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PUBLICATIONS NIST Technical Note 1524

Trapped Ions and Laser Cooling, VI

l99 Hg+ S^D

Edited by

James C. Bergquist

John J. Bollinger

Wayne M. Itano

David J. Wineland

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Selected publications of the Ion Storage Group

NIST Time and Frequency Division

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NIST Technical Note 1524

Trapped Ions and Laser Cooling, VI

Selected publications of the Ion Storage Group

NIST Time and Frequency Division

Edited by

James C. Bergquist

John J. Bollinger

Wayne M. Itano David J. Wineland

Time and Frequency Division

Physics Laboratory

National Institute of Standards and Technology

325 Broadway

Boulder, Colorado 80305

February 2002

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National Institute of Standards and Technology Technical Note 1524

Natl. Inst. Stand. Technol. Tech. Note 1524, 208 pages (February 2002)

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CONTENTS

LIST OF ADDITIONAL PUBLICATIONS vi FREQUENCY STANDARDS AND METROLOGY

1

"Lasers for an optical frequency standard using trapped Hg+ ions," B.C. Young,

F.C. Cruz, D.J. Berkeland, R.J. Rafac, J.C. Bergquist, W.M. Itano, and D.J. Wineland,

Trapped Charged Particles and Fundamental Physics, D.H.E. Dubin and

D. Schneider, eds., Proc, AIP Conf. 457, Woodbury, NY: AIP Press (1999), pp. 337-342. TN-1

2. "High-resolution, high-accuracy spectroscopy of trapped ions," D.J. Berkeland, J.D.

Miller, F.C. Cruz, B.C. Young, R.J. Rafac, X.-P. Huang, W.M. Itano, J.C. Bergquist,

and D.J. Wineland, Atomic Physics 16, W.E. Baylis and G.W.F. Drake, eds., Proc, AIP

Conf. 477, Woodbury, NY: AIP Press (1999), pp. 29-41. TN-7

3. "Hg+ optical frequency standard: recent progress," B.C. Young, R.J. Rafac, J.A. Beall, F.C. Cruz, W.M. Itano, D.J. Wineland, and J.C. Bergquist, Laser Spectroscopy,

XIV Int. Conf., R. Blatt, J. Eschner, D. Leibfried, and F. Schmidt-Kaler, eds., Singapore: World Scientific (1999), pp. 61-70. TN-19

4. "Sub-dekahertz ultraviolet spectroscopy of 199 Hg+

," R.J. Rafac, B.C. Young, J.A. Beall,

W.M. Itano, D.J. Wineland, and J.C. Bergquist, Phys. Rev. Lett. 85, 2462-2465 (2000). TN-29

5. "External-field shifts of the

199Hg+ optical frequency standard," W.M. Itano, J. Res. Natl.

Inst. Stand. Technol. 105, 829-837 (2000). TN-33

6. "Direct comparison of two cold-atom-based optical frequency standards by using a

femtosecond-laser comb," K.R. Vogel, S.A. Diddams, C.W. Oates, E.A. Curtis, R.J. Rafac, W.M. Itano, J.C. Bergquist, R.W. Fox, W.D. Lee, J.S. Wells, and L. Hollberg,

Opt. Lett. 26, 1 02- 1 04 (200 1). TN-42

7. "Time measurement at the millenium," J.C. Bergquist, S.R. Jefferts, and D.J. Wineland,

Phys. Today, 37-42 (March 2001). TN-45

8. "Sub-dekahertz spectroscopy of 199Hg+

," J.C. Bergquist, R.J. Rafac, B.C. Young,

J.A. Beall, W.M. Itano, and D.J. Wineland, Laser Frequency Stabilization, Standards,

Measurement, and Applications, J.L. Hall and J. Ye, eds., Proc, SPIE 4269 (2001), pp. 1-7. TN-51

9. "Absolute frequency measurements of the Hg+ and Ca optical clock transitions with a femtosecond laser," Th. Udem, S.A. Diddams, K.R. Vogel, C.W. Oates, E.A. Curtis,

W.D. Lee, W.M. Itano, R.E. Drullinger, J.C. Bergquist, and L. Hollberg, Phys. Rev. Lett. 86,4996-4999(2001). TN-58

10. "An optical clock based on a single trapped 199Hg+ ion," S.A. Diddams, Th. Udem,

J.C. Bergquist, E.A. Curtis, R.E. Drullinger, L. Hollberg, W.M. Itano, W.D. Lee,

C.W. Oates, K.R. Vogel, D.J. Wineland, Science 293, 825-828 (2001). TN-62

1 1

"All optical atomic clocks based on a single mercury ion and calcium atoms," R.E. Drullinger,

Th. Udem, S.A. Diddams, K.R. Vogel, C.W. Oates, E.A. Curtis, W.D. Lee, W.M. Itano, L. Hollberg, and J.C. Bergquist, Proc, 2001 European Frequency and Time Forum,

(2001), pp 441-445. TN-66

111
LASER-COOLED NON-NEUTRAL PLASMAS

12. "Mode and transport studies of laser-cooled ion plasmas in a Penning trap," T.B.

Mitchell, J.J. Bollinger, X.-P. Huang, and W.M. Itano, Trapped Charged Particles and

Fundamental Physics, D.H.E. Dubin and D. Schneider, eds., Proc, AIP Conf. 457,

Woodbury, NY: AIP Press (1999), pp. 309-318. TN-71

13. "Direct observations of the structural phases of crystallized ion plasmas," T.B. Mitchell, J.J. Bollinger, X.-P. Huang, W.M. Itano, and D.H.E. Dubin, Phys. Plasmas 6, 1751-1758 (1999). TN-81

14. "Crystalline order in laser-cooled, non-neutral ion plasmas," J.J. Bollinger, T.B. Mitchell, X.-P. Huang, W.M. Itano, J.N. Tan, B.M. Jelenkovic, and D.J. Wineland, Phys. Plasmas 7, 7-13 (2000). TN-89

15. "Simulation of a method for forming a laser-cooled positron plasma," A.S. Newbury,

B. Jelenkovic, J.J. Bollinger, and D.J. Wineland, Phys. Rev. A 62, 023405-1 (-10) (2000). TN-96

16. "Stick-slip dynamics of a stressed ion crystal" T.B. Mitchell, J.J. Bollinger, W.M. Itano, and D.H.E. Dubin, Phys. Rev. Lett. 87, 183001-1-4 (2001). TO- 106

QUANTUM MECHANICS AND QUANTUM METROLOGY

17. "Quantum computation, spectroscopy of trapped ions, and Schrbdinger's cat," D.J. Wineland, C. Monroe, W.M. Itano, D. Kielpinski, B.E. King, C.J. Myatt,

QA. Turchette, and C.S. Wood, Quantum Coherence and Decoherence - ISQM - Tokyo,

Y.A. Ono and K. Fujikawa, eds., Elsevier Science B. V. ( 1 999), pp. 1 03- 1 08. TO- 1 1

18. "Trapped-ion quantum simulator," D.J. Wineland, C. Monroe, W.M. Itano, B.E. King,

D. Leibfried, C. Myatt, and C. Wood, Proc, Nobel Symp. 104: Modern Studies

of Basic Quantum Concepts and Phenomena, Phys. Scripta T76, 147-151 (1998). TO- 116

19. "Decoherence of quantum superpositions through coupling to engineered reservoirs,"

C.J. Myatt, B.E. King, QA. Turchette, CA. Sackett, D. Kielpinski, W.M. Itano, C. Monroe, and D.J. Wineland, Nature, 403, 269-273 (2000). TO- 121

20. "Experimental entanglement of four particles," CA. Sackett, D. Kielpinski, B.E. King,

C. Langer, V. Meyer, C.J. Myatt, M. Rowe, QA. Turchette, W.M. Itano, D.J. Wineland,

and C. Monroe, Nature 404, 256-259 (2000). TO- 126

21"Sympathetic cooling of trapped ions for quantum logic," D. Kielpinski, B.E. King,

C.J. Myatt, CA. Sackett, Q.A. Turchette, W.M. Itano, C. Monroe, D.J. Wineland,

and W.H. Zurek, Phys. Rev. A 61, 032310-1-8 (2000). TN-130

22. "Scalable entanglement of trapped ions," C. Monroe, CA. Sackett, D. Kielpinski, B.E. King,

C. Langer, V. Meyer, C. Myatt, M. Rowe, Q.A. Turchette, W.M. Itano, D.J. Wineland,

Atomic Physics 17, E. Arimondo, P. De Natale, and M. Inguscio, eds., Proc, AIP Conf.

551, Melville, NY: AIP Press, (2001) pp. 173-186. TN-138

23. "Heating of trapped ions from the quantum ground state," Q.A. Turchette, D. Kielpinski, B.E. King, D. Leibfreid, D.M. Meekhof, C.J. Myatt, M.A. Rowe, CA. Sackett, C.S. Wood, W.M. Itano, C. Monroe, and D.J. Wineland, Phys. Rev. A 61 , 0634 1 8- 1 - 8 (2000). TO- 1 52

IV
24. "Decoherence and decay of motional quantum states of a trapped atom coupled to engineered reservoirs," Q.A. Turchette, C.J. Myatt, B.E. King, C.A. Sackett, D. Kielpinski,

W.M. Itano, C. Monroe, and D.J. Wineland, Phys. Rev. A 62, 053807-1-22 (2000). TN-160

25. "A decoherence-free quantum memory using trapped ions," D. Kielpinski, V. Meyer,

M.A. Rowe, C.A. Sackett, W.M. Itano, C. Monroe, and D.J. Wineland, Science 291,

1013-1015(2001). TN-182

26. "Experimental violation of Bell's inequalities with efficient detection," M.A. Rowe,

D. Kielpinski, V. Meyer, C.A. Sackett, W.M. Itano, C. Monroe, and D.J. Wineland,

Nature 409, 79 1 -794 (200 1 ). TN- 1 85

27. "Experimental demonstration of entanglement-enhanced rotation angle estimation using

trapped ions," V. Meyer, M.A. Rowe, D. Kielpinski, C.A. Sackett, W.M. Itano, C. Monroe,

and D.J. Wineland, Phys. Rev. Lett. 86, 5870-5873 (200 1 ). TN- 1 89

TECHNIQUES:

28. "Diode-pumped Nd:FAP laser at 1.126 urn: A possible local oscillator for a Hg+ optical

frequency standard," F.C. Cruz, B.C. Young, and J.C. Bergquist, Appl. Opt. 37,

7801-7804(1998). TN-193

29. "Visible lasers with subhertz linewidths," B.C. Young, F.C. Cruz, W.M. Itano, and

J.C. Bergquist, Phys. Rev. Lett. 82, 3799-3802 (1999). TN-197
ADDITIONAL PUBLICATIONS

The following publications were published during the period between January 1999 and October 2001 but are not included in this Technical Note.

1. " 199Hg+ optical frequency standard: Progress report," J.J. Rafac B.C. Young, F.C. Cruz, J. A. Beall, J.C. Bergquist, W.M. Itano, and D.J. Wineland, Proc, 1999 IEEE Int. Frequency Control Symp. IEEE catalog

number 99CH36313, Piscataway, NJ: IEEE (1999), pp. 676-681.

2. "High-accuracy frequency standards using laser-cooled '"Hg

+ ions," D.J. Berkeland, J.D. Miller, J.C. Bergquist, W.M. Itano, and D.J. Wineland, Gravitational Waves and Experimental Gravity, J. Tran Thanh

Van, J. Dumarchez, S. Reynaud, C. Salomon, S. Thorsett, and J.Y. Vinet, eds., Proc, XXXIVth Rencontres

de Moriond Les Arcs, Hanoi, Vietnam: World Publishers (2000), pp. 371-378.

3. "Primary atomic frequency standards at NIST," D.B. Sullivan, J.C. Bergquist, J.J. Bollinger, R.E.

Drullinger, W.M. Itano, S.R. Jefferts, W.D. Lee, D. Meekhof, T.E. Parker, F.L. Walls, and D.J. Wineland,

J. Res. Natl. Inst. Stand. Technol. 106, 47-63 (2001).

4. "A compact femtosecond-laser-based optical clockwork," S.A. Diddams, Th. Udem, K.R. Vogel, C.W.

Oates, E.A. Curtis, R.W. Windeler, A. Battels, J.C. Bergquist, and L. Hollberg, Proc, SPIE 4269 (2001).

5. "Crystalline order in strongly coupled ion plasmas," T.B. Mitchell, J.J. Bollinger, X.-P. Huang, W.M. Itano, J.N. Tan, B.M. Jelenkovic, and D.J. Wineland, Non-neutral Plasma Physics III, J.J. Bollinger, R.L. Spencer, and R.C. Davidson, eds., Proc, AIP Conf. 498, Melville, NY: AIP Press (1999), pp. 353-366.

6. Non-neutral plasma physics, III, J.J. Bollinger, R.L. Spencer, and R.C. Davidson, eds., Proc, AIP Conf.

498, Melville, NY: AIP Press (1999).

7. "Crystalline order in strongly coupled plasmas," J.J. Bollinger, T.B. Mitchell, X.-P. Huang, W.M. Itano, J.N. Tan, B.M. Jelenkovic, and D.J. Wineland, Trapped Charged Particles and Fundamental Physics, D.H.E. Dubin and D. Schneider, eds., Proc, AIP Conf. 457 Woodbury, NY: AIP Press (1999), pp. 295-

304.

8. "Atomic ion crystals in non-neutral plasmas," J.J. Bollinger, T. Mitchell, X.-P. Huang, W.M. Itano, J.N. Tan, B.M. Jelenkovic, and D.J. Wineland, Atomic Physics 16, W.E. Baylis and G.W.F. Drake, eds., Proc,

AIP Conf. 477 Woodbury, NY: AIP Press (1999), pp. 87-99.

9. "Negative Poisson's ratios for extreme states of matter," R.H. Baughman, S.O. Dantas, S. Stafstrom, A.A.

Zakhidov, T.B. Mitchell, and D.H.E. Dubin, Science 288, 2018-2022 (2000).

10. "A laser-cooled positron plasma," B.M. Jelenkovic, J.J. Bollinger, A.B. Newbury, T.B. Mitchell, and W.M.

Itano, New Directions in Antimatter Physics and Chemistry, C. Surko and F. Gianturco, eds., The

Netherlands: Kluwer Academic Publishers (2001), pp. 1-20.

11. "Conference report: Workshop on non-neutral plasmas," J.J. Bollinger and J. Fajans, Comments on Plasma

Physics and Controlled Fusion, 18(6), 41 1-422 (1999).

12. "Crystalline plasma," W.M. Itano, McGraw-Hill Yearbook of Science and Technology, 2001, New York,

NY: McGraw-Hill (2000), p. 1 16-1 18.

13. "Quantum logic with a few trapped ions," C. Monroe, W.M. Itano, D. Kielpinski, B.E. King, D. Leibfried,

C.J. Myatt, Q.A. Turchette, D.J. Wineland, and C.S. Wood, Trapped Charged Particles and Fundamental

Physics, D.H.E. Dubin and D. Schneider, eds., Proc, AIP Conf. 457, Woodbury, NY: AIP Press (1999), pp.

vi
378-387.

14. "Searches for anomalous interactions using trapped ions," D.J. Wineland, J.J. Bollinger, W.M. Itano, J.C. Bergquist, C. Monroe, CPT and Lorentz Symmetry, V. Alan Kostelecky, ed., Singapore: World Scientific (1999), pp. 87-93.

15. "Decay of quantum superpositions into engineered reservoirs," C.J. Myatt, B.E. King, Q.A. Turchette, C.A.

Sackett, D. Kielpinski, W.M. Itano, C. Monroe, and D.J. Wineland, Laser Spectroscopy, XIV International Conf., R. Blatt, J. Eschner, D. Leibfried, and F. Schmidt-Kaler, eds., Singapore: World Scientific (1999),

pp. 237-245.

16. "Decoherence of motional states of trapped ions," C.J. Myatt, B.E. King, Q.A. Turchette, C.A. Sackett, D.

Kielpinski, W.M. Itano, C. Monroe, and D.J. Wineland, J. Mod. Op. 47, 2181-2186 (2000).

17. "Computing with atoms and molecules?" C. Monroe and D. J. Wineland, Science Spectra 23, 72-79 (2000).

18. "Superposition and quantum measurement of trapped atoms," D. J. Wineland, C. R. Monroe, C. Sackett, D.

Kielpinski, M. Rowe, V. Meyer, and W. Itano, Ann. Phys. (Leipzig) 9, 851-854 (2000).

19. "Sub-systems for optical frequency measurements: application to the 282 nm 199Hg+ transition and the 657

nm Ca line," B. Freeh, J.S. Wells, C.W. Oates, J. Mitchell, Y.-P. Lan, T. Kurosu, L. Zink, L. Hollberg, T. Zibrova, B.C. Young, and J.C. Bergquist, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 47, 513-517

(2000).

20. "Atom cooling, trapping, and quantum manipulation," C.E. Wieman, D.E. Pritchard, and D.J. Wineland,

Rev. Mod. Phys. Centenary 71, S253-S262 (1999); in More things in heaven and earth: A celebration of

physics at the millennium, Ben Bederson, ed., Springer, NY (1999), pp. 426-441.

VII
PREFACE

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

Division, National Institute of Standards and Technology, from January 1999 to October 2001. It follows

the similar collections of papers contained in the previous Tech Notes:

NBS Technical Note 1086, Trapped Ions and Laser Cooling I (June 1985)

NIST Technical Note 1324, Trapped Ions and Laser Cooling II (September 1988)

NIST Technical Note 1353, Trapped Ions and Laser Cooling III (April 1992)

NIST Technical Note 1380, Trapped Ions and Laser Cooling TV (February 1996)

NIST Technical Note 1523, Trapped Ions and Laser Cooling V (January 2001)

Papers listed on page vi were published during the period from January 1999 to October 2001, but

are not included here. Copies can be obtained on request. We hope this collection of papers will be useful

to our colleagues in this and related fields. We acknowledge our ion-trap/laser-cooling colleagues whose contributions made this work

possible. These include Jim Beall, Dana Berkeland, Flavio Cruz, Ann Curtis, Scott Diddams, Bob

Drullinger, Dan Dubin, Rich Fox, Leo Hollberg, Pei Huang, Steve Jefferts, Brana Jelenkovic, David

Kielpinski, Brad King, Brian King, Chris Langer, David Lee, Didi Leibfried, Dawn Meekhof, Volker

Meyer, John Miller, Travis Mitchell, Chris Myatt, Amy Newbury, Chris Oates, Rob Rafac, Mary Rowe,

Cass Sackett, Joseph Tan, Quentin Turchette, Thomas Udem, Kurt Vogel, Joe Wells, Chris Wood, Brent

Young, and in particular, Chris Monroe (now at University of Michigan). We gratefully acknowledge the

support of the U.S. Office of Naval Research (ONR), the U.S. Army Research Office (ARO), the U.S.

National Security Agency (NSA), the U.S. Advanced Research and Development Activity (ARDA), the

U.S. National Reconnaissance Organization (NRO), and Timing Solutions, Inc. We thank Eyvon Petty

and Edie DeWeese for assembling this collection.

James C. Bergquist

John J. Bollinger

Wayne M. Itano David J. Wineland

Boulder, Colorado

December 2001

vm
Lasers for an Optical Frequency Standard using

Trapped Hg+ Ions1

Brenton C. Young, Flavio C. Cruz,2 Dana J. Berkeland,3 Robert J. Rafac, James C.

Bergquist, Wayne M. Itano, and David J. Wineland

National Institute of Standards and Technology, Boulder, Colorado 8030S

Abstract. We are developing an optical frequency standard based on the narrow 281.5 nm transition

of trapped 199 Hg+ions. A major step toward the completion of this standard is the construction of an

isolated high-finesse Fabry-Perot cavity to stabilize the local oscillator. The cavity system that we have

assembled has enabled the creation of an optical frequency source with good short-term stability. Eventually,

this frequency source will derive long-term stability from a lock to the Hg+transition. We have recently

demonstrated an improved linewidth of 0.6 Hz (40 s averaging time) for a 563 nm dye laser locked to our

stable cavity. Additionally, we are developing solid-state laser replacements for gas and dye lasers presently

used for driving 194 nm and 281.5 nm Hg+

transitions.

INTRODUCTION

The next major advance for frequency standards probably lies in the development of optical frequency

standards. Optical frequency standards are attractive since the potential fractional frequency instability of a quantum system is inversely proportional to the transition frequency. Because optical frequencies are ap- proximately 105 times higher than the 9.2 GHz microwave transition used in cesium standards, much higher

fractional stability might be achieved in a given measurement time.

In the 1970s, Dehmelt noted that single trapped and laser-cooled ions might be nearly ideal references for optical frequency and time standards [1,2]. High resolution is possible because perturbations can be made

small and interrogation times long [1-4]. In addition, laser cooling considerably reduces first- and second-order

Doppler shifts [5,6]. Several groups are developing optical frequency standards based on a variety of ions [7-20]. Among proposed

standards that use trapped and laser-cooled ions, 199 Hg+ions are attractive because they offer both a suitable

microwave and optical transition. Figure 1(a) shows the 199 Hg+electric dipole transitions at 194 nm used

for laser cooling, optical pumping, and detection, and the electric quadrupole transition at 281.5 nm that is the reference for the optical frequency standard. Our group has recently demonstrated an accurate microwave

frequency standard based on the 40.5 GHz. F = —> F = 1, ground-state hyperfine splitting in trapped and

cooled 199 Hg+ions [21]. We expect to achieve significant gains in statistical precision, and likely in accuracy,

for an optical frequency standard that interrogates the ultraviolet transition, with a frequency over 25 000 times

that of the microwave frequency standard.

The optical standard is based on the 2 Si/2—

2 £> 5/2. 281.5 nm electric-quadrupole transition [20]. An optical

oscillator locked to this transition can have a fractional frequency instability approximately equal to 1 x 10 -15

at 1 s even for a single laser-cooled ion. However, reaching such low instabilities requires a laser (local oscillator)

whose frequency fluctuations are less than approximately 1 Hz during time intervals as long as a few seconds.

One of the main technical difficulties of working with Hg+ is the development of reliable and economical

ultraviolet laser sources. Recent advances in solid-state lasers have made possible optical sources at these

1) Work of the U.S. Government, not subject to U.S. copyright.

2) Present address: Universidade Estadual de Campinas, Campinas, SP, 13083-970, Brazil.

3) Present address: Los Alamos National Laboratory, Los Alamos, NM 87545.

CP457, Trapped Charged Particles and Fundamental Physics

edited by Daniel H. E. Dubin and Dieter Schneider

1999 The American Institute of Physics 1-56396-776-6/99/$ 15.00 337

TN-1
(a) (b)

F= 1 t= 90 ms

40.5 GHz

F =

563 nm

PZT

282 am w&*

LPF

FIGURE 1. (a) Energy level diagram of 199 Hg+

. We cool the ions using the 2 Si/2 -> 2 Pi/2 transitions at 194 nm.

Because the 2 P\/2, F=0 -» 2 Si/2 , F=0 transition is forbidden, transition p is a cycling transition. A second laser on

transition r repumps atoms that decayed to

2 Si/2 , F=0 after off-resonant excitation to

2 P\/i, F=l by the first laser. (b) Simplified schematic of the proposed optical frequency standard. A dye laser is prestabilized to a Fabry-Perot

cavity (T = 800). Further stabilization to a much higher finesse cavity (T > 50000), and eventually to a narrow

transition of trapped Hg+ion(s) should provide a highly stable frequency source. Solid lines denote optical paths and

dotted lines represent electrical connections. AD"P, deuterated ammonium dihydrogen phosphate crystal for frequency

doubling, AOM, acousto-optic modulator; T, finesse; HPF, high-pass filter; LPF, low-pass filter; PD, photodiode, PZT,

piezoelectric transducer.

wavelengths with lower initial costs and operating costs, higher reliability and efficiency, and lower intrinsic noise than for Ar+ and dye lasers. Consequently, we are developing solid-state replacements for our present

laser systems.

OVERVIEW OF THE OPTICAL FREQUENCY STANDARD

Figure 1(b) shows a simplified diagram of our proposed optical frequency standard. When interrogating a narrow atomic resonance, the laser must have a frequency width narrower than the transition linewidth

to prevent the frequency instability of the laser from limiting the performance of the frequency standard.

Consequently, one of the major steps in the development of the optical frequency standard is the construction

of an optical local oscillator with sufficient spectral purity. For the 1.7 Hz linewidth 199Hg+

transition, the

stability of the standard will not be significantly degraded if the laser linewidth is below 1 Hz for interrogation

times as long as a few seconds.

A central component of this system is a high-finesse [7 > 50 000) Fabry-Perot cavity [20], which is described

in detail in a later section. We use a dye laser at 563 nm as the optical source that is locked to this reference

cavity. (The light is frequency-doubled to 281.5 nm in a crystal close to the Hg+ trap.) Not shown in Fig. 1(b) is an iodine reference cell that we use to locate the Hg+ transition whenever changes are made to the high-finesse reference cavity.

Rather than locking the laser directly to the high-finesse cavity, we first prestabilize it to a cavity with a

finesse of approximately 800 using a Pound-Drever-Hall FM lock [22]. This prestabilization provides several advantages, including an increased locking range, a higher loop bandwidth for the lock, and improved versatility

and tunability of the laser. An intracavity electro-optic modulator (EOM) in the dye laser provides highfrequency correction of laser frequency noise. A piezoelectric transducer (PZT) behind one of the dye-laser

cavity mirrors eliminates long-term frequency drifts between the dye laser and the cavity. A loop bandwidth

338

TN-2
of approximately 2 MHz in this prestabilization stage narrows the dye laser short-term (r < Is) linewidth to approximately 1 kHz.

An optical fiber delivers light from the dye-laser table to a vibrationally isolated table that supports the

high-finesse cavity. An acousto-optic modulator (AOM) mounted on the isolated table shifts the frequency of

the incoming light to match a cavity resonance. Again, we implement the lock using the Pound-Drever-Hall

technique. The feedback loop performs corrections at low frequencies by adjusting a PZT on the prestabilization

cavity, and at higher frequencies as high as approximately 90 kHz by varying the AOM drive frequency. With

the lock enabled, the light entering the cavity has a spectral width less than 1 Hz, as we demonstrate later. Finally, the frequency-stabilized light is transported to the table holding a cryogenic Hg+ trap. The 563 nm

radiation is frequency-doubled to 281.5 nm and is focused onto the trapped ion(s). The final AOM in Fig. 1(b)

shifts the frequency of the light to match the ion transition. We plan to interrogate the transition using the

Ramsey technique [23], with a Ramsey time of approximately 30 ms. A digital servo loop will adjust the AOM

frequency to step between both sides of the central fringe, and will periodically record the values of the center frequency [20].

SOLID-STATE LASERS

The inherent frequency stability of solid-state lasers makes them attractive for metrological applications

and precision spectroscopy. In addition, a solid-state laser can be compact, reliable, and long-lived. Reliable,

commercial diode lasers do not yet exist in the uv, near the transitions needed for the Hg+ system, but highpower, near-infrared diode lasers are available. Consequently, some groups have frequency-quadrupled the

output of near-infrared diode lasers that oscillate at a single frequency and in a single spatial mode to obtain cw, single-frequency uv sources [24,25]. An alternative approach is to frequency-quadruple the output of a cw,

solid-state laser that is pumped with high-power multimode diode lasers, as has been done using Nd:YAG and

Nd:YV04 lasers [26,27]. We have taken this latter approach in developing an all-solid-state laser for driving the

199Hg+ S — D transition at 281.5 nm. The solid-state generation of 194 nm radiation for the cooling transition

employs a diode-pumped solid-state laser in addition to sum-frequency mixing. These two laser systems are described in the following subsections.

Nd:FAP laser

For the 281.5 nm light source, we plan to replace a dye laser and its multiline Ar+ pump laser with a frequency-doubled Nd3+ -doped fluorapatite (Nd:FAP) laser [28]. Nd.FAP has a lasing transition at 1.126 fim

that, when frequency-quadrupled, coincides with the Hg+ transition at 281.5 nm. The major difficulty in designing this laser is that Nd:FAP has a much stronger transition nearby at 1.063 nm [29] that must be

suppressed by the laser optics. With 680 mW of diode pump light at 808 nm, the Nd:FAP output power is approximately 90 mW at 1.126 nm. Frequency doubling in KNb03 gives approximately 5 mW at 563 nm.

For the second stage of harmonic generation, we use deuterated ammonium dihydrogen phosphate (AD*P)

to frequency-double 563 nm radiation to 281.5 nm [30]. Since less than 1 pW can be enough to saturate the narrow S — D transition [30], we simply frequency-double the radiation at 563 nm in a single-pass configuration,

approximately 25 nW is generated at 281.5 nm for 5 mW of input power at 563 nm.

Because the free-running frequency instability of this laser is dominated by low-frequency acoustical and

mechanical noise, only a moderate-speed servo system is needed to lock the laser frequency tightly to the resonance of a high-finesse cavity [20,31]. In addition, since the frequency of the Nd:FAP laser is quadrupled

to reach the atomic transition, this facilitates the first steps in a frequency chain from the optical to the

microwave. We have demonstrated tunability of this laser through the Hg+ transition. Poor stability of the

pump-diode output mode, however, has forced us to use the original dye laser source for the experimental work

described in the remainder of this paper.

Yb:YAG laser

Currently, the 194 nm cooling light is generated using a single-mode Ar+ laser at 515 nm that is frequencydoubled in /3-barium borate (BBO) to 257 nm and is then sum-frequency mixed in BBO with a diode source at

339

TN-3
S-H CO

<D T3

=3—

E

o

c

ea jO

0.5

B5en

""I to

laser #1 K

40 s averaging time

frequency span = 50 Hz

RBW =0.477 Hz

rel. drift rate = 1.2 Hz/s

0.84 Hz

fractional linewidth:

0.84 Hz/ V2 =>ix 10'

fc

- 25 Hz fc + 25 Hz

frequency (Hz)

FIGURE 2. Amplitude spectrum of the beat note between two laser beams stabilized to two independent cavities. The dashed line shows the -3 dB level. The averaging time is 40 s. A nearly uniform relative cavity drift of 1.2 Hz/s is suppressed by mixing the beat note with a swept synthesizer.

792 nm to produce light at 194 nm [32]. We plan to replace the Ar+ laser with a Yb:YAG laser at 1.03 /im [33]. With 3 W of diode pump power at 941 nm, the Yb:YAG output power is 1.2 W at 1.03 /im. We have frequencydoubled the Yb:YAG laser output in KNbG"3 to obtain over 400 mW at 515 nm, which should be sufficient power to replace the Ar+ laser. At high optical powers, the frequency-doubling conversion efficiency is limited

by losses from blue-light-induced infrared absorption [34]. We anticipate achieving a better conversion efficiency

by frequency-doubling with lithium triborate (LBO) instead of KNbCb.

HIGH-FINESSE REFERENCE CAVITY

To achieve a laser linewidth of <1 Hz for the source driving the Hg+ ion transition, we start with a highfinesse cavity that has intrinsically low sensitivity to temperature variations, and then take great care to protect

it from environmental perturbations. The separation of the cavity mirrors is set by optically contacting the

mirrors to the ends of a hollow cylinder made from a low-thermal-expansion material. The mirror substrates

are made of the same material as the cylinder. The cavity is supported inside an evacuated chamber by two

thin wires. Keeping the cavity under vacuum both avoids pressure shifts of the cavity resonance and thermally

insulates it from the environment. The temperature of the vacuum chamber is held near 30 °C, which is the point of zero coefficient of expansion for the cavity material. We protect the cavity from seismic noise

by mounting it on a passively isolated optical table. The table is suspended by strands of surgical tubing

approximately 3 m long. The fundamental vibrational mode of the suspension has a frequency of ssO.3 Hz,

which provides an isolation from floor noise that exceeds a factor of 50 in noise amplitude already at 3 Hz (some

TN-4

340
viscous damping is used). To prevent the coupling of acoustic noise into the cavity, we enclose the optical table

in a wooden box lined internally with lead foam [35]. The intracavity light heats the mirror coatings, thereby shifting the cavity resonance. To hold this shift at a reasonable value, we couple only approximately 100 /iW of 563 nm light into the cavity. Furthermore,

controlling the optical power in the cavity stabilizes this power shift. Active control of the rf power driving the

AOM stabilizes the output power from the cavity to ?»0.1%.

To characterize the cavity's short-term stability performance without referencing to Hg+ . we constructed a

second cavity and isolated table similar to that described above. Figure 2 shows the spectrum of the beat note between two independent laser beams stabilized to the two cavities. A nearly uniform relative cavity drift of

wl Hz/s is suppressed by mixing the beat note with a swept synthesizer. The width of the spectrum at its half-power point is 0.8 Hz (40 s averaging time). This implies that at least one of the lasers has a frequency

width less than 0.6 Hz at 563 nm, corresponding to a fractional linewidth of only 1 x 10" 15

. This is roughly 40

times better than previous results with only one cavity well isolated from vibrations [20], and may represent

the smallest fractional linewidth ever recorded in the optical regime.

CONCLUSIONS

We have demonstrated an optical local oscillator suitable for development of a Hg+ optical frequency standard

at 281.5 nm. The frequency source has a linewidth of less than 0.6 Hz at 563 nm (40 s averaging time),

corresponding to a fractional linewidth of 1 x 10~ 15

. We have described work on solid-state laser replacements

for gas and dye lasers presently used in the trapped Hg+ work. Future work will involve collaboration with

other researchers at NIST and JILA to develop a frequency chain for translating the frequency and the stability of our standard into the microwave regime. This work is supported by ONR and NIST.

REFERENCES

1. Dehmelt, H. G., Bull. Am. Phys. Soc. 18, 1521 (1973).

2. Dehmelt, H. G., IEEE Trans. Instrum. Meas. IM-31, 83-87 (1982).

3. Wineland, D. J., et al., J. Phys. (Pans) 42, C8-307-C8-313 (1981).

4. Fisk, P. T. H., Rep. Prog. Phys. 60, 761-817 (1997).

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

6. Wineland, D. J., and Itano, W. M., Phys. Today 40, 34-40 (1987).

7. Bergquist, J. C., ed., Proceedings of the Fifth Symposium on Frequency Standards and Metrology, Singapore: World

Scientific, 1996.

8. Sugiyama, K., Sasaki, K., Wakita, A., and Yoda, J., "Progress toward high-resolution spectroscopy of the 2 S!/2

- D5/2 transition of laser-cooled trapped Yb ," in International Workshop on Current Topics of Laser Technology,

1998, p. 56.

9. Taylor, P., Roberts, M., Barwood, G. P., and Gill, P., Opt. Lett. 23, 298-300 (1998).

10. Engelke, D., and Taram, C., Europhys. Lett. 33. 347-352 (1996).

11. Barwood, G. P., et al.. Opt. Commun. 151, 50-55 (1998).

12. Bernard, J. E., Marmet. L., and Madej, A. A., Opt. Commun. 150, 170-174 (1998).

13. Madej, A. A., et al., "Precision absolute frequency measurements with single atoms of Ba+ and Sr+ ," in Proceedings

of the Fifth Symposium on Frequency Standards and Metrology, 1996, pp. 165-170.

14. Urabe, S., et al., Appl. Phys. B 67, 223-227 (1998).

15. Knoop, M., Vedel. M., and Vedel, F., Phys. Rev. A 58, 264-269 (1998).

16. Fermigier, B., et al., Opt. Commun. 153, 73-77 (1998).

17. Peik, E., et al., "Towards an optica] clock with a laser-cooled indium ion," in International Workshop on Current

Topics of Laser Technology, 1998, pp. 23-24.

18. Nagoumey, W., Burt, E., and Dehmelt, H. G., "Optical frequency standard using individual indium ions," in Proceedings of the Fifth Symposium on Frequency Standards and Metrology, 1996, pp. 341-346.

19. Yu, N., Dehmelt, H., and Nagourney, W., Proc. Natl. Acad. Set. U.S.A. 89, 7289 (1992).

20. Bergquist, J. C., Itano, W. M., and Wineland, D. J., "Laser stabilization to a single ion," in Frontiers in Laser

Spectroscopy, 1994, pp 359-376.

21. Berkeland, D. J., et al., Phys. Rev. Lett 80, 2089-2092 (1998). 22 Drever, R. W P., et al., Appl. Phys. B 31. 97-105 (1983)

23. Ramsey, N. F., Phys. Rev. 78. 695-699 (1950)

341

TN-5
24. Zimmermann, C, Vuletic, V., Hemmerich, A., and Hansen, T. W., Appl. Phys. Lett. 66, 2318-2320 (1995).

25. Matsubara, K., et al., Appl. Phys. B 67, 1-4 (1998).

26. Hollemann, G., Peik, E., and Walther, H., Opt. Lett. 19, 192-194 (1994).

27. Kondo, K., et al., Opt. Lett. 23, 195-197 (1998).

28. Cruz, F. C, Young, B. C, and Bergquist, J. C, "Diode-pumped Nd:FAP laser at 1.126 /jm: a possible local

oscillator for a Hg+

optical frequency standard," Appl. Opt. (to be published).

29. Ohlmann, R. C, Steinbruegge, K. B., and Mazelsky, R., Appl. Opt. 7, 905-914 (1968).

30. Bergquist, J. C, Hulet, R. G., Itano, W. M., and Wineland, D. J., Phys. Rev. Lett. 57, 1699-1702 (1986).

31. Zhu, M., and Hall, J. L., "Frequency stabilization of tunable lasers," in Atomic, Molecular, and Optical Physics:

Electromagnetic Radiation, 1997, pp. 103-136.

32. Berkeland, D. J., Cruz, F. C, and Bergquist, J. C, Appl. Opt. 36, 4159-4162 (1997).

33. Fan, T. Y., and Ochoa, J., IEEE Photon. Technol. Lett. 7, 1137-1138 (1995).

34. Mabuchi, H., Polzik, E. S., and Kimble, H. J., J. Opt. Soc. Am. B 11, 2023-2029 (1994).

35. Hils, D., Faller, J. E., and Hall, J. L., Rev. Sci. Instrum. 57, 2532-2534 (1986).

^xt ^ 342

TN-6
High-Resolution, High-Accuracy

Spectroscopy of Trapped Ions

D.J. Berkeland 1

, J.D. Miller2

, F.C. Cruz3, B.C. Young, R.J. Rafac,

X.P. Huang4

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

National Institute of Standards and Technology (NIST), 325 Broadway, Boulder Colorado 80303

Abstract. Microwave spectroscopy using trapped and cooled ions can achieve precision and accuracy comparable to the best cesium frequency standards. We discuss

standards based on

199 Hg+ ions trapped in linear Paul traps: the Jet Propulsion Laboratory (JPL) standard, which uses up to 107 atoms confined near the trap axis, and the

recently evaluated NIST standard, which uses approximately ten ions laser cooled and

crystalized on the trap axis. We consider future directions in trapped ion frequency

standard work, including the use of entangled states for achieving higher precision, and

progress on trapped ion optical frequency standards. Finally, we discuss scientific and

technical applications of extremely stable frequency standards.

INTRODUCTION

Precise and accurate atomic spectroscopy can rigorously test theories of atomic

structure, quantum electrodynamics [1] and other fundamental physics [2,3], determine fundamental constants [4], and provide time and frequency standards [3]. Precise and accurate spectroscopy has two basic requirements. First, the measure- ment must reach the desired precision in a reasonable averaging time. This requires a good signal to noise ratio and a narrow transition. Second, systematic frequency

shifts and broadening mechanisms of the atomic transition being studied must be

either very small, or stable and very well measured.

As seen at this conference, many experiments with cooled and trapped ions

and with laser-cooled neutral atoms can satisfy these requirements. Here we limit our discussion to precision experiments using trapped and cooled ions [5]. The

*) Work performed at NIST is supported by the Office for Naval Research and the Army Research

Office. Work of U.S. Government; not subject to U.S. copyright.

:) Los Alamos National Laboratory, P23, MS H803, Los Alamos, NM

2

) KLA-Tencor Corp., 1701 Directors Blvd., Suite 1000, Austin, TX 78744

3> Universidade Estadual de Campinas, UNICAMP-IFGW-DEQ, CP.6165 Campinas, SP, 13083-

970, Brazil

4

> Colorado MED-Tech-RELA, Inc., 6175 Longbow Dr., Boulder, CO 80301

TN-7
systematic shifts of atomic transition frequencies in these systems can be small

and well-characterized. For example, Stark shifts are small since < E > = and

< E

2 > is small when the ions are cold. The magnetic field can be small and easy

to characterize because the ions occupy a small volume. The corresponding Zeeman

shift is typically small because usually transitions with only a second-order field

dependence are used. Because background gas pressure can be negligible, collisional

shifts and broadening can be small. The ions can be cooled using either buffer-gas

cooling or laser cooling [6], reducing Doppler shifts. The statistical precision can be high if large numbers of ions are stored, or if the shot noise limit is reached.

Finally, free precession times of several minutes have been reported [7,8], giving

extremely narrow transition linewidths.

The statistical precision of a frequency standard can be predicted quantitatively

for various cases. For example, if an atomic transition is probed using the Ramsey

technique [9] and the measurement precision is limited only by quantum fluctuations in the atomic state populations [10], the fractional precision of the frequency

measurement is given by [11]

L^Ldmeasured / \ ± —1/2 / 1 \

; = <W) = —7^= r

i

. (1)

Here, lj is the transition angular frequency, Aujmeasured is the precision with which

the frequency is measured, and the Allan deviation cry (r) is related to the fractional

frequency instability. ./V is the number of atoms used, Tr is the free precession

time between the two w/2 Rabi pulses, and r (> Tr) is the averaging time of

the measurement. Ideally, N, Tr, and r are large, although various experimental

constraints may limit these values.

Table 1 compares these parameters for two types of microwave frequency

standards—those based on ions confined in an rf Paul trap and those based on a

pulsed fountain of cesium atoms [12]. The number iVof atoms used in the trapped

ion standards is limited in part by the second-order Doppler shift due to micromotion. This motion is driven by the trap's rf electromagnetic fields, and becomes

greater as the Coulomb repulsion between the ions forces them further from the

field nodal point or line [13]. Thus there is a trade-off between using very large N

but with a substantial Doppler shift, or a negligible Doppler shift but smaller N. In

the Cs fountain standard, TV is limited in part by collisional shifts. For laser-cooled

trapped ion standards, Tr is limited by the time the ions remain cold in the ab- sence of cooling radiation. In a fountain standard it is limited to about 1 s by the

maximum practical height the atoms can be tossed (about one meter). Finally, the

averaging time r is not fundamentally limited in either type of standard.

Ions in Penning traps have been used previously to realize the first laser-cooled

frequency standard [7,16]. An important limitation to accuracy in those experiments was the uncertainty in the second-order Doppler shift due to the overall

rotational motion of the ion cloud. This motion can now be precisely controlled

[17], and it should therefore be possible to realize an rf or microwave frequency

TN-8
TABLE 1. Example parameters for trapped

ion and cesium fountain microwave frequency

standards.

Parameter Ions Cs w /(27t) 40.5 GHz ( 199 Hg+) 9.2 GHz

N « 10 [14] to 107 [15] 104 to 10 6 TR Up to 600 s [7,8] ~ 1 s

standard with accuracy comparable to what is possible with Paul traps [18]. To

date, the highest accuracies and stabilities have been obtained in linear Paul traps;

therefore, we highlight this work below.

Figure 1 shows a schematic diagram of a linear Paul trap [19-21]. In this trap,

the ions are confined axially by the two cylindrical sections (endcaps) held at static

potential Uo. The shape of these endcaps is not critical, and axial confinement can be produced by thin conducting rods located on the trap axis at both ends of the

trap [15], by small rings [20] or by segmented electrodes [21]. Two of the long, thin

rods of Fig. 1 are held at ground potential, while the other two are held at an rf

potential. This gives an oscillating electric potential that traps the ions in a radial

quadratic pseudopotential [22]. The advantage of the linear trap is that many ions can be confined near or on the electric field nodal line, where Doppler shifts from

micromotion are minimized.

MICROWAVE SPECTROSCOPY

Microwave spectroscopy with clouds of trapped ions has been used for many years

in atomic structure measurements [23]. Some recent experiments report measure- ments of g factors [24] and hyperfine constants [25-29]. Because ion traps can use

small samples, they are well-suited for measurements of hyperfine anomalies using

different isotopes, some of which are radioactive [25,26,29].

VoCOS(Qt)

n

C^lon

Uo

FIGURE 1. Schematic diagram of a linear Paul trap.

y

33

TN-9
r(2 P,/2 ) s2ns

$ 6.9 GHz

r(2 D 5 / 2 ) s90 ms

194 nm

F = FIGURE 2. Partial energy level diagram of 199 Hg+.

Taking advantage of the precision and accuracy offered by the ground state

hyperfine transition of trapped ions, several groups are working on atomic fre- quency standards based on

113 Cd+ (w = 2 tt x 15.2 GHz) [27,30] and 171 Yb+

(u>o — 2 7r x 12.6 GHz) [20,31-33]. Here we describe microwave frequency standards based on trapped 199 Hg+ ions [14,15,34]. Figure 2 shows a partial energy

level diagram of this atom. The electric dipole transitions at 194 nm are used for

state preparation and detection, and can be used for laser cooling. The ground

state hyperfine splitting frequency is 2 7r x 40.5 GHz, the highest of routinely

trapped ions. The Amp = hyperfine transition dependends only quadratically on the magnetic field, when the field is near zero. At JPL, much work has been done with clouds of 199 Hg+ ions in a linear Paul

trap [15,34]. The methods and performance of the Commonwealth Scientific and

Industrial Research Organization (CSIRO) experiments on 171 Yb+ are similar [20].

The four trap rods of the JPL linear ion trap standard (LITS) in reference [15] are evenly spaced on a circle of 1 cm radius, with two endcap rods on the trap

axis 7.5 cm apart. This trap confines clouds of up to 107 ions, which are cooled

with a helium buffer gas. States are prepared and detected with a

202 Hg lamp,

which emits broad-line radiation that partially overlaps the resonances of the two

2 S\/2->F = 1 —>2 P\/2i F = 0, 1 transitions. Because the 2 Pj

/

2 , F — 1 state can decay to the 2 5i/2, F — state, illuminating the ions with the lamp radiation

pumps them into the 2 S\/2 F = state in preparation for the Ramsey interrogation.

At the end of the Ramsey interrogation, the lamp radiation is returned to the ions,

and the detected fluorescence indicates how many ions are in the 2 S\/2 F = 1 state. Because the ions scatter only a few photons before optically pumping out

of the 2 5i/ 2 F = 1 state, and because only a small fraction of the scattered

photons are detected, the state detection efficiency is much less than unity, so the

state measurements are not quantum noise limited and Eq. (1) is not applicable.

However, because N is large, this device is very stable. With a Ramsey time Tr of

TN-10
8 s, LITS has recently demonstrated an instability of ay (r) — 3 x 10 14 r

1//2 [35].

At CSIRO, a similar standard using clouds of laser-cooled 171 Yb+ ions in a linear

trap has ay {r) = 4.7 x 10~ 14 r" 1 / 2 [20].

Although these standards are very stable, the second-order Doppler shift from

the ion motion is substantial (about 9 x 10~ 13

). At NIST, the goal is to develop

a frequency standard that is both stable and accurate. We use a linear trap such

as that shown in Fig. 1, with the endcaps approximately 4 mm apart and the

0.2 mm radius rods on a 0.64 mm radius [36]. Approximately ten ions are used,

and they are laser cooled so that they crystalize along the nodal line of the rf

electric field, near the trap axis. Using a small number of ions sacrifices precision

according to Eq. (1), but greatly reduces Doppler shifts. Groups at CSIRO [37] and

the Communications Research Laboratory (CRL) [38] have also crystalized lasercooled ions in linear Paul traps in order to improve the accuracy of their frequency

standards.

The NIST standard has other advantages. The trap is enclosed in a copper

container that forms the bottom of a liquid helium reservoir [36]. Because of the

cryogenic environment, the pressure of background neutral mercury atoms is negligible. This is critical because the background mercury pressure leads to ion loss,

presumably due to dimer formation. This ion loss limits the storage time of trapped

ions in a room temperature trap to about ten minutes.. At 4 K, the ions can be

trapped for days at a time without loss. Also, in the 4 K environment the pressure

of all other background gases is negligible, with the possible exception of helium.

This greatly reduces collisional shifts and heating from collisions. Additionally, the

black body shift, which is already over two orders of magnitude smaller than in

cesium at room temperature [39], is dramatically reduced. To ensure that the ions are on the nodal line, we minimize the micromotion observed in three non-coplanar

directions [13]. Because the rf electric field at the site of the ions in minimized, the

rf heating while the cooling lasers are off during the free precession time Tr is minimized. Finally, laser cooling (instead of buffer gas cooling) significantly reduces

Doppler shifts; we have measured the second-order Doppler shift to be less than

3 x 10- 17 [14].

The steps in operating the frequency standard are as follows. We Doppler cool

the ions by using radiation from a primary laser whose frequency is slightly reddetuned from transition p (see Fig. 2). Although this is a cycling transition, the

laser weakly couples the Si/2 , F = 1 state to the P1/2, F = I state, which decays

to the S1/2, F = state. To optically pump the ions out of this state, we overlap a less intense repumping laser beam with the primary laser beam to drive the r

transition. After a Doppler cooling period of 200 to 300 ms, we pump the ions into

the S1/2, F = state with high efficiency by applying only the primary laser for

10 ms. Then we drive the microwave transition using the Ramsey method, with

a free precession time Tr of 2 to 100 s. After this, to determine the ensemble

average of the S1/2, F — 1 state populations, we again apply only the primary

laser for about 10 ms. If an ion is found to be in the F = 1 state, it will scatter

TN-11
1E-12

1E-13

Acomeasured

(D„

1E-14

1E-15

• 7 Ions, 10 s 8 Ions, 100 s — -NIST-7 — - -Cs Fountain

JPL

* -ON • ^ ***.

-"•»

10 100 1000

Total Measurement Time x (s) FIGURE 3. Stabilities of several standards.

10000

approximately 10,000 photons before being optically pumped into the Si/2 , F —

state. Otherwise, the atoms scatter only a few photons in the same time interval,

due to the off-resonant S1/2, F= to P1/2, F=\ transition. Our detection signal is the observed fluorescence from all of the ions. Repeating this measurement process

and scanning the microwave frequency produces a Ramsey fringe pattern. We lock

the frequency to the central Ramsey fringe by stepping the frequency from one side

of the fringe to the other, while a digital servo works to keep the detection signal

constant at each step.

Figure 3 shows the instability ^(r) of the microwave oscillator when it is locked

to the central Ramsey fringe with Tr = 10 s and Tr — 100 s. For Tr = 100 s, the instability is cry (r) = 3.3 x 10~ 13 r

-1 ' 2

. The measured instability of the standard is consistently twice that expected from Eq. (1), due to fluctuations of the

laser intensity at the site of the ions. The figure also shows the stability of the

cesium beam standard NIST-7 (8 x 10-13

t

-1 / 2

) [40] and the Paris cesium

fountain standard (2 x 10-13 r

-1 /

2

) [12]. The instabilities of these three standards are comparable, while that of the JPL 199 Hg+ standard is significantly better

(<7y( T ) 3 x 10 -14 1/2)• The accuracy of the JPL standard is limited by the second-order Doppler shift to

around 10-13

[41]. The other standards in Fig. 3 have significantly greater accuracy,

and the accuracies of these standards are comparable. The fractional accuracy of

NIST-7 is 5 x 10~ 15

, limited by the distributed cavity phase shift [42]. This high

accuracy corresponds to splitting the central Ramsey fringe to about a part in 106. A natural next step toward a more accurate frequency standard is to reduce the

TN-12
linewidth. The cesium fountain frequency standard does this by increasing the

Ramsey free precession time Tr. In this standard, the accuracy is currently limited

by black body and collisional shifts to 1.4 x 10~ 15 [43]. The accuracy of the NIST

199 Hg+ standard is 3.4 x 10

-15

, and is limited by uncertainties in the Zeeman shift

[14]. The dominant shift is caused by asymmetries in currents running through the

trap electrodes, which cause a net rf magnetic field at the site of the ions.

The trapped ion and cesium fountain standard are both emerging technologies

that promise even higher accuracy. At NIST we have constructed a new, smaller

trap, which we will run with lower trap drive frequency Q. Less rf potential is needed for the same radial confinement in the trap, reducing the currents in the

trap rods and, from scaling arguments, the associated magnetic field at the site of

the ions. Also, we have improved our magnetic shielding to reduce fluctuations in

the ambient magnetic field. We expect that the next generation of this standard

will be accurate to approximately 10 -16

. FUTURE DIRECTIONS

One way to reduce the instability <ry (r) of a standard is to increase the transition

frequency loq. Therefore, many groups are working on optical frequency standards.

The laser-cooled stored ion efforts include trapped 172 Yb+ at 411 nm [44] and

3.43 fim [45],

171 Yb+ at 435 nm [46], 88 Sr+ at 674 nm [47,48],

138 Ba+ at 12.5 fim

[49], 40 Ca+ at 729 nm [50-52], 115 In+ at 237 nm [53,54], and 199 Hg+ at 282 nm [55].

Earlier work at NIST using 199 Hg+ in a room temperature trap produced a narrow

transition with structure due to Rabi oscillations [55]. The width of the central

feature was about 40 Hz at 563 nm. Our pursuit of higher resolution and a study of

systematic effects was hampered by the the limited lifetime of the ion in the room

temperature trap. However, we have now built a second cryogenic system that

will provide long ion lifetimes and more detailed investigations of this transition.

Also, we have recently made substantial improvements in the laser system; the

laser linewidth in about a one minute averaging time is now less than one hertz at 563 nm [56].

For the absolute frequency of a transition to be determined, uq must be compared

to an accepted frequency standard. We are now comparing the frequency of the

282 nm transition to a that of the narrow

1 Sq —3 P\ intercombination line of Ca at 657 nm [57,58] by mixing a CO overtone line with the fundamental 563 nm light [59].

The NIST Ca standard has been compared to the Physikalisch-Technische Bundesanstalt (PTB) Ca standard, which has in turn been compared to the 9.2 GHz

line in Cs [60]. In other laboratories, the absolute frequency of the 88 Sr+ S to D

transition at 674 nm has been measured with a frequency chain [48]. Interferometric measurements have also been made on this transition [61] and on the S to D

transition at 411 nm in 172 Yb+

[44], with frequency chain measurements in progress

[62].

Another way to reduce cry (r) is by using entangled states [63,64]. Consider

TN-13
an atomic system with two states labeled "f and J,, separated by frequency to . We use the spin 1/2 analog for these two states [65], so the total angular momentum for N ions is given by J = ]Ct'=i Sj, where Si is the spin of the z'th atom (S{ — 1/2). For uncorrelated atoms, the Ramsey spectrum is given by

< Jz > = N cos((u;o — to) Tr). The best possible uncertainty in the measured

value of uoq is the shot noise limit of Au = l/y/NTpr [10]. This uncertainty can be reduced if we use an entangled state in the following way. Suppose that the

TV-atom state at the beginning of the free precession time is given by

—iNwnt +iNunt ,— (e-s— Itita - U> +e^ II1I2 ... In>)/\/2. (2)

After the free precession time, a 7r/2 Rabi pulse is applied as in usual Ramsey

spectroscopy. The signal is obtained by measuring an operator that is the product

of the z-components of the Pauli spin matrices, — YlfLi( az)i- This gives a signal

(O) = ( — 1)^ cos(N (uj — to) Tr), where u is the frequency of the applied

radiation. The uncertainty in u) is now given by the exact Heisenberg limit Ao; =

1/Ny/Tr [63]. This method results in a factor of TV decrease in the averaging time

r required to achieve a given precision. Although similar precision gains can be

made by increasing N, the optimal value of N in high accuracy ion trap standards

might be limited by other experimental constraints, thereby making entangled state

spectroscopy advantageous.

APPLICATIONS

Improved frequency standards benefit communications and navigation [66,67]

and help in determining some fundamental physical constants [68]. Another possible application of improved frequency standards is in detecting the stochastic

background of gravitational radiation [69,70]. This background is similar to the

observed cosmic microwave background at 4 K, but has not been detected. Estimates for the density p of these gravitational waves vary by many orders of magnitude and are extremely model dependent. A limit on p may be determined by

timing the bursts of light emitted by pulsars, which are more stable for long times

than the best clocks on earth [71]. If gravitational radiation present at the pulsar

location is not correlated with the gravitational radiation present at earth, clocks

at the two locations will become decorrelated after some time r^, and the measured

instability cry (r) of the pulsar bursts will begin to increase with averaging time

r > Td. For example, using the estimates from [72], timing measurements from

the pulsar J1713+0747 should destabilize at r — Td ~ 2 y and (Ty (rd) ~ 10-14

[73]. However, because the current global time scale is stable at only the 10

-14

level on a time scale of about one year, any observed destabilization could also be from

the long-term drifts in the terrestrial time scale. Better clocks are thus required to

verify or reject these theories.

Another scientific application of better frequency standards is the laboratory measurement of possible changes in fundamental constants, such

TN-14
as the fine structure constant a. Nonlaboratory measurements include estimates from Sm isotope distributions at the Oklo mine, giving

—6.7 x 10-17 /year < a/a < 5.0 x 10 -17 /year [74]. However, this measure- ment assumes a linear change in a over time and is model dependent. Laboratory

measurements can in principle detect nonlinear changes in a over a relatively short

time. Because the hyperfine splitting frequency depends nonlinearly on the nuclear

charge Z and on a, comparing the frequency of the hyperfine constant in hydrogen

Ah to that of another alkali AAikaii gives a/a according to [75]

where Frei(aZ) is the Casimir correction factor [76]. Recently, the hyperfine structure frequency of 199 Hg+ has been compared to that of hydrogen for 140 days to

obtain | a/a \ < 3.7 x 10

_14 /year [75]. Optical atomic transition measurements

have also been proposed for detecting changes in a [77]. Clearly, any improvement

in the precision of laboratory frequency standards would tighten the limits on a/a. SUMMARY

Trapped and cooled ions are particularly suitable systems for precise and ac- curate spectroscopy. Frequency standards using trapped ions have stabilities and

accuracies that are comparable to those of the best of Cs standards and are ex- pected to continue to improve. Some directions toward improving trapped ion

frequency standards use optical transitions and entangled states. In addition to

improving technological applications, better frequency standards can test fundamental physics.

ACKNOWLEDGEMENTS

We thank Lute Maleki, Travis Mitchell, John Prestage, Don Sullivan and Matt

Young for valuable comments on this manuscript.

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TN-15
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TN-17
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TN-18
Hg+ OPTICAL FREQUENCY STANDARD:

RECENT PROGRESS*

B. C. YOUNG, R. J. RAFAC, J. A. BEALL, F. C. CRUZ,*

W. M. ITANO, D. J. WINELAND, AND J. C. BERGQUIST

National Institute of Standards and Technology, 325 Broadway,

Boulder, CO 80303, USA

E-mail: byoung@boulder.nist.gov

We report progress toward an optical frequency standard based on the 1.7 Hz

linewidth 2 S1 / 2 —* 2 Ds/2 transition of a single trapped 199 Hg+ ion. We have constructed an isolated, high-finesse, Fabry-Perot cavity for stabilization of the optical

local oscillator. A 563 nm source frequency-locked to this cavity has a linewidth

less than 0.16 Hz for averaging times up to 20 s. The measured fractional frequency

instability is 3xl0-16at 1 s. A simple scheme allows the transport of this light

through an optical fiber with negligible degradation of its spectral purity. We have

constructed small cryogenic linear traps that are designed to provide confinement

in the Lamb-Dicke regime for the optical transition.

1 Introduction

Neutral-atom and trapped-ion frequency standards based on microwave transitions have achieved fractional inaccuracies1 ' 2 near 10-15 and fractional fre- quency instabilities1 ,3 ' 4 near 4 x 10-14r

-1 /2

, where r is the measurement av- eraging time. Some of these microwave standards now have reached (or nearly

reached) the theoretical performance limit set by quantum projection noise. For an atomic standard based on an ensemble of N uncorrected atoms that

are interrogated by Ramsey's separated oscillatory field method,5 quantum

projection noise limits the fractional frequency instability to6 — «a„(T) = /T>y— . i 1 ) OJq u>ov NJ-RT

where ay {r) is the two-sample Allan deviation, uq is the frequency of the clock

transition, and Tr is the Ramsey interrogation time.

The next major advance for frequency standards probably lies in the

development of standards based on long-lived optical transitions. Because

optical frequencies are «105 times higher than the 9.2 GHz microwave transition used in cesium standards, higher fractional stability can be achieved in a •WORK OF THE U.S. GOVERNMENT. NOT SUBJECT TO U.S. COPYRIGHT.

tPRESENT ADDRESS: UNIVERSIDADE ESTADUAL DE CAMPINAS, CAMPINAS,

SP, 13083-970, BRAZIL.

TN-19
(a)

PZTj

5*" = 800

,LPF

40.5 GHz

F =

(b)

tS5~^SSn fiber lip

r = 800 M-1^ t

vc

563

lasernm ^

y . AOM 1 3PD

Figure 1. (a) Simplified energy-level diagram for 199 Hg+ . We cool the ions using the

2 Si/2—* 2 Pi/2 transitions at 194 nm. Transition p is a cycling transition. A second laser on transition r repumps atoms in 2 Si/2i -F=0 back into 2 Si/2> F=l. The microwave clock

transition is at 40.5 GHz, and the optical clock transition is at 282 nm. (b) Simplified schematic of the proposed optical frequency standard. A dye laser is prestabilized to a Fabry-Perot cavity (T = 800). Further stabilization to a much higher finesse cavity (T > 150000), and eventually to a narrow transition of a trapped Hg+ ion should provide a

highly stable frequency source. Solid lines denote optical paths and dotted lines represent

electrical connections. AD'P, deuterated ammonium dihydrogen phosphate crystal for frequency doubling; AOM, acousto-optic modulator; T, finesse; HPF, high-pass filter; LPF,

low-pass filter; PD, photodiode; PZT, piezoelectric transducer; VCO, voltage-controlled

oscillator.

given measurement time even for a smaller number of atoms. Single trapped

and laser-cooled ions might be nearly ideal references for optical frequency

and time standards.7,8 High resolution is possible because perturbations can be made small and interrogation times long.7,8 ' 9 ' 10 In addition, laser cooling

considerably reduces first- and second-order Doppler shifts. 11 Several groups

are developing optical frequency standards based on a variety of ions. 12 We

are developing an optical frequency standard using 199 Hg+ ions, which are at- tractive because they offer both microwave and optical transitions suitable for frequency standards.2 ' 13 Figure 1(a) shows the 199 Hg+ electric dipole transi- tions at 194 nm used for laser cooling, optical pumping, and detection, and the

1.7 Hz linewidth electric quadrupole transition at 281.5 nm that is the refer- ence for the optical frequency standard. For a single Hg+ ion and Tr = 30 ms,

Eq. (1) gives a quantum-projection-noise limit of <rv (r) « 10-15 r

-1 / 2

. Reaching such low instabilities requires a laser whose frequency fluctuations are <1 Hz during time intervals as long as a few seconds.

TN-20
2 Overview of the Optical Frequency Standard

Figure 1(b) shows a simplified diagram of our proposed optical frequency

standard. 13 A critical component is the high-finesse {T > 150000) FabryPerot cavity, which is described in Sec. 3.1. A dye laser at 563 nm is prestabilized to a cavity with a finesse T « 800 using a Pound-Drever-Hall FM

lock. 14 A feedback loop bandwidth of «2 MHz narrows the dye laser shortterm (<1 s) linewidth to «1 kHz. An optical fiber (not shown) delivers

light from the dye-laser table to a vibrationally isolated table that supports

the high-finesse cavity. An acousto-optic modulator (AOM) mounted on the

isolated table shifts the frequency of the incoming light to match a cavity resonance. A second FM lock performs corrections at frequencies as high as «90 kHz by varying the AOM drive frequency and at low frequencies by adjusting a PZT on the prestabilization cavity. With the lock enabled, the light

entering the high-finesse cavity has a subhertz spectral width (see Sec. 3.2). Finally, the frequency-stabilized light couples through an optical fiber to the table holding a cryogenic Hg+ trap. A simple feedback loop actively

reduces the frequency-noise contributions to the light from the fiber (see Sec. 3.3). The 563 nm radiation is frequency-doubled to 281.5 nm and is focused onto the trapped ion. AOM 2 in Fig. 1(b) shifts the frequency of the

light to match the ion transition. We plan to interrogate the transition using

the Ramsey technique5 with a Ramsey time Tr ss 30 ms. A digital servo loop will adjust the AOM frequency to step between both sides of the central

fringe, and will periodically record the values of the center frequency. 13

3 Laser Frequency Stabilization

The frequencies of several types of lasers have been locked to resonances of

Fabry-Perot cavities with imprecisions less than 0.1 Hz,15 but the frequency

instabilities of the cavity resonances were orders of magnitude greater. Previously, the narrowest published visible-laser linewidth was 10 Hz for a 1 s averaging time. 16 Recently we achieved a linewidth of 0.6 Hz for averaging

times up to 32 s. 17 Here we report a linewidth <0.16 Hz for averaging times

up to 20 s. These improvements arose chiefly from better isolation of the

cavities from mechanical vibrations.

3.1 High-Finesse Reference Cavity

The high-finesse reference cavity should be insensitive to and/or well- protected from environmental perturbations. 17 The cavity spacer and mirror

TN-21
substrates are composed of ULE,18 ' 19 a low-thermal-expansion material. The

cavity is supported inside an evacuated chamber by an aluminum V-block

with four Viton contact points. We protect the cavity from vibrational noise

by mounting the vacuum chamber on a passively isolated optical table. Dashpots filled with grease at each corner of the table provide viscous damping.

To reduce the coupling of acoustic noise into the cavity, we enclose the optical

table in a wooden box lined internally with lead foam.20 Active servo control

of the optical power transmitted through the cavity stabilizes the frequency

shift caused by mirror heating.

3.2 Measurement of Cavity Stability

To characterize the cavity's stability, we constructed a second cavity and

mounted it on a second, independent, vibrationally isolated table. 17 A laser beam is frequency-locked to each of the cavities. Some light from one of

these beams propagates from one isolated platform to the other. There, it heterodynes with light from the beam that is stabilized to the second cavity,

providing a measure of the relative frequency deviations between the two

cavities. We mix the beat note {v « 400 MHz) with a linearly swept rf source, which translates the beat-note frequency lower to facilitate high-resolution

analysis and removes a fairly uniform frequency drift (<2 Hz/s). We also can remove Doppler-shifts of the beat frequency caused by relative motion of the

isolated platforms,13,17 but that correction is usually unnecessary.

In the frequency domain, a fast Fourier transform (FFT) spectrum analyzer measures the spectrum of the beat note, as shown in Fig. 2. The width

of the spectrum at its half-power point is 0.22 Hz (20 s averaging time). The

0.19 Hz resolution bandwidth of the spectrum analyzer makes a sizable contribution to this frequency width. However, we conservatively estimate the laser linewidth by omitting the bandwidth correction. We infer that at least one

of the beams has a linewidth <0.16 Hz at 563 nm for averaging times up to 20 s. This fractional linewidth of 3 x 10~ 16

is nearly two orders of magnitude

smaller than published results for other stabilized lasers, and may represent

the smallest fractional linewidth ever measured in the optical regime.

For time-domain analysis, we first frequency-divide the beat signal by a

factor of 20, and then mix the signal down to dc. 17 The frequency division

permits a simple conversion from mixer output amplitude to relative laser phase A<p by allowing the in-quadrature condition (A</>/20 — 7r/2) < 1 to

persist for several seconds. From a time record of A<f> we compute cry (r) for r < 2.5 s. For r > 0.5 s, we perform time-domain measurements using an automated dual-mixer time-difference measurement system.21 Figure 3 shows

TN-22
1.0

o

ex 2

o

c

u 0.5 >

'M13

8

•-i to

"

| laser 1 }-^ y

PD

0.0

f - 10 Hz

J

0.22 Hz

K

f

c + 10 Hz

frequency (Hz)

Figure 2. Power spectrum of the beat note between two 563 nm laser beams stabilized to two

independent cavities. The dashed line shows the -3 dB level. The resolution bandwidth of the spectrum analyzer is 0.19 Hz, and the averaging time is 20 s. A nearly uniform relative cavity drift of 0.4 Hz/s is suppressed by mixing the beat note with a swept frequency

synthesizer. PD, photodiode.

the Allan deviations determined using these two measurement techniques,

alongside the reported cry (r) for other stable laser systems. For 30 ms < r <

100 s, the Allan deviation of our laser is approximately an order of magnitude

less than that of any other stable lasers.

3.3 Reduction of Optical Fiber Noise

Optical fibers are convenient for transporting light and avoiding alignment

instabilities. Unfortunately, fibers add considerable frequency noise to the

light.27 A high-performance fiber-noise cancellation scheme using two AOMs

has been demonstrated.27 Since we need multiple fiber links with limited table

space and wish to conserve optical power, we implemented a simpler scheme

using a single AOM, as shown in Fig. 4. The stabilized light propagates through the optical fiber and then an AOM. Some of this light retroreflects back through the AOM and the fiber,

where it heterodynes with a sample of the input fight. A phase-locked loop

servos the AOM deflection frequency so that the beat signal is phase-coherent

with a stable rf reference signal at 160 MHz. Thus, the phase-locked loop

TN-23
Figure 3. Allan deviation curves for stabilized lasers. We calculate ctv (t) for one of our

sources from an analog-to-digital sample of the beat signal (curve A) and using a dual- mixer measurement system (curve B). (We remove a linear relative cavity drift of 2.4 Hz/s.)

The dotted line shows the quantum noise limit for a Hg+ optical frequency standard (N = 1 and TR = 30 ms). Results for other stabilized lasers: (Nd:YAG) Nd:YAG lasers locked to Fabry-Perot cavities22

; (Nd:YAG/l2) iodine-stabilized Nd:YAG lasers23; (He-Ne)

methane-stabilized He-Ne lasers24

; (C02 ) C02 lasers25 locked to Os04 ; (CORE) Nd:YAG

lasers locked to cryogenic resonator oscillators. 26

stable-laser table ^lfiberk ion

f V V 3 A0M , „

o J La/4 ^l .A r^iA/4

(Af <ihz) PRS pr

A/4 j<M> n <pki<

563 nm User

polarizer

I * H— — ^>«^ ^ HfA ,- . * av^fA0M+f

N

) ?

f° +f' 282 nm

2x ( fA0M+fN>i

160 MHz

reference

I f + fA0M+ fN 2x < f + fA0M+ W

VCO

f VCO=fAOM* 80MHz

Figure 4. Fiber-noise cancellation scheme. A phase-locked loop controls the deflection fre- quency /aom of an AOM to suppress the frequency noise /n on the light caused by the

optical fiber. BS, beam splitter; PBS, polarizing beam splitter.

TN-24
impresses noise on the AOM frequency that nearly cancels the noise put on

the light by the fiber. Consequently, light that only single-passes both the fiber and the AOM has a frequency precisely 80 MHz different from the incident

light, maintaining its high spectral purity.

Unfortunately, the «80 MHz AOM also amplitude-modulates the light at a frequency nearly degenerate with our 160 MHz signal. This modulation

component, which is about 24 dB below the signal power, limits the fiber- noise cancellation. Additionally, we retroreflect light through the fiber using

linear polarization orthogonal to the incident light polarization, rather than

identical linear polarization. This conserves optical power, but there may be

a difference between the noise contributions on the two passes through the

fiber, again limiting the accuracy of the noise cancellation.

We test the capability of our scheme to transport light without significant

spectral broadening by using a fiber link similar to that in Fig. 4, except that

both fiber ends are mounted on a single optical table. We verify the noise

correction by heterodyning the single-passed light with some incident light.

The spectral purity of the 80 MHz beat signal indicates the accuracy of the

fiber-noise cancellation. When the phase lock is disabled, the frequency ex- cursions of the beat note are «20 kHz; when enabled, the phase lock largely

eliminates the fiber noise. We perform time-domain characterization of the

80 MHz beat signal as described in Sec. 3.2. The Allan deviation corresponding to the residual fiber noise (see Fig. 5) is wlO times less than <Ty (r) for the

laser light.

4 Single-Ion Frequency Reference

The high vapor pressure of Hg+ at room temperature combined with our

desire for ion storage times of several days has guided us toward cryogenic

traps. Cryogenic operation introduces a host of challenges, most prominently

the accumulation and "freezing out" of patches of charge on the trap electrodes

during the loading process. The resulting stray electric fields can add enough

additional bias to prevent the trapping of ions. Therefore, heaters must be

incorporated into the trap structure so that its temperature may be elevated

enough to permit the dissipation of any charge accumulated during the time

the Hg oven and ionizing electron beam are activated. After loading, the trap

is returned to cryogenic operation without further evidence of fluctuating bias

fields.We are experimenting with a number of heated trap structures, which,

so far, are all variations of a linear rf Paul trap geometry.28 For use in the

optical frequency standard, it is critical that the trap provide tight confineTN-25
c

o

CO ">

CDc

co

10

10

-14

15

10

10

-16

17

i i i i 1 1

1

1 1 I I ! I 1

|

1 1 1 1 I I I I

|

fiber-noise cancellation

stable laser beatnote — Hg+ quantum limit

0.001 0.01 0.1

x(s)

10

Figure 5. Fractional frequency instability introduced by our fiber link with fiber-noise cancellation. The Allan deviation measured for the incident laser light is sslO times larger than

the instability introduced by the actively corrected fiber link.

ment satisfying the Lamb-Dicke criterion that the ion's maximum excursions

|Ar| < A/27T ?s 45 nm.29 ' 30 Otherwise, the transition strength of the optical

carrier sensitively depends on the vibrational amplitude of the trapped ion,

and fluctuations in transition strength result when the mean excitation number of the harmonic motion is large or changing. Presently, we have trapped

and cooled ions to crystallization in one trap that does not satisfy the LambDicke criterion. We recently constructed a smaller trap that should yield

stronger confinement. In the future, we may employ lithographic traps similar to those used in our group for quantum-state engineering investigations,

but modified for cryogenic operation.

5 Conclusions

We have demonstrated a laser suitable for precision spectroscopy and for optical frequency standards. It has a linewidth of less than 0.16 Hz at 563 nm

for averaging times up to 20 s. Its fractional frequency instability is 3 x 10-16

at 1 s. We have assembled a new cryogenic Hg+ trap that should provide

Lamb-Dicke confinement. When tight confinement is demonstrated, we will frequency-lock our stable laser to the ion. 13 If a simple frequency synthesis

TN-26
scheme connecting the optical transition to microwave frequencies31 ' 32 proves

feasible, we anticipate a time standard with an inaccuracy near 10-18

, and

stability surpassing the best present-day clocks.

Acknowledgments

We thank C. N. Man for use of her high-finesse cavity design and R. Lalezari,

M. Lauer, and D. Willis of Research Electro-Optics for its fabrication; W. D. Lee and C. Nelson for use of their digital servo software in stabilizing one of our isolation tables; F. Walls for assistance with the dual-mixer measurement

system; and J. Wells, S. Jefferts, M. Young, and M. Lombardi for useful comments on the manuscript. This work is supported by the Office of Naval

Research, the Army Research Office, and the National Institute of Standards

and Technology.

References

1. A. G. Mann et al., in Proceedings of the 1998 IEEE International Fre- quency Control Symposium (IEEE, New York, 1998), pp. 13-17.

2. D. J. Berkeland et al, Phys. Rev. Lett. 80, 2089 (1998).

3. P. T. H. Fisk, M. J. Sellars, M. A. Lawn, and C. Coles, IEEE Trans.

Ultrason. Ferroelectr. Freq. Control 44, 344 (1997).

4. J. D. Prestage, R. L. Tjoelker, and L. Maleki, in Trapped Charged Particles and Fundamental Physics, AIP Conference Proceedings 457, edited

by D. H. E. Dubin and D. Schneider (AIP, New York, 1999), pp. 357-364.

5. N. F. Ramsey, Phys. Rev. 78, 695 (1950).

6. W. M. Itano et al., Phys. Rev. A 47, 3554 (1993).

7. H. G. Dehmelt, Bull. Am. Phys. Soc. 18, 1521 (1973).

8. H. G. Dehmelt, IEEE Trans. Instrum. Meas. IM-31, 83 (1982).

9. D. J. Wineland et al., J. Phys. (Paris) 42, C8 (1981).

10. P. T. H. Fisk, Rep. Prog. Phys. 60, 761 (1997).

11. D. J. Wineland and W. M. Itano, Phys. Today 40, 34 (1987).

12. J. C. Bergquist, ed., Proceedings of the Fifth Symposium on Frequency

Standards and Metrology (World Scientific, Singapore, 1996); see also Proceedings of the 1999 European Frequency and Time Forum, to be published.

13. J. C. Bergquist, W. M. Itano, and D. J. Wineland, in Frontiers in Laser

Spectroscopy, proceedings of the International School of Physics "Enrico

Fermi": Course 120, edited by T. W. Hansen and M. Inguscio (NorthHolland, Amsterdam, 1994), pp. 359-376.

TN-27
14. R. W. P. Drever et al, Appl. Phys. B 31, 97 (1983).

15. Ch. Salomon, D. Hils, and J. L. Hall, J. Opt. Soc. Am. B 5, 1576

(1988).

16. J. Dirscherl, B. Neizert, T. Wegener, and H. Walther, Opt. Commun.

91, 131 (1992).

17. B. C. Young, F. C. Cruz, W. M. Itano, and J. C. Bergquist, Phys. Rev.

Lett. 82, 3799 (1999).

18. Commercial products are identified in order to adequately specify the

experimental procedure. Such identification does not imply recommendation or endorsement by the National Institute of Standards h Technology.

19. S. F. Jacobs, Opt. Acta 33, 1377 (1986).

20. D. Hils, J. E. Faller, and J. L. Hall, Rev. Sci. Instrum. 57, 2532 (1986).

21. S. Stein et al, in Proceedings, 36th Annual Frequency Control Symposium

(U.S. Army Electronics Command, Ft. Monmouth, NJ, 1982), pp. 314-

320.

22. N. M. Sampas, E. K. Gustafson, and R. L. Byer, Opt. Lett. 18, 947

(1993).

23. J. L. Hall et al, in 1998 Conference on Precision Electromagnetic Measurements Digest, edited by T. L. Nelson (IEEE, New York, 1998), pp. 151-152.

24. S. N. Bagayev, A. K. Dmitriyev, P. V. Pokasov, and B. N. Skvortsov, in Proceedings of the Fifth Symposium on Frequency Standards and Metrology, edited by J. C. Bergquist (World Scientific, Singapore, 1996), pp. 289-296.

25. 0. Acef, Opt. Commun. 134, 479 (1997).

26. S. Seel et al, Phys. Rev. Lett. 78, 4741 (1997).

27. L.-S. Ma, P. Jungner, J. Ye, and J. L. Hall, Opt. Lett. 19, 1777 (1994).

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

Sci. Instrum. 67, 129 (1996).

29. R H. Dicke, Phys. Rev. 89, 472 (1953).

30. J. C. Bergquist, W. M. Itano, and D. J. Wineland, Phys. Rev. A 36,

428 (1987).

31. Th. Udem, J. Reichert, R. Holzwarth, and T. W. Hansen, Phys. Rev.

Lett. 82, 3568 (1999).

32. B. Freeh et al, "Sub-systems for optical frequency measurements: Application to the 282 nm 199 Hg+ transition and the 657 nm Ca line," in Proceedings of the 1999 European Frequency and Time Forum, to be

published.

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Volume 85, Number 12 PHYSICAL REVIEW LETTERS 18 September 2000

199i Sub-dekahertz Ultraviolet Spectroscopy of Hg,+

R.J. Rafac, B.C. Young,* J. A. Beall, W.M. Itano, D.J. Wineland, and J.C. Bergquist

National Institute of Standards and Technology, Boulder, Colorado 80303

(Received 8 May 2000)

Using a laser that is frequency locked to a Fabry-Perot etalon of high finesse and stability, we probe

the 5d ]0 6s 2 S]/2(F = 0) «-» 5d9 6s22D5 /2 (F = 2) AmF = electric-quadrupole transition of a single laser-cooled 199 Hg+

ion stored in a cryogenic radio-frequency ion trap. We observe Fourier-transform

limited linewidths as narrow as 6.7 Hz at 282 nm (1.06 X 10 15 Hz), yielding a line 2 * 1.6 X 10 14

. We perform a preliminary measurement of the 5d9 6s2 2 £>5 /2 electric-quadrupole shift due to interaction with the static fields of the trap, and discuss the implications for future trapped-ion optical frequency

standards.

PACS numbers: 32.30Jc, 06.30.Ft, 32.80.Pj, 42.62.Fi

Precision spectroscopy has held an enduring place in physics, particularly in the elucidation of atomic structure and the measurement of fundamental constants, in the development of accurate clocks, and for fundamental tests of

physical laws. Two ingredients of paramount importance

are high accuracy, that is, the uncertainty in systematic fre- quency shifts must be small, and high signal-to-noise ratio,

since the desired measurement precision must be reached

in a practical length of time. In this paper, we report

the measurement of an optical absorption line in a single

laser-cooled 199 Hg+ion at a frequency vq> = 1.06 X

10 15 Hz (wavelength ~ 282 nm) for which a linewidth

kv = 6.7 Hz is observed, yielding the highest

Q = vq/Av ever achieved for optical (or lower fre- quency) spectroscopy. We also report a preliminary

measurement of the interaction of the upper state electric- quadrupole moment with the static field gradients of

the ion trap, which is expected to contribute the largest

uncertainty for a frequency standard based on this system.

"

In spectroscopy and for clocks, fluctuations in fre- quency measurement are usually expressed fractionally:

<x v (t) = Afmeas (r)/^o, where r is the total measurement

time. When the stability is limited by quantum fluctuations

in state detection, ay {r) = C(27TVQ)~ 1 {Nrpiobe T)~ 1/2

, where TV is the number of atoms, Tprobe is the transition probe time (typically limited by the excited-state lifetime or the stability of the local oscillator), and C is a constant of order unity that depends on the method of interrogation. For many decades, the highest accuracies and the greatest

stabilities have been achieved by locking a microwave oscillator to a hyperfine transition in an atomic ground state [1-5]. Since the fractional instability ay {r) is inversely

proportional to the transition frequency, greater stability can be attained using transitions at higher frequencies

such as those in the optical region of the electromagnetic

spectrum. Only recently have lasers of sufficient spectral

purity become available to probe the narrow resonances

provided by transitions between long-lived atomic states [6-12]. The stable laser source of Refs. [6,11] and the

relative freedom from environmental perturbation afforded

by ion trapping enable the high resolution reported here.

Combined with novel, highly compact, and accurate

laser frequency measurement schemes [13,14], a trappedion optical frequency standard would appear to have

significant advantages over present-day atomic clocks. A partial energy-level diagram of 199Hg+

is shown in

Fig. 1. The 282 nm radiation used to drive the 2S\/2 «-»

2 £>5/2 transition is produced in a nonlinear crystal as the

second harmonic of a dye laser oscillating at 563 nm. The frequency of the dye laser radiation is electronically

served to match a longitudinal mode of a high-finesse

Fabry-Perot cavity that is temperature controlled and sup- ported on an isolation platform [6,11]. The stabilized laser

light is sent through an optical fiber to the table holding

the ion trap. Unavoidable mechanical vibration of this fiber broadens the laser spectrum by nearly 1 kHz. The

fiber-induced phase noise is sensed and removed using a method [11] similar to that described in Ref. [15]. Finally, the frequency of the 563 nm light is referenced to the

electric-quadrupole transition by first frequency shifting in an acousto-optic modulator (AOM) and then frequency

doubling in a single pass through a deuterated ammonium

dihydrogen phosphate crystal.

The design of the linear cryogenic ion trap used in these

measurements borrows heavily from our previous work

x(

2 P

i/2

) = 2ns

Pi/2 F

! J* $6.9 GHz t(

2D

5/2

) = 90ms

194 nm

282 nm

40.5 GHz

F =

FIG. 1. Partial energy level diagram of 199Hg+ with the transitions of interest indicated.

2462 0031-9007/00/85(12)/2462(4)$15.00 © 2000 The American Physical Society

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Volume 85, Number 12 PHYSICAL REVIEW LETTERS 18 September 2000

[4,16,17]. A single 199Hg atom from a thermal source is ionized by an electron beam and trapped in the harmonic

pseudopotential formed by the rf and static potentials of

a linear quadrupole trap. The trap operates in analogy to a quadrupole mass filter having its ends "plugged" with

static fields. The trap electrodes are constructed from Aumetallized alumina tubes 530 /xm in diameter. The elec- trode axes are held parallel and coincident with the vertices of a square of side 1.37 mm by an alumina structure that

facilitates electrical connection and mounting in the vac- uum chamber.

One pair of diagonally opposite electrodes is segmented

by laser micromachining prior to metallization to permit

application of the axially confining static potential. The

remaining pair of electrodes is driven by a cryogenic copper helical resonator coupled to a 8.6 MHz signal source. Under typical operating conditions, a single 199 Hg+

ion

exhibits secular motion at 1.45, 1.86, and 1.12 MHz in the

x, v, and z (axial) directions, respectively, as inferred from

the vibrational sideband spectrum of the ion [18]. In addition, biasing electrodes are mounted outside the trap rods

to cancel any stray static electric fields that may be present.

Previous experiments using 199 Hg were performed at room temperature and at a pressure of approximately

10~ 7 Pa [19]. Under those conditions, the background

gas pressure was large enough that the ion would be lost due to chemical reaction after only a few minutes. To

circumvent this, the ion trap is housed in a liquid He

vacuum Dewar like that described in [17]. Engineering

particulars of the liquid He cryostat allow low-frequency

(<100 Hz) vibratory motion of the trap structure rela- tive to the optical table. Uncorrected, these vibrations

contribute 50-1000 Hz of Doppler broadening to the

laser line. We eliminate the majority of this broadening

using an additional stage of Doppler cancellation, where

the correction signal is derived from optical heterodyne

detection of a motion-sensing beam reflected from a mirror rigidly affixed to the trap [19]. The resulting

cancellation is not ideal, because the sensing beam is steered by additional optical elements and its path deviates

slightly from overlap with the probe beam near the trap. Measurements indicate that this optical path difference can contribute as much as 2 Hz to the spectral width of

the 282 nm probe laser in the reference frame of the ion. The ion is laser cooled to near the 1.7 mK Doppler limit by driving the

5

d

10 6s 2 S1 / 2 (F = 1) «-» 5d 10 6p 2 Pl/2 (F =

0) cycling transition at 194 nm (Fig. 1) [18]. Because of

weak off-resonant pumping to the 2 S\/2 {F = 0) state, we

employ a second 194 nm source phase locked to the first with a 47 GHz offset that returns the ion to the groundstate F = 1 hyperfine level. We tolerate the complication

of hyperfine structure, since only isotopes with nonzero

nuclear spin can have first-order magnetic-field-insensitive

transitions that provide immunity from fluctuations of the ambient field. This significantly relaxes the requirements

for control and /or shielding of environmental magnetic

sources.

TN-30

We monitor the ion and deduce its electronic state using

light scattered from the cooling transition. Fluorescence at 194 nm is collected by a five-element uv-grade fused sil- ica f/\ objective located inside the cryostat. The scattered

light is imaged outside the Dewar, spatially filtered with a 75 lltsx aperture, and relayed with a second lens to an imaging photomultiplier tube having =5% quantum efficiency

at 194 nm. Transitions to the 2 Ds/2 state are detected us- ing the technique of "electron shelving," which infers the presence of the atom in the metastable level through the ab- sence of scattering from the strong laser-cooling transition [18,20]. A metastable-state detection efficiency near unity can be achieved, because the absorption of a single 282 nm

photon suppresses scattering of many photons from the 194 nm transition for a period determined by the lifetime

of the 2 D$/2 state. The radiation from the 194 and 282 nm

sources is admitted to the trap sequentially using mechanical shutters and an AOM, which prevents broadening of

the quadrupole transition by the cooling radiation. Typical

count rates are 2000 Hz for a single ion cycling at the half- power point of the cooling transition, compared to only

20 Hz combined laser scatter and photomultiplier thermal

background when the ion is shelved in the metastable level. Spectra of the recoilless "carrier" component of the

2 Si /2 (F = 0) <- 2 D5/2 (F = 2) AmF = transition were

obtained for a range of probe times and laser intensities by laser cooling for 30 ms, preparing the ion in the F —

ground state by blocking the repumping laser, and then in- terrogating the quadrupole transition. The spectra are built up from multiple bidirectional scans of the 282 nm probe

laser frequency. Since the frequency drift of the probe

laser is not precisely compensated, nor is it constant, we

incorporate a locking step in between pairs of positiveand negative-going frequency sweeps about the center of

the quadrupole resonance. In the locking sequence, we

step the frequency of the probe laser alternately to the ap- proximate half maximum on either side of the quadrupole

resonance, probe for a fixed time rservo , and then look for

transitions to the 2 D$/2 level. Typically, 48 measurements

are made on each side of the resonance during each lock

cycle. The asymmetry between the number of quantum

jumps detected on the high- and low-frequency sides of

the resonance is used to correct the frequency of a synthesizer used to compensate for cavity drift. In this fashion,

variations in the frequency of the 282 nm laser for times

exceeding several seconds are reduced. Using the locking

step alone with Tser/0 = 40 ms, the error signal from the lock indicates a fractional frequency instability ay {r) =

1.5 X 10-15for times comparable to the length of the combined servo and scanning cycle (15-60 s). This is worse than the quantum-projection-noise-limited stability

(~5 X 10

-16

) but is consistent with fluctuating quadratic

Zeeman shifts arising from variations in the ambient magnetic field. The trap was not magnetically shielded during

the measurements reported here. Spectra are plotted in Fig. 2 for a variety of probe

times Tprobe = 20-120 ms, with rs rprobe The

2463
Volume 85, Number 12 PHYSICAL REVIEW LETTERS 18 September 2000

FIG. 2. Quantum-jump absorption spectra of the

2 S\/2{F — 0) *-» 2 D5/2 (F = 2) AmF = electric-quadrupole

transition. A/282 is the frequency of the 282 nm probe laser detuning, and Pg

is the probability of finding the atom in the ground state. The four plots correspond to excitation with 282 nm pulses of different lengths: (a) 20 ms (averaged over 292 sweeps), (b) 40 ms (158 sweeps), (c) 80 ms (158 sweeps),

and (d) 120 ms (46 sweeps). The linewidths are consistent with the Fourier-transform limit of the pulse at 40(2), 20(1),

10(1), and 6(1) Hz.

linewidths of all the spectra are transform limited by the

finite probe time, decreasing to 6.7 Hz in the uv at 120 ms,

the longest time used. The carrier transition amplitude is a function of the initial value of the vibrational quantum

number n [see, for example, Eq. (31) of Ref. [21]]. Since

for our trapping parameters (n) ~ 35 at the Doppler

cooling limit, it is not possible to transfer the electron to the 2£>5/2 state with unit probability. The observed signals are in good agreement with the theoretical expectation,

and the signal loss for Tprobe = 120 ms [Fig. 2(d)] is consistent with applying the probe for a time that exceeds

the natural lifetime of the 2 D5 /2 state by 33%. This

result corresponds to a fractional frequency resolution of

6.3 X 10-15

, which we believe is the smallest reported

for excitation with an optical or microwave source. The

result is surpassed only by Mossbauer spectroscopy in ZnO, where a fractional linewidth of 2.5 X 10~ 15 was

achieved using a nuclear source of 93 keV gamma rays

[22], which is not practical for use in an atomic clock. The measurements reported here illustrate the potential of

a single-ion optical frequency standard and confirm the phase stability of our probe laser system, validating the

results of the heterodyne comparison of the two reference

cavities [6,11]. An excellent review of the performance of

similar systems can be found in Ref. [12]. Figure 3 shows a typical power-broadened spectrum

with consequent Rabi "sidebands," using rprobe = 10 ms

and Tservo = 40 ms. A least-squares fit to the data indi- cates that it is consistent with a pulse area of 2.41(7)77

(relative to n ~ 0) and a mean vibrational quantum number (n) = 73(9). This is twice the value of (n) expected at 2464

250

FIG. 3. Quantum-jump absorption spectrum of the powerbroadened electric-quadrupole transition. A/282 is the frequency

of the 282 nm probe laser detuning, and Pg

is the probability of

finding the atom in the ground state. The data are averaged over 348 frequency sweeps with Tpr0 be = 10 ms and rservo = 40 ms. The solid line is a least-squares fit to a Rabi line profile with

pulse area 2.41(7)7? and (n) = 73(9). The quantum-projectionnoise-limited uncertainty is indicated by representative error bars plotted every third point.

the Doppler cooling limit; the larger result is likely due to saturation of the cooling transition. In Fig. 4 we plot the central three Ramsey fringes of

a time-domain separated-oscillatory-fields interrogation of

the 2 Si/2 <-* 2 D5 /2 transition with 5 ms pulses separated

by a 20 ms free-precession interval. In operation as a fre- quency standard, the Ramsey method offers 1.6X reduction in linewidth (and a corresponding increase in stability) for an equivalent measurement period.

282

FIG. 4. The central three fringes obtained by time-domain

Ramsey interrogation of the electric-quadrupole transition. A/282 is the frequency of the 282 nm probe laser detuning,

and Pg

is the probability of finding the atom in the ground

state. The data are averaged over 328 frequency sweeps with

Tservo = 40 ms. The solid line is the expected Ramsey signal

for 5 ms pulses separated by 20 ms of free precession; the = 10% background arises primarily from suboptimal setting of the discriminator levels in our detection electronics. The

quantum-projection-noise-limited uncertainty is indicated by

representative error bars plotted every third point.

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Volume 85, Number 12 PHYSICAL REVIEW LETTERS 18 September 2000

It should be possible to reduce the uncertainties of all systematic shifts in this system including the secondorder Doppler (time-dilation) shift and static or dynamic

Zeeman and Stark shifts to values approaching 10-18

. The

electric-quadrupole shift of the 2Ds/iF = 2 {mp = 0)

state arising from coupling with the static potentials

of the trap is expected to be the limiting systematic

contribution in future measurements of the absolute value

of the 2Si/2 *-* 2 Ds/2 transition frequency, because of the

difficulty in determining with certainty the magnitude

and configuration of the static fields of the trap. In

principle, it is possible to eliminate this shift by measuring

the quadrupole transition frequencies for each of three mutually orthogonal orientations of a quantizing magnetic

field of constant magnitude; the mean value is then the unperturbed 2 S\/2 +•+ 2D$/2 transition frequency. We fol- low this procedure for several values of the total magnetic

induction and for all possible directions in a suitable

coordinate system (e.g., [1 1 1], [1 —1 —1], [—1 —1 1], where [1 1 1] is elevated 45° from the trap axis and 45°

from the vertical plane). Using the ion as a magnetometer,

the magnitude of the field produced by three sets of coils

is calibrated by the measurement of the frequency of

the F = — 2 (Am; = 2) magnetic field-dependent

electric-quadrupole transition. For an ideal linear

trap with secular frequencies co x = a) y and co z = 2tt

(1 MHz), we calculate a shift that varies between +7.5

and — 15 Hz depending on field orientation. The estimate

of the atomic quadrupole moment is based on Hartree- Fock wave functions for the ground state of Hg+

, which

yield the matrix element {5d\r 2 \5d) = 6.61 X 10~ 21 m2

[23]. In our apparatus we observe shifts departing from

the mean frequency by +12(7), +9(13), and -27(7) Hz

for the three field orientations at 282 nm, in reasonable

agreement with the simple model. The quoted uncertainties arise from imprecision in our ability to set the

absolute magnitude and direction of the magnetic induction. Thus the fractional frequency uncertainty is of the order of 10 , and we might reduce this below 10 in the present apparatus with straightforward improvements

to the magnetic shielding and control. Uncertainties

approaching 10 -18 should be obtainable through stringent control of magnetic fields in combination with a spherical

rf quadrupole trap geometry that does not rely on static potentials for confinement. Greater resolution and accuracy might be more readily achieved using a different type

of transition, e.g., the weak hyperfme-induced electric- dipole transitions like those between the low-lying 1 Sq

and 3 Pq states of the singly ionized species of the Group

iriA elements of the periodic table [12,24,25], particularly

in cases where first-order magnetic-field independent

transitions are available (albeit at nonzero field) [26]. This research was partially supported by the Office of

Naval Research and through a Cooperative Research and

Development agreement with Timing Solutions, Corp.,

Boulder, Colorado. We would like to thank C. Oates,

J. Ye, and D. Sullivan for their careful reading of this manuscript.

*Present address: Jet Propulsion Laboratory, Pasadena, CA

91109.

[1] A. Clairon et al, in Proceedings of the 5th Symposium

on Frequency Standards and Metrology (World Scientific, Singapore, 1996), pp. 49-59.

[2] R. L. Tjoelker et al, in Proceedings of the 5th Symposium on Frequency Standards and Metrology (Ref. [1]), pp. 33-38.

[3] P.T.H. Fisk et al, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 44, 344 (1997).

[4] D.J. Berkeland et al., Phys. Rev. Lett. 80, 2089 (1998).

[5] S. R. Jefferts et al. (to be published).

[6] B.C. Young et al., Phys. Rev. Lett. 82, 3799 (1999).

[7] Ch. Salomon et al, J. Opt. Soc. Am. B 5, 1576 (1988).

[8] N.M. Sampas et al, Opt. Lett. 18, 947 (1993).

[9] O. Acef, Opt. Commun. 134, 479 (1997).

[10] S. Seel et al, Phys. Rev. Lett. 78, 4741 (1997).

[1 1] B. C. Young et al, in Proceedings of the 14th International Conference Laser Spectroscopy (World Scientific, Singapore, 1999), pp. 61-70.

[12] A. A. Madej and J. E. Bernard, in "Frequency Measure- ment and Control," edited by Andre N. Luiten, Springer

Topics in Applied Physics (Springer-Verlag, Berlin, to be

published).

[13] J. Reichert et al, Phys. Rev. Lett. 84, 3232 (2000).

[14] S. A. Diddams et al. (to be published).

[15] L.-S. Ma et al, Opt. Lett. 19, 1777 (1994).

[16] M.G. Raizen et al, J. Mod. Opt. 39, 233 (1992).

[17] M.E. Poitzsch et al, Rev. Sci. Instrum. 67, 129 (1996).

[18] J.C. Bergquist et al, Phys. Rev. A 36, 428 (1987).

[19] J.C. Bergquist et al, in Frontiers in Laser Spectroscopy,

Proceedings of the International School of Physics "Enrico

Fermi," Course CXX (North-Holland, Amsterdam, 1994),

p. 359.

[20] H. G. Dehmelt, Bull. Am. Phys. Soc. 20, 60 (1975); J. Phys.

(Paris) Colloq. 42, C8-299 (1980).

[21] D.J. Wineland and W.M. Itano, Phys. Rev. A 20, 1521

(1979).

[22] G.J. Perlow et al, J. Phys. (Paris) Colloq. 35, C6-197

(1974).

[23] A.D. McLean and R.S. McLean, At. Data Nucl. Data

Tables 26, 197 (1981).

[24] H.G. Dehmelt, IEEE Trans. Instrum. Meas. 31, 83 (1982).

[25] E. Peik et al, in Proceedings of the 13th European Fre- quency and Time Forum and the 1999 IEEE Frequency

Control Symposium, Besancon, France (Institute of Electrical and Electronics Engineers, Piscataway, NJ, 1999),

pp. 682-685.

[26] D.J. Wineland and W.M. Itano, Bull. Am. Phys. Soc. 27,

864 (1982).

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2465
Volume 105, Number 6, November-December 2000 Journal of Research of the National Institute of Standards and Technology

[J. Res. Natl. Inst. Stand. Technol. 105, 829-837 (2000)] External-Field Shifts of the 199Hg+ Optical

Frequency Standard

Volume 105 Number 6 November-December 2000

Wayne M. Itano

National Institute of Standards and

Technology,

Boulder, CO 80305

itano@boulder.nist.gov

Frequency shifts of the

l99Hg+ 5d w

6s 2 S m

(F = 0,MF = 0) to 5d96s 2 2D5/2 (F = 2, Mf = 0) electric-quadrupole transition at 282 nm due to external fields are calculated, based on a combination of measured

atomic parameters and ab initio calculations. This transition is under investigation

as an optical frequency standard. The

perturbations calculated are the quadratic Zeeman shift, the scalar and tensor

quadratic Stark shifts, and the interaction between an external electric field gradient and the atomic quadrupole moment. The

quadrupole shift is likely to be the most

difficult to evaluate in a frequency standard and may have a magnitude of about 1 Hz

for a single ion in a Paul trap.

Key words: atomic polarizabilities; electric quadrupole interaction; mercury ion; optical frequency standards; Stark shift; Zee- man shift.

Accepted: November 1, 2000

Available online: http://www.nist.gov/jres

1. Introduction

It has long been recognized that a frequency standard could be based on the 282 nm transition between the ground 5d w

6s 2S 1/2 level and the metastable 5d96s 2 2D5/2 level of Hg+ [1]. The lifetime of the upper level is 86(3) ms [2], so the ratio of the natural linewidth Az^ to the

transition frequency v is 2 X 10" 15

. (Unless otherwise

noted, all uncertainties given in this paper are standard

uncertainties, i.e., one standard deviation estimates.)

Doppler broadening can be avoided if the transition is excited with two counter-propagating photons, as originally proposed by Bender et al. [1] and subsequently

demonstrated by Bergquist et al. [3]. However, optical Stark shifts are greatly reduced if the transition is driven

instead with a single photon by the electric-quadrupole

interaction. In this case, Doppler broadening can be

eliminated if the ion is confined to dimensions much

less than the optical wavelength, as was first demonstrated by Bergquist et al. [4].

Recently, the (F = 0, MF

;= 0) to (F = 2, MF = 0) hyperfine component of the 199Hg+ 5d w

6s 2 S 1/2 to 5d96s 2 2D5/2 single-photon transition has been observed

with a linewidth of only 6.7 Hz by Rafac et al. [5]. A

laser servo-locked to this transition is an extremely sta- ble and reproducible frequency reference. New develop- ments in optical frequency metrology [6, 7] may soon make this system practical as an atomic frequency stan- dard or clock. While the (F = 0,MF = 0) to (F =2, MF = 0) hyperfine component has no linear Zeeman shift, it does have

a quadratic Zeeman shift that must be accounted for. In

addition, there is a second-order Stark shift and a shift due to the interaction between the electric-field gradient

and the atomic electric-quadrupole moment. None of

these shifts has yet been measured accurately, so it is useful to have calculated values, even if they are not very

precise. Also, it is useful to know the functional form of

829

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Volume 105, Number 6, November-December 2000 Journal of Research of the National Institute of Standards and Technology

the perturbation, even if the magnitude is uncertain. For

example, the quadrupole shift can be eliminated by aver- aging the transition frequency over three mutually or- thogonal magnetic-field orientations, independent of the

orientation of the electric-field gradient.

2. Methods and Notation

The quadratic Zeeman shift can be calculated if the hyperfine constants and electronic and nuclear g -factors are known. Similarly, the quadratic Stark effect can be

calculated from a knowledge of the electric-dipole oscil- lator strengths. The quadrupole shift depends on the atomic wavefunctions. Some of these parameters have

been measured, such as the hyperfine constants and some of the oscillator strengths. There are also published calculations for some of the oscillator strengths.

Here, we estimate, by the use of the Cowan atomicstructure codes, values for parameters for which there

are neither measured values nor published calculations. The Cowan codes are based on the Hartree-Fock ap- proximation with some relativistic corrections [8]. The

odd-parity configurations included in the calculation were 5d w

np (« = 6,7,8,9), 5J 105/, 5d96s6p, 5d96slp,

5d96s5f, and 5d8 6s 2 6p. The even-parity configurations were 5d i0ns (n = 6,7,8,9,10), 5d i0nd (n = 6,7,8,9), 5d96s 2

, 5d96sls, 5d96s6d, and 5d96p

2

. Recently, Sansonetti and Reader have made new measurements of the spectrum of Hg+ and classified many new lines [9]. They also carried out a least-squares adjustment of the energy parameters that enter the Cowan-code calculations in order to match the observed energy levels. We

use these adjusted parameters in our Cowan-code calculations. As one test of this method of calculation, we esti- mated the weakly allowed 10.7 fim 5d l %p 2P1/2 to 5d96s2 2D3/2 electric-dipole decay rate. This decay is al- lowed only because of configuration mixing, since it requires two electrons to change orbitals. The calculation shows the decay to be due mostly to mixing between

the 5d l06p and 5d96s6p configurations. The calculated

rate is 111 s

-1

; the measured rate is 52(16) s

_1

[2]. Another test is the electric-quadrupole decay rate of the 5d96s 2 2D5/2 level to the ground level. The calculated rate

is 12.6 s

_1

, and the measured rate is 1 1.6(0.4) s-1

. Similar calculations have been carried out by Wilson [10]. Let Ho be the atomic Hamiltonian, exclusive of the hyperfine and external field effects, which are treated as perturbations. For convenience, we denote the eigenstates of H corresponding to the electronic levels 5d l06s 2 S]/2 and 5d96s 2 2D5/2 having Jz eigenvalue Mj by

IS 1/2 Mj) and ID 5/2 Mj), respectively.

The corresponding eigenvalues of H are denoted

W(S, 1/2) and W(D, 5/2). An arbitrary eigenstate of H

with eigenvalue W(y, J) and electronic angular mo- mentum J is denoted \yJMj). Since 199Hg+ has in addition a nuclear angular momentum /, where /= 1/2, the complete state designation is \y JFMF ), where F is the

total angular momentum, and MF is the eigenvalue of Fz

.

3. Quadratic Zeeman Shift

In order to calculate the energy shifts due to the hyperfine interaction and to an external magnetic field B = Bz, we define effective Hamiltonian operators

H's and //D that operate within the subspaces of hyperfine sublevels associated with the electronic levels 5d X06s 2S 1/2 and 5d96s 2 2 D5/2 , respectively:

H's = hAsIJ + £/(S)aib JB + g',nB I-B, (1)

Hb = hADI-J + gj(D)iLB J-B + g'/tml-B, (2)

where As and Ad are the dipole hyperfine constants, gj(S) and gj(D) are the electronic g -factors, gl is the nuclear g -factor, h is the Planck constant, and /xB is the Bohr magneton. All of the parameters entering H's and

Hq are known from experiments, although a more accurate measurement of gj(D) would be useful. The

ground-state hyperfine constant As has been measured

in a

199Hg+ microwave frequency standard to be 40

507.347 996 841 59 (43) MHz [11]. The excited-state hyperfine constant AD has been measured recently by an

extension to the work described in Ref. [5], in which the difference in the frequencies of the IS 1/2 0) to ID 5/2

2 0) and the IS 1/2 0) to ID 5/2 3 0) transition frequencies was determined to be 3AD = 2 958.57(12) MHz

[12], in good agreement with an earlier, less precise

measurement by Fabry-Perot spectroscopy [13]. The

ground-state electronic g -factor gj(S) was measured in

I98Hg+ by rf-optical double resonance to be 2.003 174

5(74) [14]. The excited-state electronic g -factor gj(D) was measured in 198Hg+ by conventional grating spec- troscopy of the 398 nm 5d l06p 2P3/2 to 5d96s 2 2D5/2 line to be 1.198 0(7) [15]. The difference in gj(S) or gj(D)

between I98Hg+ and 199Hg+

is estimated to be much less than the experimental uncertainties. The nuclear g -fac- tor g', is -5.422 967(9) X 10~4 [16]. The measurement was made with neutral ground-state 199Hg atoms, so the diamagnetic shielding factor will be slightly different from that in the ion. However, this is effect is negligible, since the magnitude of g'j is so small compared to gj(S) or gj (D).

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The determination of gj(D) could be improved by

measuring the optical-frequency difference between

two components of the 282 nm line and the frequency of

a ground-state microwave transition at the same magnetic field. Since the uncertainty in the quadratic Zee- man shift is due mainly to the uncertainty in gy(D), it is useful to see how accurately it can be estimated theoretically. The Lande g -factor for a

2D5/2 state, including the correction for the anomalous magnetic moment of the

electron, is 1.200 464. The Cowan-code calculation shows that the configuration mixing does not change

this value by more than about 10~6

, i.e., 1 in the last place. There are several relativistic and diamagnetic corrections that modify gj(D), one of which, called the Breit-Margenau correction by Abragam and Van Vleck

[17], is proportional to the electron mean kinetic energy. The other corrections are more difficult to calculate. The Cowan-code result for the mean kinetic energy of an electron in the 5d orbital of the 5d96s 2 configuration

is T= 19.32 hcRoz, where i?„ is the Rydberg constant. Using this value, we obtain a theoretical value of gj(D),

including the Breit-Margenau correction, of 1.199 85,

which disagrees with the the experimental value by

1.85 X 10~3

, which is 2.6 times the estimated experimental uncertainty of Ref. [15]. If we calculate gj(D)

for neutral gold, which is isoelectronic to Hg+

, by the same method, we obtain a value which differs from the accurately measured experimental one [18] by

(7 ± 2) X 10"5

. Thus, the error in the calculated value

for gj(D) of 199Hg+ might be less than 1 X 10

-4

, but it is impossible to be certain of this, since there are uncalculated terms. Measurements of the 199Hg+ optical clock frequency at different values of the magnetic field should result in a better experimental value for gj(D) in the near future. For low magnetic fields (5 less than 1 mT), it is sufficient to calculate the energy levels to second order

in B . To this order in B , the energies of the hyperfineZeeman sublevels for the ground electronic level are

W(S, 1/2, 0, 0, B)=W(S, 1/2)

[g/(S) ~ g',}VlB 2

3MS

4/zAs

(3)

W(S, 1/2, 1, ±l,B) = W(S, 1/2) +

^ MS) + gl]fJLBB

2

For the 5d96s 2 2D5/2 level we have

W(D, 5/2, 2, 0, B) = W(D, 5/2)

[gj(D) ~ g'AVlB 2 l2hAD

W(D, 5/2, 2, ±1, B)= W(D, 5/2) -

hAs

lhAD

lhAr

[7gj(D) - gl]^B 2[gj(D) - gl]

2fdB : 21hAr

W(D, 5/2, 2, ±2, B)= W(D, 5/2)

lhAr

[7gy (D) - gl]fiB B 5[gj (D) - g}]

2fiiB2 108/MD

W(D, 5/2, 3, 0, B) = W(D, 5/2) + 5hAD

2„2d2 [gj(D) ~ g'.YjJLJB

12/zAD

W(D, 5/2, 3, ±1, B) = W(D, 5/2) + 5hAD

W(D, 5/2, 3, ±2, B) = W(D, 5/2) + 5hAD

W(D, 5/2, 3, ±3, B) = W(D, 5/2) + 5hAD

(5)

(6)

, (7)

, (8)

(9)

[5g7 (D) + g/>B fl 2{gj (D) - gffusB 2 6 27hAD

' U ;

[5gy (D) + g/>B fi 5[gj (D) - grfniB 2

3 108/jAd

' K '

W(S, 1/2, 1, 0, B) = W(S, 1/2) + ^

igj(S) - gl]

2 n£B>

4MS (4)

[5gj (D) + gl]fiB B

2

(12)

Here, W(y, J, F, MF , B) denotes the energy of the state \yJFMF ), including the effects of the hyperfine interaction and the magnetic field.

831

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At a value of B of 0. 1 mT, the quadratic shift of the

IS 1/2 0) to ID 5/2 2 0) transition (optical clock transi- tion) is —189.25(28) Hz, where the uncertainty stems mainly from the uncertainty in the experimental value of gy(D). In practice, the error may be less than this if the magnetic field is determined from the Zeeman split- tings within the ID 5/2 F MF ) sublevels. The reason is that an error in gj(D) leads to an error in the value of B

inferred from the Zeeman splittings, which partly compensates for the gj(D) error. If instead we use the calculated value of gj(D), the quadratic shift for B = 0.1 mT

is —189.98 Hz, where the uncertainty is difficult to estimate.

4. Quadratic Stark Shift

The theory of the quadratic Stark shift in free atoms

has been described in detail by Angel and Sandars [19]. The Stark Hamiltonian is HE =-fiE, (13)

where /x is the electric-dipole moment operator,

M ?2r" (14)

and E is the applied external electric field. In Eq. (14),

r, is the position operator of the ith electron, measured

relative to the nucleus, and the summation is over all electrons.

First consider an atom with zero nuclear spin, such as

198Hg+

. To second order in the electric field, the Stark

shifts of the set of sublevels \yJMj) depend on two

parameters, aSC aiar(y, 7) and a,ensor(y, 7), called the

scalar and tensor polarizabilities. In principle, when

both magnetic and electric fields are present but are not

parallel, the energy levels are obtained by simulta- neously diagonalizing the hyperfine, Zeeman, and Stark Hamiltonians. In practice, the Zeeman shifts are nor- mally much larger than the Stark shifts, so that HE does

not affect the diagonalization. In that case, the energy

shift of the state \yJMj) due to HE is AW£ (y, 7, Mj, E)=-\ ascalar(y, J)E2

- \ atensor(y, 7)

[3Mj- 7(7+1)]

7(27- 1) (3E}-E2

). (15)

Treating HE by second-order perturbation theory leads

to the following expressions for the polarizabilities [19]:

ascalar(y,7)- 3(2y+

° 1) 2

8fre ^ l(y7ll)Lt ( "lly'7')l

2 3(27+l)^W(y',7')- W(y,J) , (16)

tttensoXy, 7) = 8ire 107(27- 1) 3(27 + 3)(7+l)(27+ 1)

X T f-lV-'i

1 l 21 Ky7ll^>lly'7')l2 X

£.

( 1}

l7 7 7'W,7')-W(y,7)- < 17>

The summations are over all levels other than Iy7).

Equations (16) and (17) can be rewritten in terms of the

oscillator strengths fyj,yjC*scalar(y,7) =

4Tre e

2

/?

2 me E

JjLiL.

yj1[W(Y,J')-W(y,J)) ,(18)

2*2

,r(y, 7) =

4tt6o^ n 307(27- 1)(27+1)

x i<-i>'i;;,

2

}

(27+3)(7+ 1) fyw

[W(y',J')-W(y,J)\ , (19)

where me is the electron mass. The tensor polarizability

is zero for levels with 7 < 1, such as the Hg+ 5d m6s 2Sy2 level. For an atom with nonzero nuclear spin /, the quadratic Stark shift of the state \yJFMF ) is AW£ (y, 7, MF , E)=-\ ascaiar(y, 7, F)E2

~ \ atensor(y, 7, F)

[3MF - F(F+l)]

F(2F- 1) (3£2 - E

2

). (20)

We make the approximation that hyperfine interaction does not modify the electronic part of the atomic wavefunctions (the / 7-coupling approximation of Angel and

Sandars [19]). This approximation is adequate for the present purpose, which is to evaluate the Stark shift of

the

199Hg+ optical clock transition. Obtaining the differential Stark shift between the hyperfine levels of the ground state, which is significant for the 199Hg+ mi- crowave frequency standard [11], requires going to a

higher order of perturbation theory [20]. In the /7-coupling approximation [19],

Q!scalar(y, 7, F) = ascalar(y, 7),

a.ensor(y,7,F) = (-l)/+y+F

F(2F- l)(2F+l)(27+3)(27+l)(7+l)

]

1) J

(21)

1/2 (2F+3)(F+ 1)7(27

7 F i] 01*TM '^

' '' (22)

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Volume 105, Number 6, November-December 2000

Journal of Research of the National Institute of Standards and Technology

Equations (18) and (19) were used to evaluate the polar izabilities for the Hg+ 5d w

6s 2S 1/2 and 5d96s 2 2D5/2 levels. For the calculation of arScaiar(S, 1/2), the oscillator strengths for all electric-dipole transitions connecting

the 5d w6s configuration to the 5d l0np (n = 6,7,8) and

5d96s6p configurations were included. These were

taken from the theoretical work of Brage et al. [21]. The

final result is ascalar (S, l/2)/(4ireb) = 2.41 X 10"24 cm3

, which compares very well with the value of

2.22 X 10~24 cm3 obtained by Henderson et al. from a combination of experimental and calculated oscillator strengths [22]. For the calculations of ascaiar(D, 5/2) and

atensor(D, 5/2), the oscillator strengths for electric-dipole

transitions to the 5d l0np (n = 6,7,8), 5d l05f, and

5d96s6p configurations were taken from Brage et al. [21]. The oscillator strengths for electric-dipole transi- tions to the 5d96s7p and 5ds6s 26p configurations were

taken from the Cowan-code calculations. The results were aScaiar(D, 5/2)/(4ire ) = 3.77 X 10"24 cm3 and

a,ensor(D, 5/2)/(4tt€o) = -0.263 X 10"24 cm3

. Evaluating

Eq. (22) for F = 2 and F = 3 in the 5d96s 2 2D5/2 level, we

obtain a,enSor(D, 5/2, 2) = fatensor(D, 5/2) and a,ensor(D, 5/2, 3) = alensor(D, 5/2). The tensor polarizability is much smaller than the

scalar polarizabilities and in any case does not contribute if the external electric field is isotropic, as is the case for the blackbody radiation field. The net shift of

the optical clock transition due to the scalar polarizabilities is 5[aSC aiar(S, 1/2) — ascaiar(D, 5/2)]£2

. In frequency

units, the shift is -1.14 X 10"3 E

2 Hz, where E is ex- pressed in V/cm. The error in the coefficient is difficult to estimate, particularly since it is a difference of two

quantities of about the same size. However, the total shifts are small for typical experimental conditions. If the electric field is time-dependent, as for the blackbody

field, the mean-squared value (E 2) is taken. At a temperature of 300 K, the shift of the optical clock transition due to the blackbody electric field is —0.079 Hz. The

mean-squared blackbody field is proportional to the fourth power of the temperature. For a single, laser- cooled ion in a Paul trap, the mean-squared trapping

electric fields can be made small enough that the Stark

shifts are not likely to be observable [23].

5. Electric Quadrupole Shift

The atomic quadrupole moment is due to a departure

of the electronic charge distribution of an atom from

spherical symmetry. Atomic quadrupole moments were

first measured by the shift in energy levels due to an

applied electric-field gradient in atomic-beam reso- nance experiments [24, 25].

The interaction of the atomic quadrupole moment

with external electric-field gradients, for example those generated by the electrodes of an ion trap, is analogous

to the interaction of a nuclear quadrupole moment with

the electric field gradients due to the atomic electrons. Hence, we can adapt the treatment used for the electric- quadrupole hyperfine interaction of an atom [26]. The

Hamiltonian describing the interaction of external electric-field gradients with the atomic quadrupole moment

is

HQ = VE(2 >-6> (2) = 2(- D'VE«> S(l

(2) (23)

q=1

where VZ?(2) is a tensor describing the gradients of the

external electric field at the position of the atom, and

6>(2) is the electric-quadrupole operator for the atom. Following Ref. [26], we define the components of

VE(2) as V^2) =-i^, (24)

V£<?)= + y^¥k = t-^? 3+jE (25) -' - 6 dz " 6 -

z ' K '

V£l2j=-^|c?± £± , (26)

where E± = EX ± iEy and d ± = jx ± i-^. The operator components 6>^

2) are defined in terms of

the electronic coordinate operators as ®^=-^10z2 -r2

),

m=-e^^zj(xj±iyj),

m=-e^l^(Xj ± iyj )

2

,

(27)

(28)

(29)

where the sums are taken over all the electrons. The

quadrupole moment 0(y, J) of an atomic level ly J) is defined by the diagonal matrix element in the state with

maximum My.

0(y,J) = (yJJ\0tP\yJJ). (30)

This is the definition used by Angel et al. [24]. In order to simplify the form of V£(2)

, we make a

principal-axis transformation as in Ref. [27]. That is, we

express the electric potential in the neighborhood of the atom as

833

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Volume 105, Number 6, November-December 2000 Journal of Research of the National Institute of Standards and Technology

<p(x \ y \ Z ') = A [(x'

2 + y'2 - 2z'2 ) + e(x' 2 - y' 2)]. (31)

The principal-axis (primed) frame (x', y', z') is the one

in which <P has the simple form of Eq. (31), while the laboratory (unprimed) frame (x,y,z) is the one in which the magnetic field is oriented along the z axis. The tensor components of VZs (2) in the principal-axis

frame are obtained by taking derivatives of <P(x', y', z')' VE?y =-2A,

V£l2,>' = 0,

eA.

(32)

(33)

(34)

In the principal-axis frame, Hq has the simple form

HQ = -2A0P' + yjl eA^0P' + 0^. (35)

As long as the energy shifts due to HQare small

relative to the Zeeman shifts, which is the usual case in practice, Hq can be treated as a perturbation. In that case, it is necessary only to evaluate the matrix elements

of Hq that are diagonal in the basis of states \yJFMF ), where F is the total atomic angular momentum, includ- ing nuclear spin /, and MF is the eigenvalue of Fz with

respect to the laboratory (not principal-axis) frame. Let

to denote the set of Euler angles { a , (3 , y] that takes the

principal-axis frame to the laboratory frame. To be explicit, starting from the principal-axis frame, we rotate the coordinate system about the z axis by a, then about

the new y axis by (3, and then about the new z axis by

y so that the rotated coordinate system coincides with

the laboratory coordinate system. We can set y = 0, since the final rotation about the laboratory z axis, which is parallel to B, has no effect. The states \yJFm)'

defined in the principal-axis frame and the states \yJF/x) defined in the laboratory frame are related by

\yJFmy=^D^((o)\yJFfi), (36)

where D^2(&>) is a rotation matrix element defined in the passive representation [28, 29]. The inverse relation

is

\yJFn) = ^D$'(a>)\yJFmy. (37)

In order to evaluate the diagonal matrix elements of

Hq in the laboratory frame, it is necessary to evaluate matrix elements of the operators 0^y

, defined in the

principal-axis frame. These matrix elements are of the form

(yJF(x\0?y \yJF(x)

= ^D^(to)D^\(o) , {yJFm'\0TM\yJFmy, (38)

= (y JF\\0a)

\\y 7F)2 {-\)F

~m

m'm

= (- \)F-»-q{y JF\\0m\\yJF)

(39)

1 —m q mj (40)

= (-\f-»-i( y JF\\0a) \\yJF) 2 (2£+l)

Km m' n n' F 2 F\(F F K\(F F K\

K —m q m/X/ji —/jl n/\m —m n;

(41)

= (-l)F-»-i(yJF\\0a)\\yJF)(_

F 2QF

^JD^(to),

(42)

where Eq. (39) follows from the Wigner-Eckart theo- rem, and Eqs. (40), (41), and (42) follow from Eqs.

(4.2.7), (4.3.2), and (3.7.8) of Ref. [28], respectively.

The required rotation matrix elements are, from Eq.

(4.1.25) of Ref. [28] (with correction of a typographical

error),

£>$*(<«>) = 5(3 cos

2 /3- l), (43)

D$l(io) = Jl sin

2 /3(cos 2a + i sin 2a). (44)

The 3-j symbol in Eq. (42) is

F 2 F

-fl fJL

= (-l)F-" 2[3/i,

2 - F(F+ 1)] [(2F+3)(2F + 2)(2F+ \)2F(2F - 1)]

1/2 '

(45)

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Volume 105, Number 6, November-December 2000 Journal of Research of the National Institute of Standards and Technology

The diagonal matrix elements of HQ

in the laboratory

frame are

(yJFMF \HQ\yJFMF )

-2[3MF - F(F + l)]A(y7FH6>(2) lly7F) " [(2F + 3)(2F + 2)(2F + 1)2F(2F- 1)]

1/2

X [(3 cos

2 j8 - 1) - e sin2 /3(cos

2 a - sin2 a)]. (46)

It is simple to show, by directly integrating the angular factor in square brackets in Eq. (46), that the average

value of the diagonal matrix elements of Hq, taken over

all possible orientations of the laboratory frame with

respect to the principal-axis frame, is zero. This also follows directly from the fact that the quantity in square

brackets is a linear combination of spherical harmonics.

It is less obvious that the average, taken over any three mutually perpendicular orientations of the laboratory z quantization axis, is also zero. This result is proven in Appendix A. This provides a method for eliminating the quadrupole shift from the observed transition frequency.

The magnetic field must be oriented in three mutually

perpendicular directions with respect to the trap elec- trodes, which are the source of the external quadrupole

field, but with the same magnitude of the magnetic field. The average of the transition frequencies taken under

these three conditions does not contain the quadrupole

shift. The reduced matrix element in Eq. (46) is, in the //-coupling approximation,

(y(/7)FH6>(2) lly(/7)F)

= (-l)'+y+'(2F+l)

7 2 7

F I F

J 2 J

-7 7 0(7,7), (47)

where / is included in the state notation in order to specify the order of coupling of / and 7. For the partic- ular case of the 199Hg+ 5d96s 2 2D5/2 level, the reduced

matrix elements are

(D 5/2 211 ©

(2, IID 5/2 2) = 2 J^ <9(D, 5/2), (48)

(D 5/2 3II6>

(2) IID 5/2 3) = 2 x/y @(D, 5/2),

Since the Cowan-code calculation shows that there is very little configuration mixing in the 199Hg+ 5d96s 2 2D5/2 level, &(D, 5/2) can be reduced to a matrix element

involving only the 5d orbital:

X

0(D,5/2) = ^(5d2d5/2 , rrij

= 5/2l3z

2 - r2 \5d2d5/2 , rrij = 5/2), (50)

= ^(5d,m, = 2\3z

2 - r2 \5d,m, = 2), (51)

/4tt = e J-j- (5d, m, = 2IF2, (6>, <f>)\5d, m, = 2), (52)

= e^j^j-(5d\r2 \5d)

/•2ir rir Yi2 (d,4>)Y2, (d,<t>)Y2, 2(d,cf>)smeddd<t>, (53)

Jo Jo

= M5d\r2 l5d)(^ll)(lll), (54)

= -^{5d\r2\5d). (55)

The apparent sign reversal in Eq. (50) relative to Eqs.

(27) and (30) is due to the fact that the quadrupole

moment is due to a single hole in the otherwise filled 5d

shell rather than to a single electron . According to the Cowan-code calculation,

<5dlr2l5</> = 2.324 a

2

, = 6.509 X 10~ 17 cm2

, (56)

where a is the Bohr radius.

^ Since the quadrupole shifts are zero in the 5d w

6s 2S m

level, the quadrupole shift of the

199Hg+ optical clock

transition is due entirely to the shift of the ID 5/2 2 0)

state, and is given by

<D 5/2 2 01/ZqID 5/2 2 0)

= |A6>(D, 5/2)[(3 cos

2 - 1)

— e sin

2 )6(cos2 a — sin2 a)],

= - ^Ae(5d\r2 \5d)[(3 cos

2 /3 — 1)

— e sin2 /3(cos2 a — sin 2 a)], « - 3.6 X 10

_3/jA[(3 cos

2 j8 - 1) - e sin2 0(cos2 a - sin2 a)]Hz, (59)

where A is expressed in units of V/cm2

. Thus, for typical values A = 103 V/cm2 and lel^ 1, the quadrupole shift is on the order of 1 Hz.

(57)

(58)

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Volume 105, Number 6, November-December 2000 Journal of Research of the National Institute of Standards and Technology

6. Appendix A. Angular Averaging of the

Quadrupole Shift

For the purpose of describing the quadrupole shift, the orientation of the laboratory (quantization) axis with

respect to the principal-axis frame is defined by the angles /3 and a. In the principal-axis coordinate system,

a unit vector along the laboratory z axis is defined in terms of j8 and a by

Z = (sin /3 cos a, sin /3 sin a, cos /3). (60)

We wish to show that the angular dependence of the quadrupole shift is such that the diagonal matrix ele- ments given by Eq. (46) average to zero, for z along any

three mutually perpendicular directions. An arbitrary set of three mutually perpendicular unit vectors e u e2 , and e3 can be parameterized by the set of

angles 9, cf>, and \p in the following way:

e x = (sin 9 cos <f>, sin 9 sin <f>, cos 9), (61)

e2 = (cos (j> cos 9 cos \\> — sin 4> sin \p, sin <f> cos 9 cos i//

+ cos <fi sin t/f, —sin 9 cos \p), (62)

e3 = (—cos (f) cos sin \\> — sin cf> cos if/, — sin </> cos sin t//

+ cos <f> cos i/*, sin sin ip). (63)

It can be verified by direct computation that c,-C/= 8//. The quadrupole shift can be evaluated for each of

these three unit vectors substituted for z [Eq. (60)] and

the average taken. First consider the average of the quantity (3 cos

2 (3 — 1) that appears in Eq. (46): We use the

fact that cos /3 is the third component of z , so the average is:

(3cos2 j8- l) = cos

2 0+sin2 cos2

i/f + sin

2 sin2

t/> - 1, (64)

= cos (65)

2 + sin2 - 1,

= 0, (66)

for arbitrary 9, (f>, and if/. Similarly, the average of

the other angle-dependent term in Eq. (46),

sin2 /3(cos2 a — sin2 a), is calculated by making use of

the fact that sin /3 cos a is the first component off, and

sin (3 sin a is the second:

(sin2 j8(cos2 a — sin2 a))

- 3[sin2 9 cos

2 4> — sin

2 9 sin2 4>

+ (cos (f> cos 9 cos t// — sin (f> sin ij/)2

— (sin (f> cos 9 cos \p + cos 4> sin i//) 2

+ (cos <f> cos 9 sin i// + sin c/> cos t//)2

— (sin <fi cos sin i/> — cos (/> cos if/) 2

],

= 0,

(67)

(68)

for arbitrary 9, <f>, and tp. Hence, the matrix elements of

Hq given by Eq. (46) average to zero for any three mutually perpendicular orientations of the laboratory

quantization axis.

Acknowledgments

We thank Dr. C. J. Sansonetti for making available the

results of Ref. [9] prior to publication. We acknowledge

financial support from the U.S. Office of Naval Research.

7. References

[1] P. L. Bender, J. L. Hall, R. H. Garstang, F. M. J. Pichanick, W. W. Smith, R. L. Barger, and J. B. West, Bull. Am. Phys. Soc.

21, 599 (1976).

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

Phys. Rev. Lett. 59, 2732 (1987).

[3] J. C. Bergquist, D. J. Wineland, W. M. Itano, H. Hemmati, H.-U.

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

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

[5] R. J. Rafac, B. C. Young, J. A. Beall, W. M. Itano, D. J. Wineland, and J. C. Bergquist, Phys. Rev. Lett. 85, 2462 (2000).

[6] J. Reichert, M. Niering, R. Holzwarth, M. Weitz, Th. Udem, and

T. W. Hansch, Phys. Rev. Lett. 84, 3232 (2000).

[7] S. A. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R. T. Udem, and T. W. Hansch, Phys. Rev. Lett. 84, 5102 (2000).

[8] R. D. Cowan, The Theory of Atomic Structure and Spectra, Univ. California Press, Berkeley, CA (1981).

[9] C. J. Sansonetti and J. Reader, Phys. Scripta 63, 219 (2001).

[10] M. Wilson, Atomic Spectra and Oscillator Strengths for Astro- physics and Fusion Research, J. E. Hansen, ed., North-Holland, Amsterdam (1990).

[11] D. J. Berkeland, J. D. Miller, J. C. Bergquist, W. M. Itano, and

D. J. Wineland, Phys. Rev. Lett. 80, 2089 (1998).

[12] J. C. Bergquist, unpublished experimental results. [13] O. Loebich and A. Steudel, Z. Phys. 166, 56 (1961).

[14] W. M. Itano, J. C. Bergquist, and D. J. Wineland, J. Opt. Soc. Am. B 2, 1392 (1985).

[15] Th. A. M. Van Kleef and M. Fred, Physica 29, 389 (1963).

[16] B. Cagnac, Ann. Phys. (Paris) 6, 467 (1961).

[17] A. Abragam and J. H. Van Vleck, Phys. Rev. 92, 1448 (1953).

[18] W. J. Childs and L. S. Goodman, Phys. Rev. 141, 176 (1966).

[19] J. R. P. Angel and P. G. H. Sandars, Proc. Roy. Soc. A 305, 125

(1968).

836

TN-40
Volume 105, Number 6, November-December 2000 Journal of Research of the National Institute of Standards and Technology

[20] W. M. Itano, L. L. Lewis, and D. J. Wineland, Phys. Rev. A 25, 1233 (1982).

[21] T. Brage, C. Proffitt, and D. S. Leckrone, Ap. J. 513, 524 (1999). (Updated tables of oscillator strengths were taken from the website http://aniara.gsfc.nasa.gov/sam/sam.html.)

[22] M. Henderson, L. J. Curtis, R. Matulioniene, D. G. Ellis, and

C. E. Theodosiu, Phys. Rev. A 56, 1872 (1997).

[23] D. J. Berkeland, J. D. Miller, J. C. Bergquist, W. M. Itano, and

D. J. Wineland, J. Appl. Phys. 83, 5025 (1998).

[24] J. R. P. Angel, P. G. H. Sandars, and G. K. Woodgate, J. Chem.

Phys. 47, 1552 (1967).

[25] P. G. H. Sandars and A. J. Stewart, Atomic Physics 3, S. J. Smith

and G. K. Walters, eds., Plenum Press, New York (1973) p. 429.

[26] N. F. Ramsey, Molecular Beams, Oxford Univ. Press, London

(1956) Chap. III. [27] L. S. Brown and G. Gabrielse, Phys. Rev. A 25, 2423 (1982).

[28] A. R. Edmonds, Angular Momentum in Quantum Mechanics,

Princeton Univ. Press (1974). (Earlier printings include some

incorrect equations involving the rotation operators. See Ref.

[29].) [29] A. A. Wolf, Am. J. Phys. 37, 531 (1969).

About the author: Wayne M. Itano is a physicist in the Time and Frequency Division of the NIST Physics Laboratory. The National Institute of Standards and Technology is an agency of the Technology Administration,

U.S. Department of Commerce.

837

TN-41
102 OPTICS LETTERS / Vol. 26, No. 2 / January 15, 2001

Direct comparison of two cold-atom-based optical frequency

standards by using a femtosecond-laser comb

K. R. Vogel, S. A. Diddams, C. W. Oates, E. A. Curtis, R. J. Rafac, W. M. Itano, J. C. Bergquist, R. W. Fox,

W. D. Lee, J. S. Wells, and L. Hollberg

Time and Frequency Division, National Institute of Standards and Technology, Boulder, Colorado 80305-3328

Received August 15, 2000 With a fiber-broadened, femtosecond-laser frequency comb, the 76-THz interval between two laser-cooled optical frequency standards was measured with a statistical uncertainty of 2 x 10" 13 in 5 s, to our knowledge

the best short-term instability thus far reported for an optical frequency measurement. One standard is based on the calcium intercombination line at 657 nm, and the other, on the mercury ion electric-quadrupole

transition at 282 nm. By linking this measurement to the known Ca frequency, we report a new frequency

value for the Hg+ clock transition with an improvement in accuracy of — 105 compared with its best previous measurement. OCIS codes: 120.3930, 320.7090, 190.4370.

State-of-the-art optical frequency standards based

on cold atoms and ions exhibit excellent frequency

stability and have the potential for achieving high

reproducibility and accuracy. Such frequency ref- erences should find application in precise tests of

fundamental physics and in next-generation atomic

clocks. Incorporating these optical-based standards

into a clockwork, however, has proved troublesome

because their large frequencies (>300 THz) could not be conveniently converted into countable microwave

signals. Several optical-frequency measurements

have been made with harmonic chains used to multiply the frequency of the 9.2-GHz cesium microwave

standard, 1-6 but these chains are complex and their

operation requires significant resources. A para- digm-changing simplification was pioneered by Udem

et al.

7,8 when they used the wide frequency-domain

comb output of a femtosecond (fs) mode-locked laser

to measure the absolute frequency of the Cs D\ line and the hydrogen ls-2s transition,9 the most accurate measurement to date of an optical frequency. A

further refinement, by Diddams et al. , 10 expanded the

available comb spectrum to an optical octave by broadening the fs-laser output in a microstructure fiber,

leading to a direct connection between microwave

and optical frequencies. A recent comparison of two

independent, fiber-broadened fs laser combs that

measured the same frequency interval verified that a

precision and reproducibility of <5.1 X 10 -16 can be

attained with this measurement method.11 In this Letter we report a high-precision comparison of two promising cold-atom optical frequency

standards by use of a fs-laser frequency comb. One

standard is based on a 2-mK collection of ~107 neutral

40 Ca atoms, and the other probes a single 199Hg+ ion

that is laser cooled to near the Doppler limit. At

657 nm, a cw frequency-stabilized diode laser is locked

to the central Ramsey -Borde fringe obtained by

four-pulse excitation of the Ca 1 Sq-z P\ intercombination transition (vCa = 456 THz, Lv = 400 Hz). 12 This

system has demonstrated a frequency instability

of 4 X 10" 15 r" 1/2 (t is the averaging time) when

it is probing subkilohertz fringe linewidths. 13 For

the present measurements the Ca spectrometer was

operated with 2.9-kHz linewidths, which gave an

estimated short-term instability of <2 X 10~ 14 t

_1//2

. The oscillator in the Hg+ standard is a frequency-narrowed cw dye laser at 563 nm that has a linewidth

of —0.16 Hz for a 20-s integration time.14 This

light is frequency doubled to 282 nm to interro- gate the 2 Si/2-2 Z?5/2 electric-quadrupole transition

(^Hg+ = 1065 THz, A^ = 1.7 Hz) of a Hg+ ion that

is confined in a linear, cryogenic, rf ion trap. Rabi

linewidths as narrow as 6.7 Hz at 282 nm have been

observed with this system. 15 For these measurements

the laser was stabilized to the Hg+ ion with a linewidth

of 40 Hz at 282 nm, and the instability under these

conditions is estimated to be <3 X 10-15 r

-1/2

. The fs-comb frequency measurement system shown

in Fig. 1 measures the 76-THz interval between the

657- and the 563-nm light which is transported from

each stabilized laser by a 10- and a 130-m optical

fiber, respectively. No attempt has been made to actively cancel the fiber-added noise, 1416 which we

measured to average as <4.4 x 10 -14 r

_1/' 2 for these

optical frequencies. The frequency comb is produced

by a Kerr-lens mode-locked Ti Sapphire laser that has

a bandwidth of 42 nm (FWHM) centered at 810 nm

and operates with a repetition rate of =98 MHz. To

control the frequency spacing between comb modes we

detect the ninth harmonic of the repetition rate with

a signal-to-noise ratio (S/N) of >70 dB in a 100-kHz

Counter DBM BPF PD

Grating

IoKHAIhD

(H-Maser) -»*0 882 MHz

Mode spacing

control (98 MHz Mode-locked Laser

Microstructure

^£) Optical Fiber

Ca @ 657 nm -Q

H»+/2 @ 563 nm O

-/

-/

7

Fig. 1. Block diagram of frequency measurement: DBM,

doubly balanced mixer; PD, photodiode; BPF, bandpass fil- ter; TO's, tracking oscillators.

TN-42
January 15, 2001 / Vol. 26, No. 2 / OPTICS LETTERS 103

bandwidth (BW) and then mix it with an 882-MHz

reference signal from a frequency synthesizer. We

phase lock the repetition rate of the mode-locked laser

to this reference frequency by using the phase-dependent mixer output to control the horizontal tilt of

the high-reflector mirror, which is situated after a

dispersion-compensating prism pair.

7-9 The internal

clock of the synthesizer is phase locked to a hydrogen maser [H-Maser; ay

{r) = 2 x 10~ 13 r~ 1/2

], whose

frequency is known with an uncertainty of ±4 X 10~ 15 by comparison with a Cs primary standard.

To extend the comb bandwidth we launch the fs-laser pulses into a 5 cm-long microstructure optical fiber

that has a core diameter of =1.7 /xm and zero groupvelocity dispersion near 770 nm.

17 Self-phase modulation and other nonlinearities in the fiber produce an

output spectrum from approximately 500 to 1100 nm. The cw light from both stabilized lasers is also cou- pled into the microstructure fiber to ensure good spatial mode overlap with the fs-laser light. The fiber

output is dispersed by an optical grating and imaged

onto slits such that only a few modes in the vicinity

of the cw light frequencies are selected. A photodiode after each slit detects the rf heterodyne beat note

(5i, 82) between the cw light and a comb mode with a S/N of -25 dB in a 100-kHz BW, with the background

limited by the shot noise of the cw light power. For

accurate frequency counting, a tracking oscillator is phase locked to each beat note to provide regenerated

signals with >50-dB S/N in a 100-kHz BW. These

tracking oscillators consist of a low-phase-noise, volt- age-controlled oscillator that is phase locked with an — 100-kHz BW to track the incoming signal in such a way that the broadband, background noise pedestal be- low the beat note is not reproduced.

In these measurements the comb-mode spacing

pTep is locked tightly to the H-maser frequency, but

the offset frequency of the fs-laser comb remains

uncontrolled. So, although the comb modes are spaced equidistantly, knowledge of the absolute fre- quency of an individual comb mode is limited by the

frequency jitter of the fs laser (—10 MHz). As we

are concerned only with a frequency difference, we

can remove this noise that is common to both beat

notes by mixing together the correlated signals from

the two tracking oscillators. 10 The mixer output is either a stable sum or difference signal 8 = S\ ± 82,

which is counted to yield the frequency interval

Au = V2 ^Hg+ _

^Ca = NvTep ± 8. We determined

the integer number N and the sign choice for 8

unambiguously by comparing Av with our previous

± 10-MHz measurement of this frequency difference, 18 and we verified our choices by making measurements

for different repetition rates. The inset in Fig. 2 shows a typical time record

of the frequency fluctuations of 8, counted with a

5-s gate time. Points that exhibit obvious cycle slip errors, which are due predominantly to a tracking

oscillator's losing lock, are eliminated in the data

sorting. We calculate the Allan deviation, shown in

Fig. 2, for various averaging times by juxtaposing the

5 s gate-time data; the deviation indicates that the

measurement precision averages as (34 Hz)r-1/ 2 for

the duration of the measurement. Three frequency

sources (z'ca, vUg* , ^rep) contribute to the short-term

instability of this measurement, and from these data

alone we cannot attribute the noise unambiguously

to a specific source. Nonetheless, by assuming that

all the noise comes solely from a given source, we

can place an upper limit on its short-term Allan deviation. From this datum we infer an upper limit of

^7 x 10" 14 t

-1/2 for the fractional frequency instability

of the two optical standards, although each probably

has a significantly better stability.

13-15 Combining

the best estimates for the (normalized) instabilities of the microwave and two optical references, as well

as for the optical fiber delivery, we arrive at a calcu- lated instability of 3.5 X 10" 13 t

-1/2 for the 76-THz

interval, in good agreement with the measured value

of 4.5 X 10 _13 t" 1/2

. There are likely additional

degradations of the stability as a result of noise in the

microwave detection of the repetition rate and (or) in

the synthesizer electronics that multiply the frequency

of the H-maser reference. The results of running the fs-comb measure- ment system on four separate days over a 6-week

period are plotted in Fig. 3 as the frequency off- set from the weighted mean, which is Av =

76 374 564 455 429 (40) Hz. Each of these points

represents the weighted mean of the data runs on an

individual day, corrected for the second-order Zeeman

shifts for both Ca and Hg+

, which are determined to an

uncertainty of <10-14for each day. The Ca system

contributes to the majority of the 40-Hz uncertainty;

the Hg+ reference supplies a <10-Hz contribution (at

563 nm), limited by the present measurement of the

electric-quadrupole shift. 15 The largest systematic

error (—30 Hz) stems from uncertainty in our knowledge of the angular overlap of the counterpropagating

probe beams in the Ca spectrometer, which leads to a residual first-order Doppler shift when the cold Ca

ensemble has a transverse drift velocity.4 It is note- worthy that the uncertainties for the data of May 5

and May 25 are dominated by systematic effects. The run-to-run measurements on those days are

consistent with =10 Hz, as illustrated for the May 25

data in the Fig. 3 inset, which show only the statistical uncertainty for each run. Given the relatively small

100-=

X

G

o

*3 •2 10^

3 1-=]

40

-40^ |^lHfM^v

| 150x5s

1 10 100 Averaging Time x (s)

Fig. 2. Allan deviation of a typical measurement

record. Curve, (34 Hz)t~ 1/2

; inset, the corresponding

time record. From these data we place upper limits of <t,(t) < 45, 7.4, 6.4 X 10~ 14 t" 1/2 for the short-term

instability of the microwave reference, vCa , and ^Hg

+

, respectively.

TN-43
104 OPTICS LETTERS / Vol. 26, No. 2 / January 15, 2001

*

*

CO

3

1 1 Apr 00 21 Apr 00 5 May 00 25 May 00

I

Fig. 3. Frequency deviation (Hz) of A^ = lh vng

+ — J'ca- Each data point is the weighted mean of an individual day's

data runs. Inset, data runs for May 25 collected over a 50-min period, with each run averaging 230 s. Error bars

in the inset reflect only statistical uncertainties.

statistical contribution (~2 Hz on a given day) to the total uncertainty, we believe that fs-laser-based

frequency metrology gives us, for the first time, a

practical tool for evaluating systematic shifts at an

inaccuracy approaching 10" 15 for these high-accuracy

optical standards.

The uncertainties given for the two optical refer- ences were estimated for conditions during these ex- periments only, and no serious attempt was made to minimize systematic effects. Nevertheless, it is en- couraging that our results for the frequency differ- ence have a standard deviation of the mean of 60 Hz

over a 6-week period. This consistency shows that all three components, the frequency-measurement system

and the two optical standards, are reproducible at this

level. Thus we offer an improved value for the Hg+ clock transition frequency by summing our measured

Hg+ /2-Ca interval with the absolute frequency of the

Ca 657-nm clock transition measured in Ref. 4. We

obtain vng+ = 2 x 532 360 804 949559(124) Hz, where

the uncertainty is dominated by the 120-Hz uncertainty in the Ca measurement. This is an 80,000-fold

improvement over the best previous frequency measurement of the 282-nm clock transition. 18 Work is currently under way to self-reference the

frequency offset of our fs comb, 19 and we anticipate

confirmation of the Hg+ and Ca frequencies with a

direct rf-to-optical measurement. In fact, locking a mode of the self-referenced comb to one of the opti- cal standards can achieve an all-optical connection be- tween Hg+ and Ca that eliminates any dependence

on the H-maser microwave reference. This interval

measurement should then average at the stability of

the optical standards alone, allowing for a more precise determination of systematic shifts. The capability to intercompare three high-performance frequency

standards (Hg+

, Ca, and Cs) has powerful advantages,

such as permitting absolute frequency stabilities to be

determined and the fidelity of fs-comb measurements

to be tested. Indeed, an optical clock is realized by

the repetition-rate output of an ail-optically referenced

comb, and microwave sources with frequency instabilities near 10~ 15 t

-1/ 2 should be obtainable, provided

that the repetition-rate signal can be extracted with

a suitably high S/N.

The authors thank J. L. Hall, J. Ye, D. Jones, S. Cundiff, Th. Udem, F. Walls, R. Holzwarth, and T. Hansch

for useful discussions and are especially grateful to J. Ranka and R. Windeler for supplying the microstructure optical fiber. This research was partially

supported by the U.S. Office of Naval Research. K.

Vogel's e-mail address is kvogel@boulder.nist.gov.

References

1. D. A. Jennings, C. R. Pollock, F. R. Petersen, R. E. Drullinger, K. M. Evenson, J. S. Wells, J. L. Hall, and

H. P. Layer, Opt. Lett. 8, 136 (1983).

2. C. R. Pollock, D. A. Jennings, F. R. Petersen, J. S. Wells,

R. E. Drullinger, E. C. Beaty, and K. M. Evenson, Opt.

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3. H. Schnatz, B. Lipphardt, J. Helmcke, F. Riehle, and

G. Zinner, Phys. Rev. Lett. 76, 18 (1996).

4. F. Riehle, H. Schnatz, B. Lipphardt, G. Zinner, T. Trebst, T. Binnewies, G Wilpers, and J. Helmcke,

in Proceedings of the 1999 Joint Meeting of the EFTF-IEEE IFCS (IEEE, New York, 1999), p. 700.

5. D. Touahri, O. Acef, A. Clairon, J.-J. Zondy, R. Felder,

L. Hilico, B. de Beauvoir, F. Biraben, and F. Nez, Opt.

Commun. 133, 471 (1997).

6. J. E. Bernard, A. A. Madej, L. Marmet, B. G. Whitford,

K. J. Siemsen, and S. Cundy, Phys. Rev. Lett. 82, 3228

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7. Th. Udem, J. Riechert, R. Holzwarth, and T. W.

Hansch, Opt. Lett. 24, 881 (1999).

8. Th. Udem, J. Reichert, R. Holzwarth, and T. W.

Hansch, Phys. Rev. Lett. 82, 3568 (1999).

9. M. Niering, R. Holzwarth, J. Reichert, P. Pokasov,

Th. Udem, M. Weitz, T. W. Hansch, P. Lemonde, G

Santarelli, M. Abgrall, P. Laurent, C. Salomon, and A.

Clairon, Phys. Rev. Lett. 84, 5496 (2000).

10. S. A. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K Ranka, R. S. Windeler, R. Holzwarth, Th.

Udem, and T. W. Hansch, Phys. Rev. Lett. 84, 5102

(2000).

11. R. Holzwarth, Th. Udem, T. W. Hansch, J. C. Knight,

W. J. Wadsworth, and P. St. J. Russell, Phys. Rev.

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13. C. W. Oates, E. A. Curtis, and L. Hollberg, Opt. Lett. 25, 1603 (2000).

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Itano, D. J. Wineland, and J. C. Bergquist, in Proceedings of XIV International Conference on Laser Spectroscopy (World Scientific, Singapore, 1999), pp. 61-70.

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TN-44
Sz&Drniinc) WSTvs' ''O^mhhn'lii J Time Measurement at the

Millennium

In 1714, stimulated by

another naval disaster

attributed to inaccurate navigation, the British Parliament passed the Longitude

Act. This act created a prize of £20 000—an enormous sum

at that time—to be given to the inventor of an accurate method for determining longi- tude. It had long been realized that longitude could be cal- culated from the positions of the Sun or stars if the time were known accurately. What was needed was an accurate, seaworthy clock. John Harrison met the challenge by

developing a rugged mechanical clock, accurate within a few seconds over several months. After years of bureaucratic foot-dragging, he eventually received the prize. 1 Navigation has continued to be one of the principal

applications of accurate clocks. Indeed, we already take

for granted the GPS (Global Positioning System) whose

phenomenal navigational accuracy is based on clocks that

keep time within 3 nanoseconds. However, navigation has

not always provided the stimulus for better clocks. In some cases, such as the development of the atomic clock,

navigation was the beneficiary of advances elsewhere in physics.

For modern-day physicists, the pursuit of better

clocks provides a natural means for studying various

aspects of nature, including the fundamental constants and the interaction of radiation and matter. Those of us who measure time are mindful of the practical applications, but we are also strongly driven by scientific considerations and the desire to apply clocks to other interesting

measurements. Perhaps we will one day find that clocks,

whose frequencies depend differently on the basic forces,

diverge in time, signaling a fundamental change in how

we perceive nature.2 Although a unit of time can be constructed from other

physical constants, time is usually viewed as an arbitrary

parameter to describe dynamics. The frequency of any

periodic event, such as the mechanical oscillation of a pen- dulum or the quantum oscillation of an atomic dipole, can

be adopted to define the unit of time, the second.

For centuries, the mean solar day served as our unit

of time, but Earth's period of rotation is irregular and

slowly increasing. In 1956, the International Astronomical Union and the International Committee on Weights

and Measures recommended adopting Ephemeris Time—

based on Earth's orbital motion around the Sun—as a more accurate and stable basis for the definition of time.

James Bergquist, Steven Jefferts, and David Weneland all work in the time andfrequency division ofthe National Institute of

Standards and Technology in Boulder, Colorado.

The latest clocks use a single ion to

measure time with an anticipated

precision of one part in 10 18.

James C. Bergquist, Steven R. Jefferts

and David J. Wineland

Four years later, the two

organizations' recommendation was formally ratified by

the General Conference on

Weights and Measures.

Until the definition of the

second in terms of atomic

time in 1967, much of the

work of the National Bureau

of Standards (NBS, the fore- runner of NIST—the National Institute of Standards and

Technology) and other standards laboratories was devoted

to developing reliable secondary standards, such as lumped-element circuits and quartz crystals, whose reso- nant frequencies could be calibrated relative to Ephemeris Time (1 s was defined as 1/31 556 925.9747 of the year 1900). l Frequencies derived from resonant transitions in

atoms or molecules offer important advantages over macroscopic oscillators. Any unperturbed atomic transition is identical from atom to atom, so two clocks based on

such a transition should generate the same time. Also,

unlike macroscopic devices, atoms do not wear out. And,

at least as far as we know, they do not change their properties over time. These features were appreciated by Lord

Kelvin, who suggested using transitions in hydrogen as a

time-keeping oscillator. However, it wasn't until the mid20th century that technology made these ideas possible.

The first atomic clock was developed in 1949 by

Harold Lyons of NBS, and was based on the inversion

transition in ammonia, which occurs at a frequency of

about 24 GHz. In the mid-1950s, Louis Essen and John

Parry of Britain's National Physical Laboratory made a

significantly more stable and accurate atomic clock based

on the ground-state hyperfine transition in cesium. As

NBS and other national laboratories developed cesium

standards, pressure mounted for an atom-based definition

of time.

This change occurred in 1967 when, by international

agreement, the second was defined as the duration of 9

192 631 770 periods of the radiation corresponding to the

transition between the two hyperfine levels of the ground

state of the 133 Cs atom. 1 This definition made atomic time

agree with the second based on Ephemeris Time, to the

extent that measurement allowed. Atomic clock recipe

The basic idea of most atomic clocks is straightforward.

First, identify a transition between two nondegenerate

eigenstates of an atom. Then, create an ensemble of these

atoms—in an atomic beam or storage device, for example.

Next, illuminate the atoms with radiation from a tunable

source that operates near the transition frequency f . Sense and control the frequency where the atoms absorb

maximally. When maximal absorption is achieved, count

© 2001 American Institute of Physics, S-O031-9228-0103-O20-4 TN-45 March 2001 Physics Today 37
Quantum Limits to Spectroscopy

A

resonance curve for N atoms is generated by sweeping

the probing radiation's frequency / and recording the outcomes of a meter operator O. In a frequency standard, the

relevant signal is the change of this measurement outcome

with change of frequency—that is, d(0)/df. The uncertainty

of a frequency determination is therefore given by

Afm = AO/\{dO/dfi\, where (AO)2 is the variance of repeated measurements of O.

In Ramsey spectroscopy4 with separable (unentangled) atoms, the state ^ after the first tt/2 pulse is as shown below

in the upper panel. After the second tt/2 pulse, we measure

the number of atoms N+ in the excited state | + ) (that is, O = N+

). In this case, Afm/f = l/(2Trf TN1/1), the standard quan- tum limit due to projection noise.3 Now suppose we replace the first Ramsey pulse with an

operation that creates the (entangled) state "$ shown in the lower panel. In this case, after the second (normal) tt/2 pulse, we want to measure parity (that is, O = P +

; P+ = +1 if the number of | + } atoms is even, and — 1 if the number of | +

)

atoms is odd). For this case, A_/^//q = 1/(2tt/ TN), the Heisenberg limit. This gain in precision is due to the fact that we

have created a state where the energy between the two states of the superposition is Ntimes larger than the separable-state

case. In practice, this means the time to reach a certain meas- urement precision would be reduced by the number N of

entangled atoms, an important improvement because clock

outputs are typically averaged for weeks or months to reduce measurement uncertainty. Experimenters are now beginning

to create such states for spectroscopy.

¥_(|_> + C-AV| + ».'(|_> +e-'<V| +» . . . (|_> +e-^| +»/2N/2

-

N<Or

Ml— > + ell

-x

.

.-;--

-—•—

+ »/V2

Entangled state "superatom"

OeeP+

the cycles of the oscillator: A certain number of elapsed

cycles generates a standard interval of time.

It is not quite as straightforward to put the recipe into

practice. To sense the atomic transition, a difference in populations between the states of the selected transition must

already exist or be created. Optimal performance is obtained

when all the atoms are prepared in one of the states. Changes to the difference in state populations caused

by the tunable "clock" radiation can be probed by a number of schemes. For example, changes in state populations

could be detected by absorption of the clock radiation (as was done in the first ammonia clock) or by looking at state-sensitive light scattering. If a number of absorption

measurements are performed for a range of frequencies of the clock oscillator, an absorption feature, or resonance

curve, can be recorded. Because the degree of absorption is not very sensitive

to small changes in frequency of the oscillator near the

center of the resonance curve, the atoms are usually irra- diated at two frequencies near the maximum slopes of the curve on opposite sides of its center. If the resonance curve

is symmetric and the two frequencies are adjusted so that

the absorptions on opposite sides of the resonance are the same, then the arithmetic mean of the two frequencies

corresponds to the condition for maximum absorption. In

practice, the determination of the frequency correspon- ding to maximum absorption is limited by noise. With

care, we can reduce electronic and other sources of noise

so that the measurement precision is limited only by what

is known as quantum projection noise. 3 Quantum projection noise can be viewed as arising

from the statistical nature of projecting a superposition of two states into one state when a measurement is made.

Whenever an atom is prepared in a coherent superposition of two states, any single attempt to measure the state composition will reveal only one of the states, not a mixture. The average of repeated measurements (or over many atoms in an atomic ensemble) will produce the

desired point on the resonance curve with a precision that

increases as the square root of the number of measure- ments (or as the square root of the number of atoms in the

ensemble; see the adjacent box). This quantum measure- ment noise limits our ability to steer the average frequency of the oscillator to the center of the atomic resonance. The fractional frequency uncertainty hfjf of our measurement can be expressed as

Afm = C

/o 2<nf TjNM

C

2TTf y[NT~T

where C is a constant of order unity; T is the interrogation

time (that is, the time during which the clock radiation is applied before each measurement); N is the number of atoms; and Mis the total number of measurements. In the

last expression, t is the total measurement time (over

many measurements), and we have assumed that the time

between interrogations is small compared to T. The stabil- ity of the clock is maximized when kfm/f is minimized.

Clearly, it is important to use a high transition frequency, a large number of atoms, and a long interrogation time.

With this prescription in mind, the makers of atomic

clocks seek an atomic transition for which &fjf can be

made small enough that the desired measurement precision is reached in a practical length of time. Equally

important is the degree to which the measured frequency

matches the unperturbed resonance frequency f of the atom. Accuracy is maximized when the uncertainty A/"a of deviations from the ideal are minimized, or the fractional frequency inaccuracy Afjf is minimized. For that reason,

stable and accurate atomic clocks use an atom for which

the environmental perturbations to the measured transi- tion frequency are small or easily accounted for. Environmental perturbations can include frequency shifts to the

clock transition caused by magnetic fields or collisions between the atoms. Frequency shifts due to the atoms' motion can also Doppler-broaden and shift the observed absorption fea- ture. Trapping the atoms eliminates the first-order Doppler shift (proportional to velocity) because the mean

velocity of a trapped atom is zero. However, care must be

taken that the trapping fields do not uncontrollably per- turb the atom's frequency.

Cooling the atoms suppresses both Doppler broadening

38 March 2001 Physics Today TN-46 http://www.physicstoday.org
and the second-order

Doppler or time-dilation

shift. With the simplest form

of laser cooling, Doppler cool- ing,4 temperatures around

0.001 K can be reached,

which are sufficient to suppress the time-dilation shift to about 1 part in 1018 (see the article by Wineland and

Wayne Itano, PHYSICS TODAY,

June 1987, page 34). However, deeper cooling is needed to prevent an untrapped sample

of atoms from spreading dur- ing the application of the

radiation (see the article by

Claude Cohen-Tannoudji and

Bill Phillips, Physics Today,

October 1990, page 33). Cesium clocks

Even without cooling or trapping, the ground-state hyperfine transition in cesium

exhibits many of the desired

attributes for making an

atomic clock. The resonance

frequency is high, the perturbations to the hyperfine

energy can be made relatively small, and first-order Doppler shifts can be largely

ehminated.4 NIST (and earli- er NBS) and the national

laboratories of many countries worldwide have con- structed a series of cesium

clocks, called primary fre- quency standards. In most

instances, each successive

standard has provided a more accurate realization of the definition of the second.

Unfortunately, the hyperfine structure of cesium is not a simple two-level sys- tem. Each hyperfine level is composed of several magnetic

substates whose degeneracies

are removed in the presence

of an external magnetic field. Most cesium clocks, including the first six primary standards

at NBS, are atomic beam devices that use magnets to reject

all magnetic substates except for the F = 4, mF = and F = 3, mF = hyperfine states. The transition frequency between

these two states (corrected to zero field) is used as the best approximation of the second because it is the least sensitive

to the applied magnetic field. Although magnetically selected atomic-beam clocks are simple and rugged, their accuracy and stability suffer, in part, from inefficient state selection and short interro- gation times (limited by the time of flight of the thermal

atoms through the apparatus).

Around 1950, Alfred Kastler developed the concept of optical pumping and proposed using it to perform state preparation and detection. Optical pumping has been

used in rubidium atomic standards for many years,4 but

became practical for cesium beam standards only with the

FIGURE 1. A SCHEMATIC VIEW of a cesium fountain frequency

standard. A sample of 107 cesium atoms is first cooled to

1 /jlK at the intersection of the laser beams (1) and then launched upward at low velocity through the state preparation (2) and

clock (4) cavities. The atoms reach apogee (3) and, about 1 s later, fall back through the clock cavity (4). Light scattered by

the probe laser is used to measure the number of atoms that changed their hyperfine state as a result of the interaction with

the microwaves.

development of solid-state lasers in the late 1970s.

Optical state preparation

and detection of the

ground-state hyperfine lev- els can produce thermal

beams with larger useable

beam flux and correspondingly smaller kfjf than

the magnetically selected beam clocks. It also avoids

the S-field inhomogeneity

associated with magnetic

state selection. The first optically

pumped cesium standard

at NIST, NIST-7, became

the US time and frequency

standard in 1993. In NIST-7,

atoms from a thermal

source first pass through a

laser beam that optically

pumps all atoms into the

F = 3 hyperfine level. The

atoms next transit a

1 -meter-long region in

which the clock radiation is applied. Finally, any atoms

making the clock transition

into the F = 4 level are

detected by state-sensitive

fluorescence in a second

region crossed by another

laser beam. The mean

interrogation time of the

atoms in NIST-7 is about

7 ms, which produces a res- onance linewidth of about

70 Hz. This standard,

which remains in operation

today, attains values of

Afjf of about 5 x 10"15 for t « 1 day.

Had atomic cesium

clocks reached their stabili- ty and accuracy limit by the

mid-1990s? Not quite!

Already in the mid-1950s,

Jerrold Zacharias had sug- gested a vertical geometry,

or atom fountain, as a way

to reach higher resolution

(see the letter by Robert Naumann and Henry Stroke,

PHYSICS TODAY, May 1996, page 89). He reasoned that if the atoms were launched vertically, then the slower atoms

emerging from the oven would be decelerated, stopped by

gravity, and fall back toward the source. The speed of most

of the atoms in the thermal distribution emitted by the

source would carry them to the top of the apparatus, where

they would simply stick. If the beam tube is a few meters

long, the mean transit time for the returning atoms is about 1 s, and the measured width of the hyperfine reso- nance about 1 Hz. Zacharias built a fountain apparatus,

but saw no signal and abandoned the experiment. It was

later determined that the faster atoms colhde with the

slower atoms and scatter them out of the beam.

Laser cooling brought fife back to the fountain concept because all the atoms in the beam could be slowed. In

the late 1980s, Steven Chu's group at Stanford University

http://www.physicstoday.org TN-47 March 2001 Physics Today 39
SowH

fo =2fo-fo= nfr

nf< +lfrt

/o FREQUENCY

2/o

FIGURE 2. THE BASIC CONCEPT of the optical frequency divider. If an integer

number n of phase-coherent frequencies, equidistantly spaced by/, can be made

to precisely span the frequency interval/ — 2/, then the frequency/ can be

directly measured by measuring/ because/ = 2f —f = nfT. Pulsed lasers can now produce a comb of equally spaced spectral components, or teeth, that strad- dle an octave in the visible region of the spectrum, making possible the conversion of any optical frequency to a countable radio or microwave frequency.

succeeded in using laser-cooled sodium atoms to demonstrate the first fountain clock.5 In 1991, a group led by

Andre Clairon and Christophe Salomon at the Laboratoire Primaire du Temps et des Frequences in Paris built the first cesium frequency standard configured as a fountain. 6 Now, many laboratories have built, or are building,

cesium fountain clocks—including NIST, which put one

into operation in 1998.7 The NIST cesium fountain, NIST-F1 (see figure 1), uses six laser beams tuned near the D2

, F = 4 — F'= 5

cycling transition (A = 852 nm) to laser-cool an approximately spherical sample of about 107 cesium atoms in

optical molasses. The atoms in the F = 3 hyperfine ground

state are not cooled in the molasses and are lost from the

sample. The remaining ball of cold atoms is then launched

vertically at a velocity of about 4 m s

_1 by inducing a fre- quency difference between the vertical beams that causes

the cold atom reference frame to move upward. The atoms

drift upward through a microwave state preparation cavity whose frequency is adjusted to cause essentially all of the atoms in the F = 4, mF = state to undergo a transition

to the F = 3, mF = state. When the atoms exit this cavity

they are irradiated by a pulse of light that is tuned to res- onance with the D2

, F = 4 — F' = 5 transition. Any F = 4

atoms remaining in the sample are pushed aside by the

absorbed photons, leaving only the F = 3, mF = atoms. The ascending ball of atoms next transits a second

microwave cavity, where the atoms are irradiated by the

clock radiation. As the atoms continue upward, they exit the upper microwave cavity and enter a drift region.

Approximately V2 s after launch, their motion is arrested

by gravity, and they start to fall. The descending atoms

reenter the microwave cavity, where they experience a

second dose of the clock radiation. This method of excita- tion—two pulses of radiation separated by a drift region

(or simply by time)—is termed the Ramsey method after

its inventor, Norman Ramsey.4 Finally, the atoms enter

the detection region, where the population of atoms in F = 4

is determined optically (the number of atoms remaining

in F = 3 is also measured for normalization). Then, when

another ball of atoms is cooled and launched, the clock

cycle is repeated. In NIST-Fl, where the noise is dominated by the

short-term frequency fluctuations of the microwave

source and not by quantum projection noise, M~JfQ reaches about 2 X 10-15 for t = 1 day. The quantum noise limit

for a cesium fountain is below this value. By using a more

stable microwave source and a larger number of atoms,

the Paris group demonstrated that it was possible to operate at the quantum projection noise limit reaching about

6 x 10"16 for t = 5.5 hours.3 With values of A/"

a

//" approaching 10"15

, the cesium

fountain standards are currently the

world's most accurate clocks. Their largest source of uncertainty arises from collisions between the cold cesium atoms, which

cause a density- (and temperature-)

dependent shift of the hyperfine frequency.

Unfortunately, then, a compromise must be

made between stability and accuracy. But it is possible to increase both the

stability and accuracy at the same time—by

increasing the interrogation time and low- ering the atomic thermal velocity. These

two avenues are difficult to pursue in the

presence of gravity, so, with NASA's support, NIST, along with the Jet Propulsion

Laboratory and the University of Colorado,

is currently constructing a laser-cooled cesium clock for flight aboard the microgravity environment of the Inter- national Space Station. The project, known as PARCS (Primary Atomic Reference Clock in Space) combines the tra- ditional beam geometry with the cold source of the atomic

fountain. The European Space Agency has a similar effort, ACES (Atomic Clock Ensemble in Space). In the absence of

gravity, it will be possible to make short beam tubes and

still achieve long interrogation times by launching slowly

moving samples of cold atoms. PARCS is projected to attain hfjf of about 5 x 10~17

, due in part to a lower cofli- sional shift and extraordinarily long (by terrestrial stan- dards for neutral beams) interrogation times of up to 10 s. Long interrogation times can also be obtained by

atom trapping.4 Neutral atoms can be trapped by a number of means, including magnetic or optical-dipole forces. These traps rely on trading an atom's kinetic energy for internal energy. Unfortunately, this exchange of energy

typically perturbs the clock transition so that high accu- racy is precluded. However, traps for atomic ions act

directly on the charge and, thus, cause minimal perturbations to the internal structure. 8 Because trapping times can be extremely long, very high resolutions can be

obtained and linewidths less than 0.001 Hz have been

observed. (An atomic fountain would have to be a few

100 km tall to achieve the same resolution.) For high

accuracies, relatively small numbers of trapped ions are

desirable, which partially offsets the advantages of long

interrogation times. Nevertheless, trapped-ion clocks based on microwave hyperfine transitions have achieved

stabilities and accuracies comparable to those of cesium

fountain clocks. 9

Optical frequency standards

Perhaps the most promising route to better clocks is to use optical transitions, simply because clock stability is proportional to frequency f . As early as the 1960s, shortly after the demonstration of the first working laser, researchers began to investigate optical atomic and

molecular transitions that might be suitable references

for clocks. 10 However, an enormous barrier loomed: No

practical device was fast enough to count the optical

cycles to generate a unit of time.

Recently that stumbling block was all but eliminated

with the demonstration—by Theodor Hansen and his group in Garching, Germany, and by John Hall, Steven

Cundiff, and their group in Boulder—that pulsed lasers can span an octave from the infrared to the ultraviolet with a grid of equidistant marker frequencies that are all phase coherent (ref. 11 and PHYSICS TODAY, June 1999,

page 19). As depicted in figure 2, the spectrum of the pulse

train from a mode-locked femtosecond laser corresponds to

40 March 2001 Physics Today TN-48 http://www.physicstoday.org
Figure 3. Single 199 Hg+ ion optical clock. Partial energy-level diagram of 199Hg+

is shown in the upper panel.

Cooling, state preparation, and state detection are done on the strongly allowed 2 S 1/2 —• 2P1/2 transition at 194 nm. Any clock

transition to the 2D5/2 level at 282 nm (the allowed electric- quadrupole transition) is marked by the abrupt cessation of

scattered 194-nm fluorescence photons. Absorption spectra of the 2S V2 {F = 0) — 2D5/2 (F = 2), AwF = clock transition are shown in the middle and bottom panels. Af2g2 is the frequency

of the 282 nm probe laser detuning and P is the probability of finding the atom in the ground state subsequent to the application of the clock radiation. In the middle panel, the

signal is averaged over 292 sweeps; in the bottom, 46 sweeps

are made. The observed linewidths are consistent with the Fourier-transform limit of the interrogation times. The laser oscillator is then steered to coincide with the center of the resonance curve. (Adapted from ref. 16.)

a comb of phase-locked frequency components whose spac- ing is precisely given by the pulse repetition rate fT

. Fre- quency fluctuations of the pulsed laser can be suppressed,

for example, by locking one tooth of the comb to the fre- quency fQ of a stable laser and locking another tooth to the harmonic 2f of the stable laser. 12 In this case, the frequency

position of each tooth in the comb and the spacing between

teeth is fixed with a stability and accuracy given by the sta- bilized laser. If a microwave or radio frequency oscillator is then locked to the pulse repetition rate, its frequency will exhibit the same stability and accuracy as the stabilized ref- erence laser. Moreover, when fT is compared to a cesium

standard, the absolute frequency f

= nfT of the optical stan- dard can be determined (n is an integer that can be deter- mined from a relatively crude measurement of A = c/f ). A powerful technique for approximating a nearly

ideal optical atomic reference system has been the cou- pling of ion-storage devices with laser cooling. 8,10' 13 For

example, it is now possible to suspend a single ion in an

ultrahigh vacuum, reduce its kinetic energy to the zero- point state, and localize its position to submicrometer

dimensions. Such conditions provide a significant decoupling of the internal states of the ion from perturbations

caused by collisions and second-order Doppler shifts. Although a single trapped ion possesses the intrinsic advantages of a nearly isolated quantum system, detect- ing the absorption of single photons at the clock transition

remains a challenge. The quantum projection noise limit can be obtained only if each clock transition is observed.3 - 8' 13 These transitions are observed by a double- resonance technique, which is very similar to the detec- tion scheme used in the atomic fountains. There, the

absorption of a single clock photon is indicated by the presence or absence of scattered light on a strongly

allowed transition (usually the laser-cooling transition)

that shares a common level with the clock transition, as shown in figure 3. Extremely weak, dipole-forbidden transitions, as well as weakly allowed dipole transitions, have

been detected with efficiencies approaching unity. Many

of the single-ion systems presently under investigation

promise instabilities and inaccuracies approaching 10~18

(refs. 8, 13). A final ingredient needed for the optimal realization of an optical frequency and time standard is a laser whose

linewidth is narrower than that of the atomic resonance. This is a stringent requirement. If the width of the atomic resonance is 1 Hz, then the laser linewidth must be less

Observable

fluorescence (A=194nm)

Clock

transition A=282 nm;

0=1.06xl0 15 Hz)

^1/2

1.0

or o.

0.7

0.6

• *ru"* *******&*%

Lm t\r\ tit

1

l\

w\

•

1

^robe= 20ms

-75 75

^ 0.!

0.7

0.6

- 6.5 Hz

rprobe = 120 ms

-7 7 A/282 (Hz)

than 1 Hz for the interrogation time. Typically, the fre- quency of the laser is prestabilized to the resonance of a

single longitudinal mode of a high-finesse, stable refer- ence cavity that is well isolated from external perturbances. 14 Many groups have demonstrated subhertz relative stabilization of their laser source to such a reference

cavity, but the absolute frequency stability of the stabilized laser can be only as good as the absolute frequency

(mechanical) stability of the reference cavity. To ensure

that the frequency of a visible laser does not change by more than 1 Hz, the length of a meter-long cavity must

not change by more than 1 femtometer, the size of an

atomic nucleus!

A number of evacuated, temperature-stabilized and

vibrationally isolated reference cavity laser systems have

been built over the years specifically to probe narrow

http://www.physicstoday.org TN-49 March 2001 Physics Today 41
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atomic resonances. Laser linewidths of a few hundred hertz have been observed by several groups. Although extremely narrow by most standards, these line widths would limit the performance of a clock based on a 1-Hz-wide atomic reso- nance. At NIST in Boulder, we were recently able to achieve

subhertz linewidths, principally by better isolating our cavities. Measurements of the beat frequency between two

independent, cavity-stabilized laser systems revealed their linewidths to be below 0.2 Hz at an oscillation frequency of 530 THz (A = 563 nm) for averaging times up to 20 s.

15 As an example of an optical clock, the harmonic radi- ation at 282 nm generated by one of these lasers was

applied to the electric-quadrupole allowed, S-D clock-transition in 199Hg+

. In work completed last year, we obtained a linewidth of approximately 6.7 Hz, which was the trans- form limit of the interrogation time. 16 At 120 ms, the inter- rogation period was 33% longer than the natural lifetime. We used a stabilized, self-referenced, pulsed laser to measure the frequency of the S-D clock transition in

issHg* with an uncertainty Afjf < 10"14 (ref. 17). A single

tooth of the pulsed laser (fT ~ 1 GHz) was locked to the

Hg^stabilized laser to produce an all-optical time and fre- quency standard based on a single stored 199Hg+ ion. We

expect bfjfo and Afjf to reach at least 10~17 for t > 104 s. Similar strategies based on narrow optical transitions in other atomic and ionic systems are being pursued at many

laboratories worldwide. The clock generation of the future

may have arrived with the millennium.

The NIST ion-clock work is supported by the Office of Naval

Research. References

1. D. Howse, Greenwich Time, Oxford U. Press, Oxford, England (1980). J. Jespersen, J. Fitz-Randolph, From Sundials

to Atomic Clocks, US Government Printing Office, Washington, DC (1999).

2. T. Damour, A. M. Polyakov, Nucl. Phys. B 423, 532 (1994). S. Karshenboim, Can. J. Phys. 78, 639 (2000).

3. W. M. Itano et al., Phys. Rev. A 47, 3554 (1993). G. Santarelli

et al., Phys Rev. Lett. 82, 4619 (1999).

4. W. M. Itano, N. Ramsey, Scientific American 269 (1), 56

(1993).

5. M. A. Kasevich et al., Phys. Rev. Lett. 63, 612 (1989).

6. A. Clairon et al., Europhys. Lett. 16, 165 (1991).

7. S. R. Jefferts et al., Metrologia (in press).

8. H. G. Dehmelt, IEEE Trans. Instrum. Meas. IM-31, 83

(1982). D. J. Wineland, Science 226, 395 (1985).

9. P. T. H. Fisk, Rep. Prog. Phys. 60, 761 (1997). J. D. Prestage

et al. in Frequency Measurement and Control, A. N. Luiten,

ed., Springer-Verlag, New York (2000). D. J. Berkeland et al., Phys. Rev. Lett. 80, 2089 (1998).

10. See, for example, Proceedings of the 3rd Symposium on Fre- quency Standards and Metrology, J. Phys. Colloq. 42 (1981). Proceedings of the 5th Symposium on Frequency Standards

and Metrology, J. C. Bergquist, ed., World Scientific, Singa- pore (1996).

11. J. Reichert et al., Phys. Rev. Lett. 84, 3232 (2000). D. J. Jones

et al., Science 288, 635 (2000). S. A. Diddams et al., Phys.

Rev. Lett. 84, 5102 (2000).

12. J. Ye et al. Opt. Lett. 25, 1675 (2000).

13. A. A. Madej, J. E. Bernard, in Frequency Measurement and

Control, A. N. Luiten, ed., Springer-Verlag, New York (2000).

14. J. L. Hall, M. S. Taubman, Jun Ye, in Handbook of Optics,

2nd edition, vol. 4, Chap. 27, McGraw-Hill, New York (2000), and references therein.

15. B. C. Young et al., Phys. Rev. Lett. 82, 3799 (1999). B. C. Young et al., in Laser Spectroscopy, XTV International Conf., Innsbruck, June 1999, R. Blatt, J. Eschner, D. Leibfried, F. Schmidt-Kaler, eds., World Scientific, Singapore (1999), p. 61.

16. R. J. Rafac et al., Phys. Rev. Lett. 85, 2462 (2000).

17. Th. Udem et al., preprint at http://xxx.lanl.gov/

physics/0101029

42 March 2001 Physics Today

TN-50
Sub-dekahertz spectroscopy of

199Hg+ [t]

J.C. Bergquist*3

, R.J. Rafac3, B.C. Youngb, J.A. Bealf, W.M. Itano

3

, and D.J. Wineland3

'National Institute of Standards and Technology, MS 847, 325 Broadway, Boulder, CO 80305

bJet Propulsion Laboratory, MS 298-100, 4800 Oak Grove Drive, Pasadena, CA 91109

ABSTRACT

Using a laser that is frequency-locked to a Fabry-Perot etalon of high finesse and stability, we probed the 5d

106s2S\n (F = 0, mF = 0) <-> 5tfbs

2 2D5/2 (F -2,mF - 0) electric-quadrupole transition of a single laser-cooled

199Hg+ ion stored in a cryogenic

radio-frequency ion trap. We observed Fourier-transform limited linewidths as narrow as 6.7 Hz at 282 nm (1.06 x 1015 Hz).

The functional form and estimated values of some of the frequency shifts of the S 1/2 «-* D5/2 "clock" transition (including the quadrupole shift), which have been calculated using a combination of measured atomic parameters and ab initio calculations, are given.

Keywords: Lasers, atomic spectroscopy, cooling and trapping, atomic frequency standards, optical clocks.

1. Introduction

For spectroscopy and clocks, fluctuations in frequency measurement are expressed fractionally as ay(r) = Avmeas(r)/v , where x

is the total measurement time. When the measured stability is limited by quantum fluctuations in state detection, ay{x) =

C(27uv )'\N Tprobe t)' m, where C is a constant of order unity, N is the number of atoms, and Tprobe is the interrogation time. For

atomic clocks, the highest accuracies and the greatest stabilities have been achieved by locking a microwave oscillator to a hyperfine transition in the electronic ground state (see, for example, the contribution by A. Clairon in these Proceedings).

Since the fractional frequency instability ay{x) of a frequency and time standard is inversely proportional to its frequency, a promising route to realizing clocks with stability significantly higher than present-day standards is to use optical frequencies.

However, there are significant obstacles that so far have thwarted the development of an optical clock. Principal among these

are the requirements of a spectrally pure and stable laser, a narrow reference transition that can be probed with a high signal- to-noise (S/N) ratio, and a device fast enough to count optical frequencies. If high accuracy is also desired, then either perturbations to the reference transition must be small or the uncertainty in measuring the frequency shifts of the reference

transition in the perturbed system must be small.

Recently, the major technical barriers to the development of an optical clock were eliminated. Two years ago, lasers suitable

for probing sub-hertz atomic linewidths were demonstrated,

1 ' 2 and in the past year, precision, optical-frequency measuring

devices based on mode-locked femtosecond lasers have emerged.3,4' 5 It has long been recognized that a frequency and time

standard based on an optical transition in a single ion that is tightly confined in a benign environment and virtually at rest could be made both highly stable and accurate.

6 In this paper, we summarize the measurement of the 5d]06s2S m (F -0,mF =

0) «-» 5(f6s

2 2D5/2 (F = 2,mF - 0) electric-quadrupole-allowed transition (X ~ 282 nm) in a single, laser-cooled

199Hg+ ion for which a linewidth Av ~ 6.7 Hz is observed.

7 The frequency of a laser can be locked to this transition to provide an extremely

stable and reproducible frequency reference. Elsewhere8 (and in these Proceedings: see S.A. Diddams et at, "A compact

femtosecond-laser-based optical clockwork") we report an absolute measurement of the frequency v of this

2S\n <->• D5/2 "clock" transition, v = 1 064 721 609 899 143(10) Hz, which represents the most precise measurement of an optical

frequency ever made. We also report recent calculations

9 of perturbations to the (F -0,mF = 0) to (F = 2,mF - 0) hyperfine

component of the

2S J/2 «-»

2D5/2 transition that include the quadratic Zeeman shift, the scalar and tensor quadratic Stark shifts, and the interaction between an external static electric field gradient and the atomic quadrupole moment. The quadrupole shift

is likely to be the most difficult to evaluate in such a frequency standard and may have a magnitude as large as 1 Hz for a

single ion in a spherical rf quadrupole trap.

*email: berky@boulder.nist.gov

TN-51
2. Spectroscopy of 199Hg+

2.1 The atomic system, probe laser, and trap

A partial energy-level diagram of

199Hg+

is shown in Fig. 1. The 282 nm radiation used to drive the S-D clock transition is produced in a nonlinear crystal as the second harmonic of a dye laser oscillating at 563 nm. The frequency of the dye-laser's radiation is made to match the resonance of a single longitudinal mode of a high-finesse Fabry-Perot cavity that is temperature-controlled and supported on an isolation platform.

1,2 The frequency-stabilized laser light is sent through a singlemode optical fiber to the table holding the ion trap. Mechanical and acoustical vibrations of the fiber broaden the laser spectrum by about 1 kHz. The phase and frequency noise caused by the journey through the fiber is sensed and removed

using a method2 that is similar to that described in references 10 and 11. Finally, the frequency of the 563 nm light is shifted away from the cavity resonance and on to the ion resonance by passing the 563 nm radiation through an acousto-optic crystal

and then doubling the frequency of the shifted light by harmonic generation in a deuterated ammonium-dihydrogenphosphate crystal.

Two types of cryogenic ion traps have been used in our most recent measurements; a linear quadrupole trap

7 and a spherical

Paul trap. For both systems, a single mercury ion is loaded into the trap by ionizing a mercury atom from a thermal source

with a pulsed electron beam. Under typical operating conditions, the radial secular frequency of the trapped ion varies between 1.2 and 1.5 MHz. Biasing electrodes are mounted outside the trap electrodes to cancel any stray static electric fields that may be present. Since the electrodes in both traps are gold coated and can be heated, oxidation is precluded and any

charging of the electrodes is minimized.

10 9 i n 7 The ion is laser-cooled to near the 1.7 mK Doppler limit by driving the 5d 6s S m (F = 1) «-» 5d bp Pi /2 (F = 0) cycling

transition at 194 nm (Fig. 1). Because of weak off-resonant pumping into the

2Si /2 (F = 0) state, we employ a second 194 nm

source, phase-locked to the first with a 47 GHz offset, that returns the ion to the ground state F = 1 hyperfine level. We

tolerate the complication of a re-pumper, since only isotopes with nonzero nuclear spin can have transitions that are first- order insensitive to magnetic field fluctuations. This provides immunity from fluctuations of the ambient field and

significantly relaxes the requirements for control and shielding of the magnetic environment. Previous experiments using

mercury were performed at room temperature and at a pressure of approximately 10"7 Pa.

10 Under those conditions, the background gas pressure was large enough that when the ion was irradiated with the 194 nm light, it would be lost due to chemical reaction after a few minutes. Partly as a means to help reduce this loss, the ion trap is now housed in a vacuum

system held at liquid-helium temperature. Single ion hold times have now exceeded 100 days, and any loss of an ion has been

caused either by a deliberate or accidental action of the operator. Vibrations of the supported trap structure relative to the

optical table are sensed and removed with an additional stage of Doppler cancellation, where the correction signal is derived

from optical heterodyne detection of a motion-sensing beam reflected from a mirror that is rigidly fixed to the trap support

structure. ' The cancellation is not ideal, because the sensing beam is steered by additional optical elements and its path

deviates slightly from overlap with the probe beam near the trap. Measurements indicate that this optical path difference can

contribute as much as 2 Hz to the spectral width of the 282 nm probe laser in the reference frame of the ion. We monitor the ion and deduce its electronic state using light scattered from the cooling transition. Fluorescence at 194 nm is collected by a five-elemental objective located inside the cryostat. The scattered light is imaged outside the dewar, spatially

filtered, and then relayed with a second lens to either an imaging tube or to a side-on photomultiplier tube. Transitions to the D state are detected using the technique of "electron shelving," which infers the presence of the atom in the metastable level through the absence of scattering from the strong laser-cooling transition.

6' 12' 13 Radiation from the 194 and 282 nm sources is admitted to the trap sequentially to prevent broadening of the quadrupole transition by the cooling radiation. Typical count

rates using the more efficient photomultiplier tube are 12 000 Hz for a single ion when the frequency of the 194 nm source is detuned below resonance with the cooling transition by about Vz the natural linewidth. The background rate is only about 25

Hz when the ion is shelved in the metastable level. Hence, each clock transition to the D level can be detected in only a few

milliseconds with near unit efficiency.

Spectra of the recoilless component of the

2Si /2 (F = 0, mF = 0) «->2 D5/2 (F -2,mF = 0) clock transition were obtained for a range of probe times and laser intensities by laser cooling for a short period, preparing the ion in the F - ground state by

blocking the repumping laser, and then interrogating the quadrupole transition. The spectra are built up from multiple bi- directional scans of the 282 nm probe-laser frequency. Since the frequency of the probe laser can drift, we incorporate a locking step in between pairs of positive- and negative-going frequency sweeps through the quadrupole resonance. During

the locking sequence, we step the frequency of the probe laser alternately to the maximum slope on either side of the S-D

TN-52
T— F=1 199Hg+

2D5/2 F=3

F=2

serve V fluorescence \ \ (X=194nm) \ \

"Clock"

/^ Transition

\ (X=282 nm)

r-V F=l

\.

>V2 _JL

6.5 El

F =

"clock" transition (@ v„^1.06xl015 Hz

"I",**- 120ms PlV^JVq * 1.5xlO" 15 (r=15-60s) -7 7 Av282 (Hz)

Figure 1: The simplified optical level diagram of 199Hg+

is depicted on the right. Quantum-jump absorption spectra of the Si /2 (F = 0, mF = 0) «-»

2 D5/2 (F = 2, /nF = 0) clock transition for two different probe times are shown on the left. Av2 g2 is the frequency detuning of the 282 nm probe laser, and Pg is the probability of finding the atom in the ground state after the probe radiation is applied. In the top spectrum

(averaged over 292 sweeps), the probe-pulse period was 20 ms; in the bottom spectrum (averaged over 46 sweeps), the probe period was

120 ms. The observed linewidths are consistent with the Fourier-transform limit of the probe period at 40(2) Hz and 6(1) Hz respectively.

quadrupole resonance, probe for a fixed time Tservo, and then look for transitions to the D level. Typically, we make 48

measurements on each side of the resonance during each lock cycle before steering the mean frequency of the 282 nm

radiation to line center. Our servo is a simple integrator that works to minimize the asymmetry between the number of

detected transitions on the high- and low-frequency sides of the resonance. In this fashion, variations in the frequency of the 282 nm laser for times exceeding several seconds are reduced.

Two examples of the spectra are plotted in Fig. 1. The probe time for the lower resolution spectrum is 20 ms and the probe

time for the higher resolution signal is 120 ms. In both cases, the linewidths are transform limited by the finite probe time at 40(2) Hz and 6(1) Hz, respectively. The carrier transition amplitude is a function of the vibrational quantum number n. For

the spectra shown in Fig. 1, the trapping parameters gave <n> ~ 35 at the Doppler cooling limit (<n> ~ 20 for the spherical

trap at the Doppler cooling limit). Hence, it is not possible to transfer the electron to the D level with unit probability. The

observed signals are in good agreement with the theoretical expectation, and the signal loss for rprobe = 120 ms is consistent with applying the probe time for a period that exceeds the natural lifetime of the

2D5/2 state by 33 %. In future experiments, we plan to reduce <n> to less than 1 toward improving the signal amplitude and S/N ratio.

3. External field shifts to the quadrupole transition

While the

2Si /2 (F = 0, mF - 0) <-»

2D5/2 (F = 2,mF = 0) hyperfine component has no linear Zeeman shift, it does have a quadratic Zeeman shift. In addition, there is a second-order Stark shift and a shift due to the interaction between a static electric-field gradient and the D-state atomic electric-quadrupole moment. None of these shifts has yet been measured

accurately but their values have recently been calculated by one of us.

9 While these values may be imprecise, it is useful to know the functional form of the perturbation. For example, the quadrupole shift can be eliminated by averaging the S-D

TN-53
transition frequency over three mutually orthogonal magnetic-field orientations, independent of the orientation of the electricfield gradient, and is zero in lowest order for the

2Si /2 (F - 2, mF = 0) *->2D3/2 (F = 0,mF = 0) hyperfine component in

20l Hg+

.

The quadratic Zeeman shift can be calculated if the hyperfine constants and electronic and nuclear g-factors are known.

Similarly, the quadratic Stark shift can be calculated from a knowledge of the electric-dipole oscillator strengths. The

quadrupole shift depends on the atomic wavefunctions. Some of these parameters have been measured, such as the hyperfine

constants and some of the oscillator strengths. In Ref. 9, values for parameters for which there are neither measured values nor published calculations are estimated using the Cowan atomic-structure codes.

15 The Cowan codes are based on the Hartree-Fock approximation with some relativistic corrections. The odd-parity configurations used in these calculations were 5dl0np (n = 6,7,8,9), 5d105f, 5(fds6p, 5<f6slp, 5d96s5f, and 5d%6s

26p. The even-parity configurations were 5dl0ns (n =

6,7,8,9,10), 5dl0nd (n = 6,7,8,9), 5^6^, Scfdsls, Sctdsdd, and 5/6/>2

. Recently, Sansonnetti and Reader have made new

measurements of the spectrum of Hg+ and classified many new lines. They also carried out a least-squares adjustment of the energy parameters in order to match the observed energy levels. Those adjusted parameters were used for the calculations in Ref. 9.

3.1 Quadratic Zeeman shift In order to calculate the energy shifts due to the hyperfine interaction and to an external magnetic field B = B z , we define

effective Hamiltonian operators Hs and HD that operate within the subspaces of hyperfine levels associated with the

electronic levels 5d106s S\a and 5J965

2 2D5/2 respectively:

Hs = hAJ •J+gJ(S)pBJ -B + g'^-B

HD =hADI -J+gJJ^fijJ -B + g'^sI-B,

where As and AD are the dipole hyperfine constants, g/(S) and gj(D) are the electronic g-factors, g ) is the nuclear g-factor, h

is the Planck constant, and p& is the Bohr magneton. All of the parameters entering Hs and HD are known from experiments,

although a more accurate measurement of the excited-state electronic g-factor would improve our estimate of the quadratic

Zeeman shift of the

2D5/2 level. So far, the best determination of g/D) (= 1.1980(7)) is derived from a conventional grating

spectroscopic measurement of the 5di06p

2P3/2 to 5cf6s2 2D5/2 at 398 nm in198Hg+

. 17 The difference in g/S) or gy(D) between

1 8Hg+ and 199Hg+

is estimated to be much less than the experimental uncertainties. The determination of g/D) could be

improved by measuring the optical frequency difference between two components of the 282 nm line and the frequency of a ground-state hyperfine transition at the same magnetic field. Since the uncertainty in the quadratic Zeeman shift is due

mainly to the uncertainty in gy(D), it is useful to improve the experimental value and/or make an accurate theoretical estimate. The Cowan-code calculation gives a theoretical value of gy(D) of 1.19985,

9 which disagrees with the present

experimental value by about 2.6 times the stated measurement uncertainty. From a similar comparison of the calculated value of g/(D) for neutral gold (which is isoelectronic to Hg+

) to its accurately measured experimental value, we might expect that the error in the calculated value of gy(D) of

199Hg+ to be less than 1 x

10"4

. However, uncalculated terms may decrease the

theoretical precision.

For low magnetic fields (< 1 mT), it is sufficient to calculate the energies of the Zeeman sub-levels of the

2Si /2 ground

electronic level and those of the D5/2 electronic level to second order in B. For a magnetic field value of 0. 1 mT, the quadratic

shift of the

2S [/2 (F = 0, mF = 0) *-> 2D5/2 (F = 2,mF = 0) clock transition is -189.25(28) Hz, where the uncertainty is dominated by the uncertainty in the experimental value of gy(D). If instead we use the calculated value of gy(D), the quadratic

shift for B = 0.1mT is -189

3.2 Quadratic Stark shift The theory of the quadratic Stark shift in free atoms has been described thoroughly by Angel and Sandars.

18 The Stark Hamiltonian is HE = -n-E,

where u is the electric-dipole moment operator,

H - -eYfi,

shift for B = 0.1 mT is -189.98 Hz, where the uncertainty may be less than 0.02 Hz but is difficult to estimate.

9

TN-54
and E is the applied external electric field. In the latter expression, r, is the position operator of the ith electron, measured

relative to the nucleus, and the summation is over all electrons. For an atom with zero nuclear spin, and to second order in the

electric field, the Stark shift of the set of sublevels \yJMj> depend on two parameters, a%c!Aai (y, J) and atensor (y. •/)» called the

scalar and tensor polarizabilities. Here, y designates the electronic level, J the electronic angular momentum, and Mj is the eigenvalue of Jz

. In principle, when both magnetic and electric fields are present but not parallel, the energy shifts of the

levels are obtained by simultaneously diagonalizing the hyperfine, Zeeman, and Stark Hamiltonians. In practice, the Zeeman

shifts normally dominate the Stark shifts, so HE does not affect the diagonalization. We also note that the tensor polarizability

is zero for levels with J < 1, such as the ground electronic level in Hg+

. For an atom with nonzero nuclear spin /, we make the approximation that the hyperfine interaction does not modify the electronic part of the atomic wavefunctions . This

approximation is adequate for the evaluation of the Stark shift of the S-D optical clock transition in

199Hg+

.

In Ref. 9, the polarizabilities for the Hg+ 5di06s

2S 1/2 and 5a^6s2 2D5/2 levels were evaluated. The tensor polarizability is much

smaller than the scalar polarizabilities and, in any case, contributes nothing if the external field, such as the blackbody

radiation field, is isotropic. The net shift of the optical clock transition due to the scalar polarizabilities is i/2[asc naI (S,l/2) -

«iensor (D,5/2)]£2

. In frequency units, the shift is -1.14 x

10"3 E

2 Hz, where E is expressed in V/cm. The error in the coefficient

is difficult to estimate, particularly since it is a difference of two quantities of nearly equal magnitude. However, the total

shift of either state is small for typical experimental conditions. If the electric field is time dependent, as for the blackbody

field, the mean square value of the field is taken. At a temperature of 300 K, the shift of the S-D clock transition due to the blackbody field is -0.079 Hz. Since the blackbody electric field is proportional to the fourth power of the temperature, in our

cryogenic environment the Stark shift due to blackbody radiation is negligible. Finally, for a single, laser-cooled ion in a Paul

trap, the mean square trapping fields at the site of the ion can be made small enough that the Stark shifts are not likely to be

observable.

1

3.3 Electric quadrupole shift The atomic quadrupole moment arises from the departure of the electronic charge distribution of an atom from spherical

symmetry. The interaction of the atomic quadrupole moment with external field gradients, such as those that might be

generated by the electrodes of an ion trap, is analogous to the interaction of a nuclear quadrupole moment with the electric

field gradients due to the atomic electrons. The Hamiltonian describing the interaction of external electric-field gradients with

the atomic quadrupole moment is HQ =VEi2) -0(2)

,

where Ws <2) is a tensor describing the gradients of the external electric field at the position of the ion, and 6>(2) is the electric- quadrupole operator for the atom. As long as the energy shifts due to HQ

are small relative to the Zeeman shifts, which is usually the case in practice, HQ can be treated as a perturbation. In that case, it is necessary only to evaluate the matrix

elements of HQ

that are diagonal in the basis states \yJFMF>, where F is the total atomic angular momentum, including the nuclear spin /, and MF is the eigenvalue of Fz with respect to the laboratory frame, where the magnetic field is oriented along

the z axis. The diagonal matrix elements of HQ

in the laboratory frame are

9

<yJFMF\ HQ\yJFMF> =

-2[3M 2F - F(F + l)]A(yJFh (2) yF)

[(2F + 3)(2F + 2)(2F + 1)2F(2F-1)] 1/2 [(3cos

z /?-l)-£sin z y9(cos2 tf-sin z tf)]. ( 1)

It is relatively straightforward to show, by directly integrating the angular factor in square brackets in the above equation, that the average value of the diagonal matrix elements of HQ

, taken over all possible orientations of the laboratory frame with

respect to the principal-axis frame,

20

is zero. It is less obvious, but nevertheless true, that the average taken over any three mutually perpendicular orientations of the laboratory z quantization axis is also zero.

9 This provides a method for eliminating

the quadrupole shift from the observed transition frequency. The magnetic field, with constant magnitude, must be oriented in three mutually orthogonal directions; the average of the clock transition frequencies taken under these three conditions does

not contain the quadrupole shift.

The reduced matrix element in Eq.(l) is, in the /./-coupling approximation,

TN-55
(Y(lJ)F\\© {2) \\y(U)F) = (-!)>

7 27

-7 7

v J

&(Y,J),

where / is included in the state notation in order to specify the order of coupling of/ and 7. For the particular case of the

199Hg+ 5d*6s2 2D5/2 level, the reduced matrix elements are

(D5/2 2|e (2) |D5/2 2) = 2J—0(D,5/2),

II II V 5

(D 5/2 3|0 (2) |d 5/2 3) = 2J—0(D,5/2)

,

II II V 5

Since the Cowan-code calculation shows that there is very little mixing in the

l99Hg+ 5cf6s2 2D5/2 level, 6>(D,5/2) can be

reduced to a matrix element involving only the 5d orbital:

9

e(D,5/2)=-<5d z d5/2 ,mj =5/2bz 2 -r

2 \5d2 d5/2 , m} =5/2>,

= - —<5d\r 2 \5d>.

In this case, the matrix element is negative because the quadrupole moment is due to a single hole in the otherwise filled 5d

shell rather than to a single electron. Again, according to the Cowan-code calculation,

<5</|/-

2 |5J>= 2.324 al = 6.509xl0~17 cm

2

,

where a is the Bohr radius. Since the quadrupole shifts are zero in the 5di06s

2Ss,2 level, the quadrupole shift of the

199Hg+

optical clock transition is due entirely to the shift of the |D 5/2 2 0> state, and is given by

<D5/2 2O|/7 Q

|D5/2 2O>=-^0(D,5/2)[(3cos 2 /3-l)-£sin 2 /?(cos

2 a-sin 2 a)],

o

i i = Ae <5d\r 2 \5d >[(3cos 2 fi -l)-£sm 2 P(cos

2 a-sm 2 a)],

= -3.6xl0"3 M[(3cos 2 /3-l)-£sin 2 /?(cos

2 a-sin 2 tf)]Hz,

where A is expressed in units of V/cm2. Thus, for typical values A ~ 10

3 V/cm2 and | s \ < 1, the quadrupole shift is on the order of 1 Hz.

9 We also note that states with zero total angular momentum have no quadrupole shift. Thus, the2S m (F = 2, mF - 0) «->

2D3/2 (F = 0,mF = 0) hyperfine component in201 Hg+

, for which the nuclear spin I is 3/2, has no quadrupole shift. Similar strategies to eliminate the quadrupole shift could be used on other potential optical-clock ions such as Ca+

, Sr

+

, Ba+

, and Yb+

. ACKNOWLEDGMENTS

We acknowledge the support of the Office of Naval Research. This work was also supported through a Cooperative Research

and Development Agreement with Timing Solutions, Corp., Boulder, Colorado. We thank Dr. C.J. Sansonetti for making

available his results (Ref. 16) prior to their publication. We would also like to thank R. Drullinger, M. Lombardi, and D. Smith for their careful reading of this manuscript.

TN-56
REFERENCES

[t] Work of U.S. government. Not subject to U.S. copyright.

1. B.C. Young, F.C. Cruz, W.M. Itano, and J.C. Bergquist, "Visible lasers with subhertz linewidths," Phys. Rev. Lett. 82, pp. 3799-3802, 1999.

2. B.C. Young, R.J. Rafac, J. A. Beall, F.C. Cruz, W.M. Itano, D.J. Wineland, and J.C. Bergquist, "Hg+ optical frequency

standard: recent progress," in Laser Spectroscopy XIV International Conference, ed. by R. Blatt, J. Eschner, D. Leibfried, and

F. Schmidt-Kaler (World Scientific, Singapore, 1999) pp. 61-70.

3. Th. Udem, J. Reichert, R. Holzwarth, and T.W. Hansch, "Absolute optical frequency measurement of the cesium Di line with a mode-locked laser," Phys. Rev. Lett.%2, pp. 3568-3571, 1999.

4. D.J. Jones, S.A. Diddams, J.K. Ranka, A. Stentz, R.S. Windeler, J.L. Hall, and S.T. Cundiff, "Carrier-envelope phase

control of femtosecond mode-locked lasers and direct optical frequency synthesis, " Science 288, pp. 635-639, 2000.

5. S.A. Diddams, D.J. Jones, J. Ye, S.T. Cundiff, J.L. Hall, J.K. Ranka, R.S. Windeler, R. Holzwarth, Th. Udem, and T. W.

Hansch, "Direct link between microwave and optical frequencies with a 300 THz femtosecond laser comb," Phys. Rev. Lett. 84, pp. 5102-5105,2000.

6. H.G. Dehmelt, "Coherent spectroscopy on single atomic system at rest in free space II," J. Phys. (Paris) Colloq. 42, pp. C8-299 to C8-305, 1981.

7. R.J. Rafac, B.C. Young, J. A. Beall, W.M. Itano, D.J. Wineland, and J.C. Bergquist, "Sub-dekahertz ultraviolet spectroscopy of 199Hg+

," Phys. Rev. Lett. 85, pp. 2462-2465, 2000.

8. Th. Udem, S.A. Diddams, K.R. Vogel, C.W. Oates, E.A. Curtis, W.D. Lee, W.M. Itano, R.E. Drullinger, J.C. Bergquist,

and L. Hollberg, "Absolute Frequency Measurements of the Hg+ and Ca optical clock transitions with a femtosecond laser," submitted to Phys. Rev. Lett.

9. W.M. Itano, "External-field shifts of the

199Hg+ optical frequency standard," J. Res. NIST, in press, 2001.

10. J.C. Bergquist, W.M. Itano, and D.J. Wineland, "Laser stabilization to a single ion," in Frontiers in laser spectroscopy,

proceedings ofthe international school ofphysics "Enrico Fermi": course 120, ed. by T.W. Hansch, and M. Inguscio

(North-Holland, Amsterdam, 1994), pp. 359-376.

11. L.S. Ma, P. Jungner, J. Ye, and J.L. Hall, "Delivering the same optical frequency at two places: accurate cancellation of

phase noise introduced by an optical fiber or other time-varying path," Opt. Lett. 19, pp. 1777-1779, 1994.

12. H.G. Dehmelt, "Proposed 10

14 Av < v laser fluorescence spectroscopy on Tl+ mono-ion oscillator," Bull. Am. Phys. Soc. 20, p. 60, 1975.

13. J.C. Bergquist, W.M. Itano, and D.J. Wineland, "Recoilless optical absorption and Doppler sidebands of a single trapped

ion," Phys. Rev. A 36, pp. 428-430, 1987.

14. D.J. Wineland and W.M. Itano, "Laser cooling of atoms," Phys. Rev. A 20, pp. 1521-1540, 1979.

15. R.D. Cowan, The theory ofatomic structure and spectra, Univ. California Press, Berkeley, CA, 1981.

16. C.J. Sansonetti and J. Reader, "Spectrum and energy levels of singly-ionized mercury (Hgll)," Phys. Scripta, (in press).

17. Th. A.M. Van Kleef and M Fred, "Zeeman effect measurements in neutral and singly ionized mercury," Physica 29, pp. 389-404, 1963.

18. J.R.P. Angel and P.G.H. Sandars, "The hyperfine structure Stark effect: I. Theory," Proc. Roy. Soc. A 305, pp. 125-138,

1968.

19. D.J. Berkeland, J.D. Miller, J.C. Bergquist, W.M. Itano, and D.J. Wineland, "Minimization of ion micromotion in a Paul

trap," J. Appl. Phys. 83, pp. 5025-5033, 1998.

20. L.S. Brown and G. Gabrielse, "Precision spectroscopy of a charged particle in an imperfect Penning trap," Phys. Rev. A

25,2423-2425, 1982.

TN-57
Volume 86, Number 22 PHYSICAL REVIEW LETTERS 28 May 2001

Absolute Frequency Measurements of the Hg+ and Ca Optical Clock Transitions

with a Femtosecond Laser

Th. Udem,* S.A. Diddams, K.R. Vogel, C.W. Oates, E. A. Curtis, W.D. Lee,

W. M. Itano, R. E. Drullinger, J. C. Bergquist, and L. Hollberg

Time and Frequency Division, National Institute of Standards and Technology, 325 Broadway, Boulder, Colorado 80305

(Received 4 January 2001)

The frequency comb created by a femtosecond mode-locked laser and a microstructured fiber is used

to phase coherently measure the frequencies of both the Hg+ and Ca optical standards with respect to the SI second. We find the transition frequencies to be /Hg = 1 064721 609 899 143(10) Hz and /Ca =

455 986240494 158(26) Hz, respectively. In addition to the unprecedented precision demonstrated here,

this work is the precursor to all-optical atomic clocks based on the Hg+ and Ca standards. Furthermore,

when combined with previous measurements, we find no time variations of these atomic frequencies within the uncertainties of |(d/Ca/dr)//cal ^ 8 X 10" 14 yr" 1 and |(d/Hg M)//Hgl ^ 30 X 10~ 14 yr

_1

.

DOI: 10.1103/PhysRevLett.86.4996 PACS numbers: 06.30.Ft, 32.30.Jc, 32.80.Pj

Optical standards based on a single ion or a collection of

laser-cooled atoms are emerging as the most stable and ac- curate frequency sources of any sort [1-5]. However, be- cause of their high frequencies (—500 THz), it has proven

difficult to count cycles as required for building an optical

clock and comparing to the cesium microwave standard. Only recently, a reliable and convenient optical clockwork

fast enough to count optical oscillations has been realized [6-8]. Here, we report an optical clockwork based on

a single femtosecond laser that phase coherently divides down the visible radiation of the Hg+ and Ca optical fre- quency standards to a countable radio frequency. By this means we determine the absolute frequencies of these optical transitions with unparalleled precision in terms of

the SI second as realized at National Institute of Stan- dards and Technology (NIST) [9]. Indeed, for the Hg+

standard, the statistical uncertainty in the measurement is essentially limited by our knowledge of the SI second at ~2 X 10~ 15

. The high precision and high demonstrated

stability of the standards [1,4] combined with the straightforward femtosecond-laser-based clockwork suggest Hg+ and Ca as excellent references for future all-optical clocks. Additionally, the comparison of atomic frequencies over time provides constraints on the possible time variation of fundamental constants. When combined with previous

measurements, the current level of precision allows us to place the tightest constraint yet on the possible variation of optical frequencies with respect to the cesium standard. The Hg+ and Ca systems have recently been described

elsewhere [1,4,10,11], so we summarize only the basic

features. The heart of the mercury optical frequency

standard is a single, laser-cooled 199 Hg+

ion that is stored

in a cryogenic, radio frequency spherical Paul trap. The

2 S]/2 (F = 0,Mf = 0) ~ 2 D5/2 (F = 2,MF = 0) electric- quadrupole transition at 282 nm [Fig. 1(a)] provides

the reference for the optical standard [1]. We lock the frequency-doubled output of a well-stabilized 563 nm dye

laser to the center of the quadrupole resonance by irradiating the Hg+

ion alternately at two frequencies

near the maximum slope of the resonance signal and on

opposite sides of its center. Transitions to the metastable

2 Ds/2 state are detected with near unit efficiency since

the absorption of a single 282 nm photon suppresses

the scattering of many 194 nm photons on the strongly

allowed 2 S\/i — 2 P\/2 transition [12,13]. Because the

fractional frequency instability of the probe laser is <10~ 15 for measurement times 0.1 s < t < 10 s [11],

usually 48 measurements are made on each side of the resonance prior to correcting the average frequency of

the 282 nm source. The resonance probe period was

either 10 or 20 ms, but each measurement cycle was

longer by the time used (for example) for state preparation

(15 ms) and detection (10 ms), as well as the decay time

(td ~ 90 ms) of the metastable state if an excitation was made. A new interrogation cycle is begun when

the 194 nm fluorescence intensity rises above a preset threshold level. This reduces dead time, since it would

otherwise be necessary to wait more than the lifetime of

2Py2 3^ = o 194 nm 7~ F \282n

/,.,

-F = (a) 2Si/2

(b)

-60 -40 -20 20 40 60

cavity detuning [Hz]

FIG. 1. (a) Partial level scheme for 199 Hg+

. The 194 nm radiation is used for Doppler cooling, state preparation, and detection. The 282 nm transition from the ground state

2 S\/2(F = 0, Mf =

0) to the metastable 2 £>5 / 2 (F = 2,MF = 0) state provides the reference for the optical clock frequency, (b) A typical spectrum

of the 282 nm clock transition obtained under lock conditions

is shown. Here, the excitation pulse length was 20 ms, and the measured linewidth is Fourier transform limited to about 20 Hz

at 563 nm (40 Hz at 282 nm).

4996

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Volume 86, Number 22 PHYSICAL REVIEW LETTERS 28 May 2001

the metastable state to ensure that the ion has returned to the ground state. If an asymmetry between the number

of excitations detected on the high- and low-frequency

sides is found, then the frequency of the probe radiation

is adjusted to minimize the asymmetry. In this way, we

steer the frequency of the 282 nm source to the center of

the S — D quadrupole resonance with an uncertainty that

is less than 2 X 10

-15

for averaging times t < 30 s and

that decreases as r

_1/2 for r > 30 s [14]. In Fig. 1(b) we

show an example of a normalized spectrum that was obtained from multiple bidirectional scans through the reso- nance during the lockup, where the probe time was 20 ms. The calcium standard is based on a collection of ~107

laser-cooled 40Ca atoms held in a magneto-optic trap. The

423 nm 'So *-* 1 P\ transition is used for Doppler cooling and trapping the atoms to a residual temperature of

~2 mK, while the 657 nm x S {Mj = 0) — 3 Pi(Mj = 0)

clock transition (400 Hz natural linewidth) is used for the frequency standard [Fig. 2(a)]. We excite the clock transition with a four-pulse Borde-Ramsey sequence (pulse

duration = 1.5 /as) with light from a continuous wave

(cw) frequency-stabilized diode laser. Using a shelving detection technique similar to that employed in the Hg +

sys- tem, near-resonant 423 nm pulses (5 /is duration) are used

before and after the 657 nm excitation to determine the

fraction of atoms transferred from the ground state. Figure 2(b) shows Borde-Ramsey fringes taken at a resolution of 960 Hz. This system has demonstrated a fractional fre- quency instability of 4 X 10~ 15 t~'/2

, when probing subkilohertz linewidths [4]. For the measurements presented

here the Ca spectrometer was operated with linewidths

ranging from 0.96 to 1 1 .55 kHz which are integer subharmonics of the recoil splitting.

The recent introduction of mode-locked lasers to optical frequency metrology greatly simplifies the task of optical frequency measurements [6-8,15-17]. The spectrum

emitted by a mode-locked laser consists of a comb of regular spaced continuous waves that are separated by the pulse

repetition rate fr

. The frequency of the nth mode of the comb is given by /„ = nfr + f [18,19], where f is the frequency offset common to all modes that is caused

T 3Pi 423 nm T-

\ /

\ 657 nm W

COEo

TM 22.0- o

J

21.6-

'oS

21.2-

(b) VW (a)

' S° -3-2-10123

cavity detuning [kHz]

FIG. 2. (a) Simplified diagram of the relevant energy levels

in the Ca standard, (b) Optical Borde-Ramsey fringes with a 960 Hz (FWHM) resolution. The total averaging time to generate this figure was 20 s.

by the difference between the group and the phase velocity inside the laser cavity. Whereas fr can be measured

by direct detection of the laser output with a photodiode,

f is measured by heterodyning the harmonic of a mode

/„ = nfr + f from the infrared wing of the comb with

a mode fjn = 2nfr + f from the blue side of the comb

[7,8]. While an octave spanning comb can be produced di- rectly from a mode-locked laser [20], launching the longer

pulses from a commercially available femtosecond laser into an air-silica microstructure fiber [21,22] also produces

a frequency comb that spans an octave. Via nonlinear processes in the fiber, additional equally spaced and phasecoherent modes are added to the comb. It has been demonstrated that this process of spectral broadening preserves

the uniformity of spacing and spectral fidelity of the comb

to at least a few parts in 1016 [8]. We couple approximately 200 mW average power from

a femtosecond Ti:sapphire ring laser (fr ~ 1 GHz) [23] into a 15 cm piece of a microstructure fiber that has a

1.7 fim core and a group velocity dispersion that vanishes near 770 nm [21]. This power is sufficient to in- crease the spectral width of the laser from 1 3 THz to more

than 300 THz, spanning from ~520 to —1170 nm. The

infrared part of the comb from the fiber (A = 1060 nm) is split off by a dichroic mirror and frequency doubled into the green portion of the visible spectrum with a 2 mm long

KNbC>3 crystal. Following an adjustable delay line that matches the optical path lengths, the frequency-doubled

light is spatially combined with the green part of the origi- nal comb using a polarizing beam splitter. A second ro- tatable polarizer projects the polarization of the combined

beams onto a common axis so that they can interfere on a photodiode. This polarizer is also used to adjust the rela- tive power of the two beams for optimum signal-to-noise

ratio in the heterodyne signal. A small grating prior to the photodiode helps to select only that part of the fre- quency comb that matches the frequency-doubled light,

thereby reducing noise from unwanted comb lines [19]. We phase lock both f and fr to synthesized frequencies

derived from a cavity-tuned hydrogen maser. Control of fr is achieved with a cavity folding mirror that is mounted on

a piezotransducer, while f is controlled by adjusting the 532 nm pump beam intensity with an electro-optic modulator [8]. When f and fr are both phase locked, the fre- quency of every mode in the comb is known with the same

precision as the reference maser. The cw light from the Hg+

(563 nm) and Ca (657 nm)

spectrometers is transferred to the mode-locked laser sys- tem via two single mode optical fibers that are 1 80 m and

10 m long, respectively. Approximately 2 mW of cw light

from each fiber is mode matched with the appropriate spectral region of the frequency comb to generate a beat sig- nal ft, with a nearby mode. This beat note is amplified

and measured with a radio frequency counter. The optical

frequency is then expressed as /opt = f + mfr + fb,

where m is a large integer uniquely determined for each

system from previous coarse measurements of

/

opt .

4997

TN-59
Volume 86, Number 22 PHYSICAL REVIEW LETTERS 28 May 2001

We detect cycle slips in both of the phase locks by

monitoring fr and f with additional counters [24]. We

selectively discard any measurement of /opt for which the measured f or fr deviate from the expected value by more

than 1/xgate, where rgate is the counter gate time in sec- onds. We avoid miscounts of ft, by using an auxiliary counter to record the ratio r between fb and fb/4, where

the division by 4 is implemented digitally. Any measure- ments of fb where the auxiliary counter gives a result that does not satisfy (r — A)fb < 10/Ygate are discarded. We

rely on the assumption that the two counters recording fb and r, if in agreement, do not make the same mistake. For

each data point the three additional counters (fr , f , and

r) are started before the counting of fb and operated with

50 ms longer gate times to ensure temporal overlap.

Figure 3 summarizes the frequency measurements of

Hg+ made between 16 August and 31 August 2000,

and Fig. 4 summarizes the Ca measurements made from

26 October to 17 November 2000. All measurements

are corrected for the second-order Zeeman shift and

for the offset of the reference maser frequency. The

uncertainty for the Zeeman correction is <1 X 10~ 15

for the Hg+ system and <2.5 X 10~ 15 in the Ca system.

The frequency of the maser is calibrated by comparing

to the local NIST time scale (5 hydrogen masers and

3 commercial cesium clocks), which in turn is calibrated by the local cesium fountain standard (NIST-F1 [9]), as well as international cesium standards. This resulted in a fractional uncertainty in the frequency of the reference maser of about 1.8 X 10~ 15 for the measurements. The weighted mean of our measurements of the Hg+

clock transition is /Hg = 1 064721 609 899 143 Hz, where

the statistical uncertainty of 2.4 Hz is near the fractional frequency instability of the reference maser (~2 X 10~ 13

at 1 s, decreasing to ~4 X 10-16at a few days). We have

not yet made a full evaluation of the systematic uncertainties of the Hg +

standard; however, we believe that 10 Hz

I

JU

CD

CM 20

1—

o>

oo 10

05o

CO U

i—

CMh-* -10

coo,_ -20

S: ^ -30

;-TT —

i

i 1 I'.

.

II

-

Aug 16 Aug 21 Aug 26 Aug 31 A chronological record of the average daily frequency

l99 Hg+ clock transition measured on six days over a

FIG. 3 of the 15 day period representing 21 651 s of total measurement time The error bars represent statistical fluctuations. The dashed lines represent an estimated systematic uncertainty of ±10 Hz in the Hg + system in the absence of a full evaluation.

4998

is a conservative upper bound for the total systematic uncertainty. The largest systematic uncertainty is due to the

interaction between the atomic quadrupole moment of the

2 £>5/2 state and a static electric field gradient. In the currently used spherical Paul trap, no static field gradient is deliberately applied. A potential difference between the

ring and the end cap electrodes of as large as 0.5 V due to contact potentials or stray charges, for example, would result in a quadrupole shift of only 1 Hz [25]. The magnetic

field is evaluated before and after each run by measuring

the frequency of one or more of the magnetic field de- pendent electric-quadrupole transitions. This results in an

uncertainty of the second-order Zeeman shift of the clock

transition of less than 0.5 Hz. The accuracy of this cali- bration has been verified by varying the magnitude of the magnetic field for successive runs. The blackbody radi- ation shift of the clock transition would be —0.08 Hz at 300 K and is considerably lower in the cryogenic trap en- vironment of approximately 4 K. At the Doppler cooling

limit, the second-order Doppler shift due to thermal motion is —0.003 Hz. Finally, the second-order Doppler shift due to residual micromotion caused by the trapping field

is estimated to have a magnitude no greater than 0.1 Hz. We anticipate that the uncertainties of all systematic shifts

in the Hg+ system can be reduced to values approaching

1 X 10-18

[1,25]. For the Ca data shown (Fig. 4), an additional correction

is applied each day to account for a frequency shift caused

by residual phase chirping on the optical Ramsey pulses

produced by amplitude modulating an acousto-optic

modulator (AOM). The phase chirping produced a resolution dependent frequency shift on the order of 100 Hz

for 11.5 kHz wide fringes but only 10 Hz for 0.96 kHz

NX

oo

LO

o

CM

co

00

CDin

in

150

100

-50 -

-100 -

-150 Oct 26 Nov 5 Nov 15

FIG. 4. The filled squares are the measured Ca frequencies on ten days over a 23 day period representing 38 787 s of total measurement time. The inner and outer error bars for each day

represent the statistical and total uncertainties, respectively. The

dashed lines show the 26 Hz systematic uncertainty assigned

to the mean. The open triangle is the Physikalisch-Technische Bundesananstalt (PTB) measurement reported in Ref. [26], and

the open circle is the Ca frequency calculated from the present Hg + result and our previous measurement of the 76 THz gap between Ca and Hg +

[27].

TN-60
Volume 86, Number 22 PHYSICAL REVIEW LETTERS 28 May 2001

wide fringes. On each day, the Ca frequency was measured

for —30 min at each of several fringe resolutions, and the zero intercept of a linear fit to the data was used as the corrected frequency. On the last three days of measurements, we were able to reduce this shift by a factor of ~3

with improvements to the rf pulses that drive the AOM's.

With horizontal beams we can achieve <20 Hz gravityinduced frequency shift, which can be readily quantified

and eliminated simply by reversing the pulse sequence. The statistical uncertainty for each day's measurement

(typically 8 Hz) is smaller than the uncontrolled systematic

uncertainties in the Ca frequency. The largest systematic

uncertainty stems from incomplete knowledge of the angular overlap of the counterpropagating beams in the Ca

spectrometer, combined with a transverse drift velocity of

the cold Ca ensemble. This leads to a residual first-order Doppler shift with a magnitude <15 Hz (except on

16 November 2000, where a large drift velocity led to a —52 Hz uncertainty). Other significant uncertainties include our lack of knowledge or control of electronic

offsets and baseline asymmetries (<12 Hz), wave front curvature (<10 Hz), and cold-atom collisional shifts (<10 Hz). Taking all known systematic uncertainties in quadrature gives a confidence level of —26 Hz for the measured mean value as indicated by the dashed lines in Fig. 4. Again, for the Ca measurement we find that the

stability is limited by the maser. Figure 4 also shows the good agreement between our measurement and the most recent value measured with a harmonic frequency chain [26], which provides a degree of

confidence in the reproducibility of the Ca standards. An

additional measure of the Ca frequency can be made by

using the present absolute measurement of Hg+ and our

earlier measurement of the 76 374 564 455 429(40) Hz gap

between /Hg /2 and the Ca standard [27]. This yields a value /Ca = 455 986240494143(40) Hz in good agree- ment with the present absolute measurement of /ca

- Finally, these results also provide data on the relative time variability of atomic frequencies. Karshenboim has

recently reviewed the implications of such comparisons

and their contribution toward constraining the possible

time variation of fundamental constants [28]. In this regard

Hg + and Ca are two of the most interesting cases to study.

Comparing our present measurement of /ca to measure- ments made by PTB in 1997 [26] gives (dfcJdt)/fc» =

(+2 ± 8) X 10-14 yr ] . Similarly, combining this result with our May 2000 measurement of /ng with respect

t0 /ca [27] provides an initial baseline constraint on the time variation of /Hg of (dfHg /dt)/fns = (-7 ± 30) X

10

-14

yr

_1

. Here we use the defined unit of time based on the frequency of the Cs hyperfine interval and assume

that any time dependence is slow and dominantly linear over the relevant time scale. At our present level of preci- sion we find no evidence of any relative time variation be- tween these three frequency standards, two optical and one

microwave.

The authors are most grateful to T Parker for providing

the crucial maser calibration and to A. Bartels of GigaOptics GmbH for his valuable assistance with the femtosecond laser. We are also indebted to R. Windeler of Lucent Technologies for providing the microstructure optical fiber. We further acknowledge many illuminating dis- cussions with D. Wineland, J. Hall, S. Karshenboim, and

F. Walls. This work was partially supported by the Office

of Naval Research and through a cooperative research and

development agreement with Timing Solutions, Inc., Boulder, CO.

Note added.—Since the submission of this work a new

measurement of the Ca clock transition was conducted at PTB with a femtosecond system similar to the one described here [29]. Although less precise, this most re- cent PTB measurement is in agreement with the value we

report.

Present address: Max-Planck-Institut fur Quantenoptik,

Hans-Kopfermann-Strasse 1, 85748 Garching, Germany.

[1] R. Rafac et al, Phys. Rev. Lett. 85, 2462 (2000).

[2] J.E. Bernard et al, Phys. Rev. Lett. 82, 3228 (1999).

[3] H. Schnatz et al, Phys. Rev. Lett. 76, 18 (1996).

[4] C. W. Oates et al, Opt. Lett. 25, 1603 (2000).

[5] J. von Zanthier et al, Opt. Lett. 25, 1729 (2000).

[6] S. A. Diddams et al, Phys. Rev. Lett. 84, 5102 (2000).

[7] D. J. Jones et al, Science 228, 635 (2000).

[8] R. Holzwarth et al, Phys. Rev. Lett. 85, 2264 (2000).

[9] S. R. Jefferts et al. (to be published); S. R. Jefferts et al, in Proceedings of the IEEE International Frequency Control Symposium (IEEE, Piscataway, NJ, 2000), pp. 714-717.

[10] C. W. Oates et al, J. Phys. D 7, 449 (1999).

[11] B. Young et al, Phys. Rev. Lett. 82, 3799 (1999).

[12] H. Dehmelt, Bull. Am. Phys. Soc. 20, 60 (1975).

[13] J.C. Bergquist et al, Phys. Rev. A 36, 428 (1987).

[14] D.J. Wineland et al, The Hydrogen Atom, edited by T. W.

Hansch (Springer, Berlin, Heidelberg, 1989), pp. 123-133.

[15] Th. Udem et al, Phys. Rev. Lett. 82, 3568 (1999).

[16] J. Reichert et al, Phys. Rev. Lett. 84, 3232 (2000).

[17] M. Niering et al, Phys. Rev. Lett. 84, 5496 (2000).

[18] A.I. Ferguson, J.N. Eckstein, and T. W. Hansch, Appl.

Phys. 18, 257 (1979).

[19] J. Reichert et al, Opt. Commun. 172, 59 (1999).

[20] R. Ell et al, Opt. Lett. 26, 373 (2001).

[21] J.K. Ranka et al, Opt. Lett. 25, 25 (2000).

[22] W.J. Wadsworth et al, Electron. Lett. 36, 53 (2000).

[23] A. Bartels, T. Dekorsy, and H. Kurz, Opt. Lett. 24, 996

(1999).

[24] Th. Udem et al. Opt. Lett. 23, 1387 (1998).

[25] W.M. Itano, J. Res. Natl. Inst. Stand. Technol. 105, 829

(2000).

[26] F. Riehle et al, in Proceedings of the Joint European Fre- quency and Time Forum and the IEEE International Fre- quency Control Symposium (IEEE, Piscataway, NJ, 1999),

pp. 700-705.

[27] K.R. Vogel et al. Opt. Lett. 26, 102 (2001).

[28] S. Karshenboim, Can. J. Phys. 78, 639 (2000).

[29] J. Stenger et al, Phys. Rev. A 63, 021802 (2001).

4999

TN-61
An Optical Clock Based on a

Single Trapped 199Hg+

Ion

S. A. Diddams,1 * Th. Udem,1 ! J- C. Bergquist, 1 E. A. Curtis, 1 - 2

R. E. Drullinger, 1 L Hollberg, 1 W. M. Itano, 1 W. D. Lee, 1

C. W. Oates, 1 K. R. Vogel,1 D. J. Wineland1

Microwave atomic clocks have been the de facto standards for precision time

and frequency metrology over the past 50 years, finding widespread use in basic

scientific studies, communications, and navigation. However, with its higher

operating frequency, an atomic clock based on an optical transition can be much

more stable. We demonstrate an all-optical atomic clock referenced to the

1 .064-petahertz transition of a single trapped 199Hg+ion. A clockwork based on a mode-locked femtosecond laser provides output pulses at a 1 -gigahertz

rate that are phase-coherently locked to the optical frequency. By comparison

to a laser-cooled calcium optical standard, an upper limit for the fractional frequency instability of 7 X 10" 15

is measured in 1 second of averaging—

a

value substantially better than that of the world's best microwave atomic

clocks.

Since the development of the first atomic

clocks around 1950, it was recognized that the

stability and accuracy of standards based on

atomic transitions would benefit from choosing

transition frequencies as high as possible. However, because it was not possible to count cycles of an oscillator at an arbitrarily high frequency

in order to generate time, most time standards have been based on hyperfine transitions in atoms that occur at microwave frequencies where the period of one oscillation corresponds

to roughly 0.1 to 1 ns. In fact, since 1967 the

internationally accepted definition of the sec- ond has been based on the 9,1 92,63 1,770-Hz

ground-state hyperfine transition in Cs, and Cs- based clocks are currently the world's most

accurate time standards with a fractional fre- quency uncertainty of about 1 X 10~ 15 (1-4). Although the stability and accuracy of an opti- cal clock based on a spectrally narrow atomic

transition in the visible/ultraviolet (UV) region

might be better, the frequency is about 10 15 Hz, where the period of one oscillation is on the order of 1 fs. Until recently, it was not practical to count such high frequencies without loss of

cycles, but the introduction of mode-locked

femtosecond lasers has now made it possible to conveniently and accurately divide optical fre- quencies to countable microwave or radio fre- quencies (5-10). The all-optical atomic clock

reported here is based on a single, laser-cooled, trapped I99Hg+

ion and a femtosecond laser comb that provides the phase-coherent clock1 Time and Frequency Division, National Institute of Standards and Technology, 325 Broadway, Boulder, CO 80305, USA. department of Physics, University of Colorado, Boulder, CO 80309, USA. *To whom correspondence should be addressed. E- mail: sdiddams@boulder.nist.gov

fPresent address: Max-Planck-lnstitut fur Quantenoptik, Hans-Kopfermann Strasse 1, 85748, Carching, Germany.

work. Our optical clock demonstrates a frac- tional frequency instability of <7 X 10~ 15

with 1 s of averaging—a value substantially

better than that of the world's best microwave

atomic standards—and promises accuracy that would be difficult to achieve with atomic clocks based on microwave transitions. This optical clock provides two distinct outputs: a comb of

stable and accurate optical frequencies throughout the visible and near infrared, and a pulsed

electronic signal at 1 GHz that is phase-coherent with the optical frequency comb. The quantum-limited instability of the Hg+

ion optical clock is expected to be around 1 X 10~ I5 t~ 1/2

, where t is the averaging time measured in seconds, with an ultimate fractional frequency

uncertainty approaching 10~ 18 (11). Ultrastable and accurate optical metrology

tools can be expected to provide an even finer- grained view of the physical world, much as precision spectroscopy in the past 50 years has opened the door to an improved understanding of many fundamental aspects of atoms and

molecules (72, 13). Ofparticular interest will be

the continued application of optical frequency

standards in spectroscopy and the improved

determination of the fine structure constant a

and the Rydberg constant /?„ (14). As measure- ment stability and accuracy improve, metrologists may find themselves in the unique position of being able to observe physical "constants" evolve in time (15). Indeed, laboratory tests on

the possible divergence of clocks based on dif- ferent atomic transitions already provide some

of the most stringent constraints of the variation of a. Other experiments of fundamental importance for which precision clocks/oscillators are of value include searches for variations in the

isotropy of space, a preferred reference frame, and Lorentz and charge-parity-time (CPT) symmetry violation (16-20). From a technological

standpoint, there is little dispute that stable and

accurate microwave atomic clocks have greatly improved navigation and communications. It is likely that optical clocks of the future will have

a similarly important impact.

Atomic clock basics. All clocks consist of two major components: some device that produces periodic events or "clock ticks," and some means for counting, accumulating, and

displaying each tick. For example, the swing of

a pendulum provides the periodic events that are counted, accumulated, and displayed by means of a set of gears driving a pair of clock

hands. Similarly, in a quartz watch, the mechanical vibrations of a small quartz crystal are

electronically detected, accumulated, and dis- played to generate time. Atomic clocks add a

third component: the resonance of a well-isolated atomic transition, which is used to control

the oscillator frequency. If the frequency of the

oscillator is made to match the transition fre- quency (i.e., the oscillator is locked to the atomic transition frequency) between two nondegenerate and unperturbed atomic states, then the time generated can have improved long-term

stability and accuracy. For an atomic clock based on a microwave transition, high-speed

electronics count and accumulate a defined number of cycles of the reference oscillator to mark a second of time. The basic concepts are

the same for an atomic clock based on an

optical transition at a much higher frequency. In

this case the oscillator is a laser locked to an

optical transition, but no electronic device exists that can count the very fast optical oscillations. For this purpose, a specialized frequency

divider (commonly called a frequency chain) is required. Until very recently, optical-to-micro- wave frequency chains have been complicated,

large-scale devices, requiring significant re- sources for operation (21). However, as de- scribed below, a femtosecond laser- based clockwork greatly simplifies this problem of

directly counting the optical frequency.

All other factors being equal, a higher transition frequency can produce a more stable frequency standard. This is the principal advantage of an optical atomic clock over a micro- wave clock because the operating frequency is ~ 100,000 times higher, providing a finer divi- sion of time and thus potentially higher precision. This is seen in the Allan deviation a (t) which provides a convenient measure of the

fractional frequency instability of a clock as a

function of the averaging time t (22). For an

oscillator locked to an atomic transition of fre- quency v and linewidth Av,

o\,(t) Av,

(1)

where Avrmsis the measured frequency fluc- tuation, Nis the number of atoms, and 7 is the cycle time (i.e., the time required to make a determination of the line center) with t > T.

www.sciencemag.org SCIENCE VOL 293 3 AUGUST 2001 825

TN-62
This expression assumes that technical noise

is reduced to a sufficiently small level such

that the quantum-mechanical atomic projection noise is the dominant stability limit (23, 24). In this limit cr (t) decreases as the square

root of the averaging time for all clocks, so a 10-fold decrease in the short-term instability leads to a 100-fold reduction in averaging

time t to reach a given stability and uncertainty. This point is particularly important if one ultimately hopes to reach a fractional frequency uncertainty of 10~ 18

, which is the anticipated level for optical clocks. In this case, oyV) < 1 X 10" 15 t" 1/2 is clearly

desirable to avoid inordinately long averaging times. Principles of the optical clock. Fundamental ideas and technical developments

principally in three areas have brought us to the point where we are now able to demonstrate an optical frequency standard representative of clocks of the future: (i) the idea (25, 26) and demonstration of laser cooling of atoms (27, 28), (ii) the frequency stabilization of lasers (29-32), and (iii) the concept (33) and demonstration (5-10) that femtosecond mode-locked lasers combined with nonlinear

fibers can provide a simple, direct, and phasecoherent connection between radio frequencies and optical frequencies. Although most

of these concepts have existed for some time, and preliminary demonstrations of optical clocks have even been made (34-36), only now have the techniques and tools advanced

to the levels required for optical frequency

standards to move beyond the benchmark

results of the microwave standards. Trapped

ions have been laser-cooled into the LambDicke regime to the zero-point energy limit of a trap potential (37), dramatically reducing

the Doppler shifts and providing near-stationary, relatively unperturbed atomic references

for extended observation times (77). Similarly, large numbers of neutral atoms that are

suitable references for optical clocks are routinely laser-cooled and then trapped in magneto-optic traps (38—43). Stabilized lasers have now been demonstrated with linewidths

less than 0.2 Hz (32, 44) and with the center frequency reproducibly controlled for extend- ed times at the levels of IA/7/1 < 10" 15 (45). And finally, a practical method for measuring

optical frequencies based on femtosecond la- ser technology has been developed (5-8) and

its measurement uncertainty tested to a few

parts in 10 16 (9). These advances are exploited in our opti- cal clock, which consists of a stable continu- ous wave (CW) laser oscillator that is fre- quency doubled and locked to a narrow UV

transition of a single trapped and laser-cooled

199Hg+

ion. Thus stabilized, the frequency of

the laser light is coherently divided down to lower frequencies by means of a femtosecond mode-locked laser that ultimately produces

an electronic output at a frequency of 1 GHz.

A simplified representation of the coherent

relation between the optical frequency stan- dard and the microwave pulses is shown in Fig. 1. The envelope of the pulse train is made synchronous with the optical phase of

the CW laser with, in our case, —532,361

optical cycles between pulses (46). The pulses thus provide "clock ticks" that are coher- wmmmmmmm -2fs

( MM I h'-.oi ! 'i.ripui |||||||E : ''Tg||||||

fs Pulse Train

(Clock Output)

1 ns

-«—*

time

Fig. 1. Illustration of the time-domain relation between the atomically stabilized 532-THz CW laser

field and the mode-locked femtosecond laser pulses. The two lines at the bottom show the near

delta-function optical pulse train with a fr = 1-GHz repetition rate and the CW laser output

without the optical oscillation resolved. When the system is locked as an optically referenced femtosecond clockwork, the pulse repetition rate is phase-coherently related to the 532-THz CW

field. This is indicated schematically in the upper portion of the figure where an expanded view of the lower two traces shows how the phase of the pulse envelope is related to the phase of the CW

laser field. The net result is that the frequency of the pulse repetition rate is simply a rational

fraction of the Hg+

transition frequency.

ently connected to the Hg+

transition. The Hg+ standard and optical clock- work. Figure 2 shows the optical clock in more detail, consisting of the optical frequen- cy standard and the femtosecond laser-based

optical clockwork. The 199Hg+

optical fre- quency standard has been described in detail elsewhere (32, 45), so we provide only the most relevant details. The 2S 1/2 (F = 0, MF =

0) <-» 2D5/2 (F = 2, MF = 0) electric-quadrupole transition at 282 nm provides the refer- ence for the optical frequency standard. The

natural linewidth of the S^D resonance is about 2 Hz at 1.064 PHz, and recently a Fourier-transform-limited linewidth of only

6.7 Hz (Q = 1.5 X 10 14

) has been observed

(45). The "local oscillator" for the standard is the output of a well-stabilized 532-THz (563

nm) dye laser (32, 44) that is frequency doubled and locked to the center of the 1 .064- PHz (282 nm) S <h> D resonance. The short- term (1 to 10 s) fractional frequency instability of the probe laser is <5 X 10~ 16

. This

short-term laser instability is low enough that information gathered from probing the ion

transition can be integrated for about 10 s before it is used to steer the average laser frequency. The net result is a ct (t) < 2 X

10~ 15 for averaging times up to —30 s, at which point a (t) begins to average down as

The Hg+standard provides high accuracy and stability, but for distribution purposes and to realize a countable clock output, we

must phase-coherently convert the optical

signal to a lower frequency. The clockwork

that divides the 1 .064-PHz optical frequency

to a countable microwave frequency fT is based on a femtosecond laser and a novel

microstructure optical fiber. The Ti: sapphire femtosecond ring laser emits a train of pulses

(compressible to ~25-fs duration) at the nominal repetition rate of fT = 1 GHz (47). The frequency-domain spectrum of the pulse

train is a uniform comb of phase-coherent

continuous waves separated by fT

. The fre- quency of the nth mode of this comb is fn =

nfT + f (48, 49), where f is the frequency

offset common to all modes that results from

the difference between the group- and the phase-velocity inside the laser cavity. If the frequency comb of the laser covers an entire octave, then/ can be measured by frequency

doubling an infrared mode (n) and heterodyning it with an existing mode (2ri) in the

visible portion of the comb (7, 9). The het- erodyne signal yields the frequency differ- ence 2(nfT + f ) - (2nfr + fj = /o

. Only

recently, with the arrival of microstructure

silica fibers (also called photonic crystal fi- bers), has the required octave-spanning spec- trum been attained with high repetition rate, low-power femtosecond lasers (50, 51). The

unique dispersion properties of the microstructure fiber provide guidance in a single

826 3 AUGUST 2001 VOL 293 SCIENCE www.sciencemag.org

TN-63
spatial mode (~l.7-u.rn diameter) with zero group velocity dispersion near 800 nm (50). Because temporal spreading of the pulse is minimized, peak intensities in the range of hundreds of GW/cm2 are maintained over a considerable propagation length, thus providing enhanced spectral broadening due to self- phase modulation. With ~200 mW (average

power) coupled into a 15-cm piece of microstructure fiber, the total spectral width is broadened from —15 to ~300 THz (spanning

from ~520 nm to —1170 nm).

In addition tof , a second heterodyne beat fh

is measured between an individual comb

element fm = mfT + fQ (m is an integer) and

the 532-THz local oscillator of the Hg+

stan- dard. As shown in Fig. 2, two phase-locked

loops (PLL) are used to control f and fb. thereby fixing the clock output fT

. PLL-1

forces/ = $fT by controlling the pump power

of the femtosecond laser (9). Similarly,

PLL-2 changes the cavity length of the femtosecond laser with a piezo-mounted mirror, such thatfb = afT

. The constants a and (3 are integer ratios implemented with frequency

synthesizers that use fT l\ 00 as a reference. In

this manner, the frequencies of both PLLs are phase-coherently linked to fT such that all oscillators used in the clock are referenced to the 532-THz laser oscillator itself. When/

and^ are phase-locked, every element of the femtosecond comb, as well as their frequency

separation^, is phase-coherent with the laser locked to the Hg+

standard (52). With no

other frequency reference as an input, we

realize all aspects of a high-accuracy, highstability optical atomic clock: a stable local

oscillator (the laser) locked to a narrow atomic reference, and a pulsed microwave output

that can be recorded with a counter. High-stability output. With both PLLs

closed, the — 1-GHz microwave output has

the value of fT = fnJ(m ± a ± (3). If we

choose the signs of beats f and fr such that a = — (3, then fT would be an exact sub- harmonic of/^ . The stability of the 532-THz

laser (given above) should be transferred to each element of the femtosecond comb, in addition to/r

. We obtain^ from the bandpassfiltered photocurrent generated with —5 mW

of the broadened comb light incident on a

p-i-n photodiode. We have measured the in- stability of/"

r by subtracting it from the output of a synthesizer that is referenced to a hydro- gen maser for which <r (1 s) ~ 2.5 X 10~ 13

. The stability of this difference frequency is then analyzed with both a high-resolution counter and a dual-mixer time-measurement

system (53). Both results are consistent with

the resolutions of the respective measure- ments and the maser stability, demonstrating

that the 1-s instability of/~r is at least as good

as that of the hydrogen maser. Before we can conclusively state that a microwave signal with stability matching that

of the optical standard can be obtained from

the optical clock, fT needs to be compared to an oscillator with stability substantially better than that of the hydrogen maser. This could be either the microwave output of a second

optical clock, or the high- stability output of a cryogenic microwave oscillator (54). Nonetheless, lacking these we can verify the exceptional stability of the comb in the optical

domain and thereby infer the expected stability of fT

. This is done by comparing one

element of the optical comb to the Ca optical standard that operates at 456 THz (657 nm). For this measurement the femtosecond comb

is phase-locked to the 199Hg+

standard as described above. For short averaging times,

the femtosecond comb is effectively controlled by the stable Fabry-Perot cavity of the

Femtosecond Laser +

Microstructure Fiber

1

Optical Standard (f Hg )

| -»®

f, PLL 2

fb =(Xfr

Clock Output

f

r = f Hg-Hm + a + P)

f,-100

Fig. 2. Schematic of the self-referenced all-optical atomic clock. Solid lines represent optical beams,

and dashed lines represent electrical paths. Photodiodes are designated by PD. The femtosecond

laser, having repetition rate fr

, combined with the spectral broadening microstructure fiber produces an octave-spanning comb of frequencies in the visible/near infrared, represented by the

array of vertical lines in the center of the figure. As shown above this comb, the low-frequency

portion of the comb is frequency-doubled and heterodyned against the high-frequency portion in PD 1, yielding the offset frequency f that is common to all modes of the comb. Additionally, an

individual element of the comb is heterodyned with the optical standard laser oscillator (/H = 532

THz) that is locked to the clock transition frequency of a single 199 Hg+

ion. When detected on PD

2, this yields the beat frequency fb

. Two phase-locked loops (PLL) control^ and fb with the result that the spacing (fr

) of the frequency comb is phase-locked to the Hg+

optical standard. Thus, /r is the countable microwave output of the clock, which is readily detected by illuminating PD 3 with

the broadband spectrum from the frequency comb. See the text for further details.

Fig. 3. Measured stability of the heterodyne

signal between one el- ement of the femtosecond comb and the Ca optical standard at 456-THz (657 nm). The

femtosecond comb is phase-locked to the 532-THz laser oscillator. The black triangles are the stability data without cancellation of the additive fiber noise, which is represented by

the dashed line. The red squares are the measured stability with active cancellation of the

fiber noise and improved stability in the Ca standard. These re- sults are about an order

of magnitude better

10

<

10

3

6

-s*^^^

4 ^"^^^

2

14

o

- c_ „ __. <_ ' -J .' 2.

' ... . ..A .-Zl^s^. i

! - ~~

... -t i i—r—t ; i fi -i— i

i

; FT^ 6- f. . L _J 1 _i. IJ § M . - ' ...1— L ' 1 ' u

, . fc_ _i. ..L L..J.-LU i ....- .! . ..X. .1.. . LL.L.L 4_ l j A

i A z : rr ,;-:

2_ Tit ! f

1

: S : L

: %

15 MHn 1 .J !

1 8-

6- J 1

j

i 1

i i 4

1 1 1 1 1 1 1 1

1 1 1 2 3 4 5 6 2 3

1 1 1 1 1 1 4 5 61 2 10 100

Averaging Time (s)

than the best stability reported with a Cs microwave standard, which is designated by the solid line (24).

www.sciencemag.org SCIENCE VOL 293 3 AUGUST 2001 827

TN-64
532-THz laser oscillator, and for t > 30 s the

control shifts to the 199Hg+

ion. We detect,

filter, and count the heterodyne beat signal

between a single element of the comb at 456

THz and a frequency-stabilized diode laser locked to the 1 SQ

<-» 3P: intercombination

transition of a laser-cooled ensemble of Ca

atoms (43, 55). The Borde-Ramsey technique

(56) is used for locking the diode laser to the 456-THz clock transition with resolutions ranging from 0.96 to 11.55 kHz, which are

integer subharmonics of the recoil splitting. When the Ca standard is operated with 0.96- kHz resolution, the Allan deviation of the heterodyne signal between the Hg+

-stabi- lized comb and the Ca standard is shown as the triangles in Fig. 3. For t < 10 s, the Allan

deviation averages down roughly as 9 X

10-15 t~ 1/2

, which is consistent with the expected instability of the Ca standard in its present configuration. We also observe a monotonic degradation in the 1-s instability of the heterodyne beat frequency as the sta- bility of the Ca standard is degraded by using lower resolution Ramsey fringes. However,

for t > 10 s, fluctuations introduced by the 180-m-long optical fiber that transmits the 532-THz light to the femtosecond system

begin to pose a limitation. We have measured

the fiber-induced noise by double-passing the

light through the optical fiber, and the aver- age fractional frequency fluctuations are in- dicated by the dashed line in Fig. 3. Further- more, for t > 30 s, the instability of the Hg+

standard is anticipated to contribute to the measurement at approximately the same level as the Ca standard. Nonetheless, the measured stability improves with averaging to -1.5 X 10" 15 at 100 s. More recently, we have implemented active cancellation (44, 57) of this fiber noise and have further improved the signal-to-noise ratio in the Ca spectroscopy. Data taken under these conditions reveal a fractional frequency instability of 7 X 10~ 15 at t = 1 s. These results are plotted as the square data points in Fig. 3. In this case, we cannot place great significance in the

stability for t > 1 s for two reasons. First, the Allan deviation for averaging times t > 1 s is calculated from the juxtaposition of 1-s averages. Such data analysis is known to result in biases for certain noise processes (58). Second,

although not generally the case, for this specific data the 532-THz laser oscillator was not locked to the 199Hg+

ion, and therefore it was

necessary to subtract out the smooth and predictable drift (~ 1 Hz/s) of the Fabry-Perot cavity to which this laser is stabilized. However,

neither of these affect the measured 1-s Allan

deviation, which provides an upper limit for the short-term (1-s) instability of 7 X 10~ 15 for the optical comb. Again, this 1-s instability is consistent with that of the Ca standard in its present configuration. Similar stability in the ~ 1 -GHz clock output remains to be verified.

Conclusion. We have constructed an optical clock based on the 1.064-PHz (282 nm)

electric-quadrupole transition in a laser- cooled, single 199Hg+

ion. The optical fre- quency is phase-coherently divided down to provide a coherent microwave output through

the use of a mode-locked femtosecond laser and a microstructured optical fiber. The

short-term (1-s) instability of the optical out- put of the clock is measured against an inde- pendent optical standard to be <7 X 10~ 15

. This optically referenced femtosecond comb

provides a countable output at 1 GHz, which

should ultimately be usable as a higher accu- racy reference for time scales, synthesis of

frequencies from the radio frequency to the UV, comparison to other atomic standards, and tests of fundamental properties of nature.

References and Notes

1. P. Laurent et a/., in Laser Spectroscopy XIV, R. Blatt, J. Eschner, D. Leibfried, F. Schmidt-Kaler, Eds. (World

Scientific, Singapore, 1999), pp. 41-50.

2. S. R. Jefferts, D. M. Meekhof, J. H. Shirley, M.

Stepanovic, T. E. Parker, in Proceedings of the 2000

IEEE International Frequency Control Symposium,

Kansas City, 7 to 9 June 2000 (IEEE, Piscataway, NJ, 2000), pp. 714-717.

3. S. Weyers, R. Schroder, A. Bauch, in Proceedings of the 15th European Frequency and Time Forum, Neuchatel, Switzerland, 6 to 8 March 2001 (Fondation Suisse pour la Recherche en Microtechnique, Neuchatel,

Switzerland, in press).

4. S. Weyers, U. Hiibner, R. Schroder, Chr. Tamm, A. Bauch, Metrologia 38, 4 (2001).

5. Th. Udem, J. Reichert, R. Holzwarth, T. W. Hansch,

Phys. Rev. Lett. 82, 3568 (1999).

6. J. Reichert et al., Phys. Rev. Lett. 84, 3232 (2000).

7. D. J. Jones et al., Science 288, 635 (2000).

8. S. A. Diddams et al., Phys. Rev. Lett. 84, 5102 (2000).

9. R. Holzwarth et al., Phys. Rev. Lett. 85, 2264 (2000).

10. J. Stenger et al., Phys. Rev. A 63, 021802R (2001).

11. A. A. Madej, J. E. Bernard, in Frequency Measurement and Control, A. N. Luiten, Ed. (Springer-Verlag, Berlin, 2001), pp. 153-194.

12. See, for example, J. L. Hall, Colloques Internationaux du C.N.R.S. No. 217—Methodes de Spectroscopie Sans Largeur Doppler de Niveaux Excites de Systemes

Moleculaires (CNRS, Paris, 1974), pp. 105-125.

13. T. W. Hansch, A. L Schawlow, C. W. Series, Sci. Am.

240, 72 (March 1979).

14. See, for example, the contributions of F. Biraben et al.

(pp. 17-41) and Th. Udem et al. (pp. 125-144), in The Hydrogen Atom: Precision Physics of Simple Atomic Sys- tems, S. V. Karshenboim, F. S. Pavone, F. Bassani, M.

Inguscio, T. W. Hansch, Eds. (Springer-Verlag Berlin, 2001).

1 5. S. V. Karshenboim, Can. J. Phys. 78, 639 (2000).

16. A. Brillet, J. L Hall, Phys. Rev. Lett. 42, 549 (1979).

17. M. P. Haugen, C. M. Will, Phys. Today 40, 69 (1987).

18. D. Hits, J. L Hall, Phys. Rev. Lett. 64, 1697 (1990).

19. V. A. Kostelecky, Ed. CRT and Lorentz Symmetry

(World Scientific, Singapore, 1999).

20. D. Bear, R. E. Stoner, R. L Walsworth, V. A. Kostelecky,

C. D. Lane, Phys. Rev. Lett. 85, 5038 (2000).

21. See, for example, C. O. Weiss, G. Kramer, B. Lipphardt, H. Schnatz, in Frequency Measurement and Control,

A. N. Luiten, Ed. (Springer-Verlag, Berlin, 2001), pp. 215-247.

22. J. A. Barnes et al., IEEE Trans. Instrum. Meas. 20, 204

(1971).

23. W. M. Itano et al., Phys. Rev. A 47, 3554 (1993).

24. G. Santarelli et al., Phys. Rev. Lett. 82, 4619 (1999).

25. T. W. Hansch, A. L Schawlow, Opt. Commun. 13, 68

(1975).

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

27. D. J. Wineland, W. M. Itano, Phys. Today 40, 34 (1987).

28. C. Cohen-Tannoudji, W. D. Phillips, Phys. Today 43, 33 (1990).

29. J. L Hall, Science 202, 147 (1978).

30. Ch. Salomon, D. Hils, J. L Hall,/ Opt. Soc. Am. B, 8, 1576 (1988).

31. S. N. Bagaev, Yu. D. Kolomnikov, V. N. Lisitsyn, V. P. Chebotayev, IEEE J. Quant. Electr. 4, 868 (1968).

32. B. C. Young, F. C. Cruz, W. M. Itano, J. C. Bergquist,

Phys. Rev. Lett. 82, 3799 (1999).

33. J. N. Eckstein, A. I. Ferguson, T. W. Hansch, Phys. Rev.

Lett. 40, 847 (1978).

34. V. P. Chebotayev et al., Appl. Phys. B 29, 63 (1982).

35. C. O. Weiss, G. Kramer, B. Lipphardt, E. Garcia, IEEE J. Quant. Electr. 24, 1970 (1988).

36. J. Ye, J. L Hall, S. A. Diddams, Opt. Lett. 25, 1675 (2000).

37. F. Diedrich, J. C. Bergquist, W. M. Itano, D. J. Wineland, Phys. Rev. Lett. 62, 403 (1989).

38. T. Kurosu, F. Shimizu, Jpn. J. Appl. Phys. 29, L2127

(1990).

39. Th. Kisters, K. Zeiske, F. Riehle, J. Helmcke, Appl. Phys. B 59, 89 (1994).

40. F. Ruschewitz et al., Phys. Rev. Lett. 80, 3173 (1998).

41. H. Katori, T. Ido, Y. Isoya, M. Kuwata-Gonokami, Phys. Rev. Lett. 82, 1116 (1999).

42. K. R. Vogel, T. P. Dinneen, A. Gallagher, J. L Hall, in Proceedings of the Joint Meeting of the European

Frequency and Time Forum and the IEEE International Frequency Control Symposium, Besancon, France, 13

to 16 April 1999 (IEEE, Piscataway, NJ, 1999), pp. 692-695.

43. C. W. Oates, F. Bondu, R. W. Fox, L Hollberg,/ Phys. D 7, 449 (1999). 44. B. C. Young et al., in Laser Spectroscopy XIV, R. Blatt,

J. Eschner, D. Leibfried, F. Schmidt-Kaler, Eds. (World

Scientific, Singapore, 1999), pp. 61-70.

45. R. Rafac, et al., Phys. Rev. Lett. 85, 2462 (2000).

46. The coherent relation between the pulse envelope and

the 532-THz CW laser oscillator is established by first phase-locking the rate f at which the pulse carrier advances with respect to the envelope from pulse to

pulse. For illustrative purposes in Fig. 1, we have chosen

this rate to be f = /r /10, such that every 10th pulse is identicaL The pulse envelope is then made phase co- herent with the CW laser oscillator by detecting and

controlling the heterodyne beat between the CW laser and the carrier field of the pulsed laser. The frequency domain description is given in the text. 47. A. Bartels, T. Dekorsy, H. Kurz, Opt. Lett. 24, 996 (1999).

48. A. I. Ferguson, J. N. Eckstein, T. W. Hansch, Appl. Phys. 18, 257 (1979).

49. j. Reichert, R. Holzwarth, Th. Udem, T. W. Hansch,

Opt. Commun. 172, 59 (1999).

50. J. K. Ranka, R. S. Windeler, A. J. Stentz, Opt. Lett. 25, 25 (2000).

51. W. J. Wadsworth et al., Electron. Lett. 36, 53 (2000).

52. We count the phase-locked beats f and fb with a highresolution counter and verify that they fluctuate about

their respective phase-locked values by less than 100 mHz in 1 s. This implies that the stability of the 532-THz

laser oscillator is transferred to the femtosecond comb

with a relative uncertainty <2 X 10" 16 in 1 s. 53. S. Stein et al., in Proceedings of the 1982 IEEE International Frequency Control Symposium, Philadelphia, PA, 2

to 4 June 1982 (IEEE, Piscataway, NJ, 1982), pp. 24-30.

54. S. Chang, A. G. Mann, A. N. Luiten, Electron. Lett. 36, 480 (2000).

55. C. W. Oates, E. A. Curtis, L. Hollberg, Opt. Lett. 25, 1603 (2000).

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(1994).

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59. We are grateful to A. Bartels (GigaOptics GmbH) for assistance with the femtosecond laser. We are also indebted to R. Windeler (Lucent Technologies) for providing the microstructure optical fiber. We ac- knowledge many illuminating discussions with J. Hall,

S. Cundiff, j. Ye, S. Karshenboim, and F. Walls. Contribution of the National Institute of Standards and Technology—not subject to U.S. copyright.

29 March 2001; accepted 11 June 2001

Published online 12 July 2001; 10.1126/science.1061171

Include this information when citing this paper.

828 3 AUGUST 2001 VOL 293 SCIENCE www.sciencemag.org

TN-65
ALL OPTICAL ATOMIC CLOCKS BASED ON

A SINGLE MERCURY ION AND CALCIUM ATOMS

RE. Drullingcr, Th. Udcm*. SA. Diddams, K.R. Vogcl" C.W. Oates, EA. Curtis, WX>. LecT

, W.M. Itano, L. Hollberg, and J.C. Bergquist

National Institute of Standards and Technology, Boulder Colorado, USA

•Max-Planck-Institut fur Quantenoptik, Hans-Kopfcrmann-Str.l, Garching Germany

••Precision Photonics, Boulder, Colorado, USA

t Research Electro Optics, Boulder Colorado, USA

ABSTRACT

The frequency comb created by a femtosecond mode-locked laser and a microstructure fiber is used to phase-cohcrcnth/ measure the

frequencies of both the Hg* and Ca optical frequency standards with respect to the SI second as realized at NIST. We find the

transition frequencies to befa - 1 064 721 609 899 143(10) Hz

and/o. = 455 986 240 494 1 58(26) Hz, respectively. This work

begins to reveal the high stability and accuracy potential of

optical atomic clocks based on the Hg* and Ca standards. Furthermore, when combined with previous measurements, we

find no time variations of these atomic frequencies within the

uncertainties of W^W/h} <2x 10

' M

Y1"1 and K3&.WJGJ

<

8 x 10-14 yr

1

. Keywords: Optical, Atomic Clock, femtosecond, comb

INTRODUCTION

Optical frequency standards based on laser-cooled atoms and ions promise superior stability and accuracy over existing microwave

standards [1-5]. However, because of their high frequencies (- 10 15 Hz or PHz), it has proven difficult to count cycles as

required for building functioning clocks. Only recently, a reliable and convenient clockwork fast enough to count optical

oscillations has been realized [6-9]. Here, we report on work

toward an all-optical- clock based on a femtosecond laser that phase-coherentry divides down the frequency of the visible

radiation from either a Hg* or Ca optical-frequency standard to a countable radio frequency. We have measured the absolute frequencies of these optical transitions in terms of the SI second

as realized at NET [10]. Indeed, for the Hg* standard, the

uncertainty in the measurement is essentially limited by our knowledge of the SI second at -2 * 10"". Additionally, the comparison ofatomic frequencies over time provides constraints on the possible time variation of fundamental constants. We now

have measurements ofthe absolute Hg* frequency taken over a

six month interval that differ by less than their statistical uncertainty and much less than their systematic uncertainty.

THE OPTICAL STANDARDS

TheHg* and Ca systems have recently been described elsewhere

[1,4, 11-13], so we summarize only their basic features. The

mercury optical frequency standard is based on a single, laser- cooled lwHg+ ion that is stored in a cryogenic, radio-frequency,

spherical Paul trap. The ion is cooled and detected by driving the

*s\n • *Pw cycling transition at 194 nm. The 25w (F =0, MF - 0)

- ^yj (F =2 ,

M

F

*= 0) electric-quadrupole transition at 282 nm

[Fig.l] provides the reference for the optical standard [1]. We

lock the frequency-doubled output of a 563 nm dye laser with

sub-hertz linewidth[12] to the quadrupole resonance. Transitions

to the mctastable ^^ state are detected with near unit efficiency since the absorption of a single 282 nm photon suppresses the

scattering of many 194 nm photons on the strongly allowed 2Sia

-

iPia transition [14, 15 ]. In figure 2 we show an example of a

normalized spectrum that was obtained from multiple,

bidirectional scans through the resonance, where the probe time was 20 ms. Most often, the frequency was locked to resonance with a 10 ms interrogation period, which yielded a fractional frequency stability of 2 xlO" x

' m for an averaging time t measured in seconds [16].

The calcium standard starts with a collection of-107 laser-cooled

46 Ca atoms held in a magneto-optic trap. The 423 nm1 5, - P{ transition is used for trapping and Dopplcr-cooling the atoms to a residual temperature of-2 mK. The lSt (M, - 0) - IP, (A<, - 0)

weakly allowed electric-dipole transition (400 Hz natural linewidth) at 657 nm is the "clock" transition for this frequency

standard [Fig. 3]. We excite the transition with a four-pulse

Bordc-Ramsey sequence (pulse duration =1.5 us) with light from

a continuous-wave (CW), frequency-stabilized diode laser. Using a shelving detection technique similar to that employed in the Hg* system, near-resonant 423 nm pulses (5 us duration) are used before and after the 657 nm excitation to determine the

fraction of atoms transferred from the ground state. Figure 4

shows the Bordc-Ramsey fringes taken at a resolution of 960 Hz.

This system has demonstrated a fractional frequency stability of

4 x 10'1S rw when probing sub-kilohcrtz linewidths[4]. For the measurements presented here the Ca spectrometer was operated with linewidths ranging from 0.96 to 11.55 kHz, which are

integer submultiplcs of the recoil splitting.

THE OPTICAL COMB

The recent introduction of mode-locked lasers to optical frequency metrology greatly simplifies the task of optical- frequency measurements [6 -8, 17 -19 ]. The spectrum emitted by a mode-locked laser consists of a comb of regularly spaced

continuous waves that are separated by the pulse repetition rate_£. The frequency ofthe n" mode of the comb is given by/, = nfr + ft [20, 21], where fj is the frequency offset common to all modes.

This offset is caused by the difference between the group- and the

phase-velocity inside the laser cavity. fr can be measured by direct detection of the laser's output with a photodiode. f, is measured

by heterodyning the 2

nd harmonic of mode f. = nfr +f, from the

infrared wing ofthe comb with mode.4, = 2nfr +f, from the blue

side ofthe comb [7, 8]. This self-referenced technique requires

that the optical comb span at least an octave in frequency space. While an octave-spanning comb can be produced directly from

a mode-locked User [22], launching the longer pulses from a

TN-66 441
commercially-available femtosecond laser into an air-silica microstructure fiber [23, 24] also produces a frequency comb mat

spans an octave. Nonlinear processes in the fiber produce the

additional equally spaced and phase-coherent modes to the transmitted light It has been demonstrated that this process of

spectral broadening preserves the uniformity of spacing and

spectral fidelity of the comb to at least a few parts in 10 1 *

[8]. We couple approximately 200 mWaverage power from

a femtosecond, Titsapphire ring laser (fr = 1 Ghz) through a 15 cm piece of microstructure fiber that has a 1.7 um core and

group-velocity dispersion that vanishes near 770 nm [23]. This power density is sufficient to increase the spectral width of the

laser from 13 THz to more man 300 THz, spanning from -520 nm to -1170 nm. The infiared part ofthe comb from the fiber (k - 1060 nm) is split off by a dichroic mirror and frequency- doubled into the green portion of the visible spectrum by means

of a KNb03 crystal 2 mm long. Following an adjustable delay

line that matches the optical path lengths, the frequency-doubled

light is spatially combined with the green part of the original comb

to produce a beat note at/. We phase-lock both/, and/, to synthesized frequencies derived from a cavity-tuned hydrogen

maser that acts as the transfer standard to the NIST realization of

the SI second [10]. Control of/ is achieved with a piezo transducer driving a cavity mirror, while / is controlled by

adjusting the 532 nm pump beam' s intensity with an electro-optic modulator [8]. When both/ and/ are phase-locked, the frequency of every mode in the comb is known with the same

accuracy as that ofthe reference maser.

The CWlight from the Hg* (563 nm) and Ca (657 nm) standards

is transferred to the mode-locked laser system via two single- mode optical fibers that are 130 m and 10 m long, respectively. Approximately 2 mW of CW light from each fiber is modematched with the appropriate spectral region of the frequency

comb to generate a beat signal fj, with a nearby mode. This beat note is amplified and measured with a counter. The optical frequency is then expressed as_^ = mfr +/„ +fit where m is a

large integer uniquely determined for each system from previous

coarse measurements of f^.

RESULTS

Figure 5 summarizes the frequency measurements ofHg+ made

between August 2000 and February 2001, while figure 6 summarizes the Ca measurements made from October 26 to November 17, 2000. The frequency of the maser is calibrated by

comparing to the local NIST time scale (5 hydrogen mascrs and

3 commercial cesium clocks), which in turn is calibrated by the

local cesium fountain standard (NIST-F1 [10]). The fractional

uncertainty in the frequency ofthe reference maser relative to the SI second was about 1.8 * 10'15 for these measurements.

The weighted mean of our measurements of the Hg+ clock

transition is J^ =1 064 721 609 899 143 Hz. The statistical uncertainty ( = 4 * 10"15) of our limited duration measurements is essentially the result ofthe reference-maser's short-term stability (—2 x 10"! V1/2

). Because we have not made a full experimental

evaluation of the Hg+ standard, we assign a very conservative value of 10 Hz for the total systematic uncertainty. The

dominant contribution to the uncertainty of the S-D transition frequency is the electric-quadrupole shift of the *D5n state arising

from coupling with the static potentials of the trap. In our

spherical Paul trap, where the confinement of the ion uses no

static applied fields, the maximum quadrupole shift should be less than 1 Hz (or fractional frequency shift <10'15) [25]. In principle,

it is possible to eliminate the quadrupole shift by averaging the S

-D transition frequencies for three mutually orthogonal

orientations of quantizing magnetic field of constant magnitude.

In the present experiment, we have measured the S-D frequency

for various field values, but we have made no attempt to eliminate

the quadrupole shift by using three orthogonal fields of constant magnitude. No shift of the resonance frequency is observed

within the precision of these measurements even under strongly

varying conditions of magnetic field. We anticipate that the

uncertainties of all systematic shifts in the Hg* system can be

reduced to values approaching 1 x 10"1* [1, 25].

For the Ca data shown [Fig. 6], an additional correction is applied each day to account for frequency shift caused by residual phase

chirping on the optical Ramsey pulses produced by amplitude

modulating an acoustooptic modulator (AOM). The phase

chirping produced resolution-dependent frequency shifts on the

order of 100 Hz for fringes 1 1 .5 kHz wide but of only 10 Hz for

fringes 0.96 kHz wide. On each day, the Ca frequency was

measured for —30 minutes at each of several fringe resolutions, and the zero-intercept of a linear fit to the data was used as the

corrected frequency. On the last 3 days of measurements, we

were able to reduce this shift by a factor of—3 with improvements

to the RF pulses that drive the AOMs. The statistical uncertainty

(typically 8 Hz) for each day's measurement is smaller than the

uncontrolled systematic uncertainties in the Ca frequency. The

largest systematic uncertainty stems from incomplete knowledge

ofthe angular overlap ofthe counter-propagating beams in the Ca

spectrometer, combined with transverse drift velocity of the cold Ca ensemble. This leads to residual first-order Doppler shift with

magnitude < 15 Hz (except on November 16, where a large drift velocity led to an uncertainty of -52 Hz). Other significant uncertainties include our lack of knowledge or control of

electronic offsets and baseline symmetries (< 12 Hz), wavefront

curvature (< 10 Hz) , and cold-atom collisional shifts (< 10 Hz).

Taking all known systematic uncertainties in quadrature gives a

confidence level of -26 Hz for the measured mean values

indicated by the dashed lines in figure 6.

Figure 6 also shows the good agreement between our measurement and the most recent value measured with a harmonic frequency chain [26], which provides a degree of

confidence in the reproducibility of the Ca standards. It is also in reasonable agreement with the very recent PTB measurements made with a femtosecond comb [27]. An additional measure of

the Ca frequency can be made by using the present absolute measurement ofHg* and our earlier measurement ofthe 76 374

564 455 429(40) Hz gap between/^, and the Ca standard [28]. This yields a value of/& =455 986 240 494 143(40) Hz in good

agreement with the value from the present direct measurement

Finally, these results also provide data on the relative time

variability of atomic frequencies. S. Karshenboim has recently

reviewed the implications of such comparisons and their

TN-67
contribution toward constraining the possible time variation of

fundamental constants [29]. In this regard Hg* and Ca are two

of the most interesting cases to study. Comparing our present measurement ofJc, to measurements made by PTB in 1997 [26]

gives (dfa/dtyfe. < 8 * 10'14 yfl

. Similarly, our August 2000 to February 2001 measurements anf^ provides an initial baseline constraint on the time variation of (dfj^fdi)//^ s 2 * 10"'5 yr"

1

. Here we use the defined unit of time based on the frequency of

the Cs hyperfine interval and assume that any time dependence is slow and dominantiy linear over the relevant time scale. We

believe this represents the tightest laboratory test yet of the time

variability ofthese disparate transitions. ACKNOWLEDGMENTS

We acknowledge the support of the Office of Naval Research. This work was also supported through a Cooperative Research and Development Agreement with Timing Solutions, Corp., Boulder, Colorado. This work is a contribution of the National

Institute of Standards and Technology, and is therefore not

subject to United States copyright

REFERENCES

[1] RRafac,B.C. Young, LA. Beall, WJvt Itano, D.J. Wineland

and J.C. Bergquist Phys. Rev. Lett 85, 2462 (2000).

[2] JEBemard, A A. Madej, L. Marmct B. G. Whitford, K. J. Siemscn, and S. Cundy , Phys. Rev. Lett 82, 3228 (1999).

[3] RSchnatz, B. Lipphardt J. Helmcke, F. Richie, and G. Zinner.Phys. Rev. Lett 76, 18 (1996).

[4] C.W.Oates, E. A. Curtis, L. Hollberg, Opt Lett 21 1603

(2000).

[5] J.von Zanthicr, Th. Becker, M Eichenseer, A Yu. Nevsky, Ch. Schwedes, E. Peik, R Walther, R Holzwarth, J. Reichcrt, Th. Udem, T. W. Hansen, P. V. Pokasov, M. N. Skvortsov, S. N.

Bagayev, Opt Lett 25, 1729 (2000).

[6] SADiddams, David J. Jones, Jun Ye, Steven T. Cundiff, and JohnL. Hall, Phys. Rev. Lett 84, 5102 (2000).

[7] DJJoncs, SA. Diddams, JJL RankaA. Stentz, R.S. Windelcr IX. Hall and ST. Cundiff, Science, 288, 635 (2000).

[8] RFfcfewarth, Th. Udem, and T. W. Hansen, Phys. Rev. Lett

85, 2264 (2000).

[9] SA. Diddams, Th.Udem, KJL Vogel, C.W. Oatcs, EA.

Curtis, RS. Windeier, A Battels, J.C. Bergquist and L. Hollberg, Proc SPIE Conf. on Laser Frequency Stabilization, Standards, Measurement, and Applications, San Jose, CA January 2001.

[10] SRJcfferts, et al., submitted to Metrologia; S.RJeffcrts, et al, TRFF. International Frcq.Control Symp., 714 (2000).

[11] C.W.Oates, F. Bondu, RW. Fox and L. Hollberg, JPhysD,

1449(1999).

[12] B.Young, F. C. Cruz, W. M Itano, and J. C. Bergquist

Phys. Rev. Lett 82, 3799 (1999).

[13] J.C. Bergquist, RJ. Rafac, B.C. Young, JA. Beall, W.M

Itano, and DJ. Wineland, Proc SPIE Conf. on Laser Frequency

Stabilization, Standards, Measurement and Applications, San

Jose, CA January 2001.

[14] RDehmelt Bull. Am. Phys. Soc. 20, 60 (1975).

[15] J.CBcrgquist Wayne M. Itano and D. J. Wineland, Phys.

Rev. A 36, 428 (1987).

[16] DJ.Wineland, et al., The Hydrogen Atom ed. by

T.Wfiansch (Springer,Beriin,Heidelbcrg 1989) pp.123-133.

[17] Th.Udem, J. Reichcrt, R Holzwarth, and T. W. Hansen,

Phys. Rev. Lett 82, 3568 (1999).

[18] JReichcr, MNiering, R Holzwarth, M Weitz, Th. Udem,

and T. W. Hansen, Phys. Rev. Lett J4, 3232 (2000).

[19] MNiering, R Holzwarth, J. Reichcrt, P. Pokasov, Th. Udem, MWeitz, and T. W. Hansen , Phys. Rev. Lett 84, 5496

(2000).

[20] AIFcrguson, J.N. Echstein, T. W. Hansen, Appl. Phys. UL

257 (1979).

[21] JReichcr, R Holzwarth, Th. Udem and T. W. Hansen, Opt

Comm. 172, 59 (1999).

[22] F.XJCartner, et al., OptLctt (in press). [23] JXRank, R S. Windelcr, A J. Stentz , Opt Lett 21 25

(2000).

[24] WJ.Wadsworth, J.C. Knight A Ortigosa-Blanch, J. Arriaga,

E. Savestre, P.StJ. RusselL ElectronLettJ6, 53 (2000).

[25] WJvLItano, J-ResJNIST (in press).

[26] FRichk, H Schanatz, BLipphardt G Zinncr, T. Trebst T. Binnewies, G. Wilpcrs, J. Helmcke, Proceedings ofEFTF-IEEE

IFCS, 700 (1999).

[27] J. Stenger.T. Binnewies, G. Wilpers, F. Riehle, H.R. Telle, J.K. Ranka, RS. Wmdeler, and AJ. Stentz, Phys Rev A 63

021802(2001).

[28] K.RVogel, S. A Diddams, C. W. Oates, E. A Curtis, R J. Rafac, W. M Itano, J. C. Bergquist R W. Fox, W. D. Lee, J. S. Wells, L. Hollberg , Opt Lett, 26, 102 (2001).

[29] S. Karshenboim, CanJRhys. 78, 639 (2000).

TN-68 443
2 D F'l * F.o A ^6-9GH2 tfa ) s90ms

194

282 nm

1/2 \ / t40.5 GH2

F ^-^ *

Fig. 1: Partial level scheme for lwHg\ The 194 nm radiation is used for Doppler cooling, state preparation and detection. The

282 nm transition from the ground state *Sm (F -0, Mp -0) to the mctastable J£>w (F -2 , Mf -O) state is the "clock"

transition.

JO

C9o

-60 -30 30

Detuning [Hz]

Fig. 2: A typical spectrum of the 282 nm transition . Here, the

excitation pulse length was 20 ms and the measured linewidth

is Fourier transform limited at about 20 Hz.

22.4

1 ^i

423 nm

\ 65: y

3p<

657

T

nm

1 5„ -2-1012

cavity detuning [kHz]

Fig. 3: Simplified diagram of the relevant energy levels in the Ca standard.

Fig. 4: Optical Borde-Ramsey fringes with a 960 Hz (FWHM)

resolution. The total averaging time to generate this figure was

20 s.

TN-69
30

20

10

N

I

COoi

^r

o>

o>

CD

O)o

CO

CN

h>. TTCDo -20 -\

•10-

-30 -

1

51800 51850

MJD

1^

51900 51950

NI

co

§5o

CN

COCDo>m

Oct 26 Nov 5 Nov 15

Fig. 5: A chronological record of the average daily frequency of

the '**Hg* clock transition measured in August 2000 and

February 2001 . The error bars represent the statistical fluctuations. The dashed lines represent our assigned 10 Hz

uncertainty in systematic effects.

Fig. 6: The filled squares are the measured Ca frequencies on

ten days over a 23-day period representing 38 787 s of total measurement time. The inner and outer error bars for each day

represent the statistical and total uncertainty, respccuvely. The

dashed lines show the 26 Hz systematic uncertainty assigned to the mean. The open triangle is the PTB measurement [26] while the open circle is the Ca frequency calculated from the present Hg* result and our previous measurement of the 76 THz gap between Ca and Hg* [28].

TN-70 445
Mode and Transport Studies of Laser-Cooled Ion

Plasmas in a Penning Trap*

T. B. Mitchell, J. J. Bollinger, X.-P. Huang and W. M. Itano

Time and Frequency Division, National Institute of Standards and Technology, Boulder, CO 80303

Abstract. We describe a technique and present results for imaging the modes of a laser-cooled plasma

of 9 Be+ ions in a Penning trap. The modes are excited by sinusoidally time-varying potentials applied to

the trap electrodes, or by static field errors. They are imaged by changes in the ion resonance fluorescence produced by Doppler shifts from the coherent ion velocities of the mode. For the geometry and conditions

of this experiment, the mode frequencies and eigenfunctions have been calculated analytically. A comparison between theory and experiment for some of the azimuthally symmetric modes shows good agreement.

Enhanced radial transport is observed where modes are resonant with static external perturbations, such as those caused by misaligning the trap with respect to the magnetic field. Similarly, the plasma angular mo- mentum can be changed through the deliberate excitation of azimuthally asymmetric modes. The resultant torque can be much greater than that from the "rotating wall" perturbation, which is not mode-resonant.

INTRODUCTION

Non-neutral plasmas consisting exclusively of particles of a single sign of charge have been used to study

many basic processes in plasma physics [1], partly because non-neutral (as opposed to neutral or quasi-neutral)

plasmas can be confined by static electric and magnetic fields and also be in a state of global thermal equilibrium

[2,3]. A particularly simple confinement geometry for non-neutral plasmas is the quadratic Penning trap, which

uses a strong uniform magnetic field Bq = Bqz superimposed on a quadratic electrostatic potential

«'.*)—af^-TJ- (I)

Here m and q are the mass and charge of a trapped ion, and u)z is the axial frequency of a single ion in the

trap. The global thermal equilibrium state for a single charged species in a quadratic Penning trap has been

well studied [3,4]. For sufficiently low temperatures, the plasma takes on the simple shape of a uniform density

spheroid. An interesting result is that all of the electrostatic modes of a magnetized, uniform density spheroidal

plasma can be calculated analytically [5,6]. This is the only finite length geometry for which exact plasma

mode frequencies and eigenfunctions have been calculated for a realistic thermal equilibrium state. In this manuscript we describe a technique for measuring these frequencies and eigenfunctions, and compare

theory predictions and experimental results for some of the magnetized plasma modes. We also discuss several potential applications for the modes in Penning trap experiments. In general, the mode frequencies depend on

the density and shape of the plasma spheroid. Therefore measurement of a mode frequency provides a nondestructive method for obtaining basic diagnostic information about the plasma. This is especially important

in anti-matter plasmas [7,8], where conventional techniques for obtaining information about these plasmas

involve ejecting the plasma from the trap. Measurement of the damping of the modes can provide information on the plasma's viscosity [9,10]. Other applications arise from the fact that the modes can strongly influence the dynamical behavior of trapped plasmas. For example, certain azimuthally asymmetric modes can have zero frequency in the laboratory frame and be excited by a static field error of the trap. These zero-frequency modes

can strongly limit the achievable density in a Penning trap [11]. Similarly, the plasma angular momentum can

Work of the U.S. Government. Not subject to U.S. copyright.

TN-71
be changed through the deliberate excitation of azimuthally asymmetric modes [12,13], and the applied torque can be much greater than that from the "rotating wall" perturbation [14,15], which is not mode-resonant.

Previous experimental mode studies on spheroidal plasmas have been limited to frequency measurements on a small class of modes. With laser-cooled Be+ ion plasmas, some quadrupole mode frequencies have been

measured and agree well with theory [6,11]. Mode frequencies have also been measured on spheroidal cryogenic

electron plasmas [16], 0.025-0.5 eV electron and positron plasmas [17], and room temperature Ar + ion plasmas

[18]. In these cases qualitative agreement with theory was observed and the modes provided some basic

diagnostic information. However, deviations from the model of a constant density spheroid in a quadratic trap

limited the comparison with the ideal linear theory. Here, in addition to measuring mode frequencies, we also measure the mode eigenfunctions. The eigenfunctions permit direct identification of the modes. In addition,

they contain much more information than the frequencies and therefore may be useful for observing nonlinear

effects such as mode couplings. Mode eigenfunctions have been measured for low frequency, z-independent

(diocotron) modes on cylindrical electron columns [19]. In that work, the mode measurements were important

in identifying two coexisting modes.

EXPERIMENTAL APPARATUS

Figure 1 shows a schematic of the apparatus [20,21] used for the mode measurements. The trap consists of a

127 mm long stack of cylindrical electrodes at room temperature with an inner diameter of 40.6 mm, enclosed

in a 10~ 8 Pa vacuum chamber. A uniform magnetic field Bo = 4.465 T is aligned parallel to the trap axis within

0.01°, and results in a

9 Be+ cyclotron frequency f2 = qBo/m = 2n x 7.608 MHz. The magnetic field is aligned

by minimizing the excitation of zero-frequency modes produced by a tilt of the magnetic field with respect

to the trap electrode symmetry axis [6,11]. Positive ions are confined in this trap by biasing the central ring electrode to a negative voltage — Vq with respect to the endcaps. Because the dimensions of the Be+ plasmas

(< 2 mm) are small compared to the diameter of the trap electrodes, the quadratic potential of Eq. (1) is a good

approximation for the trap potential. For most of the work reported here, Vq was set at 2.00 kV which results in uz — 1-k x 1.13 MHz and a single-particle magnetron frequency u>m = [fl — (fl

2 — 2u>1)^}/2 = 2n x 84.9 kHz.

We create a Be+ plasma by ionizing neutral Be atoms in a separate trap (not shown) and then transferring

axial cooling beam

FIGURE 1. Schematic of the experimental apparatus. Azimuthally symmetric m = modes were excited by applying

in-phase or 180° out-of-phase sinusoidal potentials to the trap endcaps.

TN-72
the ions to the main trap. For the work discussed here, the number of ions was typically 6 x 104

. While

the total charge in the trap is conserved after loading, the relative abundance of contaminant, heavier-mass

ions increases, presumably due to reactions between Be+ ions and background neutral molecules. Because we analyze our experimental results using an existing theory [5] for the electrostatic modes of a single-species

plasma, we took data for mode studies only with relatively clean clouds (<3% impurity ions). The plasmas

were cleaned approximately every 30 minutes by transferring the ions to the load trap where, with a shallow

3 V deep well, contaminant ions were driven out of the trap by exciting their axial frequencies. Cleaning

therefore results in a decrease in the number of trapped ions. Over a 12-14 hour period, the number of ions is reduced by a factor of 2. Because the mode frequencies and eigenfunctions in a quadratic trap are independent

of the number of ions, the mode measurements described here are not affected. The trapped Be4" ions are Doppler cooled by two laser beams at wavelength A « 313.11 nm. The main cooling

beam is directed parallel to Bo as shown in Fig. 1, and a second cooling beam propagating perpendicular to Bo

(not shown and turned off during measurements) is also used to compress the plasma by applying a radiation pressure torque [3,11]. For mode eigenfunction measurements the axial cooling-laser frequency is fixed about

one natural linewidth (~ 20 MHz) below the transition frequency. Ions which, due to excitation of a mode,

have an axial velocity vz < therefore fluoresce more strongly than ions with vz > 0. The ion temperature was not measured; however, based on previous work [3], we expect T < 20 mK.

An //5 imaging system detects the Be+ resonance fluorescence scattered perpendicularly from the axial cooling beam (waist « 0.5 mm, power « 50 /xW) to produce a side-view image of the Be+ ions. The side-view

image is obtained with a photon-counting camera system which records the spatial and temporal coordinates

of the detected photons. This data is processed to obtain the mode eigenfunctions by constructing side-view

images as a function of the phase of the external drive used to excite the modes.

ELECTROSTATIC MODES OF A SPHEROIDAL PLASMA

A constant-density, spheroidal plasma model is a good approximation for our work. In thermal equilibrium,

a Penning trap plasma rotates as a rigid body at frequency ur , where u;m < ur < Q — ujm, about the trap's z

axis [2,4]. In this work the rotation frequency was precisely set by a rotating dipole electric field [14,15]. As the

ions rotate through the magnetic field they experience a Lorentz force which provides the radial confining force of the trap. This u;r -dependent confinement results in an cjr -dependent ion density and plasma shape. At the

low temperatures of this work, the plasma density is uniform over distances large compared to the interparticle

spacing (~ 10 /xm) and is given by no = eomu>

2 /q2 where u/p — [2wr (f2 — ojt ))* is the plasma frequency. With

the confining potential of Eq. (1), the plasma is spheroidal with boundary z

2 /' z^ + x2 /ro + V2 lro = ! The

spheroid aspect ratio a = zq/tq is determined by uir [3,4]. We have neglected the effect of image charges,

because the plasma dimensions are small compared to the trap dimensions.

The modes of these spheroidal plasmas can be classified by integers (I, m), where / > 1 and < m < I [5,6]. For an (I, m) mode with frequency cj; m [22] the perturbed potential of the mode inside the plasma is given by

a symmetric product of Legendre functions,

¥m oc Plri (C1 /d)Pr{C2 )ei{m'

t, -UJ "nt)

. (2)

Here £i and £2 , discussed in Ref. [5] , are scaled spheroidal coordinates where the scaling factor depends on the

frequencies u)T , CI, and u>im, and d is a shape-dependent parameter which also depends on these frequencies. In

general, for a given (l,m) there are many different modes. In this paper we report measurements of the mode

frequencies and eigenfunctions of several magnetized plasma modes, which are defined as those modes with

frequencies \uiim\ < |fi — 2u>r \ [5,6]. For uiT <C fi/2, these modes principally consist of oscillations parallel to the magnetic field at a frequency on the order of uz

. In the experiment we detect the axial velocity of a mode.

In the linear theory, this is proportional to dty lrn /dz.

We excite azimuthally symmetric (m = 0) plasma modes by applying sinusoidally time-varying potentials to the trap electrodes. Even-/ (/, 0) modes are excited by applying in-phase potentials to the endcaps (even

drive), while odd-/ (I, 0) modes are excited by applying 180° out-of-phase potentials to the endcaps (odd drive).

Azimuthally asymmetric (ra 7^ 0) modes can be excited by applying potentials to the compensation electrodes,

which have 6- fold azimuthal symmetry. In Refs. [6,11] quadrupole (/ = 2) mode frequencies were measured by

observing the change in the total ion fluorescence from the plasma, averaged over the phase of the drive, which

occurred when the drive frequency equaled the mode frequency. However, in order to observe such a change,

the mode excitation must be large enough so that either the fluorescence from an ion nonlinearly depends on

TN-73
its velocity or there is some heating of the plasma by the mode. The large amplitude drive required by this technique decreases the precision of the mode measurements. The technique described here entails reducing the drive amplitude until the change in the phase-averaged

ion fluorescence is negligible, and detecting the mode's coherent ion velocities by recording side-view images

as a function of the phase and frequency of the external drive [23]. These Doppler images provide direct measurements of the mode's axial-velocity eigenfunction [24]. In addition, an accurate measurement of the

mode's frequency (both real and imaginary parts) can be obtained from measurements of the mode amplitude

as a function of drive frequency. High order modes have been excited and detected with this technique, such

as the (11,0) and (12,1) modes. Imaging is not required for the (1,0) and (1,1) modes because there is no

spatial variation in their eigenfunction. The driven mode amplitude and phase of these center-of-mass modes

can therefore be obtained by coherently detecting the spatially integrated fluorescence as a function of the

phase of the external drive [25].

EXPERIMENTAL RESULTS

Mode Frequency And Eigenfunction Measurements

In Fig. 2 we plot measured mode frequencies, along with the theoretical predictions, for several azimuthally

symmetric magnetized plasma modes as a function of ujt for ujz /2ir — 1.13 MHz and fl/2ir = 7.608 MHz. Many

different mode frequencies at various values of uz have been measured with the Doppler imaging technique,

and on very clean clouds agreement between the observed and predicted mode frequencies is typically better than 1%. However, as the percentage of impurity ions increases, the shift between the measured frequency and

the value predicted by the single-species theory also increases. Both positive and negative frequency shifts have

been observed. We think that these frequency shifts are caused by changes in the cloud shape which perturb

the spheroidal geometry of the single-species cloud, arising because impurity ions centrifugally separate from

the Be+ [26].

3500 1

'

1

'

1 '

1

'

1 •

1

'

1 r

•-(9,0)

d— (7,0) * (2,0)

*-

(8,0)

500 1000 1500 2000 2500 3000 3500 4000

Rotation Frequency (kHz)

FIGURE 2. Plots of the frequencies of several 771 = magnetized plasma modes as a function of rotation frequency for fi/27r=7.608 MHz and ujz /2ir=1.13 MHz. The solid lines are the theoretical predictions and the symbols are experimental measurements. Only the highest frequency (9,0) plasma mode and the second highest frequency (8,0) plasma mode

are plotted.

TN-74
Amplitude 1 -n Phase n

FIGURE 3. (a) Phase-coherent sideview image data obtained on a plasma with uir /2ir= 1 MHz while driving a

(2,0) mode at W2,o/27r=1.656 MHz. The magnetic field and axial laser beam point up. The ion cloud dimensions are

2zo = 0.76 mm and 2r = 0.24 mm, and the density n = 2.70 x 109 cm

-3

. Comparison of the amplitude (b) and

phase (c) extracted from the (2, 0) mode in (a) with the predictions of linear theory. The theory predictions for (b) and

(c) are on the right. From Ref. [23].

Figure 3 illustrates phase-coherent detection and Doppler imaging of the (2, 0) mode. This is one of the

simplest modes that is not merely a center-of-mass oscillation of the plasma. In this mode the plasma stays

spheroidal but the aspect ratio (and density) oscillate at 0*2,0 For ur < £2/2, the oscillation in r is very small, so the mode principally consists of oscillations in zq at 0*2,0 • Ions above the z = mid plane oscillate 180° out

of phase with ions below z = 0. Figure 3(a) shows one of a sequence of 18 side-view images taken as a function of the phase of the mode

drive at a>2,o/27r = 1.656 MHz. A movie of the entire sequence is included in Ref. [23]. The plasma's rotation

frequency was set to lot /2tx = 1 MHz and the rn = even drive rms amplitude was 7.07 mV. In the images, the

magnetic field and the axial laser beam point up. As expected for the (2,0) mode, the detected fluorescence in the upper half of the plasma is bright when the lower half is dark and vice versa. We analyze the data of Fig.

3(a) by performing a least-squares fit of the intensity at each point to Ao + ^2,0 cos(w2,o£ + ¥>2,o)- Figures 3(b)

and 3(c) show the resultant images of the measured mode amplitude A2,o(x,z) and phase ip2,o(x,z). These

are compared with the theoretically predicted values of these quantities. Because the plasma is optically thin,

the theoretical predictions were obtained by integrating dfy lrn /dz over y. The amplitude of the theoretical prediction is scaled to match the experiment, and both amplitudes are normalized to 1. From the fitted values of ^2,0 and Ao we can estimate the coherent-ion mode velocities if the dependence of the ion fluorescence on velocity (through Doppler shifts) is known. For the low temperatures of this experiment

a good approximation is to assume a Lorentzian profile with a full width at half maximum of 19 MHz due to the natural linewidth of the optical cooling transition. With the 20 MHz detuning used in this measurement, we estimate for the data of Fig. 3 that the maximum coherent mode velocity, which occurs at z = ±zq, is ~1.5 m/s. The spatial and density changes in the plasma spheroid for this excitation are too small to be

resolved (Az/z , An/n < 10 -3

). Therefore the observed variation in the fluorescence intensity is entirely due

to Doppler shifts induced by the coherent ion velocities of the mode.

We have measured the mode eigenfunctions of a number of different azimuthally symmetric modes including

the Z=2,3,4,5,7, and 9 modes. Like the data of Fig. 3, good agreement with the predicted eigenfunction

amplitude and phase distribution is obtained in the limit of low laser power and drive amplitude. Surprisingly

high-order odd modes could be excited with the odd drive on the trap endcaps. Figure 4(a) shows one

TN-75
Amplitude

FIGURE 4. (a) Phase-coherent sideview image data obtained on the plasma of Fig. 3 with uj r /2n— 1.00 MHz while

driving a (9,0) mode at a;9,o/27r=2.952 MHz. Comparison of the amplitude (b) and phase (c) extracted from the (9, 0) mode in (a) with the predictions of linear theory. The theory predictions are on the right. From Ref. [23].

of a sequence of 18 sideview images obtained with the highest frequency (9, 0) mode excited by a drive at

ujg

t o/2n — 2.952 MHz. For a given (/,0), the highest frequency magnetized plasma mode does not have any

radial nodes. Figures 4(b) and 4(c) show the fitted amplitude and phase from the sequence, along with

the predictions from theory. Similar high-order even (1, 0) modes are more difficult to excite. The mode

eigenfunctions of some of the azimuthally asymmetric (m—1 and m—2) modes, such as the (1,1), (2,1), (3,1)

(4,1), (6,1), (8,1) and (3,2) modes, have also been imaged. In general, the qualitative agreement with the

predictions of theory is good.

Figure 5 shows images from a plasma with u)r /2ir = 638 kHz driven by an even drive at 1.619 MHz. This case demonstrates the utility of the Doppler imaging diagnostic. These data were initially taken during a survey of the (2, 0) mode eigenfunction as a function of the plasma's rotation frequency. Analysis of the phase-coherent

data revealed additional, higher-order structure. An examination of the predictions for the mode frequencies revealed that at this particular rotation frequency, as shown in Fig. 2, both the (2, 0) mode and an (8, 0) mode

with a radial node have similar frequencies. Characteristics of both modes are seen in the data. However,

subsequent measurements of the (2, 0) mode frequency near this crossing indicated that any frequency shifts due to a nonlinear coupling with the (8, 0) mode are less than a few kilohertz. The (2, 0) mode driven in Fig. 3 occurs near a crossing with a (9, 0) mode (see Fig. 2). In this case no evidence for an excitation of a (9, 0)

is observed, presumably because it is an odd mode which does not couple to even drives, and because there is little or no mode coupling between the (2,0) and (9,0).

Doppler imaging also provides a technique for measuring the damping of plasma modes. This is done by

sweeping the frequency iupert of the sinusoidally time-varying perturbation through a mode frequency, while

measuring the mode's resultant amplitude and phase. If the perturbation amplitude is kept low to avoid large

amplitude effects the system can be modeled as a damped harmonic oscillator driven by a periodic external

force, which has a characteristic lineshape for its amplitude response and a phase difference of it above and

below resonance [27]

.

Figure 6 shows a measurement of the (2, 0) mode amplitude and phase response. The axial laser intensity was reduced in an attempt to make mode damping from viscous dissipation dominant over that from laser cooling, and the z > upper half of the plasma was blocked off to permit phase-coherent detection with- out spatial discrimination. At each perturbation drive frequency upert the fluorescence intensity was fitted to

TN-76
Amplitude 1

mii^k /*\

wit'' ' J^B ^t

-

\: till

m* ri& ^H F %

v..» \|i/

Phase

(b)

FIGURE 5. (a) Phase-coherent sideview image data obtained on a plasma with uv/27r= 638 kHz while driving with an even drive at 1.619 MHz. At this rotation frequency there is a crossing of the (2, 0) mode and an (8, 0) mode with a

radial node. Comparison of the amplitude (b) and phase (c) extracted from the data in (a) with the predictions of linear theory. The predictions of both the (2,0) and (8,0) modes are given. For this plasma 2zq = 0.70 mm and 2ro = 0.29 mm. From Ref. [23].

1

i i 1

'

1.0- -

§ 0.8- "J 1

5-0.6-

O

CM* <

-

Amplitude

o

o

. i

*.

\''V

r*f* • V •• *." '

-

"V *

i

i .

J ' ' . . , .

3-i

:..

'

' f .

' ' 1 i i

i

. . .

2^-

." .

' .-'

" * - ' ^J~^

,,—

• .' ' •.. ' #- . T3 ". " • ^ Jx- J;

ro 1 - • \ . - •\ .

'

' m / . o

-e- 0- - • • .' o

". '" *

CJ e- - . -

" ^ - Q> -1 - L _ _

* . * * . "•""

• _ W - * \ * - * * 03 i " . ' /

* .c tf%' Q.

-2-

-3-

•

1

1

' 1 i

i i

i i i i 1620 1625 1630 1635

(a) Drive Frequency to 12% (kHz)

1640 1620 1625 1630 1635

(b) Drive Frequency co

perl/27i (kHz)

1640

FIGURE 6. Measurement of the (2,0) mode amplitude (a) and phase (b) as a function of perturbation drive frequency.

The lines are fits to theory curves for the velocity of a driven, damped harmonic oscillator [27].

TN-77
Aq + ^2,0 cos(u)per tt + (p2fi)- Figure 6(a) shows the measured amplitude ^2,0 along with a 4-parameter fit to the

damped harmonic oscillator lineshape ao + a\ujpert/ (^2,0 v^pert — <^2,o)

2 + A2

), where A is the damping coeffi- cient. Figure 6(b) shows the measured phase (^2,0 along with a 3-parameter fit to ifo — arctan((wpert — ^2,0)/^).

The two fits give damping coefficient values of A = 1405 and 1409 s _1

. These are consistent with the rates of viscous damping seen in simulations [9].

Angular Momentum Transport From Resonant Modes

In principle, the confinement time of non-neutral plasmas in Penning traps is infinite because angular mo- mentum conservation in an ideal, cylindrically symmetric trap places a constraint on the radial transport of the plasma [28]. In practice radial transport of the plasma always occurs, and at rates which, with ultrahigh vacuum, are greater than can be explained by collisions of the plasma with the neutral background gas. Because the rates of this "ambient" transport increase with increasing static field errors in the trap, it is thought

to be caused by couplings between the confining field asymmetries and the plasma [29]. Although ambient transport is at present poorly understood, progress has been made on the related but

simpler mechanism of mode-resonant transport. Here, torques are imparted because azimuthally asymmetric

plasma modes can have zero frequency in the laboratory frame and hence be excited by the static field errors

[6,11,30-32]. Because these modes need to have negative (backward) frequencies in the rotating frame of the

plasma to come into resonance with a static field error, any torque they exert will slow the plasma down and

hence increase transport. An analysis based on the second law of thermodynamics yields the same result [28]. When the trap walls are well away from the plasma and the ambient field errors are small, as in our

experiment, it is particularly easy to study mode-resonant transport. Reference [11] demonstrated that torque

and heating of the plasma occur when one of the (2,1) plasma modes is resonant with a static field error produced by a tilt between the trap symmetry axis and the magnetic field. The presence of additional heating

resonances at lower rotation frequencies was also noted. We have used Doppler imaging to identify these resonances and find that they arise from (I, 1) modes which come into resonance with the tilted-field error. We have also established that they exert a torque when they are resonant. Experimentally this tilt can be

applied either mechanically by tilting the trap electrodes, or electrically with m — 1 perturbations applied to the compensation electrodes; we find no difference in the transport caused by the two methods.

Figure 7 is a plot of rotation frequency (as determined by side-view images [3]) versus time for a plasma when

the trap has been electrically tilted from its aligned value by an amount equivalent to ~5 x 10~ 4 radians of mechanical tilt. Radial transport, which is measured here by decreases in rotation frequency (corresponding in

this experimental regime to increases in the plasma radius), is enhanced by roughly a factor of 10 as compared

with the aligned case. The rotation frequencies where transport is especially rapid can be identified with the

mode resonances indicated on the plot. The lines show the predictions from theory for where the indicated

mode has u>/ m = —uir , and Doppler imaging was used to verify the identity of these resonant zero-frequency

modes. We find that the tilted-field error couples to modes with m = 1 and odd axial symmetry. Since with a

single-species cloud only the (2, 1) is predicted in linear theory to couple with a tilted-field error, the transport

displayed in Fig. 7 might require the presence of impurity ions. In ion traps these are usually present at some

level, and comprised ~ 20% of the cloud of Fig. 7. Because small-amplitude static field errors can be so effective in causing outward transport, it is not surprising that the process can be usefully inverted by actively driving modes which travel faster than the cloud's rotation. With the laser-cooled Be+ plasmas, we have demonstrated mode-resonant inward transport with the (1,1),

(2,1), (3,1) and (2,2) modes. The (1,1) is particularly easy to excite, as only an axially uniform rotating dipole

field is required, and is useful for driving clouds into the regime where ur approaches fl c. The mode-resonant

technique can do this in a few seconds, while doing the same thing with laser torque takes many seconds, and

with the rotating wall perturbation at least several minutes. We note that the (1,1) mode requires an effect to break the separation between the center-of-mass and the internal degrees of freedom of the plasma. In our

work, a small number of impurity ions could do this. In comparison with the rotating wall technique for controlling an ion cloud's rotation frequency [14,15], the

mode-resonant technique is less precise. The mechanism by which the rotating wall is believed to work with

strongly correlated plasmas is that the plasma comes into equilibrium with rotating distortions of its surface which are imposed by the perturbation. Hence, the torque from a perturbation applied at frequency /rw goes

to when Jrw = fr , and changes sign about this point. In contrast, the torque imparted by a driven (l,m)

mode usually has only one sign and is experimentally observed to depend sensitively upon such parameters

TN-78
N

X

o

c

IT

CD

o

o

CO

/uu 1

i i i 1i i 1i i i

I

i i i Ii i i

600

500

(6,1)

;

i

""l/Wvu (8,1)

400 v^_

^V (4,1) 300

\^w (6,1)

"

200 T

V^"^ (8,1)

100

I , , , 1 200 400 600

Time (s)

800 1000

FIGURE 7. Measured plasma rotation frequency vs. time obtained when the trap was electrically tilted ~5 xl0~

radians from its aligned value. The lines are predictions from theory for where the indicated modes have zero frequency

in the lab frame.

as the temperature of the plasma. As a consequence the rotation frequency at which the cloud comes into equilibrium with the mode, which is determined by a balance between the inward mode torque and the outward

ambient torque, is difficult to calculate in advance and is experimentally observed to change with variations in the cooling laser power or frequency.

However, an important advantage of mode-resonant coupling is that it can be used to transfer angular

momentum to hot (uncorrelated: T <C 1) plasmas. The phase-locked rotating wall control [14,15] described

above has only been demonstrated with laser-cooled plasmas. In contrast, mode-resonant coupling has been

used to increase angular momentum in non-neutral plasmas with temperatures up to 5 eV (which is where

ionization of neutrals begins to change the density profile). Reference [12] demonstrated inward radial transport

of a hot (T=0.9 eV) spheroidal electron cloud through the use oi (1,1) modes. This transport was accompanied

by a heating of the electrons, since there were no cooling processes in the experiment. At higher magnetic

fields the heating can be balanced by cyclotron radiation cooling; steady-state confinement of uncorrelated

electron plasmas in a 4 T field through the application of azimuthally asymmetric modes has recently been

demonstrated [33]. SUMMARY AND FUTURE DIRECTIONS

We have described a technique, Doppler imaging, for studying the mode properties of laser-cooled ion

plasmas. In general, for the magnetized plasma modes of spheroidal plasmas discussed here, good agreement

is obtained between linear theory and the experimental measurements we have made to date. In the future the technique should be a useful tool for studying deviations from the linear theory such as large amplitude

frequency shifts, non-linear corrections to the mode eigenfunction, and mode coupling. Because the width of the resonant lineshape of the mode amplitude as a function of the drive frequency provides a measurement of the mode damping, lineshape measurements may be able to provide information on the collisional viscosity of the strongly correlated plasma, about which little is currently known. Enhanced radial transport is observed

when modes are resonant with static external perturbations, and future work may permit a quantitative

TN-79
comparison to be made between experiment and theory for this basic transport process. Finally, we described

how the plasma angular momentum can be usefully changed through the deliberate excitation of azimuthally

asymmetric modes.

ACKNOWLEDGMENTS

We thank D. H. E. Dubin, C. F. Driscoll, E. M. Hollmann and D. J. Wineland for useful discussions, and

L. B. King and B. M. Jelenkovic for their comments and careful reading of the manuscript. This work is supported by the Office of Naval Research.

REFERENCES

1. Fajans, J., and Dubin, D. H. E., Non-Neutral Plasma Physics II, New York: AIP, 1995.

2. Malmberg, J. H., and O'Neil, T. M., Phys. Rev. Lett. 39, 1333-1336 (1977).

3. Brewer, L. R., et al., Phys. Rev. A 38, 859-873 (1988).

4. Bollinger, J. J., Wineland, D. J., , and Dubin, D. H. E., Phys. Plasmas 1, 1403-1414 (1994).

5. Dubin, D. H. E., 66, 2076-2079 (1991).

6. Bollinger, J. J., et al., Phys. Rev. A 48, 525-545 (1993).

7. Greaves, R. G., and Surko, C. M., Phys. Plasmas 4, 1528-1543 (1997).

8. Holzscheiter, M. H., et al., Phys. Lett. A 214, 279-284 (1996).

9. Dubin, D. H. E., and Schiffer, J. P., Phys. Rev. E 53, 5249-5267 (1996).

10. Dubin, D. H. E., Phys. Rev. £53, 5268-5290 (1996).

11. Heinzen, D. J., et al., Phys. Rev. Lett. 66, 2080-2083 (1991).

12. Mitchell, T. B., Ph.D. thesis, University of California at San Diego, 1993, pp. 58-61.

13. Huang, X.-R, et al, Phys. Rev. Lett. 78, 875-878 (1997).

14. Huang, X.-P., Bollinger, J. J., Mitchell, T. B., and Itano, W. M., Phys. Rev. Lett. 80, 73-76 (1998).

15. Huang, X.-R, Bollinger, J. J., Mitchell, T. B., and Itano, W. M., Phys. Plasmas 5, 1656-1663 (1998).

16. Weimer, C. S., Bollinger, J. J., Moore, F. L., and Wineland, D. J., Phys. Rev. A 49, 3842-3853 (1994).

17. Tinkle, M. D., Greaves, R. G., and Surko, C. M., Phys. Plasmas 2, 2880-2894 (1995).

18. Greaves, R. G., Tinkle, M. D., and Surko, C. M., Phys. Rev. Lett. 74, 90-93 (1995).

19. Driscoll, C. F., Phys. Rev. Lett. 64, 1528-1543 (1990).

20. Tan, J. N., Bollinger, J. J., Jelenkovic, B., and Wineland, D. J., Phys. Rev. Lett. 75, 4198-4201 (1995).

21. Itano, W. M., et al., Science 279, 686-689 (1998).

22. Here wjm is the mode frequency in a frame rotating with the plasma. For the m = modes discussed here this distinction is not necessary because their frequency is the same in either the laboratory or rotating frame.

23. Mitchell, T. B., Bollinger, J. J., Huang, X.-P, and Itano, W. M., Opt. Exp. 2, 314-322 (1998).

24. Information on the mode eigenfunction can be obtained from the side-view images even when there is a change

in the phase-averaged ion fluorescence. However, the images may no longer provide a linear measure of the mode

axial velocity.

25. Thompson, R. C, et al, Phys. Scripta24-33 (1997).

26. Larson, D. J., et al, Phys. Rev. Lett. 57, 70-73 (1986).

27. Landau, L. D., and Lifshitz, E. M., Mechanics and Electrodynamics, Oxford: Pergamon Press, 1972, pp. 68-70.

28. O'Neil, T. M., and Dubin, D. H. E., Phys. Plasmas 5, 2163-2193 (1998).

29. Driscoll, C. F., Fine, K. S., and Malmberg, J. R, Phys. Fluids 29, 2015-2017 (1986).

30. Driscoll, C. F., and Malmberg, J. R, Phys. Rev. Lett. 50, 167-170 (1983).

31. Keinigs, R., Phys. Fluids 27, 1427-1433 (1984).

32. Eggleston, D. L., O'Neill, T. M., and Malmberg, J. R, Phys. Rev. Lett. 53, 982-984 (1984).

33. Anderegg, F., Hollmann, E. M., and Driscoll, C. F., "Rotating Field Confinement of Pure Electron Plasmas Using

Trivelpiece-Gould Modes," submitted to Phys. Rev. Lett.

TN-80
PHYSICS OF PLASMAS VOLUME 6, NUMBER 5 MAY 1999

Direct observations of the structural phases of crystallized ion plasmas'

T. B. Mitchell,ta)J. J. Bollinger, X.-P. Huang, and W. M. Itano Time and Frequency Division, National Institute of Standards and Technology, Boulder, Colorado 80303

D. H. E. Dubin

Department of Physics, University of California at San Diego, La Jolla, California 92093

(Received 13 November 1998; accepted 18 December 1998)

Laser-cooled 9 Be+

ions confined in a Penning trap were directly observed, and the images were

used to characterize the structural phases of the ions. With the ions in two-dimensionally extended

lattice planes, five different stable crystalline phases were observed, and the energetically favored

structure could be sensitively tuned by changing the areal density of the confined ions. Qualitatively

similar structural phase transitions occur or are predicted to occur in other planar single-component

systems with a variety of interparticle interactions. Closed-shell structures were observed with small

ion clouds that were spherical or prolate, and crystals with long-range order were observed in the

centers of clouds with large numbers of ions. These experimental results are in good agreement with

theoretical predictions for the strongly coupled one-component plasma. [S1070-664X(99)92605-8]

I. INTRODUCTION

The one-component plasma (OCP) has been a model of

condensed matter in statistical physics for over 60 years, and

it is used to describe such diverse systems as dense astro- physical matter

1 and electrons on the surface of liquid

helium.2 Laser-cooled trapped ions3 are an excellent experimental realization of the OCP. The phase structures of spatially homogeneous,4' 5 cylindrical (extended in one dimension only),

6' 7 and thin planar (two-dimensionally extended,

with up to 5 planes)8 OCPs have been explored recently. In

this paper, images of ions that were confined in these geometries in a Penning trap are presented and used to further characterize the structural phases. The observed structures agree well with the theoretical predictions for the strongly

coupled OCP.

The OCP model consists of a single charged species embedded in a uniform, neutralizing background charge. In Paul9 or Penning9' 10 traps, which are used to confine charged

particles, a (fictitious) neutralizing background is provided

by the confining potentials. The thermodynamic properties of

the infinite classical OCP are determined by its Coulomb

coupling parameter,

r=

i 47re a ws kB T'

(1)

which is the ratio of the Coulomb potential energy of neighboring ions to the kinetic energy per ion. Here, e is the permittivity of the vacuum, e is the charge of an ion, k B is Boltzmann's constant, T is the temperature, and a ws is the Wigner-Seitz radius, defined by 4Tr(a ws)

3 /3= l/n , where

n is the ion number density. The onset of short-range order

for the infinite OCP is predicted11 at r^2, and a phase

*Paper F3I1.4 Bull. Am. Phys. Soc. 43, 1702 (1998).

tInvited speaker

"'Electronic mail: travis.mitchell@nist.gov

transition to a body-centered cubic (bcc) lattice is predicted11 "" 13 at T^llO. With an OCP in a planar geometry, boundary effects are predicted to cause the formation of

a variety of additional structural phases, such as the hexagonal close-packed (hep) and face-centered cubic (fee) phases.

14-16 Qualitatively similar structural phase transitions occur or are predicted to occur in other planar systems with

varied interparticle interactions, such as plasma dust

crystals,

17' 18 colloidal suspensions, 19 semiconductor electron systems,

20 and hard spheres.21 In the case of small

cylindrically- or spherically-shaped plasmas, concentric

shells are predicted to form. The crystallization of small numbers (total number N

<50) of laser-cooled ions into Coulomb clusters

23 was first observed in Paul traps.

24' 25 With larger numbers of trapped

ions, concentric shells

22' 26 were directly observed in Penning27 and Paul traps.

6' 7 ' 28 Recently, Bragg diffraction has been used to detect bcc crystals (the predicted infinite volume ordering) in large spherical (N>2.7 X 10 , radius

r >60aws) ion plasmas confined in a Penning trap.

4' 5

In this paper we present measurements taken from direct images of the central (r=0) structure of pancake-shaped

(lenticular) ion plasmas (aspect ratio a=z /r <0.1, where

2z is the plasma center's axial extent). This region has a

disk-like geometry with constant central areal density <r (charge density per unit area projected onto the z = plane),

which facilitates a comparison with planar theory. We observed five different stable crystalline phases and found that the energetically favorable central structure could be tuned by changing <x . Both continuous and discontinuous structural phase transitions were observed. We also present direct images of the cylindrical shell structures observed in small

ion clouds and of the three-dimensional (3D) periodic crystals observed in large spherical clouds, and compare our results with those obtained previously.

4' 5 ' 27

1751 TN-81

Downloaded 25 May 2001 to 132.163.135.22. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp
1752 Phys. Plasmas, Vol. 6, No. 5, May 1999 Mitchell et al.

Strobe

Rotating

wall

perturbation \ w

Compensation

electrodes (6x60°)

FIG. 1. Schematic side view of the cylindrical Penning trap with its side- and top-view imaging optics. The insets show the variables used to characterize the intra- and interlayer structure. The side-view inset also shows the

central region of a lenticular ion plasma with three axial lattice planes.

II. EXPERIMENTAL APPARATUS

The 9 Be+

ions were confined radially in a cylindrical

Penning trap (Fig. 1, inner trap diameter 40.6 mm) by a uniform magnetic field B = 4.465 T in the z direction. The

ions were confined axially by a potential difference of V

(usually -1.50 kV) applied between the center and end elec- trodes of the trap. Near the trap center this axial potential is quadratic and has a value of l/2(m/e)o>2 z

2

, where the axial frequency o>

z /2tt=978 kHz for 9 Be+ when V =-l-50 kV.

The radial electric fields of the trap, as well as the ion space

charge, cause the ion plasma to undergo an EXB drift and

thus rotate about the trap axis. In thermal equilibrium, this rotation is at a uniform frequency u> r

. The radial binding

force of the trap is determined by the Lorentz force caused

by the plasma's rotation through the magnetic field. Thus,

low <o r results in a weak radial binding and a lenticular plasma with a large radius. For 104 trapped ions with wr /27T=68.5 kHz (typical for our work on thin planar

clouds), the ion plasma has a density of 2.1 X 108 cm

-3

with 2

r

^1.3 mm and an aspect ratio a = 0.05. The rotation fre- quency was controlled by phase-locking the plasma rotation

to an applied "rotating-wall" electric field.

29' 30 At low co r , an increase in o> r increases both the plasma density and z , providing a way to sensitively adjust a . The ions were cooled3 by an axial laser beam propagating along the z axis and tuned 10 to 20 MHz lower in fre- quency than a hyperfine-Zeeman component of the 2s S 1/2 —>2/>

2 P3/2 resonance at 313 nm with a natural linewidth of

19 MHz. The laser power was —50 /aW and was focused at the ion plasma to a diameter of —0.5 mm. There was also a perpendicular cooling beam, derived from the same laser,

Phase I 1 plane

single plane hexagonal

' .- ..'--

.'.

- -

i i 'fez

Phase IV 3 planes

staggered rhombic 0=76.2° a= 19.7 pm

c, = (a^aj 0.51

cr a

2«2D = 0.92 m

'

Phase V 3 planes

staggered hex (hep-like)

FIG. 2. Top-view (x,y) images of the five structural phases observed in the experiment, with lines showing a fit of the central ions to the indicated

structure.

which had a —70 ^ra waist and variable power. The theoretical cooling limit is 0.5 rnK, and an experimental upper bound of T< 10 mK has been measured;31 for a density of

72 = 2X108 cm

-3

, these limits give a range of 160<T

<3150. A series of lenses form side- and top-view images of

the ions, with viewing directions perpendicular and parallel

to the magnetic field, respectively, on either a gateable

charge-coupled device (CCD) camera, or on an imaging photomultiplier tube. The resolution of the optical systems is —4

/xm, while typical interparticle spacings are — 15 /xm.

III. RESULTS AND ANALYSIS

A. Structure in thin planar OCPs: 1 to 5 planes

The side-view image inset in Fig. 1 shows the central region of a lenticular ion plasma with three axial lattice

TN-82
Phys. Plasmas, Vol. 6, No. 5, May 1999 Mitchell et al. 1753

TABLE I. Primitive and interlayer displacement vectors in the (x,y) plane for the observed phases, where x —ai/|a,|.

Phase Symmetry Stacking Lattice type ai a2 c2 c3

I hexagonal single plane (fl.0) (a cos 60°, a sin 60°) in square staggered bcc(OOl) (a, 0) (0,a) (a,+a2 )/2 (0,0) IV rhombic staggered bcc(llO) (a.0) (a cos 6, a sin 0) (ai + a2)/2 (0,0) V hexagonal hep-like hep (a,0) (a cos 60°, a sin 60°) (a! + a2 )/3 (0,0) vfcc hexagonal fcc-like fcc(lll) (fl,0) {a cos 60°, a sin 60°) (a,+a2)/3 2(ai+a2)/3

planes. It is representative of the flatness and radial extent (< 10% of r ) of the plasma regions used to study the planar

OCP. At a large radius, curvature of the planes can cause the side-view images of axial plane positions to blur. This effect was prevented in the planar measurements reported here by

using clouds with sufficient amounts (up to 50%) of nonfluorescing impurity ions. Because these heavier ions are centrifugally separated to larger radii than the 9 Be+

, the regions of the plasma where curvature begins to be significant can be

filled with these ions, which are sympathetically cooled by

the 9Be+

. 32' 33 With good alignment of the trap with the magnetic field (<1 mrad), the ion plasma rotation becomes phase-locked

with the rotating-wall perturbation.29' 30 Direct images of the

rotating ion structures can then be made by gating the topview CCD camera synchronously with the rotating-wall perturbation for brief gate times (<2% of the plasma rotation

period). Total exposure times of — 3X104 rotation periods were used in typical images (Fig. 2). For our study of the ion

lattice structure we limited our analysis to the central region,

where regular ordering of the ions was observed. At a larger

radius we observed an increased blurring (due to the plasma

rotation), occasional lattice distortions, and, ultimately, the

transition to the regions filled by heavier ions. The observed structure of the central crystallized region

depended on the central areal density cr of the plasma.

Within a layer, the lattice is characterized by the primitive

vectors aj and a2 (which are observed to be equal in magnitude, |a1 | = |a2 |=a), or, equivalently, by a and the angle 6

(=£90°) between the primitive vectors. The interlayer structure is characterized by the axial positions z n of the n lattice planes (measured by the side-view camera) and the interlayer

displacement vector in the (x,y) plane c„ between layers 1 and n. Hence, the equilibrium (x,y) positions of ions in

axial planes 1 and n are given by K1 (i,f) = iai+ja2 and

Rn (/,/) = ia1 +_/a2 + c„, where i, j are integers. These variables

are shown in the insets of Fig. 1. Three different types of intralayer ordering were observed: hexagonal (0=60°), square (0=90°), and rhombic

(90° > 03=65°). The square and rhombic layers stack in a staggered fashion, with the upper ions immediately above the

centers of the parallelograms below, resulting in an interlayer

displacement vector c2 =(a1 + a2

)/2. Hexagonal layers also

stack with ions above the centers of the triangles below, but

this stacking can occur in two distinct ways: 3c2 =a1 + a2 and

3c2/2=a1 +a2

. With hep-like stacking, the ions in every

other plane lie directly above each other (abab...), while

with fcc-like stacking, the ions in every third plane are so

aligned (abcabc. ..). When there were 3 or more hexagonal

layers, both types of stacking were observed. The following sequence of phase structures, with their

lattice parameters and types defined in Table I, were ob- served as the central areal density a was increased from

where order was first observed: (I) one-layer hexagonal —> (III) two-layer staggered square —» (IV) two-layer stag- gered rhombic —* (V) two-layer staggered hexagonal. At a

critical density, a third layer was formed, resulting in (HI) a three-layer staggered square. The process then repeated with

minor variations, such as phase III becoming less common.

In the labeling of the phases, we have used classifications from previous theoretical studies of quantum20' 34 and

classical

16 electron bilayer systems. Phase II, which is a

stable phase of the bilayer systems where the interlayer dis- tance is fixed, is not listed here because it is unstable for the planar OCP, where this distance can vary. We have performed an analytical calculation of the energies of these phase structures for the planar OCP. The

theory uses Ewald summation techniques in order to mini- mize the energy of n parallel lattice planes confined in a harmonic potential of the form <f> e = \l2{mle) u>

z z

2 Because

this potential is identical to the confinement potential of a Penning trap in the a—>0 planar limit,

35 the theory should

predict the structures that are observed in the central region of the plasmas of the experiment. However, since only mini- mum energy states are determined, the theory provides the

lattice structure only for the case of zero temperature (T = 00).The energy minimization was performed holding fixed n and the areal density a of the ions. Consistent with the ex- perimental observations, each lattice plane was assumed to have the same structure, consisting of a 2D lattice described

by primitive vectors ax and a2. Since the area a 1 a 2 sm ^ of the 2D primitive cell is equal to n/cr, the parameters that were

varied in order to minimize the energy were a 2 /a }

, 8, cn and

z n The functional form of the energy is given in Ref. 36. There were several local minima in the energy function,

corresponding to different minimum energy lattices. A numerical search was performed to find the true global mini- mum with respect to the parameters listed above. Finally, the search was repeated for several values of n at a given value

of a, and the value of n that provided the lowest energy was

kept. The search was then repeated for a range of values of

a. The correlation energy per particle Ecorr (in units of e-/4TTe a WS 2D) is shown in Fig. 3 as a function of the

TN-83
1754 Phys. Plasmas, Vol. 6, No. 5, May 1999 Mitchell et al.

-0.88

-0.89

-0.8955

-0.8956

W -0.8957

-0.8958

-0.8959

r , i I r i , . I t 1—]

phase IV - phase V

toe

1 i i i i 1 i i i 1 i i i

0.0 0.5 1.5 2.0

FIG. 3. The correlation energy per particle Ecorr (in units of e

1 IAtr

e

a ws2D) for the minimum energy planar lattices as a function of normalized areal

density.

dimensionless parameter cra WJ 2D' where a Ws2D = (3e 2/4Tre moj2

) 1/3 is the Wigner-Seitz radius in the 2D

limit. The correlation energy is the total energy of the lattice minus the energy of a uniform slab of charge with the same

areal density a. Each successive cusp in the energy function corresponds to the addition of a new lattice plane. The spacing between the planes is shown in Fig. 5(a). As a increases,

the number of lattice planes increases while the spacing be- tween the planes is nearly constant. The topology of the lattice planes varies as a varies. Just

as in the experiments, the minimum energy state has one of 4

separate topologies, depending on the value of a. The struc- tures are listed in Table I as phases I, HI, IV, and Vfcc . In

addition, there is a fifth stable phase, phase V, which has

hep-like stacking. Phase V has only a slighdy higher energy

than phase Vfcc in thin planar OCPs, and if its energy were

plotted in Fig. 3 the line would overlay that of phase Vfcc . The rhombic phase, phase IV, was missed in two previ- ous publications on the minimum energy states of this system.

15 '36 This is because the previous publications did not perform a full numerical minimization with respect to all of

the parameters of the model. Instead, only phases I, HI, V,

and Vfcc , along with a few other symmetric phases, were

considered. Although stability of these phases was checked

by evaluating the normal modes of the lattices, it was not recognized that phase III becomes unstable to the rhombic

phase IV deformation, since this instability was suppressed

by the periodic boundary conditions. Fortunately, the energy

of the rhombic phase is only slightly lower than that of phase

HJ, so the conclusions of the previous works concerning the phase diagram of the system are only slightly altered. In

particular, the maximum size of approximately 60 lattice planes required to observed bulk bec behavior is unaffected.37 The transition to bulk behavior is shown in Fig. 4, which

displays local minimum energy states for larger values of a. There is a competition between the energies of the rhombic

phase IV and phase Vfcc which implies that the lattice structure displays sensitive dependence on a up to a-a

2ws2 D~ 20-

21, corresponding to approximately 60 lattice planes. However, beyond this range of <xa

2 j2D this phase is only slightly

FIG. 4. The transition to the bulk bec structure: correlation energy per

particle Ecorr for larger values of the normalized areal density.

deformed from a bec lattice with the (110) plane oriented

parallel to the surface, the deformation becoming smaller as a increases further. This can be seen in Fig. 6 below, where

the intralayer angle of the rhombic phase IV is seen to ap- proach the bcc(110) lattice value of 70.5° as a increases. The predictions of the analytic planar OCP theory,

which has no free parameters, were compared directly with

the observations by identifying the areal density a of the planar OCP with the directly measured central areal density

cr of the lenticular plasmas. For a quantitative analysis of

the observed lattice structure, we performed a least-squares

fit of the positions of the ions in the central region (lines in Fig. 2) to the relevant phases (shown in Table I). Using the

best-fit values of the primitive vector length a and the intra- layer angle 6 and the observed number of lattice planes n, we calculated the central areal density a = n/(a 2 sm9).

The agreement for n = 1 to 5 between the planar OCP

theory and experiment, with measurements taken on differ- ent plasmas with N< 104

, is good (Fig. 5). This is somewhat

surprising since the energy differences between competing

lattices are extremely small, on the order of lO~ 2 e

2/4TT€ a ws2 o per particle. As the central areal density

is increased, the lattice planes move further apart axially

[Fig. 5(a)]. Eventually, it becomes energetically favorable to form an additional lattice plane. However, although the phase Vfcc was predicted to be slightly more favorable than phase V (by energies on the order of 10~ i e

2 /4TTeQ a WS 2v Per

particle), we rarely observed Vfcc ( ~ 10% of the time). These

and other minor discrepancies from theory may be due to the

finite radial extent or the nonzero temperature of the ion plasma; we observed a similar preference for hep stacking in molecular dynamics simulations of small (jV = 3000) lenticular ion plasmas with r=500.

For the dependence of the angle 6 (between the primitive vectors aj and a2) on central areal density a [Fig. 5(b)], the trend was that, when a new lattice plane was formed, 6 changed discontinuously from —60° to a higher value. As

<r of the crystal was further increased, 6 smoothly de- creased to ~65° until there was a second discontinuous transition to a hexagonal structure. This second transition has been predicted to become continuous, with 6 assuming all values 60° s=0s£9O°, in liquid (T<80) bilayer systems.

38

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Phys. Plasmas, Vol. 6, No. 5, May 1999 Mitchell et al. 1755

-| 1-

0.5 1.0

~>—r1.5

°0 ats2O

FIG. 5. (a) The interlayer structure (plane axial positions and displacement

vectors) and (b) intralayer angle of the central region as a function of the

central areal charge density. The lines show predictions from theory, and

symbols show experimental measurements. Also indicated is whether the

lattices have an interlattice displacement vector c2 characteristic of the hexagonal phases (circles, thick lines) or the square and rhombic phases

(squares, thin lines). Lengths have been normalized by a ws2 u= 10.7fan.

The lines indicate the minimum energy structures predicted

by the analytic theory. At central areal charge densities near phase boundaries, both phases could be observed. In these

regions, the phase which materialized after the crystal was

formed was initially random, but persisted if the ions were

not heated. Where there was not a strong preference for one

phase over the other, both were plotted.

B. Structure in planar OCPs: 5 to 45 planes

By increasing the number of lattice planes n, we have

been able to study the transition from the several plane re- gime where surface energy contributions are important, to one where bulk energy contributions become more important. The basic result as we increased the number of planes is that bands of alternating phase IV and phases V or Vfcc were

observed, until ~ 30 planes were formed. Above this point,

phase IV with 6 within a few degrees of 70° and (z n — z n -\)la within a few percent of 0.82 was almost exclusively seen. When 0=2arctan(l/v

/2) = 7O.53° and (z„

T3

C 64

62 E 60

Rotation Frequency (kHz)

r/A

9»:i0a planes 10»jl1o 24 a' 25 o planes 25Q;26D -

-1—'—i—'—i——i—'—

3.3 3.4 3.5 3.6 3.7 -/h ~i——i—'—i—'—i—'—

B.5 8.6 8.7 8.8 8.!

°o a

?«so

FIG. 6. Intralayer angle 8 of the central region for two ion plasmas with

9 — 11 planes (left) and 24—>26 planes (right). The symbols are experimental measurements as in Fig. 5, plotted versus rotation frequency. The lines are from the T=0 theory plotted as a function of the central areal density, with dashes indicating that the structure is not predicted to be energetically favorable. The four transitions in n used to align the two abscissas are also shown.

—z„_j)/a=yz/3 = 0.816, phase IV is equivalent to bcc(110)

ordering, which is the predicted infinite volume ordering.

The increased axial extent of the plasmas required a modification in how the images of the structure were obtained. For plasmas with 2z greater than the depth of field of the top-view f/2 objective lens ( ~ 80 fim), the coolinglaser beam directed perpendicularly to B was used to illumi- nate a section of the plasma within the depth of field [see

Figs. 7(b) and 8 below]. The fluorescence from the ions outside this region illuminated by the parallel beam could be

made negligible by chopping the parallel beam with an optical wheel and gating the CCD camera on only when the

parallel beam was blocked. This chopping technique was

also occasionally used with the side-view images in order to improve their spatial resolution. Figure 6 plots the measured and predicted intralayer

angle 6 of the central regions, for parameters near where the planar plasma has 10 and 25 axial lattice planes. Because the experimental uncertainties in a and 9 cause the calculated <x to have an unacceptably large scatter for a large number of

planes, the experimental measurements have been plotted as a function of the plasma rotation frequency, which increases monotonically with <x . To match the observations with the

theory, the range of the plotted rotation frequencies (top

axis) was adjusted to make the frequencies where the n—*n + 1 transitions shown in the figure were observed coincide

with the theory predictions.

Good agreement with the analytic theory continued to be

seen with 10 planes: Phases IV and V or Vfcc alternated as the energetically preferred structure, and the intralayer angle 6 of phase IV showed the expected decrease as the central

areal density of the crystal is increased. With a larger numbers of planes, however, phase IV occurred more often than

TN-85
1756 Phys. Plasmas, Vol. 6, No. 5, May 1999 Mitchell et al.

TABLE II. Incidence of phase IV (bcc-like) in planar OCPs (%).

n planes Experiment T=0 Theory

51020

25

334560

50

5276

90 >97

>97

40

35

31

3126

34

100

predicted (Table II). A plausible explanation for why this occurs is the ions' finite temperature. Dubin and O'Neil38 have determined the free energy of various lattice types for the planar OCP as a function of both <r and T. In Fig. 9 of

Ref. 37 they plot which of the two main lattices, fcc(lll)

and bcc(110), has the lower free energy for the regimes

200<r<5000. They found that as T was decreased, the

areal density above which the bcc(110) phase is exclusively

favored decreases. For example, when r = 600, bcc(110) is exclusively favored for n > 30.

C. Shell structure in small OCPs

When the rotation frequency is increased the plasma

changes shape from oblate to prolate, and the lattice planes near the plasma boundary bend in order to conform with the curvature of the boundary. The minimum energy structure at these regions consists of concentric shells, each made up of

imperfect 2D hexagonal crystal sheets.

22' 26 The shell curvature results in a loss of correlation between shells since the 2D lattices on different shells get out of phase as one moves

along the shell surfaces.

37 With small plasmas where boundary effects are important, shell structure can be the minimum

energy structure of the cloud; that is, the shell structure exists throughout the interior as well as near the plasma boundary.

Shell structure has been previously imaged in both Penning and if trap experiments.7 ' 27 ' 28 The Penning trap experiment showed differences with the simulations in that the

shells were observed to be open ended cylinders (parallel to the z axis) rather than the predicted closed spheroidal shells. In order to investigate this discrepancy with the present trap, which includes the capability of taking side-view images, we

returned to the geometry of the previous experiments. Figure 7 shows side- (a) and top-view (b) images of shell structure obtained on a plasma of N= 9000 Be+

ions and 15,000 impurity ions. Neither image was strobed with the rotating-wall

perturbation.

In Fig. 7(a) the parallel beam was chopped at 1 kHz, and

the side-view camera was gated on only when the parallel

beam was gated off. The perpendicular laser beam was then

translated up and down throughout the plasma, and the re- sultant ion fluorescence was integrated over many transla- tions of the beam, producing a slice of the shell structure cutting through the plasma near the r — axis. The viewing

optics are at 60° with respect to the perpendicular beam [Fig. 7(c)], which, along with the small offset of the beam from

the r = axis, produced an image with the shell structure well defined on the right hand side of Fig. 7(a). In Fig. 7(b)

- - ISlB

>i9 P5i

HK^.-t*-; iSll$B v/^l i few*?*--;-.

;

igjiH

Ww$M$bs5§

perpenicul^be^

\60°

-.

side-view

optics

FIG. 7. Side-view (a) and topview (b) images of an ion cloud with rotation frequency wr =2TrX 63 kHz confined into a cylindrical shape due to the presence of centrifugally separated, heavier impurity ions. The viewing

angle of the side-view optics is shown (top-view) in (c).

the perpendicular laser beam was held fixed near the central (z—0) section of the plasma and the ion fluorescence pro- duced by both the parallel and perpendicular laser beams

were integrated on the top-view camera. However, because

the camera was also focused on the central section of the plasma, this figure essentially shows a cross section of the

shell structure near z — . The ion plasma of Fig. 7 had 9 shells of 9 Be+

ions that were cylindrical near the middle (z—0) section of the plasma. The cylindrical shells were produced by the presence

of heavier mass contaminant ions, which centrifugally separated and produced a cylindrical boundary to the lighter Be+

ions. Figure 7(a) shows a resolved spatial structure in the

axial direction and Fig. 7(b) in the radial directions. However, in agreement with the simulations, the cylindrical shells are not open ended, but are closed by a curved shell structure. The curvature of the shells near the ends of the plasma was also observed in top-view images like Fig. 7(b) when the perpendicular laser beam was directed near an axial end of

the plasma. The reason for the frequent observation of open- ended cylindrical shells in the earlier Penning trap experi- ments is unknown, as we were not able to produce open- ended cylindrical shells in this work. For example,

misaligning the trap symmetry axis with the magnetic field by up to 0.3 mrad had no apparent effect on the shell structure.

D. Crystal structure in large OCPs

The formation of a structure in spherical plasmas with

many ions has been studied experimentally using Bragg

diffraction.

4' 5 Long-range order (3D periodic crystallization) was found to emerge in plasmas with N>50 000 ions (radius

r ^31a ws ). Bulk behavior (bcc crystals exclusively) was

observed in plasmas with /V>270 000 (r *=>60a ws ), and

lower limits to the crystal diameters of 17 and 28a ws were

TN-86
Phys. Plasmas, Vol. 6, No. 5, May 1999 Mitchell et al. 1757

•*.X*;m^h:*h«*

•• ' JJutJ£jP*fti&''t*'

i**'%

tHk&^&yw

FIG. 8. Top-view images of a spherical cloud with 100 000 Be+

ions and 60

000 impurity ions, with lines showing a fit to the bcc lattice structure with

the indicated orientation. The length of the perpendicular beam in the image

is 60a ws and the axial extent is &a„s

, where a„s = 8.7 ftm.

estimated from the widths and intensities of the Bragg peaks,

respectively. Theoretically, simulations have been limited in

size to a 20 000 ion spherical plasma, in which was found

only short range order in the form of approximately 20

spherical shells.

39 We have used top-view images, strobed by the rotating- wall perturbation, of a large ion cloud to further investigate

the structure in this regime. Of this cloud's 160 000 ions, ~ 100 000 were

9 Be+

ions with a radius of — 35a WJ . Figure

8 shows examples of such images, along with lines showing

a fit to a bcc lattice structure. Both a bcc and fee structure were observed, with bcc occurring —70% of the time. A

variety of orientations were observed, but these were usually

consistent with viewing along a symmetry axis of the crystal.

Evidence for preferred orientations has been previously

noted.5 The region with the strongest regular order of individual

ions was found to be a cylinder along the r = axis of the plasma. A typical diameter was ~\6a ws , and it extended

axially throughout the sphere until significant curvature of

the end planes was encountered. The transition from a crystal structure to concentric shells was not abrupt, as can be seen

in Fig. 8, where the structure observed out to r = 30a ws from

the center is not regularly spaced.

IV. DISCUSSION

We have measured the correlations of strongly coupled

Be+

ion plasmas in a variety of different geometries with

direct observations. The planar geometry, in particular, per- mits a detailed comparison with theoretical calculations, and,

in general, the agreement between the observations and

theory is good. We have measured the energetically preferred structures in lenticular plasmas for up to 45 lattice planes and find a gradual transition from surface-dominated

to bulk behavior to occur when the ion plasma extends —30

planes axially. Ions in a trap have been proposed as a register for a quantum computer.40 Work in this area has focused on a

string of a few ions in a linear Paul trap.

41 A single lattice

plane of ions as shown in Fig. 2 could provide an alternative 2D geometry of trapped ions for studies of quantum computing or entangled quantum states. In an approximately spherical plasma of 160 000 ions, we have observed 3D periodic crystals with long-range order. The crystals occupied the inner cylindrical core of the plasma, and outside the crystal there was a complicated transition to the shell structure. We have not observed the ther- modynamic liquid-solid phase transition predicted for the bulk OCP, as our measurements have concentrated on the

correlations obtained at the coldest temperatures (therefore

maximum F) where the ion fluorescence is maximum. The

study of this phase transition in the bulk and 2D geometries

is an interesting direction for future studies with this system.

ACKNOWLEDGMENTS

This research was supported by the Office of Naval Research and the National Science Foundation (D. H. E. D.). We thank B. E. King, L. B. King, and D. J. Wineland for useful comments, and B. M. Jelenkovic and J. N. Tan for technical assistance. This manuscript is a work of the U.S. government; it is not subject to U.S. copyright.

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TN-88
PHYSICS OF PLASMAS VOLUME 7, NUMBER 1 JANUARY 2000

SPECIAL TOPICS SECTION

Crystalline order in laser-cooled, non-neutral ion plasmas

J. J. Bollinger, T. B. Mitchell, X.-P. Huang, W. M. Itano, J. N. Tan,a)

B. M. Jelenkovic,b) and D. J. Wineland

Time and Frequency Division, National Institute of Standards and Technology, Boulder, Colorado 80303

(Received 17 May 1999; accepted 25 May 1999)

Laser-cooled trapped ions can be strongly coupled and form crystalline states. In this paper we

review experimental studies that measure the spatial correlations of Be+ion crystals formed in Penning traps. Both Bragg scattering of the cooling-laser light and spatial imaging of the laser-induced ion fluorescence are used to measure these correlations. In spherical plasmas with more than 2 X 105 ions, body-centered-cubic (bcc) crystals, the predicted bulk structure, are the only

type of crystals observed. The orientation of the ion crystals can be phase locked to a rotating

electric-field perturbation. With this "rotating wall" technique and stroboscopic detection, images

of individual ions in a Penning trap are obtained. The rotating wall technique also provides a precise control of the time-dilation shift due to the plasma rotation, which is important for Penning trap frequency standards. [S1070-664X(00)01501-9]

I. INTRODUCTION

In this paper we review recent progress on the study of

strongly coupled ion plasmas in Penning traps. It is similar to the review in the conference proceedings of Ref. 1 and contains more background material on Bragg scattering results than Ref. 2, which focuses on results obtained from real images of the ion crystals.

Trapped ions are a good example of a one-component

plasma (OCP). A OCP consists of a single charged species

immersed in a neutralizing background.3 In an ion trap, the trapping fields provide the neutralizing background.4 Examples of OCPs include such diverse systems as the outer

crust of neutron stars

5 and electrons on the surface of liquid

helium.6 The thermodynamic properties of the classical OCP

of infinite spatial extent are determined by its Coulomb coupling constant,

3

r=

i 47re„a wsA: B r'

(1)

which is a measure of the ratio of the Coulomb potential energy of nearest neighbor ions to the kinetic energy per ion. Here, e is the permittivity of the vacuum, e is the charge of

an ion, & B is Boltzmann's constant, T is the temperature, and

aws is the Wigner-Seitz radius, defined by 47r(a ws)

3 /3 = l/« , where n is the ion density. For low-temperature

ions in a trap, n equals the equivalent neutralizing background density provided by the trapping fields. Plasmas with

T>1 are called strongly coupled. The onset of fluid-like behavior is predicted at r»=2,3 and a phase transition to a

"'Present address: Department of Physics, Harvard University, Cambridge,

Massachusetts 02138.

b)On leave from the Institute of Physics, University of Belgrade, Belgrade,

Yugoslavia.

body-centered-cubic (bcc) lattice is predicted at r = 170. 3 ' 7 From a theoretical perspective, the strongly coupled OCP has been used as a paradigm for condensed matter for decades. However, only recently has it been realized in the laboratory.8

Experimentally, freezing of small numbers (/V<50) of

laser-cooled atomic ions into Coulomb clusters was first observed in Paul traps.

9-11 With larger numbers of trapped

ions, concentric shell structures were observed directly in Penning12 and linear Paul 1314

traps. The linear Paul traps provided strong confinement in the two dimensions perpendicular to the trap axis and very weak confinement along the trap axis. This resulted in cylindrically shaped plasmas

whose axial lengths are large compared to their cylindrical diameters. Cylindrical-shell crystals that are periodic with

distance along the trap axis were observed. The diameter of

these crystals was limited to ~ 10a ws in Ref. 13 and

~30a ws in Ref. 14, presumably due to rf heating, 15 which is produced by the time-dependent trapping fields and increases with the plasma diameter. These plasma diameters appear to be too small to observe the three-dimensional (3-D) periodic

crystals predicted for the infinite, strongly coupled OCP.

Strong coupling and crystallization have also been observed

with particles interacting through a screened Coulomb potential. Examples include dusty plasma crystals

16 and colloidal suspensions. 17 ' 18 Because Penning traps use static fields to confine charged particles, there is no rf heating. This has enabled ion plasmas that are large in all three dimensions to be laser cooled. For example, we have laser cooled ~ 106 Be+

ions in an approximately spherical plasma with diameter ~200aWs- With these large ion plasmas we have used Bragg scattering of the cooling laser light to detect the formation of bcc

crystals,

19' 20 the predicted state for a bulk OCP with T

> 170. In addition, we have studied the spatial correlations in

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Phys. Plasmas, Vol. 7, No. 1, January 2000 Bollinger et al.

rotating quadrupole

field (top-view) + +

side-view camera

f/5

XO

vw

\compensation

electrodes (6x60°)

axial coolin" beam

FIG. 1. Schematic view of the cylindrical trap with real space imaging

optics for the side-view camera and Bragg diffraction detection system for

the axial cooling beam. The size of the plasma is exaggerated. The cross

section of the rotating quadrupole field (in the x-y plane) is shown in the

insert. From Ref. 2 1

.

planar, lens-shaped plasmas with axial thickness :£l0a Ws- These plasmas consist of extended, two dimensionally peri- odic lattice planes. The importance of the plasma boundary

in this case results in different crystalline structures depending on the details of the plasma shape.

A potential drawback of the Penning trap versus the rf trap is that the ions rotate about the trap magnetic field, and

this has previously prevented the imaging of the ion crystals

as done in Paul traps. This is because the rotation, created by

the E x B drift due to the radial electric and the trap magnetic fields is, in general, not stable. For example, fluctua- tions in the plasma density or shape produce fluctuations in the ion space charge fields that change the plasma rotation. However, we are able to phase lock the rotation of the laser- cooled ion crystals to a rotating electric field perturbation.21 ' 22 The success of this "rotating wall" tech- nique enables us to strobe the cameras recording the ion

fluorescence synchronously with the plasma rotation and obtain images of individual ions in the plasma crystals.

23

Figure 1 is a schematic of the cylindrical Penning trap we use to confine 9Be +

ions. The trap consists of a 127 mm

long vertical stack of cylindrical electrodes with an inner diameter of 40.6 mm, enclosed in a room temperature, 10

_8 Pa vacuum chamber. The uniform magnetic field B

= 4.46T is aligned parallel to the trap axis within 0.01° and

produces a

9Be+

cyclotron frequency CI = 2itX1.61 MHz. A

quadratic, axially symmetric potential (mo)2/2e)(z 2 — r

2 l2)

is generated near the trap center by biasing the central elec- trodes to a negative voltage — V . At V = 1 kV, the singleparticle axial frequency ti> z = 2-7TX799 kHz and the magnetron ExB drift frequency a> m = 2 -ttX 42.2 kHz. The trapped

Be+ions are Doppler laser cooled by two 313 nm laser beams. The principal cooling beam (waist diameter —0.5

mm, power —50 /xW) is directed parallel to B . A second,

typically weaker cooling beam with a much smaller waist (—0.08 mm) is directed perpendicularly to B (not shown in Fig. 1). This beam can also be used to vary the plasma rota- tion frequency by applying a torque with radiation pressure. With this configuration, ion temperatures close to the 0.5 mK

Doppler laser-cooling limit are presumably achieved. However, experimentally we have only placed a rough 10 mK

upper bound on the ion temperature.

24 For a typical value of « = 4xl08 cm

-3

, this implies T>200.

Two types of imaging detectors were used. One is a charge-coupled device (CCD) camera coupled to an electronically gatable image intensifier. The other is an imaging

photomultiplier tube based on a microchannel-plate electron

multiplier and a multielectrode resistive anode for position

sensing. For each detected photon, the position coordinates

are derived from the current pulses collected by the different electrodes attached to the resistive anode. This camera there- fore provides the position and time of each detected photon.

However, in order to avoid saturation, we placed up to 20 dB

of attenuation in front of this camera to lower the detected photon counting rate to less than —300 kHz.

In thermal equilibrium, the trapped ion plasma rotates without shear at a frequency u> r where u)m <ior <il — com . 25 ' 26 For the low-temperature work described

here, the ion density is constant and given by n = 2e mwr (fl-wr)/e 2

. With a quadratic trapping potential the plasma has the simple shape of a spheroid, z

2lz\ + r

2/rl=l, where the aspect ratio a=z /r depends on

o) r. 24' 26 This is because the radial binding force of the trap is determined by the Lorentz force due to the plasma's rotation through the magnetic field. Thus low oj r results in a lenticular plasma (an oblate spheroid) with a large radius. As tu r increases, r shrinks and z grows, resulting in an increasing

a. However, large o) r (o> r >fl/2) produces a large centrifugal acceleration that opposes the Lorentz force and lenticular plasmas are once again obtained for co r ~fl- co m . In our work, torques from a laser or a rotating electric field are used

to control u> r

, and therefore the plasma density and shape.

The plasma shape is observed by imaging the ion fluores- cence scattered perpendicularly to B with an f/5 objective.

(See Fig. 1.) All possible values of io r from ut m to £l — (om

have been accessed using both methods of applying a

torque.

22' 27' 28 Azimuthally segmented compensation elec- trodes located between the main trap electrodes are used to apply the rotating electric-field perturbation. Both a rotating

quadrupole (see the inset in Fig. 1) and rotating dipole field (not shown in Fig. 1) have been used to control u) r

. Below we explain how the rotating quadrupole field provides precise control of u> r

.

II. BRAGG SCATTERING

A. BCC crystals

An infinite OCP with T a 170 is predicted to form a bcc

lattice. However, the bulk energies per ion of the faceTN-90
Phys. Plasmas, Vol. 7, No. 1, January 2000 Crystalline order in laser-cooled, non-neutral ion

FIG. 2. Bragg diffraction patterns from a plasma phase locked to a rotating quadrupole field (wr =2TrX 140kHz,n =4.26X 108 crrT 3 ,a«l.l). (a) A 1

s time-averaged pattern. The long rectangular shadow (highlighted by solid

lines) is from the deflector for the incident beam; four line shadows (highlighted by dashed lines) that form a square are due to a wire mesh at the exit window of the vacuum chamber. The small open circle near the center of the

figure marks the position of the undeflected laser beam, (b) Time-resolved

pattern obtained nearly simultaneously with (a) by strobing the camera with

the rotating field (integration time =5 s). A spot is predicted at each inter- section of the rectangular grid lines for a bcc crystal with a [110] axis aligned with the laser beam. The grid spacings were determined from the n

calculated from o) r and are not fitted. From Ref. 22.

centered-cubic (fee) and hexagonal-close-packed (hep) lat- tices differ very little from bcc (< 10~ 4

). 29 Because some of

the fee and hep planes have lower surface energies than any

of the bcc planes, a boundary can have a strong effect on the preferred lattice structure. One calculation29 estimates that the plasma may need to be 5:1 00<a ws across its smallest dimension to exhibit bulk behavior. For a spherical plasma

this corresponds to ~ 105 ions. We used Bragg scattering to measure the spatial correlations of approximately spherical plasmas with N>2x 105

trapped Be+ions.

19' 20 The cooling-laser beam directed along

the trap axis was used for Bragg scattering, as indicated in Fig. 1. First the plasma shape was set to be approximately

spherical. (In early experiments this was done with the per- pendicular laser beam; more recent experiments used the rotating wall.) The parallel laser beam was then tuned approximately half a linewidth below resonance, and a Bragg

scattering pattern recorded (—1-30 s integration). The

plasma was then heated and recooled, and another Bragg

scattering pattern was recorded. Because the 313 nm wavelength of the cooling laser is small compared to the interion separation (—10-20 /xm), Bragg scattering occurs in the for- ward (few degree) scattering direction. In order for a dif- fracted beam to form, the incident and scattered wave vectors

k, and ks must differ by a reciprocal lattice vector (Laue

condition).30 In a typical x-ray crystal diffraction case, satis- fying the Laue condition for many reciprocal lattice vectors, requires that the incident radiation have a continuous range

of wavelengths. Here the Laue condition is relaxed because

of the small size of the crystal, so a crystalline Bragg diffraction pattern is frequently obtained, even with monochromatic

radiation. Figure 2(a) shows a time-averaged diffraction pattern obtained on a spherical plasma with N—7.5X 105

. The multiple concentric rings are due to Bragg scattering off different planes of a crystal. A concentric ring rather than a dot pattern

is observed because the crystal was rotating about the laser beam. In general, many different patterns were observed,

8.

i VF VT 2 -Js

——i—i—

A

B :

:C

D1 u.

2 4 6 8 10 12 14 16 9 aWS

FIG. 3. Histogram showing the numbers of peaks (not intensities) observed

as a function of <?a Ws (defined in the text) for 30 time-averaged Bragg

scattering patterns obtained on two different spherical plasmas with N>2

X 105

. The dotted lines show the expected peak positions for a bcc crystal, normalized to the center of gravity of the peak at A (corresponding to Bragg

reflections off {110} planes). From Ref. 20.

corresponding to Bragg scattering off crystals with different orientations. Figure 3 summarizes the analysis of approximately 30 time-averaged patterns obtained on two different spherical plasmas with N>2X 105

. It shows the number of Bragg peaks as a function of the momentum transfer q = |ki -kl-| = 2*sin(egcM /2) ( = & 6^ for 6>scatt «l), where/: — 2tt/\ is the laser wave number and scatt is the scattering

angle. The density dependence of the Bragg peak positions is removed by multiplying q by a ws , which was determined

from a> r. The positions of the peaks agree with those calculated for a bcc lattice, within the 2.5% uncertainty of the angular calibration. They disagree by about 10% with the values calculated for a fee lattice. The ratios of the peak

positions of the first five peaks agree to within about 1% with

the calculated ratios for a bcc lattice. This provides strong

evidence for the formation of bcc crystals in spherical plas- mas with N>2X 105 ions. This result is significant because

it is the first evidence for bulk behavior in a strongly coupled

OCP in the laboratory.

B. Rotating wall By strobing the camera recording the Bragg scattering pattern synchronously with the plasma rotation, we should be

able to recover a dot pattern from the time-averaged concentric ring pattern in Fig. 2(a). Initially we used the time de- pendence of the Bragg scattered light to sense the phase of

the plasma rotation.

20' 31 More recently we used a rotating

electric field perturbation to phase lock the ion plasma

rotation.

21,22 Consider the rotating quadrupolar perturbation shown in the inset of Fig. 1. This z-independent perturbation produces

a small distortion in the shape of the spheroidal plasma. In

particular, the plasma acquires a small elliptical cross section normal to the z axis. (In our work the distortion created by

the rotating quadrupole field was typically less than 1% of

the plasma diameter.) The elliptical boundary rotates at the applied rotating wall frequency a> w

. An ion near the plasma

boundary experiences a torque due to this rotating boundary.

If the ion is rotating slower than a> w , the torque will speed it up. If it is rotating faster than co w , the torque will slow it TN-91
10 Phys. Plasmas, Vol. 7, No. 1, January 2000 Bollinger et al.

down. Through viscous effects, this torque is transmitted to the plasma interior. Therefore, if other external torques are small, the rotating wall perturbation will make cj r equal u> w . Crystallized plasmas behave more like a solid than a liquid or gas. Because the viscosity is high, the whole plasma will tend to rotate rigidly with its boundary. In particular, the orientation of the ion crystals can phase-lock to the rotating

quadrupolar perturbation if the frequency difference between

(xi r and u> w is small. To check for phase-locked control of u> r

, we strobed the camera recording the Bragg scattering pattern in Fig. 2(a) with the synthesizer used to generate the rotating wall signal.

Specifically, once each 2tt/u> wperiod, the rotating wall sig- nal gated the camera on for a period :S0.02(27r/a> w ). The

resulting Laue dot pattern in Fig. 2(b) shows that the plasma

rotation was phase locked to the rotating electric-field perturbation. The dot pattern provides detailed information on

the number and orientation of the crystals that contributed to the Bragg scattering signal. For example, the pattern in Fig.

2(b) was due to a single bcc crystal with a [1 10] axis aligned

along the laser beam. For phase-locked operation of the rotating wall, other external torques must be small. For ex- ample, a misalignment of the trap magnetic field with the

trap electrode symmetry axis of >0.01° prevented phase- locked control of the plasma rotation. In our work, alignment

to £0.003° was obtained by minimizing the excitation of

zero-frequency plasma modes.27 ' 28

In addition to the rotating quadrupole perturbation,

phase-locked control was also achieved with a uniform rotating electric field (a "dipole" field). In fact, under many

circumstances a uniform oscillating field worked equally

well. In these cases the corotating component of the oscillat- ing field controlled the plasma rotation while the perturbing

effects due to the counter-rotating component were minimal.

The simplicity of the oscillating dipole field makes it a convenient tool for controlling wr

. However, in a quadratic trap, control of co r with a uniform rotating or oscillating electric

field requires an effect that breaks the separation of center- of-mass and internal degrees of freedom of the plasma. In our work this is done by impurity ions that experience a

different centrifugal potential than the 9Be+

ions.

22

III. REAL-SPACE IMAGES

Bragg scattering measures the Fourier transform of the

spatial correlations of the trapped ions. It provides a picture of these correlations in reciprocal-lattice space. With phase- locked control of u> r

, real-space imaging of individual ions

in a Penning trap becomes possible. To obtain real-space

images with high resolution, we replaced the Bragg scatter- ing optics (see Fig. 1) with imaging optics, starting with an

f/2 objective, which formed a real, top-view image of the ion plasma. The combined resolution limit of the optics and camera was less than 5 /xm near the optimal object plane of

the f/2 objective. This is less than the ~10 /xm resolution

limit required to resolve individual ions. However, the depth

of field of an f/2 objective for 10 /xm resolution is —80 /xm. For lenticular plasmas with 2z s80/U,m, all of the ions within the plasma were resolvable. For plasmas with 2z

1.2 mm 0.5 mm

FIG. 4. Real-space images of an N~ 1.8X 105 ion plasma phase locked with an oscillating dipole field at o> r = 2 ttX 120 kHz. (a) Time-averaged side- view image showing the overall plasma shape. The bright line of fluores- cence through the plasma center is due to a laser beam directed perpendicularly to B . The plasma shape is approximately spherical. The presence of heavier-mass ions, which centrifugally separate from the 9 Be +

ions, pro- duces the straight vertical boundaries in the image, (b) Strobed top-view image, obtained simultaneously with (a), showing the presence of a bcc

crystal in the plasma center. The distance scales in (a) and (b) are different, as noted.

>80/U.m. the cooling-laser beam directed perpendicularly to B was used to illuminate a section of the plasma within the depth of field. Figure 4 shows side-view and top-view images of an approximately spherical plasma with N~ 1.8X 105

. The fluo- rescence from the perpendicular laser beam used to highlight

a small region of the plasma is clearly visible. In the top- view image, a square grid of dots is observed near the plasma

center. The measured spacing between nearest neighbor dots

is 12.8±0.3 /xm, in good agreement with the 12.5 /xm spacing expected for viewing along a [100] axis of a bcc crystal

with density determined by the u> r set by the rotating field. Real-space imaging provides direct information on the location and size of the crystals. In Fig. 4 the crystal was located

in the radial center of the plasma and was at least 230 /xm

across, or at least one-quarter of the plasma diameter. For lenticular plasmas with 2z ~80/u,m, all of the ions within the plasma are resolved without the use of the perpendicular laser beam. Lenticular plasmas are obtained with io r slightly greater than u> m

. For small plasmas (N

:£ 2000 ions) we were able to use the rotating-dipole electric

field to lower wr and obtain a single plane while maintaining

long-range order in the top-view images. Figure 5(a) shows a

FIG. 5. Strobed top-view images of a small (N~ 300) Be+ ion plasma

phase-locked with a rotating dipole field at (a) o> r =2TrX65.7 kHz and (b) 66.5 kHz. Below are unstrobed side views showing the axial lattice planes. Heavier-mass ions are located outside the 9Be+

ions.

TN-92
Phys. Plasmas, Vol. 7, No. 1, January 2000 Crystalline order in laser-cooled, non-neutral ion 11

top- and side-view image of such a plasma. Near the plasma

center a 2-D hexagonal lattice is observed, the preferred lat- tice for a 2-D system. Here each dot is the image of an

individual ion. Starting with a single plane like that shown in Fig. 5(a), we studied the structural phase transitions that occur as (o r is increased.

23 With increasing a> r , the radial confining force of

the Penning trap increases, which decreases r . At a particular point, there is a structural phase transition near the plasma

center from a single, hexagonal lattice plane to two lattice planes where the ions form a square grid in each plane, as shown in Fig. 5(b). Further increases in u> r increase the number of ions per unit area of each plane as well as the spacing

between the planes. During this process the square lattice planes smoothly change into rhombic lattice planes and

eventually there is a sudden transition to hexagonal lattice planes. Further increases in u> r eventually produce a structural transition to three square lattice planes, and the basic

pattern repeats. The structure of the crystallized ions depends sensitively on the projected areal density a of the plasma. The side- and

top-view images were analyzed to characterize the phase

structure. Within a layer, the structural order is characterized by the primitive vectors a] and a2 (which are observed to be

equal in magnitude) and the angle 6 (=S90°) between them. The interlayer order is characterized by the axial positions z n of the n lattice planes (measured by the side-view camera)

and the interlayer displacement vector cn between layers 1 and n. Hence, the equilibrium positions in the (x,y) plane of

ions in axial planes 1 and n are given by R1 =: ia1 +ja2 and

Rn = /a1 +ya2 +cn, where i,j are integers. Three different types of intralayer ordering are observed: hexagonal (6 = 60°), square ((9 = 90°), and rhombic (90° > 03=65°). The

observations were compared to the results from Dubin,23 who performed an analytic calculation of the energies of lat- tice planes that are infinite and homogeneous in the (x,y) direction but are confined in the axial direction by a har- monic external electrostatic confinement potential, cf> e = l/2(m/e)coz z

2

. Since this potential is identical to the confinement potential of a Penning trap, as seen in the rotating

frame in the a^0 planar limit, the minimum energy phase

structures predicted by the theory should match the struc- tures observed in the central regions of the oblate plasmas of

the experiments.

Figure 6 displays the agreement between theory and ex- periment for the interlayer quantities, with measurements

taken on different plasmas with 7V< 104

. Lengths have been

normalized by Gt WS2_ D =(3e 2 /4-7re /Ma>

2

)

1/3 = 10.7 /xm,

which is the Wigner-Seitz radius in the planar limit. As the

central areal density is increased, the lattice planes move

farther apart axially in order to match their average density to the neutralizing background. Eventually it becomes energetically favorable to form an additional lattice plane. The symbols indicate whether the lattices had an interlattice displace- ment vector c2 characteristic of the hexagonal phase

(triangles) or the square and rhombic phases (squares).

Figure 7 displays the agreement between experiment and

theory for the dependence of the angle 6 (between the primitive vectors) on central areal charge density a. The trend is

o a,^ (central areal charge density)

FIG. 6. Interlayer structure (plane axial positions and displacement vectors)

as a function of normalized areal charge density. The lines are the predictions of theory, and the symbols are experimental measurements.

that when a new lattice plane is formed, 6 changes discontinuously from =60° to a higher value. As the central areal density of the crystal is further increased, 6 smoothly decreases to =65° until there is a second discontinuous transi- tion to a hexagonal structure. This latter transition has been

predicted32 to become continuous in liquid (T<80) bilayer

systems. The lines indicate the minimum energy structures predicted by the 2-D theory.

IV. DISCUSSION

With Bragg scattering and spatial imaging, we have

measured the correlations in both highly oblate and spherical

strongly coupled 9Be+ion plasmas. The planar geometry

permits a detailed comparison with theoretical calculations. We have measured the preferred lattice structures for up to

five lattice planes in lenticular plasmas and obtain good

agreement with theory. By increasing the number of planes

(by adding more ions to the plasma), the transition from

90

85

§? 80

-o

.22CO 75

2.70

10

&T? ***-.

0.5 1.0 1.5 2.0 a a^2D (central areal charge density)

FIG. 7. Intralayer angle 8 structure as a function of normalized areal charge

density. The lines are the predictions of theory, and the symbols are experi- mental measurements. Representative error bars are included with some of

the measurements. TN-93

Downloaded 25 May 2001 to 132.163.135.22. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp
12 Phys. Plasmas, Vol. 7, No. 1, January 2000 Bollinger et al.

FIG. 8. Five-fold Bragg scattering and real-space patterns obtained by strob- ing the intensified CCD camera synchronously with the rotating electric

field perturbation, (a) A Bragg scattering pattern obtained on an N~\.2

X 105 ion plasma phase locked with a rotating dipole field at wr =2ir X 166.84 kHz. Here V = 500 V and a = 2.6. (b) Real-space image of a lenticular plasma consisting of four horizontal planes in the plasma center. The

rotating dipole field was used to set wr =2-rrX74.35 kHz.

In addition to enhancing studies of Coulomb crystals, the phase-locked control of u> r has improved the prospects of a microwave frequency standard based on a hyperfine-Zeeman

transition of ions stored in a Penning trap. This is because

the time-dilation shift due to the plasma rotation is one of

the largest known systematic shifts in such a standard. In Ref. 35, the potential frequency stability and accuracy of a microwave frequency standard based on 106 trapped ions is discussed. For ions such as

67Zn+ and 201 Hg+

, fractional frequency stabilities ;SlO-14/r1/2 with time-dilation shifts due to the plasma rotation of ~fewXlO-15 are possible. Here r is the measurement time in seconds. With phase- locked operation of the rotating wall, we think it should be

possible to stabilize and evaluate the rotational time-dilation

shift within 1%. Therefore the inaccuracy due to this shift would contribute a few parts in 10

-17

.

surface-dominated to bulk behavior in the planar geometry

can be studied. Ions in a trap have been proposed as a register for a quantum computer.33 Work in this area has fo- cused on a string of a few ions in a linear Paul trap.

34 A

single lattice plane of ions as in Fig. 5 could provide a 2-D

geometry of trapped ions for studies of quantum computing

or entangled quantum states. In spherical plasmas with more than 2 X 105 ions, we

have observed the formation of bcc crystals, the predicted

state for the infinite strongly coupled OCP. The crystals oc- cupied the inner quarter of the plasma diameter. Outside the

crystal there was a complicated transition to a shell structure. In this system we have not observed the thermodynamic

liquid-solid phase transition predicted for the bulk OCP. Our

measurements have concentrated on the correlations obtained at the coldest temperatures (therefore maximum Y)

where the ion fluorescence is maximum. The phase transition may take place in the present system, but we have experimentally missed detecting it, or possibly larger crystals (for example, where the number of ions in the crystal is large

compared to the number of ions in the shells) may be re- quired in order for a sharp phase transition to be exhibited. We have observed structures for which we do not have a good current theoretical understanding. Figure 8(a) shows an approximate five-fold Bragg scattering pattern that was observed a number of times under different experimental cir- cumstances. A five-fold Bragg scattering pattern is characteristic of a quasicrystal. However, more sets of dots would

be present in a true quasicrystalline Bragg scattering pattern. We now think that the five-fold Bragg scattering pattern of

Fig. 8(a) is due to a structure like that shown in Fig. 8(b). Figure 8(b) is a top-view image of a lenticular plasma that consisted of four horizontal planes. Even though it is difficult to distinguish individual ions in this figure, it is possible to see that there are five distinct regions where the ions resided

in vertical planes. The planes from these different regions

form a five-sided structure that would produce a Bragg scat- tering pattern like Fig. 8(a). (With the small crystals and

forward Bragg scattering angles of this work, each set of

vertical planes produces two Bragg peaks.) Once formed,

this five-fold structure was stable and persisted for reasons

that we do not understand.

ACKNOWLEDGMENTS

We thank S. L. Gilbert and R. J. Rafac for their comments and careful reading of the manuscript. We gratefully

acknowledge the support of the Office of Naval Research.

'J. J. Bollinger, T. B. Mitchell, X.-P. Huang, W. M. Itano, J. N. Tan, B. M.

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25 R. C. Davidson, Physics of Non-neutral Plasmas (Addison-Wesley, New

York, 1990), pp. 39-75.

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

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31

J. N. Tan, J. J. Bollinger, B. Jelenkovic, W. M. Itano, and D. J. Wineland,

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TN-95
PHYSICAL REVIEW A, VOLUME 62, 023405

Simulation of a method for forming a laser-cooled positron plasma

A. S. Newbury,* B. M. Jelenkovic,+

J. J. Bollinger, and D. J. Wineland

Time and Frequency Division, National Institute of Standards and Technology, Boulder, Colorado 80303

(Received 13 January 2000; published 18 July 2000)

We have simulated the trapping and cooling of moderated positrons in a Penning trap in which the positrons

lose energy through collisions with a simultaneously stored laser-cooled 9 Be+ plasma. Once the positrons are trapped, they cool through sympathetic cooling with the

9 Be+ plasma. After the positrons cool, their motion

parallel to the magnetic field reaches a state of thermal equilibrium with the 9 Be+

ions and they rotate about

the trap axis at the same frequency as the 9 Be+

ions . Therefore, a centrifugal separation will occur, forcing the positrons to coalesce into a cold column along the trap axis. A simulation which, in part, utilizes Monte Carlo

techniques, indicates a capture efficiency of as high as 0.3% for 300 K moderated positrons passing through a

9 Be + plasma with a density of 1010 atoms cm

-3 and a column length of 1 cm. This capture efficiency leads to the positron capture rate of ~ 1000 positrons per second, assuming a 100 mCi positron source and 10

-3

for the efficiency for moderating positrons from the source. The resulting dense reservoirs of cold positrons may be

useful for antihydrogen production and for reaching a plasma state in which the mode dynamics must be

treated quantum mechanically.

PACS number(s): 32.80.Pj, 52.25.Wz

I. INTRODUCTION

With advances in the use of positron moderators to pro- duce low-energy positron beams [1-12], and in the trapping of non-neutral plasmas [13,14], attention has been focused on trapping and cooling positrons in electromagnetic traps [2-4,9-12,15-17]. Cold positron reservoirs are useful for positron-normal matter interaction studies, such as the study of resonances in low-energy positron annihilation on molecules [4]. With sufficiently high trapping rates, cold positrons can be released from electromagnetic traps to produce

cold beams of high brightness for a number of different ex- periments [4,6,18]. A dense gas of positrons at sufficiently

low temperature also provides an example of a plasma with

quantized normal modes [15,16,19]. Finally, by passing cold

antiprotons through a reservoir of cold positrons, one could

form antihydrogen through three-body recombination [20-

22].Several groups have successfully trapped positrons in electromagnetic traps. Schwinberg, Van Dyck, and Dehmelt

used resistive cooling of the positrons in a Penning trap to achieve trapping of small numbers [23]. Gabrielse, Haarsma,

and Abdullah have combined this method with a 3 mCi

source and a positron moderator to trap ~3X 104 positrons

at a rate exceeding 103 per hour [3]. More recently this group

has been able to trap more than 106 positrons in 17 hours

through a different method where apparently positronium in a high Rydberg state created on the surface of the moderator

is field-ionized in the trap [24,25]. Conti, Ghaffari, and

Steiger have also trapped positrons in a Penning trap by in- jecting slow positrons into the trap while ramping the trap

electrostatic potential [12,26]. Mills has discussed accumulating positrons in a magnetic bottle to produce a slow pos-

*Permanent address: MIT Lincoln Laboratory, Lexington,

MA 02420. Permanent address: Institute of Physics, University of Belgrade,

Belgrade, Yugoslavia.

itron beam [27]. Demonstration of positron trapping in a magnetic mirror by cyclotron-resonance heating has been re- cently demonstrated [28]. The largest number of trapped positrons (-3X108

) has been reported by Surko and co- workers [4,18,29,30]. These experiments employed

collisional cooling of positrons with a room-temperature

buffer gas of N2 to provide trapping and cooling. By removing the buffer- gas, the base pressure is reduced to 3 X 10

-10

Torr, resulting in a positron lifetime of about one

hour. With a 90 mCi positron source, a trapping rate of

3 X 108 positrons in 8 min and a trapping efficiency of moderated positrons greater than 25% were achieved.

In this paper we explore the possibility of capturing and

cooling positrons in a Penning trap through collisions with a simultaneously stored laser-cooled plasma of 9 Be+

ions. Slow positrons become trapped through Coulomb collisions with the 9 Be+ plasma. Once trapped, the positrons will then be sympathetically cooled by the 9 Be+ plasma, which can be

laser-cooled to temperatures as low as 0.5 mK [31,32]. Sympathetic cooling refers to the cooling of one species through

Coulomb interactions or collisions with a second, directly cooled species [32,33]. Since this technique employs high vacuum, positron annihilation will be suppressed, permitting

long trap lifetimes. One of the simplest methods to study the transport of

positrons in a

9 Be+ plasma is the Monte Carlo method. Unlike collisions between neutral atoms, Coulomb collision deflections at large distances are important, with each of these "distant collisions" producing a small scattering and velocity change. In Monte Carlo simulations one can treat the problem of Coulomb collisions through the cumulative effect of a large number of small angle scattering; we have used

Monte Carlo simulations to calculate the scattering angle of

the moderated positron after each pass through the 9 Be+

plasma. The simulations were based on the expression for

the probability distribution for scattering of a positron into an

angle 6 after a large number of collisions, assumed to occur

as a positron passes through the 9 Be+ plasma [34,35]. Recently, Nanbu used a Monte Carlo method to derive a simple

1 050-2947/2000/62(2)/023405( 1 0)/$ 1 5.00 TN-96

62 023405-1 ©2000 The American Physical Society
NEWBURY, JELENKOVIC, BOLLINGER, AND WINELAND PHYSICAL REVIEW A 62 023405

laser

Cu moderator

slow e

fast e

+

y Potential diagram

FIG. 1. Schematic diagram of a cylindrical Penning trap and of

axial potentials. V , moderator potential; Vg

, grid voltage; SEZ

, positron axial kinetic energy above the grid voltage; Vr

, Vc, and VEC , potentials on other trap electrodes.

analytical expression for the probability density function for

a deflection angle after many Coulomb collisions [36]. We

also have done a number of calculations using expressions

from Ref. [36] to calculate the scattering angles for positrons

after passing through the plasma and obtained good agree- ment between the two data sets. The basic method for capturing and cooling positrons us- ing a

9 Be+ plasma, outlined previously in Ref. [16], is dis- cussed in Sec. II. In addition, we describe here some details of a hypothetical experiment designed to trap positrons. In

Sec. Ill we have increased the scope of the discussion by

including the effects on the capture efficiency caused by the energy distribution of moderated positrons, the finite size of

the positron source, the radial electric field within the plasma, and 9 Be+

recoil. The trap geometry, the plasma parameters, and the positron behavior described in Sees. II and

III are used in the modeling of positron trapping. The Monte

Carlo method used to calculate the efficiency of the proposed

method [16] is described in Sec. IV. The results of the simulations and a discussion of the results are presented in Sec. V.

II. BASIC METHOD

The model assumes the

9 Be+

ions are first trapped in a

cylindrical Penning trap contained in a room-temperature vacuum enclosure with an axial magnetic field of 6 T. Figure

1 illustrates the simple Penning trap design considered in modeling the capture of positrons. In this magnetic field, a laser-cooled 9 Be+ plasma in thermal equilibrium can reach a uniform density n Q of up to 10 10 atoms cm

-3

[14,37]. This

high density can be reached by using torques due to a laser beam [37] or due to a rotating electric-field perturbation

[38,39] to control the plasma's angular momentum. A lowenergy positron traversing this plasma along the magneticfield direction will scatter off the

9 Be+

ions via the Coulomb

interaction. The positron's parallel momentum (along the magnetic-field direction) can thus be converted to perpendicular momentum. If sufficient momentum is converted, the positron's momentum along the magnetic field can be reduced so that it will not leave the trap. In Ref. [16], it was

assumed that if the positron were initially captured in this way, it would lose enough energy through cyclotron radiation to be permanently trapped. However, this is true only for

a small fraction of positrons because typically, before the positron can lose enough energy through cyclotron radiation,

its energy can be redistributed back by Coulomb collisions along the trap axis and it will escape [40]. In the work described here, we now include this escape process and the cooling effects of 9 Be+

recoil, which initially provides a more efficient cooling mechanism than cyclotron radiation. In the Penning trap, the plasma rotates around the magnetic-field axis at a frequency a> r

. The technique of

"Doppler" laser cooling [31,41,42] reduces the temperature of the plasma to less than 10 mK. The Debye length of the plasma can be expressed as X D — (kB T€ /n q

2

) m, where k B

is Boltzmann's constant, q is the ion's charge, and e is the permittivity of free space. At temperatures near 10 mK, the Debye length is small compared to plasma dimensions [31]. In this limit, the density of 9 Be +

ions can be expressed as a function of the 9 Be+ cyclotron frequency ft, the plasma

rotation frequency co r

, and the 9 Be+ mass MBe , as n = 2e MBe a> r (ft — u> r )lq 2 [14,19,31]. The maximum achievable density is the Brillouin density, which occurs when o) r — D./2 and has been achieved in laser-cooled plasmas with up to a few hundred thousand 9 Be+

ions in magnetic fields up to 4.5 T [37,39]. For 9 Be+

ions confined in a 6 T magnetic field, this limit is n — 1010 cm

-3

. We will assume that the magnetic field is uniform along the length of the trap. As illustrated in Fig. 1, high-energy positrons from a

22Na

source are injected from the right into the trap, on the trap

axis, through a cylindrical endcap. The positrons have a beta- decay endpoint energy of 545 keV, and these high-energy

positrons will not significantly affect the ions in the plasma

(a discussion of the interactions between hot positrons and

plasma is contained in the Appendix). After passing through

the plasma, the positrons strike a room-temperature crystal moderator. The positrons will thermalize by interacting with

electrons and phonons in the crystal. In this "reflection ge- ometry," a small fraction (up to 10

-3

) will avoid annihilation in the crystal and emerge as a beam of slow positrons

[1,5,43] which then enter the trap [44]. At the surface of the

crystal the moderated positrons have an energy determined

by the crystal temperature. In addition, they are accelerated

in the direction normal to the crystal surface by the work

function <$> of the crystal [1]. For the method of trapping

positrons discussed here, the narrow distribution of thermal- energy positrons at the surface of the crystal is important. Measurements show that positrons emitted from a Cu(lll)

single-crystal moderator can have a narrow energy distribu- tion whose width is reasonably consistent with thermal broadening given by the temperature of the moderator crystal

[45]. In our calculations, for the purpose of the crystal work

function, we assume the use of a Cu(lll) crystal moderator.

After the positrons are emitted from the moderator crystal, their axial kinetic energy is assumed to be further re- duced by a conducting screen with good transmission (the retarding grid of Fig. 1), which has a potential a few tenths of a volt above the moderator potential. If the moderator and

023405-2

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SIMULATION OF A METHOD FOR FORMING A LASER- PHYSICAL REVIEW A 62 023405

retarding grid potentials are equal, the axial energy of positrons as they pass through the grid is <& + Ez

, where E[ is the axial component of the positron's thermal energy at the

crystal surface. The positrons will then enter the 9 Be + plasma with relatively little kinetic energy. At these low energies ( — few eV), positron annihilation on the 9 Be+

ions is made negligible by Coulomb repulsion. By adjusting the potential of the rightmost cylindrical electrode, we can ensure

that the moderated positrons are reflected at the end of the plasma farthest from the moderator and pass through the plasma twice. During each pass, some of the axial energy can

be converted to perpendicular energy through Coulomb collisions with the

9 Be+

ions , thereby preventing them from

escaping back through the retarding grid. Positrons that re- main trapped for many passes will lose enough energy

through 9 Be+recoil to remain permanently trapped. The

positrons that are not trapped are assumed to strike the moderator or grid and annihilate. Once the positrons are trapped within the laser-cooled

9 Be+ plasma, they will be cooled through a combination of

sympathetic cooling through 9 Be+

-e + Coulomb collisions and cyclotron radiation. After the positrons are cooled by the

9 Be+ plasma, both positrons and 9 Be+

ions will undergo

uniform rotation at the same frequency (o r and the positrons

will be forced to the center of the rotating plasma because of

their smaller mass [33,46]. In the limit of zero temperature,

the edges of each plasma will be sharp, and the plasmas will separate, with the positrons forming a column of uniform

density along the trap axis. If the 9 Be+ plasma density is significantly below the Brillouin limit, the densities for confined plasmas of e + and 9 Be+

are expected to be approximately equal and the plasma separation quite small [15,46].

This implies good thermal coupling and possible positron

axial temperatures less than 10 mK. The discussion of

strongly magnetized plasma equilibria by Glinsky et al. [47]

indicates that the positron plasma axial and cyclotron degrees of freedom will be strongly decoupled in a 6 T magnetic field. Therefore, cyclotron radiation may keep the positron cyclotron temperature in near-thermal equilibrium with

the trap electrodes. Here we assume the electrodes are maintained at room temperature, but the equilibrium cyclotron

temperature could be reduced, for example, by cooling the electrodes to 4 K with a liquid helium bath or to lower temperature with a dilution refrigerator.

One way to experimentally detect the presence of trapped

positrons could be by imaging the near-resonant 313 nm

fluorescence of the 9 Be+ plasma and looking for the absence

of 9 Be+ions in the center of the plasma [16]. Other ions with charge-to-mass ratios higher than 9 Be+

, such as

4 He+

, H^" , and 9 Be2+

, will also be trapped in the center of the plasma. These ions will not fluoresce at 313 nm and will therefore mimic the positron signature on the imaging tube. We anticipate that these ions could be distinguished from the positrons through their resonant response to radiation applied

at the cyclotron frequency. The size of the "hole" in the

9 Be+ plasma will yield an estimate of the number of trapped

positrons. With the imaging technique we estimate we can

detect the presence of a single "string" of a few tens of

positrons trapped on the axis within the 9 Be+ plasma [48].

III. POSITRON TRAPPING

Positrons within the Cu(lll) moderator crystal rapidly thermalize to a Boltzmann velocity distribution [1]. Within

the crystal, the positron velocity distribution P(u,) will con- form to

P(vi)*e -me v jIlkgT (1)

where the subscript i indicates the velocity direction (i =x,y,z), m e is the positron mass, and T is the temperature

of the crystal moderator. Positrons emitted from the moderator are accelerated in the direction perpendicular to the moderator surface by the crystal work function <$> . As indicated

in Fig. 1, the positron velocity is primarily along the magnetic-field axis (z) since the crystal surface is oriented perpendicular to that axis. Immediately outside the crystal the slow positrons will have an axial kinetic energy distribution,

P(Ez )dEz *e -£,-<&„/* o'kB TdE (2)

for £z ^4> . This distribution combines the probability of

effusion from the moderator surface [7,49] with the acceleration at the surface due to the work function. Equivalently, we

can assume the positron axial velocity at the crystal surface

is selected from the distribution

P(v z )*v z em< v

z /2kBT

, (3)

and then accelerated by the potential <J> . The grid potential can be adjusted so that the positrons have small excess axial energy (SEZ

) with respect to the grid. Before reaching the plasma, the positrons will be accelerated by the plasma potential, Vp (r)= —n qr

2/(4e ). [For simplicity, we have as- sumed that we adjust the moderator and electrode potentials

to make the plasma potential along the trap axis Vp (r = 0) = 0.] We have modeled the initial collisions of the moderated

positrons with the 9 Be+ions in the weakly magnetized ap- proximation where the effects of the magnetic field on the

collisions are neglected. This approximation is valid as long

as the positron's cyclotron rotation is less than one cycle

during the time of a collision [47]. The number of cyclotron

orbits during a collision can be defined as k = D,c t, where fl f is the positron cyclotron frequency and r is the binary colli- sion time. Therefore, we consider collisions where /c<l

[47]. For example, the minimum collision time is r=blv,

where b = q

2/2Tre me v2

is the collisional distance of closest approach. For an energy of 0.1 eV, v — 1.9X107 cm s-1

, making 5-1.4X10-6 cm. At 6 T, ft f -l.lxl012 s"

1

, yielding k — O.OS. We calculate the initial capture of positrons after one pass by using the distribution for multiple small-angle Coulomb

scattering [34,35]. (The cross section for multiple small- angle Coulomb scattering is typically larger than the cross

section for a single large-angle scattering [50].) Below, we

define a "pass" through the plasma as a pass back and forth

(or from left to right and back in Fig. 1) ending with the

023405-3

TN-98
NEWBURY, JELENKOVIC, BOLLINGER, AND WINELAND PHYSICAL REVIEW A 62 023405

positron traveling towards the moderator crystal. As seen in Fig. 1, the positron can only leak out of the trap the way it entered. The angular distribution for multiple small-angle Coulomb scattering can be simulated by calculating an energy- dependent rms scattering angle \l{6

2

) according to the Rutherford scattering formula such that [50]

{02 y- n lq A

i 2 2 4 2ire me v In (4)

Here / represents twice the plasma length, and bj^ and b^

are the maximum and minimum impact parameters, respectively. We use b max =v/Clc

, where Cl c

is the positron cyclotron frequency and v is the magnitude of the positron velocity. The quantity b max is the maximum impact parameter for which we can use the weakly magnetized approximation. It

is derived by setting the parameter k— 1 . For the parameters

used in the above discussion of k, 6 max =0.17 fim, which is more than an order of magnitude smaller than the mean ion spacing in the Be+ plasma. We use & min =F/2 to limit the

scattering to small angles [50]. The probability of multiple

scattering in one pass through an angle 6 can then be ap- proximated by a Gaussian distribution in solid angle

[34,35,51],

e

2 P5 ( 6>)rffl oc s in(0)exp| —\ddd<f>. (5)

This distribution is valid for multiple angle scatterings where

each is less than 10° [35]. The capture of positrons within the 9 Be+ plasma is di- vided into two processes. The first process is based on Coulomb collisions and traps the positrons temporarily. After a

single pass a positron can be trapped if the amount of axial energy converted into perpendicular or cyclotron energy is greater than the excess axial kinetic energy of the positron. Because of the difference in the positron and 9 Be+ masses,

positrons will actually lose very little energy by passing once

through the plasma. If initially trapped, the positron will continue to make passes through the plasma until it either es- capes the trap or becomes permanently trapped. To escape

the trap, a positron which is "initially captured" needs to convert its perpendicular energy back to axial energy. The second process permanently traps the positrons by

depleting their excess energy primarily through 9 Be+

recoil cooling. A positron with energy E scattering off the ions in the plasma through an angle 6 will lose an energy

- : i|^-jsinz(0/2)

^Be/

(6)

to

9 Be+

recoil. Because of the large mass difference be- tween a

9 Be + ion and a positron, a positron will have to make many transits through the plasma in order to lose its excess energy. But once sufficient positrons are trapped,

other positrons can lose axial energy through e +

-e +

colli- sions. The trapping efficiency under these collisions is expected to be higher than for the e + -9 Be+

collisions because

of the larger energy loss due to positron recoil. This en- hanced recoil cooling is not taken into account here.

IV. SIMULATION

The Monte Carlo simulation proceeds as follows. For

each positron, an initial radial coordinate is chosen according

to a flat distribution over the active area of the source. The

initial velocities out of the moderator in the x and y directions are chosen using velocity distribution functions P(v t ) of Eq. (1). The z component of the positron velocity was

obtained using a modified Boltzmann distribution [Eq. (3)]. Equivalently, the velocity v z at the surface of the crystal is determined from the equation v z = v$[ — ln(l—Rn)] m, where

Rnis a random number between and 1. Here v^ = \J2kT/m, where Tis the temperature of the crystal. We use E\^=mv\l2 to denote the axial kinetic energy. At the moderator surface the positron is further accelerated in the axial direction by the surface work function. We have used the Cu(lll) work function <l> = 0.4 eV in the Monte Carlo

simulation. Figure 1 illustrates the electrical potential experienced by

the positrons as they travel from the moderator, held at V , through the grid at potential Vg

, and into the 9 Be+ plasma.

The moderated positrons with an axial energy above the re- tarding grid potential, SEz [ = q(V -Vg

) + ^ + Ef], follow

the magnetic-field lines and accelerate into the plasma. Their

radial coordinate r with respect to the trap symmetry axis does not change until they undergo a large number of colli- sions inside the plasma, since their cyclotron radius is less than 10 -4 cm. At low temperature, the electric potential in- side the plasma is approximately independent of the axial coordinate, and is given by Vp(r)= —n qr

2/(4e ). Coulomb scattering caused by one pass through the plasma is described by two angles, &s

, the magnitude of the deflection angle, and <f> s

, the orientation of the scattering

around the deflection cone. The scattering angle 6S was cal- culated assuming the distribution given by Eq. (5). This as- sumption leads to an expression for the scattering angle 8S = yj(6

2 )[-\\\{R e)\ where ( 6

2

) is the rms scattering angle

given by Eq. (4). Since the positron has no preferred azi- muthal orientation, 4>s was obtained at the end of the pass from a uniform distribution 2ttR^. Here R e and i?^ are random numbers between and 1. At the end of each pass, the new values of the v x , v

y

, u z were calculated from 8S ,

cf) s

, and the change of energy [Eq. (6)]. A test was then made

to determine if the positron was permanently trapped. If this

is not the case, the positron will either make another pass or

is lost. By repeating these "runs," we determine the percent- age of moderated positrons trapped within the plasma. Typically, the Monte Carlo runs had 1.5 X 105 positrons.

V. RESULTS AND DISCUSSION

Results for the efficiency of trapping positrons in a

9 Be+ plasma for particular conditions are shown in Fig. 2. In this case the plasma radius was 0.1 mm, and the density was

10 10 cm

-3

. We chose the moderator potential to be V

023405-4

TN-99
SIMULATION OF A METHOD FOR FORMING A LASER- . . . PHYSICAL REVIEW A 62 023405

100-

80-

S- 60 H

cdO)

raS 40 -| o

53 Q. 20

0-

J i I i I i L.

CD D—DDI Moderator T : 300 K —•—permanent —a— temporary —O— integral EDF

3.35 3.36 3.37 3.38 3.39 3.40 3.41 3.42 3.43 3.44 3.45 Retarding grid potential (V)

FIG. 2. Fraction of moderated positrons entering the trap (inte- gral EDF) along with the fraction of temporarily and permanently

captured positrons as a function of the retarding grid voltage for V =3 V. The crystal work function was 0.4 eV and the temperature was 300 K. The 9 Be+ plasma parameters were as follows: density n = 1010 cm

-3

, length /= 1 cm, and radius r = 0.l mm.

Squares, integral energy distribution of positrons; circles, permanently trapped; triangles, temporarily trapped.

=3 V, while the grid potential Vg was varied around V + <J> = 3.4 V. The percentage of temporarily trapped (tri- angles) and permanently trapped (circles) positrons is plotted

as a function of Vg. Also shown (squares) is the fraction of

positrons entering the trap vs the retarding grid potential.

This curve is the integral distribution of the axial component

of kinetic energy, Ez

. For the results of Fig. 2, the temperature of the crystal was taken to be 300 K. The maximum

efficiency for positron trapping occurs at Vg = 3.4 V and was —0.4% for permanent and 24% for temporary trapping

only. The results of the trapping efficiencies for the crystal

cooled to 100 K are shown in Fig. 3. The width of the energy

distribution is reduced by a factor of 3. Such thermal narrowing has been confirmed in experiments [7]. The efficiency for permanent trapping increases to —2.5%. Figures 2 and 3

indicate, as mentioned previously, that the energy spread of

the moderated positrons is important for the trapping method

simulated here. Experimental studies have reported near- thermal energy spreads for metal, single-crystal moderators

[7,45]. In practice this condition may not be straightforward

to obtain. Figure 4 shows the trapping efficiency for an en- ergy spread of the moderated positrons corresponding to T

= 2000 K. The factor of 7 increase in the positron energy

spread of Fig. 4 over Fig. 2 has resulted in a factor of 40

decrease in the efficiency for permanently trapping positrons. In Fig. 5 we show the fraction of captured positrons for

different bias potentials V of the moderator while holding

the crystal temperature (300 K) and plasma parameters (density, length, and radius) constant. For each value of V in Fig. 5, the grid voltage was set to the value Vg =V

+ 0.4 V which maximizes the trapping efficiency. The data

100 HI

80-

2^ 60-

CD

CD

CD Q.

40-

20-

04>

-a a a—o-i Moderator T: 100K

••— permanent

a— temporary

integral EDF

3.36 3.38 3.40 3.42 Retarding grid potential (V)3.44

FIG. 3. Same as in Fig. 2 but for a moderator temperature of 100 K.

show that the efficiency for permanent trapping has a maximum of —0.45% when V is at about 4 V. While we do not have a detailed understanding of the location of this maximum, we can describe some effects which could produce it. Once temporarily trapped, a positron will leave the plasma if

it enters the loss cone dc about the z axis defined by sin(0c) = \J8ET IET, where ET

is the total positron kinetic energy in the plasma and SET

is the excess kinetic energy the positron must lose to be trapped. A positron with a larger kinetic energy and the same excess energy has a smaller loss cone, which tends to increase the efficiency of trapping positrons

with increasing energy. However, the energy loss per pass of

a positron decreases with energy. This increases the number

of passes required to permanently trap a positron (see discus- sions below), and will tend to decrease the trapping effi- ciency with increasing energy.

100-

80

S- 60 -| CDO)

CO

CD Q.

40-

20-

Moderator T: 2000 K —•— permanent —A— temporary —D— integral EDF

0- •-•-

3.35 3.40 3.45 3.50 3.55 3.60 Retarding grid potential (V)

3.65

FIG. 4. Same as in Fig. 2 but for a moderator temperature of 2000 K.

023405-5

TN-100
NEWBURY, JELENKOVIC, BOLLINGER, AND WINELAND PHYSICAL REVIEW A 62 023405

50

40

g> 30 EL

Q.

<D 20

COo

°- 10H

x100

permanent ^— temporary

50

10 15 20 Moderator potential (V)

FIG. 5. Capture percentage of permanently trapped (circles) and

temporarily trapped (triangles) positrons as a function of V . The

moderator work function was 0.4 eV and grid potential was always

VG =V + OA.(n =lOm cm~ 3

, 1=1 cm,andr = 0.1 mm.)

40-

D3C

'a. a.2

|

o

a.

30

20

10'

—•— permanent —a—temporary

x100

Density (10

9 cm*

3

)

—

r10

FIG. 7. Capture percentage of permanently trapped (circles) and

temporarily trapped (triangles) positrons as a function of 9 Be+ plasma density = 4.4 V).

(r = 0.1 mm, 1=1 cm, V = 4 V, and V

Shown in Figs. 6 and 7 are the variations in efficiencies for trapping positrons as the 9 Be+ plasma length and density were changed. The percentage of temporarily trapped positrons is increasing as a square root of both length and density. The probability for temporarily trapping positrons de- pends on the final scattering angle 6, which in turn is proportional to Vin. The percentage of permanently trapped

positrons is increasing linearly with n and /, possibly because

the energy loss for permanent trapping [Eq. (6)] varies as the square of the rms scattering angle.

The efficiency for trapping positrons decreases with in- creasing plasma radius, as shown in Fig. 8, for the same

80

70'

60- ^ 50-

'5. Q.W 40-

2 30-| ~mo

0. 20

10'

J I I 1 I i_

• permanent

fit to permanent

-a— temporary

x100

i 1 1

1 1 1 1 1 1 1 1 1

1 11

1

1 1 r 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Length (cm)

FIG. 6. Capture percentage of permanently trapped (circles) and

temporarily trapped (triangles) positrons as a function of 9 Be + plasma length (n =1010 cm" 3

, r = 0.1 mm, V = 4 V, and Vg = 4.4 V).

reason that it decreases with increasing moderator potential

V for V >4 V (see Fig. 5). At larger radii, the positrons

will have correspondingly larger energies entering the plasma. Throughout this manuscript we assume that the source radius is equal to the plasma radius. The number of round trips in the plasma before the positron either exits back through the grid or is permanently

trapped varies with the positron excess energy, values of V

and Vg

, and the plasma radius. The histogram in Fig. 9

shows the fraction of trapped positrons vs the number of

passes the positrons made through the 9 Be+ plasma before

40

35

30' TM 25 C

'5. Q. 20 H

o 15-

coo

O- 10-

5-

_i I i I i I i

0.0 0.2 0.4 —r~

0.6 —f—0.8 —J—1.0 1.2 1.4 1.6 Radius (mm)

FIG. 8. Capture percentage of permanently trapped (circles) and

temporarily trapped (triangles) positrons as a function of the radius of the 9 Be + plasma (n =1010 cm" 3

, /=1 cm, V = 4 V, and V^ = 4.4 V). We assume the positron source radius equals the plasma radius.

023405-6

TN-101
SIMULATION OF A METHOD FOR FORMING A LASER- PHYSICAL REVIEW A 62 023405

W 2Q o

-*—

*

w

§.10

-o

<DQ.

Q.

CO 0- iz»- 40 o

CDo>

(0 30 Co

^ 20

CD Q_

10-

10 15 20 25 30

b)

JZZl

80 100 120

10 passes

FIG. 9. Histogram showing the fraction of permanently trapped

positrons vs the number of passes the positrons made through the

9 Be + plasma before being trapped, for n = 1010 cm

-3

, / = 1 cm, r =0.\ mm, and T

(b) V =10 V, VG =10.4 V.

300 K. (a) V = 4 V, VG = 4A V;

being permanently trapped. The results are shown for a plasma radius of 0.1 mm and for two moderator potentials,

V = 3 V (Vg = 3A V) [Fig. 9(a)] and V =10 V (Vg = 10.4 V) [Fig. 9(b)]. Although the positrons have the same

excess energy as they pass the grid in Figs. 9(a) and 9(b), they spend different times in the trap before being captured

because they enter the plasma with different kinetic energies. Similar increases in the number of passes were obtained

when the plasma radius was increased to 0.5 mm. The

plasma potential decreases as — r

2 and therefore positrons

entering the plasma at larger radius have higher energies.

We can estimate the overall capture rate given the results of the simulation by including an estimate of both the source and the moderator efficiencies. A 100 mCi source will iso- tropically produce positrons at a rate of 3 X 109 s~ ' . Only a

fraction of the emitted positrons will reach the moderator

crystal. We expect the positron flux at the Cu crystal to be — 4X108 s _1

[24]. Assuming a moderator efficiency of

10~ 3 and the trapping efficiency of 0.3%, we get a trapping

rate of about 1300 positrons per second. Using the method outlined in this paper, it should be possible to achieve a low-temperature, high-density positron

plasma. In a magnetized, uncorrected plasma, the antihydro- gen recombination rate should scale as n

2 T~ 9' 2 [20]. In a

correlated plasma (plasma exhibiting liquidlike and solidlike behavior), this dependence will likely be modified. Further- more, a pressure of 1.3 X 10~ 8 Pa ( 10" 10 torr) may provide

positron lifetimes longer than 5 days (see the Appendix).

Since the Brillouin limit to the plasma density increases as

the square of the magnetic field, it is possible to increase

these trapping efficiencies further by going to larger magnetic fields. We note that other electrode geometries can replace the transparent retarding grid. Any geometry which provides a

potential hill between the moderator and the 9 Be+ plasma

can mimic the effects of the retarding grid. In an experiment,

a geometry other than a grid is desirable because azimuthal

asymmetries in the retarding grid potential near the plasma

might limit the ultimate 9 Be+ plasma density [14,37]. In this manuscript we assume that the 9 Be+

ions recoil from positron impact as if they were free particles. In fact, laser-cooled ion plasmas are often strongly coupled and exhibit liquidlike or solidlike behavior where an ion is bound in a local potential well. However, because the collision time of

the weakly magnetized collisions considered here is fast compared to the period of any of the ion's plasma-mode

frequencies, in considering their recoil we may treat the

9 Be+

ions as if they were free particles. In addition to the importance of achieving a relatively

low-temperature thermal energy spread of the moderated

positrons, perhaps the largest uncertainty in an experiment

designed along these lines is the ability to produce highdensity, laser-cooled 9 Be+ plasmas of sufficient length.

While large-number plasmas (=109 ions [52]) and highdensity plasmas (n — 10 10 cm

-3

[37]) have been achieved in Penning traps, the combination of these two parameters has not yet been experimentally realized. In recent experiments we have been able to reach the Brillouin limit with ~ 106

ions in a 4.5 T and 6 T magnetic field [39,53]. It may also be

possible to "stack" a series of shorter plasmas in separate

traps along the magnetic field, thereby maintaining high density and increasing the effective column length. However,

even with a modestly sized single plasma, it should be possible to trap a sufficient number of positrons to evaluate the effectiveness of this technique.

ACKNOWLEDGMENTS

Support for this research was provided by the Office of

Naval Research. We thank Travis Mitchell and Tom Heavner

for their comments on the manuscript.

APPENDIX

It is important to discuss interactions between the energetic positrons from the 22Na source and the cold 9 Be+

/e +

plasma. Here, we examine 9 Be+loss due to positron impact

ionization, plasma heating caused by the positron beam, and

the loss of trapped positrons due to interactions with back- ground gas. For simplicity we assume a 200 keV monoenergetic positron beam (the peak energy of the

22Na beta-decay

distribution) from an isotropic 2 mCi source and a

9 Be+ plasma of 1 cm length, with a 1 mm diameter and a density

10 ions cm" 3 (-8X107 of 10 ions).

The probability of an individual scattering event between

a positron and a

9 Be+ ion can be expressed as P = n al'

,

where n is the ion number density, a is the event cross

section, and /' is the effective path length through the plasma. Since positrons are emitted from the source isotro023405-7 TN-102
NEWBURY, JELENKOVIC, BOLLINGER, AND WINELAND PHYSICAL REVIEW A 62 023405

pically, many will have initial velocities perpendicular to the magnetic-field direction. These positrons will spiral along

the magnetic-field lines. Spiraling through the plasma will increase the path length of these particular ions through the plasma. We can eliminate the positrons with the largest ef- fective path lengths by electrically retarding the positron

beam from the

22Na source. By placing a potential hill of 1400 V between the source and the plasma, we can prohibit

the positrons with the longest path lengths from making it to the plasma. Eliminating only —9% of the positrons in this manner, we reduce the average path length through the plasma to 2.4/. The calculations below for 9 Be+ heating and

loss assume this effective path length.

1.

9 Be+

loss The energetic positrons from the 22Na source can doubly

ionize the

9 Be+ plasma through electron impact ionization.

9 Be2+ will remain trapped but can be only sympathetically

cooled; its presence can decrease the cooling capacity of the ion plasma by reducing the number of laser-cooled ions. The

cross section for second ionization of Be+ through 200 keV

electron impact is approximately [54]

fTT/2 da

EH = 2-nnNv E(d)—si

JeM ad

smddd, (A3)

o-(Be + + e"^Bez + + 2e")^3.1X 10 17 cm2

. (Al)

We assume the electron-impact and positron-impact ionization cross sections are approximately equal for positrons of

this high energy [55]. Using this cross section and a total flux of positrons Re + of 1.3X107 s , the number density of

9 Be+

ions n , and the average path length through the plasma 2.4/, we can estimate the loss rate of 9 Be+

ions as

RBe2+ = 2.4ln <jR e +*a ll s" (A2)

At this rate, 7% of the initial 8X107 9Be+ions would be

lost in about 6 days.

Another mode of 9 Be+loss is through high-energy posi- tron annihilation on the 9 Be+

ions. The cross section for positrons with 200 keV of kinetic energy to annihilate on

9 Be+

is approximately 3X10-25 cm2 [56]. Thus loss of

9 Be+ through this mechanism caused by positrons from the

22Na source is negligible.

2.

9 Be+ plasma heating

High-energy positrons passing through a cold dense 9 Be+ plasma can heat the plasma via Coulomb collisions. Since our plasma is simultaneously laser cooled, it is necessary that the rate of laser cooling be larger than that of the positron

heating. To estimate the heating rate, we perform a calculation of nonrelativistic scattering. Since the heating from

positron-positron collisions dominates over collisions be- tween positron and 9 Be+

, we estimate the heating rate due to trapped positron recoil. A high-energy positron scattering

through an angle 6 will impart an energy E(0) = (l/2)m e v

2 sin2(0) to the trapped positron. We can estimate

the rate of plasma heating EH by integrating,

where n is the density of positrons in the beam, v is their velocity, N is the number of positrons in the plasma, and 6M

is the minimum scattering angle for which the weakly magnetized approximation is valid [47]. This corresponds to an impact parameter approximately equal to the positron beam

radius. In this limit, we estimate a heating rate of EH —A.\ x 10~ 8 eV/s for each positron in the plasma. If we assume a positron column plasma 1 mm in diameter and 1 cm in length containing 8X107 positrons, the plasma heating rate

will be 3.3 eV/s.

Since the heating rate scales as E~ 1 '2

, where £, is the energy of incident positrons, the heating from moderated

positrons incident on the plasma is significantly higher than

that from the unmoderated ones. Taking into account the moderator efficiency, the overall heating from these positrons is comparable to that of the unmoderated ones.

It is necessary to compare this heating rate to EL , the rate

at which energy is removed from the plasma through laser cooling. We assume a 313 nm laser beam directed perpendicularly to the magnetic field with a 25 /im waist perpendicular to the magnetic axis and 250 /xm along the axis, centered on the ion plasma. The laser intensity is adjusted to give a resonant scatter rate of 10 MHz for an ion at the center of the beam. We assume a

9 Be+ cloud of 1 cm in length and

1 mm in diameter rotating at wr = 27r(5 MHz). Laser cool- ing is most efficient using a laser beam propagating along the

trap z axis because the Doppler shift associated with the plasma rotation is absent. Experimentally, this would be difficult to realize in the apparatus described here because the positron source and moderator also lie on the z axis. We

estimate the laser-cooling rate using Eq. (17) of Ref. [28]. We find that for a laser detuning of 20 MHz and a

9 Be+

plasma temperature of 1 K, EL — -1000 eV/s. Since \EL \ >\Eff\, the plasma heating from positron impact should not

significantly affect the plasma equilibrium.

3. Positron loss We can estimate the rate at which trapped positrons are

lost due to background collisions by scaling the results of

Murphy and Surko [29]. In their experiment, positrons were

trapped and cooled through collisions with a roomtemperature background gas of nitrogen [29]. The trap life- time was limited to 40 s because of annihilation and positronium formation on the 1.3X10 -4 Pa (10~ 6 torr) N2 background. Background gas pressures in room-temperature

Penning traps approach 1.3X10" 8 Pa (10

-10

torr). If we

assume that the cross sections for annihilation on other back- ground gases are similar to that of N2 [57], our trap lifetime should approach 5 days, long enough to accumulate a sig- nificant number of positrons.

We have also estimated the number of positrons ejected

from the trap due to large-angle scattering by positrons from

the positron source and the moderator. The Rutherfordscattering cross section for these collisions is quite small and

the trap loss rate is lower than that of background collisions.

023405-8

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SIMULATION OF A METHOD FOR FORMING A LASER- PHYSICAL REVIEW A 62 023405

[1] P.J. Schultz and K.G. Lynn, Rev. Mod. Phys. 60, 701 (1988), and references therein.

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Scientific, Singapore, 1988), p. 63. [16] D.J. Wineland, CS. Weimer, and J.J. Bollinger, Hyperfine Interact. 76, 115 (1993).

[17] H. Boehmer, in Slow Positron Beam Techniques for Solids and

Surfaces, edited by E. Ottewitte and A. Weiss, AIP Conf. Proc. 303 (AIP, New York, 1994), p. 422.

[18] S.J. Gilbert, C Kurtz, R.G. Greaves, and CM. Surko, Appl.

Phys. Lett. 70, 1944 (1997). [19] J.H. Malmberg and T.M. O'Neil, Phys. Rev. Lett. 39, 1333

(1977).

[20] G. Gabrielse, S.L. Rolston, L. Haarsma, and W. Kells, Phys.

Lett. A 129, 38 (1988). [21] M.E. Glinsky and T.M. O'Neil, Phys. Fluids B 3, 1279 (1991).

[22] G. Gabrielse, W. Jhe, D. Phillips, W. Quint, C. Tseng, L. Haarsma, K. Abdullah, J. Grobner, and H. Kalinowsky, Hyperfine Interact. 76, 81 (1993).

[23] P.B. Schwinberg, R.S. Van Dyck, and H.G. Dehmelt, Phys.

Lett. 81A, 119 (1981).

[24] D.S. Hall, Ph.D. thesis, Harvard University, Cambridge, MA,

1997.

[25] G. Gabrielse, J. Estrada. S. Peil, T. Roach. J.N. Tan, and P. Yesley, in Non-neutral Plasma Physics III, edited by J.J. Bollinger, R.L. Spencer, and R.C Davidson (AIP, New York,

1999), p. 29. [26] B. Ghaffari and R.S. Conti, Phys. Rev. Lett. 75, 3118 (1995).

[27] A.P. Mills, Jr., in Positron Scattering in Gases, edited by J.W. Humberston and M.R.C McDowell (Plenum Press, New York,

1984), p. 121.

[28] H. Boehmer, M. Adams, and N. Rynn, Phys. Plasmas 2, 4369

(1995).

[29] T.J. Murphy and CM. Surko, Phys. Rev. A 46, 5696 (1992).

[30] CM. Surko, S.J. Gilbert, and R.G. Greaves, in Non-neutral Plasma Physics III, edited by J.J. Bollinger, R.L. Spencer, and

R.C. Davidson (AIP, New York, 1999), p. 3. [31] L.R. Brewer, J.D. Prestage, J.J. Bollinger, Wayne M. Itano, D.J. Larson, and D.J. Wineland, Phys. Rev. A 38, 859 (1988).

[32] For reviews, see D.J. Wineland, Wayne 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, 1985), p. 3; W.D. Phillips, J.V. Prodan, and

H. Metcalf, ibid., p. 338, and references therein. [33] D.J. Larson, J.C. Bergquist, J.J. Bollinger, Wayne M. Itano, and D.J. Wineland, Phys. Rev. Lett. 57, 70 (1986). [34] J.D. Jackson, Classical Electrodynamics (John Wiley and

Sons, New York, 1975).

[35] W.R. Leo, Techniques for Nuclear and Particle Physics Ex- periments (Springer-Verlag, New York, 1987), p. 44. [36] K. Nanbu, Phys. Rev. E 55, 4642 (1997).

[37] D.J. Heinzen, J.J. Bollinger, F.L. Moore, Wayne M. Itano, and

D.J. Wineland, Phys. Rev. Lett. 66, 2080 (1991). [38] X.-P. Huang, J.J. Bollinger, T.B. Mitchell, and W.M. Itano, Phys. Rev. Lett. 80, 73 (1998).

[39] X.-P. Huang, J.J. Bollinger, T.B. Mitchel, W.M. Itano, and

D.H.E. Dubin, Phys. Plasmas 5, 1656 (1998). [40] We thank T. M. O'Neil, U. C. San Diego for pointing out this mechanism for positron loss. [41] D.J. Wineland, R.E. Drullinger, and F.L. Walls, Phys. Rev.

Lett. 40, 1639 (1978); W.M. Itano and D.J. Wineland, Phys. Rev. A 25, 35 (1982).

[42] W.M. Itano, L.R. Brewer, D.J. Larson, and D.J. Wineland,

Phys. Rev. A 38, 5698 (1988). [43] D.T. Britton, P.A. Huttunen, J. Makinen, E. Soininen, and A. Vehanen, Phys. Rev. Lett. 62, 2413 (1989).

[44] Alternatively, the positron source and moderator could be kept on axis behind one endcap by using a thin crystal film trans- mission moderator. However, the transmitted positron energy

spread is heavily dependent on the foil quality and thickness. [45] C.A. Murray and A.P. Mills, Jr., Solid State Commun. 34, 789

(1980); R.J. Wilson, Phys. Rev. B 27, 6974 (1983).

[46] T.M. O'Neil, Phys. Fluids 24, 1447 (1981).

[47] M.E. Glinsky, T.M. O'Neil, M.B. Rosenbluth, K. Tsuruta, and

S. Ichimaru, Phys. Fluids B 4, 1156 (1992).

[48] S.L. Gilbert, J.J. Bollinger, and D.J. Wineland, Phys. Rev.

Lett. 60, 2022 (1988).

[49] F. Reif, Fundamentals of Statistical and Thermal Physics

(McGraw-Hill, New York, 1965), p. 273.

[50] X. Krall and X. Trivelpiece, Principles of Plasma Physics

(McGraw-Hill, New York, 1992), p. 1156.

[51] In Ref. [16], a distribution /

5 (0)rf0ocexp(0 2

/<6>

2 ))</6> was as- sumed whereas the distribution P(6)d6<xexp(62/(0

2))sin Odd

should have been used. Assuming small angles (sin 0=0), Eq.

023405-9

TN-104
NEWBURY, JELENKOVIC, BOLLINGER, AND WINELAND PHYSICAL REVIEW A 62 023405

(9) of Ref. [16] should read 7/c = exp(-62/Ad2

s

), where A o] = {d2

). This does not significantly affect the results stated in Table 1 of Ref. [16], except for the last value of r)c , which is overestimated.

[52] C.F. Driscoll and J.H. Malmberg, Phys. Rev. Lett. 50, 167

(1983); F. Anderegg, E. Sarid, and C.F. Driscoll, Bull. Am.

Phys. Soc. 38, 1971 (1993).

[53] B.M. Jelenkovic, T.B. Mitchell, J.J. Bollinger, X.-P. Huang,

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Soc. 43, No. 8, 1653 (1998).

[54] H. Tawara and T. Kato, At. Data Nucl. Data Tables 35, 167

(1987).

[55] E. Morenzoni, in Collisions ofAntiparticles with Atoms, edited by D. Berenyi and G. Hock (Springer-Verlag, Berlin, 1991), p. 173.

[56] H.S.W. Massey, E.H.S. Burhop, and H. B. Gilbody, Electronic and Ionic Impact Phenomena (Oxford University Press, London, 1974), Vol. V, p. 3125.

[57] P.A. Fraser, in Advances in Atomic and Molecular Physics 4 (Academic Press, New York, 1968), p. 63.

023405-10

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Volume 87, Number 18 PHYSICAL REVIEW LETTERS 29 October 2001

Stick-Slip Dynamics of a Stressed Ion Crystal

T.B. Mitchell, 1 J.J. Bollinger, 2 W.M. Itano,2 and D.H.E. Dubin3

' Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716

'Time and Frequency Division, National Institute of Standards and Technology, Boulder, Colorado 80305

3 Department of Physics, University of California at San Diego, La Jolla, California 92093

(Received 24 May 2001; published 10 October 2001)

We study the control of the rotation of a laser-cooled ion crystal in a Penning trap by a rotating electric

field perturbation. We show that application of a small torque produces sudden angular jumps or "slips" of the crystal orientation spaced by intervals when the crystal is phase locked or "stuck" relative to the

rotating perturbation. The distribution of angular slips is described by a power law, where the power-law

exponent depends on the applied torque. We believe this system is driven by a constant force and small

perturbations or thermal effects trigger the slips.

DOI: 10.1 103/PhysRevLett.87.183001

Non-neutral plasmas confined in Penning-Malmberg

traps are used in a variety of experiments including plasma

physics [1], Coulomb crystal studies [2,3], precision spec- troscopy [4], antimatter research [5], and storage of highly

charged ions [6]. Recently there has been a great deal of interest in using a rotating electric field perturbation

to control the global E X B rotation of these plasmas

[5,7,8]. For crystallized ion plasmas, phase-locked control of the plasma rotation has been demonstrated [8], which

has important implications for atomic clocks [4] and for quantum computation with trapped ions [9]. In this Letter we study the limits to phase-locked control due to the

application of a small torque produced by the radiation pressure of a weak laser beam. We observe sudden

angular jumps or "slips" of the crystal orientation spaced

by intervals when the crystal orientation is phase locked

or "stuck" relative to the rotating perturbation. Stick-slip

behavior similar to that observed here is found in many

different and diverse systems: for example, in studies of friction between two surfaces [10,11], in experiments on avalanches and slips in granular systems [12-14],

and as the underlying process in spring-block models of

earthquakes [15,16]. Many of these systems, including

the study presented here, exhibit a power-law distribution of the slip amplitudes, indicative of an underlying critical point [17,18].

Our work uses the Penning-Malmberg trap at NIST to store —15 000 9 Be +

ions. The ions interact by unscreened

Coulomb repulsion and are Doppler laser cooled [19] to millikelvin temperatures, where their thermal energy is small compared to the Coulomb potential energy between

nearest neighbor ions. Under these conditions the ions are strongly coupled [20] and form a Coulomb crystal (a classical Wigner crystal) [2,3]. Structurally similar Coulomb

crystals are believed to exist in dense astrophysical matter, such as the interior of white dwarfs and the outer

crust of neutron stars [21]. Observations of power-law

statistics of soft gamma-ray events have recently been

interpreted as evidence that the Coulomb crystal comprising the outer crust of a magnetized neutron star can

PACS numbers: 32.80.Pj, 05.65. +b, 52.27.Gr, 52.27.Jt

undergo very large-scale slips ("starquakes") [22,23]. The

measured power-law exponents of the neutron starquakes

lie within the range of exponents we measure here (see

Fig. 3). Figure 1(a) shows the experimental setup [3,8]. The

9 Be+

ions were confined radially by a uniform magnetic field B = 4.465 T (cyclotron frequency 0. c /2tt =

7.608 MHz) in the z direction and axially by a potential

difference of V = —500 V applied between the center and end electrodes of the trap. Near the trap center the trap

(a) Topview camera

FIG. 1. (a) Schematic of the cylindrical Penning trap and the top-view imaging system. The side-view imaging system is not shown, (b) Strobed top-view image of a five-axial plane 9 Be +

ion crystal with a bcc structure, similar to those used in this study, (c) Side-view image (unstrobed) of the same ion crystal. The diameter (2rBe ) of the 9 Be + ions is 495 p.m. Ions of greater mass are located at r > rBe but do not fluoresce in the laser beam. The rotation axes are indicated.

183001- 003 1-9007/01/87(1 8)/ 183001(4)$! 5.00 183001-1

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Volume 87, Number 18 PHYSICAL REVIEW LETTERS 29 October 2001

potential is quadratic and given by ma> 2 {z

2 — r2 /2)/(2e),

where the axial frequency oj z /2tt = 565 kHz for 9 Be +

. Here r and z denote the cylindrical radial and axial coordinates. Because of the axial magnetic field and

the radial components of the ion space charge and trap

electric fields, the ion crystal rotates at a frequency u> r about the trap symmetry (2) axis. In addition to

9 Be +

ions, ions of greater mass ("heavy ions") such as BeH + and BeOH+

are created by reactions with 9 Be+

ions and

background neutral molecules. For the work discussed

here, typically 20% to 50% of the plasma consisted of

heavy impurity ions. These ions are sympathetically

cooled to temperatures similar to the 9 Be+

ions and, due

to the rotation, centrifugally separate to larger radii where

they crystallize.

We applied an electric field perturbation rotating about

the z axis at frequency co ip to control a> r [8]. The rotat- ing perturbation applies a torque on the radial boundary of

the plasma (on the nonfluorescing, heavy ions) by creating a small-amplitude traveling wave. The torque due to this wave is then transferred to the plasma interior through the strong interparticle forces, which act to bring the plasma

to the same rotation frequency as a> Tp [24]. We observe

similar stick-slip motion with both dipole and quadrupole

rotating fields. However, most measurements, including

those we report here, were taken with a dipole rotating

field. The radial binding force of the trap is due to the Lorentz force produced by the plasma's rotation through

the magnetic field. Therefore, changing a> r changes the

radial binding force of the trap and provides a sensitive way to adjust the overall shape and structural phase of

the plasma. In this work, co r — o> rp = 2tt X 22.8 kHz,

which produced a disk-shaped plasma consisting of five axial planes and a bcc-like crystal structure in the plasma

center [3]. Because co r « fl c , the ion motion in a di- rection perpendicular to the magnetic field is determined

principally by E X B guiding center dynamics [25]. The main cooling-laser beam (A = 313 nm) was di- rected along the z axis. This beam's power was —50 /xW,

and it was focused to a —0.5 mm waist at the ion crystal. A

second cooling beam [1 beam in Fig. 1(a)], derived from

the same laser, was directed perpendicularly to z and had a —70 /Am waist and — 1 ,u.W power. Both the perpendicular and parallel cooling lasers were required to form a well defined crystal in the disk-shaped plasmas discussed here. The J. beam is normally directed through the radial center (r = 0) of the crystal in order to minimize its applied

torque while providing a low Doppler-cooling temperature

[19]. In this experiment, we offset the I-beam position

slightly (5-30 yum) from the plasma center to produce a torque on the 9 Be +

ions in the same direction as the plasma

rotation [26]. The torque from the 1 beam was larger than any other ambient torque due to, for example, asymmetries

in the trap construction or background gas drag.

A series of lenses formed side- and top-view images

of the ion fluorescence, with viewing directions perpendicular and parallel to the magnetic field, respectively, on

either a gateable charge-coupled device (CCD) camera or an imaging photomultiplier tube. The resolution of the

optical systems was —4 /xm, while typical interparticle

spacings were —15 itm. By detecting the ion fluorescence synchronously with the rotating perturbation drive, images

of the individual ions which make up the Coulomb crystals were obtained. Figure 1(b) shows a strobed, top-view

CCD camera image accumulated over 40 s of a five-axial plane crystal in the bcc structural phase. The ion positions

are well localized in the plasma center; however, at larger

radii they are blurred. To investigate the blurring we used the imaging photomultiplier tube in the top-view position to record the positions and detection times of the fluorescence photons. Runs

consisted of 125 ms intervals of data recorded each second

over long periods of time (up to 5000 s). Images similar

to those in Fig. 1(b) were created for each 125 ms inter- val by constructing 2D histograms of the ion fluorescence

in the frame of the rotating perturbation. The orientation 9cry of the central crystallized region in the rotating frame was determined (modulo tt due to the bcc crystal bilateral symmetry) with an uncertainty of —0.002 tt rad [27]. In Fig. 2 we plot &C ry(t) for two runs which differ mainly

in the amount of I-beam torque. Over long time scales the I-beam torque produces a slightly faster rotation (a rotational "creep") of the 9 Be +

crystal relative to the rotating

perturbation. For example, in run 2 Aco = co r — co Tp ~

2tt X 8 mHz. Over shorter time scales, as shown in the

inset in Fig. 2, much of this crystal rotation takes place

with sudden jumps in cry , slips, whose time scale is too

fast to be captured by the top-view diagnostic. Intermittent behavior appears to be a common feature in the plastic deformation (creep) of many materials. (See [28] for

50- Magnified 2/> ~ 8000 -

CO

CD

Q>D5O

"°_ 6000-

c"

40i 30

20

10

•/

2 c) 10 20 30 40 50

cu 4000-

1 ^ 2000 -

CTJ° 0- 1 | 1

1 1 1 1 500 1000 1500 Time (sees)

2000 2500

FIG. 2. Crystal orientation 9cry in the frame of the rotating

perturbation for two data runs with different I-beam torques. The torque is greater in run 2. The lines are from a linear regression fit. The inset shows a magnified plot of the first 60 s of data.

1 8300 1 -2 1 8300 1 -2

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Volume 87, Number 18 PHYSICAL REVIEW LETTERS 29 October 2001

references.) Let A#cry denote the angular displacement be- tween two successive measurements of dcry . The statistics of A0cry consists of a normal distribution (from measure- ment error) centered about zero with a width of —0.002tt, and infrequent larger slips. Because of the known sign of

the J. -beam torque and the it ambiguity mentioned above, we choose A0cry to lie in the range [0, tt). To separate

statistically significant slips from measurement error we

further require 0.007tt < A0cry < 0.97tt. We find that

statistically significant slips account for greater than 90%

of the measured change in 9ciy . The _L-beam torque is applied to all the 9 Be+

ions in the radial interior of the crystal. The rotating perturbation,

however, applies its torque on the outer radial boundary

of the heavy ions. We therefore believe the stress due to the competition between these torques is greatest in the region of the heavy ions and anticipate that the slips of

Fig. 2 are due to ion motion between the radial boundary of

the 9Be+

ions, r% c , and the overall radial boundary of the plasma. This is supported by the top-view images, which

show most slips occurring as approximate rigid rotations of the 9 Be+

ions and also by simulation work discussed

below. Because the slips occur at a radius greater than

/"Be, and rBe varied from run to run, we characterize a slip

amplitude As \i p by the linear distance A#cry rB e

. Figure 3 shows the distribution /G4s ij p

) of slips for

the two data runs shown in Fig. 2. Because we cannot distinguish between slips with amplitude v4 s ij p or As iip + mrrBc , where n is an integer, we fit to the function /s , ip « X"='oUsiiP + n>rrrnti )~ y to determine

the agreement of the data with a power-law distribution. Here ncut is a cutoff that could depend on the system

size, creep rate, or other factors. We obtain a good fit for

0.01 -

i i

1 1 i i i i i

• 2 : a 1

1E-3^ • 4*^"tL_ -

. , : *-2.2n E

.3. N

£- 1E-4^

o2.0-

1.1.8 \

I

Q. W1.6-

5 1 k?r 5 X T^ < 1 *" 1E-5.

J31.4- »1.2- ki I ^i

5£i.oJ \5"

10 20 30 40 50 60 70 80 90

1E-6i Creep Rate Au rBe (urn / sec)

'i '

i . i

-.

10 100 A (Slip Amplitude, urn)1000

FIG. 3. Distribution /(Asl i p

), where /04s ii P )^As |j p

is the fre- quency of slips between i4 S | ip and A% \ ip + dA iUp , for the two

data runs shown in Fig. 2. The lines are fits to a power law with

a cutoff as described in the text. The inset shows the measured

power-law exponent y versus applied torque, as parametrized

by the creep rate, for all of the data runs.

any ncut but find that x

2 g°es through a weak minimum

at rtcut = 6. We use ncut = 6 in the following analysis. This results in measured ys slightly larger (<10%

and within the uncertainty of the fit) than those resulting

from ncul = 1. In the inset in Fig. 3 we plot the measured

power-law exponent y as a function of the creep rate AwrBc for ten data runs with the same rotating perturbation strength but different J. -beam torques. We find that y decreases as the creep rate, a measure of the applied

J. -beam torque, increases. Decreases in the stick-slip

exponent with increased drive have been observed in some

systems [11,29,30], but not in others [28]. Most experiments exhibiting stick-slip behavior are per- formed with "constant-velocity driving" where the force

is applied through an effective elastic coupling [11-13].

The driving force of the system is something like F{t) =

K[Vt — x(t)], where "x" is the "position" of an element

in the system (for example, the position of a bead or slider block in a chain), K is the effective spring constant coupling the applied force to each element in the system, and

V is the constant average velocity that is imposed on the system. Stick-slip motion occurs for small V and K and a

critical point exists in the limit V — and K —* [18,29].

If the system gets stuck, it will eventually slip again be- cause the driving force increases linearly until slip occurs. However, this experiment is performed under conditions more similar to "constant-force driving" since the J. -beam

radiation-pressure force is constant in time and applied di- rectly to the 9 Be+

ions. In constant-force driving the sys- tem undergoes a depinning transition at a critical force Fc and moves with constant average velocity proportional to (F — Fc )& for F > Fc and critical exponent (3 [29]. Inspection of Fig. 2 shows that the time intervals be- tween successive slips (the waiting periods) are typically

many seconds. An analysis of the waiting periods shows

an approximately Gaussian distribution with mean waiting

periods ranging from 4 s for the highest _l_-beam torques to 12 s for the lowest J. -beam torques. These waiting periods

are long compared to any known dynamical time scales due

to internal modes of the system. In constant-force driving,

if the system gets stuck for such a long period, it should

permanently stick, which is not what we observe. One possibility is that the slips could be excited by a perturbation.

By deliberately modulating the amplitude of the cooling

and torquing lasers we have established that the amplitude

noise present in these beams is not high enough to trigger

slips. We have performed molecular-dynamics simulations

with 1000 ions (40% 9 Be+

, 60% heavy) with the goal of better understanding the source of the slips. About

four months of computation time has been required to simulate the equivalent of ~ 1 s of experimental time. The simulations have produced one event which can be

interpreted as a slip. In that event a rearrangement of a small number of heavy ions in the vicinity of a lattice defect produced a sudden change in the orientation of the

183001-3 183001-3

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Volume 87, Number 18 PHYSICAL REVIEW LETTERS 29 October 2001

crystal. Because it appears that the rearrangement of a few

heavy ions can trigger a slip, thermal fluctuations of the

ions may be responsible for starting a slip. Once started, the slip eventually stops because the driving force of the 1 beam is not sufficient to sustain continuous motion.

In summary, we have observed stick-slip motion in the

rotational control of laser-cooled ion crystals in a Penning trap. We believe this system is constant-force driven

and may be an experimental example of a subcritical state [18] where the slips are triggered by thermal fluctuations or by other unidentified perturbations (such as collisions with neutral background atoms). The trapped-ion crystal

system discussed here possesses most of the features of a

self-organized critical (SOC) state [17,18]. Therefore fur- ther investigations of the stick-slip behavior over a wider

range of control parameters (J. -beam torque, temperature,

and rotating perturbation strength) could be useful for understanding the applicability of the SOC concept to real physical systems. Finally, minimizing the occurrence of

the slips is important for some applications [4,9]. This can be done by minimizing the _L-beam torque, either through

active control of the J. -beam position or by appropriate tai- loring of the I-beam profile [31]. Increasing the strength

of the rotating perturbation should also decrease the fre- quency of slips due to small ion rearrangements. Two runs

taken with half the rotating perturbation strength of the data set analyzed here showed an increase in the number

of slips and rotational creep of the ion crystal.

This research was supported by the Office of Naval Research and the National Science Foundation (D. H. E. D.).

We thank S. Zapped, R. L. Kautz, J. M. Kriesel, J. P. Schiffer, and D. J. Wineland for useful comments and B. M. Je- lenkovic and X.-P. Huang for technical assistance.

[1] Non-Neutral Plasma Physics III, edited by J.J. Bollinger, R. Spencer, and R. Davidson (AIP, New York, 1999).

[2] W.M. Itano, J.J. Bollinger, J.N. Tan, B. Jelenkovic, X.-P. Huang, and D.J. Wineland, Science 279, 686 (1998).

[3] T. B. Mitchell, J. J. Bollinger, D. H. E. Dubin, X.-P. Huang,

W.M. Itano, and R.H. Baughman, Science 282, 1290

(1998).

[4] J.N. Tan, J.J. Bollinger, and D.J. Wineland, IEEE Trans.

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[5] R.G. Greaves and CM. Surko, Phys. Rev. Lett. 85, 1883

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[6] L. Gruber, J. P. Holder, J. Steiger, BR. Beck, H.E. DeWitt, J. Glassman, J.W. McDonald, DA. Church, and

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M. Pfeifer, A.-L. Barabasi, T. Vicsek, and P. Schiffer, Phys. Rev. Lett. 84, 5122 (2000).

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[16] R. Burridge and L. Knopoff, Bull. Seismol. Soc. Am. 57, 341 (1967).

[17] P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987).

[18] A. Vespignani and S. Zapperi, Phys. Rev. E 57, 6345

(1998).

[19] W. M. Itano, L. R. Brewer, D. J. Larson, and D. J. Wineland,

Phys. Rev. A 38, 5698 (1988).

[20] S. Ichimaru, H. Iyetomi, and S. Tanaka, Phys. Rep. 149, 91 (1987).

[21] H.M. Van Horn, Science 252, 384 (1991).

[22] B. Cheng, R.I. Epstein, R.A. Guyer, and A.C. Young,

Nature (London) 382, 518 (1996).

[23] E. G6gii§, P.M. Woods, C. Kouveliotou, J. van Paradus,

M. S. Briggs, R. C. Duncan, and C. Thompson, Astrophys.

J. 532, L121 (2000).

[24] D.H.E. Dubin and T.M. O'Neil, Rev. Mod. Phys. 71, 87

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(1988).

[26] D.J. Wineland, J.J. Bollinger, W.M. Itano, and J.D.

Prestage, J. Opt. Soc. Am. B 2, 1721 (1985).

[27] The crystal images occasionally showed defect lines that moved slowly across the crystal and prevented a definitive measurement of 6cry . This occurred more frequently with

larger torque. In this case shorter time series separated

by intervals when 8cry could not be unambiguously deter- mined were appended to each other without introducing

discontinuities in cry at the junctures.

[28] M.-C. Miguel, A. Vespignani, S. Zapperi, J. Weiss, and

J.-R. Grasso, Nature (London) 410, 667 (2001).

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63, 104104 (2001).

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[31] The J. -beam torque can be reduced with a beam that has a large waist and frequency dispersion across the waist that matches the ion Doppler shifts due to the plasma rotation.

183001-4 183001-4

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Quantum Coherence and Decoherence - ISQM - Tokyo 98

Y.A. Ono and K. Fujikawa (Editors) 6

1999 Elsevier Science B.V. All rights reserved. 103

Quantum Computation, Spectroscopy of Trapped Ions,

and Schrodinger's Cat *

D.J. Wineland, C. Monroe, W.M. Itano, D. Kielpinski, B.E. King, C.J. Myatt, Q.A.

Turchette, and C.S. Wood

National Institute of Standards and Technology (NIST), Boulder, CO, 80303

We summarize efforts at NIST to implement quantum computation using trapped ions,

based on a scheme proposed by J.I. Cirac and P. Zoller (Innsbruck University). The use

of quantum logic to create entangled states, which can maximize the quantum-limited

signal-to-noise ratio in spectroscopy, is discussed.

1. INTRODUCTION

The invention by Peter Shor [1] of a quantum algorithm for factorizing large numbers

has stimulated a host of theoretical and experimental investigations in the field of quantum

information [2]. In the area of quantum computation, various schemes have been proposed

to realize experimentally a model quantum computer [2]. In the ion storage group at NIST, we are trying to realize such a device based on the proposal by Cirac and Zoller [3]. In the Cirac-Zoller scheme, qubits are formed from two internal energy states, labeled

| I) and | t), of trapped atomic ions. If the ions are laser cooled in the same trap, they

form a crystalline array whose vibrations can be described in terms of normal modes.

The ground and first excited states of a selected mode can also form a qubit. This qubit can serve as a data bus, since the normal modes are a shared property of the ions. An

individual ion in the array can be coherently manipulated and coupled to the selected normal mode by using focused laser beams [3]. A universal logic operation, such as a

controlled-not (CN) logic gate between ion qubit i and ion qubit j, is accomplished by

(1) mapping the internal state of qubit i onto the selected motional qubit, (2) performing

a CN between the motional qubit and qubit j, and (3) mapping the motional qubit state back onto qubit i. Each of these steps has been accomplished in the NIST experiments

with a single ion [4,5]. We are currently devoting efforts to: (1) scaling quantum logic

operations to two or more ions (Sec. 5), (2) applying quantum logic to study fundamental

measurement problems on EPR and GHZ-like states, and (3) applying quantum logic to fundamentally improve the signal-to-noise ratio (SNR) in spectroscopy and atomic

clocks. In this paper we briefly discuss this last application. We are aware of similar

efforts to implement trapped-ion quantum logic at IBM, Almaden; Innsbruck University; Los Alamos National Laboratory; Max Planck Institute, Garching; and Oxford Universitjr

.

"Contribution of NIST; not subject to U.S. copyright

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104

2. ENTANGLED STATES FOR SPECTROSCOPY

A collection of atoms (neutral or charged) whose internal states are entangled in a

specific way can improve the quantum-limited SNR in spectroscopy. This application of quantum logic to form entanglement is useful with a relatively small number of atoms

and logic operations. For example, for high-accuracy, ion-based frequency standards [6], a relatively small number of trapped ions (L < 100) appears optimum due to various

experimental constraints; with L = 10 — 100, a significant improvement in performance

in atomic clocks could be expected. In contrast, factoring a number which cannot easily

be factored on a classical computer would require considerably more ions and operations. In spectroscopy experiments on L atoms, in which the observable is atomic population, we can view the problem in the following way using the spin-1/2 analog for two-level atoms. The total angular momentum of the system is given by J = £2£=i ^»> where S, is the spin of the ith atom (Si = 1/2). The task is to measure u>o, the frequency of transitions

between the | |) and | T) states, relative to the frequency ur of a reference oscillator. We

first prepare an initial state for the spins. Typically, spectroscopy is performed by applying

(classical) fields of frequency ur for a time Tr according to the method of separated fields by Ramsey [7]. We assume the same field amplitude is applied to all atoms (the phases

might be different) and that the maximum value of Tr is fixed by experimental constraints

(Sec. 3). After applying these fields, we measure the final state populations; for example,

the number of atoms L± in the | 4-) state. In trapped-ion experiments, this has been

accomplished through laser fluorescence detection with nearly 100% efficiency, which we

assume here (see the discussion and references in Ref. [5]). In the spin-1/2 analog,

measuring L± is equivalent to measuring the operator Jz , since L± = JI—JZ where I is the identity operator. The SNR (for repeated measurements) is fundamentally limited

by the quantum fluctuations in the number of atoms which are observed to be in the

| 4-) state. These fluctuations can be called quantum projection noise [8]. Spectroscopy is typically performed on L initially unentangled atoms (for example, ^(1 = 0) = Yli=i I i)«)

which remain unentangled after the application of the Ramsey fields. For this case, the

imprecision in a determination of the frequency of the transition is limited by projection

noise to the "shot noise71 limit (Aaj) m(. aj = I/t/LTrt where r 2> Tr is the total averaging

time [8]. If the atoms can be prepared initially in particular entangled states, it is possible

to achieve (Au)mM3 < 1/^/LTrt.

In optics, squeezed states have been shown to improve the SNR in interferometers

beyond the shot noise limit [9,10]. In 1986, Yurke [11] showed how particular entangled

states, if they could be created, could be used as inputs to Mach-Zehnder interferometers

to approach the Heisenberg limit of SNR. In 1991, Kitegawa and Ueda [12] showed how the

Coulomb interaction between electrons in the two arms of an electron interferometer might

be used to improve the SNR beyond the shot-noise limit. Because of the formal identity

of Mach-Zehnder interferometers and Ramsey spectroscopy [131, similar ideas might be

applied to the spectroscopy problem. Reference [13] showed how a Jaynes-Cummingstype coupling between trapped-ion internal states and a normal mode could be used to improve the SNR in spectroscopy beyond the shot-noise limit. The scheme in Ref. [13] has the advantage that the appropriate states can be generated by acting on all the ions at once (thus not requiring focused laser beams), but has the disadvantage that these states

TN-111
105

are entangled with the motion, thereby requiring small motional decoherence. Reference

[14] investigated the use of the generalized GHZ state, sometimes called the maximally

entangled state, in spectroscopy. This state has the form

^ax=^('u)iU)2 ---U>i + e^|t)i|t>2---|t)A (1)

where 4>{t) = 4>o — Lu t. After application of the Ramsey radiation, we measure the

operator O = H*=i ^zi - The resulting signal gives the exact Heisenberg limit of SNR

((Au)mMJ = 1/L^/Trt where r 3> Tr) in spectroscopy (and interferometry)

.

The state i/Wr can be generated in a straightforward way by the application of L CN

gates [3]. An alternative method was suggested in Ref. [14] and in Refs. [5] and [15] methods to generate ifimax with a fixed number of steps (independent of L) are discussed. For all of these methods, the the motion is entangled with internal states during the

creation of mox, but is not entangled afterwards. Therefore, once t/Wr is created, the

motion can lose coherence without affecting the entanglement of the internal states.

2.1. Schrodinger's Cat

As L becomes large and more macroscopic, states like ipmax become more like Schrodinger's

cat in that they represent coherent superpositions between widely separate regions of a

large Hilbert space; for example, | f)i| fH • • •

I T)i <==*y "live cat;" | 4-)i | J.) 2 •• | i)z •*=> "dead cat". As has been emphasized in many discussions, as L becomes large the co- herence between the two components of the cat becomes harder and harder to preserve

[16]. This is apparent in Eq. (1) because if, for example, u fluctuates randomly, the two

components of i/ifflaI will decohere relative to each other L times faster than for .one ion

(ii'fflai for L = 1). Trapped ions are interesting because it may be possible to make L very

large without significant decoberence. This is the same property that makes trapped ions

interesting as possible frequency standards. For example, in Refs. [17] and [18], coherence

times for individual ions (X = 1) exceeding 10 minutes were obtained.

3. Applicability

In the above, we have assumed that Tr is fixed, limited by some independent experimental factor. This assumption is warranted in many trapped-ion atomic clock experiments,

where, for example, we want to limit the heating that takes place with laser cooling ra- diation absent. (During application of the Ramsey fields the cooling radiation must be

removed to avoid perturbing the clock states.) Additionally, we may want to lock a local

oscillator to the atomic reference in a practical time [6,19], thereby limiting Tr. However, the use of entangled states may not be advantageous, given other conditions. For example, Huelga, et al. [20] assume that the ions are subject to a certain dephasing

decoherence rate (decoherence time less than the total observation time). In this case, there is no advantage of using maximally entangled states over unentangled states. The

reason is that since the maximally entangled state decoheres L times faster than the states of individual atoms, when we use the maximally entangled state, Tr must be reduced by

a factor of L for optimum performance. Therefore, the gain from using the maximally

entangled state is offset by the required reduced value of Tr.

TN-112
106

Reference [5] discusses another case of practical interest. In atomic clocks, the fre- quency of an imperfect "local" oscillator, whose radiation drives the atomic transition,

is controlled by the atom's absorption resonance. Depending on the spectrum of this oscillator's frequency fluctuations (when not controlled) the use of entangled states may

or may not be beneficial.

4. Implementations

If we are able to create, with good fidelity, the state t^max (Eq. (1)), how do we perform

spectroscopy? First, we note that ipmax is the state we want after the first Ramsey 7r/2 pulse. Therefore, if we were to follow as closely as possible the Ramsey technique, we

would take xpmax and apply a 7r/2 pulse of radiation at frequency uQ to make the input

state for the Ramsey radiation. However the first Ramsey 7r/2 pulse would only reverse

this step; therefore, it is advantageous to take the creation of ^' mai as the first Ramsey n/2

pulse. The second Ramsey pulse (after time Tr) can be applied directly with radiation at frequency ur. The phase of this pulse (on each ion) must be fixed relative to the phases

of the radiation used to create tpma.x . In general, the relation between these phases and

<f>o (Eq. (1)) will depend on the relative phases of the fields at the positions of each of the

ions [5,21]. This will lead to a signal S = (0) oc cos(LAu>Tr + 4>/) where Au = u>r — uq and where (j>j depends on all of these phases.

We can extract u>q (relative to ur) by measuring (0) as a function of Tr, with Au fixed. This can be further simplified by measuring the signal for two values of Tr, Tr2 3> Tri,

where (0) ~ 0. Unfortunately, if the measured signal has a systematic bias as a function

of Tr, an error in the determination of Au> will result. This might happen, for example,

if the ions heat up during application of the Ramsey radiation and a loss of signal occurs due to a reduced overlap between the ions and the laser used for fluorescence detection

of the states. This problem could be overcome by measuring (0) for two values of u>r, ljri and ujr2 such that uri — u>q ~ — (u>r2 — uq) (determined by the above method),

and two values of Tr, Tr\ <gC Tr2. We then iterate the following steps: (1) we make

(0((uri — wo)Tri)) ~ (0(((jJr2 — ld )Tri)) by adjusting the phase of the final n/2 pulse

to make <f>j -4 0. This will take a negligible amount of time since Tri <C Tri. (2) We

make (0((uri — uj )Tr2)) — (0((ljr2 — uo)Txi)) by adjusting ujri and/or ur2 to force uri — uQ —>• — (ijJr2 — u>o). This gives u relative to ujr even if (O) has a systematic bias

as a function of Tr. An alternative solution is suggested by Huelga, et al. [20]. After Tr, instead of applying

a n/2 pulse of radiation at frequency ojr, we apply the time-reversed sequence of operations

which created TJ? ma.x- This has the advantage of cancelling out all of the CN phases that

contribute to 4>q and maps the signal (oc cos(LAujTr)) onto a single ion (whereupon Sz is measured for that ion). This also reduces the problem of detection efficiency to one ion

rather than L ions. The disadvantage of this technique is that for large values of Tr, the

motional mode used for logic will, most likely, have to be recooled. This would require

sympathetic cooling with the use of an ancillary ion which, to avoid the decohering effects of stray light scattering on the logic ions, might have to be another ion species [5]. A more serious limitation to the accurate determination of u is that, in practice,

V'mox wiU be realized only approximately an^ the state produced by the logic operations

TN-113
107

will also be composed of states other than the | ^)i\ j} 2

• • •

] t)z, and I 4-)i| 4)2 • •| 4)l

states; these other states will have a definite phase relation to the | t)i| T)2 " '

'

I t)i an(3

I 4)i I 4)2 • • •

I 4)x states. Consequently, in general, the signal produced with either implernenation will be of the form

L S = YCp cos(pAuTH + £p ). (2)

To accurately determine Au>, it will be necessary to Fourier decompose S. Since this will take more measurements, the advantages of using entangled states will be reduced.

In spite of this, in some applications, it will be useful to determine changes in u>o with

respect to some external influence. For example, we might want to detect changes in u

caused by changes in an externally applied field. In this case, as long as \CV \ -C 1, for

all p < i, we derive the benefits of entangled states (assuming the decoherence time is longer than Tr/X) by measuring changes in S for a particular value of Tr.

5. Experiments

As usual, our enthusiasm for implementing these schemes far exceeds what is accomplished in the laboratory; nevertheless, some encouraging signs are apparent from recent experiments. In Ref. [22], all motional modes for two trapped ions have been cooled to the ground state. The non-center-of-mass modes are observed to be much less susceptible

to heating, suggesting the use of these modes in quantum computation or quantum state engineering. In Ref. [21], we describe logic operations which enabled ^mQI for L = 2 to be

generated with modest fidelity (~ 0.7). For small L, it is only necessary to differentially

address individual ions to create tl' max and for L — 2, general logic can be realized even

if the laser beams cannot be focused exclusively on the individual ions [21]. For general

logic on more than two ions, two avenues are being pursued. For modest numbers of ions

in a trap, the Cirac-Zoller scheme of individual addressing with the use of focused laser beams is the most attractive. Current efforts are devoted to obtaining sufficiently strong

focusing to achieve individual ion addressing in a relatively strong trap where normal

mode frequencies are relatively high (~ 10 MHz) in order to maximize operation speed.

Alternatively, general logic on many ions could be accomplished by incorporating accu- mulators [5], and using differential addressing on two ions at a time. This idea might be

realized by scaling up a version of a linear ion trap made with lithographically deposited

electrodes as we have recently demonstrated [16,23]. Concurrently, efforts are being de- voted to the investigation (and hopefully, elimination) of mode heating [5] for different electrode surfaces and dimensions.

6. Acknowledgments

We gratefully acknowledge the support of the U.S. National Security Agency, U.S. Army

Research Office, and the U.S. Office of Naval Research. We thank J. Bollinger, R. Blatt, D. Sullivan, and M. Young for helpful comments on the manuscript.

TN-114
108

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

Rev. A46, R6797 (1992); D.J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heinzen, Phys. Rev. A50, 67 (1994).

14. J.J. Bollinger, D. J. Wineland, W. M. Itano, and D. J. Heinzen, Phys. Rev. A54,

R4649 (1996).

15. J. Steinbach and C.C. Gerry, quant-ph/9806091.

16. See, for example, S. Haroche, Physics Today 51 (7), 36 (1998).

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

Trans, on Instrum. and Measurement 40, 126 (1991).

18. P.T.H. Fisk, M.J. Sellars, M.A. Lawn, C. Coles, A.G. Mann, and D.G. Blair, IEEE

Trans. Instrum. Meas. 44, 113 (1995).

19. J.J. Bollinger, J.D. Prestage, W.M. Itano, and D.J. Wineland, Phys. Rev. Lett. 54,

1000 (1985).

20. S.F. Huelga, C. Macchiavello, T. Pelizzari, A.K. Ekert, M.B. Plenio, and J.I. Cirac,

Phys. Rev. Lett. 79. 3865 (1997).

21. Q.A. Turchette, C.S. Wood, B.E. King. C.J. Myatt, D. Leibfried, W.M. Itano, C. Monroe, and D.J. Wineland, submitted (quant-ph/9806012).

22. B.E. King, C.S. Wood, C.J. Myatt, Q.A. Turchette, D. Leibfried, W.M. Itano, C. Monroe, and D.J. Wineland, Phys. Rev. Lett. 81, 1525 (1998).

23. C. Myatt, et a/., NIST Ion Storage Group, unpublished.

TN-115
Physica Scripta. Vol. T76, 147-151, 1998

Trapped-lon Quantum Simulator

D. J. Wineland,* C. Monroe, W. M. Itano, B. E. King, D. Leibfried, C. Myatt and C. Wood

National Institute of Standards and Technology, Boulder, CO, 80303, USA

Received September 12, 1997 ; accepted November 17, 1997

pacs Ref: 0365Bz, 0367Lx

Abstract

Coherent manipulations involving the quantized motional

and internal states of a single trapped ion can be used to

simulate the dynamics of other systems. We consider some

examples, including the action of a Mach Zehnder interfer- ometer which uses entangled input states. Coherent manipulations can also be used to create entangled states of

multiple trapped ions; such states can be used to demonstrate fundamental quantum correlations.

1. Introduction

Stimulated, in part, by the interest in quantum computation

and quantum communication [1], a number of papers have

investigated the possibility of synthesizing or "engineering"

arbitrary quantum states of trapped ions (for recent reviews,

see Refs. [2] and [3]). To the extent that this can be accomplished for a large number of trapped ions, such a system

would allow general quantum computations, including the

factorization of large numbers [1, 4]. By anybody's reckoning, factorizing large numbers is a daunting task. Therefore,

it is desirable that a quantum computer, or a system which

can generate arbitrary entangled states, have wider applicability. Various possibilities have been explored in the recent

literature [5-11]. Some of these proposals extend the ideas

of Feynman who considered whether or not one quantum

system could be used to simulate the behavior of another

quantum system [12]. In this spirit, we discuss some simple

examples of how a single trapped ion might be used to simulate the behavior of other quantum systems, such as entangled particles acted on by a Mach Zehnder interferometer. We also briefly discuss how the states of multiple

trapped ions can be entangled; these states can be employed

in fundamental demonstrations of quantum measurements.

2. Coupling of a two-level trapped ion to its motion

We first consider a single ion trapped in a 3-D harmonic

well with oscillation frequencies cox , coy and coz along three

cartesian axes. This situation is closely approximated by a

single ion confined in a Paul (rf) trap. We will be interested

in two internal states of the ion which we label as | f) and

| j), and which are separated in energy by hco . We apply a

(classical) radiation field or fields (typically laser fields) of

the form

E(x, t) = E cos (k ' x — cot + (j>). (2.1)

• e-mail: dwineland@nist.gov

The Hamiltonian which describes the (resonant) coupling

between the ion's internal states and its motion (provided by

E(x, t)) can be written in the rotating-wave approximation

as

*'[* 6t + <t>l if, = hQ(S + f e" + h.c, (2.2) where Q is the coupling strength (Rabi frequency), S + is the

raising operator for the internal states (S+ | j.) = 1 1», x is the ion's position relative to its equilibrium position, 5 = co — co , and (j> is a phase factor of the field [3]. In eqs (2.1) and (2.2), k is the wavevector of the field for single photon

transitions or k is the difference between the two wavevectors when two-photon stimulated-Raman transitions are used [13]. Similarly, co is the frequency of the applied field for single-photon transitions, or is the difference in fre- quencies of the two applied fields when stimulated-Raman

transitions are used. The exponent £ is equal to 1 when

internal state transitions are involved and e = when the

internal state is unchanged (stimulated-Raman transitions). In an interaction picture of the ion's motion, this Hamiltonian becomes [3]

H, = fiQ(S + )

c e-^-v Yl exp (i[ij/a,. e

_io* + a] e

io*)])

j=x, y, z

(2- 3) + h.c, where a,- and a] are the lowering and raising operators for harmonic motion in the jth direction, and rj x = k • xx is the Lamb-Dicke parameter in the x direction, where x =

y/hKlmcOx) (m is the ion mass), and similarly for rj y and r\ z

. Now, assume that Q is small enough, and that cox , co

y and

coz are incommensurate so that we can (resonantly) excite only one spectral component of the possible transitions induced by this interaction. For a particular resonance condition 5 Leo, ly co

y

lz coz

(lj integers), and in the Lamb-Dicke limit, we find

#, ~ ha e^s+y nj = x,y.z i_K\h\

(iVjaj) 101

I IV.

(iM.at)ior + h.c (2.4)

The two mode case where e = lz = 0, lx , ly # is considered

by Drobny and Hladky [14], and in a different excitation scheme by Steinbach, et al. [15]. If the Lamb-Dicke limit is not rigorously satisfied, we must consider higher-order terms in the expansion of the exponentials of eq. (2.3) [16];

specific examples are discussed by Wallentowitz and Vogel

[17], and Steinbach, et al. [15]. These nonlinear terms appear as corrections to the Rabi frequencies for Fock

states and have been observed in the experiments of Ref. [18].

Physica Scripta T76

TO-116
148 D. J. Wineland et al.

3. Simulations of processes in optics

Referring to eq. (2.4), the carrier and first red and blue (z- motion) sidebands on internal state transitions (e.g., e = 1,

lx = ly = 0, lz = 0, + 1) are used in experiments to cool the

ion to the ground state of motion [13, 19], for quantum

logic [20], and to generate nonclassical motional states [2, 18]. The upper and lower sidebands (lz = +1) correspond

to emission and absorption of single vibrational quanta or "phonons" associated with internal state changes; this is directly analogous to the emission and absorption of single

photons into a cavity by an atom inside. An interesting

system which could be simulated with these couplings is a "phonon maser" which provides vibrational amplification

by stimulated emission [21]. The case £ = 0, lx — ly = 0,

|/z | = 1 has been used to create coherent [18] and Schrodinger cat [22] states of motion. Coherent states of ion

motion correspond to coherent states in optics. The case

e = 0, lx = ly = 0, | lz | = 2 has been used to create squeezed

states of ion motion [18]. A realization of the Hamiltonian

if, oc S+(at

) 2 + h.c. (e = l,lx = ly = 0, lz = -2) for ions has

been reported by Leibfried, et al. [23]. This is similar to the

case of two-photon excitation in cavity QED analyzed by

Buck and Sukumar [24] and Knight [25]. An example of an

interesting new case would perhaps be the realization of three-phonon downconversion (e.g., e = 0, lx = 3, ly = — 1,

lz = 0) ; this is accomplished by driving a two-mode reso- nance using stimulated-Raman transitions where the differ- ence in frequencies of the two laser beams is equal to a>

y — 3a>x

. This case corresponds to three-photon downconversion in quantum optics (see Refs. [15], [26], and ref- erences therein). A suggestion to realize a Hamiltonian

proportional to al a\ + h.c, (e = 0, lz = 0, lx = 2, ly = — 1) is discussed by Agarwal and Banerji [27].

Clearly, a very large number of possibilities could, in principle, be realized just for a single ion ; moreover, the number

of possibilities increases dramatically if we consider all modes of motion for multiple trapped ions. The only limi- tation on how high | lj \ in eq. (2.4) can be is that Q be

chosen sufficiently small that couplings to other (unwanted)

resonances are avoided. This will require that decoherence

be small enough to see the desired dynamical behavior

before coherence is lost. Finally, the analogy to optics dis- cussed here should not be surprising since a single ion's motion (for one mode) and a single mode of the radiation

field are both described by quantized harmonic oscillators.

4. Mach Zehnder interferometer with entangled states

Realization of the various Hamiltonians indicated in eq.

(2.4) can lead to simulation of various devices of practical

interest. As an example, we can simulate the action of a Mach Zehnder interferometer for various input states. We

consider HI to act on two modes of ion motion ; to be specific, we will assume these are the x and y modes. The

analogy with a Mach-Zehnder interferometer for bosons is that the two input modes to the boson interferometer are replaced by the x and y modes of ion oscillation. The (50/50)

beamsplitters in the boson interferometer are replaced by an

operator [28-30]

B ± = exp [±in(alay + ax aJ)/4]

Physica Scripta T76

(4.1)

This operator can be realized by applying the interaction in

eq. (2.4) with £ = lz = 0, and lx = -ly = 1 for a time given

by Qr\xr\ yt = n/4. A differential phase shift between the two

arms of the interferometer can be simulated by shifting the

relative phases of the fields in eq. (2.4) between successive applications of B+ . In a particle (e.g., boson) interferometer, one typically measures the number of particles in either one

or both output modes. For single ions, the experiments so

far have only one convenient observable, the internal state of the ion (either | j> or ||». Nevertheless, we can fully

characterize the action of the phonon interferometer by

repeating the experiment many times and measuring the

density matrix of the output state [23, 31].

It will be interesting to characterize the action of the

interferometer for various nonclassical input states. One

interesting input state is the two-mode Fock state \nxyx \n

yyy [32]. This state could be prepared by applying

the Fock state creation techniques described in Ref. [18]

sequentially to the ion's x and y modes. This state is inter- esting because it has been shown that one could approach

the Heisenberg uncertainty limit in a Mach Zehnder inter- ferometer by measuring the distribution of bosons in the output modes [32-34]. The observable is the variance of the number of particles detected in one of the output ports

when the arms of the interferometer are of approximately

equal length. As the difference in length of the arms deviates from equality, the variance increases sharply. An alternative technique for studying the action of a beamsplitter on the two-mode Fock states has been suggested by Gou and

Knight [35] when cox = co

y

. Here, a beamsplitter could be

simulated by first preparing \nx}x \n

y ) y along two orthogonal axes and then probing along two other axes (x' and y') which are rotated (in the xy plane) with respect to the first. This technique could also be used to analyze, for example,

the (| 0>x

.

1 2}y

. + 1 2>x,

| O^y)/^/! state from an initially pre- pared 1 1>X | l>y state [35]. Another interesting input state to consider is the state Bl(\ N>x 1 0>y + 1 0>x | iV>

y)A/2. (Equivalent^, the state after the first beam splitter is (\N}x \0>y + j 0>JC

| iV>

y)/^/2.) This

state has been shown to yield exactly the Heinsenberg

uncertainty limit for an interferometer for any value of N

and any difference of the lengths of the arms [8]. The

observable is the parity of the number of particles measured

in one of the output ports. For example, we could measure

the number of particles N(x) in the x output port. The result of this measurement is assigned the value (— 1)

N(X)

. For a single ion, the state after the first beamsplitter could

be prepared from the 1 1> 1 0>x 1 0>, state by the following

two steps:

(1) Apply a n/2 pulse on the iVth blue sideband of mode x

(e = l,lx = -N,ly = lz = 0); this creates the state (| |> 1 0>x + IT>|Jv>I)|oyN

/2.

(2) Apply a n pulse on the Nth. blue sideband of mode y

( e = 1, lx = lz = ! ly =-N); this creates the state IT>(]^>x|0>y +|0>x |N>y)/72.

After the second beamsplitter, we have a state which can

be written as

^fma. = I T> £ CJ nx> x | N - nx\

(4.2) ni=0

We now want to measure nx , record the value N(x), and

assign the value ( — l)

N(x) to the overall measurement. EffecTN-117
Trapped-Ion Quantum Simulator 149

tively, this assignment can be accomplished if we can find 1 interaction M which provides the mapping R

MW(iai] = 1 1> I Cnx e^| nx}x \ N - nx\

+ ll> E Cnx e^\nx)x \N-nx} y n^ odd

After this mapping, we need only measure the internal

state; if the ion is found in the 1 1> we assign the value +

1

to the measurement; if the ion is found in the 1 1> state, we

assign the value —1. The mapping M can be achieved by

applying radiation with k\\\ at the carrier frequency (e = 1,

t = 2nm ±nx n where m

the Rabi frequency for ll>*-HT) which depends on nx due to terms in the expansion of exp (ifc • x) which are nonlinear in x. We find [3]

lx = ly — lz = 0) and insuring Q,

is an integer. Here Qni is

Q„ n t ~ Qt e-**2/2 1 nr\l 1 + (4.4)

2 Therefore, if we satisfy Q exp ( — rjl/2)t = 2nm and r\ d

+ f*/4) = (2w!)~\ we achieve the desired mapping as long

as the contribution to the phase from the term proportional

to nx in this equation is small compared to n. Therefore we

require m ^> iV2/8 or, equivalently, r\ x<^ 2/N.

If JV is large and/or the Lamb-Dicke parameter is very

small, creating the state ITXI^X|0>y + 1 0>x | Nyj/^/I by

steps (1) and (2) above may be very slow. If we use an auxiliary internal state, we can speed up this process by employing first-order sidebands. To be specific, we will assume we

can realize a coupling of the form of eq. (2.4), between the

| ]"> state and auxiliary state which we label | A}. We assume

state \A} is lower in energy than state ||> so that SA+ \A} = 1 1>. A particular realization of states ||>, ||>,

and \A} is described in Ref. [20]. As an example, starting

with the state l!>|0>x|0>„, we can create the state

(I TXI 3>x 1 0>, + 1 0>x 1 3X)A/2 with the following steps (and

appropriate choices of (f>)

:

\l>\0yx \0X-(S+(n/2) + h.c.)-*

IT>(|3>X |0>,+ 0>J3>A (4.5)

In this expression, the notation SA + ax(%) + h.c. means the operator SA+ ax + h.c. is applied for a time sufficient to

drive a n pulse, etc. From this, it is straightforward to see

(4.3) how to generate the state | fXI #>x I &>y + 1 0>x I -N>,)A/2

for N odd. For N even, we can, for example, substitute two

carrier transitions for two sideband transitions in the above

steps. One final example of a two-mode interferometer which

directly yields Heisenberg 1/N phase sensitivity is a "beamsplitter" which creates that state (I i> l-^XIO), + | t>|0>x |iV>

y)/v

/

2). This state (for N even) could be

created as in eq. (4.5) except the last two operations are replaced by the operation SA+ (n) + h.c. For example, to

create the state (| |> 1 2>x 1 0>y + | i> 1 0>x 1 2>y)/v

/2), we

replace the last two steps of eq. (4.5) by

v/2

(| Ay 1 2>x I OX + | i> 1 0>x 1 2>.) - {SA+ {7i) + h.c.) ^

(IT>|2>,|0>, + |i>|0>x |2>A (4.6)

For N odd, we can, for example, substitute two carrier transitions for two sideband transitions in the above steps. If an

auxiliary state is not available, this state can be created by

first making the initial dual Fock state (| |> + \l})\N/2yx \N/2yy/y/2 with the methods described in

Ref. [18]. (In this example, we assume N is even.) Next, we

apply N/2 rc-pulses alternating between the two interaction Hamiltonians H^ = Qrjx ny S + ax ay + h.c. and H2 =

Qr\x

r\ y S + ax a\ + h.c. In this way, the ion is stepped through

the sequence

x/2

1 1> | JV/2>X | JV/2>

y - (S + (tt/2) + fe-c.)

(1 1> I N/2>x | N/2yy + 1 1> | N/2>x | N/2%) - (HJ

V~2

(IT> + U»|O>x |o>,-(sA+(70 + fe.c.) (|T>|tf/2 + l>x |iV/2-lX

(\A} + \iy)\0yx \0yy -(SA+ al(n) + h.c.)

1 1 -^(IT>|i>x + ll>|0>j|0>,-(s^ +flx(«) + fe.c.)^ -/=

(I Ay I 2>x + I j> 1 0>x) I OX - (S + al(n) + h.c.) •

+ |i>|iW2-l>x

|tf/2 + l>„)-(ff2)

(\l>\N/2 + 2>x \N/2-2>y

+ \iy\N/2-2yx \N/2 + 2yy)-(H1 )

:{\A>\ 2>x 1 0>y + 1 1> 1 0>x 1 1>,) - (S + ay(n) + h.c.)

\ Ay 1 2>x 1 0>, + I i> 1 0>x 1 2>,) - (SA+ 4(n) + h.c.)

72

T> I 3>x 1 0>y + I i> 1 0>x 1 2>J - (S + aj(«) + h.c.)

-(H2)->-^(li>|JV>x |0>y + |T>|0>x |JV>A (4.7)

The interactions H1 and H2 follow from eq. (2.4) with

£ = 1, Zx = -1, ly = l, lz = 0, and e = 1, lx = 1, ly = -1,

lz = respectively. The fcth pulse has Rabi frequency

Qr\xr\ y s/iN/2 + k){N/2 - k + 1) in the Lamb-Dicke regime.

After a relative phase is accumulated in the two "paths" of

the interferometer (simulated by adjusting the phase of the

Physica Scripta T76

TN-118
150 D. J. Wineland et al.

laser pulses as discussed above), then the steps in eqs (4.7) or

(4.5) and (4.6) are reversed. Upon measuring the probability

of occupation in state ||> or ||>, the interference fringes exhibit 1/N phase sensitivity.

If the Lamb-Dicke criterion is not satisfied, the two components of the wavefunction superposition may experience

different Rabi frequencies during each pulse, leading to undesired evolution. However, as long as r\x = ri y

it can be

shown that the system will evolve as in eq. (4.7), even when

the Lamb-Dicke criterion is not satisfied [3].

5. Quantum correlations The coherent manipulations on single ions discussed above

can be extended to multiple ions [4]. As one step towards

this goal, a controlled-not quantum logic has been demonstrated [20] between qubits formed with the ground and

first excited state for one mode of motion (\nx = 0yx =

|0>m> \ nx = I)* — I l)m) and tQe i°n

' s internal states

(|J,> = |0>, |T> = |1>). The controlled-not gate exhibited

the logic

i£i>m|e2>-Hei>m|ei © e2 > (5.1) where © signifies addition mod 2. For a collection of ions

in a trap, we select a particular mode (say the center-of-mass mode along the axis of a linear Paul trap) to comprise the motional qubit. By first mapping the internal state of ion i onto the motional qubit (which is shared by all ions), per- forming the logic in eq. (5.1) between the motional qubit and

ion j, followed by reversing the first mapping step, we can

realize a controlled-not logic operation between ion i and

ion; [4]

l«i>il«a>j-H«i>il«i © h>j- (5.2) Application of these gates to small numbers of trapped

ions can lead to interesting experiments which may shed

light on the viability of local hidden-variables theories. For

example, for two ions, starting with the state |0> 1 |0> 2 we

can apply a x/2 pulse to the internal states of ion 1 followed

by a controlled-not between ions 1 and 2

|0> 1 |0> 2 --^(|0> 1 + |1> 1 )|0> 2

|0> 1 |0>2 + |l> 1 |>a). (5.3)

If the states of the resulting entangled particles are detected

outside of each other's fight cones, then, for particular sets of measurements, we may derive Bell's inequalities [36]

which local hidden-variables theories must obey, but which

quantum mechanics violates. The experiments performed by

Aspect and co-workers [37] (and more recent versions - see

Ref. [38] and A. Zeilinger and P. Kwiat, these proceedings)

provide strong evidence against local hidden-variables theories. The Aspect et al. experiments used polarization measurements on entangled pairs of photons. The detection of

the photons' polarization states occurred outside each

others' light cones. Thus, the measurement on one photon

could not have affected the other measurement, which

closed possible "loopholes" in the proof of quantum mechanics over other explanations.

Physica Scripta T76

However, some loopholes still remain open. Since the photon detection in the Aspect et al. experiments was not 100% efficient, the group had to make assumptions that the photons they measured were a "fair" sample of the whole

population of events. Thus, their experiments do not rule out the (seemingly implausible) possibility of local hiddenvariables theories in which the hidden variables cause some

sub-ensemble of the photon pairs to preferentially interact with the measurement apparatus. In the system of two ions, we may detect the state of

either ion with nearly 100% efficiency through the use of

"electron-shelving" (for a discussion, see Ref. [3]). On the other hand, it may be difficult to perform measurements on two ions outside each other's light cone. Such a measure- ment would require separating the ions by a distance larger

than the speed of light times the measurement time. In principle of course, the ions could be first entangled and then placed in different traps which could be separated by large distances before measurements were performed. Alternatively, it may be possible to entangle distant pairs of ions using

optical fibers [11]. Nonetheless, an experiment with two

entangled ions confined in the same trap could be viewed as complementary to those of Aspect and others: the photon

experiments definitively close loopholes of causality, and the ion experiments could close loopholes due to detection inef- ficiency. Such experiments have the additional appeal of studying EPR on massive particles (E. Fry, these proceedings). EPR states of atoms have recently been

created in an atomic beam using the methods of cavity

QED (S. Haroche, these proceedings); if detection efficiency can be improved, these experiments could also close loopholes due to detection inefficiency. Finally, even though

measurements of quantum correlations between entangled

ions cannot be easily performed outside each other's light cone, one can argue strongly that the ions cannot transfer information by any known mechanism. Therefore, if the observed correlations violate Bell's inequalities, the correlations are established by some new force of nature or are, in fact, inherent in the structure of quantum mechanics.

An intriguing possibility for ions is the possibility of

making "GHZ states" [39, 40]. For three ions, the GHZ

state has the form

^ = ^= (|0> 1 10> 2 |0> 3

-J- e^| 1>! [ 1>2 ] 1> 3 ). (5.4)

This state can be made starting with the state

1 0>i 1 0>2 1 0>3 , applying the first two steps shown in eq.

(5.3), and following with a controlled-not gate between ions

1 and 3 [4]. For such a state, a single measurement can

distinguish between the predictions of quantum mechanics

and those of any local hidden-variables theory [39, 40].

Aside from these possibilities, Bell states, GHZ states, and

Schrodinger-cat states are highly entangled, and are thus of

inherent interest for the study of uniquely quantum behavior. As the experiments improve, it will be interesting to push the size of entangled states to be as large as possible.

The question is not whether we can make states which have

the attributes of Schrodinger cats, but how big can we make

the cats? Certain theories which address the measurement

problem will be amenable to experimental tests, for example, quantitative limits on spontaneous wavefunction

TN-119
Trapped-Ion Quantum Simulator 151

collapse theories [41, 42] can be established. The isolation from the environment exhibited by trapped ions, coupled

with the control possible over their quantum state and high

detection efficiency make them an interesting laboratory for the study of fundamental issues in quantum mechanics.

Acknowledgements

This work is supported by the U.S. National Security Agency, Office of Naval Research, and Army Research Office.

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Rev. A62, 403 (1989). Monroe, C, Meekhof, D. M., King, B. E., Itano, W. M and Wineland, D. J., Phys. Rev. Lett. 75, 4714 (1995). Wallentowitz, S., Vogel, W., Siemers, I. and Toschek, P. E., Phys. Rev. A54, 943 (1996). Monroe, C, Meekhof, D. M., King, B. E. and Wineland, D. J., Science

272, 1131 (1996). Leibfried, D. et al, J. Mod. Opt 44, 2485 (1997). Buck, B. and Sukumar, C. V, Phys. Lett. 81A, 132 (1981). Knight, P. L., Physica Scripta T12, 51 (1986). Banazek, K and Knight, P. L., Phys. Rev. A55, 2368 (1997). Agarwal, G. S. and Banerji, J., Phys. Rev. A55, R4007 (1997). Yurke, B., McCall, S. L. and Klauder, J. R., Phys. Rev. A33, 4033

(1986). Lai, W. K, Buzek, V. and Knight, P., Phys. Rev. A43, 6323 (1991). Sanders, B. C. and Milbum, G. J., Phys. Rev. Lett. 75, 2944 (1995).

Leibfried, D. et al, Phys. Rev. Lett 77, 4281 (1996). Holland, M. J. and Burnett, K, Phys. Rev. Lett 71, 1355 (1993). Bouyer, P. and Kasevich, M., Phys. Rev. A56, R1083 (1997). Kim, T., Phister, O., Noh, J., Holland, M. J. and Hall, J. L. (1997), unpublished. Gou, S.-C. and Knight, P., Phys. Rev. A54, 1682 (1996).

Bell, J. S., Physics 1, 195 (1964). Aspect, A., Dalibard, J. and Roger, G, Phys. Rev. Lett 49, 1804

(1982).

Tittel, W. et al, Europhys. Lett. 40, 595 (1997). Greenberger, D., Home, M. A. and Zeflinger, A., in "Bell's Theorem,

Quantum Theory, and Conceptions of the Universe" (edited by M.

Kafatos) (Kluwer Academic, Dordrecht). Greenberger, D., Home, M A. and Zeilinger, A., Phys. Today 46, No.

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Physica Scripta T76 TN-120
articles

Decoherence of quantum

superpositions through

coupling to engineered reservoirs

C. J. Myatt*, B. E. King*, Q. A. Turchette, C. A. Sackett, D. Kielpinski, W. M. Itano, C. Monroe & D. J. Wineland

National Institute of Standards and Technology, Div. 847.10, 325 Broadway, Boulder, Colorado 80303, USA

The theory of quantum mechanics applies to closed systems. In such ideal situations, a single atom can, for example, exist simultaneously in a superposition of two different spatial locations. In contrast, real systems always interact with their environment, with the consequence that macroscopic quantum superpositions (as illustrated by the 'Schrodinger's cat' thoughtexperiment) are not observed. Moreover, macroscopic superpositions decay so quickly that even the dynamics of decoherence

cannot be observed. However, mesoscopic systems offer the possibility of observing the decoherence of such quantum

superpositions. Here we present measurements of the decoherence of superposed motional states of a single trapped atom. Decoherence is induced by coupling the atom to engineered reservoirs, in which the coupling and state of the environment are

controllable. We perform three experiments, finding that the decoherence rate scales with the square of a quantity describing the amplitude of the superposition state.

One of the fundamental properties of quantum mechanics is the

principle of superposition, a principle whose introduction was

considered a "drastic" measure by Dirac 1 . The fact that quantum

superpositions do not exist in the macroscopic world hinders our

intuition and leads to the apparently strange behaviour dictated by

quantum mechanics. A famous example of this was posed by

Schrodinger in 1935 (ref. 2) who pointed out that quantum

mechanics would predict bizarre situations such as a cat being

simultaneously dead and alive. The existence of superpositions

prescribed by quantum mechanics is valid for systems that are

closed, that is, free from external influences. In contrast, real systems

always couple to these external influences, the environment, which

is typically composed of an extremely large number of degrees of freedom. Lack of knowledge about the environment is expressed by

averaging (mathematically tracing) over the possible states of the environmental degrees of freedom. This leads to an evolution of the density matrix of the system, in which the quantum superpositions

are continuously reduced to classical probability distributions, a process generally known as decoherence (see, for example, refs 3-5).

One approach to describing decoherence is to treat the environment

as a reservoir of quantum oscillators, each of which interacts with

the quantum system in question. An example of such a reservoir- system interaction is the ensemble of empty electromagnetic field modes, each represented by a quantized harmonic oscillator, inter- acting with an atom in order to induce spontaneous emission. As a quantum superposition is made larger, decoherence tends to act more quickly. For truly macroscopic superpositions, such as that of

'Schrodinger's cat', decoherence occurs on such a short timescale

that it is almost impossible to observe quantum coherences. However, mesoscopic systems present the possibility of studying, in a controlled way, the process of decoherence and the transition from

quantum to classical behaviour.

In the past few years, techniques have been realized to generate mesoscopic superpositions, also called 'Schrodinger cats', of motional states of trapped ions6 and of photon states in the context of cavity QED (ref. 7), where decoherence through coupling to

•Present addresses: Research Electro-Optics, 1855 South 57th Court, Boulder, Colorado 80301, USA

(C.J.M.); NIST. Atomic Physics Division (842). 100 Bureau Drive, Stop 8424, Gaithcrsburg, Maryland

20899-8424, USA (B.E.K.).

NATURE| VOL 403|20 IANUARY 2000 1 www.nalure.com

ambient reservoirs and the sensitivity of the rate of decoherence to the size of cat were observed. Here we extend the investigations

beyond the ambient reservoirs and 'engineer' the state of the

reservoir, as well as the form of the system -reservoir coupling.

One way this can be achieved for a system of trapped ions is by

applying noisy potentials to the trap electrodes, simulating a hot

resistor (reservoir) connected to the trap electrodes, with controllable temperature and spectrum. For a range of two-component

superposition states, we demonstrate the expected exponential

dependence of the decoherence rate on the separation of the components in Hilbert space. We also present the first, to our

knowledge, study of decoherence into an engineered quantum

reservoir, using laser cooling techniques to generate an effectively zero-temperature bath8,9

.

Theoretical predictions

Decoherence of specific mesoscopic quantum superpositions, with

a variety of couplings to a reservoir, has been investigated extensively in theory3 "5,810" 12

. The model in these studies is a system

harmonic oscillator coupled to a bath of environment quantum

oscillators. (These and other sources of decoherence in the context

of trapped-ion experiments have been more recently discussed

theoretically in refs 13-16.) As an illustration, we consider the system oscillator to be in a superposition of coherent states. A

coherent state

17 of a harmonic oscillator is a gaussian wavepacket

which oscillates back and forth while retaining its shape. In quantum mechanics, a coherent state is represented by a state vector |tv), where a = \ct\e'

e is a complex number whose magnitude

|cv| is a dimensionless amplitude of the wavepacket's motion and

whose phase 6 is the phase of the oscillation at some initial time

t = (the phases of all subsequent coherent manipulations are set relative to this initial phase). Coherent states are analogous to

classical trajectories of a harmonic oscillator, approximated by a marble rolling back and forth in a bowl. A superposition of coherent

states, \\p) = N(|a,) + \a2 )) where N is a normalization factor, can be visualized as a marble rolling in a superposition of two

trajectories. We consider the system oscillator to couple to the reservoir through an interaction proportional to the product of the ampli- tude of motion of the system oscillator and the amplitude of

269

TN-121
articles

fluctuations of the reservoir. For brevity, we call this an amplitude

reservoir. In the classical analogy, a hot amplitude reservoir behaves

as if the bowl is subject to random displacements of its centre, resulting in a random force on the marble. For a superposition of coherent states coupled to such a reservoir, a simple scaling law may

be stated: the rate of decoherence (here a dephasing between the |aj)

and |a2) components of |^)) scales as the square of the separation of

the wave packets, |a, — a2 \

2

. In an idealized case where, first, the superposition is created, then the amplitude reservoir is coupled to the system for a time f, and then the coupling is turned off, the remaining coherence between the two wave packets is

4

:

C(r) = exp[-|a,-a2 |

2 £f] (1) Here £ is a coupling constant between the reservoir and the system. The larger the size (|a, — a2 |) of the superposition, the faster the decoherence. Another basis of quantum states for the harmonic oscillator is the energy eigenstates, also known as Fock or number states. The Fock

state \n) has energy hco(n + 1/2) and represents a state of n units of quantized vibration, where n 3= is an integer. Fock states have no

classical analogue, as they are delocalized in position and uniformly

distributed in phase. A superposition of two Fock states

\\p) — (\n i )+ \n2 ))/y/2 loses coherence when the modes of the

reservoir couple linearly to the energy of the oscillator, which is equivalent to averaging over a gaussian distribution of phase shifts of the oscillator. We denote this case a phase reservoir. The

coherence between the two Fock states decays at a rate that scales as the square of the difference between the Fock indices, |n, — n2 \ 2

, given by4

:

C(f) = exp[-|« l -«2 |

2 /cf] (2) Here k is a coupling constant.

Trapped ions

In the experiments described here, a linear Paul trap, similar to the one described in ref. 18, confines single

9 Be

+ atomic ions in a harmonic potential, for which we isolate the axial motion at frequency to = 2ir X 11.3 MHz. Within the ion's electronic ground-state hyperfine manifold we restrict our attention to two

states, the \F = 2, m? = — 2) state, which we label ||), and the

\F = 1, mf = - 1) state, which we label |t), separated in energy by

fkii , where to = 27t X 1.25 GHz, and where F and mr are the quantum numbers associated with the total angular momentum

of the atomic state. The ion is cooled to the n = ground state of motion, denoted |0), and optically pumped to the ||) state with

resolved-sideband stimulated Raman cooling". Thus, the initial state for all the experiments is |1)|0). We drive coherent stimulated Raman transitions with a pair of

laser beams detuned approximately 12 GHz from the atomic

resonance near 958 THz (\=313nm). We use three types of Raman transitions, determined by the beam geometry and differ- ence frequency of the two beams: ( 1

) Motion-independent spin-flip transitions (| l)|n)«-» |T )!»)) Here, the Raman beams are co- propagating and the difference frequency is set to co . (2) Sideband

transitions (| \ )\n) «-»

| f )\n + An)). Here, the beams are orientated with their difference wavevector pointing along the trap axis and

their difference frequency set to a motional sideband at w + coAm.

(3) Motional displacement transitions. Here, the beams are orientated with their difference wavevector pointing along the trap axis and

their difference frequency set to the trap frequency to. This approxi- mates the harmonic-oscillator displacement operator D(a), where

the operator is defined4,5 by the relation D(a)\0) = \a). The dis- placement |a| is proportional to the duration of the laser pulse, and

6 is set by the phase of the applied laser field

613

. In general, a

depends on the internal state of the ion. We can efficiently detect the ||) internal state of the ion by

applying circularly polarized laser light resonant with the transition !!)<- |e), where \e) is a short-lived excited electronic state that usually decays back to ||) by emitting a photon". In contrast, the

transition |t) «-» \e) is out of resonance, and an ion in the |1) state scatters negligible light.

High-temperature amplitude reservoir The motion of a trapped ion couples to uniform electric fields E

through the potential U = — qx E, where x is the displacement of

the ion from its equilibrium position (proportional to the ampli- tude of motion) and q is the charge of the ion. This coupling is independent of the ion's internal state. Our engineered amplitude

reservoir consists of random uniform electric fields applied along

the axis of the trap, oscillating near the ion's axial-motion frequency

co. We generate axial fields in the trap by applying voltages to one of

the trap electrodes. A commercial function generator produces

i 1

10°l

C/l tno

o

CDO)

lZ

° \ °

1 1

ll)u1

X *

*^a

« |Aaj = 1.4 o |Aor| = 2.0

a |Ao| = 2.5

|Aaj = 3.1

s

o \

) 200

\ v

400 600 800 1 |Aapt(ns)000

10"1

* |Aor| = 1.0 o |Aa| = 2.1 a |Aor| = 2.6

|Aa| = 3.1 v |Aor| = 4.1 < |Aor| = 5.2

*

, ,

i N , 0.6

|Aa| 2<l/2 )(V2

)

Figure 1 The decoherence of 'Schrbdinger-cat' states coupled to an amplitude reservoir.

In the main figure, each point is the measured contrast of the interference fringes after noisy potentials were applied to the trap electrodes. The fringe contrast at (I/

2

) = is scaled to unity in order to make comparisons between different values of lAal. The size of the superposition, lAcvl, varies linearly with the pulse time for Raman transition type (3).

270

The applied mean-squared voltage (V 2

) is scaled by lAal

2

. The solid line is a fit to an

exponential. Inset, fringe contrast versus time of interaction with ambient fields is plotted.

Again, the fringe contrast is scaled to unity at t = for comparison between different values lAal. The solid line is a fit to an exponential.

TN-122 NATURE VOL 403 20 JANUARY 2000 www.nalure.com
articles

pseudo-random voltages which are applied through a band-pass

filter centred near w, defining the frequency spectrum of the

reservoir. In all the experiments reported here, we measure the coherence

of the quantum superpositions with single-atom interferometry,

analogous to that used in our previous work6. For example, to observe the effects of the amplitude reservoir, the motional state of

the ion is split into a superposition of two components, each

associated with a different internal state of the ion, forming a

state like that of the Schrodinger cat

6

. The superposition is then coupled to the reservoir, and finally the perturbed superposition is recombined by reversing the steps which initially created it. We

repeat the experiment many times, measuring the internal state of

the ion as a function of the relative phase ofthe creation and reversal

steps, and the contrast of the resulting interference fringes characterizes the amount of coherence remaining after coupling to the

reservoir. In more detail, we first form a cat state of the form:

the ion in the ||) state is

2

lfc> = (li>K>+IT>|a,»/>/2 (3)

This is created by driving a Raman transition (type (1)) to generate an equal spin superposition, ||)|0)— (||) + |T))|0)/\/2, followed by

a Raman transition (type (3)), with laser polarizations set such that

a, = — a!|/2 in equation (3). A uniform electric field oscillating near the trap frequency w

(applied in the experiment for 3 (is) results in the displacement

operator D((3) acting equally on both \[) and |T), giving:

l\U- IV'c) = (IDI/3 + «i> + e»" |T)|j3 + a^ly/l (4)

Here <f>m — Im/3Aa* and Aa = o;

1 — a,. We probe the coherence by

reversing the steps taken to generate the cat state. We first reverse the motional Raman transition (type (3)), resulting in the state

l^>-»l^) = (ll> + e

2 *"IT»li8Vv/2 (5) A final pulse on the motion-independent spin-flip transition (1), with phase 5 relative to the first pulse on transition (1), leads to interference fringes with a residual phase shift 20m . Averaging

over the gaussian random variable /3, the probability of finding

10°

10-'

] *\

1

^s^a

D

s. O

I ' 1

o \ V

a

* |0> + |1) ° |0> + |2> ° |0) + |3> v |1> + |2>

0.05 0.1 0.15 0.2 0.25 03 0.35 0.4 (An)2<V2

> (V2

)

Figure 2 Decoherence ot superpositions of Fock states coupled to the phase reservoir. The data points are the measured fringe contrast. The fringe contrast is normalized to

unity at (V2 ) = 0. The mean squared voltage applied to the trap electrodes is scaled by

the squared size of the superposition lA/il 2

. The solid line is a fit to an exponential.

A=rO e coso) (6)

Interference fringes are generated by recording Pj while sweeping 5. The variance a of /3 is proportional to the mean-squared voltage noise (V2) (proportional to the temperature of the simulated

resistor). A plot of the interference- fringe contrast as a function of the applied mean-squared voltage, scaled by the squared 'size' of

the cat state | Aa| , is shown in Fig. 1. Decay curves were recorded for a variety of superposition sizes |Aa|, and all the data agree with a

single exponential. In addition to the engineered reservoir of the applied voltage

noise, the ion also interacts with ambient fluctuating electric fields, which we expect to have the character of an amplitude reservoir. To

examine this 'natural' decoherence, we ran the experiment outlined above without any applied voltage noise, and with a variable time f between the creation of the cat state and the recombination. The

fringe visibility as a function of |Aor|

2 r is shown in the inset to Fig. 1. The decay curves are normalized to unity at f = 0. The decay of the

fringe visibility is exponential, and the decay constant 7=

6.7 X 10

~ 3 (xs _1

is consistent with the measured heating rate

13 of 7 ~ 5.9 X 10

~ 3 (xs~ ' for this apparatus. The effects of this ambient

reservoir were negligible during the time (3 u.s) that the engineered

amplitude reservoir was coupled to the ion.

High-temperature phase reservoir A phase reservoir coupled to the ion is simulated by random

variations in the trap frequency w, changing the phase of the ion

oscillation without changing its energy. We realize this coupling

experimentally by modulating the trap frequency. A random voltage noise source is passed through a low-pass filter network with a cutoff frequency well below w to maintain adiabaticity. The fluctua- tions in potential are applied symmetrically to the trap electrodes so

as to produce linear field gradients and negligible uniform fields. This in turn perturbs the trap frequency, by <5to(r). When integrated over the time (20 (xs) of the applied noise, the ion's motion is phaseshifted by 4> = /5o>(r)df. This technique yields a gaussian-distrib- uted ensemble of phase shifts with variance a

2 proportional to the applied mean-squared voltage noise (V2

). Motional decoherence caused by a phase reservoir is clearly illustrated with a superposition of two Fock states. We generate superpositions of Fock states of the form li^) = |s)(|«)+ \n'))l\/2, where s — 1 or |, with pulses on the Raman motional sidebands (case

(2) above) as in ref. 20. The trap frequency is then perturbed by the

Figure 3 Implementation of an engineered zero-temperature reservoir. The states 11)1 n) and \\)\n - 1) are coupled by driving Raman motion-sensitive transitions (case (2) in the

text). The state IT) is coupled to the environment by applying a weak optical pumping

beam. The circles represent the superposition generated before applying this zero- temperature reservoir. The arrows show how the population in the I1)I2) state is driven

to the IT)I1) state, and subsequently to the 11)11) state, through spontaneous Raman

scattering.

NATURE|VOL 403 1 20 JANUARY 2000 1 www.nature.com TN-123 271
articles

o

o

<uen

Figure 4 Decoherence of a Fock state superposition into the engineered zero- temperature reservoir. Fringe contrast is plotted as a function of the time the system is

applied random potentials, and the Fock states of the superposition

acquire a relative phase factor e'^", where An = n — ri . The steps that created the superposition are then reversed, with a relative phase difference 5 between the creation and reversal pulses, leading

to a probability of detecting the ion in the \[) state

21

:

Pi=-[l + e

|An|V/2

cos<5] (7)

Interference fringes are recorded by varying 5 as in the Schrodingercat interferometer. The fringe contrast is plotted as a function of

|An| 2(V2) in Fig. 2. As with the Schrodinger-cat states and amplitude

reservoir, the data were fitted by a single exponential in |Am| 2(V)

2

.

Zero-temperature reservoir A third type of engineered reservoir requires a quantum mechanical

description. This is a bath of laser cooling light plus optical

spontaneous emission, an engineered (nearly) zero-temperature

reservoir following the suggestion of Poyatos et al.

s

. Our implementation, shown in Fig. 3, is essentially a continuous Raman

cooling technique. A pair of Raman beams (case (2)), tuned to the

first red sideband, couples the states IDIh)*-* ||)|« — 1). Concurrently, an optical pumping beam causes spontaneous Raman

transitions from ||) to ||) through an unstable excited state \e), which decays at rate T. The Raman coupling strength is characterized by the Rabi frequency Qnb, a function of the intensity and

detuning of the Raman beams' 3

. If the Rabi frequency of the optical

pumping beam is Od , then we can define an effective damping rate for the |f) state of y — D

/r, valid for our case ofOd <C T. From the diagram in Fig. 3 we see that all populations are driven towards the

state |1)|0), the defining property of a zero-temperature reservoir. By

varying the strength of the Raman and optical pumping couplings, we can control the reservoir parameters.

In the experiment, we examine the time evolution of the coher- ence of the Fock state superposition \j/ = ||)(|0)+ |2))/\/2 for varying lengths of reservoir-interaction time. The interferometry

is the same as in the study of the phase reservoir, where the Fock

superposition is created, coupled to the reservoir, recombined, and

probed, generating interference fringes. The data are shown in Fig.

4. Each data point represents the contrast of the fringes after the system interacts with the reservoir. We show two cases, y < Qrhs and

7 > Qrsh . In the former case, the coherence between the |0) and |2) state disappears and reappears over time, with an overall decay of

the fringe contrast. The underlying effect is population transfer back and forth (Rabi flopping) between the states |1)|2) and |f)|l) with a coupling of the ||)jl) state to the outside environment through

coupled to the zero temperature reservoir. The only difference in the two cases shown was

the intensity of the optical pumping beam (see Fig. 3).

spontaneous Raman scattering. In effect, we have restricted the size of the environment (here the manifold of ||)|n) states, weakly

coupled to the outside environment) to an extent where we can

reverse the effects of decoherence (of the \j/ = |i)(|0)+ \2))ly/l

state) in a way similar to that proposed in ref. 22. This is also a

striking example of non-exponential decay23 in a context that is investigated in ref. 24. For the case y > Onb , the fringe contrast decreases monotonically to zero. Even in the case of monotonic

decay, a deviation from exponential is observed, a manifestation of

the quantum Zeno effect

24,25

. Although the data with y < Orib illustrate how coherence transferred to the environment can be recovered, an alternative explanation would say that by transferring the |I)|2) component of the superposition to the ||)| l) state, we gain 'which-path' information in our interferometer—the paths being the ||)|0) and |J)|2) parts of

the superposition. The oscillation in which-path information is analogous to that illustrated by other experiments26,27

.

Conclusions

The decoherence caused by the engineered high-temperature reser- voirs described above can be explained by ensemble-averaging over random classical fields applied to the ion21,28

. From previous

experiments20

, we know that we can undo the effects of this decoherence by applying, in each experiment, a pulse of radiation

that reverses the 'random' displacement. Similarly, the experiments

here could also be carried out by coupling a hot resistor (with

appropriate spectral filtering) between the trap electrodes (our case would correspond to a limit where the temperature T —> °° and the damping resistance R — 0) (ref. 10). However, even in this case we

could, subject to both practical and fundamental measurement

uncertainties, record the voltages applied to the electrodes and

reverse the effects of the random noise in each experiment. If we

choose to ignore any knowledge of the electrical potentials applied

to the trap electrodes, we can account for the observations just as well by considering the ion to be coupled to a large number of quantum oscillators, forming a heat bath. In the latter case, the state of the ion is entangled with that of the environment oscillators. After tracing over the environment variables, we are left with a reduced system, involving only the ion. The behaviour is the same as

that obtained in the former case, in which the decoherence is caused by a deliberately applied external potential, but the environment is not considered to be a dynamical system itself

3,4

. Loosely speaking,

the effect of an environment oscillator in the latter case is replaced

by that of a single Fourier component of the electrical potential in the former case. Therefore, in the high-temperature limit simulated

272

TN-124

NATURE [ VOL 403 1 20 JANUARY 2000 1 www.nature.com
articles

by the first two experiments, one need not consider the entangle- ment with the environment because the environment noise can be

sensed (classically) and its effects reversed. This is in contrast to the decay of ion motion into a zero-temperature reservoir described

above, similar to that seen in cavity-QED experiments7

. In this case,

after the quantum system couples to the environment through

spontaneous emission, a measurement of the environment is not

sufficient to reverse the effects of decoherence. The methods of engineering reservoirs that are presented here begin to broaden the field of experimental investigations of decoherence. With control over the reservoir parameters combined with

non-classical motional states of trapped ions, detailed comparisons

between theory and experiment are possible. Here we have simulated the decoherence caused by coupling a charged atom to a hot

resistor (reservoir) by applying noisy voltages to the ion-trap

electrodes. The cases considered demonstrate a quadratic depen- dence of the rate of decoherence on the size of the superpositions,

demonstrating the difficulty in generating truly macroscopic superpositions, such as that of 'Schrodinger's cat'. As a practical matter, these 'high-temperature' sources of noise are important because

they currently limit the performance of a trapped-ion quantum

computer13. We have also simulated a zero-temperature reservoir by

using laser cooling to damp the ion motion. Extensions of the technique used to generate this zero-temperature bath should

permit some interesting system-bath interactions that would be

difficult to realize in any other way. One possibility is generating a squeezed reservoir, where all initial states asymptotically relax to a squeezed state of motion8

. Other couplings can be tailored to relax the system into a 'Schrodinger-cat' state

29,30

.

Received 24 September; accepted 30 November 1999.

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Phys. Rev. Lett. 75, 4011-4014 (1995).

20. Meekhof, D. M-, Monroe, C., King, B. E., Itano, W. M. 8c Wineland, D. J. Generation of nonclassical motional states of a trapped atom. Phys. Rev. Lett. 76, 1796-1799 (1996).

21. Myatt, C. ). et al. in Laser Spectroscopy 13 (ed. Blatt, R., Eschner, ]., Leibfried, D. 8c Schmidt-Kaler, F.) (World Scientific, Singapore, in the press).

22. Raimond, J. M., Brune, M. & Haroche, S. Reversible decoherence of a mesoscopic superposition of

field states. Phys. Rev. Lett. 79, 1964-1967 (1997).

23. Wilkinson, S. R. et al. Experimental evidence for non-exponential decay in quantum tunnelling. Nature 387, 575-577 (1997).

24. Thompson, R. C, Hernandez-Pozos, J.-L., Hoffges, J., Segal, D. M. 8c Vincent, J. R. in Trapped Charged

Particles and Fundamental Physics (eds Dubin, D. H. E. 8c Schneider, D.) 388-392 (American Institute of Physics Conf. Proc. 457, AIP Press, Woodbury, New York, 1999).

25. Itano, W. M., Heizen, D. )., Bollinger, J. J. 8c Wineland, D. ). Quantum Zeno effect. Phys. Rev. A 41, 2295-2300(1990).

26. Chapman, M. S. etal. Photon scattering from atoms in an interferometer: coherence lost and regained.

Phys. Rev. Lett. 75, 3783-3787 (1995).

27. Durr, S., Nonn, T. 8c Rempe, G. Fringe visibility and which-way information in an atom inter- ferometer. Phys. Rev. Lett. 81, 5705-5709 (1998).

28. Itano, W. M. etal. Quantum harmonic oscillator state synthesis and analysis. Proc. SP/E 2995, 43-55

(1997).

29. de Matos Filho, R. L. 8c Vogel, W. Even and odd coherent states of the motion of a trapped ion. Phys.

Rev. Lett. 76, 608-61 1 ( 1996).

30. Garraway, B. M-, Knight, P. L. 8c Plenio, M. B. Generation and preservation of coherence in dissipative quantum optical environments. Phys. Scr. T76, 152-158 (1998).

Acknowledgements We thank the US National Security Agency, Army Research Office, and Office of Naval

Research for support. We thank R Zoller, H. Mabuchi and W. Zurek for discussions. We

thank them, D. Leibfried, M. Rowe, D. Sullivan and M. Lombardi for comments on the manuscript.

Correspondence and requests for materials should be addressed to D.J.W.

(e-mail: dwineland@nist.gov).

NATURElVOL 403 I 20 IANUARY 2000lwww.nalure.com TN-125 273
letters to nature

Experimental entanglement

of four particles

C. A. Sackett*, D. Kielpinski*, B. E. King*t, C. Langer , V. Meyer*,

C. J. Myatt*$, M. Rowe*, Q. A. Turchettel , W. M. Itano*, D. J. Wineland * & C. Monroe"

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

t Atomic Physics Division, NIST, Gaithersburg, Maryland 20899, USA

$ Research Electro-Optics, 1855 S. 57th Ct., Boulder, Colorado 80301, USA

Quantum mechanics allows for many-particle wavefunctions that cannot be factorized into a product of single-particle wavefunctions, even when the constituent particles are entirely distinct. Such 'entangled' states explicitly demonstrate the non-local character of quantum theory1 , having potential applications in highprecision spectroscopy2

, quantum communication, cryptography

and computation3

. In general, the more particles that can be

entangled, the more clearly nonclassical effects are exhibited4,5 —

and the more useful the states are for quantum applications. Here we implement a recently proposed entanglement technique6 to generate entangled states of two and four trapped ions. Coupling

between the ions is provided through their collective motional

degrees of freedom, but actual motional excitation is minimized.

Entanglement is achieved using a single laser pulse, and the method can in principle be applied to any number of ions. Most experimental demonstrations of entanglement to date have

relied on the selection of data from random processes, such as the preparation and detection of photon pairs in parametric downconversion7"9 or of atoms in a thermal beam 10

. All methods of this type suffer from inescapable signal degradation when entanglement

of larger numbers of particles is attempted, as the probability of randomly generating the appropriate conditions decreases exponentially. For instance, in the experiment of ref. 7, two-photon

entangled states could be generated and detected at a rate of roughly

1,000 per second, three-photon states at a rate of 30 per hour, and

four-photon states at an extrapolated rate of several per year. Trapped ions have been suggested as a system in which such effects might be avoided", and we have demonstrated two-particle entan- glement in a deterministic way

12

. By 'deterministic' we mean that the desired state could be produced with a high degree of certainty at a user-specified time 13

, which is necessary for avoiding the degradation described above. However, that experiment relied on the

particular behaviour of two ions in a quadrupole radio-frequency

trap, and could not easily be applied to larger numbers of particles. The entanglement technique proposed by Molmer and

Sorensen614 can be understood by considering a pair of spin-half

charged particles confined together in a harmonic potential. The

energy levels of this system are illustrated in Fig. 1, where hu is the

internal energy splitting of each particle, and v is the oscillation frequency of a particular collective mode of the particles in the trap. Using laser-cooling and optical-pumping techniques 15

, the particles are initially prepared in their spin-down internal state and in the ground state of their collective motion: |^) = 1 11 0). By applying

optical fields oscillating at w + v — 8 and u> - v + 8, the two-step

transition from J XX 0) to |fl 0) is driven. For sufficiently large 6, the intermediate states |U l)and|H l) are negligibly occupied, so that no

motional excitation occurs. The resulting interaction hamiltonian,

in the rotating-wave approximation and the Lamb-Dicke limit, is then

Rabi frequency Q and the Lamb-Dicke parameter is rj. For an

excitation involving momentum transfer hk and a total particle mass of M, 77 is given by (frtf/lMv)i . Entanglement is achieved by

applying H for a time t = ir/20, making the spin wavefunction

1^2) = (ITT) - i|li)W2 - This spin state is in fact created for any

initial motional state |«), so long as the Lamb-Dicke criterion

tj 2 « -C 1 is satisfied. In order for the intermediate states |T1 l) and | XT l) to be negligibly

occupied, the detuning 6 must be large compared to the transition linewidth rjO. However, it is clear from the expression for Cl that the entanglement speed is maximized for small 8, and in fact the technique can still be applied for 8 = r)Q (refs 14, 16). Although

motional excitation does then occur to some degree, for select values of 8 the excitation vanishes at precisely the time that the entangled spin state is created. The condition for this to occur is

8/vn = 2y/rrt (2)

for any integer m, and the maximum excitation during the pulse then has mean quantum number n — \l2m. Our experiment is operated with m = 1

.

As discussed in ref. 6, the entanglement method is scalable in the

sense that precisely the same operation can be used to generate the N-particle entangled state

l^) = (ITT...T) + 'N+, III...l))/v/2 (3)

ifNis any even number, while for Nodd, \\p^) can be generated using one entanglement pulse accompanied by a separate independent

rotation of each particle's spin.

If the ions are uniformly illuminated, the Molmer and Sorensen scheme614 requires that they all participate equally in the inter- mediate motional excitation, which implies that the only suitable mode for arbitrary N is the centre-of-mass mode. However, this mode has a practical disadvantage because in our experiments

fluctuating ambient electric fields cause it to heat at a significant

rate. Although for large 5 the entanglement operation is largely independent of the motion, so that heating is unimportant, in the small-5 case it is necessary that motional decoherence be avoided. Modes involving only relative ion motion couple to higher

moments of the field, so heating of these modes is negligible 15

. For N = 2 and N = 4, such modes do exist in which each particle

participates with equal amplitude 17. In both cases, they are symmetric 'stretch' modes, in which alternating ions oscillate out of

|tto>-

H

nn

(ITTXUI + IUXTTI + IUX1TI + I1TXUI) 0)

with Q = r)

2 Q

2 /8 when the single particle |1)«— |T) transition has

|U0>-

Figure 1 Entanglement scheme for two particles. Each ion is initially prepared in the ll>

internal state, and the collective motion of the pair is cooled to its ground state I0>. Laser

fields oscillating near w + v and u - v couple the 111) and ITT) states as shown. By

detuning the single transition frequencies by a small amount 5, the populations of the li|1 > and ITU) states are kept small. Then, by driving the double transition for the appropriate

time, the entangled state (| ft)- /] 11 »/v 2 is created. For four ions, the same procedure

generates the state (| TTTT > + 'I UU ))/;2. We note that in the actual experiment, each

of the single transitions shown is itself a two-photon Raman transition, driven by a pair of

laser beams; the entire process therefore consists of a four-photon transition.

256 TN-126 i © 2000 Macmillan Magazines Ltd NATURE | VOL 404 | 16 MARCH 2000| www.nature.com
letters to nature

phase. We use these modes here. Excitation of the centre-of-mass mode still affects the experiment, as the motion in spectator modes

modifies the coupling strength to the mode of interest'

415

. For this reason, we initially cool both the centre-of-mass and stretch modes

to near their ground state. The experiment was performed using 9 Be

+ ions confined in a miniature linear radio-frequency trap

18

, with the N ions lying in a

line along the trap's weak axis. Two spectrally resolved ground-state

hyperfine levels comprise the effective spin-half system, with

fl>= \F = 2,mF = -2) and \])=[F=l,mF = - l), where F is the total angular momentum quantum number, and hmF

is the

projection of the angular momentum along the quantization axis defined by an externally applied magnetic field. The hyperfine

splitting u /2ir is approximately 1.25 GHz. Coherent coupling

between |1) and ||) is provided by stimulated Raman transitions. The two Raman laser beams have a wavelength of 3 1 3 nm with a difference frequency near co , and are perpendicular, with their difference wave-vector lying along the line of ions. They are detuned —80 GHz blue of the 2PU2 excited state, with intensities giving

Qllir = 500 kHz. For both the two- and four-ion experiments, the

desired stretch-mode frequency vllir was 8.8 MHz, giving q = 0.23/N'72

. The two driving frequencies required for the entan- glement operation are generated by frequency-modulating one of

the Raman beams using an electro-optic modulator.

After the entanglement operation, the ions are probed by illumi- nating them with a circularly polarized laser beam tuned to the 2S1/2 (F = 2, mF = — 2) <-* 2Pi/2 (F = 3, mF = — 3) cycling transition. Each ion in ||> fluoresces brightly, leading to the detection of ~15 photons per ion on a photomultiplier tube during a 200-(xs detection period. In contrast, an ion in ||) remains nearly dark. Because the number of photons detected from a spin-down ion

fluctuates according to Poisson statistics, in a single experiment the number of spin-down ions can be determined with only a limited

accuracy. For the data reported, each experiment was repeated 1,000 times under the same conditions, and the resulting photonnumber distribution fitted to a sum of poissonians to determine

the probabilities Pj for j ions to be in ||). The results are given in Table 1, and show that in both cases, the probabilities for all N ions

to be in the same state are large compared to the probabilities for the other cases. This is characteristic of the states |i/- w), although the fact

Table 1 Characterization of two-ion and four-ion states N Po P, P2 P3 P4 ''111)

2 0.43 0.11 0.46

4 0.35 0.10 0.10 0.10 0.35 0.385

0.215 N is the number of ions, P, denotes the probability that/ ions were measured to be in |1), and pm) denotes the amplitude of the density matrix element pi,..tj„.i. Uncertainties in pm and the A/ = 2

populations are ±0.01 , and uncertainties in the A/ = 4 populations are ±0.02.

that the middle probabilities are non-zero indicates that we do not

generate the entangled states with perfect accuracy. In order to prove that we are generating a reasonable approximation to \\Pn)> it is necessary to prove that the populations of |T...j)

and ||...|) are coherent. In terms of the density matrix for the system, p, we must measure the far off-diagonal element pj.-.U-.l' whose amplitude will be abbreviated p^. This can be achieved by

applying a simple analysis pulse to the ions before observing them.

If the Raman difference frequency is set to to (and the frequency

modulator turned off), each ion i undergoes ordinary Rabi oscilla- tions, evolving according to the hamiltonian

H,=^(e'1?),(l|,+e-'*U),(f|,) (4)

where </> is the phase of the difference frequency relative to that of

the entanglement pulse. This hamiltonian is applied for time irl2Q

(a 7r/2 pulse), and the parity

/7=|Viypy (5)

>=0

is observed while </> is varied. As seen in Fig. 2, for Nions 77 oscillates as cos N(f>, and the amplitude of this oscillation is in fact 2p(tj) (ref. 2). The resulting values are given in Table 1. From the data shown in the table, our state preparation fidelity

F - <iMpI*n> = ^Pm + Pa>) + P(Si) (6)

can be determined, where P(t) is the population of |t..-t) and P(J) is the population of |]...l). For N = 2 we achieve F = 0.83 ± 0.01, while for N = 4, F = 0.57 ± 0.02.

180 270 Phase (degrees)

Figure 2 Determination of pm. a, Interference signal for two ions; b, four ions. After the entanglement operation of Fig. 1 , an analysis pulse with relative phase <A is applied on the

single-ion 1 1 ) —• 1 1 ) transition. As $ is varied, the parity of the A/ ions oscillates as

cos N<t>, and the amplitude of the oscillation is twice the magnitude of the density-matrix element pIU) . Each data point represents an average of 1 ,000 experiments, corresponding

to a total integration time of roughly 10s for each graph.

NATURE|VOL404| 16 MARCH 2000 1 www.nature.com «v« © 2000 Macmillan Magazines Ltd TN-127257
letters to nature

The fact that p^ is non-zero is still insufficient to guarantee

entanglement. To be explicit, a system with density matrix p exhibits N-particle entanglement only if no decomposition p = E*pi 1^X^*1

exists with all the {{^k)} factorizable into products of wavefunctions

that depend on fewer than N particles. For example, if |\t), =

(It); + |i);)/>/2 is a state of ion /', then the four-particle state

|^)i I $)i 1 <A)3 1^)4 is not entangled, but still has p(I1) = 1/16. We note

that these are the types of states studied in liquid-state nuclear magnetic resonance experiments". Alternatively, for \4) n =

(ll>.IT>2 + II>,|i>2 yV2 and |tf>34 = (ll>,lT>4 + ll>3li>«y>/2, the

state |iA)i 2 |'/')34 exhibits two-particle, but not four-particle entanglement, and has p,n) = 1/4. To establish that we are actually observing N-particle entanglement, consider an arbitrary factorizable wavefunction

\tF )=[a\\...l)x + b\l...l)x + ...][c\ })Y + d\l...l)Y + ...]

(7)

where X and Y refer to two distinct subsets of the N particles, with

|T Ox indicating the state with all particles in subset X spin-up, and

similarly for the other terms. Normalization of the factor wavefunctions requires \a\

2 + \b\2 =£ 1 and |c|2 + \d\2 =£ 1, which can be combined and rewritten as

(\a\-\c\)2 +2\ac\+(\b\-\d\f+2\bd\^2 (8)

Since the squared terms on the left are positive, equation (8) implies

that \ac\ + \bd\ =£ 1, and in turn that (\ac\ + \bd\f =£ 1. Expanding

the square yields the desired relation20

: P(I) +PU) + 2p(U) = 2F^l (9) where P(I) = \ac\

2

, P(1) = \bd\ 2, and p(U) = \abcd\ are the previously

defined quantities. Since equation (9) holds for any separable

wavefunction, it must also hold for any separable density matrix. Both our N = 2 and N — 4 experiments give F > 1/2, so the states they produce exhibit N-particle entanglement.

Quantifying the amount of entanglement present is a more

difficult question. A variety of measures of entanglement have been proposed, but most are difficult to calculate even

numerically21,22

. For N = 2, there is an explicit formula for the "entanglement of formation" £ as a function ofp (ref. 23). Although we have not reconstructed the entire two-particle density matrix,

the populations measured place sufficient bounds on the unmeasured elements to determine that E ~ 0.5. This indicates that roughly two pairs of our ions would be required to carry the same quantum information as a single perfectly entangled pair. In the four-ion case, no explicit formula for entanglement is known. The data do indicate that our density matrix can be

expressed approximately as

p = 0.43|^X^4 |+0.57Pincoh (10) where |i/- 4 ) is the desired state and p Lncoh is completely incoherent

(that is, diagonal). These coefficients are determined directly from

the value of p( jj) in Table 1, together with the fact that no evidence

for other off-diagonal matrix elements was observed. To determine a measure of entanglement, however, it is necessary to decompose p as a sum of ^4) and a 'worst-case' factorizable matrix pF , which can be accomplished as

p = 0.13|^4 X^I+0.87pf

. (11) Note that equations (10) and (11) both describe the same physical

state, but that in equation (11), pF consists of a specific mixture of two- and three-particle entangled states that is highly unlikely to occur in our experiments. In either description, it is clear that our

state-preparation accuracy is limited. The source of decoherence in our experiments is not entirely

clear, but evidence suggests that it is related to intensity fluctuations

in the Raman laser beams, a problem we are working to understand

and correct

24

. The presence of decoherence, and the fact that it affects the four-ion experiment more strongly than the two-ion one,

illustrates the need to carefully define the sense in which our

entanglement operation is 'scalable'. Any entanglement experiment

is more sensitive to decoherence as the number of particles involved

is increased, unless sufficient accuracy can be achieved for error- correction schemes to be usefully applied. Such schemes are thought

to require an error rate of the order of 10~4 per operation25

, and we

are certainly far from this regime. However, even if such a level of

fidelity were to be achieved, applications such as quantum computing still require very large entangled states to be generated in a reasonable amount of time and using a reasonable amount of

resources. The method demonstrated here is important in this regard, since it uses only a single operation and requires a time

that scales roughly as N

l 2

. In the language of quantum information science, we have realized a four-quantum-bit logic gate. This system is relevant for the future development of quantum information technology, as such states may be used to implement quantum error-detection schemes26 or to make rudimentary demonstrations of quantum algorithms27,28

. Entanglement of four particles is also interesting in its own right, as such states can show strong violations of local realism5

. Even the

two-particle Bell's inequality measurement would be interesting to implement, as the near- perfect detection efficiency for ions would

eliminate the "fair sampling" hypothesis which has been required in other experiments29

. In addition to improved fidelity, applications such as these do require the ability to perform individual manipulation and detection of each ion, but this is not expected to be a severe experimental challenge30

.

Received 1 1 February; accepted 15 February 2000.

1

.

Bell, J. Speakable and Unspeakable in Quantum Mechanics (Cambridge Univ. Press, 1 987).

2. Bollinger, I., Itano, W. M., Wineland, D. & Heinzen, D. Optimal frequency measurements with maximally correlated states. Phys. Rev. A 54, R4649-R4652 ( 1996).

3. Lo, H.-K-, Popescu, S. & Spiller, T. (eds) Introduction to Quantum Computation and Information (World Scientific, Singapore, 1997).

4. Pan, J.-W., Bouwmeester, D., Daniell, M., Weinfurter, H. & Zeilinger, A. Experimental test of quantum

non-locality in three-photon Greenberger-Horne-Zeilinger entanglement. Nature 403, 515-519

(2000).

5. Mermin, N. D. Extreme quantum entanglement in a superposition of macroscopically distinct states. Phys. Rev. Lett. 65, 1838-1840 (1990).

6. Molmer, K. & Serensen, A. Multiparticle entanglement of hot trapped ions. Phys. Rev. Lett. 82, 1835-

1838(1999).

7. Bouwmeester, D., Pan, J.-W., Daniell, M., Weinfurter, H. 8c Zeilinger, A. Observation of three-photon

Greenberger-Horne-Zeilinger entanglement. Phys. Rev. Lett 82, 1345-1349 (1999).

8. Kwiat, P. et al. New high-intensity source of polarization-entangled photon pairs. Phys. Rev. Lett. 75, 4337-4341 (1995).

9. Tittle, W., Brendel, J., Zbinden, H. 8c Gisin, N. Violation of Bell inequalities by photons more than 10km apart. Phys. Rev. Lett. 81, 3565-3566 (1998).

10. Hagley, E. ctal. Generation of Einstein-Podolsky- Rosen pairs ofatoms. Phys. Rev. Lett. 79, 1-5 (1997).

11. Cirac, ). 8c Zoller, P. Quantum computations with cold trapped ions. Phys. Rev. Lett. 74, 4091-4094

(1995).

12. Turchette, Q. et al. Deterministic entanglement of two trapped ions. Phys. Rev. Lett. 81, 3631-3634

(1998).

13. Law, C 8c Kimble, J. Deterministic generation of a bit-stream of single-photon pulses./. Mod. Opt. 44, 2067-2074 (1997).

14. M0lmer, K. 8c Sorensen, A. Entanglement and quantum computation with ions in thermal motion.

Pre-print quant-ph/0002024 at (xxx.lanl.gov) (2000).

1 5. King, B. et al. Cooling the collective motion of trapped ions to initialize a quantum register. Phys. Rev.

Lett. 81, 1525-2528(1998).

16. Milburn, G. Simulating nonlinear spin models in an ion trap. Pre-print quant-ph/9908037 at

(xxx.lanl.gov) (1999).

17. James, D. Quantum dynamics of cold trapped ions with applications to quantum computation. Appl.

Phys. B 66, 181-190(1998).

18. Turchette, Q. et al. Heating of trapped ions from the quantum ground state. Pre-print quant-ph/ 0002040 at (xxx.lanl.gov) (2000).

19. Fitzgerald, R. What really gives a quantum computer its power? Phys. Tocfay 53, 20-22 (2000).

20. Bennett, C et al. Purification of noisy entanglement and faithful teleportation via noisy channels.

Phys. Rev. Lett. 76, 722-725 (1996).

21. Vedral. V., Plenio, M., Rippin, M. 8c Knight, P. Quantifying entanglement. Phys. Rev. Lett. 78, 2275-

2279 (1997).

22. Lewenstein, M. 8c Sanpera. A. Separability and entanglement of composite quantum systems. Phys.

Rev. Lett. 80, 2261-2264 (1998).

23. Woolters, W. K. Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 2245-2248 (1998).

24. Wineland. D. et al. Experimental issues in coherent quantum-state manipulation of trapped atomic

ions. /. Res. NIST 103, 259-328 (1998).

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25. Preskill, I. Battling decoherence: the fault tolerant quantum computer. Phys. Today 52, 24-30(1999).

26. Vaidman, L, Goldenberg, L. & Wiesner, S. Error prevention scheme with four particles. Phys. Rev. A

54, R1745-R 1748 (1996).

27. Shor, P. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum

computer. S1AM I. Comput. 26, 1484-1509 (1997).

28. Grover, L. Quantum mechanics helps in searching for a needle in a haystack. Phys. Rev. Lett. 79, 325-

328(1997).

29. Eberhard, P. Background level and counter efficiencies required for a loophole-free Einstein- Podolsky-Rosen experiment. Phys. Rev. A 47, R747-R750 (1993).

30. Nagerl, H. et al. Laser addressing of individual ions in a linear ion trap. Phys. Rev. A 60, 145-1 48

(1999).

Acknowledgements We thank J. Bollinger and E. A. Cornell for comments on the manuscript. This work was

supported by the US National Security Agency, Army Research Office and Office of Naval

Research.

Correspondence and requests for materials should be addressed to C. M.

(e-mail: monroe@boulder.nist.gov).

Quantum distribution of protons

in solid molecular hydrogen

at megabar pressures

Hikaru Kitamura t+, Shinji Tsuneyuki*, Tadashi Ogitsu

& Takashi Miyake

* Institute for Solid State Physics, University ofTokyo, Roppongi, Tokyo 106-8666, Japan

t Department ofSpace Physics and Astronomy, Rice University, Houston,

Texas 77251-1892, USA

$ Applied Theoretical and Computational Physics Division, Los Alamos National

Laboratory, Los Alamos, New Mexico 87545, USA

§ National Center for Supercomputing Applications, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA

\\ Joint Research Center for Atom Technology, Tsukuba, Ibaraki 305-0046, Japan

Solid hydrogen, a simple system consisting only of protons and

electrons, exhibits a variety of structural phase transitions at high

pressures. Experimental studies1 based on static compression up

to about 230 GPa revealed three relevant phases of solid molecular

hydrogen: phase I (high-temperature, low-pressure phase), phase

II (low-temperature phase) and phase III (high-pressure phase).

Spectroscopic data suggest that symmetry breaking, possibly

related to orientational ordering1,2

, accompanies the transition

into phases II and III. The boundaries dividing the three phases

exhibit a strong isotope effect

3

, indicating that the quantummechanical properties of hydrogen nuclei are important. Here we

report the quantum distributions of protons in the three phases of

solid hydrogen, obtained by a first-principles path-integral molecular dynamics method. We show that quantum fluctuations of protons effectively hinder molecular rotation—that is, a quantum

localization occurs. The obtained crystal structures have entirely

different symmetries from those predicted by the conventional

simulations which treat protons classically. The structures of these broken-symmetry phases have been

extensively investigated both by experimental 1 3 and by theoretical

4 '' studies, though the results are still controversial. In most

theoretical studies, even when they precisely compare total energies of model structures on the basis of electronic-structure calculations, quantum fluctuations of protons are usually neglected, or roughly

discussed, perhaps due to some technical reason. But as shown by

our previous study on impurity muonium (a bound state comprising

a positive muon and an electron) and hydrogen in crystalline

silicon 1 ", quantum states of light particles may exhibit distributions

significantly different from those expected from stability analyses based on the classical potential-energy surface. It is therefore

desirable to examine quantitatively the quantum-mechanical

properties of protons in solid hydrogen, especially when we are concerned with its symmetry breaking under high pressure. Natoli

et «/." treated both electrons and protons with quantum Monte

Carlo methods; their simulations were, however, performed with

trial wavefunctions based on the fixed static configuration of the protons, and were restricted to the molecules centred on the hexagonal close packed (h.c.p.) lattice sites. The first-principles path-integral molecular dynamics (FP- PIMD) method enables us to incorporate quantum-mechanical

properties of protons in conventional Car-Parrinello-type first- principles simulations. The basic formalism was first presented by

Marx and Parrinello 12. Recently this scheme was further developed

by the present authors'", and we use this in the present study. We

consider a supercell containing N (= 64) atoms, which is subject to the periodic boundary conditions. To represent the quantum

properties of protons in the path-integral formalism, imaginary

time @h (where is the inverse temperature in units of Boltzmann

constant) is divided into P finite time slices; each proton is thus represented by a polymer consisting of P 'beads' interacting via intrapolymeric harmonic forces'

3

. Exchange effects between protons are neglected in this study. Assuming the Born-Oppenheimer

approximation, interatomic forces between protons at each atomic

configuration and at each time slice are determined by electronic- structure calculation based on the density-functional theory (DFT).

The resultant classical system of NP particles is simulated by

molecular dynamics (MD) at constant volume and temperature

with the aid of the Nose-Hoover chain thermostat 14. In the evaluation of the exchange-correlation potential in the DFT calculation, we adopt the generalized gradient approximation (GGA) 1 '. The interaction between a proton and an electron has been

described by the norm-conserving non-local pseudopotential of

the Troullier-Martins type

16

; the electronic wavefunctions have been expanded in plane waves with a cut-off energy of 50 Ry. Integration over the first Brillouin zone has been achieved by

sampling 8 uniform Ac-points. Energy eigenvalues of the lowest 32 occupied bands have been explicitly calculated, since the system should remain an insulator in the relevant pressure range

1

. The

ground-state electron density has been obtained by minimizing the

total energy functional through the conjugated-gradient method17

. We have verified, through static DFT calculations with these parameter conditions, that the potential energy surface associated with molecular rotation in the Pcal x structure obtained earlier by

Figure 1 Illustration of the Pca2, structure projected onto the x-y plane. Thick arrows

represent molecules on the a-plane and thin arrows depict those on the p-plane, both

pointing towards the positive-z hemisphere. These two planes are apart from each other by c/2 in the z-direction. The quantity measures the angle between a molecular axis and

the c(z)-axis of the crystal: <p is the corresponding azimuthal angle. The Cmc2y structure

is obtained by setting <p = 90°.

NATURE | VOL 404 | 16 MARCH 2000 1 www.nature.com *5 © 2000 Macmillan Magazines Ltd TN-129

259
PHYSICAL REVIEW A, VOLUME 61, 032310

Sympathetic cooling of trapped ions for quantum logic

D. Kielpinski, B. E. King, C. J. Myatt, C. A. Sackett, Q. A. Turchette, W. M. Itano, C. Monroe, and D. J. Wineland

Time and Frequency Division, National Institute of Standards and Technology, Boulder, Colorado 80303

W. H. Zurek

T-6 (Theoretical Astrophysics), MS B288, Los Alamos National Laboratory, Los Alamos, New Mexico 87545

(Received 27 August 1999; published 16 February 2000)

One limit to the fidelity of quantum logic operations on trapped ions arises from heating of the ions' collective modes of motion. Sympathetic cooling of the ions during the logic operations may eliminate this source of errors. We discuss the benefits and drawbacks of this proposal, and describe possible experimental

implementations. We also present an overview of trapped-ion dynamics in this scheme. PACS number(s): 03.67.-a, 32.80.Pj

I. INTRODUCTION

One of the most attractive physical systems for generating

large entangled states and realizing a quantum computer [1]

is a collection of cold trapped atomic ions [2]. The ion trap quantum computer stores one or more quantum bits (qubits)

in the internal states of each trapped ion, and quantum logic

gates (implemented by interactions with externally applied

laser beams) can couple qubits through a collective quantized mode of motion of the ion Coulomb crystal. Loss of

coherence of the internal states of trapped ions is negligible

under proper conditions but heating of the motion of the ion

crystal may ultimately limit the fidelity of logic gates of this type. In fact, such heating is currently a limiting factor in the National Institute of Standards and Technology (NIST) iontrap quantum logic experiments [3,4]. Electric fields from the environment readily couple to the motion of the ions, heating the ion crystal [3-7]. If the ion

trap is much larger than the ion crystal size, we expect these

electric fields to be nearly uniform across the crystal. Uniform fields will heat only modes that involve center-of-mass motion (COM motion), in which the crystal moves as a rigid

body. Motional modes orthogonal to the COM motion, for

instance, the collective breathing mode, require field gradients to excite their motion. The heating of these modes is therefore suppressed [4]. However, even if quantum logic

operations use such a "cold" mode, the heating of the COM

motion can still indirectly limit the fidelity of logic operations. Since the laser coupling of an internal qubit and a motional mode depends on the total wave-packet spread of

the ion containing the qubit, the thermal COM motion can

reduce the logic fidelity [3,4]. In this paper, we examine sympathetic cooling [8] in a

particular scheme for which we can continuously laser cool

the COM motion while leaving undisturbed the coherences

of both the internal qubits and the mode used for quantum

logic. In this method, one applies continuous laser cooling to only the center ion of a Coulomb-coupled string of an odd

number of ions. One can address the center ion alone if the center ion is of a different ion species than that composing

the rest of the string [9]. Alternatively, one can simply focus

the cooling beams so that they affect only the center ion. In

either case, the cooling affects only the internal states of the

center ion, leaving all other internal coherences intact. If the

logic operations use a mode in which the center ion remains

at rest, the motional coherences in that mode are also unaf- fected by the cooling. On the other hand, the sympathetic

cooling keeps the COM motion cold, reducing the thermal wave packet spread of the ions. In the following, we will discuss the dynamics of an ion string in which all ions are

identical except the center ion, assuming heating by a uni- form electric field. Our results give guidelines for implementing the sympathetic cooling scheme. Similar results would apply to two- and three-dimensional ion crystals [10-

13].

II. AXIAL MODES OF MOTION

We consider a crystal of N ions, all of charge q, in a linear radiofrequency (RF) trap [10,11]. The linear RF trap is essentially an RF quadrupole mass filter with a static confining

potential along the filter axis z- If the radial confinement is sufficiently strong compared to the axial confinement, the ions will line up along the z axis in a string configuration

[10,1 1]. There is no RF electric field along z, so we can write

the axial confining potential as 0(z) = qa o z

2 /2 for a a constant. The potential energy of the string is then given by

1 4 2 q

2 V(z 1 ,...,z n )=^a ^z,+^^ \Z~Zj

(1)

for Zj the position of the ith ion in the string (counting from

the end of the string). The first term in the potential energy

expresses the influence of the static confining potential along

the z axis, while the second arises from the mutual Coulomb

repulsion of the ions. For a single ion of mass m, the trap frequency along z is just wz = \Jc\a /m.

We can compute the equilibrium positions of the ions in the string by minimizing the potential energy of Eq. (1). Defining a length scale / by /

3 = (\/(4ire a ) and normalizing the ion positions by m, = z,// gives a set of equations

for the u, as

61032310-1

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D. K1ELPINSKI et al. PHYSICAL REVIEW A 61 032310

i-l «,-S ;+.S j=\ {u-UjY y=/+i (uj-uj) = 0, i=l,...,N,

(2)

which has analytic solutions only up to N=3. Steane [1] and

James [14] have computed the equilibrium positions of ions

in strings with N up to 10. The potential energy is independent of the mass, so the equilibrium positions of ions in a

string are independent of the elemental composition of the

string if all the ions have the same charge.

In a real ion trap the ions will have some nonzero temperature and will move about their equilibrium positions. If the ions are sufficiently cold, we can write their positions as

a function of time as z i (t) = fui + q i (t), where q t (t) is small enough to allow linearizing all forces. We focus on the case of an odd number of ions N, where all ions have mass m,

except for the one at the center of the string which has mass M. The ions are numbered l,...,N, with the center ion la- beled by n c = (N+ l)/2. Following James [14], the Lagrangian for the resulting small oscillations is

J 2 *

2 +y<

i = i

d

2 1 V

N -

2 2iJ= i dZidZj {?,}=o

lilj ( 3)

N \A 1 N

1 = 1 i¥=n c

(4)

where

A«=<

1 N 1+22

-2-

k*i

1

(5)

i*jWe define a normalized time as T=(oz t. In treating the case of two ion species, we write Li = M/m for the mass ratio of the two species and normalize the amplitude of the ion

vibrations q t {t) as Q, = <7,Vqao, i*nc , Qn =qnc yjqa fx. The Lagrangian becomes

^m —z X A'ijQtQj. (6) * U=l

where

A';l

' A u

i,j¥=nc

Ajjlyj/x i or j = n c ,i=£j (7)

generalizing the result of James [14]. The Lagrangian is now cast in the canonical form for small oscillations in the coordinates Q,(t). To find the nor- mal modes, we solve the eigenvalue equation

A'-v (k) =CJv

(k) k=l,...,N (8)

for the frequencies £k and (orthonormal) eigenvectors u

(/c) of

the N normal modes. Because of our normalization of the Lagrangian (6), the t,k are normalized to co

z and the v

(k) are expressed in terms of the normalized coordinates <2,(0- In terms of the physical time t, the frequency of the &th mode is £k

(o

z

. If the kth mode is excited with an amplitude C, we

have

9,(0=Re[Ci;|(*) e'(&"«f+ **)] i*n

1 q n (t) = Re

(9)

(10)

in terms of the physical coordinates g,(0-

We can solve for the normal modes analytically for N

= 3 . Exact expressions for the normal-mode frequencies are

£i =

£3 =

13 1 -^+—(21-^441 -34^+169/^)

1/2

^2=V3,

1 y^+ Y^;(21+ V441-34M + 169/* 2

)

1/2

normalized to co

z

. The mode eigenvectors are

;n>: NAl,—(13-5£i), 1 ,

7(3) :

v w= N2 (l,0, -1),

N3 1,-^(13-5^|), 1

(11)

(12)

(13)

(14)

(15)

(16)

in terms of <2,(0- Here A^ , N2, and A7 3 are normalization

factors. In the case of three identical ions (/x=l), we can

express the mode eigenvectors in terms of the Qt (t) as u

(,) = (1,1,1)/V3, u

(2) = (l,0,-l)/>/2, and v m=(l,-2,l)/yJ6.

The mode eigenvectors, in this special case, also give the ion

oscillation amplitudes in terms of the physical coordinates q t (t). For three identical ions, then, pure axial COM motion

constitutes a normal mode. (This result holds for an arbitrary

number of identical ions.) We also note that the center ion does not move in mode #2; hence the frequency and eigen- vector of mode #2 are independent of fx. For any odd number N of ions there are (N— l)/2 modes for which the center

ion does not move. These modes will likewise have frequencies and eigenvectors independent of ll. Moreover, they have

v

(k)_ m = -v'n

k)+m and so they are orthogonal to the COM

motion and do not couple to uniform electric fields. The

center ion moves in the other (A/ +l)/2 modes, and unless ll—\, each of these (N+l)/2 modes has a component of

axial COM motion and therefore couples to uniform electric

fields.

032310-2

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SYMPATHETIC COOLING OF TRAPPED IONS FOR PHYSICAL REVIEW A 61 032310

10

#2

\#3 (a)

\\\

\

\

\ (b)

#4

#2

"""~^4»

\#1

\

0.01 0.1 1 10 100 001 0.1 10 100

#6 \^ (c)

#4 -^

#2 \#3

""^

#1\

#8.... \^9 (d)

.. .«£. #4

#2 ~~^4|3

"•..#1

FIG. 1. Normalized axial mode frequencies as a function of

ll for (a) 3, (b) 5, (c) 7, and (d) 9

ions.

0.1 1 100 0.1

For N= 5 and higher, the normal mode frequencies de- pend on ll in a complicated way. However, it is easy to find the frequencies numerically. Figure 1 shows the mode fre- quencies for /V = 3, 5, 7, and 9 as a function of fx for 0.01 </u,<100. The modes are numbered in order of increasing

frequency (at ll = 1), and are normalized to <o

z

. In each case,

the lowest-lying mode has all ions moving in the same di- rection and consists of pure COM motion for fx=\. The

even-numbered modes correspond to the (/V— l)/2 modes

for which the center ion does not move. Their frequencies

are therefore independent of ll. For both very large and very

small fx the modes pair up, as shown in Fig. 1 . For each pair

there is some value /x> 1 for which the modes become degenerate. The relative spacing between modes in a pair is also smaller in the large-yu, limit than in the small-//, limit.

If the static confining potential of the trap is not perfectly

harmonic, the normal modes of motion will exchange energy

with each other. This problem is addressed in Ref. [3] (Sec.

4.1.8). These effects are small if all modes are reasonably

cold. Moreover, mode cross-coupling is a resonant process which requires the mode frequencies to be related as sums or

differences. Avoiding such resonances by tuning the trap potentials reduces the rate of energy transfer by a large factor. The Rabi frequency of the laser-ion interaction sets the speed of quantum logic gates and the linewidths of transi- tions between vibrational states of the ion crystal. If we per- form quantum logic on a normal mode whose frequency is too close to that of another "spectator" mode, we must re- duce our Rabi frequency, and thus our gate speed, to avoid

driving transitions on the spectator mode; otherwise we suffer a loss of fidelity. Only the lowest-frequency mode is well separated from all other modes for ll very large. We will

^

show below (Sec. IV) that this mode is undesirable for use in quantum logic. Hence, in order to maximize gate speed, it is best to use a cooling ion that is of the same mass as, or

lighter than, the logic ions. In this case mode 2 is well separated from all other modes, as shown in Fig. 1.

III. TRANSVERSE MODES OF MOTION

We now consider the motion of the ions transverse to the

z axis. The ions experience an RF potential ^cos(fir)(j^ —

y

2)/2 for a suitable choice of axes x and y perpendicular to

z, where fi is the frequency of the RF field and x is a

constant. The static confining potential can be written (qa /2)(z

2 — ax2 — (1 — a)y 2

) at the position of the ions (with a a constant), so there is also a transverse static electric field. To analyze the ion motion, we work in the pseudopotential approximation [15], in which one time averages the motion over a period of the RF drive to find the ponderomotive force on the ion. If the static potential is negligible, the RF drive gives rise to an effective transverse confining potential of j m u)

2Q (x 2 + y

2

) , where a) r0 = q^/( y2fi/n) for an

ion of mass m. If we include the effects of the static field, the transverse potential becomes \m{u>2x

2 + u>

2y

2 ), where <ax

1 — aa) z /o)r0 , o)y = u) r0 \jl — (1 — a)ioj(or0 . Below we will assume a = 1/2, so that <x> y =i»x

. In any case, the transverse potential is that of a simple harmonic oscillator, as we saw also for the axial potential. However, the transverse

potential depends directly on the ion's mass, so the center

ion of a string feels a different trap potential than the others

for ll =£ 1

.

We define e= cj r0 /co. , so that a>x = <w. \le

2 — 1/2. Then the

032310-3 TN-132
D. KIELPINSKI el al. PHYSICAL REVIEW A 61 032310

normalized Lagrangian for the motion along x is

'-ss-SHi.'i**' <i7)

where Xi =xl Tq^o for i¥=n c and Xn =x, JqoQ/x are normalized ion vibration amplitudes along x. Here

Bai,j¥=n c B\y Bij/yffi i or j = nc ,i¥=j (18)

and

BU =

2 f. \u-uk \ 3 l=J>J* nc

Jt#i

(19)

fc^i

i±j.

We can describe the normal mode frequencies and oscilla- tion amplitudes in terms of the eigenvectors and eigenvalues of B'jj Just as for the axial case above. The normalizations of

the time and position coordinates remain the same as in the

axial case. In the previous section, we assumed that the radial con- finement of the ions was strong enough that the configuration of ions in a string along the z axis was always stable. However, for sufficiently small e, the string configuration be- comes unstable. The stable configurations for different val- ues of e can be calculated [16,17], and several of these configurations have been observed for small numbers of ions

[10,11]. Rather than review the theory of these configurations, we will simply find the range of validity of our smalloscillation Lagrangian for the string configuration. The string

will remain stable for all e greater than some es = es (/x); es also varies with N. On the boundary between stable and unstable regions, the frequency of some mode goes to zero. Recalling that the determinant of a matrix is equal to the product of its eigenvalues, we see that es (/x) is the maximum value of e satisfying detB'(e,/x) = for jx fixed. Figure 2 shows es (/x) as a function of /x for 3, 5, 7, and 9 ions. In each case, there is a cusp in €s (/jl) corresponding to the crossing of the two largest solutions to detS'(e,/i) = 0. The

position of the cusp varies with the number of ions, but lies between ,u = 0.1 and /x=l for /V=S9. The positions of the cusps are labeled with arrows in Fig. 2. For /x greater than

the value at the cusp, e<es (/x) corresponds to instability of

the zigzag mode, so that the string breaks into a configuration in which each ion is displaced in the opposite direction

to its neighbors [16,17].- For /x smaller than the value at the

100 :

OJ

FIG. 2. Trap anisotropy at instability of the string configuration

as a function of ^ for 3, 5, 7, and 9 ions. Arrows indicate the cusps

discussed in the text.

cusp, es is independent of /x. In this regime, e< es creates an

instability in a mode similar to the zigzag mode, except that the center ion remains fixed. We can proceed to calculate the frequencies of the trans- verse modes for values e>es (/x). Again, these frequencies

are normalized to the axial frequency of a single ion of mass

m. Figure 3 shows the transverse mode frequencies for 3, 5,

7, and 9 ions as a function of jx, where e is taken equal to l.les (jx). The modes are numbered in order of increasing

frequency at /x=l (all ions identical). In this numbering

scheme, the central ion moves in odd-numbered modes but

not in even-numbered modes. The frequencies of the even- numbered modes appear to depend on /x because they are calculated at a multiple of es (/x); for constant e these fre- quencies are independent of /x. The cusps in the mode fre- quencies in Fig. 3 arise from the cusps of es (/x) at the cross- over points between the two relevant solutions of detS' = 0. Mode frequencies plotted for a constant value of e do

not exhibit these cusps. As in the case of axial motion, the mode frequencies form pairs of one even- and one oddnumbered mode for small /x. However, for large /x all but one of the transverse modes become degenerate. The only

nondegenerate transverse mode in this case is the zigzag

mode. In general, the modes are most easily resolved from

their neighbors for /x=l, as in the case of axial motion.

Increasing e reduces the frequency spacing between nearly

degenerate modes. At e=l.les (/x) and fx=l, for instance, the fractional spacing between the cold transverse mode of 3

ions and its nearest neighbor is 0.20, but for e= 1.5es (/x) the same spacing is 0.09. The near degeneracy of the modes for large or small fx and for el

s significantly greater than 1 limits the usefulness of these modes because of possible mode cross-coupling, just as for the axial modes. Resolving a particular transverse mode requires operating the trap near the point at which the

string configuration becomes unstable, i.e., e near e (/x). In

this regime, the collective motion of the ions is quite sensi032310-4

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SYMPATHETIC COOLING OF TRAPPED IONS FOR PHYSICAL REVIEW A 61 032310

(b)

Z'*5 #2

„„„*}i*L.

#1 —^

#2 \ #1\

\

0.01 100

10 - (d) \#8.#5 \ jr

#8.#7 | ^^T

/m-wi

#1

#4,#3 ,#2

1

• #2 \

\#1

FIG. 3. Normalized frequencies of the transverse modes as a

function of fi with e=l.le (fi) for (a) 3, (b) 5, (c) 7, and (d) 9

0.01 0.1 1 10 100 0.01 0.1 1 10 100

tive to uncontrolled perturbations, which may pose signifi- cant technical problems for using a transverse mode in quan- tum logic operations.

IV. MODE HEATING

Stochastic electric fields present on the ion trap electrodes, for instance, from fluctuating surface potentials, can

heat the various normal modes of motion incoherently. For

ion trap characteristic dimension dlrap much larger than the

size of the ion crystal dions , these fields are approximately

uniform across the ion crystal, so they couple only to the COM motion. The (A7 — l)/2 even-numbered modes are or- thogonal to the COM motion, so they are only heated by

fluctuating electric-field gradients. The heating rates of these modes are reduced by a factor of at least (dions /dtrap ) 2<^l

as compared to the heating of the other modes [4]. In the following, therefore, we will neglect the effects of fluctuat- ing field gradients, so that the even-numbered modes do not

heat at all. The analysis of Sees. II and III shows that the motion of a

crystal of N ions is separable into the 3

A

7 normal modes,

each of which is equivalent to a simple harmonic oscillator. Hence we can quantize the crystal motion by quantizing the normal modes. The kth normal mode gives rise to a ladder of energy levels spaced by ft^o), , with 3N such ladders in all. If we now write the uniform electric-field power spectral

density as 5£(o>), we can generalize the result of Ref. [18] to give

for the heating rate of the kth mode, expressed in terms of

the average number of quanta gained per second. Recall that v\k)

is the oscillation amplitude of the ith ion in the kth normal mode, expressed in the normalized coordinates. It is useful to normalize the heating rate in Eq. (20) to the heating

rate of the lowest-lying axial mode of a string of identical

ions. This normal mode consists entirely of COM motion

and we write v^OM'= l/\[N for all ions. The normalized heat- ing rate of the kth mode is then

»*

n COM

,(*) (21)

j*"c

_, _ q

2 S£(&o>z )

* 4mh£k

(o

z

,(*) f+2 .<*> (20)

;=i

where we have assumed that the spectral density SE((o) is constant over the frequency range of the normal modes, i.e., SE(a>z) = SE(£k

<i>z). Figure 4 shows plots of the normalized heating rates of

the axial modes for N = 3, 5, 7, and 9 as a function of fi. Figure 5 is the same, but for the transverse modes, with e = l.les

. The numbering of modes on the plots of heating

rate matches the numbering on the corresponding plots of

mode frequency (Figs 1 and 2). In both axial-mode and transverse-mode plots, the even- numbered modes have the center ion at rest, while the center

ion moves for all odd-numbered modes. We see from Figs. 4

and 5 that the modes for which the center ion is fixed can

never heat, while all the other modes always heat to some

extent for /n¥= 1 . We will refer to these modes as "cold" and

"hot" modes, respectively. If the ions are identical, only the modes with all ions moving with the same amplitude (COM

032310-5 TN-134
D. KIELPINSKI et al. PHYSICAL REVIEW A 61 032310

(b)

0.1 •

X^*5

..'#3

v.

#3

#5

0.01

mm

0.01 0.1 1 10 100 0.01 0.1 10 100

(0 01 - i—

#1 - ^ vw

s> r

/

/ \ / JB

CO o.oi - <D :

\ ' ""

\ ' /

0.001 ;

•Al if

0.01 0.1

1 #1

#3. / "~ld)

0.1 - ^\#9 /

/ #5

0.01 #3 if #7 *S_ """N, \

r

#9

0.001 • #7 •nM

FIG. 4. Normalized heating

rates for the axial modes as a function of fj. for (a) 3, (b) 5, (c) 7, and (d) 9 ions.

10 100 0.01 0.1 10 100 n n

modes) can heat. There are three such modes, one along x, one along y, and one along z. In interpreting Figs. 4 and 5, it

is important to recall that the normalized heating rate defined

in Eq. (21) is inversely proportional to the mode frequency. For instance, the /j. dependence of the heating rate of the highest-frequency transverse mode can be largely ascribed to variations in the mode frequency, rather than to changes in the coupling of the mode to the electric field.

V. PROSPECTS FOR SYMPATHETIC COOLING

Heating reduces logic gate fidelity in two ways. The logic

mode itself can be heated, but by choosing a cold mode, we

can render this effect negligible. On the other hand, the Rabi

frequency of the transition between logic-mode motional

states depends on the total wave-packet spread of the ion involved in the transition [3,4]. Heating on modes other than

the logic mode can thus lead to unknown, uncontrolled changes in this Rabi frequency, resulting in overdriving or underdriving of the transition. The purpose of sympathetic

cooling is to remove this effect by cooling the center ion and

thus all hot modes.

In the foregoing, we have chosen to consider only the case of a crystal of an odd number of ions, with the cooling

ion at the center. We now see that this is the only case

suitable for sympathetic cooling, since only in this case do we find both (a) cold modes for arbitrary jx and (b) isolation of motion of the cold modes from motion of the cooling ion. As long as the crystal is symmetric under inversion in z, the normal modes must be either symmetric (hot) or antisymmetric (cold) under inversion in z. If the cooling ion is not at the center of the crystal, the crystal symmetry is broken and

cold modes will only appear for particular values of fx. The

case of a crystal of an even number of ions, with two cooling

ions at the center, will again yield cold modes. However,

excitation of any mode will cause the cooling ions to move,

so that the cold modes are not well isolated from the sympathetic cooling in this case. For sympathetic cooling to be useful, we must find a cold mode suitable for use in quantum logic. The cold mode must

be spectrally well separated from any other modes in order to maximize gate speed. We can use the lowest-lying cold axial mode as the logic mode for ^£3. In this mode, called the breathing mode, the center ion remains fixed and the spacings between ions expand and contract in unison. Unless the

trap is operated very close to the instability point of the

string configuration, the breathing mode is better separated

from its neighbors than are any of the cold transverse modes. For /iiS:3 any cold mode, either axial or transverse, is nearly

degenerate with a hot mode. In this regime one must make a

specific calculation of mode frequencies in order to find the best-resolved cold mode. Even so, the cold axial modes are again better separated from their neighbors than are the cold

transverse modes, except for e very close to €s (/jl). It seems

best to select a cold axial mode as the logic mode in most

cases. By selecting our laser-beam geometry appropriately, we

can ensure that the Rabi frequency of the motional transition on the axial mode used for logic depends chiefly on the spread of the ion wave packet along z. In this case, heating of

the axial modes will affect logic-gate fidelity, but heating of

the transverse modes will have little effect. If the mass of the

central ion is nearly the same as that of the others (/i«l),

only the lowest axial mode will heat significantly, and we

032310-6

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SYMPATHETIC COOLING OF TRAPPED IONS FOR . .

.

PHYSICAL REVIEW A 61 032310

1

- ~\#3 (a) #i

<u

ro °- 1 '#1 V- 1 O)

c

(0 0.01 ;

#3 (b)

0.1

|! l »••

0.01 -

0.001

#1

0.01 0.1 1 10 0.1 100

c

to

FIG. 5. Normalized heating

rates for the transverse modes as a function of /a with e=l.\e (/j.) for (a) 3, (b) 5, (c) 7, and (d) 9

ions.

can continuously cool this mode by cooling only the central

ion, ensuring that all ions remain in the Lamb-Dicke limit

[3]. If /a is not near 1, we must cool all (N+ l)/2 hot modes

(again by addressing the central ion) to keep all ions in the Lamb-Dicke limit. The analysis above indicates that, all other things being

equal, we are best off if our substituted ion is identical to, or

is an isotope of, the logic ions. However, sympathetic cool- ing can still be useful if the two ion species have different masses. For example, we can consider sympathetic cooling

using the species 9 Be+ and 24Mg+. Linear traps constructed

at NIST have demonstrated axial secular frequencies of over

10 MHz for single trapped 9 Be +

ions. For three ions with

24Mg+

as the central ion, o>

z (Be +

) = 27rX 10 MHz yields a spacing of 1.6 MHz between the cold axial breathing mode

and its nearest neighbor. If we reverse the roles of the ions

[cuz (Mg+

) = 27rX 10 MHz], the spacing increases to 6.2 MHz. The transverse modes are much harder to resolve from

each other. For three ions with 24Mg+in the center, we

require wr0 (Be +

) = 27rX27.6 MHz to obtain e = 1 . 1 es

, and

the spacing between the cold transverse zigzag mode and its nearest neighbor is only 560 kHz. Reversing the roles of the

ions, we find e= l.le^ at wr0 (Mg+

) = 27rX 14.7 MHz with

a spacing of 1.1 MHz. For this combination of ion species,

the cold axial breathing mode seems most appropriate for

10 100

logic. For a string of 3 or 5 ions, sympathetic cooling would

require driving transitions on 2 or 3 axial-mode sidebands,

respectively. From this example we see that sympathetic

cooling can be useful even for ion mass ratios of nearly 3

to 1.

VI. CONCLUSION

We have investigated a particular sympathetic cooling

scheme for the case of an ion string confined in a linear RF

trap. We have numerically calculated the mode frequencies

of the axial and transverse modes as functions of the mass

ratio /j. and trap anisotropy e for 3, 5, 7, and 9 ions. We have

also calculated the heating rates of these modes relative to the heating rate of a single ion, assuming that the heating is driven by a uniform stochastic electric field. The results in- dicate that the scheme is feasible for many choices of ion species if we use a cold axial mode as the logic mode. The

optimal implementation of the scheme employs two ion species of nearly equal mass. However, a demonstration of sympathetic cooling using 9 Be + and 24Mg+ appears well within

the reach of current experimental technique.

ACKNOWLEDGMENTS

This research was supported by NSA, ONR, and ARO.

[1] A. Steane, Appl. Phys. B: Lasers Opt. 64, 623 (1997).

[2] J.I. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995).

[3] D.J. Wineland et ai, J. Res. Natl. Inst. Stand. Technol. 103, 259 (1998).

[4] B.E. King el ai, Phys. Rev. Lett. 81, 1525 (1998).

[5] S.K. Lamoreaux, Phys. Rev. A 56, 4970 (1997).

[6] D.F.V. James, Phys. Rev. Lett. 81, 317 (1998).

[7] C. Henkel and M. Wilkins, e-print quant-ph/9902009.

[8] D.J. Larson el ai, Phys. Rev. Lett. 57, 70 (1986).

[9] We assume that the cooling transitions of the center ion are

032310-7 TN-136
D. KIELPINSKI et al. PHYSICAL REVIEW A 61 032310

well resolved from all relevant transitions of the logic ions. [14] D.F.V. James, Appl. Phys. B: Lasers Opt. 66, 181 (1998).

[10] M.G. Raizen et al, Phys. Rev. A 45, 6493 (1992). [15] P. K. Ghosh, Ion Traps (Oxford, Clarendon, 1995).

[11] H. Walther, Adv. At., Mol., Opt. Phys. 31, 137 (1993). [16] R. Rafac et al, Proc. Natl. Acad. Sci. USA 88, 483 (1991).

[12] M. Drewsen et al, Phys. Rev. Lett. 81, 2878 (1998). [17] J.P. Schiffer, Phys. Rev. Lett. 70, 818 (1993).

[13] T.B. Mitchell et al, Science 282, 1290 (1998). [18] D.J. Wineland et al, Fortschr. Phys. 46, 363 (1998).

032310-8

TN-137
Scalable Entanglement

of Trapped Ions

C. Monroe, C.A. Sackett, D. Kielpinski, B.E. King,

C. Langer, V. Meyer, C.J. Myatt, M. Rowe,

Q.A. Turchette, W.M. Itano, and D.J. Wineland

Time and Frequency Division, National Institute of Standards and Technology1 Boulder Colorado 80303

Abstract. Entangled states are a crucial component in quantum computers, and are

of great interest in their own right, highlighting the inherent nonlocality of quantum

mechanics. As part of the drive toward larger entangled states for quantum computing, we have engineered the most complex entangled state so fax in a collection of four trapped atomic ions. Notably, we employ a technique which is readily scalable to much

larger numbers of atoms. Limits to the current experiment and plans to circumvent

these limitations are presented.

INTRODUCTION

At the heart of quantum mechanics lies the principle of superposition, where

physical properties of a system can exist in two or more states simultaneously.

When a system is composed of more than one degree of freedom, superpositions can be prepared where distinct degrees of freedom are perfectly correlated, yet

the state of each degree of freedom is by itself in superposition. The prototypical

example is Bohm: s version [1] of the Einstein-Podolsky-Rosen paradox [2], where a spin-zero particle decays into a pair of spin-1/2 daughters, resulting in the singlet state

|<P

, |t)lU)2-U)l|t>2 m

This state is entangled, since it cannot be expressed as a direct product of states representing each particle. When one of the subsystems in such a state is measured,

11 This work was supported in part by the U.S. National Security Agency and the Advanced

Research and Development Activity under contract MOD7037.00, the U.S. Army Research Office, and the U.S. Office of Naval Research.

CP551. Atomic Physics 17. edited by E. Anmondo. P. DeNatale. and M. Inguscio 2001 American Institute of Phvsics 1-56396-982-3

173

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the other subsystem is also determined, even when the particles are not in physical contact or outside each other's lightcones. In general, entangled states such as I^epr) highlight the nonlocal character of quantum mechanics. Quantitatively,

this is usually expressed in terms of Bell's inequality violations [3], where measured

correlations between the entangled subsystems can be shown to be incompatible

with what would be expected under conditions of local realism.

Although the correlation in the above state cannot be used for superluminal

communication, it can be harnessed for enhanced communication rates over what

can be obtained classically [4,5]. Furthermore, such states are useful in a variety of quantum communication schemes such as quantum cryptography [6] and quantum

"teleportation" [7-10].

Entangled states of larger systems are a defining feature of a quantum computer.

Here, for example, a collection of N spin-1/2 particles are prepared in an arbitrary

entangled state of the form

|*qc) = ao|000. .

. 0) + ajlOOO ... 1) + ... + 02w_i|lll • •• 1), (2)

where |0) and |1) refer to the two spin states of each particle, and the ak are the amplitudes of the number k being stored by the register of particles. By choosing

appropriate entangled states and making appropriate state measurements of the

particles, quantum computers can solve certain problems much faster than any

classical computer [11,12]. The reason quantum computers are mere speculation at

this point is that \$qc) is very difficult to produce in the laboratory.

SCALABLE ENTANGLEMENT WITH TRAPPED IONS

Nearly every demonstration of entanglement to date has relied upon a random or selection process which prohibits scaling to large numbers of particles. This can be

quantified in terms of the entanglement efficiency parameter e, or the probability per unit time that a perfect entangled pair is created [13]. The probability of

realizing a perfect N-particle entangled state typically scales as e cA\ where c is of order unity and depends on the particular experiment.

The first measured Bell's inequality violations were seen in atomic cascade ex- periments involving the entanglement of a pair of spontaneously-emitted photons

[14,15]. Spontaneous parametric downconversion is now a popular source of entangled photons, where typically ultraviolet photons traverse a nonlinear crystal and

downconvert into a pair of polarization-entangled infrared beams [16,17]. Unfortunately, the probability of each input photon being converted leads to an efficiency

e ~ 10

-4

, so the probability of entangling larger numbers of photons becomes very

small. (Nevertheless, by waiting long enough, three-photon entangled states were

recently observed from simultaneous downconversion into two pairs [18]). Experiments in cavity-QED have recently shown entanglement of two atoms [19] and two atoms with a photon [20], where a thermal (random) source of atoms traverse a common microwave cavity. In these experiments, e — 0.005. Experiments with

174

TN-139
optical parametric oscillators can also entangle the continuous quadratures of two

optical field modes [21]. Although this source has near-unit entanglement efficiency,

scaling to larger numbers of degrees of freedom appears difficult.

The Cirac-Zoller Scheme

In 1995, Cirac and Zoller showed that a collection of trapped ions may be suitable

for storing large-scale entangled states such as \Vqc) [22]. In their proposal, each

atomic ion stores a quantum bit (qubit) of information in a pair of electronic energy

levels, and a collective mode of harmonic vibration is used to entangle any pair of ion qubits. By applying laser beams to an individual ion in the collection, its internal qubit state can be mapped onto the collective ion motion, and subsequent

quantum logic gates can be applied between the motion and a second ion, effectively

entangling the two ions. The entanglement can be extended to any number of ions by repeating these steps on other pairs of ions. When accompanied by single-ion

rotations, the Cirac-Zoller scheme allows the creation of an arbitrary entangled

state [Eq. (2)], and therefore forms a set of universal quantum logic gates.

The basic elements of the Cirac-Zoller scheme were demonstrated on a single

trapped ion in 1995 [23]. A variation of this scheme was later used to entangle a

pair of trapped ions [24] with entanglement efficiency e ~ 0.8, representing the first scalable entanglement source with near-unit efficiency.

The M0lmer-S0rensen Scheme

Instead of entangling the ions sequentially, Molmer and S0rensen showed how to create the A-ion entangled state

|^ v =

lt)ilt) 2 -.|t)^ + e">w U)iU)2 -U)^

(3)

with a single pulse of laser radiation [25]. The M0lmer-S0rensen operation applied

to any pair of qubits in a collection of ions (accompanied by single ion rotations)

allows the creation of any entangled state [Eq. (2)], and thus forms a set of universal

quantum logic gates alternative to the Cirac-Zoller scheme [26]. We have employed

the Molmer-Sorensen sci :ne to create the entangled state of Eq. (3) for TV = 2 and

A

T = 4 trapped ions [27]. ;n both cases, the entanglement efficiency was e ~ 0.8, as discussed below.

The Molmer-Sorensen entanglement technique can be understood by considering

a pair of identical spin-1/2 charged particles confined together in a harmonic potential [28]. The energy levels of this system are illustrated in Fig. 1, where hu) is the internal energy splitting of each qubit, and v is the oscillation frequency of a

particular collective mode of the particles in the trap.

175

TN-140
T

(£>n

•5 »• / \

/ \ 5

n+l =TF --T- n —

y*-

Wn

\/

ITT)

lit) + g'O-li?)

V2

|U>

FIGURE 1. Entanglement scheme for two particles. Each ion is initially prepared in the | 4-1) internal state, and the collective motion of the pair initially contains exactly n quanta. Laser

fields equally illuminating the two ions and oscillating near wo + v + <5 and u — v — 6 couple

the | 44-) and | ft) states as shown. For sufficient detuning 5, the populations of the middle

states are kept small. By driving the double transition for the appropriate time, the entangled

state (| ft) + e

"* 2

l 44-))/v/2 is generated. For four ions, the same procedure generates the state (ltttt) + e*^|44ii))/v/2.

The ions are initially prepared in the | 4-1) internal state, and we assume the

ions are in a collective motional eigenstate \n). By simultaneously applying optical

fields near the first upper and lower motional sidebands (oscillating at u + v + S and l) — v — 5 respectively) with equal illumination on the two ions, the two-step

transition from | 44) |n) to | tt)|n) is driven through the intermediate states

Wint)-.

4.)lt)|n±l)+ei »-lt)ll)|n±l)

x/2

(4)

where cf>_ is the phase difference of the field at the two ion positions. For sufficiently

large detuning 5 from the sidebands, these intermediate states are negligibly oc- cupied, so that the motional state is not altered. We also assume 5 <?C v so that

intermediate states involving other motional modes are not involved in the coupling. As shown in Fig. 1, there are two paths from | 44-) to | tt)i and their respective couplings are given by the product of the two resonant sideband Rabi

frequencies divided by the detuning ±6 from the relevant virtual intermediate level. For the upper- then lower-sideband path (arrows on left side of Fig. 1), this coupling is (rig-Jn + 1 ) 2

/<5, and for the lower- then upper-sideband path (arrows on

right side of Fig. 1), it is -{rjgy^n) 2 /5, where g is the single ion resonant carrier Rabi frequency and 77 is the Lamb-Dicke parameter of the motional mode involved.

(These expressions are valid only in the Lamb-Dicke limit T] 2 (n+ 1) <C 1.) Adding

the couplings from these two paths results in a net Rabi frequency from | 4-4-) to

I tt) of SI = rfg2 /5, independent of the motional state \n) within the Lamb-Dicke

regime. The net interaction Hamiltonian is proportional to CTp ) cr[2)

, where a^ is

176

TN-141
the transverse Pauli spin-1/2 operator of ion i. Entanglement is achieved by simply

applying these beams for a time r = 7r/2Q, creating the desired spin state m> - iw^iw (5)

where 0_ is the sum of the field phases at the two ion positions.

1 Fast Entanglement

In order for the intermediate states |^int)± to be negligibly occupied, the de- tuning S must be large compared to both single-spin sideband Rabi frequencies

T}g\/n + 1 and T]gy/n, meaning the entangling operation must be much slower than

the resonant sideband operations. (This is the characteristic slowdown of driving

higher-order transitions through virtual levels.) However, it is possible to violate

this condition and still generate the state |^2 ) [29,30]. In this case, the intermediate

states |^,„t)± are occupied during the operation (and the motional state becomes

entangled with the spins), but this occupation can vanish at exactly the moment

the desired entangled spin state |\P 2 ) is created. Without regard to the the spin

states, we find that for arbitrarily small S (and within the Lamb-Dicke limit), the

motion evolves during the operation as a coherent superposition of its original state pm {0) and an oscillating displaced state [29]

p*s (t) = v Pm{0)V (6)

where T>(a) is the displacement operator with phase space argument a [31]. The

overall motion is thus in a "Schrodinger Cat" -type superposition state [32], with

maximum separation in phase space 2JQ/5. The phase space trajectory of the

displaced component p^

s {t) follows a circle from its original state with radius yjQ/S,

returning to the initial motional state pm (0) at times tm = 27rm/<5, where the

positive integer m is the number of complete circular cycles of the displacement [29]. Setting the entanglement pulse time r defined above equal to tm, we find that the

condition for a return to the initial motional state following the entanglement step is Q./5 = l/(4m). The entangling time can thus be rewritten as r = TTy/rn/(r}g), which

is only a factor of 2y/m slower than an analogous resonant sideband transition. To maximize the speed of the Molmer-Sorensen operation in the experiment, we

operate with m = 1.

2 Scalable Entanglement

Surprisingly, the Molmer-Sorensen entangling scheme is scalable in the sense that

precisely the same operation can be used to generate the TV-particle entangled state

177

TN-142
of Eq. (3) for any even number of ions N. (For N odd, \$n) can be generated

using one entanglement pulse accompanied by a separate independent rotation of each particle's spin.) The M0lmer-S0rensen interaction is proportional to J^, where

Jx is the transverse spin operator for the effective spin- TV/ 2 particle. Physically,

this interaction simultaneously flips all pairs of ions in the collection. Through the

properties of angular momentum rotations [25], this results in the desired entangled

state |\Ev). In scaling to larger numbers of ions, the only difference (for a given

motional mode frequency) is that the operation is \/N times slower, since the LambDicke parameter is proportional to 1/y/N. In addition, the phase which appears in Eq. (3) is the sum of the field phases at each ion position.

If the ions are uniformly illuminated, the Molmer-Sorensen scheme requires that

they all participate equally in the intermediate motional excitation, which implies

that the only suitable mode for arbitrary N is the center-of-mass mode. However,

this mode has a practical disadvantage that fluctuating ambient electric fields cause

it to heat at a significant rate [33]. For large 5, the entanglement operation is independent of the motion, so that heating is unimportant, so long as the ions

remain in the Lamb-Dicke regime [29]. In the small-5 case however, motional

decoherence of the Schrodinger-Cat state discussed above must be avoided. Modes

involving only relative ion motion couple to higher moments of the field, so heating

of them is negligible [34]. For N = 2 and TV = 4 ions, such modes do exist in which each particle participates with equal amplitude [35]. In both cases, they

are uniform "stretch" modes, in which alternating ions oscillate out of phase; we

use these modes here. Excitation of the center-of-mass mode does still affect the

experiment, as the ion eventually can get heated out of the Lamb-Dicke regime. For this reason, we initially sideband cool both the center-of-mass and uniform

stretch modes to near their ground state. We note that other modes of motion can

also be used for entanglement, as long as the laser intensity on each ion is adjusted

to compensate for the difference in mode amplitude of that ion, resulting in equal

sideband couplings for all ions.EXPERIMENT

The experiment was performed using 9 Be+ ions confined in a miniature linear RF

trap [33], with the N ions lying in a line along the trap's weak axis. Two spectrally

resolved ground-state hyperfine levels compose the effective spin-1/2 system, with

|4-) = \F = 2,mF = -2), | t) = \F = l,mF = -1). The hyperfine splitting

between these states is cj /27t ~ 1.25 GHz.

Coherent coupling between | 1) and | t) is provided via stimulated Raman transitions. The two Raman laser beams have a wavelength of A ~ 313 nm with a

difference frequency near u> . Their wave-vectors are perpendicular with their dif- ference wave-vector lying along the line of ions with magnitude 5 k = 2it\/2/\.

They are detuned ~80 GHz blue of the 2Pi/2 excited state, with intensities giving

g/2ir ~ 500 kHz.

178

TN-143
The Raman beam frequencies can also be tuned to coherently flip the spins while

simultaneously affecting the collective motional state of the ions. For modes considered here (having equal amplitudes of motion for all the ions), the spin-motional

coupling is determined by the Lamb-Dicke parameter tj = 5k(h/2Nmiu) 1 ^2 of the

mode with frequency v, where m\ is the mass of a single particle in the collection.

Fig. 2 displays a stimulated-Raman absorption spectrum of four trapped ions in a

linear array, with the four axial modes as well as higher-order features clearly visible. For both the two- and four-ion experiments, the desired stretch-mode frequency was

set to v/2-n ~ 8.8 MHz, giving a Lamb-Dicke parameter of t)Str = 0.23/Nl/2. For

the M0lmer-S0rensen operation, the two driving frequencies required to generate a coupling near the first blue- and first red-sidebands are generated by frequency

modulating one of the Raman beams using an electro-optic modulator. The spectral

positions of the relevant difference frequencies of the Raman beam pairs is indicated

by the two arrows and dashed vertical lines in Fig. 2.

£ 60 -

cuoc

a>o

CO

CDo

3

40 -

20

Raman Detuning 5R (MHz)

FIGURE 2. Raman absorption spectrum of four ions confined in a linear crystal and Doppler

laser-cooled. The ordinate is the detuning 5r of the Raman beams' difference frequency from

the carrier and the abscissa shows the average counts of ion fluorescence per experiment (200^is integration time), proportional to the number of ions in the state | 4-) (the ions are initially prepared in state | 4444))- The carrier appears at 5r = 0, and the first sidebands of the four

axial normal modes of motion (labelled by letters a-d) appear at 5p. = ±3.62, ±6.23, ±8.67 MHz,

and ±11.02 MHz in agreement with the theoretical frequency ratios 1 : \/3 : 2.410 : 3.051.

Several higher order sidebands also appear at sums and differences of harmonics of the normal

mode frequencies, as indicated. The sideband asymmetry (upper sidebands are always stronger)

indicates cooling to the quantum regime with not-too-many thermal phonons. The two arrows and

the dashed lines, just outside the first upper and lower stretch sidebands, indicate the frequencies used for the four ion Malmer-Sorensen scheme.

After an interaction with the stimulated Raman beams, the ion internal states

179

TN-144
TABLE 1. Characterization of two-ion and

four-ion states. Pj denotes the probability that j ions were measured to be in | 4-), and |p-f...T,4.--4-l de- notes the coherence between |t . . . f) and \i i). Uncertainties in the N = 2 measurements are ±0.01; uncertainties in the N = 4 populations are ±0.02. A

r Po Pi Pi Pz Pa |/>T—T,4— -4-1 24

0.43 0.11 0.46 0.35 0.10 0.10 0.10 0.35

0.385

0.215

are measured by illuminating them with a circularly-polarized laser beam tuned to the 2S\/2{F = 2,mp = —2) -H 2P3 / 2

(-r7 = 3,ttif — —3) cycling transition. Each

ion in | 4-) fluoresces brightly, leading to the detection of ~15 photons/ion on a photomultiplier tube during a 200 /lis detection period. In contrast, an ion in | t)

remains nearly dark. For a single ion, we are able to discriminate between | f) and

| I) with approximately 99% accuracy, as shown in the histograms of Figs. 3a and

3b. This accuracy is limited by off-resonant optical pumping which causes the dark

state | 4-) to eventually partake in the cycling transition and fluoresce [24]. This 1%

error rate could be improved considerably by appropriately weighting the photon

counts by their arrival time, as this optical pumping will contaminate later counts more so than earlier counts. Fig. 3c shows a histogram of four ions prepared in an initial state with incoherent populations in all five possible states of excitation

without distinguishing the individual ions. Here, the number of ions in state | 4-) can be determined with an accuracy of about 80% on any a given experiment,

although this number could be improved to better than 95% by weighting the counts as discussed above. These statistical detection errors can be averaged away

by repeating the experiment many times and fitting the resulting photon-number

distribution to a sum of Poissonians to determine the probability distribution Pj

of having exactly j ions in the state | |) [24].

N-particle entanglement results

Following the M0lmer-S0rensen entangling procedure, the probability distribution Pj is measured. The results are given in Table 1, and show that in both cases, the probabilities for all N ions to be in the same state are large compared to the

probabilities for the other cases. This is characteristic of the state \^n) [Eq. (3)], although the fact that the middle probabilities are nonzero indicates that we do

not generate the entangled states with perfect accuracy. In order to prove that we are generating a reasonable approximation to \$ n),

it is necessary to prove that the populations of | ftTT) and | -U44-) are coherent.

In terms of the N-spin density matrix pN , we must measure the far off-diagonal

180

TN-145
.Q

CO n

o

(a)

10 20 30 10 20

# photons collected in 200jis

0.06 -i

25 50 75 100 125

# photons collected in 400jis

150 175

FIGURE 3. Measured probability distribution of detected fluorescence counts of a single

trapped ion in (a) state [ f) and (b) state | !) after 200 us of integration (1000 measurements),

(c) Measured probability distribution of detected fluorescence counts of four trapped ions after 400 fis of integration (1000 measurements). The lines are least-squares fits to reference distributions for having anywhere from ions (leftmost curve) to 4 ions (rightmost curve) in state | i), providing relative probabilities Pj of j ions in state | 4-)

element /3-f--T,J-4- This can be achieved by viewing the first entanglement pulse as the first pulse in a Ramsey experiment [36], and applying a second (non-entangling)

7r/2 pulse to the ions before observing them, closing the Ramsey interferometer.

The relevant observable after this modified Ramsey experiment is the parity of the

number of ions in state | 4-) [37]

j=0

(7)

As the parity is measured while <p is varied, the resulting Ramsey fringes oscillate as

181

TN-146
cos N(j> for TV ions, as seen in Fig. 4. The amplitude of the fringes is just twice the

(a)

tc/2 k 3rc/2 phase shift (f>

(b)

tc/2 7i 3tc/2 phase shift

FIGURE 4. Determination of pt-.-T.i--4 f°r ( a) two ions an<^ 0*) f°ur ions - -^ter tne entangle- ment operation, a non-entanling n/2 pulse with relative phase <f> drives the | 4) +-»

| T) transition

in each ion. As <p is varied, the parity of the N ions oscillates as cos Ncp, and the amplitude of the oscillation is twice the magnitude of the density-matrix element Pt—T.i-4- Each data point

represents an average of 1000 experiments, corresponding to a total integration time of roughly

10 s for each graph.

desired coherence 2|pT ...T4...il- This compression of the Ramsey fringes by a factor of N is the basis for extracting Heisenberg-limited signal-to-noise in spectroscopy of entangled states, where the frequency uncertainty Aw is limited by the TV-particle

Heisenberg uncertainty relation AtoAt > 1/7V for observation time At [37]. Fig. 5 shows an analog of this effect in a Mach-Zender interferomer.

The measurements of |pt...T4--.il are listed in the last column of Table 1 for both

2- and 4-ion cases. The fidelity of our state generation, or the overlap between the

idealized state |¥/v) in Eq. (3) and the observed density matrix, is

TN = (|¥*)|p*|¥tf) = ^T^ + IPT-t4...il (8)

For N = 2 we achieve T2 = 0.83 ±0.01, while for TV = 4, TA = 0.57 ±0.02. In both

cases the fidelity is above 0.5, indicating A

r-particle entanglement [27].

182

TN-147
c«ei ooooI

\

\

^

FT

even

or odd

# counts?

FIGURE 5. Mach-Zender interferometer analog for four-particle entanglement observation. Four photons propagate through a "super-beamsplitter," which sends the photons in a super- position of all going through and all reflecting. One arm contains a phase shifter, and the two

paths are recombined on a normal beamsplitter. The parity of the number of photons received in one of the output ports is measured as the interferometer phase is scanned. Because the photons

all take the same path, the measured phase shift is amplified by a factor of four (fringe period = t/2), providing enhanced interferometric sensitivity.

Quantifying the amount of entanglement is a more difficult question. A variety of measures of entanglement have been proposed, but most are difficult to calculate even numerically [38,39]. For N = 2, Wootters has given an explicit formula for the entanglement of formation, £(p2) [40]. Although we have not reconstructed

the entire two-particle density matrix, the populations measured place sufficient bounds on the unmeasured elements to determine that £{p2) ~ 0.5. This indicates

that roughly two pairs of our ions would be required to carry the same quantum

information as a single perfectly entangled pair. In the four-ion case, no explicit formula for entanglement is known. The data

does indicate that our density matrix can be expressed as

pA ^ 0.43|* 4 )(*4l + 0.57p!

ico/i (9) where |^4 ) is the desired state of Eq. (3) and p\

ncoh

is completely incoherent (ie., diagonal). The coefficients of Eq. (9) are determined directly from the value of

PtttT.l ui m Table I, together with the fact that no evidence for other off-diagonal

matrix elements was observed. (Other coherences involving less than four ions would have given fringes varying as cosd, cos2(j). or cos3<fi in the measured populations Pj{d>) and parity II(<p)). A measurement of |pf...t,i-4l > does not by itself guarantee TV-particle entan- glement. For instance, consider the four-particle states

l*A>

/|U4) + rTTT)\ oVl^ + IT)

x/2 ; v/2

(10)

and

183

TN-148
,*B) = (!«>) 8 (U>^J)).

An equally weighted statistical mixture of $a aQd $b exhibits only three-particle

entanglement, yet has |/7-f...t,4...4.| = 0-25 (larger than our observed value) without

any other coherences. A similar mixed state with pairs of two-particle entangled

states also has |p-f...t,4...4.| = 0.25 without other coherences. However, these states significantly differ from |ty 4 )(\I/ 4 | along the diagonals, so the observed populations

Pj following the entanglement procedure (Table 1) can set an upper bound on how

much these states can contribute to the measured density matrix. We decompose

p4 as a sum of the desired state |^4 )(^'4| and a "worst-case" factorizable density

matrix pf which includes mixed states such as above. We find that an upper bound

on the amount of four-particle coherence in pf is

Ptm,uu(77iax) = Min P ,Pi,Y + Min(Pu P3 ) (12)

From the data in Table (1), we find that p^A

,

|

{max) =0.15, leaving the remainder of the observed four-particle coherence (0.065) to be unambiguously associated

with the four-particle entangled state |\I/ 4 ). This gives the worst-case decomposition

p4 = 0.13|VO (^1 + 0.87pf

(13)

where pf contains mixtures of particular two- and three-particle entangled states (such as Eqs. (10) and (11)) which are very unlikely to occur in the experiment.

OUTLOOK

The data on two-ion and four-ion entanglement are consistent with an entanglement efficiency of t zz 0.8. Although this represents the only demonstrated source

of 4-particle entanglement and uses a scalable method, the imperfect contrast of

(Fig. 4) indicates that even this efficiency will limit how many particles can be

entangled in this experiment. It may be required to achieve entanglement efficien- cies e > 0.9999 in order to implement fault-tolerant error correction schemes which

may allow entanglement of arbitrarily large numbers of particles [41]. Several technical noise sources degrade the observed efficiency, including laser intensity and beam-pointing noise, nonuniform illumination of the ions during the Molmer-Sorensen operation, and magnetic field noise. The chief limitation in the current experiment appears to be stochastic heating of the ions to outside the

Lamb-Dicke regime. The center-of-mass (CM) motion of the ions is observed to heat at a rate of (nCM ) « 0.02N ps

_1

[33], so after a 10 ps four-ion entangling

operation, {tl C m) approaches ^ 10 thermal quanta. This invalidates the LambDicke criterion rj^. M (nCM ) <§; 1, and severely limits the fidelity of the operation.

Molmer and S0rensen have shown [29] that the expected fidelity of the entangled

state 1$ N ) of N ions is

184

TN-149
F*l-N{N-l)rfCM {nCM)\ (14)

to lowest order in the center-of-mass Lamb-Dicke parameter t\Cm with N > 1 and (ticm) > 1- The factor' N(N - l)/2 comes from the number of pairs of N ions which are simultaneously flipped during the M0lmer-S0rensen entangling

operation. We find for the TV = 4 experiment, the above expression is consistent

with the observations.

The source of ion heating has not been pinpointed, but it appears to be related to fluctuating microscopic potentials on the electrodes. The observed heating is not a fundamental limitation, as it has been observed to be orders of magnitude

smaller under some conditions [33]. Moreover, by trapping multiple ion species and

continuously laser-cooling one, the other qubit ions can be sympathetically cooled

to remain in the Lamb-Dicke regime while not disturbing the qubit coherence [42]. Producing entangled states of very large numbers of ions (tens or hundreds)

for relevance to large-scale quantum computing will require a different approach.

This is because a trap confining more than several ions will have low oscillation frequencies, and mode cross-coupling from the complicated mode structure will be

unavoidable. A promising path to large numbers is to use a multiplexed ion trap structure of many separated ion traps [43]. Here, entangling operations are done

only in traps holding a few (2 — 5) ions, and the ions are be shuttled between

traps to extend the entanglement to larger numbers. Because the quantum bits are stored in magnetic dipole (hyperfine) internal states and the ions are moved around

with electric fields acting on their charge, the coherence of the qubits should not be

disturbed. Peeling away an ion from or introducing an ion to other ions in a trap will obviously introduce a significant amount of motional energy, but this energy can be

removed again by trapping multiple species and relying on sympatheic cooling to return the motion to well inside the Lamb-Dicke regime for subsequent entangling

operations.

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29. A. Sarensen and K. Maimer, quant-ph/0002024 (2000).

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186

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PHYSICAL REVIEW A, VOLUME 61, 063418

Heating of trapped ions from the quantum ground state

Q. A. Turchette,* 1 D. Kielpinski, B. E. King,* D. Leibfried, 5 D. M. Meekhof, C. J. Myatt,f M. A. Rowe, C. A. Sackett, C. S. Wood," W. M. Itano, C. Monroe, and D. J. Wineland

Time and Frequency Division, National Institute of Standards and Technology, Boulder, Colorado 80303

(Received 14 October 1999; published 17 May 2000)

We have investigated motional heating of laser-cooled 9 Be +ions held in radio-frequency (Paul) traps. We

have measured heating rates in a variety of traps with different geometries, electrode materials, and characteristic sizes. The results show that heating is due to electric-field noise from the trap electrodes that exerts a stochastic fluctuating force on the ion. The scaling of the heating rate with trap size is much stronger than that expected from a spatially uniform noise source on the electrodes (such as Johnson noise from external circuits), indicating that a microscopic uncorrelated noise source on the electrodes (such as fluctuating patch-potential

fields) is a more likely candidate for the source of heating.

PACS number(s): 32.80.Pj, 39.10,+j, 42.50. Vk

I. INTRODUCTION

Cold trapped ions have been proposed as a physical

implementation for quantum computation (QC) [1], and ex- periments on one [2-5] and two [6,7] ions have demonstrated proof of the principle. Work is currently underway to extend these results. In ion trap QC, ion-laser interactions prepare, manipulate, and entangle atomic states in ways de- pendent on the quantum motional state of the ions. A limit- ing factor in the fidelity of an operation is uncontrolled heat- ing of the motion during manipulations. Heating leads to decoherence of the quantum superposition states involved in the computation [8,9], and can ultimately limit the number of

elementary gate operations that can be strung together.

Speculations have been made about the mechanisms that lead to heating [8,10-14], but measurements are scarce since

the necessary sensitivity can be achieved only through laser cooling to near the ground state of motion. Additionally,

systematic studies of the dependence of heating rate on vari- ous trap properties are difficult, since often this requires the construction and operation of an entirely new trap apparatus,

which may have different values of properties not under

study. Indeed, the data presented here pose several interpretational difficulties for this reason. Heating of a single trapped ion (or the center-of-mass motion of a collection of trapped ions) occurs when noisy

electric fields at the position of the ion couple to its charge,

giving rise to fluctuating forces. If the spectrum of fluctua- tions overlaps the trap secular motion frequency or its micromotion sidebands, the fluctuating forces can impart signifi- cant energy to the secular motion of the ion. Here, we

express the heating rate as the average number of quanta of energy gained by the secular motion in a given time. There

are several candidates worth considering for sources of the noisy fields that give rise to heating. Some of these are

*Electronic address: quentint@reoinc.com

f Present address: Research Electro-optics, Boulder, CO.

^Present address: NIST, Gaithersburg, MD.

^Present address: University of Innsbruck, Austria. "Present address: ILX Lightwave, Boulder, CO.

1 050-2947/2000/6 1 (6)/0634 1 8(8)/$ 1 5.00

[8-14] Johnson noise from the resistance in the trap elec- trodes or external circuitry (the manifestation of thermal

electronic noise or black body radiation consistent with the boundary conditions imposed by the trap electrode structure), fluctuating patch potentials (due, for example, to ran- domly oriented domains at the surface of the electrodes or adsorbed materials on the electrodes), ambient electric fields from injected electronic noise, fields generated by fluctuating currents such as electron currents from field-emitter points on the trap electrodes, and collisions with background atoms. Only the first two mechanisms will be considered here since

the remaining mechanisms (and others) are unlikely contributors [8] or can be eliminated by comparing the measured

heating rates of the center of mass and differential modes of two ions [6]. As will be shown below, the Johnson noise and

patch-potential mechanisms give rise to heating rates, which

scale differently with the distance between the ion and the trap electrodes.

II. TWO MODELS FOR SOURCES OF HEATING

A. Preliminaries The heating rate caused by a fluctuating uniform field can

be derived as in Savard et al. [15] and agrees with a classical calculation [8]. The Hamiltonian for a particle of charge q and mass m trapped in an harmonic well subject to a fluctuating, uniform (nongradient) electric field drive e{t) is

H(t) = H -qe(t)x, (1)

where H = p

2/2m + mcoll x2 /2 is the usual, stationary har- monic oscillator Hamiltonian with trap frequency co m . From

first-order perturbation theory, the rate of transition from the ground state of the well (|n = 0)) to the first excited state

(|/i = l» is [15]

fo^=^r dre'^ T{e{t)e{t+r))\{0\qx\\)\2

. (2)

Evaluating the motional matrix element gives

61 063418-1 ©2000 The American Physical Society

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Q. A. TURCHETTE el al. PHYSICAL REVIEW A 61 063418

To-. 0^1 4mht

SE

(tom), (3)

where SE((o) = 2jZ aD dre io' T(e(t)€(t + t)) is the spectral

density of electric-field fluctuations in units of (V/cm)2Hz_1

. For an ion trapped by a combination of (assumed noiseless) static fields and inhomogeneous rf fields (Paul trap) the heating rate can be generalized to [11]

n =

4mh< (4)

where n is the rate of change of the average thermal occupation number, to m is now the secular frequency of the mode

of motion under consideration, and Cl T

is the trap rf drive frequency. The second term on the right-hand side (RHS) of

Eq. (4) is due to a cross coupling between the rf and noise

fields; it will not be present for the axial motion of a linear

trap, which is confined only by static fields. Even for motion

confined by rf pondermotive forces, this second term will be

negligible in the absence of spurious resonances in SE

(to) or

increasing SE(u)) (since o)

2

n /D,

2-~lO~ 4

) and is neglected in what follows [8]. We differentiate two sources of the noise that gives rise to heating. The first is thermal electronic noise in the imperfectly conducting trap electrodes and elsewhere in the trap

circuitry. Though this source of noise is ultimately microscopic in origin, for our purposes here it can be treated ad- equately by use of lumped circuit models. Thermal noise has been considered in the context of ion-trap heating in several places [8,11-14]. The second source of noise considered

here is due to "microscopic" regions of material (small

compared to the size of the trap electrodes) with fluctuating,

discontinuous potentials established, for example, at the in- terface of different materials or crystalline domains. We call

this patch-potential noise, and its microscopic origin leads to manifestly different heating behavior from that for the ther- mal electronic noise case. Static patch potentials are a well- known phenomenon, but little is known about the highfrequency (MHz) fluctuating patches, which are required to account for our observed heating rates [16-18].

B. Thermal electronic noise Heating rates in the case of thermal electronic noise (Johnson noise) can be obtained simply through the use of

lumped-circuit models, which are justified by the fact that the wavelength of the relevant fields (at typical trap secular

or drive frequencies) is significantly larger than the size of

the trap electrodes. Such an analysis has been carried out elsewhere [8,11,19], and only the major results will be

quoted here. Resistances in the trap electrodes and connecting circuits give rise to an electric-field noise spectral density

SE (io) = 4k B TR((x))/d 2

, where d is the characteristic distance from the trap electrodes to the ion, T is the temperature (near room temperature for all of our experiments), k B is Boltzmann's constant, and R(co) is the effective (lumped-circuit)

resistance between trap electrodes. The heating rate is given

by

q

2 k B TR(com ) mha)„,d~

(5)

A numerical estimate of the heating rate for typical trap pa- rameters gives 0.1/s <n R <l/s [8,11], which is significantly

slower than our observed rates. As a final note, the lumpedcircuit approach is convenient, but not necessary. In the Appendix, we present a microscopic model that is valid for

arbitrary ion-electrode distances and reproduces Eq. (5) for

all the traps considered here (and for all realistic traps where

d>S, where S is the skin depth of the electrode material at the trap secular frequency).

C. Fluctuating patch-potential noise To derive the heating rate for the case of microscopic

patch potentials we use the following approximate model.

We assume that the trap electrodes form a spherical conducting shell of radius a around the ion. Each of the patches is a

disc on the inner surface of the sphere with radius r

p <a and

electric potential noise Vp (to). Alternatively, each patch is assumed to have power noise spectral density S v

(u)). The

electric field noise at the ion due to a single patch is Ep (co)= — 3Vp (co)r

2 /4a 3 in the direction of the patch. There

are N**=4Ca< lr

2 such patches distributed over the sphere

with coverage C=£l. Averaging over a random distribution of patches on the sphere, we find that the power spectral

density of the electric field at the ion (along a single direction) is

SEJat)\

This gives a heating rate

3C5y(a))r;

4a4 (6)

_, 3<7 Cr

p S v

(ojm ) \6mfn

(7)

in which the association d — a is made. Note the difference in scaling with electrode size between Eqs. (5) and (7). The

thermal electronic noise model gives a scaling n^d~2

, while the patch-potential model gives « P«cf" 4

. In fact, a d~ A dependence also arises from a random distribution of fluctu- ating charges or dipoles.

III. MEASUREMENTS

A. Measuring the heating rate To determine the heating rate, we first cool the ion to near

the ground state. In sufficiently strong traps, this is achieved

simply by laser cooling with light, red detuned from a fast cycling transition (y«wra , where y is the radiative line- width of the upper state) propagating in a direction such that

063418-2

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HEATING OF TRAPPED IONS FROM THE QUANTUM . PHYSICAL REVIEW A 61 063418

its k vector has a component along the direction of the mode

of interest. In weaker traps, additional sideband Raman cool- ing is utilized to cool to the ground state [3]. Typical starting values of n, the average number of thermal phonons in the mode of interest, are between and 2. After cooling and optically pumping the ion to its internal ground state (denoted ||)), we drive Raman transitions be- tween atomic and motional levels [2-4]. Tuning the Raman

difference frequency Aw to the kth motional blue sideband

(bsb) at Aa) = w +bm drives the transition |l)|n)

<->|T)|n + &) where |T).||) refer to the internal (spin) states of the atom that are separated by w . The kth red sideband

(rsb) at Aw= w — kcom drives ||)|'j) <-> |T)l /l — ^)-

'^rie mea" surement utilizes asymmetry in the strengths of the red and

blue motional sidebands to extract n. The strengths of the sidebands are defined as the probability of making a transi- tion ||)<->|T). which depends on the occupation number of

the motional levels. The strengths are probed by a Raman

pulse of duration t tuned to either kth sideband. The probability P| of remaining in ||) after probing is measured and

7 rsb bsb. the strengths Ik =l-PLl$b and l

k -l-.Pj.bsb are extracted. For thermal motional states, the strengths of the red and blue sidebands are related by [3,8]

rrsb 'k = 2 Pn

m = k sin2 ft.

- \ k 00 n -\ 2 ^m sin

2 Q. m + kmt l+n m=o

,bsb l+n,

(8)

(9)

(10)

where ft m +k,m = ^m,m + k

' s tne Rabi frequency of the kth sideband between levels m and m + k, and Pm = n

m

l{\ + n) m + l is the probability of the mth level being occupied

for a thermal distribution of mean number n. The ratio of the sidebands Rk = I

Tk b/Ib

k sb

is independent of drive time t and

immediately gives the mean occupation number n,

l I I

1

p

t "*v**^ rsb \F

\

" bsb ^^ c:

- ;

Au>

T\

1

! 33S /"A * « - SI pz

- P4 rsb

' bsb

1 i I —i 1 0.00 0.05 0.10 0.15 020

l[ms]

FIG. 1 . An example of heating rate data. The main graph shows

n { = (n)) vs i*, the delay between cooling and probing. The insets show Raman spectra from which n is extracted, according to Eq.

(11). For the insets, P^ is the probability that the ion remains in the

I J.) state after application of a Raman probe of fixed duration with

difference frequency Aw; rsb, red motional sideband; bsb, blue motional sideband. The sidebands shown are the first sidebands. The

data are for trap 5 from Table I at 5 MHz secular frequency and n= 12±2/ms.

(Rk)

uk l-(Rk) uk (11)

which is valid even if the Lamb-Dicke criterion is not satisfied. In principle, k should be chosen to be the positive inte- ger nearest to n in order to maximize sensitivity. In practice we use k = 1 , 2, or 3 in most cases. Note that Eq. (1 1) is valid only for thermal states; this is adequate since Doppler cooling leaves the motion in a thermal state [4,20], as does any

cooling to near the ground state.

In order to determine the heating rate n, delays with no

laser interaction are added between the cooling cycle and the probing cycle. An example of a data set at a fixed trap secular frequency is shown in Fig. 1. The error bars are deter- mined as follows: The raw data of Raman scans over the

TABLE I. Summary of traps. The size column is approximately the distance between the ion and the nearest electrode surface. Cl T

is the trap rf drive frequency. The heating rate (n) is for a trap secular frequency of 10 MHz, which in the case of traps 3a and 3b had to be extrapolated from data at lower trap secular frequencies. The two numbers quoted for 3a and 3b are for two different versions of the trap. See the

text for a further discussion of the parameters.

Trap Type Materia] Size (/J.m) n r

/27r [MHz] n (ms ') Refs.

1 circular ring Mo 170 250 1 [3,4] 2 elliptical ring Be 175 250 10 [6.7] 3a circular ring Mo 175 150 10, 10 2

3b circular ring Mo 395 150 0.5,10 ^ 4 linear Au J&r 160 150 2.3 [9] 5 linear Au J£6 ISO 230 3.5 [22] 6 linear Au f 3*5 220 230 1.1 [22]

CoirVTfc-fvo *

063418-3

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Q. A. TURCHETTE el al.

trap 3bv—'

trap 3a 77

ring I electrode fork electrode

FIG. 2. Schematic diagram of the electrodes of trap 3 (from

Table I). The distance between traps 3a and 3b is 1.7 mm. Not to

scale.

sidebands (such as is those shown in the insets of Fig. 1) are

fit to Gaussians, from which the depths of the sidebands are

extracted, with error on the parameter estimate calculated assuming normal distribution of the data. The errors from the

rsb and bsb strengths are propagated through Eq. (11) for an

error on n . The error bars shown are 1 a, and include only

statistical factors. These errors are incorporated in the linear

regression to extract n with appropriate error. Many such

data sets are taken for various types of traps and at different secular frequencies.

B. The traps

The measurements of heating rate in this paper extend

over a five year period and utilize six different traps. The

traps are summarized in Table I. The traps are described in the references listed; here only a brief discussion is included. The "ring" traps are approximate quadrupole configurations

consisting of a flat electrode (125 /xm thick) with a hole

drilled through it (the ring) and an independent "fork" elec- trode (100 fxm thick) that forms end caps on either side of

the electrode, centered with the hole, similar to the trap shown in Fig. 2. In trap 1, the ring and endcap electrodes were at the same average potential; in traps 2 and 3 a static bias field could be added between the fork and ring to change

the distribution of binding strengths along the three principle axes of the trap. The size of these traps is stated as the hole

radius, with the end cap spacing approximately 70% of the hole diameter. For the elliptical ring trap (trap 2) the stated

size is the radius along the minor axis and the aspect ratio is 3:2; the fork tines are parallel to the major axis of the ellipse.

Traps 3a and 3b were drilled into a single flat electrode with

a single graded fork electrode (see Fig. 2). The rings are

circular and the size stated is the radius. This was the trap used for the size-scaling measurements. The heating in all of

the ring traps was measured in a direction in the plane of the

ring electrode, parallel to the tines of the fork electrode. Traps 4, 5, and 6 are similar linear traps with geometry in- dicated in Fig. 3. Trap 6 was made slightly larger than traps 4 and 5 by increasing the space between the two electrode

wafers. Heating was measured along the axial direction,

which has only a static confining potential. The size quoted

in Table I for the linear traps is the distance between the ion and the nearest electrode. All traps are mounted at the end of

a coaxial X/4 resonator for if voltage buildup [21]. Typical

resonator quality factors are around 500 and rf voltage at the

PHYSICAL REVIEW A 61 06341 —J L— 0.2 mm

FIG. 3. Schematic diagram of the electrodes of the linear traps

(traps 4, 5, and 6 from Table I). The traps are formed by evaporating gold (approximately 0.75 /xm thick) on an alumina substrate. The outer segmented electrodes are the endcaps, while the long unbroken electrodes carry rf. The axial direction (labeled z) is parallel to the rf electrode. The two separate trap wafers are spaced by 200 yum for traps 4 and 5 and 280 fim for trap 6 (spacers not shown). Schematic diagram not to scale.

open end is approximately 500 V with a few watts of input power. In all traps except for traps 3a and 3b the resonator is inside the vacuum chamber with the trap. In traps 3a and 3b,

the resonator is outside the chamber, with the high-voltage rf applied to the trap through a standard vacuum feedthrough.

Since we believe that surface effects are an important factor in heating, we cleaned the electrode surfaces before using

a trap. When trap electrodes were recycled, they were first cleaned with HC1 in order to remove the Be coating deposited by the atomic source. For the molybdenum traps an elec- tropolish in phosphoric acid was then used. For the beryllium

electrodes electropolishing in a variety of acids was ineffective, so abrasive polishing was used. Finally, the traps were

rinsed in distilled water followed by methanol. The gold

electrodes of the linear traps were cleaned with solvents after being evaporatively deposited on their alumina substrates. The time of exposure of clean trap electrodes to the atmosphere before the vacuum chamber was evacuated was typically less than one day. The traps were then vacuum baked at —350° C for approximately three days.

C. Data Our longest-term heating measurements were made on

trap 1. In Fig. 4 we plot the heating rate as a function of date of data acquisition for a fixed trap frequency (11 MHz). The

heating rate is on the order of 1 quantum per millisecond with a basic trend upwards of ~ 1 quantum per millisecond per year. Over this time the electrodes were coated with Be

from the source ovens, but beyond this, nothing was changed

in the vacuum envelope, which was closed for this entire period of time. The cause of the increase in heating rate is unknown, but may be related to increased Be deposition on

the electrodes. Be plating on the trap electrodes could be a source of patch-potential noise. Figure 5 shows heating rates in the linear traps (traps 4, 5, and 6) and the elliptical ring trap (trap 2) as a function of trap secular frequency. The frequency dependence of the heating

063418-4

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HEATING OF TRAPPED IONS FROM THE QUANTUM . .

.

PHYSICAL REVIEW A 61 063418

100

tO <£> <D

g> o> o">

date

FIG. 4. Heating rate in ring trap 1 vs time. The secular fre- quency for all measurements was ~ 1 1 MHz. The solid line shows

a trend, which does not account for the weights of the data points.

rate is expected to scale as SE

((om)/a>m [Eq. (4)]. For ex- ample, a trap electrode with a fiat noise spectrum [Se

((d) = const] will have a heating rate that scales as a)~ . The

actual spectrum of fluctuations is impossible to know a priori, but in principle the data can be used to extract a spec- trum over a limited frequency range given the model leading

to Eq. (4). For the three linear traps, the heating rate data are most consistent with a o>~

2 scaling, implying SE

<xco~

l

. This

(a) 100 =*

10 -

1 -

o.i b_

1

-i 1 1—i—

r

11

M trap 2 O trap 4

&-=

3 4 5 6 7 8 9 10

(b)

trap secular frequency, 0^ /2jt [MHz]

[ ]

1

'

10 &

A

$

m

-

i

A

CD

CD - A 7

trap 5 A trap 6

1

"l 1 5 6 7 8 9

trap secular frequency, <om I2n [MHz]

FIG. 5. Heating rates vs trap secular frequency in (a) the elliptical ring trap 2 and in the microlinear trap 4 and in (b) linear traps 5 and 6. The only intended difference between traps 5 and 6 is the

size. In all four data sets, the secular frequency was varied by

changing a static potential only.

o.i t±

(a) D small trap (3a) • big trap (3b) 1 i

1 !-

-1 1 I I L Iltl

100 J.

i—i—i—i

1

m

1—i

+

10

(b) small trap (3a) • big trap (3b) m

1

0.1 9 I Hd m

-

0.01 i -

ii 0.001 + . i i i i i , t

trap secular frequency, a)m

lln [MHz]

10

FIG. 6. Data from trap 3, showing heating rates vs trap secular frequency, (a) Data set number 1. The two points on the small trap

at wm/27r=5.3 MHz were taken with Raman cooling to n(t=0)~0 and with Doppler cooling only to n(t = 0)~2. Note

that they give comparable results, as they should. The dashed lines show a>~ ' scaling, (b) Data set number 2. The small trap data were

taken with an rf voltage of —400 V and the big trap with —600 V. The secular frequency was changed by tuning the dc bias between

fork and ring electrodes.

does not greatly assist us in identifying a physical mechanism for the heating. For example, pure Johnson noise will have a flat spectrum, low-pass-filtered Johnson noise will have a spectrum that decreases with increasing frequency,

and the spectrum of fluctuations in the patch-potential case is entirely unknown. In addition to the theoretical ambiguity,

there is evidence in other data sets of different frequency

scalings (though they are always power-law scalings). This measurement certainly cannot be used to pinpoint a heating

mechanism; it is presented here only for completeness.

The data of Fig. 5(b) provide a first indication of the scaling of the heating rate with trap size. Trap 6 is about 1.3 times larger than trap 5, while its heating rate (at 10 MHz)

was a factor of 3 slower. This indicates that the dependence

of heating rate on trap size is stronger than d~ 2

, but is consistent with d~ 4

. Of course, this comparison is to be taken with some caution, since these are two separate traps measured several weeks apart, and therefore likely had different microscopic electrode environments. However, a comparison

is warranted since the traps were identical apart from their

sizes. In particular, all the associated electronics were the same and the rf drive voltage was very nearly the same. In

fact, the rf voltage was slightly larger for the measurements

on trap 6, which showed the lower heating rate. This is important to note because we observe a slight dependence of

the heating rate on the applied rf trapping voltage. Though

063418-5

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Q. A. TURCHETTE el al. PHYSICAL REVIEW A 61 063418

FIG. 7. Expected heating rate vs distance from electrode for thermal electronic noise, from

Eqs. (A6), (A7), and (4). The numerical parameters are those for 9 Be + with molybdenum elec- trodes at 10 MHz secular frequency and room

temperature.

io- i<r -icr distance from plane [m]

i<r

we have only a qualitative sense of this dependence at present, it seems that heating rates increase with rf voltage, up to a point, at which the effect levels off. This rf-voltage

dependence is observed along directions where the ion is confined both by static fields and by pondermotive fields. It may not be unreasonable that the increased rf voltage in- creases the intensity of the noise source (possibly due to an

increase of temperature of the electrodes), even when it does

not affect the trap secular frequency, as in the axial direction of the linear traps. Trap 3 was designed to give a controlled measure of the heating rate as a function of trap size, while all other parameters were held fixed. The trap electrodes were made from

the same substrates, the electrodes were subjected to the same preuse cleaning, the traps were in the same vacuum

envelope, driven by the same rf electronics (simultaneously),

and data for both traps were acquired with minimal delay. For direct comparison at the same secular frequency in both

traps, it was necessary to change the applied rf voltage since

a)m

<xl/d2

. (A static bias between ring and endcap can be

added, as discussed above, but this was not sufficient to measure heating at identical secular frequencies for the same rf drive.) There are two data sets to be discussed for this trap, shown in Fig. 6. In the first set, shown in Fig. 6(a), we have data points at two different secular frequencies for the "small trap" (trap

3a) and one point for the "big trap" (trap 3b). The heating

rates of the small trap are comparable to the heating rates for other traps and show a u>~ ' scaling of the heating rate. The

single point on the big trap is at a lower secular frequency,

yet has a much slower heating rate. In fact, if we extrapolate

the data from the small trap to the same secular frequency

(using o>~'), the heating rate is over an order of magnitude

lower in the big trap. The ratio of the heating rate in the small trap to that of the big trap is 20±6. This is a much

stronger scaling than that predicted by a Johnson noise heating mechanism [Eq. (5) predicts a d

2 ~4.8 scaling], but is consistent with the scaling in the patch-potential case [Eq.

(7) predicts a d

4 ~23 scaling]. When these data are used to predict an exponent for the size scaling, the result is d

38±0- 6

. For the second data set, shown in Fig. 6(b), the trap was

removed from the vacuum enclosure, given the usual cleaning (as discussed above), and replaced for the measurements.

In this data set, the trap behaved quite differently from all other traps, with heating rates significantly below those of

other traps. Also, SE must have been a strong function of co for this trap since the scaling with trap frequency was rather pronounced. The scaling with size was also strong: the heat- ing rate was 16000 times smaller in the big trap. When these

data are used to predict an exponent for the size scaling, the

result is d n±2. Needless to say, it is difficult to draw general

conclusions from the data for this particular trap, but the difference in heating rates between the two traps seems to strongly indicate, again, that Johnson noise is not the source of the heating. We cannot be sure why this trap had such

anomalous heating behavior, but we speculate that it is due

to a less-than-usual deposition of Be on the trap electrodes

prior to the measurements, because the trap was loaded with

minimal exposure to the Be source atomic beam. At this point it is useful to compare the present results to heating rates in other experiments. There are two other measurements. The first was done with 198Hg+

[10]. For that experiment wm/27r=3 MHz and d = 450 yu.m and the heating

rate was 0.006/ms. Accounting for scalings with trap fre- quency («"') [23] and mass (m~ ] ), these results are consistent with the present results for a size scaling of d~ A

. Another measurement has been made with Ca+

[5]. For

that experiment a> m

/2-n-=4 MHz and d***700 /xm and the heating rate was 0.005/ms. Compared to the present experi- ments and the Hg experiment, this is also consistent with a d~ A scaling, although it is certainly unlikely that all systems

had the same patch field environment.

063418-6

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HEATING OF TRAPPED IONS FROM THE QUANTUM PHYSICAL REVIEW A 61 063418

IV. CONCLUSIONS AND OUTLOOK

We have measured heating from the ground state of

trapped ions in different traps. The magnitude of heating

rates and the results of the size-scaling measurements are inconsistent with thermal electronic noise as the source of

the heating. The results do not indicate any strong dependence on trap-electrode material or on the type of trap potential (pondermotive or static). The rf voltage applied to the electrodes may play a role in heating, in as much as it may

have an influence on patch potentials.

Since we have not identified the mechanism for the observed heating, it is difficult to say what path should be taken

to correct it. If fluctuating patch potentials on the surface of

the electrodes are the cause, then further cleaning may be

appropriate. Additionally, better masking of the trap elec- trodes from the Be source ovens may help.

The results coupled with those of other experiments

[10,5] strongly indicate that bigger traps have smaller heat- ing rates. This is not a surprise, but the strength of the scal- ing may be. With little sacrifice in the trap secular frequency

(which ultimately determines the fastest rate of coherent manipulation) a dramatic decrease in the heating rate vs logic

gate speed appears possible using larger traps.

ACKNOWLEDGMENTS

We acknowledge support from the U.S. National Security

Agency, Office of Naval Research, and Army Research Office. We thank Chris Langer, Pin Chen, and Mike Lombardi

for critical readings of the manuscript. We thank J. Beall for help in construction of the linear traps (Fig. 3).

APPENDIX: THERMAL ELECTRIC FIELDS

We are interested in the thermal electric field power spectral density SE

(r,a)) generated from a specified volume of

conductor. The conductor can be decomposed into a web of

resistors each carrying current spectral density Sr = 4kB T/Rj (where we assume k B T>ha)). The resistance along the ith direction of an infinitesimal volume element is Rj = dl/(adA), where a is the conductivity, dl is the length

along i, and dA is the cross-sectional area. A Fourier component of current /,-(«) through the volume dV=dldA gives

rise to an electric dipole Pi {(i>) = Ii (a>)dlla>. Thus the equivalent spectral density of electric dipole of the infinitesi- mal resistor is isotropic: SP

{<i)) = 4kB Tcr dV/co2

. The electric field from an electric dipole P(r' ,co) oscillat- ing at frequency id and position r' is

E,(r,u>)= 2 Pj(r',<a)Gij(r,r',<o). (Al)

In this expression, Gtj (r,r' ,co) is a Green function matrix,

representing the ith component of electric field at position r due to the y'th component of a point dipole at r', which

satisfies the appropriate boundary conditions of the geometry. The electric-field spectral density at position r is an

integral over the dipoles in the conductor volume:

SE (r,a>)=—— \ a(r') 2 |G,7

(r,r',a>)| 2 dV .

(A2)

The Green function satisfies G,j(r,r',a>) = G,

; (r',r,a>), so

the above integral can be interpreted as the Ohmic power

absorbed by the conductor from the electric fields generated

by a point dipole at position r. By energy conservation, this must be equivalent to the time-averaged power dissipated by

a point dipole at r, which is related to the imaginary part of

the Green function matrix Gy(r,r,<w) [24]. This simplifies

Eq. (A2), leaving the fluctuation-dissipation theorem

2k T

SE

(r, (o) = —L- 2 Im G,.(r,r,w). (A3)

' 0) j=x,y,z

Agarwal solved Maxwell's equations for G,; (r',r,w) for the simple geometry of an infinite sheet of conductor, filling the space z^O with the conductor-vacuum interface in the

z = plane [25]. Although this idealized geometry is far from any real ion-trap electrode structure, rough scalings of

the thermal fields can be relevant to real ion-trap geometries.

From Ref. [25], the Green function matrix for this problem is diagonal with axial (z) and radial (p) components

, f°°<7

3 (wQ e-w\

Gzz (z,z,a>) = G

free (a)) + i — -2——]e2,w°t dq,

(A4)

Gpp

(z,z,o>) = G^ree (<o)

k

2 --f2Jo

Wn £ — W

WnE + W

+

vv \w + w

,2iw z °

z dq. (A5)

In the above expressions, e((u) = e +/a-/e w is the di- electric function of the conductor (in the low frequency

limit), k = (o/c, and wave vectors vv and w (generally complex) are defined by w^ = k

2 — q2 and w2 = k2 s — q2 with Im wq^O and Imw^O. The free-space Green's function G

free {u)) has imaginary part Im G^ree (o)) — k

3/67re and

gives rise to the isotropic free-space black body electric-field fluctuations when substituted into Eq. (A3).

The above integrals are significantly simplified in the "quasistatic" limit, where kz< 1 and the conductivity is sufficiently high so that k 8<4l, where 8= ^2c 2 eqIu>ct is the skin depth of the conductor. Despite these conditions, no

restriction is placed on the value of zl 8. We break the above

integrals into two pieces. The first piece / has q^k with vv

real. In the quasistatic limit, this piece can be shown to cancel the free-space contribution to the transverse Green function Im Gpp

(z,z,(o), while doubling the free-space contribution to the axial Green function Im Gzz (z,z,oj). Physically,

the presence of the conductor negates the transverse free- space black body field while it doubles the axial black body

field due to a near-perfect reflection. The second piece of the integrals /^ has q^k with w imaginary. These pieces of the

integral can be solved to lowest order in kz and kS. Com063418-7 TN-158
Q. A. TURCHETTE el al. PHYSICAL REVIEW A 61 063418

bining terms and substituting the results into Eq. (A3), the thermal electric-field spectral density is

2k B T<o2 k B T / 1 II z SE (z,co)=- t + rl-+\/-+-

37re c

3 4t7o-z 3 \2 V4 tf J

1/2

k,T 1 I z

4

1/2

(A6)

(A7)

These expressions show that the thermal electric-field noise scales as 1/z

3 for z<8 [14], but scales as 1/z

2 for z >S [8,11]. At large distances z> \[S/k (with kz<l), the

axial field noise settles toward twice the free-space black body value while the radial field vanishes. This result is also reported in Ref. [26]. The behavior is shown in Fig. 7, where

Eqs. (A6) and (A7) have been substituted into Eq. (4), giving

the expected thermal heating rate for a

9 Be ion trapped with

molybdenum electrodes at room temperature. Note that the predicted heating rate at trap sizes typical in our experiments

is significantly slower than the 0.1-1 quanta/s rate predicted

in [8,1 1]. This difference comes from the choice of the value of the resistance in Eq. (5), which was chosen in [8,1 1] as an

absolute upper limit. When interpreting these results, only the rough scaling should be considered. Realistic ion-trap electrode geometries

are more complicated than a single infinite conducting plane,

involving a more closed electrode structure. This generally

requires a full numerical solution to the relevant boundary

value problem. Moreover, we are usually interested in the

electric-field fluctuations at the center of the trap, where

these fluctuations will be substantially different from those above an infinite plate.

[1] J. I. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995).

[2] C. Monroe, D. M. Meekhof, B. E. King, W. M. Itano, and D.

J. Wineland, Phys. Rev. Lett. 75, 4714 (1995).

[3] C. Monroe, D. M. Meekhof, B. E. King, S. R. Jefferts, W. M.

Itano, D. J. Wineland, and P. Gould, Phys. Rev. Lett. 75, 401

(1995).

[4] D. M. Meekhof, C. Monroe, B. E. King, W. M. Itano, and D.

J. Wineland, Phys. Rev. Lett. 76, 1796 (1996).

[5] Ch. Roos, Th. Zeiger, H. Rohde, H. C. Nagerl, J. Eschner, D.

Leibfried, F. Schmidt-Kaler, and R. Blatt, Phys. Rev. Lett. 83, 4713 (1999).

[6] B. E. King, C. S. Wood, C. J. Myatt, Q. A. Turchette, D.

Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, Phys. Rev. Lett. 81, 1525 (1998).

[7] Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D.

Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, Phys. Rev. Lett. 81, 3631 (1998).

[8] D. J. Wineland, C. Monroe, W. M. Itano, D. Leibfried, B. E. King, and D. M. Meekhof, J. Res. Natl. Inst. Stand. Technol.

103, 259 (1998).

[9] C. J. Myatt, B. E. King, Q. A. Turchette, C. A. Sackett, D.

Kielpinski, W. M. Itano, C. Monroe, and D. J. Wineland, Nature (London) 403, 269 (2000).

[10] F. Diedrich, J. C. Bergquist, W. M. Itano, D. J. Wineland,

Phys. Rev. Lett. 62, 403 (1989).

[1 1] D. J. Wineland, C. Monroe, W. M. Itano, B. E. King, D. Leibfried, D. M. Meekhof, C. Myatt, and C. Wood, Fortschr. Phys.

46, 363 (1998).

[12] S. K. Lamoreaux, Phys. Rev. A 56, 4970 (1997).

[13] D. F. V. James, Phys. Rev. Lett. 81, 317 (1998).

[14] C. Henkel and M. Wilkens, Europhys. Lett. 47, 414 (1999).

[15] T. A. Savard, K. M. O'Hara, and J. E. Thomas, Phys. Rev. A

56, R1095 (1997).

[16] J. B. Camp, T. W. Darling, and R. E. Brown, J. Appl. Phys. 69, 7126 (1991).

[17] C. Kleint, Ann. Phys. (Leipzig) 10, 309 (1963). [18] G. W. Timm and A. Van der Ziel, Physica (Utrecht) 32, 1333

(1966).

[19] D. J. Wineland and H. G. Dehmelt, J. Appl. Phys. 46, 919

(1975).

[20] S. Stenholm, Rev. Mod. Phys. 58, 699 (1986). [21] S. R. Jefferts, C. Monroe, E. W. Bell, and D. J. Wineland,

Phys. Rev. A 51, 3112 (1995). [22] C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer,

C. J. Myatt, M. Rowe, Q. A. Turchette, W. M. Itano, D. J. Wineland, and C. Monroe, Nature (London) 404, 256 (2000).

[23] There is no way to know if w~ ' is the proper way to scale for

trap frequency, but it is a reasonable first assumption.

[24] L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1 3rd

ed. (Pergamon, Oxford, 1980), Chap. XII. [25] G. S. Agarwal, Phys. Rev. A 11, 230 (1975).

[26] C. Henkel, S. Potting, and M. Wilkens, Appl. Phys. B: Lasers

Opt. B69, 379 (1999).

063418-8

TN-159
PHYSICAL REVIEW A, VOLUME 62, 053807

Decoherence and decay of motional quantum states of a trapped atom

coupled to engineered reservoirs

Q. A. Turchette,* C. J. Myatt/ B. E. King,* C. A. Sackett, D. Kielpinski, W. M. Itano, C. Monroe,§ and D. J. Wineland

Time and Frequency Division, National Institute of Standards and Technology, Boulder, Colorado 80303

(Received 28 February 2000; published 16 October 2000)

We present results from an experimental study of the decoherence and decay of quantum states of a trapped atomic ion's harmonic motion interacting with several types of engineered reservoirs. We experimentally

simulate three types of reservoirs: a high-temperature amplitude reservoir, a zero-temperature amplitude res- ervoir, and a high-temperature phase reservoir. Interaction with these environments causes the ion's motional

state to decay or heat, and in the case of superposition states, to lose coherence. We report measurements of the decoherence of superpositions of coherent states and two-Fock-state superpositions into these reservoirs, as well as the decay and heating of Fock states. We confirm the theoretically well-known scaling laws that predict

that the decoherence rate of superposition states scales with the square of the "size" of the state. PACS number(s): 42.50. -p, 03.67.Lx, 03.65. -w, 32.80.Pj

I. INTRODUCTION

The decoherence and decay of quantum states coupled to a variety of reservoirs have been investigated extensively in theory [1-8]. The model in these studies is typically a sys- tem harmonic oscillator coupled to a bath of environment

harmonic oscillators. One of the most interesting results has been the realization that macroscopic superposition states de- cay at extremely fast rates. As an illustration, consider a charged harmonic-oscillator system in a superposition of co- herent states ipx\ ai ) + \a2 ) coupled to the noisy electric

field E of the environment. The interaction potential is V = —qx-E, where x is the position and q is the charge of the

particle. Regardless of the temperature of the reservoir,

(E2 )?t 0, so that the system is always subject to some level of noise from the environment. For a superposition of coherent states coupled to a reservoir of fluctuating fields, a simple

scaling law may be stated [3]: the rate of decoherence scales

as the square of the separation of the wave packets, |Aa| 2 — \a y — a2 \ 2

. In an idealized case, the superposition is created, the amplitude reservoir is coupled to the system for a time t, and the coupling is then turned off. The remaining

coherence C between the two wave packets, expressed for instance as the magnitude of the off-diagonal components of

the system density matrix, is

C(f) = C(0)exp ia 12 y

(1)

where y is a coupling constant between the reservoir and the system. For the same interaction, the energy H of the system

decays like

H(t) = H(0)exp[-yt], (2)

*Electronic address: quentint@boulder.nist.gov

Present address: Research Electro-Optics, Boulder, CO.

^Present address: NIST, Gaithersburg, MD.

s Present address: Department of Physics, University of Michigan, Ann Arbor, MI.

at a rate independent of the size of the initial state. For su- perpositions of macroscopic-sized wave packets, the quan- tum coherence may be obliterated in a time over which the energy of the system does not change appreciably.

Significant attention has been paid to the role of decoher- ence in the classical/quantum correspondence and in funda- mental issues of physics. Recently, decoherence has also been studied in a more pragmatic role: as a primary impedi- ment to quantum computing. Quantum computation relies on

entanglement of large quantum states to perform efficient calculations. Such states, separated by large distances in Fhlbert space, will be very susceptible to the detrimental effects of decoherence. Any system will interact with its environment, so the nature of the interaction and the time scales over which it acts are of critical import in implementations of quantum information processing. Since trapped ions are a leading technology for investigations in quantum computing,

a study of decoherence in an ion trap quantum computer

system [9], as presented here, is particularly relevant. Over the past several years, techniques have been described to generate mesoscopic superpositions, called "Schrodinger cats," of motional states of trapped ions [10]

and of photon states in cavity quantum electrodynamics [11]. In both cases decoherence through coupling to ambient res- ervoirs was observed, and sensitivity to the size of the superposition was demonstrated [10,11]. In more recent work with

trapped ions [12], we extended the investigations beyond

ambient reservoirs and "engineered" the form, bandwidth,

and strength of coupling to the reservoir. We observed the quantitative scaling laws of the decoherence of superpositions of coherent states and superpositions of Fock states into a variety of reservoirs for a range of parameters. This paper

is a more detailed discussion of the experiments of Ref. [12] with additional theoretical development and additional data. The paper is arranged as follows. In Sec. II we review the theory of the damping of a harmonic oscillator coupled to several types of reservoirs. In particular, we derive specific

formulas for the time evolution of the density matrix for several types of system-reservoir interactions. In Sec. Ill we

discuss the experimental apparatus and techniques, and in Sec. IV we analyze the data.

1050-2947/20OO/62(5)/053807(22)/$ 15.00 TN-160

62 053807-1 ©2000 The American Physical Society
Q. A. TURCHETTE ex al.

II. THEORETICAL SUMMARY

A. Damping of harmonic oscillators

1. Thermal amplitude reservoir We begin by reminding the reader of some formal results

in the theory of damping of harmonic oscillators coupled to two types of baths. In this section we consider the case in which the reservoir is a bath of oscillators, and the coupling

is to the position of the oscillator via the interaction Hamiltonian

H,= h^ (r,£,a f + r*£Ja), (3)

where T, is the coupling rate of the ith bath oscillator to the system, £, is the lowering operator of this reservoir oscillator, and a is the lowering operator of the system oscillator. This is known as an amplitude coupling. If furthermore, the reservoir oscillators are in thermal equilibrium at temperature T, then the system exchanges energy with the reservoir

in a process leading to overall thermal equilibrium. The time

evolution of this process is described by the following master

equation for the system density matrix formed by tracing over the reservoir degrees of freedom [5]:

p(r) = — («+ l)(2apa* — a*ap — pa^a)

+ —n(2a f pa — aa^p — paa i ), (4)

where y is the system decay rate and n is the average number of quanta in the reservoir at the resonance frequency of

the system oscillator a> :

-fioin lk R T n =

l-e - hcoQ IkfcT (5)

PHYSICAL REVIEW A 62 053807

Pnm(t) =

1 m\n(n,m)

N(t)

l+N(t) jf \l + N(t)

yt \l

— yt/2 \ n + m — 2j \+N{t)

x2 |1

; = o l + N(t)

X

n + l—j\lm + l—j\ln\lm

n~j l\ m-j l\j)\j

X Pn + l-j,m + l-j(®)>

where pnm (0) is the initial density matrix and

N{t) = n{\-e~ y')

(7)

(8)

can be interpreted as the average number of quanta in the system at time t for an initial ground-state system [p(0) = |0)(0|]. The form of Eq. (7) is qualitatively illuminating: the evolution of a given density-matrix element depends only on neighbors along its own diagonal. We note that this cal- culation has been performed in Refs. [13] and [14]. We used

Ref. [13] to derive Eq. (7); in this reference the final form

[Ref. [13], Eq. (37)] seems to be written incorrectly, though

the correct result can be derived from earlier equations. In Ref. [14] the result is written in terms of the Q function, and

agrees with Ref. [13], and therefore not with our result. In some cases, we are interested in the time evolution of

the diagonal elements only, which are given by:

Pnn(t) : 1 2

N(t)

1+N(t) j=b\l+N(t)l \l+N(t

,-yt/2 \2n-2j

(9)

/ = 1-

l+N(t)j \ n-j

I ]Pn + l-j,n + l-j(Q)- (10)

As stated above, the zero-temperature reservoir is a special case of the amplitude reservoir. We write the solution in this case as:

and k Bis Boltzmann's constant. The zero-temperature reser- voir (« = 0) is a special case of the thermal amplitude reser- voir in which the inescapable quantum noise dominates the

classical thermal noise of fluctuating oscillators.

a. Amplitude damping, Fock states. The solution of Eq.

(4) is nontrivial, and has been discussed at some length in the

literature (which is reviewed in Ref. [8]). We first consider

its solution in a Fock-state basis. In this basis, the decomposition of p{t) is given by

P(0 = 2 pnm {t)\n)(m\. (6)

Based on calculations presented in Ref. [13], we state the solution to Eq. (4) with the expansion of Eq. (6) as:

pnm (t) = e- ( " +m) y"2 X {l-e'") 1

1 =

X

n + l n m + r

\pn+ , m+l(0). (11)

For the diagonals only,

Pnn(t) = en "?, (l-e-y'Y

1 =

n + l Pn + Ln + I (0). (12)

b. Amplitude damping, coherent states. In this paragraph we consider the solution of Eq. (4) in a situation appropriate

to coherent-state evolution. We follow exactly the results of

Ref. [4]. For coherent states the density matrix is expanded

as:

053807-2

TN-161
DECOHERENCE AND DECAY OF MOTIONAL QUANTUM .

.

p(t) =

) d

2

Jxd2 vP(jx,v*)j^-, (13)

where P{(jl,v*) is the positive-P representation [5-7]. The

solution to Eq. (4) is written [4]

p(t) = j

d

2nd2 vP (fi,v*)

x(v\nre ~

y' n+2Nu) m)PT(t)&(v), (H)

where P (fi,v*) describes the initial system density matrix

through Eq. (13),

PHYSICAL REVIEW A 62 053807

yt/2

/* =

l + 2N(t) {[{l+N(t)],x + N(t)v}, (15)

{[l+N(t)]v+N(t)fM}, (16) l+2N(t)

and p T{t) is a thermal state density matrix with N(t) quanta:

P T(t) : 1

l + JV(0»=o

N(t)

l+N(t)

\m)(m\. (17)

N(t) is defined in Eq. (8). f) is the displacement operator

t>(a)\0) = \a). The form of this solution makes it difficult to see the simple result for the off-diagonal elements of the density matrix as presented in the Introduction. The appropriate experimental measurement will reveal it when we apply this formal result below.

2. Phase reservoir Here we consider a reservoir whose coupling preserves

the energy of the system, with interaction Hamiltonian

H,=h^ (TiB^a + Tf B]a^a). (18)

This is known as a phase-damping reservoir. For the system-reservoir interaction described by Eq. (18)

and a reservoir in thermal equilibrium at a temperature cor- responding to a mean occupation number n [as in Eq. (5)], the master equation is given by [6]:

p(t) = -[2^apat a-(a f a) 2 p-p(a^a)2

]. (19)

Here K = K(2n + 1), is the system decay rate with K the coupling to the reservoir of oscillators.

a. Phase damping, Fock states. For initial Fock states, the

solution of Eq. (19) is straightforward. Using the expansion

of Eq. (6),

Pnm=- 2^ n ~ m ^ Pnr, (20)

P„ m (>) = e~ ( "/2)( "- m)2 'pmn (0). (21)

with solution

Here it is clear that the off-diagonal density-matrix elements decay exponentially at a rate proportional to the square of

their distance from the diagonal.

b. Phase damping, coherent states. For the coherent state basis, the solution of Eq. (19) is rather complicated. We will forego its solution in favor of the technique presented in the next section.

B. Interferometry

This section presents an experimentally motivated ap- proach to calculating the decay of quantum coherences. Here

the off-diagonal matrix elements are measured as the contrast of an ion interferometry experiment. This section is adapted

from Ref. [15].

1. Ion states and transitions

First we review a few details necessary to understand the ion interferometer analysis. For the purposes of this section, a trapped ion has an external degree of freedom that is a

perfect one-dimensional (ID) harmonic oscillator (along z) of frequency a>. The internal degree of freedom is assumed

to be a perfect two-state system, the states of which we label

in analogy with a spin- 1/2 system by ||) and |T). The states

1 1 ) and |t) are coupled to the motional harmonic-oscillator

states \n) via a classical optical laser field. The interaction Hamiltonian has resonant couplings between internal states

I 5 ) (l 5 ) = ll) or It)) and motional states \n) whose matrix

elements are [16,17]

(s' ,n'\H,\s,n) = hn{s' ,n'|o-+^"7(<' + " +)

+ o--ei *'l+ "

f) \s,n), (22)

where cr+ (<r_) is the atomic raising (lowering) operator, a

f

(a) is the harmonic-oscillator creation (annihilation) operator, H is the laser-atom coupling strength, or Rabi frequency,

and 77= Skzo is the Lamb-Dicke parameter, where Sk is the

z component of the applied field wave vector and z = \Jh/2mu) is the harmonic-oscillator characteristic length

[16].The excitation spectrum of the ion consists of a carrier transition (|l)|n)<->|T)l")) ar>d motional sideband transi- tions (||)|n)<-^|t)|w)/? :

it m). The sideband transition fre- quencies differ from that of the carrier by a frequency of (n-m)co, where \n — m\ is the order (first, second, etc.) of

the sideband. With resolved sideband cooling [18] and optical pumping we prepare the ion in the ||)|0) state before each repetition of the experiment. At the end of any experiment, there is one observable: the probability that the ion is found in the ||) state, denoted by

P 1 . The measurement is not sensitive to the motional state of the ion, so a trace over the motion is required to calculate

this probability. The final density matrix at the end of an

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Q. A. TURCHETTE et al. PHYSICAL REVIEW A 62 053807

experiment is written pf

, and in general is not separable

between spin and motion parts [19]. The detection signal is given by

/

, | = (l|TrMP/|l} = E (n\a\pf

\i)\n). (23)

n

For pure states pf=\ipf)(ipf\, this reduces to

^1=2 IUK»I*/>I

5 (24)

In experiments in which a coherence is measured, we employ a type of Ramsey interferometry. Generically, the pro- cedure is to embed the coupling to the reservoir between the

initial and final "beam splitters" of a Ramsey interferometer. The beam splitters create and undo a motional quantum

state. As the coherence of the quantum state is degraded (due

to interaction with the reservoir) the contrast of the Ramsey

fringes decreases, giving a measure of the remaining coherence.

2. Amplitude reservoirs

a. Schrodinger cat interferometer and amplitude reservoir. The first type of interferometer that we employ uses the Schrodinger cat state [10]

1 ^)=-^(H)|ai ) + |T)|a T )), (25)

where |f) and \{) are internal states of the ion, and la^t)

are coherent states of the ion's motion in the harmonic well of the trap. As will be seen, the action of an amplitude res- ervoir on this Schrodinger cat interferometer clearly illus- trates the scaling of the decoherence as a function of the size of the superposition.

A diagram of the Schrodinger cat interferometer is pre- sented in Fig. 1. Initially the ion is prepared in the state

|<A) = ||)|0). Next, we generate the spin superposition \ip) = (||) + |T))|0)/V2, as shown in panel 1. The motional su- perposition is created by applying a spin-dependent optical

dipole force that approximates the coherent displacement operators [10,19] P(ap and P(aT ) acting on the states | J.) and

|t). The resulting state (with constant phase factors suppressed), shown in panel 2, is that of Eq. (25). These two

operations constitute the first "beam splitter," or Ramsey

zone of the interferometer. During the Ramsey waiting time,

the amplitude reservoir [with coupling shown in Eq. (3)] is applied, by placing noisy fields on the trap electrodes (this will be discussed in Sec. Ill C 1 below). The application of

the amplitude reservoir for a fixed time results in a random

displacement /? that displaces both spin states equally. Since

V(\)\ K) = enm[X * K] \X + K), (26)

where \k) and |X + /c) are coherent states, after application of the operator V(f3) to the wave function from Eq. (25)

1 ) 7c/2 on spin Imcc I

3) displacement

g.

2) displacement

Aa

4) recombine

v_y ©

FIG. 1. Pictorial representation of the Schrodinger-cat interfer- ometer with coupling to an amplitude reservoir. The coherent states are represented by minimum uncertainty circles in phase space, while the different spin states are indicated by the hatching. In panel

1, a spin superposition |<A) = (|I) + \]))\0)/\/2 is created with a 77-/2 pulse on the carrier transition. The cat state |^f) = (|l)|a|) + |f)|a|))/V2 is completed with a spin dependent dipole force, as shown in panel 2. The effects of coupling to an engineered amplitude reservoir are shown in panel 3. A random displacement by f3 in panel 3 displaces both spin states equally, but with different phase

factors. After the reversal of the initial displacement in panel 4 the motion may again be factored out, but leaving a relative phase shift 2 Im[/3Aa*] between the spin components. An ensemble average over /? leads to a loss of contrast (decoherence) that scales with the

size of the superposition |Aa|.

|^)=-^(||)|a1 + ye) + ,''m!^*]|T)|aT + /S)), (27)

where Aa=a| — a^ and an overall phase factor has been

removed. The motional superposition is reversed by applying

V( — <x\) and T>{ — a^), resulting in

!«/,)=— (|i> + *

2 ' ta"*"*)|T»li8>. (28)

A final 7i/2 pulse transforms the spins as

1 II)- V^

(|i) + e _ iT»,

lTH-7|(lT>-eil»,

(29)

(30)

and yields the state

1 |0)=-[(l+«'<*+2tal/»A«*]>)||)

+ (e

2nm[/3* a

*

] -eiS)\l)]\j3), (3D

where S is the Ramsey phase, i.e., the phase difference be- tween the initial and final tt/2 spin flips. These last two steps constitute the second, recombining beam splitter of the inter- ferometer.

In a single experiment the probability that the ion is in the

||) state is

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DECOHERENCE AND DECAY OF MOTIONAL QUANTUM . PHYSICAL REVIEW A 62 053807

^i(y8) = |(||<A)|

2 = ^[l+cos(^+2Im[y8Aa*])].

(32)

If j3 were fixed on each repetition of the experiment, as the Ramsey phase S is swept, the signal would be a perfect cosine with phase dictated by the displacement f3. The deco- herence (and loss of contrast in this picture) enters, because

/? is a random variable that fluctuates to a different value on

each repetition of the experiment, giving rise to a randomphase shift of the fringe that averages the contrast away from

unity. In the experiment, we force (3 to be a Gaussian dis- tributed random variable, as, for example, realized by a hot

resistor coupled to the trap electrodes. We take both Re /? and Im f3 to have a standard deviation a. An ensemble aver- age of P [{/B) yields the form of the Ramsey fringes:

1 P^M+ e -2|Aa|V

cos S]

.

(33)

We see that the contrast of the fringes is exponentially sensitive to (the square of) the "size" |Aa| of the superposition. This scaling [3,7,20] is one of the key results that is demonstrated experimentally in a later section.

(b) Fock state interferometer and amplitude reservoir. A

second type of interferometer generates a superposition of two Fock states with the first beam splitter using the tech- nique described in Ref. [17]. For example, consider the in- terferometer that generates the superposition (|0) + |2))/>/2

and then recombines the two parts of the wave function to produce interference fringes. Starting from the state ||)|0), a

it/2 pulse on the first blue sideband generates the state l^) = (U)|0) + lT)|l})/\/2- A it pulse, tuned to the first red

sideband, drives the transition |f)|l)—->||)|2) while leaving

the population in the ||)|0) state unperturbed. This combination of pulses generates the superposition

1 lfA>=-gll>(|0> + |2». (34)

The second beam splitter of the interferometer is realized by

reversing the above procedure—driving a tt pulse on the first red sideband followed by a 7r/2 pulse on the first blue side- band. When the phase of the second tt/2 pulse is swept, the spin state at the end of the experiment oscillates between ||)

and |T). For the experiments reported here, we use two pulses to generate the beam splitters in the Fock state interferometer. For the first beam splitter, the first pulse is a tt/2 pulse on the n'th order blue sideband, generating the state

1 l^)=-7|(U)|0) + |T>|n')). (35)

A second pulse selectively drives only one of the spin components, resulting in the Fock state superposition

TABLE I. The stimulated Raman pulses required for the first beam splitter of the Fock state interferometer where the initial state

is always |1)|0). To reverse the superposition in the second beam

splitter, first the tt pulse is applied, followed by the tt/2 pulse with

a phase added with respect to the first tt/2. RSB and BSB stand for red sideband and blue sideband, respectively.

tt/2 pulse tt pulse State

carrier 1

st RSB |I)(|0) + |1))/V2

1 st BSB I s

' RSB |I)(|0) + |2))/V2

2

nd BSB 1st RSB |I)(|0) + |3))/V2

1

st BSB 2nd BSB |T)(|1) + |2))/V2

\^)=j=\s)(\m) + \n)), (36)

where \s) denotes one of the spin states | J.) or ||). The four

pairs of Fock states generated in the experiments, along with

the pulses used to generate the superpositions, are shown in Table I. The Fock state interferometer can be coupled to an amplitude reservoir. For simplicity, we restrict ourselves to Fock state superpositions of the form

l«A)=-gll)(|0) + |m)). (37)

It is an extended exercise to calculate results for the entire interferometer in the manner of the preceding case (coherent

states in amplitude reservoir). The complication is twofold:

First, displacing Fock states leads to cumbersome expressions. Second, there is a problem with the beam splitters: the pulses of the second beam splitter are tt and 7r/2 pulses

strictly for the transitions ||)|»i)<->|t)|/n— 1) and

||)|0)<->|T)|m — 1), respectively. Thus applying these pulses to an ion in a motional state that has changed from its original form does not result in full spatial overlap. It turns out that while the calculation is complicated, the details are not important to the actual experiment, which is performed

only for small m, and in the limit yt<\. Rather than present

the full calculation, we will merely state a simple result based on the master-equation approach. For states of the type

shown in Eq. (37), the time evolution of the relevant density- matrix elements predicted by Eq. (7) reduces to the simple

form

1 PO.n 2{l+nyt) 1 +m

(38)

for yt< 1 . The measured contrast is simply twice this matrix

element. For the range of m and n yt studied in the experiment, this expression differs by less than 5% from an exact

treatment of the imperfect spatial overlap.

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Q. A. TURCHETTE et al. PHYSICAL REVIEW A 62 053807

I ) 7C/2 on spin Im a i Re a

3) phase shift

2) displacement

4) recombine

FIG. 2. Pictorial representation of the Schrodinger-cat interfer- ometer with coupling to a phase reservoir. The coherent states are represented by minimum uncertainty circles in phase space, while

the different spin states are indicated by the hatching. In panel 1, a

spin superposition |^) = (||)+|T))|0)/v/2 is created with a ttI2 pulse on the carrier transition. The cat state |<A) = (|J.)|«j) + |t)|a T ))/v2 is completed with a spin dependent dipole force, as shown in panel 2. Coupling to a phase reservoir is depicted in panel

3. The random-phase shift in panel 3 prevents the correct reversing of the initial creation of the cat, resulting in a loss of contrast due to both the phase shift <p as well as the reduced spatial overlap.

3. Phase reservoirs

(a) Schrodinger cat interferometer and phase reservoir. A

second reservoir that we apply is a phase reservoir. A phase

reservoir is realized by adiabatically changing the trap fre- quency for a period of time and then restoring the original

trap frequency. If the deviation of the trap frequency is Sio(t), then the phase shift of the ion oscillation in time T is

Jo

<p = Sa)(t)dt. (39)

The potentials required to change the trap frequency are applied to the trap electrodes during the Ramsey time between

the beam splitter pulses (see Sec. Ill C 2). The effect of a phase reservoir on a Schrodinger cat state

is diagrammed in Fig. 2. The generation of the Schrodinger

cat state \i//) = (\i)\a i ) + \'\)\a^))/^/2 is accomplished as discussed previously. A phase shift of the ion oscillation by

the random variable <p introduces complex phase factors to the coherent states,

|^>=-^(U>|a1 e'*> + |T>|aT e

l>». (40)

Due to these phase factors e

l<l>

, the motional superposition is not correctly reversed, as shown in Fig. 2, panel 4. The sec- ond beam splitter consists of the displacements T>( — a^) and

T>{ — «|) followed by a second ttI2 pulse on the carrier, yielding

^>=~[e'1a

< lW(ii>+eiT>)K(e ,<A -i)>

In a single run of the experiment, the probability to find the ion in the | J. ) state is

P i (<t>) = \(l\>P)\

2 =2{l-TMP[-\ba\ 2 (l-cos<t>)]

Xcos[<5+(|ajJ 2 — |a

T | 2 )sin (p + 2(l-cos0)ImafaT ]}. (42)

The phase shift <p is taken to be a Gaussian distributed random variable with standard deviation a. The average of

Pi(cp) is

1 2\

l V2^Jexp

2 a

2 -|Aa| 2 (l-cos0)

Xcos[( | aj

2 — |«||2 )sin 4>]

Xcos[S+2(l -cos (p^mafa^dcp} , (43)

where the odd part of the integrand vanishes. We note that in general there is a phase shift to the fringes even though the

distribution of <p is symmetric. In general this integral does

not simplify. However, in the case of experimental interest, a^a^ (specifically, aj=— 2aj) in which case lma*a^

= 0. We further make the small-angle approximation in the integrand, which is valid if cr

2 <il, when the term e~^

dominates; or, if |Aa|>l and a^s-n-, when the term

g

-|Aa| (i-cos0) dormnates. In these regimes, the integral can

be evaluated analytically to yield the signal

1 1-cos S

VT+|AajV

^exP (Kl 2 - M

2

) 2 ^

2

2(l + |Aa| 2

cr

2

) (44)

This approximate expression provides a good guide to the behavior and can be used for a simplified comparison to the

data. We note that in the experimentally accessible regime of |Aa|~l-4 and cr=£2, Eq. (44) is accurate to about 10%

even though the small angle approximation is not strictly valid. (b) Fock state interferometer and phase reservoir. We

start in the state [\n)+ |m)]||)/v/2. We again denote the ion

oscillation phase shift by (p, and the phase shifted superposition is

\if/)=-^\i)(e- l ""f'\m) + e-'^\n)). (45)

In the second beam splitter a — tt pulse again entangles the spins with the motion, as

!«/,)= _(g -'-»'*||)|o> + e-"'*|T>|/i'». (46)

_,_^ t |« T

l 2 sin <^(|^)_ e '^jt>)|Q- T (e'^— !)>]- (41) A final tt/2 pulse generates the state

053807-6

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DECOHERENCE AND DECAY OF MOTIONAL QUANTUM PHYSICAL REVIEW A 62 053807

])7t/2bsb

n = 0,l

3) phase shift

2) 71 rsb

B = 0,

4) -Jt rsb

FIG. 3. Pictorial representation of the action of a phase reservoir on a Fock state interferometer. Here we consider the interferometer

that generates the state \iff) = \l)(\0)+\2))/yJ2. A -rr/2 pulse on the

first blue sideband generates the state |^) = (|J.)|0) + |T)|l))/v/2

depicted in panel 1, where the different spin states are indicated by

the shading. A tt pulse tuned to the first red sideband drives the

transition |T)|1) —Hi) 1 2) and generates the desired superposition.

The random motional phase shift (f> in panel 3 adds a relative phase

factor, |i/') = |l)(|0) + e" 2 ^|2))/ N/2, where the phase is scaled by

the difference in indices of the Fock states (2 in this case). A second

tt pulse generates the state |^) = (| j)|O) + <?" 2,

' |T)|l))/\/2 (panel

4). The result of the final ttI2 pulse depends on the phase (p. Ran- dom fluctuations in <f> result in decreased contrast of the Ramsey

fringes.

|0)=-[(e-'«*- e '<*-»«)|i)|O)

+(e-' n *+e''(<5 - m *) )iT)k)].

The projection onto | J. ) yields

(47)

P l (cf>) = \(l\ tp)\

2 = -{l+cos[S+(n-m)4,]}. (48)

Again, the phase shift introduced between the beam split- ter pulses results in a shift of the fringes. The random-phase

shifts 4> are assumed to be Gaussian distributed with stan- dard deviation a, and the ensemble average of P^(cf>) is

^1 =

2 H +<? _(n ~'" )V/2

cos S]. (49)

The decoherence scales exponentially with the square of the difference in the Fock state indices n—m, so that "larger"

superpositions lose coherence exponentially faster. Compare

the scaling of the decay of the fringe contrast as a function of

n — m in Eq. (49) with the decay of the off-diagonal elements

of the density matrix pmn given in Eq. (21). The results are equivalent. (See Fig. 3.)

C. Connection of averaged interferometer approach

and master equation

It is reasonable to ask in what sense the interferometer experiments of Sec. II B directly probe the relevant off- diagonal matrix elements as calculated in Sec. II A. In the case of Fock states interacting with the phase reservoir (Sees.

II A 2 and II B 3), the connection is quite simple, in other

cases, the connection is not as straightforward.

/. Spin coherence

In this section we argue that the contrast of the interfer- ence fringes is a measure of the coherence of the combined

spin and motional quantum state, and that the spin part can

be negligible or can be removed. To address this issue, we

consider the interferometer experiment in the case of no

added noise. We take the example of the cat state interfer- ometer discussed in Sec. II B and write the cat state of the system after the first 7i72 and displacement pulses (assumed

to be short) as a density matrix [the wave function for C = 1 is given in Eq. 25]:

P= 2

L [ki>II)(II(«il + kT )lT)(TK« T

l

+ C|a i )U)(T|(a T | + C*|a T )|T)(l|(a i |], (50)

where C=Cs Cm quantifies the coherence that has contributions from both the spin (Cs ) and motion (Cm ). This separation of the contrast into a product of spin and motion contri- butions is valid for reservoirs in which the spin and motion

are not coupled, such as the amplitude, natural, and phase

reservoirs. Moreover, separate experiments verify that the spin is not disturbed by or during the time of application of our engineered phase, amplitude, and zero-temperature res- ervoirs, so that in these experiments the fringe contrast is a simple probe of the change in motional coherence. The only

reservoir experiments in which the spin decoherence is not

negligible are the natural reservoir experiments, since for all of our applied reservoirs the time of application is short. For

the natural reservoir experiments, we are forced to simply

wait a time t for the ambient noise to act on the system. Note

that the separation of coherences is valid if the spin under- goes only phase decay, with no random spin-flip transitions, which is verified in our experiments. To within experimental

errors, we can adequately characterize spin decoherence as exponential, CJ = exp(— ysi). In the second part of the cat state interferometer, the sec- ond displacement is applied, undoing the first displacement,

leaving

p=

2

1 [|o)|I)<1|(o| + |o)|t)(T|(o|

+ C|0>||>(T|<0| + C*|0>|T><1|<0|]. (51)

The final 7772-pulse with phase S transforms the spins as in Eqs. (30), and the detection of | J. ) with trace over the motion

leaves:

l-Re(C)cos<S

(52)

Thus the contrast of the Ramsey fringes depends directly on

the coherence.

2. Cats, amplitude noise The interferometer calculations presented in Sec. II B predict the outcome of particular experiments. The master053807-7

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Q. A. TURCHETTE et al. PHYSICAL REVIEW A 62 053807

equation solutions in Sec. II A do not. In this section, we

take the solution of the master equation, in terms of the time

evolution of a density matrix and derive a prediction for an

interferometer experiment. The result is shown to be equivalent to the average-over-classical-variables approach in Sec.

II B. We do this for the Schrodinger cat interferometer subject to amplitude noise. The solution to the master equation was given in Sec. II A 1. It is a straightforward matter to add

the spin part to this solution, which must be done for the interferometer calculation. Starting from Eq. (14), we divide

the initial density matrix [written as an initial P function

FoC/Lt.v*)] into four parts, corresponding to the initial state

l<A) = (ll)ki) + lT)|a T )) / V2: PQ {V,V*) = P$\p,V*) + P$\lJi,V*) + P<$\^V*) + Ptf\fi,v*) (53)

with

P

( l)

( fi,v*)=-S(fJL-ai )S(v-a [ )\l)(l\, (54)

Pg\fi,v*)=-8(fL-a ] )8(v-a ] )\1)(1l (55)

/'( 3) (/i , J,*) = -(aT |a l )

<5(M-ai

)

<5( l.-aT )|l)(T|,

P

{o

4) (»,v*) = [p£\v,v*)y.

From Eq. (14) this leads to four contributions to p(t)

p(0 = p

(1) (» + p(2) (0 + p(3) (0 + p<4) (0

with

p^(t)=-V(ai )pT(t)VHa l )\i)([\,

p

{2 \t)=-V(a1 )pT(t)VHa1 )\l)(l\>

(56)

(57)

(58)

(59)

(60)

> (3) (»

= -(ai\ai y-^y'» + 2N^V(al )p T(t)VHa 1 )\l)ai

P

(4) (0 = [p(3) (0] f

.

(61)

(62)

where a^ are defined as in Eqs. (15) and (16). To complete

the experiment, we apply a second displacement, which is given by the operator

|j>U|P(a,e-'V|T><Tl^(«Te-'^ (63)

where is the phase of the second displacement with respect

to that of the first. We then apply the final tt/2 Ramsey pulse

that transforms the spins as in Eq. (30) and measure the probability that the ion is in the spin-down state, as discussed

in the context of Eq. (23). Thus we are interested only in the component of p given by Pj — (l|p/|l), where pf

is the density matrix after the final displacement and Ramsey pulses.

Furthermore, since we do not measure the motional state, a

trace over the motional degrees of freedom is required:

P1 (0 = TrII p1 (0 = 2 Hp|(0l«>- (64)

n

The quantity p| is given by

,<»/ ,< : > 0)< M)i

Pi {t) = p\

l \t) + p\»{t) + p\»{t) + p^{t) (65)

with

p\

i \t)=-V{ai e-<»mai )pT{t)V\ai )V\a[ e-'»),

(66)

,(2)/,W

pY

, (t)=-V(a ] e-' t')V(a ] )pT(t)V r (a ] )V\a] e-' ),

(67)

XVia^prit^ia^VHa^-' ), (68)

p\

4) (t) = [p\

3) (t)V. (69)

In what follows, we assume that ajj are real. Performing

the trace, we find that T^p'/^Tr^pf^ 1/4 and that Tr,^3 ' 4

' gives rise to the interference. Using the identity

T>(K)V(\) == eMk\*-k*\)V(k+\), (70)

substituting for pT from Eq. (17) and using the closure rela- tion Sm |m)(m| = 1,

Trn p

(

, 3) (?

1 l+N(t)

i(aj-ffj)(aj + aj)sin 8

x? W (71)

where /3=a^e l6 -\-a^ — a^e l8 —oty and £ is a Laguerre

polynomial which comes from (m\V{ K)\m) = exp(-iK|2/2)£m(|Ac| 2

). Using 2 m

em £m (x) = (l - e) -1 exp[— ec/(l — e)] and further simplifying,

1 1 -a,) 2 [l+n(l-<? y') + t P [ (t)=2-2 e ~

(ai

Xcos[<S+e -y,/2(a2 -a2 )sin0]

- yia

cos e\

(72)

For all of our experiments we can consider the hightemperature, short-time limits: n> 1 and yt< 1 . In this limit,

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DECOHERENCE AND DECAY OF MOTIONAL QUANTUM . PHYSICAL REVIEW A 62 053807

0.2 mm

FIG. 4. Schematic diagram of the electrodes of the linear trap (not to scale). The trap electrodes are formed by evaporating gold

onto alumina substrates. The outer segmented electrodes are the endcaps, while the long unbroken electrodes carry rf. The axial (£) direction is parallel to the rf electrode. The two separate trap wafers

are separated by 200 fim with alumina spacers (not shown).

1 2 - p ,(/)= —{1 — e~

<- a

l~ al

) n +n rf+cos6')

Xcos[<S+(l-yr/2)(a2 -a2 )sin0]}. (73)

Normally, the second displacement is the reverse of the first (0=7r) and the Ramsey fringes in Eq. (33) are recovered

with the association nyr—>2cr2

. The scaling with the square of the size of the initial state is evident. The constant term in the exponential measures a residual overlap between the initial state and the final state, present even in the infinite time

limit. Similar procedures to this for other types of initial states and reservoirs yield results that agree with those from the interferometer averaging over random displacements/phases

approach given in Sec. II B.

III. EXPERIMENT

A. Ion trap

We use a miniaturized version of the linear Paul trap [21]. A diagram of the trap structure is shown in Fig. 4. It consists of two alumina wafers separated by 200 yum with alumina

spacers (not shown). Each wafer has a slot 2 mm long by

200 iJ.m wide. Gold is deposited onto the edges of the slot to form electrodes. One side of the slot is the rf electrode, and

the other side is divided into three segments to which differ- ent static potentials are applied to make a static well along

the z axis. The segmented electrodes are formed by cutting two side slots of width 20 /xm. The side slots are separated

by 200 /im, forming two "endcaps" and one "middle"

electrode. When the wafers are sandwiched together with the spacers, the four interior edges of the main slots approximate

the four wires of a linear quadrupole trap. The microfabri- cated structure described here allows miniaturization of the

linear trap and correspondingly high-trap strength.

The linear trap is mounted at the end of an rf quarter- wave coaxial resonator that supplies the required rf voltage

[22]. The trap and coaxial resonator are contained within an UHV enclosure, as in Ref. [22]. The two rf electrodes are

2,-2= li>

t 1— 1,-1=|T> 1 LL2.-1 /

2s

ZS 112

FIG. 5. Simplified level diagram of 9 Be+

(not to scale). The

transitions labeled D are: Dl: optical pumping (also called red Doppler), D2: detection/cooling, D3: repumping. The transitions labeled R are the two arms of the two-photon stimulated-Raman transition. A is the detuning between the 2p 2 P1/2 and the Raman virtual level and u> /2tt= 1.25 GHz is the Is 2 S ]/2 hyperfine splitting. The fine- structure splitting of the P state is 197 GHz. The transition wavelength from the S to P state is 313 nm.

connected to the tip of the coaxial center conductor. Each of

the static-potential segments are connected to UHV

feedthroughs through a pair of 200 kHz RC low-pass filters in series. The coaxial resonator is 30 cm long and partially

filled with alumina to lower its resonance frequency to Hr

/277-= 1 13 MHz. With 8 W of rf power, the radial secular frequencies are approximately 13 MHz. The axial potential is controlled by the static potentials applied to the segmented

electrodes. Positive potentials are applied to the endcaps to confine positive ions in the middle of the trap. For the data presented in this paper, we restrict our attention to the axial mode of motion. With 30 V applied to the endcap segments

and the middle segments and rf electrodes held at static ground potential, the axial frequency is u>J2tt = 11.3 MHz.

B. Raman transitions and cooling the ion The level structure of 9 Be+

is shown in Fig. 5. We select two states from the ground-state hyperfine manifold: the \F — 2,m F — -2) state, which we label ||), and the \F=\,m F = — 1) state, which we label ||). The ion is Doppler cooled by applying circularly polarized light detuned ~ 8 MHz be- low the strongly-allowed (17277-= 19 MHz) cycling transi- tion 25 2 5 1/2 |2,-2)^2p 2 /

> 3/2 |3,-3) (beam D2 of Fig. 5). Due to imperfect a~ polarization the ion optically pumps to the 1 1,— 1 } state with a probability ~ 1 X 10~ 4

. To prevent a

loss of cooling from these decay events, we apply laser light resonant with the transition 2s 2 5 1/2 |l,— 1

) —>2/7

2 P U2\2, -2). This transition is labeled Dl in Fig. 5. After Doppler

cooling, the mean occupation number of the axial mode of motion is (n)= 1. We efficiently detect (Fig. 6) the | J. ) internal state of the ion by applying Doppler beam D2 with no Dl light [18]. The

fluorescence collection system has a//1.2 aperture, and the

light is transferred onto the photocathode of a photomultiplier tube (PMT). The PMT quantum efficiency is =30%.

After all losses at windows and filters, the overall scattered- photon detection efficiency is measured to be «»1.5X 10~ 3

. For detection, the laser is at the same detuning as for Doppler cooling and the intensity is kept well below saturation /~0.1/sa,

. During a 200-ytiS-detection period, we collect an

053807-9

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Q. A. TURCHETTE et al. PHYSICAL REVIEW A 62 053807

COE

8.X

CO o

CMW

COo

c

CO

I—3

ooO

1000

200' r 1 w

100- r

! -i

, ri K 10 20 30

Photons collected in 200 jxs

FIG. 6. Detection efficiency. The graphs show histograms of photons collected in a 200 /us interval for 2000 repetitions of the experiment for the two initial states ||) and ||). Note that if a discriminator were placed at the bin associated with n = 3 photons,

then, on a per experiment basis, the state of the ion would be

distinguishable with a 97% quantum efficiency.

average of 10 photons if the ion is in the ||) state, and 0.1 photons if it is in the 1 1

) state. The photon distributions are Poissonian. By averaging the number of photons detected in successive repetitions of a given experiment and normalizing

to the count rate when the ion is prepared in a control ex- periment in the |j) state (by simple optical pumping), we

determine the probability P±

In previous experiments using miniature spherical quadruple ion traps (see e.g., Ref. [24]), ion internal state dis- crimination was limited by sidebands on the 2s 2 S 1/2 F=1 —»2p

2Py2, F=2 transition due to ion micromotion at fre- quency ft r

. Thus a laser tuned to transition D2 would

weakly drive an ion in the | T ) state. The miniature linear trap addresses the issue of micromotion in two ways. First, there

is negligible micromotion associated with the confinement

along the axis of the trap. Second, the close proximity of the

static-potential electrodes permits accurate cancellation of

stray static fields, resulting in negligible sidebands on the

transitions from micromotion. This permits state discrimination of 97% in a single experiment. This is shown in Fig. 6. Even in the absence of micromotion, detection efficiency is limited by off-resonant optical pumping of the | f

) state by

the detection laser [23]. We drive coherent stimulated Raman transitions with a

pair of laser beams (such as Rl and R2 shown in Fig. 5) detuned A/27r=10 to 20 GHz from the 2s 2 S m-^2p 2 P m

transition near A. = 313 nm. The Raman beams are derived

from a single dye laser beam that is frequency doubled, and

the difference frequency between the two Raman beams is

generated with several acousto-optic modulators in series. We employ three types of Raman transitions, distinguished

by beam geometry and difference frequency [16,25]. The

beam geometry sets the projection of photon momentum

onto the axis of the trap (z), characterized by the wave vector difference Sk = (k l — k2 )-z. The coupling of this photon

momentum to the motion of the ion is parametrized by the Lamb-Dicke parameter rj=Skzo, where z = ^hl2mb>*°=l nm is the harmonic-oscillator characteristic length for axial motion.

The three types of Raman transitions are as follows, (i) Copropagating beams (^^O) drive a motion-independent

"spin-flip" transition, |f)|«)<-+||)|«). The frequency differ- ence between the Raman beams is set equal to the frequency

difference between the states, o) = (oh e+ wz , where

a> h f/27T=l.25 GHz is the hyperfine splitting and <jj zI2tt «12 MHz is the Zeeman splitting due to an applied magnetic field |B| ==0.57 mT. (ii) Beams oriented 45° to the trap axis and 90° to each other result in a wave vector difference pointing along the trap axis with Sk — 2\J2Trl\ and couple to the axial motion. For the conditions of this experiment, rj ^=0.20. These beams at difference frequencies w + wAn,

couple the states | j)|n) and |f)|n-l- An). In the Lamb-Dicke

limit, (77

2 (n)<§l), the strength of the sideband coupling is proportional to 77^"'. The Raman beams in cases (i) and (ii) are linearly polarized, with one beam polarized parallel to the quantization axis (Aw = 0) and the other polarized per- pendicular (Am = ±1). Note that only the cr~ component of

the latter beam is required to drive the Raman transition, but

the use of linear polarized light reduces the Stark shifts rela- tive to the case of using circularly polarized light, (iii) Beams

oriented as in (ii), with difference frequency set equal to the

axial secular motion frequency o> approximate the harmonicoscillator displacement operator V(a), defined by the rela- tion X>(a)|0)= I a), and |a) is a coherent state. In the LambDicke limit, the displacement \a\= rjCl-pt is proportional to the duration of the laser pulse t and the Rabi frequency ftp,

and is set by the phase and polarization of the applied laser

fields. In this case, the two Raman beams are linearly polar- ized such that the polarization vectors are mutually orthogonal to both each other and to the quantization axis. This

cancels Stark shifts to a high degree and results in the two

spin states being simultaneously displaced in opposite direc- tions such that aj= — a^/2.

The ion is first prepared in the ground state of motion

with resolved sideband Raman cooling [18]. Raman cooling

consists of repeatedly driving the first red-Raman sideband

of the axial-motion-sensitive transition, case (ii) with An

— - 1 , followed by optical pumping to the | J, ) state after each Raman pulse. The optical pumping is achieved by ap- plying beams Dl and D3 (see Fig. 5). The rate at which the ion undergoes the red sideband Raman transition depends on

the initial state of motion as ft rst, a \fn. The duration of the successive Raman pulses is progressively lengthened, so as

to successively drive -n pulses on the n,—»n,-— l,n,— 1—>n, -2, etc., transitions, where n, is the total number of cooling

pulses. The final pulse of this sequence is selected to drive a

tt pulse on the |l)|l)—> |T)|0) transition. After each Raman

053807-10

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DECOHERENCE AND DECAY OF MOTIONAL QUANTUM PHYSICAL REVIEW A 62 053807

2.5

i i i y I i (a) /

l^-*"

2 ;/ &^^^ "

1.5 Jj^^^^^ -

1 "

0.5 o 1 us drive - ^" 1 1 1 1 1 a 4 us drive

rt 0.02 0.04 0.06 0.08 E [arb. units]

0.1

3

t[ns]

0.12 0.14

2

i i i i ^A-' -

1.5

(b) S^^^

- S 1

- o^-^~

_S———~"

-

0.5

i i i

O |i> " D |T>

FIG. 7. Calibration of displacement amplitude

for (a) uniform electric-field drive and (b) laser

drive. The y axis is in terms of the coherent-state amplitude a, determined as described in the text.

pulse, we drive the D\ and D2> transitions for 5 /as, optically

pumping any population in the ||) state, or the (2,— 1) state to the ||) state. Typically, after 6 cycles of Raman cooling,

the ion has a probability of 95% to be in the ground state U)|o>.

C. Ion interferometry: techniques

In the experiments reported here, we measure the coher- ence of quantum superpositions with single-atom interferometry, analogous to that used in our previous work [10] and

discussed in Sec. II B. The motional state of the ion is split into a superposition of two parts. The "beam splitters" of

our interferometers consist of a pair of stimulated Raman

pulses. We employ two types of interferometers, both dis- cussed in Sec. II B.

1. Ion interferometry: Cat states The Schrodinger-cat interferometer starts with the usual ground-state (|i)|0)) preparation. The first pulse is then a

tt/2 pulse on the motion-independent transition (i), with

drive time typically T^^O.S /is, which generates an equal

spin superposition, ||)|0)—>(|j.)+lt))|0)/\/2 as indicated schematically in Fig. 1, panel (1). We drive Raman transition (iii) to excite the motion associated with each spin state into a coherent state, resulting in the state of Eq. (25). We call this the displacement pulse

[Fig. 1, panel (2)]. We vary the length of the Raman drive pulse in order to vary |aj and |aJ. We independently measure |aj and \a^\ in the following manner [17] [which is also discussed in the context of Eqs. (85) and (86), below].

The ion is prepared in a coherent state, and then Raman

transition (ii), tuned to the first blue sideband, is driven for a variable time t p

. After this "probe" Raman transition, we

determine the probability P [{tp

). The resulting curve may be decomposed into sums of sinusoids that correspond to A/i = 1 transitions between Fock states. The Fock state distribution is then fit to that of a coherent state, thereby extracting

\a\ [17]. The values of |aj and \a^\ are measured separately, by preparing the ion in either the ||)|0) state or in the |T)|0) state, applying the coherent drive pulse, and then probing and detecting as just described. Curves of \a^

T

| vs

the length / of the displacement pulse are shown in Fig. 7(b). The ratio |aj/|a|| = 2.0±0.1, averaged over the data set, is consistent with the expected value of 2. The slope

d\Aa\/dt= TjCl is consistent with the expected value. The Schrodinger-cat interferometer is completed by ap- plying a second coherent displacement pulse 180° out of phase from the first displacement pulse [Fig. 1, panel (4)]. The phase of the displacement is set by the phase of an

oscillator that drives an acousto-optic modulator that is in one of the Raman beams. All phases are referenced to a master oscillator by phase locking to that oscillator. If the motion was unperturbed during the time between the dis- placement pulses, then the inverse displacement would re- store the motion associated with each spin state to the ground

state of motion. The wave function would return to \ip) = (U) + lT))|0)/\/2. A second tt/2 pulse on the motionindependent spin-flip transition, with phase S relative to the

first 7r/2 pulse, interferes with the spin components. The

probability P^S) varies sinusoidally with S. To record F|( S), we step the frequency of the oscillator controlling the

7r/2 pulses and bin the fluorescence counts at each step. Here, S is the accumulated phase difference caused by the frequency difference between the oscillator frequency v and

v as in normal Ramsey spectroscopy. We typically repeat

the experiment 100 times at each step. Note that the resulting

fringe spacing is much smaller than the width of the Rabi

pedestal, meaning that the predominant effect is a change in phase, without introducing significant inaccuracy in the tt/2 pulse. We typically record three oscillations of Pj . The

fringes are fit with a sinusoid A + B cos(o»— (Oq)T, where T is the time between spin pulses. The contrast of the fringes is

053807-11 TN-170
Q. A. TURCHETTE el al. PHYSICAL REVIEW A 62 053807

defined as BIA. For the basic interferometer just described,

with no perturbations of the motion between the "beam

splitter" pulses, the contrast is typically 0.6-0.8. This contrast differs from unity for several reasons. (1) Errors in the

7r/2 spin-flip pulses, primarily due to laser power fluctuations. (2) Fluctuating magnetic fields that reduce the spin

coherence through fluctuating Zeeman shifts. (3) Motional

heating during and between the displacement pulses, with

further small contributions due to inaccuracies in reversing

the initial displacement pulse. These effects are expected to be random and should not influence the additional decoher- ence caused by the applied reservoir (see Sec. II C).

2. Interferometry: Fock state interferometer The Fock state interferometer beam splitters combine a

7r/2 pulse with a it pulse on transitions of either type (i) or

(ii), as shown in Table I. As an example, consider the generation of the superposition ||)(|0)+ |2))/ N/

/2. A 7r/2-pulse on the first blue sideband, transition type (ii) with An= + 1 and duration T^a^\ lis, drives the transition ||)|0) -+ (ll)|0) + lt)|l))/V2. A 77 pulse on the first red sideband,

transition type (ii) with An= — 1 and duration T„TM*1.5 /xs, drives the transition |f)|l)—>|J.)|2). This second pulse does

not affect the population in the ||)|0) state. The result of

these two pulses is the desired state |(A) = ||)(|0) + |2))/ v

/2. The second beam splitter recombines the two parts of the wave function by driving a second it pulse on the red side- band followed by a 7r/2 pulse on the blue sideband. By

sweeping the frequency of the blue sideband oscillator we

generate Ramsey fringes in the probability P i . Contrast

(without coupling to the reservoir) is limited by the same

mechanisms as in the Schrodinger-cat interferometer.

D. Engineering reservoirs: techniques

1. High-temperature amplitude reservoir As discussed in the Introduction, the motion of a trapped

ion couples to environmental electric fields E through the

potential U= -qx-E, where x is the position of the ion rela- tive to its equilibrium position (proportional to the amplitude

of motion) and q is the charge of the ion. If E is due to a reservoir of fluctuating field modes, then E« 'Z

j ei (b i + b]),

where b t

, b] are the lowering and raising operators of the

field modes (as in Sec. II A 1). A classical coherent drive (for which E is a narrow-band sine-wave) applied to the trap electrodes at the ion axial motional frequency co results in a displacement of the motional state proportional to the size and duration of the applied field. This is shown in Fig. 7(a)

for two different durations of the applied field. We simulate

the effects of the high-temperature amplitude reservoir dis- cussed in Sec. II A 1 by applying random electric fields along the axis of the trap, whose spectrum is centered on the ion axial motion frequency a>. To generate the required axial fields, we apply a potential

to one of the endcap electrodes. This generates both an axial

field and a small radial field. We ignore the effect of the

radial field since we are insensitive to motion in the radial direction. This field also modulates the axial potential and

gives rise to an accompanying (random) phase shift; however, for the experimental conditions this phase shift is negligible. We start with a 10 MHz bandwidth noise source that

rolls off at 6 dB/octave starting at 10 MHz. We filter the noise such that the resulting noise spectrum has 3 dB points

at 9 MHz and 11.5 MHz, and the noise rolls off at 6 dB/

octave above and below these points. This spectral noise is applied to the trap electrodes through a network of RC lowpass filters designed to prevent such fields from reaching the

static trap electrodes. The filters effectively limit ambient

noise from reaching the electrodes, but large deliberately applied fields can still be effective. The noise is applied for 3 /xs between the Ramsey zones. The total time for the Ramsey experiment (the time between the 7r/2 spin-pulses) is about r=20 lis and is held fixed. For these filters, trap ge- ometry and application time, it can be shown that this type of

drive correctly simulates an infinite-bandwidth, amplitudeonly reservoir.

2. Phase reservoir A phase reservoir, as described in Sec. II A 2, is simulated

by random variations in the trap frequency a>, changing the phase of the ion oscillation without changing its energy. We

realize this coupling experimentally by adiabatically modulating the trap frequency. The random potential fluctuations

are filtered through a low-pass network with a cutoff fre- quency well below w to maintain adiabaticity and to avoid any power at the trap secular frequency, which may cause amplitude noise. The potential fluctuations are applied to one

of the middle electrodes, which is symmetric with respect to the axial direction, so as to produce field gradients and mini- mize any applied uniform axial fields further reducing the

possibility of changing the motional energy. Applying a potential to just one middle electrode does result in a radial

field, but as before, we isolate the axial motion and ignore any radial motion.

In the phase reservoir, the ion follows the change in trap frequency in the sense that there is no change in the energy

of the trap motion, however the phase is either advanced or retarded relative to the unperturbed motion. We start with a 10 MHz pseudorandom noise source and filter the noise, cre- ating a spectrum that is flat between 1 kHz and 100 kHz with

a roll-off of 12 dB/octave above 100 kHz and 6 dB/octave

below 1 kHz. The phase deviation of the ion motion is re- lated to the voltage deviation on the electrodes. The trap axial frequency is related to the voltage difference between

the endcap and middle segments by the relation u)

z <xJVq.

The deviation of the trap frequency is given by So)/ojz = SV74V since only one electrode is driven. The phase shift of the ion motion is the integral of the frequency deviation over the time of the noise pulse Tn

, S(f>=j "Sa) z dt = jl"a>,(SV/4V )dt. Typically 7n =10 lis. We cannot di- rectly measure the voltage on the trap electrodes since some

of the filters between our noise generator and the electrodes

are within the vacuum enclosure. However, on a mock up of

the trap electrodes and the associated filters, we measured the voltage deviations with an integrating circuit. These measurements verified that the phase deviations are Gaussian

053807-12

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DECOHERENCE AND DECAY OF MOTIONAL QUANTUM PHYSICAL REVIEW A 62 053807

1.6 4- !

I

1.4 -

1.2 ^^^^L

1.0 ~**>*******^

0.8

.^-""""^ -

0.6 — ^^-*"'"^ —

0.4 -

0.2 %***^* —

0.0 + I I +

0.00 0.05 0.10

t[ms]

0.15 0.20

FIG. 8. (Color) Heating of the motional state. At t = the ion is prepared in a state near the ground

state |0). As time passes, the ion

heats out of the ground state, as measured by the energy in the motional state, which is proportional

to the average occupation number

distributed, with a width that varies with the applied noise

voltage, and that negligible voltage noise is added at the trap secular frequency. In separate experiments, we verified that the change in motional energy was negligible during the application of the phase reservoir.

3. Natural reservoir The third type of reservoir is not engineered, but is the ambient, or "natural" noise in our trap. We have previously

observed heating in miniaturized Paul traps [26]. We have

further determined that this ambient noise is dominated by

uniform field fluctuations rather than field-gradient fluctua- tions [27]. We do not currently have an explanation of the heating mechanism. However, thermal or Johnson noise,

background gas collisions, and several other sources are too small to explain the heating rate. We do know that this ambient noise source adds energy to the ion (amplitude noise),

and that the state of the ion evolves into a thermal state with

steadily increasing temperature. The suspected cause is fluc- tuating patch potentials at the trap electrodes, a problem that

is exacerbated by the small size of our trap [26]. To observe

the effects of this heating on quantum coherences, we simply

run the Ramsey interferometer experiment with varying time

between the Ramsey pulses.

We can also observe the effect of the heating on the en- ergy of the motional state. We find that due to our unknown

environmental source of noisy electric fields, the ion heats from the ground state at a rate linear in time with d(n)ldt

~5.7±1 quanta per ms. Shown in Fig. 8 are data for the mean quantum number in the motion as a function of delay

time. That is, the ground state is prepared and the average quantum number is measured after a fixed delay time t. At

each point the motional state is thermal as determined by the method of Refs. [17] and [26].

4. Zero-temperature reservoir The fourth type of engineered reservoir was a zero- temperature reservoir realized by laser cooling. Some theoretical results for this reservoir were mentioned briefly in

Sec. II A la. The experimental scheme is outlined in Fig. 9 and follows a proposal by Poyatos et al. [28]. The goal is to simulate decay of the states ||)|n) into a T=0 reservoir. To

do this, we simultaneously drive on the first red sideband

transitions ||)|n)—> |T)I« — 1) at a Rabi frequency Orsb and

the red-Doppler (Dl from Fig. 5) from |f) to the P v2 l evel at a rate Cl D (<T). This is effectively a continuous Raman

FIG. 9. Engineering a T=0 reservoir with laser cooling: sche- matic diagram of the levels. The top panel shows the full three-

(internal) level system (dressed by the motional states), while the bottom panel shows the equivalent two-level system. The red Doppler beam with resonant Rabi frequency Cl D

is used to give the |T)

state an effective linewidth y=yeff=Q,

2D

/r. Simultaneously the red sideband (rsb) is driven coherently at rate fi^ to drive population

out of the | J.) state. The sum of the two applied fields simulates a variable bandwidth zero temperature reservoir.

053807-13

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Q. A. TURCHETTE et al. PHYSICAL REVIEW A 62 053807

cooling process (in contrast to the stepwise Raman cooling

process described at the end of Sec. Ill B) which eventually

relaxes the system to a state in which all population is in the ground state |0)||) [29,30]. The |f ) level is given an effective decay rate y^—D^lT . We begin by preparing an initial

state | j)(|0) + |2))/\/2. The point of the experiment is to follow the time evolution of the coherence of this state

(Poj,2| as we ' arjel the relevant off-diagonal element of the density matrix) as it decoheres during the cooling process. To gain a qualitative understanding of the experiment we can

consider a reduced problem, as follows. We are simulating a 7=0 reservoir by simultaneously

driving (resonantly) on the red sideband and red-Doppler

transitions. The red-Doppler drive gives the ||) state an ef- fective linewidth yeff (see Fig. 9). The system density matrix

obeys the following time evolution:

7eff p = i[p,H] + —[2a_pcr+ -(T + (T_p-p(T + (T_] (74)

Consider two limiting cases. In the first g>yeff, so that \ ± have an imaginary part. Here the time evolution of the co- herence is an exponentially decaying cosine, reflecting the

fact that the coherent drive is faster than the decay rate:

e ~ W4 Pm i, oi (')-» —~—cos( Vmgr). (82)

In the second case g<ye ff, so that X ± are strictly real. In

this case,

Pml.Ol (0-

1 exp 2g l m 1 YeS

(83)

and we have simple exponential decay of the off-diagonal

density-matrix element. It is instructive to compare this to the T=0 limit of the amplitude reservoir presented in Sec.

II A la. From Eq. (11), with pom (0)=l/2

with interaction-picture Hamiltonian

H = g(a'a- + aa+ ),

1

(75)

where g — il Tsb is the Rabi rate of the red sideband drive (assumed real) and a+ (<x_) is the raising (lowering) operator of the atom. Here we have neglected the effects of recoil heating, valid for rj <\. This is simply the master equation

for a driven, decaying two-state atom dressed by coupling to a harmonic-oscillator system [familiar to cavity QED as the Jaynes-Cummings model [31] with atomic (but no cavity)

decay]. We consider the experiment in which we make an

initial state

<Ao = ll>(|0)4»)/V2 (76)

and we measure the coherence that we call pm j,oi • Equation

(74) leads to:

PmlM~~ l ^mSPm-\},0[

. I— Teff Pm-lT,0j- -

'V>HSPmi,01 _ —Pm -l|,01

(77)

(78)

The solution to this set of differential equations [with initial conditions specified by Eq. (76)] is

lyjmg pm -,r.oi- 2(x + _ x

[e i+t-^-n

(79)

1 Pml.01

:

with

2(X + -X_) K + + f)e^M + fU

Yeff . 1 /Teff 4 ~2 4mg-

(80)

(81)

P0r, 7

2

-expi-TM-* (84)

With the identification y/2—>2g 2 /yef{ , Eqs. (83) and (84) are

identical. This is the anticipated result, since the effective linewidth of the |f ) state is simply the square of the Rabi

frequency divided by the bare decay rate, and in the case of

the red sideband the Rabi frequency scales as g 4m.

IV. DATA

To reiterate, the experiments consist of a Ramsey type interferometer experiment in which the ion is coupled to one

of the engineered reservoirs between the initial and final "beam splitter" pulses. The experiment proceeds as follows. The ion is first Doppler cooled and then Raman cooled, pre- paring the ion in the ||)|0) state. The state preparation

pulses are then applied to generate the desired superposition

(first beam splitter). A period of time elapses during which

the ion is coupled to the reservoir. After this time, we reverse

the sequence of pulses used in state preparation (second

beam splitter). The resulting internal state of the ion depends on the frequency and phase of the oscillators used in the preparation and reversal pulse sequences. The ion internal

state is detected as described in Sec. Ill B. The contrast of

the resulting Ramsey fringes is a measure of the motional

state coherence, which decreases with increasing interaction with the reservoir, either in time or in strength (applied volt- age noise).

A. Schrodinger-cat decoherence

1. Cat states, applied amplitude noise The effect of coupling the Schrodinger-cat interferometer

to the engineered amplitude reservoir is shown in Fig. 10. The reservoir is simulated by driving particular trap elec- trodes with a noisy potential V near the trap secular fre- quency as discussed in Sec. Ill D 1. The amplitude reservoir

053807-14

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DECOHERENCE AND DECAY OF MOTIONAL QUANTUM PHYSICAL REVIEW A 62 053807

10" *V

*

'

. "sSo o\ * \0

o

*

N. D

>t *

N. V • |ta| -1.0 o |Aa|-2.1 o |Aa| - 2.6

. |Aa|-3.1 N » V |Aa| = 4.1 < |Aa| - 5.2

' ' ' '

N. Q

1 N 1

> i

10"

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

lAol2^) [volts3

!

FIG. 10. Decay of Schrodinger-cat state coherences with applied

amplitude noise. Note the universal scaling of the states to an exponential with decay constant proportional to |Aa| 2

. The reservoir was applied for 3 /xs.

10"

1 I 1 1 • 1

*^\»

|Aa| = 1.4 " ° |Aa| = 2.0

|Aa| = 2.5

o |Aa| = 3.1 9k

ov

1 1 1 1

o \

1 a—i 100 300 400

lAaft&is]

500

FIG. 11. Decay of Schrodinger-cat state coherences in natural amplitude noise. Note again the universal scaling of the states to an

exponential with decay constant proportional to |Aa| 2

, and that the decay constant is the same as that derived from the heating data of

Fig. 8. This is discussed further in the text.

is coupled to the ion for 3 /ms between the beam splitter pulses of the interferometer. The variance <r2 of j3 (Sec.

II B 2a) is proportional to the mean-squared voltage noise (V2

). Figure 10 shows a plot of the interference fringe contrast as a function of the applied mean-squared voltage, scaled by the squared "size" of the cat state |Aa| 2

. Decay

curves were recorded for a variety of superposition sizes |Aa|, and all the data agree with a single exponential, as in Eq. (33). The initial contrast for each value of |Aa| is nor- malized to unity at (V2

) = 0.

2. Cat states, natural noise The natural noise that gives rise to the heating described

in Sec. Ill D 3 causes decoherence. For this experiment, we

simply wait for some amount of time between the Ramsey

zones and measure the fringe contrast as a function of this time. The results are shown in Fig. 11. As in the previous

case, the results are consistent with a single exponential. This

time, the decay constant has meaning relative to another ex- perimental quantity: it should be the same as that measured

for the heating. For example, the mean quantum number in the system is given by Eq. (8), from which y is interpreted as

the heating rate of Sec. Ill D 3. The decay constant relevant

to the decoherence measured in Fig. 11 is given by Eq. (73), with the same coupling constant y appearing. The measured

numbers for y agree reasonably well. We find a decay rate of

y/2tT = 5.7± 1 quanta/ms from the heating data (Fig. 8) and

y/2Tr=7.5 ±0.7 quanta/ms from the decoherence data (Fig.

11). Discrepancies can easily arise due to variations in heat- ing rate from day to day and other systematic errors that are not included in the error quoted on 7, such as miscalibration of the size of the coherent states A a. In addition, there is some spin decoherence during the 80 /xs of Ramsey time used for the longest data points (see the discussion of Sec.

II C). The contrast is normalized to unity at f = 0.

3. Cat states, phase noise

Results from coupling of the Schrodinger-cat interferometer to the engineered phase reservoir are shown in Fig. 12. The theory has only one free parameter, which is a scaling to convert voltage noise to phase deviation (see below for a discussion of this). The theory curves are from Eq. (44), all with the same scaling parameter. Note that there is no simple

universal scaling law for the functional form of the decoher- ence as there is in the case of Schrodinger cats subject to amplitude noise. The contrasts at (

V

2

) = are normalized to

unity. The phase noise reservoir is applied for —10 fis.

0.02 0.04 0.06 0.08 0.1 (V

2

) [volts2

]

FIG. 12. Decay of Schrodinger-cat state coherences in an applied phase noise reservoir. Several sizes of cats are shown. The

reservoir was applied for 20 /xs.

053807-15

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Q. A. TURCHETTE et al. PHYSICAL REVIEW A 62 053807

IU 1

1

Q

*o

V

|o> + |i) : |0>+|2)

|0) + |3) |1>+|2>

\o • v\ a

a

10-1

I

V

oV

a

•

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

(An)

2 <V2

> [volts2]

FIG. 13. Decay of Fock state coherences in applied phase noise. Note the universal scaling of the states to an exponential with decay

constant proportional to | An| 2. The reservoir was applied for 10 ixs.

B. Decoherence of superpositions of Fock states and decay of Fock states

1. Fock states, phase noise Coupling the Fock state interferometer to phase noise results in the simple behavior predicted by Eq. (21). This is confirmed in the data of Fig. 13 in which the fringe contrast of the interferometer is measured for different values of the applied phase noise reservoir. The data for different initial Fock state superpositions are scaled by the squared "size"

(An) 2 of the superposition in order to show the universal

scaling law of Eq. (21). The scaling of the x axis of the graph

is a free parameter in the theory. In principle, the size of the noise measured in radians could be known from the variance of the applied noise (V2

). However, in practice, unknown

geometrical factors in converting voltage on a subset of the

trap electrodes to change in trap secular frequency, prevents

a direct comparison. A rough estimate, nonetheless, gives a correspondence of about tt radians of phase noise at 0.1 V

2

, which is in rough agreement with the fit parameters from the

data. The voltage noise was applied for a 10 //.s interval.

2. Fock states, amplitude noise: decoherence and decay

The coupling of superpositions of two Fock states to an

amplitude reservoir is particularly relevant to quantum logic

with trapped ions: Fock state superpositions are generated

during quantum logic gates, and the natural decoherence is due to coupling to a hot thermal amplitude reservoir. Decoherence of superpositions of Fock states and heating (decay) of Fock states limit the fidelity of quantum operations.

a. Decoherence. To simplify analysis, we restricted our

attention to superpositions of the form (\0) + \n))l \J2 in which Eq. (38) in Sec. II B 2b applies to a good approximation. Data for n= 1,2, and 3 are shown in Fig. 14, where the amplitude reservoir is the natural noise on the trap electrodes discussed in Sec. Ill D 3. Using Eq. (38) the fits Fn for states (|0) + |n))/>/2 are given by Fn = exp(-yst)/(l+sn

t) ]+n with

\ \*^\

• |0)+|1> o |0> + |2> |0) + |3)

\°

'

o Vo

D

*\^

^vj

50 150

t[ns] 250

FIG. 14. Decay of Fock state coherences in natural amplitude

noise. The theory curves are to the functional form of Eq. (38) with an additional exponential decay to account for spin decoherence. The horizontal scaling is forced to be the same for the three states, and the data are normalized to unit contrast at f = 0.

sn the scaling factor forced to be the same for all n. The

exponential decay accounts for the loss of spin coherence

due, for example, to fluctuating magnetic fields, with ys /27r ~0.01 /xs~ ]

. The best overall fit to the data gives s„ = 5/ms,

which is consistent with the decay rates quoted in Sec. IV A 2.

b. Decay of populations; purity and entropy. In order to study this case more completely, we performed measure- ments directly on the populations (diagonal elements of the system density matrix). Starting from a Fock state \n), we

applied the amplitude reservoir and observed the evolution of the system. The procedure was as follows. We generated

the state \n) [17], waited a time r, during which the Fock

state \n) would evolve to a distribution over Fock states {|n')}. We then drove transitions on the blue sideband (bsb) for various Raman probe times t p

. Following this, P ^ was

measured from the fluorescence signal. The resulting signal has contributions from all possible transitions ||)|«')

—If )\n' + 1), each of which is at a well-defined frequency.

The bsb curve thus generated is described by:

Pl(tp )=\ + 1L \pn[cos{2ajtp)e-ynh-] (85)

with the Rabi frequency given by

n„ = |</i|e- ,''(fl+at)

|ii + l>l = ^o«"^y2('»+l)"

(86) where C are Laguerre polynomials. The phenomenological

decay rate yn — — (n+ l)°'

7 yo is used to model decay in the data due to Raman beam intensity fluctuations and other sources [17]. We perform a singular value decomposition

(svd) [32] on the data to extract the populations Pn of the various |«)-states at the end of the reservoir application time

t. The time evolution of the populations is shown in Fig. 15

053807-16

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DECOHERENCE AND DECAY OF MOTIONAL QUANTUM . . PHYSICAL REVIEW A 62 053807

0.15

(V2

) [volts2

]

FIG. 15. (Color) Decay of Fock states in a hot amplitude reservoir. Each graph shows the time evolution of the populations P„ in state \n) of the motional state for an experiment starting in the initial state i//(0) indicated. [The initial states are not perfectly generated, as can be seen by P„(0).] The populations are given in Eq. (10), from which the solid lines are derived.

for different initial starting states t/f(0) = \n ). The theory curves are given in Eq. (10) which describe the evolution of

the diagonals of the density matrix. From these data, the expected qualitative effects are evident. The theory for all the data of Fig. 15 has only a single free parameter, which is the scaling of the x axis. Additionally, we use the svd populations from the initial experimental state for the initial state of

the calculations [p nn (0) from Eq. (10)]. An interesting way to further quantify the progression to

thermal equilibrium of the system is via the purity and en- tropy of the (mixed) states. The time evolution of the purity

is shown in Fig. 16 and of the entropy in Fig. 17. It should be noted that we use the terms entropy and purity in an approxi- mate sense: the svd is in practice only useful to n=»4, so we

assign a cutoff to the calculations on the data. Additionally,

only the diagonal elements P„ — pnn are used. This is valid for initial Fock states and thermal states (to which the ground

state evolves under the amplitude reservoir) since the coher053807-17

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Q. A. TURCHETTE et al. PHYSICAL REVIEW A 62 053807

FIG. 16. Evolution of purity for initial Fock

states in hot amplitude reservoir. Each graph shows the time evolution of the purity of the motional state for an experiment starting in the initial state indicated. The purity is defined in Eq.

(87), from which the solid curves are derived. The dotted lines are an untruncated theory, whereas the solid lines are truncated at a maxi- mum value of n = n„„Y

.

ences are absent. However, imperfections in state preparation could lead to unwanted coherences in the experimental

states. In practice, we assume that such effects are small, but we have not experimentally verified this assumption. Some

of the discrepancies between theory and data may result from

this. The definitions under these constraints are, for the purity,

theory that is compared to the data is also truncated at n max . This has only a minor effect on the purity calculation, but does change the qualitative result of the entropy, which for a

limited basis set saturates much more quickly and to a lower

value than it otherwise would. The predicted entropy and

purity curves (with no truncation, that is, as n max

—><x>) are shown as the dotted lines in their respective figures.

H0-Tr P

2 (r)-2 P

2

n (t)

n =

(87)

and for the entropy,

n max 5(0=Trp(/)lnp(r)-2 Pn (t)]nPn (t), (88)

n =

where « max is the value of n at which the series is truncated. As stated above, since preparation of individual Fock states

is not perfect, it is necessary to take into account the imperfections in the initial-state populations, especially as the initial state gets large. To match the data to the theory, the

initial diagonal elements of the density matrix are measured

and used as an initial condition in the theory curves. The

C. Zero-temperature reservoir

a. Decoherence. Here we discuss the effect of the variable-bandwidth zero-temperature reservoir presented in Sec. Ill D 4 on a superposition of Fock states. As before, we

prepare the superposition state, apply the reservoir, and undo

the state preparation forming a Ramsey interferometer to measure the coherence. In this case, the only state considered was the superposition state ||}[|0) + |2)]/>/2.

We distinguish the two limiting cases discussed in Sec.

Ill D 4. In the first, the effective decay rate yeff is much

faster than the coherent Rabi rate nrsb . This regime leads to exponential decay of the coherence and is a simulation of a broad bandwidth T-0 reservoir. In the other, the effective decay rate is slow compared to the coherent Rabi rate. Here,

the system is driven coherently before appreciable decay sets

053807-18

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DECOHERENCE AND DECAY OF MOTIONAL QUANTUM PHYSICAL REVIEW A 62 053807

FIG. 17. Evolution of entropy for initial Fock

states in hot amplitude reservoir. Each graph shows the time evolution of the entropy of the motional state for an experiment starting in the

initial state indicated. The entropy is defined in Eq. (88), from which the solid curves are derived. The dotted lines are an untruncated theory, whereas the solid lines are truncated at a maxi- mum value of n = n„„ .

in. Thus the coherent drive [on the red side band (rsb)] is capable of completing several Rabi oscillations of the spin.

The evolution of fringe contrast shows that the coherence

goes to zero (as a tt pulse transfers most of the population

out of 1 1 ) 1 2

) into 1 1 ) 1 1 ) and therefore destroys all coherence

between |2) and |0)) but then comes back (with opposite

sign) as a reservoir pulse with area greater than it is applied.

During this time, the coherence is decaying due to dissipation on ||). The evolution between the two regimes is shown

in Fig. 18. Note that the first graph has almost no decay. Here the red Doppler was absent (yeff«nD = 0) so that the

residual decay is dominated by natural heating and Rabi fre- quency fluctuations. In the subsequent graphs, either the strength of the red Doppler laser, or the Rabi frequency of

the rsb drive, or both were changed in such a way as to increase the ratio of dissipation to coherent drive, ye ff^rsb- The experiment has an additional parameter that must be

accounted for. The application of the red Doppler beam

causes a Stark shift of the level ||). In the data of Fig. 18 this effect was corrected for during the experiment simply by

changing the detuning of the Raman beams to compensate

for the level shift; this must be done anew for each value of nrsb . It is more convenient to perform the experiment not

compensating for the Stark shift A. The theory for this is only slightly more complicated than that associated with Eq.

(80), which can be generalized to:

1 PmL0i =yexp(-yefff/4) aRt Cl Rt cos- n,

sinA = A +

'7eff

n«=V4me 2 +A2

.

, (89)

(90)

(91) A progression of increasing ye ff^/? with Stark-shift detunings is shown in Fig. 19.

b. Decay. For completeness, we also consider the decay

subject to the zero temperature reservoir of the diagonals of

the density matrix (with no detunings, A = 0). We measure

the time evolution of these populations starting from an initial state of ip(0) = \ l)\2) . The predicted time evolution is given by Eq. (12). The data are shown in Fig. 20. One will immediately notice the negative "populations." This results from population left in the ||) state after the continuous

053807-19

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Q. A. TURCHETTE et al.

1.

PHYSICAL REVIEW A 62 053807

10 20

tftis]

10 20

t[us]

FIG. 18. Decay of Fock state coherences coupled to an engi- neered 7=0 reservoir. The points

in each graph are the contrast of a Ramsey fringe that measures the coherence between the |0) and |2) motional states as the time t of continuous Raman cooling (7=0

reservoir) is varied. The graphs have progressively larger ratios of r=yeff/s:(a)r=0.23,(b)r=l.l,

(c) r=2.3, and (d) r=8. The

Stark-induced detunings are cor- rected experimentally, that is, the position of the RSB and its drive

laser are made coincident. The

theory curves are fits to Eq. (80), from which r is quoted.

FIG. 19. Decay of Fock state coherences in 7=0 reservoir. Same as Fig. 18, except the Stark- induced detunings are not cor- rected experimentally, but are ac- counted for in the theory curves, which are fits to Eq. (89). From

the fits, yeff/g= (0.1,0.7,1-3,2.7, 2.2) and A/g = (0,0.5,0.4,l,2) for graphs (a)-(e), respectively. See

the text for a discussion.

053807-20

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DECOHERENCE AND DECAY OF MOTIONAL QUANTUM . .

.

PHYSICAL REVIEW A 62 053807

10 30 40

t[us]

30 40

t[us]

fi 0.5

Q.O

Q. $

10 20 30 40

t[jxs]

10 20 30 40

tins]

50

70

70

Q n=0 U

n-1

I> n=2

60 70

70

FIG. 20. (Color) Decay of Fock state populations in a T= reservoir starting from ip(0) = |2). The strength of the red Doppler (which

provides dissipation) is increased starting from the upper graph. Theory curves are from Eq. (12) with a modification to account for residual population left in | T ) at the end of the continuous cooling cycle. See the text for a discussion.

Raman cooling. For weak enough red Doppler power, the optical pumping does not entirely depump the ||) state. This

appears as a negative contribution to the flopping state analysis seen in the first two panels of Fig. 20 [see Eq. (85)].

V. CONCLUSION

We have presented a detailed study of the interaction of

several types of quantum states with several types of engineered reservoirs. The results compare favorably with theoretical predictions, and for several cases, we have confirmed

predicted scalings for a range of experimental conditions. ACKNOWLEDGMENTS

We acknowledge support from the U. S. National Security

Agency, Office of Naval Research and Army Research Office. We thank Chris Langer, Mary Rowe, Mike Lombardi,

and Don Sullivan for critical readings of the manuscript.

053807-21

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Q. A. TURCHETTE el al. PHYSICAL REVIEW A 62 053807

[1] W.H. Zurek, Phys. Today 44, 36 (1991), and references

therein.

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[8] V. Buzek and P. L. Knight in Progress in Optics XXXIV

(Elsevier, Amsterdam, 1995), pp. 1-158.

[9] J.I. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995). [10] C. Monroe, DM. Meekhof, B.E. King, and D.J. Wineland,

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[11] M. Brune, E. Hagley, J. Dreyer, X. Maitre, A. Maali, C. Wunderlich, J.M. Raimond, and S. Haroche, Phys. Rev. Lett. 77, 4887 (1996).

[12] C.J. Myatt, B.E. King, Q.A. Turchette, C.A. Sackett, D. Kielpinski, W.M. Itano, C. Monroe, and D.J. Wineland, Nature (London) 403, 269 (2000). [13] V. Perinova and A. Luks, Phys. Rev. A 41, 414 (1990). [14] M.S. Kim and V. Buzek, Phys. Rev. A 46, 4239 (1992).

[15] C. J. Myatt, B. E. King, Q. A. Turchette, C. A. Sackett, D.

Kielpinski, W. M. Itano, C. Monroe, and D. J. Wineland, in Laser Spectroscopy, XIV International Conference, Innsbruck,

June, 1999, edited by R. Blatt, J. Eschner, D. Leibfried, F. Schmidt-Kaler (World Scientific, Singapore, 1999), pp. 237-

245.

[16] D.J. Wineland, C. Monroe, W.M. Itano, D. Leibfried, B.E. King, and D.M. Meekhof, J. Res. NIST 103, 259 (1998).

[17] D.M. Meekhof, C. Monroe, B.E. King, W.M. Itano, and D.J. Wineland, Phys. Rev. Lett. 76, 1796 (1996).

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Itano, D.J. Wineland, and P. Gould, Phys. Rev. Lett. 75, 4011

(1995).

[19] S. Wallentowitz and W. Vogel, Phys. Rev. A 58, 679 (1998).

[20] W.M. Itano, C. Monroe, D.M. Meekhof, D. Leibfried, B.E. King, and D.J. Wineland, SPIE Proc. 2995, 43 (1997).

[21] M.G. Raizen, J.M. Gilligan, J.C. Bergquist, W.M. Itano, and

D.J. Wineland, Phys. Rev. A 45, 6493 (1992).

[22] S.R. Jefferts, C. Monroe, E.W. Bell, and D.J. Wineland, Phys. Rev. A 51,3112 (1995).

[23] B. E. King, Ph.D. thesis, University of Colorado, 1999 (unpublished). [24] Q.A. Turchette, C.S. Wood, B.E. King, C.J. Myatt, D. Leibfried, W.M. Itano, C. Monroe, and D.J. Wineland, Phys. Rev.

Lett. 81, 3631 (1998).

[25] D.J. Wineland, C. Monroe, W.M. Itano, B.E. King, D. Leibfried, D.M. Meekhof, C. Myatt, and C. Wood, Fortschr. Phys.

46, 363 (1998).

[26] Q. A. Turchette, D. Kielpinski, D. Leibfried, D. M. Meekhof,

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[27] B.E. King, C.S. Wood, C.J. Myatt, Q.A. Turchette, D. Leibfried, W.M. Itano, C. Monroe, and D.J. Wineland, Phys. Rev.

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053807-22

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REPORTS

A Decoherence-Free Quantum

Memory Using Trapped Ions

D. Kielpinski, 1 * V. Meyer, 1 M. A. Rowe,1 C. A. Sackett, 1 W. M. Itano, 1 C. Monroe,2 D. J. Wineland 1

We demonstrate a decoherence-free quantum memory of one qubit. By en- coding the qubit into the decoherence-free subspace (DFS) of a pair of trapped

9Be +

ions, we protect the qubit from environment-induced dephasing that

limits the storage time of a qubit composed of a single ion. We measured the

storage time under ambient conditions and under interaction with an engi- neered noisy environment and observed that encoding into the DFS increases the storage time by up to an order of magnitude. The encoding reversibly transfers an arbitrary qubit stored in a single ion to the DFS of two ions.

IT) -» (Ut> + '1tl»/V2

T4) -* (I Tl ) + i\ IT »/V2 (2)

A quantum memory stores information in superposition states of a collection of twolevel systems called "qubits." Quantum computation, which may provide a substantial speedup in factoring large numbers (7) and in searching databases (2), works by operating on information in the form of such superpositions. Robust quantum memories are there- fore essential to realizing the potential gains of quantum computing (3). However, inter- action of a quantum memory with its envi- ronment destroys the stored information, a process called "decoherence" (4, 5). Many

proposed quantum memories decohere be- cause of an environment that has the same

coupling to each qubit (6-12). Encoding the stored information into a decoherence-free

subspace, or DFS, allows the memory to retain information, even in the presence of

this type of decoherence (10-12). The DFS

states are invariant under the coupling to such

an environment, protecting the encoded "logical qubit" from the decoherence affecting

general superpositions of the "physical

qubits" that make up the full state space. DFSs have been shown to require an asymptotically small overhead for large systems

(10) and to support universal fault-tolerant quantum logic (75, 14). These properties suggest that DFSs will be intrinsic to future quantum computing architectures. Logic

gates on DFS-encoded qubits have been pro- posed in the context of cavity quantum elec- trodynamics (15) and solid state quantum

logic schemes (14). Also, a recent experi- ment has demonstrated the immunity of a DFS of two photons to collective noise (16). Here we demonstrate the immunity of a DFS

of two atoms to collective dephasing and implement a technique for encoding an arbitrary physical qubit state into the DFS.

^ime and Frequency Division, National Institute of Standards and Technology, Boulder, CO 80305, USA. department of Physics, University of Michigan, Ann

Arbor, Ml 48109, USA. To whom correspondence should be addressed. E- mail: davidk@boulder.nist.gov

Our physical qubits are

9Be+

ions confined along the axis of a miniature linear radio-frequency (RF) trap (17). We choose two 9Be+

hyperfine states, denoted ||) and

If), as our physical qubit basis states. We

detect logic states by applying laser light resonant with a

9Be +

cycling transition (18). The detection laser causes ions in ||) to fluoresce, whereas ions in If) fluoresce negligibly, allowing discrimination of logic

states with high efficiency (19). Applying

nonresonant laser beams BR (blue Raman)

and RR (red Raman) with frequency differ- ence a)BR - Wj^ equal to the If) <-» ||) transition frequency w drives stimulated Ra- man transitions between \i) and If). This

"carrier" transition implements rotation of a

single physical qubit, one of the fundamental quantum logic gates (19). The corresponding

evolution is

||)-»cos8||> + e'* sinB |t>

|T)->cos0|f> - e-'* sin ||) (1)

where is proportional to the carrier drive duration and <fj is the phase difference be- tween the BR and RR beams at the position of the ion, referred to as the "ion phase." For two ions, we write the ion phases (}>. for each

ion as 4> p 4>2 (20). The experiments reported here use two

trapped ions. The ions are strongly coupled

by the Coulomb interaction, so that the motion of the ions along the axis decomposes

into symmetric and antisymmetric normal modes at frequencies of 5.0 and 8.8 MHz,

respectively. The wave vector difference be- tween BR and RR lies along the trap axis, so we can drive transitions that couple the inter- nal and motional states of the ions. Driving

these transitions as described in (27), we

perform the two-ion logic gate of Sorensen and Molmer (22), which entangles the two

ions. The evolution under this gate is

III) -> (|II) + 'ITT)VV2

ITT) -> (IU) + <H!»/V2

We can also realize the inverse of this gate by

performing the gate three times in succession. In general, the evolution under the twoion gate depends on the ion phases, but we

choose a phase convention in which cj>, =

<|>2 = during application of the two-ion

gate, yielding the evolution of Eq. 2. This

gate, in combination with single-qubit rotations, implements universal quantum logic

(23), in the sense that these gates suffice to transform any superposition of the states III), Itl). lit), and lit) i nto anv otner

superposition of those states. The DFS realized here is spanned by

k|»_> = (lit) - «IH»/V2 and kK) =

(lit) + 'ltl)VV2, which form the basis

states for our logical qubit. These states are

clearly invariant under collective dephasing; the transformation If) —> e

,? |f), applied simultaneously to both ions, leaves any superposition of \>\i_) and t\i + ) invariant. Such collective dephasing is expected

to be a major source of qubit decoherence

for quantum information processing using

trapped atoms. The encoding method demonstrated below reversibly transfers infor- mation between one physical qubit (one

ion) and one decoherence-free logical

qubit. The encoding works even if we have no information about the initial state of the physical qubit. This fact is essential for the use of our method in quantum information

processing, in which the state to be encoded may be entangled with the state of another system. To demonstrate the general character of

the encoding method, we show that the encoding works for arbitrary states of form

||)(a||> + 6| T»

2 + | 6|2 =1 a,b complex (3) To prepare a state of this form, we first initialize the ions in the logic state |||) by

optical pumping. We then drive the carrier transition of Eq. 1 on both ions simultaneously, once with = 3, and again with = (3 and dp,, shifted by tt. We set <t> 2 = a for both

pulses. The final state has a = cos 2(3, b =

e'a

sin 2(3. In a classical picture of spin with P = tt/8, the first drive takes j j to \v The

second drive reverses the sense of rotation on

ion 1 while keeping it the same on ion 2, so

the second drive takes V* to |—> The net

effect is to rotate ion 2 alone, without changing the state of ion 1 (24). We encode the state of Eq. 3 into the DFS in two steps. First, we apply the inverse of the two-ion gate of Eq. 2, yielding a(lj|) -

* I T T » + 6(llT)-'lTl))- Then, we drive the

carrier with = tt/4, 4>. = tt/2, 4>2 = to

TN-182

www.sciencemag.org SCIENCE VOL 291 9 FEBRUARY 2001 1013
Reports

obtain |v|jdfs) = a |»|> + ) + b | i|;_). The infor- mation stored in the physical qubit of ion 2 is now encoded in the logical qubit of the DFS. In the experiment, we take \a\ = \b\, though our method permits preparation and encoding of any state of the more general form. To read out the encoded information, we

reverse the carrier pulse in the encoding and

apply the two-ion gate of Eq. 2 to decode

I i)i DFS> into \l)(a ||) + b If)). After decoding, we rotate ion 2 as in preparing the state of Eq.

3 but with the phase on ion 2 changed to a'. We then measure the probability P2 of find- ing both ions in | j). P2 varies sinusoidally with a - a', and the magnitude of oscillation

is equal to the coherence C of ion 2 (25). Because we set \a\ = \b\ = \l\/2, ideally C = 1. Departures from C = 1 measure the

effects of both decoherence and imperfect

logic. We verified that C is independent of a, thus showing that our encoding method

works even if we have no information about

the phase of the input state. To study the effects of decoherence on the DFS-encoded state, we leave a fixed delay time between the encoding and decoding

steps and apply an engineered noisy environ- ment for some fraction of the delay time. The

engineered environment consists of an offFig. 1. Decay of the DFS-encoded state (circles) and the test state (crosses) under engineered

dephasing noise. The noise is applied for a fraction of the delay time of —25 u.s between encoding and decoding. Coherence

data are normalized to their val- ues for zero applied noise. The fit lines are exponential decay

curves. The test data are predict- ed to decay exponentially for white noise, so we exclude the

point with zero applied noise from the fit. The DFS data are fit for comparison. The decay rate of the test state is 0.18 ± 0.01

u.s _1

, whereas the decay rate of the DFS state is 0.0035 ± 0.0050

u.s _1

, consistent with zero decay.

Fig. 2. Decay of the DFS-encoded state

(circles) and the test state (crosses) under ambient decoherence. We vary the

delay time between encoding and de- coding to give the ambient noise a vari- able time to act. Coherence data are normalized to their values for zero applied noise. The fit lines are exponential

decay curves for purposes of compari- son and are not theoretical predictions. The decay rate of the test state is (7.9 ± 1.5) X 10 -3

u-s"

1

, whereas the de- cay rate of the DFS state is (2.2 ± 0.3) X 10 -3

u-s -1

. Because the coherence

time of the DFS-encoded state is much

longer than that of the test state, we

see that the chief source of ambient

decoherence is collective dephasing.

resonant laser beam with a randomly varying

intensity. The beam induces a shift of w

through the ac Stark effect, causing the If) component of each ion to acquire a random

phase relative to the ||) component. The two

ions are nearly equally illuminated by the beam, so the random phase is nearly the same

on both ions, leading to collective dephasing.

The DFS state should resist the dephasing

effect of this environment. The coherence of

ion 2 in the test state ||>(| |) + <?'*

If ))fy/%

however, should be sensitive to collective dephasing. We measure the decay of the test state by simply turning off the encoding and

decoding sequences in the procedure used to measure the decay of the DFS-encoded state. We applied decoherence to the test and encoded states during a delay time of —25 u,s (Fig. 1). The coherence without applied noise

is —0.69 for the test state and —0.43 for the encoded state; they depart from 1 because of

imperfect logic gates and detection, due in

part to laser intensity fluctuations and heating of the ions (17, 21). For white-noise intensity fluctuations of the Stark-shifting beam, we

expect C to decay exponentially for the test

state, as shown by the fit line. The small decay rate of the test state between and

2.5-p.s noise duration arises because the inNoise Duration [|is]

Delay Time |^s]

tensity fluctuations of the noise beam have

finite bandwidth (dc to 100 kHz) (6). We

therefore extract the decay rate of the test state by excluding the point with zero applied

noise from the fit. We also fit the coherence

of the DFS state to an exponential decay for comparison. The decay rate of the test state is 0.18 ± 0.01 u,s _1

, whereas the decay rate of

the DFS state is 0.0035 ± 0.0050 \is~\

consistent with zero decay. To investigate the eventual decay of the DFS state, we increased

the delay time to —200 u,s and applied deco- herence for up to 100 u,s during the delay

time. The DFS state coherence dropped by

50% for 100-u.s applied noise relative to its value for the same delay time and no applied

noise, but we think that this decay was not due to collective dephasing. The decay was

Gaussian, rather than exponential, and is consistent with the effect of differential dephasing due to small departures from equal illu- mination by the Stark-shifting beam. We

think that ambient sources of differential dephasing are much weaker than the differential dephasing caused by the Stark-shifting beam.We also measured the storage times of the encoded and test states under ambient conditions in our laboratory (Fig. 2). Here we

measure the coherence as a function of the delay time between encoding and decoding to give the ambient noise a variable time to act, rather than leaving a fixed delay time and

applying noise for some fraction of the delay

time. The coherence data for this case are normalized in the same way as the data with

applied noise. The decoherence of the test state is dominated by ambient fluctuating

magnetic fields whose frequencies lie primarily at 60 Hz and its harmonics. These fields randomly shift w through the Zeeman effect. Because these fields are roughly uniform

across the ion string, they induce collective dephasing similar to that created by the en- gineered environment. We empirically find the decay of both test and encoded states to be roughly exponential, as indicated by the fit lines. The decay rate of the test state is (7.9 ±

1.5) X 10~ 3

jjls -1

, whereas the decay rate of

the DFS state is (2.2 ± 0.3) X 10~ 3 |xs _l

. Although Fig. 1 presents only normalized

contrasts, the data show that the unnormalized contrast of the DFS state is higher than

that of the test state for delay times exceeding

150 jjls. The DFS-encoded state maintains coherence much longer than the test state, so we conclude that collective dephasing from

magnetic field noise is the major ambient

source of decoherence for the test state (26). The loss of coherence of the encoded state is consistent with degradation of the decoding

pulses (19, 22, 27), due to heating of the ion motional state over the delay time (17). We have demonstrated reversible encoding of a qubit stored in one ion into a DFS of

1014 9 FEBRUARY 2001 VOL 291 SCIENCE www.sciencemag.org

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Reports

two ions. The DFS-encoded state can store a qubit at least 10 times as long under applied

noise as a single ion can, and appears immune

to collective dephasing. Under typical ambient conditions, the DFS encoding also improves storage time considerably, showing

that collective dephasing is indeed the limit to quantum memory using our physical qubits. Even without normalizing for the overhead

incurred in encoding and decoding, the en- coded state retains more coherence than the

test state for long storage times in ambient

conditions. The DFS encoding therefore currently provides an improved single-qubit

quantum memory for ion-trap quantum computing applications. The loss of coherence

incurred in encoding and decoding is a drawback to our scheme, but in the future, practical quantum computing will in any case re- quire logic gates of a much higher fidelity than those used in this work. We therefore expect that, once the technical problems of

ion heating and laser fluctuations are solved,

the scheme presented here should be a practical method for long-term storage of qubits with near-perfect fidelity. Our results suggest applications in quan- tum communication and large-scale quantum

computing. Single photons have already been shown to transmit quantum information over

long distances with high fidelity (8, 9), and

the information in a single photon can be mapped onto a single atom (28, 29). With our encoding technique, the quantum information

received by a single ion can be mapped into a DFS for robust storage. Our encoding tech- nique will also be essential in scaling up

ion-trap quantum computers. In one model of

large-scale ion-trap quantum computing (79), qubits reside in a large array of interconnect- ed ion traps. To perform one- or two-qubit

logic gates, the relevant ions are moved into "accumulator" regions where they interact with lasers that drive the gates. One obstacle

to this quantum computing architecture is that the magnetic field strength must be well- characterized across the entire device. Otherwise, the ions will constantly accumulate un- known relative phase during transport, lead- ing to decoherence. Encoding into the DFS

solves this problem, because the phase of a

logical qubit in the DFS does not depend on

the local magnetic field strength as long as

the field strength is the same at each physical

qubit. Thus, we can entangle two logical

qubits, move them far apart, and perform

operations on them in separate accumulators

without losing phase information.

References and Notes

1. P. W. Shor, in Proceedings of the 35th Annual Symposium on Foundations of Computer Science. S. Goldwasser, Ed. (IEEE Computer Society, Los Alamitos, CA,

1994). p. 116.

2. L K. Crover, Phys. Rev. Lett. 79, 325 (1997).

3. A. Steane, paper presented at the Fifth International

Conference on Quantum Communication, Measurement, and Computing, Capri, Italy, 3 to 8 July 2000.

4. W. H. Zurek, Phys. Today 44 (no. 10), 36 (1991).

5. D. Ciulini et a/., Decoherence and the Appearance of

a Classical World in Quantum Theory (Springer, Berlin, 1996).

6. C. M. Palma, K.-A. Suominen, A. K. Ekert. Proc. R. Soc. London Ser. A 452, 567 (1996).

7. P. Zanardi, F. Rossi, Phys. Rev. Lett. 81, 4752 (1998).

8. Fortschr. Phys. 46 (no. 4-8) (1998).

9. Fortschr. Phys. 48 (no. 9-11) (2000).

10. D. A. Lidar, I. L Chuang, K. B. Whaley, Phys. Rev. Lett. 81, 2594 (1998).

11. P. Zanardi, M. Rasetti, Phys. Rev. Lett. 79, 3306

(1997).

12. L M. Duan, C. C Cuo, Phys. Rev. A 57, 737 (1998).

13. D. A. Udar, D. Bacon, K. B. Whaley, Phys. Rev. Lett. 82, 4556 (1999).

14. D. Bacon, J. Kempe, D. A. Lidar, K. B. Whaley, Phys.

Rev. Lett. 85, 1758 (2000).

15. A. Beige, D. Braun, B. Tregenna, P. L Knight, Phys. Rev.

Lett. 85, 1762 (2000).

16. P. C. Kwiat, A. J. Berglund, J. B. Altepeter, A. C. White,

Science 290, 498 (2000).

17. Q. A. Turchette etal., Phys. Rev. A 61, 063418 (2000)

(available at http://publish.aps.org/abstract/PRA/ v61/e063418).

18. The hyperfine states corresponding to I J. > and I f > are the \F = 2, mF = -2) and If = 1, mF = -1)

sublevels of the ground a S 1/2 state, respectively. The

cycling transition is excited by left circularly polarized

light and connects I 1 > to the If = 3, mf = -3)

sublevel of the excited *P3/2 state. 19. D. J. Winetand et at,/ Res. NIST 103, 259 (1998).

20. For two ions, we effect common-mode changes of <f>, and <t> a by changing the phase of the RF synthesizer

that controls the difference frequency cu BR - wRR . By

changing the oscillation frequency of the trap, we can

control the ion spacing precisely, allowing us to make

differential changes to <t> n and 4>2. These two tech- niques combined allow us to control <b, and 4> 3 independently for two ions. 21. C A. Sackett et at.. Nature 404, 256 (2000).

22. A. Sarensen, K. Maimer, Phys. Rev. A 62, 022311

(2000) (available at http://publish.aps.org/abstract/ PRA/v62/e022311).

23. Phys. Rev. Lett. 82, 1971 (1999).

24. Individual optical addressing of the ions can prepare

the desired state with only one carrier pulse [H. C. Nagerl et at, Phys. Rev. A 60, 145 (1998)], but it is technically more challenging in our experiment be- cause our ions are separated by only —2.5 u.m. 25. The decoding and final rotation can give rise to coher- ence in ion 2 even if the state before decoding is completely incoherent We avoid this eventuality by

detuning the RF synthesizer used for the initial and final rotations of ion 2 from the synthesizers used for en- coding and decoding. The phases a and a' then change from shot to shot of the experiment, and the spurious coherences average out over many shots. No trace of

this effect then remains in the data. 26. We can also reduce the dephasing from this mechanism by using the 2 S

1/2 hyperfine sublevels If = 1, mF = 0) and If = 2, mf = 0) to represent the two

levels of a qubit In a separate experiment, we measured the coherence time of superpositions of these two states at a static magnetic field of —1.6 x 10~3 T to be an order of magnitude longer than the coherence time between the qubit basis states used

here. Magnetic field dephasing still limits the coher- ence time between the mf = states through the quadratic Zeeman shift. We expect the DFS encoding

to improve coherence time in this case as well, because the encoding protects against purely collective dephasing to all orders. A practical ion-trap quantum computer will probably use the mf =

states as qubit basis states but will also encode

qubits into the DFS.

27. B. E. King et a/., Phys. Rev. Lett. 81, 1525 (1998).

28. X. Maitre et at, Phys. Rev. Lett. 79, 769 (1997).

29. J. I. Cirac, P. Zoller, H. J. Kimble, H. Mabuchi, Phys.

Rev. Lett. 78, 3221 (1997).

30. This work was supported by the U.S. National Security Agency and the Advanced Research and Devel- opment Activity under contract MOD-71 71.00 and

by the U.S. Office of Naval Research. We thank J. Bergquist and A. Ben-Kish for comments on the manuscript. This paper is a contribution of the National Institute of Standards and Technology and is not subject to U.S. copyright

10 November 2000; accepted 22 December 2000

Published online 5 January 2001; 10.1126/science.1057357

Include this information when citing this paper.

Magnetization Precession by

Hot Spin Injection

W. Weber,1 * S. Riesen, 1 H. C. Siegmann2

As electrons are injected at various energies into ferromagnetic material with their spin polarization vector perpendicular to the axis of the magnetization, we observe precessional motion of the spin polarization on the femtosecond time scale. Because of angular momentum conservation, the magnetization vector must precess as well. We show that spin injection will generate the precessional magnetization reversal in nanosized ferromagnetic bits. At reasonable injected current densities this occurs on the pi- cosecond time scale.

Electrons injected into ferromagnetic material experience exchange coupling to the magnetization and spin-dependent scattering, leading to excitations of the magnetiYaboratorium fur Festkorperphysik. ETH Zurich, CH8093 Zurich, Switzerland. 2 Stanford Linear Accelerator

Center, Stanford University, Stanford, CA 94309, USA. *To whom correspondence should be addressed. E- mail: weber@solid.phys.ethz.ch

zation (1-4). By injecting currents of high

density, these excitations have been observed through the occurrence of spin waves (5-7), permanent changes of the micromagnetic structure (8, 9), or even a reversal of the magnetization (10—12). However, to date, the injection of electrons from a ferromagnetic emitter through nanocontacts occurs continously or in pulses

that are long compared with the relaxation

TN-184

www.sciencemag.org SCIENCE VOL 291 9 FEBRUARY 2001 1015
letters to nature

Experimental violation of a Bell's

inequality with efficient detection

M. A. Rowe , D. Kielpinski , V. Meyer4

, C. A. Sackett'. W. M. Itano*,

C. Monroet & D. J. Wineland'

* Time and Frequency Division, National Institute of Standards and Technology,

Boulder, Colorado 80305, USA

t Department of Physics, University ofMichigan, Ann Arbor, Michigan 48109, USA

Local realism is the idea that objects have definite properties

whether or not they are measured, and that measurements of

these properties are not affected by events taking place sufficiently far away

1

. Einstein, Podolsky and Rosen2 used these reasonable assumptions to conclude that quantum mechanics is incomplete.

Starting in 1965, Bell and others constructed mathematical

inequalities whereby experimental tests could distinguish

between quantum mechanics and local realistic theories1,3" 5. Many experiments1,6"" have since been done that are consistent with quantum mechanics and inconsistent with local realism. But

these conclusions remain the subject of considerable interest and

debate, and experiments are still being refined to overcome

'loopholes' that might allow a local realistic interpretation. Here we have measured correlations in the classical properties of massive entangled particles ( 9Be+ ions): these correlations violate a form of Bell's inequality. Our measured value of the appropriate

Bell's 'signal' is 2.25 ± 0.03, whereas a value of 2 is the maximum

allowed by local realistic theories of nature. In contrast to previous measurements with massive particles, this violation of

Bell's inequality was obtained by use of a complete set of measurements. Moreover, the high detection efficiency of our

apparatus eliminates the so-called 'detection' loophole.

Early experiments to test Bell's inequalities were subject to two

primary, although seemingly implausible, loopholes. The first might be termed the locality or 'lightcone' loophole, in which the

correlations of apparently separate events could result from

unknown subluminal signals propagating between different regions of the apparatus. Aspect 16 has given a brief history of this issue, starting with the experiments of ref. 8 and highlighting the strict relativistic separation between measurements reported by the Innsbruck group'

5

. Similar results have also been reported for the Geneva experiment 1417

. The second loophole is usually referred to as the detection loophole. All experiments up to now have had

detection efficiencies low enough to allow the possibility that the subensemble of detected events agrees with quantum mechanics

even though the entire ensemble satisfies Bell's inequalities. Therefore it must be assumed that the detected events represent the entire ensemble; a fair-sampling hypothesis. Several proposals for closing

this loophole have been made18 "24

; we believe the experiment that we

report here is the first to do so. Another feature of our experiment is that it uses massive particles. A previous test of Bell's inequality was

carried out on protons25

, but the interpretation of the detected

events relied on quantum mechanics, as symmetries valid given

quantum mechanics were used to extrapolate the data to a complete

set of Bell's angles. Here we do not make such assumptions.

A Bell measurement of the type suggested by Clauser, Home,

Shimony and Holt5 (CHSH) consists of three basic ingredients

(Fig. la). First is the preparation of a pair of particles in a repeatable

starting configuration (the output of the 'magic' box in Fig. la). Second, a variable classical manipulation is applied independently

to each particle; these manipulations are labelled </>, and <t> 2

. Finally, in the detection phase, a classical property with two possible outcomes is measured for each of the particles. The correlation of

these outcomes

<?(0i,</>2) = Wsam(,(0,,4>2 )-Nd ,(0, N. NA (1)

is measured by repeating the experiment many times. Here Nsame and Ndiffcrent are the number of measurements where the two results were the same and different, respectively. The CHSH form of Bell's inequalities states that the correlations resulting from local realistic theories must obey:

B(a„51 ,^2 ,72)=|q(51 ,72 )-q(a1 ,72)| + \q(8 ] ,02 ) + q(al ,(32 )\^2 (2) where c^ and 5j (j32 and y2 ) are specific values of 0! (<j>2). For

example, in a photon experiment 15

, parametric down-conversion

prepares a pair of photons in a singlet Einstein-Podolsky-Rosen

(EPR) pair. After this, a variable rotation of the photon polarization

is applied to each photon. Finally, the photons' polarization states, vertical or horizontal, are determined. Our experiment prepares a pair of two-level atomic ions in a repeatable configuration (entangled state). Next, a laser field is applied to the particles; the classical manipulation variables are the

Classical V

r Manipulation

1 ^

A

Measure 1

"Magic" box

J V >

\ (V

1

A

\

Manipulation 2

(<M i> Measure 2

j

| Classical Detection laser

Cy> ^i

Figure 1 Illustration of how Bell's inequality experiments work. The idea is that a 'magic

box' emits a pair of particles. We attempt to determine the joint properties of these

particles by applying various classical manipulations to them and observing the

correlations of the measurement outputs, a, A general CHSH type of Bell's inequality experiment, b, Our experiment. The manipulation is a laser wave applied with phases <£, and 02 1° i° n 1 anu ion 2 respectively. The measurement is the detection of photons emanating from the ions upon application of a detection laser. Two possible measurement

outcomes are possible, detection of few photons (as depicted for ion 1 in the figure) or the

detection of many photons (as depicted for ion 2 in the figure).

NATURE | VOL 409 1 15 FEBRUARY 2001 1 www.nature.com filt ©2001 Macmillan Magazines Ltd TN-185 791
letters to nature

phases of this field at each ion's position. Finally, upon application of a detection laser beam, the classical property measured is the number of scattered photons emanating from the particles (which

effectively measures their atomic states). Figure lb shows how our

experiment maps onto the general case. Entangled atoms produced

in the context of cavity-quantum-electrodynamics26 could similarly be used to measure Bell's inequalities. The experimental apparatus is as described in ref. 27. Two 9 Be

+

ions are confined along the axis of a linear Paul trap with an axial centre-of-mass frequency of 5 MHz. We select two resolved levels of the 2S]/2 ground state, \\)=\F = 2,mF = —2) and If)—

\F = 1, mF = — l), where F and mF are the quantum numbers of

the total angular momentum. These states are coupled by a coherent

stimulated Raman transition. The two laser beams used to drive the

transition have a wavelength of 313 nm and a difference frequency

near the hyperfine splitting of the states, w = 27r X 1.25 GHz. The

beams are aligned perpendicular to each other, with their difference wavevector Ak along the trap axis. As described in ref. 27, it is possible in this configuration to produce the entangled state

\h) = ~y= (ITT>-|U» (3)

The fidelity F = (i/'jIpIi^,), where pis the density matrix for the state we make, was about 88% for the data runs. In the discussion below we assume |i/<2) as the starting condition for the experiment.

After making the state \\j/ 2), we again apply Raman beams for a pulse of short duration (~400ns) so that the state of each ion; is

one bright

n

I I T

40 80 120 160 Number of photons

two bright

40 80 120 Number of photons160

Figure 2 Typical data histograms comprising the detection measurements of 20,000

experiments taking a total time of about 20 s. In each experiment the population in the IT)

state is first coherently transferred to the | F = 1,Mf = +1)to make iteven less likely to fluoresce upon application of the detection laser. The detection laser is turned on and the number of fluorescence photons detected by the phototube in 1 ms is recorded. The cut between the one bright and two bright cases is made so that the fractions of two equal

distributions which extend past the cut points are equal. The vertical arrows indicate the

location of the cut between the (1) bright and 1 (2) bright peaks at 25 (86) counts.

a, Data histogram with a negative correlation using 0, = 3vr/8 and <j> 2 — 3tt/8. For these data A/ = 2,200, /V, = 15,500 and N2 = 2,300. b, Data histogram with a

positive correlation using <t>, = 37r/8 and <j> 2 = - tt/8. For these data A/ = 7,700,

W, = 4,400 and N2 = 7,900. The zero bright peak extends vertically to 2,551

.

transformed in the interaction picture as

| T;)—4= (l !,>-fe

- *li,-»; I J,) U| l;>-"^||,-» (4) V 2 V 2 The phase, <f>p is the phase of the field driving the Raman transitions (more specifically, the phase difference between the two Raman

beams) at the position of ion; and corresponds to the inputs 0, and

<f> 2 in Fig. 1. We set this phase in two ways in the experiment. First, as an ion is moved along the trap axis this phase changes by Ak-Axj. For example, a translation of X/y/2 along the trap axis corresponds

to a phase shift of 2?r. In addition, the laser phase on both ions

is changed by a common amount by varying the phase, <£5

, of the radio-frequency synthesizer that determines the Raman difference frequency. The phase on ion j is therefore 4 = </> s + Ak-Xj (5)

In the experiment, the axial trap strength is changed so that the ions move about the centre of the trap symmetrically, giving Ax, = — Ax2

. Therefore the trap strength controls the differential phase, A</> = <j>

l — <j> 2 = Ak(x, — x2 ), and the synthesizer controls

the total phase, <t>,ot = </>,+ 4>2 = 2<£ s. The calibration of these

relations is discussed in the Methods. The state of an ion, |J) or ||), is determined by probing the ion with

circularly polarized light from a 'detection' laser beam27

. During this detection pulse, ions in the |1) or bright state scatter many photons,

and on average about 64 of these are detected with a photomultiplier tube, while ions in the ||) or dark state scatter very few photons. For two ions, three cases can occur: zero ions bright, one ion bright, or two ions bright. In the one-ion-bright case it is not necessary to know which ion is bright because the Bell's measurement requires

only knowledge of whether or not the ions' states are different. Figure 2 shows histograms, each with 20,000 detection measurements. The three cases are distinguished from each other with

simple discriminator levels in the number of photons collected with

the phototube.

An alternative description of our experiment can be made in the language of spin-one-half magnetic moments in a magnetic field (directed in the z direction). The dynamics of the spin system are the same as for our two-level system

28

. Combining the manipulation

(equation (4)) and measurement steps, we effectively measure the spin projection of each ion; in the r

7 direction, where the vector fj is in the x — y plane at an angle <j>j to the y axis. Although we have used quantum-mechanical language to describe the manipulation and measurement steps, we emphasize that both are procedures completely analogous to the classical rotations of wave-plates and measurements of polarization in an optical apparatus. Here we calculate the quantum-mechanical prediction for the

correlation function. Our manipulation step transforms the starting

Table 1 The four sets of phase angles used for the Bell's experiment

Experiment input *1 <t>2 A0 <t>ia

a,02 -jr/8 -jr/8 -ir/4

<*172 -tt/8 3tt/8 -t/2 +ir/4

6,ft> 3t/8 -tt/8 + 7T/2 + 7T/4 6,72 3tt/8 3*78 +3tt/4

Table 2 Correlation values and resulting Bell's signals for five experimental

runs

Run number q(a,,/32) Q(ai, 72) 9(6,. 2) 9(«i. 72> B[a ,,6,, ft, 72)

1 0.541 0.539 0.569 -0.573 2.222

2 0.575 0.570 0.530 -0.600 2.275

3 0.551 0.634 0.590 -0.487 2.262

4 0.575 0.561 0.559 -0.551 2.246

5 0.541 0.596 0.537 -0.571 2.245

The experimental angle values were a, = - (?r/8). 6, = 3!r/8. 2 = - (t/8). and >2 = 3i/8 The

statistical errors are 0.006 and 01 2 lor the q and S values respectively. The systematic errors (see

text) are 03 and 0.06 lor the q and S values respectively.

792 TN-186 £8 ©2001 Macmillan Magazines Ltd NATURE | VOL 409 1 15 FEBRUARY 2001 | www.nature.com
letters to nature

state, \\[/ 2 )> to m=-4={d + eiWl+*2) )(| T T>- e-'

( *' +02)

| I D)

2%/2

»"(1 i(0i+*2 ) )(e~'*2 IT i> + e-'*'|l T»} (6)

Using the measurement operators Nsamc = Ntot [| f T)(T T 1+

1 1 i)(l 1 1] and Ndifferent = Ntot [| T IXT 1 1 + li TXi T IL the correlation function is calculated to be

q(0„02 ) = i[2|l+e"'(0I+02)|2 211 _ i(*l+*2)| 2

] = COS(0, + </> 2 )

(7) The CHSH inequality (equation (2)) is maximally violated by

quantum mechanics at certain sets of phase angles. One such set is a, = -(tt/8), 5, = 3ir/8, j32 = -(tt/8) and y2 = 3tt/8. With these phase angles quantum mechanics predicts

B -

IT 37T It 37T = 2V2 (8)

This violates the local realism condition, which requires that £=£2.

The correlation function is measured experimentally at four

sets of phase angles, listed in Table 1 . The experiment is repeated

JVtol = 20,000 times at each of the four sets of phases. For each set of

phases the correlation function is calculated using

(N + N2 ) - N,

q=

* —- (9)

"'tot Here N , Nx and N2 are the number of events with zero, one and two

ions bright, respectively. The correlation values from the four sets of phase angles are combined into the Bell's signal, B(a 1 ,5,,^2 ,72 )> using equation (2). The correlation values and resulting Bell's signals from five data runs are given in Table 2. So far we have described the experiment in terms of perfect

implementation of the phase angles. In the actual experiment,

however, au 5,, B2 and y2 are not quite the same angles both

times they occur in the Bell's inequality. In our experiment the dominant reason for this error results from the phase instability of

the synthesizer, which can cause the angles to drift appreciably

during four minutes, the time required to take a complete set of measurements. This random drift causes a root-mean-squared error

for the correlation function of ±0.03 on this timescale, which

propagates to an error of ±0.06 for the Bell's signal. The error for the Bell's signal from the five combined data sets is then ±0.03,

consistent with the run-to-run variation observed. Averaging the

five Bell's signals from Table 2, we arrive at our experimental result, which is

B

IT 3w IT 37T = 2.25 ± 0.03 (10)

If we take into account the imperfections of our experiment

(imperfect state fidelity, manipulations, and detection), this value

agrees with the prediction of quantum mechanics. The result above was obtained using the outcomes of every

experiment, so that no fair-sampling hypothesis is required. In

this case, the issue of detection efficiency is replaced by detection

accuracy. The dominant cause of inaccuracy in our state detection comes from the bright state becoming dark because of optical

pumping effects. For example, imperfect circular polarization of

the detection light allows an ion in the |1) state to be pumped to ||), resulting in fewer collected photons from a bright ion. Because of such errors, a bright ion is misidentified 2% of the time as being

dark. This imperfect detection accuracy decreases the magnitude of

the measured correlations. We estimate that our Bell's signal would

be 2.37 with perfect detection accuracy. We have thus presented experimental results of a Bell's inequality

measurement where a measurement outcome was recorded for every experiment. Our detection efficiency was high enough for a

Bell's inequality to be violated without requiring the assumption of

fair sampling, thereby closing the detection loophole in this experi- ment. The ions were separated by a distance large enough that no known interaction could affect the results; however, the lightcone

loophole remains open here. Further details of this experiment will be published elsewhere.

Methods

Phase calibration The experiment was run with specific phase differences of the Raman laser beam fields at each ion. In order to implement a complete set of laser phases, a calibration of the phase on

each ion as a function of axial trap strength was made. We emphasize that the calibration method is classical in nature. Although quantum mechanics guided the choice of

calibration method, no quantum mechanics was used to interpret the signal. General arguments are used to describe the signal resulting from a sequence of laser pulses and its dependence on the classical physical parameters of the system, the laser phase at the ion, and the ion's position.

In the calibration procedure, a Ramsey experiment was performed on two ions. The first 7r/2 Rabi rotation was performed identically each time. The laser phases at the ions' positions for the second tt/2 Rabi rotation were varied, </>[ for ion 1 and 2 f° r 'on 2. The

detection signal is the total number of photons counted during detection. With an

auxiliary one-ion experiment we first established empirically that the individual signal depends only on the laser phase at an individual ion and is C + Acos0r Here C and A are the offset and amplitude of the one-ion signal. We measure the detector to be linear, so that the detection signal is the sum of the two ions' individual signals. The two-ion signal is therefore

C + Acos0, + C + Acos<t>2 = 2C + 2Acos -(0, + <t> 2

) cos -(0, - 0, (H)

By measuring the fringe amplitude and phase as 0, = (0, + 0>)/2 is swept, we calibrate 0. — 4>i as a function of trap strength and ensure that 0, + 0, is independent of trap

strength. We use the phase convention that at the ion separation used for the entanglement

preparation pulse the maximum of the correlation function is at 0, = 0, = (or A0 = IO1 = 0). Our measurement procedure begins by experimentally finding this condition of 0, = 0, = by keeping A0 = and scanning the synthesizer phase to find

the maximum correlation. The experiment is then adjusted to the phase angles specified above by switching the axial trap strength to set A0 and incrementing the synthesizer phase to set IO1 .

Locality issues The ions are separated by a distance of approximately 3 u.m, which is greater than 100

times the size of the wavepacket of each ion. Although the Coulomb interaction strongly couples the ions' motion it does not affect the ions' internal states. At this distance, all known relevant interactions are expected to be small. For example, dipole-dipole

interactions between the ions slightly modify the light-scattering intensity, but this effect is negligible for the ion-ion separations used29

. Also, the detection solid angle is large enough that Young's interference fringes, if present, are averaged out

30

. Even though all known interactions would cause negligible correlations in the measurement outcomes, the ion separation is not large enough to eliminate the lightcone loophole. We note that the experiment would be conceptually simpler if, after creating the entangled state, we separated the ions so that the input manipulations and measurements

were done individually. However, unless we separated the ions by a distance large enough

to overcome the lightcone loophole, this is only a matter of convenience of description and

does not change the conclusions that can be drawn from the results.

Received 25 October; accepted 30 November 2000.

1. Clauser, J. F. & Shimony, A. Bell's theorem: experimental tests and implications. Rep. Prog. Phys. 41, 1883-1927(1978).

2. Einstein, A., Podolsky, B. 8c Rosen, N. Can quantum- mechanical description of reality be considered complete? Phys. Rev. 47, 777-780 (1935).

3. Bell, J. S. On the Einstein-Podolsky-Rosen paradox. Physics 1, 195-200 (1965).

4. Bell, J. S. in Foundations ofQuantum Mechanics (ed. d'Espagnat, B.) 171-181 (Academic, New York,

1971).

5. Clauser, J. F, Home, M. A., Shimony, A. 8c Holt, R. A. Proposed experiment to test local hiddenvariable theories. Phys. Rev. Lett. 23, 880-884 (1969).

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7. Fry, E. S. & Thompson, R. C. Experimental test of local hidden-variable theories. Phys. Rev. Lett. 37, 465-468 (1976).

8. Aspect, A., Grangier, P. 8c Roger, G. Experimental realization of Einstein-Podolsky-Rosen-Bohm Gedankenexpenmenf. a new violation of Bell's inequalities. Phys. Rev. Lett. 49, 91-94 (1982).

9. Aspect, A., Dalibard. J. 8c Roger, G. Experimental test of Bell's inequalities using time-varying

analyzers. Phys. Rev. Lett. 49, 1804-1807 (1982).

10. Ou, Z. Y. 8c Mandcl, L. Violation of Bell's inequality and classical probability in a two-photon

correlation experiment. Phys. Rev. Lett. 61, 50-53 ( 1988).

11. Shih. Y. H. 8c Alley, C O. New type of Einstein -Podolsky- Rosen- Bohm experiment using pairs of light

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letters to nature

quanta produced by optical parametric down conversion. Phys. Rev. Lett. 61, 2921-2924 (1988).

12. Tapster, P. R., Rarity, ). G. & Owens, P. C. M. Violation of Bell's inequality over 4 km of optical fiber.

Phys. Rev. Lett. 73, 1923-1926 (1994).

13. Kwiat, P. G., Mattle, K., Weinfurter, H. & Zeilinger, A. New high-intensity source of polarization- entangled photon pairs. Phys. Rev. Lett. 75, 4337-4341 (1995).

14. Tittel, W., Brendel, J., Zbinden, H. & Gisin, N. Violation of Bell inequalities by photons more than 10km apart. Phys. Rev. Lett. 81, 3563-3566 (1998).

15. Weihs, G. et al. Violation of Belt's inequality under strict Einstein locality conditions. Phys. Rev. Lett. 81,5039-5043(1998).

16. Aspect, A. Bell's inequality test: more ideal than ever. Nature 398, 189-190 (1999).

17. Gisin, N. & Zbinden, H. Bell inequality and the locality loophole: active versus passive switches. Phys.

Lett. AIM, 103-107(1999).

18. Lo, T. K. 8c Shimony, A. Proposed molecular test of local hidden-variable theories. Phys. Rev. A 23, 3003-3012(1981).

19. Kwiat, P. G., Eberhard, P. H., Steinberg, A. M. 8< Chiao, R. Y. Proposal for a loophole-free Bell

inequality experiment. Phys. Rev. A 49, 3209-3220 (1994).

20. Huelga, S. F„ Ferrero, M. & Santos, E. Loophole-free test of the Bell inequality. Phys. Rev. A 51, 5008-

5011 (1995).

21. Fry, E. S., Walther, T. & Li, S. Proposal for a loophole free test of the Bell inequalities. Phys. Rev. A 52, 4381-4395 (1995).

22. Freyberger, M., Aravind, P. K., Home, M. A. & Shimony, A. Proposed test of Bell's inequality without a detection loophole by using entangled Rydberg atoms. Phys. Rev. A 53, 1232-1244 (1996).

23. Brif, C. & Mann, A. Testing Bell's inequality with two-level atoms via population spectroscopy. Europhys. Lett. 49, 1-7 (2000).

24. Beige, A., Munro, W. J. 8c Knight, P. L. A Bell's inequality test with entangled atoms. Phys. Rev. A 62, 052102-1-052102-9(2000).

25. Lamehi-Rachti, M. & Mittig, W. Quantum mechanics and hidden variables: a test of Bell's inequality by the measurement ofthe spin correlation in low-energy proton-proton scattering. Phys. Rev. D 14, 2543-2555(1976).

26. Hagley, E. era/. Generation of Einstein-Podolsky-Rosen pairs of atoms. Phys. Rev. Lett. 79, 1-5 (1997).

27. Sackett, C. A. et al. Experimental entanglement of four particles. Nature 404, 256-259 (2000).

28. Feynman, R. P., Vernon, F. L. 8c Hellwarth, R. W. Geometrical representation of the Schrodinger

equation for solving maser problems. /. Appl. Phys. 28, 49-52 (1957).

29. Richter, T. Cooperative resonance fluorescence from two atoms experiencing different driving fields. Optica Ada 30, 1769-1780 (1983).

30. Eichmann, U. et al. Young's interference experiment with light scattered from two atoms. Phys. Rev.

Lett. 70, 2359-2362 (1993).

Acknowledgements We thank A. Ben-Kish, J. Bollinger, J. Britton, N. Gisin, P. Knight, P. Kwiat and I. Percival

for useful discussions and comments on the manuscript. This work was supported by the US National Security Agency (NSA) and the Advanced Research and Development

Activity (ARDA), the US Office of Naval Research, and the US Army Research Office. This paper is a contribution of the National Institute of Standards and Technology and is not

subject to US copyright.

Correspondence and requests for materials should be addressed to D.J.W.

(e-mail: david.wineland@boulder.nist.gov).

require some form of manual intervention 10 " 18

. Here we report a

structural polymeric material with the ability to autonomically

heal cracks. The material incorporates a microencapsulated heal- ing agent that is released upon crack intrusion. Polymerization of

the healing agent is then triggered by contact with an embedded

catalyst, bonding the crack faces. Our fracture experiments yield

as much as 75% recovery in toughness, and we expect that our approach will be applicable to other brittle materials systems

(including ceramics and glasses). Figure 1 illustrates our autonomic healing concept. Healing is accomplished by incorporating a microencapsulated healing agent

and a catalytic chemical trigger within an epoxy matrix. An

approaching crack ruptures embedded microcapsules, releasing

healing agent into the crack plane through capillary action. Poly- merization of the healing agent is triggered by contact with the embedded catalyst, bonding the crack faces. The damage-induced

triggering mechanism provides site-specific autonomic control of

repair. An additional unique feature of our healing concept is the use of living (that is, having unterminated chain-ends) polymerization

catalysts, thus enabling multiple healing events. Engineering this self-healing composite involves the challenge of combining polymer

science, experimental and analytical mechanics, and composites

processing principles.

We began by analysing the effects of microcapsule geometry

and properties on the mechanical triggering process. For example,

capsule walls that are too thick will not rupture when the crack approaches, whereas capsules with very thin walls will break during

processing. Other relevant design parameters are the toughness and

the relative stiffness of the microcapsules, and the strength of

the interface between the microcapsule and the matrix. Micromechanical modelling with the aid of the Eshelby-Mura equivalent inclusion method19 has been used to study various aspects of the complex three-dimensional interaction between a crack and a microcapsule. An illustrative result from these studies is presented

in Fig. 2a, which shows the effect of the relative stiffness of the microcapsule on the propagation path of an approaching crack. The crack, the sphere and the surrounding matrix are subjected to

a far-field tensile loading, a„, perpendicular to the crack plane.

Autonomic healing of

polymer composites

S. R. White", N. R. Sottost, P. H. Geubelle', J. S. Moorel, M. R. Kesslert,

S. R. Sriram+, E. N. Brownt & S. Viswanathan*

* Department of Aeronautical and Astronautical Engineering, t Department

of Theoretical and Applied Mechanics, $ Department of Chemistry, University of

Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA

Structural polymers are susceptible to damage in the form of

cracks, which form deep within the structure where detection is difficult and repair is almost impossible. Cracking leads to mechanical degradation 1 " 3 of fibre-reinforced polymer composites; in microelectronic polymeric components it can also lead to electrical failure4

. Microcracking induced by thermal and mechanical fatigue is also a long-standing problem in poly- mer adhesives5

. Regardless of the application, once cracks have formed within polymeric materials, the integrity of the structure

is significantly compromised. Experiments exploring the concept

of self-repair have been previously reported6 " 8

, but the only

successful crack-healing methods that have been reported so far

Crack

m^Catalyst

Microcapsule-^

•Polymerized

- healing agent •

Figure 1 The autonomic healing concept. A microencapsulated healing agent is embedded in a structural composite matrix containing a catalyst capable of polymerizing

the healing agent, a, Cracks form in the matrix wherever damage occurs; b, the crack

ruptures the microcapsules, releasing the healing agent into the crack plane through

capillary action; c, the healing agent contacts the catalyst, triggering polymerization that bonds the crack faces closed.

794 TN-188 £$©2001 Macmillan Magazines Ltd NATURE|VOL 409 j 15 FEBRUARY 2001 | www.nature.com
Volume 86, Number 26 PHYSICAL REVIEW LETTERS 25 June 2001

Experimental Demonstration of Entanglement-Enhanced Rotation Angle Estimation

Using Trapped Ions

V. Meyer, 1 M. A. Rowe, 1 D. Kielpinski, 1 C.A. Sackett,2 W.M. Itano,

1 C. Monroe,3 and D.J. Wineland 1

1 Time and Frequency Division, National Institute of Standards and Technology, Boulder, Colorado 80305-3328

2 Physics Department, University of Virginia, Charlottesville, Virginia 22904

^Department of Physics, University of Michigan, Ann Arbor, Michigan 48109

(Received 26 February 2001)

We experimentally investigate three methods, utilizing different atomic observables and entangled

states, to increase the sensitivity of rotation angle measurements beyond the "standard quantum limit" for nonentangled states. All methods use a form of quantum mechanical "squeezing." In a system of two

entangled trapped 9 Be +

ions we observe a reduction in uncertainty of rotation angle below the standard quantum limit for all three methods including all sources of noise. As an application, we demonstrate

an increase in precision of frequency measurement in a Ramsey spectroscopy experiment.

DOI: 10.1103/PhysRevLett.86.5870

Entanglement has played an important role in elucidat- ing fundamental, and sometimes apparently mysterious,

aspects of quantum mechanics [1]. It is an integral part of

quantum information processing [2], where potential applications include efficient algorithms for problems that are computationally hard on classical computers. Entanglement can also provide increased sensitivity in quantumlimited measurements; here we report experimental

measurements of rotation angle in an atomic ensemble

where the observed uncertainty is smaller than can possibly

be obtained without entanglement.

Generally, we assume a quantum system where an observable O depends on a system parameter £. Using measurements of 0({) to determine £, the uncertainty in our

determination of £ for a single measurement is given by

H =

\d(d)/d{\

' (1)

where ( A O)2 = (O2

) — (O)2 is a measure of the rms fluc- tuations in repeated measurements of O. The specific

problem we investigate is efficient estimation of spin rotation angle. We consider a system of TV spin- 1/2 particles

with total angular momentum J = X,=i S,-, where 5, is the spin of the jth particle. For each spin, ||), and It), are spin eigenstates with respect to a chosen axis. Rotations of the entire system are characterized by the operator R = exp(-/^7 • u) for £ an angle and u the axis of

rotation. To make the best estimate of £, we want to pre- pare an input state and choose an observable O that will minimize 8£. This problem is analogous to measurements

of path-dependent phase differences in a Mach-Zehnder

interferometer [3-5] or transition frequencies in spectro- scopy [6-8]. For nonentangled spin- 1/2 particles, the states which

minimize S£ are (angular momentum) coherent states [9]. Coherent states can be obtained from the state l^o) =

li>i li>2 - - • ll>/v = \J = N/2,mj = -N/2) by an overall

rotation. In this case O = J± minimizes 8g, where J±

is the angular momentum operator perpendicular to (J)

PACS numbers: 42.50.Dv, 03.65.Ud, 32.80.Pj, 39.30. +w

in the plane of rotation (Fig. 1). Then, 8g = A7± /|(7)|

[10]. Assuming no additional sources of error, for coherent

states we obtain 8£ = 8£c = l/\/77, the standard quan- tum limit (SQL) [3-8].

We examine three proposed methods to reduce 8£

using entangled states. The first uses states well described

by "spin squeezing" [3,4,6], as depicted in Fig. lb. Here, we take O = J± and 8g = A7i/|(7~)| as for coherent

states. A second method [11-13] has been discussed in the context of a Mach-Zehnder interferometer for bosons,

where the angle £ to be measured is the interferometer

phase offset due to unequal arm length. In the spin context

here, a state of the form 1^) = |7 = N/2,mj = 0) is rotated and subsequently measured with the variance

operator V = J

2 — (Jz )2

. A third method [8] uses

states of the form |¥> = [|7, -7) + e^\J, +7)]/V2 =

[ID, ll>2 • • ID/v + e'P It), |T>2

•

• • ID*]/VI that are rotated and subsequently measured with the parity operator O =

P = n, = i °"zii where o\, = 2SZ j/h is the Pauli spin operator in the z direction for the /th particle. In experiments

FIG. 1. (a) For spins in a coherent state the uncertainty dis- tribution of the perpendicular spin components A7± is uniform and forms a circle of radius y/J/2, whereas for entangled spins

(b) the distribution of AJL can be "squeezed" in one direction, forming an ellipse. To measure AJ± , we rotate the spin J into the x-y plane and measure A7Z

, observing its oscillation with

respect to the phase angle <f>. From this and the measured value of K7)| we determine 86 (Fig. 2).

5870 0031 -9007/01 /86(26)/5870(4)$15.00 © 2001 The American Physical Society

TN-189
Volume 86, Number 26 PHYSICAL REVIEW LETTERS 25 June 2001

81-1-T-*-

! 0.9 -

0.8 - SQL

0.7-

0.6

• anti-squeezing

/ squeezing

-7t/4 jc/4

~\ 1

ji/2 3jc/4

<!> [rad]

- 1.6

7 - 1.4

1.2

1.0

0.8

FIG. 2. The ideal phase sensitivity of the state |^) =

cos(77-/10) |U) + i sin(ir/10) Iff) is shown as the dashed curve

[Eq. (4)]. The full circles are the values determined from 10000

experiments (=l ms per experiment) per data point. Error bars

are smaller than the marker size. The dotted line marks the standard quantum limit 89c = 1/V2, which is independent of

<f> [7].

with two ions, we use each of these methods and measure

a value of 8£ < 8£c. We quantify these results with

the parameter £r = 8g/8gc [6], related to the entanglement of the system [14]. The minimum possible value of €R is l/y/N (8£ = 1/A0, the Heisenberg limit [3-8].

The experiments use two 9 Be+

ions that are confined

along the axis of a miniature linear radio frequency trap

[15]. The spectrally resolved \F = \,mp = — 1) — IT) and \F = 2,ttif = —2) = \[) 2 Si/2 ground-state hyperfine levels of 9 Be+ form the basis of an effective spin- 1/2

system. Coherent superpositions between ||) and |f) are generated by laser-driven two-photon stimulated-Raman

transitions [16]. Defining the quantization axis to be the z axis, these operations are equivalent to the spin rotation R

for h in the x-y plane:

li) cosf|l>-sinf«-«*|T>,

IT> — cosflT) + sinfe'^IO,

(2a)

(2b)

where <f> is the laser phase and 9 is proportional to the duration the laser pulses are applied (Fig. lb). Using

stimulated-Raman transitions that couple to the ions' motion [17,18], we can also realize the entangling operation:

|U>— |¥> = cosa|U> + isinalff), (3) where a is proportional to the laser pulses' duration. At

the end of each experiment, we detect the number of ions

in the |i) or |f) state with state-sensitive fluorescence [19]. For spin squeezing, we take |^) with values of a #

M77/4 (M odd) in which case (/) = (Jz ) = — cos(2a)

does not vanish. In the experiment, we extract (J) by ap- plying the rotation of Eqs. (2), varying 9, and recording

(Jz ) ("Rabi flopping"). In general A/x depends on 0, as indicated in Fig. lb. To determine A7X (0), we rotate |\^) into the x-y plane by driving a

" 7r/2-pulse" on both ions

[Eqs. (2) with 9 = 77-/2] and measure A/z for different values of cf>. This operation preserves the expectation val- ues of (J) and &J± in the rotated frame. Therefore we

can regard the experiment as measuring the precision of

our £ = 9 = tt/2 rotation of the spins for the initial state |^); this precision will be optimized for certain values of

4> . For the ideal case, we have

sof^ A-'iW VEJ -sin(2a)sin(2</>)]

1(7)1 I cos(2a)|

We performed experiments for several values of a,

obtaining the highest sensitivity for a = 77/ 10 (Fig. 2). From the Rabi flopping curves we measure |(7)| to be

0.768(2) in this case (ideally, we expect 0.809). The highest measured sensitivities achieved are <50min = 0.65(1), or gR = 0.92(1). [Ideally <50min = 0.561 (& = 0.794)].

An increase in sensitivity over the SQL for two nonentangled spins (86c = 1/V2) is visible for a range of

about 7r/4 rad. The discrepancy with the theoretically

expected minimal values is caused primarily by imperfect

entangled-state preparation. Note that ideally, for two

spins, <50mj n —» 1/2 (the Heisenberg limit) as a —< 77-/4. However, then (J) —- so that any added noise prevents

achieving this limit. The improvements in phase sensitivity over the SQL

arise from spin squeezing. For angular momentum states, the uncertainty relation (A7,)2 (A7y

) 2 > ^h 2 \{Jk )\ 2 allows

for one of the variances (A/,)2 to be reduced (squeezing) at the cost of (A7y

) 2 increasing (antisqueezing). 1(7^)1 must also shrink. The Hamiltonian H = ^72 which carries the state |U) into |¥) [Eq. (3)] [4,17,20] establishes a

correlation between the spins of both particles that results

in spin squeezing [4]. The redistributed variances of the

spins are indicated as an ellipse in Fig. lb and allow us

to obtain values of £r < 1 when A/, shrinks more than

\(Jk)\- Recently, observations of spin squeezing have been

reported in Refs. [21] and [22], but the results were not

cast in terms of measured 8£ or £r; therefore, a direct comparison is precluded. For the maximally entangled state I^m) = (III) +

i|TT»A/2[a = 7r/4inEq. (3)], or the state |7 = \,mj =

0), (J) = [23] so that the above method is experimentally inaccessible. In these cases, the parity operator P

or the variance V can be used to increase the sensitivity

of a phase measurement. For two ions, \J = l,ntj =

0) = (lit) + IU))/V2 can be obtained from \VM) by a rotation [Eqs. (2) with 9 = tt/2 and <f> =

77-/4]. F°r two ions, we have V = (1 + P)/2 — Qz )2

, so that, except for an offset and scale factor, the measured values of V and P are the same. Therefore we

can explore the second and third methods cited in the introduction with the same experiment. Here, we cast the experiment in terms of I^m) and the parity operator P. We view the experiment as performing a 77-/2 rotation of the state I^m) [Eqs. (2) with 9 = tt/2] and

desire to determine { = 4> with maximum sensitivity.

To do this, we prepare the state I^m), perform a 7r/2

rotation for various values of (f>, and measure P{4>), in which case 8cf> = AP/\d(P)/dcf)\. The measurements

5871

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Volume 86, Number 26 PHYSICAL REVIEW LETTERS 25 June 2001

are displayed in Fig. 3b. The amplitude of the observed

sinusoidal oscillation is 0.845(2) rather than the theoretical maximum of 1, due primarily to imperfections

in state creation. Because of these imperfections, 8(f) also depends on 4> as shown in Fig. 3c. Ideally, we have

8(f) = 1/2 (£/? = 1/V2), the Heisenberg limit, independent of 4> [8]. In the experiment, we observe 8(f) < 8(f) c for a limited range of <p values. The minimal uncertainty observed is 8(f) = 0.59(1) < 1/V2 = 8(f> c [ijR =

0.83(1)]. Note that the period of the oscillation of (P)

with respect to (f) is half (in general \/N [8]) that of the period when the observable is Jz (Fig. lb). This results

in a relative increase in \d(0)/d(f)\, which is the main

reason we can find values of {;r < 1. The analog of

A 1.0

t 0lV

0.5 0.0-

-0.5-

-1.0 /

/

/ /

/ Niw

A 1.0 -

ID.V

0.5-

T

T b

0.0 -

-0.5 -

-1.0 -

.. . . •!

*• 1 r~ i i i -1.0-

(0 i. 0.9 - t C

0.8 -

SQL

,. I ?. 0.7 — • •

0.6 - • • • • • •• • m

0.5 -

Heisenberg Limit

I I I I -71/2 -7C/4 7C/4 [rad]

FIG. 3. In (a), we plot the theoretical value (P((f>)) for an

initial state \^M ) = (|U) + i'|tt))/V2 which has been rotated through angles 9 = it/2, (f> [Eqs. (2)]. The bars represent the variance (A.P) 2 of the parity. In (b), we show the corresponding

measured values from 10000 experiments (experiment duration — 1 ms) per data point and a fit to the expected sinusoidal de- pendence, (c) The resulting phase sensitivity 8(f) = AP/\'-j^-\,

as determined from the data of (b). For the idealized experi- ment 8(f) = 1/2, independent of <f> [8], the Heisenberg limit. The dotted line at 1/V2 represents the SQL. All figures share

the same abscissa [shown in (c)].

this experiment using entangled photon pairs has been

reported in Ref. [24]. These methods are of interest for improved frequency

measurements [6,8]. Here, we consider use of the parity

operator P in combination with maximally entangled

states [8]. In Ramsey separated-field spectroscopy [25], if the state after the first Ramsey pulse is of the form |^) =

[ll)i ll)2

• • • \{)n + e'P |t), |T>2 • IT)*]/ V2, the transition frequency between the states ||) and ||) can be determined

with a precision of S(a> — coq) = S£/T = I /(NT) [8]. In this expression, T is the time difference between the

first and the second Ramsey pulse and 8(u> — coq) is the uncertainty in the measured frequency difference between

the atoms' transition frequency coq and the frequency co of the applied radiation. Therefore, here phase sensitivity translates into frequency sensitivity through the relation 8£ = 8 (co — o)o)T. The gain of a factor of l/y/N

compared to spectroscopy of atoms in coherent states is of

particular interest for precision spectroscopy, which has come close to the SQL [7,26].

To demonstrate the use of entangled states for spectroscopy, we prepare an initial state of two ions of the form

\Vr) — (lit) + ltl))/V2- The Ramsey experiment was

performed on |]) «-» It) transitions using stimulated Raman

excitation with o)/2tt detuned by about 10 kHz from

u>q/2tt = 1.25 GHz. We varied the time T between

the two Ramsey pulses to change the phase difference

£ = (o) — o)q)T for the experiments; that is, we de- termine a) — coo by varying T and measuring P. In

Fig. 4 the maximal gain in precision is a factor of 1.14(1)

[£r = 0.88(1)], compared to an idealized Ramsey experiment using two ions in a coherent state where preparation

and detection are perfect. In summary, we have demonstrated a fundamental

increase in sensitivity of rotation angle measurement

benefiting from entanglement. The reported measure- ments include all sources of noise (no noise subtraction)

— 1-7 -

J2 1.5-

P 1.3-

?1.1-

.£. 0.9 -

0.7-

0.5 -

SQL .

I •**• Heisenberg Limit

8 10 ~r12

~i r

14 16 T[us]

18

FIG. 4. Uncertainty in frequency determination in a Ramsey

experiment using the input state I^r) = (lit) + |tl))/V2. We

vary the time T between the Ramsey pulses for fixed frequency

detuning \co - o) \/2ir = 10 kHz. We performed 10000 ex- periments (experiment duration —1 ms) per data point. The

dotted line represents the SQL for two ions in a coherent spin

state [7]. The dashed line marks the Heisenberg limit.

5872

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Volume 86, Number 26 PHYSICAL REVIEW LETTERS 25 June 2001

and demonstrate a sensitivity better than that which can be obtained without entanglement under ideal conditions. The most important limitation of our experiments is the imperfection of initial state preparation induced by heating

of the ions to higher motional states [15], which reduces

the degree of entanglement. Maintaining the ions close

to their motional ground state should significantly reduce

these effects [17]. In an application to a Ramsey spec- troscopy experiment, we achieved an increased precision in frequency measurement compared to an idealized experiment using unentangled particles. This may be of signifi- cance for the construction of more precise atomic clocks. We thank J. J. Bollinger and B. E. King for helpful comments on the manuscript. This work was supported by

the U.S. National Security Agency (NSA) and the Advanced Research and Development Activity (ARDA) un- der Contract No. MOD-7171.00, the U.S. Office of Naval

Research, and the U.S. Army Research Office. This paper

is a contribution of the National Institute of Standards and

Technology and is not subject to U.S. copyright.

Note in proof. —Since our submission, numbersqueezed atomic states have been reported [27]. Such

states are analogous to the \J = \,nij =0) state of our experiment [3-5] and are an important step toward

sub-shot-noise atom interferometry [12].

[1] See, for example, J.S. Bell, Speakable and Unspeakable

in Quantum Mechanics (Cambridge University Press, Cambridge, 1987); N.D. Mermin, Phys. Today 38, No. (4), 38

(1985).

[2] See, for example, M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge

University Press, Cambridge, 2000).

[3] B. Yurke, Phys. Rev. Lett. 56, 1515 (1986); B. Yurke, S.L. McCall, and J.R. Klauder, Phys. Rev. A 33, 4033

(1986).

[4] M. Kitagawa and M. Ueda, Phys. Rev. A 47, 5138 (1993).

[5] B.C. Sanders and G.J. Milburn, Phys. Rev. Lett. 75, 2944

(1995).

[6] D.J. Wineland, J.J. Bollinger, W.M. Itano, F.L. Moore,

and D.J. Heinzen, Phys. Rev. A 46, R6797 (1992); D.J. Wineland, J.J. Bollinger, W.M. Itano, and

D.J. Heinzen, Phys. Rev. A 50, 67 (1994).

[7] W.M. Itano, J.C Bergquist, J.J. Bollinger, J.M. Gilligan, D.J. Heinzen, F.L. Moore, M.G. Raizen, and

D.J. Wineland, Phys. Rev. A 47, 3554 (1993).

[8] J.J. Bollinger, W.M. Itano, D.J. Wineland, and

D.J. Heinzen, Phys. Rev. A 54, R4649 (1996).

[9] F. T. Arecci, E. Courtens, R. Gilmore, and H. Thomas,

Phys. Rev. A 6, 2211 (1972).

[10] More precisely, 8£ = aretan(Ayx /|<7>|) (Fig. la). Fol- lowing convention, we use instead 8£ = &J±/\{J)\ given

by Eq. (1). In typical situations, where measurements of O are repeated M times (M ~» 1), A7± —» AJ± /y/M and

this distinction is not important.

[11] M.J. Holland and K. Burnett, Phys. Rev. Lett. 71, 1355

(1993).

[12] P. Bouyer and M.A. Kasevich, Phys. Rev. A 56, R1083

(1997).

[13] T. Kim, O. Pfister, M.J. Holland, J. Noh, and J. L. Hall, Phys. Rev. A 57, 4004 (1998).

[14] A. S0rensen, L.-M. Duan, J. I. Cirac, and P. Zoller, Nature (London) 409, 63 (2001).

[15] Q.A. Turchette, D. Kielpinski, B.E. King, D. Leibfried,

D.M. Meekhof, C.J. Myatt, M.A. Rowe, C.A. Sackett, C.S. Wood, W.M. Itano, C. Monroe, and D.J. Wineland,

Phys. Rev. A 61, 063418 (2000).

[16] C. Monroe, D.M. Meekhof, B.E. King, S.R. Jefferts, W.M. Itano, D.J. Wineland, and P. Gould, Phys. Rev. Lett. 75, 4011 (1995).

[17] A. S0rensen and K. M0lmer, Phys. Rev. A 62, 022311

(2000).

[18] C.A. Sackett, D. Kielpinski, B.E. King, C. Langer,

V. Meyer, C.J. Myatt, M. Rowe, Q.A. Turchette,

W.M. Itano, D.J. Wineland, and C. Monroe, Nature (London) 404, 256 (2000).

[19] M.A. Rowe, D. Kielpinski, V. Meyer, C.A. Sackett, W.M. Itano, C. Monroe, and D.J. Wineland, Nature (London) 409, 791 (2001).

[20] B.C. Sanders, Phys. Rev. A 40, 2417 (1989).

[21] J. Hald, J.L. S0rensen, C. Schori, and E.S. Polzik, Phys.

Rev. Lett. 83, 1319 (1999).

[22] A. Kuzmich, L. Mandel, and N. P. Bigelow, Phys. Rev. Lett. 85, 1594 (2000).

[23] In the context of Fig. 1, \Wm) is a circle of unit radius in the plane defined by the z and (x — y)/\fl directions and

centered at the origin. [24] A. Kuzmich and L. Mandel, Quantum Semiclass. Opt. 10, 493 (1998).

[25] N. F. Ramsey, Molecular Beams (Oxford University Press, London, 1963), p. 124.

[26] G. Santarelli, Ph. Laurent, P. Lemonde, A. Clairon,

A.G. Mann, S. Chang, A.N. Luiten, and C. Salomon,

Phys. Rev. Lett. 82, 4619 (1999).

[27] C. Orzel, A.K. Tuchman, M.L. Fenselau, M. Yasuda, and M.A. Kasevich, Science 291, 2386 (2001).

5873

TN-192
Diode-pumped Nd:FAP laser at 1.126 fxm: a possible

local oscillator for a Hg+

optical frequency standard

Flavio C. Cruz, Brenton C. Young, and James C. Bergquist

We report the efficient operation of a continuous-wave, single-frequency, diode-pumped Nd:FAP laser at 1.126 |xm. When frequency quadrupled, such a laser might be used as a local oscillator for an optical

frequency standard based on the single-photon 2S1/2-zDs/2 electric quadrupole transition of a trapped

and laser-cooled 199Hg+ion. Since the frequencies of the atomic transition and the laser are harmonically related, this scheme helps to simplify the measurement of the S-D clock transition frequency by a

phase-coherent chain to the cesium primary frequency standard. OCIS codes: 020.7010, 140.3480, 140.3580, 190.4160, 300.6520.

Trapped and laser-cooled ions have great potential

for optical frequency and time standards. 1- 6 High

resolution is possible because perturbations can be

made small7 and interrogation times long. 89 In addition, laser cooling reduces first- and second-order

Doppler shifts to extremely low levels. 10 Among

proposed standards that use trapped and laser-cooled

ions,6 a

199Hg+

ion standard is attractive because it offers both a microwave transition9 and a narrow

optical transition that promise high performance.

The optical standard is based on the 2S1/2-2D5/2 , 281.5-nm electric quadrupole transition. 11 Optical

frequency standards are attractive since the potential fractional frequency instability of a standard is inversely proportional to its frequency. An optical

clock can have a fractional frequency instability less than 1 x 10~ 15 at 1 s even for a single laser-cooled

ion. However, for reaching such low instabilities it is necessary to have a laser (local oscillator) whose

frequency fluctuations are less than 1 Hz for times as long as a few seconds.

The inherent frequency stability of solid-state la- sers makes them attractive for metrological applications and precision spectroscopy. In addition, a

solid-state laser can be compact, reliable, and longlived. Reliable, commercial diode lasers do not yet

When this research was performed, all the authors were with the Time and Frequency Division, National Institute of Standards and

Technology, 325 Broadway, Boulder, Colorado 80303. F. C. Cruz

is now with the Universidade Estadual de Campinas (UNICAMP),

Instituto de Fisica "Gleb Wataghin," CP.6165, Campinas, SP

13083-970, Brazil. The email address for B. C. Young is byoung@

boulder.nist.gov. Received 24 June 1998.

exist in the UV, near the transition needed for the

Hg+ system, but high-power, near-infrared diode la- sers are available. Consequently, some groups have

frequency-quadrupled the output of near-infrared di- ode lasers that oscillate at a single frequency and in a single spatial mode to obtain cw, single-frequency

UV sources. 12- 13 An alternative approach is to frequency-quadruple the output of a cw, solid-state

laser that is pumped with high-power multimode di- ode lasers, such as has previously been done with

Nd:YAG and Nd:YV04 lasers. 14 Here we report the

development of a diode-pumped solid-state Nd:FAP

laser that operates in a single spatial and singlefrequency mode at 1.126 |xm. The frequency of the

output radiation from this laser was quadrupled and

might be used as a local oscillator in an optical fre- quency standard based on the S-D quadrupole transition of trapped and laser-cooled Hg+ ions at 282 nm. Because the free-running frequency instability

of this laser is dominated by low-frequency acoustical

and mechanical noise, only a moderate-speed servo system is needed to lock the laser frequency tightly to the resonance of a high-finesse cavity. 11- 15 In addition, since the frequency of the Nd:FAP laser is quadrupled to reach the atomic transition, we have

completed the first steps in a frequency chain from

the optical to the microwave.

The spectral analysis and lasing performance of

Nd:FAP at 1.06 |xm have been reported else- where. 1617 The absorption spectrum of Nd:FAP,

like Nd:YAG, has a strong line near 808 nm, approximately 2.5 nm wide, which is well suited to optical

pumping by a diode laser. In addition to a strong

emission peak at 1.06 |xm, weaker features near 1.12

and 0.93 (xm are also observed. We make the Nd:

20 November 1998 / Vol. 37, No. 33 / APPLIED OPTICS 7801

TN-193
563 run, .282 nm

> > XTAL 0C2 AD*P

Fig. 1. Experimental setup for the frequency-quadrupled, diode- pumped Nd:FAP laser: DPL, diode pump laser at 808 nm; IC1,

flat input coupler for the Nd:FAP laser (HR at 1126 nm, T « 90%

at 808 nm); PZT, piezoelectric transducer; Ml and M2, 15-cm

radius-of-curvature mirrors (HR at 1126 nm); OD, optical diode; OC1, flat output coupler (T = 0.4% at 1126 nm, T ~ 50% at 1064

nm); IC2, flat input coupler for doubling cavity (T = 1.7% at 1126

nm); XTAL, KNb03 or LiNb03 doubling crystal; OC2, flat output

coupler (HR at 1126 nm,T- 80% at 563 nm for KNb03 or 94% for LiNb03 ); AD:|! P, deuterated ammonium dihydrogen phosphate.

FAP laser oscillate at 1.126 |xm by strongly suppressing the higher-gain oscillation at 1.06 |xm. Our

crystal is cut, polished, and oriented at Brewster's

angle to avoid the need for any antireflection coatings. The c axis lies in a plane parallel to the polished surfaces and is oriented perpendicular to the

polarization of the radiation of both the pump laser

and the Nd:FAP laser. The thickness of the crystal

is 2 mm, and the diameter is 6.3 mm. The Nd con- centration gives approximately 80% absorption of the

light from a diode laser at 808 nm. Absorption is nearly 100% when a narrow-band Ti:sapphire laser is used. We largely eliminate thermal problems for pump powers as high as 1 W by heat sinking the

crystal in a tightly fitting copper support piece.

Maximum output powers are achieved for cavity

waists co between 80 and 100 |xm. The diode laser

used for pumping emits as high as 1 W at 808 nm,

which is coupled into a fiber with a 100-|xm core diameter. Approximately 750 mW is available after

the fiber. A 3.5-cm focal-length lens collimates the

beam diverging from the fiber, and a 6.7-cm lens

focuses the light through the Nd:FAP crystal.

The 1.126-(jLm laser is built as a ring cavity formed

by two flat mirrors (input and output couplers) and

two 15-cm radius-of-curvature mirrors that are highly reflecting (HR) at 1.126 |im (Ml and M2 in

Fig. 1). The Nd:FAP crystal is positioned midway

between the 15-cm mirrors at a nearly optimum cavity waist of o) = 80 |xm. The flat input coupler IC1

transmits 90% of the pump light at 808 nm and is HR

at 1.126 |a,m. The output coupler OC1 was coated to transmit 0.4% of the radiation at 1.126 |a-m and 50%

at 1.06 |xm, which eliminates lasing on the strong line at 1.06 |xm. The pump beam is focused into the

crystal with a beam waist of oj = 100 fxm. The

300 400 500 600 pump power (mW)

Fig. 2. Output power of the Nd:FAP laser at 1.126 |xm as a

function of pump power from a Ti:sapphire laser.

astigmatism introduced by the crystal is compensated by the off-axis orientation of the 15-cm mirrors.

Unidirectional oscillation is enforced by an optical

diode formed by a terbium- gallium garnet Faraday

rotator and a quartz compensating plate. The laser

oscillates at 1.126 jxm in a single-frequency TEM00 mode without any frequency-selective elements in- side the cavity. However, a thin etalon improves the

mode stability and makes it possible to tune the

wavelength. Tuning of the laser from approximately 1.118 to 1.128 |xm was verified by use of an

interferometric wavemeter and an optical spectrum

analyzer. When frequency quadrupled, the Nd:FAP

laser reaches the Hg+ S-D transition at 281.5 nm. The tuning range also includes the A2 £(ra" = 1) <— X2 H(n' = 0) transition of OH molecules at 281.93 nm,

which could possibly be used for remote sensing of

OH in the atmosphere. 18 The output power of the 1.126-|jim laser as a function of pump power from the Ti:sapphire laser is plotted in Fig. 2. The slope efficiency is approximately

40%. For an input power of 780 mW, an output

power of 260 mW is achieved, corresponding to an

overall efficiency of 33%. When pumped by the di- ode laser, the efficiency is reduced to approximately

20% owing to the poorer spatial mode quality of the

diode laser beam and to the lower absorption of the

pump power by the crystal.

We double the fundamental radiation at 1126 nm

using either KNb03 or LiNb03

. Noncritically

phase-matched harmonic generation in KNb03 , which would be highly efficient, is predicted to occur

for 1126-nm light at a phase-matching temperature19 of 268 °C. Unfortunately, this is precluded since

KNb03 experiences a phase transition near 220 °C. However, less efficient type-I angle-tuned critical phase matching at room temperature is still possible

for &-cut KNb03 at a phase-matching angle of pm «= 23.9°. The calculated effective nonlinear coefficient

deff is 11.8 pm/V and the walkoff angle is approximately 60 mrad, which implies an efficiency per unit

crystal length of 2 x 10"^ W" 1 cm" 1

. For 2-mmlong crystals we measure single-pass efficiencies t\ slightly higher than 1 X 10"4 W"1 under optimum

focusing conditions, which is less than predicted.

We increase the harmonic power by using

7802 APPLIED OPTICS / Vol. 37, No. 33 / 20 November 1998

TN-194
antireflection-coated KNb03 in an external build-up

cavity (Fig. 1). The cavity is locked to resonance with the frequency of the laser by the HanschCouillaud polarization technique.20 The powerenhancement cavity is formed by two flat mirrors and

two spherical mirrors (7.5-cm radius of curvature)

that produce a waist of 14 |jim, nearly optimum21 for the 2-mm-long KNb03

. The input coupler IC2

transmits 1.7% of the radiation at 1126 nm. This

transmittance gives nearly 100% coupling into the

cavity for 100 mW of incident radiation when light at the second harmonic is generated. The output coupler OC2 is one of the 7.5-cm mirrors, which is HR at 1126 nm and 80% transmitting at 563 nm. The

other mirrors are HR at 1126 nm. The crystal is positioned in the tight focus between the two curved

mirrors, and the fundamental beam is mode matched

into the softer waist between the two flat mirrors.

We keep the cavity angles as small as possible to minimize the astigmatism introduced by the curved

mirrors. The cavity losses (excluding the input coupler) are 0.1% and the crystal losses are 0.3%, as determined from the measured finesse and build-up

factors of the cavity with and without the crystal.

The crystal losses are dominated by surface scattering. For an input power of 100 mW at 1126 nm and

an enhancement factor of 150, we achieve an output

power of 10 mW at 563 nm. We also verified that generation of the second harmonic of 1126-nm radiation is possible in congruent

LiNb03 at 7/pm ~ 142 °C. At this temperature we

observe no indication of optical damage. At 1126

nm, deff is 4.3 pm/V, which gives n « 1.9 x 10"3 W

_1

for our 11.3-mm-long crystal. However, we measure

a single-pass efficiency of only 8.0 X 10~4 W

_1

for optimum focusing conditions (w «* 20 |xm). We

again use a power-enhancement cavity, but the size

of our oven restricts us to focusing mirrors with

longer radii of curvature. Hence the waist in the

crystal is larger than optimum, limiting t) to approximately 3.7 x 10~4 W

_1

. The power-enhancement

factor in the presence of the crystal is only 76, largely

because of scattering from the Brewster-cut crystal

surfaces. Even so, we again attain an output power

of approximately 10 mW at 563 nm. This corre- sponds to approximately 22 mW of harmonic radiation generated in the crystal when the power is corrected for the 94% transmittance of the output

coupler and the 60% transmittance of the Brewster

face of the crystal. Since the usable power is nearly

the same in both cases, we find it easier and more

practical to use KNb03

. For the second stage of harmonic generation we use

deuterated ammonium dihydrogen phosphate (AD*P)

to double 563 to 281.5 nm.22 Since less than 1 pW can

be enough to saturate the narrow S—D transition,22 we

simply double the radiation at 563 nm in a single-pass

configuration (Fig. 1). Type-I noncritical phase

matching is possible for doubling 563 nm with AD*P at Tpni = 128 °C. The measured single-pass efficiency is approximately 10~3 W

_1

for our 3-cm-long crystal.

Approximately 0.1 |xW is generated at 281.5 nm for 10

(a)

50

frequency (kHz)

- (b)

"| 1 1 1 r~

50

frequency (kHz)

Fig. 3. (a) AM and (b) FM noise spectral densities in a 1-kHz bandwidth for the Nd:FAP laser at 563 nm. The peak at approx- imately 50 kHz in (a) caused by relaxation oscillations.

mW of input power at 563 nm. If more power is needed, then another option could be (3-barium borate

(BBO) in an external build-up cavity. Type-I angletuned critical phase matching at 6pm «= 44° is possible

with deS « 1.64 pm/V. The walkoff angle is 77 mrad,

which gives r\ «= 1.3 x 10~4 W

_1

for a 6-mm-long

crystal.

Figure 3 shows the power spectral densities of AM

and FM noise for the radiation from the Nd:FAP laser

at 563 nm. Also shown are the shot noise and detector noise. For the AM noise measurements part

of the laser power is directed onto a fast photodetector. The amplitude noise spectrum of the Nd:FAP

laser is obtained by a fast Fourier transform of the

detected signal. The FM noise measurements are

similar, except that a Fabry—Perot cavity is used as a

frequency discriminator, converting frequency fluctu- ations into intensity fluctuations. The spectral dis- tributions of amplitude and frequency noise of the

Nd:FAP laser are dominated by contributions at low

frequencies (<12 kHz). The pump diode laser, the

elements in the Nd:FAP laser cavity, and the components in the external doubling cavity were all mounted in aluminum supports connected by Invar

bars to improve the passive stability. A broad peak

caused by relaxation oscillations at several tens of

kilohertz (depending on the pump power) is observed

in the spectral density of AM noise, which is shot20 November 1998 / Vol. 37, No. 33 / APPLIED OPTICS 7803

TN-195
noise limited above 100 kHz. The relaxationoscillation peak is nearly absent in the FM spectral

density at 563 nm. Both the AM and FM noise are lower when the Nd:FAP laser is diode pumped than

when it is pumped by a Ti:sapphire laser. The free- running rms frequency excursions of the Nd:FAP la- ser are measured to be smaller than 100 kHz for periods of a few seconds. Therefore we anticipate

that a moderate-speed servo system would be suffi- cient to lock its frequency to the resonance of a quiet

reference cavity. When the frequency of the laser is locked to the cavity, one can measure the laser line- width by heterodyning two independent systems11 and by probing the narrow S—D atomic transition in

199Hg+

. The quadrupled Nd:FAP laser should also

facilitate the implementation of a frequency chain to the cesium standard at 9.2 GHz. One can directly

compare the Nd:FAP fundamental frequency to a sec- ondary frequency standard such as the methanestabilized He—Ne laser at 3.39 |xm by using a recently

demonstrated 3:1 optical divider.23

This study is partially supported by the Office of

Naval Research. F. C. Cruz acknowledges the support of Fundacao Coordenacao de Aperfeicoamento de

Pessoal de Nivel Superior (Brazil). We thank R. Byer, T. Y. Fan, and D. Nabors for providing the

Nd:FAP crystal and for their useful comments.

References and Notes

1. H. G. Dehmelt, "Mono-ion oscillator as potential ultimate laser frequency standard," IEEE Trans. Instrum. Meas. 31, 83-87

(1982).

2. E. Peik, G. Hollemann, and H. Walther, "Laser cooling and

quantum jumps of a single indium ion," Phys. Rev. A 49, 402-

408 (1994).

3. P. Taylor, M. Roberts, G. P. Barwood, and P. Gill, "Combined

optical-infrared single-ion frequency standard," Opt. Lett. 23, 298-300 (1998).

4. L. Marmet, A. A. Madej, K. J. Siemsen, J. E. Bernard, and B. G.

Whitford, "Precision frequency measurement of the 2S1/2—

2D5/2 transition of Sr+ with a 674-nm diode laser locked to an

ultrastable cavity," IEEE Trans. Intrum. Meas. 46, 169-173

(1997).

5. S. Urabe, M. Watanabe, H. Imajo, and K. Hayasaka, "Laser

cooling of trapped Ca+ and measurement of the 3

2D5/2 state

lifetime," Opt. Lett. 17, 1140-1142 (1992).

6. See, for example, Proceedings of the Fifth Symposium on Fre- quency Standards and Metrology, J. C. Bergquist, ed. (World

Scientific, Singapore, 1996).

7. D. J. Wineland, J. C. Bergquist, R. E. Drullinger, H. Hemmati, W. M. Itano, and F. L. Walls, 'Laser cooled, stored ion exper- iments at NBS and possible applications to microwave and

optical frequency standards," J. Phys. 42, C8-307-C8-313

(1981).

8. 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); J. J. Bollinger, D. J. Heinzen, W. M. Itano,

S. L. Gilbert, and D. J. Wineland, "A 303-MHz frequency stan- dard based on trapped Be +

ions," IEEE Trans. Instrum. Meas.

40, 126-128 (1991).

9. D. J. Berkeland, J. D. Miller, J. C. Bergquist, W. M. Itano, and

D. J. Wineland, "Laser-cooled mercury ion frequency standard," Phys. Rev. Lett. 80, 2089-2092 (1998).

10. D. J. Wineland, J. C. Bergquist, W. M. Itano, F. Diedrich, and

C. S. Weimer, "Frequency standards in the optical spectrum,"

in The Hydrogen Atom, G. F. Bassani, M. Inguscio, and T. W.

Hansen, eds. (Springer-Verlag, Berlin, 1989), pp. 123-133.

11. J. C. Bergquist, W. M. Itano, and D. J. Wineland, 'Laser sta- bilization to a single ion," in Frontiers in Laser Spectroscopy,

T. W. Hansch and M. Inguscio, eds. (North Holland, Amster- dam, 1994), pp. 359-376.

12. C. Zimmermann, V. Vuletic, A. Hemmerich, and T. W. Hansch,

"All solid state laser source for tunable blue and ultraviolet radiation," Appl. Phys. Lett. 66, 2318-2320 (1995).

13. K. Matsubara, U. Tanaka, H. Imajo, K. Hayasaka, R. Ohmukai, M. Watanabe, and S. Urabe, "An all-solid-state tunable 214.5-nm continuous-wave light source by using two-stage fre- quency doubling of a diode laser," Appl. Phys. B 67, 1-4 (1998).

14. See, for example, G. Hollemann, E. Peik, and H. Walther,

"Frequency-stabilized diode-pumped Nd:YAG laser at 946 nm

with harmonics at 473 and 237 nm," Opt. Lett. 19, 192-194

(1994); K. Kondo, M. Oka, H. Wada, T. Fukui, N. Umezu, K

Tatsuki, and S. Kubota, "Demonstration of long-term reliabil- ity of a 266-nm, continuous-wave, frequency-quadrupled solid- state laser using |3-BaB2 4

," Opt. Lett. 23, 195-197 (1998).

15. M. Zhu and J. L. Hall, "Frequency stabilization of tunable

lasers," in Experimental Methods in the Physical Sciences,

F. B. Dunning and R. G. Hulet, eds. (Academic, San Diego,

1996), Vol. 29C, pp. 103-136, and references therein.

16. R. C. Ohlmann, K. B. Steinbruegge, and R. Mazelsky, "Spectroscopic and laser characteristics of neodymium-doped cal- cium fluorophosphate," Appl. Opt. 7, 905-914 (1968).

17. X. X. Zhang, A. B. Villaverde, M. Bass, G. Lutts, and B. H. T. Chai, "Spectroscopy and lasing performance of Nd3+ doped

Ca5(P04) 3F," in Growth, Characterization, and Applications of Laser Host and Nonlinear Crystals II, B. H. T. Chai, ed., Proc. SPIE 1863, 35-38 (1993).

18. D. D. Davis, W. S. Heaps, D. Philen, M. Rodgers, T. McGee, A. Nelson, and A. J. Moriarty, "Airborne laser induced fluores- cence system for measuring OH and other trace gases in the

parts-per-quadrillion to parts-per-trillion range," Rev. Sci. In- strum. 50, 1505-1516 (1979); G. L. Vaghjiani and A. R. Ravishankara, "Kinetics and mechanism of OH reaction with

CH3OOH," J. Phys. Chem. 93, 1948-1959 (1989).

19. I. Biaggio, P. Kerkoc, L.-S. Wu, P. Giinter, and B. Zysset, "Refractive indices of orthorhombic KNb03

. II. Phase- matching configurations for nonlinear-optical interactions," J. Opt. Soc. Am. B 9, 507-517 (1992).

20. T. W. Hansch and B. Couillaud, "Laser frequency stabilization by polarization spectroscopy of a reflecting reference cavity," Opt. Commun. 35, 441-444 (1980).

21. G. D. Boyd and D. A. Kleinman, "Parametric interaction of focused Gaussian light beams," J. Appl. Phys. 39, 3597-3639

(1968).

22. J. C. Bergquist, R. G. Hulet, W. M. Itano, and D. J. Wineland,

"Observation of quantum jumps in a single atom," Phys. Rev.

Lett. 57, 1699-1702 (1986).

23. O. Pfister, M. Miirtz, J. S. Wells, L. Hollberg, and J. T. Murray,

"Division by 3 of optical frequencies by use of difference- frequency generation in noncritically phase-matched RbTiOAs04

," Opt. Lett. 21, 1387-1389 (1996).

7804 APPLIED OPTICS / Vol. 37, No. 33 / 20 November 1998

TN-196
Volume 82, Number 19 PHYSICAL REVIEW LETTERS 10 May 1999

Visible Lasers with Subhertz Linewidths

B. C. Young, F. C. Cruz,* W. M. Itano, and J. C. Bergquist

Time & Frequency Division, National Institute of Standards and Technology, 325 Broadway, Boulder, Colorado 80303

(Received 5 January 1999)

We report a visible laser with a subhertz linewidth for use in precision spectroscopy and as a

local oscillator for an optical frequency standard. The laser derives its stability from a well-isolated, high-finesse, Fabry-Perot cavity. For a 563 nm laser beam locked to our stable cavity, we measure a linewidth of 0.6 Hz for averaging times up to 32 s. The fractional frequency instability for the light locked to the cavity is typically 3 X 10" I6 at 1 s. Both the linewidth and fractional frequency instability are approximately an order of magnitude less than previously published results for stabilized lasers. [S003 1-9007(99)09 100-0]

PACS numbers: 42.55. -f, 06.30.Ft, 07.10.Fq, 42.60.Da

Stable and spectrally narrow lasers are important for

optical frequency standards and also for measurements

of fundamental constants, high-resolution spectroscopy,

and fundamental tests of physics [1,2]. It has already

been demonstrated that the frequencies of several types

of lasers can be locked to resonances of Fabry-Perot

cavities with precisions better than 0.1 Hz [3]. What

remained to be demonstrated, however, was that the resonance frequency of a reference cavity could have

a stability better than 1 Hz. Previously, the narrowest

reported visible-laser linewidth was 10 Hz for a 1 s averaging time [4]. Here we report a linewidth of 0.6 Hz

for averaging times up to 32 s. Previous results from

our group (21 Hz linewidth for 70 s averaging times

[5]) were limited by cavity length changes caused by

vibrational noise. The improvement reported here derives

from modifications that include cavities with more stable material properties and better isolation of the cavities from

mechanical vibrations. The central component of our stabilized laser is a highfinesse (J7 > 150000) Fabry-Perot cavity [5]. We use a dye laser at 563 nm as the optical source that is locked

to this reference cavity. Rather than locking the laser

directly to the high-finesse cavity, we first prestabilize it to a cavity with a finesse of =800 using a Pound-DreverHall frequency modulation (FM) lock [6]. An intracavity

electro-optic modulator (EOM) in the dye laser provides

high-frequency correction of laser frequency noise. A

piezoelectric transducer (PZT) behind one of the dyelaser cavity mirrors eliminates long-term frequency drifts between the dye laser and the low-finesse cavity. A

loop bandwidth of =2 MHz in this prestabilization stage narrows the dye-laser short-term (r < 1 s) linewidth to =1 kHz.

An optical fiber delivers light from the dye-laser

table to a vibrationally isolated table that supports the high-finesse cavity. An acousto-optic modulator (AOM)

mounted on the isolated table shifts the frequency of

the incoming light to match a cavity resonance. Again, we implement the lock using the Pound-Drever-Hall

technique. The feedback loop performs corrections at frequencies as high as —90 kHz by varying the AOM

drive frequency and at low frequencies by adjusting a PZT

on the prestabilization cavity.

The high-finesse cavity must have intrinsically low

sensitivity to temperature variations and must be well

protected from environmental perturbations. The spacer between the cavity mirrors is composed of a low-thermalexpansion material. While we have used both Zerodur

and ULE spacers [7,8], the data reported here were

collected using ULE spacers. Our spacer is cylindrical

with a 15 cm outer diameter and a 24 cm length. The

spacer is tapered at both ends to provide greater stiffness than that of a cylinder with the same length and mass. A 1 cm diameter core is drilled through the center of

the cylinder. A second core is drilled from the side of

the cylinder into the center to allow evacuation of the

intracavity region. The cavity mirrors are made with ULE

substrates and are optically contacted onto the ends of the

spacer. The cavity is supported inside an evacuated chamber by

an aluminum v-block. Four cylindrical pieces of Viton

embedded in the v-block serve as the contact points with

the cavity. The Viton contacts help to provide highfrequency vibration isolation, damping for mechanical

motion, and thermal isolation for the cavity (the thermal

time constant between the cavity and the vacuum chamber

is =14 h). Evacuating the chamber reduces shifts of the cavity resonance caused by index-of-refraction changes in the intracavity volume and thermally insulates the cavity

from the environment. The temperature of the vacuum

chamber is held at =30 °C, which is near the temperature

at which the coefficient of thermal expansion for the cavity spacer is zero. The residual temperature fluctuations of the vacuum chamber are =10 mK.

We protect the cavity from vibrational noise by mounting the vacuum chamber on a passively isolated optical

table. The table is suspended by vertical strands of surgical tubing stretched to =3 m. The fundamental stretch mode and the pendulum mode of the suspension both have

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Volume 82, Number 19 PHYSICAL REVIEW LETTERS 10 May 1999

frequencies of —0.3 Hz. This suspension system reduces

the amplitude of the transmitted vibrational noise at fre- quencies greater than 3 Hz by more than a factor of 50. In many systems the cavity is isolated by using a pendulum suspension inside the vacuum chamber, but our table suspension is sufficiently good so that we can semirigidly

mount the cavity inside the vacuum chamber. Dashpots

filled with grease at each corner of the table provide vis- cous damping (damping time constant =1 s). Proximity

detectors at three corners of the table serve as sensors for

stabilizing the table position relative to the floor. A servo system drives heaters surrounding the surgical tubing so

as to stabilize the table position with a time constant of — 100 s. To prevent the coupling of acoustic noise into the cavity, we enclose the optical table in a wooden box

lined internally with lead foam [9]. The intracavity light heats the mirror coatings, thereby

shifting the cavity resonance by —1 Hz//u.W . We minimize this shift by coupling only 100 /aW to 200 /aW of

563 nm light into the cavity and controlling the circulating

optical power by monitoring the light transmitted through

the cavity. Active control of the rf power driving the AOM (also used for frequency control) stabilizes the output power from the cavity to —0.1%.

To characterize the cavity's stability, we constructed

a second cavity and vibrationally isolated table similar

to that described above. The dye-laser output is split into two parts, each of which couples through an optical

fiber onto one of the reference cavity tables. Since the prestabilized dye-laser light has a linewidth of — 1 kHz and

the rf lock to the reference cavity has a loop bandwidth

of —50 kHz, very little correlation exists between the frequency fluctuations in the two beams at frequencies

:S10 kHz, which contribute dominantly to the linewidth. Consequently, it is appropriate to consider beams locked

to the two cavities as independent sources when measuring

noise at frequencies £10 kHz.

We expect negligible correlation between environmental perturbations of the two reference cavities. The masses

of the two vibrationally isolated tables differ by a factor of 4, and the spring supports for one table are attached to the ceiling, while the spring supports for the other table are supported by posts resting on the laboratory floor. These differences cause the coupling of mechanical vi- brations to differ for the two tables. Additionally, the separate vacuum chambers, temperature control systems,

and acoustic-isolation boxes provide further decoupling

between the two cavities. The largest common-mode vari- ations probably arise from long-term drifts of the room

temperature. Temperature-induced effects should occur dominantly at very low frequencies, so they should not degrade the short- and medium-term stabilities observed

here. (Long-term drifts will be removed when the laser is locked to an atomic reference.) Part of the beam locked to one of the cavities is split off and travels from one isolated platform to the other. There,

it is heterodyned with part of the beam that is stabilized

to the second cavity, providing a measure of the relative frequency deviations between the two optical beams. The

beams locked to the two cavities differ in frequency by

=400 MHz. Mixing the beat note with a precision rf source translates the beat-note frequency lower to facilitate high-resolution analysis. A fairly uniform drift of the beat- note frequency (£2 Hz/s), which may be caused by a slow

temperature drift of one of the cavities, is removed by

mixing the beat note against an rf source with a matching

linear frequency chirp.

Residual relative motion between the two isolated optical tables makes Doppler-shift contributions to the beat- note frequency width. We measure these contributions

using an auxiliary beam sent between the tables [5]. On

one of the tables, a beam is split into two parts —one

of which is given a precise frequency offset for use as

a frequency reference. The second beam travels from

one table to the other along a path very near (=1 cm)

the beam path used for the intercavity beat note and then

retroreflects back to the initial table. The heterodyne beat

note between the auxiliary beam and the reference beam

contains frequency fluctuations from relative table motion

twice as large as the corresponding fluctuations on the in- tercavity beat signal. We digitally divide the frequency

of this Doppler-shift beat signal by a factor of 2. The re- sulting signal can then be mixed with the intercavity beat

signal to remove contributions to the noise from the rela- tive table motion. Typically, relative table motion is not

the dominant source of frequency noise, so we often omit

this correction. A similar Doppler-cancellation technique

will be critical for avoiding linewidth degradation while

transporting light from the vibrationally isolated table to an atomic reference [5,10].

We have analyzed the beat signal between the beams

locked to the two cavities using both frequency-domain

and time-domain techniques. We use a fast Fourier trans- form (FFT) spectrum analyzer to measure the spectrum of

the beat note, as shown in Fig. 1. The width of the spec- trum at its half-power point is 0.9 Hz. Correcting for the 0.477 Hz resolution bandwidth of the spectrum analyzer, we infer that at least one of the beams has a frequency

width less than 0.6 Hz at 563 nm for averaging times up

to 32 s. This fractional linewidth of 1 X 10

-15

is more

than an order of magnitude smaller than previously published results. An additional measurement in the frequency domain in- volves using a spectrum analyzer as a frequency discri- minator [11]. The resolution bandwidth of the spectrum

analyzer is set wide enough to include all noise frequencies of interest. The frequency span is zeroed, and the center frequency is set at the —3 dB point of the beat

note. This arrangement converts frequency noise of the beat note into amplitude noise on the intermediate fre- quency output from the spectrum analyzer, which is then analyzed on an FFT spectrum analyzer. This technique

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Volume 82, Number 19 PHYSICAL REVIEW LETTERS 10 May 1999

directly reveals the noise power spectral density. Hence,

it was a particularly useful diagnostic in the early stages of

the work as we searched for the dominant contributions to the laser linewidth. Conversion of the noise power spectral density to a laser linewidth gives a result consistent with the more direct linewidth measurement. For time-domain measurements, we mix the beat sig- nal down to dc and record the phase evolution using

a computer-based data acquisition system. Figure 2(a)

shows a typical time record. The extreme values of the mixer output correspond to phase differences between the two beams of A0 = rad and tt rad. Typically, A0

varies less than tt rad for times on the order of 1 s, consistent with our subhertz linewidth measurement. Unfortunately, the relative phase differs sufficiently from the in-quadrature condition (A(f> = n/2 rad) so that the conversion from beat-signal amplitude to A cf> is nonlinear and

indeterminate at the extrema. We can, however, frequency

divide the beat note by a factor of 20 and measure similar time records of A0/2O [see Fig. 2(b)]. In this case,

the in-quadrature condition is maintained for time durations long enough to facilitate easy determination of the phase noise for averaging times less than a few seconds. For time intervals S:0.5 s, we perform time-domain measurements using an automated dual-mixer time-difference

measurement system [12]. The fractional frequency instabilities, or Allan deviations cry (r), determined using these two measurement techniques are plotted in Fig. 3, alongside the reported Allan deviations for a number of other

stable laser systems. For the 30 ms to several second time

scale relevant for our proposed optical standard [5], the Allan deviation of our laser is approximately an order of

magnitude less than that of the other stable lasers. We have made preliminary investigations of the sensitivity of the laser-beam frequencies to various environmental perturbations of the reference cavity or laser

f„ + 25 Hz

frequency (Hz)

FIG. 1. Power spectrum of the beat note between two laser beams stabilized to two independent cavities. The averaging

time is 32 s. The resolution bandwidth of the spectrum analyzer

is 0.447 Hz. A nearly uniform relative cavity drift of 2.4 Hz/s

is suppressed by mixing the beat note with a swept synthesizer.

(PD) photodiode.

locking system. For example, we have examined the

tilt and acceleration sensitivity of the cavities by using

solenoids to apply sinusoidal driving forces to the vibrationally isolated optical tables. Tilting a cavity modifies

the optical resonance frequencies because of distortions of the cavity caused by the redistribution or reorientation of the support forces. Accelerations modify the resonance

frequencies because the support forces change to balance

inertial forces, again distorting the cavity. For fixed drive

amplitude, inertial forces vary with the square of the drive

frequency. Cavity distortions due to accelerations dominate in our system at drive frequencies sO.l Hz.

Driving one of the tables at a low frequency of 0.01 Hz

tests predominantly the tilt sensitivity of the cavity reso- nance. For table tilts as large as 20 /trad, frequency shifts are less than 2 Hz. Consequently, driving the table with

similar amplitude at a higher frequency of 1 Hz mainly

probes the acceleration sensitivity of the cavity systems.

At higher motional frequencies, we observe considerable

broadening of the beat-note frequency width. Much of this broadening is caused by Doppler shifts from the component of the table motion along the direction of the beam

traveling between the two tables. Generally, applying the Doppler-shift correction described earlier reduces the beat- note width by a factor of 5 to 10. Assuming the residual broadening is caused by inertial forces on the cavities gives an acceleration sensitivity of —100 kHz/(m/s2

). Theoretical estimates of the sensitivity of our cavity to accelerations suggest that the dominant shifts should be

caused by vertical accelerations, which is consistent with

FIG. 2. Two time records of the beat signal between two

stable 563 nm laser beams. The waveform sampling rate is 2 kHz. (a) The extrema of the mixer output of ± 1 correspond

to relative phases A<£ = rad and tt rad. For clarity, an 8 Hz,

first-order, low-pass filter on the mixer output suppresses some

high-frequency noise components, (b) Frequency division of

the beat signal by 20 facilitates conversion from the mixer

output amplitude to a phase for sample periods of many

seconds, and allows calculation of the Allan deviation between

2 ms and 2 s. The signal bandwidth exceeds 50 kHz.

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Volume 82, Number 19 PHYSICAL REVIEW LETTERS 10 May 1999

10"

_ \k

1 1 _ / Nd:YAG/ 2

:

; Nd:YAG ^4 He-Ne A

1 \^o «S_ *{ 1

• A\ 'S< ~-y

co2 ;

: CORE :

- r- v> — -J-.-" ""' B -

1 . ii.i.l

-

10"' 10 10" 10

1

T(S)

102

103 10*

FIG. 3. Allan deviation curves for stabilized lasers. We

calculate <T

y {j) for one of our sources from an analog-to-digital

sample of the beat signal (curve A) and using a dual-mixer measurement system (curve B). The dotted line shows the quantum noise limit for a Hg+

optical frequency standard with one ion and a 30 ms Ramsey interrogation time [13]. Results for other stabilized lasers: (Nd:YAG) Nd:YAG lasers locked to Fabry-Perot cavities [14]; (Nd:YAG/I2 ) iodinestabilized Nd:YAG lasers [15]; (He-Ne) methane-stabilized He-Ne lasers [16]; (C02 ) C02 lasers locked to Os04 [17]

(see comparable results in Ref. [18]); (CORE) Nd:YAG lasers locked to cryogenic resonator oscillators [19].

our observations for various motional modes of the tables. The estimated sizes of the shifts agree within an order of

magnitude with the experimental results. Our laser is suitable for precision spectroscopy and for

optical frequency standards. It has a linewidth of less than

0.6 Hz at 563 nm for averaging times up to 32 s. Future

efforts will involve more detailed study of the sensitivity

of the laser frequency to environmental perturbations such

as temperature, acoustic noise, optical-power fluctuations,

and laser-beam misalignment. The frequency instability

of our laser is still approximately an order of magnitude

higher than the linewidth limit set by the lock to the Fabry-Perot cavity. Consequently, further improvement

may be achieved. Although the results presented here

are sufficiently good for the Hg+

optical standard that we are assembling [5], lasers with lower linewidths could

be beneficial for other standards based on longer-lifetime

metastable states in atoms or ions. One possible route

to more reliable laser operation is the development of an

all-solid-state replacement for the dye laser [20]. Finally, we are assembling a cryogenic Hg+

ion trap [21] for our proposed optical frequency standard. This system,

coupled with a frequency chain to microwave frequencies,

may eventually provide a time standard with an accuracy

near 10" 18

. We thank C. N. Man for the use of her high-finesse

cavity design and R. Lalezari, M. Lauer, and D. Willis of Research Electro-Optics for its fabrication; W. D. Lee

and C. Nelson for the use of their digital servo soft- ware in stabilizing one of our isolation tables; F. Walls

for assistance with the dual-mixer measurement system;

and D.J. Wineland, L. Hollberg, J. Hall, M. Young, and

D. Sullivan for useful comments. This work is supported

by the Office of Naval Research and the National Institute of Standards and Technology.

*Present address: Universidade Estadual de Campinas,

Campinas, SP, 13083-970, Brazil.

[1] Laser Spectroscopy XII, edited by M. Inguscio,

M. Allegrini, and A. Sasso (World Scientific, Singapore,

1996).

[2] Proceedings of the Workshop on the Scientific Applications of Clocks in Space, edited by L. Maleki (Jet Propulsion Laboratory, Pasadena, 1997).

[3] C. Salomon, D. Hils, and J. L. Hall, J. Opt. Soc. Am. B 5, 1576 (1988).

[4] J. Dirscherl, B. Neizert, T. Wegener, and H. Walther, Opt.

Commun. 91, 131 (1992).

[5] J.C. Bergquist, W. M. Itano, and D.J. Wineland, in Frontiers in Laser Spectroscopy, Proceedings of the International School of Physics "Enrico Fermi": Course

120, edited by T. W. Hansch and M. Inguscio (NorthHolland, Amsterdam, 1994), pp. 359-376.

[6] R. W. P. Drever et ai, Appl. Phys. B 31, 97 (1983).

[7] Commercial products are identified in order to adequately

specify the experimental procedure. Such identification does not imply recommendation or endorsement by the National Institute of Standards and Technology.

[8] S.F. Jacobs, Opt. Acta 33, 1377 (1986).

[9] D. Hils, J. E. Faller, and J. L. Hall, Rev. Sci. Instrum. 57, 2532 (1986).

[10] L.-S. Ma, P. Jungner, J. Ye, and J. L. Hall, Opt. Lett. 19, 1777 (1994).

[11] J. Hough et al, Appl. Phys. B 33, 179 (1984).

[12] S. Stein et al, in Proceedings, 36th Annual Frequency

Control Symposium (U.S. Army Electronics Command,

Ft. Monmouth, NJ, 1982), pp. 314-320.

[13] D. J. Wineland et al, in Frequency Standards and Metrology: Proceedings of the Fourth Symposium, edited by

A. De Marchi (Springer-Verlag, Berlin, 1989), pp. 71-77.

[14] N. M. Sampas, E. K. Gustafson, and R. L. Byer, Opt. Lett. 18, 947 (1993).

[15] J. L. Hall et al., in 1998 Conference on Precision Electromagnetic Measurements Digest, edited by T. L. Nelson (IEEE, New York, 1998), pp. 151-152.

[16] S.N. Bagayev, A. K. Dmitriyev, P. V. Pokasov, and

B. N. Skvortsov, in Proceedings of the Fifth Symposium on Frequency Standards and Metrology, edited by J.C. Bergquist (World Scientific, Singapore, 1996),

pp. 289-296.

[17] O. Acef, Opt. Commun. 134, 479 (1997).

[18] V. Bernard et al., IEEE J. Quantum Electron. 33, 1282

(1997).

[19] S. Seel et al., Phys. Rev. Lett. 78, 4741 (1997).

[20] F.C. Cruz, B.C. Young, and J.C. Bergquist, Appl. Opt.

37,7801 (1998).

[21] D.J. Berkeland et al., Phys. Rev. Lett. 80, 2089 (1998).

3802

TN-200

U.S. GPO 2002-0-773-018/40207
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