) temperature of the ions is

reduced to a value where the second order Doppler shift is dominated by the

velocity of ions due to cloud rotation at angular frequency oj[3,13,15]. This

rotation is an essential part of ion storage in the Penning trap since it

provides the inward Lorentz (q(vxB)/c) force which overcomes the radially outward

force due to the trapping fields and space charge. We have <Av-q2/vq> = -

(ujr^2.) /5c^ where the cloud is a uniformly charged spheroid of height 2Zj,]_ and

radius r^-]_[13]. From Ref . 13, the density of ions is given by vlq = muj(n^-uj)/2TTq^

where Q^ = qBg/mc is the ion cyclotron frequency. From these two expressions,

the total number of ions in the cloud is [3]

N = 3.10 X 1013b(T)<-Avj52/^0>^ ^cl^^cr^c^^^ (^)

where B is in teslas, z^j. ^^'^ ^cl ^" centimeters and r^ = (5 <-Avq2/'^0^^^ c/Q.^.

For a given value of <Avj^2/^0'*» l^^S^ B, Zj,]_ and r^i (low density) are desirable.

N is independent of M (for the typical case r^ « r^^.)* ^® TM^y choose B to make

the clock transition field independent to first order[l,9].

5. Single Ions

From Eq. 1, if ijjq and Tj^ are large enough, stability can be high enough for

practical use, even with N = 1. For single ions, many systematic effects become

negligible[ 1-6] . For example, from our recent experiments on single Hg"*" ions[5],

we estimate uncertainties in the second order Doppler shift correction to be less

than 1 part in 10^^. What will actually be the limiting systematic effect will

depend on practical considerations and which ion is used. For the sake of

definiteness we consider two cases of interest at Boulder and list what we think

are the limiting systematic effects.

5.1 ^^^Hg"*", 5d^°6s 2si(F=o, mp=0) -> Sd^Gs^ ^05/2(^=2, mp=0) optical quadrupole

transition (vq = 1.07 x 10^^ jj^).

(1) Magnetic field sensitivity. From the Breit-Rabi formula, we obtain (vq(B)-

vq(B=0))/vq = - 1.8 X lO'^B^ where B is expressed in teslas. It should be

possible to obtain fractional precision of better than 1 part in 10^°.

(2) D5/-) state quadrupole interaction. The quadrupole moment of the D5/0 state

can couple to static electric field inhomogeneities to cause a shift of this

level. In a suitable (Cartesian) coordinate system, the atomic quadrupole moment

TN-36
interacts with a potential of the form cp = A(x^+y^-2z-) + 5A(x^-y^). In the

specific case where 5A = and B (quantization axis) is parallel to z, then Avq = -1.8 Hz for 1 V applied to the ring electrode for the trap of Ref. 5. The

presence of contact potentials on the trap electrodes makes the direction of z (in (p) uncertain but Avq averages to zero for three measurements of vq where the

directions of B are mutually orthogonal. [ 16]

5.2 ^^^Hg"*" ground state (F=0, mp=0) *-> (F=l, m-p=0) hyperfine transition. (\>q =

40. 5 GHz)

(1) Magnetic field sensitivity. (vq(B) - vo(B=0))/vq = 0.24 B^ where B is

expressed in teslas. For single ions, magnetic field homogeneity requirements

are not stringent.

Table I: Expected performance in particular systems. In rows 1 and 2, N is

calculated from Eq. 4. Eq. 1 is used to determine Oy(l s). For N > 1, the ion

cloud is assumed to be spherical (radius = tj,^). B is chosen to minimize field

sensitivity. For Tj^ > 1 s, quoted stabilities are referred to 1 s.

SYSTEM <-Avq2/^0> ^(T) r^iCcm) N Tr(s) a (1 s)

^Be hyperfine

Vq s 303 MHz 10

(Penning trap) [9]

15 0.819 0.5 2.0x10^ 100 1.2x10 -13

^O^Hg"'' hyperfine

Vq = 25.9 GHz 10"^6

(Penning trap)[l]

0.534 0.5 4.1x10^ 100 1.9x10 15

199jjg+ hyperfine

Vq = 40.5 GHz <10"^5

(rf trap)

= 100 3.9x10 -13

^^'^Hg''" optical

quadrupole

Vq = 1.07x10^5 <10'

Hz (rf trap)[5]

18 = 0.025 9.4x10 16

6. Examples

Table I lists some possibilities of interest at Boulder. Implicit in the Table

is the use of laser cooling. Sympathetic cooling [9,17] may be necessary for N >

1. Reasonable short term stability is obtained for single ^^^Hg"*" ions using the

ground state hyperfine transition. Systematic effects should be more easily

TN-37
controlled using single ions. [1-6] Fluctuations in cloud density and size (and

therefore detected fluorescence) could degrade stability over that shown in the

first two rows. For example, laser induced instabilities similar to those cited

in Refs. 15 and 18 have been observed in all Penning traps at NBS; these

instabilities must be suppressed to achieve OyCx) given by Eq. 1.

7 . Acknowledgements

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

Scientific Research for support. We thank D. Heinzen and L. Hollberg for helpful

comments on the manuscript.

''' Present address: Max-Planck Inst. Quantenoptik, Garching.

1. D.J. Wineland, W.M.Itano, J.C. Bergquist, F.L. Walls, Proc. 35th Ann. Symp.

Freq. Control (1981), p. 602 (copies available from Electronic Industries

Association, 2001 Eye St., N.W., Washington, B.C. 20006)

2. D.J. Wineland, et. al., J. de Phys., Colloque C8, 42 (1981) p. C8-307

3. D.J. Wineland, in Prec. Meas. and Fundamental Constants II, B.N. Taylor and

W.D. Phillips, eds., Nat. Bur. of Stand. (U.S.) Spec. Publ. 617 (1984), p. 83; D.J. Wineland, W.M. Itano, J.C. Bergquist, J.J. Bollinger and J.D.

Prestage, Ann. Phys. (Fr.) 10, 737 (1985)

A. H.G. Dehmelt, IEEE Trans. Instrum. Meas. IM-31 . 83 (1982); H.G. Dehmelt, in

Advances in Laser Spectroscopy , ed. by F.T. Arecchi, F. Strumia, and H. Walther (Plenum, 1983) p. 153; H.G. Dehmelt, these proceedings

5. J.C. Bergquist, et. al., these proceedings; F. Diedrich et al., submitted

for publication

6. D.J. Wineland, Science 226 , 395 (1984) and references therein

7. L. Cutler, these proceedings

8. G. Werth, these proceedings

9. J.J. Bollinger, et. al., these proceedings

10. J. A. Barnes, et. al., IEEE Trans. Instrum. Meas. IM-2Q , (1971)

11. Ref. 1 is in error by a factor of 72

12. H.G. Dehmelt, Adv. At. Mol. Phys. 3, 53 (1967)

13. D.J. Wineland, J.J. Bollinger, W.M. Itano, and J.D. Prestage, J. Opt. Soc.

Am. B2, 1721 (1985); L.R. Brewer, et al. , Phys. Rev. A38, 859 (1988)

14. L.S. Cutler, R.P. Giffard, and M.D. McGuire, Appl. Phys. B36, 137 (1985)

15. W.M. Itano, et. al. , these proceedings

16. W.M. Itano et. al. , to be published

17. D.J. Larson, J.C. Bergquist, J.J. Bollinger, W.M. Itano, and D.J. Wineland,

Phys. Rev. Lett. 57, 70 (1986)

18. R.C. Thompson, G.P. Barwood, and P. Gill, Opt. Acta 33, 535 (1986)

TN-38
From: Proc. 4th Symposium on Freq. Standards and Metrology,

Ancona, Italy, Sept. 1988; A. Demarchi, ed. (Springer Verlag,

Heidelberg)

Frequency Standards Utilizing Penning Traps*

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

Time and Frequency Div., National Institute of Standards and Technology

Boulder, CO 80303

1. Introduction

Ion traps have a number of characteristics that make them attractive candidates

for frequency standards [1]. Ion confinement times from a few hours to a few

days permit long interrogation times and therefore narrow linewidths and large

line Q's. In addition, ion traps typically do not suffer from the usual

perturbations associated with confinement. For example, collisions of hydrogen

atoms with the confining walls of a hydrogen maser can produce fractional

frequency shifts greater than 1 x lO"^-'- on the masing transition [2]. Stark

shifts due to the confining electrostatic field of a Penning trap can, in many

cases, be made less than 1 x 10"^^ [3,4].

The largest systematic shift and uncertainty in a frequency standard using

many stored ions is typically the second order Doppler (time dilation) frequency

shift Avjj2' For a clock transition of frequency vq, ^\>-q2 is given by

<Avi)2/vo> = - <^^>/(2c2) (1)

-» where c is the speed of light, v is the ion velocity, and < > denotes an average

over the ions in the trap. Without a means of achieving low ion temperatures,

this shift can be unacceptably large. For example, a 300 K ^Be ion has a second

order Doppler shift of -4.6 x LO"-*-^. Laser cooling [5,6] can be used to reduce

the ion temperature to much less than 1 K. For Be ions, this results in second

order Doppler shifts smaller than -1.5 x 10"^^. In our laboratory, up to 10^ Be"*" ions stored in a Penning trap have been laser cooled to temperatures less than 1

K. The Penning trap is therefore a suitable candidate for a laboratory microwave

frequency standard where the Dicke confinement criterion [7] is easily satisfied

and where it is desirable to store many ions in a trap in order to obtain a signal-to-noise ratio that gives a useful fractional frequency stability. The

number of ions used in a Penning trap frequency standard is determined from a tradeoff of signal-to-noise considerations and the second order Doppler frequency

shift due to the ExB rotation of the ions[3,8]. We note that in an rf (Paul)

trap the number of ions that can be laser cooled has been typically small (< 50)

"Contribution of the U.S. Government, not subject to U.S. copyright.

TN-39
due to the problem of rf heating[9].

At the previous (Third) Symposium on Frequency Standards and Metrology, the

results of an initial experiment done at the National Bureau of Standards (NBS) on ^^Mg"*" ions stored in a Penning trap were reported[ 10] . In this experiment a 12 mHz linewidth on a 292 MHz magnetic field- independent hyperfine transition in

the ground state of ^^Mg"*" was obtained[ 11 ] . Below we svunmarize the Penning trap

frequency metrology work done in our laboratory since the previous symposium. In

the next section the results of a frequency standard based on up to 2000 laser

cooled "fie"^ ions stored in a Penning trap are discussed. The inaccuracy of this

frequency standard was on the order of 1 x 10~-'--'[4]. It was limited by the

second order Doppler shift of the ions due to the ion heating during the clock

interrogation period when the cooling laser was turned off. The final section of

this paper discusses our current work using sympathetic cooling[12] to maintain a cold Be"*" cloud and obtain a 0.9 mHz linewidth on the Be"*" clock transition. With

sympathetic cooling we project an order of magnitude improvement in the Be''" frequency standard. In addition to the work done at NBS discussed here,

frequency metrology work with ions stored in Penning traps is being conducted by

groups at Orsay, France[13] and Teddington, England[14].

2. Laser-Cooled ^Be Frequency Standard

The Penning trap uses a uniform magnetic field to confine ions in a direction

perpendicular to the magnetic field. The ions are prevented from leaving the

trap along the magnetic field direction by a static electric field, typically

provided by three hyperbolic electrodes[ 1] . Figure 1 shows the hyperfine

structure of the ground state of ^Be as a function of magnetic field. At a magnetic field of 0.8194 T the hyperfine transition v-^ (see Fig. 1) depends only

quadratically on the magnetic field deviation AB according to Av-j^/v^ = -0.017(AB/B)'^. In spite of its relatively low 303 MHz transition frequency, v-^ provides the basis for a frequency standard. (We have chosen to work with Be , because it is easy to trap and laser cool. Heavier ions such as Hg offer large

microwave transition frequencies but are technically much harder to work with.)

A narrow-band radiation source (power = 20-60 |iW) tuned to the low frequency side

of the 2s ^Skimi = -3/2, mj = -1/2) ^ 2p '^V^/l^'^^^' "^Z^) (A = 313 nm)

transition of ^Be was used to cool and spatially compress the ions and optically

pump them into the (-3/2, -1/2) ground state[ 11 , 15] . The resonance fluorescence

induced by this cooling laser was used to detect the ions. The size, density,

and temperature of the ion clouds were determined by use of a similar radiation

source as a probe[15]. Typical clouds ranged from a few hundred to 2000 ions

with cloud diameters from 300 to 500 [im and densities of about 3 x 10' ions/cm-^.

Cyclotron and axial temperatures of less than 100 mK and a second order Doppler

IN-40
shift due to the cloud rotation of less than 3 xlO"^'^ were obtained with the

cooling laser applied continuously.

(Ml, Mj)

F=1

F=2

''-!..

(-3/2, 1/2) (-1/2, 1/2)

(1/2, 1/2)

(3/2, 1/2)

(3/2, -1/2)

(1/2, -1/2)

(-1/2, -1/2)

(-3/2, -1/2)

Figure 1. Hyperfine structure (not drawn

to scale) of the ^Be"*" 2s ^Si ground state

as a function of magnetic field. v^^ is

independent of magnetic field to first

order at B = 0.8194 T.

The v-j^ transition was detected by optical-microwave-rf triple resonance. Microwave radiation tuned to the electron spin-flip resonance transferred half of

the ion population from the optically pumped (-3/2, -1/2) state to the (-3/2,

+1/2) state. Some of the (-3/2, +1/2) state population was transferred to the

(-1/2, +1/2) state by application of rf near the 303 MHz v^^ transition frequency.

As a result of the microwave mixing this resulted in an additional decrease in

the (-3/2, -1/2) state population and therefore a decrease in the observed

fluorescence. The cooling laser and mixing microwaves were typically on for a period of 3 s, during which the "Be ions were prepared in the (-3/2, -1/2) and

(-3/2, +1/2) states. The cooling laser and mixing microwaves were then turned

off in order to avoid light and ac Zeeman shifts during the v^ interrogation

period. Ramsey's method of separated oscillatory fields was used to interrogate

V]^. Typically an rf pulse of duration t = 0.5 s was applied, followed by a free

precession interval of duration T = 19 s and a second rf pulse of duration t = 0.5 s coherent with the first pulse. After the interrogation period, the laser

and microwaves were turned back on, and the signal was obtained from the absence

of fluorescence during the first 0.3 s of this time interval. The (t,T) = (0.5

s, 19 s) interrogation resulted in a linewidth Av-j^ = 25 mHz and a Q i vj^/Av^^ of

1.2 X lO^*^ on the 303 MHz v-^ transition frequency.

A synthesized rf source was locked to the v-^ transition. A description of the

servosystem is given in reference [4]. With about 1000 ions in the trap, a

- 1 1 -- fractional frequency stability of a^(x) = 2x10 x ^ for measurement time x in

seconds was obtained with the (0.5 s, 19 s) Ramsey interrogation. The systematic

uncertainty of 1x10"^^ was dominated by the second order Doppler frequency shift

[4]. During the 20 s interrogation period when the cooling laser was turned off.

TN-41
the ion temperature increased from less than 1 K to 35 K. This heating produced

a second order Doppler shift of 4x10"'--' on the v-j^ transition and was measured

with about 25% uncertainty. The rapid heating that occurred during the

interrogation period was probably caused by asymmetry- induced transport [16].

With the cooling laser off, axial asymmetries of the trap can increase the total

canonical angular momentum of the ions, resulting in an increase in the ion cloud

radius. As the ion cloud expands, electrostatic potential energy of the ions due

to the space-charge and trap electric fields is converted into thermal energy of

the ions.

3. Sympathetic Cooling

An improvement in Be frequency standard performance requires a reduction in the

observed heating. This can possibly be done by improvement of the trap axial

symmetry. With this in mind we have constructed a Penning trap with care taken

to minimize axial asymmetries. Rather than the usual hyperbolic electrodes, the

trap was constructed with hollow cylindrical electrodes [17]. In addition, in

order to avoid asymmetric contact potentials due to plating of the trap

electrodes from the Be oven when ions were loaded into the trap, a second trap,

mounted below the experimental trap, was used to load the ions. The ions were

then transferred to the clean experimental trap. The expansion of the Be"*" ion

cloud in the cylindrical Penning trap was less than in the previous Be Penning

trap. However, the expansion was eliminated by using a second ion species to

sympathetically cool and compress [12] the Be ions. Specifically, •^°Mg ions were loaded into the same trap with the ^Be ions. The Mg ions were continuously laser-cooled and spatially compressed by a 280 nm (power = 50-80 pW)

radiation source. Through the Coulomb interaction with the Mg"*" ions, the Be

ions were then cooled and compressed. Due to the ion cloud rotation, the Mg

ions were observed to centrifugally separate and form a doughnut surrounding the

Be ions

.

The 313 nm source was used to optically pximp and detect the Be"*" ground state.

It had only a small effect on the Be"*" ion temperature and cloud size and, as in

the previous experiment, was turned off during the Ramsey interrogation period.

Because the 280 nm Mg"^ cooling laser beam is nonresonant with any Be"*" transitions, it was left on continuously during the Ramsey interrogation period

without seriously perturbing the v-^ transition frequency. In this way a cold,

steady state cloud of Be"^ ions was obtained independently of the 313 nm radiation

source, which permitted the use of very long interrogation times.

In the previous triple resonance experiment (optical-microwave-rf ) only half

of the stored Be ions were used to measure the v-^ transition due to the

microwave mixing. The stable field of the superconducting magnet used in the

present experiment permitted the following quadruple resonance procedure which

TN-42
used all of the stored ions to measure the \>-^ transition. The Be"*" ions were optically pumped into the (3/2, 1/2) ground state by tuning the 313 nm source to

the 2s 2si(3/2,l/2) -» 2p 2p3/2(3/2,3/2) transition. The 313 nm source was turned

off and the ions were transferred to the (1/2,1/2) and then the (-1/2,1/2) state

by successive 0.2 s Rabi tt pulses. After Ramsey's method of separated

oscillatory fields was used to interrogate Vj^, the ions remaining in the

(-1/2,1/2) state were transferred back to the (3/2,1/2) state by reversing the

order of the previous two Rabi it pulses. The 313 nm source was turned back on, and the signal was obtained from the absence of the Be"*" fluorescence during the

first few seconds after the 313 nm source was turned on. In addition to

increasing the number of ions used to measure the \>-^ transition, this procedure

increased the signal by a factor of 4. The long {-111, +1/2) ^ (3/2, 1/2) laser

repumping time also allowed the signal counting time after the 313 nm source was turned back on to be increased. This increased the signal-to-noise ratio by

increasing the change in the number of detected photons for each ion that made

the clock transition.

With sympathetic cooling, long interrogation times were possible. Figure 2 shows the signal obtained with a (1 s, 550 s) Ramsey interrogation. The 0.9 mHz

linewidth gives Q = 3.3xl0-'-^ on the v-j^ clock transition. If used to servo a local oscillator, this linewidth along with the estimated signal-to-noise ratio

should result in a fractional frequency stability of Oy(x) = 3.8x10~^^t~2

(x > HOG s). This stability is consistent with the stability of our reference

oscillator and is about a factor of 8 or 9 larger than the stability estimate

based on the number of stored ions [8]. Shorter (1 s, 5 s) Ramsey interrogations

resulted in a signal-to-noise ratio within a factor of 2 of the estimate based on the number of stored ions.

Preliminary estimates of all known systematic uncertainties for the transition

frequency of Fig. 2 are less than 1x10"^^. With the use of an imaging tube, the

cloud rotation frequency and radial distribution of the Be ions were determined.

The second order Doppler shift due to the cloud rotation was calculated to be

-AxlO"^-*. From previous observations, we believe the sympathetic cooling

provided Be temperatures « 1 K and second order Doppler shifts less than

IxlO"^^. The ac Stark shift due to the 280 nm source is estimated to be less

than 1x10"^-*. Due to the rotation of the earth, there is a systematic shift of

2.46(0.05)xl0~^^ [4]. All other known systematic shifts are estimated to be less

than 1x10-15,

In the future we plan to evaluate the use of sympathetically cooled Be ions

to steer a local oscillator. With a passive hydrogen maser [2] as a source for

the local oscillator, we may be able to obtain a fractional frequency stability

of a (x) = 2xl0"l^x"2 (x > 1000 s), limited by the stability of the passive

TN-43
hydrogen maser. An inaccuracy of less than 1x10 '• , limited by the second order

Doppler shift, appears feasible.

15000

z

UJ

OO

o

3

0.0000 0.0013

FREQUENCY - 303016377.265

(Hz)

Figure 2. Signal obtained with 3200 Be ions on the vj^ field independent

transition with a 550 s Ramsey free precession period. The data are the result

of one sweep. The sweep width is 2. A mHz and the frequency interval between

points is 0.2 mHz. The dots are experimental and the curve is a least squares

fit. A signal-to-noise ratio of 12 is estimated from the fit.

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

Scientific Research and the U.S. Office of Naval Research. We thank C. Weimer

and D. Heinzen for carefully reading the manuscript.

References

1. See for example D.J. Wineland, W.M. Itano, and R.S. Van Dyck, Jr.: Adv. At.

Mol. Phys. r9, 135 (1983) and references therein

2. F.L. Walls: Proc. of the IEEE 74, 142 (1986)

3. D.J. Wineland, W.M. Itano, J.C. Bergquist, and F.L. Walls: in Proc. of 35th

Ann. Freq. Control Symposium (USAERADCOM, Ft. Monmouth, NJ 07703) p. 602

(1981); D.J. Wineland in Precision Measurement and Fundamental Constants II , ed. by B.N. Taylor and W.D. Phillips (Natl. Bur. Stands. (U.S) Spec. Publ.

617, 1984), p. 83

4. J.J. Bollinger, J.D. Prestage, W.M Itano, and D.J. Wineland: Phys. Rev.

Lett. 54, 1000 (1985)

5. S. Stenholm: Rev. Mod. Phys. 58, 699 (1986)

6. D.J. Wineland and W.M Itano: Physics Today 40, 34 (June 1987)

7. R.H. Dicke: Phys. Rev. 89, 472 (1953)

8. D.J. Wineland et al.: these proceedings

TN-44
9. D.A. Church and H.G. Dehmelt: J. Appl. Phys. 40, 3A21 (1969)

10. D.J. Wineland, J.C. Bergquist, R.E. Druilinger, H. Hemmati, W.M. Itano, and

F.L. Walls: J. Phys. (Paris) Colloq. A2, C8-307 (1981)

11. W.M. Itano and D.J. Wineland: Phys. Rev. A24, 136A (1981)

12. D.J. Larson, J.C. Bergquist, J.J. Bollinger, W.M. Itano, and D.J. Wineland:

Phys. Rev. Lett. 57 70 (1986)

13. F. Plumelle, M. Desaintfuscien, M. Jardino, and P. Petit: Appl. Phys. B41

183 (1986)

lA. R.C. Thompson, G.P. Barwood, and P. Gill: Appl. Phys. BA6, 87 (1988)

15. L.R. Brewer, J.D. Prestage, J.J. Bollinger, W.M. Itano, D.J. Larson, and

D.J. Wineland: Phys. Rev. A38, 859 (1988)

16. C.F. Driscoll, K.S. Fine, and J.H. Malmberg: Phys. Fluids 29, 2015 (1986)

17. S.L. Gilbert, J.J. Bollinger, and D.J. Wineland: Phys. Rev. Lett. 60, 2022

(1988)

TN-45
Volume 57, Number 14 PHYSICAL REVIEW LETTERS 6 October 1986

Observation of Quantum Jumps in a Single Atom

J. C. Bergquist, Randall G. Hulet, Wayne M. Itano, and D. J. Wineland

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

(Received 23 June 1986)

We detect the radiatively driven electric quadrupole transition to the metastable ^^5/2 state in a

single, laser-cooled Hgll ion by monitoring the abrupt cessation of the fluorescence signal from the

laser-excited ^51/2^ ^/'i/2 first resonance line. When the ion "jumps" back from the metastable D

state to the ground S state, the S—' P resonance fluorescence signal immediately returns. The statistical properties of the quantum jumps are investigated; for example, photon antibunching in the emission from the D state is observed with 100% efficiency.

PACS numbers; 32.80.Pj, 42.50.Dv

Recently, a few laboratories have trapped and radiatively cooled single atoms^"^ enabling a number of

unique experiments to be performed. One of the ex- periments now possible is to observe the "quantum

jumps" to and from a metastable state in a single atom

by monitoring of the resonance fluorescence of a strong transition in which at least one of the states is coupled to the metastable state. When the atomic

electron moves to the metastable state, the fluores- cence from the strongly driven transition disappears.

When the electron drops back into the strongly driven

transition, the fluorescence abruptly returns. Thus the

strong transition fluorescence is a monitor of the

quantum state of the atom. Several years ago, Dehmelt had proposed this optical double-resonance

scheme (terming it electron shelving) as an amplification mechanism to detect a weak transition in singleatom spectroscopy.^ This technique has been used for several years^ in high-resolution spectroscopic studies

of samples of many laser-cooled ions, achieving quantum amplifications of 10^ and higher. In 1981, elec- tron-shelving amplification was used to perform

optical-optical double resonance in a single, laser- cooled, trapped ion.'^ While the signal-to-noise ratio in that experiment was not sufficient to see quantum

jumps directly, the fact that the atomic fluorescence

would be bistable was noted. More recently, the statistics of quantum switching in a single atom have been

theoretically treated in some detail first by Cook and

Kimble^ and subsequently by several other authors.*"'^

In this Letter we report the clear experimental demonstration of quantum jumps in a single laser-cooled

i98j^g+

jQpj stored in a miniature radio-frequency trap. The basic idea for quantum switching and the associated statistics is illustrated with the three-level system

shown in Fig. 1 for the Hgii ion. In Hgii there is a "strong" resonance transition from the 5c/'°6s^5'i/2

ground state to the 5J"^6p^fi/2 state near 194 nm. The lifetime of the ^P\i2 state has been measured else- where to be 2.3 ±0.3 ns.'^ Additionally, there is a "weak" electric quadrupole transition from the ^51/2

ground state to the ScPds^^D^ji state near 281.5 nm. The lifetime of the metastable D state has recently

been measured to be about 0.1 s.'^'^^ A laser tuned

just below resonance on the highly allowed S-P transi- tion will cool the ion and, in our case, scatter up to 5x 10^ photons/s. By collecting even a small fraction

of the scattered photons, we can easily monitor the

quantum state of the atom. If only the strong transi- tion is radiatively driven, then for averaging times long

compared to the Pyj state lifetime and mean S—'P

excitation time, a steady fluorescence level is expeced,

corresponding to the atom rapidly cycling between the

S and P state. If radiation to drive the weak transition

is also admitted, then the atom will occasionally be

driven into the metastable state and the fluorescence

from the strong transition will abruptly disappear. The

cessation of the scattering of many photons on the

strong transition for one photon absorbed on the weak

transition permits unit detection efficiency of the tran- sition to the metastable state.^-'^ Some time later, the

atom returns spontaneously or is driven out of the D

state back to the ground state, which causes a sudden

return of the fluorescence on the strong transition. The random on/off "telegraphic" signal provides a di- rect indication of the quantum state of the ion. The experimental setup is largely the same as that

5d^°6p

3/2

194 nm

281.5 nm

FIG. 1. Simplified optical energy-level diagram for Hg II.

Work of the U. S. Government

Not subject to U. S. copyright

TN-46

1699
Vol IMi; 57, NlMHI K 14 PHYSICAL REVIEW LETTERS 6 0CT0HFR 1986

used in the Iwo-pholon spectroscopic study ol the

2,S'|/2-'^^5/2 transition in Hgli reported earlier,'"' In the

present experiment, one ion (on some occasions, two

ions) was loaded^ into the trap and cooled to helow 25

mK by 2-5 /aW of sum-frequency-generated radiation near 194 nm (spot size w',)= 10 /xm) tuned just below

the ^S\i2-^P\i2 first resonance transition. The lluores- cence light scattered by the ions was detected at right

angles to the 194-nm beam with an overall detection

efficiency of about 5x 10"^ Peak signal counts ex- ceeded 2x lOVs against a background counting rate of

200/s. This high counting rate permitted us to monitor the fluorescence at a 10-ms sampling rate with a reasonable signal-to-noisc ratio. The fast sampling

rate is necessary because of the lOO-ms lifetime of the

^D5/2 stale. Radiation from a frequency-stabilized ring dye laser near'^ 563 nm was doubled to 281.5 nm in order to drive the ^-^'1/2-^^5/2 electric-quadrupole-allowed transition directly.'* The power of the 281.5-nm radiation

could be adjusted to as much as 20 )laW. The beam

was focused at the center of the trap to a spot size wq

of approximately 25 /^im. A magnetic field of approximately 1 mT (10 G) was applied parallel to the electric

field vector of the 281.5-nm radiation and perpendicular to its direction of propagation. The selection rule

for the electric-quadrupole-allowed transitions to the

various Zeeman states for this configuration is ^mj= 1. The frequency of the 281.5-nm radiation was

70

tuned to resonance = — L*-f — — ' ' with the ^S1/2 Dill inij

2 mj=+-) Zeeman component for the

quantum switching results reported here. Also, the resonance signal obtained from the P\/2 fluorescence

counts during scanning of the 281.5-nm laser over this component revealed a linewidth of less than 8 MHz. If this width is due to Doppler broadening, the ion temperature (in the pseudopotential well) is less than 25

mK and the ion is estimated to be confined to a volume characterized by a linear dimension of less than 0.25 ixm.

In order to observe the quantum Jumps, we monitor

the strong fluorescence at 194 nm while simultaneously admitting the 281.5-nm radiation. A computer

strobes and displays the delected 194-nm fluorescence counts accumulated in a counter at 10-ms intervals for running times of 40 s. These data are then stored and

the process repealed, but now without the 281.5-nm

radiation, or, in some cases, with the 281.5-nm laser detuned from resonance. We repeat the entire se- quence numerous times for different power levels of

the 281.5-nm light and for different detunings of the 194-nm radiation. An example of 4 s of a sequential

set of data with the 281.5-nm radiation (0.3 /iW) first off and then on is shown in Figs. 2(a) and 2(b),

respectively.

For the general case of coherent excitation, it is

(0E9

k ||itV|ff/V'*^'|p^rM»'*i^^^^^^ (a)

F%i i hh

Time (s)

FIG. 2. Samples of the quantum-jump data, 'a) The

fluorescence counts detected with no resonance radiation exciting the weak Si/2-f)y2 transition. The few jumps ob- served are likely due to collisions with background llg atoms

and radiative decay to the Dy2 state, (b) Both the 194- and

281.5-nm radiation are applied simultaneously Compared

to (a), the interruptions to the fluorescence signal are more

frequent, (c) Two ions are trapped and cooled. The inter- ruptions to the fluorescence signal show two levels corre- spondmg to loss of fluorescence from one or both ions. The

sampling rate for all the data is 10 ms per point; the length

of each sample is 4 s.

necessary to examine the off-diagonal terms of the density matrix in order to include possible coherence

effects.*-'"' However, for the conditions of our experiment, thai is, for times longer than the inverse of the excitation and spontaneous emission rales on the strong transition, and when the excitation and emission rales on the strong transition exceed those on the

weak transition, the dynamics of the quantum-jump

process can be described by effective two-state rate equations for coherent or incoherent excitation.* As a consequence there exist simple probabilities per unit

time R+ and /?_ that the electron makes an upward

or downward jump on the weak transition. Using the

theory of Ref. 8 adapted to arbitrary detuning, and

with the additional condition relevant to our experiment that the radiation driving the strong transition is well below saturation, we find that the Einstein A coefficient for the spontaneous decay of the weak transi- tion is related to these probabilities by

AODsn)==R--R + (P

From data similar to those of Fig. 2(bl, we plotted the

distribution of off limes r^ff and the distribution of on

times Tpn- Theory''-* predicts that the probability density for the time duration of the off (and on) intervals

1700 TN-47
Volume 57, Nlmher 14 PHYSICAL REVIEW LETTERS 6 October 1986

is given by

for time off, and by

(2)

(3)

for lime on. Thus, /?+ and /?_ are found from exponential least-squares fits to the data, and from these

and Eq. (1) we determine the A coefficient for the metastable ^Z)2/5 state. Unfortunately, the data analysis is complicated by the background events as indicated in Fig. 2(a). Although we have not unambiguously

determined their origin, two contributions are collisions with background mercury atoms which temperorarily heat the ions^ and radiative decay from the

^Pi/2 state to the ^0^/2 state. The lifetime of the ^Djn

state has recently been measured to be about 20 ms. According to theory'^ it decays with nearly equal probability to the lower-lying '^0^2 state and to the ground

state. We estimate'^ the probability that a mercury ion

in the ^P\/2 state will decay to the ^Dy2 state as 3x 10~^. In the present experiment there is no direct measure of the background pressure at the trap, but

estimates based on ion-pump current are consistent

with the observed frequency of background events. Estimated recooling rates are also consistent with the

data.

If the background interruptions in the fluorescence

signal were due solely to collisions, one would expect

their rate of occurrence to be independent of the 194-

nm scattering rate. The average duration of the off

times after a collision would be a function of the 194-

nm intensity and detuning, since these parameters

would affect the recooling rate. If the background in- terruptions were due solely to radiative decay, one would expect their rate of occurrence to be proportional to the 194-nm scattering rate, but their average

duration to be fixed, since this would depend only on

the meiasiable decay rates. The present data indicate

that both processes may be present in that both the

rate and the average duration of off times vary with

scattering rate. Even with these background events, we find the lifetime for spontaneous emission from

the '^Dy2 level given by Eq. (1) to be 90 ±30 ms, which is in agreement with the earlier measure- ments 15.16 Also of interest is the two-time intensity-correlation

function'"'^ for the 194-nm fluorescence:

C(r) = if{r)I(i + T)). (4)

This expression can be written as^

(( 7j =(/) 2+ ((/^)-</)2)exp[ -(/?+ + /?_)-].

(5)

FIG. 3. Plots of the 194-nm fluorescence intensity correlation function C{t) and the 281.5-nm emission correlation function ^d(t). Photon antibunching in the 281.5-nm emission is inferred from the observation that gotr)—'0 as r— 0.

where

(/) = /o^_/(/?++/?_),

{l')-{iy = I^R + R_/(R^ + R_)\

(6)

(7)

In Fig. 3 we plot the two-time intensity-correlation

function for a typical run. The values of ^+ and /?_

obtained from a least-squares fit agree with those

derived from the distributions of off and on times.

Other statistical properties can be derived from the quantum switching data. For example, if one neglects

the background interruptions to the fluorescence signal, each upward (downward) transition in the fluores- cence can be assumed to mark the emission (absorption) of a ^£>5/2-'^5'i/2 photon. Let ^^(t) denote the

probability that the (assumed) emission of a 281.5-nm

photon is followed by the (assumed) emission of

another 28L5-nm photon at a time t later, normalized

to 1 at T = oo. For our experimental conditions, theory

predicts that g£)(T) = 1 -exp[ -(/?+ + /?_ )r], and

this is in agreement with the data. In Fig. 3 we plot

g£)(T) from some of our quantum switching data. The

fact that goi''')^"^ at 7 = implies the existence of

photon antibunching"''^ in the 281.5-nm radiation

from the ^0^2 state. Because of the quantum amplification in the S/' scattering loop the photon antibunching is detected with nearly 100% efficiency.

Finally, in Fig. 2(c) we show quaiiiuni switching iui the case of two laser-cooled and trapped ions (estimated separation — 2.5/um). There are three distinct lev- els of fluorescence corresponding to (a) a maximum

when both ions are in the S'-f scattering loop, (b) an

intermediate level when one ion is shelved in the D

state and only one ion is scattering, and (c) no fluores- cence in the rare cases when both ions are shelved in the D state. TN-48

1701
Volumes?, Number 14 PHYSICAL REVIEW LETTERS 6 October 1986

In summary, we have demonstrated quantum jumps

or switching in a single atom. We have analyzed the

statistics and found agreement with earlier published

values for the lifetime of the 5cfi6s^^Dy2 state in Hgll. It is interesting to speculate about a single atom

in which the upper state on the weak transition is ex- tremely long lived and excited by adiabatic rapid passage. ^° In this case the random nature of the excitation is eliminated and one could realize a single-atom

switch or flip flop.

The authors are pleased to acknowledge the expert

technical assistance of C. Manney during parts of this experiment, and the help of D. J. Larson in the earlier work to obtain single-ion cooling. We are appreciative

to H. J. Kimble for helpful discussions of "quantum

jumps" and to J. J. Bollinger, R. E. Drullinger, and

L. Hollberg for comments on the manuscript. We

gratefully acknowledge the support of the U.S. Air

Force Office of Scientific Research and the U.S. Office

of Naval Research. We note that a Letter by Nagourney, Sandberg, and Dehmelt^' describing quantum

jumps in a single Ba"*" atom has appeared subsequent

to the submission of our Letter.

'W. Neuhauser, M. Hohenstatt, P. Toschek, and

H. Dehmelt, Phys. Rev. A 22, 1137 (1980). 2D. J. Wineland and W. M. Itano, Phys. Lett. 82A, 75

(1981).

^W. Nagourney, G. Janik, and H. Dehmelt, Proc. Natl. Acad. Sci. U.S.A. 80, 643 (1983).

"^G. Janik, W. Nagourney, and H. Dehmelt, J. Opt. Soc. Am. B2, 1251 (1985). 5H. G. Dehmelt, Bull. Am. Phys. Soc. 20, 60 (1975), and

J. Phys. (Paris), Coiloq. 42, C8-299 (1981), ^D. J. Wineland, J. C. Bergquist, W. M. Itano, and R. E. Drullinger, Opt. Lett. 5, 245 (1980); D, J. Wineland, W, M.

Itano, J. C. Bergquist, J. J. Bollinger, and J. D. Prestage, in Atomic Physics, edited by R. S. Van Dyck, Jr., and E, N. Forlson (World Scientific, Singapore, 1985), Vol. 9. ^R. J. Cook and H. J. Kimble, Phys. Rev. Lett. 54, 1023

(1985). 8H. J. Kimble, R. J. Cook, and A. L. Wells, Phys. Rev. A

34, 3190 (1986).

^T. Erber and S. Putterman, Nature 318, 41 (1985).

'Oj, Javanainen, Phys. Rev. A 33, 2121 (1986),

"A. Schenzle, R. G. DeVoe, and R. G. Brewer, Phys. Rev. A 33, 2127 (1986).

'^C. Cohen-Tannoudji and J. Dalibard, Europhys. Lett. 1, 441 (1986).

'3D. T. Pegg, R. Loudon, and P. L. Knight, Phys. Rev, A

33,4085 (1986).

''*P. Eriksen and O. Poulsen, J. Quant. Spectrosc, Radial. Transfer 23, 599 (1980).

15J. C. Bergquist, D. J. Wineland, W. M. Itano, H. Hemmati, H.-U. Daniel, and G. Leuchs, Phys. Rev. Lett. 55,

1567 (1985).

•6C. E. Johnson, Bull. Am. Phys. Soc. 31, 957 (1986),

''D, J, Wineland, W, M, Itano, J. C. Bergquist, and P. L. Walls, in Proceedings of the 35th Annual Symposium on Fre- quency Control, Philadelphia, 1981 (Electronic Industries Association, Washington, DC, 1981), p. 602.

'^D. J. Wineland, J. C. Bergquist, R, E, Drullinger,

H, Hemmati, W. M. Itano, and F. L. Wails, J. Phys. (Paris),

Coiloq. 42, C8-307 (1981).

>9R, H, Garstang, J. Res, Natl, Bur, Stand. Sect. A 68, 61 (1964),

20See, for example, A. Abragam, Principles of Nuclear

Magnetism (Clarendon, Oxford, 1961), p. 65. 2'W. Nagourney, J. Sandberg, and H. Dehmelt, Phys. Rev.

Lett. 56,2727 (1986).

TN-49

1702
PHYSICAL REVIEW A VOLUME 36, NUMBER 6 SEPTEMBER 15, 1987

Quantum jumps via spontaneous Raman scattering

Randall G. Hulet and D. J. Wineland

Time and Frequency Division, National Bureau of Standards, Boulder, Colorado 80303

(Received 27 April 1987)

A single laser, which is used to induce and detect spontaneous Raman transitions, can be used

to observe quantum jumps in a single atom. The population dynamics of a particular system, consisting of two ^5 1/2 ground-state levels and four 'P3/2 excited-state levels spht by a magnetic field,

is analyzed for a laser tuned near a particular transition. We find that the statistics of the fluores- cence emitted by this system are described by the same formalism developed for the three-level V

configuration irradiated by two light sources. Over a wide range of observation times, the fluores- cence intensity will be two valued, either off or on, as has been verified for the V configuration.

Some surprising and elegant features of this new system are described.

The fluorescence emitted from a single atom can exhibit phenomena which are unobservable in a collection

of many atomic emitters. For example, the fluorescence

emitted by a driven two-level system reveals a quantum

phenomenon known as "photon antibunching."''^ Antibunching is demonstrated by measuring the time correlation between fluorescence photons emitted by a single

atom. It is found that the probability of detecting a photon instantaneously after detecting the previous one

is zero, that is, they are antibunched. This is a purely

quantum eff'ect with no classical analog.

A three-level manifestation of quantum photon emission was first analyzed by Cook and Kimble^ and is based on Dehmelt's "electron shelving" scheme.'* Figure

1 shows a three-level system in the V configuration. The

ground state (0) is strongly coupled to an excited state

(1) which has a spontaneous decay rate A y, and is weakly coupled to another excited state (2) which decays at a

rate A 2 ( A^ » Aj). Two radiation sources separately

drive the 0-«->-l and 0-<-^-2 transitions while only the

fluorescence from level 1 is monitored. On a time scale which is long compared to A f ', but less than or on the

order of AjK the fluorescence is expected to display

periods of constant intensity interrupted by periods of zero intensity while the atomic electron is "shelved" in

Q2, A2

FIG. 1. The K configuration of energy levels. Two excited

states are coupled to the ground state with two diff'erent lasers. The spontaneous decay rate of level / is denoted by /4,. Cl, denotes the Rabi frequency of the Oi->-/ transition. It is as- sumed that A\ »A2- Quantum jumps in the intensity of the fluorescence emitted from level 1 have been obsered unc'ar these conditions (Refs. 13-16).

36

level 2. The on-off" switching of the fluorescence indi- cates a discontinuous jump, or "quantum jump," be- tween level 2 and the strongly fluorescing system consist- ing of level and level l.^'^~'^ Recently, three groups

reported the observation of this phenomenon in a single

confined ion'^~'^ and a fourth group is able to infer the

eff'ect from the time correlation of fluorescence from a weak atomic beam.'^

Kimble et al. have shown that the excitation of the

weak transition (0-«-^2) is a rate process for the V level configuration." The population dynamics of the three

levels can be reduced to those of an eff'ective two-level

system for times which are long compared to /I f '. R ^

is defined as the rate of excitation out of the strongly

fluorescing level-O-level-1 system and into level 2, while

/? _ is the rate back out of level 2. The rate equation

analysis yields the mean fluorescence on-time {T^n),

off"-time (Toff), and the statistical distribution of the

fluorescence as a function of the two Rabi frequencies,

spontaneous decay rates, and laser detunings.

In this paper we analyze the population dynamics of a

single atom in a magnetic field, when a single laser is tuned near one of the principal transition resonances. We assume that the ground state has a ^51/2

configuration and the excited state is ^P^/i- Additionally, the eff"ects of hyperfine structure are assumed to be

negligible. This configuration is realized by some singly

ionized alkaUne-earth atoms, ^'^Mg"'" or ^^Mg"*", for ex- ample. We find that for the time scale of interest, the

evolution of the six ^S,/-, and ^P-in levels is also de- 1/2 scribed by a two-level rate equation and therefore, obeys

the same statistical functions derived for the three-level V configuration. Thus this system should also exhibit

quantum switching. We find that the rates R _,_ and R _

for this level configuration, in contrast to the V

configuration, are quite simple and, as we will show, possess a surprising and elegant feature.'^ Figure 2 schematically shows the Zeeman levels of the

^Si/2 and ^Pi/2 states. It is assumed that the field is sufficiently weak that the levels are well described by the

L5-coupling scheme. For convenience, the levels are la- Work of the U. S. Government

2758 TN-50 Not subject to U. S. copyright
36 QUANTUM JUMPS VIA SPONTANEOUS RAMAN SCATTERING 2759

beled from 1 to 6, starting with the lowest ground level. The energy difference between levels / and j is denoted

by iicOij=fi{a)j—(Oj). A laser of frequency co is assumed

tuned near co^, the ^5'i/2, mj = —j 3/2' m , — —

( l-<-^3) resonance frequency. If A denotes the detuning

from resonance then ^ =^31 + A. M is assumed small

compared to the energy separation between Zeeman levels, which is in turn assumed to be small compared to the ^^3/2 and ^5 1/2 energy difference. The laser Hght is assumed to be linearly polarized with polarization per- pendicular to the magnetic field axis. The only dipole

decay allowed for level 3 is back to level 1. Therefore,

the fluorescence from level 3 continues until an off- resonance transition from level 1 to level 5 is induced.

There is a small probability of a nonresonant excitation

to this level due to the finite linewidth of the transition.

If level 5 becomes populated, the atom may decay to ei- ther ground state: back to level 1 (with j

probability),

where the l<-^3 cycHng continues, or to level 2 (with j

probability), where the electron remains until another

transition, again far from resonance, may remove it. The dipole selection rules allow transitions from level 2

'2 _1

FIG. 2. The energy-level structure of the ^S^/j and the ^Pi/i

states of an atom in a magnetic field. A laser of frequency o) is assumed to be tuned near the level 1 to level 3 transition fre- quency which gives rise to a (on resonance) Rabi frequency of

il. The laser polarization is assumed to allow only Amy = + 1 transitions. The detuning from resonance is denoted by A. The total spontaneous decay rate of each excited level is y and

the energy separation between excited levels is given by Ha/!.

The dashed arrow indicates off-resonance excitation of the 1—>5 transition by the laser. The wavy arrows denote the dipole-allowed spontaneous decays from levels 3 and 5. Off- resonance excitation of the 2-^-4 and 2-^6 transitions and

spontaneous decay from levels 4 and 6 are not shown in the

figure. It is assumed that a »A, 7, n.

(1)

to either level 4 or to level 6. Once in level 4, the atom

may decay back to level 1 (with |

probability) to resume

the strong fluorescence cycling. From level 6, the atom

may decay only to level 2. The other Zeeman ground

level, level 2, is the "shelf level which removes the elec- tron from the strongly fluorescing system. Thus, as in the three-level V configuration, which uses two light sources, the fluorescence from this six-level, one-laser

system is expected to alternate between periods of darkness and light. As opposed to other realizations of quantum jumps, the shelf level in this case "decays" via spon- taneous Raman transitions since spontaneous radiative

decay from level 2 to level 1 is neghgible.

We use the density-matrix formulation to determine

the steady-state level populations and to show that the two-state rate approximation is valid during the observation time of interest. In the rotating-wave approximation, the density-matrix equations are

p 1

1 = n( Imcr ,3 + Ima 15 /V3 ) + 7 (P33 + 2p44/3 +P55 /3 ) ,

P22 = ft(Im(T24/'^^3 + Ima26) + r(p44/3 + 2p55/3+p66) , P33=-mma,3-rp33 ,

P44= — nimCT24/V^3 — 7P44 , P55=-nimai5/V3-7p55 , P66=-ftIma26-rP66 .

d-,3 = [-/(6J-a)3i)-jr]a,3 + j/n(p33-pi,+P3*5/'^^3) ,

0-15 = [-'(«-«5l)-Tr]0-15 + y'^[(p55-Pll)/^3+p35],

(T24 = [-/(w-«42)-yr]^24 + y'^[(p44-P22)/'^3+p|6],

d26=[-/(«-6>62)-Tr]0-26+T'fi(p66-p22+P46/'^3) ,

P35 = (/&)53-7)p35 + jzn(a,5-af3/A/3) ,

P46 = ('^64-r)P46 + T''^(^26/'^3-a24) ,

where D. = ftii =EQdy^/fi is the Rabi frequency of the

l-<-^3 transition, y is the total spontaneous decay rate of

each excited level, Ima,y denotes the imaginary part of a coherence and the asterisk denotes complex conjugation.

The other Rabi frequencies are determined by the rela- tive size of the dipole-matrix elements for light linearly

polarized perpendicular to the quantization axis: c?,5=c?i3/v/3, d24=d2(,/^i, and 0^26= 'i3- There are only six nonzero coherences .'or our choice

of laser polarization. The four aij are due to direct laser coupling of a ground and an excited level, while the two

Pij represent the stimulated Raman coherences between

pairs of excited levels coupled by the laser via a ground

level.

It is most compact to express the detunings, co — cojj in terms of A (A =a; — wji) and the magnetic field splitting

of the levels. In the LS-coupling scheme, the Lande' g

factors are g{-Si/2) = 2 and g("/'.v2^ = T f^^'<^ ^^^^ the electron g factor equal to 2 for this calculation). The de- tunings become

TN-51
2760 RANDALL G. HULET AND D. J. WINELAND 36

6) —6)5i=A —a ,

fu —£L)42 = A +a/4 ,

(y —a)62 = A — 3a/4 .

In the above expressions co^i = a)M =cc, where

ha = ^^BB /l) is twice the Zeeman energy sphtting be- tween adjacent ^^3/2 levels due to the magnetic field B

and /i^ is the Bohr magneton. We will assume hereafter

that a»A,n,)'.

Solving the set of algebraic equation p,, =d-y =p/y =0,

and using the conservation equation ^iPii='^ yields the

steady-state values of the populations p„. Thus to the

first nonzero order of y /a or Vl/a,

16 r^-f4A^+n2

Pu- 17 )/2^4A2 + 2a2

pil- 117 '

16 n^

Pl,i17 Y^^A^^^2Q}

AQ}

(2) P44~ 5\a^

P55== 4n2

51a^

AQ}

Pee 153a^

Terms of order (A /a) have been neglected.

We follow Cook et al.^ by defining new population

variables

p_=Pll+P33+P55 . P+=P22+P44+P66=1-P- •

Therefore, in steady state, p_«|f + 0(A/a) , p+«^+0(A/a) .

The ratio p_/p_,. = 16 + (A/a) is the ratio of the mean

duration of the fluorescence on periods to the off periods. A simple rate equation analysis confirms that the ratio of the ground-state level populations is 16 in the low-intensity limit (fl«Y) and for A = 0. '^ However, we find that this ratio, unlike that for the three-level V configuration, is largely independent of laser intensity

and is only weakly dependent on detuning (for y,n,A«a). The surprising feature here is that this ratio is independent of saturation and power broadening

effects. As fl is increased from zero, pn decreases as population is transferred to level 3. We might expect,

therefore, that the probability of transferring population

from level 1 into the shelving level should diminish.

However, Eqs. (2) indicate that the population of level 5, from which the shelving level (level 2) is populated via spontaneous decay, increases as fi^ with no effect due to saturation of the l*-'3 transition. The apparent explanation of this phenomenon is that population is transferred

P +

16.200

16.000

15.800

A = -1 - ^

A =

- ^

-

A = 1 -

i

1 1 ^^

0.010 0.100 1.000 10.000 100.000 Q (Units of y)

FIG. 3. The ratio of the steady-state population in levels 1,

3, and 5 (p_ system) to the steady population in levels 2, 4, and

6 (p+ system) is plotted vs fl for various detunings, A, and for a = 1215 (all in units of 7). P-/p+ is equal to the ratio of the mean duration of the fluorescence on periods to the off periods.

There is little effect due to saturation of the 1-^.3 transition and only a slight deviation from the value of 16. The downturn in p_ /p+ for very large il is due to the power-broadened

linewidth of the transition becoming comparable to the separation between excited levels.

from level 3 to level 5 via the Raman coherence P35. The increase in P35 with increasing CI exactly compensates for the effect of the decreasing population in level

1. We have used a computer to numerically solve Eqs. (1) for the exact steady-state solutions using various values

of the parameters A and ft. Figure 3 is a plot of p_/p_,_ versus ft for several values of A. The ratio remains re- markably close to the value of 16 for A = and ft«a,

and even for ft»7. The ratio is offset from the value

16 when A^O by approximately — 160(A/a).

Figure 4 demonstrates the importance of including

coherence effects in the calculation. Figure 4 is a plot of

P-/p+ versus ft when the coherences between excited

states P35 and p46 are neglected. P35 and p^(, are due to stimulated, off-resonant Raman coupling. Since this

0.010 0.100 1.000 10.000 Q (Units of y)

100.000

FIG. 4. Same as Fig. 3, except that the stimulated Raman

coherences pjj and p^t have been neglected in the calculation. Saturation of the l-<->3 transition causes p_/p+ to deviate from

16 when fl^y. This demonstrates that the Raman coherences

are responsible for the lack of intensity dependence of p^/p + . TN-52
36 QUANTUM JUMPS VIA SPONTANEOUS RAMAN SCATTERING 2761

coupling is small for ft «r, P-/p+ remains close to 16

in the low-intensity regime. However, P-/p+ rapidly

deviates from 16 when ft is increased beyond y, indicating the increasing role the Raman coherences assume in correctly describing the population evolution as ft in- creases. The dynamics of the population evolution is governed

by rates from two widely differing time domains: (Da

"short" time of order y

' ' and (2) a much longer time in which population is transferred between the p_ and p +

systems. An examination of the &jj equations of Eqs. (1) reveals that the short-time behavior is exponentially

damped in a time 7 "

' . Therefore, for times of interest which are greater than y

~

' we can ignore the short-time

behavior. We would like to show that the evolution of

population between the p_ and p_^_ levels of this system

is similar to that of the V configuration. (The population dynamics also determine the statistics of the emitted

photons.) To do so, we must cast the dynamical equations in the form of a two-state rate equation. If we

identify i? + as the rate out of p_, the strongly fluoresc- ing ( — ) system, and R _ as the rate back into it, the

two-state rate equations are p_ = -R+P-+R-p+ , < p+=R+p--R-P+ .

Population is transferred between the two systems via

spontaneous radiative decay. To switch from the — sys- tem to the + system the atom must decay from level 5 to level 2. The branching ratio for this transition is }.

Similarly, the reverse process requires a decay from level 4 to level 1, also with }

probability. Therefore,

and

/?+P_=(|)rp

R

8ftV

9a ^ + 0(A/a)

55

and

The correctness of these values for R _ and R ^ has been

checked by numerically integrating Eqs. (1) and verifying

that the evolution towards steady state is indeed

governed by these rates. The decay of the short-time be- havior in a time t »y~^ was also verified. The calculations which we have described assume that a laser is tuned near the l<-^3 transition. However, we

note that an entirely analogous set of equations and re- sults are obtained by tuning the laser near the 2*->6 transition. The observation of quantum jumps in this system provides an excellent opportunity for comparison with

theory. The very weak dependence of p_ /p^

(=R _ /R _^_) on all of the parameters which can fluctu- ate in an experiment means that the fluorescence statis- tics may be very accurately measured and compared

with calculation. Additionally, the switching rate can be

carefully normalized by observing a portion of the laser beam transmitted through the apparatus. An experiment is now underway at the National Bureau of Stan- dards (NBS) to observe these effects. A single ^"^Mg"^ ion

has been confined by an electromagnetic (Penning) ion

trap while a nearly resonant laser drives the

3^Si/2,my = -|<-^3^P3/2,my = -|(l*^3) transition. This new apparatus has substantially better sensitivity

than a previous experiment which was able to detect a

single Mg"^ ion but which did not have the sensitivity to detect quantum jumps. '^ For a magnetic field of 1.4 T,

the Zeeman sphtting of the excited states is i-a~j( 12157) = (2i7)26.1 GHz for ^'^Mg + , where

7 = (27r)43 MHz. Therefore, the regime of a »7, ft, A

is quite readily achieved experimentally.

R-p+=(j)rp44

Using the expressions [Eqs. (2)] for the p„ gives

R + =-^+0{A/a)

The authors gratefully acknowledge the support of the U.S. Air Force Office of Scientific Research and the U.S.

Office of Naval Research. One of us (R.G.H.) acknowledges the support of the National Research Council.

We thank R. J. Cook and L. Hollberg for their helpful

suggestions concerning the manuscript.

'H. J. Kimble, M. Dagenais, and L. Mandel, Phys. Rev. Lett. 39,691 (1977). ^See, for example, J. D. Cresser, J. Hager, G. Leuchs, M. Rateike, and H. Walther, in Dissipative Systems in Quantum

Optics, Vol. 27 of Topics in Current Physics, edited by R.

Bonifacio (Springer-Verlag, Berlin, 1982) and references

therein; F. Diedrich and H. Walther, Phys. Rev. Lett. 58, 203 (1987). ^R. J. Cook and H. J. Kimble, Phys. Rev. Lett. 54, 1023 (1985). ^H. G. Dehmelt, J. Phys. (Paris) 42, C8-299 (1981).

^T. Erber and S. Putterman, Nature (London) 318, 41 (1985).

^J. Javanainen, Phys. Rev. A 33, 2121 (1986). ^A. Schenzle, R. G. DeVoe, and R. G. Brewer, Phys. Rev. A

33, 2127 (1986). ^C. Cohen-Tannoudji and J. Dalibard, Europhys. Lett. 1, 441

(1986). ^D. T. Pegg, R. Loudon, and P. L. Knight, Phys. Rev. A 33. 4085 (1986). 'OA. Schenzle and R. G. Brewer, Phys. Rev. A 34, 3127 (1986). "H. J. Kimble, R. J. Cook, and A. L. Wells, Phys. Rev. A 34. 3190(1986).

'2p. Zoller, M. Marte, and D. F. Walls, Phys. Rev. A 35. 198

(1987). '^W. Nagourney, J. Sandberg, and H. Dehmelt, Phys. Rev.

Lett. 56, 2797 (1986).

'•j. C. Bergquist, R. G. Hulet, W. M. Itano, and D. J. WineTN-53
2762 RANDALL G. HULET AND D. J. WINELAND 36

land, Phys. Rev. Lett. 57, 1699 (1986). '^R. G. Hulet and D. J. Wineland, Bull. Am. Phys. Soc. 31,

'-'^Th. Sauter, W. Neuhauser, R. Blatt, and P. E. Toschek, Phys. 938 (1986). Rev. Lett. 57, 1696 (1986). '^D. J. Wineland, J. C. Bergquist, W. M. Itano, and R. E. •^M. A. Finn, G. W. Greenlees, and D. A. Lewis, Opt. Com- Drullinger, Opt. Lett. 5, 245 (1980). mun. 60, 149 (1986). i^j) j wineland and W. M. Itano, Phys. Lett. 82A, 75 (1981).

TN-54
RAPID COMMUNICATIONS

PHYSICAL REVIEW A VOLUME 37, NUMBER 1

JUNE 1988

Precise test of quantum jump theory

Randall G. Hulet,* D. J. Wineland, J. C. Bergquist, and Wayne M. Itano Time and Frequency Division, National Bureau of Standards, Boulder, Colorado 80303

(Received 16 November 1987)

Quantum jumps due solely to spontaneous Raman scattering between the Zeeman sublevels of

a single ^''Mg'^ ion have been observed in the fluorescence emitted by the ion. A theory of quan- tum jumps for this system predicts that coherences between excited levels cause the ratio of the mean duration of the "fluorescence-on periods" to the mean duration of the "fluorescence-off periods" to be independent of laser intensity. The measured value agrees with the predicted one

to within the measurement precision of 2%. The distribution of the durations of the off periods

also agrees with theory.

A few groups have recently observed "quantum jumps"

between the energy levels of a single atomic ion as signaled by an abrupt change in the fluorescence emitted by

the ion.'"^ In another experiment/ quantum jumps were

inferred from the statistics of the fluorescence emitted by

neutral atoms in a weak beam. In these experiments, the

quantum jump was a transition (either stimulated or spontaneous) into or out of a metastable excited state. In this paper, we report the observation of quantum

jumps due solely to spontaneous Raman transitions into and out of ground-state sublevels. As opposed to the earlier experiments, we quantitatively compare the predictions of the theory with the experiment. A particular real- ization of the process^ is illustrated in Fig. 1. It employs

just one radiation source which is tuned between a pair of

Zeeman levels of an atom or ion with a ^iS'1/2 ground state

mi

\ 6

\ 5 -\ 4

2Si

^'3^1^/31'

h

"/.

FIG. 1. The energy-level structure of the 51/2 and Pyi states of an atom in a magnetic field (not to scale). A laser of frequen- cy CO is assumed to be tuned near the S\n, mj = — j^-'Pi/i,

fnj = — J transition frequency wo. The energy separation of adjacent />3/2 iSxn) sublevels is ha/2 (3/ja/4). The indicated Lorentzian widths and detuning A, which are much less than a, have been exaggerated for clarity.

and an excited '^P^n state (for example, an alkali atom or

singly ionized alkaline earth atom). The frequency (o of

the radiation is tuned near the 5i/2, mj= — j -^P-m,

ntj = — J transition frequency coq. The atom cycles nearly continuously between these levels since the dipole selec- tion rules allow spontaneous decay only to the original

ground level. A steady stream of fluorescence photons,

which are readily detected, is emitted by the atom during

this period. However, if the radiation is linearly polarized

with the polarization direction perpendicular to the direction of the magnetic field, the 51/2 'nj= — j '--Pyj,

nij = J transition is also allowed, although it is far from

resonance. This transition is indicated by the dashed ar- row in Fig. 1. A spontaneous decay from the upper

mj = J level can then leave the atom in the my = y ground level. This spontaneous Raman transition into the mj = J ground level takes the atom out of the

nij = — J '-^mj = — 2" cycling loop, causing the emitted

fluorescence to suddenly stop. The off"-resonant 5 1/2, mj = + J *-*Py2, my = - y — 5 1/2, my = - y spontane- ous Raman transition (not shown in Fig. 1) will return the atom to the cycling loop where it will resume scattering.

In a previous paper, ^ we calculated the behavior of this system when the magnetic field strength is large enough

that a^y, ft. A, where ha/2 is the energy separation be- tween adjacent excited-state sublevels, 3 ha/4 is the separation between the ground-state sublevels, y is the excit- ed-state spontaneous decay rate, ft is the on-resonance Rabi frequency for the 51/2, my = — y — Py2, my = — y

transition, and A = ft) — <uo (see Fig. 1). We have used a single ^"'Mg'*' ion confined by the static magnetic and electric fields of a Penning trap^ to demonstrate this process. The magnetic field for the data

presented here was 1.39 T which gives a = (2;r)52

GHz=l200y, where 7 = (2;r)43 MHz. Therefore, the condition a» y, ft, A is quite readily achieved in our ex- periment. For the trap used in this work (Fig. 2), we achieved a large collection solid angle ( > 1% of 4;r sr), compared to an earlier apparatus,** by splitting the ring electrode into two halves in the plane perpendicular to the magnetic field direction.^ The relative trap dimensions were chosen,

with the aid of a computer, to make the potential nearl\

quadratic at the trap center.'" The pressure of back22 4544 TN-55 Work of the U.S. Government

Not Subject to U.S. Cop> right
RAPID COMMUNICATIONS

il PRECISE TEST OF QUANTUM JUMP THEORY 4545

ELECTRON RIAMENT END CAP 100

END CAP

FIG. 2. Cross-sectional drawing of the Penning trap used in the experiment. The shaded regions are ceramic support rods. The end-cap separation is 1.38 cm and the inner diameter of the

ring electrode is 1.63 cm. The two ring electrodes and the two

compensation electrodes were grounded and the two end caps were at -1-12 V for this experiment.

4 8 12 16 20

Off duration (ms)

FIG. 3. A histogram of the duration of fluorescence-off periods for a data run. The number of off periods which have a particular duration are plotted vs the duration. This data run had 512 off periods in 20 s. The solid line is a fit of the data to Eq. (2). Inset: A portion of a data run showing quantum jumps

in fluorescence F as a function of time.

ground gas in the apparatus was less than 10 ~ Pa. The 351/2 to 3/* 3/2 transition wavelength for Mg"*" is 280 nm. We generated up to 200 /iW of 280-nm radiation by frequency doubling the output of a single-mode,

continuous-wave dye laser in a deuterated ammonium

dihydrogen-phosphate crystal. The frequency width of

this radiation was much less than the natural width y of

the transition. The laser beam was focused near the center of the trap with a beam waist wq —45 /jm. The

ions were laser cooled when A < 0, and since the beam was

directed at an angle of 74° relative to the magnetic field axis all. of the ion's degrees of motion were cooled directly.

The equilibrium temperature of the ion was only a fraction of 1 K when A= — 7/2. '

' Under these conditions, the Doppler broadening of the transition frequency was much

less than y, as verified by a scan of the laser frequency,

and the ion was confined to dimensions which were much

less than wo. In order to laser cool in a Penning trap, it is necessary to position the center of the laser beam radially

outward from the trap axis.

'

' With a displacement of ap- proximately 20 /im and our value of wq, the saturation pa- rameter n Vy^ was equal to 1 at the position of the ion for a laser power of approximately 50 /iW. From the

definitions of Ref. 6, the ion spends twice as much time in the ground my = — y sublevel as in the excited mj = — j

level when n=y and A=0. The fluorescence photons were collected in a direction perpendicular to both the magnetic field and laser beam directions by focusing onto

a photomultiplier tube. The maximum detected photon

count rate was 2x lOVs for a single ion. Data exhibiting quantum jumps are displayed in the inset in Fig. 3. The horizontal axis is divided into time in- tervals of 0.5 ms duration. The vertical axis indicates the number of photons counted during each time interval and

the points representing each interval are connected by

straight lines. The sudden changes in fluorescence level are due to quantum jumps between the ground-state sublevels via spontaneous Raman transitions. For other realizations of quantum jumps, the dynamics

of the population evolution and, therefore, the statistics of

the emitted fluorescence are governed by effective twostale rate equations.''^ That is, there are rates R+ and

R - for the atom to make transitions out of or into the en- ergy levels which comprise the strongly fluorescing or "on" system and the system of levels which comprise the

"off' system. (R + )~^ and (/?-)"' are equal to the mean duration of the fluorescence on periods, (To,,), and

off" periods, (Tofr), respectively. Although the system considered here is substantially different than those investi- gated before, it was shown in Ref. 6 that the dynamics of

this system should also be governed by effective two-state

rate equations. The expressions for the rates R - and R +, ^+=<7'on)"'=nV/(l8a^) ,

R-={To{r)'^=Sil^y/(9a^) ,

(1)

are simple because of the off-resonance nature of the excitation, in contrast with the previously considered sys- tems.'^ A consequence of this type of dynamics is that the dis- tribution of fluorescence-on and fluorescence-off" times

should be exponential. In Fig. 3, we plot the duration of

fluorescence-off" periods versus the number of periods in the data run which are off" for this duration. The solid line

is a fit to the equation

H^ofT(f)=(7"off>-'exp(-r/(7"off)) (2)

The data are consistent with a distribution of this form.

An important prediction of Ref. 6 is that the ratio of the mean duration of the fluorescence-on periods to the mean

duration of the fluorescence-off" periods should be a value

which is only very weakly dependent on the variable pa- rameters of the system, such as laser power and frequency. This is in contrast with the rather strong dependence

of this ratio on system parameters in the other experiTN-56
RAPID COMMUNICATIONS

4546 HULET, WINELAND, BERGQUIST, AND ITANO 12

TABLE I. A summary of the data for seven runs with different values of the saturation parameter

ft Vy^. Also shown are the calculated results when the stimulated Raman coherences between excited

states are either included or neglected. Each data run contained 60000 time intervals whose length was

either 0.25, 0.33, or 0.5 ms. (rofr) ranged from 9.5 ms for run 1 to 2.3 ms for run 7. Data runs in which

the laser frequency varied excessively have been omitted from the table.

(Ton)

Calcul.ited (ro„)Measured

(ro„)

Run ftV -My

<rofr) (without coherence) (with coherence)

(T-off)

1 0.65 ±0.09 0.67 ±0.03 19.1 ±0.2 16.09 15.76±0.84

2 1.82±0.16 0.22±8:i8 25.8 ±0.5 16.03 16.02 ±0.67

3 1.89±0.I6 0.20±8:ig 26.1 ±0.6 16.03 15.52±0.58

4 2.02 ±0.1

0.84 ±0.05 21.7±0.3 16.11 16.58±0.59

5 2.45 ±0.25 1.07 ±0.05 21.0±0.2 16.13 16.13±0.70

6 2.61 ±0.20 0.60 ±0.05 24.4 ±0.6 16.08 17.35 ±0.70

7 2.64 ±0.1

0.50 ±0.06 25.2±0.6 16.07 16.26 ±0.69

merits. ' * Reference 6 shows that, for a» y, n, A,

<ron)A7'off> = 16 + 0(Va) . (3)

We might intuitively e.xpect that (ron)A7'off) would in- crease by a factor of 2 when the 51/2, my == — j — P3/2,

mj = — \ transition is saturated, since approximately

half the population which was available in the mj = — j

ground level to be transferred to the inj = j ground level

is "stranded" in the excited state. However, the stimulated Raman coherence between the mj = — j and mj = j

excited-state sublevels, through the my = — y groundstate sublevel, compensates for the depletion of population

in the ground state.*' We have investigated this phenomenon by measuring

(ToJ/iToff) for various values of the saturation parameter

ftVy^. Table I shows the results of analyzing seven

different data runs with saturation parameters ranging be- tween 0.65 and 2.64. The value of the saturation parameter and its uncertainty were obtained by fitting the distri- bution of fluorescence-off time durations to Eq. (2) to find

(Toff) and its uncertainty. Equation (l) then relates (Toft) to ft. This procedure gives results which are consistent

with estimates of the expected saturation parameter from

independent measurements of the laser power and the

laser beam waist wq. The rate of fluorescence emitted by

the ion during the fluorescence-on periods depends on the average population in the exited tnj = — j level, which

depends on the detuning A. ^ Therefore, if the count rate

for A =0 is known, then A for each data run is determined

by the mean detected count rale for the fluorescence-on

periods of that run. The count rate for A =0 is found from

the maximum detected rate when the laser is scanned

through the resonance.

'"^'''' The uncertainty given for A in Table I is due to the uncertainty in the count rate for A=0. The fourth column gives the expected (ron)A7'oflr^ ratio if the coherences betvveen excited states are neglected.^ The fifth column is the expected value of the ratio when the coherences are included in the calculation. Since A.-^O, there is a small departure from the value of

16. The uncertainty in this calculated ratio is small compared to the uncertainty in the measured value of

iTon)/iTofi-) and is therefore not reported. Finally, the last

column is our measured value for (ron)A7'oflr) for each

run. This quantity was extracted from the data by fitting the distribution of detected counts per time interval for each run to a sum of two Gaussian curves, one corre- sponding to the fluorescence-off periods and the other to the fluorescence-on periods. The ratio (ron)Aro(T) is then

equal to the ratio of the areas under the Gaussian curves. The uncertainty given is estimated from the uncertainty in determining the fitting parameters. The mean for the seven runs is 16.23 and the sample standard deviation is 0.60. Combining the seven runs (weighted equally) gives

(ron)Aron) = 16.23 ±0.26. A comparison of columns 5 and 6 reveals that our data are quite consistent with the

result of the coherent theory, which for the given set of

values of A gives a mean value for the ratio of 1 6.08. In this experiment, spontaneous Raman transitions in- duced by a single laser are responsible for quantum jumps

into and out of the long-lived "shelving' level.' The

shelving level is a ground-state sublevel rather than an

electronic excited metastable state as in the previous

demonstrations. The properties of this level structure have allowed the theory of quantum jumps, in which the dynamics of the jumps are described by effective two-level

rate equations, to be tested with high precision. This ex- periment is a realization of "single-atom optical pumping." Ordinarily, the effects of optical pumping are evi- dent only in the time-averaged values of the level populations. However, in single-atom optical pumping, the dynamical evolution of the level populations is revealed by

quantum jumps in the fluorescence intensity. Finally, the existence of stimulated Raman coherences between excit- ed states has been shown to play an important role in both

the population dynamics and in the steady-state values of

the level populations.

The authors thank C. Weimer for help in constructing

parts of the apparatus and E. C. Beaty for performmg

computer calculations for the trap design. We also thank

L. Hollberg and F. Diedrich tor helpful comments on the manuscript. The authors gratefully acknowledge the si:pport of the Air Force Oflice of Scientific Research and the Oflice of Naval Research. One of us (R.G.H.) thanks the National Research Council for support. TN-57
RAPID COMMUNICATIONS

21 PRECISE TEST OF QUANTUM JUMP THEORY 4547

'Present address: Physics Department, Rice University, Houston, TX 77251. 'W. Nagourney, J. Sandberg, and H. Dehmelt, Phys. Rev. Lett. 56,2797 (1986).

2j. C. Bergquist, R. G. Hulet, W. M. Itano, and D. J. Wineland,

Phys. Rev. Lett. 57, 1699 (1986). ^Th. Sauter, W. Neuhauser, R. Blatt, and P. E. Toschek, Phys. Rev. Lett. 57, 1696 (1986).

"^M. A. Finn, G. W. Greenlees, and D. A. Lewis, Opt. Commun.

60, 149 (1986). 5r. G. Hulet and D. J. Wineland, Bull. Am. Phys. Soc. 31, 938

(1986). ^R. G. Hulet and D. J. Wineland, Phys. Rev. A 36, 2758 (1987).

'See, for example, 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).

*D. J. Wineland and W. M. Itano, Phys. Lett. 82A, 75 (1981).

^M. H. Prior and H. A. Shugart, Phys. Rev. Lett. 27, 902

(1971).

'OE. C. Beaty, J. Appl. Phys. 61, 2118 (1987).

"W. M. Itano and D. J. Wineland, Phys. Rev. A 25, 35 (1982).

'2h. J. Kimble, R. J. Cook, and A. L. Wells, Phys. Rev. A 34, 3190 (1986), and references therein. '^R. E. Drullinger, D. J. Wineland, and J. C. Bergquist, Appl.

Phys. 22, 365 (1980).

'''W. Nagourney, G. Janik, and H. Dehmelt, Proc. Natl. Acad.

Sci. U.S.A. 80, 643 (1983).

TN-58
Volume 59, Number 24 PHYSICAL REVIEW LETTERS 14 December 1987

Radiative Decay Rates in Hg '*' from Observations of Quantum Jumps in a Single Ion

Wayne M. Itano, J. C. Bergquist, Randall G. Hulet/""^ and D. J. Wineland

Time and Frequency Division, National Bureau of Standards, Boulder, Colorado 80303

(Received 17 September 1987)

Radiative decay rates connecting the lowest four energy levels of "^Hg"*" have been derived solely from an analysis of the fluctuations (quantum jumps) of the laser-induced fluorescence of the 194-nm

first resonance transition of a single ion confined in a Paul trap. The natural linewidth of the 194-nm

first resonance transition was also measured. The measured decay rates and branching ratios are in sa- tisfactory agreement with theory.

PACS numbers: 32.70.Fw, 32.70.Cs, 32.80.Pj

Recently it has become possible to observe for extended periods the fluorescence from a single, isolated ion,

kept in place by the electric fields of an ion trap. This

capability has made it possible to study quantum jumps

(sudden transitions between atomic energy levels) '"^ and

nonclassical photon statistics (photon antibunching).^"

In this Letter we report measurements of five radiative

decay rates connecting the lowest four energy levels of

'^^Hg"*". These measurements depend on the observation

of transition between the energy levels of a single ion, as exhibited by the changes in its fluorescence. This would

not be possible with a sample of many ions observed

simultaneously. Related methods have previously been

used to measure lifetimes of metastable levels in Ba"*" (Refs. 1 and 2) and Hg"^ (Ref. 3). An unusual feature of the techniques used in the pres- ent work is that decay branching ratios are measured

without our observing the decay photons directly. It is not necessary to calibrate the detector efficiency, and it is necessary only to observe one wavelength. Instrumental

noise, such as that due to changes in the laser intensity or that due to statistical fluctuations in the numbers of

photons detected, is unimportant compared to the in- herent quantum mechanical randomness in the behavior

of a single atom. This property is shared by the quantized optical double resonance technique recently used to observe narrow resonances in a single Hg"*" ion.^ The lowest four energy levels of Hg"*" are shown

in Fig. 1. After the ion is excited from the ground

5d^%s ^5 1/2 level to the 5d^%p ^Pui level, it usually de- cays back to the ground state by electric dipole (£ 1 ) ra- diation, but has a small probability of undergoing an E 1 decay to the metastable 5d^6s^^Di/2 level. This probability is small because it requires configuration mixing in order to occur (the nominal configurations differ in two

orbitals). In addition, the transition rate is proportional

to the cube of the energy difTerence between the ^P\/2

and the ^Dy2 energy levels, which is small compared to that between the ^7*

1/2 and the ^5"1/2 energy levels. From

the ^Di/2 level, the ion decays either directly to the ground state, by electric quadrupole (El) radiation, or

to the metastable 5d^6s^^Ds/2 level, by magnetic dipole

(A/1) and E2 radiation. According to calculations,^ the E2 decay rate to the ^Ds/2 level is negligible compared

to the M\ rate. The ^Ds/2 level decays to the ground

state by £2 radiation. According to simple estimates,^ the magnetic quadrupole (Af2) and electric octupole

(£3) allowed decay rate from the ^P\/2 level to the ^Dsn

level is much too small to observe in the present experi- ment. The apparatus has been described previously.^'* The

Hg"*" ions were confined in a radio-frequency (Paul) trap

under high-vacuum conditions. The ions were loaded

into the trap by ionization of Hg atoms inside the trap

volume with an electron beam. A beam of narrow-band cw 194-nm radiation of about 5 )uW was focused to a waist size wq of approximately 5 /jm at the position of the ions. The frequency of the 194-nm radiation was

tuned below the resonance, in order to provide laser cool- ing of the ions. The 194-nm fluorescence photons were

collected by a system of lenses and focused on the Cs-Te

photocathode of a solar-blind photomultiplier tube. The

numbers of photons counted in consecutive time intervals

11 ^m

194 nm

excitation

282 nm

FIG. 1. Diagram of the lowest four energy levels of Hg"*". The decay rates from all three excited levels were determined

by observation of only the 194-nm fluorescence.

2732

Work of the U. S. Government

Not subject to U. S. copyright

TN-59
Volume 59, Number 24 PHYSICAL REVIEW LETTERS 14 December 1987

c/o

c

Z3O

o

(DO

c

(DO

CO0)o

150

100

150

100 -

150

100

400

Time (ms)

800

FIG.

(middle

straight

2. Intensity

trace), and

lines.

of 194- one ion nm laser-induced fluorescence as a function of time for samples of three ions (highest trace), two ions (lowest trace) in the trap. The integration time per point was 1 ms, and the points were connected by

were recorded by a computer.

Figure 2 shows traces of the fluorescence intensities as functions of time when three ions, two ions, and one ion were in the trap and irradiated with radiation near the

^S\i2-\.o-^P\/2 transition frequency. When a single ion was in the trap, as in the lowest trace in Fig. 2, the 194- nm fluorescence intensity had two observable states. When the ion was cycling between the ^5 1/2 level and the

'^P\I2 level (the "on" state), the fluorescence intensity

had a constant value; when it was in either the '^D^n or

the '^Dsn metastable level (the "ofT' state), it was zero. When more than one ion was in the trap, as in the

highest and the middle traces of Fig. 2, the fluorescence

intensity at a given time was proportional to the number

of ions that were in the on state at that time, so that it took one of several discrete values. The total radiative decay rate of the ^P\/2 level (7/.) was determined by measurement of the width of the 194-nm transition. The decay rate yp, in units of inverse seconds, is equal to the natural linewidth, in units of ra- dians per second. Only the low-frequency half of the resonance was observed, since the ion heated rapidly

when the frequency of the 194-nm radiation was above

the resonance center. ^'^ The frequency of the 194-nm

radiation was swept slowly enough that the transitions between the on and off states were clearly resolved, and a smooth curve was interpolated through the on-state data. The elTect of power broadening was removed by an ex- trapolation to zero intensity of the 194-nm radiation. A

similar method has been used previously by Nagourney,

Janik, and Dehmelt to measure the natural linewidth of

the 280-nm resonance line of Mg"^. '° The temperature was measured in a separate experiment to be a few millilkelvins, ^ for the range of frequency detunings required

for this line-shape analysis. This temperature was low

enough that the Doppler broadening was not significant.

The result was //> = (3.8 ± 0.6) x 10^ s ~'. The radiative decay rate from the '^P\i2 level to the

^Z)3/2 level [/(Z*— D)\ was determined by measurement

of the rate of transitions from the on state to the ofi" state

for a single ion as a function of the 194-nm fluorescence

intensity. The transition rate /?+ from the on state to the off state was obtained by division of the number of

transitions in a run by the time that the ion was in the on

state in that run. This rate is equal to the product of y{P-^D) and pOP\/2), the mean population of the

^P]/2 level when the ion is in the on state. The fluores- cence intensity in the on state is also proportional to P(^P\/2y P^^P\n) could be varied by variation of either the intensity or the frequency detuning from resonance

of the 194-nm radiation. It was found experimentally

that R-ir was proportional to the observed photon counting rate in the on state. A calibration of the value of

p0P\/2) was made by measurement of the fluorescence

intensity near resonance for different intensities of the

incident 194-nm radiation. The value of P(^P\/2) at res- onance is equal to (2+l/5')~', where S, the saturation

parameter, is a quantity proportional to the incident

194-nm intensity. Figure 3 is a plot of Y, the observed

counting rate at resonance, as a function of A', the 194-

TN-60 2733
Volume 59, Number 24 PHYSICAL REVIEW LETTERS 14 December 1987

c

D

o

o

(Uo

c

0)o

w

(Uo

0.5 1.0 1.5 2.0 2.5 3.0 194 nm intensity (arbitrary units)

FIG. 3. Fluorescence intensity at resonance of a single Hg"*"

ion as a function of the intensity of the 194-nm radiation

(dots) and a least-squares fit (curve).

nm intensity in arbitrary units. The solid curve is a

least-squares fit of the data by the function Y=Ai2

+B/X)~\ The value of POPm) for a given 194-nm

intensity and detuning is the observed counting rate di- vided by A. The value of yiP—^ D) was determined by

division of the value of /?+ for a particular run by the value of p0P\/2) for the same run. The result was y{P—> D) =52 ±\6 s~'. The uncertainty in the deter- mination of A from the least-squares fit and the uncertainty in the determination of R+ contributed about

equally to the uncertainty in yiP^^ D).

The average behavior of the ion from the time it de- cays to the ^Z)3/2 level until it returns to the ground state

is governed by the rate equations for the probabilities P\,

Pi, and Pi of being in the ^Z)3/2 level, the ^Dsn level, and the ^5 1/2 level, respectively:

dP^U)|dt = -y^P^U),

dPiiOldt =f2YiPi(t) - yiPzit),

dPiU)ldt =/, yiPi (r) + yiPjit).

(1)

(2)

(3)

Here y\ and 72 are the total radiative decay rates of the

^Dyi level and. the "^D^n level respectively. The probabilities for the ^Dyi level to decay to the ^51/2 level or to the ^Dsn level are f\ and fi (/i+/2 = l). The initial conditions are P|(0) = 1 and ^2(0) =/'3(0) =0. The

quantity observed experimentally is the probability dis- tribution W^off(T) of the duration of off periods as a function of the duration t. This quantity is proportional to dPi(t)/dt evaluated at / =t, which is the average rate at which the ion returns to the ground state after a period

T. Solving Eqs. (l)-(3) yields the expression

W^off(r)«/2y2exp(- )'2t) + (/iyi ~y2)exp(-yiT). (4)

The distribution of about 20000 off periods is shown in Fig. 4, together with a least-squares fit by a sum of two

D

o

Q.

E

1000

100

100 200 300

Duration of off period (ms)

FIG. 4. Distribution of fluorescence-off periods for a single ion (dots) and a least-squares fit (curve). These data represent an analysis of approximately 40000 quantum jumps (up or down).

decaying exponentials. The values of the parameters obtained from the fit are yi = 109±5 s~', y2 = l\.6±0A

s~\ and /i =0.491 ±0.015. The partial decay rates of

the ^Dy2 level are /,yi =53.5 ± 2.0 s"' and /271 =55.5

±4.0s"'.

In order to place a limit on the effects of collisions on

the measured lifetimes, data were collected at higher

pressure. No shift of the decay rates of the metastable

levels was observed, even when the pressure of the background gas, consisting mostly of Hg, was increased by

over a factor of 10 above the normal level. The relative pressure was estimated from the time required to create an ion in the trap by electron-impact ionization and also from the decrease in the confinement time of the ion in the trap. The latter measurement was based on the observation that ions were lost from the trap when both

194-nm resonance radiation and neutral Hg atoms were

present. We think that this process was caused by reactions of excited-state ions with neutral Hg atoms. The

kinetic energy given to an ion by such a reaction could

have been sufficient to cause it to be lost from the trap. All other perturbing effects, such as transitions out of the metastable levels induced by the 194-nm radiation or the trap electric fields, are estimated to have been negligible.

The present measurement of yp agrees with a previous

beam-foil measurement of (4.3 ± 0.7) x 10^ s~' by Erik- sen and Poulsen" but disagrees somewhat with a previ- ous optical-absorption measurement of 2.65 x 10^ s~' by

Bruneteau et al. '^ An unpublished calculation by AlSalameh and Silfvast'^ (AS) gives a value of 5x10^

s~', in fair agreement with the present measurement.

Other theoretical values are given in Ref. 11. No previ- ous measurements of y(P—*D) have been reported.

The present value is in good agreement with the value of

2734 TN-61
Volume 59, Number 24 PHYSICAL REVIEW LETTERS 14 December 1987

55 s~' calculated by AS. '^ No previous measurements

of the partial decay rates /i/i and/271 of the ^Dm level have been reported. Johnson'"* has reported a value for the total decay rate y\ of 50 ±5 s~', which disagrees

with the present value. Garstang^ has calculated the de- cay rate /i 71 to be 54 s~', in very good agreement with

the present value. The other decay rate f2y\ has been

calculated by Garstang^ to be 42 s~' and by AS'^ to be

75.6 s~'. The present value is in fair agreement with

both calculations. Two previous measurements of the

^Z)5/2 level decay rate 72 have been reported by Bergquist

et al.^-^ The values obtained by the two methods were

1 1 ± 2 s '

' (Ref. 8) and 1 1 ± 4 s ~' (Ref. 3), in good

agreement with the present value. A previous measure- ment by Johnson ''* of 72 = 10.2 ±0.5 s~' disagrees

slightly with the present value. The values of 72 calculated by Garstang^ (9.5 s~') and by AS'^ (12.6 s"')

are in fair agreement with the present value. We acknowledge the support of the U. S. Air Force

Office of Scientific Research and the U. S. Office of Naval Research.

<a)present address: Physics Department, Rice University,

Houston, TX 77251.

'W. Nagourney, J. Sandberg, and H. Dehmelt, Phys. Rev.

Lett. 56, 2797 (1986). ^Th. Sauter, W. Neuhauser, R. Biatt, and P. E. Toschek,

Phys. Rev. Lett. 57, 1696 (1986); Th. Sauter, R. Blatt, W. Neuhauser, and P. E. Toschek, Opt. Commun. 60, 287

(1986).

^J. C. Bergquist, R. G. Hulet, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 57, 1699 (1986).

'F. Diedrich and H. Walther, Phys. Rev. Lett. 58, 203

(1987).

^J. C. Bergquist, W. M. Itano, and D. J. Wineland, Phys. Rev. A 36, 428 (1987), ^R. H. Garstang, J. Res. Natl. Bur. Stand. Sect. A 68, 61 (1964).

^B. W. Shore and D. H. Menzel, Principles of Atomic Spectra (Wiley, New York, 1968), p. 440.

*J. C. Bergquist, D. J. Wineland, H. Hemmati, H.-U.

Daniel, and G. Leuchs, Phys. Rev. Lett 55, 1567 (1985). ^R. E. Drullinger, D. J. Wineland, and J. C. Bergquist, Appl.

Phys. 22, 365 (1980).

'°W. Nagourney, G. Janik, and H. G. Dehmelt, Proc. Natl. Acad. Sci. U.S.A. 80, 643 (1983).

"p. Eriksen and O. Poulsen, J. Quant. Spectrosc. Radiat. Transfer 23, 599 (1980).

'^A. M. Bruneteau, A. M. Icole, C. Rouille, A. Poquerusse,

and H. J. Doucet, Phys. Lett. 46A, 309 (1974).

'^D. Al-Salameh and W. Silfvast, unpublished calculations. "•C. E. Johnson, Bull. Am. Phys. Soc. 31, 957 (1986).

TN-62 2735
PHYSICAL REVIEW A VOLUME 38, NUMBER 1 ^^^ ' ~^Q

JULY 1, 1<

Photon antibunching and sub-Poissonian statistics from quantum jumps

in one and two atoms

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

Time and Frequency Division, National Bureau of Standards, Boulder, Colorado 80303

(Received 22 February 1988)

Antibunching and sub-Poissonian statistics of the 11-^m photon emission from the 5d^°6p^Pi/2

to 5d^6s^^Dy2 transition in one and two trapped Hg"*" ions have been inferred from the quantum

jumps of the laser-induced fluorescence of the first resonance transition at 194 nm. Each downward step in fluorescence was assumed to mark the emission of an ll-/jm photon. Signals from two ions were consistent with the assumption that quantum jumps in the two ions occurred in- dependently. Mandel's Q parameter was determined to be approximately —0.25 for one and two

ions.

Studies of collections of a few atoms or of individual

atoms reveal properties which are not easily observed with

samples of many atoms. One such property is antibunching of the emitted photons. A closely related, though distinct, phenomenon is sub-Poissonian statistics in the number of photons emitted in a given time interval. Photon

antibunching has been observed from atoms in a dilute atomic beam by Kimble, Dagenais, and Mandel' and by

Rateike, Leuchs, and Walther,^ and from atomic ions in a Paul trap by Diedrich and Walther.^ Sub-Poissonian photon statistics have been observed from single atoms by

Short and Mandel"* and by Diedrich and Walther.^ SubPoissonian photon statistics have also been observed in Franck-Hertz light, ^ in parametric down conversion,^ and

from squeezed-light generators.^

Another effect that might be observed in collections of a few atoms is the onset of cooperative atom-field processes, which are known to occur with large numbers of atoms.

Sauter, Blatt, Neuhauser, and Toschek have reported observing simultaneous quantum jumps of two and more Ba "*" ions at a rate higher than that expected from random

coincidences.^ They attributed these multiple quantum

jumps to a collective interaction of the atoms and the radi- ation field. According to a calculation by Lewenstein and

Javanainen,' collective effects occur only if the atoms are separated by much less than the transition wavelength.

We have previously reported an indirect observation of

photon antibunching. "^ Transitions from the ^5i/2 ground

state to the ^Ds/2 metastable level of a single Hg"*" ion, in- duced by a 282-nm radiation source, were observed by

noting the sudden cessation of the 194 nm laser-induced

fluorescence of the ^5'i/2 to ^P\/2 transition (see Fig. 1). When the fluorescence suddenly reappeared, we assumed

that the ion had returned to the ground state by emitting a 282 nm photon. The correlations between the times of

these assumed emissions showed antibunching.

In the work reported here, the statistical properties of

1 1 )um photons emitted from one and two Hg"*" ions were

studied. These photon emissions were inferred from the

state changes of the Hg"*" ion; these state changes were in- dicated by the abrupt changes in the 194 nm ion fluores- cence. This is similar to the way in which the state of a Rydberg atom has been used to infer properties of the

photon field in a cavity. '

' This indirect method of detect- ing the photon emissions was the same as in Ref. 10, but a

diff'erent transition was involved. Some advantages of this detection method are high efficiency (about 95%) and low

background. Collisions, which can either cause or mimic

state changes of the ions, '° were previously shown to be

negligible. Moreover, such independent random events would tend to decrease the antibunching and subPoissonian statistics observed here. The four lowest energy levels of Hg"*" are shown in Fig.

1. The radiative decay rates and branching ratios have

been measured previously. '^~'^ In our experiment, the

ion is subjected to coherent 194 nm radiation, which

drives it from the ground 5d^^6s^Si/2 level to the 5d^^6p^P\/2 level. Most of the time, it decays directly

back to the ground state, at a rate y4~4xl0^s~'. However, about once in 10^ times, it decays instead to the

metastable 5d^6s^^Di/2 Isvel and emits an 1 1 jum photon.

The rate of this transition is 73=52 ± 16 s~'. The ^Z)3/2 level decays at a rate >'i=109±5 s~'. The probability

for the ^Di/2 level to decay directly to the ground state is

194 nm

excitation

FIG. 1. Diagram of the lowest four energy levels of Hg*.

Only the 194-nm fluorescence was observed directly.

IS 559 TN-63 Work of the U.S. Government
560 WAYNE M. ITANO, J. C. BERGQUIST, AND D. J. WINELAND M

/i =0.491 ± 0.015. The probability for it to decay to the metastable 5d^6s^^Ds/2 level is /2 = l~/i- The ^Ds/2

level decays to the ground state at a rate 72 = 11.6 ±0.4

The density-matrix equations for this system are

dp\i/dt = -y\p\i + Y3P44, (la)

dp22/dt=f2np\i-Y2p22, (lb)

dpn/dt =/i yipi 1 + y2P22+ r4P44-i(p}4-p4})MEit)/h ,

(Ic)

dp44/dt = — {y3 + y4)p44'^i(p34~ P4i^flE(.t)/h , (id)

dp34/dt=[i(04}— y ( 73+74 )]p34 + /(p44— P33 )//£(?)//! .

(le)

The state labels 1, 2, 3, and 4 refer to the ^Z>3/2 level, the

^Ds/2 level, the ^iSi/2 level, and the ^P\/2 level, respectively. Here, 0)43 is the ^51/2 to ^P\/2 transition frequency, ju

is the electric dipole matrix element for that transition,

and the applied electric field is Eit)=Ecos(o)t). The

rates y\, 72, and 73 are all much less than 74. Therefore,

the ^5"

1/2 and ^7*

1/2 populations quickly reach a steady

state with respect to each other. For times much greater

than the lifetime of the ^Pm level, the system is effectively a three-level system, described by rate equations for the population variables P]=p\\, P2=P22, and

^o=P33+P44- We derive these equations similarly to the way Kimble, Cook, and Wells'^ reduced the densitymatrix equation for a three-level system to rate equations

for a two-level system. The three-level equations are

dPi/dt = -y^P^ + yoPo,

dP2ldt=f2y\P\-y2P2,

dPjdt =/, y^P^ + 72/^2 - 707*0

,

where

70=73nV(4A2-F742 + 2ft2),

a=nE/h , A = (0 ~ CO 42 .

(2a)

(2b)

(2c)

(3a)

(3b)

(3c)

Let g (t) be the intensity correlation function of the

ll-/im field generated by the radiative decay from the

"Pih level to the ^Dy2 level, normalized such that

g^'^'(oo) = \_ Photon antibunching occurs if g^^Ho) < 1. The function g (t) is proportional to Po(t), where

Po('^) is obtained from the solution of Eqs. (2a)- (2c),

subject to the initial conditions Po(0)=0, P|(0) = 1, and

P2(0) =0. This is because the average rate at which ^P\/2

to ^Z>3/2 transitions occur is the product of 73 and the

probability that the ion is in the ^P\/2 level (which for given n and A is proportional to Pq). The initial conditions signify that at time t=0, the ion has just decayed to the ^Di/2 level. The solution of the equations yields

where

y± = j K70+ 71 + 72) ± [(70+ 71 + 72) ^ -4/2/071

-47172-47072] '''^1

, (5a)

C± = ± 7+ (/i 7± - 72)7172(7+ - 7- )]

.

(5b)

All of the constants required to calculate ^^^^(r) are fixed by the radiative decay rates of the Hg "*" ion, except for the

transition rate 70 from the fluorescence-on state (cycling

between the ^S\/2 and ^f 1/2 levels) to the fluorescence-off

state (either the ^Di/2 or the ^Ds/2 level). This rate was

determined by dividing the number of downward quantum

jumps in fluorescence by the time that the ion was in the

fluorescence-on state. Since g^^Ho)=0, this system exhibits photon antibunching.

The apparatus has been described previously."^'*'' The

Hg "*" ions were confined in a radio-frequency (Paul) trap

under high-vacuum conditions. Ions were created inside the trap volume by electron impact ionization of neutral Hg atoms. A 5-/iW beam of narrow-band, cw 194-nm ra- diation was focused to a waist wq of approximately 5 /im

at the position of the ions. The frequency of the 194 nm

radiation was tuned below the atomic resonance, so the

ions were laser cooled. The 194 nm photons emitted by

the ions were collected by a lens system and focused onto

a photomultiplier tube. The net detection efficiency was

approximately 5x10"'*. Peak signals were approximately

50000 counts/s per ion. The data consisted of recordings

of the numbers of photons detected in a series of as many

as 100000 consecutive 1-ms time intervals. An example

of the data is shown in Fig. 2 of Ref. 12. Each one-ion

data set was analyzed by considering the ion to be in the fluorescence-on state if the number of detected photons was above a discriminator level and in the fluorescence-off

state if it was not. This discriminator level was set to minimize the number of false transitions due to noise. Each

transition from the fluorescence-on state to the

fluorescence-off state was assumed to indicate the emisOl

CM

100

40 60

T(ms)

100

g ^Ur) =1 — C + exp(— 7+t) — C-exp(— 7-r) , (4) lines are calculated from Eq. (4)

FIG. 2. Plots of ^"'(r) for (a) one Hg+ ion and (b) two

Hg"^ ions. The dots represent the experimental data. The solid

TN-64
RAPID COMMUNICATIONS

2S PHOTON ANTIBUNCHING AND SUB-POISSONIAN 561

sion of an 11 //m photon. Analysis of simulated data,

which included noise, indicated that about 5% of the transitions would have escaped detection, because they were

preceded by or followed by a quantum jump in the other

direction within a time interval of about 1 ms. The transi- tions from the fluorescence-off to the fluorescence-on state were not studied in detail, since it was not possible to tell from the 194 nm fluorescence whether a particular transi- tion resulted from the emission of a 198 or 282 nm photon.Data were obtained also with two Hg "*" ions, separated

by approximately 3 nm. The 194 nm fluorescence

switched among three distinct levels, depending on whether 0, 1, or 2 ions were in a metastable level. Discriminator

levels were set to distinguish among the three different

states. The time resolution was 2 ms. Each downward

quantum jump was assumed to indicate the emission of an

11 /im photon from one of the ions. With an imaging

photon-detector,'^ one could determine which ion had emitted the photon.

Figure 2 shows observed and calculated plots of g (t)

for (a) one ion and (b) two ions. The two-ion calculation was based on the assumption that photon emissions from

one ion were not correlated with those from the other one.

In that case, gi^HT) = jU+gl^Hr)], where the subscript indicates the number of ions. The agreement of the

experimental and calculated curves is an indication that the ions were independent, for time scales greater than

about 2 ms. The values of yo were about 15 s "' for the one-ion data and about 12 s ~' for the two-ion data.

If the ions are independent and are subjected to the same 194 nm radiation field, then certain relationships

hold among the probabilities po, p\, and p2 for zero, one,

or two ions to be in the fluorescence-on state. Let pon and

/>off~l ~Pon be the probabilities for a single ion to be in the fluorescence-on or the fluorescence-off" state. Then,

Po = (pof{),^ P\ =2ponPoff, and p2 = {pon)} If the transi- tions of the ions were correlated in some way, then deviations from these relationships might occur. For a particular 100 s run, the measured values were /)o~0.1836,

Pi =0.4887, and p2 =0.3277. For a value of Pon "^0.57224, obtained by least-squares adjustment, the calculated values are po =0.1 830, pi =0.4896, and

P2 =0.3275. Similarly good agreement was obtained for the other runs. Another test of the independence of the two ions was a comparison of the number of apparent double quantum

jumps to the number that would be expected from random

coincidences, given the finite time resolution of the apparatus. In 400 s of two-ion data, 11 double quantum

jumps and 5649 single quantum jumps downward in fluorescence were noted. Data generated by a Monte Carlo simulation, in which the two ions were independent, were analyzed in the same way. The simulated data had

15 double quantum jumps and 5430 single quantum

jumps. Thus, the rate of double quantum jumps was consistent with independence of the two ions. The experimental probability Pni^t) of detecting n 1

/im photons in a period A/ can be derived from the quantum-jump data, where A/ is any integer multiple of

the experimental time resolution. Mandel's Q parameter,'* which measures the deviation of the distribution from Poisson statistics, is Q = {a^-{n))l{n), (6)

where o^ and in) are the variance and the mean of the dis- tribution. Negative values of Q indicate sub-Poissonian

statistics. The theoretical value of Q is

'*

Q

= 2N_

TAt r'^^'jo''^^[^*'^(^)-i]. (7)

where TV and T are the total number of photons emitted

and the total time. Figure 3(a) shows PniAt) for one ion,

with At =200 ms and r=400 s. The Poisson distribution

for the same mean ((«) = 1.45) is also shown. For this data set, the measured value of Q, calculated from Eq.

(6), is -0.253 ±0.025; the theoretical value obtained by

inserting g'^^Hr) into Eq. (7) is -0.242. Figure 3(b)

shows PniAt) for two ions, with At =200 ms, and 7 = 400

s. The measured value of Q is -0.242 ±0.025; the

theoretical value is —0.243. The Poisson distribution for the same mean ((n) =2.84) is also shown. Previous measurements of Q in single atoms have yielded the values -2.52x10"^ (Ref. 4) and -7x10"^ (Ref. 3). The

value observed here is larger because the detection

efliciency is greater.

It should be pointed out that the 11 /im photons have

not been observed directly. Rather, their existence has

been inferred from observations of the 194 nm fluores- cence and the density-matrix equations. The 194 nm field does not itself display nonclassical properties, except on

time scales of the order of the ^P\i2 radiative lifetime,

which we do not observe. Other measurements of photon

antibunching'"^ and sub-Poissonian photon statistics^ ~^

require fewer assumptions about the nature of the emitter.

1000

800 ^ 600

a- 400

200

750

600

§ '^50

Q- 300

AV

1 ion

(a)

v..=a^-—

• (

150J;.

: /~'°~

\.

2 ions

(t>)

'^.

FIG. 3. Plots of the distributions of the numbers of 11 -/jm

photons emitted in 200-ms time intervals from (a) one Hg"*" ion and (b) two Hg^ ions. The filled circles represent experimental

data. The open circles represent Poissonian distributions with

the same means as the expcnmenial distributions. TN-65
RAPID COMMUNICATIONS

562 WAYNE M. ITANO, J. C. BERGQUIST, AND D. J. WINELAND ia

The basic elements of the density-matrix equations have,

however, been confirmed by the agreement between the calculated radiative decay rates and the rates deduced

from the quantum-jump data, '^ and we are confident in applying them to the present experiment.

We wish to acknowledge support from the U.S. Air Force Office of Scientific Research and the U.S. Office of

Naval Research. We acknowledge helpful comments

from L. Mandel. We wish to thank F. Diedrich and

J. Cooper for carefully reading the manuscript.

'H. J. Kimble, M. Dagenais, and L. Mandel, Phys. Rev. Lett. 39, 691 (1977); M. Dagenais and L. Mandel, Phys. Rev. A

18,2217 (1978). ^F.-M. Rateike, G. Leuchs, and H. Walther, results cited by

J. D. Cresser, J. Hager, G. Leuchs, F.-M. Rateike, and

H. Walther, in Dissipative Systems in Quantum Optics, edit- ed by R. Bonifacio, Topics in Current Physics, Vol. 27

(Springer, Berlin, 1982), p. 21.

3f. Diedrich and H. Walther, Phys. Rev. Lett. 58, 203 (1987).

''R. Short and L. Mandel, Phys. Rev. Lett. 51, 384 (1983).

^M. C. Teich and B. E. A. Saleh, J. Opt. Soc. Am. B 2, 275

(1985).

*J. G. Rarity, P. R. Tapster, and E. Jakeman, Opt. Commun.

62,201 (1987).

''Squeezed States of the Electromagnetic Field, edited by H. J. Kimble and D. F. Walls IJ. Opt. Soc. Am. B 4, No. 10

(1987)]. *Th. Sauter, R. Blatt, W. Neuhauser, and P. E. Toschek, Opt.

Commun. 60, 287 (1986).

^M. Lewenstein and J. Javanainen, Phys. Rev. Lett. 59, 1289

(1987).

'Oj. C. Bergquist, R. G. Hulet, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 57, 1699 (1986).

"See, for example, D. Meschede, H. Walther, and G. Miiller, Phys. Rev. Lett. 54, 551 (1985); R. G. Hulet, E. S. Hilfer, and D. Kleppner, ibid. 55, 2137 (1985); W. Jhe, A. Anderson,

E. A. Hinds, D. Meschede, L. Moi, and S. Haroche, ibid. 58, 666 (1987).

'2w. M. Itano, J. C. Bergquist, R. G. Hulet, and D. J. Wineland, Phys. Rev. Lett. 59, 2732 (1987).

'^P. Eriksen and O. Poulsen, J. Quant. Spectrosc. Radiat. Transfer 23, 599(1980).

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

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

(1985).

'5C. E. Johnson, Bull. Am. Phys. Soc. 31, 957 (1986).

'6h. J. Kimble, R. J. Cook, and A. L. Wells, Phys. Rev. A 34, 3190 (1986).

''D. J. Wineland, J. C. Bergquist, W. M. Itano, J. J. Bollinger,

and C. H. Manney, Phys. Rev. Lett. 59, 2935 (1987).

'«L. Mandel, Opt. Lett. 4, 205 (1979).

TN-66
Reprinted from Journal of the Optical Society of America B, Vol. 2, page 1721, November 1985 Copyright ® 1985 by the Optical Society of America and reprinted by permission of the copyright owner.

Angular momentum of trapped atomic particles

D. J. Wineland, J. J. Bollinger, Wayne M. Itano, and J. D. Prestage

Time and Frequency Division, Naiional Bureau o^ Standards, Boulder, Colorado 80303

Received April 10, 1985; accepted July 23, 1985

In axially symmetric atomic-particle traps, the angular momentum of the particles about the symmetry axis is conserved in the absence of external torques. Changes in this angular momentum owing to laser scattering are

discussed.

O

o

n

C/2

(Ws1. INTRODUCTION

By now, one has become fairly familiar with the idea that the momentum carried by light can be transferred to atomic (or macroscopic) particles, thereby significantly altering the velocity distribution of these particles. The status of experiment and theory will be discussed in the accompanying pa- pers of this issue; further information can be found in other

recent papers.

^"^

In the experiments on atomic deflection and laser cooling, one is concerned primarily with the changes in linear mechanical momentum imparted to the particles by light.

However, for the case of trapped particles, one must in general be concerned with the angular momentum of the trapped particles and therefore with the changes in angular

momentum of the trapped particles by light. The situation discussed in this paper is illustrated in Fig. 1. If the trap

(trapping fields are not shown) is symmetric about a particular axis, here assumed to be the z axis, then one would expect

the 2 component of angular momentum of the particles, L^, to be conserved in the absence of laser scattering. Here, L^ is the total canonical angular momentum given by

^^ " S

'- " E ^''' ^ P'^^'

p, = m,v, + q,A(x,)/c,

(la)

(lb)

where /;,, p„ m„ qi, x„ and v, are the z component of angular

momentum, the linear momentum, mass, charge, position,

and velocity of the ith particle, respectively; N is the number

of particles; c is the speed of light; and A is the vector

potential. From the geometry of Fig. 1, it is clear that the

laser scattering can be used to increase |Lj| indefinitely,

independently of the laser tuning. This effect would be

interesting to study, but of course in experiments employing

laser cooling it is clearly a situation to be avoided. In practice, changes in angular momentum will also be influenced

by various mechanisms, such as collisions with background

gas and deviations of the trap from axial symmetry.

Assuming that the laser beam is perpendicular to z, the magnitude of the average change of mechanical angular mo- mentum per photon-scattering event is given by hd/X, where

d is the distance of the laser beam from the z axis, h is

Planck's constant, and X is the wavelength of the photons.

Changes in the internal angular momentum of atomic particles will be of order h. Typically, 2ird/\ » 1; therefore we

will neglect the internal angular momentum of the particles.

As in previous treatments,'''^ for simplicity we will neglect

second-order relativistic effects and saturation. We will also assume that the collision rate between trapped particles

is high enough and the laser scattering rate low enough that

the particles can be assumed to be in thermal equilibrium.

These conditions may be violated, but the qualitative fea- tures of the problem should still apply.

To make the problem more tractable, we will discuss only

angular-momentum effects in ion traps, specifically, the rf (Paul) and Penning traps.^"^^ This seems reasonable, since many experiments have been done with these traps. The

qualitative features of these traps should carry over to neutral traps. The primary difference would seem to arise from

the absence of long-range Coulomb repulsion between particles and the fact that the field momentum that is due to the

ions' net charge [vector-potential term in Eq. (1)] is absent.

Finally, in this paper angular-momentum changes are as- sumed to arise from laser scattering, background gas, and

trap asymmetries; however, such effects may come from

scattering by particle beams,'^ positive or negative feedback on the charged-particle motions, ^'^'^ or angular-momentum

transfer from plasma waves. '•''^ In Section 2, we will assume that ion trapping is accomplished by a combination of rf and Penning traps. For such

a combined trap, single-particle trajectories have been dis- cussed by others^ "; here we will use the pseudopotential

approach.^'^'^"" Assuming that the ion cloud (or nonneutral

plasma'-'"''') rotates at a frequency a' about a particular symmetry axis, we derive relations between a' and the ion cloud's

density, potential, dimensions, and angular momentum.

These expressions can then be applied to the rf or the Penning trap in various limits. In Section 3 we discuss torques

applied to the cloud from laser scattering for a particular

ion-cloud-laser-beam interaction geometry. Backgroundgas damping is discussed in Section 4. Changes in angular

momentum owing to trap asymmetries are treated separately for the rf and the Penning traps in Sections 5 and 6, respectively. Finally, in Section 7. we make a few remarks

about angular momentum in neutral-particle traps.

0740-:V2-.M 'Sr. '1 1 IT.' I lii:^l):M)0 19Sr> Opiiial S.H i(>t\ cl" Amorii-.i x-h;
1722 J. Opt. Soc. Am. BA'ol. 2. No. U/November 1985 Wineland et al.

o

o

0) Cm

«M

*s

trapped atomic

particles

/=">

Input laser beam

Fig. 1. Schematic representation of trapped atomic particles. The

trap (not shown) is assumed to have axial symmetry about the z axis.

If a laser beam is directed parallel to the > axis but does not intersect the z axis, then atomic particle-photon scattering imparts angular

momentum (L^) to the sample, which can cause it to spin about the z

axis.

2. ION TRAPS

For further simplicity, we will assume idealized models for the rf and Penning traps and assume only one trapped ion

species. Both traps use axially symmetric electrodes whose

surfaces are figures of revolution (hyperboloids of revolution) as shown in Fig. 2. For rf trapping, when the drive (micromotion) frequency ^ is much larger than secular fre- quency Wx, Wy, or ujj (defined in Section 5), the ions can be

considered to be confined in the axially symmetric pseudopotential^"^' of the form

!)7.(r, z) = ar^ + /32^ (2) where r and z are cylindrical coordinates and a and ^ are functions of the trapping parameters:

/3 = 4(7Vo"/mfiV + 2C/o/?2

(3a)

(3b)

Vq is the amplitude of the rf voltage, Uq is the amplitude of the dc voltage applied between the ring and the end caps

(here, t/o positive means that the end caps are at higher

potential than the ring electrode), and ^^ = Tq^ -I- 2zo^, where

Tf) and 2() are defined in Fig. 2. We note that the pseudopotential [the first terms in Eqs. (3a) and (3b)] always acts to give a radially inward force (independent of the sign of q)

while the force that is due to the dc applied potential can be

in either direction. For the rf trap we want qa, g/3 > 0; the

resulting potential well is equivalent to that provided by a uniform-background charge distribution whose boundary is an ellipsoid of revolution.'^ From Poisson's equation, V^^

= -47rp, we see that the density inside this fictitious charge

distribution is given by p' = —(2a + P)/2ir. The potential for the Penning trap''"' is usually given by Eqs. (2) and (3) with

V() = 0; in this case a magnetic field B = BqZ is required for trapping. For the sake of generality we will assume that Bq

7^ and Vo r^ except where noted below. The traps have symmetry about z, and, therefore, in the absence of external torques, L^ is conserved. In this case, the ion canonical angular momentum and ion energy enter

the thermal-equilibrium distribution function on an equal

basis. The thermal-equilibrium distribution function can

be written as'"'"'^

2-Kk^^T

M-l exp[-(/y -c'/J/Ze^r],

where

H = mv^/2 + (?0(x)

is the ion energy and

/j = mv„r + qAi,{x)r/c

(4)

(5a)

(5b)

is the ion canonical angular momentum, m, q, and v are the

ion mass, charge, and velocity, and y = | v | . vs and Ao are the

6 components of the velocity and the vector potential, and r

is the radial coordinate of the ion in cylindrical coordinates.

4) = 4>i + (pT + </>ind is the total potential, which is written as the sum of the potential that is due to ion space charge cpi, the applied trap potential 0^, and the potential 0ind that is due to the induced charges on the trap electrodes, no, T, and

o) are determined by the total number of ions, energy, and

angular momentum of the system, and ks is Boltzmann's

constant. We choose the symmetric gauge where A(x) = B

X x/2, so that Ae = Br/2. We can therefore write the distri- bution function as'^

/ m \3/2

fix, v) = n(x)(---- ) exp[- V2m(v - u>re)VkBT], (6a) \2TrksTJ

where n(x) is the ion density given by

n(x) = Mq exp{ q<p(r, z) - V2ma;( A i2„ -l- w ]r^ Ik^T

(6b)

We assume that the trap potential is adjusted to make <^ =

at the origin, in which case no is the ion density at the center

of the trap, fie = I Qfi/'^cl is the cyclotron frequency. With

the assumption that u « c, we have neglected the diamagnetic correction to the magnetic field that is due to the motion of the ions. For densities near the maximum density

(discussed below) this assumption may break down for elec- trons. Equations (6) have been analyzed for the case of electrons

in a Penning-type geometry (axially symmetric but nonquadratic trap potential). '^'^ We can easily extend the analysis

of Refs. 18 and 19 to the hyperbolic Penning and rf traps. The distribution function is a Maxwellian velocity distribuBgZlrequlred for Penning trap)

end caps

ring

'j>(r,2)=A(r^-2z^

A = _y-

^o'+2^o U = Uo^-VoCOsQt

(Vq = (or Penning trap)

electrode surfaces generated

by it'(r,z) = const.

Fig. 2. Schematic representation of the electrode configuration for the Paul (rf) or Penning trap. Electrode surfaces are figures of revolution about the 2 axis and areequipotentialsof (/)(r, z) = A(r'^ —

2z'^) (cylindrical coordinates are used with the origin at the center of the trap). Typical dimensions are ^2o = ro s 1 cm. Typical

operating parameters are the following: for the Paul trap, Vo = 300

V, i1l2ir s 1 MHz; for the Penning trap, t/o s 1 V, B s 1 T.

TN-68
Wineland et al. Vol. 2, No. 11/November 1985/.J. Opt. Soc. Am. B 172.3

tion superimposed upon a rigid rotation of frequency a; [positive (negative) values of w denote rotation in the +(— ) d direction]. For the Penning trap, this rotation is caused by

the E X B drift and the pressure-gradient-induced drift [similar to E X B drift, where the term (dn/dr}r must be

included w^ith Err]."^^ In thermal equilibrium, these two

effects in combination give rise to a rotation that is constant with radius. If the rotation is not rigid, then shear exists in the cloud; the resulting friction forces drive the cloud to thermal equilibrium. We emphasize that rigid rotation does not require a quadratic trap potential but only axial symmetry and thermal equilibrium. From the expression

for the ion density, we arrive at Poisson's equation:

1^/ a^N

r dr \ dr / = —iirqriQ

X expj - q<t>- V2/nu)( pj-f], + w F /joT -4V, (7)

where we must include the fictitious charge density, p', from

the pseudopotential. In Ref. 19 it was shown that the general solution to Poisson's equation leads to axially symmetric

ion densities that are constant out to a certain radius and

then fall to zero in a universal way in a distance approximately equal to the Debye length, Xo, given for a nonneutral

plasma by

>^D = kBT/4Trnoq\ (8) Although Ref. 19 specifically dealt with electrons in a Penning-type trap, Poisson's equation looks essentially the same

for the rf trap, and therefore the general conclusion above is valid for both cases. (Recall that we are neglecting the micromotion.) Specific solutions are discussed in Ref. 19

for the Penning-type geometry and in Ref. 21 for the rf geometry.

As an example, for T = 300 K and no = lO'^/cm^, we have Xq s 0.38 mm. At higher densities and lower temperatures, the

general features of the density distribution noted above have

been directly observed in Penning-type traps.^^'^^ At higher

temperatures and lower densities (where space charge be- comes negligible and Xq ^ ion-cloud dimensions) the density

distributions become Gaussian; this has been directly ob- served in rf traps. 24'2^ Because of the laser-cooling experiments, it is interesting

to consider the limiting case when Xq « cloud dimensions,

i.e., when T —- 0. From the right-hand side of Eq. (6b), we

can see that in the cloud interior, q4) —miUcq/\q\ + a;)torV2 —

in order for the density to be finite. Therefore, in the

interior we have

</) = 0/ + 07- + (Aind = m(jc{n^q/\q\+ a))r-/2q. (9) The lack of z dependence in is to be expected since at low

temperatures the outward z force from space charge would

be expected just to cancel the restoring z force of the trap. In the case when the trap potential is quadratic [Eq. (2)], Eq.

(9) can be written as

</), = [mw(il^q/\q\ + w)/2q - a\r^ -^z^ - 4>;„a (10a) = -%7rqno(ar- -I- 62^) - 0i„d, (10b)

where Eq. (10b) is used to define a and b. From Poisson's

equation, 2a + b = 3. For a stable cloud of finite size, we

must have a, 6 > 0; otherwise the ions are not confined. If the potential from the induced charges can be assumed constant over the ion cloud (this approximation is valid if the

cloud's dimensions are small compared with r^ and zo), then

(pi is the potential distribution for a uniformly charged ellip- soid of revolution'^** of density no whose spatial extent along

the z axis we define as 2zci and whose diameter in the 2 =

plane we define as 2rei. For simplicity we will therefore assume that 0ind is constant over the ion cloud. If the di- mensions and density of the ion cloud are large enough so that 0jnd is not constant over the ion cloud, then the cloud

becomes elongated along z and expanded near the 2 =

plane. For a = b = I the ion cloud is a sphere (i.e., Zc\ = rd). For a > I > b the cloud is a prolate ellipsoid (Zc\ > r<.|), where'^'^''

a = V|l/[2(1 - fe/)] - ln[(l + /?p)/(l

b = k;\\n[{l + kp)/{l-kp)]/2kp-l\,

^p)]/4fepl: (11a)

(lib)

and where /?p = [1 - (rci/zc\}^^'^ and kp' = 3(1 - /jp2)Ap2. For a < 1 < 6 the cloud is an oblate ellipsoid (rd > 2ci), where

a = V[sin"'(/Jo)/2^o " d - ^o^)'^V2], (12a)

b = /eo'[(l - K'^r''^ - sm-\k^)/k^], (12b)

and where /eo = [1 - iZc\/rci)^V'- and /jq' = 3(1 - ko'^V'Vko^. From Eq. (10a) and Poisson's equation, we can express no

as a function of w and the trapping potentials:

no(aj, «, (3) =

27rq'

-w{^cQ/\q\ +'^) + — (2a + /3) m

(13)

For the pure Penning trap, where Ko = and 2a + p = 0, we

obtain the result of Ref. 23. The usual case for the rf trap is with Qc, <j^ = 0, and we obtain the result of Refs. 6 and 7. In

practice, the minimum temperature and therefore the maximum density in an rf trap might be limited by rf heating,2'*'2"-28 where the condition X^ « rd, Zc\ may not be obtained. If a and /3 are assumed fixed we find that the density

is maximum when w = —Q^q/^qIno(max) = m

2Trq'

9.'-/4 + ^{2a + 13) m

(14)

For the pure Penning trap (2a + ji3 = 0), we find --^ that no(max) = Bo'^/STtmc-. Also, the parameter a of Eq. (10b) is largest (i.e., 2d/rd is largest) for oj = —V-cq/'^q]. For the pure

rf trap (fi^. = 0), this implies that w = as expected, since spinning in either direction causes a centrifugal distortion that tends to make the ellipsoid more oblate. Equation (13) can be used to find w if we assume that the cloud has a

certain density no. We find that

a; = ~V.^.q/2\q\ ± [(i\/2)'- + q(2a + i3 - 27rqn^,)/m] 1/2 (15) In the pure Penning trap (2(v + d = 0), the solution for which

|a;| < i]</2 can be identified with the E X B drift magnetron

motion and the solution for which ju^ > 12^/2 can be identified with the cyclotron frequency that is shifted by the radial

electric field (see Section 6). For n > no(max) the ion orbits

in the Penning trap are exponentially increasing."

We note that Eq. (15) is not valid in the limit of a single

ion, where one may want to substitute no — 0. Implicit in

n

a

W>-^ '^o

o

crqsr

TX-r-.n
1724 J. Opt. Soc. Am. BA'ol. 2. No. ll/November 1985 Wineland et al.

o

VIo

"c

as

Eq. (15) is the assumption that the axial force on an ion is zero. Equivalently. we assume that the cloud consists of many ions and that A/j « r^i, z^x. In the limit of one ion in the

trap, the axial force on an ion is due to the applied trapping

fields (zero only at trap center). In addition, the requirement that a > for clouds of finite size restricts the density

/!o to be greater than a minimum density given by no(min) =

.V2-q. At this minimum density, a - 0, and the cloud has

infinite extent in the 2 = plane. The requirement that a >

also restricts w to values ranging from co = —qQc/'2\q\ ~

[(V.,/2y- + 2qa/m]^''^tow = -qV.,/'2\q\ + [(nj2y- + 2qa/m]^'^.

Finally, we can evaluate L; for an ellipsoidal cloud of ions

to find

L, = n,i mr-u^ + r — Ap(r) d^x

= ^Nm(i\.q/2\q\ + is,)rj. (16)

where Jv d'^x = 4Trr^.{-z,.\/3 is the cloud volume.

A convenient dividing line between rf and Penning traps

would seem to be when the radial electric force is absent, i.e., a = 0. Therefore, in what follows, we will define the rf-trap

mode as the case when the radial force that is due to the

electric trapping fields is inward {qa > 0) and the Penningtrap mode as the case when the radial force that is due to the

electric trapping fields is outward {qa < 0).

3. LASER SCATTERING

The problem that we wish to investigate is whether it is possible to observe angular-momentum changes induced by

laser scattering in an ion cloud. For brevity, we will consider

only a laser-beam/ion-cloud geometry in which these effects might be fairly strong; other interaction geometries could be

similarly treated. Consider the case of Fig. 3. We will assume that the

spatial width of the laser beam is small compared with r^\ and 2,.i and small enough that we can neglect the variation in first-order Doppler shift across the beam that is due to the cloud rotation. If the cloud is in thermal equilibrium, then

a- is constant over the cloud; therefore the average >' velocity of ions within the laser beam is l\ = wr cos = wd. For

simplicity, we will assume that|a;| « 7, where 7 is the natural

linewidth of the atomic transition to which the laser is nominally tuned. In this limit, photon scattering may be as- sumed to occur at an instant of time, and the momentum

imparted to an ion is given by Ap = ft(k — kj), where k and kj are the photon wave vectors of the incident and scattered photons (|k| = |ks| = 2ir/\).'^-^ Therefore the average torque

from laser scattering is given by

^' \dt

/(x)

hv

a{k V, T)^dn(x)dV (17a) A

where /(x) is the intensity (power per unit area) of the laser beam, a is the photon-scattering cross section, and n(x) is the ion density. In the derivation of Eq. (17a), we have

averaged over all angles of the reemitted photon. In general, the angular distribution of reemitted photons will not be

isotropic [e.g., see Eqs. (8a) and (8b) of Ref. 5], but the torque that is due to photon reemission when averaged over

angles will be zero, neglecting small relativistic corrections.'' In the limit that the laser-beam dimensions are small compared with Tel, 2ci. and d, and the density is nearly constant

over the cloud (Ad « Zd, r^), then Eq. (17a) becomes

T^ = (—) = hiirN^^n^aiVy, T)d(r,,2 - d^)''VX, (17b)

where Ni is the number of incident laser photons per second

and we note that a has a line shape given in general by a Voigt profile^" that is first-order Doppler shifted by a fre- quency cod/A. As we noted earlier, if there is no damping,

I (L;)| will eventually increase. In principle the parameter a

[Eq. ( 10b)] goes to zero; of course the edge of the cloud would

touch the ring electrode before this condition is reached.

In practice, damping of <Lj) must be considered; it appears that this should come primarily from two sources: background neutral gas and imperfections in the trap elec- trodes that destroy axial symmetry and therefore prevent

( L. ) conservation. Motional radiation damping should also be considered for electrons, ^^-^^^ but this is expected to be a much weaker effect for ions. (In any case it would mimic

background-gas damping.) Background-gas damping

should occur in a similar way for the rf and Penning traps. However, since changes in L^ owing to trap asymmetries are expected to be qualitatively different for the two traps, they

are treated separately.

laser beam

Trap axis ( B out ol paper)

ion cloud

l-i^;, 1. l,HMTljeiim jnii cIdikI iDicriution geometry. Tht laser

lii'iim ihii.Hsiimfd In he in ihe ,- = plane; the radius of the ion cloud

if r,|. For ihe rl' trap, la.ser scallerinn will lend to t-aiise the cloud to

••(•in in llie (Hreclion shown. For the I'enninn trap, the direclion ol riiiaiion <hown (w > 0) is apprnpriale lor nejzaiKe ions: for posilive ions ihf roliilioii i>. rcverned lu.' < ••).

4. BACKGROUND-GAS DAMPING

An estimate of background -gas damping can be derived

from measurements of ion mobility. The mobility K is defined''- from the expression l\i = KE, where E is the electric

field impressed on the ions in a drift experiment and vj is the average drift velocity of the ions through some background

gas. The drifting ions impart momentum to the gas at a rate dp

dl = F = qE = qvJK. (18)

i'luTclorc, lor the case of trapped ions, the torque th-

(l;inips ion rotat inn is

<'/'i,k«) = Nirdpldt) = -Nq{r-o.^)/K = -N - ^^ i\.;-u), 5 K

(19) TN-70
Wineland et at. Vol. 2, No. 11/November 1985/J. Opt. Sue. Am. B 1725

where the last equality is valid in the low-temperature limit, Xn « Tcl, 2cl5. rf TRAP [qa > 0) We can get an idea of the magnitude of the change of L^ that

is due to trap asymmetries by assuming that the pseudopotential has the form

the laser beam). We can then combine Eqs. (17b), (19), and

(22) to obtain in steady state

(r, z) = ar^ + 0z' + i(x' - y^). (20)

Assuming B = 0, a single ion in this trap has the characteristic motions

X = Xy COS(a)^i + 6^),

y = y^ cos,(Wyt + dy), z = z^ cos{(j:^t + dj, (21a)

where the frequencies w^, Wy, and Wj are given by

u)/ = 2q(a + ()/m, w/ = 2q(a - ()/m, co/ = 2q0/m.

(21b)

For l^l « |a|, an ion initially in a nearly circular orbit defined by x = f sin u^t, >- = f cos coyt will find itself in a

circular orbit of opposite sense after a time tt/Iwj; — w^.

Hence, in this case, we estimate

T, = (^^y^-LJr, r = 7r/2|a>,-a,J. (22)

For a cloud of ions we could expect approximately the same dephasing time if space charge could be neglected, i.e., the ions behave independently. For a "cold" cloud, where

the trapping force is essentially canceled by space charge, the situation is quite different. If we neglect ion-ion colli- sions, then ions drift through the cloud (which is now ellip- soidal in the x-y plane) and reflect from the edge of the ion- charge distribution. However, as an estimate we will still assume the validity of Eq. (22), where Wx and Wy are given by

the free-space values of Eqs. (21). To determine if spinning of the cloud induced by a laser beam could be observed, we have experimentally investigated a cloud of '9^Hg+ ions confined in a small rf trap.^^ In this experiment Tq = 455 ^m, 2o = 322 nm, 9.12-k = 20 MHz, Vq =

400 V, and Uq = —12 V. For a single ion in this approximately spherical well (a s /3) we would expect the secular frequency w^ s a);^ = Wy = (2qalmy'^ = (27r)1.19 MHz. For

typical conditions, where r^x = 30 ^m, d =15 ^m, laser-beam

waist size wq = 15 ^m, and laser wavelength X = 194 nm (^Si/2 ~* '^P\n transition), we can ask under what conditions we

would expect the cloud to spin at a frequency w = (27r)0.5 MHz. We choose this frequency because the corresponding

first-order Doppler shift along the beam direction is about

250 MHz, which would be easily observable on the '•^Si/2 —

^Pi/2 transition. If we assume that \o « rd, Zd, then we can

use Eq. (13) to obtain mq s 1.68 X lO'^/cm^. The ions are

observed'* ' to be near room temperature (through the Doppler width on the '^Sm —- ^Pi/2 transition); hence we calculate

that Xd = 9.2 ^m, and thus Eq. (13) should give a reasonable

estimate of the density. From Eqs. (10) and (12) we then

find that Zeis 25.8^^1. In Eq. (17b) we write N/^a = N.^Tra-o-. where Ng = Nia/wcod^ is the scatter rate per ion (for ions in

\^) =<T^L + n,, + T,)=0.

\ at /Total (23)

For our specific example (a!/27r = 0.5 MHz) we find that

^[iV, X 3.0 X 10^ - n(He) X 5.3 X 10"- - Ai'^, X 2.3 X 10"'| = 0, (24)

where we have assumed that the background gas is helium of density n(He) [K = 19.6 cm^ V~' sec"' at standard pressure^"*) and we define Ai';^ v -\^x ~ ^\\ I2t^. If we first assume

that the pressure of helium gas is small and Aciy = 1 kHz, we

require that N^ s 7.7 X 10"/sec (close to saturation) in order

to spin the cloud at a)/27r = 0.5 MHz. For the more typical

scatter rates in this experiment, N, :S lOVsec, we require that b^Vxy ^ 13 Hz in order to see the same u. This would

require a very small trap asymmetry in order for the effect to be seen, and in fact no spinning above |a;l = (27r)50 kHz was

observed, implying that Ai^x> ~ 130 Hz for Ns = lOVsec. Note that if ^i^^y = and p(He) = 6.65 X lO'^ Pa (5 X IQ-^

Torr), we need N^ = 3.1 X 10'^ to obtain \Ji = (27r)0.5 MHz.

Since Nj > 3 X lOVsec and the helium pressure was less than

6.7 X 10~3 Pa in these experiments, we could determine that

the lack of spinning was not due to background-gas damping.

We might expect to see a smaller dephasing (i.e., Av^y

smaller) for traps with weaker pseudopotential wells (ajj, Wy

smaller), but it would still be necessary to maintain the high

scatter rates (more difficult now because of the reduced

densities). Of course one might become sensitive to smaller

values of w by probing narrower-linewidth transitions. In the low-temperature limit where the ion cloud becomes

strongly coupled,23.35 the dephasing effects might also be

expected to be smaller if conditions of laminar flow or convection exist. Addition of an axial magnetic field may also help to prevent dephasing of L, (see Appendix A). Finally, we remark that by adjusting Uo/Vq it is possible to make the pseudopotential of Eq. (20) axially symmetric about either the X or the v axis, and therefore it should be possible to observe spinning about either axis experimentally. This

possiblity was investigated in the Hg"*" experiment above,

but Ng was not large enough and Uq/Vo was not sufficiently stable for such spinning to be observed. Therefore it should be possible (but perhaps difficult) to observe spinning in a rf trap. It is clearly a situation that is easily avoided. It has already been noted''-^'' that it is desir- able in laser-cooling experiments to avoid axial symmetry in order to avoid recoil heating of the ions in a direction per- pendicular to the laser beam.

6. PENNING TRAP (qa < 0) Angular momentum imparted to the cloud by light may be

difficult to observe in an rf trap, however. In Penning traps

it plays an important role. Since the radial electric trapping

force and that which is due to space charge are outward, the only way to provide trapping is from a radial inward Lorent^

force that is due to a rotation of the cloud as a whole. Therefore, even in the absence of a torque applied by the laser, the cloud must rotate at a frequency ul-. It is easy to see that for

(6O

sr

n'

o

o

Vio

nV
1726 J. Opt. Soc. Am. BA^ol. 2, No. 11/November 1985 Wineland et al.

be

o

o

o

Gu

the Lorentz force to be inward, co must be negative for q

positive and to must be positive for q negative. From Eq.

(16), we see that for the usual case, where |co| < 12^/2, the total angular momentum is in the opposite sense to the mechanical angular momentum; that is, it is mostly angular momentum from the field [x X A term of Eqs. (1) or the term

including Q,. in Eq. (16)]. The case for a single trapped ion in a Penning trap inter- acting with a laser beam has been treated in some detail in Ref. 5. From the equation of motion

mv = -qV(/) -I- q(v X B)/c, (25)

we determine that the motion in the z direction is harmonic

with frequency oj^ = {2(3q/m)^^'^, and the motion in the x-y

plane can be written in complex form:

X + iy = Tj expii(JJ^t) + r, expd'ojgf ), (26)

where a;i,2 = -9.cq/2\q\ ± [(9^/2)- + 2qa/mY'~ and where r^ and ro are complex. The solutions for coi and W2 are different from Eq. (15) for the reasons noted in Section 2. Note that

these solutions are valid independently of the value of Qc or a and thus apply for either the Penning or the rf trap (see Appendix A). From Eq. (26), it appears that the wi and ai2 components

occur in an equal way. However, making the redefinitions^

X + iy = r„ exp(iwj) + r^ exp{iui\t),

^ _Q_

I V2 - [(«c/2)^ + 2qa/m\

«/ = - T^ \aj2 + [( V2)- + 2qa/m]

kl

l/2i

1/21

(27)

(28)

(29)

we can make the following observations: In the limit a —

we see that w^ —* and Wf' — —qQ,cl\q\\ therefore the x-y

motion is pure cyclotron motion. We identify the first term

in Eq. (27) as the E X B drift magnetron motion of the guiding center of the cyclotron motion. The cyclotron fre- quency, w/, is now shifted by the radial electric fields. From

the expression for the total energy in the x-y planc^

£,,(total) = m{q%l2\q\ -I- wJ(o:^rJ - w/r;-^), (30) we see that the energy in the Wc' motion is positive while that

in the w^ motion is negative (because it is mostly potential energy, qar^ < 0). As is shown in Ref. 5, momentum transfer or damping in the direction opposite the a)m(a;<.') motion causes r^ (r^) to increase (decrease). As a consequence, to decrease both /•„ and r,., as is usually desired in laser-cooling experiments, we

must cause momentum transfer simultaneously against the

cyclotron motion and in the same direction as the magnetron

motion.'' Equivalently, we must reduce the cyclotron ener- gy (this is similar to laser cooling of harmonically bound

particles'") and increase the magnetron energy. For negative ions this can be accomplished by the laser-beam configuration shown in Fig. ?>. The laser frequency c/, should be tuned slightly below the rest frequency i-o (/'o = c/X), which

ha.s been first-order Doppler shifted by the magnetron rotation, i.e., ;'/, $ /'ii( 1 + \u.\dlc). Vrom the above considerations

we also see why ion confinement in a Penning trap is normally a situation of unstable equilibrium. If any damping is present, then r^, increases until the ions strike the ring elec- trode. From similar considerations, the cyclotron motion

should be damped by background-gas collisions. We also note a qualitative difference between the Penning

and the rf traps. For the Penning trap, by using the tech- nique of laser cooling, it is experimentally possible^^ to achieve an amplitude for the cyclotron motion, r^ much

smaller than the amplitude of the magnetron motion, r^. In

this low-temperature limit the ion motion in the x-y plane is nearly circular, and L^ is conserved. If the trap potential is the distorted one given by Eq. (20), the solution for the x-y

equations of motion is not a superposition of two circular motions as in Eq. (27) but a superposition of two elliptical

motions (see Appendix A). Of course, in the limit that the asymmetry parameter e goes to zero, these two elliptical solutions turn into the circular cyclotron and magnetron

solutions of Eq. (27). In the low-temperature limit, the ion motion is elliptical (distorted magnetron motion), but fluc- tuations in Lj are small [expression (AlO)]. This is to be

contrasted with the rf trap, in which a distortion of the type

in Eq. (20) causes a qualitatively different behavior for the motion and Lj can change significantly after a time t [Eq.

(22)]. The above considerations should apply when there is a cloud of ions except that jw^l [i.e., |w| < Q.c/2 in Eq. (15)] is shifted to higher values by space charge and [ojcI [i.e., \(j\ >

f2c/2 in Eq. (15)] is shifted to lower values. If we assume

perfect axial symmetry and no damping, then it is interest- ing to consider the situation of Fig. 3 (appropriate for negative ions). First, if the laser beam is directed on the side of

the trap axis opposite that shown, then a drag is applied to the magnetron motion that can increase the magnetron radi- us to large values or even expel the ions from the trap. This

effect has been experimentally observed for Mg"*" (Ref. 37)

and Be+ ions. Now suppose the laser beam is directed as shown in Fig. 3. Further assume that the cyclotron motion

is laser cooled {T —* 0) and that the cloud as a whole rotates

at a frequency \ujm\ — l^ "^ Q.c/2. As we continue to add

angular momentum from the laser scattering, then we expect

|co| and the density aiq to increase while the cloud radius, r^\, decreases. This happens until \J[ = Qj2 (i.e., jco^ = |a)f1 =

i2(/2), at which point the maximum density is reached [Eq.

(14)]. As more angular momentum is added by the laser, the

rotation frequency [wj becomes larger than i1j2 (i.e., |a)| =

l^c'l is now the cloud rotation frequency) and the density

starts to decrease again and the cloud radius expands. In

principle this continues to happen until tlK cloud radius reaches the ring electrode and ions are lost. So far, the only experiment in which such effects were

quantitatively studied is the one on ^Be'*'.^^ In this experiment, small clouds of ^Be"^ ions were cooled and compressed,

but the densities were limited to a value (2 X 10^/cm'') that was about an order of magnitude less than the maximum

density given by Eq. (14). For these densities, the mean

spacing between ions, n~^''-\ was about 37 tim while at the

cyclotron temperatures achieved {T^. = 10-50 mK), r<, s i ^m. We can first ask if the experimental limit was due to background-gas damping. For steady state, Eq. (23) must

still apply (assuming here that T, = 0). For the particular

ion cloud of the first row in Table 1 of Ref. 23, we had wo = 60

TN-72
Wineland et al. Vol. 2, No. 11/November 1985/J. Opt. Soc. Am. B 1727

/um, X = 313nm, Tci = 160|um, 2ci = 60^1X1, d ^ 50fim,no = 2.1 X 10Vcm3, T^ = 144 mK, N s 142, and w„ = (27r)37.3 kHz.

[For a single ion in the trap, Wm = (27r)20.5 kHz.] From Eq.

(17b), we calculate that|rj = hN, X 7.3 X 10". From Fig. 6

of Ref. 34, we take Ko(Be+, He) = 25 cm2 V-i sec-i; therefore, Eq. (19) gives | Tbkgl = ^n(He) X 7.7 X 10"^. From the experiment we could estimate that Ng = lOVsec. It was very

unlikely that n(He) was larger than lOVcm"* based on heating data after the laser was turned off."'^ Therefore the limit on density was not due to background-gas damping.

It seems likely that the density was limited by imperfections in the trapping fields, which might lead to angular

momentum's being coupled into the system.^^'-'^ Experimentally this density-limiting effect appeared very nonlinear in that the densities obtained were limited at a value that was fairly independent of laser power. (In the work reported in Ref. 23, varying the laser power by a factor of 10 had no

measurable effect on the ion density.) Moreover, with the

laser off the ions would rapidly heat at first''* (^20 K/sec

during the first 0.1 sec), but this heating would slow down

(~1 K/sec from 10 to 20 sec), and the ions would remain in the trap for many hours. A possible mechanism for the density limit and the nonlinearity of the heating can be

explained by assuming that the trap-symmetry axis is slightly tilted about the x axis by an angle 6 with respect to the magnetic field. If we assume that the z axis is along the magnetic field, then the potential [assuming that Vq = in Eqs. (3)] is

, ^0

[2z^ -r'^ + 3(y2 - z'^)sin^ 6 -3zy sin 26]. (31)

From this expression and the corresponding equations of motion, coupling between the z and y motions is clear. The

frequencies of motion can be found from the eigenvalue

problem (Ref. 40 and Appendix A), and one can also solve for the eigenvectors. Here we only point out that if the tilting is considered a perturbation, then there is an oscillating electric field in the y direction owing to the z motion

E= - — = 32t/osin26/f y dy(32)

that can drive the x-y motion. For a single particle in a Penning trap, and 6 « 1, this driving term at frequency s Wj

is nonresonant with w^ and w,.'; therefore no energy transfer between modes will occur. However, in a cloud of ions, the

Urn and W(.' resonances are broadened by ion-ion collisions. Moreover, as no increases because of compression, a;„, in- creases as noted above. In addition, the ion axial frequency

w/ (for T 9^ Q) is reduced by space charge from the free- space value. In the -'Be^ experiment discussed above,^'' an

estimate of w/ for T t^ can be obtained by assuming that

uj//2w = u^/42cb where mvz^/2 = knTJi. That is, we assume

that the ion drifts across the ellipsoidal cloud (in the z direction) at its thermal velocity and reflects from the edge

(z s 2(.|). Clearly this is only a crude approximation since ion-ion collisions and collective effects will play a more important role. ''''' However, using this model, we estimate

that Wj'/cl),,, s 1.3, and therefore we have near-resonant coupling between the axial motion and the magnetron rotation. At lower densities and higher temperatures, w.'/uj,,, becomes

larger and resonance is avoided. The apparent resonant

conditioM causes r,„ to increase rapidly, which results in an

increase in the ion cloud radius. Because of the resonant

nature of this expansion, the process is called resonant particle transport. '''^^ A rapid increase in the cloud radius can

cause rapid heating from the decrease in electrostatic energy

as the cloud expands. This may account for the rapid heating observed immediately after the laser was turned off.^* Thus it appears that distortions in trap symmetry may also be important for the Penning traps, particularly when high

densities and low temperatures are desired. For the problem discussed here, the existence of such distortions may

explain why we are unable to continue adding angular mo- mentum to the ion cloud by laser scattering.

7, NEUTRAL-PARTICLE TRAPS

The same general considerations should apply to neutralparticle traps; the problem seems qualitatively simpler in that q = 0. Because of the short range of interparticle

collisions, the situation for axially symmetric neutral traps

should be similar to the case of ions in a rf trap in the low- density limit discussed above. Dephasing of angular mo- mentum would be expected to be similar to that described in Section 5. Hence, as noted there, a small degree of trap

asymmetry is probably required in order for spinning of the trapped sample to be observed.

8. CONCLUSIONS

In this paper we have been concerned primarily with angular

momentum imparted to trapped particles by light scattering. As we noted earlier, this angular momentum could come from other sources, such as background gas, trap asymmetries, particle beams,^^ positive or negative feedback on

charged-particle motions^^'^^ (including radiative damping),

and angular-momentum transfer from plasma waves.^'*'^^ Such considerations may modify the results of certain ex- periments such as those on electron- (or ion-) atom collisions."^"'*'' For the case of ions in a Penning trap, it should

be possible to compress the ion cloud at the center of the trap

with a neutral-particle beam, although the cyclotron and

axial degrees of freedom will be heated. Therefore, in atomic-particle traps, one must in general

take into account the angular momentum of the particles

about a particular axis. If the trap has symmetry about this axis then these angular-momentum considerations are important in both the rf and the Penning ion traps and in neutral traps. We emphasize that it is the axial symmetry of the trap that is important for these angular-momentum considerations, not the adherence to the quadratic electric potentials that have been assumed, for simplicity, throughout

this paper. If this axial symmetry is violated then it may in fact be difficult to observe angular-momentum effects in the

rf or neutral traps. In Penning traps, such deviations from

trap axial symmetry owing to electric-field distortions are

not so important; in fact, angular-momentum effects, in- cluding the angular momentum imparted by laser scattering, play an important role in these experiments.

APPENDIX A

in Sections .'"> and (5 we noted that if the electric potential is not nxiallv s\iiiinetric. there is a significiint (lifference in the

3-

3^

o

1-^

cm

rN-73
^J3

o

u

0)

'S

u

1728 J. Opt. Soc. Am. B/Vol. 2, No. 11/November 1985

behavior of L, for the pure rf trap (B = 0) and the Penning

trap. To see how this difference might arise, we first examine the .v-\- motion when the trap potential is given by Eq.

(20) and ^ and a are arbitrary. The equations of motion for a single ion are

Wineland et al.

X + 2q(a + ()x/m = qV.^y/\q\,

y + 2q{a - ()y/m = -q^^x/\q\.

(Al)

(A2)

If we assume a solution x = Re[xo exp{-iwt)], y = Re[yo

exp(— (a,'f)], we can write Eqs. (Al) and (A2) in matrix form:

M(co) P

= 0, (A3)

where

M(cu) =

-w- + 2q{a + t)/m i«qfi,/| q\ -i(^qn^/\q\ -w^ + 2q(a - ()/m_

(A4)

The normal mode frequencies are obtained from det M{w) =

0, which has four roots, ± w+ and iw-, where

2(0,-^)2 = n^2 ^ 4qa/m ± (Q/ + SqaO/Zm + IGq^eVm^) 2 2/^2\l/2

(A5)

or

2{wJ'' = Q/ + 2co/ ± {U; + 4a)/12/ + io^^V", (A6)

where we have made the redefinitions oir^ = 2qa/m and co^^ =

2qe/m = (oj^^ - Wy!^)/2. From Eq. (A3), we can find the

following relationships between components of the eigenvectors:

('A - «t/„

\XoJ.=±u,,

U+ = C?[0^2 _ 2^^2 ^. (j^^4 ^ 4^^2j2^2 + 4c^^4)l/2|/2^^J2jq|,

(A7)

and

(-) -= ^iU_,

U_ = q[9.^^ - 2a;,2 - («/ + 4u>,%^ + 4^;Y^]/2^-nJiq\.

(A8)

Therefore, the general solution is

X = r+ cos(w+( + 0+) + r_ cos(aj_( -I- 0_),

y = -(y+r+ sin(w+f + /?+) - (7_r_ sin(aj_^ + dj. (A9)

In the limit il, ' 0, these .solutions are equivalent to those of Section 5 (Eqs. (21a) and (21b)] for the pure rf trap. Iff =,0, then \U±\ -* 1, and we obtain the solutions of Section 6, where |aj_| = |a>m| and |oo+| = laid- For ( 7^ 0, the solution

(Eqs. (A9)) is a superposition of two elliptical motions at frequencies cij+ and w-. For ( small, i.e., co, « w^, ilc, we find that

2\l/2l (AlO)

In general, L^ is a complicated function that can oscillate around zero. However, if we assume that only the (a,+) or the (a,-) mode is excited, then we obtain a simple expression:

^.± = mr^' U Y

[^^^ + 4u;//Q/+ 4a>,V«/)'/2

+ {Uj - l)sin2(a)i( + e^) + 2a,//i]/]. (All)

If the elliplicity of the orbit is small, i.e., (( f7±( -1( « 1, then

Lz has small fluctuations about a nonzero value. In the Penning trap, if we can assume the usual condition ( Wr/Uj[ «

I, then we find that the magnetron orbit (w- solution) has

|( U-\-l\ = co/Vw/. In the rf trap, as ilc ^ 0, then L''+ --• .and \U-\ —* °°; that is, the normal mode orbits are highly

elliptical. However, if we assume that Wr » ^c » '»'o then,

from expression (AlO), (|t/±(— 1( s co//a,rftc s (00^ — coJi/Qc- Therefore, if we superimpose a magnetic field along the z axis so that 0, » |ujx ""

<^Ji, then the normal mode orbits would be nearly circular. Based on the observations in Section 6 for the Penning

trap, we might therefore make the following conjecture for a cloud of ions in either the Penning or the rf trap: If the x-y

modes are nearly circular and if u)+, w-, and the axial fre- quency (now shifted by space charge) are sufficiently differ- ent that mode coupling is suppressed, then the dephasing or damping of L^ might be small. Therefore in the rf trap we

might expect to observe spinning more easily if we superimpose a magnetic field such that fi^ » \^x ~ (^Ji- For our

i98Hg+ example of Section 5, if Ai';tj, ^ 1 kHz this would

require that B » 130 G.

ACKNOWLEDGMENTS

We greatfully acknowledge the support of the U.S. Office of Naval Research and the U.S. Air Force Office of Scientific Research. We also thank J. C. Bergquist and L. R. Brewer

for critical comments and helpful suggestions on the manuscript.

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TN-74
Wineland et al. Vol. 2, No. 11/November 1985/J. Opt. Soc. Am. B 1729

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17. L. D. Landau and E. M. Lifshitz, Statistical Physics (AddisonWesley, Reading, Mass., 1974).

J. H. Malmberg and T. M. O'Neil, "Pure electron plasma, liquid, and crystal," Phys. Rev. Lett. 39, 1333-1336 (1977).

S. A. Prasad and T. M. O'Neil, "Finite length thermal equilibria

of a pure electron plasma column," Phys. Fluids 22, 278-281

(1979). T. M. O'Neil and C. F. Driscoll, "Transport to thermal equilibri- um of a pure electron plasma," Phys. Fluids 22, 266-277 ( 1979).

21. L. S. Cutler, R. P. Giffard, and M. D. McGuire, "Mercury-199

trapped ion frequency standard: recent theoretical progress and experimental results," in Proceedings of the 37th Annual

Symposium on Frequency Control, 1983. (Copies available from Systematics General Corporation, Brinley Plaza, Rt. 38, Wall Township, N.J. 07719), pp. 32-36; "Thermalization of '^^Hg ion macromotion by a light background gas in an rf quadrupole trap," Appl. Phys." B 36, 137-142 (1985).

22. J. S. deGrassie and J. H. Malmberg, "Waves and transport in the pure electron plasma," Phys. Fluids 23, 63-81 (1980).

23. J. J. Bollinger and D. J. Wineland, "Strongly coupled nonneutral ion plasma," Phys. Rev. Lett. 53, 348-351 (1984). R. D. Knight and M. H. Prior, "Laser scanning measurements of the density distribution of confined ^Li"*" ions," J. Appl. Phys.

50,3044-3049(1979).

H. Schaaf, V. Schmeling, and G. Werth, "Trapped ion density

distribution in the presence of He-buffer gas," Appl. Phys. 25, 249-251 (1981).

12

13

14.

15

16

18

19

20

24

26. J. B. Jeffries, S. E. Barlow, and G. H. Dunn, "Theory of space- charge shift of ion cyclotron resonance frequencies," Int. J. Mass Speclrom. Ion Phys. 54, 169-187 (1983).

27. D. A. Church and H. G. Dehmelt, "Radiative cooling of an

electrodynamically contained proton gas," J. Appl. Phys. 40, 3421-3424 (1969).

28. H. G. Dehmelt, "Stored ion spectroscopy," in Advances in Laser

Spectroscopy, F. T. Arecchi, F. Strumia, and H. Walther, eds. (Plenum, New York, 1983), pp. 153-187.

29. D. J. Wineland, "Spectroscopy of stored ions," in Precision Measurement and Fundamental Constants II, B. N. Taylor

and W. D. Phillips, eds., Natl. Bur. Stand. (U.S.) Spec. Publ. 617

(1984), p. 83-92.

30. J. T. Davies and J. M. Vaughan, "A new tabulation of the Voigt

profile," Astrophys. J. 137, 1302-1305 (1963).

31. H. G. Dehmelt and F. L. Walls, "Bolometric technique for the rf spectroscopy of stored ions," Phys. Rev. Lett. 21, 127-131

(1968).

32. E. W. McDaniel, Collision Phenomena in Ionized Gases (Wilev,

New York, 1964), pp. 426-482.

33. J. C. Bergquist, D. J. Wineland, W. M. Itano, H. Hemmati, H.- U. Daniel, and G. Leuchs, "Energy and radiative lifetime of the

5d^6s'^ ^0,5/2 state in Hg II by Doppler-free two-photon laser spectroscopy," Phys. Rev. Lett, (to be published).

34. L. M. Chanin and M. A. Biondi, "Mobilities of mercury ions in helium, neon and argon," Phys. Rev. 107, 1219-1221 (1957).

35. S. Ichimaru, "Strongly coupled plasmas: high density classical plasmas and degenerate electron liquids," Rev. Mod. Phys. 54, 1017-1059(1982).

36. W. Neuhauser, M. Hohenstatt, P. E. Toschek, and H. G. Dehmelt, "Localized visible Ba"*" mono-ion oscillator," Phys. Rev. A

22,1137-1140(1980).

37. R. E. Drullinger, D. J. Wineland, and J. C. Bergquist, "Highresolution optical spectra of laser cooled ions," Appl. Phys. 22, 365-368 (1980).

38. J. J. Bollinger, J. D. Prestage, W. M. Itano, and D. J. Wineland,

"Laser cooled atomic frequency standard," Phys. Rev. Lett. 54, 1000-1003 (1985).

39. C. F. Driscoll and J. H. Malmberg, "Length dependent contain- ment of a pure electron-plasma column." Phys. Rev. Lett. 50, 167-170 (1983).

40. L. S. Brown and G. Gabrielse, "Precision spectroscopy of a charged particle in an imperfect Penning trap," Phys. Rev. A 25, 2423-2425(1982).

41. M. D. McGuire and E. N. Fortson, "Penning-trap technique for studying electron-atom collisions at low energy," Phys. Rev.

Lett. 33,737-739 (1974).

42. G. Graff, F. G. Major, R. W. H. Roeder, and G. Werth. "Method

for measuring the cyclotron and spin resonance of free elec- trons," Phys. Rev. Lett. 21. 340-342 (1968).

43. G. Graff and M. Holzscheiter. "Method for trapped ion polarization and polarization detection." Phys. Lett. 79A, 380-382

(1980).

{see overleap

o

3

TN-75
Volume 57, Number 1 PHYSICAL REVIEW LETTERS 7 July 1986

Sympathetic Cooling of Trapped Ions: A Laser-Cooled Two-Species

Nonneutral Ion Plasma

D. J. Larson,'*' J. C. Bergquist, J. J. Bollinger, Wayne M. Itano, and D. J. Wlneland

National Bureau oj Standards, Time and Frequency Division, Boulder, Colorado 80303

(Received 31 March 1986)

Sympathetic cooling of trapped ions has been demonstrated in an experiment where '^^Hg"^ ions were confined in a Penning ion trap with laser-cooled 'Be"*" ions. '^^Hg''' temperatures below 1 K

were achieved. Ion plasma sizes, shapes, and rotation frequencies were measured. Dramatic

changes in the 'Be"^ plasma were observed when the '^^Hg"^ ions were introduced into the trap. These observations are consistent with the prediction of centrifugal separation for ions of different charge-to-mass ratios.

PACS numbers: 32.80.Pj, 52.25.Wz The techniques of trapping and laser cooling have

led to dramatic improvements in recent measurements

and future prospects for precision spectroscopy.''^

However, the number of species which can be directly

laser cooled is limited by the availability of tunable,

narrow-band sources and by more fundamental problems such as complicated level structure (as in molecules) or the absence of optical structure (as in protons and negative atomic ions). In addition, the

laser-cooling process itself strongly perturbs the levels of an atom or ion. The cooling laser must be turned

off during the interrogation time to avoid light shifts (ac Stark shifts) and broadening many orders of magnitude larger than achievable linewidths. In the most

accurate experiment to date,^ the linewidth of a radio

frequency transition in "^Be"*" was limited (^0.025

Hz) by the necessity to restrict interrogation times

( ^ 20 s) in order to minimize heating that occurred

when the cooling laser was turned off. In principle,

linewidths substantially less than 0.001 Hz should be

possible on transitions in trapped ions. Sympathetic cooling,'''*'^ where one species is cooled by interaction with a second, directly cooled

species, offers a solution to these problems. In the

present experiments, spectroscopy of continuously

cooled '^^Hg'*' ions was made possible by confining

them in a trap together with laser-cooled ^Be"*" ions. The 313-nm radiation used to cool the Be"*" ions did not seriously perturb the Hg"*" ions since this wavelength is far from any re.sonance in Hg"*". We es- timate that the fractional light shift in the ^"'Hg"*" hyperfine separation would be less than 5x 10~'^ under

conditions similar to those in the present experiment.

Although sympathetic cooling was noted previously

in experiments with different isotopeG of magnesium

ions'''* we report here the first measurements of temperatures and spatial distributions for a two species,

magnetically confined, nonneutral plasma. In addition, the present experiments have demonstrated that

radial diffusion of the sympathetically cooled species is prevented. One interesting feature of such a plasma,

that the species with higher mass-to-charge ratio

moves radially to the outside of the lighter (or more

highly charged) species,^- ^ was observed. The present

results appear to have a direct bearing on proposed ex- periments to store and cool antiprotons at very low energies^"'° and may be important in other experiments

where two or more species of different charge-to-mass

ratio are simultaneously present in the trap.'' Different considerations lead to separation of species in rf traps.

'^ The experiments were carried out using a Penning

ion trap'^ with hyperbolically shaped electrodes. The

endcap-to-endcap spacing was 0.51 cm and the innerring diameter was 0.83 cm. The trap and vacuum sys- tem were at room temperature for all of the measure- ments except one where the trap was cooled to 77 K. A magnetic field near 1.43 T was used for all of the

measurements. Be ''"-ion plasmas (or "clouds") were

created in the trap by use of a beam of electrons to ionize beryllium atoms which were evaporated from a tungsten filament wrapped with beryllium wire. Hg''" ions were loaded into the trap by ionizing mercury va- por leaked into the vacuum system through a variable

leak valve while the Be"*" ions were being laser cooled.

Three laser systems were used to cool and probe the

ions. A frequency-doubled dye laser beam with a power of approximately 50 /uW at 313 nm was directed

through a hole in the ring electrode and used to cool

the Be''" ions on the Is^S^j to Ip^Pyi-, ^y= +T to my=+y, transition.' The photons backscattered by

the Be"*" ions were collected and counted. This signal was used for all of the measurements on the Be"*" ions. A second 313-nm source was directed through another

hole in the ring. When tuned to the 25^Si/7 '1/2 to lp^Py2^ my= +- to mj= —J, transition, this probe

beam depopulated the my = + y state and thus caused

a decrease in the scattered light from the cooling

beam.' The ion-cloud sizes and shapes were measured

by observing this signal for different positions of the

probe beam. The ion-cloud rotation frequencies were

obtained through the Doppler shift by measuring the

frequency of the probe laser which produced max- imum depopulation.''* A third laser system'^ produced

70 © 1986 The American Physical Society TN-76
Vol LIMI 51, NL'MUI K I PHYSICAL REVIEW LETTERS 7 Jli V 1986

a few microwatts of tunable power at 194 nm. The positions and velocities of the Hg"*" ions in the trap were measured by collecting and counting the 194-nm photons scattered when the frequency was tuned through

the 6.v^5'|/2 to 6/7 ^f1/2, mj= - y to mj= + j transi- tion in the ion. Microwaves tuned to the Inij = I transition in the ground state prevented depletion of the

mj= - J ground state by optical pumping."" For both

Be"*" and Hg"*" the temperatures were determined by

measuring the Doppler contribution to the frequency

width of the observed signal as the appropriate laser was tuned through resonance.'"' An example of the

signal from a Hg"^-ion cloud whose cyclotron temperature has been sympathetically cooled to near 1 K is shown in Fig. 1

A cold, single-species ion cloud in a Penning trap

with hyperbolic electrodes should assume a nearly

spheroidal shape with constant density inside the cloud

and an abrupt drop m densi'y at the edge '"* '^- '^ The

eccentricity of the spheroid is determined by the trapping voltage, magnetic field, and rotation frequenc\

Any cloud m thermal equilibrium will exhibit uniform

rotation. The rotation frequency is a measure of the density and of the angular momentum of ihe ions.'

"''

It can be partially controlled by varying the torque applied to the cloud by the cooling laser. Observed rota- tion frequencies were well below those corresponding

to the maximum stable density of ions in the trap, the

Brillouin density.'" The mechanism which limits the frequency is not clearly understood, but it may depend

upon trap asymmetries.'**''^'^'' For precision spectros- copy, high rotation velocities and the accompanying

Doppler shifts should be avoided by working at low

densities.

(1) o

(

o Q)

=3 CO u to n t.:

fl) =J > (

ni O

to

-' £) O

100 -

./•

- • •

•

•

•

• •

50

• •

• •• V •

•

200 400 600

Relative Laser Frequency (MHz)

FIG 1. Observed nuorcsccnce from a llg''-i(in cloud

cooled to approximately 1 K by inlcraclion wiih a laser- cooled Be"" cloud. The horizontal axis is ihc rrequciuN of a

laser tuned through the 6v'.S'|/: < in, = - y) lo fi/'"/',-. (wy=+y) transition in Hg* The rooni-lcmperalurc

Doppler width of ihis transition is .ihoui 1 4 Cil I/.

A dramatic change in the rotation frequency (and

thus the density) and the shape (away from

spheroidal) of the Be"^ cloud occurred when Hg"^ ions were introduced into the trap. With only Be"^ in the

trap, rotation frequencies up lo nearly 200 kHz could

be achieved. The single-particlc rotation (magnetron)

frequency was measured to be only 9.4 kHz with -1.5

V on the ring, so the observed frequencies were due

primarily to the space-charge fields. With Hg"^ and

Be"^ in the trap, the largest observed rotation frequency was 21 kHz. The measured size and shape of a Be"^ cloud with and without Hg"*" is shown in Fig. 2(a).

Adding the Hg"^ ions decreased the rotation frequency

by a factor of 2 A larger change in density and shape

of a Be"*" cloud is shown in Fig. 2(b). Figures 2(a) and

2(b) are the smallest and largest clouds examined. In

all cases the Be"*" clouds were prolate spheroids

without the Hg^ and oblate and n(jn>pheroidal with

Hg"^ present The Hg"*" clouds appeared disk shaped

with diameter-to-length ratios of 6 lo 1 or larger

Clearly, the Hg"*" ions applied a substantia! torque to the Be"*" ions. For ah of the Hg"*" clouds measured,

the axial bounce frequency (equal to twice the tixial length divided by the rms axial ve!ocit> ) was approximateU' equal to the rotation frequency This condition

is consistent with axial asymmetries of the fap and

resultant resonant particle transport limiting the rota- tion frequency. ''"^-^^ No definitive evidence is avail- able, however, to exclude other mechanisms-' from

limiting the rotation frequency with the exception of

drag due tc> background gas. Cooling of the trap elec- trodes to 77 K, which substantial!),- lowered the residual rnercurv pressure, caused no change in measured ro- tation frequencies. Despite the presence of a torque on the Hg^ ions, Ihey remained in the trap for several

hours unless the cooling was interrupted. If the cooling laser was blocked or tuned off resonance the Hg"*"

ions were lost in several minutes, demonstrating that

Hg"*" confinement was dependent on the torque applied to the Hg"^ ions by the laser via the coupling to the Be"*" ions. Measured Hg"*" temperatues ranged from 0.4 to 1.8 K and Be"*" temperatures ranged from less than 0.05 to 0.2 K. The temperature difference between the two

species may be a result of the thermal coupling

between the directly laser-cooled Be'*' ions and the in- directly cooled Hg"*" ions being weak, due to centrifugal separation. Also, the axial temperature of the Be"*" ions should be higher than the transverse temperature

because of photon recoil,'"* and this ccnild cause the

lig'*"-ion temperature to take on some intermediate

value. Given the torque due to the Hg"*" ions, the counteracting torque due to the laser on the Be"^ ions, and

the temperature difference between the two clouds, it

IS not unreasonable to expect some shear. In a uniTN-77 '1
f

VoLi \ij: 57, Nlmhf.r 1 PHYSICAL REVIEW LETTERS 7 July 1986

Be"^ alone Hg+ +Be

1^-"'*

(a)

Be'

(b^

y 400

200- —1—1—

z

-i—'

200 400

Scale in '/m

(C)

I'ICi 2. i;i) Mcitsured Be"*- and I Ig"*" -ion-cloud shapes.

The magnciic field i^ alcng the raxis. The Be"^ cloud, which

coniained about KOO ions, is snown with and without the

[Ig"^ present The measured rotation frequencies were 41 kHz without Hg"* [corresponding to a density,

/;{Be'^) = 3 9x 10" cnTM and 20 kliz with Hg"* present

(/iCBe-^ ) = 1 9x 10' cm', /;(Hg' )= i.6x 10' cm The

solid lines drawn through the data points conform to the

theoretically predicted and separately confirmed spheroidal

shape lor Be"* alone but are drawn without consideration of

theoretical shapes for Be"* and Hg"^ together. The edges of

the clouds were measured with an uncertainty of about 25 /nm (b) Measured cloud shapes for a Be"*" cloud which contained about 12 000 ions and a Hg"* cloud which was larger

in radial extent than the aperture through the ring electrode (diameter "^ 2100 jj.m) The measured rotation frequencies were 73 kH/ without llg^ I /;( Be"* ) = 7,0x 10' cm--'] and

18 kHz with Hg^ i«(BeM= 1.8x10' cm-\ /;(Hg+) = 1.5xMV cm""-') (c) Theoretically predicted shapes for cold Be"*" and Hg"* clouds under condition.^ similar to those

in (a).

ffjrmly rotating cloud, the Doppler shift and thus the

observed optical resonance frequency should be a linear function of the radial distance from the center of

the cloud. [ igure 3 shows the observed resonance fre- quency for a Be'*' cloud, confined in the trap together

INI X 180 ^

o

120 c

<D

-)CT bO

CI) U_

<_y C

n3co

(n -bO

0) CC> -120

•*-• CO

CD -180

-400 -200 200 400

Probe Beam Transverse Position (p<m) FIG. 3. Measured resonance frequency in Be"*" as a function of the transverse position (perpendicular to the trap

axis) of the probe laser beam. These measurements, done

with Hg"*" ions present in the trap, placed a limit on the amount of shear in the Be"*" cloud. The statistical error in the data points is smaller than the size of the points,

with a Hg"*" cloud, as a function of the transverse position of the probe laser beam. Fitting the data in Fig. 3 with a rotation frequency aj = cti;. + Aaj(/-/rci), where /-d is the cloud radius, implies that IAoj/ojJ < 0.3. Indeed, to the extent that the data suggest that shear is present, it appears that the outer edge of the cloud is rotating faster than the middle. No evidence for shear was observed in a similar measurement on Hg"*" ions

or in the comparison of the frequencies between the

two species.

The measured shapes of the Be"'"- and Hg''"-ion

clouds were compared to numerical calculations based on a zero-temperature theory which balances magnetic, space-charge, and external electric forces.^-

*^'^'' The calculation used measured values for the trapping

voltage, magnetic field, rotation frequency, and rela- tive cloud diameters to predict shapes. A zero- temperature theory should be applicable when the ratio of the Debye length to the dimensions of the ion

cloud is small. In the present experiments this ratio was typically less than O.L The theoretical prediction

of complete radial separation of the ion clouds was not

definitively tested since all of the probing of ion position was done in a direction perpendicular to the axis

of the trap. Changes in collection efficiency for the

194-nm fluorescence as the beam position was moved

prevented the use of count rate as a measure of the

Hg"*" cloud thickness along the laser-beam direction.

The theory^' ^^ suggests that the heavier Hg"*" ions

should be found in a torus around the lighter Be''' ions

72 TN-78
Vol UMl- 57, NUMIUK 1 PHYSICAL REVIEW LETTERS 7 Jlly 1986

as shown in Fig. 2(c).

These experiments have demonstrated the effective- ness of sympathetic cooling and point the way toward

applications such as use in high-resolution spectroscopy and antiproton accumulation and storage. They

have also allowed observation of some of the characteristics of a cold, magnetically confined, two-species

nonneutral plasma. Significant features include the

dramatic change in the shape of the Be"*" -ion cloud

when the Hg"*" ions are introduced into the trap, the

temperature difference between the laser-cooled Be"*" ions and the sympathetically cooled Ug"*" ions, and the

substantial torque exerted on the Be"^ cloud by the

Hg"*" ions. A few features, including the source of the

torque on the Hg"*" ions, are not clearly understood

and may limit the number of ions that can be stored in a trap. Such techniques may be applicable to cooling

neutral particles in a irap.^'' We would like to thank J. Prestage, R. Hulet, and

C. Manney for their contributions to this work. We

thank T. O'Neil, C. F. Driscoll, L. Brewer, and

F. Walls for helpful comments on the manuscript. We

gratefully acknowledge the support of the U.S. Office

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

Scientific Research.

''^'Permanent address; Department of Physics, Llniversit> of Virginia, Charlottesville, VA 22901.

'For reviews, see D J. Wjneland, W. M. Itano, J. C. Bergquisl, J. J. Bollinger, and J. I). Preslagc, in Aio'nic Flnsics y, edited by R. S. Van Dyck, Jr., and F. N. Forison

(World Scientific. Singapore, 1985), p. 3, W I) Phillips.

J. V. Prodan, and H. Metcalf, ihiil., p. 338, and rctcrcnces

therein. ^The Mechanical Effects of Light, edited by P. Mc\sirc

andS. Stenholm, J.Opl. Soc. Am B 2, No. 11 (1985)

3J, J Bollinger, J. D. Prestage, W. M. Itano, and I) J Wineland, Phys. Rev. Lett. 54. 1000 (1985).

'*R. E. Druilinger, D. J. Wineland. and J. C. Bcrgquisi.

Appl. Phys. 22, 365 (1980). 5D. J. Wineland, R. E. Druilinger, and F L Walls, Phys

Rev. Lett. 40, 1639 (1978)

^T. M. O'Neil, Phys. Fluids 24, 1447 (1981) 'M. Geva, M. Krishnan. and J L llirshficld. J. -Appl

Phys. 56. 1398 (1984)

8H. G. Dehmelt cr ai. Bull. Am. Phys. Soc. 24. 757

(1979); G. Torelli. in ProcccJinj^is ol ihc I ijih futro/wan Symposium on Nuck'on Anli-Nuclcon Inicraciions, Bicssanonc. Italy. 23- 2H June 19Si). edited by M Crcsli dslilulo Na/ionale

di Fisica Nucleare, Padua. Italy. 1980). p. 43.

"^G. Gabrielse, H. Kalinowsky. and W. Kells. in Fhysus

with Antiproions at LEAR in the ACOL Era. edited by F. Gastaldi ct al. (Editions Frontieres. Gif'-sur-Yvetle. France.

1985). p. 665; W. Kells. IEEE Trans. Nucl. Sci. 32. 1770

(1985).

'OC. F. Driscoll, in Fiocccclin.us oj the Anti-Muttci laiility

Workshop (Univ. of Wisconsin Press, Madison, 1985);

M. H Holzscheiter, ihul. I'D. A. Church, in Ref, 1, p 137

'^C. Flory, L. Cutler, and R GilTard. to be published

'Hi. G. Dehmelt. Adv. At. Mol. Phys. 3. 53 (196^). and 5. 109 (1969); D J. Wineland. W ,VL Iiano. and R. S Win

Dyck, Jr.. Adv. Al. Mol. Phys. 19. 135 (1983)

'*J. J. Bollinger and I) J Wineland, Phys Rev Lcii. 53. 348 (1984).

'^H. Hemmati. J. C. Bergquisl. and W\ M luino. Opt. fell

8. 73 (1983). '^W. M. llano, J. C. Bergqul^l, and I) J Wineland. J. Opt.

Soc. Am. B2. 1392 (1985)

>^S A. Prasad and T. M ()"Nei!. Ph\s. Iluid-, 22. 2^<S (1979).

'8D. J. Wineland. J. J. Bollinger. W M llano, and J 1) Prestage. J. Opt. Soc. Am B 2, \il\ '1985)

'"^J, 11. Malmberg. C. I- Driscoll. and V\ 1), While. I'h>>. Scr. 7 2. 288 (1982).

^*^R. C. Davidson. Theory ojWnnnciiiral I'Icisdm'^ '.Beniamin. Reading, MA, 1974). p. 4

2lT. M. O'Neil and C. F. [)r:scoll. Phvs. Fluids 22, 266

(1979).

^2C. F. Dri.scoll and J. II. Malmberg. Phys Re\ Lett 50. 167 (1983).

2-^), L. Eggleslon. T. M. O'Ne'i. ;ind J. II M.ilmherg.

Phys. Rev. Lett. 53.982 (1984)

^*J. J. Bollinger, C Manney. and 1) ,! Wlnelind. lo be

published.

2>W V\. Wing. Prog. Quantum Fleciron 8. 181 '19X4)

TN-79

73
This paper published In Intense Position

Beams , edited by E.H. Ottewltte and W. Kells

(World Scientific, Singapore, 1988), p. 63.

Ion Trapping Techniques: Laser Cooling

and Sympathetic Cooling*

J.J. Bollinger, L.R. Brewer, J.C. Bergquist, Wayne M. Itano

D.J. Larsont, S.L. Gilbert and D.J. Wineland

Time and Frequency Division

National Bureau of Standards

Boulder, Colorado 80303

ABSTRACT

Radiation pressure from lasers has been used to cool and

compress "Be"*" ions stored in a combination of static

electric and magnetic fields (Penning trap) to temperatures

less than 10 mK and densities greater than 10' cm"-^ in a magnetic field of l.A T. A technique called sympathetic

cooling can be used to transfer this cooling and

compression to other ion species. An example of ^""Hg"^

ions sympathetically cooled by laser cooled "fie ions is

given. The possibility of making an ultracold positron

source vi^ sympathetic cooling is also discussed.

Ion traps have been used in low energy atomic physics

experiments for a period of roughly 30 years. ^»"^^ During the past 10

years, however, the technique of laser cooling and compression has

greatly increased the potential use of stored ions in a number of

applications. One example is time and frequency standards. With the

technique of laser cooling and compression, ion temperatures less

than 10 mK, densities a factor of 10 less than the Brillouin limit

(defined below), and essentially indefinite confinement times have

been obtained. ^»^'' Unfortunately, elementary charged particles

(electrons, positrons, protons, etc.) can not be directly laser

*Work of the U.S. Government, not subject to U.S. copyright.

tPermanent address: Dept. of Phys., Univ. of Virginia,

Charlottesville, VA 22901

TN-80
cooled. In addition the niimber of atomic ion species that can be

directly cooled by a laser is limited. Sympathetic cooling, ~'*^^ where one charged species is cooled by the Coulomb interaction with a second, directly cooled species, provides a means of transferring the

desirable properties of a laser cooled ion species to a species, such

as positrons, which can not be laser cooled directly.

The idea of sympathetic cooling is general and can, in

principle, be used with any type of trap. In this paper, though, the

idea of sympathetic cooling is examined only in the case of two

different charged species in a Penning-type trap. The Penning trap'^

uses a uniform magnetic field B = Bz to confine ions in a direction

perpendicular to the z axis. The ions are prevented from leaving the

trap along the z axis by an electrostatic potential, (prji, typically

provided by three electrodes. In the general discussion that

follows, unless explicitly stated, it is only assumed that the trap

potential, (prp, is symmetric about the z axis (i.e. cp^ = <p>j(r,z) where

r,z are cylindrical coordinates). This includes the Penning trap

typically used in atomic physics experiments where the electrodes

have hyperbolic shapes which give rise to a quadratic trap

potential. ^'^-' It also includes the traps of the University of

California at San Diego (UCSD) group which use cylindrical tubes as the trap electrodes. °~^^-'

In the next section, the technique of laser cooling and

compression is described and some results of measurements on laser

cooled ^Be"*" ions are summarized. The idea of sympathetic cooling is

then described in more detail and the results of an experiment where

Hg"^ ions were sympathetically cooled by laser cooled Be"*" ions are

given. We conclude with a discussion of the possibilities of

sympathetically cooling positrons and speculate at some possible

applications of very cold positrons.

TN-81
LASER COOLING AND COMPRESSION

The technique of laser cooling^ ^"•'•^^ utilizes the resonant

scattering of laser light by atomic particles. By directing a laser

beam at the ion plasma we can decrease the thermal velocity of an ion

in a direction opposite to the laser beam. The laser is tuned to the

red, or low frequency side of the atomic "cooling transition"

(typically an electric dipole transition). Some of the ions moving

in a direction opposite to the laser besim propagation will be Doppler

shifted into resonance and absorb photons at a relatively high rate.

Ions moving with the laser beam will be Doppler shifted away from the

resonance and the absorption rate will correspondingly decrease.

When an ion absorbs a photon its velocity is changed by an amount

Lv = -h^/m (1)

due to momentum conservation. Here Av is the change in the ion's

velocity, k is the photon wave vector where |k| = 2v/X,X is the

wavelength of the cooling radiation, m is the mass of the ion, and

Ziih is Planck's constant. The ion spontaneously re-emits the photon

in a symmetric way. In particular, averaged over many scattering

events, the reemission does not change the momentxim of the ion. The

net effect is that for each photon scattering event, the ion's

average velocity is changed by an amount shown in Eq. 1. To cool an atom from 300 K to mK temperatures takes typically 10'^ scattering

events. The theoretical cooling limit, due to photon recoil

effects^ ^~^-^'' is given by a temperature equal to A^/CZkg) where y is

the radiative linewidth of the atomic transition in angular frequency

units and kg is Boltzmann's constant. For a linewidth y = 2ii'19.4

MHz, which is the radiative linewidth of the Zs-^Si -» 2p^P3/2

transition in "Be"*" (X = 313 nm) used for cooling, the theoretical

minimiim temperature is 0.5 mK.

TN-82
Laser scattering can also be used to change the angular momentum

and compress the ion "cloud" or plasma. For simplicity, in the

discussion of this paper it is assumed that particles with charge q>0 are trapped. In thermal equilibrium the plasma in a Penning trap

undergoes a uniform rotation at an angular frequency -w.^^"^^^ The

minus sign indicates that for positively charged particles, the

rotation is in the -9 direction. The z component of the canonical

angular momentum for an individual particle is

9

1^ = mver + -fl^. (2)

The two terms in Eq. 2 are the plasma mechanical angular momentum and

the field angular momentum. The total z component of the angular

momentum of the plasma is^"^

L^ = m(n/2 - u))N < r^ >. (3)

Here N is the total number of ions, Q = qB/(mc) is the cyclotron

frequency and <r^> is the mean squared radius of the plasma. For all

of the work described in this paper, w « Q and

L^ . E|N_ <^2> > . (4)

Suppose the cooling laser beam is directed at the side of the plasma

which is receding from the laser beam due to the plasma rotation.

Because the rotation of the positive ions is in the -^ direction, the

torque of the laser on the ions will also be negative. Consequently

angular momentum is removed from the plasma and according to Eq. 4

the radius of the plasma must decrease. As the radius decreases, the

density of the plasma increases. In principle, the plasma compression continues until w = Q/2 where the maximum density, known as the

Brillouin density occurs. The Brillouin density, ng, is given by

ng = — . (5)

8TTq2

TN-83
In practice, the plasma is compressed until the torque due to the

cooling laser is balanced by an external torque. Collisions with

background gas particles and axial asymmetries of the trap are possible sources of external torques. The group at UCSD has observed

that the axial asymmetry of their cylindrical traps plays an important role in the electron confinement time.-*-^* With the use of

the plasma compression, it is possible to stop the normal processes

which lead to an increase in the plasma radius and obtain a steady

state plasma indefinitely. In our laboratory we routinely maintain a steady state plasma of laser cooled ^Be"*" ions for a period of a day.

We have used an optical double resonance technique-^ »^»^°^ to

measure the shape, density, and temperature of clouds of a few

hundred to a few ten thousand laser cooled 'Be ions stored in a Penning trap. Densities of 10' - 10° ions/cm-^ at a magnetic field of

1.4 T were obtained on both the large and small clouds. The

Brillouin density for "Be"^ ions in this magnetic field is 6 x 10°

ions/cm-^. We believe that axial asymmetries of the trap were

responsible for limiting the ion densities to about an order of

magnitude less than the Brillouin density. ^»^»^"^ Ion temperatures

less than 100 mK were routinely obtained. On smaller clouds

consisting of a few hundred ions, ion temperatures less than 10 mK

were measured.

SYMPATHETIC COOLING

The idea of sympathetic cooling is to use the Coulomb

interaction to transfer the long term confinement, high densities,

and low temperatures of a laser cooled ion species (for example,

"Be"*") to a charged species (for example, positrons) which can not be

directly laser cooled and compressed. O'Neil has investigated the

equilibrium distribution of a nonneutral multispecies ion plasma for

TN-84
the case where the plasma is assumed to have infinite extent along

the z axis. '-^ For simplicity consider a two species ion plasma

consisting of ion species with like charge q but different masses

m2>mj^. Thermal equilibrium requires that both ion species undergo a uniform rotation at the same frequency, w. '' The uniform rotation

tends to produce a centrifugal separation of the two ion species. In

particular, the heavier ion species tends to reside outside the

lighter species.

For the case of an infinitely long two species plasma, the

lighter species forms a column of plasma centered on the z axis with

outer radius h-^. The heavier species forms a cylindrical shell

outside the lighter species with inner radius a2 and outer radius b2.

In the limit of zero temperature, the edges of the plasma become

sharp and the separation between the two species is complete with

bj^ < a2 < b2. The separation d = a2-bj^ between the two species

depends on h-^ and the density ratio n2/n2^ of the two species and is

given by the expression

d = bi{(n2/ni)-2-l}. (6)

The density ratio depends on the rotation frequency according to

(m2-mj^)w^

n2/ni = 1 , (7)

mj^a)(Qj^-u))

2iTq^

ni =

For a finite length plasma, the situation is similar. The heavier

species tends to form a "doughnut" around the lighter species (see

Fig. 1(c) ). The gap, d(z), between the two species now depends on the z coordinate. Zero temperature calculations of plasma shapes for

TN-85
a two component nonneutral plasma in a harmonic Penning trap ^°^ show

that in general d(0) < d(z) and d(0) depends in a complicated way on

bj^, b2» n2/nj^, and the voltage applied to the trap. Eq. 6 provides a lower estimate for d(0). For a large range of the input parameters,

this actually turns out to be a good estimate of d(0).

For finite temperatures, the edge of the plasma is not perfectly

sharp, but falls to zero over a distance characterized by the Debye

length, Xj) = [kBT/(4iTnq^)]^.^^^ Here n is the density of the

particular ion species and T is the temperature. One expects the

thermal coupling between the two different ion species to be strong

if the gap d(0) is small compared to either of the species' Debye

lengths. If d(0) is large compared to the Debye lengths, the thermal

coupling may be weak. The exact strength of the thermal coupling is a difficult theoretical problem to estimate.

We experimentally tested the idea of sympathetic cooling by

loading ^'°Hg in the same trap with laser cooled ^Be"*" ions."-'

Without the Hg"*", Be"*" rotation frequencies up to 200 kHz were

observed. A dramatic change in the rotation frequency and shape of

the Be"*" cloud occurred when Hg"^ ions were introduced into the trap

(see Fig. 1). With Hg"*" and Be"*" in the trap, the largest observed

rotation frequency was 21 kHz. Clearly the Hg"'" ions applied a substantial torque to the Be"*" ions. The source of this torque is not

clearly understood. The Hg"*" ions remained in the trap for at least

several hours unless the cooling was interrupted. If the cooling

laser was blocked the Hg"*" ions were lost in several minutes

demonstrating that the long term confinement of the Hg"*" was dependent

on the torque applied to the Hg"*" ions by the cooling laser via the

coupling to the Be"*" ions. Measured Hg temperatures ranged from 0.4

to 1.8 K and Be"*" temperatures ranged from less than 0.05 to 0.2 K.

The difference in the temperatures may result from the thermal

coupling between the Be"*" and Hg"*" ions being weak. Because we were

TN-86
only able to measure the plasma boxindaries from a perspective along a direction perpendicular to the z axis (see Fig. 1), we were \inable to

experimentally verify the existence of a gap between the two ion

species. Future experiments are planned which should enable us to

study the two species nonneutral plasma in more detail. In

particular, by viewing the plasma in a direction along the z axis, a measure of the plasma bo\indaries in the z = plane can be obtained.

This should enable us to experimentally check the presence of a gap

between the two ion species. In sxonmiary, the effectiveness of

sympathetic cooling and compression was demonstrated. Long

confinement times, high densities, and low temperatures were achieved

on the trapped Hg"'' ions.

ULTRACOLD POSITRON SOURCE

Cooling and compression of positrons by sympathetic cooling with

laser cooled ^Be"*" ions appears to be experimentally possible. The

gap between the species in this case where m2/mj^ » 1 can be quite

small. For oj < Q2 « ^1» ^q. 7 can be rewritten n2/n^ = 1 - oj/Q2'

For a rotation frequency of w = 0.01Q2(Be^) (the condition achieved

in the Be"*", Hg"*" experiment), n2/nj^ = 0.99. According to Eq. 6 this

means the gap between the positrons and Be"^ ions can be quite small

and the thermal coupling high. It also follows that the density of

the positrons will be approximately limited to the density of the Be"*" ions. In particular, positron densities will be limited to values

much less than the Brillouin density for positrons. The principal

advantages of sympathetic cooling and compression of positrons, then, are long term confinement and low temperatures. Positron

temperatures less than 10 mK appear possible. Two potential

applications of such a source of ultracold positrons are as follows.

In a magnetic field B > 6 T, positron densities greater than

10^^ cm"-^ are potentially achievable by sympathetic cooling with Be"*"

TN-87
ions. A positron plasma with densities n > lO-*-^ cm"-^ and

temperatures T < 10 mK is an interesting object to study. In

addition to being strongly coupled, the cyclotron and plasma

frequencies are quantized, ^^'' making a quantum mechanical description

of the plasma dynamics necessary. Another application of ultracold

positrons may be in antihydrogen production. A recent proposal-*-' for

making antihydrogen uses the three body recombination

p + e +e ^ H + e.

The rate of this three body process has a T~"'^ temperature

dependence. Production of antihydrogen would therefore be greatly

enhanced by using ultracold positrons.

We gratefully acknowledge the support of the U.S. Office of

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

Research. We thank Carl Weimer and Jerome Helffrich for carefully

reading the manuscript.

REFERENCES

1. Dehmelt, H.G., Adv. At. Mol. Phys. 3, 53 (1967) and 5, 109

(1969).

2. Wineland, D.J., Itano, W.M. , and Van Dyck, Jr., R.S., Adv. At.

Mol. Phys. 19, 135 (1983).

3. Bollinger, J.J. and Wineland, D.J., Phys. Rev. Lett. 53, 3A8

(198A).

A. Brewer, L.R. , Prestage, J.D., Bollinger, J.J., and Wineland,

D.J., in "Strongly Coupled Plasma Physics," edited by F.J.

Rogers and H.E. DeWitt (Plenum, New York, 1987), p. 53.

5. Drullinger, R.E., Wineland, D.J., and Bergquist, J.C., Appl.

Phys. 22, 365 (1980).

TN-88
6. Larson, D.J., Bergquist, J.C., Bollinger, J.J., Itano, W.M. , and

Wineland, D.J., Phys. Rev. Lett. 57, 70 (1986).

7. Penning, F.M., Physica (Utrecht) 3, 873 (1936).

8. Malmberg, J.H. and deGrassie, J.S., Phys. Rev. Lett. 35, 577

(1975).

9. Malmberg, J.H., Driscoll, C.F., and White, W.D., Physica Scripta

T2, 288 (1982).

10. Driscoll, C.F., Fine, K.S., and Malmberg, J.H., Phys. Fluids 29,

2015 (1986).

11. Wineland, D.J., and Itano, W.M., Phys. Rev. A 20, 1521 (1979).

12. Itano, W.M. and Wineland, D.J., Phys. Rev. A 25, 35 (1982).

13. Wineland, D.J. and Itano, W.M., Phys. Today 40, 34 (June 1987);

Stenholm, S., Rev. Mod. Phys. 58, 699 (1986).

14. Malmberg, J.H. and O'Neil, T.M., Phys. Rev. Lett. 39, 1333

(1977).

15. Prasad, S.A. and O'Neil, T.M., Phys. Fluids 22, 278 (1979).

16. Wineland, D.J., Bollinger, J.J., Itano, W.M., and Prestage,

J.D., J. Opt. Soc. Am. B2, 1721 (1985).

17. O'Neil, T.M., Phys. Fluids 24, 1447 (1981).

18. Bollinger, J.J., Manney, C, and Wineland, D.J., to be

published.

19. Kells, W.P., Proceedings of this conference.

TN-89
Be* alone Hg+ +Be''

(a)

(b)

400--

200-

,r

+ +

200 400

Scale in fim

(C)

Fig. 1. (a) Measured Be"*"- and Hg"*"- ion- cloud shapes. The magnetic

field is along the z axis. The Be"*" cloud, which contained about 800

ions, is shown with and without the Hg"*" present. The measured

rotation frequencies were Al kHz without Hg''" [corresponding to a density, n(Be'*")=3.9xlo7 cm"^] and 20 kHz with Hg"*" present

[n(Be"*')=1.9xlO'' cm"^, n(Hg'*')=1.6xlO^ cm"^]. The solid lines drawn

through the data points confoinn to the theoretically predicted and

separately confirmed spheroidal shape for Be"*" alone but are drawn

without consideration of theoretical shapes for Be"*" and Hg"*" together.

The edges of the clouds were measured with an uncertainty of about 25

Mm. (b) Measured cloud shapes for a Be"*" cloud which contained about

12 000 ions and a Hg"*" cloud which was larger in radial extent than

the aperture through the ring electrode (diameter = 2100 Mm). The

measured rotation frequencies were 73 kHz without Hg"*" [n(Be''')=7.0xlO'

cm"^] and 18 kHz with Hg"*" [n(Be+)=1.8xlO^ cm"^, n(Hg+)=1.5xlo7 cm"^].

(c) Theoretically predicted shapes for cold Be"*" and Hg"'" clouds under

conditions similar to those in (a). The scale in (c) also applies to

(a) and (b).

TN-90
PHYSICAL REVIEW A VOLUME 38, NUMBER 2 JULY 15, 1988

Static properties of a non-neutral ^Be'''-ion plasma

L. R. Brewer,* J. D. Prestage, J. J. Bollinger, Wayne M. Itano, D. J. Larson,^ and D. J. Wineland

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

(Received 4 January 1988)

We report measurements of the static properties of laser-cooled non-neutral 'Be ^ -ion plasmas

stored in Penning traps under a variety of experimental conditions. We have measured the shape,

rotation frequency, density, and temperature of the ions as functions of the Penning-trap potential

and laser-cooling configuration. Two different traps were used. In one trap, we were able to measure the ion temperature in directions both perpendicular and parallel to the trap magnetic field. In the other trap, "'Hg"*" ions were stored simultaneously with the ^Be"*^ ions and their effect on the 'Be+ ions was measured. The experimental measurements are compared with theoretical predictions. Within experimental error, the ion plasmas rotate without shear and exhibit approximate

thermal equilibrium. For a single stored species, a static model of the ion plasmas, based on

thermal equilibrium, is formulated. When the ion-plasma dimensions are small compared to the

trap dimensions, this model predicts that the ion plasmas are spheroidal and have an aspect ratio which is a function of the plasma rotation frequency, and the single-particle cyclotron and axial fre- quencies. This is verified by measurement. The ion densities and temperatures measured in our ex- periments show that the plasmas are strongly coupled and should exhibit liquid and solid properties.

I. INTRODUCTION

In this paper, we discuss some of the static properties

of non-neutral ^Be"'"-ion plasmas stored in Penning traps. In particular, we describe both experimental and theoretical investigations of the shape, rotation frequency, density, and temperature of the ^Be"*" ions. Some aspects of

these investigations have been briefly reported in Refs. 1-3. They are interesting from the standpoint of basic

plasma physics and may have other applications as well. For example, uncertainties in very-high-resolution spec- troscopy of stored ions may be dominated by inaccuracies

in the measurement of the second-order Dopplerfrequency shift. Therefore, a precise knowledge of the

velocity distribution of the ions in the plasma would be

desirable. The 'Be"*" ions are cooled by a laser to temperatures

less than 1 K. The number of ions stored in these experiments ranges from a few hundred to about 40 000. Even

with these small ion numbers, the low temperatures obtained by the laser cooling result in Debye lengths

(defined below) which are small compared to the stored

plasma dimensions. The ^Be"*" ion "clouds" can be considered plasmas in the sense that they satisfy this cri- terion.* In Sec. II we begin by describing the Penning trap, its electrostatic potential, and the characteristic motion of

ions in the trap. We examine the single-particle distribu- tion function, which gives the probability of finding a

particle with a specific position and velocity, assuming

axial symmetry of the trap and thermal equilibrium of

the ions. We then examine the distribution function in the hmit that the temperature T =0 and solve for the ion

density, electrostatic potential, and shape. We find a re- lationship between the plasmas' rotation frequency and

aspect ratio. Next, we discuss laser cooling and its effect

on the angular momentum of the ion plasma. Finally, we

discuss the one-component plasma, a particularly well- studied theoretical model, which is relevant to our experi- ment. Section III presents a description of the experimental apparatus, including the two traps used in the ex- periments, the loading procedure for these traps, the optical excitation scheme, and the experimental layout. The

experimental diagnostics for measuring the plasma shape,

rotation frequency, and temperature are then discussed. In Sec. IV the experimental data are presented and compared with theoretical predictions. Data on plasma

shear, shape, and rotation frequency versus aspect ratio are discussed as well as low-temperature data for small

plasmas. Finally, we conclude and briefly discuss possi- ble future experiments.

II. THEORY

A. Confinement geometry

The quadrupole Penning trap,^~* shown schematically

in Fig. 1, is composed of two "endcap" electrodes and a "ring" electrode which are biased with respect to each

other by a static electric potential. The symmetry axis of

the trap is assumed to be parallel to a static magnetic

field B. The magnetic field provides confinement of the

ions in directions perpendicular to the magnetic field. The ions are prevented from leaving the trap along the

magnetic-field lines by the applied electrostatic potential.

This configuration is similar to that used by Malmberg

and De Grassie."* In our apparatus, electrodes which are hyperboloids of revolution give rise to an applied trap potential (in cylindrical coordinates)

(f).j mco. 4q

-dz' (1)

where m and q are the ion mass and charge, and the axial

38 859 TN-91 • 1988 The American Physical Society
860 L. R. BREWER et al. 38

B = Bz

Endcap —-|

FIG. 1. Penning trap. The two endcap electrodes are biased with respect to the ring electrode by a voltage Vq. The quadratic trap potential is a consequence of the shape of the electrodes, which are hyperboloids of revolution. We neglect the distortion of the potential from the effects of truncation and holes in the

electrodes. The motion of the ions in a direction perpendicular

to the trap axis is constrained by the trap magnetic field. Ions

are prevented from escaping along the magnetic field lines by

the quadrupole electric field. Two coordinate systems are used

in this paper: the Cartesian and cylindrical coordinate systems.

frequency co^ is defined by the equation

w. m (ro+2zo)

(2)

Here Vq is the electric potential applied between the ring

and endcaps, and Fq and Zg are the characteristic trap di- mensions as shown in Fig. 1. The motion of a single

trapped ion is composed of a harmonic oscillation along

the z axis at frequency o^, and two circular motions in the x-y plane. The ion undergoes a circular ExB drift at a frequency co^ (this is sometimes called the magnetron

frequency). In addition, the ion undergoes a circular

motion at a frequency typically near the cyclotron fre- quency fi = qB /{mc), where c is the speed of light. For

simplicity we assume that q >0 throughout this paper. This frequency is shifted from the cyclotron frequency by

the trap electric fields and is equal to fl —w„. Typically

the trap is operated with co^ «il. For a sample of many

ions, the ExB drift frequency co^ is replaced by a rota- tion frequency for the plasma as a whole. Since the rota- tion frequency is partly determined by space charge, measurement of this frequency allows a determination of the

ion density as discussed in the following sections.

B. Thermal equilibrium

We assume that the ^Be"*" ions are in thermal equilibrium due to the Coulomb interaction between ions. Be- cause the Penning trap is axially symmetric, in thermal

equilibrium the single-particle distribution function

(Boltzmann distribution) can be written (in cylindrical

coordinates) as^'^~"

/=«o

m

l-rrhgT

3/2exp

H+(ol,

(3)

H=mv^/2 + q(f>(r,z) , (4)

(5)

Here H is the Hamiltonian for an ion in the plasma in the

laboratory frame of reference, /^ is the canonical angular

momentum about the z axis for an individual ion of the plasma, and kg is Boltzmann's constant. The parameters

CO and T, the temperature, are determined by the total an- gular momentum and energy of the system. In addition,

Ug is the velocity of an ion in the 6 (azimuthal) direction,

Aq IS the component of the vector potential A in the d

direction, and (j) is the total electrostatic potential. If we

assume (^ = at the center of the trap, then aiq is the density of the plasma at the center of the trap. In Eq. (3), we

chose to use a plus sign in front of co because for positively charged ions this convention leads to o) > 0. For a more unified presentation where trapping particles of ei- ther sign is being considered, see Ref. 2. Inserting Eqs.

(4) and (5) into Eq. (3), we find

f=n(r,z)

where

m

lirkgT

3/2exp > k„T (6)

n(/-,z) = noexp{ —[q<f>{r,z) + jmco([l —o))r'^]/kBT}

(7) The density as a function of the coordinates r and z is given by n(r,z). Bounded solutions exist for co^ <co <Cl — o)^.^ The velocity distribution in Eq. (6) describes

a Maxwell-Boltzmann velocity distribution superimposed on a rigid-body rotation with frequency -co. The negative sign means that the rotation is in the —6 direction. In particular, the plasma rotates without shear. This re- sult relies only on thermal equilibrium and axial symmetry of the confinement geometry.

The single-particle distribution function of Eqs. (3)-(5)

neglects any correlated behavior of the ions. For the densities and temperatures that can be achieved in our exper- iment, correlations in the spatial positions of the ions are expected (see Sec. II E). In general, the theory developed

in the remainder of this section from Eqs. (3)-(5) will not provide a correct description for short length scales, in particular, for length scales less than the interparticle

spacing. For small, correlated plasmas with dimensions

of a few interparticle spacings, the theory may only give an approximate, qualitative description of the plasma

state. Nevertheless, Eqs. (3) -(5) should give a correct description of the plasma state for length scales longer

than the coherence length of the correlations (typically, a few interparticle separations). Since the spatial resolution

obtained in the experiments reported here is larger than

the interparticle spacing, the theory developed here

should be adequate.

In the T=0 limit, for the plasma density [Eq. (7)] to remain finite in the plasma interior, the following condition must hold:

q<f>(r,z) + \m(o{n-co)r^ =

TN-92 (8)
38 STATIC PROPERTIES OF A NON-NEUTRAL 'Be + -ION PLASMA 861

The potential (f> is independent of z. This is just the state- ment that there can be no force along a magnetic-field

hne in a zero temperature plasma. If the expression for

<j){r,z) obtained from Eq. (8) is substituted into Poisson's

equation, we see that the density is constant and is given

by2.io,n

«n =

moi(€i — (o)

Itrq'

(9)

In this paper we are interested in the case where the trap

potential ^j is given by the quadratic potential of Eq. (1). However, we note that Eq. (9) holds independently of the

form of ^7- as long as it is axially symmetric. For finite temperatures, Eq. (7) and Poisson's equation imply that the ion density is constant up to the edge of the plasma

and then drops off in a distance characterized by the Debye length,

^D —

ksT

4vrrto9'

1/2

(10)

The picture obtained in the hmit r=0 of a constant density plasma with sharp boundaries is valid for temperatures low enough so that A^, « plasma dimensions.

Most of the plasma temperatures measured in our experiment were of order 1 K or less. Typically, the density of

the trapped ^Be"*" plasmas was 2x10^ ions/cm\ These

values gave a Debye length A^ of 15 jum or less, whereas

the measured plasma dimensions were typically larger by more than an order of magnitude. Therefore the T =Q

limit is appropriate.

The total electrostatic potential (f> is composed of three terms:

(l> = <f>T + 4>l+<l>md (11)

Here ^j is the potential due to the trapped ions and 0ij,d is the potential due to the charges induced on the elec- trodes by the trapped ions. The potential (f)^„^ can be

neglected when the trap dimensions are large compared

to the plasma. This condition was achieved in the experiments and is assumed below. Solving for the ion potential using Eqs. (1), (8), and (11), we find that inside the

plasma.

<t>I=4> — ^T ' (12) ^r ,^ . 2,.1„2 ^^'^'

(f), = [(o( ^ — oi) —(o^/2]r

2q 2q

(13)

where Eq. (13) serves to define a and b, and 2a +6 =3 as required by Poisson's equation. The last expression in Eq. (13) is just the potential in the interior of a uniformly

charged spheroid.^ A spheroid is the shape obtained by

rotating an eUipse about one of its principal axes.

C. Aspect ratio versus rotation frequency

From the definition oft in Eq. (13) and the expression

for «o in Eq. (9), we find that

b =

2,o}l

2o}(£l —(o) (14)

For a uniformly charged spheroid of diameter 2r^,| in the

z=0 plane and axial extent 2z^\, the coefficient b in the

potential is given in terms of the aspect ratio z^., /r^,, by

the following expressions.^'^ When the plasma or ion

cloud is a prolate spheroid (z^,, > r^., ), 6=/c..lln[(l+^.)/(l-/c,)]/2/c,-lj ,

/c^. = 3(l-/c/)//:/ (15)

2il/2

fc^ = [l-(/-„/z„)^]

When the plasma is an oblate spheroid (r^^ >z^

6=/co,[(l-/c^)-'/2_sin-'(/co)//co] ,

fcn'=3(l kl)'^'/kl , (16)

/Co = [l-(z„/A-„)^]2il/2

Equations (14)-(16) relate the aspect ratio of the plasma

to its rotation frequency co. These equations are valid for uncorrected plasmas with X^ much less than the plasma

dimensions which are much less than the trap dimensions. They should also give an approximate description

for correlated plasmas, with the additional condition that the plasma dimensions are much larger than the coher- ence length of the spatial correlations.

D. Laser cooling and angular momentum

Laser cooling'^" '^ uses the resonant scattering of laser

light by atomic particles. The laser is tuned to the red, or low-frequency side of the atomic "cooling transition"

(typically an electric dipole transition). Some of the ions

with a velocity component opposite to the laser beam

propagation (k-v<0) will be Doppler shifted into reso- nance and absorb photons at a relatively high rate. Here,

k is the photon wave vector ( | k | =2tt/X, where k is the

wavelength of the cooling radiation). For the opposite

case (k-v>0), the ions will be Doppler shifted away from

the resonance and the absorption rate will decrease. When an ion absorbs a photon, its velocity is changed by

an amount

Av = ^k/m (17)

due to momentum conservation. Here Av is the change

in the ion's velocity, m is the mass of the ion, and 27rfi is Planck's constant. The ion spontaneously reemits the

photon symmetrically. In particular, averaged over many scattering events, the reemission does not change

the momentum of the ion. The net effect is that for each

photon scattering event, the ion's average velocity is changed by an amount shown in Eq. (17). To cool an atom from 300 K to millikelvin temperatures takes typically 10"* scattering events. The theoretical cooling limit,

due to photon recoil effects, '^~'^

is given by a temperature equal to fiy /(Ik^ ) where y is the radiative linewidth

of the atomic transition in angular frequency units. For a linewidth y = 27rXl9.4 MHz, which is the radiative

TN-93
862 L. R. BREWER et al 38

linewidth of the 2s^S^/2 =2p^P^/2 transition in ^Be +

(A. = 313 nm), the theoretical minimum temperature is 0.5 mK.

Laser scattering can also be used to change the angular

momentum of the plasma. Equation (5) shows that the z component of the canonical angular momentum for an

individual ion in the plasma is

L = mvf,r +

qBrl

Ic The two terms in Eq. (18) are the plasma's mechanical

angular momentum and the field angular momentum.

The total z component of the angular momentum of the

plasma is

L^ = ^ dz ^ lirr dr ^ d\ f{r,z,\)l^

composed of a single species of charge embedded in a background of neutralizing charge of uniform density.

Malmberg and 0'Neil'° have shown that the static properties of magnetically confined non-neutral plasmas are the same as those for a one-component plasma. In our

experiments, the identification of the neutralizing background charge can be made by writing ^ = ^j-\-(^j [from

Eq. (12)], in Eq. (7). With Eqs. (1) and (13), the terms

(18) q^T{r,z)-\-\m(o{Cl —(o)r^ can be interpreted as the potential energy of the ions due to a hypothetical spheroid

of uniform negative charge. The hypothetical spheroid

has the same aspect ratio as the ion plasma and has a charge density equal to —qriQ.

The one-component plasma is characterized by a coupling constant

= m{a/l-(o)N{r'^) (19)

Here A'^ is the total number of ions and {r^) is the meansquared radius of the plasma. For all of the work de- scribed in this paper, o) «fl and

L,~ —-

(/•^)>0 (20)

Therefore the total angular momentum is dominated by

the field angular momentum. Suppose the cooHng laser is directed normal to the z axis but at the side of the plasma

which is receding from the laser due to the plasma rotation. Because the rotation of the positive ions is in the — direction, the torque of the laser on the ions will also be negative. Consequently, angular momentum is re- moved from the plasma and according to Eq. (20) the ra- dius of the plasma must decrease. In general, the plasma

is compressed until the torque due to the cooling laser is balanced by another external torque. As the radius decreases, the density of the plasma increases. Even in the

absence of external torques, there is a limit to how far the

plasma can be compressed. From Eq. (9), the maximum

density, known as the Brillouin density, occurs when the

rotation frequency o) = Vi/2. The Brillouin density is given by

"max n 2 877-9' (21)

Collisions with background gas particles produce an

external torque which increases the angular momentum

of the plasma. This is one of the effects that could limit the compression of the plasma. External torques due to

axial asymmetries of the trap are also limiting effects. Driscoll, Fine, and Malmberg observed that the axial asymmetries of their cylindrical traps play an important

role in determining the trap confinement time. '^ Axial

asymmetries are also expected to be a limiting effect in the experiments reported here.^

E. One-component plasma

In the frame of reference rotating about the trap axis with frequency —oi, the ion plasma behaves like a neutral one-component plasma. A one-component plasma'^ is

r=

ask^T

' (22)

which is a measure of the nearest-neighbor Coulomb en- ergy divided by the thermal energy of a particle. The

ion-sphere or Wigner-Seitz radius a^ is defined by

T^ai«o = l (23)

For r> 1, the plasma is said to be strongly coupled. '^ When F > 2, the plasma should exhibit liquidlike behavior characterized by short-range order. Theoretical cal- culations predict'^ that at r= 178, the plasma undergoes

a phase transition to a crystal-like structure. Because the

static thermodynamic properties of the one-component

plasma are the same as those for a magnetically confined

non-neutral plasma, the magnetically confined plasmas of

a Penning trap are also characterized by the coupling F. In particular, at F=178, a liquid-sohd phase transition

may take place. At a magnetic field of 10 T, the Brillouin

density for ^Be"*" is 3xlO'° ions/cm^. This density and

the 0.5-mK cooling temperature limit gives a theoretical

hmit on the coupling of F= 1.7X 10'^. Consequently, the

possibility of obtaining couplings large enough to observe

a liquid-solid phase transition looks promising.

Because this is a first-order phase transition, the plasma may remain in a metastable fluidlike state if it is rapidly cooled below the transition temperature. Ichimaru

and Tanaka have investigated the lifetime of the metastable supercooled state. '^ For a ^Be"*" plasma with density

of 10'° ions/cm^, they estimate that at F = 400 the metastable state has a minimum lifetime with a value between

20 and 2X 10^ s. At larger values of F, the metastablestate lifetime increases rapidly and at F~ 1000 the super- cooled plasma may undergo a dynamic glass transition. In the absence of heating effects, the laser cooHng rate for cold ^Be+ ions ( T < 10 mK) can be much larger than 100

K/s. Consequently, it appears that the laser cooling of

the ions can be done rapidly compared to the r = 400

metastable lifetime. This would permit the investigation

of the dynamic glass transition. The correspondence be- tween the magnetically confined non-neutral plasma and

the one-component plasma rigorously exists only for static properties. The possibility of a dynamic glass transi- tion in a one-component plasma is therefore a suggestion

TN-94
38 STATIC PROPERTIES OF A NON-NEUTRAL 'Be+-ION PLASMA 863

of what might happen in the magnetically confined nonneutral plasma.

in. EXPERIMENTAL TECHNIQUES

A. Traps

Two different Penning traps were used in this experi- ment. The first trap (trap I) was constructed from gold

mesh (for the endcaps) and molybdenum wire (for the

ring electrode) so that a large fraction of the resonance

fluorescence from the ions could be collected by an ellip- soidal mirror and focused onto a phototube. Trap I was

used in experiments on ^Be"*" ions only. Trap II was used

to observe both '^^Hg""" and ^Be+ ions. It was construct- ed of solid molybdenum with 0.25-cm-diam holes in the

ring electrode to allow entrance and exit of the laser beams directed in the x-y plane and to allow the ions' fluorescence to be collected by a phototube. In addition,

small holes were located in the endcap electrodes of trap

II for the purpose of loading the ions into the trap. Both

traps had dimensions ro = 0.417 cm and Zo=0.254 cm

(see Fig. 1). In order that the vacuum region be relatively

contaminant-free, both traps were baked out at temperatures between 300 and 400 °C for several days. Base pres- sures of about 10~^ Pa (?=10~"^ Torr) were maintained

in both traps by sputter-ion pumps. This allowed the

ions to be trapped for hours even without laser cooling.

Typically, the trap voltages Vq in both traps were varied

between 1 and 4 V. The magnetic field B was 1.4 T for the work reported here.

B. Loading

Ions were loaded into the traps as follows. In both

traps, a 75-/im-diam, coiled-tungsten electron filament was located behind one endcap. During loading, the elec- tron filament was biased negatively with respect to the

endcap. Electrons emitted from the filament spiraled

along the magnetic field lines with enough kinetic energy

to ionize atomic beryllium (ionization potential =9.32

eV) in the trap. The electron energy inside the trap near

the endcap electrodes typically exceeded the ionization

potential of ^Be""" by only 0.5 to 1 eV. This helped

prevent the creation of other ions with higher ionization

potentials. The beryllium oven was located behind the

ring electrode in trap I and behind an endcap electrode in trap II. Both ovens consisted of a 75-/im-diam tungsten

filament around which was wrapped a 50-|Um-diam beryllium wire. Beryllium atoms were evaporated by heating

the tungsten filament with a current. The tungsten wire

provided strength for operation of the ovens in the magnetic field. To create ^Be"*^ ions in the trap, the beryllium oven and the electron filament were turned on for ap- proximately a 15-s duration after the trap potential was

momentarily reversed to clear any other ions out of the

trap. The trapping voltage was then briefly raised to a value sufficiently high to remove any heavier ions with

mass-to-charge ratio > 15 (atomic mass units divided by

the proton charge) that may have been formed in the

trap. After this, the ions were laser cooled to temperatures less than 1 K. In practice, for the laser powers and

ion numbers reported in this paper, this cooling was rapid, taking a few seconds to accomplish. The '^^Hg"*" was

loaded into trap II by ionizing neutral '^^Hg vapor which was leaked into the vacuum system.

C. Laser excitation scheme

The energy-level structure of the ^Be"*^ ground state and lowest ^^3/2 state are shown in Fig. 2. The cooling

laser (polarization perpendicular to B) is tuned slightly to the red of the transition between the 2s^S^/2inij,nij) = (+|,+y) and the 2p ^^3/2* +y> +}) states. Ions are

optically pumped by the cooling laser into the 2s^5,/2(+y,+7) State with |f«94% efficiency^°'^' [see Eq. (25)]. That is, for laser intensity below saturation, ||

of the ions reside in the 25 ^5,/2( +y, +y) state. The

cooling laser had a wavelength of 313 nm and a power of

up to 50 |LtW. The 313-nm laser light was obtained by

frequency-doubling 626-nm light from a single-mode dye

laser in a temperature phase-matched crystal of rubidium

dihydrogen phosphate (RDP). The dye-laser frequency was locked to an iodine reference line using a Dopplerm, + 3/2 +3/2

+ 3/2 +1/2

2P^P3/2

Observed Resonance

Fluorescence

FIG. 2. Hyperfine structure of 'Be"^ in a magnetic field, showing the transitions pertinent to the analysis of the experiment. The cooling transition is from the 2x^S|/2('"/,'"j) = ( + i. + i' state to the 2p~Pxr.( + j, + ^)

state. The probe depopulation transition is from the 25^S,/2( + i. + i) state to the 2p^Pini + ^,- {) state. The

figure is not drawn to scale. The frequency splittings between

the 2s -S,/2 I Aw, | = 1, | A'"./ 1 =0 states are on the order of 300 MHz. Only the Ip-P^r.' '"/ = + !

states are shown be- cause the frequency splittings between the 2p 'Py,: \ Aw; j — 1,

I Awy I =0 states are on the order ol" 1 MHz.

TN-95
864 L. R. BREWER et al. 38

free saturated absorption technique. The ions' resonance

fluorescence (i.e., the scattered Hght) from the cooling

transition was detected in a photomuhipher tube. The

fluorescence is proportional to the number of ions in the

2s ^S,/2(+|,+j) state. A second laser, called the probe laser, could be used to remove ions from the 2s ^S,/2( +y, +y) state by reso- nantly exciting them to the 2p^P^/2{-\-\,—\) state (the

"depopulation transition") from which they decayed with

a probability of y to the 2s ^S,/2( + y, — y) state. This

produced a decrease in the observed resonance fluores- cence. After the probe laser was turned off", the cooling

laser then optically repumped^° the ground state in about

1 s. The power of the probe laser was « 1 /zW and any

direct fluorescence due to excitation with this laser was

negligible compared to that of the cooling laser. Figure 3 shows the resonance line shape when the probe laser's frequency was scanned through the depopulation transition. The depopulation resonance was used to measure

the shape of the plasma, its rotation frequency, and temperature as described below. The density and number of

ions were then derived from these measurements.

D. Laser-beam configuration

The laser-beam configuration for trap I is shown in Fig. 4. The cooling laser beam passed through a 50%

beam splitter. The portion of the beam transmitted by

the splitter entered the ion cloud perpendicularly to B,

and the other beam, referred to as the diagonal cooling

beam, entered the trap between the ring and one endcap

at an angle of t/'=55° with respect to the z axis. The

probe laser beam passed through a telescope which was

used to translate the beam spatially. This was done by

O

cr

LU LU O C/3

LU —I

LU £

tr -I OS

a!"

- \ --

^'

- i 1

tD

Background

- 1/ -

" 1.5 GHz ! PROBE-LASER FREQUENCY

FIG. 3. Ion fluorescence from the cooling laser is plotted as a function of the probe laser frequency. The background signal,

due to light scattered from the trap electrodes, not ion fluores- cence, is the light the photomultiplier receives when the magnetic field is shifted so that the ions are detuned far from reso- nance. The quantity /?(0) [Eq. (26)] is determined experimentally from the ratio a /h.

Splitter^ U?

Input Focusing

Lens

Cooling Beam —

• B = Bz

Chopper

Probe Telescope

-Probe Beam

313nm

FIG. 4. A schematic diagram of the optical layout for trap I. The cooling laser beam passes through a 50% beam splitter, producing the diagonal cooling beam and a perpendicular cool- ing beam. The probe laser beam is translated spatially using one

lens of the telescope. The diagonal cooling beam is chopped be- cause the diagonal beam mirrors, which are mounted inside the vacuum system, scatter too much light. Resonance fluorescence

light from the perpendicular cooling beam is detected only

when the diagonal cooling beam is off.

translating one of the probe telescope lenses in a plane

perpendicular to the probe beam direction. The probe

beam can be directed either at i/' = 90° or t/' = 55° by means

of the splitter. In order to direct the laser beams along

the diagonal of the trap, mirrors were mounted inside the vacuum system next to the trap. The cooling laser light

scattered from these mirrors was larger than the signal count rate. Consequently the diagonal cooling light beam

was chopped at 1 kHz and resonance fluorescence from

the perpendicular cooling beam was detected when the

diagonal cooling beam was off. In trap II, the experimental setup was similar except that there was only cooling

and probing of the ion cloud in a direction perpendicular

to B. Both the cooling and probe laser beams were fo- cused to waists at the position of the plasma which were

small compared to r^^ and z^|.

E. Diagnostics

1. Saturation

The width and amplitude of the depopulation reso- nance (Fig. 3) were used to determine the temperature

and shape of the ion cloud. It is easy to saturate the depopulation transition, that is, to remove most of the

ions from the 2s ^S,/2( + {, -l-y ) state and broaden the resonance. This effect must be taken into account in or- der to determine the true full width at half maximum of

the depopulation resonance. In the following discussion

it is assumed that the ions have been optically pumped

into the m/ = +1 states. This pumping was reported in TN-96
38 STATIC PROPERTIES OF A NON-NEUTRAL 'Be + -ION PLASMA 865

Refs. 20 and 21. First, we note that the cooling laser nonresonantly drives ions from the 25 ^5'|/2( +y, — y)

ground state to the 2p^Pj/2^+j,+j) state where they

decay with y

probability to the 2s ^5,/2( +y» +y) ground

state. This rate is very small compared to the

25 25,/2(+|,+{) =2p^P3/2(+|,+|) scattering rate because of the large detuning (about 13 GHz at 1.4 T). Let Rp denote the rate for this repumping process. In addition, the cooling laser depletes the 25 ^Si/2( +y, +y)

ground state by nonresonantly driving the depopulation

transition. The detuning of the cooling laser in this process is approximately four times greater than its detuning

in the repumping process. If a Lorentzian line shape far from resonance is assumed, this results in a depletion rate of Rp /16. The details of this pumping have been verified

in Ref. 22. Finally, let a(Vp — Vq) be the depletion rate of

the 25 ^S,/2( + y. + y) state due to the probe laser. Here

V is the probe laser frequency and v'q is the center fre- quency of the depopulation resonance. Because of

Doppler shifts, v'q will depend on the location of the

probe laser in the plasma and the plasma rotation fre- quency. (See Sec. Ill E 3). Since the probe laser intensity

is very small, we neglect optical pumping effects on other

transitions driven nonresonantly by the probe laser. We

use a rate equation analysis to determine the full width at half maximum of the depletion rate a(v^— Vq) from the

observed fluorescence signal (Fig. 3). Let A'^^ be the

number of ions in the 25 ^S, /,( +1, +t) state. If the 2' cooling laser is well below saturation, we have

dt = -a(^v )N^+Rp(N-N_

16

-A^ (24)

where Av =v^— Vg. In steady state, dN _^_ /dt =0, and

the number of ions in the 25 ^Si/2( + y, +y) ground state

is

^4- =

17 16a(Av^

17i?„

(25) + 1

The fluorescence is proportional to the number of ions

N_^,. From Eq. (25) we can see that N_^_ =( 16/17)A^ when

the probe laser is off [a(AVp) = 0]. We normalize this baseline fluorescence ( a (16/17)A'^) to a value of one when the probe laser is off. The fractional decrease in fluorescence I3( Av^ ) for a given detuning of the probe

laser is

I3(^vp) = \

1 16a(Av^

16 «(Avp)

17 /?„

+ 1

1 +

- 16a(AvJ

(26)

MR,

From Eq. (26) we can solve for a( Av^ )/Rp and obtain

aiAVp)

R„

/3(Avp)

if[l-^(Av,)]

(27)

This is the equation for the magnitude of the depletion

rate a in terms of the magnitude of the observed reso- nance l3{Avp). We define the frequencies at half max- imum of the depletion rate a to be Vp(±). At these points a(±) = a[Vp(±)-v'Q] = \a{0). From Eq. (26), the

values of j8 at V. ( ± ) are

16 a(±) 16 a(0)

17 2R„

1 +

16a( + : 1 +

16 a(0)

(28)

llRp 17 2Rp

By substituting Eq. (27) into Eq. (28), we obtain

p(+)=-m_ . (29)

' 2-)5(0) ^

'

Therefore, the full width at half maximum of the de- pletion rate a is equal to the observed frequency

difference between the points I3{ + ) and /?( — ) determined

from Eq. (29).

2. Plasma shape

The probe laser was used to determine the shape, rota- tion frequency, and temperature of the ^Be"*" plasma. The

shape of the ion plasma was determined from the depletion rates a obtained for different spatial positions of

the probe laser beam. The area under the curve given by

a( Avp ) is proportional to the number of ions intersected

by the probe beam. This assumes that there is good spatial mixing of the ions, so that the ions which are affected by the probe laser beam then migrate to the position of

the cooling laser beam in a time which is short compared

to Rp~^- This assumption was verified by comparing the method described here with the spatial maps obtained by

observing the fluorescence from a second, low-intensity

laser beam (intensity much less than the cooling laser beam), which is tuned through the cooling transition^-^

(also, see Sec. IV E). Under the conditions of thermal

equilibrium, the width of the a(Av ) resonance should be

independent of the probe beam position, as long as the waist Wq of the probe laser beam is small compared to the

plasma dimensions. We define the waist of the ^robe

beam in terms of the laser intensity / =lQe ' " . Typically the waist of the probe beam varied from 15 to 44 nm and was more than a factor of 10 smaller than the plasma dimensions. Consequently a(0), the magnitude of

the aiAVp) resonance, is proportional to the number of

ions intersected by the probe beam. In addition, in the T —0 limit, the density of the ions is constant and a(0) is proportional to the length of the probe beam which inter- sects the ion plasma. In both traps we determined the plasma shape with the probe laser beam directed perpendicularly to the y-z plane (see Fig. 5). Because of the ro- tational symmetry of the trap about the r axis we assume

that the profiles of the plasma along the .v and y axes are

equal, that is, the plasma has a circular cross section in the x-y plane.

TN-97
866 L. R. BREWER et al. 38

(a)

^ Probe Beam

Probe Beam

FIG. 5. Plasma with rotation frequency co is shown in (a) the x-y and (b) the y-z planes. The probe laser beam is directed

parallel to the x axis and passes a distance d from the plasma z

axis. The sign of c? is positive on the side of the plasma which is receding from the probe beam due to the plasma rotation. The

plasma shape is mapped by determining the region of intersec- tion of the probe beam with the ion plasma. The dimensions

2Z(.i and Ir^ of the plasma are indicated.

The dimensions of the plasma along the y and the z axes, 2/-j,| and Iz^^, were measured as follows. The plasma

dimension along the z axis was determined by scanning

the probe laser frequency at various points along this

axis. By plotting a(0) versus the probe laser-beam posi- tion we could construct, within a scalar factor, a profile

of the plasma along the z axis. This profile was then

fitted to an ellipse and the dimension of the plasma along

the z axis was determined. An example of this is shown

in Fig. 6. This process was then repeated for the y axis. The probe laser beam was also used to determine the

coordinates of other boundary points of the plasma in the

y-z plane by varying its position along different lines parallel to the y and z axes.

3. Rotation frequency and density

The rotation frequency co was determined by measuring the center frequency of the depletion rate a as a function of the y position of the probe laser beam (see Fig. 5). For a cloud without shear, the change in the center of

this resonance is expected to vary linearly with the>' posi- tion of the probe laser beam because of the first order

Doppler shift of the depopulation transition, as observed

in the laboratory. Figure 5(a) shows the plasma rotating

about the z axis. The laser beam passes through the point

-1200 -CI -r^cl PROBE POSITION ON Z AXIS (^m )

1200

FIG. 6. Depletion rate a(0) is plotted as a function of the probe laser beam position along the z axis for J =0. The data have been fitted to an ellipse. This determines the dimension

2Z(,| of the plasma along the z axis and the plasma shape in the x-z plane, to within a scale factor.

X =Q, y =d, parallel to the x axis. The Doppler-shifted,

depopulation transition frequency v'q, in the laboratory

frame, is

LTT (30)

Here Vg is the depopulation transition frequency in the

rest frame of the ion, k is the laser light wave vector, v is the velocity of an ion, and (k-v) denotes the average

value of k-v over ions in the probe laser beam. In the ab- sence of shear, the average of k-v over the ion velocities

is constant along the path of the laser beam (neglecting

the finite waist of the laser beam). For any point along

the probe laser beam a distance r from the plasma origin

[see Fig. 5(a)], we have V-y = k(or sm6 = ka>d. Let c?i and dj be two different y positions of the probe

beam and let v'Q(d^) and v'oidj) be the corresponding

transition frequencies. The difference in the depopulation

transition frequencies is

AVo = Vo(c?, ) — Vo(c?2 277 ko)(di —dj )- 277-k(oM

(31)

Solving for the rotation frequency of the plasma, we find that

(0 =

Avq

-X . (32)

The density is then determined from Eq. (9). We emphasize that by measuring the rotation frequen- cy of the ion plasma by the first-order Doppler shift of

the probe laser beam, we have measured the frequency of an internal degree of freedom of the plasma. In contrast,

if we try to measure the rotation frequency by applying a low-frequency, spatially uniform electric field near the frequency (o, the excitation is at the center-of-mass fre- TN-98
38 STATIC PROPERTIES OF A NON-NEUTRAL 'Be + -ION PLASMA 867

quency co^, the single-particle magnetron frequency (see discussion in Refs. 24 and 25).

4. Temperature

There are several contributions to the width of the depletion resonance a. The resonance is a Voigt profile, that is, a convolution of Gaussian and Lorentzian

profiles. The Lorentzian full width at half maximum of

the contribution is just the radiative linewidth Av^ = y /Itt— 19.4 MHz. The full width at half maximum of

the Gaussian

terms

contribution Avq is composed of three

Av(; = (Av^ -t-Avy- +Av;2x1/2

(33)

Here Avj- and Avy- are the Doppler widths due to the

assumed thermal distribution of velocities in the ion- cyclotron motion and the ion motion along the z axis, re- spectively. They are separated because the cyclotron and

axial temperatures may be different. The contribution

Av„ is the broadening of the resonance Hne due to the

finite width of the laser-beam waist; this occurs because

the Doppler shift for ions changes from one side of the

probe waist to the other. We have assumed that the laser beam is Gaussian. In general, for a probe laser beam at an angle ip with respect to B (see Fig. 4), the contributions due to Doppler broadening are

(34)

The contribution due to the finite laser-beam waist is

Av^ = i(;o^(2 ln2)^ sinj/;/^

(35)

(36)

Shear in the plasma would contribute to a further

broadening, since the ion's first-order Doppler shift would vary with the ion's position along the probe beam

in the plasma. We neglect the eff"ects of shear here and in the data analysis for the temperatures presented later. If shear is present, then the calculated temperatures will be

too high.

The temperature T^ is measured with the probe laser beam directed perpendicularly to B. The Gaussian contribution Avq to the resonance is determined by first deconvoluting the natural linewidth from the full width

at half maximum of a. This can be done with the use of

tables^^ or by computer algorithm. The term Av,^ is subtracted in quadrature from the Gaussian width and the

temperature is given by

T^ = (Avl - Avl)mX^ ASkslnl) (37)

In trap I the axial (or parallel) temperature is obtained by

probing at an angle of i/'=55° and using the measured

Gaussian contribution, Av(;(t/' = 90°) obtained with the probe beam directed perpendicularly to B. In this case

mk\Av^c^r{j = 55')-sm\55'')Avi(^ = 90'')] T. = ^^— -. (38) " 8A:aCOS^(55°)ln2

In trap II we could only cool and probe in a direction

perpendicular to B and therefore we could only measure

IV. RESULTS

A Shear

It was important to test for shear in the plasma be- cause its absence was a fundamental assumption of the

theory presented in Sec. II and the data analysis of this paper. Earlier data taken by Larson et al. ^ indicated the

possible presence of shear when both ^Be"^ and '^^Hg'''

ions were loaded into trap II. We attempted to measure

shear in ^Be"*" plasmas in trap II both with and without

simultaneously loaded '^^Hg''" ions. Figure 7 shows data

taken on a ^Be"^ plasma without '^^Hg"*" ions. The depopulation resonance center frequency Vq is plotted versus the probe beam position d. According to Eq. (32), the

500

IDOUl

-500

-800

25

800 PROBE POSITION (^m)

-25

o

o

o o

o o \\0

o o -^ ^

• o

'

-800 800 PROBE POSITION (^m)

FIG. 7. Depopulation resonance center frequency ^'o is plot- ted in the upper graph vs the probe beam position d for a plas- ma of 30000 ^Be+ ions with a density of 2.8 X 10' ions/cm'.

The resonance frequency vq is plotted relative to the unshifted

transition frequency Vq in the rest frame of an ion. The solid

line is a least-squares linear fit. The residuals are plotted below,

along with a fit which includes shear. The result of the fit is ^=—0. 1±0. 1. The data were taken at a magnetic field o( 1.4 T

and a trap axial frequency of w. /27r=212 kHz.

TN-99
868 L. R. BREWER et al 38

plot should be linear with a slope determined by the plasma rotation frequency. Any deviations from a linear re- lationship would provide evidence for shear. The data were fitted using the following treatment. We assumed

that the shear, if present, was Hnear with radius, so that the rotation frequency as a function of radial position was

given by

250

CO =0) 1 +

IL (39)

where r is the cylindrical radial coordinate, r^i is the ra- dius of the plasma, and ^ is a coefficient providing a measure of the size of the shear. The Doppler-shifted, depopulation transition frequency Vq [see Eq. (30)] is then

given by

Vn = Vn 1 + i+l:

</-) (40)

where d is the distance (along the y axis) of the probe

laser beam from the center of the plasma (see Fig. 5), and

( r ) is the average cylindrical radius for ions in the path

of the probe beam. We have chosen d to be positive on

the side of the plasma which is receding from the laser due to the plasma rotation. If we neglect the finite waist

of the probe beam, (r) can be calculated analytically

from a one-dimensional integral with the result

(r) (d/r^^f

2[1 id/rj^]

In [\-(d/r,^Yy+l

d I //-.

(41)

Equation (41) was substituted into Eq. (40) and the result- ing equation was fitted to the data of Fig. 7 with the re- sult ^=—0.1+0.1. Therefore within experimental error

there was no shear in this plasma. A correct treatment of

the Doppler shift due to shear involves integrating the

Voigt resonance profile along the path of the probe laser beam. In general this will result in an asymmetric line. The center of gravity of this line, however, will be given

by Eqs. (40) and (41). This more detailed calculation was

done for typical conditions obtained in the lab. For a

^Be"*' plasma with radius r^,, = 500 /im, rotation frequency

of 19 kHz, and a shear coefficient ^ =0.33 (see Fig. 8), the

asymmetry of the resulting resonance profile was small,

producing less than a 0.5-MHz deviation in the resonance

center. The broadening of the Voigt profile due to this shear was less than 5%, which would increase the ap- parent temperature of the plasma by a small amount.

A dramatic change in the ^Be"*" plasma was observed

when '""^Hg^ ions were added to the trap.^ Without

'^^Hg"*" ions in the trap, rotation frequencies as large as 200 kHz were measured. With '^''Hg^ ions in the trap,

rotation frequencies on the order of 20 kHz were obtained. Clearly the ''"*Hg"* ions applied a substantial

torque to the ^Be+ ions. In addition, in the T =0 limit, the '""^Hg"*" ions are expected to centrifugally separate

and be located outside of the ^Be^ ions.^^ A direct measurement of this separation was not possible in Ref. 3, due to the restricted probe laser-beam directions, al-

-250 500

-500 500 PROBE POSITION (/i,m)

FIG. 8. A plot similar to Fig. 7 of data from Larson et al. (Ref. 3). The 'Be^ resonance frequency v'q is plotted vs d for a plasma of 7000 ^Be"*^ ions (density of 2.0 X 10' ions/cm^) stored simultaneously with "*Hg+ ions. The data show a well-defined departure from a linear fit, with |;=0. 33+0. 12. The data were

taken at a magnetic field of 1.4 T and a trap axial frequency of

«,/277- = 212kHz.

though the separation was indirectly verified by the

agreement between the measured plasma shapes and the

shapes calculated from a r=0 theory. The torque applied by the '^^Hg"*" ions on the ^Be"*" ions opposes the

torque due to the cooling laser, and, if centrifugal separation takes place, this torque is applied on the edge of the

^Be"*" cloud, while the cooling laser torque is applied in the interior of the ^Be^ cloud. Shear, if present, may

therefore be larger in a two-species plasma of ^Be"'" and

'^^Hg+ ions and is expected to consist of ^Be"*" ions rotat- ing more slowly at the plasma edge than the center. Fig- ure 8 shows data taken by Larson et al.^ of the ^Be"*" depopulation resonance center frequency versus the

probe position with both ^Be"*" and '^^Hg"*" ions in the

trap. A least-squares fit of Eqs. (40) and (41) to the data

determined |; = 0.33±0. 12. The condition ^>Q indicates that the ^Be"*" ions are rotating faster at the plasma edges

than at the center, contrary to what was expected. Fig- ure 9 shows more recent data for another ^Be^ and

'^^Hg+ plasma. The fit to the data resulted in

|^ = 0.06±0. 1 and shows that in the context of the model

described by Eq. (39), there was no shear in this plasma

greater than the limits set by the experimental error. An effect other than shear may be responsible for the TN-100
38 STATIC PROPERTIES OF A NON-NEUTRAL 'Be + -ION PLASMA 869

300

800

800 PROBE POSITION (jj,m)

FIG. 9. A plot similar to Fig. 7. The ^Be"*^ resonance fre- quency Vq is plotted vs d for a plasma of 25 000 'Be"*" ions (density of 1.7x10^ ions/cm^) stored simultaneously with ''^Hg''" ions. The result of the fit is ^ = 0.06+0. 1. The data were taken

at a magnetic field of 1.4 T and a trap axial frequency of

6J, 7277 = 21 2 kHz.

the results of these measurements. The graph suggests

that the rotation frequency of the plasma is decreased as the amplitude of the probe resonance is increased. This

effect has the right sign to explain the "reverse shear" of

Fig. 8. The measurement of Fig. 10 was done on the plasma of Fig. 9. The effect is large enough to create about a 10% change in the rotation frequency as the probe laser beam is moved from the center to the edge of the plasma,

comparable to the limits of shear set by the data of Fig. 9. This measurement was unfortunately not done on the plasma of Fig. 8. Consequently we cannot say whether

the size of this effect would have been large enough to ful- ly explain the apparent shear observed there. (The

dependence of the rotation frequency on the cooling laser torque may have been different for the two plasmas of

Figs. 8 and 9 because the number of ^Be"^ ions, the ion

density, and possibly the number of '^^Hg"*" ions differed

in the two plasmas.) In summary, we looked for shear in the rotation of the

^Be"*" ions in a single-species plasma consisting of only

^Be"*" ions and in two-species plasmas consisting of ^Be"*" and ^^^Hg"^ ions. Shear, if present, may be larger in the

two-species plasmas because of the large torque applied

by the '^^Hg"*" ions on the ^Be"*" ions. In Figs. 7 and 9 no

shear was observed at the limits \ ^ \ < 0. 1. The fit to the data of Fig. 8 gave a statistically nonzero and positive

value for the coefficient ^, but we believe other effects may account for this reverse shear. While the basic ex- perimental technique appears capable of establishing limits on any possible shear of | ^ | <0. 1, systematic effects of the probe laser on the plasma rotation must be carefully studied before this hmit is set. At the level of

\ ^\ < 0. 3 we believe we have experimentally demonstrated that the ^Be"*" ions are undergoing a uniform rotation,

in agreement with the predictions of thermal equilibrium.

^>0 result of Fig. 8. As the probe laser is scanned

through the depopulation transition, ^Be"*" ions are re- moved from the cooling transition. Therefore, there are fewer ions available to resonantly scatter cooling laser

light. The cooling laser exerts less of a torque on the

plasma and the rotation frequency is expected to be

lower. The probe laser removes more ions from the cool- ing laser transition when the probe laser beam is positioned at the center of the plasma than when the probe

beam is at the edge of the plasma. According to this theory, the edge of the plasma would appear to rotate fas- ter than the center of the plasma. To test this possibility,

the probe beam was fixed spatially near the edge of the

plasma and a sequence of measurements of the depopulation resonance center frequency v'q was taken with

different probe laser powers and hence depopulation sig- nal ampUtudes 13(0) (see Sec. Ill E 1). For each measure- ment the magnitude Ivq — VqI of the Doppler shift of the

depopulation resonance was determined. The difference

5 I Vq — Vq I in the Doppler shifts between two sequential

measurements was plotted as a function of the difference

8^(0) in the corresponding depopulation resonance amplitudes in order to eliminate contributions due to the drift of the probe laser frequency with time. Figure 10 shows

X

o

2i

o

0.0

6^(0)

0.8

FIG. 10. Shift 5 I Vq— Vq I in the magnitude of the depopulation resonance Doppler shift as a function of the change 5/5(0) in the depopulation resonance amplitude. The data were taken with c^«±0.9r^,| on the plasma of Fig. 9. The graph indicates

that the rotation frequency of the plasma decreased as the amplitude of the depopulation resonance increased TN-101
870 L. R. BREWER et al. 38

B. Plasma shape and aspect ratio From the discussion of Sec. II B, we expect the 'Be"*" plasmas to have constant density (for length scales larger

than the interparticle spacing) with sharp boundaries

consisting of a drop in density over a short distance compared to the plasma dimensions. Figure 6 and Sec. Ill E

2 indicate how the plasma boundaries in the y-z plane were experimentally determined. Figure 1 1 shows the re- sults of these measurements for a ^Be"*" plasma. According to Eq. (13), the plasma should have an elliptical shape

in the y-z plane in the Hmit kp « plasma dimensions «

trap dimensions. The agreement between the predicted

elliptical shape and the measured boundaries in Fig. 1 1 is good. Because of the axial symmetry of the trap and the

plasma rotation about the z axis, the cross section of the

plasma in the x-y plane is expected to be circular. Our

inability to direct the probe laser beam parallel to the z axis prevented us from verifying this experimentally.

For a given single-particle axial frequency o^ [Eq. (2)] and cyclotron frequency fl, Eqs. (14)-(16) of Sec. II

C

predict that there is a relationship between the plasma ro- tation frequency a and the plasma aspect ratio z^\ /r^^ for plasmas of sufficiently large size. This relationship was

experimentally tested by measuring the aspect ratio and

rotation frequency of ^Be"*" plasmas with 2000 to 40000

ions. The aspect ratio was determined from the plasma

shape measurements. Plasmas with different aspect ratios and rotation frequencies were obtained with different positions (and therefore torques) of the cooling laser beam.

As the cooling laser beam was moved from near the center to the edge of the plasma, the rotation frequency

and the aspect ratio z^^^/r^^ increased. In Fig. 12 we show

a plot of 1)Co\/[2(o(CI — cl))'\ versus the plasma aspect ratio

400

E

COX

<

>-

-400

-400 400

Z AXIS (/im)

FIG. 1 1. Boundaries of a 'Be^ plasma in the y-z plane. The

# data points were measured and the O data points were deter- mined by assuming rotational symmetry about the plasma z axes. The ellipse was determined from the end points along the

>> and z axis. The plasma consisted of 5000 ^Be ^ ions (density of

4. 7x 10^ ions/cm'). The data were taken on trap I at a magnetic field of 1.4 T and axial frequency of w, /27r=:290 kHz.

(N

CNJN

3

FIG. 12. Relationship between the plasma rotation frequency

and the plasma aspect ratio is shown. The quantity

3fc)^ /[ 2«( fi — &) ) ] is calculated from the measured values of o),

(o^, and n and plotted vs the measured aspect ratio z^^/r^^. The

solid, theoretical curve is determined from Eqs. (14)-(16) and is independent of the trap voltage and magnetic field. Data were

obtained for ^Be+ plasmas with 2000 to 40000 'Be+ ions. Three

different sets of data from two different traps are plotted. The

first set ( O ) was taken in trap I with 2 V trap voltage

(0)^/2 77- =290 kHz). The second set (D) was taken at a trap voltage of 1.5 V («^/277-=212 kHz) on trap II. The third set (A) was taken at trap voltages of 0.9 V (ay^/lv^MO kHz) and

1.5 V on trap II. All data were taken at a magnetic field of 1.4 T.

for data taken on both traps I and II and at different trap

voltages. The data should lie on a single curve, independent of the trap potential and magnetic field. The

theoretical prediction of Eqs. (14) -(16) is given by a soUd

curve in Fig. 12. The agreement between the data and

theory is good and confirms a prediction of the zero- temperature theory.

C. Temperature

The temperature of the plasmas was determined from

the width of a( Av^ ). Much of the data from trap I was

taken using a chart recorder and analyzed manually from

the recording. For each depopulation resonance, the

fractional decrease in fluorescence on resonance fi(0) was

obtained from the point of minimum fluorescence, as in- dicated in Fig. 3. The full width at half maximum points

of a were then determined using Eq. (29). This width was

then deconvoluted using a Voigt table to give the Gaussian contribution, as described in Sec. HIE 4. Frequency

calibration for the probe laser was provided using a spec- trum analyzer with a free spectral range of 3 GHz. The

temperatures T^ and T^n were then determined from Eqs.

(37) and (38) by measurements made with the probe laser beam both perpendicular and at an angle 55° with respect

to the magnetic field. Data on trap II and the rest of the data from trap I were taken with a computer. The depletion rate a(Avp)

TN-102
38 STATIC PROPERTIES OF A NON-NEUTRAL 'Be + -ION PLASMA 871

was calculated using Eq. (27). Most of these data consist- ed of measurements on large plasmas, where the natural

linewidth of the depopulation transition was a small frac- tion of the Voigt resonance width. Consequently, the

data were well fitted by a Gaussian curve. A graph of

aiAVp) and a Gaussian fit is shown in Fig. 13. The full width at half maximum of a was obtained from the fit. This width was then deconvoluted using a computer algorithm to obtain the contribution Av^; [see Eq. (33)]. The

temperature Tj^ was then determined using Eq. (37).

D. Discussion

In Table I we show some examples of data taken in traps I and II with perpendicular cooling only. The first row of data was taken in trap I. The parallel temperature was measured to be an order of magnitude greater than

the perpendicular temperature. The cooling laser beam,

directed perpendicular to the magnetic field, only in- directly cooled the parallel (z) motion of the ions by collisional coupling of the parallel motion with the cooled

perpendicular motion. The parallel motion was directly

heated by the recoil of the scattered photons. '^~'^ This

may explain the observed temperature difference between

the perpendicular and parallel motion. From Table II, it

is clear that cooling at an angle of 55°, in addition to per- pendicular cooling, decreases the parallel temperature.

The remaining rows of data in Table I were taken on

larger plasmas than in Table II. Typically there were

30 000 ions in these plasmas, as opposed to the small plas- mas, which typically had a few hundred ions. The per- pendicular temperatures for the larger plasmas were

greater than for the smaller plasmas. The coupling pa- rameter r was not calculated for the large plasmas in Table I because it was not possible to measure the parallel temperature. As suggested by the trap-I data, this temperature was probably larger than the perpendicular

temperature. We were able to achieve higher rotation

frequencies and therefore higher densities in trap II than

CO

2.0

1.5

>-cr< 1.0 a:

DO

Ql 0.5 \- <

7^ <

-0.5

Q- o.O( )QooooQoa

He- 320 MHz PROBE-U\SER FREQUENCY

FIG. 13. Graph of a(Avp) as determined from the measured

depopulation curve /SiAv^) along with a Gaussian fit. The full width at half maximum of this resonance is 71.3 MHz. This

corresponds to a temperature of 0. 11 ±0.05 K.

in trap I. Some of the shapes of the plasmas in trap 11 were prolate, whereas in trap I the plasma shapes were

oblate. Both of these facts may be due to the higher de- gree of axial symmetry present in trap 11.^ Diagonal cooling data were taken on small 'Be^ plas- mas to try to achieve conditions of maximum density and

minimum temperature, that is, the strongly coupled re- gime. Using trap I we were able to cool ^Be^ plasmas

both in a perpendicular direction and at an angle of

xp = 55'' with respect to the magnetic field. We were also able to probe in both directions and extract temperatures

in both the perpendicular and parallel directions to the

trap magnetic field. Table II shows the plasma dimensions, density, number of ions, rotation frequency, per- pendicular and parallel temperatures, and coupUng pa- rameter r of several small plasmas. To provide a conservative estimate, F was calculated using the higher temperature Ty.

The Brillouin density at a field strength of 1.4 T is 5.8x10^ ions/cm^. The measured densities of Table II were typically of the order 2x10^ ions/cm^. This is a factor of 30 less than the theoretical limit. The mechanism which limits the density is not well understood but

it may have to do with axial asymmetries in the Penning

trap.

2- '^ The theoretical laser coohng limit discussed in Sec.

II D is 0.5 mK. While the uncertainty in the temperature

measurement of Table II is large enough to include this Hmit in some cases, the temperatures measured were consistently higher by about an order of magnitude. The

0.5-mK limit was derived for the case of a single ion in a

trap. For an ion plasma, the temperature limit for laser cooling perpendicular to the trap magnetic field depends on the distance of the cooling laser from the center of the

plasma, the rotation frequency, and the saturation of the cooling transition. This can account for the larger temperatures we measured and will be discussed more

thoroughly in a future publication. By changing the configuration of the laser cooling, we should be able to more closely approach the theoretical limit. For example, laser cooling along the z axis should permit a temperature Hmit of 0.5 mK. A laser beam with a component

perpendicular to the magnetic field is required to contain

the ion plasma in the radial direction, but in principle it could be fairly weak and not significantly perturb the temperature.

The largest coupling parameter measured was F = 340. The uncertainty in this measurement was due largely to the uncertainty in the temperature measurement for the

axial direction, which in this case was 2.4±6.0 mK. The

measurement error is statistical and the lower bound on

the measured temperature is more realistically given by

the theoretical cooling limit of approximately 0.5 mK.

This temperature uncertainty results in a range of values

for F from 100 to a maximum of 2000. Therefore the coupling may be large enough for the plasma to be crystalline. The lowest temperatures were measured on plas- mas of several hundred ions. These plasmas cannot be

truly called infinite, three-dimensional plasmas. Since

boundary conditions of the ion plasmas may be important

in our measurements on small ion numbers, the results TN-103
872 L. R. BREWER et al. 38

J TABLE I. Experimental data for perpendicular cooling only. The first row of data is for trap I. The

remaining rows of data are for trap II. The error convention is as follows: 1.8(10)= 1.8+1.0. Here 22<.| and 2r(.| are the axial and the radial extent of the plasma. The coupling parameter T is derived from

T\\. For both traps I and II the magnetic field was 1.4 T, corresponding to a cyclotron frequency of n/27r=2.4MHz.

Vo 22„ 2r„ «o N a/l-rr T, 7-11 r

(V) (/xm) (/xm) (lOVcm^) (\(f) (kHz) (K) (K)

2.0 160(30) 290(30) 2.6(7) 0.018(9) 27(7) 0.005(2) 0.13(7) 6.1

1.5 1402(36) 1250(36) 2.85(4) 3.27(31) 29.2(4) 0.13(4) 0.9 1692(36) 1234(36) 2.26(4) 3.05(28) 23.1(4) 0.11(5)

1.5 1723(36) 718(36) 6.33(15) 2.94(43) 65.7(16) 1.30(10)

are probably best compared to a theory somewhere be- tween an infinite plasma theory and a theory for Coulomb clusters of ions (see discussion of Sec. IV E). The temperature T^ was also determined for several plasmas as a function of the probe laser beam position d. Within the signal-to-noise reported for the T^ temperatures in Table I, the temperature was constant. This, to- gether with the lack of observable shear, indicated that the plasma was approximately in global thermal equilibri- um.

E. Future measurements and conclusion

Information about the spatial correlations in a plasma

can be obtained by measuring the static structure factor. '^ The static structure factor is the spatial Fourier

transform of the radial distribution function and is exper- imentally observed in the diffraction pattern resulting

from the scattering of coherent light by the ions. It should be possible to directly measure the static structure

factor of a correlated ^Be"*" plasma with Bragg scattering

from a laser.

^*

It may be possibe to obtain some information about the

spatial correlations of ions in the plasma by directly imaging the ion-resonance fluorescence from the cooling

laser beam. Recently, several molecular dynamics simulations of a plasma of a few hundred to a few thousand

stored ions have been conducted. ^^ A result of these

simulations is that the ions tend to reside in concentric

shells. The shells exist even for couplings F < 178 but be- come more pronounced and clearly separated with larger

coupling. The image of the ion-resonance fluorescence

from the cooling laser beam should consist of bright areas corresponding to the intersection of the laser beam and

the ion shells and dark areas where no ions are present.

This assumes that the cooling laser-beam waist is small

compared to the plasma dimensions. The separation be- tween the bright and dark areas should be on the order of

the interparticle spacing, approximately 20 /zm for the

ion densities reported here. Because of the cylindrical

symmetry of the shells, the image is not destroyed by the

plasma rotation.

By "tagging" ions with the probe laser used in a pulsed

mode and monitoring the cooling laser fluorescence versus time,^^ we should be able to measure the ion

diffusion between spatially separated probe and cooling

laser beams. This may provide a technique to tell whether the plasma is a solid or liquid. If the plasma is a soHd,

the dilfusion will not take place and a decrease in the

cooling laser fluorescence will not be observed.

In conclusion, we have measured static properties of

beryllium ion plasmas stored in quadratic Penning traps

under a number of diff"erent experimental conditions.

Within experimental error, we have found that there is no shear present in the ion plasmas; this suggests that the

ions are nearly in global thermal equilibrium. Near

TABLE II. Experimental data on small 'Be"*" plasmas from trap I. The cooling was done at ^=0°

and = 55° simultaneously. The error convention is as follows: 1.8(10)= 1.8±1.0. The coupling pa- rameter r is derived from T\<^. The trap magnetic field was 1.4 T. This corresponds to a cyclotron fre- quency of il/2ir=2A MHz.

Vo 2z„ 2r„ «o N (o/2tt T, n r

(V) (|um) (/xm) dOVcm') (kHz) (mK) (mK)

2.0 130(30) 450(30) 2.4(6) 330(170) 25(6) 2.3(5) 10(5) 80

2.0 150(30) 450(30) 1.9(5) 300(150) 20(5) 1.8(10) 7.4(70) 100

2.0 130(30) 480(30) 2.2(6) 350(170) 23(6) 0.9(15) 8.9(60) 90

2.0 160(30) 450(30) 2.6(7) 450(230) 27(7) 2.4(10) 6(6) 130

2.0 160(30) 260(30) 2.8(7) 150(80) 30(7) 2.7(30) 2.4(60) 340

4.0 80(30) 390(30) 3.9(10) 250(120) 40(10) 2.9(20) 20(12) 50

1.0 190(30) 360(30) 1.2(3) 150(80) 13(3) 2.9(10) 4.7(60) 130 TN-104
38 STATIC PROPERTIES OF A NON-NEUTRAL 'Be

-ION PLASMA 873

thermal equilibrium has also been verified by measuring

T^ as a function of position. Differences in T^ and Tn are expected for perpendicular cooling due to scattered photon recoil heating. This may be reflected in the data tak- en with trap I. Measured plasma shapes are spheroidal,

which is in agreement with a theory that assumes that the

plasma is in thermal equilibrium with dimensions small

compared to the Penning trap dimensions, but large compared to the coherence length of any spatial correlations.

The data on the aspect ratio versus rotation frequency

also agrees with this theory. The temperature, density,

and rotation frequency were measured for several plas- mas under varying conditions. Measured coupHng pa- rameters of 100 or greater for small plasmas of a few hundred ions indicate that these plasmas are strongly coupled

and may be in a liquid and possibly solid state. For the study reported here, the measurement of the

size of the plasma and therefore the ion number and total angular momentum was quite laborious. However, for

the conditions obtained here, this can be simplified in the

future. The rotation frequency co requires only two probe

transition measurements as long as no shear is present.

The plasma dimension Zr^,, could be easily measured by

imaging the resonance fluorescence scattered by the ions onto a position sensitive detector. The frequencies il and

co^, which are constant and independent of the plasma

shape or size, could be measured using fluorescence detection.^'* From r^., and Fig. 12, z,., is then determined.

ACKNOWLEDGMENTS

We gratefully acknowledge the support of the U.S.

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

Scientific Research. One of us (D.J.L.) would like to ac- knowledge partial support of the National Science Foundation. We thank D. H. E. Dubin and S. L. Gilbert for their helpful suggestions.

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

^Present address: Jet Propulsion Laboratories, California Insti- tute of Technology, 298-104, 4800 Oak Grove Dr., Pasadena,

CA 91109. ^Permanent address: Department of Physics, University of Virginia, Charlottesville, VA 22901.

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D. J. Wineland, Phys. Rev. Lett. 57, 70 (1986).

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At. Mol. Phys. 19, 135(1983).

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(1979). "*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); S. Stenholm, Rev. Mod. Phys. 58, 699 (1986).

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(1981). 22r. G. Hulet, D. J. Wineland, J. C. Bergquist, and W. M.

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TN-I05
Volume 59, Number 26 PHYSICAL REVIEW LETTERS 28 December 1987

Atomic-Ion Coulomb Ousters in an Ion Trap

D. J. Wineland, J. C. Bergquist, Wayne M. Itano, J. J. Bollinger, and C. H. Manney

Time and Frequency Division, National Bureau of Standards, Boulder, Colorado 80303

(Received 29 September 1987)

We observe small numbers of laser-cooled Hg "*" ions, which are confined in a Paul radio-frequency ion

trap, to crystallize into regular arrays or clusters. We investigate the structure of these clusters by direct imaging, optical spectroscopy, and numerical calculations. The spectroscopy of such "pseudomolecules"

is unique in that we can probe individual atoms of the molecule separately. The ratio of Coulomb potential energy per ion to A:b7' in these clusters is observed to be as high as 120.

PACS numbers: 32.80.Pj, 35.10.-d, 36.40.+d

The success of various techniques for cooling of

charged atomic and elementary particles in traps and

storage rings has stimulated considerable interest in the

observation of the expected spatial ordering of a collec- tion of such particles. The study of ordering not only has

a direct bearing on the dynamics of these stored particles

but has applicability to the study of cluster formation,

Wigner crystallization, strongly coupled plasmas, and

spectroscopy.'"*

In this experiment, we investigated, both by direct imaging and by spectroscopic methods, the structure of

small clusters or crystals of laser-cooled atomic ions

which were confined in a Paul radio-frequency (rf) trap.'

Photographic images of one, two, and three ions in a Paul trap have previously been reported, '° but the resolution achieved did not allow investigation of possible or- dering. Indirect evidence for ordered structures of

trapped atomic ions was reported by Diedrich et at.

'

Imaging of ordered structures of macroscopic charged

particles stored in a Paul trap was reported in the experiments of Wuerker, Shelton, and Langmuir. '^ The ordering observed here was similar but occurred under substantially different experimental conditions. One advantage of the present experiments is that the charge and mass of the particles were well defined. In addition, the

structure of clusters of atomic ions can be spectroscopically probed. As a demonstration, we have measured the

vibrational frequency of the diatomic pseudomolecule

('98j^g + )2 by measuring its optical spectrum. This ex- periment is unique in that we were able to probe a specific atom by fixing the pseudomolecule in space. Small numbers of '^^Hg"*" ions were stored in a miniature Paul trap, '^ which has properties equivalent to those

of a hyperbolic trap with dimensions tq—466 /im and

zq — 330 /im. ''' An oscillating voltage Kac with peak amplitude between 145 and 325 V at a frequency of 23.26

MHz was applied between the ring and end-cap elec- trodes for trapping. The ions were laser cooled by 1-2

/iW of cw 194-nm radiation (bandwidth <2 MHz),

whose beam waist was varied between 5 and 15 /xm at the position of the ions. This radiation was tuned near

the 5d^%s'^SM2—^5d^%p^P\n first resonance line, which has a natural linewidth of 70 MHz. Some of the

194-nm fluorescence from the ions was focused onto the

photocathode of a resistive-anode photon-counting imaging tube as shown schematically in Fig. 1 . The optics for

this imaging was provided by a three-stage fused-quartz

lens system with an aberration-compensated first stage.

The first lens was apertured to give an / number of 4.5. The overall magnification of the lens system was 60 x. The positions of the photons detected by the imaging

tube could be displayed on an oscilloscope in real time or

stored by a computer in order to make time exposures. The resolution was limited by the imaging tube to about

a factor of 3 worse than the diffraction limit. In Fig.

IMAGE PLANE

(photocathode surface)

LENS

SYSTEM

FIG. 1. Schematic drawing of the trap electrodes (to scale) and imaging system (not to scale). The end-cap to end-cap

separation along the z axis is approximately 625 /im. The

overall magnification of the lens system is 60 x. The laser beam used to illuminate the ions also enters the trap at an angle of 54.7° with respect to the z axis and is perpendicular to the observation axis shown.

Work of the U. S. Government

Not subject to U. S. copyright

IN- 106

2935
Volume 59, Number 26 PHYSICAL REVIEW LETTERS 28 December 1987

2(a) we show the trap coordinate system as viewed by

the imaging tube. If we apply a positive static potential

Vl to the lower end cap in Fig. 1 and hold the static ring

voltage Vr at ground potential, a single ion'^ moves in the positive z direction by a known amount. '^ This

determined the length scale in the images (after we ac- counted for viewing angle) and therefore also determined

the magnification.

The harmonic pseudopotential well of the trap' is characterized by the single-ion resonance frequencies Vz and Vr for the motion in the axial and radial directions,

respectively. For Vl ""O. these frequencies take the

form'

vI'vIq-IBVr,

(la)

(lb)

2 ions

Vp = Vp = 1.70 V

Vr =

(d)

(g)

Vr =

(b)

3 ions

Vp = 1.34 V

(e)

4 ions

M

Vp = 1.59 V

Vp = 1.69 V

(f)

%

Vr = 1.94 V

(h) (i)

FIG. 2. Images of clusters of two, three, and four trapped

and laser-cooled Hg"*^ ions for the condition where v^o —2v,o=500 kHz. The coordinate system of (a) applies to the remaining pictures except for (c). The tick marks on the axes

are separated by 10 /im. (c) A three-dimensional intensity plot for two ions pinned along the z axis (.Vr — 1.95 V) and separat- ed by 7.4 /im in the plane perpendicular to the direction of observation. The variation in the intensity for different ions is due, in part, to variations in intensity of the laser beam at the

sites of the different ions. Peak counting rates were about 2000

s~'/ion. Transitions between the observed configurations can be induced by the variation of the ring voltage Vr, and thereby

the variation of the relative strengths of the axial and radial pseudopotential wells.

2936

where v^o and v,o (vzo"2v;.o) are proportional to VacVz and Vr were measured from the sideband spectra of an

optical transition'^ and by direct electric field excitation

of the motional resonances.'^ From these measurements we obtained 5=0.0567(15) MHz^A'. This agrees fairly well with the calculated value 0.0596 MHzVv based on

the designed trap dimensions.

If two ions were loaded into the trap with V/^ " K/. — 0, and the 194-nm source was slowly swept into resonance,

the ions were observed to first "swarm" around the center of the trap with orbit radii up to a few tens of micrometers. As the laser detuning was further reduced,

thereby increasing laser cooling, the ions oriented or "pinned" themselves along a preferred direction, separated by a distance d. Here the inward pseudopotential

force on each ion is balanced by their mutual Coulomb

repulsion. Since Vr "0, the pseudopotential well in the

axial direction is four times as strong as in the radial direction and the ions lie in the x -y plane. Although the trap was designed to have axial symmetry, the preferred

spatial alignment of the cold ions indicates an asymmetric potential due to trap imperfections, contactpotential variations, or some other trap asymmetry. For

convenience, we choose this preferred direction to be the X axis as indicated in Fig. 2(a).

If Vr is increased until Vz < Vr, we expect the ions to align along the z axis since this is the configuration

which minimizes the potential energy. This configuration is shown in Fig. 2(b). For the conditions of Fig. 2 (v;.o— 250 kHz), we expected this transition to occur

at Vr — \.\\ V and observed it to occur at K^ = 1.17 V. The transition appeared to occur gradually over a range

of about 30 mV rather than suddenly; this was probably

related to the source of trap asymmetry in the x -y plane.

For two "^Hg"*" ions pinned along a certain direction,

the interparticle spacing is theoretically given by

d^'i.29vd^'^ /im, where vj (in megahertz) is the

single-ion oscillation frequency along the direction of

separation. For the conditions of Fig. 2(b), we predicted

t/— 8.5 /im and measured 7.5(1.1) nm.

In Figs. 2(d)-2(i), we show various configurations for three and four ions. We also observed ordered structures

for five or more ions. For three or four ions, when Vr

was equal to zero, the ions appeared to lie the same dis- tance from the z axis but were not pinned, as was the case for two ions. Instead, they appeared to circulate about the z axis [see Figs. 2(d) and 2(g)]. This interpre- tation is supported by the observation that the size of the

circular image could be made to increase by more than a factor of 2 by appropriate positioning and tuning of the 194-nm laser beam. This is not surprising since the trap

asymmetry in the x~y plane was relatively weak and

since it was fairly easy to position the laser to apply a radiation-pressure torque about the z axis. Three stable configurations were observed for three and four ions. For large values of Vr, the ions were aligned along the z

TN-I07
Volume 59, Number 26 PHYSICAL REVIEW LETTERS 28 December 1987

axis as in Figs. 2(f) and 2(i). In the intermediate

configuration for three ions [see Fig. 2(e)], one ion ap- peared to lie on the negative x axis and the others

equidistant from the x-y plane and on a line parallel to the z axis which intersects the positive x axis. In the in- termediate configuration for four ions [see Fig. 2(h)],

two of the ions appeared to lie in the x-y plane pinned

along the x axis, and two ions were positioned above and

below the x-y plane along the z axis. We have verified the ordered structures in Figs. 2(d)-2(i) by calculating

the configurations which minimize the potential energy

for the specified trap parameters.

We also observed clusters where an ion apparently oc- cupied a normal cluster position but did not fluoresce. This "phantom ion," which may have been '^'Hg"*", or perhaps an impurity ion like HgOH"*", switched positions

with "^Hg"*" ions of the cluster if the cooling was weak.

Such clusters of atomic ions can be viewed as pseu- domolecules whose optical spectra would be expected to show rovibronic structure. As a demonstration, we have

measured the vibrational structure of the diatomic pseu- domolecule (Hg"*")2 by probing its optical structure. By

pinning the molecule along the z axis as in Fig. 2(b), we

could probe the optical structure of one of the atoms of

the molecule by detecting the fluorescence of only this atom. In particular, we measured the structure of the

5d ^%s ^Suiimj = y ) ^ 5d%s ^ ^Dsnimj =-j) electric quadrupole transition (A. = 282 nm) of one of these

Hg"*" ions. We first adjusted V^c and Vr to make Vr = 2vz=413

kHz for a single ion in the trap. We than observed the quadrupole transition on this single trapped ion by the following method.'^ The 194-nm source was first tuned

to give an easily detectable value of fluorescence. A ra- diation source near the quadrupole transition frequency was focused onto the ions and slowly swept through reso- nance. While being swept, the 282- and 194-nm sources were alternately chopped to avoid ac Stark shifts on the quadrupole transition from the 194-nm source. Each

time the 282-nm source was shut off" and the 194-nm ra- diation readmitted, the absence of 194 nm fluorescence from the ion indicated that it had made the transition to the ^Ds/2 level. The presence of 194-nm fluorescence in- dicated that the ion had remained in the ^5'i/2 state. The

spectrum of the quadrupole transition could be taken by

our plotting the probability of our observing 194-nm

fluorescence as a function of the tuning of the 282-nm

source. Figure 3(a) shows the central part of the spec- trum for a single trapped '^^Hg"^ ion. The carrier is identified by the invariance of its frequency upon variation of Kac and Vr. The resolution achieved was limited

by the 282-nm source spectral width; the natural

linewidth of the quadrupole transition is about 2 Hz. The sidebands are Doppler-eff"ect- generated FM side- bands due to the harmonic oscillations of the trapped ion

at frequencies v^ and v,. '^ They can also be viewed as

1.0 _ 0.9 o

_|,0.8

0.7

0.6

^V vTJ11N+

^

x-^ k_'

0) N N

t 1 CM K) o

o N "

-1500 -1000 -500 500 1000 1500

Frequency detuning (kHz)

(b)

FIG. 3. Absorption spectra of the electric quadrupole transi- tion ^Si/2(my--f-T)^2£)5/2(my-T) for individual "^Hg"^

ions, (a) The central part of the spectrum for a single trapped

ion. Adjacent to the carrier are Doppler-effect- generated

sidebands due to the harmonic motion of the ion in the trap. Trapping conditions were adjusted to satisfy Vr==2vz==473

kHz. The identifications below the lines show the order of the sideband. For example, the identification (z,r — z) denotes

absorption at the frequencies vq+Vz and vq-J-v, — Vj, where vo— 1.07x10" Hz is the carrier frequency, (b) The quadrupole spectrum for one of two ions, trapped in the same orientation as in Fig. 2(b). The trapping conditions were the same as

for (a). The new lines are due to the stretch vibration mode of

the ions at frequency v'z =^Vz.

the vibration modes of the pseudomolecule composed of

the ion plus the trap. For optimum laser cooling, the maximum amplitude of the ion's motion for the conditions of this experiment was about 0.18 /^m. '^''^ When two ions are confined in the trap and aligned

along the z axis, an additional vibrational mode is al- lowed; this is the stretch mode between the two ions. For

small amplitudes of motion, this mode is expected to have a frequency v^ =>/3v2. By using the double reso- nance method described above, we have observed the quadrupole spectrum of one of the trapped Hg"*" ions by

detecting the fluorescence of only that ion. This spec- trum is shown in Fig. 3(b). It is possible to attribute all of the new lines to the internal vibration mode at fre- quency v'z and its combinations with v^ and Vr. By

fitting the observed spectra (assuming v^=2vz), we

derive the frequency of this mode to be v'^ = 1.725(25)vz,

which is in agreement with the theory. It should also be possible to observe this internal vibrational mode

by direct excitation with spatially nonuniform electric

TN-108

2937
Volume 59, Number 26 PHYSICAL REVIEW LETTERS 28 December 1987

fields.'^ Finally, by measuring the width of the Doppler

envelope of the quadrupole resonance,'^ we determined

that the temperature of the ions was less than 8 mK.

This gave a ratio of Coulomb potential energy per ion to A:Br greater than 120. The imaging and spectroscopic techniques described

here should prove useful in the study of spatial ordering

under various experimental conditions. '"^ With careful

control of the trap parameters, such as static field offsets,

it should be possible to distinguish different cluster patterns whose minimum energies differ by small

amounts ^'^ and to study the transitions between these patterns for different experimental conditions. Careful

control of the balance between heating and cooling,"

and spectroscopic measurements of the temperature,

should allow the study of the transition from ordered to disordered structures. These studies may be aided by observation of the mobility of ions "tagged" by optical

pumping or ions of different species in substitution as observed here. In a Paul trap, the amplitude of an ion's micromotion' due to the oscillating electric fields is proportional to the distance that the ion is from the center of

the trap. Therefore rf heating is expected to be more of a problem for large-number clusters. This problem does

not occur for particles trapped in static electric and magnetic fields.

We gratefully acknowledge the support of the U.S.

Office of Naval Research and the Air Force Office of

Scientific Research. We thank R. Drullinger, S. Gilbert,

J. Helffrich, and C. Weimer for suggestions on the

manuscript. At the time of this submission, we learned

by private communication that the authors of Ref. 1

had succeeded in observing images at the same time as the work reported here.

'J. H. Malmberg and T. M. O'Neil, Phys. Rev. Lett. 39, 1333 (1977); J. J. Bollinger and D. J. Wineland, Phys. Rev.

Lett. 53, 348 (1984); H. Totsuji, in Strongly Coupled Plasma

Physics, edited by F. J. Rogers and H. E. Dewitt (Plenum,

New York, 1987), p. 19; D. H. E. Dubin and T. M. O'Neil, to

be published.

2S. Ichimaru, Rev. Mod. Phys. 54, 1017 (1982). ^D. Habs, in "Frontiers of Particle Beams" (SpringerVerlag, Berlin, to be published); E. N. Dement'ev, N. S. Dikanskii, A. S. Medvedko, V. V. Parkhomchuk, and D. V. Pestrikov, Zh. Tekh. Fiz. 50, 1717 (1980) [Sov. Phys. Tech. Phys.

25, 1001 (1980)1. *A. Rahman and J. P. Schiffer, Phys. Rev, Lett. 57, 1133

(1986); J. P. Schiffer and O. Poulsen, Europhys. Lett. 1, 55

(1986).

^J. Mostowski and M. Gajda, Acta Phys. Pol. A 67, 783

(1985); E. V. Baklanov and V. P. Chebotayev, Appl. Phys. B

39, 179 (1986).

*J. Javanainen, J. Opt. Soc. Am. B (to be published).

''R. Casdorff, R. Blatt, and P. E. Toschek, to be published. ^R. S. Berry, T L. Beck, H. L. Davis, and J. Jeilinek, Ace. Chem. Phys. (to be published).

^H. G. Dehmeh, Adv. At. Mol. Phys. 3, 53 (1967), and 5, 109 (1969); D. J. Wineland, W. M. Itano. and R. S. Van

Dyck, Jr., Adv. At. Mol. Phys. 19, 135 (1983).

'°W. Neuhauser, M. Hohenstatt, P. Toschek, and H. Dehmelt, Phys. Rev. A 22, 1 1 37 ( 1 980).

"p. Diedrich, J. Krause, G. Rempe, M. O. Scully, and

H. Walther, in Proceedings of the Fourth International Conference on Mulliphoton Processes, Boulder, Colorado,

/9S6, edited by S. Smith and P. L. Knight (Cambridge Univ.

Press, New York, 1987).

'^R. F. Wuerker, H. Shelton, and R. V. Langmuir, J. Appl.

Phys. 30, 342 (1959).

'3j. C. Bergquist, D. J. Wineland, W. M. Itano, H. Hemmati,

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

(1985).

'"E. C. Beaty, J. Appl. Phys. 61, 21 18 (1987).

'5j. C. Bergquist, W. M. Itano, and D. J. Wineland, Phys. Rev. A 36, 428 (1987).

'^D. J. Wineland, W. M. Itano, J. C. Bergquist, and R. G.

Hulet, Phys. Rev. A 36, 2220 (1987).

'"'F. G. Major and G. Werth, Appl. Phys. 15, 201 (1978);

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

Phys. Rev. Lett. 41, 233 (1978); D. J. Wineland, J. J. Bollinger, and,W. M. Itano, Phys. Rev. Lett. 50, 628, 1333(E)

(1983).

'*D. J. Wineland and H. G. Dehmelt, Int. J. Mass Spectrom.

Ion Phys. 16, 338 (1975), and 19, E251 (1976).

2938 TN-109
Volume 60, Number 20 PHYSICAL REVIEW LETTERS 16 May 1988

Shell-Structure Phase of Magnetically Confined Strongly Coupled Plasmas

S. L. Gilbert, J. J. Bollinger, and D. J. Wineland

Time and Frequency Division, National Bureau of Standards, Boulder, Colorado 80303

(Received 4 March 1988)

We report the observation of shell structure in 'Be"*" nonneutral plasmas (ion clouds) confined in a Penning trap. Clouds containing up to 15000 ions (density = 10* cm~^) were laser cooled to temperatures of about 10 mK. Under these conditions, the ions are strongly coupled and exhibit liquidlike and

solidlike behavior through the formation of concentric shells. The shells were observed by direct imaging

of the laser-induced ion fluorescence for values of the Coulomb coupling constant F ranging from about

20 to 200. The shell structure is compared with theoretical predictions.

PACS numbers: 32.80.Pj, 52.25.Wz

The ability to cool collections of electrons or ions to temperatures where the Coulomb repulsion may produce

a spatially ordered state has stimulated interest in several fields of physics. Detailed studies have been

made in two-dimensional Coulomb systems' and in charged particles interacting through a shielded

Coulomb potential.^ In work more closely related to that described here, crystal-like structures of small numbers (<100) of macroscopic charged particles^ and

atomic ions'* in Paul rf traps have been observed. The

experiments on atomic ions appeared to be limited to small numbers because of the presence of rf heating,^

which elevates the ion temperature in larger samples.

Possibilities also exist for the observation of ordered

structures of ions in storage rings.^'^ In Penning-type traps, where ions are confined by static electric and magnetic fields, rf heating does not take

place and large collections of ions can be cooled to temperatures less than 10 mK. Several molecular-dynamics

simulations on collections of a hundred to a few

thousand ions confined in storage rings and traps have

recently been completed. ^"'° Under conditions of strong

coupling where the Coulomb potential energy between

nearest neighbors is greater than the ion thermal energy,

the ions are predicted to reside in concentric shells. Experimentally, strong coupling has been reported in sys- tems of ions" and electrons'^ confined in Penning-type

traps, but no evidence for spatial correlation was obtained. Here we report direct evidence for spatial correlation in magnetically confined ions. In clouds of laser- cooled ''Be"'' ions we observed structures containing from

one shell ( = 20 ions) to sixteen shells (= 15000 ions).

The larger clouds may be approaching the infinite volume regime in which a bcc structure is predicted. '^ The static thermodynamic properties of an ion cloud

confined in a Penning trap are identical to those of a one-component plasma (OCP).''' An OCR consists of a

single species of charge embedded in a uniform-density

background of opposite charge. For the system of ions in a Penning trap, the trapping fields play the role of the neutralizing background charge. An OCP is characterized by the Coulomb coupling constant, '^''^

r=qyasksT, (1)

which is a measure of the nearest-neighbor Coulomb en- ergy divided by the thermal energy of a particle. The

quantities q and T are the ion charge and temperature.

The Wigner-Seitz radius Os is defined by 4;ra//io/3 = l, where —qrio is the charge density of the neutralizing

background. An infinite OCP is predicted '^ to exhibit

liquidlike behavior (short-range order) for f > 2 and

have a liquid-solid phase transition to a bcc lattice at r = 178. For a finite plasma consisting of a hundred to a few

thousand ions, the boundary conditions are predicted to have a significant effect on the plasma state. Simulations involving these numbers of ions have been done for the conditions of a spherical trap potential. ^"'° These

simulations predict that the ion cloud will separate into concentric spherical shells. Instead of a sharp phase

transition, the system is expected to evolve gradually

from a liquid state characterized by short-range order

and diffusion in all directions, to a state where there is diffusion within a shell but no diflfusion between the

shells (liquid within a shell, solidlike in the radial direction), and ultimately to an overall solidlike state."' These conclusions should apply to a nonspherical trap

potential as well if the spherical shells are replaced with

shells approximating spheriods. Preliminary independent

investigations'^'^ of the nonspherical case support this conjecture.

We have investigated this interesting system using

^Be"*" ions trapped in the cylindrical Penning trap shown

schematically in Fig. 1. A magnetic field B=BziB = 1.92 T) produced by a superconducting magnet

confined the ions in the direction perpendicular to the z

axis. A static potential Vq between the end and central cylinders confined the ions in the z direction to a region near the center of the trap. The dimensions of the trap

electrodes were chosen so that the first anharmonic term

of the trapping potential was zero. Over the region near

the trap center, the potential can be expressed (in cylin2022

Work of the U. S. Government

Not subject to U. S. copyright TN-UO
Volume 60, Number 20 PHYSICAL REVIEW LETTERS 16 May 1988

IMAGER

PERPENDICULAR

COOLING BEAM

DIAGONAL COOLING BEAM

FIG. 1. Schematic drawing of the trap electrodes, laser beams, and imaging system (not to scale). The overall length

of the trap is 10.2 cm. The trap consists of two end cylinders

and two electrically connected central cylinders with 2.5-cm

inner diameters. Ion clouds are typically less than 1 mm in both diameter and axial length. The diagonal cooling beam

crosses the cloud at an angle of 5 1 ° with respect to the z axis.

drical coordinates) as <i>=AVo(2z^—r^) where A

=0.146 cm~^. A background pressure of 10 ~^ Pa

(« 10""' Torr) was maintained by a triode-sputter ion pump. The ions can be characterized by a thermal dis- tribution where the "parallel" (to the z axis) temperature Til is approximately equal to the "perpendicular"

temperature T± (from the cyclotron motion). This

thermal distribution is superimposed on a uniform rotation of the cloud"''* (frequency co) which, at the low

temperatures of this experiment, is due to the Ex B drift, where E is the electric field due to the trap voltage and

the space charge of the ions. The ions were laser cooled and optically pumped into the 25^5'i/2(A//= 2",My = y ) state by our driving the

2s^S\/2ij,j)-*2p^Py2(j,j) transition slightly be- low the resonant frequency." The 313-nm cooling radi- ation («i 30 /iW) could be directed perpendicular to the

magnetic field and/or along a diagonal as indicated in Fig. 1. In addition to cooling the ions, the laser also applied an overall torque which could either compress or expand the cloud. '

' This allowed us to control the cloud

size by choosing the radial positions (and thus the torques) of the perpendicular and diagonal beams.

About 0.04% of the 313-nm fluorescence from the de- cay of the ^P}/2 state was focused by //lO optics onto the photocathode of a resistive-anode photon-counting imaging tube. The imager was located along the z axis, about

1 m from the ions. The imaging optics was composed of

a three-stage lens system with overall magnification of

27 and a resolution (FWHM) of about 5 lum (specifically, the image of a point source when referred to the

position of the ions was approximately 5 /xm in diameter). Counting rates ranged from 2 to 15 kHz. Positions of the photons arriving at the imager were displayed in real time on an oscilloscope while being integrated by a computer.

A second laser (power =: 1 /jW, beam waist =30

/im) was used to map the shell structure of the cloud

spatially. This probe laser was tuned to the same transi- tion as the cooling laser and was directed through the

O

(q)

• II ItlOXIJ «••»>• »M<tl*MMM<

(b)

200 Atm ^ ^

DIAGONAL

" COOLING

-PROBE

(c) PERPENDICULAR ~ COOLING

FIG. 2. Images of shell structures, (a) A single shell in a cloud containing approximately 20 ions. Trap voltage Ko = 14 V and cloud aspect ratio Or (axial length/diameter) =6.5.

This image was obtained from the ion fluorescence of the per- pendicular and diagonal cooling beams, (b) Sixteen shells (probe-beam ion fluorescence only) in a cloud containing about 15000 ions with l'o = 100 V and a,^0.8. (c) Eleven shells plus a center column in the same cloud as (b), with Ko = 28 V and

a, = 2.4. This image shows the ion fluorescence from ail three

laser beams. Integration times were about 100 s for all images.

cloud perpendicularly to the magnetic field. With the probe laser on continuously, the cooling laser could be

chopped at 1 kHz (50% duty cycle) and the image signal

integrated only when the cooling laser was off. Different

portions of the cloud could be imaged by the translation

of the probe beam, in a calibrated fashion, either parallel or perpendicular to the z axis. Images were also obtained from the ion fluorescence of all three laser beams.

The probe beam was also used to measure the cloud

rotation frequency (o and ion temperature." For

these measurements, the probe laser was tuned to the 25251/2(1,1) —2/? 2/'3/2(},-^) "depopulation"

transition. Ions in the 2p ^Pyi^ i , ~ 2" ) state can decay

to the 2s^S\i2i\ ,— 2) state, temporarily removing some of the ion population from the 25^51/2(2",})

state. This results in a decrease in the cooling fluores- cence as the probe laser is scanned through the depopulation transition. We determined o by measuring the change in the Doppler shift of the depopulation signal as the probe beam was translated perpendicular to the z

axis. The density aiq could then be calculated "'* from

no=mco(n—a))/2Kq~, where il=qB/mc is the cyclo- tron frequency. With use of no and the cloud size obtained from probe-beam images, the total number of ions was calculated. The ion temperature was derived from

the Doppler broadening contribution to the width of the depopulation signal." The temperature T± was measured with the probe beam perpendicular to the magnetic

field and Tw was measured by our sending the probe

beam along the magnetic field direction (not shown in Fig. 1). In the latter case, crossed polarizers were used

to reduce the intensity of the probe beam at the imager.

TN-llI 2023
Volume 60, Number 20 PHYSICAL REVIEW LETTERS 16 May 1988

Zp Oxm) 200 /xm + ilMHM -15

Mioiii -40

HfOli* -65

ilMMM -90

MlUlM -115

'ii«Mi' -140

*(t>>> -152

Ui -165 ^ It n -178

-100 100 -100 100

XO*m)

FIG. 3. Intensity plots along the imager x axis (parallel to the probe beam) through the center of the ion cloud with cor- responding images (above), (a) r = 180-?8 (7=6-2 mK,

«o = 7xlO' ions cm~^). Cloud aspect ratio ar = 3.5. (b) r = 50:^^8 (r = 33±l]mK, «o = 2xloMonscm"^), a. = 5. The

clouds contained about 1000 ions in both cases.

FIG. 4. Data showing evidence for concentric cylindrical

shells. On the right is a series of images obtained with the probe beam for different z positions Zp of the probe beam

(lower half of the cloud only). Intensity plots for Zp = — 40 ^im and Zp = — 178 pm are shown on the left. The cloud aspect ratio Or was about 1.9.

From the measured values of the temperature T and the density «o, r was calculated with Eq. (l). In the case of

Tx^Tw, the larger temperature was used in the calculation of r. We have observed shell structure in clouds containing

as few as 20 ions (one shell) and in clouds containing up

to about 15000 ions (sixteen shells). Images covering

this range are shown in Fig. 2. We measured the coupling constant F for several clouds containing about 100

ions. Drift in the system parameters was checked by our

verifying that the same images were obtained before and

after the cloud rotation frequency and ion temperatures were measured. Figure 3 shows examples of shell struc- tures at two different values of F. The first image is an example of high coupling (F=: 180) showing very good

shell definition in an intensity plot across the cloud. The

second image is an example of lower coupling (F»50)

and was obtained with cooling only perpendicular to the magnetic field. Variations in peak intensities equidistant

from the z axis are due to signal-to-noise limitations and

imperfect alignment between the imager x axis and the probe beam.

We obtained three-dimensional information on the

shell structure by taking probe images at different z positions. We found that there were two types of shell struc- ture present under different circumstances. The first type showed shell curvature near the ends of the cloud,

indicating that the shells may have been closed

spheriods. Shell closure was difficult to verify because of

a lack of sharp images near the ends of the cloud where

the curvature was greatest. This may have been due to the averaging of the shells over the axial width of the probe beam. In the other type of shell structure, it was

clear that the shells were concentric right circular

cylinders with progressively longer cylinders near the

center. An example of these data is shown in Fig. 4. Other evidence for cylindrical shells can be obtained

from the fact that shells in the diagonal-beam images

occur at the same cylindrical radii as those from the per- pendicular beams. This can be seen in the three-beam

images such as that shown in Fig. 2(c). Systematic

causes of these two different shell configurations have not

yet been identified. One comparison which can be made between the

theoretical calculations and our experimental results is the relationship between the number of shells and the

number of ions, TV,, in a cloud. For a spherical cloud,

approximately (7V,/4)'^^ shells are predicted.^ For the nearly spherical cloud of Fig. 2(b) (A^,^ 15000) this formula predicts 15.5 shells and we measure 16. At

present, it is difllicult to make further quantitative comparisons between our data and the theoretical calculations. For example, there is substantial uncertainty in our measurement of F due to uncertainty in the temperature measurement. Our data do agree qualitatively with

the simulations with the exception of the presence, in some cases, of an open-cylinder shell structure as op- posed to the predicted closed spheroids. Schiffer has suggested'^ that shear (that is, different rotation frequencies) between the shells may account for this discrepancy. In our experiment, shear could be caused by

differential laser torque or the presence of impurity

ions. '^ For the data here, we have determined that the

rotation frequency does not vary by more than 30%

across the cloud. This is comparable to the limits dis- cussed by Larson and co-workers.'^''* Future improvements will allow a more complete comparison of the data and the simulations, such as the rela2024 TN-112
Volume 60, Number 20 PHYSICAL REVIEW LETTERS 16 May 1988

tionship between shell definition and F for a variety of

conditions. Also, we have observed that it is possible, us- ing the probe depopulation transition, to tag ions in different parts of the cloud and see the presence (and

lack) of ion diffusion. We plan to use this technique in a pulsed mode and obtain information on diffusion times

within and between shells. Finally, with more laser power, even larger clouds (10^ ions or more) can be

cooled to these temperatures (10 mK). This may permit

observation, by Bragg scattering," of the predicted

infinite volume structure. We gratefully acknowledge the support of the U.S.

Office of Naval Research and the Air Force Office of

Scientific Research. We thank D. H. E. Dubin and J. P. Schiffer for useful discussions and C. Manney and

W. Itano for help with image processing. We also thank

G. Dunn, W. Itano, and C. Wieman for helpful suggestions on the manuscript.

'a. J. Dahm and W. F. Vinen, Physics Today 40, No. 2, 43

(1987). ^D. J. Aastuen, N. A. Clark, and L. K. Cotter, Phys. Rev.

Lett. 57, 1733 (1986); J. M. di Meglio, D. A. Weitz, and P. M.

Chaikin, Phys. Rev. Lett. 58, 136 (1987); C. A. Murray and

D. H. Van Winkle, Phys. Rev. Lett. 58, 1200 (1987).

'R. F. Wuerker, H. Sheiton, and R. V. Langmuir, J. Appl.

Phys. 30, 342 (1959).

"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).

^D. A. Church and H. G. Dehmelt, J. Appl. Phys. 40, 3421

(1969); H. G. Dehmelt, in Advances in Laser Spectroscopy,

edited by F. T. Arecchi, F. Strumia, and H. Walther (Plenum,

New York, 1983), p. 153.

6j. P. Schiffer and O. Poulsen, Europhys. Lett. 1, 55 (1986). ^D. Habs, in "Frontiers of Particle Beams" (SpringerVerlag, Berlin, to be published).

**A. Rahman and J. P. Schiffer, Phys. Rev. Lett. 57, 1133

(1986); J. P. Schiffer, to be published. ^H. Totsuji, in Strongly Coupled Plasma Physics, edited by

F. J. Rogers and H. E. Dewitt (Plenum, New York, 1987), p. 19.iOd. H. E. Dubin and T. M. O'Neil, Phys. Rev. Lett. 60, 51

(1988). "J. J. Bollinger and D. J. Wineland, Phys. Rev. Lett. 53, 348

(1984); L. R. Brewer, J. D. Prestage, J. J. Bollinger, and D. J. Wineland, in Ref. 9, p. 53.

'2j. H. Malmberg, T M. O'Neil, A. W. Hyatt, and C. F. Drisccll, in Proceedings of the 1984 Sendai Symposium on Plasma Nonlinear Phenomena, edited by N. Sato (Tohoku

University, Sendai, Japan, 1984), p. 31.

''S. Ichimaru, H. lyetomi, and S. Tanaka, Phys. Rep. 149, 91 (1987), and references therein.

''*J. H. Malmberg and T. M. O'Neil, Phys. Rev. Lett. 39, 1333 (1977).

'^J. P. Schiffer, private communication.

'^D. H. E. Dubin, private communication.

'^D. J. Larson, J. C. Bergquist, J. J. Bollinger, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 57, 70 (1986).

'*L. R. Brewer, J. D. Prestage, J. J. Bollinger, W. M. Itano, D. J. Larson, and D. J. Wineland, Phys. Rev. A (to be published).

TN-113 2025
From: Proc. of the ONR Conference on Non-neutral Plasmas, National

Academy of Sciences, Washington, D.C., March, 1988; Non-neutral Plasma

Physics , ed by C.W. Roberson and C.F. Driscoll, A. I. P. Conf. Proc. 175,

Amer. Inst, of Physics, New York, 1988, p. 93.

LIQUID AND SOLID ION PLASMAS

D.J. Wineland, Wayne M. Itano, J.C. Bergquist, S.L. Gilbert,

J.J. Bollinger, and F. Ascarrunz

National Bureau of Standards, Boulder, CO 80303

ABSTRACT

Experiments on strongly-coupled nonneutral ion plasmas,

performed at the National Bureau of Standards, are summarized.

We first discuss strong coupling of small numbers (< 100) of

macroscopic and atomic ions confined in Paul (electrodynamic)

traps, in which crystalline structures are observed. We then

discuss experiments in which shell structure is observed for up

to 10^ atomic ions confined in static electric and magnetic

fields. In our experiments, we have progressed from working with

very small numbers of ions up to an intermediate value. Future

experiments are suggested, including some where infinite volume

behavior might be observable.

INTRODUCTION

This paper briefly summarizes the work on strong coupling in

nonneutral ion plasmas at the National Bureau of Standards at

Boulder. This work has grown out of a project whose goal is very

high resolution atomic spectroscopy on samples of ions contained

in an "ion trap." Since spectral lines are, in general, shifted

and broadened by Doppler effects associated with the motion of

the ions, we desire an accurate characterization of the velocity

distributions of the nonneutral plasmas (or "clouds") of ions

contained in these traps

.

Our interest in the strong coupling problem was in part stimulated by a 1977 paper^ of Malmberg and O'Neil who discussed the

possibility of achieving liquid and solid like behavior of a pure

electron nonneutral plasma confined by static electric and

magnetic fields. Their confinement device has typically been

called a Penning trap^ by atomic physicists. At about this time, we demonstrated the method of laser cooling'' on atomic ions in a Penning trap.'' Since the achievable temperatures are in the

millikelvin range and below, we realized that strong coupling

could probably be achieved in such a sample of ions. In addition, at low temperatures, the resonant light scattering cross section for atomic ions can be on the order of 10"^° cm^ . This

is to be compared to the Thomson cross-section for electrons

which is approximately 10'^* cra^ . Therefore, for atomic ions,

the large cross sections for light scattering and long confinement times (many hours) made it possible to take pictures and

"movies" of the plasmas. This sensitivity is dramatically

illustrated by the pictures of single ions.^'^-^

The static thermodynamic properties of an ion cloud confined

in a trap are identical to those of a one-component plasma

(OCP).^'^ An OCP consists of a single species of charged

TN-114
particles embedded in a uniform-density background of opposite

charge. For the system of ions in a trap, the trapping fields

play the role of the neutralizing background charge. An OCP can

be characterized by the Coulomb coupling constant,^-

r ^ qV(a,kBT), (1)

which is a measure of the nearest neighbor Coulomb energy divided

by the thermal energy of a particle. The quantities q and T are

the ion charge and temperature. The Wigner-Seitz radius a^ is

defined by 47rag^no/3 = 1, where -qnQ is the charge density of the

neutralizing background. An infinite volume OCP has been

predicted^ to exhibit liquid-like behavior (short range order)

for r > 2 and a liquid-solid phase transition to a bcc lattice at

r - 178.

The occurrence of spatial ordering when F > 1 can be understood qualitatively. At low enough temperature, where F > 1, the

ions do not have enough kinetic energy to overcome the Coulomb

repulsion. Therefore they cannot approach each other very

closely and become more evenly spaced. Even though this basic

idea is understood, interest in this subject has continued to

grow, possibly due to the richness of phenomena expected^ and the

difficulty of achieving this condition experimentally. Moreover,

strong coupling has been studied or is expected in many fields

besides plasma physics, for example in: 2-D configurations of

electrons or ions near the surface of liquid helium,®-^ charged

particles in liquid suspension which act through a shielded

Coulomb potential, ^° electrons in solids (Wigner crystallization), astrophysics (Curtis Michel, this conference) and perhaps

somewhat surprisingly, in particles which are confined in high

energy storage rings and cooled by electron cooling. -^^

ION TRAPS

Atomic physicists have most often used two kinds of devices in

ion trapping work, the Penning trap and the Paul (radio -frequency) trap^ whose electrodes are shown schematically in Fig. 1. The Penning trap is essentially the same device as that used in

other nonneutral plasma studies . ^^ - ^^ In atomic physics experiments, the electrodes are often made to conform to equipotentials

of a quadratic potential (Fig. 1). This insures that motion of

single ions (or the center-of-mass motion for a single species)

is harmonic. Thus the "hyperbolic" traps can be used as charge

to-mass analyzers. Ions in a Penning trap can be thought of as comprising an OCP where the static magnetic and electric fields

play the role of the neutralizing background charge.^' ^^ The Paul (rf) trap typically uses the same electrode con- figuration as shown in Fig. 1. Trapping is provided by the

ponderomotive potential or pseudopotential , resulting from the

application of spatially inhomogeneous , time varying electric

TN-115
B=BoZ (required for Penning trap)

Endcap —

Ring

Endcap

^=y/(rl + 2zl)

V= Uo + Vocosilt

(Vo = for Penning trap)

Fig. 1 Schematic representation (cutaway view) of the electrode

configuration for the Paul (rf) or Penning trap. Electrode

surfaces are figures of revolution about the z axis and are equipotentials of (^(r , z)=A(r^ -2z^ ) , For this equation, cylindrical coordinates are used with the origin at the center of the

trap. Typical dimensions are JZzq = rg = 1 cm. Typical operating parameters are the following: for the Paul trap, Vq=300 V,

Q/ln - 1 MHz; for the Penning trap, Ug^l V, B=l T.

fields.^ For the hyperbolic trap (Fig, 1) this ponderomotive

potential takes the form^'^*

where

^P = ar2 + ^z'

a = qV2/(mn2^*) + n/^^

(2)

(3a)

^ = 4qV2/(mn2|*) - 2Uo/^2 (3b)

^^ = Tg + 2zo , m is the ion mass, and the other parameters are defined in Fig. 1. The center-of-mass oscillation frequencies of

ions in the quadratic potential of Eq. 2 are usually called the

secular frequencies.^

Equation 2 is equivalent to the potential inside of a uniformly charged spheroid.^* Therefore, ions confined in a Paul

trap may be thought of as comprising an OCP where the uniform

neutralizing background charge density has the shape of a uniformly charged spheroid consistent with Eq. 2. This picture

is valid when the micromotion of the ions at the drive

frequency can be neglected. For large numbers of ions this is

not always justified as discussed below.

TN-116
The Penning and Paul traps can be superimposed; the general

solution was given by Fischer. ^^ OCP properties have been discussed in Ref. 14.

MACROSCOPIC PARTICLE PAUL TRAP EXPERIMENTS

The Paul trap was experimentally demonstrated in 1959.^^ In

the experiments of Wuerker, et al., macroscopic aluminum particles (diameter - 20 ^ra) were stored. The charges on these

particles were quite high (- 5x10^ electron charges) leading to r » 1 and the observation of regular crystalline arrays. The

key to these experiments is the high value of q. Since T a q^

large values of T can be obtained even though the interparticle

spacings are macroscopic (~1 mm) and temperatures were near room temperature

Fig. 2. Cluster of alumina particles (center part of picture) in

a Paul trap, Endcaps are not visible in this picture; inner

diameter of ring electrode is 2.5 cm, Q/2if = 60 Hz.

We have performed experiments similar to those of Ref. 15. In

Fig. 2, we show a suspension of approximately 25 alumina particles (clumps of grinding powder) which have crystallized into a regular array. The inner part of the ring electrode (diameter =

2.5 cm) is visible but the endcap electrodes are obscured from

view. For this picture, we had Vq - 350 V at Q/ln = 60 Hz.

Although the electrode surfaces were not hyperbolic, their shapes

were chosen by computer calculation ^^ to make the potential near

the center of the trap nearly quadratic. The dimensions of an equivalent hyperbolic trap are Xq = 1.36 cm = 72 Zq . The

background gas pressure for this trap was about 13 Pa (100 mTorr)

of air which maintained the particle temperature near 300 K. From either the measured secular frequency along the z axis of

symmetry, or from the potential applied to one endcap in order to

counteract gravity (the z axis is vertical) we determined that

the charge- to-mass ratio is q/m - 10"^° x q/m(proton) . From the

TN-117
separation of two charged particles in the x-y plane (Ug = 0) , we determined that the mass of the particles was approximately 10" ^g

and therefore the particle's charge was about q = 6 x 10* x q(proton) . (For Uq = 0, the balance of the pseudo potential

(inward) force by the Coulomb repulsion (outward) is charge

independent, but mass dependent). This gives a value of T ~ 10^,

For Fig. 2, the particles were illuminated by a laser but a simple lamp would suffice.

MACROSCOPIC VS. ATOMIC PARTICLE EXPERIMENTS

Under fairly simple experimental conditions, very high values

of r can be obtained in the macroscopic particle experiments

because of the high values of charge obtained. It is therefore

reasonable to ask why we would like to use atomic ions where the

charge is much smaller, typically q = q(proton). One reason is

that for a given size of the ion sample with respect to the trap

dimensions, the number of ions N^ is proportional to q" ^ (see

Appendix A). Therefore, if we desire samples of many particles,

small charge on the particles is advantageous. Perhaps more

important is that in the macroscopic particle experiments, q and m are difficult to control precisely so that comparison of the

observed crystals with theory is more difficult to make. Also,

even though the macroscopic particles have high charge, their mass is relatively high and their charge-to-mass ratio q/m

relatively small compared to atomic particles (typically about 10

orders of magnitude smaller) . This consideration is important

for Penning trap confinement where the maximum density given by

the Brillouin density^ ^ is equal to n^a^ ~ B^/(87rmc^). For the

macroscopic particles of Fig. 2, we estimate m = 10"^ g, so that

even for B = 10 T, r\aax - ^ '^ 10"'' cm"^ which is extremely small

(not to mention the problems with stray electric fields) . For

atomic particles, n^^x ~ 10^° cm"''. Thus, magnetic confinement

of large numbers of particles can only be accomplished with

atomic or molecular ions or electrons. Finally, atomic ions have

internal structure which can be spectroscopically probed to give

information on plasma dynamics.^® Therefore, even though the

macroscopic particle traps can provide interesting information on

the strong coupling problem, many experiments require atomic ions or electrons.

STRONGLY COUPLED ATOMIC IONS IN PAUL TRAPS

With the practical density and charge limitations imposed on atomic ions in Penning and Paul traps, very low temperatures must

be realized in order to obtain F » 1. This has been achieved by

laser cooling.^ Here we describe an experiment using Hg"*" ions at

NBS.^^ A similar experiment has been performed at the Max Planck

Institute for Quantum Optics in Garching.^^

TN-118
IMAGE PUNE

(photocathode surface)

LENS

SYSTEM

Fig. 3 Schematic drawing of the trap electrodes (to scale) and

imaging system (not to scale) for Hg"^ cluster experiments. The

end-cap to end-cap separation along the z axis is approximately

625 urn. The overall magnification of the lens system is 180X.

The laser beam used to illuminate the ions also enters the trap

at an angle of 54.7° with respect to the z axis and is perpendicular to the observation axis shown.

Small numbers of ^^^Hg* ions were stored in a miniature Paul

trap, which has properties equivalents^ to those of a hyperbolic

trap with dimensions Tq =« 466 /im and Zg = 330 pm. An oscillating

voltage Vg with peak amplitude between 145 and 325 V at a frequency of 23.26 MHz was applied between the ring and end-cap

electrodes for trapping. The ions were laser cooled by 1-2 ;iW of cw 194-nm radiation^" (bandwidth < 2 MHz), whose beam waist was varied between 5 and 15 /im at the position of the ions. This

radiation was tuned near the 5dS°6s ^S^ -» 5dS°6p ^Pj^ first

resonance line, which has a natural linewidth of 70 MHz. Some of

the 194-nm fluorescence from the ions was focused onto the

photocathode of a resistive-anode photon-counting imaging tube as shown schematically in Fig. 3. The optics for this imaging was provided by a three-stage fused-quartz lens system with an aberration-compensated first stage. The first lens was apertured

to give an f number of 4.5. The positions of the photons

detected by the imaging tube could be displayed on an oscilloscope in real time or stored by a computer in order to make time

exposures

TN-119
In Fig. 4(a) we show the orientation of the trap as viewed by

the imaging tube. In Fig. 4(b), two ions are made to lie in the

x-y plane by making a < /9 in Eq. 2. Although the trap was designed to have axial symmetry, the preferred spatial alignment

of the two cold ions indicates that an asymmetric potential

exists which could be due to trap imperfections, contactpotential variations, or some other trap asymmetry. For con- venience, we choose this preferred direction to be the x axis as indicated in Fig. 4b.

In the remainder of Fig. 4 we show images obtained for a few cases of up to 16 ions. We can characterize the ponderomotive

potential as in Eqs . 2 and 3 or, equivalently , by the single ion

resonance frequencies v^ and v^ for the motion in the axial and

radial directions respectively. (In Fig. 4, we assume 1/3^=1/ =

i/j. . ) We chose this latter method in Ref. 18 and for Fig. 4 since

the measured oscillation frequencies^^ are a more direct means of

characterizing the trap than the applied potentials and trap

dimensions. We find a = 27r^mi/^/q and ^ = lir^iai/^/q where a and y9 are from Eq. 2. When v^/v^ is made large enough, all of the ions are forced by the strength of the potential in the z direction to

lie in the x-y plane as in Figs. 4b and d. When y^/v^ is small

enough, the ions lie along the z axis as in Fig. 4c. This

allowed us to count the ions. Below the images in Figs. 4d

through 4g, we show the configurations obtained theoretically^^

for each value of the number of ions N^ obtained by minimizing

the function

Ni Ni

E^ = q I (arf + ySz^ ) + q2 ^ Ir^ - r^ \-

(4)

1=1 i<j

where a and ^ are determined from the experimentally measured

values of u^ and u^ . The circular images or rings in Figs. 4e through g are due to

ion circulation about the z axis. For a small asymmetry in the

x,y plane, we can modify Eq. 2 to be approximately

0p = a^x2 + ayy2 + yflz^ . (5)

When Qjj<Qy<a2 , and N^^ = 2, the potential energy of Eq. 4 is

minimized for two ions along the x axis as in Fig. 4b. The

strength of this alignment is given by AE^ , the maximum variation

of E^ for 2 ions constrained in the x-y plane. This is given by

the difference between E^ for ions on the x axis and E^ for ions

constrained along the y axis. For 3 or more ions in the same

ring, the maximum variation in E^ is considerably smaller than

for two ions. Therefore, the ions more easily rotate in the

azimuthal direction even though they must be nominally equally

spaced in the ring. The number of ions in Fig. 4e could be

TN-120
ion

cluster

uz = 0.749 MHz

UT = 0.380 MHz

(c)

-\; = 6

t^z = 0.154 MHz

ur = 0.421 MHz

(d)

Ni = 6

uz = 0.780 MHz

ur = 0.347 MHz

(e)

'\- = O

i^z = 0.308 MHz

Ur = 0.376 MHz

(f)

uz = 0.439 MHz

Ur - 0.304 MHz

(g)

;Vj = 16

Uz = 0.626 MHz

Ur = 0.479 MHz

Fig. 4 Images of Hg* ion "clusters" in a Paul trap. In (a), the

trap electrodes are shown schematically in the same orientation

as for the remainder of the pictures, but at reduced magnification (inner diameter of the ring electrode is ~ 0.9 mm). The

coordinate system in (b) applies to the rest of the images. The

tick marks on the axes are 10 ^m from the origin. In (d) through

(g) , we display numerical solutions for the expected cluster

shapes using the experimental values of u^ and u^. . The "rings" are caused by ion circulation about the trap z axis of symmetry.

In (d) , a non-fluorescing impurity ion occupies a particular ion

site in the cluster and helps to pin the ions in the radial

plane.

TN-121
verified by aligning them along the z axis and counting. For

Figs. 4f and g, the number of ions could be assigned by varying

v^/Uj. and comparing the observed images (with rings) with the

simulations. In principle, we could have aligned and counted 9

and 16 ions along z, but the ratio v^^/v^ required to do this was

so low, that ions would have evaporated from the trap along the z axis

.

In Fig. 4d, we see the approximate configuration expected for

6 ions but one of the ions appears to be missing. In addition,

the ions are pinned in the x-y plane rather than forming a ring

as in Fig. 4f. These observations are consistent with a nonfluorescing or "phantom" ion that occupies the missing cluster

site. This ion may be another isotope of Hg"*" . For example,

fluorescence from ^^^Hg"*" would be extremely weak because of

ground state hyperfine optical pumping. It might also be an impurity molecular ion like HgOH"^ formed from the exothermic

reaction

Hg+ + H2O -» HgOH* + H. (6)

In any case, an impurity ion with charge- to-mass ratio different

from ^^^Hg"*" will make AE^ for ions in the same ring be larger

than AE^ if all the ions are ^^^Hg"*". This can happen because of

the differences in a and ^ for ions of different charge-to-mass

ratio. Thus we would expect pinning to be more likely with the

impurity ion present. In the experiments of Ref. 19, the trap

asymmetry in the x-y plane was probably larger than observed

here. This apparently made AE^ larger for ions in the same ring,

resulting in pinning of like ions.

For these configurations, the agreement between data and

simulations appears to be fairly good.^^ To estimate F, we use

for the background density ng the expression^*

Trm "° = 2^ ^2"""^^ = -^ ^2^' " <^ ^^^

For the conditions of Fig. 4(g), Eq. 7 implies ng = 3.8 x 10^.

In these experiments , the temperature was extracted from the

Doppler broadening of the ion's spectral lines. ^^ This broadening contributes significantly to the width of ^S^ -> ^05^2

quadrupole transition in i98j|g+ is , 2<. From these measurements,

we determined that the secular motion temperature for two ions in

the trap, when u^ - 2»/j =473 kHz, was less than 8 mK. If we

take 8 mK as a typical upper limit for the ions in fig. 4g then

we obtain F > 525.

Finally, in Fig. 4g, we see that 16 ions approximately lie on

3 rings. The larger ring, containing 8 ions, lies in the x-y

plane and the smaller rings, each containing 4 ions, lie in

planes above and below but parallel to the x-y plane. We may

also view these ions as lying on the surface of a spheroidal

TN-122
shell characteristic of the structure for large numbers of

ions.ii'13'26-28

In principle, it should be possible to observe ordered

structures of much larger numbers of atomic ions in Paul traps.

In our trap, this was apparently prevented for N^ - 20 because

the small amount of laser power available for cooling was unable

to overcome the rf heating. ^^ In rf heating, the kinetic energy

of the ion's driven motion at frequency Q is coupled into the

secular motion characteristic of the pseudopotential well.

STRONGLY COUPLED ATOMIC IONS IN PENNING TRAPS

In Penning traps, ^ where ions are confined by static electric

and magnetic fields, rf heating^ ^ does not take place and large

collections of ions can be laser-cooled to temperatures less than

10 mK. The static properties of ions in a Penning trap are identical to those of a one component plasma where the background

density is given by i-i^.i''

Here, 0^. is the ion cyclotron frequency and co is the rotation

frequency of the cloud. For a finite plasma consisting of a hundred to a few thousand ions, the boundary conditions are predicted to have a significant effect on the plasma state.

Several molecular dynamics simulations on collections of a hundred to a few thousand ions confined in storage rings and

traps have recently been completed. ^ ^ - ^^"^^ • ^° • ^

^ When T > 1,

the ions are predicted to reside in concentric shells. As the

ions are cooled, instead of a sharp phase transition, the system

is expected to gradually evolve from a liquid state characterized

by short range order and diffusion in all directions, to a state

where there is diffusion within a shell but no diffusion between

the shells (like a smectic liquid crystal) and ultimately to an overall solid-like state. ^^ We have investigated^^ this interesting system using ^Be* ions

trapped in the cylindrical Penning trap shown schematically in

Fig. 5. A magnetic field B = Bz (B = 1.92 T) produced by a superconducting magnet confined the ions in the direction

perpendicular to the z axis. A static potential Uq between the

end and central cylinders confined the ions in the z direction to a region near the center of the trap.

The ^Be"^ ions were laser cooled by driving the 2s ^S^(mj =

3/2, mj = 1/2) -> 2p 2 Pg^ 2 (3/2,3/2) transition slightly below the

resonant frequency. The 313 nm cooling radiation (= 30 ^W) could

be directed perpendicular to the magnetic field and/or along a diagonal as indicated in Fig. 5. In addition to cooling the

ions, the laser also applied an overall torque which could either

compress or expand the cloud. ^''•^^ This allowed us to control

the cloud size by choosing the radial positions (and thus the

torques) of the perpendicular and diagonal beams.

TN-123
About 0.04% of the 313 nin fluorescence from the decay of the

^P3/2 state was focused by f/10 optics onto the photocathode of

the same imaging tube used in the Hg"^ ion experiments . The

imager was located along the z axis , about 1 m from the ions

.

The imaging optics was composed of a three stage lens system with

overall magnification of 27 and a resolution (FWHM) of about 5

/xm. Positions of the photons arriving at the imager were

displayed in real time on an oscilloscope while being integrated

by a computer.

Fluorescence observed

only when cooling beams off.

Image plane

(photocathode surface)

Trap electrodes.

Laser coolmg beams

/

Lens

(mag. - 27) "Probe beam"

(X ^ 313 nm)

Fig. 5. Schematic representation of apparatus for obseirvation of

shell structure of strongly coupled ^Be"*" ions in a Penning trap.

Images are made of the ions which are intersected by the probe

beam. In the upper right, an intensity plot vs x for a value of y

intersecting the center of the cloud is shown. Shell structure

results from the boundary conditions at the edge of the cloud.

A second laser (power = 1 /iW, beam waist — 30 /im) was used to

spatially map the shell structure of the cloud. This probe laser was tuned to the same transition as the cooling laser and was directed through the cloud perpendicularly to the magnetic field.

With the probe laser on continuously, the cooling laser could be

chopped at 1 kHz (50% duty cycle) and the image signal integrated

only when the cooling laser was off as shown in Fig. 5. Different portions of the cloud could be imaged by translating the

probe beam, in a calibrated fashion, either parallel or perpendicular to the z axis . Images were also obtained from the ion

fluorescence of all three laser beams. ^^ The rotation frequency w was determined from the first order

Doppler shift of the ion's optical spectrum due to the cloud

TN-124
rotation.^'' • ^^ ' ^^ From w and an independent measurement of fi^'^^

ng was determined from Eq. 8. From the Doppler broadening of the

spectrum, T was determined. ^•^ • ^^ • •'^ From Uq and size of the ion

cloud, N^ was determined. T was determined from Uq and T.

We have observed clouds containing from 1 shell (N^ — 20) up

to 16 shells (Nj^ =: 15 000) . For a spherical cloud, approximately

(Nj^/4)^/'' shells are predicted. ^^ For the nearly spherical cloud

of Ref. 29 (Ni = 15 000) this formula predicts 15.5 shells and we measured 16. At present, it is difficult to make further quantitative comparisons between our data and the theoretical calculations. For example, there is substantial uncertainty in our measurement of F due to uncertainty in the temperature measurement. Also, the spatial resolution is partially limited by

optics. Our data do agree qualitatively with the simulations

with the exception of the presence, in some cases, of an open

cylinder shell structure^ ^ as opposed to the predicted closed

spheroids. Schiffer has suggested"^ ° that shear (that is,

different rotation frequencies) between the shells may account

for this discrepancy. In our experiment, shear could be caused

by differential laser torque or the presence of impurity ions.'^^

For the data here, we have determined that the rotation frequency

does not vary by more than 30% across the cloud. This is

comparable to the limits discussed in Refs. 32 and 33.

FUTURE EXPERIMENTS

A number of future experiments will allow us to better

understand strong coupling in nonneutral ion plasmas. Increasing

the cooling power in the experiments should allow larger (N^^ »

10^) samples of ions to be cooled. This may permit, perhaps by

Bragg scattering,^"' the observation of structure characteristic

of the infinite volume regime in which a body-centered cubic

(bcc) structure is predicted.^ Detailed images in the Penning

traps are made difficult by the rotation of the cloud which

averages over azimuthal angle. In principle, it should be

possible to strobe the laser (or detection) at a frequency

harmonically related to u>, but this requires w to be sufficiently

stable. Fluctuations in ion density may make this difficult but

perhaps an individual strongly fluorescing ion (such as Mg"^

which is locked onto the rotating lattice (of Be"^ ) could serve as a "beacon" and aid in determining w. In addition, the present

experiments would benefit from improved light collection optics.

At present, the rather large uncertainty in F in the Penning

trap experiments comes from the uncertainty in our measurement of

temperature. This is because the Doppler broadening contribution

of the optical spectrum used is small compared to the radiative,

or natural broadening. This limitation can be overcome by

driving transitions whose radiative broadening is negligible as in the Hg"^ experiments .^^ For Be"^ and Mg"^ ions, two-photon

stimulated Raman transitions might be used for this purpose.^ '^^ So far, our measurements have been limited to the static

structure of nonneutral ion plasmas. Various other experiments

may provide dynamical information. For example, ions can be

TN-125
switched off by optically pumping them with an additional laser

beam into states which don't f luoresce .^-^ • ^^ • •^ •^ By appropriate

positioning of this depopulating laser beam, certain parts of the

ion cloud can be tagged allowing ion diffusion studies via the

movement (or lack of movement) of the tagged ions.^^ In this

way, relatively rapid diffusion within the shells of Be"^ ions in

the Penning trap has been distinguished from the much slower

diffusion of ions between the shells. In addition, spectrum

analysis of the scattered light from strongly coupled ions can possibly be used to study mode structure of ordered ion motion.

Such studies may be aided by the use of sympathetic cooling^

whereby two ion species are confined simultaneously in the same

trap. One ion species is laser cooled and by Coulomb interaction

cools and maintains the second ion species at constant temperature. Dynamical studies are then performed on the second ion

species

.

It would be very useful to study large numbers of particles in

a Paul trap. One clear advantage over a Penning trap is that

ions can remain fixed without rotation. This could be insured by

making the trap asymmetric in the x-y plane. For example, by

splitting the ring into sectors and applying different static

potentials to these sectors, the asymmetry could be easily

controlled. For large numbers of ions, the mechanism of rf

heating^ ^ must be overcome. This may be aided by working at

higher laser cooling power, using smaller ratios of secular to

drive frequencies (smaller q parameter. Appendix A) and making

sure that impurity ions are absent.

ACKNOWLEDGEMENTS

We gratefully acknowledge support from the Office of Naval

Research and the Air Force Office of Scientific Research. We

thank A. Bauch for helpful suggestions on the manuscript.

TN-126
APPENDIX A

In studies of strong coupling we want low temperature and high

density. In addition, we may desire large numbers to more nearly

approach the infinite volume regime.

Paul (rf) trap

From Eqs . 3 and 7, we can write, for a low temperature sample:

Ho = 3q,Vo/(47rqe2) (Al)

where the dimensionless parameter qj. = 4qVQ/(mn^^^) is the ion's

stability parameter.^ Typically, q^. < 0.3 for stable operation.

If we assume for simplicity, that Uq is adjusted to make the

cloud spherical with radius r^ ^, the total number of ions N^ is

given by

Ni = 4^nor3,/3 = V^ (q,/q) (r, J/^z ) (a2)

Assuming q^, and t^^i/^ ^^® fixed, then to make N^^ large, we want

Vq and I big and q small.

Similarly, we can derive

r = (q'Voq,/C2)i/V(kBT), (A3)

implying that for large F, we want q and Vq big and T and ^ small. For both large Nj^ and F, we want Vq big and T small; the

choice of q and ^ may depend on specific experimental con- straints .

Penning trap

If we assume a fixed value K for the ratio of the achievable

density to Brillouin density and assume Ug is chosen to make a spherical cloud (radius r^ ^ ) , we have

Ni = K B2r,3/(6mc2), (A4)

and

F = [|;|^l"qVlc.T. (A5)

Therefore, except for practical limitations on maximum possible

values of Ug,^* in the Penning trap, we can make both N^ and F

large by making B, i (« r,, ^ ) , and q big and m and T small.

TN-127
REFERENCES

1. J.H. Malmberg and T.M. O'Neil, Phys . Rev. Lett. 39, 1333

(1977).

2. See for example: E.G. Dehmelt, Adv. At. Mol. Phys. 3, 53

(1967) and 5, 109 (1969); D.J. Wineland, W.M. Itano, and

R.S. Van Dyck, Jr., Adv. At. Mol. Phys. 19, 135 (1983).

3. See for example: D.J. Wineland and W.M. Itano, Physics

Today 40(6), 34 (1987) and references therein.

4. D.J. Wineland, R.E. Drullinger, and F.L. Walls, Phys.

Rev. Lett. 40, 1639 (1978). At the same time, laser

cooling of ions confined in Paul (rf) trap was observed:

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

Phys. Rev. Lett. 41, 233 (1978).

5. W. Neuhauser, M. Hohenstatt, P.E. Toschek, and H.

Dehmelt, Phys. Rev. A22, 1137 (1980).

6. W. Nagourney, Comments on At. Mol. Phys., to be published; H.G. Dehmelt, Physica Scripta, to be published.

7. S. Ichimaru, H. lyetomi, and S. Tanaka, Phys. Rep. 149,

91 (1987) and references therein.

8. See for example: A.J. Dahm and W.F. Vinen, Physics Today

40, 43 (Feb. , 1987).

9. G. Morales, these proceedings.

10. D.J. Aastuen, N.A. Clark, and L.K. Cotter, Phys. Rev.

Lett. 57, 1733 (1986); J.M. di Meglio, D.A. Weitz, and

P.M. Chaikin, Phys. Rev. Lett. 58, 136 (1987); C.A.

Murray and D.H. Van Winkle, Phys. Rev. Lett. 58, 1200

(1987).

11. J. P. Schiffer and 0. Poulsen, Europhys . Lett. 1., 55

(1986); D. Habs, in Lecture Notes in Physics 296 . M.

Month and S. Turner, eds. (Springer-Verlag, Berlin) 1988,

p. 310.

12. J.H. Malmberg, these proceedings.

13. T.M. O'Neil, these proceedings.

14. D.J. Wineland, J.J. Bollinger, W.M. Itano, andJ.D.

Prestage, J. Opt. Soc. Am. B2, 1721 (1985).

15. E. Fischer, Z. Phys. 156, 1 (1959); R.F. Wuerker, H.

She 1 ton, and R.V. Langmuir, J. Appl. Phys. 30, 342

(1959).

16. E.C. Beaty, J. Appl. Phys. 61, 2118 (1987).

17. R.C. Davidson, Theory of Noneutral Plasmas (Benjamin,

Reading, Mass., 1974), p. 7.

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

Bollinger, and C.H. Manney, Phys. Rev. Lett. 59, 2935

(1987).

19. F. Diedrich, E. Peik, J.M. Chen, W. Quint, and H.

Walther, Phys. Rev. Lett. 59, 2931 (1987).

20. H. Hemmati, J.C. Bergquist, and W.M. Itano, Opt. Lett. 8,

73 (1983).

21. F.G. Major and G. Werth, Appl. Phys. 15, 201 (1978); W.

Neuhauser, M. Hohenstatt, P. Toschek, and H. Dehmelt,

Phys. Rev. Lett. 41, 233 (1978); D.J. Wineland, J .J

.

TN-128
Bollinger, and W.M. Itano, Phys . Rev. Lett. 50, 628,

1333(E) (1983).

22. W.M. Itano, et al., to be published. Our simulations

agree with independent ones performed by D.H.E. Dubin

(performed on up to 10 ions), private communication.

23. J.J. Bollinger and D.J. Wineland, Phys. Rev. Lett. 53,

348 (1984); L.R. Brewer, J.D. Prestage, J.J. Bollinger,

and D.J. Wineland, in Strongly Coupled Plasma Physics . edited by F.J. Rogers and H.E. Dewitt (Plenum, New York,

1987), p. 53.

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

A36. 428 (1987).

25. D.A. Church and H.G. Dehmelt, J. Appl. Phys. 40, 3421

(1969); H.G. Dehmelt, in Advances in Laser Spectroscopy . edited by F.T. Arecchi, F. Strumia, and H. Walther

(Plenum, New York, 1983), p. 153.

26. A. Rahman and J. P. Schiffer, Phys. Rev. Lett. 57, 1133

(1986); J. P. Schiffer, submitted for publication.

27. H. Totsuji, in Strongly Coupled Plasma Physics , edited by

F.J, Rogers and H.E. Dewitt (Plenum, New York, 1987), p.

19.

28. D.H.E. Dubin and T.M. O'Neil, Phys. Rev. Lett. 60, 511

(1988).

29. S.L. Gilbert, J.J. Bollinger, and D.J. Wineland, Phys,

Rev. Lett., 60, 2022 (1988).

30. J. P. Schiffer, private communication.

31. D.H.E. Dubin, private communication.

32. D.J. Larson, J.C. Bergquist, J.J. Bollinger, W.M. Itano,

and D.J. Wineland, Phys. Rev. Lett. 57, 70 (1986).

33. L.R. Brewer, J.D. Prestage, J.J. Bollinger, W.M. Itano,

D.J. Larson, and D.J. Wineland, Phys. Rev. A., to be

published.

34. C.F. Driscoll, in Low Energy Antimatter . Ed. by D.B.

Cline (World Scientific, Singapore, 1986), p. 184.

TN-129
From: Proc . 4th Symposium on Freq. Standards and Metrology,

Ancona, Italy, Sept., 1988 (Springer Verlag)

.

Quantitative Study of Laser Cooling in a Penning Trap'

W.M. Itano, L.R. Brewer, D.J. Larson,

J.J. Bollinger, S.L. Gilbert, and D.J. Wineland

National Institute of Standards and Technology

Boulder, Colorado, 80303 U.S.A.

Penning traps are currently being studied at our laboratory [1,2] and elsewhere

[3,4] for use in atomic frequency standards. A Penning trap confines ions by a combination of a static, uniform magnetic field and a static, nonuniform electric

field. The fact that the fields are static makes it feasible to laser cool more

than 10^ ions to low temperatures [5]. In contrast, the radio- frequency (Paul)

ion trap uses oscillating electric fields, which, in combination with the non- linear Coulomb coupling between ions, cause heating and make it difficult to laser

cool large numbers of ions [6]. For either trap, large numbers increase the

signal-to-noise ratio; low temperatures reduce the second-order Doppler shift [7].

Several experimental groups have reported laser cooling of ions confined in

Penning traps to temperatures below 1 K [1-5,8-10]. Typically, the measured

temperatures were one or two orders of magnitude higher than the theoretical limit

^min '^ ^Yo/(2kg) (about 1 mK), where yq ^^ *^^® natural linewidth of the optical

transition used for laser cooling [9]. The reason for this discrepancy is now understood [10]. The rotation of the ion plasma about the magnetic field axis,

due to the presence of crossed electric and magnetic fields, has a great effect on the temperature. In order for the ion plasma to be stable, it is necessary to

displace the laser beam radially so that it illuminates the side of the plasma

which is receding from the laser. The radiation pressure applies a torque to the

plasma which counteracts the torques induced by axially nonsymmetric, static

electric and magnetic fields. This radial displacement of the beam, however,

results in higher temperatures. The reason stems from the requirement that the

work done by the laser on the ions be zero for a steady state. The rate at which

work is done on an ion of velocity ^ by a force F, which, in this case, is

parallel to the direction of propagation of the laser beam, is f-"^. Because of

the radial displacement of the beam and the plasma rotation, F'\^ tends to be

positive. In order to make the average value <?'^> be zero, the frequency of the

laser must be tuned below the optical resonance. Because of the Doppler shift,

this makes f stronger for ions with ^-^ < than for ions with ?-\^ > 0. The width

of the velocity distribution must have a particular value in order for

cancellation between negative and positive contributions to make the average value

TN-130
be zero. The required width increases with the rotation frequency and with the

distance of the laser beam from the axis of rotation of the plasma. This width

corresponds to a temperature higher than Tj^j^j^. Temperatures and rotation frequencies of ^Be plasmas were measured for a wide

range of experimental parameters [10]. The measured temperatures varied from 40

mK to 2 K. These measured temperatures were in agreement with the calculation,

within an rms error of about 10%. For these measurements, the cooling laser beam was in the plane perpendicular to the magnetic field. Lower temperatures have

been attained by the use of laser beams whose direction of propagation were not

perpendicular to the magnetic field axis, but the calculation in this case is more complicated [5].

For certain ranges of laser frequency detuning and beam position, we have

observed strong oscillations in the total ion fluorescence, with periods from less

than a second to over a minute. These oscillations have been observed in all

Penning traps at our laboratory and are similar to those reported by others

[3,11]. Sometimes, a deliberate misalignment of the magnetic field direction with

respect to the trap axis was necessary in order for oscillations to occur. This was an indication that external torques were involved. An ion plasma undergoing

oscillations was observed with a photon-counting imaging tube. The change in ion

fluorescence was accompanied by a change in the radius (which is directly related

to the total canonical angular momentum) of the plasma. For the plasma to reach a steady state, the rate of both the energy and angular momentum imparted to the

plasma by external forces must be 0. Apparently, these two conditions sometimes

cannot be satisfied simultaneously, and this leads to oscillations.

We wish to acknowledge support from the U.S. Office of Naval Research and the

U.S. Air Force Office of Scientific Research.

Work of the U.S. Government. Not subject to U.S. copyright.

1. J.J. Bollinger et al.: Phys. Rev. Lett. 54, 1000 (1985)

2. J.J. Bollinger et al.: these Proceedings

3. R.C. Thompson et al.: Opt. Acta 33, 535 (1986); Appl. Phys. B 46, 87 (1988)

4. F. Plumelle et al.: Appl. Phys. B 41, 183 (1986)

5. L.R. Brewer et al. : Phys. Rev. A 38, 859 (1988)

6. J. Hoffnagle et al.: Phys. Rev. Lett. 60, 2022 (1988); R. Blumel et al.

Nature, 334, 309 (1988)

7. D.J. Wineland et al. : these Proceedings

8. R.E. Drullinger, D.J. Wineland, and J.C. Bergquist: Appl. Phys. 22, 365 (1980)

9. W.M. Itano and D.J. Wineland: Phys. Rev. A 25, 35 (1982)

10. W.M. Itano et al.: Phys. Rev. A (in press)

11. J. Yu, M. Desaintfuscien, and F. Plumelle: (submitted to Appl. Phys. B)

TN-131
Am. Phys. Fr. 10 (1985) 737-748 decembre 1985, page 737

Optical pumping of stored atomic ions (*)

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

and J. D. Prestage

Time and Frequency Division, National Bureau of Standards, Boulder,

Colorado 80303, U.S.A.

Resume. — Ce texte discute les experiences de pompage optique sur des ions atomiques

confines dans des pieges electromagnetiques. Du fait de la faible relaxation et des deplacements d'energie tres petits des ions confines, on peut obtenir une tres haute

resolution et une tres grande precision dans les experiences de pompage optique associe a la double resonance. Dans la ligne de I'idee de Kastler de « lumino-refrigeration »

(1950), I'energie cinetique des niveaux des ions confines peut etre pompee optiquement.

Cette technique, appelee refroidissement laser, reduit sensiblement les deplacements

de frequence Doppler dans les spectres.

Abstract — Optical pumping experiments on atomic ions which are stored in elec- tromagnetic « traps » are discussed. Weak relaxation and extremely small energy

shifts of the stored ions lead to very high resolution and accuracy in optical pumpingdouble resonance experiments. In the same spirit of Kastler's proposal for « lumino

refrigeration » (1950), the kinetic energy levels of stored ions can be optically pumped.

This technique, which has been called laser cooling, significantly reduces Doppler

frequency shifts in the spectra.

1. Introduction.

For more than thirty years, the technique of optical pumping, as originally proposed

by Alfred Kastler [1], has provided information about atomic structure, atom-atom

interactions, and the interaction of atoms with external radiation. This technique

continues today as one of the primary methods used in atomic physics. Moreover,

it is a tribute to Kastler's foresight that some of his ideas have only recently been

realized — for example, the relatively new technique of laser cooling is essentially

contained in Kastler's idea of « lumino-refrigeration » [1]. Experiments on optical pumping of ions have not been as extensive as those on

neutral atoms partly due to the difficulty of producing and maintaining sufficient ion populations and partly due to the difficulty of producing the required optical

radiation which for positive ions is usually in the ultraviolet region of the spectrum.

The subject of optically pumped ions was reviewed by Weber [2] in 1977; most of

the experiments reported there used buff"er gas cells to hold the ions and gas dis- charges as sources of radiation.

Since that time, a number of experiments using electromagnetic ion « traps » and laser sources for optical pumping have been carried out. In this paper we will

restrict our attention to the unique features of the traps for optical pumping experiments; the basic methods of trapping have been discussed elsewhere [3, 4].

(*) Work of the U.S. Government ; not subject to U.S. copyright.

TN-132
738 ANNALES DE PHYSIQUE N" 6

2. Properties of ion traps for optical pumping experiments.

2. 1 Relaxation. — Often in ion trap experiments, ions are stored under ultra-high vacuum conditions (p < 10"** Pa [5]; neutral density « < 2.7 x 10^/cm^ at room

temperature) for long periods of time (hours are not uncommon). As opposed to buffer gas cell experiments, this suggests that ion-neutral relaxation processes such

as electron spin relaxation are extremely weak. As an example, suppose that ionneutral relaxation proceeds at a typical Langevin rate for orbiting type collisions [6] : k = 2 X 10"^ cm^/s. For a pressure of 10"^ Pa at room temperature, the relaxation

time constant is given by t = {kn)~^ = 190 s. Using cryogenic vacuum systems,

this relaxation time could be made much longer.

Of course in samples of many trapped ions, the ions « see » or interact with each

other through the Coulomb interaction; this might lead to relaxation in various

ways. For example, electron spin relaxation in ions with ^Sj^j ground states could

be driven by the relativistic magnetic fields B^. = v x EJc, produced in ion-ion

collisions. To estimate this effect, assume B^ has a magnitude equal to that given

by the above expression where E^ is the electric field at the distance of closest approach

r (here assumed to be the impact parameter) and a duration given by 2 r/v where v

is the magnitude of the root mean square velocity between colliding ions. Then the

spin precession angle for a single ion-ion collision is given by A0=(4 nn^ BJh).{2 r/v) where ^^ is the Bohr magneton and h is Planck's constant. For ions of mass 100 u

(atomic mass units) at room temperature, the mean precession angle averaged over

all distances of closest approach (from the minimum distance of closest approach,

^0 = 9^/3 '^B ^' up to the mean spacing between ions, n~^'^, where q is the ion

charge, /cg is Boltzmann's constant and n is the ion density in cm"^) is < A0 > ^ 1 X 10"^^ «''^. We would expect the angle to increase from multiple

collisions in a random walk fashion so that dd^/dt S < A0 >^ vn^'^ = 5 x 10"^° «. Even for ion densities ofn = 10^ cm"^, such relaxation is negligible.

Relaxation might also occur in a similar way via electric dipole transitions — the

rate would be approximately (c/v)^ ("JOo/^ /^b)^ times faster or dO'^jdt = 5 x

10"''^ n

(«() is the Bohr radius). However these calculations are valid only when the transition

frequencies of interest are small compared to the inverse of the time of duration of

the collision. If the transition frequencies are higher, then the states tend to adiabatically adjust during the collision and no relaxation occurs. Another way of saying

this is that the spectrum of the E and B fields during the collision does not have

components much above a frequency v/r. For room temperature ions (M = 100 u), even at the minimum distance of closest approach this cutoff frequency is around

20 GHz. This cutoff frequency would be expected to increase as T^'^, therefore

at elevated temperatures, such relaxation might play a role with molecular rotational

(or perhaps vibrational) relaxation. In one way it is unfortunate that such relaxation

is so weak. If electric dipole relaxation were stronger, significant rotational-translational energy exchange might be expected to occur for heteronuclear molecules.

Since the temperature of ions can be sensitively monitored by the bolometric tech- nique [7, 8], such relaxation might be a way to detect rotafional/vibrational transi- tions in molecular ions. We note that significant rotational (vibrational)-translational energy exchan'ge might be expected to occur between stored molecular ions

and a neutral background gas (because of the shorter range of interaction) ; for this

case, the bolometric technique might be used to detect rotational/vibrational transi- tions in the ions even though part of the energy released in the collision goes to the

neutral.

TN-133
No 6 OPTICAL PUMPING OF STORED ATOMIC IONS 739

Relaxation of atomic ions in metastable excited levels (e.g. the lowest ^Dj^j states in Ba"^ or Hg^) can proceed via ion-neutral collisions (expected to be efficient) or via the electric fields of ion-ion collisions. For the D states of Ba^ or Hg"^, the

collisional electric field from ion-ion collisions mixes some ^P3/2 state into the

^05^2 state ; therefore during the collision the ion decays at a rate given approximately by [< 'P,^, \qr-E\ 'D,,, y/iEeP,,^) - ^('05/2))]' ^('P3/2) where E{i) and

^(i) are the energy and decay rate of state i. If as before, the collision is assumed

to have a duration 2 r/v then when averaged over impact parameters between ((q and «~^'^, the probability of de-excitation per collision is given hy { P } = 4

q'*' al n^'^ ^(^P3/2)/(A£'^ cIq i;) where AJ? = ^'(^Pj/j) - -^(^05/2), and we have assumed

< ^P3/2 I ^r • E I ^D^i2 > = ?«o I E |. Therefore the overall relaxation rate for this process is given by < Py.vn^'^; for Hg"^ ions at « = 10^/cm^ and T = 300 K,

the relaxation rate from the ^05^2 state is about 5 x 10~^/s which is negligible.

Therefore we see that at high vacuum, relaxation of the internal energy states of

ions is expected to be extremely weak. This may be regarded as an advantage if for example, the objective of an experiment is to obtain sub-Hertz linewidths in double

resonance experiments, but may be regarded as a disadvantage if the ions become

« trapped » in a metastable state. In this case a buffer gas such as He or H2 might

be added to provide the necessary relaxation [9].

2 . 2 Energy shifts. — One of the primary advantages of the stored ion technique

is that ions can be confined in a small region of space without the usual perturbations associated with confinement (e.g. shifts due to ion-neutral collisions in a buffer gas cell). At finite vacuum one must still worry about ion-neutral collisions but the resulting

shifts should be quite small. For example, the fractional shifts of the ground state hyperfine frequencies of ^^^Ba^ (Ref [10]) and i^^Hg+ (Ref [11]) due to He colli- sions have been measured to be about 5 x 10" ^ VPa and 4 x 10" ^ VPa respectively.

Therefore, at 10"^ Pa, such shifts are extremely small. Similarly, on optical transitions, frequency shifts due to ion-neutral collisions would be expected to be smaller

than the collision rates ; in the high vacuum case fractional shifts would therefore

be very small. One must also consider the frequency shifting effects of the electric fields of ionion collisions and ion-trap « collisions » [12]. The latter effect is analogous to ionneutral collisions in a buffer gas cell however the « collisions » with the electric

fields of the ion traps can be much softer. If the ions have linear shifts of energy

levels with respect to electric field {dv oz < E », then these shifts averaged in time

must vanish. We know that because if < E > / the ions would leave the trap.

Energy shifts of non-spherical electronic levels (e.g. the D states in Ba"^ or Hg"^)

due to the electric quadrupolar trapping fields of a Penning trap [3, 4] occur at a

level of a few Hz [13]; these shifts should however be precisely controllable. For

clouds of many ions in a trap, the ion cloud can also have a quadrupole moment

which can interact with an atomic quadrupole moment. Such shifts might become

important in very high resolution optical spectra for example. To estimate such an

effect, assume that we have a « cold » ion cloud in an rf trap. Its shape will be deter- mined by a balance between the forces from the trap pseudopotential and the space

charge [3, 4]. If there is no D.C. bias on the ring electrode, the potential from the

ions is then given by 0(ions) = - nqn(x^ + y^ + 4z^)/3. Using equations (3.6)

of Kopfermann [14], the quadrupole energy shift is given by A^q = q[Qxx 4>xx +

Qyy (pyy + Qzz <i>z^ whcre 2;; Is thc Ith dlagonal element of the atomic quadrupole

TN-134
740 ANNALES DE PHYSIQUE N° 6

tensor [14] and 0^^ = d^(j)/dx^ etc. Therefore we have approximately Avg = ^Eq/h = Q^ nQjh where Q is the usvial quadrupole moment [14]. For 6 = (1 A) , Avq = 3 X 10~^nHz. This shift is probably negligible except perhaps in small rf traps

where very deep pseudo potential wells and high densities can be obtained. One

must also worry about quadrupole shifts if collisions are asymmetric ; this might

be the case if for example, the x, y motion was hotter than the z motion. As an extreme

case, assume that two cold ions are trapped in a deep pseudo potential well at a

fixed separation of 1 |jm along the z axis. In this case, Avq = qQ(l)^Jh = Iq^ Q/hz^ ^ 7 kHz.

One is often concerned with quadratic Stark shifts (Sv oc < ^^ »; the source of

the electric fields may be due to the trapping fields, ion-ion interactions, optical

pumping radiation or perhaps the black body radiation field which at room temperature has a value [15] { E^ } =70 (V/cm)^. The shifting effects of optical pumping

radiation have been known for quite some time ; due to the near resonance condition

of this radiation, resulting shifts can be quite large. In high precision double reso- nance experiments it is usually necessary to probe the transitions with the pumping

radiation off (i.e. : chop the pumping light). Electric fields due to trapping [3, 4] can be of the order of 100 V/cm but in many cases they can be significantly smaller [12]. We can estimate i E^ } due to ion-ion collisions by assuming that the

electric field is due to binary collisions {E{r) — qjr^). If we average E^{r) over the

volume between two spheres defined by the minimum distance of closest approach

and the mean spacing between ions, we obtain <

£'^ > = 3 q^ njdQ. For T = 300 K,

n = lOVcm^, < £2 ^ ^ 3 (v/cm)2.

For hyperfine structure, frequency shifts due to quadratic Stark effects can be

quite small. For example, for Ba* ground state hyperfine structure (^v/vq) = 3.5 X 10~^^ < E^ > where E is expressed in V/cm [15]. For optical structure, we can

for example estimate the shift of the ^05^2 state in Hg^ or Ba"^ as 5v ==

| < '^^^12

I qr • E I 2D5/2 > YlKEi^^za^-E{^'Dii2)) = q^ al < E^ y/AEh using the same approximation as previously. For Hg"^, Sv = ( E^ (V/cm) >. (0.002 Hz). Therefore quadratic Stark shifts can be quite small ; they can be further reduced by chopping the

pumping light, cooling the kinetic temperature of the ions [12] (see below) and

reducing the ambient temperature to reduce the black-body radiation fields. For ions stored in a Penning trap, the required magnetic fields are on the order

of 1 tesla. In this case, therefore, shifts of atomic ion energy levels can be quite strong.

Of course the study of magnetic interactions appears to be quite interesting, and

certain effects such as diamagnetic shifts to atomic hyperfine structure have been

only recently measured [16]. For high resolution double resonance experiments,

magnetic field dependent shifts could be regarded as a severe disadvantage because

of field fluctuations, but in certain cases, transition frequencies become immune to

field fluctuations to first order and extremely high resolution can be obtained [17-19].

2 . 3 Ion number. — Typically, the numbers of ions in stored ion experiments are rather small. Densities are on the order of 10^/cm^ and sample sizes are < 1 cm-'. For many experiments (such as spectroscopy of molecular ions), this is a disadvantage but in some cases even single ions can be sensitively detected [20-22]. The

small densities obtained are to be compared with the much larger densities in buffer gas experiments [2] where space charge neutralization is used. In rf traps, some

space charge neutralization has been realized [23, 24] by storing ions of opposite

charge but this technique has not been widely used perhaps because of the mechanism of rf heating [7, 25].

TN-135
NO 6 OPTICAL PUMPING OF STORED ATOMIC IONS 741

Although the ion numbers are typically rather small, usable samples can be

confined to an extremely small region of space — down to less than (1 ^m)^ for the case of single ions [20, 22]. This is important because in certain high resolution

experiments on optical spectra, it is desirable to satisfy the Lamb-Dicke criterion (confinement to dimensions less than Xjl n) in order to suppress first order Doppler

broadening [13]. Small sample sizes also reduce the requirements on magnetic field homogeneity in high resolution magnetic field dependent spectra.

3. Direct optical pumping.

The essential features of optical pumping in ion traps are not different from the

experiments on neutrals or ions in buffer gas cells. Nevertheless, there are some

unique features of optical pumping on trapped ions ; here we give a simple example.

In the inset in figure 1, we show the 3s ^Si/2 ^^'^ 3p '^P3/2 level structure of ^""^Mg"^

ions in a magnetic field. If a laser (assumed to have linear polarization perpendicular

to the magnetic field) is tuned to the ( — 1 /2) -^ ( — 3/2) transition (numbers in parentheses refer to the ground ^Sj/j and excited ^Pj/j values of Mj respectively)

then we note that ions must decay to the Mj = — 1/2 ground level because of the

selection rule AMj = 0, ± 1. At first, one might guess that after many photon

scattering events, the ions are gradually pumped from the Mj= — 1/2 to Mj= +1/2

ground state level because of excitation in the wings of other allowed transitions. In fact just the opposite occurs. Although excitation in the wings of the (— 1/2) -> (+ 1/2) transition pumps ions out of the (— 1/2) ground state, excitation in the

SCATTERED LIGKT

IIflBISrTY(5s/pt)

BACKGROUND —

+ +

_L

27412 27413

MICROWAVE FREOUENCY (MHz)

Fig. 1. — Microwave-optical double resonance spectrum of ^^Mg""". Inset shows relevant energy levels of ^"^Mg"^ in a magnetic field. With the laser tuned to the transition shown,

16/17 of the ions are pumped into the ^Sj^2 (^j = ~" 1/2) ground state and a quasi two level system is formed with this ground state and the excited ^Pj^j i^i = — 3/2) state. When

incident microwaves are tuned to the (Mj = — 1/2) *-^{My = +1/2) ground state Zeeman

transition, these levels are nearly equally populated which causes a decrease in fluorescence

scattering from the ions. Transitions in other ions can be detected in a similar way (from

Ref [26]).

[Spectre de double resonance micro-onde-optique de ^'*Mg'*'. L'encadre montre les niveaux

d'energie concernes de ^*Mg^ dans un champ magnetique.]

TN-136
742 ANNALES DE PHYSIQUE N° 6

wings of the (+ 1/2) -> (— 1/2) transition also occurs which tends to pump ions

from the (+ 1/2) to (— 1/2) ground state. Since this latter transition is four times

closer in frequency to the laser than the former one, the net result is that about

16/17 of the population is pumped into the (— 1/2) ground state [26]. Of course, by using circularly polarized light, all of the population could be pumped to the (— 1/2) ground state. This pumping is very weak (the time constant for pumping

in this experiment was a few seconds) but because of the very long relaxation times

of the ions in the trap it can be very efficient. This experiment is somewhat unusual because, using linearly polarized light,

the ions are pumped into the ground state energy level that the laser is tuned to. However, noting that the ( — 1 /2) - ( — 3/2) transition takes no part in the pumping,

this is just another case of depopulation pumping (out of the ( + 1/2) ground state) using frequency selected light [27]. Similar optical pumping can also be observed

when hyperfme structure is present ; for example [18], in the case of ^^Mg* and ^Be"^. Hyperfine pumping effects have also been observed [28] in the excited ^Sj states of

Li + . Depopulation pumping in the more typical sense (i.e. depopulation of the ground

state level one is driving on) occurs if one excites for example the ^Sj/2 -^ '^^1/2 transitions in alkali-like ions. In this case one must redistribute or mix the ground

state populations in order to see additional scattering from the pumping source. In this instance one of the advantages of the traps for high resolution spectroscopy

(weak relaxation) becomes a disadvantage in terms of observation. Solutions to this problem are : (1) Provide a buffer gas for relaxation. For example, Ruster et al. [9] observed fluorescence from single Ba"^ ions in an rf trap by relaxing the ions against an H2 buffer gas. (2) If the number of ground (and metastable) states is not too

large, artificial relaxation can be provided by auxiliary microwave or laser radiation

[4, 20, 29, 30].

4. Optical pumping-double resonance. Double resonance detection of atomic ion spectra using traps is achieved in most

cases using the same techniques developed for neutrals. However, because of the weak relaxation and high scatter rates using lasers, some novel effects appear. For

example, in the ^'^Mg"^ case above, if the scattered fluorescence light from the ions

is monitored, the (Mj — — 1/2) to (Mj = +1/2) ground state Zeeman transition

(induced by microwave radiation) can be detected by observing the decrease in laser fluorescence as the microwave oscillator is swept through resonance (Fig. 1). An

interesting feature of this optical pumping-double resonance detection scheme is that each microwave photon absorbed causes a change of about AiV* = 24 .6(— 1/2, — 3/2)/17 5(— 1/2, +1/2) in the number of scattered photons where B{Mj, Mj) is the transition rate due to the laser from the Mj ground state to the M'^ excited state. For Mg* ions in a magnetic field of about 1 T, AiV* can be as high as 2 x 10^. This technique has been called « electron shelving » because the ion's outer electron

is temporarily « shelved » in a state from which the laser scattering is essentially

absent [31]. It has been used in all the optical pumping double resonance experiments on Mg"^ and Be"^ ions stored in Penning traps. This photon amplification scheme was independently developed in optical pumping experiments on neutral atomic

beams [32]. The overall detection sensitivity is increased by this photon number amplification

effect and also by the photon energy conversion. Perhaps noteworthy is the case of

TN-137
No 6 OPTICAL PUMPING OF STORED ATOMIC IONS 743

detecting absorbed 303 MHz photons in Be^ hyperfine transitions [33] where an

energy enhancement factor of AN* A(303 MHz)/A(laser) > 10^^ is obtained.

These impressive numbers are of course not realized in practice since fluorescence

collection efficiencies are typically significantly less than 100%. However, a more

important statement is that if (the absence of) enough scattered photons per ion

(theoretically > 2) are observed before repumping takes place, then the signal to noise ratio in such experiments need only be limited by the statistical fluctuations in the number of ions that make the transition [34]. This is the maximum signal to noise ratio possible.

The long relaxation times and small perturbations achieved in the traps can lead

to very high resolution and accuracy. In ^^Mg"^, the first derivative of the ground

state (Mj — — 3/2, M, = 1/2) to (Mj = — 1/2, Mj == 1/2) transition goes to zero

at a value of the magnetic field near 1.24 T. Near this field, a resonance with a width

of 0.012 Hz and a center frequency of 291.996251899(3) MHz was observed (Fig. 2)

[18]. High resolutions have also been obtained in Yb+ (Av = 0.06 Hz, g S 2 x 10^^

[35]) and Hg+ (Av = 0.85 Hz, Q ^ 5 x lO^*' [11]). Very high accuracies in stored

ion spectra have also been achieved [11, 30, 33, 34]. In the case of Be"^, the (M,, Mj) = (— 3/2, 1/2) -» (— 1/2, 1/2) ground state hyperfine transition frequency at the

magnetic field independent point has been measured with an accuracy of about one part in 10^^ [33]. From the discussion of section 2, it should be possible to obtain

much higher resolutions and accuracies.

>»

1

- o -

'(75 .^~N. ^'-S. c ci ^\^ y^ "^^

a> / Q / \

-»— c - / \ / \ - o o/ Y0.0I2 Hz-/ o\

a 7 \ /o \

CD o / \ / \ O / \ / \ —

to y °\-c/ •^^0

o

Z3 _ t _

Ll

1

291 996 251.899 Hz

1 1 1 1.88 1.89 1.90 1.91 1.92 RF Frequency - 291 996 250 ( H z

)

Fig. 2. — The (M, = - 3/2, M, = 1/2) to (M, = - 1/2, M, = 1/2) resonance in "Mg+

about 1.24 T. The oscillatory lineshape results from the use of the Ramsey separated oscilla- tory field method, implemented by applying two phase-coherent RF pulses 1.02 s long,

separated by 41.4 s (from Ref [18]). [Resonance (M, = - 3/2, M, = 1/2) -^ (M, = - 1/2, Mj = 1/2) dans ^'Mg"^ a environ

1,24 T. La forme de raie oscillante resulte de I'utilisation de la methode de Ramsey a champs

oscillants separes; on applique ici deux impulsions RF coherentes en phase de duree 1,02 s, separees par 41,4 s (Ref [18]).]

TN-138

I
744 ANNALES DE PHYSIQUE N° 6

5. Optical pumping of kinetic energy levels-laser cooling.

In 1950, Kastler proposed the idea of « lumino refrigeration » [1]. His idea was to cool the salts of the rare earth ions with an anti-Stokes transition of sufficient intensity. The basic principle was that perhaps ions embedded in a solid could be made

to undergo anti-Stokes scattering where the energy deficit would be supplied by

the energy of lattice vibrations. Partly because of different motivations (high reso- lution spectroscopy) and different technology available (the laser) this idea was

reinvented for atoms [36] and trapped ions [37] in 1975. Laser cooling of trapped ions can be described in various ways [29], here we

briefly discuss the analogy to optical pumping and anti-Stokes scattering [38]. In

figure 3 we have illustrated the energy levels of a trapped ion which we assume has an internal structure composed of two states — the ground (g) and excited (e) states. If we assume that the ion is also constrained to move along one direction in a harmonic well (frequency v^) then its kinetic energy levels are given by E^^={n + \/2) hv^. In this picture we can think of the ion bound to the trap as a kind of « super-molecule » whose energy level structure can be investigated [38].

Fig. 3. — Pictorial representation of electronic [ground (g) and excited (e)] energy states and translational energy states (denoted by integers) for an atom. When the atom is very weakly bound or unbound the spacing between translational energy levels goes to zero. The cooling can be described as anti-Stokes Raman scattering or optical pumping where the frequency (cd^) of the scattered photon is greater than the frequency (col) of the « laser » photon (from Ref. [38]). [Schema des niveaux d'energie electroniques (fondamental (g) et excite (e)) et de translation (reperes par des entiers) pour un atome. Quand I'atome est tres faiblement lie, ou libre, la separation entre niveaux d'energie de translation tend vers zero. Le refroidissement peut

etre decrit comme une diffusion Raman anti-Stokes, ou un pompage optique ou la frequence

ojs du photon diffuse est plus grande que la frequence cOl du photon « laser » (Ref [38]).]

TN-139
NO 6 OPTICAL PUMPING OF STORED ATOMIC IONS 745

Assume the ion is initially in the (g, n^) state (normally we would expect it to be

in some distribution of states). Then if we excite the (g, «;) -»• (e, n^ — An) transition

with radiation, the ion can decay to a number of lower states but (neglecting terms

of order R = h^jl.X^ m) on the average it decays to the (g, n^ — An) ground state. Therefore in a single scattering event, the ion's kinetic energy has been reduced by

hv^ A«. Laser cooling of free atoms can also be described in this picture; in this case Vy -> and the ground and excited states become continuous bands. In principle, laser cooling should work for solids but inhomogeneous broadening and non radiative relaxation would appear to cause problems. Laser cooling of trapped ions has been accomplished in experiments at the National Bureau of Standards, Heidelberg and Hamburg, University of Washington and

at Orsay [4, 12, 13, 18-22, 25, 29-31, 39]. Temperatures of < 1 K are not uncommon

for a small sample of ions ; these low temperatures significantly reduce Doppler

effects in those experiments.

6. Exchange pumping.

6 . 1 Exchange pumping of internal energy levels. — Although polarization or pimiping of trapped ions using polarized, externally injected neutral beams has

been accomplished [3, 40, 41], exchange pumping due to ion-ion collisions would

ordinarily seem to be precluded. This is illustrated by the weak relaxation discussed

in section 2 . 1 and is mainly due to the fact that the Coulomb repulsion usually

prevents the ions from getting close enough for exchange to occur. For room temperature ions, do = 200 A. At elevated temperatures, exchange pumping might be

expected to take place; for ions of energy 10 eV, cIq < I A. If it could be realized,

such exchange pumping might significantly extend spectroscopic measurements on

ions as it has for the case of neutral-neutral and ion-neutral collisions.

6.2 Exchange pumping of kinetic energy levels. — In the sense that an ion

bound in the trap forms a supermolecule as in section 5, we can think of performing

exchange pumping on the kinetic energy levels of the ions. The idea of buffer gas cooling of ions [3, 25, 29, 30, 42-51] fits within this context Here the (bound) ion«

exchange their kinetic energy with a light buffer gas (e.g. He, or Hj) whose kinetic

energy is thermalized (or « pumped » in the sense here) to the ambient temperature.

Analogous arguments could be made for radiative damping of the ions [7, 8]. Although relaxation between the internal states of ions due to ion-ion collisions

is rather weak (except at elevated temperatures) the relaxation between kinetic

degrees .of freedom is rather strong due to the long range of the Coulomb interaction. At temperatures near room temperature, this relaxation time is typically of order

10 ms. (See for example Refs. [3, 7, 8, 25, 52]). Such collisional coupling between

different degrees of freedom can be important in the laser cooling experiments

when only one or two kinetic degrees of freedom are directly cooled by the laser and the remaining degree(s) of freedom are heated by photon recoil [38, 53]. This coupling may also extend the number of ion species which can benefit from

laser cooling. That is, we could simultaneously store two kinds of ions, one which

is easily laser cooled and another one which is of spectroscopic interest and via the Coulomb collisions is also cooled [54]. This has been demonstrated in experiments on Mg^ where ^''^Mg^ was laser cooled and by collisions « sympathetically » cooled ^^Mg"*" and ^^Mg"^ [55]. We note that in subsequent experiments at NBS

it was possible to sympathetically cool ^*Mg^ by laser cooling ^^Mg^ ; this result

TN-140
746 ANNALES DE PHYSIQUE N° 6

is more definitive because the laser was tuned to the heating side of the ^^Mg"^

transition.

It is interesting to note that at very low temperature (the case for a laser cooled

cloud of ions) we can expect species of ions with different charge to mass ratio to centrifugally separate [56]. This can be seen by considering two ion species (1 and 2)

of different charge to mass ratio, which are in thermal equilibrium in a Penning

trap. In this case we can write the distribution function for the ions as a product [56] :

where y; . „^(0) U_^ exp

m ^3/2/ = /l/2{H. + coP^j)'

rri: 3/2 k^T

(v + core)2 2/cbT

n.{x) = n.{T = 0)e~'^J<'*

where 0(r) = 0(x, y, z) = 0(r, z) == 0(ions) + 0(trap)

r and z are cylindrical coordinates, and the (conserved) angular momentum (P^p

is parallel to the z axis of the trap and the magnetic field B = Bz. The ion cloud

behaves as a rigid rotor plasma [57] which rotates at frequency — co. For the dis- tribution function to be well behaved as T ^ 0, we must have

therefore the charge density inside the cloud is given by [53]

1 ^2^,. , 1 (Bco 2 m. . p = - -r— V^0(ions) = 7^— a>- An In \ c qwhere we have used the fact that V^0(trap) = 0. For the density to be single valued,

ions with different m./q- cannot overlap. For the case of a very weak axial well,

the cloud breaks up into a cylinder of uniform density pj surrounded by an annulus

of uniform density pj [56]. (We have assumed m^/^j < mjjqj.) Solutions in the more

typical case will be discussed in a future publication. A similar separation is expected

to occur in an rf trap except when there is no angular momentum in the cloud.

We note that in an experiment using lasers, we would, in general expect there to be

angular momentum in the cloud unless the laser beam exactly overlaps the center of the cloud. Qualitatively, the separation should occur because for a given total angular momentum, a friction force exists between species of different mjq ; ions

of lower mlq tend to push out ions of higher mjq and vice versa. As a potential application, spectroscopy could be done on ^Be"^ ions which are held at the center of the trap and sympathetically cooled by an outer annulus of

^'^Mg* ions which are laser cooled. The cooling laser could be applied so that it would not spatially overlap the ^Be"^ ions and therefore light shifts on the ^Be"^

TN-141
No 6 OPTICAL PUMPING OF STORED ATOMIC IONS 747

energy levels could be avoided. In this way, extremely narrow linewidths ( <^ 1 mHz)

and high accuracy spectra on Be^ (or other ions) might be obtained. For ions in an rf trap, sympathetic cooling may be limited by rf heating.

Acknowledgments.

The authors greatfully acknowledge the support of the U.S. Air Force Office of

Scientific Research and the U.S. Office of Naval Research. We thank P. L. Bender

and F. L. Walls for reading the manuscript and for helpful comments.

References

[2o:

[21

[22:

[23

[24: [25

[26:

[27

[28

[29

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(1980)" 1137. Wineland, D. J. and Itano, W. M., Phys. Lett. 82A (1981) 75. Nagourney, W., Janik, G. and Dehmelt, H., Proc. Natl. Acad. Sci. USA 80 (1983)

643. Major, F. G. and Schermann, J. P., Bull. Am. Phys. Soc. 16 (1971) 838. Audoin, C, private communication.

Dehmelt, H. G., in « Advances in Laser Spectroscopy » (F. T. Arecchi, F. Strumia and

H. Walther, Eds.). Plenum, New York (1983) p. 153. Wineland, D. J., Bergquist, J. C, Itano, W. M. and Drullinger, R. E., Opt. Lett. 5 (1980) 245. Happer, W., Rev. Mod. Phys. 44 (1972) 169. Prior, M. H. and Knight, R. D., Opt. Commun. 35 (1980) 54. Wineland, D. J., Itano, W. M., Bergquist, J. C, Bollinger, J. J. and Prestage, J. D.,

in Atomic Physics 9, ed. by R. S. Van Dyck, Jr. and E. N. Fortson (World Scientific Publ. Co., Singapore 1985) p. 3.

TN-142
748 ANNALES DE PHYSIQUE N° 6

[30] Werth, G., ibid p. 28.

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Control, 37 (1983). (Copies available from Systematics General Corp., Brinley

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Symp. Freq. Control (1981) p. 602 (copies available from Electronic Industries Assoc, 2001 Eye St., NW, Washington, DC 20006).

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340;

Graff, G., Huber, K., Kalinowsky, H. and Wolf, H., Phys. Lett. 41A (1972) 277

Graff, G. and Holzscheiter, M., Phys. Lett. 79A (1980) 380.

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TN-143
Laser Spectroscopy of Trapped Atomic Ions

Wayne M. Itano, J. C. Bergquist, D. J. Wineland

Recent developments in laser spectroscopy of atomic ions

stored in electromagnetic traps are reviewed with emphasis on techniques that appear to hold the greatest promise

of attaining extremely high resolution. Among these

techniques are laser cooling and the use of single, isolated

ions as experimental samples. Doppler shifts and other

perturbing influences can be largely eliminated. Atomic

resonances with line widths of a few parts in 10" have

been observed at frequencies ranging from the radio

frequency to the ultraviolet. Experimental accuracies of one part in 10'* appear to be attainable.

ATOMIC SPECTROSCOPY DATES FROM THE 19TH CENTURY,

when it was discovered that atomic vapors emitted and

absorbed light at discrete resonance wavelengths, charaaeristic of each chemical element. When the quantum theory of atoms was developed in the 20th century by Niels Bohr and others, it was

realized that these characteristic patterns of resonances, called

spectra, were due to the quantum nature of the atom. The atom

normally exists only in certain states of definite energy. Transitions between these allowed states are accompanied by the absorption or

emission of quanta of light, called photons. The frequency v of the

light is related to the energy change £iE of the atom by the formula hv = |A£|, where h is Planck's constant. Accurate and detailed information about atomic spectra was crucial to the development of

the modem theory of quantum mechanics.

Traditional (that is, nonlaser) optical spectroscopic methods, such

as dispersing the light emitted from a gas with a diffraction grating,

are limited in resolution by Doppler frequency shifb. Doppler shifts are the result of the motions of the atoms in a gas and cause the

612

resonance absorption or emission lines to be much broader than the

natural line widths (the line widths that would be observed if the atoms were motionless and isolated from perturbing influences such

as collisions). Under typical laboratory conditions, the Doppler

broadening results in a line width of about one millionth of the

transition frequency, whereas the natural line widths are typically at

least 100 times narrower. For example, the 280-nm first resonance

line of Mg*, which has a frequency of about 1.07 x 10'^ Hz, has a Doppler-broadened line width at room temperature of about 3 GHz, whereas the natural line width is only 43 MHz. For a

transition with a stable lower level, the natural line width (in hertz)

is the inverse of the mean lifetime of the upper level (in seconds), divided by 2Tr. The development of tunable lasers in the 1970s led to great advances in the resolution and accuracy with which optical atomic

spectra could be observed. Laser light sources have high intensity

and narrow line width. These properties make it possible to use

various nonlinear spectroscopic techniques, such as saturated ab- sorption or multiphoton absorption, that cancel the effects of first- order Doppler shifts, that is, Doppler shifts that are linear in the

velocities of the atoms (7). The natural line width and the second- order Doppler shift, which is quadratic in the atomic velocities, still remain, however. The second-order Doppler shift is a result of

relativistic time dilation. The atomic resonance is shifted like a clock, which runs at a rate that is slower for a moving atom than for an atom at rest. For typical laboratory conditions, this shift is very

small, about one part in 10'^. Sometimes, even this shift can be

troublesome. A good example is the work of Barger et al. on the 657-nm transition of calcium, which has a frequency of 4.57 x 10'''

The authors arc with the Time and Frequency Division, National Bureau of Standards, Boulder, CO 80303. TN-144 SCIENCE, VOL. 237
Hz and a natural line width of 410 Hz (2). Resonances with line widths as small as 2 kHz were observed, but they were severely

shifted and distorted by the second-order Doppler shift. In order to reduce both the first- and second-order Doppler shifts, some method of reducing the temperature of the atoms under study

is required. An atomic vapor placed in a conventional refrigeration device would quickly condense on the walls of the container. But

laser cooling, a method by which laser radiation pressure is used to

reduce the velocities of atoms, achieves the cooling without contact with material objects (3-5). The technique was proposed independendy by Hansch and Schawlow (6) for free atoms and by Wineland

and Dehmelt (7) for trapped ions. Laser cooling to temperatures on

the order of 1 mK has been demonstrated with some kinds of atoms

and atomic ions. This reduces the second-order Doppler shifts to the

level where they are not a problem. An atom or atomic ion of mass

100 u (unified atomic mass units) cooled to 1 mK has a secondorder Doppler shift of 1.4 parts in 10'^. Transit-time broadening is anoth'-r effect that limits the resolution with which spectra can be observ.J. The observed resonance line width cannot be much less than the inverse of the observation time. This is a consequence of Heisenberg's uncertainty relation applied to time and energy. In the case of the work on calcium, the observation

time was limited to about 0.3 msec by the time it took the atoms to

pass through the 21-cm-long resonance region (2). One way to

increase the observation time is by trapping the atoms with electro- magnetic fields. In some experiments, atoms are confined with

specially coated walls or by buffer gases, but collisions with the wall or buffer gas molecules shift and broaden the observed resonances. Atomic ions can be trapped for long periods by fields that do not

disturb their resonance frequencies by significant amounts. Neutral atoms have been trapped by static magnetic fields (8) and by optical

fields {9, 10), but the trapping fields tend to strongly perturb the

resonances. The work described in this article has the goal of achieving gready

improved spectroscopic resolution and accuracy. The experimental

methods involve the use offrequency-stabilized lasers to measure the

spectra of atomic ions, sometimes single ions, that are laser cooled and electromagnetically trapped. The most obvious applications of

such work are frequency standards and clocks of great accuracy; this

is the primary goal of the work done in our laboratories at the National Bureau of Standards (NBS). These experiments have

applications to other areas of physics, however, such as atomic

physics and quantum optics. One example is the recent observation

of quantum jumps (sudden changes of quantum state) of an

individual atomic ion (11-13).

Ion Traps

Ion traps confine ions by means of electric and magnetic fields

{14, 15). Two types of ion traps are commonly used for spectroscopic experiments, the Penning trap and the Paul or rf (radio frequency)

trap. The Penning trap is based on a combination of static electric and

magnetic fields. The electric fields are produced by applying an

electric potential between the ring and endcap electrodes, which are shown schematically in Fig. 1 . The electric forces repel the ions from

the endcaps and provide confinement along the axis of the trap. A

strong, uniform magnetic field is applied along the trap axis to provide radial confinement. Full three-dimensional confinement is provided by diis combination of electric and magnetic fields. For a trap with inside dimensions of about 1 cm, a typical value of the

electric potential difference is 1 volt and of the magnetic field, 1

tesla. The potential energy of an ion is lowered as it moves out

7 AUGUST 1987

B = BgZ (required for Penning trap)

Endcap

Ring

Endcap

^ = A (Jt2+y2_ 2^2) 4=l^/(ro2+2zo2) V= Uq+I^qCos Qf

(Vg = for Penning trap)

Fig. 1 . Eiearodes for a Penning or rf ion trap. The elearic potential field <}> is created by applying the voltage V between the endcap eiearodes and the ring

electrode. The uniform magnetic field B is required only for a Penning trap. [Adapted from (42) with permission from Plenum Press]

radially from the axis of the trap. Hence, even though an ion would

remain trapped forever if undisturbed, collisions with neutral back- ground gas molecules increase the radial extent of the orbit until eventually the ion collides with the ring. In practice, under conditions of high vacuum, it is not uncommon for an ion to remain

confined for days.

The rf trap uses oscillating electric fields and does not require a magnetic fi< d. The electrode structure is the same as that of a Penning trap (see Fig. 1), but the applied electric potential varies sinusoidally in time. The electric field forces an ion to oscillate in position with an amplitude proportional to the field strength. The

phase of the motion with respect to the applied field is such that the average force on the ion in the spatially nonuniform field is directed toward regions of weaker field (the center of the trap in this case). The trajectory of an ion can be separated into a part that oscillates at the frequency of the applied field, called the micromotion, and a part

that varies more slowly, called the secular motion. The oscillating

field creates an effective potential-energy well, sometimes called a pseudopotential, for the secular motion of an ion. For a trap with

electrodes like those in Fig. 1, the pseudopotential is approximately

a three-dimensional harmonic well, so the secular motion is characterized by well-defined frequencies. The electrodes of an rf trap

identical to those used in the NBS experiments with Hg^ ions (12, 16, 17) are shown in Fig. 2. The inside radius of the trap is slightly

less than 1 mm. Under typical operating conditions, an electric potential of peak amplitude 570 volts and frequency 21 MHz is applied, creating a pseudopotential well about 15 eV deep. The

frequency of the secular motion is about 1.5 MHz. In some cases, an

ion can be held in the trap for several days before it is lost, possibly

through a chemical reaction with a residual gas molecule.

Laser Cooling

It has been known for a long time that light can exert a force on a material body. In 1873, Maxwell showed that his theon,' of electro- magnetic fields predicted that a beam of light would exert a force on

a reflecting or absorbing body. This phenomenon is commonly

known as radiation pressure. Light has momentum, the momentum

densit}' being numerically equal to the energ)' densit)' divided by c, the speed of light. Radiation pressure is thus a consequence of

conser\'ation of momentum. In the early 1900s, radiation pressure was obserx'ed experimentally by Nichols and Hull in the United

States (18) and by Lcbede\' in Russia (19), and shown to be in good

agreement with theor\'. In 1933, Frisch first obscr\cd radiation pressure on the atomic scale (20). In his experiment, a beam of

ARTICLES 6n

TN-145
sodium atoms traveling through an evacuated chamber was deflected by a lamp emitting sodium resonance light of wavelength

589 nm. The average deflection of an atom was that due to a transfer of momentum equal to that of a single photon, hvlc. The force exerted by light on atoms is often divided into two

parts. These are called the light-pressure or scattering force and the gradient or dipole force {21). In some simple cases, these two forces can be clearly distinguished, but in the general case, particularly for high light intensities, the simple descriptions of the forces break down, and a quantum-mechanical description is required.

The scattering force is simply radiation pressure at the atomic

level, the force observed by Frisch {20). The average scattering force

is in the direction of propagation of the light and is equal to the product of the momentum per photon and the photon scattering

rate. The force reaches a maximum when the light is resonant with

an atomic transition. Fluctuations in the scattering force arise because the photon scatterings take place at random times and

because the direction of the reemitted photon, and hence the

direction of the recoil momentum due to this reemission, is random.

The dipole force can be understood by considering the atom to be

a polarizable body. The optical electric field induces an electric dipole moment in the atom; the induced dipole is acted on by the

optical electric field. If the light intensity is spatially nonuniform, a force is induced parallel to the gradient of the intensity. The dipole

force attracts an atom to a region of high light intensity if the frequency ofthe light is below the atomic resonance and repels it if it

is above. Sodium atoms have been trapped, by means of the dipole

force, near the focus of a laser beam timed below the first resonance

transition {10).

The laser cooling that has been demonstrated in ion traps is based on the use of the scattering force only. [It is also possible to cool by

use of the dipole force {22), as has recendy been demonstrated in an atomic beam experiment (23), but this method does not appear to have any advantage for trapped ions.] The theoretical description of

the cooling depends on whether the natural line width 7 of the resonance transition used for cooling is greater than or less than the frequencies of the oscillatory motion (wv) of the ion in the trap. The

former case is called the heavy-particle or weak-binding limit; the

latter is called the sideband-cooling or strong-binding limit {3, 5, 24). (Both -y and Wy are expressed in angular frequency units.) Consider first the heavy-particle limit (7 >> Wv, where Wy is any of the motional frequencies). In this limit, each photon scattering event takes place in a time much less than a motional cycle, so that the scattering force can be considered to be made up of a series of

impulses. Assume thr* a trapped ion is irradiated with a beam of

light tuned lower than the resonance frequency. When the velocity

of the ion is opposed to the direction of propagation of the laser beam, the frequency of the light in the frame of reference of the ion

is Doppler-shifted closer to resonance than if it had zero velocity.

Hence, the magnitude of the scattering force is higher when the atom moves against the laser beam, and its velocity is damped.

When it moves in the same direction as the light, the light is Doppler-shifted away from resonance, and the magnitude of the

scattering force is reduced. The net effea is to reduce the kinetic energy of the ion, that is, to cool it. (The ion is heated if the light is

Fig. 2. The cicarodcs of a small rf trap, shown on a coin for scale.

614

Fig. 3. False color image of a single Hg"^ ion (small isolated dot slighdy below the center) stored in an rf trap. The other shaf>es are due to light refleaed from the trap electrodes, whose orientation is the same as in Fig. 2. The largest rcfleaion is from the front surface of the ring elearode. The two

smaller reflections arc from the edges of one of the cndcap electrodes. Computerized imaging system developed by C. Manncy and J. J. Bollinger

(NBS). [Adapted from (43) with permission from the American Institute of

Physics I

TN-146

SCIENCE, VOL. 237
7 Aa/va VV Vy /\a. ;i Aa r A/ V\A = 2,500

Absorption -^ l| /

5 -5.000 \/

f -7.500 V

o ^^ e- -10.000 / «

Ruorescence- 7

1

. 1 . |l— , 1

Relative laser frequency (UHz)

Fig. 4. The absoqjtion

signal of a single Hg*

ion stored in an rf trap (upper trace). Radiofrequency modulation and demodulation

techniques were used

to reduce noise, result- ing in a signal proportional to the derivative of the absorption with

respect to laser fre- quency. The lower

trace is the fluores- cence signal, observed at the same time. The fluorescence drops off suddenly

when the frequency is above resonance, due to laser heating. The flattened top of this curve is due to the frequency modulation. Integration times per point are 50 seconds and 10 seconds in the up[)cr and lower traces, respectively. [Adapted from (17) with permission from the Optical Society of America]

higher in frequency than the resonance.) The minimum number of

photon scatterings required to cool an ion substantially, starting

from 300 K, is on the order of the ratio of the ion's momentum to a photon momentum, or about 10,000. Because of the inherent

fluctuations of the scattering force, which cause heating, the kinetic energy is not damped to zero, but eventually reaches a steady value

that depends on the degree of frequency detuning below the resonance. The detuning that gives the lowest temperature is equal

to one half of the natural line width. This minimum temperature

^min is given by the relation k^^^^ — ^7/2, where k^ is Boltz- mann's constant and h is Planck's constant divided by 2Tt. For

example, T^in for Mg^ (-7/271 = 43 MHz) is 1 mK.

Now consider the sideband-cooling (7 << cov) limit. The absorption spearum of the ion consists of an unshifted resonance line at angular frequency wo, called the carrier, and a series of discrete lines on both sides of the carrier, each having (ideally) the natural line width, and separated by multiples and combinations of the motional

frequencies. These extra lines, called motional sidebands, are the

result of the periodic frequency modulation of the light frequency,

arising from the Doppler effea, as observed by the moving ion. To

cool an ion, the frequency of the laser beam is tuned to a sideband on the low-frequency side of the carrier, for example, to wq - /'tOy, where /> is a positive integer. The ion is induced to make transitions to the upper elearonic state, with a decrease in the vibrational energy. When the ion makes a transition back" to its ground

electronic state, it may either increase or decrease its vibrational energy, but the average vibrational energy change is equal xqK -

(huio) /{2Mc^), where M is the mass of the ion. The quantit)' R,

sometimes called the recoil energy, is the kinetic energy that an

initially motionless- free ion acquires after emitting a photon of

energy Awq. If R < phai^, the ion is cooled. If R << ftwy, which is

not hard to satisfy in practice, and if the frequency width of the light

is much less than -y, then the ion can be cooled until it occupies the lowest quantum state of the trap potential most of the time. In this

limit, the fraction of the time that the ion is not in the lowest state is on the order of {-y/ui^)^ (3, 5, 24, 25). The sideband cooling

discussion can also be applied to the 7 >> Wy case (24, 26). Recently, Lindberg, Javanainen, and Stcnholm {27) have treated

theoretically the general case of laser cooling of an ion trapped in a harmonic well, for an arbitrary ratio of -y to coy and also for arbitrary

laser intensit)'. The lowest temperatures are obtained in the lowintensity limit. The first experimental observations of laser cooling were made in 1978 at NBS by Wineland, Drullinger, and Walls {28) and at die University of Heidelberg by Ncuhauser, Hohenstatt, Toschck, and

Dehmelt (26). The NBS experiment used Mg^ ions in a Penning

trap, whereas the Heidelberg experiment used Ba* ions in an rf trap. In later experiments at these and other laboratories, cooling of

trapped ions has been observed to temperatures in the range of a few

millikelvin, close to the theoretical limit. So far, only strongly allowed transitions have been used for cooling, so the theor\' for the 7 >> coy case applies.

Only a few kinds of ions have been laser-cooled. Laser cooling

requires that the ion be o'cled repeatedly between the ground state and an excited state. To be laser-cooled easily, an ion should have a strongly allowed resonance transition, at a wavelength where a tunable, continuous wave radiation source is available, and be free of

intermediate metastable levels that would interrupt the c\'cling. Even Ba*, one of the first ions to be laser-cooled, is not ideal, since

it can decay into a metastable level from the upper level of the resonance transition. A second laser is required to drive the ion out of the metastable level. A way to cool a species of ion that cannot be

laser-cooled directly is to store it in a trap together with a species

that can be cooled direcdy. As one species is laser-cooled, the other

species is cooled indirectly by Coulomb collisions. This technique,

called sympathetic laser cooling, has been demonstrated in an

experiment in which Hg^ ions in a Penning trap were cooled to less than 1 K by Coulomb coupling with laser-cooled Be^ ions {29).

Spectroscopy of Atomic Ions in Traps

A number of spectroscopic measurements have been made of nonlaser-cooled ions in traps, particularly at microwave frequencies {14, 15, 30, 31). In the more accurate of these experiments, the uncertainty was due mainly to the second-order Doppler shift. For cases

in which laser cooling can be applied, Doppler shifts are suppressed,

and spectra can be observed with extremely high resolution, nearly

free from perturbing influences. Dickc showed in 1953 that the confinement of an atom to a

Fig, 5. Lowest energy

levels of Hg*. Absorption of a \2 photon is detected by the cessa- tion of laser- induced

fluorescence at \|.

[Adapted from (72) with permission from

the American Institute of Physics]

7 AUGUST 1987

281.5 nm

Fig. 6. The 28I.5-nm

transition obscrv'ed in a single Hg* ion. The

frequcnc)' detuning of

the laser is plotted on

the horizontal axis. The intensity ratios of

the motional sidebands

to the carrier (central

line) indicate a temperature of about 6 mK. [Adapted from

(16) with permission from the American Institute of Physics]

Y 0.85

T—I—I—I—I—

vYh^ /^'^v^'V'^

_i I I I I I i I I 1 I I i I I i_ -2 2 Frequency detuning (MHz)

TN-1A7

ARTICLES 6I<
c

1 1 1 1

r" _ o _ m w^"Nv /^^

e / \ / n

/ \ /

>. /

\ / J Vo.012Hz-/ o\

c

01 9f

cu

c -°J °v</ M)"

u

wo

3

1

t 291 996 251.899 Hz

1 1 1 1.89 1.90 1.91 Frequency - 291 996 250 (Hz)

Fig. 7. Graph of a ra- dio frequency hyperfine-structure reso- nance in ^'Mg"*^ ions stored in a Penning

trap. Fluorescence in- tensity is plotted as a function of field fre- quency minus an offset of 291996 250 Hz. Arrow indicates center of resonance. The oscillator)' line shape re- sults from the use of

Ramsey's separated-oscillatory-field method. [Adapted from (38) with permission from the American Institute of Physics]

region of space smaller than the resonance-radiation wavelength

leads to a suppression of the first-order Doppler shift broadening

(32). It is easy to satisfy this condition, commonly called the LambDicke criterion, for a microwave resonance, since the wavelengths

can be several centimeters long. Microwave frequency standards such as the atomic hydrogen maser make use of this fact. In the

optical region, it can be satisfied for a single, laser-cooled ion confined in a trap. (Coulomb repulsion between ions makes it difficult to satisfy the Lamb-Dicke criterion when more than one ion

is in the trap.) The second-order Doppler shift of an ion that is laser-cooled to Tmin is around one part in 10'* even for a strongly allowed transition (high 7). If more than one ion is present in the trap, then the kinetic energies per ion, and hence the second-order Doppler shifts, are necessarily higher. In the case of the if trap, the Gaulomb repulsion

between ions leads to their being trapped away from the center of

the trap, so the oscillating electric field, and hence the micromotion,

is nonzero. In the case of the Penning trap, the presence of the other

ions leads to increased velocities of rotation around the trap axis in the presence of crossed electric and magnetic fields. It appears that, for a single ion, shifts of resonance frequencies owing to electric and

magnetic fields could be as small as a part in lO'*, since the ion is trapped in a region where the electric field approaches zero {33, 34).

Naturally, the signal-to-noise ratio suffers when the sample under

observation consists of only a single ion. However, once an ion has been trapped and cooled, its presence can be detected easily by laser- induced fluorescence, since a strongly allowed transition can scatter

as many as 10* photons per second. In fact, a single, trapped Ba^ ion

has been observed visually, through a microscope, by laser- induced

fluorescence (35). Single ions have been observed in both Penning

and rf traps (11, 13, 15, 30, 35). Figure 3 shows a false color image

of the 194-nm laser-induced fluorescence of a single Hg^ ion confined in an rf trap. This image was obtained at NBS with a position-sensing photomultiplier tube interfaced to a computer. It has even been possible to detect the resonance absorption due to a single Hg"*^ ion by a decrease of the intensity of the 194-nm beam

passing through the trap {17). The fraction of the light absorbed by

the ion was around 10"^. Figure 4 shows the absorption signal and

the fluorescence signal, measured simultaneously as the laser was

swept through the resonance. A transition with a narrow natural line width would be difficult to

detect direcdy by fluorescence, since the long lifetime of the upper

state limits the rate at which photons are emitted. A way around this problem was suggested by Dehmelt {36). Consider an atom that has both a strongly allowed transition (at wavelength \|) and a weakly

allowed transition (at wavelength X2) from the ground state. Figure

5 shows an example of such an atom (Hg^). The narrow resonance

at \2 is detected as follows: The atom is assumed to be initially in the

616

ground state. Laser light at a wavelength near X2 is pulsed on,

possibly driving the atom to the long-lived upper level of this transition. Then light at a wavelength near \i is pulsed on. If the atom had made a transition in the previous step, no fluorescence would be observed; otherwise fluorescence will be observed at an

easily detectable level. The detection of presence or absence of the A.] fluorescence is much easier than attempting to detect the one X2 photon that eventually is emitted when the long-lived state spontaneously decays. The method is called "electron shelving," since the

optical elearon is temporarily shelved in the long-lived level. So far, only a few measurements of the optical spectra of trapped

laser-cooled ions have been made. Some measurements have been

made of strongly allowed transitions in Be"^ and Mg^, with

resolutions near the natural line widths of tens of megahertz. These measurements have yielded some values for energy level splittings, including fine-structure and hyperfine-structure splittings (75, 30). A two-photon transition from the groimd 6^Si/2 to the 5^Dy2 state has been observed in a single Ba^ ion, with a resolution, limited by

the laser line widdi, of 3 MHz {37). Recently, at NBS, die 5rf'°6s

^5i/2 to 5(f6s^ ^D^/2 281.5-nm transition has been observed in a

single, trapped Hg"^ ion {16). The ion was laser-cooled, by means of

the 194-nm 6s ^S|/2 to 6p ^P\i2 transition, to near the theoretical

limit of 1.7 mK. Figure 6 shows the spectrum obtained by tuning a

laser across the narrow 5-to-D transition. The electron shelving detection method was used. Immediately after the 281.5-nm radia- tion was shut off, the 194-nm fluorescence intensity was measured.

If it was low enough to indicate that the ion had made a transition to thcD-state, the signal was defined to be 0. Otherwise, it was defined

to be I . The average of these measurements is plotted along the

vertical axis. The carrier and the two closest motional sidebands on

each side are visible. The line widths were deliberately broadened to reduce the time required to sweep the resonance. On other sweeps,

line widths of about 30 kHz were observed, a result mainly of laser frequency instability. The lifetime of the upper level is approximately 0.1 second, so the natural line width is about 1.6 Hz. The size of

the sidebands indicates that the ion has been cooled approximately

to the Lamb-Dicke regime. The root-mean-square amplitude of

motion is estimated to be 50 nm.

150

100

50

150

c 100

200 800 Time (msec)

Fig. 8. Fluorescence intensity as a function of time for samples of three ions (top trace), two ions (middle trace), and one ion (bottom trace) in an rf trap. The fluorescence changes suddenly each time an Hg"^ ion either makes a transition to a metastable D-state (step down) or back to the ground 5-state from a D-state (step up). The integration time per point is 1 msec.

SCIENCE, VOL. 237 TN-148
Microwave spectra of trapped ions are usually observed by some

form of microwave-optical double resonance. Hypcrfine splittings

and^-faaors of the ground states of several atomic ions have been

measured (15, 30, 31). So far, all of these experiments involve samples of more than one ion. Figure 7 shows an example of the kind of resolution that can be obtained with trapped ions (38). The

line width of the resonance (a hyperfine transition in ^Mg*) is only 12 mHz. The fractional resolution is about the same as for the Hg*

S-to-D optical transition. In another experiment at NBS, the frequency of a hyperfine transition in 'Be^ has been measured with an accuracy of about one part in 10'^, which is comparable to the most accurate frequency standards in existence (39). The accuracy was limited to this value by the second-order Doppler shift, because

the laser cooling had to be shut oflF while the microwave transition was being driven, in order not to perturb the resonance frequency,

and the ions heated up during this period. Sympathetic laser cooling

using Mg^ might provide a solution to this problem.

Quantum Jiunps

Recendy, Cook and Kimble (40) theoretically investigated a three-level atom like the one considered by Dehmelt in his "shelved

electron" proposal. Light at wavelengths k\ and \2 was assumed to be present at the same time. They predicted that one would observe

the k\ fluorescence to turn off and on abrupdy as the atom made

transitions (quantum jumps) to and from the long-lived upper level. This paper generated a great deal of theoretical interest, and the problem was approached from various viewpoints (41). The novelty

of this problem is that theorists are accustomed to calculating, and

experimenters are accustomed to measuring, ensemble averages

rather than following the development in time of a single quantum

system.

The experiment that is the most similar to the one treated by

Cook and Kimble was performed at NBS on a single, trapped Hg^

ion (72). Light at 194 nm (X|) and 281.5 nm (^2) was present at the same time (see Fig. 5). The 194-nm fluorescence level was observed

to be bistable, switching suddenly between a steady level and zero. The rate at which the switches occurred increased as the intensity of

the 281.5-nm light was increased. The statistical properties of the quantum jumps were consistent with theory and with previous measurements ofthe lifetime of the '^0^/2 state. If the intensity of the 194-nm light is increased, then quantum jumps are observed even without the 281.5-nm light present. [These events occurred at a low

rate in the data of (72).] The reason for these events is that the upper

level of the 194-nm transition has a small probability (about 1 in 10') of decaying to the ^Dy2 level rather than back to the ground

state. The ^Di/2 state has a lifetime of about 10 msec and about an

equal probability of decaying either to the ^0^/2 level or to the ground state. The bottom trace of Fig. 8 shows the 194-nm

fluorescence intensity of a single ion as a function of time, clearly

showing the quantum jumps. The middle and top traces in Fig. 8

show the fluorescence when two or three ions are in the trap. When

more than one ion is present in the trap, the fluorescence level jumps

between several discrete levels, depending upon how many ions are

in one of the D-states. Quantum jumps have been observed by other groups, using trapped Ba^ ions (11, 13).

Conclusions

Spectroscopy of trapped atomic ions has already achieved impressive levels of accuracy and resolution. The ultimate limits have

certainly not been reached, and no fundamental (as opposed to

technical) obstacles preventing the achievement of measurement

accuracies (in special cases) of one part in 10'* are foreseen (33, 34).

Interesting spin-offs from the quest for improved sf>ectroscopic

methods, such as the observation of quantum jumps, will no doubt

continue to appear.

REFERENCES AND NOTES

1. High-Resolution Laser Spectroscopy, K. Shimoda, Ed. f Springer- Vcrlag, Berlin, 1976); M. D, Levenson, Introduction to Nonlinear Laser Spectroscopy (Academic

Press, New York:, 1982).

2. R. L. Barger, J. C. Bcrgquist, T. C. English, D. J. Glaze, /4/^. Phys. Lett. 34, 850

(1979); R. L. Barger, Opt. Lett. 6, 145 (1981).

3. S. Stenholm, Rev. Mod. Ph/s. 58, 699 (1986).

4. W. D. PhiUips and H. J. Metcalf, Sd. Am. 256, 50 (Maix± 1987).

5. D. J. Wineland and W. M. Itano, Ph/s. Rev. A 20, 1521 (1979); W. M, Itano and

D. J, Wineland, Und. 25, 35 (1982)".

6. T. W. Hansch and A. L. Schawlow, Opt. Commun. 13, 68 (1975).

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

8. A. L. Migdall, J. V. Prodan, W. D. Phillips, T. H. Bcrgcman, H. J. Metcalf, Pha. Rfp.Lm. 54, 2596(1985).

9. S. Chu, L. HoUberg, J. E. Bjorkholm, A. Cable, A. Ashkin, ibid. 55, 48 (1985).

10. S. Chu, J. E. Bjorkholm, A, Ashkin, A. Cable, Md. 57, 314 (1986).

11. W. Nagoumev, J. Sandberg, H. Dehmelt, tbtd. 56, 2797 (1986).

12. J. C, Bergquist, R. G. Hulet, W. M. Itano, D. J. Wineland, thid 57, 1699 (1986).

13. T, Sauter, W. Neuhauser, R. Blatt, P. E. Toschek, Und., p. 1696,

14. H, Dehmelt, yl(/p. At. Mol. Phys. 3, 53 (1967); ibid. 5, 109 (1969).

15. D. J. Wineland, W. M. ltano,'R. S. Van D\'ck, Jr., ibid. 19, 135 (1983).

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

17. D. J. Wineland, W. M. Itano, J. C. Bergquist, Opt. Lett., 12, 389 (1987).

18. E. F. Nichols and G. F. Hull, Phys. Rev. 13, 307 (1901).

19. P. Lcbedev,y4nn. Phs. (Leipzig) 6, 433 (1901); ibid. 32, 411 (1910).

20. O. Frisch, Z Phys. 86, 42 (1933).

21. A. Ashkin, Science 210, 1081 (1980); V. S. Lctokhov and V. G. Minogin, Phys. Rep. 73, 1 (1981); A. P. Kazantsev, G. A. Rvabenko, G. I. Surdutovich, V.'P.

Yakovlcv, ibid. 129, 75 (1985),

22. A. P. Kazantsev, V. S. Smimov, G. I. Surdutoxich, D. O. Chudesnikov, V. P. Yakovlev,/. Opt. Soc. Am. B 2, 1731 (1985); J. Dalibard and C. Cohcn-Tannoudji,

ibid., p. 1707,

23. A. Aspect, J. Dalibard, A. Heidmann, C. Salomon, C. Cohen-Tannoudji, Phys. Rev.

Lett. 57, 1688 (1986).

24. H. Dehmelt, in Advances in Laser Spectroscopy, P. T. Arccchi, F. Strumia, H,

Walther, Eds., vol. 95 of NATO Advanced Science Institutes Series B (Plenum, New York, 1983), pp. 153-187.

25. D. J. Wineland, W. M. Itano, J. C. Bergquist, R. G. Hulet, Phys. Rev. A, in press. 26. W. Neuhauser, M. Hohcnstatt, P. Toschek, H. Dehmelt, Pirn. Rev. Lett. 41, 233

(1978).

27. M. Lindbcrg, /. Phys. B 17, 2129 (1984); J. Javanainen, M. Lindberg, S. Stenholm,/. Opt. Soc. Am. fi 1, 111 (1984).

28. D. J. Wineland, R. E. Drullingcr, F. L. Walls, Phys. Rev. Lett. 40, 1639 (1978).

29. D. J. Larson, J. C. Bergquist, J. J. Bollinger, W. M. Itano, D. J. Wineland, ibid. 57, 70(1986).

30. D. J. Wineland, W. M. Itano, J. C. Bergquist, J. J. Bollinger, J. D. Prestage, in Atomic Physics Nine: Proceedings of the Ninth International Conference on Atomic

Physics, Seattle, Washington, July 1984, R. S. Van D\'ck, Jr., and E. N. Fortson, Eds. (World Scientific, Singapore, 1984), p. 3. 31. G. Werth, in ibid., p. 28.

32. R. H. Dickc, Phys. Rei'. 89, 472 (1953).

33. H. G. Dehmelt,7£££ Trans. Instrum. Meas. IM-31, 83 (1982).

34. D. J. Wineland, Scuna 226, 395 (1984).

35. W. Neuhauser, M. Hohenstatt, P. E. Tocchek, H. Dehmelt, Phys. Rev. A 22, 1 137

(1980); W. Nagoumey and H, Dehmelt, personal communication,

36. H. G. Dehmelt, BuU. Am. Phys. Soc. 20, 60 (1975).

37. G. Janik, W. Nagoumev, H. Dehmelt, /. Opt Soc Am. B 2, 1251 (1985).

38. W. M. Itano and D, J, Wineland, Phys. Rev. A 24, 1364 (1981).

39. J. J, Bollinger, J. D. Prestage, W. M'. Itano, D. J. Wineland, Phys. Rev. Lett. 54, 1000 (1985).

40. R. J. Cook and H. J, Kimble, ibid., p. 1023. 41 J. Javanainen, Phys. Rev. A 33, 2121 (1986); A. Schenzle, R. G. DeVoe, R. G.

Brewer, ibid., p. 2127; C. Cohen-Tannoudji and J. Dalibard, Europlrrs. Lett. 1, 441

(1986); D. T. Pegg, R. Loudon, P. L. Knight, Pfm. Rn\ A 33, 4085 (1986); A.

Schenzle and R. G. Brewer, ibid. 34, 3127 (1986);'H. J. Kimble, R. J. Cook, A. L. WeUs, ibid., p. 3190; P. ZoUer, M. Marte, D. F. Walls, ibid. 35, 198 (1987).

42. L. R. Brewer, J. D. Prestage, J. J. Bollinger, D. J. Wineland, in Strongly Coupled Plasma Physics, F. J. Rogers and H. E. De Witt, Eds. (Plenum, New York, 1987).

43. D. J. Wineland and W. M. Itano, Pljys. Today 40, 34 (June 1987).

44. We grateftiUv acknowledge the support of the Air Force Office of Scientific Research and the Office of Naxal Research. This article is not subject to copvright

in the United States.

7 AUGUST 1987 TN-149 ARTICLES 617
Physica Scripta. Vol. T22, 79-84, 1988. Precise Optical Spectroscopy >A/ith Ion Traps'

Wayne M. Itano, J. C. Bergquist, Randall G. Hulet, and D. J. Wineland

Time and Frequency Division, National Bureau of Standards, Boulder, Colorado 80303, U.S.A.

Received September 24, 1987: accepted October 8, 1987

Abstract

We have used stored ion methods to improve resolution and sensitivity in optical spectroscopy. Single atomic ions have been confined by electric and

magnetic fields, cooled by laser radiation pressure to temperatures on the order of 1 mK, and probed sjjectroscopically with narrowband lasers. The

absorption resonance of a single Hg^ ion has been observed by a decrease

in the transmitted light intensity. An ultraviolet transition in Hg^ has been

observed with a linewidth of only 30 kHz. Quantum jumps to and from

metastable levels of Hg"* have been observed and used to determine radiative decay rates and to infer the existence of photon antibunching. Quantum

jumps have also been observed in single Mg* ions.

1. Introduction

Ion traps have only recently been used in obtaining high

resolution optical spectra, although their advantages have

been realized for a long time. Microwave spectra of trapped

ions, on the other hand, have been observed with high resol- ution since the 1960s [1, 2]. Elsewhere in these Proceedings,

G. Werth discusses some recent experiments in this area [3]. A few years ago, at the Ninth International Conference on Atomic Physics, the field of optical spectroscopy of trapped

ions was reviewed [4]. At that time, the highest resolution that had been achieved in the optical domain corresponded to a

linewidth of about 3 MHz at an effective wavelength of

2.1 |jm [5]. In a recent experiment to be discussed later in this Paper, an ultraviolet transition was observed with a fractional resolution about three orders of magnitude better than this

[6].Some general problems which tend to limit the resolution

with which optical spectra are observed are Doppler shifts, perturbations due to collisions or to electric and magnetic

fields, and the limited observation time. At least in a few

favorable cases, these problems can be greatly alleviated for ions stored in traps, so that we are left with the natural line- width. For a transition with a stable lower level, the natural

linewidth (in hertz) is the inverse of the mean radiative

lifetime (in seconds) of the upper level, divided by In. The Doppler shift is usually divided into a term which is linear in the atomic velocity, called the first-order Doppler

shift, and a part which is quadratic, called the second-order

Doppler shift. The high intensity and narrow frequency width

of laser light sources make it possible to apply various nonlinear spectroscopic techniques, such as saturated absorption

or multiphoton absorption, that cancel the effects of first- order Doppler shifts [7]. However, the second-order Doppler

shift, which is a result of relativistic time dilation, still remains. For typical laboratory conditions, this shift is only

about one part in 10'", but in some cases, it may seriously

* Contribution of the National Bureau of Standards. Not subject to copyright in the U.S.

limit the resolution or accuracy of the measurement. One

example is the work of Barger et al. on the 657 nm transition of calcium, which has a frequency of 4.57 x 10'"* Hz and a natural linewidth of 410 Hz [8]. Resonances with linewidths

as small as 2 kHz were observed, but they were severely

shifted and distorted by the second-order Doppler shift. If both the first- and second-order Doppler shifts are

to be reduced, the atoms must be cooled. An atomic vapor

placed in a conventional refrigeration device would quickly

condense on the walls of the container. But laser cooling, a method by which laser radiation pressure is used to reduce

the velocities of atoms, achieves the cooling without contact with material objects [9]. The technique was proposed

independently by Hansch and Schawlow for free atoms [10] and by Wineland and Dehmelt for trapped ions [11]. Laser coohng to temperatures on the order of 1 mK has been

demonstrated with some kinds of atoms and atomic ions. This reduces the second-order Doppler shifts to the level where they are presently not a problem. An atom or atomic

ion of mass 100 u (unified atomic mass units) cooled to 1 mK

has a second-order Doppler shift of 1.4 parts in 10'^

The observed resonance linewidth cannot be much less than the inverse of the observation time. In the case of the work on calcium, the observation time was limited to about

0.3 ms by the time it took the atoms to pass through the 21 cm

long resonance region [8]. In some experiments, the observation times have been increased by confining the atoms with

specially coated walls or by buffer gases, but collisions with

the wall or buffer gas molecules shifted and broadened the observed resonances. Ions can be trapped for long periods by

electric and magnetic fields. External field perturbations of

carefully chosen transition frequencies can be less than a part

in lO'** [12, 13]. Collisions are almost entirely absent in the ultra-high-vacuum environment of an ion trap. Neutral

atoms have been trapped by static magnetic fields [14-16]

and by optical fields [17], but these trapped atoms have not

yet been studied spectroscopically. In these traps, the fields responsible for trapping would also have perturbed the resonances.

2. Sensitive detection of ions A single atomic ion can easily be observed by the laser- induced fluorescence of an allowed transition, once it has

been cooled so that the Doppler broadening of the transition

is small. In this case, a single ion may scatter as many as 10** photons per second. Even if only a small fraction of the photons, around 10 * in typical experimental cases, are

collected and counted, the fluorescence can easily be observed

[18].The apparatus used at the National Bureau of Standards

TN-150
80 Wayne M. llano et al.

11 jum

194 nm

excitation

Fig. I. The lowest energy levels of Hg^ and the transition wavelengths. The

decay rates from all three excited levels were determined by observing only

the 194nm fluorescence. (From Ref. [33]).

for experiments with Hg"^ has been described previously

[19, 20]. The Hg^ ions were confined in a radiofrequency

(Paul) trap under ultra-high-vacuum conditions. In one case, a single Hg^ ion has been kept in the trap for over a week.

The lowest energy levels of Hg"*^ are shown in Fig. 1 . The

Sd^^bs -5| 2 ground level is connected to the 5d'°6p 'P, j level by a strong electric dipole transition. The 5d^6s^ 'D^^ ^"d the 5^'6r 'D^i2 levels are metastable. In order to laser cool and

optically detect the ions, a few microwatts of c.w. 194nm

radiation were required. This radiation was generated by

sum-frequency mixing the output of a frequency doubled

514.5 nm Ar"^ laser with the output of a 792 nm dye laser

in a potassium pentaborate crystal [21]. The 194nm fluor- escence was collected by a lens system and detected by a photomultiplier tube. As many as 60000 photons per second were detected from a single ion.

2.1. Single-ion absorption detection

In a recent experiment, an optical transition in a single Hg^

ion was detected by observing the decrease in the intensity

2500

Single Ion Absorption and Fluorescence

-360 -180

Relative Laser Frequency (MHz)

Fig. 2. The 194 nm resonance of a single Hg^ ion due to absorption (upper

trace) and fluorescence (lower trace), detected at the same time. (From

Ref. (22]).

Phviun Krrinlii T~'

of the 1 94 nm beam transmitted through the trap [22]. Radiofrequency modulation techniques were used in order to avoid

low frequency noise, such as that due to amplitude fluctu- ations of the 194nm radiation. A maximum of about 10 of

the light was absorbed. Figure 2 shows the absorption signal

and the fluorescence signal, taken at the same time. The

modulation and phase-sensitive detection of the absorption

resulted in a signal which was approximately proportional to the frequency derivative of the fluorescence signal. Laser

heating caused the fluorescence signal to drop suddenly to zero when the frequency was tuned above resonance. The

integration time per point on the absorption curve was 50 s.

2.2. Electron shelving

Detecting a narrow (weakly allowed) transition by observing

fluorescence from the same transition would be very difficult. A double-resonance method originally proposed by Dehmelt

[23] makes it possible to detect such transitions with very high

efficiency. The method can be applied to Hg^ (see Fig. 1 ). The

narrow, weakly-allowed, 282 nm transition is detected as

follows: The ion is assumed to be initially in the ground level. Light at a wavelength near 282 nm is pulsed on, possibly

driving the ion to the ^Z);,, level. Light at a wavelength near 194nm is then pulsed on. If the ion had made a transition

in the previous step, no fluorescence would be observed;

otherwise an easily detectable fluorescence signal would be

observed. The method is called "electron shelving", since the

optically active electron is temporarily shelved in the upper

level of the weak transition.

3. Quantum jumps

The electron shelving techniques suggests a method of

observing quantum jumps. If light near both resonance wavelengths is present simultaneously, then the fluorescence

observed from the strong transition should turn off" and on abruptly as the atom makes transitions to and from

the metastable level. The dynamics of this process was

investigated theoretically by Cook and Kimble [24] and later by others [25]. Quantum jumps of single ions were observed

by this method in Ba"^ [26, 27], in Hg"" [20], and recently in Mg+ [28]. Quantum jumps have also been observed by us in Hg^

with only the 194nm radiation present. Once excited to the

-P, 2 level, the ion usually decays back to the ground level, but

it has a probability of about 10 ' to decay to the 'Z),; level. The probability is small because the transition frequency is low and because it requires configuration mixing to occur

(the nominal configurations differ in two orbitals). From the Z)v2 level, the atom decays either directly to the ground level with probability /, or to the ^/^sn level with probability

/, = 1 — /,. The 194nm fluorescence jumps between a steady level (the "on" state) when the ion is cycling between

the ground level and the '/'1/2 level and zero (the "off"" stale)

when it is in either the 'D^j level or the 'D.,. level. Typical fluorescence data are shown in Fig. 3 for one, two,

and three ions. The fluorescence intensity jumped from one

discrete level to another and was proportional to the number

of ions in the "on" state. For these data, only the 194nm

radiation was present. The intensity was high enough that quantum jumps due to the weak "P, , level to "D, 2 level decay

occurred several times per second. tn-1 SI
Precise Optical Spectroscopy with Ion Traps 81

150

100

§ 150 o

." 1 00

^

150

iZ 100

800

Time (ms)

Fig. 3. Fluorescence intensities as a function of time for three Hg* ions (top

trace), two Hg^ ions (middle trace), and one Hg^ ion (lowest trace). The

fluorescence takes a step down when an ion makes a transition to the metastable 'D,,; level and takes a step up when an ion returns to the ground

level from a metastable level. The integration time per point is 1 ms and the points are connected by straight lines. (From Ref. [33]).

3.1. Quantum jumps of two and three ions Sauter et al. have reported observing multiple quantum

jumps, that is, simultaneous quantum jumps of two or more

ions, when several Ba^ ions were stored in the same trap

[29]. They attributed this phenomenon to a cooperative

interaction between the atoms and the radiation field. We

have examined our data to test the statistical independence of

the quantum jumps of two or more simultaneously trapped

ions.One test was based on measuring the fractions of the time

that the ions spent in each of the possible fluorescence levels. The probability distribution of fluorescence intensities was

plotted for the data from one, two, and three ions. The

distribution for one ion is shown in Fig. 4. The counts at low

intensity (left side of plot) are due to stray scattered light when

the ion is in the "off' state; the peak at high intensity (right

side) is due to the fluorescence of the ion in the "on" state. The solid curve is a least-squares fit of the data to a sum

of a Poisson distribution (for the left peak) and a Gaussian

2.5

25 50

Counts/bin

Fig. 4. Fluorescence intensity distribution for one Hg* ion. The numbers of

1 ms time bins in which a given number of photons was detected are plotted

as dots. The solid curve is a least-squares fit to the data.

O

tnc

JO

I-(D XIE

50

Counts/bin

100

Fig. 5. Fluorescence intensity distribution for two Hg* ions (dots) and a least-squares fit (solid curve).

distribution (for the right peak). The standard deviation

(sigma) of the fitted Gaussian is about 8% larger than the square root of the mean, which is the expected value due to

statistical fluctuations. This is probably due to intensity

and frequency fluctuations of the 194nm source. The ratio of

the area under the Poisson distribution to that under the Gaussian distribution is a measurement of the ratio of the time that the ion was in the "off"" state to the time that it was

in the "on" state (0.71 for this data set). Plots of the two-ion

and three-ion intensity distributions and least-squares fits are shown in Figs. 5 and 6. The areas under the peaks of the

distributions are proportional to the amounts of time that the fluorescence was at each of the various discrete levels. The

curves fitted the data well, even though the peaks were not completely resolved. Let /7off and p„„ be the probabilities that a single ion is in the "off"" state or the "on" state, respectively. Consider

two ions subjected to the same 194nm intensity and acting

independently of each other. In this case, the probability

/Jo that both are in the "off"' state is equal to (Poir)"; the probability p, that one ion is in the "oflT' state and the other

in in the "on" state is equal to 2p„„p„f^ = 2p„^{\ — Po„): and

the probability p2 that both are in the "on" state is equal to [Ponf- Similar relationships hold for three ions. The least- squares fit to the two-ion data yielded the values po = 0.139.

/?i = 0.472, and p. = 0.389. If/?, is (arbitrarily) chosen to calculate the value of p^^, then the prediction, based on the assumption of independence, is that po — 0.145 and

O

enc

XIE

50 100 150

Counts/bin

Fig. 6. Fluorescence intensity distribution for three Hg* ions (dots) and a least-squares fit (solid curve). TN— 152

M dUi
82 Wayne M. Itano et al.

1000

en D

o

'i_CDQ.

0)E

13

100 r

10 r

100 200 300

Duration of off period (ms)

Fig. 7. Distribution of fluorescence-off periods for a single Hg"^ ion as a function of the duration of the periods (dots) and a least-squares fit. (From

Ref. [33]).

P2 = 0.383. For the three-ion data, let /?„(« = 0, 1, 2, 3) be

the probability that n ions are in the "on" state. The least- squares fit yielded the values p^ = 0.053, p\ = 0.244, p, =

0.485, and p^ = 0.218. If pj is used to calculate the value of

Pan, the prediction based on independence is that/jg = 0.061,

/?, — 0.286, and pj = 0.433. Thus, the two- and three-ion

data agreed fairly well with the assumption of independence

of the ions. The discrepancies from the predictions were most

likely due to effects of intensity and frequency variations of

the 194nm radiation source, which are not included in the simple model used to fit the data.

In another test of independence, a search through the two-ion data was made for double quantum jumps. Over 500

consecutive jumps, which took place in a period of 20 s, were

tabulated. A total of five apparent double quantum jumps

were found, which is approximately the number that would

be expected due to random coincidences within the finite resolving time of the instrumentation (1 ms).

3.2. Measurements of radiative decay rates The radiative decay rates of the 'D^p level and the ^D^^ level are reflected in the probability distribution W^^^ of the durations of the "off" periods for a single ion irradiated by

194 nm radiation. A calculation based on the rate equations

for the probabilities of being in the various levels yields

W^ofT(T) oc [/jyjexp (-y.T) -I- (/.y, - yj) exp (-y,T)]. (1) Here y, and yj are the total radiative decay rales of the ^D^n

level and the ^D^^j level respectively.

The experimental fluorescence-off distribution was least- squares fitted to eq. (1) to obtain values for y,, jj, and/,.

Figure 7 shows the data and the least-squares fit. The values

obtained from the fit are y, = 109 ± 5s"', y2 = 11.6 ±

0.4s"', and /, = 0.491 ± 0.015. These values are in fair agreement with calculations [30, 31]. The value of yj is in good

agreement with previous measurements [19, 20, 32], but the value of y, is about a factor of 2 higher than the only

previously reported value [32]. No previous measurements of

/i exist. The measurements will be described in more detail elsewhere [33].

1.25

1.00

p

0.75

en 0.50

0.25 -.•

0.00

• • •• • • • •• %.*•">'

20 40 60

T(ms)

80 100

Fig. 8. Intensity correlation function of the 1 1 urn radiation field generated

by the radiative decay from the 'P, 2 level to the ^Z),,; level in a single Hg*

ion, showing the photon antibunching. The data from several runs were

averaged.

3.3. Photon antibunching

The two-time intensity correlation function of a light field is defined by C(t) = (.I(t)I{t + t)>, where /is the intensity. A

field is said to exhibit photon antibunching if C(0) < C(x).

No classical field can have this property, so a quantum theory

is required to explain it. The quantum jumps of a single Hg*

ion from the "on" state to the "off" state, as in the lowest

trace of Fig. 3, are assumed to mark the emission of an 11 \xm

photon (see Fig. 1), even though these photons have not been observed directly. The probability density ^(t) that the emission of an 11 \xm photon is followed a time t later by the emission of another one is proportional to C(t) for the 1 1 pm

field. We expect ^(t) to go to zero for small t, since the ion has

to decay to the ground level and be excited to the "P, , level again before it can emit another 1 1 um photon. The function

g{-z) (normalized to 1 as t -^> 00) was calculated for several one-ion runs and averaged to obtain the graph shown in Fig. 8, which clearly shows photon antibunching. A similar

graph, showing photon antibunching in the 282 nm radiation

emitted when light at both 194nm and 282 nm was applied,

has been shown previously [20].

3.4. Quantum jumps in Mg'^

Quantum jumps of a single ^""Mg"^ ion in a Penning trap were

investigated recently [28]. A single 280 nm radiation source was present and was tuned close to the njj = — 1 12 to mj = — 3/2 Zeeman component of the transition from the 3,9 ^5|;2 ground level to the 3/7 ^P-^i level. The fluorescence intensity dropped to zero when the ion was driven to the mj =

-I- 1/2 sublevel of the ground level by a spontaneous Raman

transition, and returned to its previous level when another

spontaneous Raman transition drove the ion back to the mj = — 1/2 sublevel. A trace of the fluorescence intensity as a function of time is shown in Fig. 9. The data are in agreement with the theoretical prediction that the ratio of the time when the fluorescence is on to the time when

it is off is very close to 16 [34]. This ratio is nearly independent of the 280 nm intensity because of the existence

of coherences between excited levels. The experimental

results provide a high precision (2%) verification of an effec- tive two-state rate equation description of the dynamics o'( the quantum jumps, TN-153

ximm^uMMJii
100

Q) to <j E o 6 CO \ <U to 50 Irt*^^

80 1 60 240 Time (ms)

320

Fig. 9. Fluorescence intensity of a single Mg* ion as a function of time, showing the sudden changes due to quantum jumps to and from the groundstate Zeeman level which is not in resonance with the incident radiation. (Adapted from Ref. [28]).

4. Suppression of Doppler broadening by confinement

In 1953 Dicke showed that the confinement of an atom to a region smaller than the resonance-radiation wavelength leads

to a suppression of the first-order Doppler shift broadening

[35]. This condition is the basis for the observation of narrow Mossbauer transitions of the nuclei in solids. It is also easy to

satisfy this condition, commonly called the Lamb-Dicke

criterion, for a microwave resonance, since the wavelengths can be several centimeters long. Microwave frequency stan- dards such as the atomic hydrogen maser make use of this

fact. In the optical domain, it can be satisfied for a single,

laser-cooled ion confined in a trap. The residual first-order Doppler broadening takes the form of discrete resonances,

displaced from the unshifted central resonance (the carrier),

by multiples and combinations of the frequencies of motion

of the ion. These extra resonances are called motional

sidebands.

4.1. High resolution spectra of Hg^

We have used the shelved-electron method to detect the weak

^5|/2 to 'Djo transition of a single Hg^ ion [6]. If the fluor- escence was high enough to indicate that the ion was cycling

between the ^S|,2 and the 'P,,! levels, the 194nm radiation was

turned ofTand the 282 nm radiation was turned on for 20 ms,

1

\—\—I—I\—I—

r

T-r

Y^ MftV mv*i K^^\^

I I I I

I I I

1 I L

2 4 Frequency Detuning (in MHz)

Fig. 10. The ^S,,, to ^0^,2 resonance due to absorption of 282nm radiation from a frequency-doubled dye laser by a single Hg*^ ion. The shelved electron method was used to detect this weakly allowed transition. The weaker

absorption lines (motional sidebands) are separated from the central reson- ance by multiples of the frequencies of harmonic motion of the trapped ion and arc due to the Doppler shift of the 282 nm radiation in the frame of the

ion. (From Ref [6]).

Precise Optical Spectroscopy with Ion Traps 83 |

1.00

0.95 -

COc

in

0)o

c

a>o

CO£ 0.90-

T3

a>N

c

CD3

o

0.85

0.80 1340 1400 1460 1520 1580

Frequency Detuning (in kHz)

Fig. 11. High resolution frequency scan of the ^5,2 to ^D^2 resonance, showing the central resonance and the first upper motional sideband. From

Ref. [6]).

which is much less than the ^Dj, level lifetime. The 194nm

radiation was turned on again, and the fluorescence photons were counted for 10 ms to see whether the ion had made

a transition to the ^D<,,2 level. The result of the measure- ment was a or a 1, depending on whether the number of

photons detected was below or above a threshold level. This

eliminated certain kinds of instrumental noise, such as intensity fluctuations of the detection laser, leaving only the inherent quantum fluctuations of the atom. Figure 10 shows

a spectrum taken in the manner just described, displaying the

carrier and two motional sidebands on each side. Figure 1

1

shows a high resolution scan of the carrier and the first upper

sideband. The resonances are approximately 30 kHz wide.

The ratio of intensities of the carrier and sideband indicates that the ion was cooled to near the theoretical minimum of

1.7mK. The limit of 30 kHz resolution was due to laser frequency fluctuations. If a suitably stabilized laser were

used, the resolution would approach the natural linewidth

of about 2 Hz. The sensitivity of the transition frequency

to fluctuations in the magnetic field could essentially be

eliminated by using a suitable Zeeman component in either "'Hg* or ^°'Hg+ [36].

Acknowledgements

We gratefully acknowledge the support of the Air Force Office of Scientific Research and the Office of Naval Research.

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84 Wayne M. Itano et al.

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21. Hemmati, H., Bergquist, J. C. and Itano, W. M., Opt. Lett. 8, 73

(1983).

22. Wineland, D. J., Itano, W. M. and Bergquist, J. C, Opt. Lett. 12, 389

(1987).

23. Dehmelt, H. G., Bull. Am. Phys. Soc. 20, 60 (1975).

24. Cook, R. J. and Kimble, H J., Phys. Rev. Utt. 54, 1023 (1985).

25. Javanainen, J., Phys. Rev. A33, 2121 (1986); Schenzie, A., DeVoe,

R. G. and Brewer, R. G., Phys. Rev. A33, 2127 (1986); CohenTannoudji, C. and Dalibard, J., Europhys. Lett. 1, 441 (1986); Pegg, D. T„ Loudon, R. and Knight, P. L., Phys. Rev. A33, 4085 (1986); Schenzie, A. and Brewer, R. G., Phys. Rev. A34, 3127 (1986); Kimble, H. J., Cook, R. J., and Wells, A. L., Phys, Rev. A34, 3190 (1986);

Zoller, P., Marte, M. and Walls, D. F., Phys. Rev. A35, 198 (1987); Nienhuis. G., Phys. Rev. A35, 4639 (1987); Porrati, M. and Pulterman.

S., Phys. Rev. A36, 929 (1987).

26. Nagourney, W., Sandberg, J. and Dehmelt, H., Phys. Rev, Lett. 56, 2797 (1986).

27. Sauter, Th., Neuhauser, W., Blatt, R. and Toschek, P. £., Phys. Rev.

Lett. 57, 1696(1986).

28. Hulet, R. G., Wineland, D. J., Bergquist, J. C, and Itano, W. M..

(submitted for publication)

29. Sauter, Th,, Neuhauser, W„ Blatt, R, and Toschek, P. E.. Opt. Commun. 60, 287 (1986).

30. Garstang, R. H., J, Res. Natl, Bur. Stand. Sect. A68. 61 (1964),

31. Al-Salameh, D. and Silfvast, W., AT&T Bell Laboratories, Holmdel,

NJ, unpublished calculations.

32. Johnson, C. E., Bull. Am. Phys. Soc. 31, 957 (1986),

33. Itano, W. M., Bergquist, J. C, Hulet, R. G. and Wineland, D. J., Phys. Rev. Lett, (in press).

34. Hulet, R. G. and Wineland, D. J., Phys. Rev, A 36, 2758 (1987).

35. Dicke, R. H., Phys. Rev. 89, 472 (1953),

36. Wineland, D. J„ Itano, W. M., Bergquist, J, C, and Walls, F. L., in Proc. 35th Annual Symposium on Frequency Control, p, 602. Electronic Industries Association, Washington, D,C., 1981.

TN-155

Phvsic^Scripi^^2
FROM: PHYSICS TODAY; JUNE 1987

Laser cooling

The mechanical forces exerted by light can dramatically lower the temperature of

a sample of atoms or ions, allowing very-high-resolution spectroscopic measurements

and ultralow-temperature atomic physics experiments.

David J. Wineland and Wayne M. Itano

In the photograph on the opposite page we see a single mercury ion held nearly

at rest in an electromagnetic "trap."

Physicists have seen individual atoms

before, in arrays imaged by field ion

microscopes and more recently by vacu- um tunneling microscopes, but what we see here is different. It graphically

demonstrates a physicist's ideal: hold- ing a single isolated atom nearly at rest

for careful examination.

A technique now commonly called

laser cooling made the photograph

possible by reducing the mercury ion's kinetic energy. This cooling not only

limited the ion's movement, but also sharpened its spectral features by re- ducing Doppler broadening, enhancing

the scattering of laser light tuned to one of its transitions. Several laboratories now use lasers to cool ions and

neutral atoms to kinetic energies corre- sponding to temperatures near a milli- kelvin. (See physics today, September

1986, page 17.) As originally proposed in 1975 by

Theodor Hansch and Arthur Schawlow

at Stanford University and independently by Wineland and Hans Dehmelt

David Wineland and Wayne Itano are physicists in tlie Time and Frequency Division of tfie National Bureau of Standards, in Boulder, Colorado.

at the University of Washington, laser cooling can substantially reduce

Doppler effects in high-resolution spec- troscopy.' The technique should eventually reduce inaccuracies in spectroscopy to 1 part in 10'** or better for single

trapped ions.^"^ More accurate atomic

clocks are an obvious prospect. Other

applications include tests of gravitational interactions. Laser-cooled neutral atomic beams may finally allow

realization of Jerrold Zacharias's 1953

proposal for an "atomic fountain" ex- periment. Here one would achieve

long observation times by directing a slowed atomic beam upward and let- ting gravity return it to near its original position. Even with moderately

slowed atomic beams, the velocity compression achieved from laser cooling

will greatly reduce the uncertainty in the second-order, or time dilation,

Doppler shift. Laser cooling has a potential role in many other experiments. It may give us ways to: Study collisions between very cold atoms or ions. Such studies should give

detailed information on interactions,

with very high energy resolution. Study atom-surface collisions at low

temperature. It may be that at suffi- ciently low temperature, atoms will "bounce" with minimal perturbation to

their structure. This would allow the

construction of nearly ideal boxes for storing atoms for use in spectroscopy

and other experiments.

Focus atomic beams. Laser beams

directed transversely to an atomic

beam act as lenses. These lenses are

dissipative in the direction normal to the atomic beam, so beam focusing is not limited by Liouville's theorem. Laser cooling may also be used to advantage in ion storage rings.

Manipulate antihydrogen, which, if produced, must be used efficiently.

Obtain unique states of condensed

matter. Observing liquid and solid plasmas is a possibility, as we discuss

below.

Observe Bose condensation of hydrogen or other atoms. This phenomenon

may eventually be observed by cooling

atoms held in a suitable "trap." In many cases laser cooling may be

the only practical way to control the

velocity distribution of a sample of ions or neutral atoms. We begin this article with an explanation of how light imparts mechanical

forces on atoms. Because these forces are the same on both ions and neutral

atoms, we will take the term "atom" to include both unless we specifically state otherwise. We then discuss how

one can use these forces to reduce the

TN-156
Single ion of mercury. The white dot at the center of this false-color image is a

laser-cooled Hg ' ion in an electromagnetic

trap. This image was made with a photoncounting imaging tube sensitive to the 194- nm fluorescence light scattered from the

ion and trap electrodes. The white overlay

shows the position of the electrodes of the

ion trap. The circular ring electrode has an

inner diameter of about 0.9 mm. To

photograph such an ion requires a minimum exposure time of about 50 ;j.sec. The effective temperature of the ion was

approximately 2 mK. The computerized

imaging system used here was developed

by Charles Manney and John Bollinger at the National Bureau of Standards. The first photographic images of single ions were

reported in reference 15.

kinetic energy of a sample. Finally, we

discuss what can be achieved in the

laboratory.

Early history. The study of the mechanical forces that light exerts on

matter has a long history. In 1873

James Clerk Maxwell used the theory

of electromagnetism to calculate the

force on a solid body due to the absorption or reflection of a beam of light. In the early 1900s quantitative measure- ments of the force exerted by light on

solid bodies and gases verified Maxwell's radiation pressure calculations."* In 1917 Albert Einstein used quantum

theory to calculate the influence of the

electromagnetic radiation field on the

motion of molecules.^ He showed that

the light pressure causes molecules to come into thermal equilibrium with a

radiation field if that field has the

Planck spectrum. Aside from showing

the consistency of quantum theory and

statistical mechanics, this calculation was important in establishing the

quantum nature of light, because it was

necessary to assume that the molecule

emits radiation as a discrete bundle

with a definite energy and momentum,

and not as a spherical wave.

In 1933 Otto Frisch did the first experiment to show directly the mo- mentum transferred to an atom by the

absorption of a photon.^ In this experiment, light from a sodium resonance lamp deflected a beam of sodium atoms. When tunable lasers became available

in the 1970s, experiments of this sort were repeated. Due to the much higher

spectral intensities available, an atom

could be made to absorb many photons one at a time, but at a high rate, resulting in larger deflections. Around

this time physicists made proposals to use intense, resonant optical fields to manipulate atoms in various ways,

such as accelerating them or trapping

them in optical potential wells. Among

those who first recognized the possible

applications of resonant laser radiation pressure on atoms were Arthur Ashkin

in the United States and several scientists in the Soviet Union.^ In his 1950

paper on optical pumping, Alfred

Kastler suggested some ways of using

light to cool or heat atoms.® These

ideas are related to laser cooling but

are difficult to realize in practice.

Optical forces on atoms

The force that light exerts on atoms

is often conceptually divided into two

parts.^ These are called the light pres- sure or scattering force and the gradi- ent or dipole force. At least in some

simple cases, one can distinguish these

forces clearly. In the general case, and

particularly for intense fields, the simple descriptions of the forces and their

fluctuations break down, and a more

fully quantum mechanical description

is required.^' '° We can understand the scattering

force as the momentum transferred to the atom as it scatters a photon. The

average scattering force is in the direc- tion of propagation of the light and is equal to the product of the momentum

fik per photon and the photon scatter- ing rate. The photon wavevector k has

magnitude 2Tr/A, where A is the wavelength of the light. The average force reaches a maximum when the light is resonant with an atomic transition. The scattering force fluctuates because

the photons scatter at random times

and because the direction of the re- emitted photon, and hence the direc- tion of the recoil momentum due to this re-emission, is also random.

To see how large the scattering force can be, let us consider a specific example. Assume that the light from a

single laser beam is resonant with the

lowest-frequency atomic transition

and that the light is intense enough to saturate the transition, that is, that

the rate for stimulated emission ex- ceeds the spontaneous decay rate. When saturated, the atom spends

about half of its time in the excited

state. Therefore the average force on

TN-157
Incident photons

Laser cooling mechanism for free or weakly bound atoms or ions. "Weakly bound" means that the oscillation period of the trapped atom

or ion is longer than the lifetime of the upper state of the cooling

transition. The frequency co^^ of incident photons is assumed to be

less than the rest frequency coq of the atomic transition. At a velocity

where the atom's transition frequency wq 's equal to (1 — kv)co^^ , the atom resonantly scatters photons. When the photon wavevector k is antiparallel to the atom's velocity v, the average reduction in the atom's velocity is Hk/m per scattering event, where m is the atom's mass.

one atom is equal to the product of the

photon momentum Hk, the rate y of spontaneous decay from the excited

state, and the probabihty of being in the excited state, or fikY/2. Note that

absorption followed by stimulated

emission imparts no net momentum to the atom because the photon resulting

from stimulated emission also has mo- mentum fik. If the wavelength is 500 nm and the

decay rate y is 10®/sec, then the scatter- ing force is about 40 000 times the force of gravity for an atom of mass 100 u, or atomic mass units. This force is the same as the electric force on a singly

charged ion in a field of about 5x 10~^

V/cm. Therefore, the scattering force can be much greater than the gravitational force, but is weak compared with

typical electric forces. In practice,

about lO"* scattering events are re- quired to change atomic velocities by as much as room temperature thermal

velocities. The basic idea of laser

cooling, then, is to have the atom

preferentially scatter photons when its momentum and the photon momentum

are antiparallel.

The origin of the gradient or dipole

force is somewhat less obvious, al- though certain of its properties do have

a classical interpretation. The atom

can be thought of as a polarizable body.

The optical electric field induces an

electric dipole moment in the atom and

then acts on that dipole moment. If the

optical intensity is spatially inhomogeneous, the dipole interaction causes a

force along the gradient of the intensity. Like the scattering force, the dipole

force has a strongly resonant character, but unlike the scattering force, it is dispersive in nature. Its sign is such

that it attracts an atom to a region of high light intensity if the frequency of the light is below the atomic resonance and repels an atom if it is above. That

is, the electric polarizability is positive

below resonance and negative above

resonance. To this extent, the atom is like a charged harmonic oscillator—an

electron bound to a positive core by a

spring, in an oscillating electric field. The charge oscillates in phase with the

external force if the frequency is below

resonance, and 180° out of phase if the

frequency is above. If the electric field

is spatially inhomogeneous, then aver- aged over one cycle of the optical

radiation a net force on the atom can

result. Some proposals for trapping

atoms with optical fields are based on

the use of the dipole force. Laser cooling of free atoms

Laser cooling based on the use of the

scattering force can be explained as

follows: Consider an atom with a

strongly allowed resonance transition. For simplicity, let this be the lowest- frequency transition, so that if this transition is excited, the atom must

decay to the ground state. A laser beam whose frequency is close to but

lower than the atomic resonance fre- quency irradiates the atom. If the

atom is moving against the laser beam,

then the frequency of the light in the

rest frame of the atom is Doppler

shifted toward resonance. Hence, the

scattering force is higher for an atom

moving against the laser beam, and the atom's velocity is damped. In this way,

the velocity of an atomic beam can be

reduced substantially.

If only one laser beam is used and the

atom is otherwise free, the atom will eventually turn around and move away

parallel to the beam. Therefore, for a

gas of atoms, if another laser beam of equal intensity and frequency but opposite direction is introduced, atoms with

their velocities in the other direction

also have their speeds reduced. The

intensities must be equal for the aver- age scattering force on a motionless

atom to be zero; if the intensities are not equal, the atoms will have a net

drift velocity. One can obtain cooling

in all directions by using three orthogonal pairs of counterpropagating laser beams. However, because of the inher- ent fluctuations of the scattering force, the velocity is not damped to zero. The theoretical minimum temperature that can be obtained is given by a balance between the dissipation and

fluctuations.^'^ Assume the laser in- tensity is below saturation, even when

tuned to resonance. We also assume

that the recoil energy R, given by {Hkf/

2m, is less than fiy. The recoil energy is the kinetic energy that an initially

motionless atom of mass m would have

due to its recoil after emitting a photon

of wavevector k. The minimum temperature T'min is achieved when the

laser frequency is tuned below the

atomic resonance frequency by an amount equal to y/2, in which case^"

kT^,„ =\fiy (1) For a decay rate y of 10^/sec, the minimum temperature is about 0.38 mK.

TN-I58

^ 1 •-.•.1 iM
DCoCOm<

-A*

coq — 2(U„ (Oq — (lis, (^0

Frequency of laser ••q^„^q^"

for cooling -A coq + 2(a„

LASER FREQUENCY a^

Absorption as a function of laser frequency. In the sideband cooling

limit, the oscillation frequency w^ of the atom in the trap is assumed to be larger than the radiative decay linewidth y. Therefore the absorption spectrum consists of a "carrier," or recoilless line, at the atom's rest frequency mq plus Doppler-effect sidebands separated by

the frequency lo^. If the laser is tuned to coq - a>^, the atom absorbs

photons of energy ^(wq - w^) and re-emits photons of average energy

fkoo - R, where the recoil energy R is (fiV.f/2m. When R is much less than fuo^ there is an energy deficit of approximately iuo^ per scattering event, causing a decrease in the kinetic energy of the atom.

The laser cooling effects just de- scribed rely only on the scattering

force. There is another cooling effect, called "stimulated cooling," which is due to the dipole force in a strong

standing wave.'^ It is effective only for low atomic velocities and requires that

the laser frequency be higher than the

transition frequency. In a high-intensity, nearly resonant field, the simplest

description of the atom-field system is in terms of dressed-atom states, which

are coherent superpositions of the

ground state atom with n -F 1 photons

in the field and the excited state atom

with n photons. The dressed-atom

states depend on the light intensity,

and hence on spatial position. The

various dressed-atom states are either attracted to or repelled from regions of high light intensity. The rate of spon- taneous decay, which leads to transi- tions between different dressed-atom

states, is proportional to the admixture

of excited atomic states in the dressed- atom state. It turns out that for positive laser detuning, the time-averaged dipole force opposes the velocity of the atom. It should be possible to decelerate or accelerate beams of atoms

by changing the relative frequencies of two counterpropagating laser beams to create a moving standing wave. Atoms

with velocities close to that of the

standing wave could be swept along

with the standing wave. An advantage

of the dipole force is that it continues to increase as the intensity increases,

while the scattering force approaches a

limiting value when the scattering rate

is about y/2. However, the minimum

achievable temperature using stimulated cooling is no lower than that

given by equation 1. Laser cooling of trapped atoms

For trapped atoms, the theoretical

limit to cooling is the same as for free atoms when the natural linewidth y of the atomic transition used for cooling is much greater than the motional fre- quency co^. of the atom in the trap. The

reason for this is that the atom is essentially free during the time re- quired to scatter a photon. This is the

usual experimental case for a strongly

allowed transition. For trapped atoms

it is not necessary to have opposed

beams with equal intensities, because

the average light pressure just dis- places the equilibrium position of the

atom slightly. It is possible to cool with

a single laser beam as long as none of the normal modes of oscillation in the

trap are perpendicular to the direction

of propagation of the beam.

In the opposite limit, where the

natural linewidth of the transition is much less than the motional frequencies, the cooling limit is different.""

The absorption spectrum of the atom

consists of an unshifted resonance line at frequency Wq, called the carrier, and

a series of discrete lines on both sides of the carrier, each having the natural

linewidth and separated by multiples

and combinations of the motional fre- quencies. These extra lines, called motional sidebands, are due to the

periodic frequency modulation of the

light owing to the Doppler effect as observed by the moving atom.

To cool a trapped atom, one tunes a narrow-band laser to a sideband on the

low-frequency side of the unshifted

resonance, for example, to a frequency

o}Q—pco,,, where/) is an integer. The

atom makes transitions to the upper

electronic state, decreasing its vibra- tional energy, by absorbing photons of energy fiojQ— pa^,). When the atom

makes a transition back to its ground

electronic state, it may, in general,

either increase or decrease its vibrational energy, but the average change

in the vibrational energy is equal to the

recoil energy R. When R is less than

pfe,, , cooling occurs. Consider a particular case. A single

ion is trapped in a nearly isotropic

three-dimensional harmonic potential

well, a situation that is approximated'^

by an ion in an rf trap, or Paul trap,

with normal-mode vibrational frequencies all approximately equal to w, . A

quantum state of the atom in the well is identified by its internal quantum

numbers and the set of harmonic oscil- lator quantum numbers | n^ ,«, ,n, | cor- responding to the well. Three laser beams propagate along the x, y and z axes, and each is tuned to the corre- sponding first lower sideband. We

assume that the recoil energy R is much less than the energy fe,., a

condition that is not hard to satisfy in practice. In the steady state, the mean

values of the quantum numbers are"

<1 (2) There are two reasons that the mean

values do not go to zero with laser

cooling. First, the average change R in the atom's vibrational energy after it

TN-159
stopping sodium atoms with light. A beam of sodium atoms approaching the camera from

the upper right at speeds exceeding 1000 m/sec is brought to a virtual standstill by a laser beam aimed in the opposite direction. In this experiment at NBS, Gaithersburg, the atoms

travel down the axis of a solenoid (whose opening appears as the dark circle at the center of the photograph) and spread out as they come to a virtual halt near the opening. ^^

emits a photon is positive. Second,

there is some probability of driving a

transition that leads to an increase in vibrational energy on absorption rath- er than a decrease, because the neighboring sidebands, although far from

resonance, still have finite intensities. The simple theory just outlined applies

when the cooling transition is not

saturated. Markus Lindberg at the

University of Frankfurt and Juha Ja- vanainen and Stig Stenholm at the

University of Helsinki have calculated

the steady state for an arbitrary ratio of natural linewidth to motional fre- quency and also for arbitrary laser

intensity.^'' As in the case of free atoms, the lowest temperatures are achieved in the limit of low intensity, so the simple theory is usually ade- quate.2"

Experiments on trapped ions

It is perhaps not surprising that the

first laser cooling experiments were on

trapped ions, because ions can be held

for long periods in high vacuum with a

fairly high degree of thermal isola- tion.'^ Therefore, unlike in the case of neutral atoms, a long time is available

to do the cooling.

The first laser cooling experiments

were done in 1978. At the National

Bureau of Standards in Boulder, Robert Drullinger, Fred Walls and Wineland demonstrated cooling by a direct observation of ion temperature, which

they determined from the currents that

ion motion induces in trap electrodes. They observed the cooling of Mg^ ions

to 40 K in a Penning trap.'^'^^ At the same time in Heidelberg, Werner Neuhauser, Martin Hohenstatt, Peter Tos- chek and Dehmelt demonstrated cool- ing of Ba"^ ions in a Paul trap by

observing the ions' increased storage

time in the trap.'^ In subsequent experiments at these

and other laboratories, physicists have

measured temperatures on the order of 10 mK or less. They have typically

determined the temperatures by measuring the contribution of Doppler

broadening to spectral lines. For

strongly allowed electric dipole transitions, this method is not very sensitive

at low temperatures because Doppler

broadening contributes only a small

fraction of the overall linewidth. For

positive magnesium-24 ions at 1 mK,

for example, the Doppler broadening of the first resonance line (at 280 nm) is only 0.5 MHz, while the natural radiative linewidth j'/27r is 43 MHz. One

realizes a more sensitive measurement

of temperature by probing a transition

with a linewidth y that is much less than the motional oscillation frequency

CO,, of an ion in the trap.^'^ Here one can

use the strength of the motional side- bands to determine the ion temperature.'^ So far, only cooling in the limit Y>oj^ (equation 1) has been demonstrated for ions, but if cooling in the

limit a; ^, ^yis used, it should be possible

to reduce the kinetic energy to nearly

the zero-point energy (equation 2). Laser cooling of trapped ions is now done

in laboratories at NBS in Boulder,

Hamburg University, the University of Washington in Seattle, the University

of Paris in Orsay, the National Physical Laboratory at Teddington in England and the Max Planck Institute at Garching in West Germany. Other

laboratories are setting up similar ex- periments.

Single Ions. Single, laser-cooled,

trapped ions are interesting because

they provide a simple system for study.

First, the oscillation frequencies of an

ion in a trap are nearly harmonic,

which makes spectra of single ions

particularly simple. In contrast, if two

or more ions are stored together in a

trap, the oscillation frequencies of the

individual ions are dominated by ionion interactions at low temperature.

Second, the lowest possible kinetic energies, given by equations 1 and 2, are obtained for single ions. This is be- cause part of the motion in the Paul

trap (the rf-driven micromotion) or in the Penning trap (the magnetron or rotation motion) is nonthermal and not

affected in the same way by laser

cooling.'^ The kinetic energy in these

nonthermal motions can be minimized

for single ions. Finally, the simplicity of single trapped ions allows a straightforward comparison of laser cooling

theory and experiment.

Single ions are also interesting from

the standpoint of spectroscopy because

the perturbations in spectral measure- ments can be extremely small or well controlled.^" Historically, the most

difficult problem in high-accuracy ion

trap spectroscopy has been minimizing

the second-order Doppler frequency

shift, which is due to relativistic time

dilation. ^"'^ Reducing this systematic effect requires low temperatures.

Experimenters have achieved temperatures of about 10 mK or less with single Ba"^ ions at Hamburg University, with

single Mg"^ and Ba"^ ions at the University of Washington, with single Mg"^

and Hg"^ ions at NBS in Boulder and

with single Mg"^ ions at Garching. At a temperature of 10 mK, an ion with mass 100 u has a fractional secondorder Doppler shift of —1.4x10"''^,

which is almost negligible. Single laTN-160
ser-cooled ions have also allowed detailed studies to be made of the interaction between atoms and radiation, as

seen, for example, in quantum "jumps"

and in photon "antibunching," an ef- fect in which the distribution of arrival

times of fluorescence photons at a

detector is nonclassical because atoms

can emit only one photon at a time.

Liquid and solid plasmas. Large

numbers, or "clouds," of trapped ions are more properly referred to as nonneutral ion plasmas." These plasmas

are interesting because, unlike fusion

plasmas, they can reach a global ther- mal equilibrium. When these plasmas

can be laser cooled, the densities are high enough—above 10^/cm^—and the

temperatures are low enough—less than 10 mK—that the plasmas become

strongly coupled and should show liq- uid and solid behavior." It should be

possible to obtain non-neutral plasmas

whose dynamics are dominated by

quantum effects—for example, a posi- tron plasma "sympathetically" cooled

with laser-cooled ions of beryllium-9, as described below.

Even though single ions give the

lowest temperatures, larger samples of laser-cooled trapped ions have already

yielded interesting spectroscopic results. Spectroscopy" in the rf and

microwave regions has featured

linewidths of 0.01 Hz and fractional

inaccuracies as small as 1 part in 10^^. Sympathetic laser cooling. Unfortunately, direct laser cooling of ions and

neutral atoms is relatively easy only

for a very few elements. Sodium is often the practical choice for neutral

atom researchers; singly ionized magnesium, which is isoelectronic with

sodium, is a favorite choice for ion

trappers. So far both groups have

avoided working with molecules. The

reason is simply that if one drives a

suitably allowed transition at a convenient wavelength in a molecule, the

molecule quickly becomes optically

pumped into a state of different vibrational or rotational quantum number

and cooling stops.

One can extend laser cooling to other

species of ions by storing two ion

species in the same trap. The species

that is easy to laser cool will cool the

second species through Coulomb colli- sions, making the second species avail- able for high-resolution spectroscopic

investigation or other experiments.

Under typical conditions, ion-ion ther- malization by Coulomb coupling takes

place in less than a second. As a demonstration of this technique, Hg^

ions have been cooled to less than 1 K

by laser-cooled ions of beryllium-9 in a Penning trap." It should be possible to extend sympathetic cooling in order to cool neutral atoms or molecules by ion-

'Optical molasses' holding approximately 2x10^ sodium atoms at a temperature of about 240 jaK. The atomic sample, which is the orange ball in the photograph, is at the intersection

of six collimated laser beams that form the optical molasses. It is made visible by the light that the atoms scatter out of the laser beams. The "ball" of atoms is roughly 7 mm in diameter. The green light is from a pulsed laser used to evaporate sodium atoms from the

solid. ^° (Photograph courtesy of Bell Laboratories.)

atom or atom-atom collisions. Experiments on neutral atoms

If neutral atoms could be trapped

easily and held in thermal isolation from the surroundings, then laser cool- ing thern would be similar to laser cooling ions. Unfortunately, neutral- atom traps with the required thermal

isolation are not very deep. For example, magnetic traps, which convert the atom's kinetic energy into internal,

Zeeman energy, at best have a depth of about 5x10"^ eV, or 8 K. This as- sumes that an atom with a -nagnetic

moment equal to one Bohr magneton is captured in a magnetic well that is 10 T

deep. Ion traps, in contrast, can be

kilovolts deep.

Most neutral-atom laser cooling ex- periments start with an atomic beam of an alkali such as sodium and slow and

cool the beam with a counterpropagating laser beam.'*^ Because sodium

atoms are emitted from an oven source with a velocity of about 1000 m/sec, the

slowing and cooling must be done very

efficiently to stop the atoms before they

strike some portion of the apparatus.

Even at maximum cooling efficiency.

this requires about 50 cm. The first neutral-atom cooling experiments were

reported by S. V. Andreev, Victor Balykin, Vladilen Letokhov and Vladimir

Minogin in Moscow and by William

Phillips and Harold Metcalf at NBS in Gaithersburg, Maryland. The Moscow

group saw beam slowing and velocity

compression due to a counterpropagating fixed-frequency laser beam. If a

fixed-frequency laser beam is used,

atoms with velocities that put them in resonance with the laser through the

first-order Doppler frequency shift are

efficiently slowed. However, these

atoms are soon slowed enough that

they are Doppler shifted out of reso- nance with the laser beam and the

slowing is greatly reduced. Similarly,

faster atoms are slowed only very

slightly. The result is that a hole is carved out of the atoms' velocity distri- bution at a velocity corresponding to the Doppler-shifted laser frequency,

and the affected atoms tend to bunch at

slightly lower velocity.

To make the cooling more efficient, one needs a way to keep the atoms in resonance with the laser while they are slowed down. The NBS group accomTN-161
plished this by continuously tuning the

atoms' frequency. Investigators from

the group directed the atomic beam

down the bore of a solenoid whose

magnetic field varied with position.

The field varied in such a way as to keep the slowed atoms, whose Zeeman

frequency shift depended on position in the magnet, continuously in resonance with the fixed-frequency laser beam.

This enabled more of the atoms to be

slowed to lower velocities. Recent ex- periments at MIT have also used this technique.

Another way to keep the atoms tuned

to the frequency of the laser is to sweep,

or "chirp," the frequency at an appropriate rate. The first demonstration of beam slowing by this technique was made in 1983 at NBS, Gaithersburg.

Physicists at the Joint Institute for Laboratory Astrophysics later used the same technique to slow and stop atoms. '^ Subsequent experiments at

Bell Laboratories,^" Bonn University,

the State University of New York at Stony Brook, the Ecole Normale Superieure in Paris and by another group at JILA^' have employed laser chirping

for cooling. The latter experiments at JILA accomplished slowing and cooling

using diode lasers, demonstrating that

cooling at a relatively low cost is practical. With these techniques,

atoms are now stopped (and even

turned around) in several of the above

laboratories. Temperatures of the

stopped atoms are typically a few tens

of millikelvins, and gravity starts to play a role because the slowed atoms

are in the apparatus long enough to fall a significant distance. The group at Bell Labs has demonstrated the lowest temperatures yet

achieved through laser cooling.^° Re- searchers there used a chirped laser to precool sodium atoms from a pulsed

laser ablation source. They then subjected these atoms to three mutually

orthogonal, intersecting pairs of coun- terpropagating laser beams of cross section about 1 cm^, tuned to achieve

minimum temperature. The region of intersection features a laser cooling

damping force in all directions. The

accompanying recoil heating causes

the atoms to undergo Brownian motion. The diffusion time for atoms to leave this "optical molasses" can be on

the order of 1 sec, which is plenty of time for many experiments. By rapidly

turning the molasses off and on, the

Bell Labs group was able to measure

the velocities of the atoms- and deter- mine that the temperature was 240 /iK

(with a probable range of 180-440 //K), which is in agreement with the limit implied by equation 1. Now that sources of slow atoms exist,

they can be used as injection sources for 8

the relatively shallow neutral-atom

traps. Phillips and his coworkers were

the first experimenters to capture

atoms in a magnetic trap.'** (Wolfgang

Paul and his collaborators had held neutrons in a magnetic trap in 1978.)

The MIT group recently used a differ- ent magnetic trap to hold atoms.^^ The

Bell Labs group has used their optical

molasses to inject atoms into a gradi- ent-force or dipole-force laser trap.^° In Moscow, Balykin and A. I. Sidorov have

demonstrated cooling in two dimensions by coUimating atomic beams.^^ The Paris group recently demonstrated

stimulated cooling by its effect on the

collimation of atomic beams.^"* In principle, one could apply laser cooling to normal solids. For example,

crystals that are doped with impurity

ions could be driven on the lower

phonon sidebands of certain transitions. Unfortunately, nonradiative de- cay from the upper level of the cooling

transition, which shows up as heat,

may dominate the cooling process. It is interesting to examine the economics

of large-scale laser cooling. If substantial cooling requires about lO'* scatter- ing events per atom, then cooling one mole of material will require more than

10^ joules of laser energy. Therefore,

laser cooling may not be practical on a

large scale, but in many cases it may be

the only way to lower or manipulate

the velocities of atomic samples. Judging by the number of laboratories now

using laser cooling or setting up experiments using laser cooling, it appears

likely that the technique will have

many interesting applications in the

future.

We acknowledge the support of the Office of Naval Research and the Air Force Office of

Scientific Research. References

1. T. W. Hansch, A. L. Schawlow, Opt.

Commun. 13, 68 (1975). D. J. Wineland,

H. Dehmelt, Bull. Am. Phys. Soc. 20, 637

(1975).

2. H. Dehmelt, in Advances in Laser Spectroscopy, F. T. Arecchi, F. Strumia, H. Walther, eds., Plenum, New York,

(1983), p. 153.

3. D. J. Wineland, W. M. Itano, J. C. Bergquist, J. J. Bollinger, J. D. Prestage, in Atomic Physics 9, R. S. Van Dyck Jr, E. N. Fortson, eds.. World Scientific, Singapore (1984), p. 3. D. J. Wineland,

Science 226, 395 (1984).

4. E. F. Nichols, G. F. Hull, Phys. Rev. 13, 307 (1901). P. Lebedev, Ann. Phys.

(Leipzig) 6, 433 (1901); Ann. Phys. (Leipzig) 32, 411 (1910).

5. A. Einstein, Phys. Z. 18, 121 (1917).

6. O. Frisch, Z. Phys. 86, 42 (1933).

7. A. Ashkin, Phys. Rev. Lett. 24, 156

(1970). G. A. Askar'yan, Zh. Eksp. Teor.

Fiz. 42, 1567 (1962) [Sov. Phys. JETP 15, 1088 (1962)]. A. P. Kazantsev, Zh. Eksp.

Teor. Fiz. 63, 1628 (1972) [Sov, Phys.

JETP 36, 861 (1973)]. V. S. Letokhov,

Pis'ma Zh. Eksp. Teor. Fiz. 7, 348 (1968) [JETP Lett. 7, 272 (1968)].

8. A. Kastler, J. Phys. Radium 11, 255

(1950).

9. The following are review articles on la- ser cooling and related phenomena: S. Stenholm, Rev. Mod. Phys. 58, 699

(1986). A. Ashkin, Science 210, 1081

(1980). V. S. Letokhov, V. G. Minogin,

Phys. Rep. 73, 1 (1981). A. P. Kazantsev,

G. A. Ryabenko, G. I. Surdutovich, V. P. Yakovlev, Phys. Rep. 129, 75 (1985).

10. J. P. Gordon, A, Ashkin, Phys. Rev. A

21, 1606 (1980). R. J. Cook, Phys. Rev. A

22, 1078 (1980). V. G. Minogin, Zh. Eksp. Teor. Fiz. 79, 2044 (1980) [Sov. Phys. JETP 52, 1032 (1980)].

11. D, J. Wineland, W. M. Itano, Phys. Rev. A 20, 1521 (1979). W. M. Itano, D.J. Wineland, Phys. Rev. A 25, 35 (1982).

12. A. P. Kazantsev, V. S. Smirnov, G. I. Surdutovich, D. O. Chudesnikov, V. P. Yakovlev, J. Opt. Soc. Am. B 2, 1731

(1985). J. Dalibard, C. Cohen-Tannoudji,

J. Opt. Soc. Am. B 2, 1707 (1985).

13. H. Dehmelt, Adv. At. Mol. Phys. 3, 53

(1967); Adv. At. Mol. Phys. 5, 109 (1969), D. J. Wineland, W. M. Itano, R. S. Van

Dyck Jr, Adv. At. Mol. Phys. 19, 135

(1983).

14. M. Lindberg, J. Phys. B 17, 2129 (1984).

J. Javanainen, M. Lindberg, S. Sten- holm, J. Opt. Soc. Am. B 1, 111 (1984).

15. W. Neuhauser, M. Hohenstatt, P. Tos- chek, H, Dehmelt, Phys, Rev. Lett. 41, 233 (1978); Phys. Rev. A 22, 1137 (1980).

16. G. Janik, W. Nagourney, H. Dehmelt, J. Opt. Soc. Am. B 2, 1251 (1985). J. C. Bergquist, W. M. Itano, D. J. Wineland,

Phys. Rev. A (1987), to be published.

17. J. H. Malmberg, T. M. O'Neil, Phys.

Rev. Lett. 39, 1333 (1977). L. R. Brewer,

J. D. Prestage, J. J. Bollinger, D. J. Wineland, in Strongly Coupled Plasma

Physics, F. J. Rogers, H. E. DeWitt, eds.. Plenum, New York (1987), to be published. D. J. Larson, J. C. Bergquist,

J. J. Bollinger, W. M. Itano, D. J. Wineland, Phys. Rev. Lett. 57, 70 (1986).

18. W. D. Phillips, H. J. Metcalf, Sci. Am.,

March 1987, p. 50. P. Meystre, S. Sten- holm, eds., The Mechanical Effects of

Light, J. Opt. Soc. Am. B 2(11) (1985).

19. W. Ertmer, R. Blatt, J. L. Hall, M. Zhu,

Phys. Rev. Lett. 54, 996 (1985).

20. S. Chu, L. W. Hollberg, J. E. Bjorkholm,

A. Cable, A. Ashkin, Phys. Rev. Lett. 55, 48 (1985). S. Chu, J. E. Bjorkholm, A. Ashkin, A. Cable, Phys. Rev. Lett. 57, 314 (1986).

21. R. N. Watts, C. E. Wieman, Opt. Lett. 11, 291 (1986).

22. V. S. Bagnato, G. P. Lafyatis, A. G. Martin, E. L. Raab, D. E. Pritchard, Phys.

Rev. Lett. (1987), to be published.

23. V. I. Balykin, A. I. Sidorov, Appl. Phys.

B 42, 51 (1987).

24. A, Aspect, J. Dalibard, A, Heidmann, C. Salomon, C. Cohen-Tannoudji, Phys.

Rev. Lett. 57, 1688 (1986). D

TN-162
Simple electrodes for quadrupole Ion traps

EartC. Beaty

National Bureau ofStandards, 325 Broadway, Boulder, Colorado 80303

(Received 23 September 1986; accepted for publication 19 November 1986)

Quadrupole traps for charged particles often involve electrodes with portions carefully

machined to the shape of hyperboloids. It is shown here that the more important features of

such traps can be achieved using electrode shapes which are much easier to fabricate. Detailed

numerical calculations are reported on some sample electrodes. The numerical method can be

easily extended to other shapes which accommodate specialized laboratory situations.

I. INTRODUCTION

Quadrupole ion traps are used in a wide variety of re- search applications. It is well known that it is not possible to construct static electric fields which will trap ions in all three dimensions. However, it is possible to build an effective trap

by either adding a magnetic field' (Penning trap) or by

making the electric field time dependent^ (Paul trap). For

many applications it is desirable to achieve a special electric- potential function which is referred to below as the "ideal."

It is generally desirable for the electric-potential function to have both cylindrical and reflection symmetry. With these symmetry restrictions, a general potential can be expressed

in the form,

Vir^) = Co + C^H^ir,!) + C^H^{r,z)

where

f^2(/,z) = (2z2-r2)/5^

H^ir^) = (8z^ - 24zV + 3/)A^ ,

(1)

H^ir^) = {\6z^

H^{r,z) = (128z^

120zV + 90z2/ - 5r^)/s^ ,

- 1792zV + 3360zV^ - 1120zV + 35/)//.

The Hj functions (spherical harmonics) are homogeneous

polynomials in z/s, r/s with the numerical coefficients chosen so that each satisfies the Laplace equation. The C, are constants (independent of position) with the units of potential, and 5 is a distance to be chosen to suit the circumstances

of the problem. The ideal potential referred to above corre- sponds to the first two terms of this expansion. A special

quality of the ideal potential is that the equations of motion

of trapped particles are particularly simple. For instance, for the Paul trap, the r and z components have separate equations of motion, making it relatively easy to specify stability

criteria. Also, in a Penning trap, the ideal potential yields

axial, cyclotron, and magnetron frequencies which are independent of the amplitude of excitation (neglecting relativistic effects) . Thus a trap with the ideal potential is sometimes

described as "harmonic." Some relatively recent reviews cover both the operations and the applications.^"*

II. THE PROBLEM

To achieve the ideal potential represented by the first two terms of Eq. ( 1 )

, three electrodes are needed, each a

hyperboloid (and therefore infinite in extent). The elec- trodes which produce the ideal potential are referred to be- low as ideal electrodes. The three electrodes, each conforming to an equipotential surface ofthe ideal potential function,

consist of a central "ring" electrode and "endcaps" on either

side. Some errors are caused by making finite electrodes; however, these can be minimized by careful treatment of the

truncation boundaries.^ The hyperbolic electrodes are diffi- cult to fabricate and for many purposes it is worth considering a much simpler shape. The purpose of this paper is to point out that it is possible to achieve a good approximation

to the ideal with very simple electrodes. The electrode structures examined below have the same

general topology as the ideal electrodes. They have cylindrical and reflection symmetry, and involve a region of space

substantially surrounded by a ring and two endcaps. To pre- serve the reflection symmetry, the two endcap electrodes must have the same potential (assuming they have the same

shape). By appropriate choice of the potentials of the two

free electrodes, it is possible to make Cg and Cj match any

preset conditions. To make the other terms small with a three-electrode system, it is necessary to choose the shapes of

the electrodes carefully. A more general version of Eq. ( 1

)

involves terms with odd powers of z and r, and also terms

involving the angle coordinate. Such terms are not included

here because it is easy to choose experimental conditions

which, in principle, make them zero. If the electrodes and

applied potentials have reflection symmetry, the odd-order

terms will all be zero, and if the electrodes have cylindrical

symmetry, the terms involving the angle coordinate all vanish. In practice, errors in fabrication or mounting (also non- uniform contact potentials) can cause these ^^erms to be nonzero. The term of order one [C^Hi{r,z)] caused by these laboratory errors can be made small by either choosing a

different origin of coordinates for the expansion, or by making the endcap potentials unequal. If the machining and

mounting are precise enough to make the third-order term

small, then it is appropriate to look for electrode shapes

which make the fourth-order term small. Some example calculations are reported below to show

that simple electrodes are capable of adequately producing

the ideal potential over a substantial volume. The basic computational method as it applies here is described in some

detail in Ref 5. The general method has previously been

used by Harting and Read*' for electron lens calculations. Only a brief summary of the method is given here. In this

2118 J. Appl. Phys. 61 (6), 1 5 March 1 987 TN-163

2118
FIG. 1. A drawing of the trap geometry examined in this paper. The elec- trodes are cylindrically symmetric about the z axis.

procedure the surface-charge density on all the electrode

surfaces is determined numerically, using the condition that the potential is known on all the electrode surfaces. Then

either the Cj or the potential can be calculated by apphcation

of Coulomb's law. The numerical representation of the sur- face-charge density involves the arbitrary division of the

electrode surfaces into A^ small areas each of which has ap- proximately uniform surface charge density across it. The

surface-charge distribution is represented by giving N

numbers, each of which provides the assumed constant sur- face-charge density on one of the surface elements. One of

the principal computational chores is the solution ofA'^ linear algebraic equations. The values of Abused were of the order

of 100. Much larger values of A'' require a large amount of

computation time and tend to cause digital truncation error

to be a significant problem. Much smaller values of A'^ cause unacceptable errors in the results.

III. RESULTS

A diagram of the example electrode shape examined in

this paper is displayed in Fig. 1. Detailed calculations of the surface-charge densities have been carried out for a variety

of the variables defined in Fig. 1 . By using a simple search

procedure it is easy to find values of the dimensions which

yield small values of C4. It is even reasonable to search for conditions which make C^ small also. In the following dis1 .0

ra O.i

o 0.6

- 0.4

0.2

0.0

O o o ^>

r = V 0.0

A 0.2 O 0.4

0.0 0.2 0.4 0.6

z/r.

1 .0

FIG. 2. Sample plot of the potential function V (r,z) vs z for three values of r

cussion, all dimensions are given relative to /"o and the length

variables are therefore treated as dimensionless.

The Cj are each linear functions of each of the electrode

potentials. The zero of potential is arbitrary and here the

potential is assumed to approach zero far from the elec- trodes. For the purposes of this paper it is not necessary to have a completely general solution. The values of C, were

calculated for a specific set of potentials on the electrodes. These special values of C, are called Aj . The reference set of

potentials are for the ring and 1 for the endcaps (as defined

here the Aj have the same units as the endcap potential).

Scaling the results to other values of the endcap potential is straightforward. These results cannot be generally scaled to other ring potentials; however, in most circumstances it is only the potential diiference between the endcaps and the

ring which is important (for a more complete discussion see Ref 5 ) . A general solution would involve generating a sec- ond set of Cj with other values for the electrode potentials,

including a nonzero value for the ring potential. The dis- tance s in Eq. ( 1 ) is assigned the value r^ . With these nor- malizations the set of Aj are functions of the shape of the

electrodes and are independent of the size or the potentials

which are applied to the electrodes in use. Table I contains results for eight cases which have been

found to yield A^ = A ^ =0. Figure 2 displays the potential

for case six as a function of axial position. The data points in the graphs are the calculated values and the solid lines are the ideal potentials as represented by the first two terms of

TABLE I. Data for eight sets of values for the variables defined in Fig. 1. The dimensions were selected to yield A^= A(, = 0. The dimensions specified have

been scaled to the condition /"o = 1- The Aj are the spherical harmonic expansion coefficients with normalized potentials on the electrodes.

^0 '•. ^1 '2 7 Ao Aj A,

1 0.6122 0.8812 0.7952 1.1020 0.5913 0.545486 0.568051 0.002312

2 0.7000 0.6856 0.8182 1,0462 0.2678 0.497635 0.489588 0.000142

3 0.7000 0.6950 0.8197 1.1147 0.2949 0.508734 0.478829 0.000103 4 0.7000 0.7095 0.8224 1.3654 0.4438 0.524285 0.463748 0.000053

5 0.7000 0.7799 0.8518 1.1000 0.4000 0.492112 0.488948 0.000623 6 0.7071 0.8435 0.8728 1.2340 0.5956 0.493999 0.476988 0.000569 7 0.8000 0.7884 0.9356 1.2030 0.4569 0.441861 0.416657 0.000025

8 0.8000 0.8841 0.9683 1.2224 0.5573 0.435476 0.417169 0.000177

2119 J. Appl. Phys., Vol. 61 , No. 6, 1 5 March 1 987 TN-164

Earl C. Beaty 2119
0.010

0.008

o 0.006

0.004

0.002

0.000 -

-0.002

1 1 1

-

. D o .

- V

-

- o

D O Case

-

- e d

o

@

o

g

io

V

o

O

o

V

XAO

12

46

7

-

•

1 1

D 8

1 0.0 0.2 0.4 0.6

z/r„

FIG. 3. Plot of Ef/z^ vs displacement from the center of the trap. E^ is the

fractional error in the electric field on the axis.

Eq. (1). Data are shown for r = 0, 0.2, and 0.4. Equation

( 1 ) with values of C, derived from Table I can be used to calculate the potential near the center of the trap. Near the

walls Eq. ( 1 ) is generally useless because too many terms

contribute significantly to the sum. The data of Fig. 2 were

calculated directly from the charge distributions without in- volving Eq. ( 1

) . The potential is reasonably close to the ideal

all the way to the endcap.

The electric field on the z axis can be relatively easily

calculated from the surface-charge distributions (the derivative of the potential can be done analytically). The electric

field on the z axis corresponding to the ideal potential is 2A jZ. The fractional error in the electric field is given by

^, = (f-2.,z)/2^,z = 16Az- + 48^z'* + 512^z^ + ... . A, A-, Figure 3 is a plot of£}/z^ vs z for several ofthe cases ofTable

I. The limit of these curves as z goes to zero is l^A^/Aj, a quantity which is small by design. The field errors displayed

in Fig. 3 are all positive because the electrode shape chosen

here has a sharp point on the z axis. This point contributes

particularly large fields in a small volume near the point.

Away from the polar axis the electric field is not necessarily

in the same direction as the ideal, making a comparison more

difficult. While the departures from the ideal are more diffi- cult to display in other regions, the data of Fig. 3 are reasonably representative. Two of the cases are not included in the

plot because case 3 is similar to case 2, and case 5 is similar to case 6. Case 7 of Table I has the best value of^ g /^ 2 suggesting

that it might be a particularly favorable choice. Actually,

any of these cases may be difficult to implement with enough

precision to make the eighth-order term the dominant error

in a laboratory situation. Case 1 has a small ring-endcap gap

providing good shielding from external fields. Cases 2, 3, and

4 have a relatively large ring-endcap gap allowing easy access with light beams, etc. Case 6, also relatively open, has

already been used in an atomic spectroscopy application.^ It may be helpful in visualizing the shapes to know that the

diagram of Fig. 1 was drawn using the dimensions of case 6 from the table. For some purposes it would have been better

to have a hole in the endcaps on the r = axis (instead of a

point), thereby reducing the electric-field anomaly and al- lowing a convenient access for ions, electron beams, etc. Such holes would not have greatly complicated the computations, and unless they were very large, they would not have

interfered with making A 4 and A ^ small. An investigation of the errors in the general method was

reported in Ref 5. Less extensive tests conducted in conjunction with the generation of the data of Table I confirm the general conclusion that the principal errors were due to rep- resenting the charge distribution on too coarse a grid. The

grid used was kept relatively coarse to speed the calculation

through the tedious search. Any small errors in the A q and

A 2 can be easily compensated for by adjusting the potentials

applied to the electrodes. Errors in the computation of ^^4 andyl (, result in errors in the five length variables of Table I. The length data given in the table are estimated to have ap- proximately four digit accuracy. Might electrodes such as those of Fig. 1 be more sensitive to misalignments than hyperbolic electrodes? An easy and useful test is to make a small change in one of the dimensions and repeat the calculation to observe the change in the Aj . Tests made in conjunction with the work of Ref 5 but not reported there yield the result that dAj/dzg = — 0.72 and

dA 4 /dzQ = 0.04 for the case ofideal electrodes with 2zo = r^ (Zq is the distance from the trap center to each of the endcaps). In changing Zq both endcaps were displaced but their shape was not changed. Such tests were carried out both

with truncated hyperbolas and with the design of Fig. 1. Some of the cases of Table I were found to have A 4 about

twice as sensitive to this displacement as the ideal electrodes

cited. Similar sensitivities were also found for truncated hyperbolas. A conclusion is that the electrodes examined here

are not significantly more sensitive to laboratory errors than

those with hyperbolic shapes. In Fig. 1 the electrodes extend beyond the boundaries of

the drawing. An assumption was made that in practice the portion off" the drawing would involve wires and mounting

brackets special to the particular installation. Thus, the impact of various ways ofhandling the remote portions was not

systematically investigated. For the results of Table I elec- trodes were extended a short distance and simply terminated. In each case the straight boundaries parallel to the axes had a length of LSkq. To test the sensitivity to this dimension, some of the calculations were repeated with both ring

and endcap extended by an additional 0.5rQ. For cases 1, 4, and 7, the changes in ^ 4 were ( — 1, — 17, — 10) X

10"'^.

respectively. Generally, increasing the size of the endcap in- creases ^4, and expanding the ring decreases it. If the ring

and endcap are close together, the region near the center of

the trap is shielded from charges on the more remote parts of

the electrodes. For cases where the intrinsic shielding is ina- dequate a reasonable step is to insert a relatively large extra electrode and empirically adjust its potential to produce op- timum performance.

2120 J. Appl. Phys., Vol. 61 , No. 6, 1 5 March 1 987 TN-165 Earl C. Beaty 2120
IV. DISCUSSION

Some general comments can be made about the use of

the data of Table I in applications needing the ideal potential. In a common form of the problem the potential must

approximate the ideal to some specified tolerance over a

specified volume. This means that Cq and Cj have specified

values, and limits are available for the other C,. To meet

these requirements with hyperbolic electrodes is straightforward with the condition that the minimum electrode size be

large enough not to intrude on the specified volume. To meet

these requirements with the electrodes of Fig. 1, it is neces- sary to have larger values of Tq and Zq (perhaps by a factor of

2 or 3 ) and therefore higher potentials.

An application where simple electrodes might be useful

is mass spectroscopy. In one kind of mass spectrometer, the

electric potentials are made time dependent in such a way

that ions with small masses have unstable orbits for one rea- son, and ions with large masses have unstable orbits for an- other reason.^ With the ideal potential the stability conditions are well defined and it is possible to create a situation in which only ions in a narrow mass range have stable orbits. Particles with unstable orbits leave the trap rather quickly.

Also some of the particles in stable orbits are removed be- cause the orbits intersect one of the electrodes. Many such

traps have been built and found to give good service. It is clearly possible to arrange the electrodes of Fig. 1 (or other

simple shapes ) such that near the trap center the orbits of the

ions and the stability conditions are not materially changed.

A failing of the simple electrodes is that near the electrode

surfaces the orbits are much more complex, blurring the dis- tinction between stable and unstable orbits. Ions near the center which fail to meet the stability conditions will move

out where the fringing field may produce stability or at least a long lifetime. The impact of this problem on mass resolution can't be easily evaluated. Fulford et al. and Mather et

al/' experimentally investigated the performance of some

traps that had cylindrical rings and plane endcaps. They

found the mass resolution to be significantly lower than they were accustomed to finding with hyperbolic electrodes. It is not clear what the specific failing of the cylindrical traps was. The design has the endcap and ring quite close together,

resulting in high electric fields in a place easily available to the trapped particles. Also, as implemented, the endcaps had

large holes such that A 4 may not have been small. In some precise atomic physics measurements the potential function must be very close to the ideal. In many of

these experiments the precise fields are achieved by having a way to compress the cloud of ions into a relatively small

volume near the center of the trap.** While the electrodes used have typically been modified hyperbolas, in some cases

extra "compensation" electrodes have been added so that C4

can be electrically adjusted to zero.'° In many of these ex- periments it is important to closely approximate the ideal

field near the trap center while giving little concern to irregularities near the electrodes. For this situation the design of

Fig. 1 is quite appropriate. It might be desirable to add elec- trodes to a design such as that in Fig. 1 to allow compensation for some mounting errors. There is a significant need for the extra electrodes to have both reflection and rotation symmetry, with the penalty that it would be possible to compensate only those errors with these symmetries.

Gabrielse and MacKintosh" have examined the elec- trostatic properties of cylindrical traps where it is possible to solve the Laplace equation using series methods. They show

that A 4 can be made zero by appropriate choice of the ratio of height to radius. They also consider the option of dividing

the cylindrical ring electrode into three parts such that the outer two can be used as compensation electrodes. This ar- rangement allows the user to empirically adjust A^ to zero

after assembly is complete. This analysis is based on very

small interelectrode gaps. While a practical design may re- quire larger gaps and other holes, the adjustment should

have enough latitude to cover these problems. This is a good

example of how extra electrodes can be used to compensate

for incomplete design information. For the case of the Penning trap the potentials normally do not vary rapidly with

time and the potentials for the extra electrodes can be de- rived from a simple resistive divider network. In Paul traps,

high-frequency potentials are needed and the divider

network is not so easy to adjust. Also, in many applications the test for small ^ 4 is very difficult.

V. CONCLUSIONS

Normally the most eff'ective way to generate a close ap- proximation to the ideal potential distribution is to use hyperbolic-shaped electrodes. This is especially true if the application demands that the close approximation extend all the way to the electrodes. The only intrinsic problem with

hyperbolic electrodes is the truncation. The truncation er- rors can be made small either by careful design"^ or by insert- ing compensation electrodes. '° Practical problems with hyperbolic electrodes are that they are difficult to fabricate and, in some applications, it is necessary to make holes

which spoil the ideal fields. The simple electrodes discussed

above can, in some applications, be adequate and much easi- er to fabricate. With electrodes requiring only simple lathe cuts it is possible to be rather free with making access holes

and special mounting fixtures, while preserving some of the more important features of the potential distribution. In cir- cumstances where the dominant errors in the potential function are caused by alignment or machining errors, the simple

electrodes may have a positive advantage.

ACKNOWLEDGMENTS

This work was supported in part by the U. S. Office of

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

'F. M. Penning, Physica 3, 873 (1936).

'E. Fischer, Z. Phys. 156, 1 ( 1950); R. F. Wuerker, H. Shelton, and R. V. Langmuir, J. Appl. Phys. 30, 342 (1959).

'H, G. Dehmelt, Adv. At. Mol. Phys. 3, 53 (1967); 5, 109 (1969); L. S. Brown and G. Gabrielse, Rev. Mod. Phys. 58, 233 ( 1986). 'D. J. Wineland, Wayne M. Itano, and R. S. Van Dyck, Jr., Adv. At. Mol.

Phys. 19, 135 (1983).

2121 J. Appl. Phys., Vol. 61, No. 6, 15 March 1987 TN-166 Earl C. Beaty 2121
'Earl C. Beaty, Phys. Rev. A 33, 3645 (1986).

*£. Harting and F.R. Read, Electrostatic Lenses (Elsevier, Amsterdam,

1976).

'J. C. Bergquist, D. J. Wineland, Wayne M. Itano, Hamid Hemmati, H.-U.

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

*P. H. Dawson, Quadntpole Mass Spectroscopy and its Applications (Elsevier, Amsterdam, 1976).

'J E. Fulford, R. E. March, R. E. Mather, J. F. J. Todd, and R. M. Waldren, Can. J. Spectros. 25, 85 ( 1980); R. E. Mather, R. M. Waldren, J. F.

J. Todd, and R. E. March, Int. J. Mass Spectrom. Ion Phys. 33, 201

(1980).

'"R. S. Van Dyck, Jr., D. J. Wineland, P. A. Ekstrom, and H. G. Dehmelt,

Appl. Phys. Lett. 28, 15 ( 1976); Gerald Gabrielse, Phys. Rev. A 27, 2277

(1983); 29,462 (1884). "G. Gabrielse and F. C. MacKintosh, Int. J. Mass Spectrom. Ion Phys. 57P, 1 (1984).

TN-167

21 22 J. Appl. Phys., Vol. 61 , No. 6, 1 5 March 1 987 Earl C. Beaty 2122
From: Proc. of the Cooling,

Condensation, and Storage of Hydrogen

Cluster Ions Workshop, SRI, Menlo

Park, Ca., Jan., 1987; ed by J.T. Bahns,

p. 181.

ION TRAPS FOR LARGE STORAGE CAPACITY

D. J. Wine land

Time and Frequency Division

National Bureau of Standards

Boulder, Colorado 80303

ABSTRACT

Ion storage in Penning-type or rf (Paul) -type traps is discussed.

Emphasis is given to low-energy, long-term confinement of high densities and

large numbers of ions. Maximum densities and numbers are estimated using a low- temperature , static model of the ion plasmas in the traps. Destabilizing

mechanisms are briefly discussed.

INTRODUCTION

The following notes are concerned with the storage of large densities and

numbers of charged particles or ions in electromagnetic traps. Certainly,

this is a small part of the problem concerned with the accumulation, storage,

and manipulation of antimatter. However, this particular problem appears to

be interesting by itself and may have interesting applications elsewhere . •^"'^ Devices such as tokamaks and tandem-mirror machines for fusion plasmas are not

discussed because we concentrate on long-term, low-energy confinement which is

desirable for antimatter storage. For brevity, only Paul (rf) and Penningtype traps'"'^ are discussed. Maximum densities and maximum numbers of ions

for one or a mixture of species with different charge to mass ratio are investigated.

GENERAL CONSIDERATIONS

We will assume that thermal equilibrium of the stored ion sample has been

achieved. This condition may not always be achieved or desirable. For

example, antihydrogen might best be made by passing antiprotons through

positrons without thermal equilibrium between these two species being

achieved. For long term storage, however, thermal equilibrium appears likely.

If we assume that the ion trap has symmetry about the z axis, the thermal

equilibrium distribution function for the ith species can be written^ "^

3/2

fi - n, ( ^^^^^ ) exp[-(H,-a,i,i)ABT]. (D

*Contribution of the National Bureau of Standards; not subject to U.S.

Copyright

.

IN- 168
m^ is the ion mass, kg is Boltzmann's constant, T is the ion temperature, and w and n^ are constants (determined below) . If a magnetic field is

superimposed along the z axis (B — Bz) , then

Hi - m,vf/2 + qi^(x) (2)

is the ion energy and

-^zi - "i^ei^i + qiAe(x)ri/c (3)

is the ion canonical angular momentum. q^^ and v^ are the ion charge and

velocity, and v^ - [v^ j. v^^^ and A^^ are the 6 components of the velocity and

the vector potential, and r^ is the radial coordinate of the ion in

cylindrical coordinates. ^ - (^j + (^.j. + <f>^^^ is the total potential, which is

written as the sum of the potential that is due to ion space charge 4)^ , the

applied trap potential ^j , and the potential, <i>i-n^, that is due to the induced

charges on the trap electrodes.^ We choose the symmetric gauge where A(x) - B

X x/2 , so that Agi - ^x^/2. We can therefore write the distribution function

as 3/2

fi(x,v) - n^Cx) ( ) exp[-hmi(Vi-u)ri^)2/kBT], (4)

2jrkBT -

where n^^ (x) is the ion density given by

ni(x) - niexp{-[qi^(r,z)+4miw(n^i - u>)rl]/V^T) . (5)

We assume that the trap potential is adjusted to make (^ — at the origin, in

which case n^ is the ion density at the center of the trap for a single ion

species. From Eq. 4, we see that the ion cloud, or non-neutral ion plasma,

rotates at frequency w. I^cil " I'^il^/TM!*^ ^^ ^^^ cyclotron frequency. The

signs of w and 0^,^ indicate the sense of circulation where we use the righthand rule. For example, 0^^ is negative for positive ions (qj^>0) and is

positive for qi<0.

In the limit T-^0, we must have c\^4>-m^^u){(j>-Q^^)r\/2 -* for f^ to be well

behaved. This condition and Poisson's equation imply that the charge density,

p, at the position of the ions be given by

p - -pl + miW(n^i-t<))/(2jrqi), (6)

where p[ is a fictitious charge density arising for rf traps since the

pseudopotential does not satisfy Poisson's equation^ (see below). By low

temperature, we mean that the Debye length, Ap , for the plasma given by^ "

\l - kBT/(4^niq2), (7)

is short compared to the plasma dimensions. From Eq . 7, we can write

TN-169
Aj) =- 6.9(T/ni)^/Z cm, (8)

where T is expressed in kelvin, n^ is expressed in cm""', and Z is the ion

charge in units of the proton charge. For T - 4 K, Z - 1, and n^ - 10® /cm-' , Ap - 14 /im. Therefore, for the low temperatures desired for long term

storage, the T-^0 limit will usually be valid.

PENNING TYPE TRAPS

By Penning- type trap, we will mean that the trap is formed by a static

magnetic field B - Bz and an axially symmetric electric trap potential, <^j , of

the form (in spherical coordinates)

4>j - 2 Ci^r^Pj. (cos $) , k

(9)

where C^ are constants and T^ is a Legendre polynomial of order k. Axial

symmetry is particularly important for the Penning trap. If the trap is not

axially symmetric, angular momentum can be coupled into the ions and they will

diffuse out of the trap. If the trap is axially sjrmmetric, angular momentum

is a constant of the motion and stable ion clouds are maintained. ^° For

spectroscopy,*'^ particularly mass spectroscopy ,^ • ^^ • ^^ we desire ^j a

r^P2(cos $) as indicated in Fig. 1. This is because the ion motions

(neglecting relativistic effects) will be harmonic. A fourth order trap where

4>j oc r*P^(cos d) would look something like that in Fig. 2.

encHc

TB (•^4»<»'e<i "for Perin.Ma "'"VTft^)

Fig. 1. Electrode configuration for the quadrupole rf (Paul) or Penning trap

Inner surfaces of electrodes are assumed to be equipotentials of ^, and the

effect of truncating the electrodes is neglected. rj, is the inner radius of

the ring electrode and 2zo is the endcap to endcap spacing.

TN-170
10*}^

Fig. 2. Sketch of fourth order Penning- type trap for positive ions. The view

shown is a cross section in the y-z plane.

An experimentally convenient geometry^ might be a "cylinder trap" as shown in

Fig. 3.

_-iT

? m////////mf/fih -

-•^- - d

Vt

Fig. 3. Sketch of cylinder Penning- type trap for positive ions

With />i'-0, Eq. 6 can be written

n^ - m^wCn^i -a))/(27rq2) . (10)

Therefore, if we assume axial symmetry, n^ is independent of 4>j • Because of

this, we will assume throughout that we are using the experimentally

convenient geometry of the cylinder trap (Fig. 3). nj^ is maximum for w -= ngi/2, which is called the Brillouin limit. ^ w may be determined by initial

loading conditions or may be altered as discussed below. If we assume a single ion species and drop the indices,

n(max) - BV(8»nnc2) « 2.7 x 10^ B(T)2/M, (11)

where B(T) is the magnetic field in tesla and M is the ion mass in u (atomic

TN-171
mass units), n(max) is independent of Z. From Fig. 3 and Eq. 11, the maximum

number of ions, N(max) , is given by

N(max) ^ 8.4 x 10^ v^hZiT)^ /R, (12)

where r and h are the plasma radius and height in cm, respectively. The

potential, (f>^ , at the center of the plasma (assuming the central cylinder is

grounded as in Fig. 3) is easily calculated to be^

^, (V) - 1.4 X 10-7 NZ(l+2in(r„/rp))/h, (13)

where <f>^ is given in volts. The magnitude of V^ must be higher than the

magnitude of this potential to confine ions along the z axis. If we eliminate

N from Eqs . 12 and 13, we obtain

r2 - 8.3 x 10-*M^,/(B(T)2z(l+2inr^/rp)). (14)

For <f>^ - 30 kV, B - lOT, rp/r^ - 0.5, and Z - 1, we find r^ « M/10.

A few examples of n(max) and N(max) for these values of c^^, , B, and r /r^ are illustrated in table I. From the above considerations and table I, large mass storage would favor ions with large values of M.

Table I. Example parameters for a cylinder Penning-type trap. Fig. 3 applies

with V-i > <f>^ - 30 kV, B - 10 T, rp/r„ - 0.5, and Z - 1.

M(u) h(cm) r (cm) n(max) (cm" •^ ) N

4.3x10^3 ^ O.yxlO'^mg

4.3x10^3 ^ 0.7xlO-*mg

4.3x10^3

1 500 0.32 2.7x10^1

1000 500 10 2.7x10^

1/1836 500 0.0075 5.0x101*

rf(PAUL) - TYPE TBIAPS

Paul or rf - type traps provide trapping of charged particles in

oscillating, spatially-nonuniform electric fields. It is also possible to use static electric fields to alter the trapping geometry somewhat,'''^ but for

simplicity, we assume trapping in pure oscillating fields here. If the

trapping potential is given by

(^T - <f>Q (r) cosfit, (15)

then, for sufficiently high (defined below), ions are confined in a pseudo

potential, ^ j, given by''

TN-172
hi =" qi|v^o(^)|V(4m,fi2) (16)

The fictitious charge density is given by V^<^^ «= -Uirpl . For spectroscopy,

the usual choice is the quadrupole rf trap where <^q a r^P2(cos 0) (spherical

coordinates).*'^ This makes the trap pseudopotential harmonic in all

directions. In this case, sufficiently high means* •

skilvo <

qzi " ~ 1- (17)

min^(rg+2zg)

Vq is the magnitude of the rf voltage applied between ring end endcaps whose

dimensions are indicated in Fig. 1.

If we assume that the "secular" motion in the pseudopotential well of Eq

.

16 can be cooled, and if we assume u> — , the density of ions for a single

species is determined by space charge repulsion and is given by * - ^ (dropping

the subscripts)

n - 8.3 X 108 q^Vo(kV)/(Z(r2+2z2)). (18)

If we neglect ^ind' ^^^ shape of the ion plasma is a spheroid of revolution.^

If we also assume r^ - 2zo , and assume that the dimensions of the ion cloud

are equal to inner trap dimensions, the maximum number of ions is given by

N(max) - 2.3 X 10^ q^ ZoVo(kV)/Z, (19)

From Eq . 17, we have

fi/2»r(MHz) - 5.7(Vo(kV)Z/(Mq,))Vzo (20)

From Eqs . 18-20, we tabulate a few example parameters in table II

Table II. Example parameters for a quadrupole rf trap.

M Vq (kv) ro (cm) n(cm"^

)

N n/27r(MHz)

1000 10 0.5

1/1836 II It

II n 0.005

M

II 0.5 20

2.8x10^ 5.8x10^

2.8x10^

6.9x10^

0.81

25

1100

5.8x10^ 11 GHz

1.2x10^1 1.3/M^

TN-173
We note the independence of n and N(inax) on M. For electrons and positrons,

the practical limits on n and N(max) may be due to the required large values

of n.

For higher order rf traps, <^o (^) might take the form given by Eq. 9.

Perhaps a more useful form of (^q is given by the potential (in cylindrical

coordinates)

^o(r,^,z) - Vor^^i cos [ (k+1) tf ]/ro k + l (21)

where 6 is the azimuthal angle and k is an integer. A sketch of such a trap

for k—2 is given in Fig. 4. Ideally, the electrode surfaces are equipotentials of Eq . 21; in Fig. 4, they are shown as cylindrical rods for

simplicity. k-1 describes the Paul quadrupole mass filter. The ends of the

trap could be closed off by bending the rods in towards the axis of the trap

or using endcaps with appropriate static potentials thereby making a cylinder

rf-type trap. Alternatively, the individual rods could be connected end-onend to make a kind of race track. ^^^

2. axis

*;'

o o

o

Fig. 4. Sketch of cylinder rf-type trap for k - 2.

Experimental examples of higher order traps are given in refs. 14 and 15

From Eqs . 16 and 21,

2k

^pi ~ *^i^

where

Cki - qiV2(k+l)V(^n»in^ro2(k^i))

(22)

(23)

^^ is cylindrically symmetric.

From Gauss's law, the density of stored ions is determined by space

charge repulsion and is given by (if we assume w - 0)

ni(r) - k2Ci,ir2(k-i)/(^q.) (24)

TN-174
Therefore, for k » 1, the ions tend to form cylindrical shells near the inner

edge of the trap electrodes where r - rg . If the ions occupy the space out to

r •= rQ , integration of Eq. 24 gives

N(max) - ^^^3^^' ^^^ (^^1 q.^

.

(25)

where q^^^^ is the stability parameter analogous to Eq . 17, and Tq is defined

in Fig. 4. We have

ixi 8 VQq,/(m,n^r^'). (26)

For k - 1, Vq/Yq - 72 and q^^i- 1 for stability. For k - 1, the ratio of

N(max) for the cylinder rf - trap to N(max) for the rf quadrupole trap of Eqs

.

15 - 20, is equal to 3h/(4zo) where h is the length of the plasma column as in

Fig. 3. For k » 1, we have

N(max)j./N(max)i..i = kTr^/S, (27)

where we have assummed that the required value of the trap stability

parameter, q^^ , is independent of k. Thus, N(max) increases approximately as kh/zo

.

ION MIXTURES

For brevity, we consider only two ion species, with charge- to -mass ratios

q^/m^ y* q2/m2 • The results are easily generalized to more species. When

qj^/q2 > 0, we consider only cylinder- type traps for simplicity. However the

qualitative aspects of the results apply to other geometries.

Cylinder Penning- type trap, (q i/q o > 0)

In Eq. 6, with p' - for the Penning- type trap, p can be single valued

only if the two ion species occupy different spatial regions. Qualitatively,

since thermal equilibrium implies that u> is the same for all ions, if qj^ •= q2

and m2 > mj^ , species 2 is forced to larger radii than species 1 because of the

larger centrifugal force acting on it. This separation has been theoretically

and experimentally studied. ^^ • ^^ In the T - limit, the separation should be

complete with a gap between the species. If we look along the z axis of the

trap, the spatial distributions appear as in Fig. 5.

The ion densities are still given by Eq. 10. The maximum density of the

outer species is again given by w - n^2/2- Using this value of w, we can find

the corresponding value of n^ . The separation of the plasmas is given by

solving the equation of motion for the rotation motion of species 2 at the

radius a, . We find

az/bi - (Pi/P2)^ • (28)

TN-175
For 32 - bj^ > Aj) , we expect the thermal contact between the two species to be

reduced. ^^

Fig. 5. Centrifugal separation of ion species in cylinder Penning or rf-type

trap. The view is along the z axis, B is into the paper, and q^^ , q2 > 0.

A possible configuration for maintaining cold trapped positrons^ in a Penning trap might be the following. Consider simultaneous trapping of

positrons and ^Be"*" ions. The ^Be"*" ions would form a hollow cylinder around

the central e"*" column. Using laser cooling, the ^Be"'' ions and therefore the

positrons could be cooled to temperatures much less than 1 K.^^ In addition,

angular momentum can be imparted to the ions in the laser cooling process to

prevent radial diffusion of both species.^' In this way, a stable, low

temperature positron sample could be maintained. It is likely that w would be

limited to the value for maximum ^Be"*" density; that is aj(max) - Q^ (^Be* )/2

.

If we assume B - 10 T this would limit the maximum positron density to nCe"*") - 5.9xlOiVcm3 .

It would be nice to realize a similar scheme for antiprotons;

unfortunately, negative ions suitable for laser cooling do not seem to be

available. Cooling of antiprotons by thermalized electrons should work^'^®"

^° . In the condition of thermal equilibrium, the antiprotons form a hollow

cylinder around the electrons. In order to prevent radial diffusion, angular

momentum of the appropriate sign might be imparted to the ions by particle

beams, ^^ positive or negative feedback on the ion motion^^-^^ or angular

momentum transfer from plasma waves .^'^ • ^^

Cylinder rf-tvpe trap ("q i/q 2 > 0)

For the rf-type trap, as summing co - 0, separation of the species with

*5i/TMi '* ^2/^2 occurs because the pseudopotential wells are unequal. This

phenomenon has been investigated^^ for the quadrupole trap configuration. A

simpler case to analyze is that of the cylinder rf-type trap where ^q is given

by Eq. 21. If we assume qi/m^ > q2/m2 , Fig. 5 again applies. Ion densities

are still given by Eq. 24 and the separation of species can be easily

determined by noting that at r - a2 , the space -charge electric field outward

from species 1 is just equal to the pseudopotential field inward acting on species 2. We find

TN-176
bi

Z1M2

Z2M1

2k

(29)

Low order traps (e.g., k - 1 (quadrupole trap)) can give rise to a significant

separation of the species.

Penning- type traps (q i/q o < 0)

Simultaneous trapping of charges of opposite sign is particularly

interesting in the context of this meeting. An important example is the

simultaneous trapping of positrons and antiprotons for the production of

antihydrogen by radiative recombination.

For the Penning- type trap, simultaneous trapping of charges of opposite

sign is precluded because it is impossible to simultaneously trap both species

along 2. Therefore, to achieve a mixture of positive and negative ions, one species must be injected through the other. A possible arrangement might be

to stack Penning traps for charges of opposite sign along the z axis in a "nested" arrangement as suggested in ref. 26. If positrons are stored in the

central trap, antiprotons could be passed back and forth between traps

adjacent to the central trap. Angular momentum transfer would tend to spread

the positrons, but this might be overcome as described in the previous

section.

rf - type traps (q ^ /q o < 0)

In the pseudopotential approximation, it is apparent that if N^^q^^ - N2q2

the ions shrink to a point! The practical limit would appear to be due to rf

heating which tends to keep the ions very hot. This problem is accentuated

for simultaneous storage of positive and negative ions since they can come very close together due to Coulomb attraction and the rf micromotions* ^ are

180° out of phase for the two species. Some experimental work has been

reported in ref. 27, where Ti"*^ and I" ions were simultaneously stored in a quadrupole rf trap. In this work, an increase in density over the maximum

value for a single stored species was observed, but overall densities were

still fairly low. Clearly more work needs to be done on this interesting

possibility.

Penning - rf trap combinations

An example of such a combined trap might be an rf-type trap for positrons

superimposed on a Penning- type trap for antiprotons. Such a scheme appears

valid for small numbers of positrons in an antiproton sample. When the number

of positrons exceeds the number of antiprotons, trapping is more likely to

occur because the positrons are held by the rf trap pseudopotential and

antiprotons are held by the attractive space charge of the positron sample.

TN-177
PROBLEMS

The analysis in this paper has considered only the static properties of

nonneutral ion plasmas held in Penning- type or rf-type traps. The calculated

maximum densities and numbers will be reduced by various effects which we might classify as plasma instabilities.

Penning- type traps

The most serious problem appears to be due to asymmetry induced

transport. Radial transport can also occur due to ion collisions with

background gas, but at very high vacuum, this process is negligible. The

basic idea of asymmetry induced transport is that if the trap is not axially

symmetric, angular momentum can be coupled into the ions which causes them to

spread radially in the trap. Eventually, the ions will strike the electrodes

and be lost. During this spreading process, the electrostatic space-charge

energy will be converted to heat. In a cylinder Penning- type trap, spreading

rates proportional to (h/B)^ have been observed,^® where h is the plasma

length (Fig. 2). Although the spreading mechanism is not understood, one possibility might be resonant particle transport .^ • ^® • ^^ Resonant particle

transport might occur if the trap geometric axis and magnetic field axis are misaligned. As the ion density increases, w becomes larger. As the

temperature becomes lower, the individual ion axial frequency, cj^ , defined as the mean axial velocity, v^ , divided by h/7r, becomes smaller. When w^ " '*'> ^ coupling exists^ '^^ which can simultaneously heat the axial motion and

increase the ion radii. At the densities required for this condition, w^

becomes less well defined due to collisions, but the same mechanism may still

apply. For oj - w^ , the densities may be considerably less than the maximum

density given by Eq. 11. For small samples, some experimental evidence for

such a resonant effect exists.^

Independently of the mechanism, such transport is likely caused by trap

asymmetries. Conversely, if axial symmetry is preserved, angular momentum is a constant of the motion and confinement is assured. ^° Therefore, axial

symmetry in the Penning- type traps appears to be at a premium.

rf-type traps

Historically, the mechanism of rf heating'' ° has prevented the attainment

of low temperatures and long confinement times in the rf traps. Therefore,

attainable densities have typically been less than those dictated by space

charge limitations (Eqs. 18 and 24). Cooling with a buffer gas allows the

space-charge limited densities to be obtained, but this option seems to be

precluded for antimatter storage.

The basic mechanism for ion- ion rf heating is that the rf driven

micromotion can impart energy to the secular motion, that is, the motion in

the pseudopotential well. In many experiments, this has limited the ion

kinetic energy to a value of about 1/10 of the well depth of the trap due to

evaporation.* • ^ ' ^° This problem is often accentuated since it is desirable to

operate the trap with large q^ values (Eq. 17) in order to obtain deep well

depths. For the quadrupole rf trap (Eqs. 15-20), the relation w^/fi - q^/i2j2)

holds, where co_ is the single ion secular frequency . * • ^ • ^ ° Therefore, for

TN-178
large values of q^ , w^ approaches and subharmonic excitation of the secular

motion becomes more likely.

For stored ions of the same sign of charge, it would appear that rf

heating could be made less by using much lower values of q^ (q^ < 0.01). This

reduces the ion micromotion amplitude at a given distance from the center of

the trap and it increases the mean spacing between ions. Both of these

effects would reduce the rf heating mechanism and thereby allow the space

charge density limit to be approached. Unfortunately, reducing q^ also

reduces n (Eq. 18), so that a reasonable compromise would have to be met to

achieve maximum stored numbers.

ACKNOWLEDGEMENTS

This work is supported by the U.S. Office of Naval Research and the U.S.

Air Force Office of Scientific Research. The author thanks J. J. Bollinger

for many useful discussions, and thanks C. Weimer and J. J. Bollinger for

helpful comments on the manuscript.

TN-179
7. T.M.

8. S.A.

9. D.J.

Soc.

10. T.M.

11. R.S.

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22. W.D. White, J.H. Malmberg, and C.F. Driscoll, Phys. Rev. Lett. 49, 1822

(1982).

23. D.L. Eggleston, T.M. O'Neil, and J.H. Malmberg, Phys. Rev. Lett. 53, 982

(1984).

IN- 180
24. J.D. Crawford, T.M O'Neil, and J.H. Malmberg, Phys . Rev. Lett. 54, 697

(1985).

25. L. S. Cutler, private communication.

26. G. Gabrielse, K. Helmerson, R. Tjoelker, X. Fei, T. Trainor, W. Kells,

and H. Kalinowsky, in Proc . of 1st Workshop on Antimatter Physics at Low

Energy, B.E. Bonner and L.S. Pinsky eds . , Ferrailab, Apr. 1986, p. 211.

27. F.G. Major and J. P. Schermann, Bull. Am. Phys. Soc. 16, 838 (1971).

28. C.F. Driscoll, K.S. Fine, and J.H. Malmberg, Phys. Fluids 29, 2015

(1986).

29. C.F. Driscoll and J.H. Malmberg, Phys. Rev. Lett. 50, 167 (1983).

30. D.A. Church andH.G. Dehmelt, J. Appl. Phys. 40, 3421 (1969); H.G.

Dehmelt, in Advances in Laser Spectroscopy . F.T. Arecchi , F. Strumia, and

H. Walther eds., Plenum, 1983, p. 153.

TN-181
IEEE JOURNAL OF QUANTUM ELECTRONICS. VOL. 24, NO. 9, SEPTEMBER 1988 1829 Thermal Shifts of the Spectral Lines in the F3/2 to /n/2 Manifold of an

Nd : YAG Laser

SHAO ZHONG XING and J. C. BERGQUIST

Abstract—We report the thermal shifts of eleven of the twelve lines from the "Fjj Stark energy levels to the *I,i/2 energy levels in an Nd : YAG laser for a temperature change from 20-200°C. The thermal

shift diflFerence between the Stark sublevels /?,, R2 in V3/2 is found to be about -0.6 ± 0.6 cm"'/100°C. Within our experimental uncertainty, all of the lasing lines either moved to longer wavelength or re- mained unchanged with increasing temperature.

Introduction

RECENT advances in high-power diode lasers have led

to extremely efficient, CW operation of neodymiumdoped crystal and fiber lasers [l]-[3]. There have also

been important strides toward low threshold operation of

miniature hybrid and monolithic neodymium-doped lasers

[1], [4], [5]. These devices have inherently much better mode quality and frequency stability than the diode lasers used as the pump sources. Additionally, several new nonlinear crystals that are used to double these lasers into the

visible have been developed with good optical quality and

large nonlinear coefficients [6]-[8].

For these reasons, interest in these devices as ultrasta- ble frequency sources near 1 and 0.5 /xm is strong. While

there are few atomic and molecular transitions near 1 /xm

with suitably narrow linewidths to which to lock a laser,

there are many such transitions in the visible. One very

attractive candidate that we are investigating is the 5d^^6s

'^Si/2-5d^6s^ '^5/2 transition in a single laser-cooled Hg"^

ion near 281 nm ( 1.126 ^m ^ 4) [9], [10]. This transi- tion has a natural linewidth of about 1.7 Hz [10], [11]. A

laser that has been spectrally narrrowed to less than 1 Hz

and tuned to this wavelength could be stabilized to this transition to better than 1 x 10~'^. The accuracy of such

a frequency standard should also exceed 1 x 10" '^ [12].

We have begun to study various neodymium-doped

crystals and fibers to determine their suitability as narrow- band radiation sources whose fourth harmonic could be

tuned to match the transition frequency in Hg"^. Unfortunately, the neodymium-doped crystal laser and bulk

glass lasers are generally not tunable by more than a few

wavenumbers at any temperature. But some of the transitions do tune by as much as several wavenumbers per

Manuscript received March 9, 1988; revised April 20, 1988. This work was supported in part by the U.S. Air Force Office of Scientific Research and by the Office of Naval Research.

S. Z. Xing is with the Changchun Institute of Optics and Fine Mechanics, Changchun, China, on leave at the Time and Frequency Division, National Bureau of Standards, Boulder, CO 80303.

J. C. Bergquist is with the Time and Frequency Division, National Bureau of Standards, Boulder, CO 80303. IEEE Log Number 882227.

100°C change in temperature (generally to lower frequencies). In this letter we report on our measurements of the thermal shifts of the spectral lines in the ^F^/2 to '*/i]/2 manifold of a neodymium-doped, yttrium-aluminumgamet (Nd:YAG) laser. Although the linewidths, spectral lineshapes, and thermal shifts of several lines of

Nd : YAG have been studied both theoretically and experimentally [13] -[20] we were able to measure the thermal

shifts of all the laser lines in this manifold with the ex- ception of the weak Rj-Yj transition near 1.0551 nm.

Thereby we were able to do a comprehensive study of the

temperature dependence of these lines in the temperature

range from 20-200°C.

Experimental Setup

The experimental setup is shown in Fig. 1. Although

we used a ring laser cavity, the oscillation of the laser was

bidirectional. The radius of curvature of the concave mirrors Ml and M2 was 10 cm, and of M3, 15 cm. M4 was

flat. To ensure that all the lines from the V3/2 manifold

to the '*/ii/2 manifold would oscillate, all the mirrors were

highly reflecting from 1.0-1.15 /xm. The 5 mm diameter

by 10 mm long Nd:YAG crystal was cut at Brewster's

angle for low loss and ease of cleaning. The crystal was

placed in the 80 /xm beam waist between mirrors Mj and

M3. The cavity was adjusted to nearly compensate for the

astigmatism introduced by the crystal. The Nd:YAG

crystal was end pumped by laser light near 750 nm that was focused through mirror M3. M^ was nearly 90 percent

transmitting at 750 nm. The pump laser was a ring dye

laser pumped by a krypton ion laser. The output radiation

from the dye laser could be tuned from 735 to 800 nm

with as much as 600 mW of power. The oven for the crystal was made of aluminum and was heated by a 15 W

thermal plug-heater. A chromel-alumel thermocouple

monitored the temperature oi the oven. The birefringent

filter (BF) consisted of a single quartz plate approximately

5 mm thick. The monochromiator is a 0.8 m focal length

scanning monochromator with a 150 lines/mm grating

blazed for an 8 /xm wavelength. A 40 /xm slit was generally used in the entrance plane of the monochromator. In the exit plane of the monochromator, a diode array with

a resolution of 25 /xm was used to receive and display the spectrum signal on an oscilloscope.

Experimental Results

The twelve possible transitions from the upper ^3/2

manifold to the lower '^lu/i manifold in Nd:YAG are

0018-9197/88/0900-1829$01.00 © 1988 IEEE

TN-182
1830 IEEE JOURNAL OF QUANTUM ELECTRONICS. VOL. 24. NO. 9. SEPTEMBER 1988

Pumping Beam

Monochromator

Ma Power

Monitor

To Scope

Diode Array -^

Fig. 1. The setup used to measure the thermal shifts of spectral lines in Nd : YAG laser. The ring laser consists of mirror, M, , Afj, M,, and M^,

a single plate birefringent filter (BF), and an Nd : YAG crystal. The Nd : YAG crystal is housed in an oven that can be heated from room

temperature to >200°C. The crystal is axially pumped by up to 600 mW

of power from a ring dye laser tunable near 750 nm. A diode array in the exit plane of the 0.8 m focal length monochromator receives and

displays the signal on a scope.

shown in Fig. 2. Customarily, the Stark sublevels are la- beled with increasing energy as /?i and /?2 in the V3/2

manifold and as F) and Y^ in the '*/ii/2 manifold. The ring

laser oscillated on all of the transitions with the exception

of the Rj-Yj transition near 1.0551 /xm. Some of the light

from the Nd : YAG laser that escaped through M4 was di- rected into the monochromator. The diode array ia the

exit plane of the monochromator made it possible to monitor the thermal shift and gain width of each transition

easily. The array had 512 diodes and a resolution of 25

ptm. To make a thermal shift measurement, the monochromator setting was adjusted until the signal was centered on the diode array. This setting was noted and the birefringent filter was then adjusted to allow another transition to oscillate. By tuning the birefringent filter over

the line to find the maximum power, we could roughly

determine line center. The monochromator setting was

changed to bring the diffracted signal for the wavelength

of this transition to the center of the diode array and this setting was noted. Several readings for all eleven transi- tions were taken at one temperature, then the oven temperature was changed, and the entire process repeated. On

most lines, the line center could be measured to only a few tenths of a wavenumber. This limitation was not due

to poor signal-to-noise ratio but rather due to large asymmetries in the line. Some of these asymmetries were

caused by fundamental asymmetric-broadening mechanisms in the crystal [13]. But the more difficult problem,

because of our inability to properly characterize it, was

the fact that the diflFerent transitions competed to oscillate. Before fully tuning across any line, the laser was likely to jump to another transition. The wavelength at which the jump occurred depended on the birefringent filter position, the mirror reflectances and losses at the various

wavelengths, and the gain profile of the diflFerent transitions. Because of this systematic uncertainty, the line centers could not be located to better than ±0.5 cm~'.

In Fig. 3 the thermal shifts of the eleven transitions that lased are plotted as a function of temperature from 20 to 200°C (solid lines). For all lines except the R1-Y4 transition at 1.0779 ^m and the Rj -* I4 transition at 1.0682

F3/2

(cm-')

11504

11418 EEEEfll-l 3. 3. a. 3. 3. E E E p

a. 3. 3. 5

00 cvj to .- in — ;-:,-,-?:oooooooo

Ye

Y5

hi/2

Y4

Y3

Y2

Y,

2515 2473

2145

2100

2025

2002

Fig. 2. A simplified optical energy level diagram showing the ''Fj -2 manifold and the *ln/2 manifold in Nd ; YAG. The twelve possible infrared

transitions from the upper two Stark sublevels (R^, Ri) to the lower six Stark sublevels ( V, - Y^) are also indicated.

200

Temperature (°C)

Fig. 3. Plot showing the variation in the wavelength for maximum laser power as a function of Nd : YAG crystal temperature for all 12 lines from

the *Fj/2 manifold to the *lu, 1 manifold. The dotted line is the deduced

temperature variation for the R2 -* K, transition at 1.0551 jzm that did

not oscillate.

^m, the shifts appear to vary linearly with temperature.

The strongly nonlinear thermal movement of these two

lines is due to the shift and broadening with temperature

of the Y4 Stark sublevel. In Table I we list the thermal

shifts of all the transitions giving an endpoint-to-endpoint

average shift for the /?,, /?2 ~* J4 transitions. From our work there is fivefold redundancy in determining the rel- ative thermal shift of the R^ and /?2 levels (in principle, there would be sixfold redundancy but this is reduced by

1 since the /?2 -* Y2 line did not oscillate). While our

results are not inconsistent with a zero relative temperaTN-183
IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 24, NO. 9. SEPTEMBER 1988 1831

TABLE I Thermal Shifts of the Transitions Between the Stark Energy

SUBLEVELS INTHE''F3/2 MaNIFOLD AND IN THE "/,

, /2 MANIFOLD, AND THE

Relative Thermal Shift of the R^ and Rj Stark Levels Shift in an Nd : YAG Laser

1.1/2

1^1 Ya Y3 Y4 Y5 Ye

Ri

WavelengthC^im) 1.0615 1,0666 1.0738 1.0779 1.1161 1.1225

Thermal shift (cn"'/100°C) -3.0 -3.3 -3.5 -6.1 0.0 0.0

Wavelength(Mm) 1.052 1.0551 1.0641 1.0682 1.1055 1.1121

Thermal shift (cm-'/100°C) -4.0 (-4.3) -4.6 -7.0 0.0 0.0

Relative thermal

shift between Rj & Rj (cn-VlOO-C)

-1.0 (-1.0) -1.1 -0.9 0.0 0.0

ture shift between /?, and Rj, there is an indication of a small linear separation of these levels with increasing

temperature. This possibility was also noted in [3]. There

is twofold redundancy in determining the relative ther- mal shift of any pair of Stark sublevels in the lower '*/,

, /2 level. These redundancies provide a self-consistency

check of the relative movement of the Stark sublevels in each manifold. Thus, we are able to predict the thermal

shift of the weak Rj -> Y2 1.0551 /xm transition to be ap- proximately —4.3 cm~'/100°C. This is plotted as the

dashed line in Fig. 2. Fig. 3 shows that for those lines with shorter wavelength (and stronger intensity), the 1.052, 1.0615, 1.0641,

and 1.0738 /xm lines, the thermal shifts are roughly linear

and the shift difference in each pair is consistent. The

1.052 fim line moves about —7.3 cm~' from room temperature to 200°C or about -4.0 cm~'/100°C. This is in agreement with the shift measured by Kushida [13]. The

1.0615 /xm line moves about — 3.0cm~' for a temperature

change of 100°C. This compares to the —2.7

cm~'/100°C thermal variation measured by Marling [14]

and to the -3.8 cm~'/100°C temperature shift measured

by Kushida [13]. Thus the relative shift with temperatures

of the /?, and i?2 Stark levels is about -1 cm"7l00°C

from this pair of transitions. Almost the same relative thermal shift is obtained for the pair consisting of the

1.0641 fim line (Rj-Y^) and the 1.0738 fim line (R^-Y^).

The 1.0641 /xm transition moves about —4.6

cm~'/100°C and the 1.0738 /xm transition moves about — 3.5 cm~'/100°C. Thus, this pair indicates a relative

shift of about -1.1 cm"'/100°C between the 7?, and R2

Stark levels. From these two pairs of lines, it would appear that the

i?2 sublevel in the '^F^/j manifold moves to a lower energy

with increasing temperature about 1 cm~'/100°C faster than the /?, level. However, for the lines at 1.1055 /im

(R2-Yi), 1.1121 /xm (Rj-Ye), 1.1161 fim (R^-Ys) and

1.1225 /xm (R^-Yf,) we measured almost no shift for a change of nearly 200°C in temperature. Thus, these lines indicate that /?i and Rj shift with temperature at nearly the same rate.

The thermal shift of the 1.0682 /xm line (/^z-Kj) and of

the 1.0779 /xm line (R^-Y^) are strongly nonlinear in this temperature range. Thus it is difficult to determine the rel- ative shift of /?, and Rj from this pair. As stated earlier, we did assign an endpoint-to-endpoiht average shift for these two lines. This result and the thermal shifts for the

remaining transitions are summarized in the table. By as- signing equal weight to each of the measured shift differ- ences, we could average our results to give a relative ther- mal shift of -0.6 ± 0.6 cm"'/100°C between Stark

sublevels ^, and Rj. The error of ±0.6 cm"7l00°C in- cludes both the statistical and the systematic uncertainties.

Discussion

There are a number of reasons the various transitions in the ''F3/2 -> ^l\\/2 manifold have different measured ther- mal shifts. The crystalline field at the site of the impurity

Nd ion is a function of the local strain. When this strain

is dynamically produced by the latiice vibrations, the in- teraction between the Nd ion and the local crystalline field causes temperature-dependent broadenings and shifts of

the energy levels of the ion. The thermal behaviors of

linewidths and line positions are frequently dominated by

this dynamic phonon-impurity interaction not only in the Nd : YAG crystal but in other impurity : crystalline hosts

as well [13]. The crystalline field strength can also be

changed by thermal expansion of the crystalline lattice,

but this appears to be a minor contribution to the thermal

shifts and broadenings in hard crystals such as Nd : YAG

[13]. When the temperature of the crystal is increased,

energy levels and hence, spectral lines, broaden as the higher phonon states are occupied. Usually, the phononimpurity interaction lowers an energy level and usually

the dependence with temperature of higher levels is larger

than for lower levels because of smaller energy denominators. As a result, the spectral lines are generally ex- pected to move to the longer wavelengths when the temperature is increased [13]. This general feature is confirmed by our experimental results. We also observed that the longer the wavelength the wider the linewidth and the more complex the thermal

shift. The shorter wavelength lines have relatively narrow

and symmetrical lineprofiles ( Aj' ~ 5 cm~' ) that broaden

only slightly with temperature. Therefore we can deduce

that both of the upper Stark levels /?2 and R\ and the lower

Stark levels y,, Y2, Y^ remain spectrally narrow in the temperature range from 20-200 °C. We also found the

shift of these transitions to be simply linear in this temperature range. The intermediate-wavelength transitions

have a wider and asymmetric line profile (Af 5: 10

cm"'), and from the arguments above, that implies that the Y^ Stark level is >5 cm~' wider than y,, Y2, and Y^. The fact that the transitions that terminate on the Y^ Stark

level move in a nonlinear fashion with temperature in our measurement range might be explained by the asymmetrical thermal broadening of the Y^ level. While some of

the thermal broadening mechanisms give rise to a symTN-184
1832 IEEE JOURNAL OF QUANTUM ELECTRONICS. VOL. 24, NO. 9. SEPTEMBER 1988

metrical line shape, other mechanisms, such as the direct,

single-resonant-phonon interaction, cause asymmetric

broadening [13]. If the line profile, first broadened asymmetrically to the red ( ~ 80°C) because of the thermal behavior of the K4 level, then the measured line shift would

appear to move strongly toward the red. If later

( ~ 150°C) the broadening becomes more symmetric (or

asymmetric to the blue) then the rate of the red shift would

decrease.

The longest wavelength transitions have the widest

linewidths (Av > 13 cm"') among all the transitions between the '*F3/2 and '*/ii/2 manifolds. Thus the Y^ and Y^ Stark sublevels are the broadest in the Vi,/2 manifold.

Hence the thermal shifts of these transitions are somewhat

obscured by the thermal broadening over our relatively

small change in temperature. Clearly these upper energy

levels in the lower Stark manifold generally move with /?|

and /?2 with change in temperature. These results are in basic agreement with earlier work on the thermal shift of some of these transitions [13]-[18].

Acknowledgment

The authors wish to thank D. Nabors, R. Byer, and T. Baer for many helpful discussions.

References

[1] B. Zhou, T. J. Kane, G. J. Dixon, and R. L. Byer, "Efficient, fre- quency-stable laser-diode-pumped NdiYAG laser," Opt. Lett., vol.

10, pp. 62-64. Feb. 1985.

[2] T. M. Baer, "Frequency doubled, diode pumped Nd:YAG laser," Proc. Soc. Photo-Opt. Instrum. Eng., vol. 610, pp. 45-47, Jan. 1986.

[3] R. J. Mears, L. Reekie, S. B. Poole, and D. N. Payne, "Neodymium-doped silica single-mode fibre lasers," Electron. Lett., vol. 21,

pp. 738-740, Aug. 15, 1985; I. P. Alcock, A. I. Ferguson, D. C. Hanna, and A. C. Tropper, "Tunable, continuous-wave neodymiumdoped monomode-fiber laser operating at 0.900-0.945 and 1.070-

l.\35 tim," Opt. Lett., vo\. 1 1, pp. 709-71 1, Nov. 1986.

[4] T. J. Kane and R. L. Byer, "Monolithic, unidirectional single-mode

NdrYAG ring laser," Opt. Lett., vol. 10, pp. 65-67, Jan. 1985.

[5] W. R. Trutna, Jr., D. K. Donald, and M. Nazarathy, "Unidirectional diode-laser-pumped Nd : YAG ring laser with a small magnetic field," Opt. Lett., vol. 12, pp. 248-250, Apr. 1987.

[6] W. J. Kozlovsky, C. D. Nabors, and R. L. Byer, "Efficient second

harmonic generation of a diode-laser-pumped CW Nd : YAG laser using monolithic MgO : LiNbOj external resonant cavities,

'

' IEEE J. Quantum Electron., vol. 24, pp. 913-919, June 1988, R. L. Byer

"Diode pumped solid state laser," Science, to be published; T. Y.

Fan, G. J. Dixon, and R. L. Byer, "Efficient GaAlAs diode-laser- pumped operation of Nd : YLF at 1 .047 ^m with intracavity doubling

to 523.6 nm," Opt. Lett., vol. 11, pp. 204-206, Apr, 1986.

[7] See, for example, J. T. Lin and C. Chen, "Choosing a nonlinear

crystal," Laser Optron., vol. 6, pp. 59-63, Nov. 1987, and refer- ences therein.

[8] T. Baer, "Large-amplitude fluctuations due to longitudinal mode coupling in diode-pumped intracacity-doubled Nd : YAG lasers," J. Opt.

Soc. Amer. B, vol. 3, pp. 1175-1180, Jan. 1986.

[9] J. C. Bergquist, W. M. Itano, and D. J. Wineland, "Recoilless optical absorption and Doppler sidebands in a single trapped ion," Phys.

Rev. A, vol. 36, pp. 428-430, July 1, 1987.

[10] J. C. Bergquist, D. J. Wineland, W. M. Itano, H. Hemmati. H.-U.

Daniel, and G. Leuchs, "Energy and radiative lifetime of the 5d''6s^ ^Di/2 state in Hgll by Doppler- free two-photon laser sf>ectroscopy," Phys. Rev. Lett., vol. 55, pp. 1567-1570, Oct. 7, 1985.

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

"Radiative decay rates in Hg-t- from observations of quantum jumps

in a single ion," Phys. Rev. Lett., vol. 59, pp. 2732-2734, Dec. 14, 1987.

[12] D. J. Wineland, W. M. Itano, J. C. Bergquist and R. G. Hulet, "Laser

cooling limits and single-ion spectroscopy," Phys. Rev. A, vol. 36,

pp. 2220-2232, Sept. 1, 1987.

[13] T. Kushida, "Linewidths and thermal shifts of spectral lines in neo- dymium-doped yttrium aluminum garnet and calcium fluorophos- pate," Phys. Rev., vol. 185, pp. 500-508, Sept. 10, 1969.

[14] J. Marling, "1.05-1.44 ^m tunability and performance of the CW

Nd^* : YAG laser," IEEE J. Quantum Electron., vol. QE-14, pp. 56-

62, Jan. 1978.

[15] A. A. Kaminskii, "High-temperature spectroscopic investigation of

stimulated emission from lasers based on crystals and glasses acti- vated with Nd'* ions," Sov. Phys. JETP, vol. 27, pp. 388-399, Sept. 1968.

[16] D. E. McCumber and M. D. Sturge, "Linewidths and temperature

shift of the R lines in ruby," J. Appl. Phys., vol. 34, pp. 1682-1684,

June 1963.

[17] W. M. Yen, W. C. Scott, and A. L. Schawlow, "Phonon-induced

relaxation in excited optical states of trivalent praseodymium in LaFj," Phys. Rev., vol. 136, pp. A271-A283, Oct. 5. 1964.

[18] T. Kushida and M. Kikuchi, "R, R' and B absorption linewidths and phonon-induced relaxations in ruby," J. Phys. Soc. Japan, vol. 23,

pp. 1333-1348, Dec. 1967.

[19] M. Bimbaum and C. F. Klein, "Stimulated emission cross section at 1.061 nm in NdiYAG," J. Appl. Phys., vol. 44, pp. 2928-2930,

June 1973.

[20] V. A. Buchenkov, I. B. Vitrishchak, V. G. Evdokimova, L. N. Soms,

A. Stepanov, and V. K. Stupnikov, "Temperature dependence of giant pulse amplification in YAG : Nd''*," Sov. J. Quantum Electron., vol.

11, pp. 702-705, June 1981.

TN-185

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4. TITLE AND SUBTITLE

Trapped Ions and Laser Cooling II

5. AUTHOR(S)

D.J. Wineland, W.M. Itano, J.C. Bergqulst, J.J. Bollinger

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions)

National Institute of Standards and Technology

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This Technical note is a collection of selected reprints of the Ion Storage Group

for the period July 1985 to September 1988.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

atomic clocks; atomic spectroscopy; frequency standards; ion storage; laser cooling;

laser spectroscopy; trapped ions

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