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PUSltOATSONS

NIST Technical Note 1523

Trapped Ions and Laser Cooling, V

Edited by

James C. Bergquist

John J. Bollinger

Wayne M. Itano

David J. Wineland

Selected publications of the Ion Storage Group

NIST Time and Frequency Division

QC

JT

nstitute of Standards and Technology

l"v Administration, U.S. Department of Commerce

.U5753

#1523

2002 c a

NIST Technical Note 1523

Trapped Ions and Laser Cooling, V

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

rTT\

;,^g

'TES '

U.S. Department of Commerce

Donald L Evans, Secretary

Technology Administration

Phillip J. Bond, Under Secretary of Commerce for Technology

National Institute of Standards and Technology

Arden L Bement, Jr., Director
Certain commercial entities, equipment, or materials may be identified in this document in order to describe an experimental procedure or concept

adequately. Such identification is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology, nor is it intended to imply that the entities, materials, or equipment are necessarily the

best available for the purpose.

National Institute of Standards and Technology Technical Note 1523

Natl. Inst. Stand. Technol. Tech. Note 1523, 203 pages (February 2002)

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CONTENTS

LIST OF ADDITIONAL PUBLICATIONS v FREQUENCY STANDARDS AND METROLOGY

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

W.M. Itano, D.J. Wineland, and D.J. Heinzen, Phys. Rev. A 54, R4649-4652 (1996). TN-1

2. "Laser-cooled mercury ion frequency standard," D.J. Berkeland, J.D. Miller, J.C. Bergquist, W.M. Itano, and D.J. Wineland, Phys. Rev. Lett. 80, 2089-2092 (1998). TN-5

LASER-COOLED NON-NEUTRAL PLASMAS

3. "Phase-locked rotation of crystallized non-neutral plasmas by rotating electric fields," X.-P. Huang, J.J. Bollinger, T.B. Mitchell, and W.M. Itano, Phys. Rev. Lett. 80, 73- 76 (1998). TN-9

4. "Bragg diffraction from crystallized ion plasmas," W.M. Itano, J.J. Bollinger, J.N. Tan,

B. Jelenkovic, X.-P. Huang, and D.J. Wineland, Science 279, 686-689 (1998). TN-1

3

5. "Doppler imaging of plasma modes in a Penning trap", T.B. Mitchell, J.J. Bollinger, X.-P. Huang, and W.M. Itano, Opt. Express 2,314-322(1998). TN- 1

7

6. "Precise control of the global rotation of strongly coupled ion plasmas in a Penning trap," X.-P. Huang, J.J. Bollinger, T.B. Mitchell, W.M. Itano, and

D.H.E. Dubin, Phys. Plasmas 5,1656-1663(1998). TN-26

7. "Direct observations of structural phase transitions in planar crystallized ion plasmas,"

T.B. Mitchell, J.J. Bollinger, D.H.E. Dubin, X.-P. Huang, W.M. Itano, and

R.H. Baughman, Science 282, 1290-1293 (1998). TN-34

QUANTUM MECHANICS AND QUANTUM METROLOGY

8. "Generation of nonclassical motional states of a trapped atom," D.M. Meekhof,

C. Monroe, B. King, W.M. Itano, and D.J. Wineland, Phys. Rev. Lett. 76,

1796-1799 (1996); erratum, 77, 2346 (1996). TN-38

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

B.E. King, and D.J. Wineland, Science 272, 1131-1136(1 996). TN-42

10. "Experimental determination of the motional quantum state of a trapped atom,"

D. Leibfried, D.M. Meekhof, B.E. King, C. Monroe, W.M. Itano, and D.J. Wineland, Phys. Rev. Lett. 77, 4281-4284 (1996). TN-48

11. "Quantum harmonic oscillator state synthesis and analysis," W.M. Itano, C. Monroe,

D.M. Meekhof, D. Leibfried, B.E. King, and D.J. Wineland, Proc, Conf on Atom Optics,

SPIE 2995, 43-55 (1997). TN-53

12. "Simplified quantum logic with trapped ions," C. Monroe, D. Leibfried, B.E. King, D.M.

Meekhof, W.M. Itano, and D.J. Wineland, Phys. Rev. A 55, R2489-2491 (1997). TN-66

111
13. "Entangled states of atomic ions for quantum metrology and computation," D.J Wineland,

C. Monroe, D.M. Meekhof, B.E. King, and D. Leibfried, W.M. Itano, J.C. Bergquist, D. Berkeland, J.J. Bollinger, and J. Miller, Atomic Physics 15, H.B. van Linden van den Heuvell, J.T.M. Walraven, and M.W. Reynolds, eds., Proc. 15th Int. Conf on Atomic Physics, Singapore: World Scientific (1997) pp. 3 1-46. TN-69

14. "Experimental preparation and measurement of quantum states of motion of a trapped

atom," D. Leibfried, D.M. Meekhof, C. Monroe, B.E. King, W.M. Itano, and

D.J. Wineland, J. Mod. Opt. 44, 2485-2505 (1997). TN-85

15. "Experimental primer on the trapped ion quantum computer," D.J. Wineland,

C. Monroe, W.M. Itano, B.E. King, D. Leibfried, D.M. Meekhof, C. Myatt,

and C. Wood, Fortschr. Phys. 46, 363-390 (1998). TN-106

16. "Trapped ions, entanglement, and quantum computing," C.J. Myatt, B.E. King, D. Kielpinski, D. Leibfried, Q.A. Turchette, C.S. Wood, W.M. Itano, C. Monroe,

and D.J. Wineland, Methods for Ultrasensitive Detection, Bryan L. Fearey, ed., Proc, SPIE 3270, 131-137 (1998). TN-134

17. "Shadows and mirrors: reconstructing quantum states of atom motion," D. Leibfried, T. Pfau, and C. Monroe, Phys. Today, 22-28 (April 1998). TN-141

18. "Complementarity and Young's interference fringes from two atoms," W.M. Itano,

J.C. Bergquist, J.J. Bollinger, D.J. Wineland, U. Eichmann, and M.G. Raizen,

Phys. Rev. A 57, 4176-4187 (1998). TN-148

19. "Cooling the collective motion of trapped ions to initialize a quantum register," 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, 1525-1528 (1998). TN-160

20. "Deterministic entanglement of two trapped ions," 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, 3631-3634 (1998). TN-164

TECHNIQUES:

21. "Grating-tuned semiconductor MOPA lasers for precision spectroscopy," J.H. Marquardt,

F.C. Cruz, M. Stephens, C.W. Gates, L. W. Hollberg, J.C. Bergquist, D.F. Welch,

D. Mehhuys, and S. Sanders, Proc, SPIE 2834, 34-40 (1996). TN-168

22. "CW second harmonic generation with elliptical gaussian beams," A. Steinbach, M. Rauner,

F.C. Cruz, and LC. Bergquist, Opt. Commun. 123, 207-214 (1996). TN-175

23. "Sum-frequency generation of continuous-wave light at 194 nm," D.J. Berkeland, F.C. Cruz,

and J.C. Bergquist, Appl. Opt. 36, 4159-4162 (1997). TN-183

24. "Minimization of ion micromotion in a Paul trap," D.J. Berkeland, J.D. Miller, J.C. Bergquist,

W.M. Itano, and D.J. Wineland, J. Appl. Phys. 83, 5025-5033 (1998). TN-187

IV
ADDITIONAL PUBLICATIONS

The following publications were published during the period between February 1996 and December 1998 but are not included in this Technical Note.

1. "An all solid-state Hg+ optical frequency standard," F.C. Cruz, M. Rauner, J.H. Marquardt, L. Hollberg,

and J.C. Bergquist, Proc, Fifth Symposium on Frequency Standards and Metrology, J.C. Bergquist, ed., Singapore: World Scientific (1996), pp. 51 1-513.

2. "Primary atomic-frequency standards: new developments," R.E. Drullinger, S.L. Rolston, and W.M. Itano, Review of Radio Science, 1993-1996, New York, NY: Oxford Univ. Press (1996), pp. 11-41.

3. "Manipulating the motion of a single trapped atom," C. Monroe, D.M. Meekhof, B.E. King, D. Leibfried,

W.M. Itano, and D.J. Wineland, Acct. Chem. Res. 29, 585-590 (1996).

4. "High-accuracy Hg"^ microwave and optical frequency standards in cryogenic linear ion traps," D.J. Berkeland, J.D. Miller, F.C. Cruz, J.C. Bergquist, W.M. Itano, and D.J. Wineland, Proc, Workshop on the

Scientific Applications of Clocks in Space, JPL Publication 97-15, Pasadena, CA: Jet Propulsion

Laboratory (1997), pp. 133-142.

5. "Experimental creation and measurement of motional quantum states of a trapped ion," D.M. Meekhof D.

Leibfried, C. Monroe, B.E. King, W.M. Itano, and D.J. Wineland, Brazilian J. Phys., 27, 178-192 (1997).

6. "Atomic physics in ion traps," C.J. Monroe and J.J. Bollinger, Phys. World 10, 37-42 (March 1997).

7. "High-accuracy frequency standards using laser-cooled Hg+ ions," D.J. Berkeland, J.D. Miller, B.C. Young, J.C. Bergquist, W.M. Itano, and D.J. Wineland, Methods for Ultrasensitive Detection, Bryan L. Fearey, ed., Proc, SPIE 3270, 138-146 (1998).

8. "High-accuracy microwave and optical frequency standards using laser-cooled Hg"^ ions," D.J. Berkeland,

J.D. Miller, F.C. Cruz, B.C. Young, J.C. Bergquist, W.M. Itano, and D.J. Wineland, Proc, Second Int. Symp. on Modem Problems of Laser Physics, Sergei N. Bagayev and Vladimir I. Denisov, eds.. Laser

Physics 8, 673-678(1998).

9. "Formation and control of coulomb crystals in trapped ion plasmas," X.-P. Huang, J.J. Bollinger, W.M.

Itano, J.N. Tan, B. Jelenkovic, T.B. Mitchell, and D.J. Wineland, Strongly Coupled Coulomb Systems,

Kalman, et al., eds.. New York, NY: Plenum, New York (1998), p. 429-432.

10. "Structure and control of coulomb crystals in a penning trap," X.-P. Huang, J.J. Bollinger, W.M. Itano, J.N. Tan, B. Jelenkovic, T.B Mitchell, and D.J. Wineland, Hyperfine Interact. 115, 41-45 (1998).

IL "Quantum state manipulation of trapped atomic ions," D.J. Wineland, C. Monroe, D.M. Meekhof B.E. King, D. Leibfried, W.M. Itano, J.C. Bergquisf D. Berkeland, J.J. Bollinger, J. Miller, Proc, Workshop on Quantum Computing, Proc. R. Soc 454, 41 1-429 (1998).

12. "Coherent quantum state manipulation of trapped atomic ions," D.J. Wineland, C. Monroe, D.M. Meekhof

B.E. King, D. Leibfi-ied, W.M. Itano, J.C. Bergquisf D. Berkeland, J.J. Bollinger, J. Miller, Proc, Symp.

on Modem Trends in Atomic Physics, Adv. Quant. Chem. 30, 41-64 (1998).

13. "Experimental issues in coherent quantum-state manipulation of trapped atomic ions," D.J. Wineland, C. Monroe, W.M. Itano, D. Leibfried, B.E. King, and D.M. Meekhof J. Res. Natl. Inst. Stand. Technol. 103

(3), 259-328 (1998) (available at WAVw.iiist.gov/jres).
it
PREFACE

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

Division, National Institute of Standards and Technology, from February 1996 to December 1998. 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 Coo/mg /K (February, 1996)

Papers listed on page v were published during the period from February 1996 to December 1998,

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, Dan Dubin, Dan Heinzen, Leo

HoUberg, Pei Huang, Brana Jelenkovic, David Kielpinski, Brian King, Didi Leibfried, John Marquardt,

Dawn Meekhof, John Miller, Travis Mitchell, Chris Myatt, Amy Newbury, Chris Gates, Max Rauner,

Andy Steinbach, Michelle Stephens, Joseph Tan, Quentin Turchette, 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), and the U.S. National

Security Agency (NSA). 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

VI
iJ
PHYSICAL REVIEW A VOLUME 54, ^^JMBER 6 DECEMBER 1996

Optimal frequency measurements with maximally correlated states

J. J . Bollinger, Wayne M. Itano, and D. J. Wineland

Time and Frequency Division, National Institute of Standards and Technology, Boulder, Colorado 80303

D. J. Heinzen

Physics Department, University of Texas, Austin, Texas 78712

(Received 16 August 1996)

We show how maximally correlated states of N two-level particles can be used in spectroscopy to yield a frequency uncertainty equal to (NT)~^, where T is the time of a single measurement. From the time-energy

uncertainty relation we show that this is the best precision possible. We rephrase these results in the language

of particle interferometry and obtain a state and detection operator which can be used to achieve a phase

uncertainty exactly equal to the l/N Heisenberg limit, where A^ is the number of particles used in the measurement. [81050-2947(96)50712-2]

PACS number(s): 42.50.Dv, 06.30.Ft, 03.65.Bz, 39.30. -Hw

Quantum limits to noise in spectroscopy [1-4] and inter- ferometry [5-13] have been a subject of fundamental, and to an increasing degree, practical interest. This is especially true

for spectroscopy on trapped atoms or ions where the number

of particles N is fixed and kept small to reduce undesired

perturbations. Naively, the uncertainty of a spectroscopic or

interferometric measurement is limited by counting statistics to be inversely proportional to A^'''^. However, a number of

proposals have shown that by introducing quantum correlations between the particles, the measurement uncertainty can be reduced so that it scales inversely with N rather than

N^'^ [2-13]. None of the ideas discussed so far, however,

have realized the fundamental limit for quantum noise in the measurement of atomic or interferometric phase, which we

show to be precisely equal to l/N. In the present work we

realize this fundamental quantum limit for any A^ with an approach that examines a different type of correlation and

measures a different operator than previously considered. We start by considering the spectroscopy of N two-level

particles. In an equivalent spin representation [1,14], let \J — N/2,Mj=—N/2) denote the state where all the particles are in the ground (spin-down) state \g) (with energy Eg) and

\N/2,N/2) denote the state where all the particles are in the excited (spin-up) state \e) (with energy E^). We show how

the state

\^m) = {\N/2,N/2) + \N/2,-N/2)}/^ (1)

can be used to measure (OQ = {Eg — Eg)/h with a frequency

uncertainty equal to (NT^

~

'

, where T is the time of a single measurement. This state is "maximally correlated" in the sense that a measurement of any one atom's energy eigenstate determines the state of all of the others. It is an A'- particle version of the two-particle states discussed in the Einstein-Podolsky-Rosen experiments [15]. The use of

I^^m) requires measuring a different operator than customary

in spectroscopy. We find a measurement operator which

yields a l/(NT) uncertainty and discuss how this measure- ment operator and \^m) can be realized with small numbers

of trapped ions. Our arguments are phrased in the language

of spectroscopy by the Ramsey technique of separated oscillating fields [16]. The Ramsey technique is formally equivalent to Mach-Zehnder interferometry [4,6]. Therefore, after obtaining our results for optimal frequency measurements, we rephrase them in terms of interferometry.

References [2] and [4] discuss the basic idea of using

correlated or squeezed spin states to improve the precision

in spectroscopy for the Ramsey technique of separated oscil- lating fields. We briefly review the idea here and start by

considering the case of A' uncorrected particles where

each particle is initially prepared in its ground state 1^). The

initial state of the composite system for this case is equivalent to the \J^N/2, Mj= -N/2) state of a y^A'/2 spin. This

initial (r=0) state has {J^)q=-N/2, {J^)o = {Jy)Q = 0, and

A7;,(0) = A7j,(0) = N"^/2. [Here (A), denotes the expectation value of an operator A at time t and AA{t)

^{AA^)]'^, where AA^^A^-{A)^.] The Hamiltonian for the equivalent spin system is H= — jl- B, where il=jXQj is the magnetic moment of the composite system and B is the applied field. Here B = Bqz + B^ , where Bq= —hcjQ/juQ (we assume yu.o<0) and B^ is an applied field used to perform

spectroscopy. We assume that Bj is perpendicular to and

rotates about the z axis according to B^=B][ — x smcot +y coswr]. In the Ramsey technique [16], Sj is applied (is nonzero) for two periods of length t^i2='n/{2£lii), where

fl^=|/AoBi|/^ is the Rabi frequency, separated by a period

of length T during which 5] = 0. It is convenient to describe

the Ramsey technique in a frame of reference rotating with

B]. With the assumption n^^lwQ— ct>|, the first pulse rotates the spin vector about Sj (the y axis in the rotating

frame) by 90°. The spin vector then precesses about the z axis during the field-free period, acquiring an angle (wq — (o)T relative to its initial direction (the -x axis) in the

rotating frame. This angle could be read out by measuring,

for example, J^ in the rotating frame. Experimentally this is done by applying the second 7r/2 pulse, which rotates the spin vector by 90° about the B^ axis, and then measuring the number of atoms in \e). This final measurement is equivalent

to measuring J^ . We obtain

(y,),^=(iV/2)cos(a>o-a>)7", (2)

54 R4649

TN-1
R4650 BOLLINGER, ITANO, WINELAND, AND HEINZEN 54

where tf=^2t^i2+T. Throughout this paper we assume T

>t^i2 SO that tf—T. Measurements of 7^ as a function of (x>, along with Eq. (2), can be used to estimate the frequency wq [We assume that wq is sufficiently well known that w can be chosen on the

central fringe of Eq. (2). This is true for atomic frequency

standards and clocks.] Because of the statistical nature of

quanmm mechanics, the number of particles which are observed to make a transition to \e) fi-om measurement to measurement will fluctuate by AJ^{tf) [1,2,4]. This produces an

uncertainty in the estimate of coq of |Aa»| = AJ,{tf)/\d{J^), /d(o\ [4]. For the initial \NI2-NI2) state of uncorrelated particles, we obtain |Aa>(uncorrelated)|

=A7y(0)/[T|(y,)o|] = A^~''^7~^ independent of o). [Experimental measurements are usually made with w — ioq ±7t/{2T), where Eq. (2) has its steepest slope. This mini- mizes the contribution of any added noise.] With correlations between the internal states of the particles it is possible to

start with a state {J) = z{J^)o such that A7j,(0)<N"^/2.

Such "spin-squeezed" states can be used to improve the resolution in Ramsey spectroscopy if /^Jy(0)/\{J^)o\

<{2jy^^=N-^'^.

Some correlated states, such as [^ m) in Eq. (1), have a mean spin vector (/)=0. In this case the previous description

of the Ramsey technique in terms of the precession of a mean spin vector is inadequate. In order to motivate how the maximally correlated state \'^m) can improve the precision of Ramsey spectroscopy, recall that for a single particle the precession angle {o)q— (x))T that is measured in the Ramsey

method is, in the rotating firame, just the phase factor

g-'(«»o-«^)7' that the excited state \e) acquires relative to the ground state \g) during the freed precession period T. Consequently, it may be possible to improve the precision of

Ramsey spectroscopy with a state which, when rotated by

the first 7r/2 pulse, is a coherent superposition of two energy

eigenstates whose energies differ by more than h{aiQ—u>). For N two-level particles the eigenstates \N/2,—N/2) and

\N/2,N/2} provide the largest energy difference, with an ac- cimiulated phase difference over the free precession period

which is A^ times greater than for a single particle. However,

because {'^;^\j\'^i^}=0, some operator other than J^ must

be detected after the final 7r/2 pulse.

Consider the operator — U'^^^a^,, where cr^. is the z Pauli spin matrix for the Jth atom. For J = N/2 this operator

is diagonal in the \J,Mj) basis with eigenvalues

(— 1)"'"^-'. It can be detected by measuring the number of

particles in either the spin-up or spin-down state. Experimentally this can be done with nearly 100% efficiency using

electron shelving and quantum jump detection [1]. If Ng

particles are measured in the spin-down state (the ground

state), the result of this measurement is assigned the value { — l)^s [17]. Suppose the initial state is exp{ i'7rJy/2}\'^j^), so that at the end of the first 7t/2 pulse

the state \'^m) is created. We want to calculate (O), =(^/|(9|^^), where

|^^) = e-'(^'2)y,^-'"<"o-"'"=|^^). (3)

Let \e)j and |^), denote the excited and ground states of the

TN-2

jth atom and Sy=ayl2, where ay, is the y PauH spin matrix

for the fth atom. The state I'^f) can be written

where = (a)o- w)r/2. With exp[-/(7r/2)5y.] = (1 -5+.-l-5_.)/^/2, Eq. (4) can be rewritten as

1 [ "

1=1

(5)

Explicit computation then yields (0), =( — l)^cos[A'(a)o

-(o)T] and, because 0^=1, (AO^), =sm\N{(Oo- (o)T]. Note that (O), has the same form as that of a single, twolevel system with frequency interval No)q. The state \^m)

can therefore be used in spectroscopy with a frequency uncertainty \A(jj\ = AO(tf)/\d{0), /(?cu| = (A'r)"' independent

of O). The Ramsey method measures Wq by measuring the free precession of A^ identical two-level particles; that is, coq is measured by observing the free time evolution of the system.

With the state I'^m) ^ frequency uncertainty of {NT)~^ is obtained. We show that this is the best precision that can be

obtained on A^ identical two-level particles which undergo

free time evoloution for a period of length T. This follows from an application of the time-energy uncertainty principle

St\LH'^)^h}lA, (6)

where {i!i.H^) is the variance of the Hamiltonian and Sp' is the variance in estimating time from a measurement on the system. (Measurements of an operator A can be used to de- termine time with an uncertainty AAl\d{A)ldt\. See Ref.

[18] for a simple proof of the time-energy uncertainty rela- tion and Ref. [19] for additional rigorous discussions.) For

the system of A^ identical two-level particles, Eq. (6) can be

reexpressed in terms of dimensionless quantities

Scp'^iAh^); (7)

where <p-ajof and /J = 2f=i{ j|e)„(e|-i|g)„(?|}- Note

that we are here considering the full 2'^-dimensional Hilbert space and not just the J=N/2 subspace discussed earlier. We

can establish an upper limit (A/i^)«A^^/4 from (A/z^) = {h^)-{hy^{h^) and {h^)^N^/4. The last inequality fol- lows because the maximum eigenvalue of h^ is N-f4. These

inequalities and Eq. (7) imply S(p^N~^. An uncertainty Sep in determining (p after a free time evolution of duration T

results in an uncertainty Scl>(j= ScpIT in the determination of

wq • From the previous discussion, <5a>o must satisfy
54 OPTIMAL FREQUENCY MEASUREMENTS WITH R4651

S(On

1 NT'

(8)

Ramsey spectroscopy is formally equivalent to MachZehnder interferometry. In Mach-Zehnder interferometry,

schemes [5-13] which use nonclassical input states to ap- proach the 1/A^ Heisenberg limit for large N have been pro- posed [20]. Examples include the illumination of one of the input ports by a squeezed vacuum [5] where experiment has shown improvement over the shot-noise limit [8], the use of

correlated input states [6,7,9], and the use of two Fock states containing equal numbers of particles as inputs [12]. Refer- ence [13] also considers the dual Fock input state, but with a phase measurement scheme that has been optimized according to quantum information theory. Most of these cases show

an asymptotic phase sensitivity proportional to 1/A^. In general, the constant of proportionality is >1. By rephrasing our

results for spectroscopy in terms of Mach-Zehnder interfer- ometry, we obtain an input state and detection scheme that achieve a sensitivity equal to l/N even for small A^. The state after the first beam splitter that is formally equivalent to I^M> is |^M)in.= { \N)a\0)h+ |0) JA^), }/ ^2, whcrc a and b

denote the modes of the two arms of the interferometer, \N} denotes the state with A^ particles, and |0) denotes the vacuum. (The_ particles can be bosons or fermions [7].) The operator O can be detected by measuring the number of

particles Ni,' in the Z>' output mode of the second beam

splitter. The result of such a measurement is assigned the value (-1)^'''.

Cirac and Zoller [21] have recently described a method

for preparing general quantum states of a string of N laser- cooled ions in a linear rf trap. The method uses a well- focused laser beam to couple the intemal states of individual

ions with a mode of the ion string. The mode is assumed to be a quantized harmonic oscillator and is initially prepared in the \n=0) state. Reference [21] discusses the steps needed to prepare the state \^m}. Coupling the intemal state of an

individual ion with a mode of the ion string without permrbing the state of the neighboring ions may be difficult. Typical

ion spacings are approximately 10 /am [22] and, because the irequency differences between the modes increase with de- creasing ion separation, small ion spacings are desirable. Here we discuss a different method for making 1"^^) th^t does not require interacting with individual ions. This

method refines and extends the techniques discussed in Refs.

[2] and [4]. In these references, ions in a linear rf trap were assumed to be prepared in either one of the \NI2,±NI2)

states. This was followed by a resonant coupling of the in- temal states of all the ions (with identical interaction

strength) with a center-of-mass (cm.) mode. This coupling

can take the form hCt{J +a + J -a^) or hCl{J +a^ + J ^a),

where a^ (a) is the raising (lowering) operator for the cm.

mode, and J + (y_) is the raising (lowering) operator for the J-NI2 ladder of atomic states. If the cm. mode is initially prepared in a coherent state or a squeezed state, Refs. [2] and

[4] show that correlated atomic states can be made. In order

to make the state \'^ m)^ we consider the second-order side- band interactions

H2 = h^'{J+a^b + J-ab^),

H2 = hn"{J+ab^ + J_a^b),

Hj = hn^{b'c + bc^),

(9)

where b and c denote the lowering operator for the second

and third cm. modes. Suppose the system is initially pre- pared in the state |y = A^/2, — 7)|0)a|l)^|0)^. Application of

a 7r/2 pulse with Hj generates the coherent superposition

{|y,-7)|o)Ji),|o),+ |7,-y+i)|i)Jo),|o),}/^.

(10) We now "shelve" the first term of Eq. (10) with a tt pulse of

the 7/3 interaction. This swaps the wave functions of the b

and c cm. modes with the result that the second term in Eq.

(10) remains unchanged but the first term becomes

\J,-J)\0)a\0)i,\l)c . A TT pulse with H2 can now be used to increase Mj by 1 in the second term of Eq. (10) without

affecting the first term. This is then followed by a tt pulse of

H'j, which further increases Mj by 1. In this manner, by

alternating H'2 and H2 tt pulses, Mj in the second term of

Eq. (10) can be increased to 7 — 1 with an « = 1 Fock state in one of the a or b modes and n=0 states in the other cm.

modes. Suppose this term is |7,7— 1)|1)q|0)^|0)c. (A similar argument follows if the n = 1 Fock state is in the b mode.)

Application of another tt pulse with Ht, results in the state

{|7,-7)|0)Jl),10),+ |7,7-l)|l)JO),|0),}/>/2.

(11)

A TT pulse with H'2 now results in the desired state I^m)|o)Ji)6|o),.

Realization of the above scheme appears feasible with a

string of ions in a linear rf trap. The second-order sideband

interactions H'2 and H'2 can be realized for the two cm.

modes corresponding to motion orthogonal to the ion string

axis. For example, if u>q is a ground-state hyperfine transition, then H'2 and H'2 can be realized by stimulated Raman

transitions tuned to Wq + w^ — w^ and tog — oj^ + w^, , respectively. (We assume w^^^o)^ and require that the laser beam

waists be large compared to the ion string.) Parametric mode

coupling has been used in mass spectroscopy experiments

[23] to generate 7/3 and exchange the states of two cm.

modes (in a classical regime). It has also been discussed in the quantum regime [24]. Preparation of n = 1 Fock states has been realized with a single trapped ion [25]. By preparing a

state where one of the ions is shelved in an auxiliary level, these single-ion techniques can be used to create an n = l Fock state for the cm. mode of a string of ions. Alternately,

a coupled trap [24] could possibly be used to transfer an « = 1 Fock state from a single trapped ion to the cm. mode of a

string of ions. Current proposals for accurate microwave frequency stan- dards based on trapped ions include linear ion traps with a small number (A'<50) of trapped ions [26]. Therefore the preparation of \^ m) ^"d its use in frequency metrology is of

important practical interest even for small A^. Models for the decay of quantum coherence predict that the coherence in

\'^ m) "T^^y decay up to A'^ times faster than for a single ion

TN-3
R4652 BOLLINGER, ITANO, WINELANfD, ANfD HEINZEN 54

[27]. In one experiment, a lower limit of 10 min has been

obtained for the coherence time of an individual ion [28]; coherence times more than an order of magnitude longer

than this are anticipated [29]. Therefore, for N<50, coher- ence times for I'^n^) on the order of 10 s or longer appear

feasible in trapped ion experiments. This is comparable to measurement times used in current trapped ion experiments

(where the length of the measurement time can be limited by

the local oscillator stability). In addition, it is long enough to provide a means to study the decoherence of a large quantum

system.

In summary, we have shown how the maximally correlated state {"^m) can be used in spectroscopy to yield a fi-e- quency uncertainty equal to {NT^'K This is the least uncertainty that can be achieved by observing the free time evolution of N two-level atoms. Preparation of 1"^^) and its use

in frequency metrology appear feasible for small numbers of

ions in a linear rf trap.

We gratefully acknowledge the support of the Office of

Naval Research and the U.S. Army Research Office. We

thank D. Leibfried, D. J. Berkeland, and M. Young for useful comments on the manuscript.

[1] W. M. Itano et al, Phys. Rev. A 47, 3554 (1993).

[2] D. J. Wineland, J. J. Bollinger, W. M. Itano, F. L. Moore, and

D. J. Heinzen, Phys. Rev. A 46, R6797 (1992).

[3] G. S. Agarwal and R. R. Pari, Phys. Rev. A 49, 4968 (1994).

[4] D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heinzen,

Phys. Rev. A 50, 67 (1994).

[5] C. M. Caves, Phys. Rev. D 23, 1693 (1981).

[6] B. Yurke, S. L. McCall, and J. R. Klauder, Phys. Rev. A 33, 4033 (1986).

[7] B. Yurke, Phys. Rev. Lett. 56, 1515 (1986).

[8] M. Xiao, L.-A. Wu, and H. J. Kimble, Phys. Rev. Lett. 59, 278

(1987).

[9] M. Kjtagawa and M. Ueda, Phys. Rev. Lett. 67, 1852 (1991); Phys. Rev. A 47, 5138 (1993).

[10] S. L. Braunstein, A. S. Lane, and C. M. Caves, Phys. Rev.

Lett. 69, 2153 (1992).

[11] S. L. Braunstein, Phys. Rev. Lett. 69, 3598 (1992). [12] M. J. Holland and K. Burnett, Phys. Rev. Lett. 71, 1355

(1993). [13] B. C. Sanders and G. J. Milbum, Phys. Rev. Lett. 75, 2944

(1995). [14] R. P. Feynman, F. L. Vernon, Jr., and R. W. Hellwarth, J. Appl. Phys. 28, 49 (1957).

[15] D. Bohm, Quantum Theory (Prentice-Hall, Englewood Cliffs, NJ, 1951), p. 611; D. M. Greenberger, M. Home, and A.

Zeilinger, in Bell's Theorem, Quantum Theory, and Conceptions of the Universe, edited by M. Kafatos (Kluwer Academic, Dordrecht, The Netherlands, 1989), p. 69.

[16] N. F. Ramsey, Molecular Beams (Oxford, London, 1963), p. 124.

[17] In order that the signal-to-noise rado of the detection of O not be degraded, the measurement error of the number of particles

in the spin-down (or spin-up) state must be <?1 atom. This is satisfied with small numbers of trapped ions. [18] A. Messiah, Quantum Mechanics (North-Holland, Amsterdam,

1961), Vol. I, pp. 319 and 320.

[19] C. W. Helstrom, Quantum Detection and Estimation Theory

(Academic, New York, 1976), Chaps. VIII.2.C and VIII.4.c; S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72, 3439

(1994).

[20] Heisenberg limited sensitivity has also been discussed in a

different approach which uses coherent input states and a nonstandard beam splitter. See J. Jacobson, G. Bjork, I. Chuang,

and Y. Yamamoto, Phys. Rev. Lett. 74, 4835 (1995).

[21] J. I. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995).

[22] U. Eichmann et al, Phys. Rev. Lett. 70, 2359 (1993).

[23] E. A. Cornell, R. M. Weiskoff, K. R. Boyce, and D. E. Prit- chard, Phys. Rev. A 41, 312 (1990).

[24] D. J. Heinzen and D. J. Wineland, Phys. Rev. A 42, 2977

(1990).

[25] D. M. Meekhof et al., Phys. Rev. Lett. 76, 1796 (1996).

[26] M. E. Poitzsch, J. C. Bergquist, W. M. Itano, and D. J. Wineland, Rev. Sci. lustrum. 67, 129 (1996).

[27] R. Alicki, Phys. Lett. 103A, 247 (1984); D. F. Walls and G. J. Milbum, Phys. Rev. A 31, 2403 (1985).

[28] J. J. Bollinger et al, IEEE Trans Instrum. Meas. 40, 126

(1991).

[29] D. J. Wineland et al, Ann. Phys. (Paris) 10, 737 (1985).

TN-4
Volume 80, Number 10 PHYSICAL REVIEW LETTERS 9 March 1998

Laser-Cooled Mercury Ion Frequency Standard

D. J. Berkeland, J. D. Miller,* J. C. Bergquist, W. M. Itano, and D. J. Wineland

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

(Received 27 October 1997)

A stable and accurate frequency standard based on the 40.5 GHz ground-state hyperfine transition in i99jjg+

JQjjg jg described. The ions are confined in a cryogenic linear Paul (rf) trap and laser cooled

to form a hnear crystal. With seven ions and a Ramsey interrogation time of 100 s, the fractional frequency stability is 3.3 (2) X 10~'^t~'''^ for measurement times t < 2 h. The ground-state hyperfine

interval is measured to be 40507 347 996.841 59 (14) (41) Hz, where the first number in the parentheses

is the uncertainty due to statistical and systematic effects, and the second is the uncertainty in the frequency of the time scale to which our standard is compared. [80031-9007(98)05392-7]

PACS numbers: 32.80.Pj, 06.30.Ft, 32.10.Fn

Atomic frequency standards [1,2] play vital roles in physics, such as defining the unit of time and other basic

units, realizing fundamental constants, and testing basic

physical phenomena [3]. A good frequency standard

requires that the uncertainty of all systematic effects be

small, and that a high measurement precision can be

reached in a practical time. Here, we describe a frequency

standard based on a laser-cooled linear crystal of ^^^Hg"*"

ions confined in a linear Paul (rf) trap, which satisfies these requirements. The uncertainty from systematic

effects (3.4 parts in 10^^) is approximately equal to the best values reported, from a cesium beam clock (5 parts in 10'^) [4], and a cesium fountain clock (2 parts in 10'^) [5], and can be significantly reduced in future experiments.

An important systematic effect for high-resolution spec- troscopy and atomic clocks is the second-order Doppler

(time-dilation) shift caused by atomic motion. Laser cool- ing can reduce this shift and has been applied to ac- curate atomic clocks based on hyperfine transitions in trapped ^Be"'' ions [6] and Cs atoms in a fountain clock

[5]. Unfortunately, for trapped ions, part of the atomic

motion is due to the trap's electromagnetic field and

is not directly affected by laser coohng. In the linear Paul trap [7], driven motion (termed "micromotion") can be significantly reduced by confining the ions near the nodal line of the rf electric field. A limiting case is a linear crystal of ions confined along the field nodal

line. For example, if Hg"*" ions are laser cooled to the Doppler limit, the magnitude of the time-dilation shift is 2 X 10"^^ [8]. Fluctuations in frequency measurements are typically

expressed by the two-sample Allan variance [9]

""y^'^ = 2(F=1) I

i{cOk}r - {(Ok + l)T)^

2 (1)

where coq is the angular frequency, and (a»^)^ is the kth

measurement of frequency averaged over time r. The

quantity o-yir) is usually called the frequency stability. If measurement of the atomic states is limited by quantum

noise [10], the frequency stability is given by [11]

o-yir)

1 (Oq^NTr

1/2

(2)

In this expression, we assume the atomic transition is driven using Ramsey's method of separated fields [12]

with time Tr between applications of radiation pulses,

a>0 is the angular frequency of the atomic transition, and

A'^ is the number of atoms (assumed constant). Atoms

or ions that have a relatively large hyperfine frequency

are particularly attractive since good frequency stability

is possible even for small A'^. Cigar-shaped clouds of

ions whose long axes coincide with the nodal line of a

linear Paul trap have been employed to realize very stable [ayir) = 5 X 10-1^-1/2] i99Hg+ [13] and i^iYb+ [14]

microwave clocks using approximately 10^ ions cooled

by a buffer gas. However, the fractional systematic

frequency uncertainty in these clocks from the secondorder Doppler shift is about 4 X 10"'"^ [15]. The standard

described here demonstrates both good frequency stability

and accuracy by using laser-cooled ions that are all confined to the field nodal line. Figure 1 shows a partial energy level diagram of

'9^Hg+. A small magnetic field [(1-5) X 10"^ T] is applied to break the degeneracy of the F = 1 states, isolating the '^S\/2 (F = 0, m/r = 0) —> (1,0) hyperfine

clock transition (coq = 2tt X 40.5 GHz). Measurements

of the Doppler-broadened width of the 282 nm electric quadrupole transition give the temperature of the ions and

their heating rate in the absence of laser cooling. Sumfrequency-generated 194 nm light beams [16] drive the indicated electric dipole transitions, which are used to cool the ions. For Doppler cooling, the frequency of a primary 194 nm beam p is tuned slightly below that of

transition p. Although this is a cycling transition, beam

p can excite off-resonantly the ions into the '^P\/i, F = I level, from which they can decay into the ^S\/2, F =

level. To maintain fluorescence, the frequency of a weaker repumping 194 nm beam r is nearly resonant with

that of transition r. Beams p and r are overlapped to form a single beam, then split into beams a and b, which

intersect at the site of the ions. To prevent pumping into

0031-9007/98/80(10)/2089(4)$15.00 © 1998 The American Physical Society

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Volume 80, Number 10 PHYSICAL REVIEW LETTERS 9 March 1998

r(^P,/2) =Y'^s2 ns

2r^ F=1

F = Pi/2 r: \ $ 6.9GHz

194 nm\\ D- F

r{^Ds/2) s 90 ms

= 3 = 2

282 nm

F = \/ } 40.5 GHz

FIG. 1. Partial energy level diagram of '^^Hg"^ in zero magnetic field.

dark states in the F = 1 manifold, beam a is linearly

polarized in the plane formed by beams a and b, and

beam b has its polarization continuously modulated (at a frequency comparable to the maximum Rabi frequency)

between right and left circular. The ions are stored in the linear Paul trap depicted in Fig. 2 [17]. To confine the ions radially, we apply a potential V = Vq cos n r to two diagonally opposite rods while

holding the remaining two rods at ground potential. Typically, Vo = 150 V and n = 27r X 8.5 MHz, giving a radial secular frequency co^ = 27t X 230 kHz. A potential

Uq = -I- 10 V is applied to the two cylindrical end caps to confine the ions axially {a>^ = Irr X 15 kHz). The ions

form a linear crystal along the nodal line of the rf electric field at the center of the trap. The trap is placed in an

enclosure whose top is also the bottom of a liquid helium

Dewar [17]. The cryogenic environment eliminates ion

loss and suppresses frequency shifts caused by collisions with background gas. To reduce Doppler and Stark shifts induced by the trap's rf electric field, we detect and minimize ion micromotion in three noncoplanar directions [18]. We use the Ramsey technique of separated oscillatory

fields to probe the clock transition [12]. First, the ions

are cooled with both beams p and r for approximately

300 ms. Next, beam r is blocked for about 60 ms to optically pump the ions into the ^Si/2, F = level. Both

beams are then blocked during the Ramsey microwave in- terrogation period, which consists of two 7r/2 microwave

pulses of duration tR = 250 ms separated by the free pre- cession period Tr, which we vary from 2 to 100 s in separate runs. Transitions to the F = 1 state are detected by

reapplying only beam p until the ion is pumped optically

Vo cos(Q t)

1.7inm

FIG. 2. Schematic diagram of the linear trap. The trap is formed by four parallel rods of radius r = 0.38 mm, and two

end caps 4 mm apart. A string of ions is depicted at the trap center.

into the F = state (=10-20 ms), while we count the number of detected scattered photons (typically about 1 50

per ion). This process completes one measurement cycle.

We synthesize the microwave frequency from a lownoise quartz oscillator locked to a reference hydrogen maser [19]. To steer the average microwave frequency

into resonance with the clock transition, we step the frequency by +A/, then —A/, about frequency f^

(=a>o/277-), and complete a measurement cycle after each step. Usually, the stepped frequencies lie near the half-maximum points of the central Ramsey fringe. On

successive measurement pairs, we alternate the signs of

the frequency steps to avoid any bias from linear drifts

in, for example, the signal amplitude. The difference

between the number of detected photons for the pair of

measurement cycles gives the error signal sm- A digital servo adjusts the average frequency according to M

fM+\ = /o + gp^M + gi X ^'"' (^) m= ] where /o is the initial value of the frequency, and the proportional gain gp and the integral gain gi are independent

of each other. Typically, the maximum value of M for a

single run is about 130. The average frequency for each

run is calculated after discarding the first four recorded fre- quencies /,, to remove initial frequency offsets. The stability of the steered microwave frequency when

TV = 7 and F/? = 100 s is ayir) = 3.3 (2) X IQ-^'^t'^/^,

for T < 2 h. Consistently, o-j,(t) is about twice the value

expected from Eq. (2), primarily due to fluctuations in the

intensity of beam p. The measured frequency stability is comparable to those of the Cs beam standard NIST-7, for which cry(T) = 8 X

10~'37--i/2

[20], and the Cs fountain

standard', for which ayir) = 2 X \Q-^^r-^'^ [21].

The average frequency for each run is corrected for the systematic effects shown in Table I. We first correct the average frequency of a run for the quadratic

Zeeman shift due to the static magnetic field Bs- From

the measured values of gj [22] and the '^^Hg nuclear

magnetic moment [23], the fractional shift is 1.219 873

(5) X XQ-^^vl, where v± (= 1.4 X 10'°5„ where 5, is expressed in teslas) is the frequency separation in hertz of the ^51/2 [(0,0) — (1, ±1)] field-dependent transitions from the clock transition. The peak-to-peak variation in the static magnetic field between the beginning and the end

of a run is, at most, 1 X 10"^ T. Since B, = 3 X 10"'^ T, an upper bound on the uncertainty in this Zeeman shift is 1.4 X 10~^^. We observe no sidebands on the Am/r =

± 1 hyperfine transitions from power line-induced 60 Hz

fluctuations in the magnetic field. The corresponding up- per limit on the Zeeman shift is <2 X 10"^° when the

static field is 3 X 10~^ T. We also correct for an ac Zeeman shift that depends

linearly on the rf power Frf delivered to the trap. The

uncertainty in this correction dominates the overall sys- tematic uncertainty of the clock frequency. This shift can be caused by magnetic fields due to currents at frequency

2090

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Volume 80, Number 10 PHYSICAL REVIEW LETTERS 9 March 1998

TABLE I. Systematic shifts of the clock transition frequency, expressed fractionally. The magnitudes are calculated using

Prf = 20 mW, 7r = 100 s, Si = 3 X 10~^ T, and Tanibient = 300 K. Here, B(,o and Ba are the magnetic field components

of 60 Hz and fl, Ip is the intensity of beam p during the Ramsey interrogation time, dp is the detuning of the frequency of beam p

from that of transition p, y is the linewidth of the cooling transition, and dS/df is the slope of the background added to the Ramsey fringe. Other symbols are defined in the text.

Magnitude of Overall uncertainty

Shift Scaling effect in effect

Quadratic Zeeman (static) +(fi?) 2 X 10-14 1.4 X io-'5 Quadratic Zeeman (60 Hz) HBlo) <2 X 10-20 <2 X 10-20

Quadratic Zeeman (H) +{Bl) 5 X 10-15 3.2 X io-'5 Blackbody ac Stark •' ambient <1.0 X 10-16 <1.0 X 10-16 Blackbody ac Zeeman [23] 1 ~ J.2

' ambient <1.3 X 10-'^ <1.3 X 10-1^ Light shift from 194 nm radiation Ip8p/{8l + \y') <3 X 10-16 <3 X 10-16

ac Stark (from trap fields) -(£rf> <2 X 10-18 <2 X io-'8 Second-order Doppler -{vVc') <3 X 10-'^ <3 X 10-'^ Background slope -{dS/df)/Tl <2 X 10-19 <2 X 10-19 Neighboring transitions UiBtRW <1 X 10-" <1 X 10-'^ Microwave chirp, leakage and spectrum asymmetries UTr 3 X 10-16 8 X 10-16 HeHum pressure shift -«He <1 X 10-1^ <1 X 10-'^

ft in the trap electrodes that are asymmetric with respect

to the trap nodal line. (In an ideal trap, these asymmetric

currents are absent.) Allowing for this asymmetry to vary

for different ion crystals, we measure the average transi- tion frequency for Prf ranging from about 17 mW (Vq =

140 V) to 68 mW (Vq = 270 V) for each ion crystal. A

fit to these data gives the frequency shift {dco /dP^i)/ ooq and the extrapolated frequency at zero rf power (oq, for that ion crystal. Typically, {dco/dPri)/ ojq, = (2.5 ±

2.1) X 10"^^ mW-i (within the error, this value is the same for each ion crystal), and the uncertainty in the extrapolated frequency averaged over the five ion crystals

used in the frequency measurement is 3.2 X 10"^^. The

additional uncertainty due to possible rf power measure- ment inaccuracies is about 3 X lO"^^. The ac Stark shift due to blackbody radiation at 300 K is 1.0 X 10-^6 [24], but should be much less in the cryogenic environment. From the measured 194 nm

intensity at the site of the ions when the 194 nm sources

are blocked, the ac Stark shift due to stray 194 nm

light present during the Ramsey interrogation time is <3 X 10~'^. Because the ion micromotion is minimized,

the velocity v of the ion motion and the electric field frf that the ions experience can be determined from the measured secular temperature [18]. We find that, after the 194 nm beams have been off for 100 s, the secular temperature T is less than 25 mK. The corresponding

electric field causes a shift of magnitude <2 X 10"^^. This temperature also corresponds to a fractional secondorder Doppler shift of magnitude <3 X 10"^^. We search for an added sloping background signal by in- creasing A/ to 4.25A//f when Tr = 10 s, and 10.25A//J

when Tr = 25 s, where A//? is the frequency separation

between Ramsey fringes. The extrapolated fractional fre- quency shift is <2 X 10~'^ when locking to the central

fringe (A/ = 0.25A//;) and Tr = 100 s. To estimate the

effects of the neighboring field-dependent hyperfine transitions, we assume the microwave field coupling strength

of the (0, 0) —^ (1, 1) transition equals that of the (0, 0) —

(1,0) transition, while the coupling strength of the (0, 0) —

(1,-1) transition is zero. The corresponding shift is dominated by the ac Zeeman shift, and is less than 1 X

10-'^ when Tr = 100 s and 5, = 3 X 10"^ T. Fre- quency shifts due to the phase chirp of the microwaves

as they are switched on and off (combined with a possible leakage microwave field present during the free pre- cession time Tr), and to asymmetries in the microwave

spectrum, scale as \/Tr. By varying Tr, we measure

the frequency shift from these combined effects to be —3

(3) X \Q-^^/Tr.

At 4 K, the partial pressure of most gases is negligible

[25], with the possible exception of helium. An upper

limit on the collision rate with helium background gas can

be inferred from the ions' temperature after the cooling

beams have been off for 100 s, where we assume that any

heating is caused by collisions with helium atoms. We

further assume that the helium collision rate for heating

is the same as that which causes frequency shifts; we

approximate these rates as the helium density ^He times

the Langevin rate [26]. Using Cutler's measurement of

the helium pressure shift for ^^^Hg"^ [27], we can estimate

a maximum shift of < 1 X 10"^^. The fits to the transition frequencies as a function of

rf power give the extrapolated frequency a»o at zero rf power. This value is reproducible over 18 days, with 42

runs and five different ion crystals. The normalized x^

for the measurements of wq from these five ion crystals is 0.77. If we assume that the frequency depends linearly on

time, a fit to the data gives a drift of -'^ "^^ ^ ""^"'^

day, consistent with zero.

-5 (9) X 10-'^ per

2091

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Volume 80, Number 10 PHYSICAL REVIEW LETTERS 9 March 1998

The hydrogen maser frequency is referenced to primary frequency standards through International Atomic

Time (TAI) [3] to obtain the average value oiq = 2tt X

40 507 347 996.84159 (13) (5) (41) Hz. The first uncertainty is due to the statistical uncertainty in the extrapolation of zero rf power, and the second to the other

systematic shifts shown in Table I combined in quadrature. The third, due to the frequency comparison, is dominated by the published uncertainty in the frequency of

TAI [28]. This value of wq is to be compared with the previous most accurate measurement, which gave ojq =

277 X 40507 347 996.9 (3) Hz [29].

The uncertainty in our measurement is limited primarily by the uncertainty in the Zeeman shift due to fields

at the trap frequency £l. It can be reduced with more

measurements of (oq, and by decreasing fl and the trap

dimensions, thus reducing Vq for a given a)r. Better magnetic shielding will reduce fluctuations in the static magnetic field, and use of a smaller, more tightly confining

trap will allow linear crystals with more ions. By monitoring each ion individually, we can determine their inter- nal states with negligible uncertainty, which will eliminate

noise due to laser frequency and intensity fluctuations. Finally, we are also investigating the use of entangled

states to reduce (Ty{T) [30]. This work was funded by ONR and ARO. We are

grateful to R. DruUinger, S. Jefferts, D. Lee, T. Parker,

and P. Walls for useful discussions. We thank F. Walls

for providing the microwave frequency synthesizer [19],

and T. Parker for frequency comparisons to TAI. We

thank P. Huang, D. Lee, M. Lombardi, D. Sullivan, and

M. Young for carefully reading this manuscript.

*Present address: KLA, Austin, TX.

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[2] P.T.H. Fisk, Rep. Prog. Phys. 60, 761 (1997).

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[5] E. Simon, P. Laurent, C. Mandache, and A. Clairon, in Proceedings of the 11th European Frequency and Time

Forum (Swiss Foundation for Research in Microtechnology, Neuchatel, Switzerland, 1997), p. 43.

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

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Heidelberg, 1989), p. 71.

[12] N. F. Ramsey, Molecular Beams (Oxford University Press, London, 1956).

[13] R.L. Tjoelker, J.D. Prestage, and L. Maleki, in Ref. [1], p. 33.

[14] P.T.H. Fisk, M.J. Sellars, M.A. Lawn, and C. Coles, in Ref [1], p. 27. [15] R.L. Tjoelker, J.D. Prestage, G.J. Dick, and L. Maleki,

in Proceedings of the 1993 IEEE International Frequency

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[19] C.W. Nelson, F.L. Walls, E.G. Ascarrunz, and P. A. Pond, in Proceedings of the 1992 IEEE Frequency

Control Symposium (IEEE, Piscataway, NJ, 1992), p. 64.

[20] W. D. Lee, J. H. Shirley, J. P. Lowe, and R. E. DruUinger,

IEEE Trans. Instrum. Meas. 44, 120 (1995).

[21] A. Clairon, S. Ghezali, G. Santarelh, Ph. Laurent, S.N.

Lea, M. Bahoura, E. Simon, S. Weyers, and K. Szymaniec, in Ref. [1], p. 49.

[22] W.M. Itano, J.C. Bergquist, and D.J. Wineland, J. Opt.

Soc. Am. B 2, 1392 (1985).

[23] B. Cagnac, Ann. Phys. (Paris) 6, 467 (1961).

[24] W. M. Itano, L. L. Lewis, and D. J. Wineland, Phys. Rev. A 25, 1233 (1982).

[25] G. Gabrielse, X. Fei, W. Jhe, L. A. Orozco, J. Tan, R. L. Tjoelker, J. Haas, H. Kalinowsky, T. A. Trainor, and

W. Kells, Am. Inst. Phys. Conf. Ser. 233, 549 (1991).

[26] J. B. Hasted, Physics of Atomic Collisions (American

Elsevier, New York, 1972).

[27] L. S. Cutler, R. P. Giffard, and M. D. McGuire, in Proceedings of the 37th Annual Symposium on Frequency

Control (Systematics General Corp., Wall Township, NJ,

1983), p. 32.

[28] Circular T, No. 113 (Bureau International des Poids et Mesures, Sevres, France, 1997), p. 4. [29] L.S. Cutler, R.P. Giffard, and M.D. McGuire, in Proceedings of the 13th Annual PTTI Applications and Plan- ning Meeting, NASA [Conference] Publication No. 2220

(NASA, Washington, DC, 1982), p. 563.

[30] J.J. Bollinger, W.M. Itano, D.J. Wineland, and D.J. Heinzen, Phys. Rev. A 54, R4649 (1996).

2092

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Volume 80, Number 1 PHYSICAL REVIEW LETTERS 5 January 1998

Phase-Locked Rotation of Crystallized Non-neutral Plasmas by Rotating Electric Fields

X.-P. Huang, J. J. Bollinger, T. B. Mitchell, and Wayne M. Itano Time & Frequency Division, National Institute of Standards and Technology, 325 Broadway, Boulder, Colorado 80303

(Received 29 August 1997)

We report the precise control of the rotation frequency of strongly coupled non-neutral plasmas by

rotating electric fields. These plasmas of up to 10^ 'Be^ ions are trapped in a Penning trap and laser cooled into crystallized structures which undergo a rigid-body rotation. Bragg diffraction shows that the crystalline lattice can be stable for longer than 30 min (—10^ rotations), and that the plasma rotation can be phase locked to the applied field without any slip. These corotating plasmas are in a novel

global thermal equilibrium whose asymmetric surface shape (triaxial ellipsoid) has been measured. [80031-9007(97)04919-3]

PACS numbers: 52.25.Wz, 32.80.Pj

Large numbers of particles with a single sign of

charge can be trapped and cooled in Penning traps [1,2], which use a combination of static electric and magnetic

fields for particle confinement. The global rotation of

these non-neutral plasmas about the magnedc field axis

is necessary for the radial confinement [3]. Active

control of this rotation prevents plasmas from spinning

down under the ambient drag from static field errors and background neutrals, and is important for a number

of experiments including Coulomb crystal studies [4,5], precision spectroscopy [6-8], measurements of particle

and energy transport [9], trapping of antimatter plasmas

[10,11], and storage of highly stripped ions [12]. As an example, the second-order Doppler (time dilation) shift due to rotational velocity in a Penning trap atomic clock can be minimized by stabilizing the rotation at a particular

frequency [7]. Radiafion pressure from laser beams has

been used to vary the plasma rotation frequency [13,14].

However, this method is limited to the few ion species

whose atomic transitions are accessible by a laser, and is not precise due to laser power, frequency, and pointing

fluctuations. Recently, rotating azimuthally asymmetric

("rotadng wall") electric fields have been used to apply

a torque on Mg"*" plasmas with temperatures ranging

from 1 K to 5 X IC* K (4 eV), resulting in steady-state

confinement and density compression [15,16]. For these uncorrelated plasmas, the stabilized rotation frequency is somewhat less than that of the rotating field, with a slip

which increases with the plasma temperature.

In this Letter, we demonstrate that rotating wall electric fields applied to a Penning trap with quadratic potentials can control the rotation of laser-cooled, crystallized

^Be"*" plasmas without slip, thus extending the applicability of this technique from uncorrelated plasmas to strongly

coupled systems. Both rotating dipole and quadrupole

fields perpendicular to the magnetic field are used to provide this precise control of the plasma rotation. We

will concentrate on quadrupole field results, since the dipole configuration apparently requires multiple species or other nonideal effects to be effective and is therefore less general. The rotating quadrupole field interacts with charges near the plasma surface, creating a small- amplitude traveling wave. The torque due to this wave is then transferred to the plasma interior through Coulomb

interactions, which act to bring the plasma to the same

rotation frequency as the applied field [17]. For temperatures up to ~10 K, side-view images show that the plasma shape, which is determined by the rotation fre- quency, can be varied by gradually changing the rotat- ing field frequency. When the plasma is sufficiendy cold

and crystalline latdces are formed, Bragg diffraction provides a more accurate measurement of the rotation fre- quency. It is observed that the latdce and its orientadon

with respect to the axial laser beam can be stable for longer than 30 min (—10^ rotadons), and its rotation can be phase locked to the rotadng field during this dme.

In essence, these corotating plasmas have reached a new

kind of global thermal equilibrium [17], where the rota- tion frequency (and hence the density) is precisely set by

an external drive. We have observed the predicted triaxial ellipsoidal surface of this equilibrium state for oblate (pancakelike) plasmas.

Figure 1 shows the apparatus and the rotating

quadrupole field. 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~^ Pa vacuum

chamber. The uniform magnetic field Bq = 4.46 T is aligned parallel to the trap axis within 0.01°, giving a

^Be"*" (charge e and mass m) cyclotron frequency O =

eBo/m = Itt X 7.61 MHz. A quadradc axisymmetric

potendal im(oj/2e)[z~ - (x- + y^)/2] is generated

near the trap center by biasing the central electrodes to a negative voltage —Vq. At Vq = 1 kV, the single-particle

axial frequency (o^ = Itt X 799 kHz and the magnetron frequency co^ = {O, - yJCl- — 2&>^ )/2 = 27r X

42.2 kHz. The rotadng quadrupole field, which has a potendal ^iy'^ — x-)cos{2a)y^t) + 2xy sm{2co^vt), is generated by applying properly phased sinusoidal voltages

of amplitude Vw to the sixfold azimuthal sectors of the compensadon electrodes [15].

73

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Volume 80, Number 1 PHYSICAL REVIEW LETTERS 5 January 1998

Rotating quadrupole

field (top-view) + +

©B„

Bragg diffraction CCD camera

strobing

side-view camera

X(^

compensation

electrodes (6x60°)

axial cooling beam

FIG. I. Schematic side view of the cylindrical trap with real- space imaging optics and Bragg diffraction detection system. Cross section of the rotating quadrupole field (in the x-y plane)

is shown in the inset.

We create ^Be^ plasmas by ionizing neutral Be atoms

in a separate trap (not shown) and then transferring the ions to the main trap for experimentation [4,14]. This

procedure can be repeated several times to accumulate

up to 10^ ions. Immediately after loading, less than 5%

are contaminant ions, but this fraction grows on a 20 h

time scale due to ^Be"^ reactions with neutrals forming

ions with smaller charge-to-mass ratio. The trapped ^Be"*" ions are Doppler cooled by two laser beams at wavelength A ~ 313.11 nm. From previous experiments [13,14], we

estimate that temperatures 7 ^ 10 mK can be obtained.

Here, k^T refers to the average ion thermal energy in a frame rotating with the plasma which is typically much

smaller than the average kinetic energy in the global

rotation (—10^ K). The axial cooling beam, directed

parallel to Bq, only cools the ion thermal motion while not

affecting the global rotation. A second beam propagating

perpendicular to Bq (not shown and turned off during

the Bragg scattering measurements) is used to vary the

rotation frequency.

When the cloud reaches thermal equilibrium at these cryogenic temperatures, it forms a uniform density spheroid, bounded by z-^/zo + U^ + y^)/rQ = 1, with a rigidbody rotation frequency cor in the range cOm < o)r <

Cl — (Om [14]. The particle density hq is determined

from (x)r according to (o^p = e^no/eom = 2wr(n — (tir)-. where cop is the plasma frequency [14]. Since the in- fluence of image charges is negligible here, the effective trapping potential in the frame rotating with the plasma, U„y„z), is cD, = {m(oy2e)[z^ + /3{x; + y;)],

74

where radial trapping strength /3 = (co^ — a>j)/2a)j =

(OriCl — o)r)/coj — 1/2 > determines the aspect ratio " — zo/ro [14,17]. Anf/5 imaging system detects reso- nantly scattered photons from the axial cooling beam (di- ameter —0.4 mm, power —50 yu,W) to produce a side-view

image of the ^e"*" ions, from which we measure a to obtain (Or and hq. For the typical condition of T ^ 10 mK and aiq S: 4 X 10^ cm~^, we obtain a Coulomb coupling parameter r = ie^/4TT€oav^s)ikBT)~^ > 200, where Wigner-Seitz

radius aws is defined by 47raws/3 — "o^^- The strong

ion-ion coupling results in the formation of crystalline

lattices, which are typically body-centered cubic (bcc),

in nearly spherical plasmas (a — 1) with ion number

N <: 2 X 10^ [4,5]. As shown in Fig. 1, Bragg-scattered

light from the axial cooling beam is detected with a second camera near the forward-scattering direction since

A « flws [4]. When the rotating fields are first applied, their fre- quency (o^^, is set close to cor so that they interact strongly

with the plasma. By measuring the photon scattering rate from the cooling beam for a fixed laser frequency, we

observe that the ion temperature does not change significantly with the application of the rotating fields. Since

the rotating quadrupole field typically causes less than 1%

shape distortion to the plasma (see later discussions), ojr can still be inferred from aspect ratio a within the 5% uncertainty of the method. With V^ sufficiently large, we

are able to vary o)r by gradually changing (Ow By tuning

the laser frequency far below the resonance or blocking

the laser for short periods of time (~1 min), we can in- crease r up to —10 K such that the plasma is weakly

correlated (T « 0.2 for no = 4 X 10^ cm"^), and still be able to control a»r. Figure 2 shows (Or versus co^

I ' >^

6

5

; HIt

Bo y^^

4

"

3

- V

^ * N = 3.8x10^ - X^ N»8.4xio' :

2

•1 L^

)r N= 4.1x10" : • N = 6.2x10"

FIG. 2. The plasma rotation frequency ot^ versus the quadrupole field rotation frequency o)^. The frequency a>r is determined from the aspect ratio a of several plasmas with 200 < Vo :^ 500 V. The inset shows a typical side-view image and its boundary fit to an ellipse (dashed line), giving a

and WrTN-10
Volume 80, Number 1 PHYSICAL REVIEW LETTERS 5 January 1998

for several plasmas with N < 7 X 10'* and a*;. -« 17/2

(cor = n/2 is the Brillouin flow with maximal density

riB = 5.9 X 10^ cm~^). The plasma rotation frequency

(Or tracks a»„, closely within the experimental accuracy, demonstrating external control of the plasma rotation by

the rotating quadrupole field with both weak and strong

Coulomb coupling. For a more accurate determination of ojr, we detect

the time dependence of Bragg-scattered light from ro- tating crystals by strobing the diffraction camera with

a gateable image intensifier [4,5]. Figure 3(a) shows a time-averaged diffraction pattern of concentric rings from

an approximately spherical plasma with A^ ~ 7.5 X 10^. Even if this pattern is from a single crystal, rings are observed because of the plasma rotation about the axial laser beam [4]. With the rotating field applied and controlling the plasma rotation, we trigger the intensifier syn- chronously with the rotating field to open the camera for 50 ns each Itt/cow period. This enables the camera to record the diffraction pattern in the rest frame of the rotat- ing field. Figure 3(b) shows such a time-resolved pattern

taken nearly simultaneously with Fig. 3(a) and accumulated over —10^ rotations. The well-defined rectangular dot pattern demonstrates that the crystal is phase locked to the rotating field. This phase-locked rotation is routinely

obtained when Bq is aligned within 0.01° of the trap axis. In addition, the crystalline lattice and its orientation with

respect to the laser beam can last longer than 30 min under this rotation control. With the rotating quadrupole field, the plasma surface

is actually a spinning triaxial ellipsoid with three principal axes differing in length [17]. This particular shape

results in a total electric field having only the radial component so that the plasma undergoes a purely azimuthal

motion with a uniform rotation frequency. Since (Or =

(i)w, the combined effective trapping potential in the rotat- ing frame becomes <i>^ = {m(ol/2e)\j} + (yS — d)xj: +

(yS 4- 5).v;], where 8 = fg V^/Vq > is the relative strength of the rotating field and fg depends only on the

trap geometry. The thermal equilibrium state in the frame

rotating with the plasma is bounded by z^/zq + x~r/^Q "^ ^r/jo = 1 with xq > yo, and the parallel and perpendicular aspect ratios {a\\ = zo/yo, «! — ^o/yo) are deter- mined by (3 and S [17]. To maximize S/(3, we reduce Vo to less than 200 V

and set w.^, close to ojm- Side-view images of the plasma

are obtained at different phases of the rotating field. In the laboratory frame, the plasma radius along the x

axis is expected to oscillate at 2ojw, with an amplitude

proportional to xq — yo. Figure 4 shows such a plasma

with a\\ =0.15 and ax = 1.4 rotating at 2.5 kHz {Vq =

50 V, Vw ~ 185 V). Excellent agreement between the theory and data is exhibited, showing that the plasma is indeed equilibrated with the rotating field. We have measured the dependence of ai on parameters

S and /3. Figure 5(a) shows ai versus 8 at fixed /3 =

0.103. Good agreement with the theory is obtained by

fitting this data for fg with the result fg ~ 6.38 X 10"^. Figure 5(b) shows ai versus /3 for fixed 8 = 0.0159

using the calibrated fg value. Excellent agreement is again

observed, showing the rapid decrease to unity for ax as

fi is increased. Under typical conditions (Vq ^ 500 V,

V^ < 50 V, and o)„ > l.lcom, giving 5 < 7 X 10""^ and

j3 > 0.1), ax — 1 is less than 1%, but this small distortion apparently generates sufficient torque to phase lock the plasma rotation. We have also examined rotation control using the dipole field with a potential o^ysinicowt) — xcos{(Oy^,t).

Theoretically, this field should not control the plasma

rotation, because for a single-species plasma in a quadratic

trap it only causes a center-of-mass orbital motion about

the trap axis and is thus decoupled from the internal plasma rotation. Experimentally, phase-locked rotation control similar to that by the quadrupole field is obtained.

In addition, because larger dipole fields can be applied.

FIG. 3(color). Bragg diffraction from a crystalline lattice (bcc with a {110} plane 1 to z axis) in a plasma phase locked

to the rotating field (w^ = (o„ = Itt X 140 kHz, no = 4.3 X

10^ cm"^, a ~ 1.1). (a) 1 s time-averaged pattern; (b) time- resolved pattern by strobing the camera with the rotating field (integration time ==5 s). Both graphs are in false color with

logarithmic photon count scales. The long rectangular shadow

is from the deflector for the incident beam; four line shadows

that form a square are due to a wire mesh at the exit window

of the vacuum chamber.

FIG. 4. Oscillation of the plasma radius rjab in 1.5 rotation

periods. The theory curve is calculated using the calibrated fg

from Fig. 5(a). The relative phase between the theory and data

is not adjusted.

75

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Volume 80, Number 1 PHYSICAL REVIEW LETTERS 5 January 1998

1.5

1.4 o

>» 1.3

Ill

1.1 h

N =4.1x10"

+ N =7.6x10*

Theory

(a)

P = 0.103 f= 6.38x10'' (fit)

0.00 0.01 0.02 0.03

S-fgV^/Vo

III

1.5

1.4

1.3

1.2

1.1

1.0

fl 1 1 r— 1 T— -T 1 ' 1

\ (b) :

\ N»4.1x10* - V + N = 7.6x10" :

- \^ Theory

5 = 0.0159 >^^^^-—-,-^ * ~

0.0 0.2 0.4 0.6

P=(u^(fi-a)„)/(o/.1/2

FIG. 5. Dependence of perpendicular aspect ratio a^ versus

(a) rotating field strength 8 and (b) radial trapping strength yS. Two plasmas are used with 50 ^ Vq — 100 ^ and 15 < V^, ^

185 V. The geometric factor /^ is calibrated in (a).

rotation stabilization in the range a>„ < w^ ~ 0-9 (H —

a»m), which includes Brillouin flow, has been achieved.

Experimental evidence suggests that the coupling between

the center-of-mass motion and plasma rotation can be

provided by the contaminant ions which have a slighdy

larger center-of-mass displacement than ^Be"*" due to centrifugal inertia. By measuring the plasma dynamic

response under sudden change of w^ [15], we have

observed that the torque from the dipole field increases with time, presumably due to the growing number of

contaminant ions. This observation, however, does not exclude the possibility that other nonideal effects can also provide the coupling.

In the future, direct imaging of individual ions in a crystallized plasma may be possible because of the phase-locked rotation. Furthermore, the increased crystal

stability improves the prospect of observing the solid- liquid phase transition. We thank D. Wineland, D. Dubin, T. O'Neil, and

C.F. Driscoll for discussions; B. Jelenkovic, J. Tan,

J. Bergquist, and L. Hollberg for technical assistance; and

D. Leibfried, C. Myatt, M. Young, and D. Sullivan for useful comments. This work is supported by the Office

of Naval Research.

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C.F. Driscoll (AIP, New York, 1988), p. 28.

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X.-P. Huang, and D. J. Wineland, "Bragg Diffraction from

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76

TN-12
ratory experiments for their penetrating power, propagation over intergalactic dis- tances is not without hurdles. A diffuse isotropic infrared background (DIRB) was

produced when the first galaxies formed. Massive stars in early galaxies produced

large amounts of dust in their winds, re- processing the visual and ultraviolet light

from the stars into infrared (IR) light. By

colliding with these ample IR photons, y

ray photons can disappear and turn into electron-positron pairs (23-25). The most numerous IR photons above the threshold

for pair production with 10-TeV 7 rays have wavelengths ~25 |xm. The mean free path (X^^) for pair creation at multi-TeV

energies is of the order of the distance d of Mrk 501. The exact value depends on the DIRB, which is difficult to measure directly because of the presence of zodiacal light

and galactic cirrus clouds. One can use the observed power law

spectrum (2) to put a limit on the maximum allowed pair attenuation, assuming

that the observed power law is the unat- tenuated spectrum emitted by the source

(consistent with the proton-based model).

In general, only contrived intrinsic spectra

would look like a smooth power law after the quasi-exponential attenuation. The

maximum allowed deviation from the power law [1 — exp(— d/X^^)] is taken to be the size of the statistical error bar at 10 TeV, yielding an optical depth t^^ = d/X^v < 0.7. This limit can be relaxed by a

factor not larger than ~2, admitting for weakly absorbed spectra that still approx- imate a power law (dashed line in Fig. 1). There is some dependence of the attenuation on the shape of the DIRB spectrum.

Useful models for the spectral shape can

be found in (23-25) and yield a similar

limit for the 25-[xm DIRB normalization

vl„ (25 jJLm) < (2 to 4) nW m"^ sf^ The

absence of 7 ray attenuation in Mrk 501 is consistent with no contribution to the DIRB other than from the optically select- ed galaxies, for which one expects ~10%

of their optical emission to be reprocessed

by warm dust, yielding vl^ (25 |xm) ~1

nW m~^ sr~' (26), but would also allow a DIRB that is stronger by a factor of 2 to 4. A DIRB of at least ~3 nW m"^ sr"' is suggested by faint IR galaxy counts and

indicates contributions from dust-en- shrouded galaxies at red shifts of ^ ~ 3 to 4 (24). Electron-based models for the 7 ray emission from Mrk 501 (9) predict deviations from a power law in the multi-TeV

range even without external attenuation

and therefore impose an upper limit on

the DIRB that is below the lower limit from faint IR galaxy counts. If both methods of estimating the DIRB (deviations

from a power law spectrum in the multiTeV range and faint IR galaxy counts) use

correct assumptions, a cutoff in the 7 ray spectrum of Mrk 501 must be present in the energy range 10 to 30 TeV.

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27. D. Pe\r/ et al.,\n Proceedings of the 25th International Cosmic Ray Conference, Durban, South Africa. August 1997, P. A. Evensonefa/., Eds. (Intemational Union of Pure and Applied Physics, in press); S. M.

Bradbury ef al., Astron. Astrophys. 320, L5 (1997). 28. I thank P. Biermann, A. Dar, J. Kirk, H. Meyer, J. Primack, W. Rhode, F. Rieger, and the referees for

their critical reading and suggestions for improve- ment of the manuscript. This research was generously supported by the Deutsche Forschungsgemeinschaft under travel grant DFG/Ma 1545/6-1.

3 September 1997; accepted 25 November 1997

Bragg Diffraction from Crystallized Ion Plasmas

W. M. Itano,* J. J. Bollinger, J. N. Tan,t B. Jelenkovic.l:

X.-P. Huang, D. J. Wineland

Single crystals of a one-component plasma were observed by optical Bragg diffraction. The plasmas contained 10^ to 10^ single-positive beryllium-9 ions (^Be'^) at particle densities of 10^ to 10^ per cubic centimeter. In approximately spherical plasmas, single

body-centered cubic (bcc) crystals or, in some cases, two or more bcc crystals having

fixed orientations with respect to each other were observed. In some oblate plasmas, a

mixture of bcc and face-centered cubic ordering was seen. Knowledge of the properties

of one-component plasma crystals is required for models of white dwarfs and neutron

stars, which are believed to contain matter in that form.

1 lasmas, the ionized states of matter, are usually hot and gaseous. However, a suffi- ciently cold or dense plasma can be liquid or solid. A one-component plasma (OCP)

consists of a single charged species embedded in a uniform, neutralizing background

charge (I). Aside from its intrinsic interest

Time and Frequency Division, National Institute of Stan- dards and Technology, Boulder, GO 80303, USA.

*To whom correspondence should be addressed. E-mail;

w/itano@nist.gov

tPresent address: Frequency & Time Systems, Beverly, MA 01915, USA. tOn leave from the Institute of Physics, University of Bel- grade, Belgrade, Yugoslavia.

as a simple model of matter, the OCP may

be a good model for some dense astrophysical plasmas (2), such as the crusts of neutron stars or the interiors of white dwarfs, where the nuclei are embedded in a degenerate electron gas. According to calculations, a classical, infinite OCP freezes into a bcc lattice when the Coulomb coupling

parameter

1 e'

4-n-€o owsKbT

is approximately equal to 170 (3). Here, Eq

is the permittivity of the vacuum, e is the charge of an ion, kg is Boltzmann's con686 SCIENCE • VOL. 279 • 30 JANUARY 1998 • www.sciencemag.org

TN-13
stant, T is the temperature, and aws is the Wigner-Seitz radius, defined by 47ra^g/3 =

I/tIq, where n^ is the particle density; T is the ratio of the Coulomb potential energy

of neighboring ions to the kinetic energy

per ion. Ion plasmas can be confined and

brought to thermal equilibrium in Penning

traps. Such systems have static thermal

equilibrium properties equivalent to those

of an OCP, where the magnetic field takes the place of the background charge (4-6). Calculations (7) and experiments (8) for approximately spherical plasmas having

N «= 10^ to lO"* ions show concentric shell structures, dominated by surface effects. Calculations by Dubin and O'Neil (9, JO)

suggest that a bcc lattice might begin to form in the center when the number of concentric shells is greater than about 30, which corresponds, for a spherical plasma,

to N =« 10^. Ordered structures of tens of thousands of ions have been observed in a radio-frequency (rf) quadrupole storage ring

{11) and in a linear rf trap (J 2) but, because

of the elongated shapes of these structures, surface effects dominated and bulk structure was not observed. Tan et al. have reported Bragg diffraction patterns from laser-cooled ions in a Penning

trap (J 3). For approximately spherical plas- mas with 200,000 ions or more, the patterns were consistent with bcc ordering but not with face-centered cubic (fee) ordering.

However, the Bragg patterns were smeared

into circles by the rotation of the plasma

about the magnetic field axis, so it was not

possible to distinguish between scattering

by a single crystal and scattering by several

crystals or to determine the orientation of the crystals. Here we report the observation of time-resolved (stroboscopic) Bragg dif- fraction patterns, from which the effect of

the plasma rotation is removed (J4). In our experiment (Fig. 1 ), the ^Be"^ ions were confined in a cylindrical Penning trap, consisting of an electrostatic quadrupolar

potential and a uniform magnetic field B =

4-465 T, parallel to the z axis. The radial electric field leads to a rotation, at frequen- cy cjj, of the plasma about the z axis. For a given N, an equilibrium state of the plasma can be parameterized by T and w^ (4-6). In the limit of low T, approached in our ex- periments, the plasmas are uniform-density

spheroids. For N = 10^, a spherical plasma

at a typical density of 4 X 10^ cm~^ has a diameter of 1.7 mm.

The ions are cooled by a laser beam propagating along the z axis and tuned slightly lower in frequency than a hyperfine-Zeeman

component of the 2s ^S,^2 to 2p ^P^n reso- nance at 313 nm. The laser power is approximately 50 |xW and is focused at the ion plasma to a diameter of about 0.5 mm. We

estimate that T S 10 mK [15, 16). For a

typical value of tIq = 4 X 10^ cm~^, this results in r S 200. A series of lenses forms an image of the diffraction pattern on an

imaging photodetector.

We used two methods to derive a timing

signal for stroboscopic detection of the Bragg diffraction patterns. The first (passive

method) is based on detecting a photon

from a diffracted beam after it has passed

through an aperture (Fig. 1). The second

(active method) is based on phase-locking

the rotation of the plasma to an applied

rotating electric field {17 , 18). Two types of imaging detectors were

used. One (the MCP-RA detector) is an

imaging photomultiplier tube (PMT) based on a microchannel-plate (MCP) electron

multiplier and a multielectrode resistive an- ode (RA) for position sensing. For each

photon, the position coordinates are derived from the current pulses collected from

the different parts of the RA. The other is a charge-coupled device (CCD) camera couPMT

Lens 3

Imaging

photodetector

Mirror irroy

, /

\ r

Lens 2

Polarizer 2 i^^ Aperture

/ \

/ \

/ V Lens 1 _L \. ' Beam deflector a;

Vacuum envelope

Polarizer 1

Laser t)eam

Fig. 1. Experimental setup. Laser light is directed through the ion plasma in the Penning trap. A

diffraction pattern is created at a plane beyond

lens 2, where rays that are parallel leaving the plasma are focused to a point. A mirror, placed

near that plane, deflects the light to an imaging

photodetector. An aperture placed inside a hole in the mirror allows diffracted light to be detected by

a photomultiplier tube (PMT). The aperture is placed off the axis of the optical system, so the PMT generates a timing signal as the diffraction pattern rotates.

pled to an electronically gateable image

intensifier. Time-integrated diffraction patterns were obtained with both the MCP-RA detector and the CCD camera. Before at- tempting to observe crystal diffraction patterns, we tuned the frequency of the laser beam from several gigahertz to ~10 MHz

below resonance, causing T to vary from

above to below the liquid-solid transition temperature. The duration of the frequency sweep was about 10 to 30 s. About 30% of

the time, we observed a pattern consisting of several sharp rings, indicating that a crystal had been formed (J3, J4). Figure 2, which is consistent with a bcc lattice rotat- ing about a (100) (fourfold symmetry) axis

{19), is an example of such a pattern. In order to compare quantitatively the observed Bragg diffraction pattern to a cal- culated one, it is necessary to know n^, which can be determined from co^ [equation 10 of Bollinger et a!. (6)]. In (J3), co^ and tXq were determined from the aspect ratio a =

zJfQ, where 2rg and Izj^ are, respectively, the radial and axial diameters of the plas- mas [equation 16 of Bollinger et al. (6)]. The uncertainty in w^ determined by fitting the side-view images is ~5%. If there are

discrete Bragg diffraction peaks, to^ can be

determined accurately (to about 0.1%) from

time correlations between scattered photons (Fig. 1). A typical correlation spectrum

is shown in figure 4(a) of Tan et cA. (14). As reported in (13), 14 time-integrated

Fig. 2. Time-integrated Bragg diffraction pattern obtained with the CCD camera. Rotation of the plasma causes the diffraction spots to be

smeared into circles. The long rectangular shad- ow is due to the laser beam deflector. The small

circular shadow is due to the hole in the mirror. The four linear shadows forming a large square

are due to a wire mesh. Here, w^ = 2tt x 1 28 kHz, Ho = 3.90 X 108cm-3, /V = 5 x 10^, a = 1.00, and 2r„ = 1.35 mm.

www.sciencemag.org • SCIENCE • VOL. 279 • 30 JANUARY 1998 687

TN-14

I
Bragg diffraction patterns were analyzed for an approximately spherical plasma having

270,000 ions. Patterns for a larger data set, in which o)^ was determined by photon

correlation, are shown in Fig. 3. The posi- tions of the peaks agree with those calculated for a bcc lattice, to within the 2.5%

uncertainty of the angular calibration. They

disagree by about 10% with the values cal- culated for an fee lattice. The ratios of the

25

Q- 20

1 V2 v'3 2 >/5

15

,:; 10

oJlUt^ 6 8 10 12 14 16

Fig. 3. Histogram representing the numbers of peaks (not intensities) observed as a function of Qaws' where q = kg - k^ is the difference between

the incident (k^) and scattered (kj photon wave

vectors. We analyzed 30 Bragg diffraction pat- terns from two approximately spherical plasmas

having 270,000 and 470,000 ions. The dotted

lines show the expected peak positions, normalized to the center of gravity of the peak at A ({11 0}

Bragg reflections).

Fig. 4. Time-resolved Bragg diffraction pattern of the same plasma as in Fig. 2. Here and in Figs. 5 and 6 the small open circle marks the position of the undeflected laser beam. A bcc lattice, aligned

along a (1 00) axis, would generate a spot at each

intersection of the grid lines overlaid on the image. The grid spacing corresponds to an angular deviation of 2.54 X 10^2 rad^ Here, to, = 2-n- x 125.6

peak positions of the first five peaks agree to within about 1% with the calculated ratios for a bcc lattice. The scatter of the data is much reduced relative to that of figure 3 of Tan etal. (13), reflecting the more accurate

cOj. determination.

In principle. Fig. 3 provides information on the orientations of the crystals. If the crystals formed with random orientations, we would expect Fig. 3 to show a

Fig. 5. A Bragg diffraction pattern with twofold symmetry. It matches the pattern expected for a bcc lattice oriented along a (1 15) direction. A dif- fraction spot is predicted at each intersection of the grid lines. The passive timing method and the MCP-RA detector were used. Here, w^ = 2tt x

,/V = 4 X 10^, a

kHz, Hq = 3.83 X 1( cm"

0.98, and 2r^ = 1 .36 mm.

A/= 5 X 105

Fig. 6. Time-resolved Bragg diffraction pattern showing a superposition of twofold and sixfold symmetric patterns. The rectangular grid connects the points for which diffraction spots are predicted for a bcc lattice oriented along a (110)

direction. An fee lattice oriented along a <1 1 1

)

direction would generate diffraction spots at the

vertices of the hexagon. The orientation of the hexagon has been adjusted to fit the data, and it differs by about 3° from that of the rectangular

grid. The active timing method and the CCD camera were used. Here, co^ = 2tt x 70 kHz, n^ =

2.15 X 108 Q^-3 A/ = 5 x 10^, and 2^^ = 2.27

greater number of diffraction peaks at C

({211} Bragg reflections) than at D ({220}

Bragg reflections), whereas it actually

shows the reverse. This data set showed a preference for alignment of the crystals

with a (100) axis along the magnetic field direction. Preliminary observations indi- cate that the degree to which the magnetic field direction coincides with the symmetry axis of the trap electrodes influences the crystal orientations. Tan et al. have noted (13) that not all of

the diffraction rings allowed for various ori- entations of a bcc lattice were seen at any

given time. This indicated that the portion

of the plasma having bcc ordering included

at most a few crystals rather than many

randomly oriented crystallites. Figure 4 is an

example of a time-resolved diffraction pat- tern obtained with the passive timing method and the CCD camera. In this case, the

diffraction spots all line up on a square grid, consistent with a single bcc crystal oriented

so that the incident laser beam is along a (100) axis. For these data, an angular cali- bration was made with an uncertainty of

less than 1% with a mask. The agreement

between the observed and calculated grid

spacing was ~1%.

In order for a diffracted beam to form,

kj and k- must differ by a reciprocal lattice vector (Laue condition) (20). In a typical x-ray crystal diffraction case, satisfying the Laue condition for many spots requires that the incident radiation have a contin- uous range of wavelengths. Here, the Laue condition is relaxed because of the

small size of the crystal, so a pattern is obtained even with monochromatic radi- ation. If the diameter of the region of the plasma having crystalline order is L, the mismatch in reciprocal space can be about

2'7t/L. The diameter of this plasma was ~1.36 mm. In Tan et al. (13), approximate

lower limits for L of 150 |xm and 240 (xm were determined from the widths and in- tensities of the Bragg peaks, respectively. For this plasma, a^^ = 8.5 [im, and the cubic lattice spacing is 17 |xm. A cube 240

fj-m wide would be about 14 lattice spacings in diameter and would contain about

6000 ions. We also observed patterns that were

consistent with single bcc crystals nearly

aligned along other directions, including

(111), (115), (012), (113), (110), and (013). A pattern consistent with a single bcc crystal oriented along a (115) direction is shown in Fig. 5. Some time-resolved patterns were observed that were not consis- tent with a single crystal but were consis- tent with two or more crystals having a fixed relative orientation. With approximately spherical plasmas

(a between 0.6 and 1.4), different diffrac688 SCIENCE • VOL. 279 • 30 JANUARY 1998 • www.sciencemag.org

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iE13S!30

tiun pdttemi weie oLseived on different cooling cycles. With more oblate plasmas, the same pattern was observed each time. A

very oblate plasma resembles the planar

geometry considered by Dubin and O'Neil

(9, W), in which a stack of bcc (110) planes was predicted to have the lowest energy when there are about 60 or more planes. For some cases with fewer planes, a stack of fee (111) planes has lower energy. In a timeresolved diffraction pattern from a plasma

having a = 0.38 (Fig. 6), the most intense

diffraction spots form a rectangular array, consistent with a bcc lattice oriented along

a (110) direction, that is, a stack of (110)

planes. Weaker diffraction spots, forming a hexagon, are also seen. These appear at the

lowest temperatures. The expected posi- tions of the spots for the {220} Bragg reflec- tions of an fee lattice oriented along a (1 1 1) direction, that is, a stack of (111) planes, are at the vertices of the hexagon overlay.

An ideal hexagonal close-packed lattice, oriented along the [001] direction, would

generate the same hexagonal spot pattern. However, it would also generate another hexagonal spot pattern at a smaller radius, which is not observed. Simulations of ion plasmas show hexagonal patterns resembling fee (111) planes on the layers nearest the surface (7). The

hexagonal diffraction pattern in Fig. 6 could be the result of scattering from surface layers, and the rectangular pattern could result from scattering from the central region.

Some spots in Fig. 6 do not match either the rectangular grid or a hexagonal lattice. They may be due to scattering from a transition region that is neither bcc nor fee. Further examination of oblate plasmas with

different thicknesses may enable the transi- tion from surface-dominated structure to bulk behavior in a finite, strongly coupled

OCP to be studied. REFERENCES AND NOTES

1. S. Ichimaru, Rev. Mod. Phys. 54, 1017 (1982).

2. H. M. Van Horn, Science 252, 384 (1991).

3. E. L. Pollock and J. P. Hansen, Phys. Rev. /\ 8, 31 10

(1973); W. L. Slattery, G. D. Doolen, H. E. DeWitt,

ibid. 21 , 2087 (1980); ibid 26, 2255 (1982); S. Ogata and S. Ichimam, ibid. 36, 5451 (1987); G. S. String- fellow and H. E. DeWitt, ibid. 41, 1105 (1990); D. H. E. Dubin, ibid. 42, 4972 (1990).

4. R. C. Davidson, Pliysics of Nonneutral Plasmas (Ad- dison-Wesley, Redwood City, CA, 1990), chap. 3.3.

5. J. H. Malmberg and T. M. O'Neil, Pliys. Rev Lett. 39, 1333(1977).

6. J. J. Bollinger, D. J. Wineland, D. H. E. Dubin, P/iys. Plasmas ^. 1403(1994).

7. A. Rahman and J. P. Schiffer, P^ys. Rev Lett. 57,

1 1 33 (1 986); H. Totsuji, in Strongly Coupled Plasma

Physics, F. J. Rogers and H. E. DeWitt, Eds. (Ple- num, New York, 1987), p. 19; D. H. E. Dubin and

T. M. O'Neil, Phys. Rev Lett. 60, 511 (1988); J. P.

Schiffer, ibid 61 , 1 843 (1 988); R. W. Hasse and V. V. Avilov, P^ys. Rev A 44, 4506 (1 991); J. P. Schiffer, in Non-neutral Plasma Physics II, F. Fajans and D. H. E. Dubin, Eds. (American Institute of Physics (AlP) Conference Proceedings 331, AlP Press, New York,

1995), p. 191.

8. S. L. Gilbert, J. J. Bollinger, D. J. Wineland, Phys. Rev Lett. 60, 2022(1988).

9. D. H. E. Dubin, Phys. Rev A 40, 1 140 (1989).

10. andT. M. O'Neil, \n Strongly Coupled Plasma

Physics, S. Ichimaru, Ed. (Elsevier, Amsterdam,

1990), p. 189.

11. G. Birt<l, S. Kassner, H. Walther, Nature 357, 310

(1992).

12. M. Drewsen, C. Brodersen, L. Hornekaer, J. S. Hangst, J. P. Schiffer, in preparation.

13. J. N. Tan, J. J. Bollinger, B. Jelenkovic, D. J. Wineland, Phys. Rev. Lett. 72, 4198 (1995).

14. A preliminary report of the time-resolved detection has appeared [J. N. Tan, J. J. Bollinger, B. Jelenkovic, W. M. Itano, D. J. Wineland, in Physics of Strongly Coupled Plasmas , Proceedings of the Inter- national Conference, Binz, Germany, W. D. Kraeft and M. Schlanges, Eds. (World Scientific, Singapore, 1996), p. 387].

15. J. J. Bollinger and D. J. Wineland, Phys. Rev Lett. 53,348(1984).

16. L. R. Brewer ef a/., Phys. Rev A 38, 859 (1988).

17. X.-P. Huang, F. Anderegg, E. M. Hollman, 0. F. Driscoll, T. M. O'Neil, Phys. Rev Lett. 78, 875

(1997).

18. X.-P. Huang, J. J. Bollinger, T. B. Mitchell, W. M.

Itano, /b/d. 80,73(1998).

19. The notation refers to the conventional cubic unit

cells for the bcc and fee lattices. A lattice plane specified by Miller indices h, k, and € is denoted by ijikt); a direction specified by components n, , Oj, and n^ Is denoted by [n^/ij/ig]; a set of planes that are equivalent by symmetry is denoted by {hk(}; and a set of directions that are equivalent by symmetry Is denot- ed by (n^n^n^). See, for example, N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders

College, Philadelphia, PA, 1976), chap. 5. 20. N. W. Ashcroft and N. D. Mennin, ibid., chap. 6. 21

.

This manuscript is a wor1< of the U.S. govemment. It

is not subject to U.S. copyright. We acknowledge

financial support from the Office of Naval Research.

29 August 1997; accepted 14 November 1997

Changes in the West Antarctic Ice Sheet Since

1 963 from Declassified Satellite Photography

Robert Bindschadler* and Patricia Vornberger

Comparison of declassified satellite photography taken in 1 963 with more recent satellite imagery reveals that large changes have occurred in the region where an active ice stream enters the Ross Ice Shelf. Ice stream B has widened by 4 kilometers, at a rate much faster than suggested by models, and has decreased in speed by 50 percent. The

ice ridge between ice streams B and C has eroded 14 kilometers. These changes, along

with changes in the crevassing around Crary Ice Rise, imply that this region's velocity

field shifted during this century.

(_)ne of the major uncertainties in the Intergovernmental Panel on Climate

Change's projection of future sea level is the uncertain behavior of the West Antarctic Ice Sheet ( J ). It was much larger during the last glacial maximum 20,000 years ago, and its retreat since then has been rapid at times (2). Current behavior does not indi- cate that it is now retreating rapidly, but

areas of rapid change have been discovered and the potential for unstable behavior re- mains under study.

The thick West Antarctic Ice Sheet is grounded on a submarine bed contained in an extensional rift basin coated with thick marine sediments and is subject to high

geothermal heat flow (3). Discharge of West Antarctic ice is dominated by rapidly moving ice streams. These ice streams

feed floating ice shelves; the transition from grounded to floating ice occurs at the "grounding line." Occasionally ice shelves ground, forming ice rises that the ice shelf must flow around. Between ice streams, the ice accumulates to form higher elevaR. Bindschadler, Code 971, NASA-Goddard Space

Flight Center, Greenbelt, MD 20771, USA.

P. Vomberger, General Sciences Corporation, 6100

Chevy Chase Drive, Laurel, MD 20727, USA.

*To whom correspondence should be addressed.

tion ridges that slowly flow laterally into the ice streams across heavily crevassed shear margins.

Declassified intelligence satellite photography (DISP) recently made available

affords a direct view of the ice sheet's configuration in the early 1960s, greatly ex- tending the limited surface observations made during the International Geophysical

Year in 1958 to 1959. Here, we analyzed

changes in the mouth of ice stream B: from

the downstream tip of ridge B/C (between

streams B and C) to the area just downstream of Crary Ice Rise (Fig. 1) (4). The DISP data were collected on 29 and

31 October 1963 (5). The DISP frames we

used were 4 inch by 5 inch (10.16 cm by

12.7 cm) contact negatives, which we

scanned at 600 dots per inch to convert them to digital form. They were collected by the cartographic camera onboard a Corona mission satellite and have a ground

spatial resolution of about 150 m. Our sec- ond data set is a mosaic of two images from

the advanced very high resolution radiometer (AVHRR) collected on 12 November

1980 and 8 December 1992. These images were obtained from the U.S. Geological

Survey (USGS) World Wide Web site as part of an Antarctic mosaic and have a

www.sciencemag.org • SCIENCE • VOL. 279 • 30 JANUARY 1998 689

TN-16
Doppler imaging of plasma modes 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

travis.mitchell@nist.gov

Abstract: We describe a technique and present results for imaging the

modes of a laser-cooled plasma of ^Be"*" ions in a Penning trap. The

modes are excited by sinusoidally time-varying potentials applied to the trap electrodes. 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.

©1998 Optical Society of America

OCIS codes: (350.5400) Plasmas; (120.7250) Velocimetry; (140.3320) Laser cooling; (300.6520) Spectroscopy, trapped ion

^ Work of the U.S. Government. Not subject to U.S. copyright.

References

1. Non-Neutral Plasma Physics II, eds. J. Fajans and D. H. E. Dubin (AIP, New York, 1995).

2. J. H. Malmberg and T. M. O'Neil, "Pure electron plasma, liquid, and crystal," Phys. Rev. Lett. 39, 1333-1336 (1977).

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in pure electron plasmas," Phys. Rev. Lett. 60, 1290-1293 (1988).

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

6. D. H. E. Dubin, "Theory of electrostatic fluid modes in a cold spheroidal non-neutral plcisma,"

Phys. Rev. Lett. 66, 2076-2079 (1991).

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

"Electrostatic modes of ion-trap plasmas," Phys. Rev. A 48, 525-545 (1993).

8. R. G. Greaves and C. M. Surko, "Antimatter plasmas and antihydrogen," Phys. Plasmas 4, 1528-1543 (1997).

9. G. Gabrielse, X. Fei, L. A. Orozco, R. L. Tjoelker, J. Haas, H. Kalinowsky, T. A. Trainor, and W. Kells, "Cooling and slowing of trapped antiprotons below 100 meV," Phys. Rev. Lett. 63,

1360 (1989).

10. M. H. Holzscheiter, X. Feng, T. Goldman, N. S. P. King, R. A, Lewis, M. M. Nieto, and G. A. Smith, "Are antiprotons forever?," Phys. Lett. A 214, 279 (1996),

11. D. J. Heinzen, J. J. Bollinger, F. L. Moore, W. M. Itano, and D. J. Wineland, "Rotational

equilibria and low-order modes of a non-neutral ion plasma," Phys. Rev. Lett. 66, 2080-2083

(1991).

12. X.-P. Huang, J. J. Bollinger, T. B. Mitchell, and W. M. Itano, "Phase-locked rotation of crystallized non-neutral plasmas by rotating electric fields," Phys. Rev. Lett. 80, 73-76 (1998).

13. D. H. E. Dubin and J. P. Schiffer, "Normal modes of cold confined one-component plasmas,"

Phys. Rev. £53, 5249-5267 (1996).

#4417 - $15.00 US Received January 13, 1998 (C) 1998 OSA 13 April 1998 / Vol. 2, No. 8 / OPTICS EXPRESS 314

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14. D. H. E. Dubin, "Effects of correlations on the thermal equilibrium and normal modes of a non-neutral plasma," Phys. Rev. E 53, 5268-5290 (1996).

15. C. S. Weimer, J. J. Bollinger, F. L. Moore, and D. J. Wineland, "Electrostatic modes as a

diagnostic in Penning trap experiments," Phys. Rev. A 49, 3842-3853 (1994).

16. M. D. Tinkle, R. G. Greaves, and C. M. Surko, "Low-order longitudinal modes of single- component pleisma.s," Phys. Plasmas 2, 2880-2894 (1995).

17. R. G. Greaves, M. D. Tinkle, and C. M. Surko, "Modes of a pure ion plaisma at the Brillouin

limit," Phys. Rev. Lett. 74, 90-93 (1995).

18. C. F. Driscoll, "Observation of an unstable m = 1 diocotron mode on a hollow electron column,"

Phys. Rev. Lett. 64, 1528-1543 (1990).

19. J. N. Tan, J. J. Bollinger, B. Jelenkovic, and D. J. Wineland, "Long-range order in laser-cooled, atomic-ion Wigner crystals observed by Bragg scattering," Phys. Rev. Lett. 75, 4198-4201 (1995).

20. W. M. Itano, J. J. Bollinger, J. N. Tan, B. Jelenkovic, X.-P. Huang, and D. J. Wineland, "Bragg

diffraction from crystallized ion plasmas," Science 279, 686-689 (1998).

21. Here ujim 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.

22. 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.

23. R. C. Thompson, K. Dholakia, J-L. Hernandez-Pozos, G. Zs. K. Horvath, J. Rink, and D. M.

Segal, "Spectroscopy and quantum optics with ion traps," Phys. Scr. T72, 24-33 (1997).

1. 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 nonneutral (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 [2-4]. 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

Here m and q are the mass and charge of a trapped ion, and Uz 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 [4,5]. 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 [6,7]. 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 azimuthally symmetric modes.

The modes have several potential applications 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 non-destructive method for obtaining basic diagnostic information about the plasma. This is especially important

in anti-matter plasmas [8-10], where conventional techniques for obtaining information

about these plasmas involve ejecting the plasma from the trap. Other apphcations arise from the fact that the modes can strongly influence the dynamical behavior of trapped

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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 be changed through the deliberate excitation

of azimuthally asymmetric modes, and the applied torque can be much greater than

that from the "rotating wall" perturbation [12], which is not mode-resonant. Finally,

the modes may provide useful information on the internal state of a plasma. For example, measurement of the damping of the modes can provide information on the plasma's

viscosity. This measurement could presumably be done in the interesting regime where

the plasma is strongly correlated [13,14]. Previous experimental mode studies on spheroidal plasmas have been limited

to frequency measurements on a small class of modes. With laser-cooled Be"^ ion plas- mas, some quadrupole mode frequencies have been measured and agree well with theory

[7,11]. Mode frequencies have also been measured on spheroidal cryogenic electron plas- mas [15], 0.025-0.5 eV electron and positron plasmas [16], and room temperature Ar"*" ion plasmas [17]. 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, ^-independent (diocotron) modes on cylindrical electron columns [18]. In that work, the mode measurements were important in identifying two coexisting modes.

2. Experimental apparatus

Figure 1 shows a sketch of the apparatus [19,20] 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~* Pa vacuum chamber. A uniform

magnetic field Bq = 4.465 T is ahgned parallel to the trap axis within 0.01°, and

results in a ^Be"*" cyclotron frequency Cl = qBo/m = 27r x 7.608 MHz. The magnetic

field alignment is accomplished by minimizing the excitation of zero-frequency modes

produced by a tilt of the magnetic field with respect to the trap electrode symmetry

axis [7,11]. Positive ions are confined in this trap by biasing the central "ring" electrode

to a negative voltage —V^ 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 the

work reported here, Vq = 2.00 kV which results in a;^ = 27r x 1.13 MHz and a single

particle magnetron frequency Um = [fi — (fi^ — 2ujI)2]/2 = 27r x 84.9 kHz.

We create a Be"*" plasma by ionizing neutral Be atoms in a separate trap (not

shown) and then transferring the ions to the main trap. For the mode work discussed

here, the number of ions was typically 6 x 10"*. While the total charge in the trap is con- served 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 [6] for the electrostatic modes of a single-species plasma, we took data 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 num-

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A

'Be+

axial cooling beam

Figure 1. Sketch of the experimental appeiratus. Modes were excited by applying

in-phcise or 180° out-of-phcise sinusoidal potentials to the trap endcaps.

ber 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 measurements described

here are not affected. The trapped Be"*" ions are Doppler-cooled by two laser beams at wavelength A « 313.11 nm. The main cooling beam is directed parallel to Bq as shown in Fig.

1, and a second beam propagating perpendicular to Bo (not shown and turned off during the mode eigenfunction measurements) is also used to compress the plasma by

applying a radiation pressure torque [4,11]. For mode measurements the axial coolinglaser 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 [4], we expect T < 20 mK.

An //5 imaging system detects the Be''" resonance fluorescence scattered per- pendicularly from the axial coohng beam (waist « 0.5 mm, power ^ 50 /iW) 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.

3. Electrostatic modes of a cryogenic plasma

The 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 uim < uJr < ^ — u}m, about the trap's z axis [2,5]. In this work the rotation

frequency was precisely set by a rotating dipole electric field [12]. As the ions rotate through the magnetic field they experience a Lorentz force which provides the radial

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confining force of the trap. This Wr-dependent confinement results in an w^-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 = eomujp/q'^ where ujp = [2uJr{Cl — ^r)]^ is the plasma frequency. With

the confining potential of Eq. (1), the plasma is spheroidal with boundary z^ /zq +

x'^/rQ + i/^/rQ = 1. The spheroid aspect ratio a = zqJtq is determined by a;^ [4,5]. 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 (/, m), where

I > 1 and < m < / [6,7]. For an {l,m) mode with firequency uim [21], the perturbed

potential of the mode inside the plasma is given by a symmetric product of Legendre

functions,

Here |i and I2 , discussed in Ref. [6] , are scaled spheroidal coordinates where the scaling

factor depends on the frequencies u)r, 17, and uim, and J is a shape-dependent parameter

which also depends on these frequencies. In general, for a given (/,m) there are many

different modes. In this paper we report measurements of the mode frequencies and

eigenfunctions of a few magnetized plasma modes, which are defined as those modes

with frequencies \u)im\ < \ft — 2Wr| [6,7]. In addition, we only discuss measurements

of azimuthally symmetric (m = 0) modes. For ujr <C 17/2, these modes principally

consist of oscillations parallel to the magnetic field at a frequency on the order of w^. In the experiment we detect the axial velocity of a mode. In the linear theory, this is proportional to d'^'^^/dz.

We excite plasma modes by applying sinusoidally time-varying potentials to the

trap electrodes. Azimuthally symmetric {m — 0) even I modes are excited by applying

in-phase potentials to the endcaps (even drive) , while odd / modes are excited by applying 180° out-of-phase potentials to the endcaps (odd drive). Azimuthally asymmetric

{m ^ 0) modes can be excited by applying time-varying potentials to the compensation

electrodes, which have 6-fold azimuthal symmetry. In Refs. [7, 11] quadrupole (/ = 2) mode frequencies were measured by observing the change in the total ion fiuorescence

from the plasma, averaged over the phase of the drive, which occiured when the drive

firequency equaled the mode frequency. However, in order to observe such a change, the

mode excitation must be large enough so that either the fiuorescence firom an ion nonlinearly depends on its velocity or there is some heating of the plasma by the mode. The

larger amplitude drive required by this technique decreases the precision of the mode

measurements.

The new technique reported 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. These Doppler images provide direct measurements

of the mode's axial-velocity eigenfunction [22]. In addition, an accurate measurement

of the mode's frequency can be obtained from the line center 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. We note that for the (1,0) and (1,1) modes, imaging is not required because there is no spatial variation in their eigenfunction. The driven mode amphtude 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 [23]

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N

X

3500

3000 -

2500 -

O 2000

c

O" 1500

CD

<DO

1000

500 -

11

1

' 1

' 1'

1

'

1 '1

'

1

. ^^"^""^^ - /^*"^^

" . 'Y

-•-(9,0)

-A- (8,0) --(7.0) --(2,0)

• ^

• ~~~-jj'~"~~-^ ^^

-

1 I.I.I ^

500 1000 1500 2000 2500 3000 3500 4000

Rotation Frequency (kHz)

Figure 2. Plots of the frequencies of several magnetized plasma modes as a function of rotation frequency for n/27r=7.608 MHz and a)i/27r=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 fre- quency (8, 0) plasma mode are plotted.

4. Experimental results

In Fig. 2 we plot several measured mode frequencies, along with the theoretical predictions, for azimuthally symmetric magnetized plasma modes as a function of a;^ for

uJz/2t: = 1.13 MHz and Q,/2-k = 7.608 MHz. Many different mode frequencies at various

values of uJz have been measured, and on clean clouds agreement between the observed

and predicted mode frequencies is typically better than 1%. In this manuscript we concentrate on describing the images obtained of the (2,0) and the highest frequency (9,0)

magnetized plasma modes. For a given (/, 0), the highest firequency magnetized plasma

mode does not have any radial nodes.

Figure 3 demonstrates the phase-coherent detection 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

ijJ2,o- For uJr <€. fl/2, the oscillation in ro is very small, so the mode principally consists of oscillations in zq at a;2,o- Ions above the z — plane oscillate 180° out of phase with

ions below z = 0. Figure 3(a) shows a sequence of 18 side-view images as a function of the phase

of the mode drive at a;2,o/27r = 1.656 MHz. The plasma's rotation frequency was set to

a;r/27r = 1.00 MHz and the m = 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 + >l2,o cos{u2fit + ^2,0)- Figures 3(b) and 3(c) show

the resultant images of the measured mode amplitude A2^q{x,z) and phase ip2fi{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

d^'^^ /dz over y. The amplitude of the theoretical prediction is scaled to match the

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(b) (c)

Figure 3. (a) Movie of sideview image data obtained on a plasma with ijOr/'iT^= 1.00 MHz while driving a (2,0) mode at u;2,o/27r=1.656 MHz. The magnetic field and axial laser beam point up. The ion cloud dimensions are 2zo = 0.76 mm and

2ro = 0.24 mm, and the density no = 2.70 x 10^ cm~^. Comparison of the ampli- tude (b) and phase (c) extracted from the (2, 0) mode in (a) with the predictions

of linear theory. The theory predictions are on the right.

(b) (c) Figure 4. (a) Movie of sideview image data obtained on the plasma of Fig. 3 with

ijJr/2-K= 1.00 MHz while driving a (9,0) mode at u;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.

experiment, and both amplitudes are normalized to one. From the fitted values of vl2,o and ^o we can estimate the coherent ion mode

velocities if the dependence of the ion fluorescence on velocity (through Doppler shifts)

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is known. For the low temperatures of this experiment a good approximation is to assume a Lorentzian profile with a 19 MHz full- width-at-half- maximum due to the

natural linewidth of the optical cooling transition. With the 20 MHz detuning used in

this experiment, we estimate for the data of Fig. 3 that the maximum coherent mode

velocity, which occurs at z = ±zo, is ~1.5 m/s. The spatial and density changes in the

plasma spheroid for this excitation are too small to be resolved (Az/zq, An/no < 10"'^). Therefore the observed variation in the fluorescence intensity is entirely due to Doppler

shifts induced by the coherent ion velocities of the mode.

1 -K K

m

Figure 5. (a) Movie of sideview image data obtained on a plasma with u!r/2n=

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 pleisma 2zo = 0.70 mm and 2ro = 0.29 mm. We have measured the mode eigenfunctions of a number of different azimuthally

symmetric {m=0) modes including the /=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 highorder odd modes could be excited with the odd drive on the trap endcaps. Figure 4(a)

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shows a sequence of 18 sideview images obtained with the (9, 0) excited by a drive at

ijjgfi/2-K = 2.952 MHz. Figures 4(b) and 4(c) show the fitted amphtude and phase from

this sequence, along with the predictions from theory. Similar high-order even (/,0) modes are more difficult to excite. Finally, Fig. 5 shows images from a plasma with ijjrjl'n = 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 fre- quencies. Characteristics of both modes are seen in the data. Measurements of the (2, 0) mode frequency near this crossing indicate that any frequency shifts due to a non-linear

coupling with the (8, 0) mode are less than a few kilohertz. We note that 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 the excitation of a (9, 0) is observed, presumably because this is an odd

mode.

5. Summary and conclusion

We have described a technique for imaging the eigenfunctions of a laser-cooled ion

plasma. In general, for the azimuthally symmetric modes on spheroidal plasmas dis- cussed here, good agreement is obtained between linear theory and experimental measure- ments made with low mode drive amplitude and laser power. The technique should be

a useful tool for studying deviations from the linear theory, such as large amplitude

frequency shifts and non-linear corrections to the mode eigenfunction. Data like that

of Fig. 5 should be useful for studying the coupling between modes. Finally, the width

of the resonant lineshape of the mode amplitude as a function of the drive frequency

provides a measurement of the mode damping. With low laser power and a sufficiently clean plasma this should provide information on the viscosity of a strongly correlated plasma.

6. Acknowledgments

We thank D. H. E. Dubin and D. J. Wineland for useful discussions, and B. Jelenkovic, R. Rafac and S. Robertson for useful comments on the manuscript. This work is sup- ported by the Office of Naval Research.

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PHYSICS OF PLASMAS VOLUME 5, NUMBER 5 MAY 1998

Invited Papers

Precise control of the global rotation of strongly coupled ion plasmas

in a Penning trap*

X.-P. Huang,''" J. J. Bollinger, T. B. Mitchell, and W. M. Itano Time & 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 1997; accepted 5 January 1998)

Rotating asymmetric electric fields have been applied to control the rotation frequency (and hence

the density) of non-neutral plasmas, which are confined in Penning-type traps and have relaxed

close to thermal equilibrium characterized by a global rigid-body rotation. "Infinite" confinement

times and density compression were first reported for uncorrected plasmas of ~ 10* Mg"*" ions with

temperatures ranging from 1 K to 5 X 10"* K (4 eV) [Huang et ai, Phys. Rev. Lett. 78, 875 (1997)]. In this paper, the rotating field technique has been applied to control strongly coupled plasmas of ~ 10^ ^Be"*" ions which are laser-cooled to millikelvin temperatures so that the plasma freezes into a solid with a crystalline lattice. Here, Bragg diffraction peaks from crystals provide an accurate way

of measuring the rotation frequency, and it is observed that the plasma rotation can be phase locked

to the applied rotating field without any slip. In essence, these corotating plasmas have reached

thermal equilibrium with the rotating field, and the azimuthally asymmetric boundaries of the equilibrium states have been measured experimentally. Both rotating dipole and quadrupole fields have been used to provide this precise control of the plasma rotation. However, the effectiveness of

the dipole field depends on the presence of multiple ion species. With the rotating dipole field, density compression to near the Brillouin limit and increase of the rotation frequency to near the cyclotron frequency have been achieved. [S1070-664X(98)91 105-3]

I. INTRODUCTION

Unneutralized plasmas with a single sign of charge are often confined in Penning-type traps'"'^ for a variety of ex- periments including plasma physics,' Coulomb crystal

studies,'*'^ precision spectroscopy,^'^ antimatter research,*'^ and storage of highly charged ions.'*' Since there is an aver- age radial electric field, these trapped non-neutral plasmas

undergo a global (ExB) rotation about the magnetic field axis. In principle, perfect confinement can be obtained in an

ideal trap with cylindrical symmetry due to conservation of

(canonical) angular momentum." In practice, background

,12-14 12,15-18 neutral molecules ex-

"" and static field asymmetries

ert an ambient drag on the rotating plasma, causing slow

expansion and eventual particle loss. Radiation pressure from laser beams has been used to balance the angular mo- mentum loss and to vary the plasma rotation frequency. '^'"'^ However, this method is limited to the few ion species whose

atomic transitions are accessible by a laser, and is not precise

due to laser power, frequency, and pointing fluctuations. Recently, azimuthally asymmetric ("rotating wall") electric

fields rotating in the same sense as the plasma have been

used to exert a torque on Mg"*" plasmas with temperatures

*Paper gTuaI2-2 Bull. Am. Phys. Soc. 42, 1876 (1997),

'''Invited speaker.

ranging from 1 K to 5 x lO"* K (4 eV), resulting in steadystate confinement and density compression.'^'"^' For these uncorrelated plasmas, the stabilized rotation frequency is somewhat less than that of the rotating field, with a slip

frequency which increases with the plasma temperature.^^

In this paper we show that rotating wall electric fields applied to a Penning trap with quadratic potentials can control the rotation of laser-cooled, crystallized ^Be"*" plasmas

without slip, thus extending the applicability of this tech- nique from uncorrelated plasmas to strongly coupled

systems.'^ Precise control of the plasma rotation is important

for some applications. As an example, the second-order Doppler (time dilation) shift due to rotational velocity in a Penning trap atomic clock can be minimized by stabilizing the

rotation at a particular frequency.^ Both axially independent

dipole and quadrupole fields in the plane perpendicular to the magnetic field have been used to provide this precise control of the plasma rotation. However, experimental evidence suggests that the effectiveness of the dipole field requires the presence of more than one ion species or other nonideal effects.The rotating field control, which relies on ion-ion inter- actions, is fundamentally different from the sideband cooling

or "axialization" techniques, where rf fields convert single1656 TN-26

Downloaded 25 May 2001 to 132.163.135.22. Redistribution subject to AlP license or copyright, see http://ojps.aip.org/pop/popcr.jsp

i
Phys. Plasmas, Vol. 5, No. 5, May 1998 Huang et al. 1657

particle magnetron motion into damped axial or cyclotron motions. ^^""^ The steady-state sideband-cooled radius of the magnetron motion depends on the strength of the rf field and

the damping rate of the motion to which the magnetron motion is being coupled. In general, the rotation control will be

of limited precision and may be effective only for low rotation frequencies where the space-charge field is weak.'^* In

contrast, the rotating fields interact with ions near the plasma

surface, creating a small-amplitude traveling wave. The

torque due to this wave is then transferred to the plasma

interior through "viscosity" or strong ion-ion Coulomb coupling, which acts to bring the plasma to the same rotation frequency as the applied field.^^ In the present experiment, side-view images show that the plasma shape, which is determined by the rotation fre- quency, can be varied by slowly changing the rotating field frequency for both weakly and strongly coupled plasmas.

When the plasma is sufficiently cold and crystalline lattices are formed, Bragg diffraction provides a more accurate measurement of the rotation frequency. It is observed that the

lattice and its orientation can be stable for longer than 30 min ( — 10^ rotations), and its rotation can be phase

locked to the rotating field during this time. In essence, these

corotating plasmas have reached a novel global thermal

equilibrium,^^ where the rotation frequency (and hence the

density) is set precisely by the external drive. For the rotating

quadrupole field we have measured the triaxial ellipsoidal surface of this equilibrium state for oblate (pancake-like)

plasmas, and found quantitative agreement with the analytical theory. For the rotating dipole field applied to plasmas

with contaminant ions we have observed the expected asymmetric distribution of the ^Be"*" ions in close agreement with

Monte Carlo simulations. Control of the global rotation has been achieved for nearly all allowed rotation frequencies

with the use of the rotating dipole field.

II. EXPERIMENTAL SETUP

Figure 1 shows the apparatus with its optical diagnostics

and schematics of the rotating dipole and quadrupole fields. 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" * Pa vacuum chamber. The uni- form magnetic field Bo = 4.46 T is aligned parallel to the

trap axis within 0.01°, giving a ^Be"^ (charge e and mass m)

cyclotron frequency fl = eBQ/m = 2'TTXl.6l MHz. An axi- symmetric trapping potential imcol/2e)[z^-ix' + y'^)/2] is generated near the trap center by biasing the central elec- trodes to a negative voltage - Vq- At Vo= 1 kV, the singleparticle axial frequency w^ i s 27rX799 kHz and the magnetron frequency 0}„ = {Cl- ^JCl^-2<ol)/2 is 2-77X42.2 kHz.

The z -independent rotating fields are generated by applying

properly phased sinusoidal voltages of amplitude V^ to the

six-fold azimuthal sectors of the two compensation electrodes, which are positioned symmetrically in the axial direction with respect to the trap center. We create ^Be"*^ plasmas by ionizing neutral ^Be atoms

in a separate trap (not shown) and then transferring the ions

to the main trap for experimentation.'^^ This procedure can be

Bragg diffraction CCD camera

rotating quadrupole

field (top-view) ++

®B»

strobing

side-view camera m

compensation

electrodes (6x60°)

axial cooling beam XO »

FIG. 1. Schematic side-view of the cylindrical trap with real-space imaging

optics and Bragg diffraction detection system. Cross sections of the rotating dipole and quadrupole field (in the x-y plane) are shown separately in the two insets.

repeated several times to accumulate up to 10^ ions. We

expect that essentially all the ions in the trap are singly

charged since the formation of doubly charged ions is energetically forbidden. While the total charge in the trap is conserved after loading, the relative abundance of contaminant

ions increases from <5% for a new cloud on a time scale of

20 h, presumably due to reactions between ^Be"*" ions and

background neutral molecules. By exciting ion cyclotron resonances, we determine that the two main contaminant ion species have mass 10 u and 26 u, where u is the atomic mass unit. These two species are likely to be BeH'"' and

BeOH"*", respectively. Unless noted, data presented in this paper are obtained on relatively clean clouds with <10%

contaminant ions. The trapped ^Be"*" ions are Doppler-cooled by two laser beams tuned slightly below the 2s ^5i/2(M/=3/2,My= 1/2) —>2/7 ^P 3/2(3/2,3/2) resonant transition frequency (wavelength \ = 313.11 nm).'^ These laser beams also optically

pump most of the ions into the 25 ^5 1/2(3/2,1/2) state. From

previous experiments,'^'^" we estimate that temperatures T

& 10 mK can be obtained. Here k^T refers to the average

ion thermal energy in a frame rotating with the plasma,

which is typically much smaller than the average kinetic en- ergy in the global rotation (—10^ K). The axial cooling

beam (waist diameter =0.5 mm, power =50 /ttW), directed

parallel to Bq as shown in Fig. 1, cools the ion thermal motion while not affecting the global rotation. A second laser beam (not shown in Fig. 1) with a much smaller waist diameter ( = 0.07 mm) is directed perpendicular to Bq and is used

TN-27
1658 Phys. Plasmas, Vol. 5, No. 5, May 1998 Huang et al.

to compress the plasma and vary the rotation frequency. This

beam is turned off during the Bragg scattering measurements.When the cloud reaches thermal equilibrium at these cryogenic temperatures, the plasma Debye length becomes

much smaller than its diameter Ir^ and axial length 2zo- The

influence of image charges is negligible here due to the small dimensions of the plasma compared to the trap radius (< 10%). Consequently, the plasma forms a uniform density

spheroid, bounded by z^/zl + (x^+y^)/rl—l, with a rigidbody rotation frequency w^ in the range a»^<Wr

<iCl — (o„)?'^ The particle density azq is determined from w^

according to (o^ = e^nQ/€f)m = 2a>r{i},-(i)r), where cOp is the plasma frequency.^'' The maximum density (Brillouin limit) ng=€QBQ/2m = 5.9x 10^ cm~^ occurs at w^— ^/2, the condition for Brillouin flow. In the frame rotating with the plasma, (Xr,y^,z), the evxB Lorentz force gives rise to a

radially confining pseudo-potential, and the effective vacuum

trapping potential becomes

.„ .2 (D =. 2e -iz^+fiixj + y^r)] .

where /3 is the radial trapping strength defined as

"-2

1 w a>r(n — Wr) 1

a;? >0

(1)

(2)

The parameter /3 determines the aspect ratio a=Zo/rQ of the

spheroid.^°'^^ We use an f/5 imaging system to detect reso- nantly scattered photons from the cooling beams and pro- duce a side-view image of the ^Be"*" ions. From this side- view image, we measure a and obtain lo^ and hq. For the typical conditions of TslO mK and Hq^

4X 10* cm~^, we obtain a Coulomb coupling parameter

r=

47760 aws/ I<^bT

1 >200 , (3)

where the Wigner-Seitz radius Cws is defined by 477^^5/3 = no This strong ion-ion coupling results in the formation

of crystalline lattices, which are body-centered cubic (bcc) in nearly spherical plasmas (a«=l) with ion number

NS:2X 10^."*'^ As shown in Fig. 1, Bragg-scattered light

from the axial cooling beam is detected with a chargecoupled device (CCD) camera near the forward-scattering

direction since the wavelength is much smaller than the lat- tice spacing (X<^avvs)-^ After passing through the trap, the

axial cooling beam is deflected away from the lenses collect- ing the Bragg-scattered light.

III. RESULTS AND ANALYSES

A. General results for both rotating fields When the rotating fields are first applied, their rotation frequency (o^^, is set close to co, so that they interact strongly

with the plasma. By measuring the photon scattering rate from the cooling beams for a fixed laser frequency, which is a function of the ion temperature, we have established that the ion temperature does not change significandy with the application of the rotating fields. Since the rotating fields

(0 /o) w m

FIG. 2. The plasma rotation frequency w^ versus the rotation frequency oi^ of the rotating fields. The frequency w^ is determined from the aspect ratio a of several plasmas with 200« Vq"^ '000 ^- Solid symbols are with the quadrupole field and X 's are with the dipole field. The inset shows a typical side-view image and its boundary fit to an ellipse (dashed line), giving a

and 0)..

typically cause less than 1% shape distortion to the plasma

(see later discussions), the plasma rotation frequency w^ can

still be inferred from the aspect ratio a within the 5% uncertainty of this method. Furthermore, because <!>, depends

weakly on the ion mass for (jj^<Ct/2 as shown in Eq. (2), a multispecies plasma essentially has the same charge distribu- tion as a pure ^Be""" plasma except for the effect of centrifu- gal separation, in which the heavier ions tend to occupy positions at larger radii.'*^ With a sufficiendy large rotating field amplitude V„ , we are able to vary the plasma aspect ratio (and hence w^) by slowly changing (o„ . Figure 2 shows co^ as determined from the side-view images versus w^, with

both the rotating dipole and quadrupole fields and for several plasmas with particle number N<1X 10* and (Or<il/2. The

plasma rotation frequency co^ tracks w,,, closely within the experimental accuracy, demonstrating external control of the plasma rotation by the rotating fields. The data of Fig. 2 were

obtained at low temperatures (TSlO mK) with the axial cooling laser on continuously. However, we are also able to control the plasma rotation with the laser tuned far below the atomic transition frequency or blocked for short periods of time (~1 min). Under these conditions, we expect signifi- cantly higher plasma temperatures (up to T~ 10 K) so that the plasma is only weakly correlated (r~0.2).

For a more accurate determination of oj^ , we use the time dependence of the laser light Bragg-scattered from the

rotating crystals.**'^ A gateable image intensifier, installed in front of the CCD camera, allows the diffraction pattern to be

recorded stroboscopically. Figure 3(a) shows a time- averaged diffraction pattern of concentric rings from an ap- proximately spherical plasma with A^=7.5X 10^. Even if this pattern is from a single crystal, rings are observed because of

the plasma rotation about the axial laser beam.'* With the rotating field applied and controlling the plasma rotation, we

trigger the intensifier synchronously with the rotating field opening the camera for 50 ns each 2tt/cl>^^ period. This en- TN-28
Phys. Plasmas, Vol. 5, No. 5, May 1998 Huang et al. 1659

FIG. 3. Bragg diffraction patterns from a plasma phase locked to a rotating quadrupole field (a;,= a;^ = 27rX 140 kHz, no=4.26X 10^ cm~\a=l.l).

(a) 1 s time-averaged pattern. The long rectangular shadow is from the

deflector for the incident beam; four line shadows that form a square are due

to a wire mesh at the exit window of the vacuum chamber, (b) Timeresolved pattern obtained by strobing the camera with the rotating field (integration time =5 s). A spot is predicted at each intersection of the rectangular gnd lines for a bcc with a { 110} plane perpendicular to the axial

laser beam. The grid spacings are determined from Wg and are not fitted.

ables the camera to record the diffraction pattern in the rest frame of the rotating field. Figure 3(b) shows such a time- resolved pattern taken nearly simultaneously with Fig. 3(a) and accumulated over ~ 10^ plasma rotations. The well- defined rectangular dot pattern demonstrates that the crystal

is phase locked to the rotating field. In this work, we could

determine phase-locked rotation when Bq is aligned within

0.01° of the trap axis and for co,^Cl/20. This alignment is obtained by minimizing the amplitude of zero-frequency

modes of the plasma. Furthermore, the structure and spacings of the crystalline lattice and its orientation with respect

to the laser beam can last longer than 30 min under this rotation control. For comparison, a particular Bragg scatter- ing pattern typically changes after ~ 1 min without the rotating field control, perhaps due to effects from the plasma

spin-down. With a weak rotating field amplitude or when the

trap axis is tilted a few hundredth of a degree with respect to Bq, we cannot obtain phase locking but are able to stabilize the plasma rotation frequency close to the rotating field fre- quency with Sl% slippage.

B. Quadrupole field control

With the rotating quadrupole field, which has a potential

°<:(>'^-x^)cos(2ci)^^/) + 2xy sin(2ci)^^,0, the equilibrium plasma

surface is actually a spinning triaxial ellipsoid with three

principal axes differing in length. ^^ Since the plasma is phase

locked with (xi^—co^^, the combined vacuum trapping potential in the rotating frame becomes

,2 o:=-

mco. 2e

[z~ + ((3-S)xl + (/3+S)y^^], (4)

where S=fg V^./Vq^O is the relative strength of the quadrupole field and /^ depends only on the trap geometry. Here

electric fields from the ambient drag and image charges are neglected. The thermal equilibrium state in the frame rotat- ing with the plasma is bounded by z^/zl + xj/xl + yj/yl- 1 with XQ^yQ. The parallel and perpendicular aspect ratios

{o!ii = zo/yo^oci—xo/yo) '"e determined by parameters yS and S. This shape produces a space-charge potential which

FIG. 4. Oscillation of the plasma radius ri^^ of the side-view image with the

rotating quadrupole field. The theory curve is calculated using the calibrated fg from Fig. 5(a). The relative phase between the theory and data is not

adjusted.

cancels O^ within the equilibrated plasma. In this equilibrium, each individual ion still undergoes a purely rotational average motion as in the ordinary thermal equilibrium state without the rotating field. To obtain an a^^ significantly above one, we maximize

the ratio S/j3 by reducing Vq to less than 200 V and setting

o}„ close to the magnetron frequency w„ . At these low rotation frequencies and with only the perpendicular cooling

beam, the contaminant ions are well mixed with the ^Be"*" ions, making the boundary of the fluorescing ^Be"*" ions coincide with that of the plasma. Side-view images of the plasma are then recorded stroboscopically at different phases

of the rotating field. The radius r^^^ of the side-view image is expected to oscillate at 2 co„, with an amplitude proportional

to XQ — yQ. Figure 4 shows a measurement of riat, on a plasma

with q;|| = 0.15 and a^=« 1.4, rotating at 2.5 kHz(Vo = 50 V,

Vj^,= 185 V). Excellent agreement between the theory and

data is exhibited, showing that the plasma is indeed equili- brated with the rotating quadrupole field. We have measured the dependence of a^ on parameters

S and /3. Figure 5(a) shows a_^ versus S at fixed yS = 0.103. Good agreement with the theory is obtained by fitting these data for fg with the result /^ = 6.38X 10"^ Figure 5(b)

shows ttj^ versus /3 for fixed <5= 0.0159 using the calibrated fg value. Excellent agreement is again observed, showing the rapid decrease to unity for aj^ as /3 is increased. Under typi- cal conditions (Vq^SOO V, V^,=s50 V, and w^^l.l w^

,

giving (5<7X10"'* and /3>0.1), a^^-l is less than 1%.

This small distortion, however, generates sufficient torque to phase lock the plasma rotation when the trap is nearly

aligned with the magnetic field.

C. Dipole field control

We have examined rotation control using the dipole field with a potential ^y sin(a)j^,0-xcos(a>„,0. Theoretically, this

field is not expected to provide rotation control for a singlespecies plasma in a quadratic trap. Instead, it drives only a "center-of-mass" orbital motion about the trap axis without

TN-29
1660 Phys. Plasmas, Vol. 5, No. 5, May 1998 Huang et al.

1.5

1.4

•^ 1.3 oX

III 6" ''

1.1

1.0

. / . N = 4.1x10* (a) ^ / : + N =7.6x10* •x

-

- Theory

>^|i = 0.103

fg = 6.38x10' (fit) - ^ 1

•

70

0.00 0.01 0.02 0.03 S-fgV^/V^

1.5

1.4 -

1.3 -

H 1.2

1.1 -

1.0

- TT 1 1 1 1-

(b)

\ N = 4.1x10* - > N = 7.6x10* : Theory

; 6 = 0.0159 >~-^,^

*~" * • :

0.0 0.2 0.4 0.6

FIG. 5. Dependence of perpendicular aspect ratio a^ versus (a) rotating

field strength S and (b) radial trapping strength /8. Two plasmas are used

with 50« Vq^ 100 V and 15« V„,« 185 V. The geometric factor /j is cali- brated in (a).

surface distoitions and is decoupled from the internal plasma

rotation. Experimentally, however, rotation control including

phase locking similar to that from the quadrupole field is obtained with the rotating dipole field as discussed in the previous sections. While the asymmetric field from image

charges induced by the center-of-mass motion could provide

the observed coupling to the plasma rotation, this field is too weak to be effective for our typical trap parameters.

Experimental evidence suggests that the coupling be- tween the center-of-mass motion and plasma rotation is pro- vided by the contaminant ions. By measuring the plasma

dynamic response under sudden changes of the dipole field frequency^' we have observed that the torque from the dipole

field increases with time, presumably due to the growing

percentage of the contaminant ions. Figure 6 shows the evolution of plasma rotation frequency as cajlir is suddenly

changed by ±5 kHz from 60 kHz (magnetron frequency

w„ = 2 77X42.2 kHz) at Ar = 0. The initial and final rotation frequencies of the plasma are equal to that of the dipole field as determined from time-resolved Bragg scattering measurements. The intermediate data points are derived from the aspect ratio of the side-view images assuming that the plasma goes through successive thermal equilibrium states during the evolution. The torque from the rotating dipole

field has clearly increased from the 1-h-old plasma with apNX

CM

50

A(iQ^/2ji=±5kHz(lh)

Att(„/2n=±5kHz(2h)

20 40 60 80

At(s)

FIG. 6. Time evolution of the plasma rotation frequency under sudden

changes of the dipole field frequency. Measurements on the same plasma 1 and 2 h after loading are plotted.

proximately 5% contaminant ions to the 2-h-old plasma with

twice as many contaminant ions. The effect of the contaminant ions on the plasma can be

understood in terms of their azimuthally asymmetric distri- bution with the rotating dipole field. For typical rotation fre- quencies and laser cooling conditions, these heavier contaminant ions separate radially from the ^Be"*" ions and form a nonfluorescent outer cylindrical layer without the rotating

field.'°'^' As the plasma undergoes a driven center-of-mass

circular motion with the rotating dipole field, the heavier

contaminant ions distribute preferentially farther from the

trap center than the ^Be"^ ions. In order for the contaminant

ions to be distributed away from the trap center and to have

rigid-body rotation, the plasma rotation fi-equency must

match the rotation frequency of the dipole field (see discussions below). This produces a coupling between the driven center-of-mass motion and the plasma rotation, enabling the

rotating dipole field control. When the plasma reaches ther1.8

1.4

1.0

0.6 -

0.2

/"

• N = 3.0x10*

Theory for pure 'Be* Theory from Fig.8/ \

^y

0.0 0.5 1.0

tCO^/27t

1.5 2.0

FIG. 7. Oscillation of the positive radial boundary x^^ with the rotating dipole field. The dashed curve is calculated for a pure 'Be* plasma and the

solid curve is obtained from the simulation shown in Fig. 8. The dipole field amplitude at the plasma is derived by fitting the data, while the relative phase between the theories and data is not adjusted.

TN-30
Phys. Plasmas, Vol. 5, No. 5, May 1998 Huang ef a/. 1661

mal equilibrium with the rotating dipole field at C0r= co^. , the average ion motion is simply a rigid-body rotation about the trap axis at u)„ . Figure 7 shows the position Xq^ of the positive radial boundary (at z = 0) from the side-view images of a plasma

containing about 12% contaminant ions and phased locked

with a dipole field rotating at 9 kHz (magnetron frequency is 277X8.4 kHz). This measurement is obtained with the same

stroboscopic technique that was used for the quadrupole field

results. For these conditions, however, the contaminant ions

are well separated from the ^Be"^ ions due to the higher

rotation frequency. For a pure ^Be"*" plasma, a periodic oscillation of Xge at a)^^, is expected due to the center-of-mass motion, as shown in Fig. 7. The data generally follow this prediction except for the "clipping" of Xge at certain phases

of the rotating dipole field. A Monte Carlo simulation on the equilibrium distribution of a two-species plasma (in the rest frame of the dipole

field) is shown in Fig. 8. The plasma is made of 1000 particles with 88% ^Be"^ (dots) and 12% contaminant ions having mass 26 u (open circles), and is driven by a rotating

dipole field resulting in the same relative displacement as in Fig. 7. The plasma has a very oblate shape which enhances

its center-of-mass displacement as shown in the side-view of

Fig. 8(b). In the ;c->' distribution shown in Fig. 8(a), the

overall charge is distributed nearly symmetrically with re- spect to the shifted plasma axis, while the ^Be"^ ions and the contaminant ions are positioned asymmetrically inside the plasma. The boundary of the ^Be"'" ions is approximately

made of two arcs: one, bordering the vacuum, is centered on

the plasma axis (the ® symbol), and the other, bordering the contaminant ions, is centered on the trap axis (the cross). This shape explains the observed clipping of x^^ in Fig. 7. The small discrepancy between the data and the solid curve

in Fig. 7 is likely due to an overestimation of the percentage

of the contaminant ions from the side-view images without

the rotating field. When contaminant ions are present, it is advantageous to use the dipole rather than quadrupole field to control the plasma rotation because greater electric field strength at the plasma surface can be obtained for the same amplitude V^

on the compensation electrodes. This is because the plasma

dimensions are typically much smaller than the trap radius and the dipole field decays much more gradually from the compensation electrodes than the quadrupole field. Rotation

control in the range w„<a)r<(n-a>„), which includes rotational equilibrium near Brillouin flow (0)^ = 0/2), has been

achieved using the rotating dipole field. For co^—Cl/l, crystalline lattices are generally not observed, and the aspect ratio a depends only weakly on the rotation frequency. However, we can infer w^ by measuring the angle of the first Bragg scattering ring, ^s^att - from ions in the shell structures. This angle is related to the Wigner-Seitz radius a^s through

^scattaws'^4.4|— , (5)

'Be* (88%)

26u(12%)

where d^^^ is in radian (see Ref. 4). The Wigner-Seitz ra- dius in turn is determined by the plasma rotation frequency.

FIG. 8. A Monte Carlo simulation of the thermal equilibrium distribution of

a two-species plasma with the rotating dipole field. The plasma is made of 1000 particles with 88% 'Be"^ and 12% contaminant ions having mass

26 u, and has a coupling parameter T = 40. The trapping parameters are the same as in Fig. 7. (a) The x-y distribution. Solid and dashed lines denote

boundaries of the 'Be* ions and the plasma, respectively, (b) The x-z dis- tribution with ten times smaller z scale.

Figure 9 shows the scattering angle ^jcatt ^s a function of the

rotation frequency of the dipole field for two plasmas with

different compositions of contaminant ions. With only

heavier contaminant ions (dots), we can control the plasma

rotation frequency up to about 0.9 (dotted vertical line) corresponding to the frequency at which ions having mass

10 u can no longer remain in the plasma. If we intentionally create some lighter ions (X's), we are able to control co^ TN-3I
1662 Phys. Plasmas, Vol. 5, No. 5, May 1998 Huang et al.

« *

Theory (©, = (0^) Max. freq. for 10 u Heavier ions only

Lighter ions created

0.0 0.5

(0^/t2

1.0

uncorrelated plasmas at higher temperatures, two- dimensional rotating fields may not be able to generate

enough torque to balance the ambient drag, and electric

fields with nonzero z components may be required.^^ Rotating field control of pure electron plasmas also needs further refinement to improve its capability.^^ In the future, direct imaging of individual ions in a crystallized plasma may be possible because of the phase-locked

rotation. In addition, the increased crystal stability improves

the prospect of observing the solid-liquid phase transition of

strongly coupled plasmas. Dynamics of the Brillouin flow is also an interesting area for investigation. Finally, it may be

possible to influence the formation of crystalline lattices with

the rotating fields and other perturbations, enabling external control of the crystal orientation.

FIG. 9. The angle of the first Bragg scattering ring from shells versus the rotation frequency of the dipole field. Data from a plasma with only heavier contaminant ions are shown as dots. X 's are data from a plasma with some

lighter ions than 'Be"^.

through all allowed rotation frequencies. This observation provides further evidence that the contaminant ions are the main coupling mechanism for the rotating dipole field. We

cannot, however, exclude the possibility that different nonideal effects can cause sufficient coupling in other experi- ments.

IV. CONCLUSIONS AND DISCUSSION

We have shown experimentally that azimuthally asymmetric electric fields rotating in the same sense as the plasma

can exert a torque that balances the ambient drag in both weakly and strongly coupled regimes, resulting in infinite confinement times. The torque from the rotating wall fields acts to bring the plasma to the same rotation frequency as the applied field. For strongly coupled plasmas with laser cooling, we obtain phase-locked rotation with the rotating field even though there is a finite ambient drag on the plasma.

This phase-locked rotation is possible presumably because

crystallized plasmas are capable of sustaining some stress without structure breakup. As a result of this observation, we

are now able to precisely control plasma properties such as density and boundary shape.

There are other methods to inject angular momentum

into the plasma and vary the rotation frequency. For ex- ample, by exciting an azimuthally asymmetric mode that travels in the same direction but faster than the plasma rotation, a),, can be increased very effectively due to the global coherent resonance.

'^"^^ The challenge of this approach is to remain on resonance with the mode as the plasma rotation frequency is increased. It may also be difficult to balance the heating from the mode excitation. In addition, this method

will probably not result in rotation control as precise as the

rotating field technique.

Substantial questions remain concerning the theoretical models which give quantitative predictions for the torque

from the rotating fields.^^'^'' In the strongly coupled regime, such models that describe the experiment are lacking. For

ACKNOWLEDGMENTS

We thank D. Wineland, T. O'Neil, and C. F. Driscoll for discussions, and B. Jelenkovic, R. Rafac, M. Young, M.

Lombardi, and D. Sullivan for useful comments on the manuscript.

This work is supported by the Office of Naval Research.

'j. H. Malmberg, C. F. Driscoll, B. Beck, D. L. Eggleston, J. Fajans, K.

Fine, X.-P. Huang, and A. W. Hyatt, in Non-neutral Plasma Physics,

edited by C. W. Roberson and C. F. Driscoll (American Institute of Physics, New York, 1988), p. 28.

^J. J. Bollinger, D. J. Wineland, and D. H. E. Dubin, Phys. Plasmas 1, 1403

(1994). 'For recent results, see Non-neutral Plasma Physics II, edited by J. Fajans and D. H. E. Dubin (American Institute of Physics, New York, 1995).

"j. N. Tan, J. J. Bollinger, B. Jelenkovic, and D. J. Wineland, Phys. Rev.

Lett. 75, 4198 (1995); J. N. Tan, J. J. Bollinger, B. Jelenkovic, W. M.

Itano, and D. J. Wineland, in Physics of Strongly Coupled Plasmas, edited by W. D. Kraeft and M. Schlanges (Worid Scientific, Singapore, 1996),

p. 387. 'W. M. Itano, J. J. Bollinger, J. N. Tan, B. Jelenkovic, X.-P. Huang, and D.

J. Wmeland, Science 279, 686 (1998). 'R. Blatt, P. Gill, and R. C. Thompson, J. Mod. Opt. 39, 193 (1992); R. C. Thompson, Adv. At. Mol. Phys. 31, 63 (1993).

^J. N. Tan, J. J. Bollinger, and D. J. Wineland, IEEE Trans Instrum. Meas.

44, 144 (1995). *G. Gabrielse, X. Fei, L. A. Orozco, R. L. Tjoelker, J. Hass, H. Kali- nowsky, T. A. Trainer, and W. Kells, Phys. Rev. Lett. 65, 1317 (1990). 'R. G. Greaves and C. M. Surko, Phys. Plasmas 4, 1528 (1997).

'°D. Schneider, D. A. Church, G. Weinberg, A. J. Steiger, B. Beck, J. McDonald, E. Magee, and D. Knapp, Rev. Sci. Instrum. 65, 3472 (1994). "T. M. O'Neil, Phys. Fluids 23, 2216 (1980).

'-J. S. DeGrassie and J. H. Malmberg, Phys. Fluids 23, 63 (1980). "D. a. Moore, R. C. Davidson, S. M. Kaye, and S. F. Paul, in Non-neutral Plasma Physics II (Ref. 3), p. 118. '^Torques from rotating neutral gas are considered in A. J. Peurrung and S. E. Barlow, Phys. Plasmas 3, 2859 (1996). "j. H. Malmberg and C. F. Driscoll, Phys. Rev. Lett. 44, 654 (1980); C. F. Driscoll, K. S. Fine, and J. H. Malmberg, Phys. Fluids 29, 2015 (1986).

'*D. L. Eggleston, T. M. O'Neil, and J. H. Malmberg, Phys. Rev. Lett. 53, 982 (1984).

''j. Notte and J. Fajans, Phys. Plasmas 1, 1123 (1994).

'^D. L. Eggleston, Phys. Plasmas 4, 1196 (1997).

"j. J. Bollinger and D. J. Wineland, Phys. Rev. Lett. 53, 348 (1984).

^°L. R. Brewer, J. D. Prestage, J. J. Bollinger, W. M. Itano, D. J. Larson, and

D. J. Wineland, Phys. Rev. A 38, 859 (1988); D. J. Heinzen, J. J. Bollinger, F. L. Moore, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 66, 2080 (1991). ^'Some preliminary results on Mg"^ plasmas are in F. Anderegg, X.-P. Huang, C. F, Driscoll, G. D. Severn, and E. Sand, in Non-neutral Plasma

Physics II (Ref. 3), p. 1.

TN-32
Phys. Plasmas, Vol. 5, No. 5, May 1998 Huang et al. 1663

^^X.-P. Huang, F. Anderegg, E. M. Hollmann, C. F. Driscoll, and T. M.

O'Neil, Phys. Rev. Lett. 78, 875 (1997). ^^A preliminary work on pure electron plasmas is in R. E. Pollock and F. Anderegg, in Non-neutral Plasma Physics II (Ref. 3), p. 139. ^"X.-P. Huang, J. J. Bollinger, T. B. Mitchell, and W. M. Itano, Phys. Rev.

Lett. 80, 73 (1998). ^^D. J. Wineland and H. G. Dehmelt, Int. J. Mass Spectrom. Ion Processes

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Phys.

'°T. M. O'Neil, Phys. Fluids 24, 1447 (1981). ^'D. J. Larson, J. C. Bergquist, J. J. Bollinger, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 57, 70 (1986). "T. B. Mitchell, Ph.D. thesis. University of California at San Diego, 1993,

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TN-33
REPORTS

Direct Observations of

Structural Phase Transitions in

Planar Crystallized Ion Plasmas

T. B. Mitchell.* J. J. Bollinger, D. H. E. Dubin, X.-P. Huang,

W. M. Itano. R. H. Baughman

Laser-cooled ^Be"*^ ions confined in two-dimensionally extended lattice planes were directly observed, and the images were used to characterize the structural phases of the ions. 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. The experimental results are in good agree- ment with theoretical predictions for the planar (infinite in two dimensions)

one-component plasma. Qualitatively similar structural phase transitions occur, or are predicted to occur, in other experimentally realizable planar systems.

The one-component plasma (OCP) has been a model of condensed matter in statistical physics for over 30 years, and it is used to describe such diverse systems as dense astrophysical matter (1) and electrons on the surface of

liquid helium (2). Laser-cooled trapped ions

(5) are an excellent experimental realization of the OCP. The phase structures of spatially homogeneous (infinite) (4) and cylindrical

(infinite in one dimension only) (5) OCPs

have been explored previously. Here, images

of individual ions that were confined in twodimensionally extended lattice planes are pre- sented and used to characterize the structural phases; the observed structures agree well with the predictions of an analytic theory for the planar OCP.

The OCP model consists of a single

charged species embedded in a uniform, neutralizing background charge. In Paul (6) or Penning {6, 7) traps, which are used to confine charged particles, a (fictitious) neutraliz- ing background is provided by the confining

potentials. The thermodynamic properties of

the infinite classical OCP are determined by

its Coulomb coupling parameter, T =

[l/(4-TTeQ)][e^/(a^gA:37)], which is the ratio of

the Coulomb potential energy of neighboring

ions to the kinetic energy per ion; e^ is the

permittivity of the vacuum, e is the charge of

an ion, k^ is Boltzmann's constant, T is the temperature, and a^^g is the Wigner-Seitz

radius [defined by 4Tr(a^g)^/3 = I/Wq, where

«Q is the ion density]. The onset of short- range order for the infinite OCP is predicted

(8) at r ^ 2, and a phase transition to a

T. B. Mitchell, J. J. Bollinger, X.-P. Huang, W. M. Itano, Time and Frequency Division, National Institute of Standards and Technology, Boulder, CO 80303, USA. D. H. E. Dubin, Department of Physics, University

of California at San Diego, La Jolla, CA 92093, USA. R. H. Baughman, Research and Technology, Allied SIgnaL Morristown, NJ 07962, USA. *To whom correspondence should be addressed, E- mail: travis.mitchell@nist.gov

body-centered cubic (bcc) lattice is predicted

{8, P) at r = 170. With an OCP in a planar

geometry (infinite in only two dimensions),

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

(]0. 11). Qualitatively similar structural phase transitions occur, or are predicted to occur, in other planar systems with varied

interparticle interactions, such as plasma dust

crystals (72), colloidal suspensions (13), semiconductor electron bilayer systems {14), and hard spheres (75). The crystallization of small numbers (total number N < 50) of laser-cooled ions into Coulomb clusters (16) was first observed in Paul traps (77). With larger numbers of

trapped ions, concentric shells (75) were di- rectly observed in Penning (19) and Paul

traps (5, 20). Recently, Bragg diffraction has been used to detect bcc crystals (the predicted

infinite volume ordering) in large and spherical (N > 2 X 10^; radius r^ > 60 a^s) '°" plasmas confined in a Penning trap (4). Measurements were taken from direct images of the central (r = 0) structure of pancake-shaped (lenticular) ion plasmas

(aspect ratio a = zjr^ < 0.1, where Iz^ is the plasma center's axial extent). The central region has a disklike geometry with

constant areal density a^ (charge density per unit area projected onto the z =

plane), which facilitates comparison with

planar theory. We observed five different stable crystalline phases and found that the energetically favorable central structure can be tuned by changing a^. Both contin- uous and discontinuous structural phase

transitions were observed. The ^Be"^ ions were confined radially in a cylindrical Penning trap (inner trap diameter, 40.6 mm) (Fig. 1) with a uniform magnetic

field B = 4.465 T in the z-axis direction. The

ions were confined axially by a potential

difference of Vq = —1.50 kV, which was

applied between the center and end electrodes of the trap. Near the trap center, this axial potential is quadratic and has a value of

l/2(m/e)a)^z^, where the axial frequency u)J

2-n- = 978 kHz for ^Be+. The radial electric fields of the trap, as well as the ion space

charge, cause the ion plasma to undergo an E

X B drift and thus rotate about the trap axis. In thermal equilibrium, this rotation is at a uniform frequency w^.. 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 w^ results in a weak radial binding and a lenticular plasma

with a large radius. For 1 0" trapped ions with

(0|./2tt = 68.5 kHz (typical for this work), the ion plasma has a density of 2.1 X 10^ cm~^

with 2ro = 1.3 mm and an aspect ratio a =« 0.05. The rotation frequency was controlled by phase-locking the plasma rotation to an

applied "rotating wall" electric field (27). At

low Wj., an increase in w^ increases both the plasma density and Zg, providing a way to sensitively adjust the central areal density of

the plasma.

The ions were cooled (i) by a laster beam

propagating along the z axis and tuned 1 to 20 MHz lower in frequency than a hyperfine- Zeeman component of the 2s S^/2 2p

'^P^/2 resonance at 313 nm with a natural linewidth of 19 MHz. The laser power was ~50 p,W and was focused at the ion plasma

to a diameter of —0.5 mm. The theoretical cooling limit is 0.5 mK, and an experimental upper bound of T < 10 mK has been measured (22). For a density of Mq = 2 X 10* cm~^, these limits give a range of 160 < F <

3150. A series of lenses formed side- and

top-view images of the ions, with viewing

directions that were 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 sys- tem is ~4 \x.m, whereas typical interparticle spacings are ~20 p,m. The side-view image insert in Fig. 1, which shows the central region of a lenticular ion plasma with three axial lattice planes, is representative of the flatness and radial extent (< 1 0% of Tq) of this region. At large radius, curvature of the planes can cause the side- view images of axial plane positions to blur. This effect was prevented in the measure- ments reported here by using clouds with

sufficient amounts (up to 50%) of nonfluorescing impurity ions. Because these heavier- mass ions are centrifugally separated to larger

radii than the 'Be"^, the regions of the plasma where curvature begins to be significant can

be filled with these ions, which are sympathetically cooled by the ^Be+ (23). With good alignment of the trap with the magnetic field (<10~^ rad), the ion plasma

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rotation is phase-locked with the rotating wall

perturbation (27). A direct observation of the

rotating ion structures was achieved for the

first time by gating the top-view CCD camera

synchronously with the rotating wall pertur- bation for brief gate times (<2%) of the plasma rotation period). Total exposure times of ~3 X lO'' rotation periods were used for

typical images (Fig. 2). For our study of the ion lattice structure, we limited our analysis

to the central region, where strong localization and regular ordering of the ions were

observed. At larger radius, we observed an

increased blurring (due to the plasma rotation), occasional lattice distortions, and, ulti- mately, the transition to the regions filled by

heavier mass ions. The observed structure of the central crystallized region depends on the central areal density a^, of the plasma. Within a layer, the

lattice is characterized by the primitive vectors a, and a^ (which were observed to be

equal in magnitude, |a,| = [HjI = a) or, equivalently, by a and the angle 9 (:S90°) between the primitive vectors. The interlayer

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 inter- layer structure. The

side-view inset also shows the central re- gion of a lenticular ion plasma with three axial

lattice planes. The rotation frequency of the

ion plasma was controlled by applying

properly phased sinusoidally time-varying

electric potentials to the sixfold azimuthally

segmented compensation electrodes.

structure is characterized by the axial positions z^ of the n lattice planes (measured by

the side-view camera) and the interlayer dis- placement vector c^ between layers 1 and n. Hence, the equilibrium (x, y) positions of

ions in axial planes 1 and n are given by

R,(i, y) = i'a, +732 and R^{i,j) = /a, -I- ja2 + c„, where / andj are integers. Three different types of intralayer ordering were observed: hexagonal (6 = 60°), square (9 = 90°), and rhombic (90° > 9 > 65°). The

square and rhombic layers stack in a staggered

fashion, with the upper ions immediately above

the centers of the parallelograms below, result- ing in an interlayer displacement vector c^ =

(a, -I- 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 = a, + 82 and 3C2/2 = a, + aj. With

hep-like stacking, the ions in every other plane

lie directly above each other (abab . . .), whereas with fcc-like stacking, the ions in every third plane are so aligned (abcabc . . .). When there were three or more hexagonal layers, both types of stacking were observed.

Strobe

Side-view camera f75

1

1'

.--A — .'Be* Vn

Rotating

wall

perturbation m

Compensation

electrodes (6x60°) h

Axial

cooling beam X©

Table 1. Primitive and interlayer displacement vectors in the (x, y) plane for the observed phases. The

primitive vector a., defines tiie x direction, and |a.,| = la^l = a. Dashes, not applicable.

Vectors Knase symmetry MacKing

ai a? ^2 C3

1 Hexagonal Single plane (a. 0) (a cos 60°, a sin 60°) — —

ill Square Staggered (a, 0) (0. a) (a, + a2)/2 (0,0)

IV Rhombic Staggered (a, 0) (a cos 6, a sin 6) (a, + a^)/2 (0,0) V Hexagonal hcp-lilce (a, 0) (a cos 60°, a sin 60°) (a, + a,)/3 (0,0) v,„ Hexagonal fcc-like (a, 0) (a cos 60°, a sin 60°) (ai + aJ/3 2(a, + a,)/3

The following sequence of phase structures, with lattice parameters defined in Table

1, were observed as the central areal density

(Tg was increased from where order was first observed: (I) one-layer hexagonal —> (III) two-layer staggered square -^ (IV) two-layer

staggered rhombic -^ (V) two-layer stag- gered hexagonal. At a critical density, a third layer was formed, resulting in a (III) three- layer staggered square. The process then re- peated with minor variations, such as phase

III becoming less common. We have fol- lowed the classifications used in previous

theoretical studies of quantum (14) and classical (77) electron bilayer systems. Phase II, which is a stable phase of the bilayer systems where the interlayer distance is fixed, is not

listed here because it is unstable for the pla- nar OCP, where the interlayer distance can

vary.We have performed an analytical calculation of the energies of these phase struc- tures for the planar OCP (24). The calculation minimizes the energy (25) of several

parallel lattice planes that are infinite and homogeneous in the (x, y) direction but are confined in the z direction by a harmonic

external electrostatic confinement potential

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.

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Fig. 3 (left). Interlayer structure (plane axial positions and displacement

vectors) of the central region as a function of normalized central areal charge density. The lines show predictions from theory, and symbols

show experimental measurements. The symbols indicate whether the

lattices had an interlattice displacement vector c^ that was characteristic of the hexagonal phases (circles) or the square and rhombic phases

(squares). Lengths have been normalized by a^^^o ~ (Se^M-n-eomo)^)^'^

90

85

80

"Tn<u2 75

05

0) o"" 70

CD

65

60 i-lU.

—e={3L,*a,)a

1^

0.5

= 10.7 |jLm, which is the WIgner-Seitz radius in the planar limit. Fig. 4 (right). Intralayer angle 6 of the central region as a function of central

areal charge density. The solid lines indicate the minimum energy

structures that are predicted by the 2D theory. The symbols indicate experimental measurements, which are from the same data sets that were used in Fig. 3. Some representative error bars that indicate the

scatter in the measurements are included.

<}>g = l/2(w/e)co^z^. Because this potential is identical to the confinement potential of a Penning trap in the a —> planar limit (26),

this theory should predict the structures that were observed in the central region of the

lenticular plasmas of the experiments. The

predictions of this two-dimensional (2D)

theory, which has no free parameters, were compared directly with our observations by

identifying the areal density of the planar

OCP with the directly measured central

areal density a^, 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 to the relevant phases

(shown in Table 1) (lines in Fig. 2). Using

the best-fit values of the primitive vector

length a and the intralayer angle 9 and

using the observed number of lattice planes

n, we calculated the central areal density CTq = n/ia^ sin 9). The agreement between the planar OCP

theory and experiment, with measurements

taken on different plasmas with N < 1 0^, was good (Figs. 3 and 4). As the central areal density was increased, the lattice planes

moved farther apart axially (Fig. 3). Eventually, it became energetically favorable to form an additional lattice plane. However,

although the phase V^^^ was predicted to be

slightly more energetically favorable than phase V, we rarely observed V^^^ (—5% of

the time). These and other minor discrepancies from theory may be due to the finite radial extent of the ion plasma; we observed

a similar preference for hep stacking in molecular dynamics simulations of small (N =

3000) lenticular ion plasmas.

For the dependence of the angle (be- tween the primitive vectors 3] and aj) on

central areal density a^ (Fig. 4), the general

trend is that, when a new lattice plane is formed, 9 changes discontinuously from ~60° to a higher value. As a^ of the crystal was fiirther increased, smoothly decreased

to —65° until there was a second discontinu- ous transition to a hexagonal structure. This second transition has been predicted to be- come continuous, with assuming all values 60° < 9 < 90°, in liquid (F < 80) bilayer

systems (27). At central areal charge densities near phase boundaries, both phases can be

observed. In these regions, the phase that materializes after the crystal is formed is initially random but tends to persist if the ions

are not heated. Where there was no strong preference for one phase over the other, both were plotted (Fig. 4). Like most materials, the hexagonal and

square phases contract in lateral directions when elongated. However, the rhombic phase

shows quite different behavior because the

intralayer angle strongly depends on the

z-axis strain; one rhombus diagonal contracts, and the second expands when the rhombic phase is elongated in the z-axis di- rection. The dimensional change for the latter diagonal corresponds to a negative value of

the Poisson's ratio (which is the ratio of the

lateral contraction to the longitudinal elongation). The present experimental observations

substantially expand the mass density range over which negative Poisson's ratios have been established [from —10 g/cm^ for cubic metals (28) and —0.1 g/cm-* for reentrant foams (29) to — 10~'^ g/cm^ for the present ion crystals].

References and Notes

1. H. M. Van Horn, Science 252, 384 (1991).

2. C C. Crimes and G. Adams, Phys. Rev. Lett. 42, 795

(1979).

3. D. J. Wineland, R. E. DruUinger, F. L Walls, ibid. 40, 1639 (1978).

4. J. N. Tan, J. J. Bollinger, B. M. Jelenlcovic, D. J. Wineland, ibid. 72, 4198 (1995); W. M. Itano etal., Science 279, 686 (1998).

5. I. Walcl, S. Kassner, C. Blrkl, H. Walther, Phys. Rev.

Lett. 68, 2007 (1992); C. Blrkl, S. Kassner, H. Walther, Nature 357, 310 (1992).

6. P. K. Ghosh, Ion Traps (Clarendon, Oxford, 1995).

7. C W. Roberson and C F. Driscoll, Eds., Non-Neutral

Plasma Physics (American Institute of Physics, New

York, 1988).

8. S. Ichlmaru, H. lyetomi, S. Tanaka, Phys. Rep. 149, 91

(1987).

9. D. H. E. Dubin, Phys. Rev. E 42, 4972 (1990).

10. J. P. Schiffer, Phys. Rev. Lett. 70, 818 (1993); D. H. E. DubIn, ibid. 71, 2753 (1993).

11. C. ColdonI and F. M. Peeters, Phys. Rev. B 53, 4591

(1996).

12. J. B. Pleper, J. Goree, R. A. Quinn,/ Vac. Sci. Technol. A 14, 519 (1996); H. TotsujI, T. KIshlmoto, C Totsuji, Phys. Rev. Lett. 78, 3113 (1997).

13. D. H. Van Winkle and C A. Murray, Phys. Rev A 34, 562 (1986).

14. S. Naraslmhan and T-L Ho, Phys. Rev. B 52, 12291

(1995): G. Goldoni and F. M. Peeters, Europhys. Lett. 37, 293 (1997).

15. M. Schmidt and H. Lowen, Phys. Rev. E 55, 7228

(1997).

16. R. Rafac, J. P. Schiffer, J. S. Hangst, D. H. E. Dubin, D. J. Wales, Proc. Natl. Acad. Sci. U.S.A. 88, 483 (1991).

17. F. DIedrlch, E. Peik, J. M. Chen, W. Quint, H. Walther,

Phys. Rev. Lett. 59, 2931 (1987); D. J. Wineland, J. C

Bergquist, W. M. Itano, J. J. Bollinger, C H. Manney,

ibid., p. 2935.

18. A. Rahman and J. P. Schiffer, ibid. 57, 1133 (1986);

D. H. E. Dubin and T. M. O'Neil, ibid. 60, 511 (1988).

19. S. L Gilbert, J. J. Bollinger, D. J. Wineland, ibid., p. 2022.

20. M. Drewsen, C. Brodersen, L Hornekaer, J. S. Hangst,

J. P. Schiffer, ibid. 81, 2878 (1998).

21. X.-P. Huang, J. J. Bollinger, T. B. Mitchell, W. M. Itano,

ibid. 80, 73 (1998); Phys. Plasmas 5, 1656 (1998).

22. L R. Brewer et at., Phys. Rev. A 38, 859 (1988).

23. J. M. O'Neil, Phys. Fluids 24, 1447 (1981); D.J. Larson

etal., Phys. Rev Lett. 57, 70 (1986).

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13 NOVEMBER 1998 VOL 282 SCIENCE www.sciencemag.org
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24. T. B. Mitchell, J. J. Bollinger, D. H. E. Dubin, X.-P. Huang, W. M. Itano, in preparation (preprint available

at www.boulder.nist.gov/timefreq/lon/penning/pop/

pop.pdO.

25. Minimization of the energy was carried out numerically

for given areal density by allowing the positions of the

planes to vary with respect to one another within the

external potential and by varying the lattice structure within the planes. We made the simplifying assumption, which was consistent with the experimental observations, that the lattice structure of each plane was identical The

areal density of the planar OCP is a control parameter

that, when varied, leads to transitions in both the number

and lattice structure of the planes. This theory focuses on

the minimum potential energy states wrfiile neglecting

finite temperature effects that may be important (partic- ularly effects that are near the continuous phase transi- tions where large structural fluctuations may occur). 26. T. M. O'Neil and D. H. E. Dubin, Phys. Plasmas 5, 2163

(1998).

27. V. I. Valtchinov, C. Kalman, K. B. Blagoev, Phys. Rev. E 56, 4351 (1997).

28. R. H. Baughman, J. M. Shacklette, A. A. Zakhidov, S. Stafstrom, Nature 392, 362 (1998).

29. R. Lakes, Science 235, 1038 (1987).

30. This manuscript is a work of the U.S. government; it

is not subject to U.S. copyright. This research was

supported by the Office of Naval Research and NSF

(D.H.E.D.). We thank B. King, R. J. Rafac, j. P. Schiffer, and D. J. Wineland for useful comments and B. M.

Jelenkovic and J. N. Tan for technical assistance.

14 August 1998; accepted 14 October 1998

Controlling Factors for the

Brittle-to-Ductile Transition in

Tungsten Single Crystals

Peter Cumbsch,* Joachim Riedle, Alexander Hartmaier,

Hellmut F. Fischmeister

Materials performance in structural applications is often restricted by a transition from ductile response to brittle fracture with decreasing temperature.

This transition is currently viewed as being controlled either by dislocation mobility or by the nucleation of dislocations. Fracture experiments on tungsten

single crystals reported here provide evidence for the importance of dislocation nucleation for the fracture toughness in the semibrittle regime. However, it is shown that the transition itself, in general, is controlled by dislocation mobility

rather than by nucleation.

Some materials are brittle and shatter like glass, whereas others are ductile and deformable. Several materials, such as the refractory metals,

steels, and semiconductor crystals, exhibit both

types of behavior (7-5) with a brittle-to-ductile fransition (BDT) at a characteristic temperature. A crack introduced into a material may

propagate as a brittle crack with an atomically

sharp crack front; alternatively, the material near the crack tip may show sufficient plasticity to slow down or arrest the crack. Crack-tip

plasticity comprises two distinct processes, nucleation of dislocations at or near the crack tip and their propagation away from the crack. Several models describe the BDT as a nucleation-controlled event {6, 7), and others focus on dislocation mobility as the controlling factor (8-10). The experimental evidence on the controlling factors of the BDT, which has been

obtained mostly from silicon single crystals, remains inconclusive. On the one hand, the temperature at which the BDT occurs [BDT

temperature (BDTT)] is strongly dependent on the strain rate, which allows an activation energy for the BDT to be determined. This

activation energy has been shown to be equal

to that for dislocation motion (2, 8, 9), which

suggests a mobility-controlled BDT. On the other hand, specimen size and the availability

Max-Planck-lnstltut fijr Metallforschung, Seestrasse

92, 70174 Stuttgart, Germany.

*To whom correspondence should be addressed. E- mail: gumbsch(3)finix.mpi-stuttgart.mpg.de

of dislocation sources have a pronounced in- fluence on the fracture toughness to the de- gree that they may even switch the material's behavior from brittle to ductile (77). We jjerformed cleavage experiments on

tungsten single crystals to determine the controlling factors of the BDT. To exclude distur- bances from grain boundaries, single crystals were chosen. Fracture toughness tests were per- formed on all four low-index crack systems. A

crack system is specified by the nominal crack

plane and the crack front direction. The four crack systems studied are therefore denoted as {100}<010>, {100}<011>, {110}<lT0>,

and {1 10}<001> in crystallographic notation. The temperature range covered by the experi- ments was between the liquid nitrogen temperature (77 K) and 650 K. The toughness tests were performed on high-purity (72) tungsten

single crystal bars (3 mm by 6 mm by 30 mm)

loaded in three-point bending. To obtain a well- defined value for the fracture toughness, a sharp

crack front was introduced by precracking at 77 K. Precrack length was evaluated from the compliance and, where possible, was also de- termined postmortem on the fracture surfaces, where faint arrest lines are visible. The tests were performed with a constant loading rate, which translates into an almost constant stress intensity rate of 0.10 ± 0.02 MPa m'^^/s (75). More details of the experimental procedure are given in {14).

Multiple tests (more than five) were con- ducted for all four crack systems at room

temperature and at 77 K. Several individual

tests at other temperatures were performed to determine the temperature dependence of the

fracture toughness (Fig. 1 ). Macroscopically,

the transition from brittle to ductile response

usually correlates with the maximum in frac- ture toughness. The temperature at this max- imum was taken as the BDTT and was deter- mined from Fig. 1 within 30 to 60 K (Table

1). The so-defined BDTTs fell into an inter- val of 1 00 K for all four crack systems, even though both {110} crack systems had a sig- nificantly higher room temperature fracture toughness than the {100} systems (Table 1). Testing at 77 K always resulted in brittle cleavage fracture on the precrack plane for all four crack systems. The load-displacement curves showed perfectly linear behavior up to the load at which fracture occurred. The frac- ture surfaces were clean except for a few

river lines. At room temperature and above, most of

the fracture specimens revealed small deviations from linearity in the load-displacement curves before final fracture, indicating a limited amount of plastic relaxation. The fracture surfaces were significantly rougher than

those of the precracks and of the specimens

tested at 77 K. Cleavage below the BDTT

was always catastrophic. No noticeable amount of slow crack growth was observed. The preexisting dislocation density and the

availability of dislocation sources can be in- creased by plastic deformation before toughness testing. Specimens with a {110}<110>

crack system were deformed by compression

along the <110> (long) axis at 400°C to a

plastic strain of e^^ = 0. 1 before precracking. At low temperatures the predeformed speci- mens are significantly tougher than the undeTable 1. BDTT and fracture toughness of tungsten

single crystals for the {100} and {110} cleavage

planes with different crack front directions. Frac- ture toughness at room temperature K^'' and at

liquid nitrogen temperature K'''"^ are mean values from at least five individual measurements (stan- dard deviation in parentheses). Fracture toughness

is in MPa m"'^.

Crack system BDTT (K) K"'' K^''*'

{100}<010> 470 8.7 (2.5) 3.4 (0.6) {100}<011> 370 6.2(1.7) 2.4 (0.4) {110)<001> 430 20.2 (5.5) 3,8 (0.4) {110}<110> 370 12.9(2.1) 2.8 (0.2)

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Volume 76, Number 11 PHYSICAL REVIEW LETTERS 11 March 1996

Generation of Nonclassical Motional States of a Trapped Atom

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

Time and Frequency Division. National Institute of Standards and Technology, Boulder, Colorado 80303-3328

(Received 11 October 1995)

We report the creation of thermal, Fock, coherent, and squeezed states of motion of a harmonically

bound 'Be^ ion. The last three states are coherently prepared from an ion which has been initially laser cooled to the zero point of motion. The ion is trapped in the regime where the coupling between

its motional and internal states, due to applied (classical) radiation, can be described by a JaynesCummings-type interaction. With this coupling, the evolution of the internal atomic state provides a

signature of the number state distribution of the motion.

PACS numbers: 42.50.Vk. 32.80.Pj. 32.80.Qk

Nonclassical states of the harmonic oscillator associated

with a single mode of the radiation field (for example,

squeezed states) have been a subject of considerable

interest. One method for analyzing these states has been

through the dynamics of a single, two-level atom which

radiatively couples to the single mode radiation field. This system, described by the Jaynes-Cummings model

(JCM) interaction [1,2], is important to the field of cavity

QED [3]. Nonclassical states of motion occur naturally on an

atomic scale, for example, for electrons in atoms and atoms

in molecules. On a macroscopic scale, the benefits of

nonclassical mechanical states, such as squeezed states, for detection of gravitational waves have been appreciated for some time [4], but so far these states have not been realized. More recently, there has been interest in the generation

and detection of nonclassical states of motion for an atom

confined in a macroscopic, harmonic trap; for trapped ions, see Refs. [5-16]. These states are of interest from the standpoint of quantum measurement concepts and may

facilitate other measurements such as sensitive detection

[5,7,13] or quantum computation [17]. In this Letter we report the generation and detection of

thermal, Fock, coherent, and squeezed states of motion

of a single ^Be~ ion confined in an rf (Paul) trap. We

detect the state of atomic motion by observing the evolution of the atom's internal levels [6,11] (e.g., collapse

and revival) under the influence of a JCM-type interaction realized with the application of external (classical) fields. Under certain conditions, the interaction Hamiltonian is formally equivalent to the JCM Hamiltonian of cavity

QED. Here, the harmonic motion of the atom replaces the single mode of the radiation field. The coupling can be

realized by applying quasistatic fields [7], traveling-wave

fields [6,10,13,15], or standing-wave laser fields [8,9,12]. In each case the coupling J-f/ = — fx E(r) between in- ternal and mofional states is induced by the atom's motion

through the spatially inhomogeneous electromagnetic field E(r), where ft is the atomic dipole operator.

In the present experiment, we drive stimulated Raman

transitions between two hyperfine ground states by applying a pair of traveling-wave laser beams detuned from

an excited electronic state [18]. The resulting interaction between these internal states \S) (denoted | [) and 1 1)) and motional harmonic oscillator states \n) and \n') in the X direction is given by matrix elements

{S',n'\:H:,\S,n) = /in(S',nV+^"''"^"'* + o-_^-"'<"^"*'|5,«) (1)

in a frame which rotates at the difference frequency of the

laser beams. In this expression, cr+ (cr-) is the raising

(lowering) operator for the internal atomic state, a^ (a) is the harmonic oscillator raising (lowering) operator, and Cl

is the Raman coupling parameter [5,13,18]. The LambDicke parameter is defined by 77 = 5k xq, where Sk is the wave-vector difference of the two Raman beams along

;c, and xq = yjH/2mco is the spread of the \n = 0) wave

function in the harmonic well of frequency a). The order n' — n of the vibrational coupling is selected by tuning the Raman beam difference frequency. For ex- ample, by tuning to the first red sideband in the Raman

spectrum, we resonantly enhance the term which drives

transitions between states | j, /?) and \\,n — \). In the Lamb-Dicke limit [8kyJ{x^) «: 1, x = xoia + a"*^)]. the exponentials in Eq. (1) can be expanded to lowest order,

resulting in the operator r]{aa~ + a^tr-), which corre- sponds to the usual JCM operator. We can easily control

the strength and duration of the interaction by varying the

intensity and time the lasers are applied. By choosing other

laser tunings, we can select other operators such as the anti- JCM operator rj{a^ cr+ + ocr-) at the first blue sideband

(which is not present in cavity QED) or the "two-phonon"

JCM operator {ri^/2) {a^a+ + a'^-a-) at the second red sideband. In this experiment, the higher-order terms in the expansion of the exponential in Eq. (1) must also be

taken into account [19]. Reference [20] has explicidy dis- cussed the consequences of these higher-order terms on the trapped ion internal and motional state dynamics.

Additional differences from cavity-QED experiments

include the methods of state generation available (described below) and the relatively small decoherence. In

all but the case of thermal states, we coherently prepare

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PHYSICAL REVIEW LETTERS 11 March 1996

the states starting with an ion which has been laser cooled

to the \n = 0) ground state of motion [18]; this cooling is crucial to preserve the consistency of the generated states. Decoherence in these experiments is small enough that the atomic population coherence survives for many Rabi cycles of the JCM interaction —the "strong coupling" regime

of cavity QED [3]. A single ^Be"^ ion is stored in a strong rf Paul trap [21]

with a pseudopotential oscillation frequency along x of

(o/Itt ~ 11.2 MHz. The ion is cooled using sideband

cooling with stimulated Raman transitions [18] between

the ^Si/2 (F = 2, mp = 2) and -S^/2 (F = I, nif = 1) hyperfine ground states, which are denoted by | i) and

1 1), respectively, and are separated by == 1 .25 GHz. This

prepares the ion in the \S =i, n = 0) state more than 95%

of the time. The Raman beam wave vectors point at 45° to the X axis with their wave-vector difference nearly

along the x axis of the trap [the Lamb-Dicke parameter

is 7/ = 0.202(5)], so the Raman transitions are highly

insensitive to motion in the y or z directions. The beams

are detuned =12 GHz to the red of the -Pi/2 excited state with approximately 0.5 mW of power in each beam, so that the Raman transition coupling is CI/Itt ~ 500 kHz, and

the vibrational structure is clearly resolved. Once the ion is prepared in the | i, 0) state, we create the desired motional state as described below. We

then acquire the signature of the motional state as follows. The Raman beams are tuned to the first blue sideband

and applied for a time t. The population of the 1 1) level

is then measured by applying cr"^ -polarized radiation on

the I i) — ^^3/2 cycling transition and detecting the fluo- rescence [18]. This experiment is repeated at a rate of a few kilohertz, while t is slowly stepped, accumulating the probability Pj(?) of occupation in | i). Fock states of motion can in principle be produced by

quantum jumps [9,16], adiabatic passage [12], or trapping

states [14]; here we use an alternate technique. Since the ion is initially cooled to the | i, 0) Fock state, we create higher-n Fock states by simply applying a sequence of

Rabi 77 pulses of laser radiation on the blue sideband, red sideband, or carrier. For example, the | T, 2) state is prepared by using blue sideband, red sideband, and carrier

TT pulses in succession, so that the ion steps through the

states 11,0), 11,1), I i, 2), and IT, 2). We create a series of Fock states, 1 1, n), and record Piit).

The expected signal is Piit) = cos^(n„,„+ir)exp(— 7„f),

where ft„,„ + i is the Rabi frequency and yn is the decoherence rate between levels \n) and \n + 1). The measured Piit) for an initial | i, n = 0) Fock state is shown

in Fig. 1(a) and fitted by this equation, yielding Hq,! =

94(1) kHz and jq = 1 1.9(4) kHz. Note that Ho,! is much

greater than yo. satisfying the strong coupling condition. We believe the decoherence is due to technical problems

primarily intensity fluctuations of the laser beams and

instabilities of the trap drive frequency and voltage amplitude. The observed increase of y„ with n [we find y„

==

Fock state

(a)

^

—i '

1

'

1

' T"

6 8 10 12 Fock state number

FIG. 1. (a) Piit) for an initial |1,0) Fock state driven by

a JCM-type interaction provided by tuning the stimulated Raman beams to the first blue sideband. The solid line

is a fit by an exponentially decaying sinusoid, (b) The

relative Rabi frequencies n„,„+i/no,i vs the prepared Fock

state number n. The lines represent the predictions of the nonlinear JCM for certain Lamb-Dicke parameters, showing

very good agreement with the known Lamb-Dicke parameter

q = 0.202(5). The 77 «C 1 line corresponds to the LambDicke limit: fi„„+]/no,i = yjn + I.

yo(n + 1)"^] is qualitatively consistent with this view.

In the Lamb-Dicke limit the Rabi frequency between levels II, n) and |t, n + 1) is n„_„+i = yjn + 1 rjCl. If the Lamb-Dicke limit is not satisfied, nonlinear effects in the interaction modify these rates [19,20]. The measured

Rabi frequency ratios nn^„ + i/no,i are plotted in Fig. 1(b),

showing very good agreement with the JCM for the trap's Lamb-Dicke parameter 77 = 0.202. When the ion's motion is not in a Fock state, Piit)

shows a more complicated structure. The motional state

is characterized by a density operator whose diagonal

elements have a number distribution P„ leading to

Piit) = -yni

(2)

For a thermal distribution P„ = N[n/i\ + TT)]", where

N is a normalization constant and n is the average vibrational quantum number. By performing Doppler cooling on the li) — P3/2 cycling transition [18], we generate a thermal state of motion [22]. The value of n can be controlled by the Doppler detuning. An example of Piit) data

for a thermal state of motion is given in Fig. 2. To demonstrate consistency with a thermal state of motion, the timedomain data are fitted by Eq. (2) with a thermal population

distribution for P„. The signal scale and n are allowed to vary in the fit. Values for the base Rabi frequency CLqj and

base decay rate yo (from which the other rates are scaled using the Fock state data) are obtained from a separate trace of Pj(r) for an initial II, 0) state, as in Fig. 1(a). For Fig. 2, the fit yields TT = 1.3 ± 0.1. The inset shows the results of

er^V^ vol.77 2IH d^n) U p^Ct) 4Z. R^ 0"^ ^^^ ^-^".-^^^

ntv^

TN-39
Volume 76, Number 1

1

PHYSICAL REVIEW LETTERS 11 March 1996

thermal

state

coherent

state

20 30

time (^s)

FIG. 2. P[{t) for a thermal state. The data (points) are fitted (line) by a superposition of Fock states with P„ given by a thermal state distribution. The fit allows n to vary, finding

1.3 ± 0.1. The inset shows the decomposition of the data onto the Fock state components (bars) with a fit (line) by the expected exponential distribution, yielding 1.5 ± 0.1.

an independent analysis (the frequency-domain analysis). In this case, we directly extract the populations of the vari- ous \n) levels. Since the Fock state parameters n„,„+i

and y„ are well characterized, the time-domain data can be decomposed into Fock-state components. Equation (2)

is linear in the P„, so we use singular-value decomposition [23] to extract the probabilities, shown in the inset to Fig. 2. The probabilities are fitted by an exponential,

yielding n = 1.5 ± 0.1. Finally, we independently measure n by comparing the size of the red and blue sidebands

[18], yielding n = 1.5 ± 0.2. A coherent state of motion of the ion corresponds

to a displaced zero-point wave packet oscillating in the

potential well. The distribution among Fock states is Poissonian, P„ = {n"e~")/n\. As predicted by the JCM,

the internal-state evolution P|(/) will undergo quantum

collapses and revivals [24]. These revivals are a purely

quantum effect due to the discrete energy levels and the narrow distribution of states [2,24].

Coherent states of ion motion can be produced from

the \n — 0) state by a spatially uniform classical driving

field [25], by a "moving standing wave" [26], by pairs of

standing waves [8], or by a sudden shift of the trap center

[5]. We have used the first two methods; for the data shown here we use the first. For the classical drive, we

apply a sinusoidally varying potential at the trap oscillation frequency on one of the trap compensation electrodes [21] for a fixed time (typically 10 //.s). In Fig. 3 we present an example of P\{t) after creation of a coherent state of

motion, exhibiting collapse and revival. The time-domain

data are fitted by Eq. (2) using a Poissonian distribution and allowing only n to vary. All other parameters are measured from a separate trace similar to Fig. 1(a). The

inset shows the probabilities of the Fock components,

extracted using the frequency-domain analysis described

above. These amplitudes display the expected Poissonian

dependence on n. The observed revival for higher-^

20 30

time (^s)

FIG. 3. /*j(?) for a coherent state, showing collapse and

revival. The data are fitted by a coherent state distribution, yielding 77 = 3.1 ± 0.1. The inset shows the decomposition of

the data onto the expected Fock state components, fitted by a

Poissonian distribution, yielding n = 2.9 ±0.1.

coherent states is attenuated due to the progressively faster decay rates of the higher-n Fock states, and for states with

Tf S: 6 we are unable to see the revival. A coherent state has a definite phase relationship

between the Fock state components. The signal Piit),

however, does not contain this phase information. To

demonstrate the phase coherence of the created states, we

apply a second pulse of classical driving force, which coherently returns the ion to the \n = 0) state, provided the pulse is of the same amplitude as the first and 180° out of

phase with the coherent state. As expected, the return of

the ion to the \n = 0) state is very sensitive to the phase

of the second pulse. However, we are able to reverse

the coherent state and return the ion to \n = 0) more

than 85% of the time, as indicated by a single frequency

component in a subsequent measurement of Piit).

A "vacuum squeezed state" of motion can be created by

a parametric drive [5], by a combination of standing- and

traveling-wave laser fields [8], or by a nonadiabatic drop in the trap spring constant [5]. Here we irradiate the |n = 0)

ion with two Raman beams which differ in frequency

by 2a), driving Raman transitions between the even-«

levels within the same hyperfine state. The interaction can

also be thought of as a parametric drive induced by an

optical dipole force modulated at 2fc> [26]. The squeeze

parameter /3 (defined as the factor by which the variance of

the squeezed quadrature is decreased) grows exponentially

with the driving time. Figure 4 shows Piit) for a squeezed

state prepared in this way. The data are fitted by a vacuum

squeezed state distribution, allowing only ft to vary. The

fit of the data in Fig. 4 demonstrates consistency with a squeezed state and finds yS = 40 ± 10, which corresponds

ton « 7.1. The population distribution for a vacuum squeezed

state is relatively flat and is restricted to the even states, P2n = N{2n)\{tanhrf"/{2"n[f, with /3 = exp(2r). A

squeezed state with /3 = 40 has 16% of the population in states above n = 20. The Rabi frequency differences of

1798 TN-40
Volume 76, Number 1

PHYSICAL REVIEW LETTERS 11 March 1996

1.0 - L squeezed state

0.8 - \

^ 0.6 - \ A y^^^'''\^ /"^'^'V^ /-

0.4 - \J\/^ ^v/ ^^^S^/

0.2 -

0.0 - V V V

r 1 1 1 . ' 5 10

time (us)

15

FIG. 4. P\{t) for a squeezed state. The data are fitted by a squeezed state population distribution, finding ;3 = 40 ± 10, which corresponds to n = 7.1.

these high-n levels are small [see Fig. 1(b)], and with the inclusion of nonlinear effects [19,20] the Rabi frequencies

begin to decrease with n after n = 20. The levels can no longer be distinguished, and the frequency-domain

analysis cannot be used to extract the level populations.

In summary, we have created thermal, Fock, coherent,

and squeezed states of motion of a trapped ion and examined these states through the evolution of the ion internal

state P[{t) induced by a (nonlinear) JCM-type interaction. This experiment demonstrates the utility of a trapped

ion for the creation of nonclassical states of motion and

investigations of the dynamics of Jaynes-Cummings-type

interactions. Given a suitable coupling, it should be possible to transfer these nonclassical state properties to other harmonic oscillators including macroscopic oscilla- tors [5]. In addition to work to reduce the decoherence,

further efforts will involve the creation of arbitrary quan- tum states of motion, including macroscopic superposition

states (Schrodinger's cat states) [3], investigation of the "two-phonon" Jaynes-Cummings model [27,28], quantum

state endoscopy [29], and quantum state tomography [30]. The same interaction can be extended to prepare correlated

internal states of two or more trapped ions for sensitive detection [7,13] or quantum computation [17]. This work is supported by the U.S. Office of Naval

Research and the U.S. Army Research Office. We ac- knowledge important contributions by J. C. Bergquist and

helpful comments on the manuscript by M. Stephens, C. S. Wood, and M. Young.

[1] E.T. Jaynes and F.W. Cummings, Proc. IEEE 51, 89

(1963).

[2] B.W. Shore and P. L. Knight, J. Mod. Opt. 40, 1195

(1993).

[3] Cavity Quantum Electrodynamics, edited by P. R. Berman

(Academic Press, Boston, MA, 1994).

[4] J.N. Hollenhorst, Phys. Rev. D 19, 1669 (1979).

[5] D.J. Heinzen and D.J. Wineland, Phys. Rev. A 42, 2977

(1990).

[6] C.A. Blockley, D.F. Walls, and H. Risken, Europhys.

Lett. 17, 509 (1992).

[7] D.J. Wineland, J.J. Bollinger, W.M. Itano, F. L. Moore,

and D.J. Heinzen, Phys. Rev. A 46, R6797 (1992).

[8] J.I. Cirac, A.S. Parkins, R. Blatt, and P. Zoller, Phys. Rev. Lett. 70,556(1993).

[9] J.I. Cirac, R. Blatt, A.S. Parkins, and P. Zoller, Phys. Rev. Lett. 70,762(1993).

[10] H. Zeng and F. Lin, Phys. Rev. A 48, 2393 (1993).

[11] J.I. Cirac, R. Blatt, A.S. Parkins, and P. Zoller, Phys. Rev. A 49, 1202 (1994).

[12] J. I. Cirac, R. Blatt, and P. Zoller, Phys. Rev. A 49, R3174

(1994).

[13] D.J. Wineland, J.J. Bollinger, W.M. Itano, and D.J. Heinzen, Phys. Rev. A 50, 67 (1994).

[14] R. Blatt, J.I. Cirac, and P. Zoller, Phys. Rev. A 52, 518

(1995).

[15] H. Zeng and F. Lin, Phys. Rev. A 52, 809 (1995).

[16] J. Eschner, B. Appasamy, and P. E. Toschek, Phys. Rev.

Lett. 74, 2435 (1995).

[17] J. I. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995); C. Monroe, D. M. Meekhof, B.E. King, W. Itano, and

D.J. Wineland, Phys. Rev. Lett. 75, 4714 (1995).

[18] C. Monroe, D. M. Meekhof, B.E. King, S.R. Jefferts, W. M. Itano, D. J. Wineland, and P. L. Gould, Phys. Rev.

Lett. 75, 4011 (1995).

[19] See Eq. (31) in D.L Wineland and W.M. Itano, Phys. Rev. A 20, 1521 (1979).

[20] W. Vogel and R.L. de Matos Filho, Phys. Rev. A 52, 4214 (1995).

[21] S. Jefferts, C. Monroe, E. Bell, and D.J. Wineland, Phys. Rev. A 51, 3112(1995).

[22] S. Stenholm, Rev. Mod. Phys. 58, 699 (1986).

[23] W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P.

Flannery, Numerical Recipes (Cambridge University

Press, Cambridge, 1992).

[24] J.H. Eberly, N. B. Narozhny, and J.J. Sanchez- Mondragon, Phys. Rev. Lett. 44, 1323 (1980),

[25] P. Carruthers and M. M. Nieto, Am. J. Phys. 7, 537 (1965).

[26] D.J. Wineland, J.C. Bergquist, J.J. Bollinger, W.M.

Itano, F. L. Moore, J. M. Gilligan, M. G. Raizen, D. J. Heinzen, C. S. Weimer, and C. H. Manney, in Laser Manipulation of Atoms and Ions, edited by E. Arimondo,

W. D. Phillips, and F. Strumia (North-Holland, Amster- dam, 1992), p. 553.

[27] RL Knight, Phys. Scr. T12, 51 (1986).

[28] R.L. de Matos Filho and W. Vogel, Phys. Rev. A 50, R1988 (1994).

[29] P. J. Bardroff, E. Mayr, and W. P. Schleich, Phys. Rev. A

51, 4963 (1995), and references therein. [30] K. Vogel and H. Risken, Phys. Rev. A 40, 2847 (1989); D. T. Smithey, M. Beck, M.G. Raymer, and A. Faridani,

Phys. Rev. Lett. 70, 1244 (1993); T.J. Dunn, I. A. Walmsley, and S. Mukamel, Phys. Rev. Lett. 74, 884

(1995); S. Wallentowitz and W. Vogel, Phys. Rev. Lett. 75, 2932 (1995); J.F. Poyatos, R. Walser, J.I. Cirac,

P. Zoller, and R. Blatt (to be published).

TN-41 1799
A "Schrodinger Cat"

Superposition State of an Atom

C. Monroe,* D. M. Meekhof, B. E. King, D. J. Wineland

A "Schrodinger cat"-like state of matter was generated at the single atom level. A trapped

^Be"^ ion was laser-cooled to the zero-point energy and then prepared in a superposition

of spatially separated coherent harmonic oscillator states. This state was created by

application of a sequence of laser pulses, which entangles internal (electronic) and

external (motional) states of the ion. The Schrodinger cat superposition was verified by

detection of the quantum mechanical interference between the localized wave packets.

This mesoscopic system may provide insight into the fuzzy boundary between the classical and quantum worlds by allowing controlled studies of quantum measurement and

quantum decoherence.

vjuantum mechanics allows the preparation of physical systems in superposition

states, or states that are "smeared" between two or more distinct values. This curious

principle of quantum mechanics ( J ) has been extremely successful at describing

physical behavior in the microscopic

world—from interactions of atoms with

photons to interactions at the subnuclear

level. But what happens when we extend

the quantum superposition principle to macroscopic systems conventionally described by classical physics? Here, superpositions introduce a great amount of conceptual difficulty, as pointed out in 1935 by the

celebrated Einstein-Podolsky-Rosen (2) and Schrodinger cat (3) paradoxes. For ex- ample, in Schrodinger's thought experiment (3), an unfortunate cat is placed in a quantum superposition of being dead and

alive (correlated with a single radioactive atom that has and has not decayed). The

The authors are in the Time and Frequency Division, MS

847, National Institute of Standards and Technology,

Boulder, CO 80303, USA. 'To whom correspondence should be addressed. TN-42 SCIENCE • VOL. 272 -

state of the system can be represented by

the entangled quantum mechanical wave

.

function.

v|,=

l©)lT)+ l©)U)

V2

(1)

where |q) and l@) refer to the states of a

live and dead cat, and I i ) and I t ) refer to the internal states of an atom that has and

has not radioactively decayed. This situa- tion defies our sense of reality because we

only observe live or dead cats, and we ex- pect that cats are either alive or dead inde- pendent of our observation (4). Schrodinger's cat paradox is a classic illustration of the conflict between the existence of quan- tum superpositions and our real-world ex- perience of observation and measurement. Although superposition states such as Schrodinger's cat do not appear in the macroscopic woild, there is great interest in the

realization of "Schrodinger cat"-like states

in mesoscopic systems, or systems that have

both macroscopic and microscopic features. In this context, the "cat" is generalized to

24 MAY 1996 1131
represent a physical system whose attributes are normally associated with classical concepts, such as the distinguishable position of

a particle (instead of the state of livelihood of

a real cat). In this article, we report the

creation of a mesoscopic "Schrodinger cat"

state at the single atom level: An atom was

prepared in a quantum superposition of two

spatially separated but localized positions. In analogy to Schrodinger's original proposition

given by Eq. 1, we created the following state:

V|; =

Ui)i r)+ Uz)! i)

(2)

where Ixj) and Ixj) denote classical-like wave packet states corresponding to sepa- rated spatial positions of the atom, and I i ) and I 1 ) refer to distinct internal electronic quantum states of the atom (5). The wave

packets are separated by a mesoscopic dis- tance of more than 80 nm, which is large

compared with the size of the individual wave packets (^7 nm) or the atomic di- mension (=0.1 nm).

Mesoscopic Schrodinger cats may provide an interesting testing ground for the controversial theory of quantum measurement (6). At the core of this historical issue is the

question of the universality of quantum mechanics. The "Copenhagen interpretation" of Bohr (7) and Heisenberg (8) holds that the measuring apparatus always involves

classical concepts, thus forcing a seemingly

arbitrary division between the quantum and

classical worlds. Einstein (2) on the other hand argued that for quantum mechanics to be complete, it should describe physical be- havior at all scales. One practical approach

toward resolving this controversy is the in- troduction of quantum decoherence, or the

environmentally induced reduction of quan- tum superpositions into statistical mixtures and classical behavior (9). Decoherence is commonly interpreted as a way of quantifying the elusive boundary between classical and quantum worlds and almost always pre- cludes the existence of macroscopic Schrodinger cat states, except at extremely short time scales (9, 10). The creation of mesoscopic Schrodinger cat states may allow controlled studies of quantum decoherence and

the quantum-classical boundary. We note

that quantum decoherence has received much interest lately because of its importance in proposals for quantum computation

(J J) and quantum cryptography {12).

Macroscopic superposition states of matter have been realized for electron {13), neutron {14), and atom {15) beam splitters, where these particles are split into superpositions of separated paths. The matter wave

packets in these experiments spread in time

because the particles are unbound. Spatially

separated superpositions of electrons within atoms have been demonstrated by exciting

electrons to Rydberg states with pulsed la- sers {16). Here, the electron wave packet is also dispersive because of its anharmonic

binding potential. There have been related proposals for the creation of macroscopic

superposition states of vibration in molecules or crystals {17) and of electrical currents flowing in superconducting rings (18). The appeal of creating a Schrodinger cat

state in a harmonic oscillator is that wave

packet dispersion can be negligible. The simple time evolution of a coherent harmonic

oscillator wave packet preserves the separa- tion of the superposition and aids in the

visualization and interpretation of experi-

(3,-3)

c (0-), d (a-)

Fig. 1. (A) Electronic (inter- nal) and motional (e>cternal) energy levels (not to scale) of the trapped ^Be"^ ion, coupled by the indicated laser beanns a through d. The dif- ference frequency of the

"carrier" Raman beams a and b is set near (ji^p/2-n «

1 .250 GHz, providing a two- photon Raman coupling be- tween the ^Si/2(/^ = 2,mp =

-2) and ^S^^^il^ = 1 , m^ =

-1) hyperfine ground states (denoted by \ i), and I t ),, respectively). The difference frequency of the "displace- ment" Raman beams b and

c is set to (o/2tt == 1 1 .2 MHz.

This excites the motion of the ion to a coherent state I ae"')^ from an initial zero-point state of motion 1 0)^

in the harmonic potential. Because of the polarization of beams b and c, they do not affect motion correlated

with the I i )i intemal state. The three Raman beams (a, b, and c) are detuned A « -12 GHz from the

^P-i/2(F = 2, mp = -2) excited state (radiative linewidth y/Z-K = 1 9.4 MHz). Detection of the internal state is accomplished by first illuminating the ion with a~-polarized "detection" beam d, which drives the cycling

^S.^,2iF = 2, m^ = -2) ^^^3/2^^ = 3, m^r = -3) transition, and then observing the scattered fluorescence.

(B) Geometry of the three Raman laser beams a, b, and c, with polarizations indicated. The quantization axis defined by the applied magnetic field B is 45° from the x axis of the harmonic trap potential.

b (CT+/CT-)

a in)

1132 SCIENCE VOL. 272 • 24 MAY 1996

ments. There have been several proposals to create mesoscopic Schrodinger cat states in a

single mode of the electromagnetic field, which is formally equivalent to a harmonic

oscillator. For instance, these states are ex- pected to evolve from the amplitude disper- sion of a laser beam propagating in an an- harmonic Kerr medium {19). In cavityquantum-electrodynamics, these states are predicted to emerge by driving a coherent

state with a Jaynes-Cummings interaction to the point of collapse {20), by continuously

pumping a single cavity mode with polarized two-level atoms {21), or by realizing a dis- persive interaction between a single atom

and a single cavity mode {22, 23). It has been proposed that Schrodinger cat states of

a single harmonically bound atom can be

created by driving the atom with a strong

laser field and relying on a measurement to project the desired superposition state {24) or by optically pumping the atom to a "dark"

state with multiple laser beams {25).

Experimental approach. In the present work, we create a Schrodinger cat state of the harmonic oscillator by forming a superposi- tion of two coherent-state wave packets of a

single trapped atom with a sequence of laser pulses. Each wave packet is correlated with a

particular intemal state of the atom. To analyze this state we apply an additional laser pulse to couple the intemal states and then

measure the resulting interference of the dis- tinct wave packets (26, 27). The key features of our approach are that (i) we control the harmonic motion of the trapped atom to a high degree by exciting the motion from initial zero-point wave packets to coherent-state wave packets of well-defined amplitude and

phase; (ii) we do not rely on a conditional measurement to project out the desired Schrodinger cat state; and (iii) wave packet

dispersion of the atomic motion is negligible.

The experimental apparatus is described elsewhere {28, 29). A single ^Be"^ ion is confined in a coaxial-resonator radio frequency

(RF)-ion trap {28) that provides harmonic

oscillation frequencies of (co^, co^, co^)/ Zir =«

(11.2, 18.2, 29.8) MHz along die principal axes of the trap. We laser-cool the ion to the quantum ground state of motion (29) and

then coherently manipulate its intemal (elec- tronic) and external (motional) state by applying pairs of off-resonant laser beams, which

drive two-photon-stimulated Raman transi- tions (29, 30). As shown in Fig. lA, the two

intemal states of interest are the stable '^S.^i2{F = 2, rrip = -2) and ^Si„(F = I, rrip = -1)

hyperfine ground states (denoted by | i ); and

I I )(, respectively), separated in frequency by

oopjp/2'TT == 1.250 GHz. Here, F and nip are quantum numbers representing the total in- temal angular momentum of the atom and its projection along a quantization axis. The Ra- man beams are detuned by A '^ —12 GHz

from the ^Pi/2(F = 2, mp = -2) excited state,

TN-43
which acts as the virtual level, providing the Raman coupling. The external motional

states are characterized by the quantized vi- brational harmonic oscillator states \n)^ in the X dimension, separated in frequency by

co^Ztt « 11.2 MHz.

When we tune the Raman beam differ- ence frequency near co^^p and apply the "carrier beams" a and b of Fig. 1, the ion experiences a coherent Rabi oscillation between

the internal states I I )^ and I | )^. By adjusting the exposure time of the carrrier beams, we can, for example, "flip" the internal state (a TT-pulse, or one-half of a Rabi cycle) or "split-recombine" the internal state (a ir/Z- pulse, or one-fourth of a Rabi cycle). Transitions on the carrier do not significantly

affect the state of motion, because beams a and b are copropagating. When we tune the Raman beam difference frequency near w^

and apply the "displacement" beams b and c of Fig. 1, the effect is formally equivalent to applying the displacement operator to the

state of motion (30). Alternatively, the dis- placement beams can be thought of as pro- ducing a "walking wave" pattern whose

time-dependent dipole force resonantly excites the harmonic motion {31). This force promotes an initial zero-point state of motion |0)^ to a coherent state (32)

|^), = exp(-|(3p/2)S„^"/(n!)^/^|n),

(3) where 3 = ae'^ is a dimensionless complex

number that represents the amplitude and

phase of the motion in the harmonic potential (33). The probability distribution of vibrational levels in a coherent state is Pois- sonian with mean number of vibrational quanta (n) = a^. The coherent state of motion is much like classical motion in a harmonic potential with amplitude 2axQ,

where Xq = (^/2moo^)'/^ = 7.1(1) nm is the

root mean square Gaussian size of the oscil- lating wave packet, m is the mass of the ion, and h is Planck's constant divided by 2tt (the number in parentheses is the standard

error in the last digit). The polarizations of the three Raman

beams a, b, and c produce it, <y'^/(r~, and a~

couplings, respectively, with respect to a

quantization axis defined by an applied 0.20- mT magnetic field, as indicated in Fig. IB. As a result, the displacement beams (b and c) affect only the motional state correlated with

the I f )j internal state, because the (T~-polarized beam c cannot couple the internal

state 1 i ); to any virtual ^Pi/i states (34). This selectivity of the displacement force provides quantum entanglement of the inter- nal state with the external motional state. Although the motional state can be thought

of as nearly classical, its entanglement with

the internal atomic quantum levels precludes any type of semiclassical analysis.

TN-44

Table 1. Raman beam pulse sequence for the generation of a Schrodinger cat state. The magnitude

(phase) of the coherent state is controlled by the duration (phase) of the applied displacement beams in steps 2 and 4. The phases of internal state carrier operations in steps 1 , 3, and 5 are relative to step 5. The states created after each step do not include overall phase factors, and the phase appearing in the

final state is 5 = |x - 2v + it.

Step Function Approximate

duration (|xs) Phase State created (see Fig. 2)(initial state: | i)||0)e)

1

2

Carrier iT/2-pulse Displacement0.5 T-10.0

3

4

Carrier TT-pulse Displacement1.0 T- 10.0 5 Carrier 7r/2-pulse

0.5

^JL [li),|0>,-;e-'Ht>ilO)J/V2 -ci)/2 [ U >, 1 0)g - /e-'*^ I t >, I ae-'*'2>J/V2

V [e'*''-^'' U )J ae-''»'\ + /e-- 1 t ), 1 0>J/V2

4,/2 [e'O'- »-) U ). I ae-'t"'2)^ + /e"^ I f >, 1 ae'*^^)j/V2

1 /2 U ),[ I ae-'^'X - d^ I ae-^^s)

J

-\I2\ t )i[|ae-'*'\ + e«|ae'*^2)J

Each Raman beam contains ===1 mW of power at =313 nm. This results in a two- photon Rabi frequency of H/Ztt = 250 kHz

for the copropagating Raman carrier beams a and b, or a ir-pulse exposure time of about 1

|JLS. We apply the displacement Raman

beams (b and c) to the ion in directions such

that their wave vector difference 8k points

nearly along the x axis of the trap. Motion in the 31 or ? dimensions is therefore highly

insensitive to the displacement beams. When we apply the displacement beams to a zero-point wave packet (correlated with the

I t )j state) for time t, we expect to create a coherent state of amplitude a = "nfl^T. Here,

y\ = 0.205(5) is the Lamb-Dicke parameter

{30) and flj/2TT =« 300 kHz is the coupling

strength of the displacement beams. After each preparation cycle (described below), we

detect which internal state (I i ); or I Dj)

the atom occupies independent of its state of motion. This is accomplished by applying a few microwatts of a~-polarized light ("de- tection" beam d of Fig. lA) resonant with

the cycling I i )( -» ^P3/2(F = 3, nip =

—3) transition [radiative linewidth 7/2-77 = 19.4 MHz at wavelength (X.) «= 313 nm]

and observing the resulting ion fluores- cence. Because this radiation does not ap- preciably couple to the I | ); state, the

fluorescence reading is proportional to the

probability P ^ the ion is in state I \, );. We collect on average about one photon

per measurement cycle when the ion is in the I \, )j state (29). Creation and detection of the Schrodinger cat state. The ion is first laser-cooled

so that the I i ); U^ = 0)^ state is occupied

=«95% of the time as described in {19). We

then apply five sequential pulses of the Raman beams (the evolving state of the system is summarized in Table 1 and Fig. 2). In step 1, a tt/2 -pulse on the carrier splits the wave function into an equal superposi- tion of states I i)JO)^ and I t)ilO)e- I^i step 2, the displacement beams excite the motion correlated with the I t )^ component to a coherent state I ae"'*'^)^. In step

3, a TT-pulse on the carrier swaps the in- ternal states of the superposition. In step

IT):

]i>i

\i/ O Ni/

^

^k>

\i/ \i/ \i/ \> v> x>

A B C D E F

Fig. 2. Evolution of the position-space atomic wave packet entangled with the intemal states I i \ and

I t )i during creation of a Schrodinger cat state with a = 3 and ((> = tt (see Table 1 ). The wave packets are snapshots in time, taken when the atom is at the extremum of motion in the harmonic trap (represented

by the parabolas). The area of the wave packets corresponds to the probability of finding the atom in the

given internal state. (A) The initial wave packet corresponds to the quantum ground state of motion after laser-cooling. (B) The wave packet is split after a Tr/2-pulse on the carrier. (C) The I t )i wave packet is excited to a coherent state by the force F of the displacement beams. Note the force F acts only on the

I t )| wave packet, thereby entangling the internal and motional systems. (D) The I i )| and I t )i wave

packets are exchanged following a -ir-pulse on the carrier. (E) The I t >; wave packet is excited to a coherent state by the displacement beam force -F, which is out of phase with respect to the force in (C). The state shown in (E) corresponds most closely to Schrodinger's cat (Eqs. 1 and 2). (F) The I J, )j and

I t >i wave packets are finally combined after a 'n-/2-pulse on the carrier.

SCIENCE • VOL. 272 24 MAY 1996 1133
4, the displacement beams excite the motion correlated with the I | )^ component

to a second coherent state I ae'*^^)^. In

step 5, a final ir/Z-pulse on the carrier combines the two coherent states. The

relative phases of the above steps are de- termined by the phases of the RF differ- ence frequencies of the Raman beams (29,

30), which are easily controlled by phaselocking the RF sources. The state created after step 4 is a super- position of two independent coherent states each correlated with an internal state of the

ion, in the spirit of Schrodinger's original

thought experiment (Eqs. 1 and 2). We

verify this superposition by recombining the coherent wave packets in the final step 5. This creates the following entangled state: l^)= U),ls_),-ilT),ls,>,

with

ls^).=-

-'^'\ ^''\

2

(4)

For cf) = TT and 8 = 0, the states I S^)^

(when properly normalized) are known as "even" and "odd" Schrodinger cats {35).

(J) Pi (<t>)

Tt 0.50

0.75jt 0.50

0.507C 0.50

0.207t 0.45

0.1 Oji

0.07j:

0.037C

0.80

0.60

0.24

0.00 0.00

Fig. 3. Evolution of the position-space wave

packet superposition correlated w/ith the I i ); in- ternal state as the phase separation ^ of the two

coherent states is varied, for a = 3 and 5 = 0. The

expected signal P ^ (<j)) is the integrated area under

these wave packets. Each trace is a snapshot in time, taken when one of the wave packets is at the

rightmost fuming point in the harmonic trap. The wave packets are maximally separated at <|) = ir [P i (<t)) «« 1/2], but they begin to overlap as <}) gets

smaller [P ^{i?) oscillates]. Finally, the wave packets destructively interfere at 4) = [P ^

(<})) = 0]. This vanishing interference signal is a signature of an odd Schrodinger cat state associated with the

I i )i state, because 5 = 0, Probability conservation is ensured by a similar but constructive inter- ference in the I t )i state.

1134

The relative populations of I i ); and

I I ); depend on the motional phase differ- ence ^ between the two coherent wave

packets because of the quantum interference between the two coherent states contained

in IS^)g. We directly measure this interfer- ence by detecting the probability P ^

(cf)) that the ion is in the I i ); internal state for a given value of cj). We continuously repeat

the experiment—cooling, state preparation, detection—while slowly sweeping the rela- tive coherent state motional phase <)). Figure

3 depicts the expected position basis wave

packet I (x| S_ )^| ^ correlated with the | | )j internal state as a function of cj) for 8 = and a = 3. The calculated wave packets in the

figure are snapshots in time, as each part of

the superposition oscillates in the harmonic

trap. The measured signal P ^ {^) is just the

a = 0.84(2)

1.0 -\

0.5

0.0

a = 2.97(6) -Ap^^

1.0

0.5

0.0

a = 6 (theory)

-7C/2 ;c/2

Coherent state phase separation <{>

Fig. 4. Measured and fit interference signal P ^

(cf)) versus the phase difference cf) of two coherent

states for 8 = 0. Curves in (A) to (D) represent measurements for various values of t (2, 3, 5, and

15 |xs, respectively). As t grows, the feature near

(j) = narrows. The lines are fits of the measure- ments to the parameter a (Eq. 4), yielding a =

0.84, 1 .20, 1 .92, and 2.97, respectively. The fit in curve (D) includes a loss of contrast and repre- sents a superposition of two Xq =» 7-nm wave

packets with a maximum separation of AoXq = 80

nm. Curve (E) is a theoretical plot for a pair of coherent states with a = 6. Each data point in (A) to (D) represents an average of =»4000 measurements, or 1 s of integration.

SCIENCE VOL. 272 • 24 MAY 1996

integral of the complete I i ); wave packet

over space and is time-independent.

Pi (4)) \{x\S-)Vix =

1 - £-«'(!- ^0^*) cos(8 + oihm^)

(5)

The wave packets of the superposition

are maximally separated in phase space for

4) = ±7T, where the signal is about 1/2 (for large a). However, as c}) approaches 0, the wave packets of the superposition begin to

overlap, finally interfering completely at 4) = 0. For large a, the signal P ^

((])) acquires

oscillations near cf) = 0, with the width of

the central interference fringe (in cf) space)

proportional to 1/a^. If the two pieces of the wave packet are not phase-coherent or if the state is a statistical mixture (8 random

between preparations) instead of a coherent

superposition of wave packets, the signal

would remain constant, P ^

(c()) = 1/2. We

experimentally set the phase 8 associated with the internal state superposition by

blocking the displacement beams (a = 0) and measuring P; = sin^(S/2).

Superpositions versus mixtures. In Fig. 4, we display the measured P |

(<j)) for 8 = and

a few different values of the coherent state amplitude a, which is set by changing the

duration t of application of the displacement

1.0

0.5 -a

0.0 -

1.0 -

0.5 -

0.0 -

Odd cat

—r"

-TZl2 7C/2

Coherent state phase separation <f

Fig. 5. Measured interference signal P|(4>) for three values of S (a = 1 .5). The top curve corre- sponds to 5 = 1 .0317 (approximate even cat state correlated with I J, )| exhibiting constructive inter- ference), the middle curve to 5 = 0.48-tt [approx- imate "Yurke-Stoler" cat state (79)], and the bot- tom curve to 8 = 0.06tt (approximate odd cat

state exhibiting destructive interference). Each

data point represents an average of =4000 measurements, or 1 s of integration.

TN-45
beams (steps 2 and 4 of Table 1). The pres- ence of the interference feature near (}> =

verifies that we are producing superposition

states instead of statistical mixtures, and the

feature clearly narrows as a is increased. We

have verified that the interference feature vanishes [P j^ (<})) = 1/2] when 8 is randomized

between preparations. In Fig. 5, we present P I

(cj))' for three different values of the phase

8 while fixing T. The shape of the interference

at (j) = indicates the parity of the cat state at

4) = ±'7T. Here, we see the transition from an

even cat (8 = tt) to the Yurke-Stoler (J 9) cat

(8 «= tt/2) to an odd cat state (8 == 0) correlated with the I I )| state. We extract the amplitude of the Schrodinger cat state by fitting the interference data to the parameter a appearing in Eq. 4. The extracted values of a agree with the independently measured value tiH^t for short displacement beam durations (t S 10

(jLs) {36). We measure coherent-state amplitudes as high as a == 2.97(6), corresponding to an average of (n) = 9 vibrational quanta in the state of motion. This indi- cates a maximum spatial separation of 4ctXo = 83(3) nm, which is significantly larger

than the single wave packet size of Xq =

7.1(1) nm. The individual wave packets are thus clearly separated in phase space. For longer displacement beam durations

(t a 10 |xs), the interference signal loses contrast, as evident in Fig. 4D. We believe

this is partly due to fluctuations of the ion

oscillation frequency co^, which causes the motional phase difference ^ to fluctuate from measurement to measurement and

wash out narrow interference features. The

measured interference signal is sensitive to fluctuations of m^ at a time scale that is longer than the time to create the cat (t^ == T = 10 |jLs) but shorter than the integrated measurement time ( = 1 s per data point in

Figs. 4 and 5). The observed loss of contrast

indicates a phase fluctuation of 8^) = 0.1

rad, which would be consistent with a frac- tional ion oscillation frequency fluctuation of 8(0^0)^ = 10"'^ in a «=100-kHz bandwidth.

Anharmonicities of the trap {28) are expect- ed to contribute to a phase dispersion of only

'^'lO"^ rad during the creation of the cat. Decoherence. When a Schrodinger cat consisting of two separated coherent states

is coupled to a thermal reservoir, the super- position decays exponentially to a statistical mixture with a rate initially proportional to

a^, or the square of the separation of the wave packets (9, 10, 27). As the separation

is made larger (more classical), the lifetime-^ of the superposition shortens. This decoher- ence process underlies the reason quantum

superpositions are not generally seen in the macroscopic world and also illustrates the experimental difficulty in preparing and

maintaining even mesoscopic superpositions. In the experiment, the quantum interTN-46

ference signal is only sensitive to decoher- ence during the period of time t^ between

the generation of the two coherent states

(steps 2 and 4 of Table 1). This is because

only the internal atomic state is detected,

and once the second coherent state is pro- duced (step 4), the internal and motional

states do not interact, even if the motion

equilibrates with an external reservoir. We

therefore expect the interference signal (Eq.

4) to exhibit a contrast of exp(— a^X.T^), where X. is the temperature-dependent re- laxation rate to the thermal reservoir (JO). The loss of contrast we observe may involve

the onset of decoherence, although it is difficult to make a quantitative comparison

because we do not know the spectrum and

effective temperature of the supposed reservoir. We note that we have previously measured a heating rate of 3(n)/5t «= 10^ s~^ (29), but because the source of this heating is not

understood at the present time, it is difficult to characterize its effect on decoherence. The next step is to deliberately induce

decoherence of the Schrodinger cat by coupling the system to "engineered" reservoirs during the interval t^. For instance, a uni- form stochastic electric field can be applied,

whose coupling to the ion could simulate a thermal reservoir at a controllable temperature. Alternatively, one can pulse the Raman

beams and controUably allow spontaneous

emission to occur during the interval t^. [similar to stimulated Raman cooling {29)]. With

this coupling, one can simulate thermal, zero- temperature, squeezed, and other reservoirs

{37). By monitoring the contrast of the

interference signal, it should then be possible to study the decoherence of the Schrodinger cat to these known reservoirs. The effects of decoherence might also be

measured by mapping the complete density matrix of the Schrodinger cat state, as pro- posed in recent tomographic schemes {38).

We finally note that our technique for preparing Schrodinger cat superpositions

of two coherent states in one dimension

can easily be extended to create superpositions of more than two coherent states and superpositions in two and three di- mensions. This technique may also be useful

for the creation of superposition states of the

collective motion of many trapped atoms.

REFERENCES AND NOTES

1

.

P. A. M. Dirac, The Principles of Quantum Meclianics

(Clarendon, Oxford, ed. 4, 1984), p. 12.

2. A. Einstein, B. Podolsky, N. Rosen, Phys. Rev. 47, 777(1935).

3. E. Schrodinger, Naturwissenschaften 23, 807

(1935); translation in (6). 4. Issues involving observation by the cat itself (by virtue

of its consciousness) are beyond the scope of this

article. For our purposes, the cat can be replaced, for example, by an object placed in a superposition of macroscopically separated positions.

5. In the literature, there is not universal agreement

SCIENCE • VOL. 272 • 24 MAY 1996

about the definition of the "cat' ' state. As opposed to the definition used in Eqs. 1 and 2 of this article, some authors prefer to define a Schrodinger cat

state as (1xt> + Ix^WVz, or a superposition of tvi'o classical-like states that are not correlated vwth the

state of a second system. In the analysis of the state

shoviTi in Eq. 2, vi^e in fact create the state corre- sponding to this altemate definition.

6. J. A. Wheeler and W. H. Zurek, Eds., Quantum Theory and Measurement (Princeton Univ. Press,

Princeton, NJ, 1983).

7. N. Bohr, Nature 1 21 , 580 (1 928); Phys. Rev. 48, 696

(1935).

8. W. Heisenberg, Z. Phys. 43, 172 (1927); translation

in (6). 9. W. H. Zurek, Phys. Today 44, 36 (October 1991),

1 0. D. F. Walls and G. J. Milbum, Phys. Rev A 31 , 2403

(1985); C. M. Savage and D. F. Walls, ibid. 32, 2316

(1985); M. J. Collett, ibid. 38, 2233 (1988).

11. D. P. DiVincenzo, Science 270, 255 (1995); I. L

Chuang, R. Laflamme, P. W. Shor, W. H. Zurek,

ibid., p. 1633.

12. C. H. Bennett, Phys. Today 48. 24 (October, 1995).

13. L Marton, J. Arol Simson, J. A. Suddeth, Phys. Rev. 90,490(1954).

1 4. H. Maier-Leibnitz and T. Springer, Z, Phys. 1 67, 368

(1962).

15. 0. Carnal and J. MIynek, Phys. Rev Lett. 66, 2689

(1991); D. W. Ke'ith, C. R. Ekstrom, 0. A. Turchette,

D. E. Pritchard, ibid., p. 2693; M. Kasevich and S. Chu, ibid. 67, 181 (1991); J. Lawall etal., ibid. 75, 4194(1995).

16. L D. Noordam, D. 1. Duncan, T. F. Gallagher, Phys. Rev A 45, 4734 (1992); R. R. Jones, C. S. Raman,

D. W. Schumacher, P. H. Bucksbaum, Phys. Rev.

Lett. 71, 2575 (1993); M. W. Noel and C, R. Stroud

Jr., ibid 75, 1252(1995).

17. J. Janszky, A.V. Vinogradov, T, Kobayashi, Z. Kis, Phys. Rev A 50, 1 777 (1 994); i. A. Walmsley and M.

G. Raymer, ibid. 52, 681 (1995).

18. A. J. Leggett and A. Garg, Phys. Rev Lett. 54, 857

(1985).

19. B. Yurke and D. Stoler, ibid. 57, 13 (1986); Phys. Rev A 35, 4846 (1987); ibid 36, 1955 (1987). 20. S. J. D. Phoenix and P. L Knight, Ann. Phys. 186, 381 (1988); J. Gea-Banacloche, Phys. Rev. Lett. 65, 3385 (1990); V. Buzek, H. Moya-Cessa, P. L. Knight,

S. J. D, Phoenix, Phys. Rev. A 45, 8190 (1992). 21

.

J. J. Slosser, P. t^eystre, E. M. Wright, Opt. Lett. 1 5, 233(1990).

22. C. M. Savage, S. L. Braunstein, D. F. Walls, ibid., p. 628.

23. M. Brune, S. Haroche, J. M. Raimond, L Davidovich,

N. Zagury, Phys. Rev A 45, 51 93 (1 992); S. Haroche and J. M. Raimond, in Cavity Quantum Eiectrody- namics, P. Berman, Ed. (Academic Press, New/

York, 1994), pp. 123-170.

24. J. F. Poyatos, J. I. Cirac, R. Blatt, P. Zoller, in preparation.

25. R. L. de Matos Filho and W. Vogel, Phys. Rev. Lett. 76,608(1996).

26. L Davidovich, A. Maali, M. Brune, J. M. Raimond, S. Haroche, ibid. 71 , 2360 (1993). 27. P. Goetsch, R. Graham, F. Haake, Phys. Rev. A 51

,

136(1995).

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

/fa/cf.,p. 3112.

29. C. Monroe etal., Phys. Rev Lett. 75, 4011 (1995). 30. D. M. Meekhof, C. Monroe, B. E. King, W. M. Itano, D. J. Wineland, ibid. 76, 1796 (1996). 31

.

D. J. Wineland ef a/., in Laser Manipulation ofAtoms

and ions, E. Arimondo, W. D. Phillips, F. Strumia, Eds.

(North-Holland, Amsterdam, 1992), pp. 562-563.

32. R. J. Glauber, Phys. Rev 131, 2766 (1963); ibid., p. 2529.

33. In the laboratory fi-ame, the coherent-state parameter p has the periodic time dependence Pl = ae''* ~

"'), w/here to is the ft-equency of the harmonic oscillator. Throughout the article, this time dependence is suppressed by working in an interaction fi-ame where

p = PLe'"'. 34. The effects of coupling to the ^P^/z excited state are small (detuning «= -210 GHz) and can be ne- glected here.

35. J. PeTna, Quantum Statistics of Linear and Nonlinear

Qpticai Phenomena (Reidel, Dordrecht, 1984).

1135

t
36. For large values of r\Clai. we expect a{j) to sat- 37. J. F. Poyatos, J. I, Cirac, P. Zoller, in preparation. useful discussions with J. I. Cirac, P. Zoller, and D. urate because of nonlinear correction factors that linn- 38. D. T. Smithey, M. Beck, M. G. Raymer, A. Faridani, Walls. We thank M. Young, S. Mechels, and D. Lee

it the amplitude of the coherent state and introduce Phys. Rev. Lett. 70, 1244 (1993); S. Wallentowitz for critical comments on the manuscript. This work is amplitude squeezing [see (37) and W. Vogel and R. L and W. Vogel, ibid. 75, 2932 (1 995); J. I. Cirac and P. supported by the U.S. Office of Naval Research and

de Matos Filho, Ptiys. Rev. A 52, 4214 (1995)]. For Zoller. in preparation. the U.S. Army Research Office, the experimental Lamb-Dicke parameter of —0.2, 39. We acknowledge key assistance from W. M. Itano, these corrections are not significant for a £ 5. D. Leibfried, J. C. Bergquist, and J. Erickson and 2 January 1996; accepted 6 March 1996

1136 SCIENCE • VOL.272 • 24 MAY 1996 TN-47
Physical Review

LETTERS

Volume 77 18 NOVEMBER 1996 Number 21

Experimental Determination of the Motional Quantum State of a Trapped Atom

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

Time and Frequency Division, National Institute of Standards and Technology, Boulder, Colorado 80303-3328

(Received 1 1 July 1996)

We reconstruct the density matrices and Wigner functions for various quantum states of motion

of a harmonically bound 'Be^ ion. We apply coherent displacements of different amplitudes and

phases to the input state and measure the number state populations. Using novel reconstruction schemes we independently determine both the density matrix in the number state basis and the Wigner function. These reconstructions are sensitive indicators of decoherence in the system.

[80031-9007(96)01713-9]

PACS numbers: 03.65.B2, 32.80.Qk, 42.50.Vk

In quantum mechanics, once the density matrix of a system is determined, all knowable information is at hand.

All the elusive quantum properties, like superpositions

and decoherence are reflected in it. Although it is well

established that the wave function or density matrix of a

single quantum system cannot be determined in general,

multiple measurements on an ensemble of identically

prepared quantum systems can reveal their density matrix. Early work on determination of the quantum state in such an ensemble was reviewed by Royer [1]. In quan- tum optics, numerous reconstruction schemes have been

proposed, based on the measurement of probability dis- tributions in different representations [2]. More recently,

proposals for determining the motional state of a trapped

atom have been published [3-6], partially inspired by the analogy between cavity QED and a trapped atom interact- ing with laser fields [7-9].

Few experiments have succeeded in determining the density matrices or Wigner functions of quantum systems.

Angular momentum density matrices were measured in collisionally produced hydrogen [10], the Wigner function

and density matrix of a mode of light was experimentally

mapped by optical homodyne tomography [11,12], and the Wigner function of the vibrational degree of freedom of a diatomic molecule was reconstructed [13]. In this Letter we present the theory and experimental demonstration

of two novel schemes that allow us to reconstruct both

the density matrix in the number state basis and the Wigner function of the motional state of a single trapped

TN-48

atom. A unique feature of our experiment is that we are able to prepare a variety of nonclassical input states [9] which can, for example, exhibit negative values of the Wigner function. To our knowledge these are the first experimental reconstructions revealing a negative Wigner

function in position-momentum space.

In order to measure the complete state of motion, we

controllably displace the input state to several different locations in phase space. Specifically, a coherent displacement [9,14] Ui-a) = UHa) = exp(a*a - aa^) (-a

is used for convenience below) is first applied to the input

motional state. Here a and a^ are the lowering and raising

operators of the harmonically bound atom (frequency

Wjc), while a is the complex parameter characterizing the coherent amplitude and phase. We then apply radiation to the atom for a time t, which induces a resonant exchange

between states \ l}\k) and |T)I^ + 1) in a JaynesCummings-type interaction [7-9]. Here | 1) and | t) denote two selected internal states, and \k} is the motional

eigenstate with energy hco^ik + 1/2). For each a and

time / the population Pi{t, a) of the | \) level is then measured by monitoring the fluorescence produced in driv- ing a resonant dipole cycling transition [9]. The internal

state at r = is always prepared to be | i), so the signal

averaged over many measurements is [15]

Pi{t,a) = ^ 1 + Y.Qk{oc)cos{2CtkM + \t)e' jk' k=0

(1) 4281
Volume 77, Number 21 PHYSICAL REVIEW LETTERS 18 November 1996

(where n^<.+i ^e the Rabi frequencies and yic their experimentally determdned decay constants). Because the Rabi frequency between | i)\k) and | T) 1^ + 1) depends on k [9], the populations Qk((x) of the motional eigenstates

after the displacement can be extracted [7-9,16]. We

repeat this scheme for several magnitudes and phases

of the coherent displacement and finally reconstruct the density matrix and the Wigner function from the measured

displaced populations Qk{a)-

To reconstruct the density matrix p in the number state base, we use the relation

Qk{a) = {k\uHa)pU{a)\k). (2)

Note that Qo{a)/7r is the g-quasi-probability distribution

[4]. Rewriting (2) we get

Qda) = j^{0\a'uHa)pUia){a^'\0) = ^(a|(a - a)'p{a^ - a*)'\a) = '

"^l""''' X i-irJ-^'{)){'j,)^{m + j)\{n + j'V.pn^r,m+j

n,m=0 j.j'=0

(a*)"-Ja"'-j"

n\ m\

(3)

To separate the contributions of different matrix elements we may displace the state along a circle, ap = |a|exp[/(7r/A^);7], (4) where p€{—N, ... ,N — 1}. The number of angles 2^^ on

that circle determines the maximum number state n^ax =

A'^ — 1 included in the reconstruction. This allows us to perform a discrete Fourier transform of Eq. (3) evaluated

at the values Up, and we obtain the matrix equations

p = -N

i^k —

2N - I yknPn,n + l , (5)

n=max(0,

with matrix elements

Jkn

— lap I \2k n\in{k,n) min(k,l + n)

k\

X (-1)

j'=Q

1

7=0

l< \2{n-j-j')+l

V(/ + n)ln\

(6)

[t + n - jV.in -/)!

for every diagonal p„,„+/ of the density matrix. To keep

the matrix dimension finite, a cutoff for the maximum n

in Eq. (5) is introduced, based on the magnitude of the input state. For an unknown input state an upper bound

on n may be extracted from the populations Qk{a). If these are negligible for ^'s higher than a certain ^max and

all displacements a, they are negligible in the input state as well, and it is convenient to truncate Eq. (5) at «max =

^max- The resulting matrix equation is overcomplete for some /, but the diagonals p„,„+/ can still be reconstructed by a general linear least-squares method [17]. The Wigner function for every point a in the complex

plane can be determined by the simple sum [16,18],

W{a) = - Ji-irQnia). (7)

In practice, the sum is carried out only to a finite /Jmax. as described above. In contrast to our density matrix method

it provides a direct method to obtain the Wigner function

at the point a in phase space, without the need to measure

at other values of a. This also distinguishes the method

from preceding experiments that determined the Wigner

function by inversion of integral equations (tomography)

[11,13]. In our experiment, the trapped atom is a single ^Be"^

ion, stored in a rf Paul trap [19] with a pseudopotential

oscillation frequency of co^/ln = 11.2 MHz [20]. The

ion is laser cooled using sideband cooling with stimulated

Raman transitions [21] between the ^S\/2 (F = 2, rrif =

—2) and ^Si/2 {F =1, m/r = — 1) hyperfine ground

states, which are denoted by | i) and | T), respectively, and

are separated by approximately 1.25 GHz.

The preparation of coherent and number (Fock) states of motion starting from the ground state is described in

[9]. The coherent displacement we need for the reconstruction mapping is provided by a spatially uniform classical driving field [14,19] at the trap oscillation frequency.

The rf oscillators that create and displace the state are phase locked to control their relative phase. Different dis- placements are realized by varying the amplitude and the phase of the displacement oscillator. For every displacement a, we record Pi{t,a). Qnict) can be found from the measured traces with a singular- value decomposition [9]. To determine the amplitude |a| of each displacement, the same driving field is applied to the \n = 0) ground state, and the resulting collapse and revival trace is fitted to that of a coherent state [9]. The accuracy of the reconstruction is limited by the uncertainty in the applied displacements, the errors in the de- termination of the displaced populations, and decoherence

during the measurement. The value of the Wigner function is found by a sum with simple error propagation rules. The density matrix is constructed by a linear least-squares

method, and it is straightforward to calculate a covariance

matrix [17]. As the size of the input state increases, decoherence and the relative accuracy of the displacements be- come more critical, thereby increasing their uncertainties.

4282 TN-49
Volume 77, Number 21 PHYSICAL REVIEW LETTERS 18 November 1996

Magniliidcs Pfwscs

FIG. 1. (a) Reconstructed number-state density matrix amplitudes pnm for an approximate \n = 1) number state. The coherent reconstruction displacement amplitude was \a\ = 1.15(3). The number of relative phases A' = 4 in Eq. (4), so /Jmax = 3. (b) (color) Surface and contour plots of the Wigner function W{a) of the \n = 1) number state. The plotted points are the

result of fitting a linear interpolation between the actual data

points to a 0. 1 by 0. 1 grid. The octagonal shape is an artifact of the eight measured phases per radius. The white contour

represents W(a) = 0. The negative values around the origin

highlight the nonclassical character of this state.

In Fig. 1 , we show the reconstruction of both the number

state density matrix (a) and Wigner function (b) of an approximate \n = \) number state. The large negative

part of the Wigner function around the origin highlights

the fact that the \n = 1) number state is nonclassical.

In contrast, the state closest to a classical state of motion

in a harmonic oscillator is a coherent state. As one ex- ample, we have excited and reconstructed a coherent state with amplitude \/3\ ~ 0.67. The experimental amplitude

and phase of the number state density matrix are depicted

in Fig. 2. The off-diagonal elements are generally smaller

for the experiment than we would expect from the theory

of a pure coherent state. In part, this is due to decoher- ence during the measurement, so the reconstruction shows

a mixed state character rather than a pure coherent state

e(p)

FIG. 2. Experimental amplitudes p„„ and phases &{p„„) of

the number-state density matrix elements of a \/3\ ~ 0.67 coherent state. The state was displaced by |a| = 0.92, for N = 4in Eq. (4).

signature. This view is further supported by the fact that farther off-diagonal elements seem to decrease faster than

direct neighbors of the diagonal. The reconstructed Wigner function of a coherent state with amplitude |yS| ~ 1.5

is shown in Fig. 3. Next we created a coherent superposition of |« = 0)

and \n = 2) number states. This state is ideally suited

to demonstrate the sensitivity of the reconstruction to coherences. The only nonzero off-diagonal elements should

be po2 and p2Q, with a magnitude of |po2l = lp2ol =

y/PooP22 ~ 0.5 for a superposition with about equal probability of being measured in the \n = 0) or \n = 2) state. In the reconstruction shown in Fig. 4 the populations poo and P22 are somewhat smaller, due to imperfections in the

FIG. 3(color). Surface and contour plots of the reconstructed Wigner function of a coherent state. The plotted points

are the result of fitting a linear interpolation between the

actual data points to a 0.13 by 0.13 grid. The approximately

Gaussian minimum uncertainty wave packet is centered around

a coherent amphtude of about 1.5 from the origin. The half width at half maximum is about 0.6, in accordance with the minimum uncertainty half width of .y/(l/2) ln(2) = 0.59. To

suppress artifacts in the Wigner function summation, we have

averaged over ^max = 5 and «max = 6 truncations, as suggested

by M. Coliett.

TN-50 4283
Volume 77, Number 21 PHYSICAL REVIEW LETTERS 18 November 1996

FIG. 4. Reconstructed density matrix amplitudes of an ap- proximate \/\fl{\n = 0) — i\n = 2)) state. The state was dis- placed by \a\ = 0.79 for A^ =4 in Eq. (4). The amplitudes of

the coherences indicate that the reconstructed density matrix is close to that of a pure state.

preparation, but the coherence has the expected value of

lp2ol = Ipoil ~ yJPOOPllIn contrast to the above, a thermal state should exhibit no

coherences. In the experiment such a state was prepared

by (only) Doppler cooling the ion [9]. The reconstruction

of the resulting thermal state with mean occupation number

n ~ 1.3 is depicted in Fig. 5. As expected, there are no coherences, and the diagonal, which gives the number

state occupation, shows an exponential behavior within the experimental errors. In summary, we have created number, thermal, coherent, and number-state superposition states of motion

of a trapped atom and determined both density matrices

in the number-state basis and Wigner functions of these

states. The methods are suitable for arbitrary quantum

states of motion, including mesoscopic superposition states (Schrodinger's cat states) [22] and could be a useful tool

to study decoherence in these states. These methods could

also be implemented in cavity-QED experiments to deterFIG. 5. Reconstructed density matrix of a 7f = 1.3 thermal

state. This state was displaced by \a\ = 0.78, for N = 4

in Eq. (4). As one would expect for a thermal state, no

coherences are present within the experimental uncertainties and the populations drop exponentially for higher n. 4284

mine the states of an electromagnetic field (using avail- able techniques) [23], or in neutral atom traps where dipole

forces could provide the drive for a coherent displacement

[9,22]. Another straightforward extension of this work in ion traps would be to perform tomography on entangled

motional and internal states of two or more trapped ions, by

combining the motional state reconstruction with Ramseytype and correlation experiments.

This work is supported by the U.S. National Security Agency, the Office of Naval Research, and the Army Research Office. D.L. acknowledges a Deutsche

Forschungsgemeinschaft research grant. D. M. M. is sup- ported by a N. R. C. postdoctoral fellowship. We thank

W. Vogel for pointing out the connection of the Wigner

function to the displaced populations and J. I. Cirac and

P. Zoller for stimulating discussions. We acknowledge

important contributions by J. Bergquist and helpful comments on the manuscript by M. Young, J. J. Bollinger,

P. Huang, and M. Holland. Note added. —After submission of this work we have

learned that Mlynek et al. have measured the Wigner

function of atoms in an interferometer [24], and that Opatmy et al. [25] propose a very similar method to reconstruct the density matrix of a light field in the number-state basis.

[I] A. Royer, Found. Phys. 19, 3 (1989).

[2] K. Vogel and H. Risken, Phys. Rev. A 40, 2847 (1989).

[3] S. Wallentowitz and W. Vogel, Phys. Rev. Lett. 75, 2932

(1995).

[4] J. F. Poyatos, R. Walser, J. I. Cirac, P. Zoller, and R. Blatt, Phys. Rev. A 53, R1966 (1996).

[5] C. D'Helon and G.J. Milbum. Phys. Rev. A 54, R25

(1996).

[6] P. J. Bardroff, C. Leichtle, G. Schrade, and W. P. Schleich

(to be published).

[7] C.A. Blockley, D. F. Walls, and H. Risken, Europhys.

Lett. 77, 509 (1992).

[8] J.I. Cirac, R. Blatt, A.S. Parkins, and P. Zoller, Phys. Rev. A 49, 1202 (1994).

[9] D.M. Meekhof, C. Monroe, B.E. King, W.M. Itano, and

D.J. Wineland, Phys. Rev. Lett. 76, 1796 (1996).

[10] J.R. Ashbum, R.A. Cline, P.J.M. van der Burgt, W.B.

Westerveldt, and J.S. Risley, Phys. Rev. A 41, 2407

(1990).

[11] D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani,

Phys. Rev. LeU. 70, 1244 (1993).

[12] G. Breitenbach, T. Miiller, S. F. Pereira, J. Ph. Poizat,

S. Schiller, and J. Mlynek, J. Opt. Soc. B 12, 2304 (1995).

[13] T.J. Dunn, I. A. Walmsley, and S. Mukamel, Phys. Rev.

Lett. 74, 884 (1995).

[14] P. Carruthers and M. M. Nieto, Am. J. Phys. 7, 537 (1965).

[15] Equation (2) of Ref. [9] is in error and should be replaced

by Eq. (1) here.

[16] In this experiment we can consider the internal atomic

state to be the detector. If we neglect noise and de- coherence in the mapping operations, the motional state information is mapped according to Eq. (1) with unit

TN-51
Volume 77, Number 21 PHYSICAL REVIEW LETTERS 18 November 1996

efficiency onto the | 1) state. Therefore, to have unit detection efficiency in the experiment, it is not necessary

to detect the | 1) state with unit efficiency. The analogy

with photon detection would be a 100% efficient detector which is read out only sporadically.

[17] W.H. Press, S.A. Teukolsky, W.T. Vetterling, and

B. P. Flannery, Numerical Recipes (Cambridge University

Press, Cambridge, 1986), Chap. 14.3. A. Royen Phys. Rev. Lett. 52, 1064 (1984); H. MoyaCessa and P. L. Knight, Phys. Rev. A 48, 2479 (1993);

S. Wallentowitz and W. Vogel, Phys. Rev. A 53, 4528

(1996); K. Banaszek and K. Wodkiewicz, Phys. Rev. Lett. 76, 4344 (1996).

S. Jefferts, C. Monroe, E. Bell, and D. J. Wineland, Phys. Rev. A 51, 3112(1995).

[20] For the parameters of this experiment, the effects of

rf "micromotion" are small and can be neglected. A

thorough treatment which includes the micromotion is described in Ref. [6]. [21] C. Monroe, D. M. Meekhof, B.E. King, S.R. Jefferts, W.M. Itano, D.J. Wineland, and PL. Gould, Phys. Rev.

Lett. 75, 4011 (1995).

[22] C. Monroe, D. M. Meekhof, B. E. King, and D. J. Wineland, Science 272, 1131 (1996).

[23] M. Brune, F. Schmidt-Kaler, A. Maali, J. Dreyer, E. Hagley, J. M. Raymond, and S. Haroche, Phys. Rev. Lett. 76, 1800 (1996).

[24] J. Mlynek (private communication).

[25] T. Opatmy and D.-G. Welsch (to be published).

TN-52 4285
Invited Paper

Quantum harmonic oscillator state

synthesis and analysis*

W. M. Itano, C. Monroe, D. M. Meekhof,

D. Leibfried, B. E. King, and D. J. Wineland

Time and Frequency Division

National Institute of Standards and Technology

Boulder, Colorado 80303 USA

ABSTRACT

We laser-cool single beryllium ions in a Paul trap to the ground (n = 0) quantum harmonic oscillator state with

greater than 90% probability. From this starting point, we can put the atom into various quantum states of motion

by application of optical and rf electric fields. Some of these states resemble classical states (the coherent states), while others are intrinsically quantum, such as number states or squeezed states. We have created entangled position

and spin superposition states (Schrodinger cat states), where the atom's spatial wavefunction is split into two widely

separated wave packets. We have developed methods to reconstruct the density matrices and Wigner functions

of arbitrary motional quantum states. These methods should make it possible to study decoherence of quantum

superposition states and the transition from quantum to classical behavior. Calculations of the decoherence of superpositions of coherent states are presented.

Keywords: quantum state generation, quantum state tomography, laser cooling, ion storage, quantum computation

1. INTRODUCTION

In a series of studies we have prepared single, trapped, ^Be"*" ions in various quantum harmonic oscillator states and

performed measurements on those states. In this article, we summarize some of the results. Further details are given

in the original reports.

-"-"^

The quantum states of the simple harmonic oscillator have been studied since the earliest days of quantum

mechanics. For example, the harmonic oscillator was among the first applications of the matrix mechanics of Heisenberg^ and the wave mechanics of Schrodinger.''' The theoretical interest in harmonic oscillators is partly due to the

fact that harmonic oscillator problems often have exact solutions. In addition, physical systems, such as vibrating

molecules, mechanical resonators, or modes of the electromagnetic field, can be modeled as harmonic oscillators, so that the theoretical results can be compared to experiments.

A single ion in a Paul trap can be described effectively as a simple harmonic oscillator, even though the Hamiltonian is actually time-dependent, so no stationary states exist. For practical purposes, the system can be treated

as if the Hamiltonian were that of an ordinary, time- independent harmonic oscillator (see, e.g., Refs. 8,9).

2. SYSTEM AND EFFECTIVE HAMILTONIAN

The effective Hamiltonian for the center-of-mass secular motion is that of an anisotropic three-dimensional harmonic

oscillator. If we choose an interaction Hamiltonian which affects only the x-motion, then we can deal with the one-dimensional harmonic oscillator Hamiltonian,

H^ = huj:,ala:^, (1)

* Work of the U.S. Government. Not subject to U.S. copyright. Send correspondence to W.M.I. E-mail: witano@nist.gov; telephone; 303-497-5632; fax: 303-497-6461

SPIE Vol. 2995 • 0277-786X/97/$10.00 TN-53 43
where cux is the secular frequency for the x-motion, and aj. and ax are the creation and aruiihilation operators for the

quanta of the x-oscillation mode. The constant term huJx/2, which results from the usual quantization procedure,

has been left out for convenience. The eigenstates of Hx are |ni), where

Hx\nx) = Uxhuj^ \n,. (2) The internal states of the ^Be"^ ion which are the most important for the experiments are shown in Fig. 1. We are mostly concerned with the hyperfine-Zeeman sublevels of the ground 25 ^5i/2 electronic state, which are denoted by

|25 ^5i/2; F, Mp), where F is the total angular momentum, and Mp is the eigenvalue of Fj. The ^Be nucleus has spin

3/2. Of chief importance are |25^5i/2; 2,2), abbreviated as H), and |25^5i/2; 1; l), abbreviated as |t)- The hyperfineZeeman sublevels of the 2p^Pj [J = 1/2 or 3/2) fine-structure multiplet also play a role, either eis intermediate states

in stimulated Raman transitions or as the final states in resonantly-driven single-photon transitions used for laser cooling or state detection. We denote these states by |2p^Pj;F, Mj?). The energy separation of |t) and ||) is huo,

where wq ~ 27r x 1.250 GHz. They are coupled by laser beams Rl and R2, through the intermediate |2p^Pi/2;2, 2) state. The frequency detuning from the intermediate state is A, where A w —27r x 12 GHz.

If we restrict the internal states to the space spanned by 1 1) and | i) , then the internal Hamiltonian can be written

hujo as H-n (3) where a^ is a PauU spin matrix whose nonzero matrix elements are(t|cr2|t) — +1 and (4-|c^|4-) = —1- The total effective Hamiltonian for the system consisting of the

internal 1 1) and | i) states and the x motional degree of fireedom is H = Hn + Hx+H,nu (4) where Hint is the effective interaction Hamiltonian coupling due to the two laser beams Rl and R2 in Fig. 1. We define an interaction- picture operator O^ in terms of a Schrodinger-picture operator O^

as

The effective interaction Hamiltonian in the interaction picture and

the rotating wave approximation is

'1/2 H-L = % ('a+e^''("-+°-)-''^* + cr_e-'''("-+"-)+''^*) , (6)

where g is the interaction strength, 5 is the detuning of the fre- quency difference of the two laser beams with respect to wq, and

77 = k^hl{2mu)x) is the Lamb-Dicke parameter, where k is the

magnitude of the difference between the wavevectors of the two laser beams, and m is the mass of the ion. The nonzero matrix elements

of a+ and cr_ are (t|cr+||) = (4,|(7_|t) = 1- The detuning 8 can be tuned to multiples oiux, ^ = [it-' — n)ux,

so as to resonantly drive transitions between \-i.,n) and |ti^')- We

refer to the 6 = resonance as the carrier, the S = —cux resonance as the first red sideband, and the 6 = +ujx resonance as the first blue

sideband.

The signal that is detected in the experiments is the probability

Pl{t) that the ion is in the | i) internal state after a particular

preparation. If we irradiate the ion at time t with circularly polarized light, resonant with the electronic transition from \l) to |2p^P3/2;3, 3), there will be a high fluorescence intensity if the |4-) state is occupied, since the selection

rules only allow that upper state to decay back to the ] 4-) state, so it can continue to scatter photons. This transition

is called a cycling transition. If the ion is in the | f) state when it is irradiated with the same light, it will scatter a

negligible number of photons. Thus, if we repeatedly prepare the ion in the same way, apply radiation resonant with

the cycling transition, and detect the fluorescence photons, the average signal will be proportional to Pi{t).

Figure 1. Internal and motional energy levels of a ^Be"*" ion. The | t) and | 4-) states are particular hyperfine-Zeeman components

of the ground ^5i/2 state, separated in energy

by huQ. Each internal state can exist in any

of a ladder of vibrational energy states \nx),

where rix = 0, 1, 2, . . ., separated in energy by

hur, where cj^ ss 27r x 11.2 MHz.

44

TN-54
3. CREATION AND PARTIAL MEASUREMENTS OF QUANTUM STATES

3.1. Fock states

The ion is prepared in the n = state by Raman cooUng.-"- (From now on we drop the x label on n.) Raman cooling

consists of a sequence of laser pulses on the red sideband, driving | i, n) to | ti''^ — 1} transitions, followed by laser pulses which recycle the ion back to the | i, n — 1) state. The probability of heating due to recoil during the recycling

step is small. At the end of the sequence, the ion is in the n = state more than 90% of the time.

A Fock state is another name for an n-state. Higher-n Fock states are prepared from the n = state by a sequence of 7r-pulses on the blue sideband, red sideband, or carrier. For example, the 1 1, 2) state is prepared by using

blue sideband, red sideband, and carrier 7r-pulses in succession, so that the ion steps through the states li, 0), |t, 1), 11,2), and It, 2).

If the atom is initially in the |4-, n) state, and the first blue sideband is driven, it will oscillate between that state and the It,''^ + 1) state. The probability of finding it in the \i,n) state at a time t is

P^{t) = - [1 +cos(2fi„,„+it)e-^"*] , (7)

where fln,n+i, the Rabi flopping rate, is a function of the laser intensities and detunings and n, and 7„ is a damping

factor which is determined empirically. Thus, the frequency of the oscillations of Pi{t) is a signature of the initial value of n. Figure 2(a) shows Pi{t) for an initial 1 1, 0) state. Figure 2(b) shows the observed ratios of Rabi frequencies

compared with the values calculated from the matrix elements of the interaction Hamiltonian [Eq. (6)].

(c)

thermal

state

1.0 - 1 2 4 6 8 10

0.8 - •

n

qT

0.6 -

0.4-

0.2 -

0.0 - . . . . 1 . ^^\f\

—I— "

e A.

' '

1 20 30 time (^s)

40 50

Figure 2. (a) P^it) for an initial 1^,0) number state driven by the first blue sideband. The solid line is a fit to an exponentially damped sinusoid, (b) Observed ratios of the Rabi frequencies Jln,n+i/^o,i for different values of the initial n. The lines represent calculated values for different values of the Lamb-Dicke parameter 77. The value of

T], inferred from other measured quantities, was 0.202 ± 0.005. (c) P\^{t) for a thermal state. The solid line is a fit of the data (dots) to a sum of Fock states having a thermal distribution. The fitted value for the mean value of n

is n = 1.3 it 0.1. The inset shows the amplitudes of the Fock state components (bars) with a fit to an exponential,

corresponding to n = 1.5 ± 0.1 (line).

3.2. Thermal states

A thermal state of the motion of the ion is not a pure state, but rather must be described by a density matrix, even though there is only one ion. The statistical ensemble which is described by the density matrix is generated by

repeatedly preparing the state and making a measurement.

TN-55 45
The ion is prepared in a thermal state by Doppler laser cooling.-'^^ The temperature of the distribution can be

controlled by changing the detim.ing of the cooling la^er. When the ion's state is not a Fock state, Pj.(i) has the form

Pdt) I + '^Pn cos(2n„,n+ii)e -7r.t n=0

(8)

where Pn is the probability of finding the ion in the state |n). Figure 2(c) shows Pi{t) for a thermal state.

3.3. Coherent states

Coherent states of the quantum harmonic oscillator were introduced by Schrodinger,^-^ with the aim of describing a

classical particle with a wavefunction. A coherent state \a) is equal to the following superposition of number states:

,..-iH'ESi">-

71=0

(9)

In the Schrodinger picture, the absolute square of the wavefunction retains its shape, and its center follows the

trajectory of a classical particle in a harmonic well. The mean value of n is n = |Qp. A coherent state can be created

from the n = state (a special case of coherent state) by applying a spatially uniform classical force (see Appendix

A). The drive is most effective when its frequency is uJx- Another method is to apply a "moving standing wave," that

is, two laser beams differing in frequency by uix and differing in propagation direction, so that an oscillating dipole

force is generated. -^^ We have used both of these methods to prepare coherent states. Figure 3(a) shows Pi{t) for a coherent state. This trace exhibits the phenomenon of collapse and revival. -^^

coherent °^ state ^" 1

20 30

time (|iS)

squeezed state

time (us)

(a) (b)

Figure 3. (a) Pi{t) for a coherent state. The solid line is a fit of the data (dots) to a sum of Fock states having

a coherent state distribution. The fitted value for n is 3.1 ± 0.1. The inset shows the amplitudes of the Fock state components (bars) with a fit to a Poissonian distribution, corresponding to n = 2.9 ± 0.1 (line), (b) Pi{t) for a squeezed state. The solid line is a fit of the data (dots) to a sum of Fock states having a squeezed-state distribution. The fitted value for /3 is 40 ± 10, which corresponds to n ^ 7.1

3.4. Squeezed states

A "vacuum squeezed state" can be created by applying an electric field gradient having a frequency of 2u;i to an ion

initially in the n = state.-''* Here, the ion was irradiated with two laser beams which differed in frequency by 2uxThis has the same effect. The squeeze parameter P is defined as the factor by which the variance of the squeezed

46 TN-56
'^> vy \A/ \J^ \L/ O Kj^

'^> \i/ \i/ \l/ \J^ \J^ \J^

(a) (b) (c) (d) (e) (f)

Figure 4. Creation of a Schrodinger cat state. See text for details.

quadrature is decreased. It increases exponentially with the time the driving force is applied. The probability

distribution of n-states is nonzero only for even n, for which,

(2n)!(tanhr)^" ^'^-^

(2^^ ' (^°)

where /3 = e^^, and N is a, normalization constant. Figure 3(b) shows Pi{t) for a squeezed state having /? = 40 ± 10.

3.5. Schrodinger cat states

The term "Schrodinger cat" is taken here to denote an entangled state which consists of two coherent states of motion

correlated with different internal atomic states. A simple example is

i^) =

'^'°'^^'-°'

. (11)

Figure 4 shows how such a state is created, (a) The ion is prepared in the | i, 0) state by Raman cooling, (b) A

7r/2-pulse on the carrier generates an equal superposition of 14-, 0) and |tjO). (c) The displacement beams generate

a force Fi, which excites the component in the 1 1) internal state to a coherent state \a). Due to the polarizations

of the displacement beams, they do not affect an atom in the | 4-) state, (d) A 7r-pulse on the carrier exchanges

the 14^) and |t) components, (e) The displacement beams generate a force F2, which excites the component in the

1 1) internal state to a coherent state lae^*^), where the phase cf) is controlled by an rf oscillator. The state here is analogous to Schrodinger's cat. (f) The ||) and |t) components are combined by a 7r/2-pulse on the carrier. At this point, the radiation is applied on the cycling transition, and the signal is recorded.

The predicted signal, for a particular value of 0, is PM =

^

[1 - ce-"'^^-^"^'^) cos{a^ sin0)] , (12)

where c = 1 in the absence of decoherence. Figure 5 shows experimental data and fits to Eq. (12) for various values

of a. In (d), c is clearly less than one, indicating decoherence, although the source is not yet determined.

4. COMPLETE MEASUREMENTS OF QUANTUM STATES

The measurements described in the previous Section determine only the n-state populations or probabilities, and

therefore do not provide a complete description of the motional states. The density matrix p does provide a complete

description of a state, whether it is a pure or a mixed state. The Wigner function W{a) also provides a complete

description. The Wigner function resembles a classical joint probability distribution for position and momentum in some cases. However, it can be negative, unlike a true probability distribution. We have demonstrated experimental

methods for reconstructing the density matrix or the Wigner function of a quantum state of motion of a harmonically

bound atom.^

TN-57 47
PiW

(a)

(b)

(c)

(d)

(e)

a = 0.84(2)

a = 1.20(3)

1.0

0.5

0.0

1.0

0.5

0.0

a = 1 .92(4)

a = 2.97(6) --Af^——

a = 6 (theory)

-n -71/2 71/2 n coherent state phase separation

Figure 5. Mecisured and fitted Pi{<p) interference signals for the Schrodinger cat state. The curves in (a) through

(d) represent measurements for different values of the time during which the displacement beams excite the coherent

states. The solid lines are fits to Eq. (12), allowing a to vary. In (d), a term representing loss of contrast was included

in the fit. The curve in (d) represents a superposition of two iq ~ 7 mn wave packets with a maximum separation

of 4axo ~ 80 nm. Curve (e) represents a calculation for a pair of coherent states with q = 6.

Both of these methods depend on controllably displacing the state in phase space, applying radiation to drive the first blue sideband for time t, and then measuring Pj.. The averaged, normalized, signal is

Pi{t,a) =

^

1 + Yl '5fc(^) cos{2nk,k+it)e-^'''

k=0

(13)

where the complex number a represents the amplitude and phase of the displacement, and Q{a) is the occupation

probability of the vibrational state \k) for the displaced state.

If the Qk{c() coefficients are derived for a series of values of a lying in a circle,

Qp = |a| exp[2(7r/A'')p], (14)

where p — —N,...,N — 1, then the density matrix elements pnm can be determined for values of n and m up to N — I. The details of the numerical procedure are given in Ref. 4. Figure 6(a) shows the reconstructed density matrix amplitudes for an approximate n = 1 state. Figure 6(b)

shows the reconstructed density matrix for a coherent state having an amplitude |/3| « 0.67. The Wigner function for a given value of the complex parameter a can be determined from the sum-^^"-^^

W(a) = -^(-l)"0„(a). (15)

n=0

48 TN-58
Magnitudes Phases

.^^-r::::::::^V

r—t^^^^^-'-V—

" 1 1 1 ;\

0.6 v_\i 1 !J j 'k\

K J

-—

i : vv .L- '

: h= iV\

0.4

IPl 0.2 ^"1--—n—^^f^^

i F—"\

"^1>

0(p)

(a) (b)

Figure 6. (a) Reconstructed number-state density matrix amplitudes ]pnm| for an approximate n = \ number state,

(b) Reconstructed amplitudes \pnrn\ and phases 9(pnm) of a coherent state.

Figure 7 shows the reconstructed Wigner function for an approximate n = 1 state. The fact that it is negative in a region around the origin highlights the fact that is a nonclassical state. Figure 8 shows the reconstructed Wigner

function for a coherent state with amplitude |/3| ~ 1.5. It is positive, which is not surprising, since the coherent state

is the quantum state that most closely approximates a classical state.

Figure 7. Surface and contour plots of the reconstructed Wigner function W(a) of an approximate n = 1 number

state. The negative values of Wio) around the origin highlight the nonclassical nature of this state.

TN-59 49
Figure 8. Surface and contour plots of the reconstructed Wigner function W{a) of a coherent state. The width is in good agreement with the expected minimum-uncertainty value.

5. DECOHERENCE OF A SUPERPOSITION OF COHERENT STATES

Decoherence, the decay of superposition states into mixtures, is of interest because it leads to the familiar classi- cal physics of macroscopic objects (see, e.g. Ref. 19). Experiments are beginning to probe the regime in which

decoherence is observable, but not so fast that the dynamics are classical. One example is the observation of the decoherence of a superposition of mesoscopic (few-photon) quantum states of an electromagnetic field having distinct phases by Brune et al.'^^ In this experiment, as in the Schrodinger cat experiment of Ref. 3, the mesoscopic system was entangled with an internal two-level system of an atom. A simpler system, which has been well-studied theoretically by others, ^^~^'*

is a superposition of two coherent

states of a harmonic oscillator, without entanglement with a two-level system. The system interacts in some way

with the environment, leading to decoherence of the superposition state. Here we carry out a simple calculation of the decoherence of a superposition of two coherent states of a harmonic

oscillator. At time t = 0, the system is in an equal superposition of two coherent states.

l*(0))^^(e' ^(°)|ai(0)) + e'^^(°V2(0)) (16)

where the notation is the same as in Appendix A, and the interaction picture is used. (It is assumed that the two

components of the state vector are nearly orthogonal. Otherwise the normalization factor is different.) The system

is subjected to a random force, which is uniform over the spatial extent of the system, for example, a uniform electric

field acting on a charged particle. In Appendix A, we show that a single coherent state, subjected to such a force,

will remain in a coherent state. It follows from the linearity of the Schrodinger equation that the state described by

Eq. (16) will, at a later time, still be in an equal superposition of coherent states, although ai, 02, ^i, and 62 will change. Each time the system of Eq. (16) is prepared, ai, 0:2, &!, and 62 will have a different time dependence, due

to the random force. We can study the decoherence as a function of time by carrying out an ensemble average. Although both qi and 02 change with time, their difference does not, since they each change by the same amount.

This can be seen from Eq. (45). We define Aa = ai — a2 = constant.

50 TN-60
Decolierence is due to the changes in the phase difference [02 (i) — &i{t)]- We define the change in the phase

difference to be

Ae{t) = [92{t) - 01 (t)] - [02(0) - 0i(O)]. (17)

Then, from Eq. (48),

Ae{t) = l Aa [ f{t')e-'^'^'dt' + Aa' f

f{t')e'^-^'dt' . ^ I Jo Jo

(18)

The double-integral term in Eq. (48) does not contribute to A9{t), since it does not depend on either a(0) or 0(0), so it makes the same contribution to 02(i) and to 0i(i). The square of A0(t) is

(A0(i))' =

\

rt rt {AaY / f{t')f{t")e

Jo Jo

t rt

/N ^^^//\^-ia..t'-iu..t"^^/^^// _^ 2|AaP

'0 ^0 f{t')f{t"y ^'^^^^"^-^"-*'-^'"-*"cit'(it"

+ {Aa*f f

f f{t')f{t

Jo Jo

'^f'^"^e^'^-^'+"^'^" dt'dt" (19)

and its ensemble average is

{Ae{t)r

t ft

2 / / / f i4.i\ j: rj.li\\ —iu)^t' —iuiit" 1,1 Till [AaY / (/(t')/(t"))e

Jo Jo + {Aa''f I I

{f{t')f{t"))e"^'''+'^-'"dt'dt"

Jo Jo

dt'dt" + 2\Aa\^ II {f{t')f{t"))e Jo Jo

f^,ull^\„iu^.t'-iu^^t"^^|^^n

(20)

where the angular brackets denote an ensemble average. Thus, ( (A0(i)) ) depends on the autocorrelation function

of the random function /(i), which is proportional to the force. If we assume that / is a stationary random variable,

then its autocorrelation function

R{T)^{f{t)f{t + T)) (21)

exists and is independent of t. According to the Wiener-Khinchine theorem, the power spectrum tt;(i/), defined for

z/ > 0, is rOO vj{y) = 4 / -R(t) cos(27rzyT)cfr. Jo

(221

A white-noise spectrum corresponds to R{t) oc 5{t), where 5{t) is the Dirac delta function. If we let R{t) = C5{t),

then

(A0(t))^

C

4 (Aa)2 f f 6{t" -t')e-''^-''-^'^'^"dt'dt" + 2\Aa\^ f f

5{t" - t')e^'^-^'

Jo Jo Jo Jo

-iuJ:ct J+l Jj.ll dt'dt

t rt 2 / / X/j-" j.l\Au>it' +iu>^t" jj.1 jj.ll + {Aay / / 5{t" -t')e

Jo Jo

ft

2 / ,-2iwxt'jj./

dt'dt'-

{Aaf I e-2^"-*'dt' + 2|Aap / dt' + {Aa*Y f e^'^-'' dt' Jo Jo Jo

(23)

The first and third integrals on the right-hand-side of Eq. (23) oscillate, but are bounded, while the second one grows

with time, so, for oj^^t ^ 1, {Ae{t)f) ^ \c\Aa\H. (24)

1/2 Hence, the decoherence time, that is, the time required for the rms phase difference ((A9{t)) ) to increase to about 1 radian, is on the order of 2/ (C|Aa|^).

Using the same approximations, we can calculate the rate of change of the amplitude ai{t) for a single coherent

state, where 2 is 1 or 2. From Eq. (45) in Appendix A,

a^{t) - a,{0) = I ff{t')e'-^-Ut', (25)

TN-61 51
and

\a,it)-a,{0)\^= f f

f{t')fit")e"^-''-"--'"dt'dt". (26) Jo Jo

If /(t) has a white-noise spectrum, as considered previously, the ensemble average is

(|a,(t) - a^{0)\^) = C f f d{t" - t')e''^'*' -'"'-'" dt'dt" = C f dt' = Ct. (27) Jo Jo Jo

In the time, t « 2/ (CjAap), required for decoherence, the rms change in a, is y2/|AQ;|. Consider the case Oi(0) = — 02(0). The energy of a single coherent state is proportional to the square of its amplitude (oc loip). The

fractional change in the energy is

A|a.P 2|a,(t)-a,(0)| ^ 2^2 ^ V2

|a^P a,(0) ~ |Aa||a,(0)| |a,(0)|2' ^ ^

which becomes small for |ai(0)|^ :$> 1. Thus, we see that coherent superpositions of macroscopic (|ai(0)p 3> 1) states decohere much more quickly than they change in energy. We can make a more quantitative statement about the decoherence rate by considering the decay of the off- diagonal density matrix elements. For an initial pure state 2~"-^/^(|ai) + |a2)), the density matrix is

p = - (|ai)(ai| + |Qi)(a2| + |a2)(ai| + |Q2)(a2|) • (29)

If Aq » 1, so that the phase changes are more important than the changes in ai and 0:2, then the off-diagonal matrix

element |q;2)(q:i| evolves to e''^^(*^|a2)(Q:i|. Let us assume that the random variable A6 at time t has a Gaussian

distribution P{A9). Given the variance of A9 from Eq. (24), P{A6) must have the form

P{Ad) = , ^ exp {--^-—\ . (30)

^ ' •J-KC\Aa\H \ C\Aa\H) ^ ' We evaluate the ensemble average of e*^^('),

..A9(t)^^ ^ r p(A^)e^^^Wd(Ae) = exp (-9\^^\ . (31)

This yields the basic result, previously derived by others,^-^"'^^ that the off-diagonal matrix element decays exponentially in time, with a time constant that is inversely proportional to lAap.

An extension of the experiment of Ref. 3, with a random electric field deliberately applied, might be used to

verify these calculations.^ A means of simulating thermal, zero-temperature, squeezed, and other reservoirs by

various combinations of optical fields has been discussed by Poyatos et al.'^^ ACKNOWLEDGMENTS

This work was supported by the National Security Agency, the Army Research Office, and the Office of Naval

Research. D. L. acknowledges a Deutsche Forschungsgemeinschaft research grant. D. M. M. was supported by a National Research Council postdoctoral fellowship.

APPENDIX A. THE FORCED HARMONIC OSCILLATOR

If a coherent state is subjected to a spatially uniform force, it remains a coherent state, though its amplitude

changes. This exact result, which is independent of the strength or time-dependence of the force, seems to have

been discovered by Husimi'^^ and independently by Kerner.-^^ For other references on the forced quantum harmonic

oscillator, see Ref. 28. The qualitative result stated above is enough to establish that applying a force to the n =

state will generate a finite- amplitude coherent state, which was required in Sec. 3.3. However, in order to study the

decoherence of a superposition of two coherent states, it is useful to have an exact expression for the time-dependent

52 TN-62
state vector, given that the state vector is an arbitrary coherent state at t = 0. Since the pubhshed results of which we are aware do not give this particular result, we give here an elementary derivation. We consider a Hamiltonian, H = Hq + V{t), where Hq = hLOxal.ax is the Hamiltonian of a one-dimensional

harmonic oscillator, and V{t) = —xF{t) is a time-dependent potential. Here, F{t) is a real c-number function of time and corresponds to a spatially uniform force, and x = XQ{ax + a\.), where xq = {?i/2muj^Y^'^ . For example, if the particle has charge q, and a uniform electric field xEx{t) is applied, then F{t) = qEx{t).

It is convenient to switch to the interaction picture, where an interaction-pictiire state vector {"^{t))^ is related

to the corresponding Schrodinger-picture state vector |^(t))'^ by

\^{t)y = e'"°'^''\'i{t)f, (32)

and an interaction-picture operator 0\t) is related to the corresponding Schrodinger-picture operator 0^{t) by

0^(i) = e'^ot/hQS(^^^^-iHot/h_

(33)

The equation of motion for an interaction-picture operator is

±0^{t) = -'-[0\t),H,], (34)

so

^4(t) = -^[ai{t),Ho]=-iUxa\t), (35)

ai{t) = 4(0)e-'-'*, (36)

4\t) = 4\0)e^-^'. (37)

T C For simplicity, we let a = a^(0) = af and a^ = a^ (0) = aj. . The Schrodinger equation in the interaction picture is

^l^(i))^ = -'-V\t)\^{t)y = if it) (ae—»' + ate-^*) |^(t))^ (38)

where /(f) = XQF{t)/h. Note that if f{t) = 0, |*(i))-^ is constant. We impose the initial condition that l'I'(O))''^ is a coherent state. We make the assumption that |*(f))^ is also a

coherent state for f > 0,

\<b{t)y = e^^W|c,(t)) = e^^(Oe-|k(t)l= V^ ^|„), (39)

where the \n) are the eigenstates of al.aj: with eigenvalue n at i = 0, and 6{t) is an arbitrary real function, so that ^{'^{t)\'^{t))^ = 1. (This assumption must be verified later by substitution of the resulting solution back into Eq. (38).) The problem reduces to finding the complex function a{t) and the real function 9{t).

The right-hand-side of Eq. (38) is

a"(f)

.^V\t)\<i!{t)y = i/(t)(ae-^'^-'-fa^e^'^^*)e^^We-5l°Wl' J^^ln

n=0 = ^/(t)e-^We-§l-(^)l^ |e--^ V "^^t^ |n-l) + e--^ V ""(^^r^ |n + 1) | (40)

TN-63 53
The left-hand-side of Eq. (38) is

|i*(t))'

dt E

n=0

Q"(t)

n) +eie{t)

' n=0

+ te'^We-5i-Wl' J] In). n=0

We equate the coefficients of \n) in Eqs. (40) and (41) and divide by a common factor to obtain

which can be rearranged as da

dt

if[t)e^

. , .dO 1 , , ., .da 1 9, .da* o, v ,, > ,, ^'''^^^dt ~ 2''^^^''^^^ d^ ~ -a2(t)— -ta2(t)/(t)e-'^^' =0

(41)

(42)

(43)

In order for Eq. (43) to be true for all n, the expression in the square brackets, which multiplies n, must equal zero. Thus, we have the first-order differential equation for a{t),

da

dt

if{t)e''^'' = 0,

which has the solution

a{t) = a(0) + i / f{t')e'^-' dt' Jo

If the expression in brackets is set to zero, Eq. (43) becomes, after division of both sides by ia{t),

dO i , , , da i , s da* , . r, s 4, , t

(44)

(45)

(46)

The imaginary part of Eq. (46) is satisfied as long as Eq. (44) is, so it provides no new information. The real part

of Eq. (46) is a first-order differential equation for 9{t):

T ~ /(t) [Re a(t) COS LJit -I- Ima(i) sin w^t]

dt = fit)

= fit)

Rea(O) cosu;xi + lnia(0) sina;xi — coswit / f {t') sin tJxt'dt' + sin uj^t / f {t') cos Uxt'dt' Jo Jo

Rea(0)coswxi + ImQ:(0)sinwit-t- / f it') sin [ux{t - t')]dt' Jo

(47)

where Re and Im stand for the real and imaginary parts. Integrating Eq. (47), we obtain

e{t) = eiO)+Rea{0) f f{t') cos ujxt'dt'+lma{0) f f{t')sinujjdt'+f dt'f{t') I dt" f{t") sin [uj^it' -t")] . (48) Jo Jo Jo Jo

Substitution back into the Schrodinger equation [Eq. (38)] verifies the solution and justifies the original assumption

[Eq. (39)]. REFERENCES

1. C. Monroe, D. M. Meekhof, B. E. King, S. R. Jefferts, W. M. Itano, and D. J. Wineland, "Resolved-sideband

Raman cooling of a bound atom to the 3D zero-point energy," Phys. Rev. Lett. 75, pp. 4011-4014, 1995.

2. D. M. Meekhof, C. Momoe, B. E. King, W. M. Itano, and D. J. Wineland, "Generation of nonclassical motional

states of a trapped atom," Phys. Rev. Lett. 76, pp. 1796-1799, 1996.

54 TN-64
3. C. Monroe, D. M. Meekhof, B. E. King, and D. J. Wineland, "A 'Schrodinger cat' superposition state of an

atom," Science 272, pp. 1131-1136, 1996.

4. D. Leibfried, D. M. Meekhof, B. E. King, C. Monroe, W. M. Itano, and D. J. Wineland, "Experimental deter- mination of the motional quantum state of a trapped atom," Phys. Rev. Lett. 77, pp. 4281-4285, 1996.

5. D. Leibfried, D. M. Meekhof, B. E. King, C. Monroe, W. M. Itano, and D. J. Wineland, "Experimental preparation and measurements of quantum states of motion of a trapped atom," J. Modern Opt. (m press) .

6. W. Heisenberg, "Uber quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen," Z. Phys.

33, pp. 879-893, 1925.

7. E. Schrodinger, "Quantisierung als Eigenwertproblem II," Ann. Phys. (Leipzig) 79, pp. 489-527, 1926.

8. P. J. Bardroff, C. Leichtle, G. Schrade, and W. P. Schleich, "Endoscopy in the Paul trap: Measurement of the vibratory quantum state of a single ion," Phys. Rev. Lett. 77, pp. 2198-2201, 1996.

9. G. Schrade, V. I. Man'ko, W. P. Schleich, and R. J. Glauber, "Wigner functions in the Paul trap," Quantum

Semiclass. Opt. 7, pp. 307-325, 1995.

10. S. Stenholm, "The semiclassical theory of laser coohng," Rev. Mod. Phys. 58, pp. 699-739, 1986.

11. E. Schrodinger, "Der stetige Ubergang von der Micro- ziir Makromechanik," Die Naturwissenschaften 14, pp. 644-666, 1926.

12. D. J. Wineland, J. C. Bergquist, J. J. Bollinger, W. M. Itano, F. L. Moore, J. M. Gilhgan, M. G. Raizen, D. J. Heinzen, C. S. Weimer, and C. H. Manney, "Recent experiments on trapped ions at the National Institute of Standards and Technology," in Laser manipulation of atoms and ions, E. Arimondo, W. D. Phillips, and

F. Strumia, eds., pp. 553-567, North-Holland, Amsterdam, 1992.

13. J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, "Periodic spontaneous collapse and revival in a simple quantum model," Phys. Rev. Lett. 44, pp. 1323-1326, 1980.

14. D. J. Heinzen and D. J. Wineland, "Quantmn-limited cooling and detection of radio-frequency oscillations by

laser-cooled ions," Phys. Rev. A 42, pp. 2977-2994, 1990.

15. A. Royer, "Measurement of the Wigner function," Phys. Rev. Lett. 55, pp. 2745-2748, 1985.

16. H. Moya-Cessa and P. L. Knight, "Series representations of quantum-field quasiprobabilities," Phys. Rev. A 48,

pp. 2479-2485, 1993.

17. S. Wallentowitz and W. Vogel, "Unbalanced homodyning for quantum state measurements," Phys. Rev. A 53,

pp. 4528-4533, 1996.

18. K. Banaszek and K. Wodkiewicz, "Direct probing of quantum phase space by photon counting," Phys. Rev.

Lett. 76, pp. 4344-4347, 1996.

19. W. H. Zurek, "Decoherence and the transition from quantum to classical," Phys. Today 44 (10), pp. 35-44,

1991.

20. M. Brune, E. Hagley, J. Dreyer, X. Maitre, A. Maah, C. Wunderhch, J. M. Raimond, and S. Haroche, "Observing

the progressive decoherence of the 'meter' in a quantum measurement," Phys. Rev. Lett. 77, pp. 4887-4890,

1996.

21. D. F. Walls and G. J. Milburn, "Effect of dissipation on quantum coherence," Phys. Rev. A 31, pp. 2403-2408,

1985.

22. J. P. Paz, S. Habib, and W. H. Zurek, "Reduction of the wave packet: Preferred observable and decoherence

time scale," Phys. Rev. D 47, pp. 488-501, 1993.

23. B. M. Garraway and P. L. Knight, "Evolution of quantum superpositions in open environments: Quantum

trajectories, jumps, and localization in phase space," Phys. Rev. A 50, pp. 2548-2563, 1994.

24. P. Goetsch, R. Graham, and F. Haake, "Schrodinger cat states and single runs for the damped harmonic

oscillator," Phys. Rev. A 51, pp. 136-142, 1995.

25. J. F. Poyatos, J. I. Cirac, and P. Zoller, "Quantum reservoir engineering with laser cooled trapped ions," Phys.

Rev. Lett. 77, pp. 4728-4731, 1996.

26. K. Husimi, "Miscellanea in elementary quantum mechanics II," Prog. Theor. Phys. 9, pp. 381-402, 1953.

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TN-65 55
Physical Review A

ATOMIC, MOLECULAR, AND OPTICAL PHYSICS

THIRD SERIES, VOLUME 55, NUMBER 4 APRIL 1997

RAPID COMMUNICATIONS

The Rapid Communications section is intended for the accelerated publication of important new results. Since manuscripts submitted

to this section are given priority treatment both in the editorial office and in production, authors should explain in their submittal letter why the work justifies this special handling. A Rapid Communication should be no longer than 4 printed pages and must be accompanied

by an abstract. Page proofs are sent to authors.

Simplified quantum logic with trapped ions

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

National Institute of Standards and Technology, Boulder, Colorado 80303

(Received 17 December 1996)

We describe a simplified scheme for quantum logic with a collection of laser-cooled trapped atomic ions. Building on the scheme of Cirac and Zoller, we show how the fundamental controlled-NOT gate between a

collective mode of ion motion and the internal states of a single ion can be reduced to a single laser pulse, and

the need for a third auxiliary internal electronic state can be eliminated. [51050-2947(97)50904-8]

PACS number(s): 03.65.Bz, 89.70. +c, 32.80.Pj

Lately, much thought has been focused on the implementation of simple quantum logic circuits for quantum computing [1] and other applications such as the generation of multiparticle entangled states for spectroscopy [2]. A promising

candidate for quantum logic is based on the work of Cirac

and Zoller [3], who showed how to construct universal

multibit quantum logic gates in a system of laser-cooled trapped atomic ions. In the simplest form of the ion trap quantum computer, two internal electronic levels of each ion

in a collection represent a quantum bit of information, and

the quantum bits are "wired" together by virtue of the ions' collective motion in the trap. Trapped ions are attractive for quantum logic applications because their internal levels can

be well isolated from the pernicious effects of quantum de- coherence [4], while at the same time, the ions strongly in- teract with one another through their Coulomb repulsion, al- lowing the formation of entangled states. Several groups have shown that any quantum computation can be built from a series of one-bit and two-bit quan- tum logic gates [5]. A fundamental entangling quantum logic

gate is the controlled-NOT (CN) gate [6,7], in which one quantum bit is flipped (rotated by tt radians) depending on

the state of a second quantum bit. Cirac and Zoller showed

how to realize a CN quantum logic gate in a collection of

trapped ions by applying several appropriately tuned laser pulses to the ions and invoking an interaction with a third (auxiliary) internal atomic level. Their scheme was adapted

to an experiment on a single 'Be ^ ion and required three

laser pulses [8]. This Rapid Communication discusses a simpler CN gate scheme between an ion' s internal and motional

states that requires only a single laser pulse, while eliminat- ing the requirement of the auxiliary internal electronic level. These simplifications are important for several reasons.

(1) Several popular alkalilike ion candidates, including

2*Mg+, ^Ca + , »»Sr + , '^^Ba^, '^^Yb^ and '^^Hg^, do

not have a third electronic ground state available for the auxiliary level. These ions have zero nuclear spin with only two Zeeman ground states (m^=±|). Although excited optical metastable states may be suitable for auxiliary levels in some

of these ion species, this places extreme requirements for the frequency stability of the exciting optical field to preserve coherence.

(2) The elimination of an auxiliary ground state removes

the possible existence of spectator internal atomic levels, which can act as potential "leaks" from the two levels spanned by the quantum bits (assuming negligible population

in excited electronic metastable states). This feature may be

important to the success of quantum error-correction schemes [9], which can be degraded when leaks to spectator

states are present [10].

(3) The elimination of the auxiliary level can dramatically reduce the sensitivity of a CN quantum logic gate to external magnetic fields fluctuations. It is generally impossible to find three atomic ground states whose splittings are all first-order magnetic-field insensitive. However, for ions possessing hyperfine structure, the transition frequency between two levels can be made first-order magnetic-field insensitive at particular values of an applied magnetic field.

1 050-2947/97/55 (4)/2489(3)/$ 1 0.00 TN-66

55 R2489 © 1997 The American Physical Society
R2490 C. MONROE et al. 55

(4) Finally, a reduction of laser pulses simplifies the tun- ing procedure and may increase the speed of the gate. For

example, the gate realized in Ref. [8] required the accurate

setting of the phase and frequency of three laser pulses, and

the duration of the gate was dominated by the transit through

the auxiliary level. For quantum logic with trapped ions, we assume that each

ion has two internal states (denoted by |i) and |T) with energy separation ^wq). We consider the center-of-mass (COM) mode of the ions' collective motion at harmonic fre- quency a>, described by the ladder of states \n) having vi- brational quantum number n = 0,l,2,... and energy

h(x){n + \ ). In quantum logic applications, we consider only

the |«) = |0) and |n) = |l) motional states. The "reduced"

CN logic gate flips the internal state of a particular ion j if and only if |n) = 1 1 ), summarized in the following truth table and realized on a single ion in Ref. [8]:

|o)li),-|o)U),.,

iO)|T),-|0)|T);,

|l)li),-|l)lT);,

|l)lT);-il)li);. (1) When this gate is surrounded by two extra operations that map and reset the internal state of another ion / onto the collective COM state of motion, the result is a more general

CN gate that flips the internal state of ion j, if and only if, ion / is in state |t), [3]. Below, we show how the reduced

CN gate of Eq. (1) can be condensed to a single laser pulse,

or, equivalently, how the more general CN gate between two

ions can be condensed to three laser pulses.

If I [ )j and 1 1 )^ are coupled by a dipole moment operator

fij , then exposing ion j to a traveling-wave electric field E(z) = EoCos(k-z— a;^r+ 0) with frequency w^, phase 0,

and wave vector k, results in the interaction Hamiltonian:

In this expression, gj = /jLj-'E^llh is the resonant Rabi fre- quency connecting \i)j to \'\)j in the absence of confine- ment, S+' {S^-) is the internal level raising (lowering) operator of the yth ion, i=Zoz{a + a^) is the COM coordinate operator of the confined motion with associated harmonic

raising (lowering) operator a^ (a) and zero-point spread

Zo — (^/2Mw)"", and M is the total mass of the ion collection. If the applied radiation frequency is tuned to

(i}i = (x>Q+{n' — n)u), thereby coupling the states |n)|i) and

I «

' ) I T ) . 'H\' ' is transformed to

>^/^'= -^5;(5V'e"'*"+'''*~"^+5'^'e -"'<''+'''' + "*) (3)

in a frame rotating at w^ , where terms oscillating faster than

gj (gj<oj,(x>Q) have been neglected. Here, 77=(kz)zo is the Lamb-Dicke parameter, which controls the amount of coupling between internal and motional states. In the case of two-photon stimulated Raman transitions through a third

TABLE I. Selected "magic" values of the Lamb-Dicke parameter T) which satisfy 1 — rj^ = (2k+ 1 )/2m. When the trapped ion is exposed to the carrier for a particular duration, the result is a "TT-pulse" (mod27T) between the 1 1)|J.)*~*I OlT) states and no net rotation (mod27r) between the |0)|J,)«->|0)|T) states.

k (rotation of m (rotation of

|n) = |l) states) |n) = |0) states) 77=[l-(2/t-M)/2w]"^

0(7r) 1(277) 0.707

2(477) 0.866

3(677) 0.913

1(377-) 2(477) 0.500

3(677) 0.707

4(877) 0.791

2(577) 3(677) 0.408

4(877) 0.612

5(1077) 0.707

3(777) 4(877) 0.353

5(1077) 0.548

6(1277) 0.645

4(977) 5(1077) 0.316

6(1277) 0.500

7(1477) 0.597

(virtual) electronic level [11,12], gj is replaced by

gj.\Sj,2^^' where gj -^ and gj2 are the individual Rabi fre- quencies of the two beams when resonantly coupled to the

virtual level and A is the detuning from the virtual level;

u>i{(t>) is replaced by the difference frequency (phase) of the beams; and k is replaced by the difference in wave vectors of

the two beams A=ki — k2. The CN quantum logic gate [Eq. (1)] can be realized with

a single pulse tuned to (x)i = (Oq (the "carrier") that couples

the states \n)\[) and \n)\'\) with Rabi frequency [13,14]

^n,n = \\{n\{m'P\i)\n)\

= g^\{nW^^'^^''%)\ = gje-^''^L,{v^), (4)

where L„(x) is a Laguerre polynomial. Specializing to the

|n) = |0) and |«) = 1 1) vibrational levels relevant to quantum

logic, we have

n (5) From Eqs. (4) and (5), the carrier Rabi frequencies de- pend nonlinearly on the vibrational quantum number n, with

the nonlinearity mediated by the Lamb-Dicke parameter 77 [15,16]. The reduced CN gate [Eq. (1)] can be achieved in a

single pulse by setting 77 so that Hj i/no,o=(2^ + l)/2m,

with k and m integers satisfying m>k^O [17]. By driving

the carrier transition for a duration r such that £i^\T={k +

7)77, or a "77-pulse" (mod 277) on the |n) = | 1) component.

TN-67
55 SIMPLIFIED QUANTUM LOGIC WITH TRAPPED IONS R2491

then ClQQT=imT. Thus the states |l)U)-^|l)|T) are swapped, while the states |0)|].) and |0)|t) remain unaffected. The net unitary transformation, in the {Oi,OT,U,lT} basis is

1

1

ie''>'(- 1)

k — m

/e"'^(-l) k — m

(6)

This transformation is equivalent to the reduced CN of Eq.

(1), apart from phase factors that can be eliminated by the appropriate phase settings of subsequent logic operations [7]. The "magic" values of the Lamb-Dicke parameter

that allow a single-pulsed reduced CN gate satisfy

L,(77^)=l-77^ = (2it+l)/2m and are tabulated in Table I for the first few values [18]. It may be desirable for the reduced CN gate to employ the |n) = |2) or |/i) = |3) state instead of the |n) = |l) state for error correction of

motional-state decoherence [19]. In these cases, the "magic" Lamb-Dicke parameters satisfy L2(77^) = 1 — 2?;^

+ 7j'^/2 = {2k+ l)/2m for quantum logic with |n) = |0) and

|2), or L^(rf") = l-3r/^ + 3r/*/2-7f^/6 = i2k+l)/2m for quantum logic with |n) = |0) and |3).

Finally, we comment on the stability requirements of the Lamb-Dicke parameter in this scheme. Since the mapping

pulses surrounding the single-ion CN gate (resulting in the general two-ion CN gate) have interaction strength proportional to 77, this scheme does not place any additional pre- mium on the stability of 77. In this simplified scheme however, the accuracy of 77 must also be maintained; otherwise

at least one of the two vibrational states will not be rotated by the correct amount. In the two-photon Raman configuration [11,12], the Lamb-Dicke parameter 77=|^|zo can be

controlled by both the frequency of the trap (appearing in Zq) and by the geometrical wave-vector difference Sk of the two Raman beams. Accurate setting of the Lamb-Dicke pa- rameter should therefore not be difficult [16]. We acknowledge support from the U.S. National Security

Agency, Office of Naval Research, and Army Research Office. We thank Brana Jelenkovic, Travis Mitchell, and Matt Young for a critical reading of the manuscript.

[I] D. P. DiVincenzo, Science 270, 255 (1995); S. Lloyd, Sci. Am. (Int. Ed.) 273, 140 (1995); A. Ekert and R. Jozsa, Rev. Mod. Phys. 68, 733 (1996).

[2] D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heinzen,

Phys. Rev. A 50, 67 (1994); J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, ibid. 54, 4649 (1996).

[3] J. I. Cirac and P. Zoller, Phys. Rev. Len. 74, 4091 (1995).

[4] W. H. Zurek, Phys. Today 44(10), 36 (1991).

[5] D. P. DiVincenzo, Phys. Rev. A 51, 1015 (1995); A. Barenco

et ai, ibid. 52, 3457 (1995); S. Lloyd, Phys. Rev. Lett. 75, 346

(1995).

[6] R. P. Feynman, Opt. News 11, 11 (1985).

[7] A. Barenco, D. Deutsch, A. Ekert, and R. Jozsa, Phys. Rev.

Lett. 74, 4083 (1995).

[8] C. Monroe, D. Meekhof, B. King, W. Itano, and D. Wineland,

Phys. Rev. Lett. 75, 4714 (1995).

[9] P. W. Shor, Phys. Rev. A 52, R2493 (1995); A. M. Steane, Phys. Rev. LeU. 77, 793 (1996); R. Laflamme, C. Miquel, J. P. Paz, and W. H. Zurek, ibid 77, 198 (1996); D. P. DiVincenzo

and P. W. Shor, ibid 77, 3260 (1996).

[10] M. Plenio and P. Knight (unpublished).

[11] J. E. Thomas, P. R. Hammer, S. Ezekiel, C. C. Leiby, R. H.

Picard, and C. R. Willis, Phys. Rev. Lett. 48, 867 (1982). [12] 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

1

(1995).

[13] K. E. Kahili and R. J. Glauber, Phys. Rev. 177, 1857 (1969). [14] D. J. Wineland and W. M. Itano, Phys. Rev. A 20, 1521

(1979).

[15] W. Vogel and R. L. de Matos Filho, Phys. Rev. A 52, 4214

(1995).

[16] D. M. Meekhof, C. Monroe, B. E. King, W. M. Itano, and D.

J. Wineland, Phys. Rev. Lett. 76, 1796 (1996).

[17] Setting il]j/ClQQ=2m/{2k+ 1 ) will also work, with the roles of the |0) and 1 1) vibrational states switched in Eq. (1). [18] For rf (Paul) trap confinement along the COM motional mode,

the Rabi frequencies of Eqs. (4) and (5) must be modified to include effects from the micromotion at the rf drive frequency

a>rf. In the pseudopotential approximation ((O'^co^), this cor- rection amounts to replacing the Lamb-Dicke parameter 77 in

this paper by 7)[l-<o/{2^l2o)^)], as pointed out in P. J. Bardroff, C. Leichtle, G. Schrade, and W. P. Schleich, Acta

Phys. Slovaca 46, 231 (1996). There is no correction if the COM motional mode is confined by static fields (e.g., the axial COM mode of a linear trap). [19] J. 1. Cirac, T. Pellizzari, and P. Zoller, Science 273, 1207

(1996).

TN-68
ENTANGLED STATES OF ATOMIC IONS

FOR QUANTUM METROLOGY AND COMPUTATION^

DJ. WINELAND, C. MONROE, DM. MEEKHOF, BE. KING, D. LEIBFRIED,

WM. ITANO, J.C. BERGQUIST, D. BERKELAND, JJ. BOLLINGER, J. MILLER

Ion Storage Group, Time and Frequency Division, NIST, Boulder, CO, 80303, USA

Asingle trap{>ed 'Be'^ ion is used to investigate Jaynes-Cummings dynamics for a two-level atomic

system coupled to hannonic atomic moticMi We create and investigate nonclassical states ofmotion

including "Schr6dinger-cat" states. Afundamental quantum logic gate is realized using the quantized motion and internal states as qubits. We e}q)lore some ofthe applications for, and problems in realizing quantum logic based on multiple trapped ions.

1 Introduction

Currently, a major theme in atomic physics is coherent control of quantum states. This

theme is manifested in a number of topics such as atom interferometiy, atom optics, the atom laser, Bose-Einstein condensation, cavity-QED, electromagnetic-induced

transparency, lasing without inversion, quantum computation, quantum cryptography,

quantum-state engineering, squeezed states, and wavepacket dynamics. A number ofthese

topics are the subjects of other presentations at this meeting.

In this paper we report related trapped-ion research on (1) the study of JaynesCummings dynamics for a two-level atomic system coupled to harmonic atomic motion, (2) the study of quantum mechanical measurement problems such as the generation of

Schrodinger-cat-like superposition states and their relation to various decoherence

phenomena, and (3) coherent quantum logic for the investigation of scaling in a quanttim

computer and for preparation of entangled states useful for spectroscopy.

2 Entanglement

An entangled quantum state is one where the wave function of the overall system cannot be

wntten as a product of the wave functions of the subsystems. In this case, a measurement

on one of the subsystems will affect the state of the other subsystems. For example,

consider a two-level atom bound in a 1 -D harmonic well. Suppose we can create the state

7 = J-(|0|„)*e'*|I>|n')), m

v/2

where the kets 1 1 ) and 1 1 ) denote the two internal eigenstates of the atom (here, we use the

spin-'/2 analog to a two-level system: o^| 1 > = +| ! ), etc. ), the second ket denotes a harmonic

oscillator eigenstate |n), and (}) is a (controlled) phase factor. If we measure the motional

eigenstate of the atom and find it to be in level n, then it must also be found in the i internal

state if we measure o^. Similarly, if we find the atom in the n' motional state, it must be

31 TN-69
32

found in the T internal state. Such correlatiaQS are at the heart of the "EPR" experiments'.

Another state we will consider below is the state for Ntwo-level atoms

T =-i:(|i),|I>,...|iV^e'>i!)Jl>,...|!V). (2)

v2

where the subscript i (= 1 , 2, .... N) denotes the ith atom. This state is "maximally

entangled" in the sense that a measurement of o^ on any atom automatically determines the value of o^ of all other atoms.

3 Jaynes-Cummings-type coupling between internal and motional states

To achieve entanglement from an initially nonentangled system, we need to provide a coupling between subsystems so that the state of one subsystem affects the dynamics of another. Coupling between spins or two-level atoms can, in principle, be

achieved through a dipole-dipole interaction (like the hyperfine coiqjling between electron and proton in the hydrogen atom). In a system of trqjped neutral atoms, dipole-dipole

coupling may be difficult to control to the desired level; for tr^ped ions the Coulomb

repulsion inhibits strong dipole coupling between ions. However, in the case of trapped

ions, the motion can be strongly coupled to the internal levels with the appUcation of

inhomogeneous (classical) electromagnetic fields. For example, we consider an atom

confined in a 1 -D harmonic potential. The atom's dipole moment fi is assumed to couple to an electric field E(x,t) through the Hamiltonian

Hj = -tiE(x,f) ni r\ ^ dE 1 oE E{x-U,t) + X + X + dx 2 dx

(3)

We have fi °^ o+ + o., where o+ and o. are the raising and lowering operators for the internal levels (in the spin-'/i analog). In Eq. (3), the positic»i x is an operator which we write as i =

x„(a + a^), where a and a^ are the usual harmonic oscillator lowering and raising operators,

and X, is the rms spread of the n=0 zero-point state of motion. As a simple example,

suppose the field is static and the motional oscillation frequency o) of the atom is equal to the resonance frequency o, of the internal state transition. In its reference frame, the atom

experiences an oscillating field due to the motion through the inhomogeneous field. Since co = (i)„ this field resonantly drives transitions between the internal states. If the extent of the atom's motion is small enough that we need only consider the first two terms in Eq. (3), H,

can be approximated as Hx^^ = hQ(o+a + a.a}) (in the interaction frame and using the

rotating wave approximation) where Q is a proportionality constant This Hamiltonian is also obtained if E is sinusoidally time varying (frequency wj and we satisfy the resonance

condition Ql + o = o)„. This type of coupling was used to couple the spin and cyclofron

motion in the classic electron g - 2 experiments ofDehmelt and coworkers^. Formally it is equivalent to the Jaynes-Cummings Hamiltonian of cavity-QED'-* which describes the

TN-70
33

coupling between a two-level atom and a single mode of the radiation field. This analogy

has been pointed out in various papers*^; for a review, see Ref 9 and fiirther references in Ref. 8.

3.1 Realization a Jaynes-Cummings-type couplingfor a trapped 'Be* ion

To controllably manipulate the internal and vibrational levels of the ion, we must (1

)

initialize the ion in a well defined internal and motional state and (2) make the vibrational

level spacing (trap frequency) much larger than any internal or motional relaxation rates. To accompUsh this, we have built an rf (Paul) ion trap which confines a single 'Be* ion with

pseudopotential hannonic trap frequencies of ((•)„ cOy, (oJ/2n = (11,19, 29) MHz along the

three principal axes of the trap'". The energy-level structure of 'Be* is summarized in Fig. 1. Because the ion is harmonically bound, the internal *Be* electronic states must include the ladder of external harmonic oscillator levels of energy E„ = ho)(n+'/2), where we have considered only the x- dimension of the oscillator (w s coj and its associated quantum number n £ n^ e (0, 1, 2,

...). The two internal levels of interest are the ^S,/^ ground state hyperfine levels |F=2,

mF=2) (denoted by 1 1)) and |F=1, mp=l) (denoted by | T )), which are separated in frequency

by (jijln ~ 1 .25 GHz. The other Zeeman levels are resolved from the | i ) and 1 1 ) states by

the apphcation of a =0.2 mT magnetic field*-".

Be

Pi/2(2,2)

(a)

'1/2

A

"7^^ ./

r / R2

\—t

IT) ^

. lu

_ 1 J.)

Fiq. 1. (a) Electronic (internal) and motional (external) energy levels (not to scale) ofthe trapped *Be'^ ion, coupled

by indicated laser beams Rl and R2. Tbe difference frequency ofthe Raman beams Rl and R2 is set near 0^271 = 1.250 GHz, providing a two-photon Raman coupling between the *Su,(F=2, in,=2) and *Su,(F=l, m,=l) hyperfine ground stales (denoted by 1 1 ) and | T ) respectively). Tbe motional energy levels are depicted by a ladder of

vibrational states separated in frequency by tbe trap frequency u/2n - 11 MHz. Tbe Raman beams are detuned A/2n = -12 GHz firm the 'Pu,(F=2,nv-2) excited state. As shown, the Raman beams are tuned to the red

sideband, (b) Detection ofthe internal stale is accomplished by illuminating the ion with a'^-polahzed "detection" beam D2, whidi drives the cycling *Siq(F=2, nv=2) - *Pj,(F=3, m,=3) transiti<m, and observing the scattered

fluorescence. The vibrational structure is omitted fixim (b) since it is not resolved . Beam DI, also o* polarized, provides spontaneous recycling fi^xn the | T ) to | stale.

TN-71
34

Strong field gradients can be obtained with laser fields (e*^ factor). In our

experiment, the field corresponding to that in Eq. (3) is provided by two laser fields which

drive stimulated-Raman transitions between the levels of interest (Rl and R2 ofFig. 1 a). (The xise of stimulated-Raman transitions has some technical advantages, but is formally

equivalent to driving a narrow single-photon transition.) Two-photon stimulated Raman

transitions between the 1 1 ) and | T ) states can be driven by tuning the difference frequency

ofRl and R2 to be near a)„,. The two Raman beams (X - 3 1 3 nm) are generated from a

single laser source and aco\isto-optic modulator, allowing excellent stabihty of their relative frequency and phase. Both beams are detuned A/27i= 12 GHz from the excited ^j^^

electronic state (radiative linewidth y/Zn = 19.4 MHz), and the polarizations are set to couple through the ^^^(F=2, mF=2) level (the next nearest levels are the ^-^ states which

are over 2O0 GHz away and can be neglected). Because A » y, the excited ^ state can be

adiabatically eliminated in a theoretical description, resulting in a coupling between the two

ground states which exhibits a linewidth inversely proportional to the interaction time. When Rl and R2 are applied to the ion with wavevector difference 5^ = ^, - l^j along the

x-direction, the effective coupling Hamiltonian in the rotating-wave approximation is given

by

The coupling strength g depends on A and the intensity of the laser beams, t] = 1 61c |xo - 0.2 is the Lamb-Dicke parameter, Xo = (h/2ma))"' = 7 nm, and 6 is the difference between

the relative frequency of the two Raman beams and (i)„. Setting 6^ to be parallel to the xaxis of the trap, yields almost no coupling between the internal states and motion in the y- and z-directions.

If 6 = a)(n'-n), transitions are resonantly driven between the levels | i ^) and 1 1 ^')

at a rate ^^. which is dependent on n and n' *

. Starting from the 1 1 ) |n) state, application of

a Rabi -n pulse coherently transfers the ion to the] t ) |n') state; this corresponds to applying

the Raman beams for a duration t such that Qn^t = 7x/2. If we apply the Raman beams for half of this time, we create the entangled state of Eq. (1). Here, we will assiime the ion is confmed in the Lamb-Dicke limit (

1 6k | <x^''* « 1 ) and will consider three fransitions. The carrier, at 6 = 0, drives transitions between states 1 1 ,n) - 1 1 ^) with Rabi frequency

Q^ = g. The "first red sideband," corresponding to 6 = -o), drives transitions between

states 1 1 ^) - 1 1 Ji-1) with Rabi frequency Q,^_^, = gnVn. This coupling is analogous to the case in cavity-QED'' where energy is coherently exchanged between the internal and

external degrees of freedom. The "first blue sideband," at 6 = +0), drives transitions between states 1 1 ,n) - 1 1 ,n+l) with Rabi frequency Q,^^^, = gr|(n+l)'''.

Preparation of the | i ) | n=0) state is accomplished by first Doppler cooling the ion

to (n) - 1 , followed by sideband laser cooling using stimulated Raman transitions". For

sideband laser cooling, Ti pulses on the first red sideband (| i)|n) ^ | l)|n-l)) are alternated with repumping cycles using nearly resonant radiation (Fig. lb) - which results (most

probably) in transitions 1 1 > | n) - 1 1 ) | n) . These steps are repeated (typically 5 times) until

TN-72
35

the ion resides in the | i ) 1 0) state with high probability (> 0.9). From the \l)\0) state, we are able to coherently create states of the form | i )?(x),

where the motional state Y(x) = S„C„e'~^|n) and the C„ are complex. We can analyze the motional state created as follows: The Raman beams are pulsed on for a time t and the probability P,(t) that the ion is in the 1 1 ) internal state is measured. The experiment is repeated for a range of x values. When the Raman beams are tuned to the first blue

sideband, the expected signal is

^iW = \

( - \

l+E^„cos(2Q^.,T). (5)

where P„ = | C^|^ is the probabihty of finding the ion in state n and Yb are experimentally

determined decay constants. The internal state | i ) is detected by applying near-resonant De- polarized laser radiation (beam D2, Fig. lb) between the | i) and ^3/j(F=3, mF=3) energy

levels. Because this is a cycling transition, detection efiiciency is near unity*-". The

measured signal P,(t) can be inverted (Fourier cosine transform), allowing the extraction of

the probabihty distribution of vibrational state occupation P„. This signal does not show the phase coherences (phase factors of the CJ, which must be verified separately*-'^ The most

complete characterization is achieved with a state reconstruction technique'^.

3.Creation ofCoherent and Schrodinger-Cat states

We have created and analyzed thermal, Fock, squeezed, coherent, SchrOdinger-cat states, and superpositions of Fock states*-'^''; here we briefly describe the creation and

measurements of coherent and Schr6dinger-cat states. A coherent state of motion

7(x) = |a) - exp(-|a|2/2)^-^|n), (6)

n-O ^

corresponds to a displaced zero-pomt wave-packet oscillating in the potential well with

amplitude 2 1 a | x^,. From Eq. (5), P , (t) for a coherent state will undergo quantum collapses

and revivals'\ These revivals are a purely quantum effect due to the discrete energy levels and the narrow distribution of states^-". We have produced coherent states of ion motion fi-om the 1 1 ) 1 0) state by applying

either a resonant (fi-equency u)) classical driving field or a "moving standing wave" of laser radiation which resonantly drives the ion motion through the dipole force*-'^. In Fig. 2, we

show a measurement ofP,(t) after creation of a coherent state of motion, exhibiting the expected collapse and revival signature. (For comparison, see the cavity-QED experiment

of Ref 4.) This data is fitted to Eq. (5) assxmiing a Poissonlan distribution, allowing only

(n) to vary. The inset shows the results of a separate analysis, which yield the probabilities

of the Fock-state components, extracted by applying a Fourier cosine transform to P,(t) at the known fi"equencies as described above. These amphtudes display the expected

TN-73
36

Poissonian dependence on n.

n 0.1 -

0.0

8 10

f'-'-^''''''^^\/\/\A^^

10 20 30 t(^s)

40 50

Fig. 2. PiCt) for a coherent state driven by the first blue sideband interaction, showing collapse and revival behavior. The data are fitted to a coherent state distribution, yielding (n) = 3.1(1). The inset shows the results of

inverting the time-domain data by employing a Fourier cosine transform at the known Rabi frequencies Q,^,, fitted to a Poissonian distribution, yielding (n) = 2.9(1). Each data point represents an average of =4000 measurements,

or 1 s of integration. A SchrOdinger-cat state is a coherent superposition of classical-like states. In

SchrOdinger's original thought experiment", he described how one could, in principle,

entangle a superposition state of an atom with a macroscopic-scale superposition of a live and dead cat. In our experiment'^, we construct an analogous state, on a smaller scale, with

a single atom. We create the state

1 T = -^(|l)|a,)+e'*ll)|a2)),

v/2 (7)

where | a,) and | a^) are coherent motional states and 4> is a (controlled) phase factor. The

coherent states of the superposition are spatially separated by mesoscopic distances much

greater than the size of the atom wavepacket which has a spread equal to x,. Analysis of this state is interesting from the point of view of the "quantum

measurement problem," an issue that has been debated since the inception of quantum

theory by Einstein, Bohr, and others, and continues today'*. One practical approach toward

resolving this controversy is the introduction of quantum decoherence, or the environmentally induced reduction of quantum superpositions into classical statistical mixtures'^. Decoherence provides a way to quantiiy the elusive boundary between classical and quantum worlds, and almost always precludes the existence of macroscopic

SchrOdinger-cat states, except at extremely short time scales. On the other hand, the

creation of mesoscopic SchrOdinger-cat states like that ofEq. (7) may allow controlled

studies of quantum decoherence and the quantum-classical boundary. This problem is directly relevant to quantum computation.

TN-74
37

In our experiment, we create a Schr6dinger-cat state ofthe single-ion 'Be"^ harmonic oscillator (Eq. (7)) with a sequence of laser pulses'^. First, we create a state of the form (

1 1 ) + e*^ 1 1 )) I n=0)l^2 with a 7i/2 pulse on the Raman carrier transition (Sec. 3.1). To

spatially separate the 1 1) and | ! ) components of the wave function, we apply a coherent

excitation with an optical dipole force which, because of the polarization of the beams used

to create the force, selectively excites the motion of only the 1 1 ) state. We then swap the | i

)

and 1 1 ) states with a Ti-carrier pulse and re^ply the dipole force with a different phase to create the state of Eq. (7). In principle, if we could make | a,^| large enough, we could

design a detector which could directly detect the (distinguishable) position ofthe particle

and correlate it with a spin measurement**. Instead, to analyze this state in our experiment, we apply an additional laser pulse to couple the internal states, and we measure the resulting interference of the distinct wavepackets. With this interferometer, we can establish the

correlations inherent in Eq. (7), the separation of the wavepackets, and the phase coherence

4> between components of the wavefiinction. These experiments are described in Ref 12. The interference signal should be very sensitive to decoherence. As the separation | Oj - ttj

| is made larger, decoherence is expected to exponentially degrade the fringe contrast^". We remark that other experiments generate SchrOdinger-cats in the same sense as

in our experiment. Examples are atom interferometers "•^°, and superpositions of electron wavepackets in atoms^' (also, see additional citations in Ref 12). However, as opposed to these experiments, the harmonic oscillator cat states ofEq. (7) do not disperse in time. This

lack of dispersion provides a simple visualization of the "cat" (e.g., a marble rolling back

and forth in a bowl which can be simultaneously at opposite extremes of motion) and should

allow controlled studies of decoherence models.

4 Quantum Logic

Interest in quantum computation in the atomic physics community was stimulated, in part, by a talk given by Artur Ekert at the last ICAP meeting". Subsequently, Ignacio Cirac and

Peter Zoller"-^^ proposed an attractive scheme for a quantum computer which would use a

string of ions in a linear trap as "qubits." This proposal has stimulated experimental efforts

in several laboratories including those at Innsbruck, Los Alamos National Laboratory, IBM,

and NIST.

Each qubit in a quantum computer could be implemented by a two-level atomic

system; for the ith qubit, we label these states 1 1)^ and | T ), as above. In general, any

quantum computation can be comprised of a series of single-bit rotations and two-bit

"controlled-NOT" (CN) logic operations'^'. We are interested in implementing these two

operations in a system of 'Be* ions. Single-bit rotations are straightforward and correspond

to driving Raman carrier transitions (Sec. 3 . 1 ) for a controlled time. Such rotations have

been achieved in many previous experiments. Next, we describe the demonstration of a

nontrivial CN logic gate with a single 'Be* ion^'.

4.1 "Conditional dynamics" and a single-ion controlled-not logic gate

The key to making a quantum logic gate is to provide conditional dynamics; that is, we

TN-75
38

desire to perform on one physical subsystem a vmitary transformation which is conditioned

upon the quantum state of another subsystem^. In the context of cavity QED, the required

conditional dynamics at the quantum level has recently been demonstrated ^•^. For trapped

ions, conditional dynamics at the quantum level has been demonstrated in verifications of

zero-point laser cooUng"'^'. Recently, we demonstrated a CN logic gate with the abiUty to prepare arbitrary input states (the "keyboard").

A two-bit quantum CN operation provides the transformation:

|€,)|e2) - |e,>|6,©e2), (g)

where £,,€2 6 {0,1 } and e is addition modulo 2. The (impUcit) phase factor in the transformation is equal to 1 . In this expression 6, is the called the control bit and e^ is the

target bit. If £[ = 0, the target bit remains unchanged; if e, = 1, the target bit flips. In the

single-ion experiment of Ref 26, the control bit is the quantized state of one mode of the

ion's motion. If the motional state is |n=0), it is taken to be a 1 6,=0) state; if the motional

state is |n=l), it is taken to be a | ei=l) state. The target states are two ground-hyperfine

states of the ion, the 1 1 ) and 1 1 ) states of Sec. 3 . 1 with the identification here | -l ) — | e^K))

and 1 1 ) -^

I Gj-l)- Following the notation of Sec. 3.1, the CN operation is realized by

applying three Raman laser pulses in succession:

(la) A "71/2-pulse" is applied on the spin carrier transition. For a certain choice of

initial phase, this corresponds to the operator V**(Tt/2) ofRef 23. (lb) A 271-pulse is applied on the first blue sideband transition between levels 1 1 ) and

an auxiliary level | aux) in the ion (the |F=2, Mp=0) level in 'Be* ). This operator

is analogous to the operator tf' of Ref 23. This operation provides the

conditional dynamics for the controlled-not operation in that it changes the sign of

the 1 1 ) |n=l) component of the wavefunction but leaves the sign of the 1 1 ) |n=0)

component of the wavefiinction unchanged.

(1 c) A 7i/2 -pulse is applied to the spin carrier transition with a 1 80° phase shift relative to step (la); This corresponds to the operator W\-tiI1) ofRef 23.

Steps (1 a) and (1 c) can be regarded as two resonant pulses (of opposite phase) in the Ramsey separated-field method of spectroscopy. We can see that if step (b) is active (thereby changing the sign of the | T ) |n=l) component ofthe wave fimction) then a spin flip

is produced by the Ramsey fields. If step (lb) is inactive, the net effect of the Ramsey fields

is to leave the spin state unchanged. This CN operation can be incorporated to provide an

overall CN operation between two ions in an ensemble ofN ions if we choose the ion

oscillator mode to be the center-of-mass (COM) mode of the ensemble. Specifically, to realize a controUed-not C^ between two ions (m = control bit, k = target bit), we first assume the COM is prepared in the zero-point state. The initial state of the system is therefore given by

TN-76
39

7 |0). (9)

C,^ can be accomplished with the following steps:

(2a) Apply a 7i -pulse on the red sideband of ion m (the assumption is that ions can be

addressed separately^). This accomplishes the mapping (a | i )„ + p | ! ) J 1 0) -

1 1 )„(a 1 0) - e'^p 1 1 )) , and corresponds to the operator U^-° ofRef 23 . We note

that in our experiments, we prepare the state (a 1 1 ) + P | ! )) 1 0) using the carrier

transition (Sec. 3.1). We can then implement the mq)ping (a| I) + p| T))|0) -

I ^)n.(*'^|0) " ^Pl !))• This is the "keyboard" operation for preparation of arbitrary

motional input states for the CN gate of steps 1 a - 1 c above. Analogous mapping

of internal state superpositions to motional state superpositions were

demonstrated in Ref 26. (2b) Apply the CN operation (steps 1 a - 1 c above) between the COM motion and ion k. (2c) Repeat step (2a).

Overall, C^ provides the mappings 1 1)„ I i),0)- | i)„l DJO), 1 1)„| t),|0) - |i).10k|0),

1 1 )n,l Ok|0) - 1 1 )„| OJO) ,and | T )„| DJO) - | T)„| i)JO) which is the desu-ed logic of Eq.

(8). Effectively, C^ works by mapping the internal state of ion m onto the COM motion,

performing a CN between the motion and ion n, and then mapping the COM state back onto

ion m. The resulting CN between ions m and k is not really dififerent from the CN described

by Cirac and Zoller\ because the operations V"''(6) and U^-° commute.

4.2 Quantum Registers and Schrodinger Cats

The state represented by Eq. (9) is of the same form as that ofEq. (7). Both involve

entangled superpositions and both are subject to the destructive effects of decoherence.

Creation of Schrodinger-cats like Eq. (7) is particularly relevant to the ion-based quantum

computer because the primary source of decoherence will probably be due to decoherence

of the |n=0,l) motional states during the logic operations.

5 Potential for, and Problems with, Trapped-Ion Quantum Logic

Quantum computation has received a great deal of attention recently because of the algorithm proposed by Peter Shor for efficient factorization^. This has important

implications for public-key data encryption where the security of these systems is due to the

inability to efficiently factorize large numbers. To accomplish quantum factorization is extremely formidable with any technology; however, other applications of quantum logic

may be more tractable.

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40

5.1 Positive Aspects ofTrapped-Ion Quantum Logic

Internal state decoherence can be relatively small in experiments on trapped ions. The ions' energy level structure is, of course, perturbed at some level by electric and magnetic fields. However, energy level shifts caused by electric fields (Stark shifts) tend to be quite small

and, in many cases, level shifts due to magnetic fields can be controlled well enough. This

is evident fi-om trapped-ion atomic clock experiments where linewidths smaller than 0.001 Hz have been achieved^ '•'^, indicating internal state coherence times exceeding 10 min.

The required laser cooling to | n=0) has been demonstrated"-^ for single ions. A

string of laser-cooled ions (Fig. 3), which could be used as a quantum register, has been

achieved in a linear ion trap'^-'^ but an immediate future task will be to achieve zero-point

cooling (for at least the COM mode) on an ensemble of ions. For a computation performed on an ensemble of ions in a trap, this need not be done extremely well. All we require is that the cooling be sufficient that the ion's COM mode is predominantly in the n=0 state, so

the "correct" answer to a computation is obtained most of the time. Similarly, although

nearly unit detection efficiency has been achieved with trapped ions"-^', the basic

requirement is that the noise in the "readouf' of the quantum register should minimize the nijmber of times the calculation is repeated.

VoCosQt

Fig. 3 . The upper part ofthe figure shows a. schematic diagram ofthe electrode configuration for a linear Paul-RF

trap (rod spacing - 1 nun)*'. The lower part ofthe figure shows an image of a string of "*Hg* ions, illuminated with 194 nm radiation, taken with a UV-sensitive, photon counting imaging tube. The spacing between adjacent

ions is approximately 10 ^m. The "gaps" in the string arc occupied by impurity ions, most likely other isotopes of

Hg*, which do not fluoresce because the frequencies of their resonant transitions do not coincide with those ofthe 194 nm 'Sh - 'P„ transition of "Tig*.

Estimates of motional decoherence times from the |n=0,l) states, due to fundamental causes, should be very long (more than 100 s)^*. However, these predictions

have not been realized experimentally and the causes of the observed decoherence"-^' must

still be found.

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41

5.2 Problems and Possible Solutions

Many problems may conspire to prevent large-scale quantum computation; some of the problems relevant to trapped ions are briefly mentioned here. More complete analyses are given elsewhere,"- **. Motional decoherence can arise from fluctuating trap fields and radiative coupling

to the environment. The integrity of the trap electrode structure is expected to play an

important role, particularly if the number of ion qubits becomes large. Lithographic

techniques for constructing electrodes^' may be useful to insiire accurate dimensional

tolerances. With these techniques, it may also be possible to incorporate accurate (Josephson) voltage standards and (superconducting) flux magnetometers into the structure. Laser power fluctuations will affect the fidelity of the rotations and logic gates (for example, n/2 rotations become t:/2 ± e rotations where e is unknown). Although the effects depend on the form of the noise and on the computational algorithm, a conservative estimate

is to assume that phase errors accumulate linearly with the number of elemental operations.

A computation requiring 1 0' elemental operations would therefore require an intensity

stability of one part in 1 0* over the time of the computation. With current algorithms,

factorization of large numbers will require even more elemental operations, so extreme laser

stabilization will be required.

We have noted some of the advantages of using stimulated Raman transitions in quantum logic^*. One apparent disadvantage is that, smce the Raman beams are detuned

from a virtual optical level, the energy levels are shifted by AC-Stark effects [see for example Ref 36]. These effects are absent in single-photon transitions as assumed m Ref

23. Therefore, if stimulated-Raman transitions are used and if the laser intensities fluctuate, additional Stark-induced phase errors will accumulate. However, we can show^* that these

errors are of the same order as those incurred from the angular errors ofthe preceding

paragraph, if the two Raman laser intensities are approximately equal.

The scheme of Cirac and Zoller" assumes that the laser beams can separately address individual ion qubits. This necessitates a tradeoff between two factors. We desire

that the ions be well separated spatially to allow a focused laser beam to address only one

ion at a time. However, we also desire to spectrally isolate individual modes of the ion motion, to insure the fidelity of the logic. The closer in frequency the "contaminating"

transitions (from couplmg to other motional modes) are, the slower the logic speed must be

to obtain isolation. To give an idea of the problem, we note that the separation of two *Be*

ions confined m a harmonic well of frequency v = Ci)/27t is given by d = 9.2 Iv"^ where d is

in micrometers and v m megahertz. For longer strings, the spacing of the central ions becomes closef*". Although the focusing predicted with the use of Gaussian beam formulas

implies the required isolation could be obtained for v up to a few MHz, in practice, stray

light intensity will undoubtedly be a problem. With the use of stimulated-Raman transitions

^'^^ one solution for this problem is to take advantage of the inherent AC Stark shifts. The

basic idea is that the (resonant) Rabi frequencies g,,g2 of the two Raman beams are made

substantially different, say gi»g2- The transition frequency for the selected qubit is therefore shifted from the frequency of adjacent ions so that the adjacent ions are relatively unaffected. Unfortunately, the sensitivity to intensity fluctuations also becomes worse by the

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42

ratio g,/g,^.

A large scale computation will require a large qubit register. This makes it extremely hard to isolate unwanted motional mode transitions from the desired one-**. As

noted in Ref. 23, the desired COM trap-axis mode frequency in a linear trap is smaller than

other trap axis modes and can therefore be relatively well isolated spectrally. However, as the number of ions in the trap increases, the radial mode frequencies will tend to overlap the COM mode frequency. Also, multi-mode excitations may become a problem when the difference frequency of the modes is close to the COM mode frequency. Therefore, a multiplexing scheme for ion qubit registers seems desirable; we discuss one possibility

below.

5.3 A 1- or 2-qubit ion accumulator

One possibility for multiplexing in a trapped-ion quantum computer is to perform all logic in minimal accumulators which hold one or two ions at a time''*. Ions would be shuffled around in a "super-register" and into and out of the accumulators which are well shielded from the other ions. The shuffling could be accomplished with interconnected linear traps with segmented electrodes; this appears possible with the use of lithographic techniques.

Smgle-bit rotations on the mth ion would be accomplished by moving that ion into an

accumulator. Logic operations between ions m and k would be accomplished by first moving these ions into an accumulator. An accumulator would hold a second species of ion

(say Mg"^) which could be used to provide laser cooling to the | n=0) level (of the mode used

for the gate) if necessary. Therefore, for logic operations, an accumulator would hold two

computational ions and the auxiliary ion. This scheme should make it easier to select ions with laser beams because it should be straightforward to address one ion while nulling the

laser intensity on the other ion, even with very high trap frequencies. The very small number of logic ions in an accumulator (1 or 2) would make extraneous mode coupling

much easier to avoid. The main problem appears to be that computational speed is reduced

because of the time required to shuffle ions in and out of the accumulator and provide laser coolmg with the auxiliary ion, if required. However, energy shifts of the ion's mtemal

structure, due to the electric fields required to move the ion, need not be severe . For

example, to move a 'Be* from rest to a location 1 cm away (and back to rest) in 1 |is would

require a field of less than 50 V/cm. Electric fields of this order should give negligible

phase shifts in qubits based on hyperfme structure^'. The phase shift caused by time dilation would be less than 1 firad.

5. 4 Perspective on Ion Quantum Computation

To be useful for factorization, a quantum computer must be able to factorize a 200 digit

decimal number. This will require a few thousand ions and perhaps 1 0' elementary

operations^. Given the current state -of-the art (one ion and about 10 operations), we

should therefore be skeptical. Decoherence will be most decisive in determining the fate of

quantum computation. Already, decoherence from spontaneous emission appears to limit the number of operations possible^^*''. The experiments can be expected to improve

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43

dramatically, but we must hope for more efficient algorithms or ways to patch them (such as error correction schemes'*) before large scale factorization is possible.

Any quantum system that might be contemplated in quantum computation must be

reproducible, stable, and well isolated from the environment. Quantum dots have the

potential advantage of large scale integration using microfabrication; however at the present

time, they suffer from lack of precise reproducibility and excessive decoherence. Trapped

ions are reproducible and relatively immune to environmental perturbations - this is the reason they are candidates for advanced frequency standards'". In principle, high

information density could be achieved by scaling down the electrodes; however, we must

then worry about excessive environmental coupling such as magnetic field perturbations

caused by impurities and/or currents in the (nearby) trap electrodes'*. Electric field perturbations will also become important. Therefore, in terms of scale, the frapped ion

system may be close to optimum.

Finally, factorization, discrete logs, and certain other mathematical computations appear to be the hardest problems that quantum logic might be applied to. One of the

applications for quantum computation that Richard Feynman originally had in mind was to simulate quantum mechanical calculations*'. This idea is being explored again with new

possibilities in mind**. Below, we consider an application to atomic measurement.

6 Quantum Logic Applied to Spectroscopy

We conclude by discussing a possible application of quantum logic in the realm of atomic

physics. This application has the advantage of being usefiil with a relatively small number

of ions and logic operations.

Entangled atomic states can improve the quantum-limited signal-to-noise ratio in

spectroscopy*-'*-'*^. In spectroscopy experiments on N atoms, in which changes in atomic

populations are detected, we can view the problem in the following way usmg the spin-'/2

analogy for two-level atoms. We assume spectroscopy is performed by applying (classical) fields of frequency cOr for a time T^ according to the Ramsey method of separated fields**. Ailer applying these fields, we measure the fmal state populations. For example, we might measure the operator N. corresponding to the number of atoms in the | i ) state. In the spm14 analog, this is equivalent to measuring the operator J^ , since N = JI - J,, where I is the

identity operator.

If all technical sources of noise are eliminated, the signal-to-noise ratio (for repeated measurements) is fundamentally limited by the quantum fluctuations in the number

of atoms which are observed to be in the | i ) state. These fluctuations can be called quantum "projection" noise*'. If spectroscopy is performed on N initially uncorrelated atoms (e.g., T(t=0) = H| l), ), the imprecision in a determination of the frequency of the

transition is limited by projection noise to (Aco),,,^ = 1/(NTrT)''^ where t » Tr is the total averaging time. If the atoms can be imtially prepared in entangled states, it is possible to achieve (Aci)),„^ < 1/(NTrT)'''. Initial theoretical investigations*-** examined the use of

correlated states which could achieve (A(j),„^ < 1/(NTrT)''' when the population (JJ was

measured. More recent theoretical investigations*^ consider the initial state to be one where,

after the fu"st Ramsey pulse, the internal state is the maximally entangled state of Eq. (2).

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44

After applying the Ramsey fields, we measure the operator = 11^0^ instead of J^ (or >J). For unit detection efBciency, we can achieve (Ag)).^ = 1/(N^TrT)''' which is the maximum

signaJ-to-noise ratio possible. For an atomic clock where T^ is fixed by other constraints,

this means that the time required to reach a certain measurement precision (stability) is reduced by a factor ofN relative to the xmcorrelated-atom case. In terms of quantum

computation, this amounts to a computation of the fimction cos(N((i) - wJT). Of course,

this computation has special significance for the measurement of co, (an intrinsic computer

parameter) but otherwise is much better suited for a classical computer! See Ref 50 for

related work.

Cirac and Zoller^ have outlined a scheme for producing the state in Eq.(2) using

quantum logic gates. Using the notation of Sec. 3 . 1

, we would first apply a Tt/2 rotation to ion 1 to create the stateT = 2-^(| l), + e**| t>,)| i)^\ i),...] i)^. We then apply the CN gate of

Eq. (8) sequentially between ion 1 and ions 2 through N to achieve the state ofEq. (2). An

alternative method for generating this state, without the need of addressing individual ions

is described in Ref 47.

Acknowledgments

We gratefully acknowledge the support of the National Security Agency, the US Office of

Naval Research, and the US Army Research Office. We thank P. Huang, M. Lombardi, C. Wood, and M. Young for helpfiil comments on the manuscript

References

t Contribution of NIST; not subject to US copyright

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TN-84
JOURNAL OF MODERN OPTICS, 1997, VOL. 44, NO. 11/12, 2485-2505

Experimental preparation and measurement of quantum

states of motion of a trapped atomf

D. LEIBFRIED, D. M. MEEKHOF, C. MONROE, B. E. KING,

W. M. ITANO and D. J. WINELAND

Time and Frequency Division, National Institute of Standards and

Technology, 325 Broadway, Boulder, CO 80303-3328, USA

{Received 5 November 1996)

Abstract. We report the creation and full determination of several quantiim

states of motion of a 'Be"*" ion bound in a RF (Paul) trap. The states are coherently prepared from an ion which has been initially laser cooled to the zero-point of motion. We create states having both classical and non-classical character including thermal, number, coherent, squeezed, and 'Schrodinger cat' states. The motional quantum state is fully reconstructed using two novel

schemes that determine the density matrix in the number state basis or the Wigner function. Our techniques allow well controlled experiments decoherence and related phenomena on the quantum—classical borderline.

1. Introduction

The ability to create and completely characterize a variety of fundamental

quantum states has long been sought after in the laboratory since it brings to the

forefront issues involving the relationship between quantum and classical physics.

Since most theoretical proposals to achieve these goals were put forward in the

field of quantum optics, it might seem surprising that some of the first experiments

succeeding in both respects were realized on the motion of a trapped atom.

However, since both the photon field of quantum optics and the motion of a trapped atom are quantum harmonic oscillators, their couplings to internal atomic

levels (described by the Jaynes-Cummings model (JCM) [1, 2]) are quite similar

[3, 4]. In addition, for the case of a harmonically-bound atom driven by a light

field, there are interactions beyond the simple Jaynes-Cummings coupling,

allowing more control over the engineering and measurement of quantum states. Section 2 will give a brief description of the interaction of a trapped atom with

light fields and outline the similarities to the Jaynes-Cummings Hamiltonian

studied in quantum optics. Our experimental setup and the cooling of the trapped

atom to the motional ground state are described in section 3. We then describe the

controlled preparation of both classical and non-classical motional states including

a 'Schrodinger-cat' type state in section 4. The complete measurement of either

the density matrix in the number state basis or the Wigner function is covered in

section 5 and we finally offer some conclusions in section 6.

j Work of the U.S. Government. Not subject to U.S. copyright.

0950-0340/97 812-00 © 1997 Taylor & Francis Ltd.

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2486 D. Leihfried et al.

P«(3.3)

(a) (b)

Figure 1 . {a) Electronic (internal) and motional (external) energy levels of the trapped

Be"*" ion, coupled by laser beams Rl and R2. The difference frequency of the Raman

beams is set near a;o/27t ~ 1 -250 GHz, providing a two photon Raman coupling

between the '^'^i/iiF — 2,mp = 2) and ^Si/2(-^ — li ^F — 1) hyperfine ground states (denoted by |J,) and |t) respectively). The motional energy levels are depicted by a ladder of vibrational states separated by the trap frequency uix/{2n) ~ 11 -2 MHz.

The Raman beams are detuned by zl/(27i) ~ — 12 GHz from the ^Pi/2 {F = 2, mp — 2) excited state. As shown, the Raman beams are tuned to the first red

sideband, {b) Detection of the internal state is accomplished by illuminating the ion

w^ith a a"*" polarized 'detection' beam D2, which drives the cycling ^Siy2 {P — 2, mp = 2) —> ^P3y'2(F = 3, mp = 3) transition, and observing the scattered

fluorescence. The vibrational structure is omitted from (b) since it is not resolved. Beam Dl, also cr^ polarized, provides spontaneous recycling from the |t) to |i) state.

2. Interaction of a trapped atom with light fields

To describe the interaction of the trapped atom with light fields we make the

following assumptions which will be justified below. First we asstime that the

internal degrees of freedom of the trapped atom are sufficiently described by a two

level system, second that the motion of the atom bound in the trap is harmonic in

all three dimensions, and finally that the vibrational level spacings (trap frequencies) and internal state transition frequencies are much larger than any internal or motional relaxation rates. Starting from these assumptions we can describe the

trapped atom as a two level system with levels labeled |J,), and |t), dressed by the

harmonic oscillator ladders of the external motion with number states {nxTiynz). We

will consider coupling to only the x dimension harmonic oscillator with number

states \nx) = \n) [n= 1, 2, ...oo, see figure 1 (a)]. To couple the motional and

internal degrees of freedom of the trapped atom, we apply two laser beams whose

difference frequency matches the separation of two energy levels, as depicted in

figure 1 (a). The beams are each sufficiently detuned from short-lived excited

electronic states, resulting in two-photon stimulated Raman transitions between

the states of interest which are formally equivalent to narrow single photon

transitions. By employing two laser beams to drive stimulated Raman transitions, we are able to combine the advantages of strong optical electric-field gradients

(allowing manipulation of the state of motion) and microwave stability of the

crucial difference frequency. In the rotating wave approximation in a frame rotating with ujq, where Hujq is the energy difference of the two internal levels, the interaction of the classical laser

field w^ith the two levels of the trapped atom is described by the Hamiltonian

Hin.t{t) = 'hg{cr+ exp [—i{8t — k-x)] -|- cr_ exp [i{6t — k-x]), (1)

TN-86
Experimental preparation and measurement of quantum states of motion 2487

where g denotes the interaction strength, a^ and a~ are the PauH spin matrices

describing the two level system and 8 the detuning of the frequency difference of

the two Raman beams uj = uj\ — 002 with respect to ujq, and k = kj — k2 is the

difference of the two Raman beam wave-vectors. In these experiments the wavevector difference was always chosen to be parallel with the x-direction of the trap, so )s.-'x. = kx and the interaction couples only the motion in ic-direction to the

internal state of the trapped atom.

In our experiment we confine a single beryllium-ion in a RF-(Paul) trap [5], so the trapping potential is not a simple harmonic potential but rather a time

dependent potential with the periodicity of the RF field. Based on the full quantum

mechanical treatment of Glauber [6], several workers have studied the time

dependence of the interaction Hamiltonian (1) [7, 8]. While Bardroff et al. [7] give a general expression for the Rabi frequencies, we restrict ourselves to an

approximate treatment in the limit where the dimensionless Paul trap parameters a and q related to the static and RF-potential [5] are much smaller than one (in our trap, a ~ 0, 5 ~ 0-14). The main change in the full treatment is a common

scaling factor in the interaction strength (Rabi frequencies). This is already

taken into account in our experiments, since we scale all our Rabi frequencies

with the experimentally determined Rabi frequency i2o,i (see below). In the

approximation of a simple harmonic oscillator, with creation and destruction

operators a"*" and a, the interaction Hamiltonian (1) reduces to

-f^eff — TT-g{cr+ exp [i'q{a^ -|- a) — i8t] + (T_ exp [—ri{a^ + a) -\- i5f]), (2)

where tj = k[7i/(2muJx)] , m is the mass and Ux is the secular frequency of the ion

in jc-direction. By tuning the frequency difference 5 to an integer multiple of the

secular frequency cjz, 6 — (n' — n)uJz, we can resonantly drive transitions from

\l, n) to II, n'). In this case iJeff is dominated by a single stationary term. The

exponent exp[ir}{a^ + a)] in H^u contains all powers of a and a^ . Since their time

dependence (in the interaction picture) is a(t)"' = a{t)"' exp(—imUxt), all contributions with m y^ n' — n oscillate rapidly and average out when cj^ is much larger than

g. The biggest stationary term in the Lamb-Dicke limit ((77(a+ a^) )

' <C 1) is proportional to

^n—n Heff ~ ng Z'

(a+(^t)'-"' + a_a"-"'), (3)

[n — n )\

if n' — n > 0, and

i?eff - %T^ ^ l'^^^"""' + <^-(«^)"""'), (4)

[n — ny.

if n' — n < 0. In the special case of m' — n = — 1 {6 < 0, first red sideband) we

obtain the familiar Jaynes-Cummings Hamiltonian ri{a+a + a^a^). By choosing

other detunings we can realize a number of couplings beyond the JaynesCummings coupling; for example a 'two-phonon' coupling (77^/2) (cr+a^-f- cr_(a') ) for n — n' = —2 (second red sideband). The coupling strength to lowest

order in 77 is given by the matrix elements (Rabi frequencies)

nOny = (i, nl^effl T, n') ~ %-!^l^ [„>(„> _ 1) . . . („^ + i)]i, (5)

TN-87
2488 D. Leibfried et al.

where n> (rz<) is the greater (lesser) of n and n' . The differences are only significant for large n or n' . A 'Rabi-7t-pulse', which transfers a pure |J., n) state to a pure

If, n') state, corresponds to applying the Raman beams for a time r, such that

^n,n'T = n/2.

3. Experimental setup and cooling to the motional ground state In our experiment, a single ^Be"*" ion is stored in a RF Paul trap [9] with a secular frequency along x oiuJx/2vi ^ 11-2 MHz, providing a spread of the ground

state wavefunction of about Axq = (x^) ~ 7nm. The ||) and ||) levels are the

long-lived ^Si/2(-f' = 2, mp = 2) and ^Si/2(-^ = 1; ^F = 1) hyperfine ground states (see figure 1). To prepare the ion in the ground state, it is first 'Doppler pre-cooled' by

two beams of a'^ polarized light detuned by approximately one linewidth

(r/27t= 19-4 MHz) to the red of the ^Si/2{F = 1 and 2)^^Fy2 transitions. This cools the ion to n~ 1, in the |J,) state. To further cool the ion we use narrower Raman transitions in order to be in the resolved sideband limit. One

cooling cycle consists of two steps. First we drive stimulated Raman transitions to the It) state by applying a pair of travelling-wave laser beams detuned from the

^Pi/2 state [10]. These Raman beam wave-vectors point at 45° to the jc-axis with

their wave-vector difference nearly along the jc-axis of the trap [r]x = 0-202(5)], so the raman transitions are highly insensitive to motion in the y or z directions. The

beams are derived from the same laser with an acousto-optic modulator, reducing

the effects of laser frequency jitter. The difference frequency can be tuned near the

ground state hyperfine splitting of a;o/27i: ~ 1-25 GHz. The beams are detuned by

approximately 12 GHz to the red of the ^Pi/2 excited state with approximately

0-5 mW in each beam, so that the Raman transition Rabi frequency Qq^i/Iu is approximately 200 kHz, and the vibrational structure is clearly resolved. For

cooling, the frequency difference is tuned to the red sideband {S = —cUx), so that one vibrational quantum is lost in the transfer to the jf) state. The time during

which we apply the red-sideband interaction is optimized to leave the internal state as close to a pure If) state as possible (Rabi-7t-pulse). In the second step of the

cooling cycle, we apply lasers tuned to the |T) "* ^P3/2 ^^d ^Si/2(-P' = 2, mp = 1) ^ ^Pi/2 that repump the atom to the ||) state. In analogy to the

Mossbauer effect, the recoil of the spontaneous emission process in this repumping

is absorbed by the whole trap structure with high probability, leaving the motional

state of the trapped atom unchanged. Five cycles of this two-step cooling scheme

prepare the ion in the |J., n = 0) state approximately 95% of the time [10]. In the experiments described below we detect the probability of being in one of

the states |t) or |J.). We detect P|, the probability of finding the | J.) -state, by

driving a cycling transition to the ^P3/2 with cr"*" polarized light and detecting the

emitted fluorescence. The ||) is not resonantly coupled to an excited state by the

light-field, so P| is proportional to the number of tries where we see fluorescence

when we repeat the experiment. Since the application of the resonant light field effectively reduces the internal state to either \]) or \l), we can consider the internal

atomic state to be a unit efficiency detector, even if we fail to detect the

fluorescence every time. The analogy with photon detection would be a 100%

efficient detector which is read out only sporadically.

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Experimental preparation and measurement of quantum states of motion 2489

4. Creation of various motional states and measurement of their number

state population

4.1. Fock states A Fock state \n) is a harmonic oscillator energy eigenstate, designated by the

number n of energy quanta. Several techniques for the creation of Fock states of

motion have been proposed, using quantum jumps [4, 11], adiabatic passage [12], or trapping states [13]; here we use an alternate technique. Since the ion is initially

cooled to the |i, 0) Fock state, we create higher-n Fock states by simply applying a sequence of Rabi-Tt-pulses of laser radiation on the blue sideband, red sideband, or

carrier. For example, the If, 2) state is prepared by using blue sideband, red

sideband, and carrier 7t-pulses in succession, so that the ion steps through the

states li, 0), IT, 1), U, 2), and |T, 2) [14]. Once the Fock state is created, the signature of the state can be found by

driving Rabi transitions on the blue sideband. Specifically, the Raman beams were

tuned to the first blue sideband and applied for a time t. The probability of finding

the ID level was then measured by applying o""*" polarized radiation on the ID —> ^P3/2 cycling transition and detecting the fluorescence as described above.

The value of t was stepped, and the data P|(i) was acquired. The rate of the Rabi

flopping, Qn,Ti+\ in equation (5), depends on the value of n of the Fock state occupied. The expected signal is

Pi{i) = i[l + cos (2r3„,„+i0 exp (-7„(0], (6) where 7„ is the decoherence rate between levels \n) and |n+ 1). The naeasured

P|(t) for an initial |D n = 0) Fock state is shown in figure 2 {a) together with a fit to equation (6), yielding f3o,i/(27t) = 94(l)kHz and 70 = 11-9(4) 10^ s^^

We created a series of Fock states |i, «) and recorded P[{t). The measured Rabi

frequency ratios i2„^„+i/floi are plotted in figure l{b), showing very good agree- ment with the theoretical frequencies corrected for the trap's finite Lamb-Dicke

parameter 77 = 0-202. The observed increase of 7„ with n (we experimentally find 7n ~ 7o(" + 1) ' ) is qualitatively consistent with our view that the decoherence is due to technical problems.

4.2. Thermal states When the ion's motion is not in a Fock state, Piit) shows a more complicated

structure. In this case.

Pi{t)-\ 1 +

^

P„ cos (2i2„,„+i t) exp (-7^0

71=0 (7)

where P„ is the probability of finding the atom in the nth motional number state. For example, a thermal distribution is found after Doppler cooling [15]. In this case, the probability of occupying the nth Fock level is P„ = W/{^ + ")"^ ]> where

n is the average vibrational quantum number. The value of n can be controlled by

the detuning during Doppler cooling. An example oi P[{t) data for a thermal state of motion is given in figure 3 . To demonstrate consistency with a thermal state of

motion, the time-domain data are fitted to equation (7) with a thermal population

distribution for P„. The signal scale and n are allowed to vary in the fit. Values for the base Rabi frequency Qq^\ and base decay rate 70 (from which the other rates are scaled using the dependence found in the Fock state data) are obtained from a

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2490 D. Leibfried et al.

(b)

3.0

-. 2.5

T 2.0

1.5

1.0

40 time 60 80 100

(jisec)

Figure 2. (a) Pi{t) for an initial |i, 0) Fock state driven by a JCM-type interaction provided by tuning the stimulated Raman beams to the first blue sideband. The

solid line is a fit to an exponentially decaying sinusoid. 96) The relative Rabi

frequencies i2„„+i/i2o,i plotted against the prepared Fock state number n. The lines represent the predictions of the nonlinear JCM for certain Lamb—Dicke parameters,

showing every good agreement with the known Lamb—Dicke parameter 77 =

0-202(5). Foif 77^1 the ratio of the Rabi frequencies is given by Qn,n+\/

Oo,i = (w + 1)^

thermal

state

20 30

time (iisec)

Figure 3. Piit) for a thermal state. The data (points) are fitted (line) to a superposition

of Fock states with P„ given by a thermal state distribution. The fit allows n to vary, yielding a value of 1-3 ±0-1. The inset shows the decompostion of the data onto

the Fock state components (bars) with a fit (line) to the expected exponential

distribution, yielding 1-5 ±0-1.

separate trace of P{{t) for an initial |i, 0) state. For figure 3, the fit yields

n= 1-3 ±0T. The inset shows the results of an independent analysis (the fre- quency domain analysis). In this case, we directly extract the populations of the

various \n) levels. Since the Fock state parameters Qn,n+i . and 7„ are well

characterized, the time-domain data can be decomposed into Fock-state components. Equation (7) is linear in P„, so we use singular value decomposition [16] to

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Experimental preparation and measurement of quantum states of motion 2491

coherent p

°^

20 30

time (;isec)

Figure 4. Piif) for a coherent state, showing collapse and revival. The data are fitted to

a coherent state distribution, yielding «= 3-1 ±0-1. The inset shows the decomposition of the data onto the expected Fock state components, fitted to a Poissonian distribution, yielding n = 2-9 ± 0-1.

extract the probabilities, shown in the inset of figure 3. The probabilities are fitted to an exponential, yielding n= 1-5±0T. A third measure of n by comparing the

size of the red and blue sidebands [10] yields n = T5 ± 0-2.

4.3. Coherent states A coherent state of motion \a) of the ion corresponds to a minimum uncertainty

wave-packet whose centre oscillates classically in the harmonic well and retains its shape. The probability distribution among Fock states is Poissonian, P„ =

\{n I a)\ = {n" exp {—n))/n\ with w = |q;| . As predicted by the JCM, the internal

state evolution P[{t) will undergo collapses and revivals [17], a purely quantum

effect [2, 17].

Coherent states of motion can be produced from the \n = 0) state by a spatially

uniform classical driving field [18], by a 'moving standing wave' [19] by pairs of

standing waves [20], or by a non-adiabatic shift of the trap centre [21]. We have

used the first two methods. For the classical drive, we apply a sinusoidally varying

potential at the trap oscillation frequency on one of the trap compensation

electrodes [9] for a fixed time (typically 10|is.) For the 'moving standing wave'

creation we use two Raman beams which have a frequency difference of only uj^. Applying these beams couples adjacent oscillator levels within a given internal

state. In the Lamb—Dicke limit this interaction is formally equivalent to applying

the coherent displacement operator to the state of motion. The Raman beams

produce an optical dipole force which is modulated at u^ [19], resonantly driving

the motion of the atom. On resonance, the magnitude of the coherent state grows

linearly with the coupling time. In figure 4 we present an example of Pi{t) after creation of a coherent state of

motion. Similar behaviour has recently been seen in the context of cavity QED

[22]. The time domain data are fitted to equation (7) using a Poissonian distribution and allowing only n to vary. All other parameters (signal amplitude, signal

offset and decoherence rate) are measured from a separate \n = 0) Fock state trace. The inset shows the probabilities of the Fock components, extracted using the

frequency domain analysis described above. The amplitudes display the expected

Poissonian dependence on n. The observed revival for higher n coherent states is

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2492 D. Leibfried et al.

1.0 -

\ 0.8^ \

e 0.6 - ^ 0.4 - \y'i^-W*iV*-<?= 0.2 - \j

n n -

• V•

20

' 1 40

\i^'^''>^!f;^^fifis^''\fts^

60 80

time (lis)

Figure 5. Pi{i) for a coherent state, driven on the second blue sideband. Since the Rabi

frequencies are almost commensurate in this interaction, Pi shows very sharp

collapse and revival features. The solid line is a fit using the exact Rabi frequencies

for finite rj. It yields n — 2-0 ±0-1.

attenuated due to the progressively faster decay rates of the higher-n Fock states, and for states with n > 7 we are unable to see the revival. We can also realize a different interaction by tuning the frequency difference of

the Raman beams to the second blue sideband; 6 = 2uJx- The interaction Hamilton

is then proportional to {rj^ /2){a+{a^) + a-o^) [see equation (3)] and the Rabi

frequencies [see equation (5)] are given by

^«,„+2 ^^Y[(« + !)(" + 2)]' ~^^(n + f). (8)

Since the last relation holds within 6% for all n, the Rabi frequencies in Pi{t) are almost commensurate, so Pj.(f) shows very sharp collapse and revival features,

similar to the two-photon cases discussed by Buck and Sukumar [23] and Knight

[24]. Our experimental result is shown in figure 5. The factor | in the above

approximation leads to an additional 37r phase shift between successive revivals. This inverts the interference feature from revival to revival. In addition there are small deviations from equation (8), because of the approximation made and the

finite Lamb-Dicke parameter rj which alters the Rabi frequencies [see figure 2 (&)].

4.4. Squeezed states A 'vacuum squeezed state' of motion can be created by a parametric drive [21],

by a combination of standing- and traveling-wave laser fields [20], or by a nonadiabatic drop in the trap spring constant [21]. Here we irradiate the \n = 0) ion

with two Raman beams which differ in frequency by 2uj, driving Raman transitions

between the even-n levels within the same hyperfine state. The interaction can also be thought of as a parametric drive induced by an optical dipole force modulated at

2uj [19]. The squeeze parameter /5 (defined as the factor by which the variance of the squeezed quadrature is decreased) grows exponentially with the driving time.

Figure 6 shows Pi{t) for a squeezed state prepared in this way. The data are fitted to a vacuum squeezed state distribution, allowing only /? to vary. The fit of the data

in figure 6 demonstrates consistency with a squeezed state and gives /3 = 40 ih 10

(16 dB below the zero point variance), which corresponds to n ^ 7.1. The probability distribution for a vacuum squeezed state is restricted to the even states, P2„ = A/'(272)!(tanhr)~"/(2"n!) , with /3 — exp (2r). The distribution is very flat; for example, with P — 40, 16% of the population is in states above n — 20. The Rabi frequency differences of these high-n levels are small (see figure 2(b)),

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Experimental preparation and measurement of quantum states of motion 2493

squeezed state

Figure 6. Pi{t) for a squeezed state. The data are fitted to a squeezed state population

distribution, yielding /3«40±10 (16 dB below the zero point variance), which

corresponds to n ~ 7-1.

and with 77 = 0-202 the Rabi frequencies begin to decrease with n after n = 20. The

levels can no longer be well distinguished by frequency to extract the level populations.

In the above cases, we have checked our state creation through the values of the

P„. This information is, of course, incomplete since it does not reveal the

coherences. Measurements completely characterizing the quantum state will be

discussed below.

4.5. A 'Schrodinger cat' state of motion

A 'Schrodinger cat' state can be taken as a superposition of classical-like states. In Schrodinger's original thought experiment [25] he describes how we could in principle transform a superposition inside an atom to a large-scale superposition of

a live and dead cat. In our experiment [26], we construct an analogous state at the

single atom level. A superposition of internal states (|f) and |J,)) is transformed into

a superposition of coherent motional states with different phases. The coherent

states of the superposition are separated in space by mesoscopic distances much

greater than the size of the atom.

This situation is interesting from the point of view of the quantum measure- ment problem associated with 'wavefunction collapse,' historically debated by

Einstein and N. Bohr, among others [27]. One practical approach toward resolving

this controversy is the introduction of quantum decoherence, or the environmentally induced reduction of quantum superpositions to statistical mixtures and

classical behaviour [28]. Decoherence is commonly interpreted as a way of

quantifying the elusive boundary between classical and quantum worlds, and

almost always precludes the existence of macroscopic Schrodinger cat states, except at extremely short time scales [28]. The creation of mesoscopic Schrodinger

cat states may allow controlled studies of quantum decoherence and the quantum/

classical boundary.

In the present work, we create a Schrodinger cat state of the harmonic

oscillator by forming a superposition of two coherent state wavepackets of the

single trapped atom with a sequence of laser pulses. The coherent states are excited

with the use of a pair of Raman laser beams as described above. The key to the

TN-93
2494 D. Leibfried et al.

i^> vy \L/ O \i/ O K>

I 71/2! ^\r I

\itn.

!'•> \i/ \i/ \1/ \J^ Vi^ K>

(a) (b) (c) (d) (e) (f)

Figure 7. Evolution of the idealized position-space atomic wavepacket entangled with

the internal states |J.) and |t) during the creation of a 'Schrodinger cat' state with a = 3 and 4> = n (displacement forces in opposite directions). The wavepackets are snapshots in time, taken when the atom is at extrema of motion in the harmonic trap (represented by the parabolas). The area of the wavepackets corresponds to the probability of finding the atom in the given internal state, (a) The initial wavepacket

corresponds to the quantum ground state of motion following laser cooling, (b) The

wavepacket is split following a |-pulse on the carrier, (c) The If) wavepacket is excited to a coherent state by the force Fj of the displacement beams. Note the force

acts only on the |f) wavepacket, thereby entangling the internal and motional

systems, (d) The H) and the \[) wavepackets are exchanged, following a 7C-pulse on

the carrier, (e) The |J,) wavepacket is excited to a coherent state by the displacement

beam force F2, which in general has a diflferent phase with respect to the force in (c) (F2 = —Fj in the figure). The state shown in (e) is analogous to a 'Schrodinger cat' state. (/) The |t) and the ||) wavepackets are finally combined following a |-pulse on the carrier.

experiment is that the displacement beams are both polarized a'^ , so that they do

not affect the \]) internal state. It is this selection that allows a superposition of internal states to be transformed into a superposition of motional states. Following laser cooling to the \i)\n — 0) state as described above, we create the

Schrodinger cat state by applying several sequential pulses of the following Raman

beams. (1) A 7r/2-pulse on the carrier splits the wavefunction into an equal

superposition of states |i)|0) and |t)|0)- (2) The displacement beams excite the

motion correlated with the If) component to a coherent state \a). (3) A 7r-pulse on

the carrier swaps the internal states of the superposition. (4) The displacement

beams excite the motion correlated with the new If) component to a second

coherent state |aexp (i4>)). (5) A final Tr/2-pulse on the carrier combines the two

coherent states (the evolving state of the system is summarized in figure 7). The

relative phases (<(> and the phases of steps 1, 3, and 5) of the steps above are determined by the phases of the RF difference frequencies of the Raman beams,

which are easily controlled by phase locking the RF sources. The state created after step 4 is a superposition of two independent coherent

states, each correlated with an internal state of the ion (i.e. for (j) = n),

\^) = {\a)n) + \-a)\i))/V2, (9) In this state, the widely separated coherent states replace the classical notions of 'dead' and 'alive' in Schrodinger's original thought experiment. We verify this mesoscopic superposition by recombining the coherent wavepackets in the final step 5. This results in an interference of the two wavepackets as the relative phase

(p of the displacement forces (steps 2 and 4) is varied. The nature of the

interference depends on the phases of steps (1), (3), and (5), and is set here to cause destructive interference of the wavepackets in the |J.) state. We directly measure this interference by detecting the probability Pi{4>) that the ion is in the ID internal state for a given value of (p. the signal for particular choices of the phases in 1, 3 and 5 is

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Experimental preparation and measurement of quantum states of motion 2495

^m

0=0.84(2)

(e)

1.0

0.5

0.0 -

a = 6 (theory)

—1

-n -jt/2 tU2 coherent state phase separation ^

Figure 8. Pi{4>) interference signal with increasing values of \oi\. The data are fitted to equation (10), yielding a = 0-84, 1-20, 1-92 and 2-97. The fit in curve {d) includes a term to account for the loss of contrast.

Pj((/)) = i[l — Cexp [—0:^(1 — cos (/>)] cos {a^ sin (/>)], (10)

where a is the magnitude of the coherent states and C = 1 is the expected visibility

of the fringes in the absence of decoherence. We continuously repeat the experiment—cooling, state preparation, detection—while slowly sweeping the relative coherent state motional phase (p. In figure 8, we display the measured Pi{(f>) for a few different values of the

coherent state amplitude a, which is set by changing the duration of application of

the displacement beams (steps 2 and 4 from above). The unit visibility of the

interference feature near (j) — verifies that we are producing superposition states instead of statistical mixtures, and the feature clearly narrows as a is increased. We

extract the amplitude of the Schrodinger cat state by fitting the interference data to the expected form of the interference fringe. The extracted values of a agree with

an independent calibration of the displacement forces. We measure coherent state amplitudes as high as a ~ 2-97(6), corresponding to an average of « ~ 9 vibrational

quanta in the state of motion. This indicates a maximum spatial separation of

A-axQ — 83(3) nm, which is significantly larger than the single wavepacket width

characterized by xq — 7-1(1) nm as well as a typical atomic dimension (~ 0-lnm).

The individual wavepackets are thus clearly separated in phase space. Of particular interest is the fact that as the separation of the cat state is made

larger, the decay from superposition to statistical mixture (decoherence) becomes

faster [29]. In the experiment, decoherence due to coupling to a thermal reservoir

is expected to result in a loss of visibility in the interference pattern of

C = exp {—a^Xt) where A is the coupling constant and t the coupling time. The

TN-95
2496 D. Leibfried et al.

exponential reduction of coherence with the square of the separation (a^ term)

undelies the reason that bigger 'cats' decay faster. In figure 8 {d), the observed loss of contrast at the largest observed separation may already indicate the onset of

decoherence. Decoherence due to radiative coupling has been observed in cavity

QED [29]. The precise control of quantum wavepackets in this version of

'Schrodinger's cat' provides a very sensitive indicator of quantum decoherence,

whose characterization is of great interest to quantum measurement theory and

applications such as quantum computing [30] and quantum cryptography [31].

5. Complete quantum state measuretnent

The controlled interaction of light and RF electromagnetic fields with the

trapped atom allows us not only to prepare very general states of motion, but also

to determine these quantum mechanical states using novel techniques. Few

experiments have succeeded in determining the density matrices or Wigner

functions of quantum systems. The angular momentum density matrices of the

substate in principal quantum number n = 3 were measured in coUisionally

produced atomic hydrogen [32]; the Wigner function and density matrix of a mode of light were experimentally mapped by optical homodyne tomography [33,

34]; the Wigner function of the vibrational degree of freedom of a diatomic

molecule was reconstructed with a related technique [35]; and, more recently

the Wigner function of an atomic beam passing through a double slit was

reconstructed [36]. Here we present the theory and experimental demonstration

of two distinct schemes that allow us to reconstruct both the density matrix in the

number state basis and the Wigner function of the motional state of a single

trapped atom [37]. For other proposals to measure the motional state of a trapped

atom, see this issue and [38, 39]. As described above, we are able to prepare a variety of non-classical input states [14] which can, for example, exhibit negative

values of the Wigner function. Also, comparing the results of the state determination with the state we intended to produce can give an idea of the accuracy of the

preparation.

Both of our measurenaent techniques rely on our ability to displace the input

state to several different locations in phase space. Specifically, a coherent displacement [14, 18] £/(— q) = iP {a) = exp {a* a — aa^) (—a is used for convenience

below) controlled in phase and amplitude is used in our schemes. We then apply

radiation on the blue sideband to the atom for a time t, which induces a resonant exchange between the motional and internal degrees of freedom (see section 2). For

each a and time t, the population Pj.(f, a) of the |J.) level is then measured by

monitoring the fluorescence produced in driving the resonant dipole cycling

transition (see section 2). For these experiments the internal state at t = is always prepared to be |J,) for the various input states, so the signal averaged over many measurements is

P^{t,a)^Ul+J2 Qkic^) '^os {2QkM^t) exp (-7,0 I. (11)

Without the coherent displacement we would just recover the previously discussed

Pl{t) signal (equation (7)) and would find the populations of the motional

eigenstates only. But since we repeat these measurements for several magnitudes

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Experimental preparation and measurement of quantum states of motion 2497

and phases of the coherent displacement, we are able to extract information about

the off-diagonal elements of the density matrix and can also reconstruct the

Wigner function from the measured displaced populations Qk[oi).

5.1. Reconstruction of the number state density matrix

To reconstruct the density marix p in the number state basis, we use the

relation

Qk{a) = {k\U\a)pU{a)\k) = {a, k\p\a, k), (12)

where |a, k) is a coherently displaced number state [40]. Hence every Qk{o:) is the

population of the displaced number state \a, k) for an ensemble characterized by

the input density matrix p. Rewriting (12) we get

Qkioc) i(0|«*t/t(a)pC7(a)(at)>) — {a\{a - a)'' p{a^ - a*f\a)

exp \—\oi\ )\o.\

k\

{a*)*N"-J _,«-;

n\m\

X (-1)-

\J' [(m+;)!(n+y')!pP«+;',m+;- (13)

To separate the contributions of different matrix-elements Pn,m, we may displace

the state along a circle,

ap = \a\ exp [i{it/N)p\, (14)

where p € {—N, . . . ^ N — \^. The number of angles IN on that circle determines

the maximum number of states n^^ = N — 1 that can be included in the reconstruction. With a full set of populations of the state displaced along 2N points on a

circle we can perform a discrete Fournier transform of equation (14) evaluated at the values ap, and we obtain the matrix equations

Q^^=^ E Qk{ap)exp[-il{n/N)p]

p=-N

=

7:=niax(0,— E Z)

(0

with matrix elements

(Q_ exp(-|Q| )|q| ^ ^

2(n-y-;')+/

j'=0 ;=0 k\

(15)

X (-l)"^-^'

[{l + n)\nV[^

[j\\j'\{l + n-j)\{n-j')\ (16)

for every diagonal pn,n+i of the density matrix. To keep the matrix dimension finite, a cutoff for the maximum n in (15) is introduced, based on the magnitude of the

TN-97
2498 D. Letbfried et al.

input state. For an unknown input state an upper bound on n may be extracted

from the populations Qk{a). If these are neghgible for k higher than a certain ^max

and all displacements a, they are neghgible in the input state as well, and it is convenient to truncate equation (15) at n^ax — ^max- The resulting matrix equation

is overcomplete for some Z, but the diagonals p„,„+/ can still be reconstructed by a general linear least squares method [16].

5.2. Reconstruction of s-parameterized quasiprobability distributions As pointed out by several authors, s-parametrized quasiprobability distributions F[a, s) have a particularly simple representation when expressed in populations of displaced number states Qk{oi) [40, 41, 42, 43].

1 _°°_ n F{a,s)^-Y,[{^ + \)/2rY.^-lfQQ,{a). (17)

For 5 = — 1 the sum breaks down to one term and F{ot, — 1) == Qo{oi)/7C gives the

value of the Q quasi-probability distribution at the complex coordinate a [44]. Also, for 5 = 0, the Wigner function F{a, 0) = W{a) for every point a in the

complex plane can be determined by the single sum

9

°° wia)^-j2i-^rQn{c^y (18)

In our reconstruction, the sum is carried out only to a finite Wmax, as described

above. Since truncation of the sum leads to artifacts in the quasi-probability

distributions [45], we have averaged our experimental data over diflFerent nmax- This smoothes out the artifacts to a high degree. In contrast to the density matrix method described in section 5.1, summing the

displaced probabilities with their weight factors provides a direct method to obtain

the quasi-probability distribution at the point a in phase space, without the need

to measure at other values of a. This also distinguishes the method from preceding

experiments that determined the Wigner function by inversion of integral equations (tomography) [33, 35].

5.3. Experimental results The coherent displacement needed for the reconstruction mapping is provided

by a spatially uniform classical driving field at the trap oscillation frequency. This

field is applied on one of the trap compensation electrodes [9] for a time of about

10)J.s. The RF oscillators that create and displace the state are phase locked to control their relative phase. Different displacements are realized by varying the

amplitude and the phase of the displacement oscillator. For every displacement a, we record Pj.(f, a). Q«(q;) can be found from the measured traces with a singular

value decomposition (see section 4.2). To determine the amplitude \a.\ of each

displacement, the same driving field is applied to the \n = 0) ground state and the

resulting collapse and revival trace is fitted to that of a coherent state (see section

4-3).

The accuracy of the reconstruction is limited by the uncertainty in the applied

displacements, the errors in the determination of the displaced populations, and

decoherence during the measurement. The value of the Wigner function is found

TN-98
Experimental preparation and measurement of quantum states of motion 2499

by a sum with simple error propagation rules. The density matrix is reconstructed

by a Hnear least-squares method) and it is straightforward to calciilate a covariance

matrix [16]. As the size of the input state increases, decoherence and the relative

accuracy of the displacements become more critical, thereby increasing their

uncertainties. In figure 9, we show the reconstruction of both the number state density matrix

[a) and surface and contour plots of the Wigner function {b) of an approximate

\n= \) number state. The plotted surface is the result of fitting a linear interpolation between the actual data points to a O'l grid. The octagonal shape is an artifact of the eight measured phases per radius. The white contour represents W{a) = 0. The negative values around the origin highlight the non-classical character of this

state. The Wigner function W{a) is rotationally symmetric within the experimental errors as confirmed by our measured values. Therefore we averaged sets of

data with the same value of |a|. The averaged points are displayed in figure 9 (c), together with a radial slice through the theoretical Wigner function for a pure

number state (dashed line) and a thermally contaminated state (solid line) assuming the ion is prepared in a thermal distribution with a probability of finding it in ground state of only «90% after cooling and prior to the preparation of the \n = \)

number state. This was independently verified to be the case in our experiment by

comparing the magnitude of red and blue sidebands after Raman-sideband cooling

to the ground state [10]. Again the large negative part of the Wigner fxinction

around the origin highlights the fact that the prepared state is non-classical. In contrast to the number state, the state closest to a classical state of motion in

a harmonic oscillator is a coherent state. As one example, we have excited and

reconstructed a small coherent state with amphtude |/5| « 0-67. The experimental

amplitude and phase of the number state density matrix are depicted in figure 10. The off-diagonal elements are generally slightly smaller for the experiment than we would expect from the theory of a pure coherent state. In part this is due to decoherence during the measurement, so the reconstruction shows a mixed state character rather than a pure coherent state signature. This view is further

supported by the fact that farther off-diagonal elements seem to decrease faster than direct neighbours of the diagonal.

The reconstructed Wigner function of a second coherent state with amplitude

|/3| PS 1.5 is shown in figure 11. The plotted surface is the result of fitting a linear

interpolation between the actual data points to a 0-13 by 0-13 grid. The approximately Gaussian minimiim uncertainty wavepacket is centred around a coherent

amplitude of about 1-5 from the origin. The half-width at half maximum is about

0-6, in accordance with the minimum uncertainty half-width of [ln(2)/2]^ ^ 0-59. To suppress truncation artifacts in the Wigner function summation (18) [45], we

have averaged over Wmax = 5 and n^^s. = 6. We have also created a coherent superposition of |m = 0) and \n = 2) number

states. This state is ideally suited to demonstrate the sensitivity of the reconstruction to coherences, the only non-zero off-diagonal elements should be po2 and piQ,

with a magnitude of |po2| = |p2o| = [pooPii^ ~ 0.5 for a superposition with about

equal probability of being measured in the \n = Q) or \n = 2) state. In the

reconstruction shown in figure 12 the populations poo and p22 are some'what

smaller, due to imperfections in the preparation, but the coherence has the

expected value of |p2o = |po2| ~ [pwPii]^-

For a known density matrix one can also find the Wigner function by

TN-99
2500 D. Leibfried et al.

Figure 9. (a) reconstructed number state density matrix amplitudes p„m for an approximate |7z=l) number state. The coherent reconstruction displacement

amplitude was |a| = 1-15(3). The number of relative phases N = ^ in equation (14), so «max — 3. (b) Surface and contour plots of the Wigner function W(a) of the

\n = 1) number state. The white contour represents W{a) = 0. The negative values around the origin highlight the non-classical character of this state, (c) The Wigner

function of the \n = 1) number state is rotationally symmetric: W{a) = W{\a\). In

this figure we show a radial slice through this function for a pure number state (dashed line) and a thermally contaminated state (soUd line) which assumes the ion

is in the ground state only «90% of the time after cooling. This was independently

verified after sideband cooling [10]. The dots are experimentally determined values

of the Wigner function, averaged for equal \a\.

TN-100
Experimental preparation and measurement of quantum states of motion 2501

Magnitudes Phases

Figure 10. Experimental amplitudes p„„ and phases &{pnm) of the number state density

matrix elements of a |/3| w 0-67 coherent state. The state was displaced by \a.\ ~ 0.92,

for iV = 4 in equation (14).

Figure 11. Surface and contour plots of the reconstructed Wigner function of a coherent state. The approximately Gaussian minimum uncertainty wavepacket is centred around a coherent amplitude of about 1-5 from the origin. The half-width at half maximum is about 0-6, in accordance with the minimum uncertainty half-width

of [ln(2)/2]^ « 0-59.

Figure 12. Reconstructed density matrix amplitudes of an approximate

1/i/2(|k = 0) — i\n = 2)) state. The state was displaced by \a\ = 0-79 for iV = 4 in equation (14). The amplitudes of the coherences indicate that the reconstructed

density matrix is close to that of a pure state.

TN-101
2502 D. Leibfried et al.

0.4

0.2

Figure 13. Comparison of tiie Wigner function of an approximate l/v^2(|w = 0) —

i\n = 2)) state transformed from our experimental density matrix data {a) with its theoretical counterpart (6).

expanding equation (18) in the number state basis,

7!=0 k,l=Q (19)

with the matrix elements given by (/ ^ n) [46]

(Z|q, n) = [n!/Z!]V-" exp {-l/l\a\^)C^"'\\a\^), (20)

where £„ " is a generalized Laguerre polynomial. Using this approach we have

generated a plot of the Wigner function using our density matrix data. The result

is shown in figure 13 together with the theoretical Wigner function for a \/ yj2{\n = 0) — i\n = 2)) state. The differences can be traced to the imperfections

TN-102

II
Experimental preparation and measurement of quantum states of motion 2503

Figure 14. Reconstructed density matrix of an 77«1.3 thermal state. This state was

displaced by |q:| = 0-78, for AT = 4 equation (14). As one would expect for a thermal

state, no coherences are present within the experimental uncertainties and the populations drop exponentially for higher n.

in the preparation also visible in figure 12; the small but non-zero values of |pii|

and |p33| and the respective coherences lead to the differences in the central feature

of the Wigner function.

Finally, we have generated a thermal state by only Doppler-cooling the ion. The reconstruction of the resulting state is depicted in figure 14. As expected,

there are no coherences, and the diagonal, which gives the number state occupation, shows an exponential behaviour within the experimental errors, indicating a mean occupation number n ~ 1-3.

6. Conclusions

The interaction of a trapped atom with classical light fields can lead to experimental situations that allow us to coherently prepare a multitude of quantum

states, both classical-like and non-classical in character. Since the interaction can

be tailored to resemble the Jaynes-Cummings model, the system is suited to realizing many proposals originally introduced in the realm of quantum optics and

cavity quantum electrodynamics. One special application is the preparation of a

state where the internal degree of freedom is entangled with two coherent states, with a separation in phase space much bigger than their spread. This state, bearing

many features of 'Schrodinger's cat', is well suited for the study of decoherence

phenomena on the boundary of quantum and classical mechanics such as the

decoherence of mesoscopic objects. Such studies are especially interesting in our system since we should be able to engineer different couplings and reservoirs by an

appropriate choice of the interaction Hamiltonian [47]. Our level of control in the preparation of the states also allows us to prepare the same state many times to a high accuracy. By extending our techniques to several simultaneously trapped ions, we should be able to controllably prepare and

manipulate their combined state and thus implement simple quantum logic

gates [48, 49]. The techniques described here for characterizing the quantum

state of motion, combined with a Ramsey-type spectroscopy on the internal states, seem to lead to a method for completely measuring the internal and motional states of several simultaneously trapped ions (for an alternative method see [50]). Apart

TN-103
2504 D. Leibfried et al.

from detecting quantum correlations in these states, this might be a useful way to fully characterize simple quantum logic gates [51].

Acknowledgments

This work is supported by the United States National Security Agency, the

Office of Naval Research and the Army Research Office. D.L. gratefully acknowledges a deutsche Forschungsgemeinschaft research grant. D.M.M. is supported by

an N.R.C. postdoctoral fellowship. We thank W. Vogel, M. Collet, P. Knight, J. I. Cirac and P. ZoUer for stimulating discussions. We acknowledge important

contributions by J. Bergquist and helpful comments on the manuscript by

M. Young, B. Esry and B. Jelenkovic.

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. Sanchez-Mondragon, J. J., 1980, Phys. Rev. Lett., 44, 1323.

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[19] Wineland, D. J., Bergquist, J. C, Bollinger, J. J., Itano, W. M., Moore, F. L., GiLLiGAN, J. M., Raizen, M. G., Heinzen, D. J., Weimer, C. S., and Manney, C. H., 1992, Laser Manipidation of Atoms and Ions, Proc. Int. School of Physics 'Enrico Fermi' Course 118, edited by E. Arimondo et al. (Amsterdam: North Holland),

p. 553.

[20] Cirac, J. I., Parkins, A. S., Blatt, R., and Zoller, P., 1993, Phys. Rev. Lett., 70, 556.

[21] Janszky, J., and Yushin, Y. Y., 1986, Opt. Comm., 59, 151; Fan Hong Yi and Z.\idi, H.

R., 1988, Phys. Rev. A, 37, 2985; Heinzen, D. J., and Wineland, D. J., 1'990, Phys. Rev. A, 42, 2977.

[22] Brune, M., Schmidt-K\ler, F., Maali, A., Dreyer, J., H.a.gley, E., Raimond, J. M.,

and Haroche, S., 1996, Phys. Rev. Lett., 76, 1800.

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[24] Knight, P., 1986, Phys. Scr. T, 12, 51. [25] Schrodinger, E., 1935, Naturwissenschaften, 23, 807.

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Experimental preparation and measurement of quantum states of motion 2505

[26] Monroe, C, Meekhof, D. M., King, B. E., and Wineland, D. J., 1996, Science, 111,

1131.

[27] Wheeler, J. A., and Zurek, W. H. (editors), 1983, Quantum Theory and Measurement

(Princeton NJ: Princeton Univ. Press).

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Raimond, J. M., and Haroche, S., 1996, Phys. Rev. Lett., 11, 4887.

[30] Ekert, a., and Josza, R., 1966, Rev. Mod. Phys., 68, 733.

[31] Bennett, C. H., 1996, Phys. Today, 48, 24.

[32] Ashburn, J. R., Cline, R. A., van der Burgt, P. J. M., Westerveldt, W. B., and

Risley, J. S., 1990, Phys. Rev. A, 41, 2407.

[33] Smithey, D. T., Beck, M., Raymer, M. G., and Faridani, A., 1993, Phys. Rev. Lett., 70, 1244.

[34] Breitenbach, G., Moller, T., Pereira, S. P., Poizat, J. Ph., Schiller, S., and

Mlynek, J., 1995, Opt. Soc. B, 12, 2304.

[35] Dunn, T. J., Walmsley, I. A., and Mukamel, S., 1995, Phys. rev. Lett., 74, 884.

[36] KuRTSiEFER, C, and Mlynek, J., 1997, Nature, 386, 150.

[37] Leibfried, D., Meekhof, D. M., King, B. E., Monroe, C. M., Itano, W. M.,

Wineland, D. J., 1996, Phys. Rev. Lett., 11, 4281.

[38] Wallentovvitz, S., and Vogel, W., 1995, Phys. Rev. Lett., 75, 2932; Poyatos, J. F., Walser, R., Cirac, J. I., ZoLLER, P., and Blatt, R., 1996, Phys. Rev. A, 53, R1966.

[39] D'Helon, C, and Milburn, G. J., 1996, Phys. Rev. A, 54, R25; Bardroff, P. J., Leichtle, C, Schrade, G., and Schleich, W. P., 1996, Phys. Rev. Lett., 11, 2198.

[40] Moya-Cessa, H., and Knight, P. L., 1993, Phys. Rev. A, 48, 2479.

[41] RoYER, A., 1985, Phys. Rev. Lett., 55, 2745.

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[43] Banaszek, K., and Wodkiewicz, K., 1996, Phys. Rev. Lett., 76, 4344.

[44] Poyatos, J. F., Walser, R., Cirac, J. I., Zoller, P., and Blatt, R., 1996, Phys. Rev. A,

53, R1966.

[45] CoLLETT, M., and Walls, D., private communication, July 1996; see also Banaszek,

K., and Wodkiewicz, K., this issue, 2441.

[46] Cahill, K. E., Glauber, R. J., 1969, Phys. Rev., Ill, 1857.

[47] Poyatos, J. F., Cirac, J. I., and Zoller, P., 1996, Phys. Rev. Lett., 11, 4728.

[48] CiR.AC, J. I., and Zoller, P., 1995, Phys. Rev. Lett., 74, 4091.

[49] Monroe, C, Meekhof, D. M., King, B. E., Itano, W. M., and Wineland, D. J., 1995,

Phys. Rev. Lett., 75, 4714.

[50] Wallentowitz, S., de Mathos Filho, R., and Vogel, W., Sept. 1996, preprint.

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Fortschr. Phys. 46 (1998) 4-5, 363-390

Experimental Primer

on the Trapped Ion Quantum Computer

D.J. WiNELAND, C. Monroe, W.M. Itano, B.E. King, D. Leibfried,

D.M. Meekhof, C. Myatt, and C. Wood

National Institute of Standards and Technology, Boulder, CO 80303

Abstract

The development of a quantum computer based on a system of trapped atomic ions is described,

following the proposal of Cirac and Zoller. Initial results on a two-bit quanmm logic gate are pres- ented, and select experimental issues in scaling the system to larger numbers of ions and gates are

treated.

PACS numbers: 32.80.Qk, 42.50.Vk, 89.80.-^h work of U.S. Government; not subject to U.S. copyright

I. Introduction

In its simplest form, a quantum computer is a collection of A^ two-level quantum systems

(quantum bits) which can be prepared in an arbitrary entangled quantum state spanning all 2'^ basis states [1, 2]. A quantum computer, unlike its classical counterpart, can thus store and simultaneously process superpositions of numbers. Once a measurement is performed on the quantum computer, tiie superposition collapses to a single number, which in some

cases can jointly depend on all of the numbers previously stored. This gives the potential

for massive parallelism in particular algorithms [3], most notably an algorithm which factorizes numbers efficiently [2, 4]. Apart from applications to this and other algorithms [5], creating multi-particle entangled states is of great interest in its own right, from the stand- point of quantum measurements [6] and, for example, for improved signal-to-noise ratio in spectroscopy [7, 8]. Unfortunately, there are very few physical systems which are amenable to the task of

quantum computation. This is becau.se a quantum computer must (i) interact very weakly

with the environment to preserve coherence of the superpositions, and (ii) interact very

strongly with other quantum bits to facilitate the construction of quantum logic gates necessary for computing. In addition to these seemingly conflicting requirements, the quantum

bits must be able to be controlled and manipulated in a coherent fashion and be read out with high efficiency.

In 1995, CiRAC and Zoller showed that a collection of trapped and cooled atomic ions can satisfy these requirements and form an attractive quantum computer architecture [9]. In

their proposal, each quantum bit is derived from a pair of internal energy levels of an

individual atomic ion. By using laser beams, the quantum bits are coupled to one another

TN-106
364 D. J. WiNELAND et al.: Experimental Primer

by virtue of the quantized collective motion of the ions in the trap, mediated by the Coulomb interaction. A reduced version of their scheme was implemented in an experiment on

a single trapped ion [10].

This paper concentrates on the trapped ion quanmm computer architecture, and

covers some of the experimental details involved in conducting simple logic operations between small numbers of trapped ions. Following a brief introduction to ion

traps and the interaction between internal and motional states of trapped ions in section 11, preliminary experiments involving one ion are reviewed in section EI, and

particular technical problems in the extension of this scheme to A^ > 1 ions are dis- cussed. Section IV considers the problems of addressing individual ions in a string

with lasers and the cross-coupling of the 3N quantized modes of motion, and section

V characterizes some expected sources of decoherence. The topics covered here are by no means exhaustive, but may give an indication of some of the key problems

which may lie ahead in the near future. We have attempted a more complete investi- gation of the problems in Ref. [11]. For more reviews of the trapped ions quantum

computer, see Refs. [12, 13]. Other notable physical systems proposed for quantum

computation not covered here include cavity-QED [14] and bulk spin nuclear magnetic resonance [15].

n. Background

A. Internal states and detection

Ions can be confined for days in an ultra-high-vacuum environment with minimal perturbations to their internal atomic structure, and collisions with background gas can be neglected. Even though the ions interact strongly through their mutual Coulomb interaction, the fact that the ions are localized necessarily means that the time-averaged value of the

electric field they experience is zero; therefore electric field perturbafions are small. Magnetic field perturbations to internal structure are important; however, the coherence time

for superposition states of two internal levels can be very long by operating at fields where the energy separation between levels is at an extremum with respect to field. For

example, a coherence time exceeding 10 minutes between a pair of ^Be"*" ground state hyperfine levels has been observed [16]. It is also possible to employ a ground and

excited (metastable) electronic state of a trapped ion as a quantum bit [9]. This option seems difficult at the present time, primarily because the energy splitting is typically in the optical region, requiring extremely high laser frequency stability to drive coherent

transitions. Figure la shows a reduced energy level diagram of a single '^Be"^ ion. Although

many other ion species would also be suitable for quantum computadon, we will concentrate on Be"^ here for concreteness and to make a connection to the experiments at NIST [10. 17]. We will primarily be interested in two electronic states, the ~S]/2{F = 2. mf — 2) and -Si/2(F = 1 . /7!f = 1

) hyperfine ground states (denoted by

|i). and If), respectively), separated in energy by licoQ. These long-lived "spin" states

will form the basis for a quantum bit. Detection of the spin states can be accomplished using the technique of quantum jumps [18]. By tuning a polarized laser beam to the ID^'P^/o transition near 313 nm (Fig. la), many photons are scat- tered if the atom is in the |J,) spin state (a "cycling" transition), but essentially no

photons are scattered if the atom is in the |]") spin state. If a modest number of

the.se photons are detected, the efficiency of our ability to discriminate between these

two states approaches 100%.

IN- 107
Fortschr. Phys. 46 (1998) 4-5 365

Be

P3/2(3.3)

Detection

beam

IT)

"Si;2(1,1)

'Pi/2(2,2)

^1=313 nm

Raman

beams,

I A

Ico

'^,2 (2,2)

U>

B.

(a) (b)

Fig 1 : (a) Electronic (internal) energy levels (not to scale) of a ''Be'^ ion. The

-S\-2{F = -• '"/•" = 2) and "Si,2(F = 1. mp = 1) hyperfine ground states (denoted by |i) and ID respectively), separated in frequency by cva/ln ^ 1.250 GHz. form the basis of a quantum bit. Detection of the internal state is accomplished by illuminating the ion with a a' -polarized "detection" beam near 313 nm, which drives the cycling

-S| 2(F = 2. /7!f = 2) — -P3/2(f" = 3. /;;/; = 3) tran.sition, and observing the scattered

fluore.scence. The excited P state has radiative linewidth y/lirr^ 19.4 MHz. (b) Energy

levels of a trapped Be* ion. including the motional states of a single mode of harmonic

motion, depicted by ladders of vibrational states .separated in frequency by the mode fre- quency 0)-. Two "Raman" beams, both detuned A :^ ojq, w-. y ( from the excited 'Pyi

state, provide a coherent two-photon coupling between states \n) |J.) and |;7') jt) by setting the difference frequency co/. to match the desired transition frequency. As shown, the Raman

beams are tuned to the first red sideband of the |1) -^ |f ) transition (a>£ = c^o — co-).

Ion traps and motional states

In Fig. 2, we show a schematic diagram of a linear Paul trap [19], consisting of four elec- trode rods. The linear trap is similar to a quadrupole mass filter [20] which is plugged at the ends with static electnc potentials. A potential Vo cos Qrt is applied between diagonally

TN-108
366 D. J. WiNELAKD et al. : Experimental Primer

VoCosQxt

' > « « «'-«

Fig. 2 : The upper part of the fig- ure shows a schematic diagram of

the electrode configuration for a

linear Paul-rf trap (rod spacing

~1 mm). The lower part of the

figure shows an image of a string of '^^Hg ions, illuminated with 194 nm radiation, taken with a

UV-sensitive, photon counting

imaging tube [22]. The spacing

between adjacent ions is approximately 10 ^m. The "gaps" in the

string are occupied by impurity

ions, most likely other isotopes of

Hg"*", which do not fluoresce be- cause the frequencies of their re- sonant transitions do not coincide with those of the 1 94 nm

^Si/T—»'Pin transition of

199Hg-*

1/2

opposite rods, which are fixed in a quadrupolar configuration, as indicated in Fig. 2. To

provide confinement along the axial z direction, static potentials Uq are applied to the end

segments of the rods as indicated. We assume that the rod segments along the z direction are coupled together with capacitors (not shown) so that the rf potential is constant as a function of z. Near the axis of the trap this creates a potential of the form

~ COS Qrt

2

1 +

R'- 1

Iz- X- yd'- (i;

where R is equal to the distance from the axis to the surface of the electrode and d is, z. characteristic axial dimension of the static electrodes. This gives rise to harmonic ponderomotive potentials [21] in the radial (x and _v) directions accompanied by static harmonic

confinement in the axial iz) direction resulting in an effective 3D harmonic confining potential U{r) = 4 mcD'^.x- + \ mcoly- + \mar:z~. with

a>X ~ (JOy (Ji>Z oj-9

(D- =

IqUo

md-

(2)

where cOp = qVo /

[2^ '- QpnR-) describes the ponderomotive portion of the potenfial, and q

and m are the charge and mass of the ion, respectively. In these expressions, it is assumed

that (jOp "C Qj^ 2 condition known as the "pseudopotential approximation" [21].

Figure 2 also shows an image of a "string" of '^^Hg"^ ions which are confined near the z axis of the trap described in Ref. [22]. This was achieved by making a;.^, oOy > a>;, thereby

forcing the ions to the axis of the trap. The spacings between individual ions in this string are governed by a balance of the force along the - direction and the mutual Coulomb

repulsion of the ions. Example parameters are given in the figure caption.

Of the IN normal modes of small oscillation in a linear trap, we are primarily inter- ested in the N modes associated with axial motion. A remarkable feature of the linear ion trap is that the axial modes frequencies are nearly independent of N, offering the

possibility that mode interference might be small, even for large numbers of ions [9]. For

two ions, the axial normal mode frequencies are at (d- and \/3aJT; for three ions they are

TN-109
Foitschr. Phys. 46 (1998) 4-5 367

o),, vS CO- and \/5.4 w,. For N > 3 ions, the 7\^th axial normal mode can be determined

numerically [12, 13, 23]. We will concentrate on the axial center-of-mass (COM) mode,

which is the lowest frequency mode and is also the most resolved in frequency from the

others. The quantum state of axial COM motion at frequency w- can be described by the ladder of vibrational eigenstates \n), of energy ha)r{n + ^) with vibrational index n de- scribing the number of "phonons" contained in the collective harmonic COM motion.

C. Coupling between internal and motional states We describe the coupling between the internal levels of a particular ion in the string with

the COM axial mode of collective motion when a classical radiation field is applied to that ion. If the internal levels |J.)- and If) of the y'th ion in a string are coupled by a dipole

moment operator // (other couphngs can be shown to behave analogously), then exposing

this ion to a traveling-wave electric field E{r) — Eq cos {k -r — coit + cp) with frequency

Ci)L, phase go, and wavevector k, results in the interaction Hamiltonian

Jf; = -fij E{r) = nQ'{S^+ + SJ_) [e'^^^-'^L^+f) +

^-Hkz-o^L'+cp)^

. (3)

In this expression, ^ = — (||«. |X) Eq/AH is the resonant Rabi frequency connecting \l)j to If)- in the absence of confinement, 5^ (5'_) is the internal level raising (lowering) operator or the ;th ion changing \i)j to \])j and vice-versa, z = Zoziae'''^--' + a^e'""'-') is the axial COM coordinate operator of the confined motion with associated harmonic raising (lowering) operator a' (a) and zero-point spread zo = [Ti/lNma).)^'' , and Nm is the total mass

of the ion collection. If the applied radiation frequency is tuned to co^ — ojq -{- (n' — n) co-, thereby coupling the states \n) |J,). and \n') |t)y, =^^/ is transformed to

y^-i = nO' {Si e"^^°^"' '+'<^ + S_ £-"/("+''' )-'?'

)

(4)

in a frame rotating at w^, where terms oscillating faster than

^

[Q' >C (jo-, ojq) have been

neglected. Here, rj = {k z) Zq is the Lamb-Dicke parameter, which controls the amount of

coupling between internal and COM motional states. When the coupling of Eq. (4) is applied between the 7th ion and the COM mode of

motion, the system evolves between the two quantum states \n) ||) and \n') |T)y with Rabi

frequency [24. 25]

- 0/ 1 /,/! e"^("+"^) |„) I = ^V/'"'-"! (?-"'/-

\ —, L^;;'-"\ (77-)

,

(5) • y /7> ! -

where n^{n^) is the greater (lesser) of n and n' and L"(x) is a generalized Laguerre polynomial. If the Lamb-Dicke criterion is satisfied, where the amplitude of the ion's motion in the direction of the radiation is much less than A/2jr (or ti^^-rj -Cl), we can evaluate ^,/ „ to lowest' order in rj to obtain

Qi, ^Qi ^^'"'"" .I'llL . (6) \n'-ny ^' ' ' TN-110
368 D. J. WiNELAND et al. : Experimental Primer

We will be primarily interested in three types of transitions — the carrier (n' = n), the first red sideband [n' = n — 1), and the first blue sideband («' = n + 1) whose Rabi frequencies,

in the Lamb-Dicke limit, are given from Eq. (6) by Q' , rjn^^^Q', and 7j{n+ 1)

'~ Q^ respectively. In practice, driving direct transitions between \n) |J.). and |n') ||)- withrf or microwave

radiation is not be feasible, as the sideband operation transition rates (proportional to

7] = 2jtzo/A) would be extremely slow due to the long wavelength of the radiation. Alternatively, optical fields can be used to drive two-photon stimulated Raman transitions between

\n) \i)j and \n') \J)j [17, 26]. As depicted in Fig. lb, two laser beams detuned by A from an

excited state of radiative width y are applied to the jth ion with their difference frequency

matched to the desired transition frequency. For sufficient detuning \A\ » y, the excited

state may be adiabatically eliminated, and the above couplings apply, with Q' replaced by

g'^g'2/A, where g'^ and g'.^ are the individual Rabi frequencies of the two beams when reso- nantiy coupled to the excited level. In addition, ojl{<p) is replaced by the difference fre- quency (phase) of the beams; and k is replaced by the difference in wavevectors of the two

Raman beams 6k = k^ —ki. Since the relevant frequency depends only on the microwave

difference between the two laser frequencies, both beams can be generated with a single

laser source and a modulator, thereby relaxing the constraints of laser frequency stabilization. The use of stimulated Raman transitions thus combines the advantages of the strong

coupHngs with the frequency stability of microwave sources [26, 27].

D. Laser cooling to the motional ground state

As a starting point for trapped ion quantum computing, the ions must be initialized in known pure states. Using standard optical pumping techniques, we can prepare the ions in the 1^), internal state. Laser cooling in the resolved sideband limit [25] can, for single ions,

generate the |/7 = 0) motional state with reasonable efficiency [17, 28]. This type of laser cooling is usually preceded by a stage of "Doppler" laser cooling [29] which typically

cools the ion to («) > 1, or an equivalent temperature of about 1 mK.

Resolved sideband laser cooling for a single, harmonically-bound atom can be explained

as follows: For simplicity, we assume the atom is confined by a 1-D harmonic well of

vibration frequency on-. We use an optical transition whose radiative linewidth y is rela- tively narrow, y -C W; (Doppler laser coohng applies when y > w-). If a laser beam (fre- quency ojl) is incident along the direction of the atomic motion, the bound atom's absorption spectrum is composed of a "carrier" at frequency wq and resolved frequencymodulation sidebands that are spaced by w-. These sidebands in the spectrum are generated

from the Doppler effect (like vibrational substructure in a molecular optical spectrum). Laser cooling can occur if the laser is tuned to a lower (red) sideband, for example, at coi = a){) — W-. In this case, photons of energy Ti{(joq — w-) are absorbed, and spontaneously

emitted photons_^ of average energy Tiwo - R return the atom to its initial internal state, where R = {Tik)~ /2m = Tiojr is the photon recoil energy of the atom. Overall, for each

scattenng event, this reduces the atom's kinetic energy by Vioj- if oj-^ » a>/?, a condition

which is satisfied for ions in strong ion traps. Since cor/co- = ?/- where ?/ is the LambDicke parameter, this simple form of sideband cooling requires that the Lamb-Dicke parameter be small. For example, in '^Be"^, if the recoil corresponds to spontaneous emission from

the 313 nm "Pi 72 —* "Si/t transition (typically used for laser cooling), (Of{/2jr c^ 230 kHz.

This is to be compared to trap oscillation frequencies in some laser-cooling experiments of

around 10 MHz [17]. Cooling proceeds until the atom's mean vibrational quantum number

in the harmonic well is given by {n)^-^„ ~ [y/2aj-)~ <C 1 [29]. As discussed above, it is convenient to use two-photon stimulated Raman transitions for sideband cooling [17, 30], but

the basic idea for, and limits to, cooling are essentially the same as for single-photon transitions.

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Fortschr. Phys. 46 (1998) 4-5 369

Although laser cooling to the \n = 0) state has been achieved with single ions [17, 28]; a prerequisite to future work is to laser cool a collection of ions (or, at least one mode of the

collection) to the zero-point state. Cooling of any of the 3N modes of motion of a collec- tion of ions should, in principle, work the same as cooling of a single ion. To cool a particular mode, we tune the cooling radiation to its first lower sideband. If we want to cool all modes, sideband cooling must be cycled through all 3A^ modes more than once, or applied to all 3N modes at once, since recoil will heat all modes. For the COM mode, the cooling is essentially the same as coohng a single particle of mass Nm; however, the recoil energy upon re-emission is distributed over all 3A^ modes. Other methods to prepare atoms

in the \n = 0) state are discussed in Refs. [31]. In Ref [32], it is shown that it is not necessary to satisfy the condition oj/? -C a»; (?/ <C 1) to achieve cooling to n = 0.

m. Quantum Logic with Trapped Ions

Several authors have shown that an arbitrary unitary operation (therefore any quantum computation) on a collection of quantum bits can be broken into a series of fundamental single

bit and dual bit quantum logic gates [33]. This is similar to classical computing, where

certain families of logic gates are universal (for instance, the single bit NOT and two bit AND gates). One such family of universal quantum logic gates consists of the single bit rotation gate and the two-bit controlled-NOT gate [34]. For brevity, we concentrate on these

two gates and how they can be implemented in a system of trapped ions. The single bit rotation gate [operator R{d, q))] simply changes the state of a single quantum bit and is characterized by the following transformation:

\i)^cos{e/2)\i)-ie'Um{d/2)\])

IT) -cos (0/2) IT) -/e-'^ sin (0/2) li),

(7)

where and 9 are parameters of the gate. This transformation is commonplace in atomic

physics and nuclear magnetic resonance, and has been widely applied to two level systems.

In the context of trapped ions, the single bit rotation gate is accomplished by tuning to the

carrier transition {coi = cuo) and applying radiation for a time T such that 6 = IQ't

(Eq. (6)). The parameter 6 describes the "rotation" between the two spin states {0 = 71 is called a ".t" pulse, etc.), and the parameter cp describes the phase of the rotation. Figure 3 depicts the observed Rabi flopping between |J.), and |T), states in a single trapped ^Be"^ ion

Pi(t)

50 100 150

t(^is)

Fig. 3: Experimental plot of the probability P[{t) of finding a single Be"^ ion in the |X)

.state after first preparing it in the jj.) state and applying the carrier coupling (Eq. (7)) for a time /, with 6/2 — Q't. P\{t) does not follow a perfect sinusoid due to decoherence, de- scribed in section IV. Each point represents an average of 4000 observations. TN-112
370 D.J. WnvfELAND et al. : Experimental Primer

according to the above transformation [35]. The ion is first prepared in the |J,), state, then

the carrier transition is apphed for a time t, and the state (|X), or |t)) is detected as described above and in Fig. 1 a. A more nontrivial quantum logic gate is the two-bit controlled-NOT (CN) gate [33, 34],

reminiscent of the classical XOR gate. This gate flips the spin of ion k, the "target bit"

(iJ-)jt ^^ \'\)k) ^ ^^'^ °^y ij^ ion J, the "control bit," is in state If). The transformation of the two quantum bits j and k are as follows:

li>,lT),-|i);|T),

IT), li),- IT); IT),

IT), IT),- IT); li),-

ClRAC and Zoller [9] showed how the above transformation could be accomplished

between two of a collection of trapped ions by utilizing a motional mode (i.e., axial COM

mode) as a "data bus" through which the quantum bits are shuttled. As described above,

the COM mode is assumed to be initially cooled to the \n = 0) ground state. The scheme is outlined as follows:'

(1) Map the state of ion 7 (spanning the states \i)j. |T)^) onto the first two axial COM

motional states (spanning the states \n = 0) , \n = \)) shared amongst all ions.

(2) Flip the spin state of ion k {\\)f. ^ |f)^.) if and only if there is a phonon in the COM

mode.

(3) Reverse step (1): map the state of the motion back onto ion 7. The central ingredient here is step (2), which is itself a "reduced" CN gate, with the first two motional states actins as the control bit and ion k actins; as the target bit

(9)

This last transformation has been realized on a single trapped ion [10]. In that experiment,

performed on a trapped ^Be~ ion, the control bit was one of the three modes of the ion's motion. The reduced CN operation between these states (step (2) above) was realized by

applying three laser pulses in succession:

(a) A 7z/2 pulse {Q^t — Ji/A in Eq. (6)) is applied on the carrier transition. For a certain choice of initial phase, this corresponds to the rotation operator R{d = Jt/2, cp). (b) A Iji pulse is applied on the first blue sideband transition between levels |T) and an

auxiliary level jaux) in the ion (the \F = 2. Mp — 0) level in ^Be"^; see Fig. 4). This

operation provides the "conditional dynamics" for the overall CN operation. It changes

the sign of the |1) |t) component of the wavefunction but leaves the sign of the |0) |T)

component of the wavefunction unchanged; that is, the sign change is conditioned on whether or not the ion is in the |0) or |1) motional state.

(c) A 71 /I pulse is applied to the spin carrier transition with a 180" phase shift relative to step (a). This corresponds to the operator R{6 = 7r/2, cp + Jt).

Steps (a) and (c) can be regarded as two resonant pulses of opposite phase in the Ramsey

separated-field method of spectroscopy [36]. If step (b) is active (thereby changing the sign

TN-113

|0) ID, - |0) li).

|0) IT), - |0) IT), |i)li),- |1)IT),

10 IT),- |i)li).
Il>

371

T)

laux)

(a) (b) (c)

Fig. 4: Energy levels of a single trapped ^Be"*" ion, including the internal hyperiine levels

IX) and It) and an auxiliary level jaux) [the "Si/2(F = 2, nif = 0) state], each dressed by

the lowest two motional quantum harmonic oscillator states \n) — \0) and |1). The con- trolled-NOT quantum logic gate results in a spin flip (|J.) =; |t) if and only if \n) = |1).

This transformation is realized with a sequence of three pulses of laser hght which couple

the states indicated by the arrows: (a) A jr/2 pulse couples states \n) jj.) to \n) |t). (b) A 2jr

pulse couples state |1) If) to |0) |aux), resulting in a sign change of any component in the

|1) It) state, (c) A —Jr/2 pulse couples states |n) ||) to \n) \]) (same as step (a) with a t: phase shift). If |n) = |0), then step (b) is inactive since it only affects to the |1) |t) state, and

the two n/2 pulses cancel, leaving the initial state unaffected. If \n) — jl), then the sign

change in step (b) causes the two 7r/2 pulses to add, resulting in a net spin flip (|J.) ^ |t)).

The result is the transformation of Eq. (9).

of the |1) It) component of the wavefunction), then a spin flip is induced by the Ramsey

fields. If step (b) is inactive, step (c) reverses the effect of step (a). Instead of the three pulses (a—c above), a simpler CN gate scheme between an ion's

internal and motional states can be achieved with a single laser pulse, while eliminating the requirement of the auxihary internal electronic level [37]. By applying a single pulse tuned

to the carrier transition, from Eq. (5), the states \n) |J.) and \n) |t) are coupled with Rabi

frequency

a,.„ = -Q \{n\ e"J^"^"'^ \n)\ = Q e'^" I'- ^„{rr) , (10)

where ^^{r]-) = L^^{r]-) is a Laguerre polynomial. Speciahzing to the |/2) = |0) and

\n) — \\) vibrational levels relevant to quantum logic, we have

;ii)

The CN gate can be achieved in a single pulse by setting rj so that ^1.1 /-^o.o = {2.k + \)/lm, with k and m positive integers satisfying m> k>Q. Setting

•^i.i/'^o.o = 2m/{2k+ 1) will also work, with the roles of the |0) and |1) motional states switched in Eq. (10). By driving the carrier transition for a duration r such that I^i.iT = (A: + j)jT, or a "jr-pulse'" (mod 27t) on the |n) = |1) component, this forces

Qq.qt = nut. Thus the states ji) |1) ^-^ |T) il) ^^ swapped, while the states ji) |0) and

It) |0) remain unaffected. This transformation is equivalent to the reduced CN of Eq. (9), apart from phase factors which can be eliminated by the appropriate settings of the phase

of subsequent logic operations.

The mapping steps (steps (1) and (3) above) can be realized by applying a jr-pulse on

the red sideband of ion j. This accomplishes the mapping |0) (a \i)j + ^ \])j) —> (a |0)

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372 D. J. WiNELAND et al.: Experimental Primer

+ ^|1))|X):. Analogous mapping of internal state superpositions to motional state superpositions of a trapped ion has also been reported in the generation of a "Schrodinger

cat" state of motion [38] and the tomographic measurement of nonclassical states of

motion [39]. To complete this section, we give an example of how a "maximally-entangled" state of N

ions might be prepared with the use of the quanmm logic gates described in this section

[9]. We desire to create the state

^ ll)lli)2li)3---|i)N + ^'nT).|T)2lT)3---lT)A. .,.. g>ME = y= I J ^)

V2

(or equivalently, a coherent superposition of the numbers and 2^—1 in binary where we

make the identification |J.) = and |t) = 1). Starting in an initial state ^init where all spins

are in the |J.) state, it is easy to show that one way to prepare Eq. (12) is to operate the following N gates on the initial state:

^ME - [CA^i.A^] [CNi.M-i] [CNi,N-2] . .

. [CNiJ [RiiJt/2, cp)] ^,ni., (13)

where CNij denotes the controlled-NOT operator with ion / as the control bit and ion j as the target bit, and the rotation operator is applied to the first ion. The state 0m£ can be

viewed as the /V-particle generalization of the entangled pair states envisioned by Einstein,

PoDOLSKY, and Rosen [40], and may fmd uses in improved atomic spectroscopy and fre- quency standards [7, 8].

IV, Packing Ions into a Trap

The reduced CN gate demonstrated in [10] involved only a single ion, and is therefore not

useful for computation. Future experimental work will concentrate on scaling the system up

by packing more ions into the trap and improving the gate fidelity, thereby allowing more

gates to be coherently performed. In this section, we survey selected experimental problems

which may arise in the scaleup. For a more extensive coverage of the scaling problem, see

Ref. [111.

A. Individual ion addressing

One major concern in extending quantum logic to larger numbers of ions is the requirement

that ions be individually addressed with laser beams for logic operations. This may be

difficult, because the high vibrational frequencies desired for efficient laser cooling and

suppression of decoherence also results in closely spaced ions. It can be shown that the minimum .separation of adjacent ions in a linear trap between the center ions scales approximately as Smm —2sN~^^-^^, with s= [q- /AjZEQinar:)^^^ , where A^ is the number of ions

[12, 13]. For '^Be~ ions with an axial COM frequency of a)-/l7z= 1 MHz, this separation

is about 10 \m\ for 2 ions, and 4 \m\ for 10 ions. The most straightforward method for individual optical addressing is to tightly focus

laser beams on the selected ion [9]. The transverse intensity distribution of a Gaussian

optical beam of power P is

IP f 2}-\

/(/-)= —^exp -— , (14) JTHn V K7'0

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Fortschr. Phys. 46 (1998) 4-5 373

where wq :^ ?l{jt NA) is the beam waist, a the radiation wavelength, and NA = tan is the numerical aperture of the beam with cone half-angle 6 (the formula for wq in the paraxial ray approximation is vahd only for NA < 1) [41]. For large numerical apertures

(NA ~ 0.5), it appears that laser beams can thus be focussed down to a spot on the order of

a wavelength, but this is difficult to realize in the laboratory. If we can reahze wq = 5 p-m

in a Gaussian beam, then at a distance 10 jam from the center of the beam this would

imply a relative intensity of about 3 • 10"'^ or a electric field amphtude (proportional to Rabi frequency) of 1.8% relative to the center of the beam. If wq = 2 [am could be obtained, the intensity (electric field) would be down by a factor of 1.3 • 10"'"^ (1.1 • 10~^).

These results are likely too optimistic, since imperfections in the surfaces of the intervening vacuum port window, multiple reflections from these windows, and diffraction will typically distribute laser intensity outside of the theoretical waist of the beam. The degree to which this occurs depends on the details of window surfaces, etc. and must be resolved

experimentally.

Tightly focussed Gaussian beams possess high transverse intensity gradients, resulting in the potential for significant intensity fluctuations at the selected ion if the relative position

of the beam with respect to the ion is not stable on the time scale of the computation. An

alternative to using tightly focussed Gaussian laser beams is to first feed the (expanded)

laser beam through a sharply defined aperture (slit or aperture), and use a lens to image the aperture at the position of the ions. With this technique, the beam intensity can be distribu- ted more smoothly around the selected ion and have very steep intensity edges (on the order of the original aperture sharpness) away from the ion, thus suppressing beam vibration problems and confining the radiation to a single ion. This technique has been used to make relatively "hard" walls for an optical dipole trap [42]. For this technique to work

well, the imaging lens must collect a large fraction of the light transmitted through the aperture or else diffraction effects will result in light intensity outside the image of the

aperture. To address individual ions, we require very small aperture images, which gives

rise to a design tradeoff. If a one-to-one relay lens is used to image a small object aperture,

effects of diffraction are enhanced. If a demagnifying lens is used to reduce a large object

aperture, then the aperture must be placed a large distance from the lens, requiring a relatively large lens. For two ions, imaging a sharp edge such as a razor blade at the space

between the ions may be sufficient. We might also consider having every other ion in a

string be a "garbage" ion which is not used in the computation, thereby increasing the

spacing between qubit ions by a factor of two (or more, if more garbage ions are used

between each qubit ion). This has the disadvantage that total number of ions (and spectator

modes) increases, aggravating the problems associated with large quantum registers. If suffi- ciently good addressing on one ion in a string can be accomplished, it may be simpler to

.shift the ions, rather than the laser beams, in order to address different ions. This could be

accomplished by applying different static potentials Uq and Uq to the end segments of the rods in Fig. 2. However, changes in Uo and ^/q would have to be coordinated to keep the COM axial frequency constant or else additional phase shifts would be introduced. Stimulated-Raman transitions have the advantage that the effective wavevector dk =^ k] — ki can

be made parallel to the axis of the trap even though each beam is at an angle with respect

to the trap axis. This would allow selection of a particular ion, while eliminating coupling

to transverse modes.

Another method of optically addressing individual ions is to cause a destructive optical

interference at the position of a specific ion, with a net coupling at the other ion(s). For

instance, if ion j is positioned at the node of a resonant standing wave laser field, the coupling between states \n) |J.)- and \n') If)- is proportional to (n'| sin [rjj{a + a'')] \n). In this case, the coupling of the stanaing wave to ion / vanishes when the laser frequency is tuned

to an even order sideband such as the carrier (n' = n). If, instead, the ion is posinoned at an antinode, the coupling is proportional to {n' \ cos [rj -{a + a^)]\n)\ thus, the coupling

TN-116
374 D. J. WiNELAND et al.: Experimental Primer

vanishes when the laser frequency is tuned to an odd order sideband, such as the first blue

•or red sideband {n' =n± 1). By choosing the angles of focussed laser beams relative to the trap axis appropriately, it should be possible to position an antinode (node) at ion j

while approximately positioning nodes (antinodes) at the ions adjacent to ion j (for equally

spaced ions). In the case of two-photon stimulated Raman transitions, we desire to place

ion j at a common node or antinode of two standing waves. Although this interference

technique should allow individual access to each of two trapped ions, it appears technically

difficult to extend this technique to more than three ions. Next, we consider the apphcation of external field gradients which shift the internal energy levels of ions depending on their position. For a magnetic field gradient to give this selectivity, we require the Zeeman spUtting between adjacent ions to be much larger than

the Rabi frequency, or Afi{d\B\/dz) s/h ':^ Q, where Au is the difference in (jj.-B)/\B\

between the two levels of interest, and s is the ion separation along the z direction. For

AfJ. = fig, i' = 10 jam, and a Rabi frequency of Q/lTt^lMHz, this requires d\B\/dz

::§> 0.1 T/cm. Field gradients of this magnitude can be achieved; however, they might intro- duce large and uncontrollable phase shifts for the other ions in a quantum register.

The laser beam itself can provide ion selectivity by employing the transverse gradient in the optical field intensity. For instance, if we desire to perform a 0-pulse on ion j without

affecting neighboring ion k, the intensity profile of the laser beam can be set so that the

ratio of field strengths (intensities for the case of two-photon stimulated-Raman transitions)

at ion j vs. ion k is d/ljtm, where m is an integer. Now if the pulse duration is set so that ion j is rotated by 9, ion k receives a rotation of Ircm. and hence returns back to its initial state (with an extra phase factor of ( — 1)'"). For the case of two-photon stimulated-Raman transitions, the laser beam can provide ion

frequency selectivity by employing the Stark shift and the transverse gradient of the optical

field. Here, for example, we could assume that the two counterpropagating Raman beams

of equal intensities and spatial profiles are offset so that beam 1 is centered on ion j, and

beam 2 is centered on adjacent ion k as depicted in Fig. 5. Let s be the fraction of peak

intensity seen by the offset ions (that is, the intensity of beam 2 at ion 7 and beam 1 at ion

k). Assume that when either beam is centered on either ion, the single photon resonant Rabi frequencies are equivalent: ^1 — g2 — g- When the beams are offset, the two-photon

Rabi frequency at each ion is Q = e'/-(g-/z!;?), where g'/A^ is the Rabi frequency ex- pected if both beams were centered on a given ion. The Stark shifts of the two ions are in opposite directions: dj = +do, 6;. = —do, where (3o = ^(1 — e)/£^l~. If we make (5o :» ^

(£ <C 1), then by appropriately tuning the difference frequency of the laser beams, we

can selectively drive transitions on either ion j ox k. Alternatively, if we desire to per- form a 0-pulse to ion j without affecting ion k in an "unrepairable" way, e can be

tuned to a particular value which results in ion k retuming to within a phase factor to

its initial state. For square pulses in time, we require 6/2{l + 6'^ /Q^) '~ = m7C, or

£- — [\ + {Imn/Oy] £ -K 1 = 0, where m is an integer. For m = 1 and ^ = jr (a :T-pulse on

ion j), this occurs for e = 0.208. Generalizing this to more than two ions becomes difficult

if the laser beams also overlap other qubit ions. This scheme places an additional premium

on laser power stability, since the light shifts are bigger than the Rabi frequencies by l/s^l^

for £ <^ 1. In addition, in both of the above schemes, employing the laser beams to differ- entially affect neighboring ions, one major drawback is that the positions and profiles of the laser beams must be accurately controlled.

Finally, we consider a method of addressing which utilizes rf micromotion. In a conventional Paul trap, generaUzed to the case of asymmetric electrodes (0 oc Vqcos (iSj-f)

x(£x-^ + (1 — e)^ — Z"), < £ < 1) [43], the rf fields vanish only at a single point in space. When multiple ions are crystalized in such a trap, each ion experiences a different rf

field, leading to different amounts of micromotion. In general, this effect causes each ion to have a unique Rabi frequency when a laser is applied, allowing the possibility of differenTN-117
Fortschr. Phys. 46 (1998) 4-5 375

Fig. 5: Differential AC Stark shift- ing of neighboring ions. Equal in- tensity counter propagating beams 1 and 2 are centered on ions j and k, respectively. A fraction e of the peak intensity /q of each beam is applied to the other ion. This results in a differential AC Stark shift of

ions j and k, allowing the possibility of individually accessing the ions by

tuning the frequency of the laser beams.

tial addressing of the ions. If we assume that each ion sees the same intensity laser field, the Rabi frequency of ion j is proportional to Jo{k ^j) [11, 44], where Jq{X) is the zeroth

Bessel function of argument X, k is the effective laser wavevector, and ^j is the amplitude

of micromotion associated with the y'th ion. We assume that ion ;" experiences a static electric field Esj (due to the Coulomb field of the other ions in addition to background static fields) which prevents the ion from occupying the trap center defined by the rf fields. For

simplicity, we assume that Esj = E^jX is along the x axis of the trap, which should be the case if the ions crystalize along the x-axis (£ < 0.5). To estimate t,, we solve the classical equations of motion in the trap for Uq = and find = ^j x = v 2 gEsj/mco^Qr^ where cOjc is the X-axis COM secular harmonic frequency. Thus, the Rabi frequency of theyth ion is

QJ QqJq

V2k,qEj

:i5)

where ^^ is the Rabi frequency in absence of micromotion. (This reduction of the Rabi

frequency due to ion motion is treated in section IV.B.l. under a different context.) Thus by

controlhng the electric field Ej at the jth ion, some degree of differential addressing of the ions is possible. For example, for three ions held along the x-axis with the middle ion

placed exactly at the rf null position, we find that the ratio of Rabi frequencies of the outer ions to the middle ion is J()(/5), where fi — 1 .52kx{q-aj^- /4jt8om) jQj. An interest- ing case for individual addressing occurs when /5 = 2.405, in which case 7o(/^) = 0' ^"d

the laser interaction with the outer two ions is effectively shut off. For three ^Be"^ ions

held in a trap with a rf drive frequency of Qj ^ 240 MHz, we find that the Rabi frequency of

the outer two ions vanishes at a COM secular frequency of a)j,/2jT ^ 6.1 MHz. Here we

assume that the Rabi frequency describes a stimulated Raman coupling between hyperfine

ground states with k^- = \/2{2jt//i) and A ^ 313 nm, relevant to recent experiments [10]. Many of the above individual addressing schemes are improved greatly when dealing

with only two ions instead of a string of many. This leads us to seriously consider systems

where quantum logic operations are performed on accumulators consisting of only two

ions, with the other ions located somewhere else. For example, pairs of ions may be held in different regions of the same trap structure [11], or quantum information may be transferred from one register of ions to another by optical means [45].

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376 D. J. WiNELA>JD et al.: Experimental Primer

B. Multimode interference

Each additional ion in a quantum register adds three more motional modes. The simplest

form of the Cirac/Zoller scheme ideally uses just a single mode as the bus. In this section, a few potential problems associated with the 3N — 1 spectator modes are considered. First,

we must generalize the interaction with electromagnetic fields discussed in section n.B to consider motion in all 3A^ modes of motion for A^ trapped ions. Here, as was assumed by

CiRAC and Zoller [9], we consider that, on any given operation, the laser beam(s) interacts with only the yth ion; however, that ion will, in general, have components of motion from

all modes. In this case Eq. (3) for theyth ion becomes

H^ = hQJ{S{ + Sl) [e'^" ^J-'^'+Vj) + h.c]

(16)

We write the position operator of the yth ion (which represents the deviation from its equilibrium position) as

Xj = UjX + UN^jy + U2N+jZ, ; e {1, 2, ,... A^}, (17)

where the Up are related to the 3A^ normal mode coordinates qk (fc G {1, 2, . . . 3N} by the following relations [46]

". = E ^k^k , qk=T. ^". , ^k = qkM e-""'-' + al e'"'^') . (18)

k=\ p=\

In this expression, qi^ is the position operator and a^ and a^ are the associated lowering and

raising operators for for the ^th normal mode, and the matrix Z)^ is the transformation

matrix between physical coordinates of the individual ions and normal coordinates of the

string. Following the procedure of section II. C, we find

//j = nO' (sL exp / J2 V'ki^k e"'""' + 4 e'^^O - K^^ - <Pj) + h.c.

j (19)

where we have kept the time dependence to allow consideration of all motional modes with

different frequencies, and 6 = oji — otQ is the detuning of the applied radiation frequency

(or difference frequency for the Raman coupling). The generalized Lamb-Dicke parameters

are t][ = [k Hyj. + k yD^^'^-' + k iDf^^-^) q^o, but for the linear trap case, motion will be

separable in the x, y, and - directions and r/^ will consist of one of these terms. We are

typically interested in coupling the internal states of a given ion/ to a selected mode of collective motion k. In this case, the Rabi frequency coupling the states jn/.) |J.) to |n[.) |T)j is

3A' , . ^'n'n, ^ ^

l.({".#^}^ «J n e"^''"'^"'' l{'V = J, «A.)|

(20)

where |{;j^^^}) denotes the state of motion of the 3A'' - 1 spectator modes excluding

mode k.

1. Effects of motion in spectator modes on logic gates (Debye-Waller factors)

From the last equation, the Rabi frequency of a particular operation will in general depend

on the motional state of the spectator modes. For instance, the conventional controlled-not

gate employs two carrier pulses (steps (a) and (c) in Sec. ITT) which ideally do not depend

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Fortschr. Phys. 46 (1998) 4-5 377

on the state of motion; this requires the Lamb-Dicke parameter rj to be small (see Eq. (10)). In the Raman configuration, rj is proportional to the difference in two wavevectors and can be made negUgible by using co-propagating beams {6k ^0). On the other hand, with sin- ale-photon optical transitions, the Rabi frequencies depend on the motion of all modes

which have a component of motion along the direction of k. We can take advantage of the motional dependence of the carrier to construct a logic gate, but in this case also, the Rabi

frequency will depend on the motion in the other modes along the direction of k or 6k. Similarly, for sideband operations, such as steps (1) and (3) of the CN scheme discussed in Sec. in, it will, in general, be impossible to have Q'^, ^ depend on only one mode of

motion. In this section, we examine the influence of extraneous modes on the Rabi frequencies Q'„, „. In a collection of A'' ions, the motion in the 3jV — 1 spectator modes reduces the Rabi

frequency in much the same way as lattice vibrations affect a single emitter or scatterer embedded in a crystal, as described by the Debye-Waller effect [47]. Typically, the motional quantum numbers of the spectator modes in Eq. (20) will be thermally distributed

(i.e., P{np) oc y"'' where y — np/[\ -)- hp) with rip the mean number of phonons in mode p), so we can calculate the rms and mean values of the Rabi frequency given this distribution. For simplicity, we assume that all spectator modes are in the Lamb-Dicke regime

{rfphy- -C 1), but see Ref. [11] for the more general case. If the frequencies and amplitudes of all modes contributing to the axial motion of ion j are assumed to be about the same, we can write rf ~ rj/N^^- and tip ~ n where rj and n are the Lamb-Dicke parameter

and mean occupation for the axial motion of a single (thermahzed) trapped ion. In this case, the fractional fluctuation in the Rabi frequency from run to run is

f5£,„^./«(i±ir. (21)

Q', \ N ^ '

This expression indicates that a large number of ions is beneficial because it tends to aver- age out the effects of motion in the A^ — 1 spectator modes. Eq. (21) is an overestimate of

the fluctuations since the spectator modes will have higher frequency than the COM mode,

leading to smaller amplitudes of mofion than assumed in this crude estimate. In any case, it

is clearly desirable to cool all modes (whose motion is parallel tp 6k) to the zero-point

state (up = 0) to suppress the effects of these Debye-Waller factor fluctuations.

2, Mode cross-coupling from static electric field imperfections

If the 37V — 1 spectator modes of oscillation are not all laser-cooled to their zero-point

energy, then energy can be transferred to the kih mode of interest. Even when the spectator

modes are cooled to the zero-point state, they can act as a reservoir for energy from the COM mode. This does not lead to heating but can cause decoherence.

Ideally, the ions are subjected to quadratic potentials as in Sec. II.B. In practice, higherorder .static potential terms are present; these terms can induce a coupling between the modes. Similar couplings are induced by the time varying fields necessary for providing

entanglement; these are discussed below. We will assume that the higher order field gradients act as a perturbation to the (harmonic) normal mode solution. Following the convention in the above introduction to Sec. IV.B, these fields will be specified by £, for

/ € { 1 . 2. . . . 3A/} where the index / specifies both the ion and direction of E. We write the

electric field at the y'th ion as

Ej = Eji + E^,^jy + E2N^jZ: j e {l, 2, . . .N} . (22)

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378 D. J. WiNELAND et al.: Experimental Primer

From Eqs. (18) and (21), we can write the equation of the ^h normal mode as [46]

(23)

In general, we can write

£,•=£,({«,}) =£,({9;}) = E,{{qj] = 0) + E 9" m=\

dEi

dqr, {%} = o

I 3L 3N

/ = 1 771 = 1

d-Ei

dqi dq„

+

H}=o

(24)

where the derivatives are evaluated at the equilibrium positions. The first term on the right

side of the last equation just gives rise to a shift of the equilibrium positions, and the second term can be absorbed into new definitions of the normal mode frequencies a;,. The

second-order term (last term shown in this equation) can resonantly couple two modes of

oscillation (/ and m) to the normal mode of interest k. We find a possible resonant term:

d-qk

dt3N

+ (^iRk = - E ^'kRiq^ m

d^E:

dqi dqr,

{<?;•} =

(25)

where the / and m mode frequencies satisfy \(Di ± aj„^\ = co^. This type of coupling

can either add to or extract energy from mode k, depending on the relative phases

of motion in the three modes. By substituting the free solution to modes / and

m {qj{t) = Qi: txp {±i{a)jt + (Pj)) into the second-order term, we find that if gk (^ = 0) = {^qk/dt)j^Q = 0, the driven solution to the amplitude of mode k initially

grows linearly with time:

kA(OI =

qf

Imw

3A' E D'kQiQ,,

k / = I

d-E.

dqi dq„,_

i'?;}=0

(26)

We illustrate with an approximate numerical example which might have been expected to play a role in the motional heating that was observed in the NIST experiments [10, 17]

(see section V.B below). In those experiments, performed on single ^Be"^ ions, the heating that was observed on the motion in the x direcfion was such that the ion made a

transition from the « = to n = 1 level in about 1 ms. For a single ion, the three nor- mal modes are just the oscillation modes along the x. v, and ; directions {q\ = x, q2=y- q? = Z'. D[ =6^). The mode frequencies were (a».,. (i>y io-^jln ~(11.2, 18.2,

29.8) MHz, thus approximately satisfying the condition co, + oj, = cO;. For sake of argu- ment, we ^ssume this resonance condition to be exactly satisfied. We consider heating of

the X motion assuming both the v and ; modes are excited. From Eq. (26), we find

|x(/)| = \qtAyA-\d^Exldyd-\t^^.^.^Q^I'(lm(X):,)\ where A,, and A- are the amplitudes of motion

in the y and z directions. Here, we neglect the fact that energy is exchanged between the

three modes; for simplicity we assume the amplitudes of the v and z motion remain fixed.

In this approximation, if Ay = A- = ^. the time it takes to excite the x motion to the same

amplitude is given by t — |2ma;^-/((y^[c)-£v/5}'<9z],^.^- = o)l- If $ = 10 nm (corresponding to

(hv) ~ (/7-) ~ 1 for the conditions of the single ^Be"^ ion NIST experiments, the field gradient required to drive the x motion to an amplitude of 10 nm [{n^) ~ 1) in the

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Fortschr. Phys. 46 (1998) 4-5 379

observed time of 1 ms is approximately d^Ex/dydz= 1000 V/mm^. It is highly unlikely

the gradient was this large for the NIST experiments, and, furthermore, the resonance

condition was only approximately satisfied. Moreover, this source of heating was easily tested by varying the initial values of Ay and A. (by varying the Doppler-cooling minimum temperature through laser detuning) and studying the heating rate of the x motion

which had previously been cooled to the zero point of motion. No dependence on the

initial values of Ay and A- was found. In any case, if all modes of motion are initially

cooled to the zero-point state this source of heating vanishes because the assumed coupling only provides an exchange of energy between modes. This places a premium on

cooling all modes to as low an energy as possible. Finally, it appears that this singleion example gives a worst case scenario since, for large numbers of ions, the force on

the generalized coordinates (right hand side of Eq. (23)) requires a high-order field gra- dient to be nonzero. These gradients are highly suppressed in the typical case where ion-ion

separation is much smaller than the distance between the ions and the trap electrodes.

3. Mode cross coupling induced by logic operations

In the preceding sections, we have assumed that when transitions are driven between

|n/;) |i.)y and |«^.) |T), involving a single mode of motion k, the other 3N — 1 spectator

modes of motion are not affected because coupling to them is nonresonant. However, when

the sum or difference frequency of two or more spectator modes is near the frequency of

the desired mode-/: transition {(x>k\n'j. — nk\), higher order couplings can entangle the

1/2/..) IJ.)^ and \n'^) |f)^- states with the spectator mode states. Equation (19) describes the general interaction Hamiltonian between the internal levels of

ion j and all 3A^ modes of motion. By expanding the exponential in Eq. (19) to all orders, we fmd

H] = TiQ'j (27)

where Q'^ = Q' exp [— \ Z/(?/J"]. This equation describes the processes of each mode / gaining or losing (i>/ — di) vibrational quanta, accompanied by ±e raising or lowering of the

internal electronic levels of ion j. We must account for all terms in Eq. (27) which do not vary rapidly in time, or terms in which the resonance condition is nearly met:

Iiih/ — c!i)w] ~ (3 = oj),{n'i. — ni^). Although detailed treatment of this problem is beyond

our intent, a couple of comments may be made.

In general, we must account for all the terms in Eq. (27) which cause significant errors

in the overall computation we are trying to carry out. For two or more trapped ions, some

combination of modes will nearly always satisfy the resonance condition. However, this may occur only for high orders of hi and di, and if the Lamb-Dicke criterion is met, the contributions are vanishingly small. The terms that will cause problems are the ones that are close to satisfying the resonance condition and are relatively low order in bi and d/. If the Lamb-Dicke criterion is satisfied, it will always be possible to avoid these spurious

couplings, but it may be at the expense of making the Rabi frequency so small (in order to avoid coupling to relatively nearby off-resonant terms) that the operations will become too

slow.

To understand this problem in the context of a simple example, we assume that a cross- mode coupling of this type occurs when two modes, p and q, have frequencies which

satisfy the condition UpCOp — riqU)^ ~ 0, oj/,., or —a>;- corresponding to possible extraneous mode coupling on the carrier, first blue sideband, or first red sideband of the logic operations (assumed to utilize mode k). This additional resonance condition yields, to lowest

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380 D. J. WiNELAND et al.: Experimental Primer

order in the Lamb-Dicke parameters, the effective Hamiltonian

np\nq\

J where the resonance conditions are given by (5 = co^ — a>o = 0, +cok, or —oJk- A specific

example is relevant to the NIST single ^Be" ion experiments. Here, mode k was the x

oscillation, and modes p and q are identified with the z and y oscillations of the single ion in the trap. In this experiment 77^. = 6k xxq, rj^. = dk yyo, rj. = dk zzq, and ojy ~ W; — Wy.

(The frequency relationship (u^ = o). — cOy is a consequence of Maxwell's equations for a quadrupole rf trap in the absence of static potentials applied to the electrodes [43]) We

assume that the desired transition is the first blue sideband of mode x{d = w,)- In this case, the resonant part of Eq. (28) becomes

H] = TlQ] [5-'+ {z?7,a|, - Vz^A^y + '^(^^ ) } + h-c] (29)

The term proportional to a\ is the desired anti-Jaynes-Cummings operator, and the term

proportional to aloy can entangle the internal state with the other spectator modes (z and y),

leading to errors. For logic operadons on a string of ions in a linear trap, we will assume that all other

mode frequencies are higher. With the use of stimulated-Raman transitions, we can make

(5fc

II z and restrict our attendon to spectator modes along the - axis. Nevertheless, as A' becomes large, nearby resonances of the type shown in Eq. (28) will become harder to avoid. These coupling terms always scale as products of Lamb-Dicke parameters. Thus if the spectator mode Lamb-Dicke parameters are small enough, or if at least one Lamb-Dicke

parameter is approximately zero, the higher order unwanted resonances may be sufficiently

suppressed. Furthermore, if the spectator modes are cooled to near the zero-point energy

{{n) <^ 1), then any couphngs in Eq. (28) with powers of the annihilation operator a^ will be absent most of the dme. Hence, in large registers, it will probably be important to cool

all modes to near the zero-point energy.

V. Decoherence

A. Internal state decoherence from spontaneous emission

The iniemal atomic states of trapped ions, which store quantum bits of informadon, must

be protected from spontaneous emission, at least for the duration of the computadon. This

excludes the possibility of "error correcuon," [48] which can tolerate a certain level of

errors due to spontaneous emission. For qubit levels coupled by single photon opdcal transitions, this may be accomplished by employing long-lived energy levels which do not have

an allowed electric dipole coupling, such as metastable electronic levels with a quadrupole

or intercombinadon coupling to the ground state. Although the interaction of these states with the vacuum (causing spontaneous emission) is reduced, their interaction with an exter- nal field for use in quantum logic operations is also reduced. This results in a fundamental

limit on the accuracy of each operation by roughly the ratio of the spontaneous emission

rate to the Rabi frequency ^ = F/Q. In the case of optical transitions, Q cannot be in- creased indefinitely, since at optical intensities beyond about lO''^ W/cm~, the atom is quickly photoionized. This amounts to inaccuracies due to spontaneous emission on the

order of g = 10~^ — 10"'' [49]. Even this limit may be too optimistic, as the two-level

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Fortschr. Phys. 46 (1998) 4-5 381

approximation breaks down before photoionization occurs, and the coupling to other electronic levels must also be considered [49]. This results in inaccuracies due to spontaneous

emission on the order of ^ = 10^^ — 10"^, depending on the particular ion used.

In the case of two-photon stimulated-Raman transitions between stable electronic ground

states, the ratio of spontaneous emission rate to Rabi frequency is approximately

^5j^ = y^J{g~ /^), where y^^ ^ Fg^/A' is the off-resonant spontaneous emission rate, g is die resonant single-photon Rabi frequency of each laser beam, and A is the demning of the Raman beams from the excited state. This results in an inaccuracy F/A due to spontaneous

emission, which is independent of optical intensity. Since Raman transitions between S

electronic ground states are effective only when the detuning A is not much greater than

the fine structure splitting of the atom [50] ; this results in an inaccuracy |sr due to sponta- neous emission in range from about lO"'^ (^Be"*") to iO~^ ('^^Hg"^), depending on the parti- cular ion used. Spontaneous emission from spectator electronic levels should not significantly affect this limit, provided that their splitting from the virtual excited state significantly exceeds A and that the single photon resonant Rabi frequencies coupling the ground states to the spectator levels are not much bigger than g [49]. These appear to be

reasonable assumptions for most candidate ions. The decohering effects of spontaneous emission can be overcome by error correction

schemes. Error correction is complicated by the fact that when spontaneous emission occurs, the atoms may decay to states outside the original set of computational basis states. However, this situation can, in principle, be detected by optically pumping the ions back to the computational basis and applying the error correction schemes [49, 51].

Spontaneous emission decoherence could, in principle, be nearly ehminated by driving

single-photon transitions between ground-state-hyperfine or Zeeman levels with rf or micro- wave radiation since spontaneous emission from these levels is negligible. This may be

accomplished by coupling the internal and motional states with inhomogeneous magnetic

fields [52]. The speed of sideband operations is limited by the size of the field inhomogeneity one can obtain. For example, consider an ion moving along the c-axis with a magnetic

field gradient dB^^jdz applied on top of a uniform magnefic field B- along the direction of

motion. In the ion's reference frame, there is an rf magnetic field which can induce transi- tions between the internal states. The interaction Hamiltonian is then

n = -ti B=^ -fiB:S:_ -

^^^^-'^^^'^'" {S^ + 5_) {a^ + a) , (30)

4

where // ~///j is the magnitude of the ion magnetic moment, typically near one Bohr magneton, -() is the spatial spread of the zero-point ion wavepacket, S± are the spin raising and

lowering operators from Sec. II. C and a and a' are the usual motional raising and lowering

operators. The coupling (last term in Eq. (30)) is analogous to Eq. (3), allowing the construction of logic gates as described above. Unfortunately, this method requires very high

magnetic field gradients dBy/dz. For instance, in order to realize an effective Rabi fre- quency of 0.1 MHz {Qc{{ ~ Wb(<95v/&) ::()/4), a magnetic field gradient of around 10 T/cm

would be necessary, where we have assumed that the mofional frequency is 1 MHz.

Although these gradients aren't unreasonable in the laboratory, the residual fields that ac- company the gradient will dramatically shift the internal electronic levels of the trapped

ions. Moreover, it would be difficult to address selected ions because of the long wavelength of the radiation relative to typical ion spacings.

B. Motional decoherence

Although the internal states of ions may be well isolated from environmental influences, the motional states of a collection of trapped ions are expected to be more susceptible to decoTN-124
382 D. J. WiNELAND et al. : Experimental Primer

herence. The Cirac/Zoller scheme for quantum logic [9] transfers quantum information tran- siently through the motional states, so any decoherence of the motion will compromise

operation of any multi bit quantum gate and must be minimized. In the NIST experiments

with a single ion, the motional state of all three modes was observed to absorb energy from

the environment at the rate of about 1 phonon (energy Hoj^, where a;;/2jr ^10 MHz) per

millisecond from the ground state [10, 17]. Although this as of yet unexplained heating rate

is not believed to be fundamental, it is of considerable interest to characterize the possible

sources. There are several potential mechanisms for motional decoherence; for instance "rf heating" from rf fields in the trap [53], collisions with background atoms, flucmating patch

and contact potentials on the trap electrodes, electron field emission from the electrodes,

and Ohmic loss of induced image currents in the trap electrodes [54, 55]. These and other

effects, considered in Ref. [11], are not expected to be major obstacles to motional coher- ence in a linear ion trap. By virtue of the electric charge of trapped ions, it is natural to consider the coupling of

spurious electric fields to the motion of the ion string. In this section we relate the size of

electric field noise to the expected transition rate from the quantum ground state of motion

in the ion trap. Such noisy electric fields might be generated from unstable trap parameters,

external radiation, thermal (Johnson) noise from resistive losses in the trap electrodes, or patch potentials from nearby surfaces. We first consider the axial COM harmonic motion of a string of A'^ ions at frequency (v-. An external uniform electric field will shift the position of the trap center; an external

electric field gradient will change the effective spring constant of the trap. Following the

approach of Savard, O'Hara and Thomas [56], who considered heating mechanisms in the context of neutral atom dipole traps, we can similarly calculate the rate Fq at which the ground state of ion motion is vacated due to noisy electric fields. We find

paxiai ^ax,al ^r Jl n —

AmhojSe{co-) -1 Se'{2co- ANmw-

:3r

where Se{co) is the electric field noise spectral density in (V/cm)^/Hz and Se'{co) is the

electric field gradient noise spectral density in (V/cm'^)~/Hz. The noise densities are defined

so that the mean-squared electric field and field gradient are {li!:)'^ ^^ dco Se{co) and

(2n-)~ j^^ da)SE'{co), respectively [56].

Note that contributions of electnc field noise to motional heating is concentrated near resonance {w-) in Eq. (31), as expected for a resonantiy driven harmonic oscillator. Contributions of electric field gradient noise to heating is concentrated near twice the resonant frequency (2cy_-), as expected for a parametrically driven harmonic oscillator. For modes of

motion with secular frequency cov derived from ponderomotive rf forces (i.e., the radial modes of a linear trap), there are additional contributions to heating from noisy fields at frequencies Qr ± w.v and gradients at frequencies Qj ± 2a) v. to lowest order in the pseudopotential approximation [{wy/QT) ^ 1] [21]. We find

prauial -^rad.al ^T ' n — " AmhWy

2 f- I

'-) 2 SeUo..) +

^

Se{Qt ± CO,) +—-!—isE'{2co_,-) + -^ Se'{Qt ± 2a;,,: IQr 4Nma)_, \ Q-

(32)

Given a source of electric field noise 5£(a') '', we may expect the corresponding gradient noise spectrum Se'{co) '' to be no larger than Se{co) /d, where d is the characteristic

ion trap electrode size. From the above equations, we find that this implies that the heating

rate from gradient noise is expected to be a factor of ^ [zo/ldf' smaller than the heating

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Fortschr. Phys. 46 (1998) 4-5 383

rate from electric field noise, where we recall that zo = {h/lNmo),) '~ is the size of the around state wavepacket of the mode of interest. This ratio is typically very small in ion

Traps (in the NIST ^Be^ experiments, zo ~ 7 nm, J ~ 250 ^im. and {zo/ldf ^2x 10-'°).

fsloise in the electric field gradient is therefore expected to be much less of a concern than

noise in the electric field, unless the electric potentials on the trap electrodes possess an

unusually high degree of symmetry. It also follows that the COM modes should be most

susceptible to heating, as the non-COM modes do not respond to uniform electric fields. For instance, in the case of two ions confined along the z axis, the COM mode along z is sensitive to resonant uniform electric fields. The "stretch" mode along z (where the two

ions' motions are out of phase) on the other hand, is only sensitive to resonant electric field gradients. Therefore, if quantum logic operations are limited by motional heating due to uniform electric fields, higher-order (non-COM) motional modes should be used for quantum logic. COM heating will still indirectly affect the fidehty of operations on other modes,

but this effect should be of higher order, as discussed in section IV.B.l.

We examine two expected sources of electric field noise, and estimate dieir effects on the heating rate of COM modes. These results are compared to the NIST experiment on a

single ^Be"^ ion [17], where we observed a heating rate Fq ^ lO-'/sec, corresponding to an

electric field noise of Se{co.x) ^4- 10~^ V/cm/Hz'^^ or electric field gradient noise of

5F(2a;.,)'/- ^ 0.07 V/cm=/Hz'/-.

1. Thermal or blackbody noise

Thermal fluctuations from lossy elements of the trap electrode strucmre may lead to electric

field noise at the ions. This is essentially the effect of blackbody radiation on the ion

mofion, altered by the "cavity" formed from the trap electrodes [11, 54, 57]. Lossy or

resistive elements of the trap electrodes will only give rise to a noisy electric field only if the effective resistive current path is asymmetrically oriented with respect to the ions; other- wise only neglegible field gradients will contribute, as discussed above. The voltage noise

spectral density from a resistor R at temperature T is 4kTR. For an electrode structure of

characteristic size d, we thus expect an electric field noise density of Se{(o) ~ 4kTR{a))/d-,

where R{aj) describes the resistance seen by the ion from the surrounding electrode structure, or, equivalently, the resistance through which currents induced by ion motion flow. Neglecting the gradient terms in Eqs. (31) and (32), this results in axial and radial mode

healing rates of [7, 1 1 ]

Nq-kTR{aj,]

md-hcDr-'^ ^

"'' ^'::''^'' (33)

and

^r' = ^^ [RM + :^ R[Qr ± co..)] . (34)

We now use this last equation to estimate the heating rate which might have been ex- pected in the NIST single ion "^Be"^ experiments [17], where all modes were confined from

ponderomotive forces. We consider two current paths in the electrode structure which

would likely have provided the largest resistances. The first is a direct path between the endcap electrodes, which were positioned around the ring electrode [11, 43]. Since this path length is only about / ^ 0.5 cm and much smaller than the wavelengths associated

with the frequencies of interest (i»v/2jr % 10 MHz and [Qr ± CL>x)/2jt ~230 MHz, we can TN-126
384 D. J. WiNELAND et al.: Experimental Primer

treat this path as a lumped resistor with resistance R{co) ^ qI/A{cd). Here, g is the resistivity of the electrode material and A{a)) is the effective cross-sectional area of the resistive path, proportional to the skin depth at frequency co. We conservatively estimate

R{cOx) ~ 0.04 Q and R{Qt ± co^) ~ 0.20 Q in the experiment [11, 43] and calculate a heat- ing rate of Fq ~ 0.7/s, dominated by the first term of Eq. (34) and much smaller than

observed.

The if and static electrodes in an rf trap are typically joined through an rf step-up trans- former. This leads us to consider a resistive path from the rf to the static electrode, as the

resistance between these electrodes can be high at frequencies near the resonant rf drive

frequency Qj. In a linear trap, these fields should not have a component along the axial modes of motion; moreover, these fields will vanish at trap center and only provide a gradient, unless there is an extreme asymmetry in the electrode structure. We thus consider noise

fields resulting from a geometrical asymmetry /3 between the ring and endcap electrodes

(0 < /3 < 1, where ^ is proportional to the electric field at the ion if a potential is applied

between the ring and endcaps; /S = for perfect symmetry). In the NIST ^Be"^ experiments

[10, 17], a quarter-wave transmision line acts as the step-up transformer [43], thus the impedance between the rf and static trap electrodes as a function of frequency co is

ZtrapCty) = Zotanh

jT (O .71

4Q y'O^^'lQ^ (35)

where Zq is the characteristic line impedance, and Q is the loaded quality factor of the transmission line of length jzc/IQt at resonance {cj = Qj). The resistance is the real part of Eq. (35), and at the frequencies of interest which might couple to the ion motion, w.v and i3r ± a>v, we have i?(a».v) ~ (jrZo/42) (a>,v/^r)'''^ ~ 0.03 Q and

R[Qt ± oj.v) ^ [Zq/uQ) {Qt/wS' ~ 34 Q. In these expressions, Zq ~ 100 Q and Q ^ 500

are the characteristic impedance and quality factor of the transformer, and cOx/2ji and

QjI2ji are taken to be 10 MHz and 230 MHz respectively [43]. Note the estimated resis- tance R{(Dx) is nearly equivalent to the value in the skin-depth model above, since this path

length is still only a small fraction of the wavelength associated with ni^. We find that the two terms in Eq. (34) contribute roughly equal amounts to a total heating rate of approximately To ~ 0.8/3~/s. Again, this rate is much smaller than the observed rate, even for

large asymmetries.

2. Noise on trap voltages

Fluctuations in the trap voltages {IJ{^ and Vq in Eq. (1)) will nominally only give rise to a noisy field gradient. However, in some cases, asymmetric fluctuations in trap parameters

can give rise to a noisy electric field. For instance, noise on the static potentials of the end

segments of a linear trap (Fig. 2) may not be common mode if the potentials are provided

from uncorrelated power supplies, or if the leads connecting to the end segments experience

different amounts of noise from pickup. In these cases, we can relate the electric field noise

density to a differential potential noise density defined by S(iU^{oji) = d-SE{a)), where dUo

is the difference in potentials between the rod end segments. From Eq. (31), this would

give rise to a heating rate of

For example, a typical power supply might have rms noise of 0. 1 mV uniformly distributed across a 20 MHz bandwidth. If the supply is filtered so that signal amplitude at frequency

TN-127
Fortschr. Phys. 46 (1998) 4-5 385

0)^ is attenuated by a factor F < 1, we find [S(,u^{2a)^)]^''^ ^ 20F (nV/Hz'/-), leading to a heating rate of the axial COM mode of Tq ~ 4 10^A'' F'/s. Here we have assumed trap

parameters (0-/2jz ^ 1 MHz and d ^ 0.025 cm.

If there exist asymmetric static patch or contact potentials on the electrodes, this could

lead to a static electric field ^static which would push the ions away from at the geometrical

trap center defined hy x = y — z = in Eq. (1). In this case, fluctuations in the potentials

Uq and Vo will be converted into a noisy electric field at the ions. If E^^mc is in the axial direction, then a noise density of the static voltage Suoi<^) would be equivalent to a field noise density of SE{a)) — {Esatic/ {Uq))' Su^lco). If we assume £static ~ (p/d, where q) is an

effecfive axial potential difference across the electrodes (due to patch potentials for in- stance), then we find from Eq. (31)

Nq^ fcp

Similar to above, this leads to a heating rate of Fq ^ A \0^N F-{cp/ {Uq))~ (s assuming the above values of the trap parameters. Controlling these potenfially troublesome

heating rates clearly points to the importance of heavily filtering the static electrodes at frequency O)^. The filtering is best accomplished as close as possible to the trap electro- des, and for reasonable filter factors of F < lO^'*, these heating sources should be negligible.

For fluctuations in the rf potential Vo, we can derive a similar heating rate of motional

modes confined by ponderomotive rf forces, which would be relevant to the NIST experi- ments. For simphcity, we consider the COM radial mode at frequency oj^ and assume the confinement is dominated by the ponderomotive force {cOp term m Eq. (2)). We consider the case of a patch field Fstanc = ^/R- The equivalent electric field noise density is then

Se{oj) = (2£static/{^o))~'S'vi,(w), which from Eq. (32) results in a heating rate

1

(JJZ ^»'"' = j^- (|y)" (^-("'-) + 1% ^-(°^ ±

-'V

• t^«>

where S\\X(x>) refers to the rf potential noise density at the trap rf electrode. To estimate the

effect of rf amplitude noise at the input, we again assume that the rf input lead is connected

to a step-up transformer. For a transformer of characteristic impedance Zq and quality factor Q > I?7-/2cy, 3> 1, input signals of frequency oj,- are essentially filtered out, and signals at Qj ± ojy are multiplied by the factor [Zo/(27r^,(2)]' Qt/co.x, where R^ is the source impedance. We thus neglect the first term of Eq. (38) and replace 5v„(^r±<^.v) by

[Zi)/{2:rRsQ)] (^r/w,)" 5v„, input (•^x ± '^.v) in the second term. If the rf source feeding the input has an effective noise figure of NF =: 10 dB above the Johnson noise of a /?,^ = 50 Q

.source impedance. we have ^v,,. input ~ \0{4kTR,) ^ 10"'^ V-/Hz. Assuming

w,./2.T ^ 10 MHz, we find {Vq) ^ 750 V and a heating rate of Fq ~ 10(<p/(Vo))"/s- Since

(P "C (V()) for typical patch potentials of < J V, this source of heating is again much smaller than observed. Moreover, in the NIST experiments, q) was varied over a wide range

without an observed dependence on heating rate. Although we have tried to characterize the more obvious sources of electric field noise, the actual electric field noise spectral density in an ion trap may have a complicated structure and may be difficult to characterize. The motional modes of a string of

ions are most susceptible to electric field noise near the motional frequency of interest, as expected. If spurious resonances should occur at these frequencies due to background

electric fields, it might be possible to avoid them by simply changing the trap paramTN-128
386 D. J. WiNELAND et al. : Experimental Primer

eters. As stated above, heavy filtering of the electrodes should provide adequate noise

suppression at these frequencies and the COM motion of ions trapped in electrode

structures with a high degree of symmetry will be less susceptible to electric field noise.

C. Induced decoherence from applied field amplitude noise

In the ion trap quantum computer, all operations can be traced back to Eq. (5), the coupling

of states |n) |X) to \n') If). The fidelity of these operations will depend, in part, on how

accurately the coupling strength Q' and the apphcation time can be set. For instance, in the

rotation gate of Eq. (7), the rotation angle d = 2Q^r, and noise in these parameters will lead

to evolution to an undesired quantum state. The effects of (Gaussian) noise on laser intensity have been treated by Schneider and Milsurn [58]. These effects show up in a well- characterized way for transitions involving Pock states. Here we briefly investigate the size of laser power fluctuations, as the Rabi frequency is proportional to the laser power in the Raman configuration. We do not examine how such errors might propagate in an extended

gate structure [9, 11, 59].

Fluctuations in the laser intensity at the site of a given ion can arise from both

fluctuations in the relative position of the beam with respect to the ion and fluctuations

in laser power. Laser/ion position stability is particularly important since the Cirac/Zoller

scheme of quantum logic assumes that ions in an array be selectively addressed, there- by requiring a high degree of control of the laser beam spatial profile (Sec. IV.A). Of

course, the simplest method for minimizing position fluctuations is to employ mechanically stiff mounts for the opfics and ion trap electrodes, and have the laser source as close as possible to the ions. A quadrant detector indexed to the trap electrodes and

placed near the ion may also be used to actively stabilize the beam position by feeding

back to a galvanometer or acousto-optic modulator. If optical fibers are used to deliver

laser beams to the ions, position fluctuations between the fiber and the ions could be

made small; however, we must also consider position fluctuafions between the laser source and the input to the fiber. If the posidon tolerances can be adequately controlled, the dominant source of intensity noise at the ion would likely be given by

fluctuations in optical power and laser mode. Here, we estimate hmits on laser amplitude noise.

If we assume the laser fields responsible for quantum logic operations are coherent

states, the fundamental noise floor is photon shot noise. For a laser beam of average power

Pu the fractional level of shot noise is

where cu is the (optical) photon frequency. Top is the time the radiation is applied, and, for

simplicity, we assume square pulse envelopes. Almost all laser sources have significant

amplitude noise well above the shot-noise limit in the 10 Hz— 10 kHz range due to acousric

vibrations which, for example, affect the laser cavity resonators. Much of this noise can be

removed by acfive power stabihzation, where a beamsplitter directs a portion of the laser power to a photodetector, and an error signal is derived and fed back to an upstream modulator or, in the case of a diode laser, directly to the amphtude of the laser source [60]. The

limiting noise of this stabilization scheme is degraded slightly by the imperfect quantum

efficiency of the photodetector as well as the beamsplitter. If the beamsplitter directs a

fraction, of the input optical power to the stabihzer (which then gives an optical power

TN-129
Fortschr. Phys. 46 (1998) 4-5 387

/'„ ~ ( 1 — e) Pq directed to the ion), and the quantum efficiency of the photo detector is

Ty^gj, the Umit of fractional power noise in the logic pulse is (assuming no added electronic

noise in the feedback loop)

""

(40)

P'^T^opVd^A^ -£)

For a laser wavelength of 313 nm, and assuming, f = 0.5, and 77^^, = 0.5, we have

dPu/Pu > 2.3 lO''^{PuTop)~^^'^. For 1 W of usable laser power and Top = 1 jis, this corre- sponds to a firactional power fluctuation of > 2.3 • 10~^.

This estimate applies only to the laser power fluctuations at the beamsplitter and assumes no additional noise is introduced between the beamsplitter and the photo detector or the beam splitter and the ions. Typically, the usable part of the laser beam must be directed

further through optics, the air, and a window to the vacuum envelope enclosing the ion

trap. Fluctuating etalon effects in the optics and air currents may therefore seriously in- crease the power fluctuations beyond Eq. (39).

Fluctuations in timing errors may also be important. Clearly, fractional fluctuations in the duration of laser pulses will correspond directly to the same fractional fluctuations in the desired value of the gate parameters (e.g., rotation angles in Eq. (7)). If we require frac- tional fluctuations of 10~^ on these parameters, then we require timing precision of 1 ps on

a 1 s pulse. Similar considerations apply to the stability of pulse envelope shapes.

For both amplitude and timing fluctuations, it may be possible to sample a portion of the beam used for logic and apply it to a "check bit" ion. The response of this ion could then

be used to monitor and control the amplitude and timing of the pulses.

VI. Conclusion

A system of trapped and cooled atomic ions is one of the few viable experimental candidates for quantum computation. Internal levels of the ions can coherently store quantum

bits for extremely long times. Preserving coherence during logic gates will clearly be more

difficult, because (1) the quantum bits are distributed through a collective motional degree

of freedom in the trap which more readily couples to the environment, and (2) technical

noise on the logic gates degrades gate operation. Nevertheless, to our knowledge, these

difficulties appear to be technical, not fundamental. The maximum number of gates which

can be applied coherently, and the maximum number of ions which can be packed into a

trap, will undoubtedly be determined by technical limits. Although it appears that the quantum factoring algorithm would be extremely difficult to implement, we conclude that an ion trap quantum computer of very modest numbers of bits and gates looks quite pro- mising.

In this paper, we have attemped to identify some of the more important experimental concerns with the ion trap system, and expect that their resolution will likely be determined

in the laboratory. We summarize with a few general observations regarding the ion trap

quantum computer.

( 1

)

The most attractive coupling .scheme appears to involve two-photon stimulated Ra- man transitions between hyperfine (or Zeeman) ground states separated by rf or microwave

frequencies. This not only relaxes the laser frequency stability requirements when compared

to single-photon optical transitions, but also allows more geometrical control of the effective wavevector dk = k\ — ki, which determines the coupling to particular motional modes.

(2) Laser cooling of all modes having a component of motion along 6k to their ground

state energy will minimize cross coupling between the modes.

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388 D. J. WiNELAND et al. : Experimental Primer

(3) The axial COM motional mode may not be the best choice for a "data bus" which

couples the quantum bits between ions. First, although the COM mode is most resolved

from the other modes, it has the lowest motional frequency, which may ultimately set a

limit on the quanmm gate speed. Second, the non-COM motional modes should be less sensitive to heating from external electric fields. (4) For a given number of ions, the optimum value of the mode frequencies will likely

involve a key tradeoff. Larger fi-equencies might minimize the effect of some sources of

motional decoherence, as the motional energy levels would be further separated. This

would also allow gate operations (and laser cooling) to proceed faster, as the Rabi fre- quency can be no larger than the mode frequency. On the other hand, smaller frequencies

will increase the spatial separation of the ions, thus easing the problem of optical addressing of individual ions in a string.

(5) Internal state decoherence will likely be small, and it does not appear that motional

decoherence will be a fundamental problem, notwithstanding the heating problem observed

in the NIST experiments. It is important to heavily filter the ion trap static electrodes at (o- to minimize environmental influences on the ion string COM motion.

Acknowledgements

We gratefully acknowledge the support of the U.S. National Security Agency, Army Research Office, and Office of Naval Research. We acknowledge useful discussions with

R Bardroff, R. Blatt, I. Cirac, T. Darling, L. Davidovich, A. Despain, D. DiVincenzo,

A. Ekert, B. Esry, N. Gisin, S. Haroche, M. Holland, M. Holzscheiter, R. Hughes,

D. James, J. Kimble, R Knight, S. Lloyd, G. Milbum, J. Preskill, W. Schleich, A. Steane,

W. Vosel, R Zoller, and W. Zurek.

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(Manuscript received: September 19, 1997)

TN-133
Trapped ions, entanglement, and quantum computing*

C. J. Myatt, B. E. King, D. Kielpinski, D. Leibfried^ Q. A. Turchette, C. S. Wood,

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

Time and Frequency Division, National Institute of Standards and Technology,

Boulder, CO 80303

ABSTRACT

A miniature, elliptical ring rf (Paul) ion trap has been used in recent experiments toward realizing a quantimi computer in a trapped ion system. With the combination of small spatial dimensions and high rf drive potentials, around 500 V amplitude, we have achieved secular oscillation frequencies in the range of

5-20 MHz. The equilibrium positions of pairs of ions that are crystallized in this trap lie along the long

axis of the ellipse. By adding a static potential to the trap, the micromotion of two crystallized ions may

be reduced relative to the case of pure rf confinement. The presence of micromotion reduces the strength

of internal transitions in the ion, an effect that is characterized by a Debye-Waller factor, in analogy with

the reduction of Bragg scattering at finite temperature in a crystal lattice. We have demonstrated the dependence of the rates of internal transitions on the amplitude of micromotion, and we propose a scheme

to use this effect to differentially address the ions.

Keywords: quantum computing, quantum logic, ion traps, laser cooling and trapping

1. INTRODUCTION

Since the development of useful algorithms for quantum computation'', there has been an

explosion of work toward realizing a practical quantum computer. One of the more attractive systems for implementation is a string of trapped ions^", and already a quanmm logic gate has been demonstrated

with a single trapped ion*. In this paper, we report on further progress toward achieving quanmm logic in a trapped ion system. We describe a miniature, elliptical ring, rf (Paul) trap used in our current experiments. The addition of a static potential can be used to reduce the micromotion of several ions confined in this type of trap. We then describe progress toward individual addressing of ions using the Debye-Waller factors due to rf micromotion. Other recent results, including cooling two ions to their collective motional ground state as well as observations of heating, will be described in a separate

publication^.

2. ELLIPTICAL RF PAUL TRAP

The ion trap used in our current experiments, shown schematically in Figure 1, is based closely on the trap used in previous experiments^. The ring and endcap electrodes are made from 125 i^m thick beryllium foil. The ring electrode was formed by punching a hole in the foil with a pinpoint and then

widening the hole into an elliptical shape using a thin tungsten wire as a file. The edges of the hole were smoothed by "flossing" the hole with the mngsten wire. The aspect ratio of the ring is approximately 3:2, and the elongated axis is approximately 525 [im long. The endcaps are made by cutting a 250 |j.m slot in a similar piece of foil.

' Work of the U. S. Government. Not subject to U. S. copyright.

^ Present address: Inst, flir Exp. Physik, Universitat Innsbruck, 6020 Innsbruck, Austria.

TN-134
ENDCAP (Be)

RING (Be)

x<- \Y

RFhigh

static ground

RF ground

static voltage

Figure 1. A schematic of the elliptical ion trap used in recent studies. Ions align along the elgongated, weak direction of the elliptical ring electrode. The endcaps may be biased with a

static electric potential to adjust the spring constants of the trap.

The rf drive is generated by feeding a copper quarter-wave coaxial line, resonant at about 240

MEiz, that is inside the vacuum system^. The rf is coupled in at the base of the resonator with a single

loop of wire, the shape of which is empirically adjusted to impedance-match the 50 D. rf source. The ring is clamped to the end of the center conductor, and the typical voltage amplitude on the ring is 550 V for I W input rf power. The endcaps may be biased with a static potential to adjust the trap

frequencies. There are, in addition, four shim electrodes (not shown) that allow us to null out stray

electric fields. The classical motion of an ion in a rf quadrupole trap is described by the Mathieu equation.*'^ the pseudopotential approximation, the solution for the motion may be broken up into a fast, small

amplitude motion at the rf drive frequency, termed the micromotion, and a slower, larger amplitude

motion that describes the position of the ion averaged over a period of the rf drive, which is called the

secular motion. The secular oscillation frequencies are observed through resonant detection^'' °. A plot of

the secular frequencies as a function of the electrical potential applied to the endcaps is shown in Figure 2. The measured frequencies have been fit with the functions

In

a> = ^(aa+{aqfll. (1)

n ^2 2 cOy= —y(l-a)a+(l-a) (/ /2 , and (2)

CO: = —^-a + q'/2 , (3)

where Q = 2;t x 238.3 MHz is the rf drive frequency, a is the geometric ellipticity parameter, and a

TN-135
o

c

o

3

cr

3O

endcap potential [-Volts]

Figure 2. The secular motional frequencies as a function of the voltage applied to the endcaps.

Note that the voltage is actually negative with respect to the ring electrode. The x symbols are for the X frequencies, the open squares are for the z frequencies, and the crosses are for the y

frequencies. The lines are fits to Equations 1 through 3.

and q are static and rf parameters in the Mathieu equation for the classical amplitude of motion . The

parameters is defined so that the electrical potential has the form O oc ax- +[l-a)y^ -z'' Thus,

for a spherical quadrupole trap, a -0.5 . Due to the asymmetry of the endcaps, the ellipticity parameter is a = 0.36 even when the ring is circular^ The elliptical ring used in the current trap enhances the

ellipticity of the fields, and we extract a = 0.26 from a fit to the data.

In the type of trap described here, the electric fields vanish at a single point. In the absence of

stray fields, a single ion in the trap is confined in the vicinity of this zero-field point. Thus, in the pseudopotential approximation (q« 1), there is almost negligible micromotion when a single ion is cooled to the zero-point state of secular motion. In the case of two ions crystallized in the trap, Coulomb

repulsion pushes the ions away from this zero-field region. Thus there is significant micromotion even when two ions are cooled to the zero point of secular motion. The kinetic energy 7„ of micromotion is

equal to the rf pseudopotential energy U^ at the ion equilibrium positions". However, the addition of a

static potential Uq can reduce the micromotion of two ions relative to the case of a pure rf trap. The ion

separation is reduced by using a static potential to strengthen the confinement along the a.\is on which the ions lie, which in turn decreases the amplitude of the rf fields at the ion equilibrium positions. The

potential energy at the "squeezed" equilibrium is a sum of the rf pseudopotential and static potential

contributions, U - U^ + Uq , and the ratio of micromotion kinetic energy to total potential energy, T^/U = U^/(U,f +UQ),is less than 1.

In the elliptical rf trap, the equilibrium positions of two cold crystallized ions lie along the

direction of the weakest spring constant. In the absence of a static potential, the weakest confinement is

TN-136
on the long axis of the elliptical ring, which we label the x axis (see Figure 1). As Figure 2 shows, the

three trap frequencies may be altered by applying a voltage to the endcaps. In particular, a negative

potential on the endcaps decreases the f confinement while strengthening the confinement in the plane of

the ring. As the ions are squeezed together, they maintain their equilibria along the x axis so long as

cOj. <C02 . To determine the maximum reduction in micromotion, consider the limiting case co^ - o)^

In this case, the ratio of micromotion kinetic energy to total potential energy is Tn/U=a. Thus the

physical geometry of the trap determines the maximum suppression of micromotion, which can be

substantial for small values ofa .

3. MICROMOTION AND DEBYE-WALLER FACTORS

Quantum logic operations on a string of ions require the ability to differentially address the ions. This can be accomplished by focusing a laser so that only one ion is illuminated and the others are in the

dark. However, the scheme outlined in the previous section to reduce the micromotion, whereby the ions are squeezed together using a static potential, excacerbates the problems of illuminating only one ion. In

this section we present an alternative method for differential addressing that does not require tight

focussing of a laser beam. In this scheme, the ions are uniformly illuminated, and the internal transition strengths are differentially modified by the motion of the ions. This sort of effect is familiar in crystallography, where the amplitude of the Bragg diffraction peak is a sensitive function of the random

motion of the atoms in the crystal. The reduction of Bragg scattering at elevated temperatures is governed

by the Debye-Waller factor.

In the case of two ions, the amplitude for driving stimulated Raman transitions between two

internal states is sensitive to the motion of the ion, and we call the motional terms that modify the Raman

transition rate the Debye-Waller factors''. For our system of trapped ions, both the thermal motion of the ion and the micromotion contribute to a reduction in the rate of stimulated Raman transitions. In the frame of the oscillating ion, the applied laser light is frequency modulated at the ion motional frequencies

due to the Doppler shift. We are here concerned with the micromotion, so the applied laser light appears

to have sidebands at the rf frequency, and the power in the central frequency is depleted. If the modulation index due to the motion is m, then matrix elements of transitions driven by the central, or

carrier, frequency are proportional'" to the Bessel function Jo{m) . This reduction of the transition

strength can be manipulated by applying external fields. In particular, an applied electric field along

the X direction displaces the ions so that the micromotion of one ion is reduced while that of the other is increased. This will increase the Rabi frequency of one ion at the expense of reducing the Rabi frequency

of the other ion. ff the ratio of the Rabi frequencies can be made equal to 2, then a pulse of radiation can

be a 71 pulse for one ion and a 27r pulse for the other ion. Alternatively, the micromotion amplitude of one

ion may be so large that Jo{m) = , allowing individual addressing of the other ion. The applied electric

field may then be reversed to address the second ion.

In the experiment, an electric field along the x direction is generated by applying a potential to the four shim electrodes. For a variety of applied fields, the Rabi frequencies of the two ions on the carrier

transition was observed ^•'°, as shown in Figure 3. The data are fitted to a simple function with two

independent frequencies. These pairs of frequencies are plotted in Figure 4 as a function of applied

voltage on the shim electrodes. This clearly demonstrates the ability to adjust the Rabi frequencies. The

implementation of quantum logic with this scheme is now under experimental investigation.

TN-137
(a)

(b)

1.2 i- 1 ' 1 1 i

1.0

y.

• -

t

0.8

' ftA. 'Hy\ 5 .-«:'..;>

0}o

o

0.6

0.4 -1j •.

f ^:-^fy.)i^ ^

m

0.2 -

v = +500

S -

nn f- 1

1 1

1 4

2 3

t[MS]

Figure 3. Rabi flopping on the carrier Raman transition for two ions. The detected flourescence

oscillates in time as the ions flop between a "bright" state and a "dark" state.^''° In the upper

trace, the ions have equal amplitudes of micromotion, and hence their Rabi frequencies are

identical. In the lower curve, the ions have different amplitudes of micromotion, and this is reflected in two different Rabi frequencies. The fit is to two decaying sinusoids. The electric potential applied to the shim electrodes is indicated.

TN-138
JL 1.5 uc3

400 450 500 550 600 650

Shim Electrode Potential [Volts]

700 750

Figure 4. As the ions are displaced by an applied voltage on the shim electrodes, the Rabi

frequencies are different for the two ions. The lines are only to guide the eye.

4. OUTLOOK

Achieving nontrivial quantum computation in a trapped-ion system requires a number of difficult steps. In this paper, we have discussed one possibility for individual addressing. Combined with our

recent achievement of cooling two ions to the ground state of their collective motion^, this puts simple

quantum logic operations on two ions within reach. Future work will revolve around implementing some

scheme for individual addressing in order to entangle the states of the ions. In the longer term, linear ion

traps will be pursued in order to achieve quantum logic with more than two or three ions.

5. ACKNOWLEDGEMENTS

We thank the U. S. National Security Agency, the U. S. Army Research Office, and the U. S. Office of Naval Research for funding this work.

TN-139
6. REFERENCES

1. P. W. Shor, in Proc. ofthe 35th Annual Symposium on the Foundations of Computer Science, edited

by S. Goldwasser, pp. 124-134, 1994.

2. L. K. Grover, Phys. Rev. Lett. 79, pp. 325-328, 1997.

3. J. I. Cirac and P. Zoller, Phys. Rev. Lett. 74, pp. 4091-4094, 1995.

4. D. J. Wineland, C. Monroe, W. M. Itano, D. Leibfried, B. E. King, and D. M. Meekhof, submitted to

J. Res. Natl. Inst. Stand. Tech.

5. C. Monroe, D. M. Meekhof, B. E. King, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 75, pp. 4714-4717, 1995.

6. B. E. King, C. S. Wood, C. J. Myatt, Q. A. Turchette, D. L. Leibfried, W. M. Itano, C. Monroe, and

D. J. Wineland, submitted to Phys. Rev. Lett.

7. S. R. JefFerts, C. Monroe, E. W. Bell, and D. J. Wineland, Phys. Rev. A 51, pp. 3112-3116, 1995.

8. H. DthmzM, Adv. At. Mol. Phys. 3, p. 53, 1967.

9. R. F. Wuerker, H. Shelton. and R. V. Langmuir, J. Appl. Phys. 30, pp. 342-349, 1959.

10. D. M. Meekhof, C. Monroe, B. E. King, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 76, pp. 1796-1800, 1996.

11. F. G. Major and H. G. Dehmelt, Phys. Rev. 170, pp. 91-107, 1968.

12. D. J. Wineland and W. M. Itano, Phys. Rev. A 20, pp. 1521-1540, 1979.

TN-140
Shadows and Mirrors:

Reconstructing Quantum

States of Atom Motion

Imagine that a pair of coins are tossed in a black box. The box reports only one of the following three results at random: (1) the outcome of the first coin (heads or tails), (2) the outcome of the second

coin (heads or tails), or (3) whether the outcomes of the

two coins matched or were

different. Our task is to construct a joint probability dis- tribution of the four possible outcomes of the coins (HH,

TT, HT, TH) based on many observations of the black box

outputs. Now suppose that after many trials, the black

box reports that each coin comes up heads two-thirds of the time when measured individually, yet the coins never match when they are compared. (Clearly the results of the coin tosses have been correlated—perhaps a joker in the black box flips the coins and then changes the out- comes appropriately.) We seek a distribution that both

reflects this correlation and obeys the marginal distribu- tions of each coin as two-thirds heads, one-third tails (see the three tables on page 23). The only way to satisfy both

requirements is to force the joint probability P(TT) of getting two tails to be negative! Mathematically, this is because P(HH) + P(TT) is observed to be zero, yet we

expect P(HH) to be greater than P(TT), because the indi- vidual coins are weighted toward heads.

The sleight of hand giving rise to negative probabilities is that we have attempted to reconstruct a joint

probability distribution without ever having observed in- dividual joint outcomes of the coins. The only measured

events are described by sums of joint probabilities such

as P(HT) + P(TH) =: 1 or P(HH) + P(HT) = %. One way to interpret the distribution of table 3 is to note that, since individual joint outcomes of the coins are inaccessible,

nothing prevents us from assigning negative probabilities

to such immeasvirable events. With this rule in mind,

this joint "quasi-probabihty" distribution may be a useful bookkeeping tool, as it not only characterizes the hidden

correlations within the black box, but also retains inforDlETRlCH LnBFRlED is a physicist at Innsbruck University in Austria. He was a guest researcher at the National Institute of

Standards of Technology in Boulder, Colorado, during the writing ofthis article. TiLMAN Pfau is a physicist at the University ofKonstanz in Germany. CHRISTOPHER

Monroe is a staffphysicist at the National Institute of

Standards and Technology in Boulder, Colorado.

Quantum mechanics allows us only one

incomplete glimpse of a wavefunction,

but if systems can be identically prepared

over and over, quantum equivalents of

shadows and mirrors can provide the full

picture.

Dietrich Leibfried, Tilman Pfau and

Christopher Monroe

mation about the marginal

probabilities of the individ- ual coins. ^ Although this example of

coins in a black box is highly

artificial, a similar situation

arises in nature when we describe the probability distri- bution of a quantum mechanical particle in positionmomentum phase space. A

classical particle occupies a

single point in phase space, and an ensemble of classical particles can be characterized by a phase-space probability

distribution. On the other hand, the Heisenberg uncertainty relationship requires that a quantum mechanical

particle be described by an area of uncertainty in phase

space no smaller than Ax Ap = hl2. If a particle's position

is known well, then its momentum is not, and vice versa. In mathematical language, the position wavefunction "^yix)

and momentum wavefunction ^p(p) are related by a Fourier transform; thus, localized position wavefunctions

lead to delocalized momentum wavefunctions, and vice versa. A probability distribution in quantum phase space must somehow incorporate this feature. Wigner distribution and 'negative probabilities'

In 1932, Eugene Wigner presented a convenient mathematical construct for visualizing quantum trajectories in phase space. ^ The Wigner distribution, or Wigner function

W{x, p), retains many of the features of a probability

distribution, except that it can be negative in some regions

of phase space. In the coin example above, practical use

of the quasi-probability distribution of table 3 is limited

to events described by sums of any two entries. Similarly,

when we apply the Wigner distribution to measurements

in quantum phase space, the probability distribution for the outcome of a measurement is obtained essentially by

convolving W{x,p) with a distribution of possible states of the measurement device, which must be distributed over an area of order h or larger. This prescription leads to a

natural connection between quantum and classical phase

space: As the measurement resolution is degraded away

from the quantum limit so that the Heisenberg uncertainty relationship plays no role, localized regions of W{x, p) (with possible negative values) become washed out, and the convolved Wigner distribution approaches the

usual classical phase-space probability distribution.

Similar to the quasi-distribution of the coins above, the Wigner distribution is not a bona fide probability distri- bution, but can be a useful bookkeeping tool that high22 April 1998 Physics Today TN-141 1998 American Institute of Physics, S-0031-9228-9804-010-1
Jnomentum

Figure 1: Wigner function for the

DOUBLE-SLIT EXPERIMENT, visualized in phase

space, a: The initial Wigner distribution representing the superposition of two Gaussian

lobes directly behind the slits. The oscillating part in the center is due to the spatial coherence between the two lobes. The

"spacelike shadow" (on the orange screen, at

left) shows the spatial marginal distribution I'^'xWP of' the state, obtained by ignoring the momentum information. The pale burgundy

shadow at rear shows the corresponding

"momentum-like shadow" I'*? (p)p. With a position-sensitive detector measuring the

spacelike shadow, we can view the initial Wigner distribution from different angles by

either rotating it (b) or shearing it (c)

.

lights the inherent anticorrelation of position and momen- tum uncertainty.

For a pure quantum state, the Wigner distribution is related to the position or momentum wavefunction by

W(X, p) : 277 J ' x-|Wix + |

e-Pds

= -1

2ttJ %\p+^\%\P ds , (1)

where we have set h-1. The Wigner distribution of a mixed quantum state is a weighted sum of either expres- sion over the appropriate wavefunctions. These expressions may not be very illuminating, and the equivalent of the Schrodinger equation describing the time evolution of W(x, p) is even less so. (However, Wolfgang Schleich and

Georg Siissmann discussed a physical interpretation of

this form of the Wigner distribution in PHYSICS TODAY,

October 1991, page 146.) Nevertheless, Wigner showed

that W{x, p) is indeed the closest thing we have to a probability distribution in quantum phase space, as it corresponds

to the phase-space probabihty distribution in the classical

Tables: Joint probability distributions of the outcomes

of tossing two coins. (1) Probability distribution given

that the two coins are tossed independently, both weighted

toward heads with P(H) = V^ and P{T) = V3. The marginal

probabilities of the outcomes of either coin (obtained by

adding the entries vertically or horizontally) result in two-thirds heads and one-third tails. (2) Probability

distribution given that the coins are tossed in a black box that reports that the coins never match—that is, P(HH) + P(TT) = 0. The off-diagonals add to 1 as required,

with p arbitrary. The marginal probabilities can no longer be two-thirds heads for both coins. (3) "Quasi-probability"

distribution under same conditions as (2) and also exhibiting marginal probabilities of each coin as two-thirds heads and one-third tails. The price paid is that one of the

entries is negative!

H T; ;|H ; T| W T

H % % Hop H Ve H

T

(1)

V9 T 1-p

(2)

T .

(3)

T^-142 April 1998 Physics Today 23
50|am

1.2

-^ 1.0 e^ 0.8 i c

o^ 0.6

exo 0.4

0.2

POSITION posmoN POSITION

Figure 2: Double-slit atom INTERFEROMETRY. a: The experimental arrangement.

Atoms from a collimated source propagate through a double slit with 8 fim separation and

strike a position-sensitive detector screen. The source produces fast-moving atoms and a range

of slower atoms, b: Diffraction pattern of the atomic matter waves plotted as a function of position and of the propagation time t^ for the atoms to travel from the double slit to the

detector. The distance from double slit to detector is d = 195 cm, for which the slow atoms

propagate long enough to produce a Fraunhofer (far-field) diffraction pattern. The fast atoms

produce the near-field shadow of the slits at the bottom. (This shadow is magnified because of the geometry of the apparatus.) c: Calculated diffraction pattern for atoms having a wide

range of velocities (and propagation times tj), showing the transition between Fresnel

(near-field) and Fraunhofer diffraction, d: Data for d = 25 cm, where the transition between

Fresnel and Fraunhofer diffraction becomes visible in the slow atoms' pattern.

limit, and also preserves the marginal probability distri- butions of position and momentum |'4'"x(x)p and |'^p(p)p:

\^^{x)\^ = \wix,p)dp and |'^p(p)|2 = jw(x,p)dx . (2)

Can the Wigner distribution W(x,p) of a quantum

particle be measured? At first glance, the answer appears

to be no. The probability distribution of any physical

observable corresponds to an integral over W(x,p), as in equation 2, so a single measurement cannot provide lo- calized values of W{x,p). But if we prepare a particle in the same quantum state in repeated experiments, we can perform a large number of measiu-ements on effectively

the same quantimi system. We can then reconstruct the

Wigner distribution by measuring various shadows or projection integrals of W{x,p) in separate experiments, or by averaging an observable whose expectation value is proportional to W{x,p) in repeated experiments.^

In the follovdng, we describe two methods for reconstructing the Wigner distribution of atomic motion in phase space from such a set of repeated measurements.

In one experiment, identically prepared atoms from a beam travel through a double-slit interferometer, and

different measurements are performed on them. In an- other experiment, a single trapped atom is repeatedly

prepared in an identical state of motion, and a different measurement is performed after each preparation. The

atoms in both experiments are prepared in nonclassical

states of phase space; thus their corresponding Wigner

distributions have features not found in classical phasespace distributions, such as negative values. Quantum shadows and the double sHt

Detecting the positions of many identically prepared atoms

yields the spatial marginal distribution |'*P'x(-'c)P as a

"spacelike shadow" of the Wigner distribution; likewise, a

momentum-sensitive detector yields the "momentum-like

shadow" ["^pCp)!^. As shown

in figure la, we can observe

shadows across different angles in phase space either by

rotating the detector's point

of view or by rotating the

Wigner distribution and keeping the detector fixed. For

example, figure lb shows the

Wigner distribution rotated

by 60° and measured with

a position detector. The

spacelike shadow on the

screen now contains information about both x and p of the

initial distribution. The

Wigner distribution can be

sheared in phase space as shown in figure Ic by allow- ing the particle to evolve

freely. A shear rotates the

spacelike shadow and gives an additional stretching,

which can easily be compensated for. Thus, we can ob- serve different shadows of the initial Wigner distribu- tion by allovring particles to evolve freely for different

times before we measure

their position.

Tomography is a general technique for reconstructing

the shape of an inaccessible object from a set of different shadows of that object. For instance, medical imaging uses

this technique to obtain a full three-dimensional picture

of the brain by piecing together various two-dimensional

shadows from x rays or nuclear magnetic resonance tech- niques. Quantum state tomography has been used to reconstruct the quantum state of light waves'* and molecular vibration,® and has also been theoretically consid- ered for the reconstruction of the Wigner distribution of atoms from an atomic beam.^ All these applications use

a mathematical device called the inverse Radon transfor- mation to generate an image of the higher dimensional

object from a full set of shadows. In this sense, quantum

mechanics places the observer in the situation of Plato's prisoner—chained in a cave so he can see only the shadows

of objects outside the cave, not the objects themselves.

However, when the objects are rotated or sheared, even

Plato's prisoner can obtain a full picture of the objects.

At the University of Konstanz, Jiirgen Mlynek's group

use this tomographic technique in sending an atomic beam

through a double-slit apparatus and reconstructing the Wigner distribution of the atoms immediately behind the

slit." The theoretical Wigner distribution in figure la

depicts the idealized quantum state of the transverse

position and momentum of each atom as it leaves the double slit. For a plane matter wave, the emerging

quantum state is a linear superposition of one atomic

wavepacket going through one slit and another such

packet going through the other. The coherence between

these two wavepackets leads to an interference pattern in the momentum distribution. The signature of this coher- ence in the Wigner distribution is the oscillating positive

and negative values between the two main lobes. In the experiment, the spatial distribution of the atoms is measured on a screen. As the atoms freely propagate between

the double slit and the screen, the corresponding quantum

24 April 1998 Physics Today TN-143
state is sheared in phase space as shown in figure Ic. Different atoms experience different shear, since they are

distributed over a broad range of velocities and therefore evolve for different times as they travel from the double

slit to the detector. A velocity-selective experiment can

therefore yield the full information about the quantum

state of the motion.

The Konstanz experiment is sketched in figure 2a. A

discharge sovirce for metastable helium atoms fires for 10

/xs, generating a double-peaked distribution of atomic

velocities consisting of slow atoms between about 1000

and 3000 m/s and fast atoms near 33 000 m/s. The

corresponding de Broglie wavelengths are concentrated near 3 picometers for the fast atoms and between 20 and

70 pm for the slow atoms. A 5 /xm wide entrance slit coUimates this beam. Farther downstream, the beam

passes through a microfabricated double-slit structure with a slit separation of 8 ^am and openings of 1 /xm. The

combination of entrance slit and double slit acts as a preparation tool for the transverse motional quantum

state of the atoms. After emerging from this preparation

tool, the atoms propagate over a distance d to a time- and

space-resolving detector When each metastable atom

strikes the detector, it releases a large amount of energy, allowing nearly every atom to be detected. The spatial

and temporal coordinates of each such event at the detector are recorded. This data provides a measurement of

spatial atomic distributions for different longitudinal velocities V in the beam, or equivalently, different propagation times t^ = d/v from the double slit to the detector. As discussed above, different propagation times t^ lead

to different views of the Wigner distribution. Another way

to look at this situation is to treat the atomic wavepacket

evolution as an optical diffraction problem, in which the shear of the Wigner distribution corresponds to the transition from the Fresnel (near-field) regime to the Fraunhofer (far-field) regime. Figure 2c shows the results pre- dicted by theory for atoms with a wide range of propagation times. In the extreme Fresnel regime, we recognize

the spacelike shadow of the two slits. With increasing t^, the wavepackets start to overlap and interfere until, for large t^, we arrive at the Fraunhofer regime in which the

diffraction pattern embodies the momentum-like shadow

of the state. In figure 2, experimental measurements of the time-resolved diffraction patterns are shown on both

sides of the theoretical plot. On the left, figure 2b corre- sponds to a distance d = 195 cm, the Fraunhofer regime

for slow atoms. It shows nicely a resolved interference

pattern that corresponds to the momentum-like shadow.

The very fast atoms produce the spacelike shadow of the double slit at the bottom of figure. This measurement

corresponds to two separate ranges of propagation times,

or view angles, of the quantum state's Wigner distribution. The other view angles are missing because their respective

atom velocities are absent from the atomic beam. To fill in these views, a second experiment is performed with the

detector screen placed only d = 25 cm behind the double

slit. Figure 2d shows the result of this experiment, which

features the transition between the spreading individual

wavepackets and the overlapping and interference of the slow atoms, in addition to the usual spacelike shadow of the very fast atoms at the bottom.

Figure 3 displays the Wigner distribution that is reconstructed by binning the data according to the differ- ent propagation times t^ and performing the inverse Radon

transformation. Figure 3a shows the Wigner distribution reconstructed from the d - 25 cm data, and figure 3b

shows the Wigner distribution derived from the d = 195 cm data. In both cases, we recognize two positive ridges

corresponding to the spatial distribution of the atoms

TN-144

J---4 ii ^,

.g^Ja^ .^ X ^-aj ^ ^ T

-4

0.5

-0.3

POSITION xOu,m)

Figure 3: Reconstructed Wigner DiSTRjBLrriONs

derived from the experimental data sets shown in figure 2d

(corresponding to li = 25 cm) (a) and figure 2b {d = 195 cm)

(b). Both reconstructions show the expected two lobes separated by the slit separation of 8 ju-m. Between the lobes, the Wigner distribution oscillates between positive and

negative values, indicating the spatial coherence and

nonclassical character of the state immediately behind the double slit.

immediately behind the double slit. These ridges are separated by 8 /xm—the spacing of the double slit. The

coherence between the two spatially separated parts of the wavefunction at the double slit leads to interference,

reflected by the oscillations in the Wigner distribution in the region between the ridges. In this region, the recon- structed Wigner distribution assumes negative values,

indicating a property that cannot be obtained by classical phase-space distributions and revealing the quantum nature of the observed ensemble of atoms. The reconstructed

Wigner distributions, determined from about 500 000 at- oms, exhibit all the features of a superposition state expected from an atom interferometer. The measured

Wigner distributions differ in some respects from what is theoretically expected, including residual shear and spurious

negative regions close to the two large positive ridges.

These artifacts occur primarily because the reconstructions are from an incomplete range of projection angles.

Quantum mirrors and a trapped atom

To reconstruct the quantum state of motion of a single

harmonically trapped atom, we turn to a more direct method that does not require the transformations involved

in the tomographic technique described above. Instead,

April 1998 Physics Today 25
the Wigner distribution at a particular point in phase

space is extracted directly by performing several different measurements on an identically prepared system. This

method is based upon a simple and powerful picture of the Wigner distribution that was first pointed out in 1977

by Antoine Royer.^

Suppose we create a mirror image of the wavefunction

^j;(^) about the point Xq and then measure the overlap of the mirror image M{Xq)^^{x) = "^^(2x0 -x) with the origi- nal 'i^J.x). Formally, this is a measurement of the expectation value m of the mirror operation !M,

mix,) = {^,\M{xon,)

Md5^;fxo + fWxo (3)

If ^J.x) is localized around x^, the mirror operation about

Xq = Xi will largely map this area onto itself, resulting in a nonzero overlap. But if we perform the mirror operation

about some other point Xg—far from Xi—the overlap with

the original wavefiinction wall nearly vanish. (See figure

4a.) Thus, we might expect that the observed overlap

m(xo) will be nonzero only for positions Xg where "i^^ix) is localized. But now suppose that the wavefunction is localized in two separated regions, centered at -x and x. If we perform the mirror operation halfway in between,

at Xq = 0, the lobes of the mirrored wavefunction will nearly coincide with the original lobes, resulting in a large

overlap. Moreover, the overlap will contain information

about the phase difference between the original and mirrored wavefunction. For instance, in figure 4b, the two

pieces of the wavefunction are 180° out of phase (a mountain and a valley), resulting in a negative value of the overlap between the wavefunction and its image.

Returning to equation 3, we note that the mirror

expectation mCxg) is proportional to the Wigner distribu- tion at zero momentum W{xq, 0). If the mirror operation

Figure 4: The quantum mirror, a: The classical-like case of a localized particle. If the initial wavefunction (solid black

line) is mirrored around Xj, where the particle is localized, the mirror image (blue dashed line) lies right on top of the

original and the overlap is large. If the mirror is at X2, the image (red dashed line) has essentially no overlap with the

original. Thus, the overlap is locahzed like the particle's probability distribution, b: The quantumlike case of a particle

with coherent amplitudes in different locations. The

wavefunction of the first excited {n = 1) energy eigenstate of a harmonic oscillator has a valley and a peak left and right of the origin (solid black line) and exhibits odd parity. Peak and

valley are interchanged on the mirror image around Xq =

(blue dashed line) and the overlap product of the two

functions (dashed green line) is zero or negative everywhere.

The value of the Wigner distribution at the origin of phase

space is the area of the green curve, and is thus maximally

negative for the n = 1 state.

could be performed about the phase-space coordinates

{xq,Pq), we might hope the Wigner distribution W(Xo, Po)

could be extracted directly from a measurement of this modified overlap. Royer made this connection and saw

that the mirror operator about the origin of phase space

is just the parity operator H. Therefore the Wigner

distribution at (xq,Pq) can be interpreted as the expectation of the displaced parity operator,

W(xo, po) = ^ (^|£'^(-Xo, -Po) n V{-Xo, -po)m , (4)

where 2?(x, p) is the coherent displacement operator, which

displaces a state across phase space by an amount {x,p)

or, equivalently, shifts the origin of phase space from

(0, 0) to (-X, -p).^ The examples of figure 4 illustrate the connection between the overlap of wavefunction mirrors

and the Wigner distribution, and figure 4b highlights a

particular case in which the Wigner distribution can take on its peculiar negative values. These negative values

occur only when the wavefunction is nonlocally distrib- uted, thereby highlighting the nonclassical or delocalized features of the quantum state. In experiments conducted by David Wineland's group

at the National Institute of Standards and Technology in Boulder, Colorado, ^° a single ^Be* ion is confined in a radio

frequency (Paul) ion trap. The trapping potential is well characterized by a three-dimensional anisotropic harmonic

oscillator. We describe the measurement of the Wigner

distribution for motion in one of the dimensions, characterized by the ladder of energy eigenstates \n) of energy

(n -I- V2)^a;, where n = 0, 1, 2, . . ., and w/27t = 11 MHz is the harmonic oscillation frequency. To reconstruct W(xo, Po)

in this system, the same quanttmi state must be prepared

over and over. First, the ion is initialized in the harmonic

oscillator ground state by laser cooUng. (See the article by Wineland and Wayne Itano in PHYSICS TODAY, June

1987, page 34.) Next, a particular motional state is prepared in a controlled fashion by applying laser pulses

and RF fields. A variety of harmonic oscillator states can

be created, including thermal, coherent, squeezed and

energy eigenstates (number states),^^ and superpositions

of these types of states, including "Schrodinger's cat"

states. ^^ The relative phases of these states of motion can

be controlled by the stable relative phases of the laser and RF fields used in their creation. The quantum mirror measurement of the Wigner

distribution reqtiires two ingredients: a coherent displacement of the state (equivalent to a displacement of the

phase-space origin), and a way to determine the expectation value of the parity operator of this displaced state.

26 April 1998 Physics Today TN-145
The displacement operator is achieved by applying an

oscillating (resonant) electric field, which couples to the

ion's harmonic motion, similar to pushing (or stopping) a

child on a swing. In principle, the state can be coherently

displaced by any amount (xq, Pq) in phase space by varjdng

the amplitude of the applied field and its phase with

respect to that of the initial state of motion. The expectation value of the parity operator after the displacement

can be determined by measuring the populations of energy

eigenstates, which, for a harmonic oscillator, are also parity eigenstates. That is, states \n) with an even or odd

number of energy quanta n have even or odd parity,

respectively. Therefore, the expectation of the parity op- erator can be deduced by simply measuring the probability

distribution P„(xo,Po) of energy eigenstates of the dis- placed state and performing an alternating sum over these

probabilities. Substituting this result in equation 4, we

find that the Wigner distribution is

W{Xo,Po)=i^i-irPniXo,Po) (5) n=0

The measurement of the motional state occupation

probabilities P„(xo, po) is tricky, because it is very difficult to detect a single ion's motion directly. Instead, features

of the motional state are encoded onto two internal elec- tronic (hyperfine) levels of the ion, labeled |l) and |f). The

occupation of these states can be detected with nearly

100% quantum efficiency by applying laser radiation that

connects one of the hyperfine levels (say |i)) to an excited

electronic state. If the ion is in state |i), it scatters thousands of photons, an event that can easily be detected.

If, on the other hand, the ion is in state |t), essentially no photons will be scattered. ^^ To encode the motional

states onto the internal states of the ion, a "mapping

interaction" is realized with laser beams. For an appropriate tuning of the lasers, the external motion is coupled

to the internal h5rperfine levels, and energy is periodically

exchanged between the two systems, similarly to energy

exchange between two coupled pendulums. The exchange

frequency (or Rabi frequency) Ct^ between \i) and |t) due

to this mapping interaction is different for each motional

eigenstate |n), and if the atom is initially in state |1), after

Figure 5: A classical-like coherent state

of a harmonic oscillator (in this case an ion of mass m in z trap) produces this experimentally

reconstructed Wigner distribution. The ion's coordinates of position x and momentum p are scaled to x' = x^rrKo/h and p' =p/^mh(jj and

further transformed to a frame ix,p) that

rotates at the harmonic trap frequency o), in which the Wigner distribution is stationary.

The center of gravity is about 2 scaled units from the origin, and the roughly circular Gaussian shape has nearly the minimum

uncertainty width allowed by the Heisenberg

uncertainty relationship (AxA^ = Vj).

the mapping interaction is applied for a time t, its probability of being in state |1) is^-"-

Pi(T) = X^"(^0,Po)COs2(n„T). (6)

This whole process—initial state preparation, displacement by (xq, Pq), mapping interaction for time r, measure- ment of Pi(t)—is repeated for different values of interac- tion time T. The motional probabilities P„(xo, Po) are then

extracted from equation 6 by Fourier transforming the measured Piir) at the known frequencies n„. Figure 5 shows the reconstructed Wigner distribution

for the single trapped ion in a classical-like coherent state, which is simply a wavepacket oscillating in the harmonic

potential without changing shape. In the laboratory

frame, the Wigner distribution rotates in phase space at the harmonic trap frequency w, but here the reconstruction

is performed in the rotating frame (rotating in phase

space), where W{x,p) is stationary. Within the limits of experimental uncertainty, the reconstructed Wigner dis- tribution is positive everywhere and the nearly Gaussian

hump has a width close to the Heisenberg limit, which is z\x Ap = V2 in the scaled coordinates (this is most obvious

in the contour plot at the bottom of the figure).

Figure 6 shows the reconstructed Wigner distribution

of the first excited energy eigenstate of the harmonic

oscillator (that is, the n = 1 Fock state). Although Fock

states of the harmonic oscillator are treated in every

introductory quanttmi mechanics textbook, the NIST ex- periments represent the first time Fock states (other than

the 72 = ground state) have been created on demand and

fully characterized. (In many quantum optics experiments, single-photon states have been produced using

down-conversion, but such states are not created on de- mand—they occur at random moments in a nonlinear

crystal—and their mode identity is not well defined.) In accord with the nonclassical nattire of this state, the Wigner distribution is negative around the origin. The

experimental reconstruction in the figure reaches approximately -0.25 at the origin of phase space, not far fi-om the theoretical value of -I/tt, which is in fact the largest

negative value the Wigner distribution (as defined in TN-146 April 1998 Physics Today 27
Figure 6: The first excited energy

eigenstate of a harmonic oscillator produces

this experimentally reconstructed Wigner

distribution. The coordinates x and p are scaled and in a rotating frame, as in figure 5. Because energy and phase are complementary,

the measured function is nearly rotationally

symmetric, providing no phase information. The measured values of the Wigner

distribution near the origin are negative and

reach a minimum value of about -0.25 at the

origin. This is close to the largest negative value possible (-l/vr) for a Wigner distribution.

equation 1) can reach in any physical system. Discrepancies with respect to theory can be traced to slight

imperfections in the preparation and are not surprising,

considering the stability required of the experimental

parameters—the reconstructions are the result of about

24 million preparations of the same state. Nevertheless,

the reconstructed Wigner distributions correspond very

closely to that of a pure quantum state. Applications for quantum trickery

The Wigner distribution W{x,p) corresponds most closely

to the idea of a phase-space probability distribution in quantum mechanics, making it a useful tool for characterizing quantimi states. We've seen that the Wigner

distribution is not a real probability distribution, because

certain joint events (such as simultaneous position and

momentum states) are inaccessible. Localized negative

values of W{x, p) emphasize this fact. To make a connection between the quantum Wigner distribution and the

"negative probabilities" derived in table 3 for flipping coins

in a black box, we're tempted to identify the joker in the

black box as Heisenberg himself, who somehow knows

what is to be measured each time, and changes the results of the coin tosses accordingly. And yet, in a sense, quantum mechanics is stranger still than this picture, for the

wavefunction or the Wigner distribution ensures the con- sistency of different measurements without any need for such a trickster behind the scenes. The recent experiments we have discussed, in which

quantum states of matter waves have been reconstructed

by mapping their Wigner distributions, were made possi- ble by advances in quantum state preparation and manipulation. These newly developed measurement tech- niques have abundant future applications. For instance,

in the fields of quantum control and quantum computing,

these techniques could be extended to provide a complete

picture of the evolution of a quantimi logic gate. Control

and diagnostics of an atomic beam at the quantum level might be a helpful tool in deposition techniques reaching

quantum-limited resolution. An intriguing prospect would

be to reconstruct the output state of laserlike sources of atoms that might evolve from the current research in Bose-Einstein condensates of dUute gases. A fundamental

application of these techniques will be the study of quantum

decoherence. The reconstruction of a quantum state as

it loses coherence may someday shed light on the elusive mechanisms at work when a wavefunction "collapses."

We thank our coworkers David Wineland, Wayne Itano, Dawn

Meekhof and Brian King at NIST; and Christian Kurtsiefer, Martin Wilkens, Ulf Janicke and Jilrgen Mlynek at the University of Konstanz. We give special thanks to Ben Stein of the American

Institute of Physics for suggesting this article and Matt Young at NIST for carefully reading the manuscript. The work at NIST

was supported by the National Security Agency, the Army Research

Office and the Office of Naval Research. One of us (Leibfried) acknowledges a German Research Foundation (DFG) research

grant. The work at the University of Konstanz was supported by

the DFG.

References

1. This example was inspired by an article by R. P. Feynman,

"Negative Probabilities," in Quantum Implications: Essays in Honor of David Bohm, B. J. Hiley, F. D. Peat, eds., Routledge

& Kegan Paul, London (1987), p. 235. See also R. P. Feynman,

Int. J. Theor. Phys. 21, 467 (1982).

2. E. P Wigner, Phys. Rev. 40, 749 (1932).

3. For general references on this subject, see Ulf Leonhardt,

Measuring the Quantum State ofLight, Cambridge U. P., New

York (1997). M. G. Raymer, Contemp. Physics 38, 343 (1997). W. P. Schleich, M. G. Raymer, eds., special issue ofJ. Mod. Opt., 44, no. 11/12 (1997).

4. D. T. Smithey, M. Beck, M. G. Raymer, A. Faridani, Phys. Rev.

Lett. 70, 1244(1993).

5. T. J. Dunn, I. A. Walmsley, S. Mukamel, Phys. Rev. Lett. 74, 884 (1995).

6. M. G. Raymer, M. Beck, D. F. McAlister, Phys. Rev. Lett. 72, 1137 (1994). U. Janicke, M. Wilkens, J. Mod. Opt. 42, 2183

(1995).

7. C. Kurtsiefer, T. Pfau, J. Mlynek, Nature 386, 150 (1997).

8. A. Royer, Phys. Rev. A 15, 449 (1977).

9. R. J. Glauber, Phys. Rev. 131, 2766 (1963).

10. D. Leibfried, D. M. Meekhof, B. E. King, C. Monroe, W. M.

Itano, D. J. Wineland, Phys. Rev. Lett. 77, 4281 (1996).

11. D. M. Meekhof, C. Monroe, B. E. King, W. M. Itano, D. J. Wineland, Phys. Rev. Lett. 76, 1796 (1996).

12. C. Monroe, D. M. Meekhof, B. E. King, D. J. Wineland, Science

272, 1131 (1996).

13. R.Blatt,PZoUer, Eur. J. Phys. 9, 250(1988).

28 April 1998 Physics Today TN-147
PHYSICAL REVIEW A VOLUME 57, NUMBER 6 JUNE 1998

Complementarity and Young's interference fringes from two atoms

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

Time and Frequency Division, National Institute of Standards and Technology, Boulder, Colorado 80303

U. Eichmann

Max-Born-Institut for Non-linear Optics and Short Pulse Spectroscopy, Rudower Chaussee 6, J 2489 Berlin, Germany

M. G. Raizen

Department of Physics, University of Texas, Austin, Texas 78712

(Received 18 November 1997)

The interference pattern of the resonance fluorescence from a J= 1/2 to J= 1/2 transition of two identical atoms confined in a three-dimensional harmonic potential is calculated. The thermal motion of the atoms is included. Agreement is obtained with experiments [U. Eichmann et al, Phys. Rev. Lett. 70, 2359 (1993)]. Contrary to some theoretical predictions, but in agreement with the present calculations, a fringe visibility greater than 50% can be observed with polarization-selective detection. The dependence of the fringe visibility on polarization has a simple interpretation, based on whether or not it is possible in principle to determine which atom emitted the photon. [81050-2947(98)01606-0]

PACS number(s): 03.65.Bz, 32.80.Pj, 42.50.Ct

L INTRODUCTION

Many variants of two-slit interference experiments, often "thought experiments," have been used to illustrate fundamental principles of quantum mechanics. Recently, Eich- mann et al. [1] have observed interference fringes in the resonance fluorescence of two trapped ions, analogous to those seen in Young's two-slit experiment. Of particular in- terest was the fact that the interference fringes appeared

when it was impossible in principle to determine which ion

scattered the photon and disappeared when it was possible to do so. This is in agreement with Bohr's principle of complementarity, which requires that the wave nature of the photon

(the interference fringes) cannot be observed under the same

conditions as its particle nature (the possibility of assigning

to the photon a trajectory that intersects just one of the ions). In contrast to many thought experiments [2], the disappear- ance of the fringes when the path of the particle can be

determined cannot be understood in terms of random classi- cal momentum kicks. The experiment contains features from some thought experiments of Scully and Driihl [3], regarding

the interference of light scattered by two multilevel atoms.

Recently, controversy has arisen over the mechanism by

which complementarity is enforced in a two-slit interference experiment. Some claim that the destruction of interference by a determination of the particle's path is always due to a random momentum transfer necessitated by the indetermi- nacy relations [4-6]. Others claim that the mere existence of

the path information can be sufficient to destroy the interfer- ence [7]. Englert et al. claim that the experiment of Eich- mann et al. supports the second position [8]. Published calculations explain some aspects of the observations of Eichmann et al. [9-13]. However, none of those

calculations include all of the factors required to make a comparison with the experimental data. Here we calculate

the scattering cross section for arbitrary directions and polarizations of the incident and outgoing light. While the results

were used in the analysis of the data in Ref. [1], the details of

the calculations were not given. The main limitation of the calculation is the use of perturbation theory, so that it is valid only for low light intensities. However, it includes the effect of thermal motion more precisely than has been done else- where, taking into account the actual normal modes of the system. Also, the actual experimental geometry is fully taken

into account, which is not always the case in the other cal- culations.

Finally, we clarify the sense in which the loss of the

fringe visibility [defined as (/max-^min)/(^max+'^min)] for certain detected polarizations is due to the existence of

"which path" information in the ions. This is an application

of the fundamental quantum principle that transition amplitudes are to be added before squaring if and only if they connect the same initial and final states.

II. EXPERIMENT

The experimental apparatus has been described previously

[1,14]. Figure 1 shows the geometry. Two '^^Hg"*" ions were

confined in a linear Paul (rf) trap by a combination of static and rf electric fields. The ions were laser cooled to temperatures of a few mK with a beam of linearly polarized,

continuous-wave light, nearly resonant with the 194-nm transition from the ground 6s ^Sxij level to the 6p ^Pia level. The laser beam diameter was about 50 /urn and the power

was 50 ytiW or less. The same beam was the coherent source

for Young's interference. Cooling in the trap resulted in strong localization of the ions, which was essential for observation of interference fringes. The trap potentials were

arranged so that a pair of ions would be oriented along the symmetry (Z) axis of the trap. The incoming photons, with wave vector kjn and polarization vector ejn , made an angle

of 62° with respect to the Z axis. The X axis is oriented so

that the X-Z plane contains kjn . Light emitted by the ions was collimated by a lens and directed to the surface of an

TN-148 57 4176
57 COMPLEMENTARITY AND YOUNG'S INTERFERENCE 4177

6p 'Pin -~(

A =194 nm ^ TV a a

's.n -Jmj = +1/2

mj = -1/2

mj = +1/2

mj = -1/2

FIG. 2. Zeeman sublevels involved in the 194-nm, 65 ^Si/2 to 6p ^P\i2 transition of '^^Hg*. The allowed tt and a transitions are labeled. The Zeeman splitting of the levels is exaggerated.

FIG. 1. Geometry of Young's interference experiment, projected onto the X-Z plane. The equilibrium positions of the two ions, represented by the filled circles, lie along the Z axis. The wave

vector kjn of the incoming photon is in the X-Z plane, making an

angle with the Z axis. The Y axis is out of the plane of the figure. The projection of the wave vector ko^ onto the X-Z plane makes an

angle 4> with kj,, . The angle that kp^, deviates from the X-Z plane in the + Y direction is O (not shown). The polarizafion vectors of the incoming and outgoing photons are €^„ and Cout •

imaging photodetector, which was used to observe the

fringes. The wave vector and polarization of an outgoing

photon are kout and ^out The projection of ko^ onto the X-Z

plane makes an angle </> with respect to kj^ . The deviation of

kout fj'om the X-Z plane in the + Y direction is <J> (not shown

in Fig. 1). The sensitive area of the photodetector included a range of from about 15° to 45° and a range of 4) from

about —15° to +15°. For polarization-selective detection, a

glass plate oriented at Brewster's angle was placed in the detection path so that nearly all of the light with eou, in the X-Z plane was transmitted into the glass, while some of the

light polarized along the Y axis was reflected to the imaging

detector. The input polarization ^i^ was varied. Another lens system formed a real image of the ions on a second imaging

detector. This image was used to determine when there were

precisely two ions in the trap.

III. TWO-ION HARMONIC-OSCILLATOR SYSTEM

In the pseudopotential approximation, the Hamiltonian for

the translational motion of the two ions in the harmonic trap

is

p p; e^ Wtrans=y- + ;r^ + nRl)+V(R2)+;; p5 ^, 2m 2m 47r6o|Ri~R2i

(1)

where R, and P, are the position and momentum of the ith

ion, e and m are the charge and mass of an ion, and

V{R)^-mcol{X'-+Y^)+-mcolz- (2)

is the potential energy of a single ion in the trap. In Eq. (2) we have made the approximation that the trap pseudopotential is cylindrically symmetric. Here R = {X,Y,Z), in the Cartesian coordinate system shown in Fig. 1. The classical

equilibrium positions of the ions, found by minimizing the

total potential energy, are R° = (c//2)Z and Rl=-{d/2)Z,

where J = (e"/2iTeomw2)"^, and it is assumed that a>^ >iOz.

For small displacements u, = Ri — Rj and U2 = R2 — R2

about the equilibrium positions, a harmonic approximation can be made. The Hamiltonian separates into terms involving

either center-of-mass (cm.) or relative coordinates and momenta defined by

u='"=(u, + U2)/2,

u^='=(u,-U2)/2,

pcm^p , p

prel=p p^ (3)

The translational Hamiltonian, in the harmonic approximation, is

Wtra„s=^^z(^r + l/2) + Aa;/,(/V^"^ +A^^,'" + 1) + hws{Nf +1/2) + ha)TiNf + Nf+ I). (4)

The number operators are defined in the usual way by N'j'"' ^{al'^yaf"' and Nf^iaf^af for i = X,Y,Z. The annihilation operators are defined in the usual way, for example.

„c.m

07 =

ma>2

cm. 1^ "Z +

^/4^mu>z

(5)

The three center-of-mass modes have the same frequencies

as those of a single ion in the trap: (o^ and o)^ . The three

relative modes include a symmetric stretch mode along the Z direction at frequency W5=V3a)2 and two tilting or rocking modes along the X and Y directions at frequency a»7-=(a>^- w^)"^. The eigenstates of //trans are the simultaneous eigenstates of the set of number

operators In'^"' ji'y""^n'^""- ,nf ,nf y^) with

values

In'x"' ji'y""^n'z'"- ,nf ,nf y^^^ eigenh[coz{n'z"' + 1/2) + a)R(n'x"'+n''y'^ + 1 ) + (Osinf + l/2) + Wr('2x' + "K'+l)]-

rV. ATOMIC LEVEL STRUCTURE

Figure 2 shows the magnetic sublevels involved in the 65 ^5 1/2 to 6p "Pi/2 transition. These levels form an approximately closed system since the probability that the 6p 'P\n

TN-149
4178 W. M. ITANO et al. 57

level radiatively decays to the Sd'^bs' 'Dyj level is only

1.4X 10~^ [15]. The rest of the time it returns to the ground

6s "5i/2 level. The 5d'^6s' ~Dy2 level has a lifetime of 9-ms

and decays with about equal probability to the ground level or to the 5d'^6s~ "£>5/2, which has a lifetime of 86-ms and

decays only to the ground level. Since the static magnetic field is small, we are free to define the quantization axis of the ions to be along the electric polarization vector €[„ of the incident light. If the static magnetic field is along some other direction, then the Zee- man sublevels defined according to the electric polarization vector are not stationary states. This does not change the

analysis as long as the Zeeman precession frequency is much

less than the inverse of the scattering time, which is approximately equal to the 6p ^Pi/2 state lifetime (2.3 ns). In the experiments described here, the magnetic field was small enough that this was always the case. Figure 3 shows a Cartesian coordinate system having its z axis oriented along e^n . The x axis is parallel to kjn . The y

axis is defined so that ix,y,z) forms a right-handed coordinate system. This coordinate system is more useful than the trap-oriented iX,Y,Z) coordinate system of Fig. 1 for describing the angular distribution of the scattered light.

V. SCATTERING CROSS SECTION

Consider the process in which two ions, initially in their ground electronic states, absorb a photon having a wave vector kjn and polarization €,„, emit a photon having a wave

vector kom and polarization tom and are left in their ground

electronic states. The ions may change their Zeeman sublevels during the process. Also, the motional state of the two

ions may change.

The electric-dipole Hamiltonian that causes the transitions

is

^ED=-DrE(Ri,f)-I>2-E(R2,/), (6)

where Dj and D2 are the electric-dipole moment operators

FIG. 3. Coordinate system for description of the direction and

polarization of the outgoing photon. The z axis is parallel to €;„ and

the X axis is parallel to kjn . The polarization vector e„ lies in the plane containing €-,„ and k^u, , while €„ is perpendicular to that plane.

for ions 1 and 2 and E(R,r) is the electric field, consisting of

a classical part, representing the incident laser beam, and the quantized free field operator

E(R,0 = e,„Re£'oe'''--""'"i'''

+ S 2 60 V

{a,e,e'^s^-alle-'^s^, (7)

where Re denotes the real part, £q is the amplitude of the

laser electric field, a^ is the annihilation operator for a photon of wave vector k^ , frequency co^ , and polarization e^ , and V is the quantization volume.

The electric-dipole Hamiltonian, in second-order pertur- bation theory, gives the cross section for the two ions to scatter a photon in a particular direction:

J S) i^MDre^Je -ik„,,,Ri ^F,.)(%.|(D,-eJ^"'in«.|>I^,)

oiq— (o^^+{Ej — Ej)/fi — i'y/2

(^I'^|(D20.-"'out•R2|^^.)(^I.^.|(D2•a,J^"'.^•«2|^I^,) w0- aji„+ {Ej-Ei)lh - iy/2

(8)

r

where Win=c|kin|, Wout~c|l^outl' ^^0 is the separation be- tween the ground and excited electronic states of an ion, y is the decay rate of the excited state, and Cj = o)l^^l6TT^c'^h^€l. The initial, final, and intermediate

states describing the electronic and motional degrees of free- dom of the system are |^,), l'^/), and 1"^^). The energies

£,, E/' and Ej are the motional energies of the ions in the

initial, final, and intermediate states. They depend on the values of the six harmonic-oscillator quantum numbers,

'rN-150

which we denote by {/ihoI- Because of energy conservation,

the frequency of the outgoing photon depends on the final state

(Oont=(^in+ iEi~Ef)/h. (9)

Thus the scattered light has a discrete frequency spectrum

and the different components could, in principle, be detected

separately. In Eq. (8) all frequency components are summed.
57 COMPLEMENTARITY AND YOUNG'S INTERFERENCE 4179

which is appropriate if the detection is frequency insensitive. The laser frequency is assumed to be nearly resonant with an

optical transition in the ion, so that only one intermediate

electronic state has to be included in the sums and we can

neglect the counterrotating terms. We ignore dipole-dipole

interactions between the ions because they were separated by

many wavelengths in the experiment. Here we specialize to the case of an ion that has no nuclear spin and has a "5 1/2 ground state and a ^Pyi excited state, like the '^^Hg'^ ions used in Ref. [1]. We denote a state in which ion 1 is in the

^Si/2, my— + 1/2 state, ion 2 is in the 'P]/2, mj= — 1/2

state, and has the harmonic-oscillator quantum numbers

{«Ho} by

|^) = |(25i/2,+ l/2),('Pi/2,-l/2)2^Ho})- (10)

There are four possible sets of initial ntj quantum numbers for the two ions and four possible final sets. There are two basic kinds of scattering processes, those that preserve

the mj quantum numbers of the ions and those that change

ntj of one ion. We treat these cases separately. The form of

Eq. (8) excludes the possibility of both ions changing their

rrij quantum numbers.

A. Both rrij quantum numbers remain the same ( 77 case)

In order to be definite, we set mj= + 1/2 for both ions,

both before and after the scattering, that is,

l^,)=l('5,/2,+ l/2),('5,/2,+ l/2)2{nHo},). (11)

l^/) = l('5,/2,+ l/2),('5,/2,+ l/2)2{«Ho}/)- (12)

We call this the tt case because it involves only tt transi- tions, that is, transitions that leave nij unchanged. Because of

the electric-dipole selection rules, the only intermediate

states that contribute nonzero terms are of the form

i%) = l(''Pi/2,+ l/2)i('S,/2, + l/2)2{'JHo}y> (13)

for the first sum and

l%)=l('5i/2,+ l/2),(-Fi/2,+ l/2)2{«H0};) (14)

for the second sum. The matrix elements connecting the initial states to the intermediate states are

<%|(D,eJe'''.^«p|^I^,) = ((2/>,,,,+ l/2)p|D,J(2Sl,2,+ l/2),)({/IH0};k''''"•''H{«H0},)

1 ^CPv2WTSrnX{nno}j\e"'-''r\{n^oh), (15)

where /? = 1 or 2, Dp^ is the z component of the D^ operator,

and i'^Pv2\\D^^^\\^Si/2) is the reduced matrix element of the dipole moment operator (the same for both ions). The angular distribution of the outgoing photon is contained in the matrix elements connecting the intermediate

states to the final states. The unit propagation vector for the outgoing photon is

kout~(sin ^ cos (p,sin # sin (p,cos i?). (16)

where -d and (p are spherical polar angles with respect to the ix,y,z) coordinate system of Fig. 3. The polarization vector €oM i"ust be perpendicular to k^^^ . We define two mutually

orthogonal unit polarization vectors, both perpendicular to

kout' by

eT^={ — cos ^ cos ^, — cos -& sin (p,sin i^) (17)

and

eg.= ( — sin (p,cos (p,0). (18)

Since only the z components of D, and D2 contribute to the matrix elements connecting the intermediate and final states

for this case, light with polarization vector €„. cannot be

emitted. With the choice of eout=6iT, the matrix elements connecting the intermediate states to the final states are

(^^|(D,.O^~''-''H%) = sind<(25,;2,+ l/2);,|D,J(2p,,2,+ l/2)p)<{nH0}/k"''-^

sin d

S

eSu2\\D^'^'Pv2){{nuo}f\e-'^^"'-^^\{nno}j)- (19)

Equation (8) for the cross section becomes

TN-151
4180

rfcr^i) sin^ -d dflr 36

j

leS.aW'YPmrl. C

W. M. ITANO el al.

({nH0}/k-"'-'''l{«H0},><{nH0};k'"'"-''>l{«H0},)

57 E

>o-(^m+ iEj-Ei)/h - iy/2

({«H0}/k-'"--^^l{«H0};)({'^H0};k"'-^^|{"H0},)

(Oq — a)j„+{Ej — Ej)/h — iy/2

(20)

The ( 1

) superscript on the cross section is to label it as the n

case. The same result would have been obtained for any of

the other three possible sets of initial mj quantum numbers,

so this is the general result for the case in which the nij values do not change. The presence of two terms in Eq. (20), which are added and then squared, is the source of Young's

interference fringes. These two terms can be identified with

the two possible paths for the photon, each intersecting one

of the two ions. The sums over intermediate harmonic-oscillator states can be done by closure if the energy denominators are constant. While they are not constant, because Ej — E^ varies, it can be

shown (see, for example, Ref. [16]) that the main contributions to the sum come from terms where l^^ — £,| is less than

or on the order of ^JRE^, where R is the photon recoil energy {nkoJ^/2m. For the Hg+ 194.2-nm transition, R = hX26.1

kHz. For Doppler cooling, £, will be on the order of hy,

where, for this transition, hy=hX70 MHz. Thus the rms

value of Ej — Ei will be on the order of /zX 1.4 MHz, which

is much less than hyl2. Therefore, while the denominators

are not strictly constant, they are nearly constant for the terms that contribute significantly to the sums.

If we neglect {Ej~Ej)/h compared to y/2 and use clo- sure to evaluate the sums, Eq. (20) simplifies to

da^'^ sin^ # iCS.aWD^'^fP \I2) d^r 36 (^^-^j2 + ^2/4 ^Ec,|({nHo}/k-'"--^>^"'"-"'|{«Ho},> + ({«Ho}>-'"--^^^'''"'-^^|{«Ho},)P

sin^#(^5ijD(^>p/>,,2)r

36 (0,0-0,^2+^2/4-^

sin^ # leSyiD^^r^P.n)]'

E C,K{nHo}/k-'^''>I^Ho},) + ({"Ho}/k-"'-''2|{''Ho},>P

36 iwo-coJ'+y'/4 fE CA{{n^o)f\e-"'-^^ + e-"^-^A{n^o])?, (21)

where q^kout" kjn. Since the branching ratio for decay of

the excited ^/'i/2 states to the ground ^51/2 states is nearly

100%, the spontaneous decay rate y is

y-

_|/'2c ||r)(l)||2p

3

\\ ^XIlW^ II ^ \I2)

Equation (21) for the cross section becomes

Jo-*^' sin^ ^

o-oA^in-wo)

(22)

Cfflout 877

xEK{«H0}/k-"''*' + ^-"'''^l{«H0},)P,

(23)

where <To = Xo^27r is the resonance cross section, Xq — 2ttc/(j)q is the resonance wavelength, and jC{(o^„~ coq) is a Lorentzian of unit height and width y:

C{W;„-lOo) =

{yl2y

(wi„-a)o)2 + (y/2)^

(24)

TN-152

In deriving Eq. (23) we have assumed that wo/^out"^ 1 • The sum over final harmonic-oscillator states can be done by closure:

^o-^" sin- ^

dCL^^i 877 a,£{co,,-coo){{nuo}i\ie'''-^' + e'^-^^~)

• 2 q

O 77 + .-"J(«i-i^2)|{„^j,). (25)

The exponentials can be combined in Eq. (25) because the components of Ri and R2 commute. The cross section can be written in terms of the equilibrium ion separation d and

the displacement coordinates Uj and U2 as

rfo-<" sin^ d

d^oui 877 c7o/:(a,i„-c.o)({«Ho},|2 + e">-'''+"'-"^^

+

^-,q.(d + u,-u,)||,^^^}_^_ (26)

The exponential factors in Eq. (26) depend on the relative coordinates of the two ions and not on their center-of-mass

coordinates.
57 COMPLEMENTARITY AND YOUNG'S INTERFERENCE . 4181

In order to compare with the experiment, we compute the cross section averaged over a thermal distribution of |{nHo}i)

initial states:

rfo-^'H sin^ d

dVL,

cro£(a,,„-a;o)[2 + e"''»(^"'<">-"2)) 877

(27)

where {A) denotes the thermal average of the operator A. For harmonic oscillators, the thermal averages have a simple

form [17,18]

{e- iq(u,-U2)\3^g-([q(ui-U2))^)/2_

(28)

While Refs. [17,18] assume a common temperature for all of

the harmonic-oscillator modes, Eq. (28) is still valid if dif- ferent modes have different temperatures. Different modes

are laser cooled at different rates depending on the direction of the laser beam. Hence the modes have different temperatures unless the energy transfer rate between them is fast [19]. The thermally averaged cross section is

da(I)

dVL^

sin- d

477 •O-oA^in-^o)

X[l+cos(q-d)e~<f'*'"i""2)]^>/2]^ (29)

which is equivalent to Eq. (1) of Ref. [1], except that it includes the sin^ -d angular dependence. The interference

fringe visibility is given by the exponential factor multiplying cos(qd). This factor decreases with increasing temperature and is analogous to the Debye-Waller factor for x-ray

scattering from a crystal. It can be rewritten as

^-([q.(u,-U2)]2)/2^exp

= exp

= exp

S(<<>4)-^(«>4|-

ficjx \ h(Oj

coth

hq\

2m coth [.

mcosy ^' 2

^T [ikBT'xJ 2mo}j XlksTfj 2mws

hq\

-coth \2k,Tfl

nrel rel prel\ ^X'^B'X Hy'^b^y hz'^b^z

miOn mo). mui^

(30)

where V2 is the temperature of the m^' mode, etc., and the approximation in the last line is valid when the mean

harmonic-oscillator quantum numbers are large. In the limit of small thermal motion or small |q| (near-forward scattering), the visibility can approach 100% (with polarized detection), in agreement with Ref. [13], but in contradiction to Ref. [11], where it was claimed that the visibility could not exceed 50%.

B. One nij quantum number changes ( o- case)

Here we consider the case in which one of the ions changes its nij quantum number in the scattering process. We call this the cr case since it involves a a transition, that

is, a transition that changes nij by ± 1 in one of the ions. There are eight cases since there are four possible initial states and two ions that could change quantum numbers.

In order to be definite, we pick the case where mj = + 1/2 for both ions before the scattering and ion 1 changes

to mj= — 1/2 after the scattering, that is,

i^,) = |(-5,/2,+ l/2),('5,p_,+ l/2)2{«Ho}/), (31)

l^/) = l('5,;2,-l/2),(-5,/., + l/2)2{«Ho}/). (32)

Only the first sum over ; in Eq. (8) contributes since only it contains D,, the dipole moment that leads to the change in

nxj of ion 1.

As in the previous case, the only intermediate states that contribute nonzero terms are of the form

l%) = l(''Pi/2, + l/2),('5,;2,+ l/2)2{«Ho},)- (33)

The matrix elements connecting the initial states to the inter- mediate states are

{^P,|(D,eJ."'^«.|^I^,) = ((-Fi/2.+ 1/2), |Di,|(25,/2,+ 1/2)1)

X({nuo}j\e"'-^^\{nno}i}

= ^(-/'i/2l|0('>fSi/2)({«Ho}>"'--^'|{«Ho}i>- (34)

In the -&= 77/2 plane, only the polarization corresponding

to €„. is emitted, but, in general, light with both Cg. and e^

contributes to the scattered intensity. We consider these two

cases separately. For ^0^1 = ^0-, the matrix elements connecting the intermediate states to the final states are TN-153
4182 W. M. ITANO et al. 57

le

''ouiRiI'vp' A

X({'^H0}/k"""'TM'^'l{«H0b)

is very similar to the tt case. The final result, analogous to Eq. (29) for the tt case, is

da(2]

d^onil Stt O-oA^in-Wo), (36)

which is independent of kou, and shows no interference

fringes. The same result would have been obtained for any of

the other three initial states since the absolute squares of the matrix elements are the same. where D^Jl j is the (1,- 1) spherical tensor component of the For €o^^^ = e^ , the matrix elements connecting the intermedipole moment operator for ion p. The rest of the calculation diate states to the final states are

-^(25j|D<"PP.;2)<{nHo}/k-"'--^'|{«Ho};),

(35)

-cos ^e''P (^^\iD,.eJe-'^o.-^^\^^)= -= <(25i/2,-l/2)i|Dl'2j(2p,,2,+ l/2),)({nHo}/k-"'ou.«.|{nHo};)

cos ^e''^

4~e eS,M'YP,n){{n^o}f\e-''°--^^{nno]j)- (37)

The final result is

da^^A cos^ 1^ dVt. Stt O-O'C('^in-Wo)' (38)

which shows no interference fringes. The same result would

have been obtained for any of the other initial states. For the cr case. Young's interference fringes are not observed because only one of the two terms inside the absolute value bars in Eq. (8) is nonzero. There is only one path for the photon, intersecting the ion whose state is changed in the

scattering process.

C. Total cross section with or without polarization-selective detection

In Ref. [1] a linear polarizer was sometimes placed before

the photon detector. For experimental convenience, the ori- entation of this polarizer was fixed, while the input polarization could be varied. To obtain the total cross section describing a given experimental situation, we sum over all final atomic states and average over all initial states. For

polarization-insensitive detection, we also sum over the polarizations of the outgoing photon.

The cross section for polarization-insensitive detection is ^^unpon /^^(i

d^. dn. + 2 da(2)dfl^ + 2 da(3)

dflr

O-Q = --£(a>,n-Wo){l+cos- 1^ 477

-Hsin^ -^[l+cos(q-d)e-<l'>("'-"2)l'V2]}.

(39)

The fringe visibility in this case cannot exceed 50%.

TN-154

The cross section for detection of light with polarization

€„ is da^

dflr.u,

da^'n Ida^'^ + 2

dCl. Jfl.

O-Q 477 + sin2 ^[l+cos(q-d)e-<['i-("'-"2)]'V2]}.

(40)

The fringe visibility in this case can approach 100% in the d=77/2 plane if the Debye-Waller factor is close to 1. The

cross section for detection of light with polarization e^. is

which is totally isotropic and shows no fringes.

D. Which-path interpretation

The presence of interference fringes in the 77 case and

their absence in the a case have a simple explanation in terms of the possibility, in principle, of determining which of

the two ions scattered the photon. Consider the sequence of

transitions in Fig. 4(a), representing the 77 case. Each box

represents the combined state of the two ions. Ion 1 is rep- resented by the diagram on the left side of a box and ion 2 by

that on the right. The ordering of energy levels is the same as

in Fig. 2. For simplicity, we neglect the translational degrees

of freedom, which lead to the appearance of the DebyeWaller factor in Eq. (29). The system begins in the state
57 COMPLEMENTARITY AND YOUNG'S INTERFERENCE 4183

lr) = l('5,/2, + 1/2), (-S„2, -1/2)2). (42)

One ion or the other absorbs a photon from the laser beam

and undergoes a 77 transition to the excited state. That ion emits a photon and undergoes a vr transition back to the ground state. The two paths, corresponding to either ion 1 or ion 2 scattering the photon, lead to the same final state. Therefore, the amplitudes for these two paths must be added

and this leads to interference. Since the final states of the ion

are the same as the initial states, it is not possible to deter- mine which of the ions scattered the photon by examining

their states. Now consider the sequence of transitions in Fig. 4(b),

representing the a case. As in the previous case, one ion or

the other absorbs a photon and undergoes a tt transition to the excited state. However, in this case, that ion undergoes a a transition when it emits a photon and changes its nij quan- tum number. The final states differ, depending on which of

the ions scattered the photon. Hence there is no interference between the two paths. It would be possible to tell which ion

scattered the photon by examining the states of the ions before and after the scattering.

The preceding analysis is valid only in the limit of low

laser intensity, so that the probability of both ions being excited at the same time is negligible and stimulated emission

can be neglected. It is not necessary that the two ions be in the same quantum state for interference to occur, only that the final combined states for the two paths be indistinguishable. For definiteness, a particular initial state [Eq. (42)] was

chosen. For each of the three other possible initial states, there is a process like Fig. 4(a) in which the ions scatter a photon and return to their original states and one like Fig.

4(b) in which one of them scatters a photon and changes its state. Processes of the former type lead to interference; those of the latter type do not.

VI. COMPARISON WITH EXPERIMENT

Figure 5 shows an image of the fringes observed for the tt case in which e^^ was perpendicular to the X-Z plane and the detector was sensitive only to light polarized parallel to e^n The dark spots are due to stray reflections of the laser beams.

When Cjn was rotated by 90° without changing the polarizer

in front of the detector (cr case), the image showed no

fringes. The image data from a single ion, which shows no

interference fringes, were used to correct the data of Fig. 5

for a slowly spatially varying detection efficiency. The data within the rectangle in Fig. 5 were summed along the vertical direction and divided by the detection-efficiency function. The normalized data points are shown in Fig. 6 together

with a least-squares fit. In this fit, as in Ref. [1], the temperatures of the stretch and tilt modes were assumed to have the

ratio expected from theory [19],

Tf/r^' = {l+[^cos\@)ry{l+[^sinHQ)]-'},

(43)

and both temperatures were allowed to vary together in the

fit. The fringe visibility in the vicinity of the X-Z plane is insensitive to the temperature of the Y motion, which is cooled indirectly by coupling to the other modes. The mean

ion separation was calculated from knowledge of the trap parameters. The dependence of Eq. (29) on the out-of-plane

angle $ is small and <J> was set to in the fit. The fitted value of Tf was 1.08±0.12 mK, or 0.92±0.10 times the Doppler-cooling limit. The fringe visibility, extrapolated to = 0, would be 100% if it followed Eq. (29). The fitted value for this parameter was (71 ±4)%. The errors represent

the standard deviations estimated from the fit. The maximum

observed visibility at the minimum value of in Fig. 6 is approximately 60%.

There are several likely causes of the difference between

the observed and predicted values of the fringe visibility.

First, the theory was derived for the limit of low intensity.

The saturation parameter was measured to be 5 = 0.078

±0.025 (see the Appendix). By itself, this would reduce the maximum visibility to (1 +.y)~' = 93% because the spectrum

of the resonance fluorescence in this polarization contains an

incoherent part [20]. Other likely causes of reduced visibility are unequal laser intensities at the two ions, imperfect polarizers, stray background light, and quantum jumps of one of

the ions to a metastable state, leaving only one ion fluorescing. Each of these effects might reduce the visibility by a few

percent.

VII. DISCUSSION

The fact that the resonance fluorescence from a two-level atom illuminated by weak, monochromatic light is coherent

with the applied field was noted by Heitler [21]. The spec- trum of the resonance fluorescence for arbitrary applied in- tensities was calculated by Mollow [22]. In the limit of low

applied intensity, the spectrum is monochromatic and coherent with the applied field (a S function). At higher intensities, the coherent component decreases in amplitude and a component not coherent with the applied field and having a width

equal to the natural linewidth appears. At very high intensities, the coherent component continues to decrease in amplitude and the incoherent component splits into three separate

Lorentzians. The existence of a coherent component in the resonance fluorescence of a single ion was confirmed directly

by Hoffges et al. by a heterodyne measurement [23]. Classically, we would expect the resonance fluorescence from two two-level atoms at fixed positions, excited by the same monochromatic field, to generate interference fringes

having 100% visibility in the limit of low applied intensity since the radiated fields are coherent with each other. At

higher applied intensities, the visibility should decrease since more of the resonance fluorescence intensity belongs to the incoherent component. Quantum treatments for two twolevel atoms have been given by Richter [24] and by Kochan

et al. [25], who predict a visibility equal to (1 +.y)~', where

s is the saturation parameter defined in Ref. [26]. This is just

the ratio of the intensity of the coherent component to the

total resonance fluorescence intensity for a single atom.

Polder and Schuurmans [20] calculated the spectrum of

the resonance fluorescence of a 7= 1/2 to J = 1/2 transition

for a single atom. The spectrum of the light having polarization e^ is like that for a two-level atom. Hence interference

fringes would be expected in the e^-polarized resonance

fluorescence from two such atoms for low applied intensity.

TN-155

ki>:l
4184 W. M. ITANO et al. 57

r^ =^

TT TT

V— -^J

a i^ (7

(b)

FIG. 4. Each box represents the combined state of the two ions. The ordering of energy levels is the same as in Fig. 2. In (a) (the tt case), one ion or the other undergoes a tt transition from the ground

to the excited state. That ion undergoes a tt transition back to the ground state. The two paths lead to the same final state of the two

ions. Hence the probability amplitudes must be added and interfer- ence is possible. In (b) (the a case), one ion or the other undergoes

a TT transition to the excited state, but the excited ion undergoes a a

transition to the ground state. The two paths lead to different final states of the two ions. Hence there is no possibility of interference.

In order for interference to occur, it is not necessary that the initial states of the two ions be the same, only that the final combined

states for the two paths be the same. TN-156

FIG. 5. Experimental fringe data for the case in which the detected light is polarized in the same direction as the incoming light (tt case). The ion separation d = A.\l /xm. The angle (p (the deviation from the forward-scattering direction) increases to the right. The decrease in visibility with increasing (f> is due to diermal motion of the ions. The dark spots are due to stray reflections of the

laser beams. The data within the rectangle were summed along the

vertical direction and least-squares fitted.

The spectrum of the light having polarization e^- does not

contain a S function. In the limit of low applied intensity, it is a Lorentzian having a width approximately equal to the photon scattering rate, which can be much less than the natural linewidth. Even for applied intensities approaching 5=1,

the coherence length is on the order of c/ y, where y is the spontaneous decay rate of the excited state. For the Hg"*"

6p '^P\i2 level, this is about 70 cm. For interference fringes

to exist, the radiation from the two atoms must be mutually

coherent. Whether or not fringes should exist in the

Co^-polarized light from two atoms is not immediately obvi- ous from a classical analysis. However, the perturbative

quantum treatment of Sec. V predicts that there should be no

interference since there is only one probability amplitude

connecting the initial and final states. The absence of inter- ference in this case is fundamentally a quantum effect, though one having more to do with the quantum nature of the atom and the existence of degenerate, orthogonal ground

2.0

0.0

I

I I r I

I I I I I

I

I I I I

I

I I I I

I

i • >

-I I I I I I

' ' I I I I I I I I I I.J I I I I I l_l I L. 20 25 30 35 40 45

(t)(deg)

50

FIG. 6. Experimental fringe data (dots) from the image of Fig. 5 and a least-squares fit (line) to the sum of the theoretical intensity [Eq. (29)] and a constant background. The fitted temperature is approximately equal to the Doppler-cooling limit.
57 COMPLEMENTARITY AND YOUNG'S INTERFERENCE 4185

states, than with the quantum nature of the electromagnetic

field. Precisely the same point was made by Scully and Driihl when they showed that interference fringes are not present in the Raman radiation emitted by two three-level atoms having

a A configuration [3]. Wong et al. [13] calculated the interference of resonance

fluorescence from two four-level atoms having a level struc- ture like that of '^^Hg "*". Their analytic calculations are for a simpler geometry than the one actually used by Eichmann

et al [1] and ignore the motion of the ions. They do, however, include the effect of the decrease in visibility due to the incoherent component of the resonance fluorescence, which

is not included in the perturbative calculation of Sec. V. The

analytic calculations of Wong et al. and the present calculations agree in the limits in which they are both valid, that is, for low applied intensities and for no ion motion. In particular, Wong et al. show that the fringe visibility can approach

100% at low applied intensities, with polarization-selective

detection. Wong et al. also made Monte Carlo wave-function

simulations, in which the motion of the ions was included

classically, and observed a decrease in visibility due to this

effect. Huang et al. [12] calculated the effect of thermal motion

on the interference fringe visibility for two two-level atoms,

each trapped in a separate harmonic well. They obtained an

expression equivalent to Eq. (1) of Eichmann et al. [1] for

this model. However, the treatment of Eichmann et al., the

details of which are given in the present article, is more

useful for the analysis of the experiment of Ref. [1] since it deals explicitly with the actual normal mode structure of the two trapped ions. Brewer has published a theory of interference in the light scattered from two four-level atoms [11]. One prediction of

this theory is that the fringe visibility cannot exceed 50%,

even with polarization-selective detection. This contradicts

the experimental results of Sec. VI shown in Fig. 6. While

the maximum visibility is about 60%, only slightly exceeding 50%, no background has been subtracted from the data and there are several known sources of decreased visibility,

including thermal motion of the ions, the incoherent component of the resonance fluorescence, and stray scattered light. The data were normalized by division by a slowly varying

detection sensitivity function, a process that cannot enhance

the visibility.

The basic flaw in Brewer's argument can be seen in Eq.

(2) of Ref. [11], where he lists the basis states for the two- atom system. The states |5)-|8) are the four states in which

both atoms are in the ground electronic state. The states 1 1 )-

1 4) are linear combinations of states in which one atom is in the ground state and one is in the excited state. However,

most of the possible states of this type are missing, apparently because of a false assumption that the allowed states must have a particular kind of exchange symmetry. For ex- ample, the intermediate superposition state shown in Fig.

4(a) is, in his notation.

1 -7=(kl^2)^n,<,l + l«1^2)'^««,2) (44)

in which the two atoms are initially in different nij states. Thus he reaches the false conclusion that the two atoms must

initially be in the same mj state in order for interference to occur. Since he misses half of the processes that lead to interference, he predicts a maximum visibility, with

polarization-sensitive detection, of 50% rather than 100%.

We conclude with some remarks regarding the principle

of complementarity. Wave and particle properties of light are complementary and hence cannot be observed at the same

time. If it is possible to determine which atom scattered the photon, the interference fringes must vanish. Feynman's

thought experiments, in which various methods of determining the path of an electron through a two-slit Young inter- ferometer lead to the destruction of interference fringes due

to a random momentum transfer, are often quoted (see Ref.

[2], pp. 1-6-1-11). However, in Chap. 3 of the same text- book, Feynman emphasizes the seemingly more fundamental

viewpoint that interference is present only if there exist dif- ferent indistinguishable ways to go from a given initial state to the same final state. His example of the scattering of neutrons from a crystal is very similar to the experiment of Eich- mann et al. If the nuclei of the atoms in the crystal have a nonzero spin, the angular distribution of scattered neutrons is the sum of a featureless background and some sharp diffrac- tion peaks. The sharp diffraction peaks are associated with neutrons that do not change their spin orientations in the

scattering. The featureless background is associated with neutrons whose spins change their orientations in the scattering. In this case, there must also be a change in the spin

orientation of one of the nuclei in the crystal. It would be

possible in principle to determine the nucleus which scat- tered the neutron, so there is no interference.

ACKNOWLEDGMENTS

This work was supported by the Office of Naval Research. U.E. acknowledges financial support from the Deutsche Forschungsgemeinschaft. Dr. J. M. Gilligan as- sisted in the early stages of the experiment and suggested the method that was used for the polarization-selective detection.

APPENDIX: CALIBRATION OF THE SATURATION

PARAMETER

For the case where an electric-dipole transition between a ^5 1/2 ground state and a 'P\a excited state is excited by

linearly polarized light, we define the saturation parameter s similarly to the way in which it is defined for a two-level system [26]. The magnetic field is assumed to be small and

the quantization axis for the ion is along the electric field. We define

nf/2

(a)o-a>J- + (7/2)2'

(Al)

and is not contained in the list. The neglect of these basis

states leads to the neglect of processes like that of Fig. 4(a),

where O, = 6""-|£'o(-Si/2||D*'^f Fi/,)!^"' is the Rabi fre- quency, and the other terms have been defined previously. In order for the perturbative analysis of Sec. V to be valid, we TN-157
4186 W. M. ITANO et al. 57

must have s<l. In the case of Hg"*", the "Pi/t state has a small (approximately 10 ~^) probability of decaying to the metastable "D3/2 state, which decays either directly to the ground state or to the metastable ^D^/2 state, which decays to the ground state. The 194.2-nm fluorescence intensity from a single ion is bistable since it has a steady level when the ion

is cycling between the "5 1/2 and 'P\/2 states and vanishes

when the ion drops to a metastable state. The fractional population of the ^P^a state, summed over both nij values,

is 5/2(1 +s) while the ion is cycling between the ^5j/2 and

^Pi/2 states. The quantum jump statistics have been dis- cussed in several previous articles [15,27,28]. For a single

ion, we define pg^ to be the fraction of the time that the ion

is cycling between the "5 1/2 and 'P1/2 states and p^ff = (1 — Pon) to be the fraction of the time that it spends in either of

the metastable states. It can be shown, from the steady-state

solutions of the differential equations for the populations

[Eqs. (2a)-(2c) of Ref. [28]], that s is related to the ratio Poff/Pon according to

12 (I+5)

Tl r2(Poff/Pon) ^^^ Poff -— -^0.36 , 731 72 +72 Tl) Pon (A2)

where the parameters yj, yi, 73, and/2 have been measured

[15] and the uncertainty in the coefficient (0.36) is about

30%, due mostly to the uncertainty in 73. For two ions, the fluorescence will be tristable since zero, one, or two ions may be in a metastable state. During an

interference fringe measurement, the number of photons detected in each successive period of a few milliseconds was

recorded. Figure 7 shows a plot of the probability distribu- tion of the 5-ms photon counts during the measurement of

Fig. 5. The three peaks correspond, from left to right, to two,

one, or zero ions being in a metastable state. The leftmost peak corresponds to the signal from stray background light since there is no fluorescence from the ions. The curve is a

least-squares fit to a sum of three Gaussians. The areas under

1500

.E 1000

0)E

3 500

50 100 Photon counts in 5 ms bin 150

FIG. 7. Plot of the probability distribution of the fluorescence

intensity for two ions, used to determine the saturation parameter s. The horizontal axis corresponds to the number of photons counted

in a 5-ms interval. The vertical axis corresponds to the number of 5-ms intervals in which a given number of photons was counted. This was measured simultaneously with the interference fringes

shown in Fig. 5. The curve is a least-squares fit to a sum of three Gaussians. The areas under the Gaussians, from left to right, are proportional to the probabilities that two, one, or none of the ions

are in a metastable state. Higher values of j correspond to higher

populations in the metastable states.

the peaks should be in the ratio Poff -^PoffPon -Pon • The ratios of the areas obtained from the fit are 0.011:0.160:0.828, so

Poff/Pon = 0-10±001 and, from Eq. (A2), 5=0.078

±0.025, so the perturbative analysis should be a good ap- proximation.

During this measurement period, the interference fringe

detection was gated off for 5 ms if the number of photons

detected in the previous 5 ms was less than 80. This helped

to prevent loss of the fringe visibility due to background

from single-ion fluorescence, which would have no interfer- ence fringes.

[1] U. Eichmann, J. C. Bergquist, J. J. Bollinger, J. M. Gilligan,

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

Lett. 70, 2359 (1993).

[2] R. Feynman, R. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison-Wesley, Reading, MA, 1965), Vol.

III.

[3] M. O. Scully and K. Druhl, Phys. Rev. A 25, 2208 (1982).

[4] P. Storey, S. Tan, M. Collet, and D. Walls, Nature (London)

367, 626 (1994).

[5] E. P. Storey, S. M. Tan, M. J. Collet, and D. F. Walls, Nature (London) 375, 368 (1995).

[6] H. W. Wiseman, F. E. Harrison, M. J. Collett, S. M. Tan, D. F. Walls, and R. B. Killip, Phys. Rev. A 56, 55 (1997).

[7] M. O. Scully, B.-G. Englert, and H. Walther, Nature (London)

351, 111 (1991).

[8] B.-G. Englert, M. O. Scully, and H. Walther, Nature (London)

375, 367 (1995); Sci. Am. (Int. Ed.) 271(6), 56 (1994).

[9] R. G. Brewer, Phys. Rev. A 52, 2965 (1995).

[10] R. G. Brewer, Phys. Rev. A 53, 2903 (1996). TN-158

11] R. G. Brewer, Phys. Rev. Lett. 77, 5153 (1996). 12] H. Huang, G. S. Agarwal, and M. O. Scully, Opt. Commun.

127, 243 (1996).

13] T. Wong, S. M. Tan, M. J. Collett, and D. F. Walls, Phys. Rev. A 55, 1288 (1997).

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

D. J. Wineland, Phys. Rev. A 45, 6493 (1992).

15] W. M. Itano, J. C. Bergquist, R. G. Hulet, and D. J. Wineland,

Phys. Rev. Lett. 59, 2732 (1987).

16] H. J. Lipkin, Ann. Phys. (N.Y.) 18, 182 (1962).

17] N. D. Mermin, J. Math. Phys. 7, 1038 (1966).

18] D. S. Bateman, S. K. Bose, B. Dutta-Roy, and M. Bhatta- charyya. Am. J. Phys. 60, 829 (1992).

19] W. M. Itano and D. J. Wineland, Phys. Rev. A 25, 35 (1982). 20] D. Polder and M. F. H. Schuurmans, Phys. Rev. A 14, 1468

(1976). 21] W. Heitler, The Quantum Theory of Radiation, 3rd ed. (Oxford

University Press, London, 1954), pp. 201 and 202. 22] B. R. Mollow, Phys. Rev. 188, 1969 (1969).
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[23] J. T. Hoffges, H. W. Baldauf, T. Eichler, S. R. Helmfrid, and Photon Interactions (Wiley, New York, 1992), p. 369. H. Walther, Opt. Commun. 133, 170 (1997). [27] W. M. Itano, J. C. Bergquist, R. G. Hulet, and D. J. Wineland,

[24] Th. Richter, Opt. Commun. 80, 285 (1991). Phys. Scr. T22, 79 (1988).

[25] P. Kochan, H. J. Carmichael, P. R. Morrow, and M. G. Raizen, [28] W. M. Itano, J. C. Bergquist, and D. J. Wineland, Phys. Rev. A

Phys. Rev. Lett. 75, 45 (1995). 38, 559 (1988).

[26] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, AtomTN-159
Volume 81, Number 7 PHYSICAL REVIEW LETTERS 17 August 1998

Cooling the Collective Motion of Trapped Ions to Initialize a Quantum Register

B. E. King,* C. S. Wood, C. J. Myatt, Q. A. Turchette, D. Leibfried,' W. M. Itano, C. Monroe, and D. J. Wineland

Time and Frequency Division, National Institute of Standards and Technology, Boulder, Colorado 80303

(Received 6 March 1998)

We report preparation in the ground state of collective modes of motion of two trapped 'Be"*" ions. This is a crucial step towards realizing quantum logic gates which can entangle the ions' internal electronic states. We find that heating of the modes of relative ion motion is substantially

suppressed relative to that of the center-of-mass modes, suggesting the importance of these modes in future experiments. [50031-9007(98)06838-0]

PACS numbers: 03.67.Lx, 03.65. -w, 32.80.Pj

In physics, quantum computation [1] provides a general framework for fundamental investigations into subjects such as entanglement, quantum measurement, and

quantum information theory. Since quantum computation

relies on entanglement between qubits, any implementation of a quantum computer must offer isolation from

the effects of decoherence, but also allow controllable and coherent interaction between the qubits. Cirac and

Zoller [2] have proposed an attractive scheme for realiz- ing a quantum computer, which is scalable to an arbitrary

number of qubits. Their scheme is based on a collection of trapped atomic ions, where each qubit (one per ion)

is comprised of a pair of the ions' internal states, while

quantum information is transferred between different ions using a particular quantized mode of the ions' collective motion. This "quantum data bus" must first be initialized

in a pure quantum state [2]: for example, its ground state

[3]. The basics of this scheme have been demonstrated

experimentally in a fundamental logic gate (a Controlled- NOT) operating between a motional mode of a single

trapped ion and two of the ion's internal states [4]. In

that work, the motional state was initialized in the ground

state by laser cooling [5]. The next step towards implementing the Cirac-Zoller scheme is to cool at least one mode of collective motion of multiple ions to the ground

state. In this Letter, we describe the first experiments to realize this goal. We also report a significant difference

between the decoherence rates of the center-of-mass and

non-center-of-mass modes of motion.

We confine ^Be"*" ions in a coaxial-resonator-based rf (Paul) trap, similar to that described in Ref. [6]. The

electrodes in this trap are made from 125-/U,m-thick

sheets of Be metal, as shown in Fig. 1. We apply a potential 0(r) = Vocosiflrt) + Uq to the (elliptical) ring electrode relative to the end cap electrodes. If several ions are trapped and cooled, they will naturally align

themselves along the major axis of the ring electrode. The electrode's elliptical shape, in combination with

Uq > 0, allows a linear crystal to be maintained while

suppressing rf micromotion of the ions along this direction [7]. With Vo =« 520 V, ilT/27T = 238 MHz, and

f/o = V, the pseudopotential oscillation frequencies are

u (o,)/27T - (4.6, 12.7, 17.0) MHz. With Uq =

18.2 V, the frequencies become (8.6, 17.6,9.3) MHz.

Figure 1 shows two ions confined in the trap and imaged

with an //3 lens system onto a position-sensitive photomultiplier tube. The ions are cooled and probed with laser beams whose

geometry is indicated in Fig. 2(a). The relevant level structure of ^Be^ is shown in Fig. 2(b). The quantization axis is defined by an applied static magnetic

field; \B\ ~ 0.2 mT. The levels of interest for quan- tum logic operations are the 2s '^Si/zlF = 2, rrifr = 2) and

2s~S\/2\F = \,mf = 1) states, abbreviated by | i) and

It), respectively. Laser beams Dl, D2, and D3 are a^

polarized and focused to nearly saturate the ions (/sat ^

85 mWcm~-). Beams Dl and D2 provide Doppler pre- cooling in all three dimensions, and beam D3 prevents optical pumping to the \F = 2, m/r = 1) state. The | i) —

2p -P3/2IF = 3,mf = 3) transition (radiative linewidth

y/277 == 19.4 MHz), driven by D2, is a cycling transi- tion, which allows us to detect the ion's electronic state

(1 1) or 1 1)) with nearly unit detection efficiency.

Beams Rl {cr^ / a~ polarized) and R2 (tt polarized)

are used to drive stimulated Raman transitions between

525 |im

3nm

FIG. l(color). Two ions trapped in an elliptical rf (Paul) trap. The ring has an aspect ratio of 3:2 and the major axis is 525 yLtm long. The slot which forms the end caps is 250 yttm across. A potential 4>{t) is applied to the ring (see text). The Be sheets are =125 /i.m thick. With an .v-axis pseudopotential

oscillation frequency cdJItt = 4.6 MHz, the ion-ion spacing

is approximately 3 jxxn.

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Volume 81, Number 7 PHYSICAL REVIEW LETTERS 17 August 1998

I i) and It), through the virtual 2p^P\/2 state [5]. These

beams are derived from a single laser, whose output is split by an acousto-optic modulator [8]. The beams are detuned by A/27r ^ 40 GHz to the red of the 2s -S\/2 —^

2/? ^P\/2 transition, and their frequency difference is tuned around the 2s ^S]/2 hyperfine splitting of (jl)qI2tt ~

1.25 GHz. (Here, ojq includes stable shifts of a few megahertz from the Zeeman and ac Stark effects.) R2 is di- rected along (-1/V2)x + (1/2) (-j + z). If Rl ± R2

as in Fig. 2, then the Raman beam wave vector difference

Sk II Jc, and the transitions are sensitive to ion motion

only in this direction. If, however, Rl is counterpropagating to R2, the transitions become sensitive to motion

in all three dimensions.

When two cold ions are held in the trap and undergo

small oscillations about their equilibrium positions, we

may solve the equations of motion using normal mode

coordinates. For two ions lying along the x axis there are two modes involving motion along this axis: the center-of- mass (COM) mode (in which the ions move together with

frequency coqom = (^x) and the stretch mode (wherein

the ions move out of phase, with frequency Wstr =

v3ft^coM)- The other motional frequencies are Wy (y center of mass), (o~ (z center of mass), yJa)^--(o\ {xy

rocking), and \l oy\ — o)\ {xz rocking).

The lower traces in Fig. 3, taken with 5k \\ x, show

an X-axis normal mode spectrum; results for the y and

z modes are very similar. We take the data with the following steps: first we turn on beams Dl, D2, and

D3 for approximately 10 yits to Doppler cool the ions

to the Lamb-Dicke regime, where the ions' confinement

is much smaller than the laser wavelength. Next, we

turn off beam D2, and leave beams Dl and D3 on for

5 /xs to optically pump both ions to the j [) state. We

then turn on only the Raman beams Rl and R2 for a time /pr, with relative detuning wq + 5pr (the "Raman

probe" pulse). Finally, we drive the cycling transition with D2 and measure the ions' fluorescence. We repeat

the experiment at a rate of a few kilohertz while slowly

sweeping 5pr. If the Raman beam difference frequency

is resonant with a transition, then an ion is driven from

I i) — If) and the D2-driven fluorescence rate drops. For a single ion, the carrier transition (5pr = 0) causes

the population to undergo sinusoidal Rabi oscillations between U) and ||) [9]. The effective Rabi frequency

is ft = g\g2/ ^ = 277 X 250 kHz, where g\, g2 are the single-photon resonant Rabi frequencies of beams Rl and

R2. (We assume A » y, a»,„ » ft, where (Om is the fre- quency of the motional mode of interest.) If 5pr = — a*^ (the first lower x sideband), then the transition couples the

states li, «.x) and 1 1, «,: — 1), where /i^ is the vibrational

level of the quantized motion along x. In the LambDicke regime, the corresponding Rabi frequency is given

by ft„,,„,-i = r]x-Jn^^ [9]. Here, r]^ = xo\5k x\

is the Lamb-Dicke parameter (= 0.23 when (Oj^/2tt =

8.6 MHz) and xq = ^h/(2m(i}x) is the spread of the

tix = wave function, with m being the ion's mass).

(Note that if the ion is in the n^ = state of motion, this lower sideband vanishes.) The first upper x sideband

transition (5pr = +0Jx) couples \[.,n^) and 1 1, ''.v + 1) with Rabi frequency ft„,,„, + i = rixyjn^ + 1 ft. In the case of M'o ions driven on the carrier transition, each ion independently undergoes Rabi oscillations

RKctV) R2(7t) 2pT

(b)

FIG. 2. (a) Laser beam geometry. The trap ring electrode is shown rotated 45° out of the page. The end cap electrodes

are omitted for clarity (see Fig. 1). A magnetic field B

of magnitude 0.2 mT defines the quantization axis along

-{\/\f2)x + (I/2)(^ - z), and laser beam polarizations are indicated, (b) Relevant ''Be"^ energy levels (not to scale), indicated by F, mp quantum numbers in the ground state. -P

fine structure splitting is =197 GHz, -S^/j hyperfine splitting is coq/Itt = 1.25 GHz, 2Pi/2 hyperfine splitting is —237 MHz,

and the ~Pi/2 hyperfine structure («:y/27r == 19.4 MHz) is not resolved. All optical transitions are near A = 313 nm, and A/27r == 40 GHz.

o3

1—I—I—r T—I—I—

r

T II T 1 I

I r 16 -14 -9-8 8 9

5p/(27i) (MHz)

14 16

FIG. 3. Spectrum of sidebands due to two-ion .r-axis normal mode motion: (from left to right) lower stretch, lower COM,

upper COM, and upper stretch. The ordinate is the detuning

of the Raman probe beam difference frequency from the

carrier transition. The abscissa shows the ion fluorescence

(proportional to the expectation value of the number of atoms in the state | i)), plus a constant background (whose approximate

level for the lower curves is indicated by the dashed line). The solid lines, meant as guides to the eye, are fits to Gaussians. The lower traces show the effects of Doppler

cooling. The upper traces, offset vertically for clarity, show the

effects of several pulses of Raman cooling on the mode which

is displayed. Vanishing lower motional sidebands indicate cooling to the ground state of motion. The peak widths are consistent with the Raman probe pulse lengths (=3 yu.s).

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Volume 81. Number 7 PHYSICAL REVIEW LETTERS 17 August 1998

between | i) and | |) with Rabi frequency Cl. Since

the laser beam waists (=20 yttm) are much larger than the ion-ion separation (=2 /mm), the ions are equally illuminated. Nonetheless, if the micromotion

of the two ions is different, then the reduction of the

carrier (and sideband) transition strengths due to the micromotion will give a different Rabi frequency for each ion [7,10]. This could be used as a means of

selectively addressing the ions [11]; however, in the present work the two ions' Rabi frequencies were

equal.

Since the sideband transitions affect the motional state, which is a shared property of both ions, such transitions produce entanglement between the ions' spins and their collective motion [12]. The system can no longer be

treated as two, independent, two-level systems and the measured fluorescence following the Raman probe is a complicated function of the probe pulse duration tpr- For

example, given an initial state U, 1, «) (where n is the

vibrational level of the COM or stretch motion along the X axis) driven on the corresponding lower sideband for a time /pr, the wave function evolves as

\Mtpr)) = 1 - 5;^[1 - cos(Grp,)] U,l,n) - /.'''-*)/-^^^^sin(Grp,)

i\U}±e''f>\li))\n - 1) V2

- ,ie ^n~ n

^^ _ ^^^((j^^^)] | ^ ^ „ _ 2), In - 1

where G = yj2(2n — \)Vliqxm and 6, (f> are the sum and

difference of the Raman beam phases at the ions. On the COM sideband [top sign in Eq. (1)], rj;,,^ = 77^,com =

r)x/y/2 (down by a factor of V2 from the single-ion case due to the extra mass of the two-ion string), whereas on the

stretch sideband (lower sign), r}^„, = rj^ ^tr = '7x/v 2^3.

The expressions for transitions on the upper motional

sidebands are similar. If, before the Raman probe pulse, the ions have probability p„ of being in the motional state

\n), the subsequently measured average fluorescence from

the cycling transition is

5(v)= X/'"(2ia,i,«i<AnM>i'

n+ Kiln - \\>Pn{tpr))\^

+ Kiln - l|.AnM)l') (2)

This signal is proportional to the expectation value of the number of atoms in the state | i). For the data shown in Fig. 3, ^pr was chosen to maximize the sideband features. The upper traces in Fig. 3 show the effects of adding

several cycles of Raman cooling [5] on one particular

X mode after the Doppler cooling but before the probe

pulse. The reduction in the mean vibrational number (n)

is indicated by the reduction in size of the lower sideband,

which vanishes in the limit (n) —> 0. The data are consistent with a thermal state of {«com) = 0.11-o!o3 or

(nstr) = 0.01 ^aoi- This implies that the COM and stretch modes are in their ground states 90^12% and 99-7% of the time, respectively. We have also simultaneously cooled

the COM and stretch modes along x, to comparable values of (n) (and have separately cooled the other four motional

modes

y and z COM, xy rocking, and xz rocking —to near their ground states). Each cycle of Raman cooling consists of (i) a pulse of

the Raman beams with their difference frequency tuned to

(1)

one of the lower sidebands (COM or stretch mode) and

(ii) optical repumping to the | i) state driven by beams Dl

and D3. The Raman transition reduces the vibrational en- ergy by Hcom, whereas the repumping, on average, heats each mode by approximately the recoil energy {«fi(Om)-

Therefore, the ion is cooled through this process. Five

pulses of Raman cooling were used for the data shown

in Fig. 3. The exact durations of the Raman pulses were

chosen to optimize the cooling rate —each pulse was ap- proximately 5 /Lis long. For an ion-trap implementation of a quantum computer,

the motional modes are most susceptible to decoherence.

The ions' motional states lose coherence if they couple to (stochastic) electric fields caused by fluctuating potentials on the electrodes. This leads to heating, which has

previously been observed in single ions [5,10,13]; in Ref. [5], the heating drove the ion out of the motional

(COM) ground state in approximately 1 ms. We have

performed similar heating measurements on the COM and

non-COM modes of motion of two ions. The results are summarized in Table I. The heating rate was determined

by inserting a delay between laser cooling and the Raman

TABLE I. Heating rates of the six normal modes of two

trapped ions. The Raman beams were counterpropagating for

the y- and c-axis data, making the Raman probe sensitive to motion in all three dimensions. Note that the COM modes are heated at a much higher rate than the non-COM modes (see

text). (The precision with which the heating rates are given for

the last five modes is limited by measurement noise.) Mode w„,/27r (MHz) S{n)/8t (ms-')

-'^COM 8.6 19115

>'COM 17.6 >10

ZCOM 9.3 >20

-^slr 14.9 <0.18

-''^Vrocking 15.4 <1

-"^^rocking 3.6 <0.5

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Volume 81, Number 7 PHYSICAL REVIEW LETTERS 17 August 1998

probe. The main results from these data are that the COM

modes are heated out of the ground state much more

quickly than the non-COM modes. This can be explained

as follows. The COM modes, in which both ions move in phase, can be excited by a uniform electric field. However,

no non-COM mode can be excited by a uniform electric

field [14] —since these modes involve differential motion

of the ions, they can be driven only by field gradients.

If the fluctuating field at the ion (along the direction of motion of the mode of interest) is E{t), an estimate

of the corresponding field gradient is E{t)/d, where d

is a characteristic internal dimension of the trap. For

stochastic fields, the COM heating rate scales as {E^{t)); the non-COM mode heating rates scale as ([-j-Ax]^)

(where Ax is the ion-ion separation), down by a factor of 10"* for the present trap. Similarly, other non-COM

modes for more than two ions can be excited only by

higher-order field gradients, leading to further reductions

in their heating.

This suggests using non-COM modes for the quantum

data bus in the Cirac-Zoller scheme. Excitation of the "spectator" COM modes along the direction to which the Raman transitions are sensitive will still alter the Rabi

frequencies, but these effects will be higher order in the Lamb-Dicke parameter [10]. In the two-ion example, in the Lamb-Dicke regime, the Rabi frequency for a first sideband transition |«i) — \n\) on (cold) mode I, given

that (hotter) mode 2 is in the state \n2), is [10]

n„„„i(n2) = ^vi^J^e-^'^'^^^^'Hi - m-nl), (3) where n\> denotes the larger oi n\ or ni, and 771 and

772 are the Lamb-Dicke parameters for modes 1 and 2, respectively. Fluctuations in the Rabi frequency of mode

1 due to fluctuations in nj therefore occur in order 772. However, for the conditions of the present experiment, even if quantum logic operations were performed using

the x-stretch mode, the x-COM mode heating would still limit the number of operations to around ten by the above

mechanism. Clearly, this heating must be eliminated in future experiments.

The two-ion cooling results presented here are comparable to our previous single-ion results [5], indicating that

rf heating should not be a concern for small numbers of

ions [10]. Comparable cooling for jV > 2 ions should not present any fundamental difficulties, as long as spurious

overlaps of motional modes are avoided.

The preparation of a pure state of motion (the ground

state) of multiple trapped ions represents the first step

towards realizing quantum logic operations on them.

Such operations should lead to the creation of arbitrary

entangled states of massive particles, including EPR- or GHZ-like spin states [15]. Unlike other systems in which

EPR states have been generated, it should be possible to reliably create these states on demand [11] rather than by

a selection process, and to detect them with nearly perfect

efficiency [16]. We acknowledge support from the U.S. National

Security Agency, Office of Naval Research, and Army

Research Office. We thank Kristan Corwin, David

Kielpinski, and Matt Young for critical readings of the manuscript.

*Electronic address: kingb@ucsu.Colorado.EDU

'Present address: Institut fiir Experimentalphysik,

Universitat Innsbruck, Innsbruck, Austria.

[1] A. Ekert and R. Jozsa, Rev. Mod. Phys. 68, 733 (1996); A. Steane, Rep. Prog. Phys. 61, 117 (1998).

[2] J. I. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995).

[3] If the "data bus" is in a thermal state with probability /'o of being in the ground state then, if we ignore all other sources of error, a given quantum computation must be

repeated \/Pq times, on average, to give a correct answer.

[4] C. Monroe et al., Phys. Rev. Lett. 75, 4714 (1995).

[5] C. Monroe et al., Phys. Rev. Lett. 75, 4011 (1995).

[6] S.R. Jefferts, C. Monroe, E. Bell, and D.J. Wineland,

Phys. Rev. A 51, 3112(1995).

[7] C. J. Myatt et al, in Methods for Ultrasensitive Detection, SPIE Proceedings Vol. 3270, edited by Bryan L. Fearey (SPIE- International Society for Optical Engineering, Bellingham, WA, 1998) p. 131; R.G. Devoe, Phys. Rev. A 58, 910-914(1998).

[8] J.E. Thomas et al.. Phys. Rev. Lett. 48, 867 (1982).

[9] D.M. Meekhof e/ al., Phys. Rev. Lett. 76, 1796 (1996).

[10] D.J. Wineland et al.. J. Res. Natl. Inst. Stand. Technol.

103, 259 (1998); Forschr. Phys. 46, 363 (1998).

[11] Q. A. Turchette et al., quant-phy/9806012.

[12] D.J. Wineland, J.J. Bollinger, W.M. Itano, and D.J. Heinzen, Phys. Rev. A 50, 67 (1994).

[13] F. Diedrich, J.C. Bergquist, W.M. Itano, and D.J. Wineland, Phys. Rev. Lett. 62, 403 (1989).

[14] D.J. Wineland and H. G. Dehmelt, J. Appl. Phys. 46, 919

(1975); D.F. V. James, Phys. Rev. Lett. 81, 317 (1998).

[15] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev.

47, 777 (1935); D.M. Greenberger, M.A. Home,

A. Shimony, and A. Zeilinger, Am. J. Phys. 58, 1131

(1990); M. Lamehi-Rachti and W. Mittig, Phys. Rev. D

14, 2543 (1976); E. Hagley et al.. Phys. Rev. Lett. 79, 1 (1997).

[16] R. Blatt and P Zoller, Eur. J. Phys. 9, 250 (1988).

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Volume 81, Number 17 PHYSICAL REVIEW LETTERS 26 October 1998

Deterministic Entanglement of Two Trapped Ions

Q. A. Turchette,* C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried,' W. M. Itano, C. Monroe, and D. J. Wineland

Time and Frequency Division, National Institute of Standards and Technology, Boulder, Colorado 80303

(Received 26 May 1998)

We have prepared tiie internal states of two trapped ions in botli the Bell-like singlet and triplet entangled states. In contrast to all other experiments with entangled states of either massive particles or photons, we do this in a deterministic fashion, producing entangled states on demand without

selection. The deterministic production of entangled states is a crucial prerequisite for large-scale

quantum computation. [50031-9007(98)07411-0]

PACS numbers: 42.50.Ct, 03.65.Bz, 03.67.Lx, 32.80.Pj

Since the seminal discussions of Einstein, Podolsky,

and Rosen, two-particle quantum entanglement has been

used to magnify and confirm the peculiarities of quantum

mechanics [1]. More recently, quantum entanglement has been shown to be not purely of pedagogical interest, but

also relevant to computation [2], information transfer [3], cryptography [4], and spectroscopy [5,6]. Quantum computation (QC) exploits the inherent parallelism of quan- tum superposition and entanglement to perform certain tasks more efficiently than can be achieved classically [7]. Relatively few physical systems are able to approach

the severe requirements of QC: Controllable coherent in- teraction between the quantum information carriers (quantum bits or qubits), isolation from the environment, and

high-efficiency interrogation of individual qubits. Cirac

and Zoller have proposed a scalable scheme utilizing

trapped ions for QC [8]. In it, the qubits are two inter- nal states of an ion; entanglement and computation are achieved by quantum logic operations on pairs of ions in- volving shared quantized motion. Previously, trapped-ion

quantum logic operations were demonstrated between a

single ion's motion and its spin [9]. In this Letter, we use

conditional quantum logic transformations to entangle and

manipulate the qubits of two trapped ions. Previous experiments have studied entangled states of

photons [10,11] and of massive particles [12-14]. These

experiments rely on random processes, either in creation of the entanglement in photon cascades [10], photon

down-conversion [11], and proton scattering [12], or in the selection of appropriate atom pairs from a larger

sample of trials in cavity QED [13]. Recent results

in NMR of bulk samples have shown entanglement of

particle spins [14,15], but because pseudopure states are selected through averaging over a thermal distribution, the signal is exponentially degraded as the number of

qubits is increased. In the preceding experiments the

efficiency of state generation will exponentially decrease

with the system size (both particles and operations). This

is because the preceding processes are selectable but

not deterministic generators of entanglement. We mean

deterministic as defined in Ref. [16] which in the present context is "the property that if the [entanglement] source

is switched on, then with a high degree of certainty

[the desired quantum state of all of a given set of

particles is generated] at a known, user-specified time." Deterministic entanglement coupled with the ability to store entangled states for future use is crucial for the

realization of large-scale quantum computation. Ion-trap

QC has no fundamental scaling limits; moreover, even

the simple two-ion manipulations described here can, in principle, be incorporated into large-scale computing by

coupling two-ion subsystems via cavities [17], or by using

accumulators [6]. In this Letter, we describe the deterministic generation

of a state which under ideal conditions is given by

i<A.(0)) = ?iiT)-^'^^in), 5 Ul/ <^ 5 (1) where |i) and |t) refer to internal electronic states of

each ion (in the usual spin- 1/2 analogy) and is a

controllable phase factor. For = or tt, |i/'(.(0)) is a good approxijnation to the usual Bell singlet ( — ) or triplet

(+) state \iPb) = [lit) + IU)]/%/2 since K«Ab |iA.(0))P =

\{ipB\ipe{Tr))\' = 0.98 and £[(A.(0)] = 0.94 where E is the entanglement defined in [ 1 8] . We also describe a novel means of differentially addressing each ion to generate

the entanglement and a state-sensitive detection process

to characterize it, leading to a measured fidelity of our

experimentally generated state described by density matrix

p- of <(A.(7r,0)|p-|iA.(7r,0)) - {(As Ip-liAl) ^ 0.70. The apparatus is described in Ref. [19]. We confine ^Be"*" ions in an elliptical rf Paul trap (major

axis ~ 525 ^tm, aspect ratio 3:2) with a potential applied

between ring and end caps of Vq cos fir t + Uq with

fLT/27T = 238 MHz, Vq = 520 V. The trap is typically

operated over the range 12 < C/q < 17 V leading to secular frequencies of {(o^,coy,(v~)/27r = (7.3,16,12.6)

to (8.2, 17.2, 10.1) MHz. The ion-ion spacing (along x)

is / ~ 2 pm.

The relevant level structure of '^Be'^ is shown in Fig. la. The qubit states are the 2s ~S\/2 \F = 2,mf =

2) = li) and 2s -S\/2 \F = X^mp = \) ^ \\) states. Laser beams Dl and D2 provide Doppler precooling and beam D3 prevents optical pumping to the |F = 2,m/r = l) state. The cycling 11) ^ 2/? -P3/2

363]

IN- 164
Volume 81, Number 17 PHYSICAL REVIEW LETTERS 26 October 1998

Ut)

2,2 H|^)

(a)

lU) J L n+l

(b)

In)

FIG. 1. (a) Relevant ^Be""" energy levels. All optical transi- tions are near A = 313 nm, A/Itt = 40 GHz, and coo/2tt =

1.25 GHz. R1-R3: Raman beams. D1-D3: Doppler cooling,

optical pumping, and detection beams, (b) The internal basis qubit states of two spins shown with the vibrational levels connected on the red motional sideband. The labeled atomic states are as in (a); n is the motional-state quantum number (note that the motional mode frequency Wjir <^ ioq}- n,+ are the Rabi

frequencies connecting the states indicated.

\F = 3,/n/r = 3) transition driven by the cr^-polarized

D2 laser beam allows us to differentiate |T) from |i) in a

single ion with —90% detection efficiency by observing

the fluorescence. Transitions |i) \n) ^ |t) \n') (where n, n' are vibrational quantum numbers) are driven by stimulated Raman pro- cesses from pairs of laser beams in one of two geometries. Additionally, two types of transitions are driven:

the "carrier" with n' = n, and the red motional sideband

(rsb) with n' = n — 1 [20]. With reference to Fig. la, the pair of Raman beams R1-LR2 has difference wave

vector 8k \\ x and is used for sideband cooling (to pre- pare lU) |0)), driving the x rsb, and to drive the 'x carrier." Beam pair R2 || R3 with 8k ~ drives the "copropagating carrier" and is insensitive to motion.

Two trapped ions aligned along jc have two modes

of motion along x: the center-of-mass (cm.) mode (at (Ox) and the stretch mode (at co^u = y/lco^) in which

the two ions move in opposite directions. We sideband

cool both of these modes to near the ground state, but use the stretch mode on transitions which involve the motion since it is colder (99% probability of \n = 0))

than the cm. and heats at a significantly reduced rate [19]. Figure lb shows the relevant states coupled on the rsb with Rabi frequencies (in the Lamb-Dicke limit)

n,+ = y/nrj'Cli; a,- = Vn + 1 rj^lt , (2)

where 77' = 77/V2V3 is the stretch-mode two-ion LambDicke parameter (with single-ion 77 ~ 0.23 for co^/Itt ~

8 MHz) and fl, is the carrier Rabi frequency of ion / [9]. On the carrier the time evolution is simply that of

independent Rabi oscillations with Rabi frequencies H,-. On the copropagating carrier, fl] = D.2 — flc- In the Cirac-Zoller scheme, each of an array of tightly

focused laser beams illuminates one and only one ion for individual state preparation. Here, each ion is equally

illuminated, and we pursue an alternative technique to attain Hi ¥= Clj. Differential Rabi frequencies can be

used conveniently for individual addressing on the x

carrier: for example, if Hi = 2D.2, then ion 1 can be

driven for a time flit = tt {Itt pulse, no spin flip) while

ion 2 is driven for a tt pulse resulting in a spin flip. For differential addressing, we control the ion micromotion. To a good approximation, we can write [21]

n, = fi,jo{\8k\^,), (3)

where 70 is the zero-order Bessel function and ^, is the amplitude of micromotion at Hy (along x) associated

with ion /, proportional to the ion's mean x displacement

from trap center. The Bessel function arises because

the micromotion effectively smears out the position of

an ion, thereby suppressing the laser-atom interaction

[21]. The micromotion is controlled by applying a static electric field to push the ions [22] along Jc, moving ion 2

(ion 1) away from (toward) the rf null position, inducing

a smaller (larger) Rabi frequency. The range of Rabi

frequencies explored experimentally is shown in Fig. 2a. We determine fl\2 by observing the Rabi oscillations of the ions (between |i) and It)) driven on the x carrier. An example with fl\ = 2172 is shown in Fig. 2b. We

-2

1.0 I _L 1 jjsxct, 1 1 K (a)

'"

0.8 Q,

- a 0.6 ^ % Q, -

Cf' 0.4 - / \ — theory

y<r ^

0.2 / 1 1 " 1 / \

0.0 V , V1 2 d[^m]

2.0

I 1 1 1 1 ^ ft i

(b)-

1.5 i li ft ft ^

" ^10

llilWklw^ 0.5 -f _ data

fit 0.0 t

27i:7i

1 1 10

t[^s]

15 20

FIG. 2. (a) Normalized x-camer Rabi frequencies 0,70^ of each of two ions as a function of center-of-mass displacement d from

the rf-null position. The solid curves are Eq. (3) where the distance between the maxima of the two curves sets the scale of the

ordinate, based on the known ion-ion spacing of / = 2.2 /zm at co^/2tt = 8.8 MHz. (b) Example of Rabi oscillations starting from

the initial state |U)|/! = 0) with fl, = 2^2. A fit to Eq. (4) determines that fl\/2TT = IVIi/Itt = 225 kHz, j/Itt = 6 kHz, and a ~ -0.05. The arrow in (a) indicates the conditions of (b).

3632

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Volume 81, Number 17 PHYSICAL REVIEW LETTERS 26 October 1998

detect a fluorescence signal S{t) = IPn + (1 + a)P[] +

(1 - a)Pn where Pa, = \mt)\kl)\-, kj £ {M}, <A(0 is the state at time r and |a| <c 1 describes a small differ- ential detection efficiency due to the induced differential micromotion. Driving on the x carrier for time / starting

from lU) |0). Sit) can be described by

Sit) = 1 + (1/2) (1 + a)cos(2n,r)e"^' + (1/2) (1 - a)cos(2a20^~*"^^"'''", (4)

where y allows for decay of the signal [20]. The

local maximum at t = 2.4 /as on Fig. 2b is the In-.n

point at which ion 1 has undergone a 277 pulse while

ion 2 has undergone a n pulse resulting in |U) |0) —

lit) |0). Driving a 77:77 pulse on the copropagating

carrier transforms lit) |0) to |ti) |0) and |ii) |0) to |tt) |0),

completing the generation of all four internal basis states of Fig. lb. Now consider the levels coupled by the first rsb [20]

shown in Fig. lb. If we start in the state 11^(0)) =

lit) |0) and drive on the (stretch mode) rsb for time t, the Schrodinger equation can be integrated to yield

l«A(0) = -

+

iCl2-

G

G2

sin(Gr)lii)ll)

-(cosG/ - 1) + 1 lit) 10)

+ e"

^2-^1-

G2

(cosG/ - 1) lti)|0), (5)

where G = (af_ + 0?-)^/^ ^^d a,-^is from Eq. (2) with n = 0. The phase factor (p = 8k • {x\ — X2) de- pends on the spatial separation of ions and the arises be- cause each ion sees different laser phases. The ion-ion

spacing varies by 5/ ~ 100 nm over the range of Uq

cited previously (</> = for Uq = \6.2> V and = 77 for Uq = 12.6 V, with dip/dUQ in good agreement with

theory). For Gt = rr and fli = 2(12, the final state is

i//ei<f)) from Eq. (1). Note that Oi = (V2 + 1)02 would

generate the Bell states (but we would not have access

to the initial state lit), since il, are fixed throughout an

experiment).

We now describe our two-ion state-detection procedure.

We first prepare a two-ion basis state |^/), apply the detection beam D2 for a time r^ ~ 500 /iS, and record

the number of photons m detected in time r^. We repeat

this sequence for A^ ~ 10"* trials and build a histogram

of the photons collected (Fig. 3). To determine the population of an unknown state, we fit its histogram to a weighted sum of the four basis histograms with a simple

linear least-squares procedure.

We observe that the jtt) count distribution (Fig. 3a) is not the expected single peak at m = 0, but includes contributions at /72 = 1 and m = 2 due to background counts. The signal in bins m > 2 (which accounts for —10%

of the area) is due to a depumping process in which

(b) |ti)'

(d) |U)

60

FIG. 3. Photon-number distributions for the four basis qubit

states. Plotted in each graph is the probability of occurrence Pim) of m photons detected in 500 /is vs m, taken over ~10'*

trials. Note the different scales for each graph.

D2 off-resonantly drives an ion out of It), ultimately

trapping it in the cycling transition. We approximately

double the depumping time by applying two additional Raman "shelving" pulses (|t) —» ^'S'i/2|F = 2, m/r =

0) — ^S\/2\F = l,mf = —1); |i) unaffected) after every

state preparation. This results in an average difference of 10-15 detected photons between an initial |i) and |t) state, as shown in Fig. 3. The distributions associated

with lit), Iti), and |ii) are non-Poissonian due to detection

laser intensity and frequency fluctuations, the depumping described previously and |i) —»• |t) transitions from

imperfect polarization of D2.

One may ask: What is our overall two-ion state- detection efficiency on a per experiment basis? To

address this issue, we distinguish three cases: (1) |tt), (2) Iti) or lit), and (3) |ii). Now define case 1 to be true when m ^ 3, case 2 when 3 < m < 17, and case 3 when

m > 17. This gives an optimal 80% probability that the

correct case is diagnosed.

We have generated states described by density operators p~ in which the populafions (diagonals of p-)

are measured to be P[] ~ P]i ~ 0.4, P^ « 0.15, and

Pj] ~ 0.05. To establish coherence, consider first the Bell singlet state (/'s which has Pji = P|i = 1/2. Since

t(/B has total spin 7 = 0, any /-preserving transfor- mation, such as an equal rotation on both spins, must

leave this state unchanged, whereas such a rotation on a mixed state with populations Pjj = P^ = 1/2

and no coherences will evolve quite differendy. We

perform a rotation on both spins through an angle 6

by driving on the copropagating carrier for a time t such that 6 = Cl^t. Figure 4a shows the time evolution of an experimental state which approximates

the singlet Bell state. Contrast this with the approximate "triplet" state shown in Fig. 4b. The data show that p- is decomposed as p- = Cli/'g ){i/'B | + (1 — C)pn,

3633

TN-166
Volume 81, Number 17 PHYSICAL REVIEW LETTERS 26 October 1998

0.8

<a 0.6 o

^A^^/^SA^i^'A^^^^^^'^W^^^-

0.8

(0 0.6 O

0.4 -

P|T+Pt

1 1 (a) "singlet"

FIG. 4. Probabilities P[^ + P]i and P^i + P^ as a function of

time f driving on the copropagating carrier, starting from (a) the "singlet" ij/e(0) and (b) the "triplet" (/'e('n') entangled states. The equivalent rotation angle is IClct (Hc/Itt = 200 kHz for these data). The solid and dashed lines in (a) and (b) are sinusoidal fits to the data, from which the contrast is extracted.

in which p„ has no coherences which contribute to the measured signal (off-diagonal elements connecting |ti)

with lit) and \]]) with |U)), and C = 0.6 is the contrast of the curves in Fig. 4. This leads to a fidelity of

{-Ab Ip = I<Ab) = (^IT + ^n + c)/2 - 0.7. The nonunit fidelity of our states arises from Raman

laser intensity noise and a second-order (in 17) effect on

ft, due to excitation of the cm. mode [19]. These effects can be seen in Fig. 2b as a decay envelope on the data [modeled by y of Eq. (4)] and cause a 10% loss of fidelity in initial state preparation [23]. The micromotion-induced selection of Rabi frequencies

as here demonstrated is sufficient to implement twoion universal quantum logic with individual addressing

[8]. To start, we arrange the trap strength and static electric field in such a way that |5^|^i = and \Sk\^2 =

ao, where 7o(«o) = 0. To isolate ion 1, note that by

Eq. (3) Hi = CLcMO) = H^ and 02 = n^Mao) = 0. To isolate ion 2, we add nr/27r = ±238 MHz to the difference frequency of the Raman beams. This drives

the first sideband of the rf micromotion so that the Jq of

Eq. (3) is replaced by 7i, resulting in Hi = flcJiiO) =

and 02 = ClcJ\iao) + 0. In conclusion, we have taken a first step which is crucial for quantum computations with trapped ions. We

have engineered entangled states deterministically; that is, there is no inherent probabilistic nature to our quantum

entangling source. We have developed a two-ion state- sensitive detection technique which allows us to measure

3634

the diagonal elements of the density matrix p- of our

states, and have performed transformations which directly measure the relevant off-diagonal coherences of p-.

We acknowledge support from the U.S. National Security Agency, Office of Naval Research, and Army Research Office. We thank Eric Cornell, Tom Heavner,

David Kielpinski, and Matt Young for critical readings

of the manuscript.

Electronic address: quentint@boulder.nist.gov.

'Present address: Univ. Innsbruck, Innsbruck, Austria.

[1] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev.

47, 777 (1935); J. S. Bell, Speakable and Unspeakable

in Quantum Mechanics (Cambridge University Press, Cambridge, England, 1987).

[2] P.W. Shor, SIAM J. Comput. 26, 1484 (1997); L. K.

Grover, Phys. Rev. Lett. 79, 325 (1997).

[3] A. Barenco and A. Ekert, J. Mod. Opt. 42, 1253 (1995); C.H. Bennett, Phys. Today 48, No. 10, 24 (1995).

[4] A. Ekert, Phys. Rev. Lett. 67, 661 (1991); C.H. Bennett,

Sci. Am. 267, No. 4, 50 (1992).

[5] J.J. Bollinger el at., Phys. Rev. A 54, R4649 (1996); S.F. Huelga et al, Phys. Rev. Lett. 79, 3865 (1997).

[6] D.J. Wineland et al, J. Res. Natl. Inst. Stand. Technol.

103, 259 (1998).

[7] A. Ekert and R. Jozsa, Rev. Mod. Phys. 68, 733 (1996); A. Steane, Rep. Prog. Phys. 61, 117 (1998).

[8] J.I. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995).

[9] C. Monroe et al., Phys. Rev. Lett. 75, 4714 (1995).

[10] S.J. Freedman and J.F. Clauser, Phys. Rev. Lett. 28, 938

(1972); E. S. Fry and R. C. Thompson, Phys. Rev. Lett. 37, 465 (1976); A. Aspect, P. Grangier, and G. Roger, Phys. Rev. Lett. 49,91 (1982).

[11] Z.Y. Ou and L Mandel, Phys. Rev. Lett. 61, 50 (1988); Y.H. Shih and CO. Alley, Phys. Rev. Lett. 68, 3663

(1992); Z.Y. Ou et al, Phys. Rev. Lett. 68, 3663 (1992);

P. Kwiat et al, Phys. Rev. Lett. 75, 4337 (1995); W. Tittel

et al, Europhys. Lett. 40, 595 (1997); D. Bouwmeester

etal. Nature (London) 390, 575 (1997).

[12] M. Lamehi-Rachti and W. Mittig, Phys. Rev. D 14, 2543

(1976).

[13] E. Hagley et al, Phys. Rev. Lett. 79, 1 (1997).

[14] R. Laflamme et al, quant-ph/9709025 (unpublished).

[15] I. L Chuang, N. Gershenfeld, and M. Kubinec, Phys. Rev.

Lett. 80, 3408 (1998); I.L. Chuang et al. Nature (London)

393, 143 (1998); D. G. Cory, M. D. Price, and T. F. Havel,

Physica (Amsterdam) 120D, 82 (1998); D. G. Cory et al. Phys. Rev. Lett. 81, 2152 (1998).

[16] C. K. Law and H. J. Kimble, J. Mod. Opt. 44, 2067 (1997).

[17] J.I. Cirac etal, Phys. Rev. Le«. 78, 3221 (1997).

[18] C.H. Bennett et al, Phys. Rev. A 53, 2046 (1996).

[19] B.E. King et al, Phys. Rev. Lett. 81, 1525 (1998).

[20] D.M. Meekhof er a/., Phys. Rev. Lett. 76, 1796 (1996);

77,2346(1996).

[21] D.J. Wineland and W. M. Itano, Phys. Rev. A 20, 1521

(1979).

[22] S. R. Jefferts et al, Phys. Rev. A 51, 3112 (1995).

[23] The reference histograms for |1|) and \][) (Figs. 3b and

3c) have had this 10% contamination from IfT) and |U) removed.

TN-167
Grating-tuned semiconductor MOPA lasers for precision spectroscopy*

J. H. Marquardt, F. C. Cruz, M. Stephens, C. W. Gates, L. W. Hollberg, J. C. Bergquist

National Institute of Standards and Technology

Boulder, CO 80303

D, F, Welch, D. Mehuys, and S. Sanders SDL, Inc. 80 Rose Orchard Way, San Jose, CA 95134

ABSTRACT

A standard grating-tuned extended-cavity diode laser is used for injection seeding of a tapered semiconductor laser/amplifier.

With sufficient injection power the output of the amplifier takes on the spectral characteristics of the master laser. We have

constructed master-oscillator power-amplifier (MOPA) systems that operate near 657 nm, 675 nm, 795 nm, and 850 nm.

Although the characteristics vary from system to system, we have demonstrated output powers of greater than 700 mW in a single spatial mode, linewidths less than 1 kHz, coarse tuning greater than 20 nm, and continuous single-frequency scanning

greater than 150 GHz. We discuss the spectroscopic applications of these high power, highly coherent, tunable diode lasers as applied to Ca, Hg*, Ij, and two-photon transitions in Cs.

Keywords: diode lasers, tapered amplifiers, injection locking, high resolution spectroscopy, MOPA

Contribution of NIST and not subject to copyright

1. INTRODUCTION

Many scientific and technical applications of tunable diode lasers require higher powers than have been available fi"om semiconductor laser sources. Recent advances in tapered gain structures have opened a new frontier in single-mode, singlefi-equency, tunable, semiconductor laser capabilities.' Tapered amplifiers (TA) can have broad gain-bandwidths, providing

tremendous potential for tunable lasers with high power. They provide roughly one order of magnitude more power than previous single-spatial-mode devices. However, these tapered amplifiers/oscillators usually operate with many simultaneous

longitudinal modes, which limits their usefulness for coherent and high-resolution applications. Using tapered amplifiers in extended cavities for wavelength control (such as optical feedback from diffraction gratings) allows good coarse tuning at high power,^ but it is difficult to maintain true single-frequency operation. Extended-cavity TAs often exhibit instabilities which can cause jumps among a few nearby longitudinal modes or result in lasing on a few modes simultaneously.

One powerful solution to these problems is to separate the function of the high-power generation from that of the highprecision single-frequency tuning. A few modem examples of this type of system can be found in the literature and include: distributed Bragg reflector (DBR) laser diode plus tapered amplifier,^ solitary diode las&i plus tapered amplifier," and

grating-tuned extended-cavity diode laser (ECDL) plus tapered amplifier. In this paper we describe the applications of

extended-cavity diode-laser plus master-oscillator power-amplifier (ECDL-MOPA) systems that are currently under way at NIST. The characteristics of these systems are discussed from a user's perspective, where our applications are primarily in high resolution spectroscopy. Other related systems and applications have been described by Zimmermann et al.^ and

Goldberg and Bums.*

2. EXPERIMENT

By using several different experimental setups we gain some insight into the general characteristics of injection-locked

MOPAs. A typical experimental setup is shown in Figure 1; it consists of a standard grating-tuned ECDL in the Littman34 1 5PIE Vol. 2834 0-8 7 94-2222-3/96/56. 00

TN-168
Meicalf (grazing incidence) configuration, which acts as the master laser to inject the TA slave. The light from the master

laser is sent through an optical isolator and spatially mode-matched to the beam emitted from the back facet of the TA. Since

the front output aperture of the TA is very asymmetric (~ 1 |jm by 100 |jm), a cylindrical lens, in addition to the standard

spherical lens is needed to collimate the output beam. (In addition the output beam can have large astigmatism which may be

compensated.)

Extended Cavity Isolator Tapered

Diode Laser Amplifier

Figure 1. General ECDL-MOPA set-up. The ECDL output is sent through an isolator and then mode-matched to the beam emitted from the rear facet of the tapered amplifier. The collimating

optics consist of one spherical and one cylindrical lens.

0.6 0.8 1.0 1.2 1.4 MOPA Current (A)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 MOPA Current (A)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 MOPA Current (A)

Figure 2. This figure shows the power versus injection current for three of our ECDL-MOPA

systems (a) 657 nm, (b) 674 nm, and (c) 792 nm. Each graph is shown with and without optical

injection power. Graph (a) and (b) show clearly a lasing threshold of the TA at high currents even

without optical injection. The thresholds of all of the systems are lowered, and the slope

efficiencies increase with increasing injection power. The optical injection power P; used in these systems is (a) 1.3 mW, (b) 6 mW, and (c) 6 mW.

SPIE Vol. 2834 I 35

TN-169
With optical injection from the master laser into the TA, the threshold of the TA decreases, resulting in more emitted power

and the slope efficiency increases to as much as 0.7 W/A for the 674 nm and 792 nm systems (Figure 2). The MOPA output

power quickly saturates with injection power from the master laser. Figure 3 shows the output power of the MOPA as a

function of injected optical power. With good mode matching of the master into the TA, the output power saturates with

about 2-3 mW of injected power. The different ECDL-MOPA systems vary slightly in this characteristic but they all saturate with less than -5 mW of injection power due to the high gain and strongly asymmetric nature of the TA.

550 r _ 500 5

^ *

E ^ ' r 450 ^"""^ o

o

/^ ^

c

400

m/

o y

u. T 350 /

300 12 3 4 5 6

Optical Iniection Power (mW)

Figiu-e 3. Dependence of output power with optical injection power for the 674 nm system. The

output power saturates quickly with as little as 2-3 mW (solid circles).

The isolation required between the master and the slave depends somewhat on system parameters such as master laser power, mode matching, and the power emitted out of the back of the amplifier. We often use another isolator on the output of the amplifier to eliminate feedback into the high power amplifier. Although good quality isolators are needed between the master and the slave for relatively low injection powers (<5 mW), this requirement may be relaxed with higher injection powers. With optical injection from the master laser we have observed a decrease in temperature of the TA due to the large increase

in optical power and high efficiency of the TA.

in

:

i. i--

;

: : : ;

Pi = OmW

'fZ.L.A

pz|::::p4Z

-'•-'*^^^'^''*=_J.-----^ > ^ ^>^ i \ \

, ^_.,.^^^^^^^^^^^^^^^^-.. i'*^^^

' \ T i \ \ \ \"' '"'

665 nm 675 nm

Wavelength

585 nr

Figure 4. Optical spectnmi analyzer trace showing the output of the ECDL-MOPA with and

without 6 mW of injection power. The vertical scale is 5 dB/div and the horizontal scale is 2 nm/div. The instrument resolution is 0.1 nm.

The MOPA output power can be quite high (up to 700 mW), in a single spatial mode, while retaining the spectral properties of the master laser. Figure 4 shows an optical spectrum of the TA at 674 nm with and without injection. With injection the broad spectrum collapses into the single strong mode of the master laser. The single-frequency output takes power away

from nearby regions and rises 25-30 dB above the background pedestal as measured with a 0.1 mn resolution bandwidth. As

the master laser is tuned, the single-mode, coherent output of the MOPA follows exactly.

36 / SPIE Vol. 2834

TN-170
The residual facet reflectance of the TA can significantly affect the tuning characteristics of the MOPA and varies somewhat

with each system. When injection current in the TA is brought above its own lasing threshold, the residual chip modes of the

amplifier become important. Since these chip modes tune with slave current and temperature, it is straightforward to reach

the desired wavelength and be centered on a "good" MOPA chip mode as well. When the TA shows no threshold behavior

(as for the 792 nm system), these chip modes can be suppressed by the optical injection. For those TAs that can operate well above their threshold, the modulation is always present. These characteristics represent two different operating regimes for the MOPA: below the threshold of the TA the system acts like an optical amplifier, while above the threshold of the TA the system behaves like an injection locked laser. Using the 674 nm system near its own lasing threshold (-1.2 A) and with -6

mW of injection power, we were able to scan continuously over 150 GHz with an output power of -100 mW.' As a demonstration of this capability. Figure 5 shows absorption of L near 674 nm. Although there is some modulation in the power, the sweep remains single mode for greater than 150 GHz. Even when the chip modes are strong, the system can be

scanned -20-30 GHz without mode jumps. This performance is comparable to that of many single-frequency tunable dye

lasers. For most high resolution spectroscopy, this scan range is more than adequate, but some applications, such as molecular spectroscopy, may need longer single mode scanning ranges. In addition, active control of the injection current of

the slave can be used to suppress the power modulation due to the residual facet reflectance.

Frequency

Figure 5. Absorption spectrum of I, near 674 nm taken with an ECDL-MOPA system (adapted

from ref (7)). A specially designed long-scanning ECDL was used as the master oscillator to obtain this spectrum. The peak absorption was about 30% for a 5 cm long cell with a temperature

of about SOX.

3. APPLICATIONS

Applications of these ECDL-MOPAs provide "real-world" tests of the practicality and usefulness of these systems. The

following examples from our laboratory are used to illustrate the performance characteristics of these systems.

The first system at 792 nm is used to generate 194 nm light for use in a Hg* frequency standard. In this experiment Hg* ions

are held in electromagnetic traps and laser-cooled to temperatures on the order of 1 mK. Single-frequency radiation at 194 nm, required for laser-cooling the ions, is generated by sum-frequency mixing of radiation at 257 and 792 nm in an external build-up cavity. Radiation at 257 nm is obtained by frequency-doubling 515 nm radiation from an Ar* laser. Singlefrequency radiation at 792 nm is generated from the ECDL-MOPA system. More than 700 mW of light at 792 nm is generated with optical injection between 3 mW and 20 mW from the master laser. Greater than 90% of the slave laser power

can be mode-matched into the sum-frequency cavity, and negligible variation of the mode quality' is observed as the slave

laser's output power or frequency is varied. The ability to couple such a high percentage of light into the external ring cavity

illustrates the excellent spatial mode these systems can provide. Two isolators (net isolation of about 60 dB) were used

between the master and the slave laser, but no isolator was used on the output of the ECDL-MOPA. This high level of

isolation was required to prevent instabilities of the ECDL-MOPA caused by a small amount of optical feedback from the

SPIE Vol. 2834/37

TN-171
build-up cavity.

Using the 792 nm system an experiment to check for the relative coherence between the injection beam and the TA output was f>erformed. We split off part of the master laser beam, frequency-shifted it in an acousto-optic modulator (AOM), and

then heterodyned this beam with the output of the TA (Figure 6). The beat signal at the modulation frequency was then

heterodyned with the signal driving the AOM, and the resulting DC signal (proportional to the relative optical phase between

the master and the slave) was observed on an oscilloscope. In this way phase variations between the master and slave lasers were observed directly. Phase locking between the master and slave laser was observed. The phase difference never exceeded :: rad for dmes of the order of minutes. Similar results have also been reported by Kikuchi et al.*

detector

Figure 6. Schematic diagram of the method used to measure the relative phase coherence between

the master laser and the slave.

cavity

fiber

Figure 7. Calcium spectroscopy set-up. The solid lines represent the laser light and the dotted

lines represent the servo loops.

A good demonstration of the narrow linewidths and high optical resolution that can be achieved with these ECDL-MOPAs is provided by the 657 nm system we use for Ca spectroscopy. The system is shown in Figure 7. ECDLl is locked to an

ultra-stable high finesse Fabry-Perot cavity (fringe width - 50 kHz FWHM) using the Pound-Drever-Hall technique. The

beam transmitted through the cavity is used to phase-lock the 657 nm ECDL2-M0PA. This ECDL-MOPA output is then

spatially filtered using an optical fiber which delivers the hght to the Ca atomic beamsystem. In addition, the output of the

fiber is intensity-stabilized by active feedback control to the MOPA current This is done to compensate for the intensity fluctuations due to polarization changes and thermal drifts of the fiber system. The very high resolution achievable with this system is evident in Figure 8. This figure shows the optical Ramsey spectra of the 657 nm calcium line. The FWHM of

38/SPIEVol. 2834

TN-\72
approximately 11.5 kHz, with a signal-to-noise ratio of about 30 in a 1 Hz bandwidth, corresp>onds to what would be a noise- hmited frequency stability 6v/v of about 8x10"'^ t"^ (where t is the averaging time). The resolution is limited here by the physical size of the windows on the atomic beam apparatus which limits the distance between interaction regions. Using the Ca lineshape and the Fabry-Perot cavity as frequency discriminators we measure the residual FM noise of the ECDL-MOPA

to be 5 1 kHz dominated by the acoustical mode coupling between the laser table and the reference Fabry-Perot cavity (at about 1-2 Hz).

20 40 60 80 100 12C 140 160 180

Relative 657 nm tuning (kHz)

Figure 8. Ramsey Mrgc of calcium using the ECDL-MOPA setup shown in Figure 7. The

FWHM of the fringe is -1 1.5 kHz, at an optical frequency of 456 THz.

20 40 60 80 100

Relative 674.1 nm tuning (MHz)

120

Figure 9. Fluorescence spectroscopy of the 6S1;,, F=4 -> 12S,;2' ^-^ two-photon transition in cesium (1 Hz bandwid±).

A third ECDL-MOPA system is used for high resolution and high sensitivity nonlinear spectroscopy in cesium atoms. The

674 nm system provides 500 mW of usable output power which is sufficient for two-photon spectroscopy on weak optical transitions without the need of a build-up cavity. The 674 nm light is scanned over the 6Si^ -> 12S,^ two-photon transition

in a cesium vapor cell healed to -80 °C; the transition is detected by the decay fluorescence from the 12Si;2 state to the 6P^^

state at 54] nm (Figure 9). The nonlinear signal is quadratic in laser power, yet the signal-to-noise ratio is excellent (>1000

in a 1 Hz bandwidth). This demonstrates that the MOPA output has low AM noise.

SPIEVo!. 2834/39

TN-173
4. LIMITATIONS

This new technology is exciting and appears to have a bright future. However, there are some limitations. One limitation is that the ECDL-MOPA systems are roughly one order of magnitude more costly to build than the standard low power

counterparts. Although more expensive they are still an order of magnimde less expensive than many other high-power

tunable laser systems such as dye and TitAljOj lasers. As with low power diode lasers, these ECDL-MOPAs are very

susceptible to optical feedback and often require good quality isolators.

5. CONCLUSIONS

With these ECDL-MOPA lasers a new era in diode laser spectroscopy is opened. Power is increased by an order of magnitude or more over conventional diode lasers while the systems retain the good spectral characteristics and tunability of

the lower power ECDL lasers. We have demonstrated the ability to use these lasers in several different experiments

including the high resolution spectroscopy of Ca, Hg*, Ij, and Cs. Although the systems have slightly different

characteristics, our experiments demonstrate that tunable, highly coherent, spatially single-mode, high power laser systems

are achievable. These lasers will undoubtedly play an increasingly important role in the high resolution spectroscopy and

nonlinear optics of the future.

6. ACKNOWLEDGMENTS

We acknowledge the support of the U.S. Air Force Office of Scientific Research.

7. REFERENCES

1. J. N. Walpole, "Semiconductor amplifiers and lasers with tapered gain regions," Opt. Quantum Electron., 28, pp. 623-645

(1996).

2. D. Mehuys, D. Welch, and D. Scifres, "1 W CW, Diffraction-Limited, Tunable External-Cavity Semiconductor Laser,"

Electron. Utt., 29, pp. 1254-1255 (1992).

3. S. O'Brien et al., "Operating characteristics of a high-power monolithically integrated flared amplifier master oscillator power amplifier," IEEE J. Quantum Electron., 29, 2052 (1993).

4. D. Mehuys, D.F. Welch, and L. Goldberg, "2.0 W CW, dif&action-limited tapered amplifier with diode injection," Elec. L€«., 28, 1944(1992).

5. 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, pp. 2318-2320 (1995).

6. Lew Goldberg and W. K. Bums, "Deep UV to mid-IR generation with laser diodes and non-linear fi-equency conversion,"

in Proc. LEOS '95, edited by T. Fan, pp. 459-460 (IEEE, Piscatoway, NJ, 1995).

7. L. Hollberg, J. MarquardL M. Stephens, and R. W. Fox, "Diode lasers and spectroscopy," in Proc. LEOS '95, edited by T. Y. Fan, pp. 327-328 (IEEE Piscatoway, NJ, 1995).

8. K. Kikuchi, C.-E. Zah, and T.-P. Lee, "Measurement and analysis of phase noise generated from semiconductor optical

amplifiers," IEEE J. Quantum Electron., 27, 416 (1991).

40 I SPIE Vol. 2834

TN-174
HLSHVIHR

15 January 1996

Optics Communications 123 ( 1996) 207-214

Optics

Communications

CW second harmonic generation with eUiptical Gaussian beams

A. Steinbach, M. Rauner ', F.C. Cruz, J.C. Bergquist

Time and Frequency Division. National Insiiiuie of Standards and Technology, Boulder, CO 80303. USA

Received 28 April 1995; revised version received 27 September 1995

Abstract We present a study of the efficiency and optimization of cw second harmonic generation by elliptical Gaussian laser beams.

Elliptical focusing slightly improves conversion efficiency and reduces crystal damage risk when heavy walk-off is present. Single-pass measurements of the efficiency for doubling 515-nm radiation in beta-barium borate (BBO) agree with theory. Thermal effects, caused by radiation absorption, limit the doubling efficiency of single-frequency radiation in an external enhancement ring cavity.

1. Introduction

Second harmonic generation (SHG) has become a very useful and widely employed technique to provide

monochromatic light sources at wavelengths that are

difficult or even inaccessible with conventional lasers. Literature on cw SHG is extensive and explores many

possible schemes toward high conversion efficiencies from radiation at the fundamental frequency into ra- diation at the harmonic. The optimization of cw SHG

using spherically focused Gaussian beams was first treated in the most general case by Boyd and Kleinman

[

1 ]

, but SHG using focused Gaussian beams with elliptical cross section offers advantages in some cases. Librechet and Simons [2] show that when critical phase-matching is required, a small increase in doubling efficiency can be expected by using an optimally

focused elliptical laser beam rather than an optimally

focused circular beam. Asymptotic and exact solutions

are found for an ADP crystal with a length of 2 cm

and for different values of ellipticity. Comparisons are

' Present address; Institut fiir Quantenoptik, Universitiit Hannover, 30167-Hannover, Germany.

made between theoretical predictions and experimental values for three choices of ellipticity. Kuizenga [3]

treats elliptical focusing for the case of a parametric

amplifier and finds that by optimizing the confocal parameters, the threshold for gain can be lowered from

that obtained for optimum spherical focusing. This re- sult is valid only when the walk-off parameter B ex- ceeds 1. He also finds that the signal and idler beams

remain almost circular while the pump beam grows

increasingly elliptical as B increases. Recently, Taira

[4,5] has reported high power generation for the sec- ond harmonic of a 5 1 5-nm argon-ion laser using elliptically focused beams. Motivated by these results, we

sought to more rigorously study the generation of sec- ond harmonic radiation using cylindrical focusing. We

examine conversion efficiencies for a wide range of

practical confocal parameters in the critical and noncritical directions and for various values of the walkoff parameter B. For heavy walk-off, we find the the diameter of the light beam in the noncritical direction must remain within a factor 2 of optimum elliptical

focusing, otherwise, less harmonic power is generated

than for optimum spherical focusing. Our study also

0030-40 1 8/96/512.00 © 1996 Elsevier Science B.V. All nghts reserved 550/0030-4018(95 ) 00626-5

TN-175
208 A. Stembach el al. / Optics Communications 123 (1996) 207-214

reveals no fundamental difference in the far-fieid mode

pattern whether the harmonic radiation is generated

by cylindrically or spherically focused light. Measure- ments of the nonlinear conversion efficiency obatined

by single-pass doubling of 51 5-nm radiation in angletuned beta-barium borate (BBO) are presented and

compared to theory. We also discuss experimental re- sults obtained by doubling the 51 5-nm radiation in a Brewster-cut BBO crystal placed in a low-loss external ring cavity using cylindrical mirrors. Although

high circulating powers are possible in the absence of

harmonic generation, absorption of the UV radiation

at 257 nm causes thermal lensing in the crystal that limits the generation of harmonic radiation at higher powers.

2. Theory

In order to calculate the second harmonic power

generated by an arbitrary elliptical Gaussian beam

passing through a uniaxial nonlinear crystal, we allow

fundamental electric fields of the form [6]:

Ei{x,y,z) =

X exp

Eoexp{\k\z)

N/(l+irjy(H-iTv)

^'5.(1 + i7-J H'2^,(l + 'iTy)

' (1)

where

r, = 2

/,

b, = w5,/:|.

Eq. ( ! ) represents an elliptical TEMoo Gaussian beam

whose focal points (/, and fy) and beam waists (woi

and vv(jv ) in the x and y transverse directions are inde- pendently adjustable, k] is the magnitude of the wave

vector inside the crystal. Here we neglect absorption

of radiation by the crystal. In analogy to the heuristic treatment of Boyd and

Kleinman, we derive the second harmonic field amplitude in the far field outside the crystal and obtain

the power of the second harmonic by integrating over

the intensity distribution (^2^/877-) |£2|":

Pi = KPflki

where

MB,Ak,^„i,), (2)

h{B,Ak,i,,^,.) = \^

P

I I ^>m:'~z)Q~4B\z'-zfi,/r- (j,(i;

yi + ir^ymnvn^/r:^

(3)

and

^. = l/b., B = fi\/fh/2.

t' differs from r by replacing z with ;'. We use the same coordinate system as in Fig. 1 of Ref. [ 1 ] where

the optical beam axis is the z direction, the origin is the point where this axis enters the crystal, and the

a:j-plane is parallel to the crystal faces. The crystal

length is /, p is the walk-off aigle in radians, and nj

is the extraordinary index of refraction at the doubled

frequency. The optic axis of the crystal lies in the xzplane, and A^ is the wave vector mismatch (2A;i — ^2)-

P\ is the power of the fundamental radiation, ^eff is the

effective nonlinear coefficient, and n\ is the ordinaryray index of refraction. The second harmonic power is proportional to the function h(B,Ak,^x^^y) which, for a given crystal

length and amount of input power, contains all vari- ables for optimization. Our numerical calculations show that harmonic power is maximized by adjusting

both beam waists to lie at the crystal's midpoint ( fx =

/v = 1/2). Further, in calculating the harmonic power

for a given set of parameters, we use the value of A^

that maximizes the harmonic p>ower. Experimentally

Ak can be optimized by temperature or angle-tuning

the crystal. Thus, once a particular crystal type is chosen, the waist size in each transverse direction is the only remaining optimizable parameter.

We can solve for second harmonic power as a function of ^x^ with ^v fixed at the value that maximizes

Ji{B,Ak,^x<^y) for a given B and A^. h is plotted

in Fig. 1 for a variety of walk-off strengths B and

for Ak = Akm, the optimum phase-match angle. Comparing these curves to the theoretical predictions for optimized spherical focusing [ 1 ] reveals perhaps the most atractive feature of cylindrical focusing: For crystals with heavy walk-off, it is possible to decrease

the focusing parameter ^x by several orders of magnitude without dropping below the harmonic power that

TN476
A. Steinbach et al. / Optics Communications 123 (1996) 207-214 209

would be obtained by optimal spherical focusing. It is thus possible to reduce the peak field intensity by several orders of magnitude without lowering conversion

efficiency. This reduces the risk of radiation damage

to the crystal as well as possible thermal effects. Optimal cylindrical focusing also improves har- monic conversion efficiency as compared to optimized

spherical focusing. For example, for B = 16 the op- timum focusing parameters are ^x = 0.25, ^y = 3.3. This generates 27% more power than that generated

for the spherical optimum at ^ = ^;t = ^y = 1 .39. In all cases where 5 =?*= 0, optimal focusing is achieved by

a softer focus than the spherical optimum in the trans- verse direction sensitive to phase matching. When

5 = 0, the cylindrical optimum reduces to the well known spherical focusing optimum given by ^ = 2.84. The curves in Fig. 1 are essentially those in Fig. 2 of Ref. [3], which give optimum focusing conditions

for parametric gain, and resemble efficiency curves

for SHG with spherical focusing in the case that both

the harmonic and fundamental power are enhanced in resonant cavities placed around the nonlinear crystal

[7-9].

Fig. 2 gives a direct comparison of harmonic power

generated by cylindrical and spherical focusing for the case of heavy walk-off. The solid curve gives relative harmonic power versus the spherical focusing parameter ^ = l/b for B = \6. The dashed curve represents

cylindrical focusing versus ^^, also for B = 16. ^v is fixed at its optimum value of 3.3. Although the inSX

Fig I /i maximized with respect to ^y for difterent wali<-off parameters B. Aim is the optimized value of A/:. For zero walk-off,

fl == 0, the maximum power is the same as in the spherical

case; same coordinate system as in Fig. 1 of Ref 1 1 ] where

the optical cylindncal focusing technique is only of advantage if energy walk-off occurs. ^^ approaching zero corresponds to the plane wave limit.

E

Fig. 2. Cylindrical versus spherical focusing for S = 16 and Lk = Aim- The focusing parameters which maximize SHG conversion efficiency are ^y = 3.3 and f.t = 0.25. For cylindrical focusing ^x can be as small as 0.01 before conversion efficiency

returns to the maximum value obtained for spherical focusing. The

dashed curve represents cylindrical focusing with ^v = 1-39.

crease in efficiency is apparent here ( 27% for B - 16),

in practice it may be preferable to decrease the peak

field intensity at the crystal surface by decreasing ^x^ since the loss in second harmonic power can be small. For example, with 5 = 16 it is possible to decrease ^^

to 0.007 before the harmonic power drops to the max- imum possible with spherical focusing. This assumes

that ^v is fixed at its optimum value of 3.3. The dotted curve in Fig. 2 shows the relative second harmonic

power obtained for ^y fixed at 1.39, which is the optimal value of ^ for spherical focusing. In this case

the somewhat softer focus in the noncritical direction only slightly lowers the nonlinear efficiency from the optimized cylindrical case. But even this nonoptimum

curve still rises above the optimum for spherical fo- cusing. Hence, ^x can be decreased without sacrific- ing conversion efficiency, and the lower fundamental

intensity reduces the likelihood of crystal damage and

the amount of thermal effects. In general, it may not be possible to attain the op- timum focusing in either the critical or noncritical direction (for example, in internal or external SHGsetups that use cylindrical elements [4]). For these

cases, it would be useful to compare second harmonic

generation for non-optimum focusing. In Fig. 3, we

show relative harmonic power generation for the case of heavy walk-off (B = 16) and various focusing pa- rameters ^x and ^v A factor 2 change in the waist

size in the noncritical direction away from optimum

reduces the maximum harmonic power for cylindrical

TN-177
210 A. Steinbach et al. /Optics Communications 123 (1996) 207-214

E

Fig. 3. Dependence of ~h upon focusing. The conversion efficiency

for 6 = 16 is plotted for different ^y corresponding to a variation of

the waist size in the sagittal (vz) plane. The optimum conversion

efficiency can be achieved with fv = 3.3. In our experiment,

cylindrical mirrors with radius of curvature of 10 cm are used. In

this case the values for ^y are on the order of 0. 1 . For comparison,

the curve for spherical focusing (•) is also shown.

_ I.O

i 0.8

i 0.6 IE^ 0.4

'I 0.2 — 0.0

-1.0

-v = -^-'' y^ N

^x="-> /

xz - plane

\ ^^ plane

_• ^

-c; 0.0 0.5 1.0 X (cm)

Fig, 4, Second harmonic intensity distribution in the far-field for B = \2. The intensity profiles are shown in th x- and y-direction

for optimized cylindrical focusing. Whereas the profile in the y-direclion is Gaussian, the intensity distributions in the walk-off

plane exhibit interference patterns in the wings. An integration over

the tangeniiai intensity profile shows that the main lobe contains 96.6% of the total power

focusing below that obtained for optimum spherical

focusing. Clearly it is desirable to remain near optimum focusing in the noncritical direction to realize the full advantages of cylindrical focusing.

The far-field intensity pattern is fundamentally the same for either spherical focusing or cylindrical fo- cusing. Fig. 4 shows the far-field intensity profiles of

the harmonic radiation along the j:-axis and the y-axis

for cylindrical focusing. The interference pattern along

the j:-axis (the walk-off direction) is due to second

harmonic light emitted at different points along the path of the fundamental beam inside the crystal. Since

the generated harmonic radiation is emitted under a

certain angle relative to the surfaces of equal phase of

the fundamental, fringes appear in the intensity distri- bution. They are also present in the spherical focusing

case. However, the amplitude of these fringes is small compared to the amplitude of the main lobe. For both

cylindrical and spherical focusing, the main lobe contains over 95% of the total power. In the absence of

energy walk-off, the generated radiation would be in phase throughout the crystal and there would be no

interference. The intensity distribution in the walk-off direction

is much narrower than the Gaussian intensity distribu- tion in the y-direction. This feature is also common to both methods of focusing. In the spherical case, beam

walk-off of the harmonically generated light enlarges

the tangential waist of the harmonic compared to the waist in the sagittal plane. In the cylindrical case, both

the asymmetric focusing and beam walk-off contribute

to the larger waist in the tangential plane. The ellipti- cal cross-section of the second harmonic beam in the

far-field is not a serious obstacle. It can be made nearly

spherical-Gaussian by means of suitable lenses or mirrors. It has been shown that approximately 89% of

the second harmonic radiation can be mode-matched

into external resonators that support spherical Gaussian modes [10].

3. Experimental results and discussion

We compared our theoretical predictions to measurements of single-pass SHG conversion efficiencies for various cylindrical focusing geometries. Light

from a single-frequency argon-ion laser at 515 nm

was doubled to 257 nm in an angle-tuned, Brewstercut BBO crystal. Combinations of cylindrical and

spherical lenses and mirrors were used to create the various TEMoo elliptical Gaussian beams. The BBO

crystal length is 6.5 mm and the walk-off angle p is 0.085 rad; hence, B = 15.6. In Fig. 5, the solid line represents the theoretical SHG conversion efficiency

for B = 15.6 as a function of ^^. ^y is fixed at 2.4, which corresponds to the 15 /xm value of wq, that was

used for all our experimental data points (this value is near the optimum value for cylindrical focusing that gives ^v = 3.3 and WQy ~ 13 /xm ). The single-pass

conversion efficiency 77 is given by P2 = "nP}, where

TN-178
A. Steinbach et al. /Optics Communications 123 (1996) 207-214 211

10'

o

10". /

—i -^ A

lO" 10-3 10- 10-' 10" 10' 10=

Fig. 5. Comparison of theoretically predicted values and experi- mental data for the single-pass conversion efficiency. In the experiment, the waist size and position in the (noncritical) yz -plane is kept fixed at 1 1.5 /xm, while the waist size in the walk-off plane

is varied by means of different lens combinations. The theoretical curve is calibrated by deriving d^^ for our BBO crystal from

a single-pass conversion efficiency measurement under optimum

spherical focusing conditions.

P2 is the output harmonic power. The theoretical peak

height depends on the value of <ieff used. The value of ^eff for our crystal was derived from a single-pass measurement under optimum spherical focusing conditions (wo ~ 20 /Ltm). We measured the efficiency

to be 7.0 X 10-^ W-' which gives c/eff ^ 0.32 pm/V

(subsequently we obtained a second BBO crystal

from another vendor and measured its single pass

efficiency to give ^eff — 0.4 pm/V). In Fig. 5 the data points represent the measurements of single-pass

efficiency as a function of ^^. The agreement between

the theoretical curve (5 = 15.6, ^;c=2.4) and the ex- perimentally obtained values is good. The error bars account for statistical error. In applications that need high second harmonic

power, cylindrical focusing can be achieved easily

with optical cavities that boost the amount of fun- damental power incident on the crystal. High power, cw 257-nm generation, obtained by doubling the fre- quency of an argon-ion laser in an intracavity setup, has been demonstrated [4] with cylindrical lenses as focusing elements. In Fig. 6 we show a simple fre- quency doubling scheme where the crystal is placed

between two cylindrical mirrors in an external ring

resonator. This configuration avoids the additional

losses introduced by intracavity lenses and the complications and expense of low-loss AR coatings for both the fundamental and the harmonic. Mirrors M2,

M3 and M4 have a high reflectivity {R > 0.998) at

J./2 ^ At* - l^ser —f2zn3-HIEtaion

"*

* Polarized in sagittal plane

• Polarized in tangential plane

Fig. 6. The experimental setup for generating high-power 257-nm

radiation. The single-frequency green light (A ~ 515 nm) of an

argon-ion laser is doubled with a Brewster cut and polished BBO

crystal placed in an external ring resonator. M 1 (input coupler) and M4 are spherical mirrors with a radius of curvature of 30 cm. In order to produce a Gaussian mode with an elliptical cross section, M2 and M3 are cylindrical mirrors with a radius of curvature of

10 cm in the plane of the drawing. M3 is highly reflecting for 515 nm and highly transmitting for 257 nm.

the 515-nm wavelength of the argon-ion laser and the transmission of the input coupler Ml is about 1.8%.

The cylindrical mirror M3 transmits 94% of the generated UV at 257 nm. Two spherical lenses, LI and

L2, constitute a telescope to establish mode-matching

into the cavity. The cavity is locked to resonance by means of the Hansch-Couillaud locking scheme [11].

Ml and M4 are spherical mirrors with matching

radii of curvature of 30 cm. M2 and M3 are cylindrical mirrors with a radius of curvature of 10 cm. The

eigenmode of the resonator is spherical-Gaussian, ex- cept in the region between the two cylindrical mirrors where the cross section of the beam is elliptical.

Since M2 and M3 act as flat mirrors for the tangential

component of the beam, the cavity is free of astigmatism other than the small amount due to the off-axis incidence on mirrors Ml and M4. The angles of inci- dence on all mirrors can be made small to minimize

most higher-order aberrations. Maximum conversion

efficiency is obtained when both the tangential and the

sagittal focus are centered inside the crystal. Therefore

the ideal alignment is symmetric about the crystal.

The lowest order TEMqo eigenmode of the resonator

exhibits comparatively soft focii {wqx — 200 /Ltm) at the middle of the crystal as well as halfway between

M4 and Ml. In the sagittal plane the cylindrical mirrors focus the beam more tightly (wov — 50 /xm).

TN-179
212 A. Steinbach et al. / Optics Communications 123 (1996) 207-214

Faster cylindrical mirrors are needed to come closer to the optimum focusing condition where wq^ — 46 fim, wov — 13 /xm for B = 15.6 and / = 6.5Z mm. A cavity using cylindrical mirrors with a radius of curvature of about 3 cm in combination with a pair of spherical

mirrors with a radius of curvature of 30 to 50 cm can

establish near optimum focusing conditions. Theoretically, an improvement in the doubling efficiency of

about 2.5 can be expected as compared to our current

setup. For Brewster cut, highly transparent crystals, linear

intracavity losses are usually dominated by light scat- tering from the crystal surfaces due to imperfect polishing and by a small transmission of the fundamental

radiation through the highly reflecting mirrors. We de- termined the round-trip linear loss term of the cavity

to be about 0,7% (for the second, more efficient BBO

crystal) from the mode-matched power enhancement

factor of about 1 15. For this measurement, the angle

of the BBO crystal is adjusted so that no harmonic ra- diation is generated. Since the transmission of the in- put coupler is 1.8%, not all of the fundamental power

is coupled into the cavity. Ideal coupling (no power

reflected from the input coupler) is obtained when the transmission of the input coupler equals the round-trip

intracavity loss term [12]. However, when the BBO

crystal is again angle tuned so that harmonic radiation

is generated, the impedance match improves as the fundamental power is increased, because the nonlinear loss term increases as a greater fraction of the cir- culating fundamental power is converted to UV light

[ 13,14]. The losses due only to the mirrors were de- termined to be 0.35% by measuring a power enhace- ment factor of about 156 for the cavity without the

crystal. Therefore the linear losses due solely to the

crystal are also about 0.35%.

We measure the fundamental power enhancement in two ways; either by sweeping the cavity through res- onance or by locking the cavity to resonance. In both

cases, the power leaking through one of the highly re- flecting cavity mirrors is compared to the power trans- mitted by the same mirror with the input coupler to the cavity removed. We expect both methods to give identical results unless there are thermal problems which

degrade the power enhancement when the cavity is locked to resonance [14]. For input powers less than 250 mW, the power enhancement factor in the funda- mental mode, measured by either method, is about 1 10

(~ 5% of the input power was coupled into higher- order modes) . As the input power is increased, the en- hancement factor (again measured by either method)

begins to decline as the loss term due to nonlinear conversion of fundamental radiation into second harmonic

light increases. However, for input powers exceeding

approximately 1 W (85-90 W intracavity), the power

amplification factors measured by the swept method

and the locked method begin to diverge. When locked

to resonance, the circulating power is less than that measured for the swept case, presumably due to ra- diation absorption that causes thermal lensing in the

crystal [14]. The divergence increases as the input

power is raised (2.1 W is the single-mode power limit of our present laser). If the crystal is angle detuned

so that harmonic generation does not occur, than the power enhancement factor remains about 1 10 for all input powers whether measured when locked to res- onance or swept . Interestingly, the conversion effi- ciency T) also drops for higher powers as the ther- mal effects become more prominent. 77 remains near

4.5 X 10~^ W~' for our cavity focusing conditions and

for power inputs up to 1 W, then begins to decrease (remember that 77 depends on the focusing parameters which change as the thermal lensing increases). At 2 W into the cavity the build-up on resonance is only 63. The amount of absorption and radiative heat- ing seems to vary widely from crystal to crystal, even

for BBO [ 15,16]. The cavity power enhancement for our poorer crystal drops from about 90 to nearly 40

at 2 W input power and we are unable to stably lock

the cavity to resonance at 2 W input power. For the

better BBO crystal, an input power of 380 mW generates 64 mW of UV power; at a fundamental input

power of 1.9 W, the cavity can still be stably locked

to resonance and approximately 500 mW of harmonic

power is produced. Although we made no long term measurements of the stability of the harmonic power,

the 500 mW was reproducible day-to-day and stable

for 30 minute periods. These and other values for dif- ferent input powers are plotted in Fig. 7a. The values

for the UV power have been corrected for the 94%

mirror transmission and the 20% Fresnel loss at the

exit face of the Brewster cut crystal. The solid line in Fig. 7a represents the theoretical expectation for the harmonic output obtained from :

TN-180
A. Steinbach etal. /Optics Communications 123 (1996) 207-214 213

Fig. 7. (a) Second harmonic power Pt as a function of fundamental

input power Pi at A = 515 nm. (b) Overall conversion efficiency

e = Pz/Pi as a function of input power P\ . The solid line curves

are denved fromEq. (4) using the measured values t; = 4.2x 10~^ W-'. L = 0.7% and T = 1.8%.

[2-/r^(2-L- v^^)]2 ' (4)

where e = Pj/ P\ denotes the overall conversion ef- ficiency, T the transmission of the input coupler, and L the cavity round-trip linear loss term [13,14]. Eq.

(4) includes the nonlinear loss factor in the last term

in the denominator. The deviation of the experimental results from the theoretical prediction above 1 W

of input power reveals the thermal effects due to ab- sorption of harmonic radiation. In Fig. 7b, the theoretical (solid line) and experimental values of e are plotted which also clearly show the deviation caused

by thermal effects in the BBO crystal. Our best values

for overall conversion efficiency slightly exceed 30%.

While better conversion efficiences can be expected

for optimum focusing and better impedance matching

at low input powers, the thermal effects caused by ra- diation absorption may worsen for the tighter focusing

and severely limit 77 at higher powers ^

.

4. Conclusion

In conclusion, we have calculated efficiency curves

for harmonic generation for the general case of cylindrical focusing for a wide range of confocal parameters and for practical values of beam walk-off. We have

shown that, for angle-tuned harmonic generation, el- liptical focusing can produce the same or slightly more

harmonic power than spherical focusing for the same

fundamental power. Also, since the intensity can be

lower for elliptical focusing, the risk of radiation damage to the crystal can, in principle, be reduced. Measurements of single-pass efficiency are in agreement

with these calculations. We have shown the the farfield intensity distribution for the harmonic radiation

is the same whether the fundamental beam is cylindrically or spherically focused. We have also built a simple, low-loss external ring cavity using cylindrical mirrors and an intracavity, Brewster-cut BBO crystal. The

cavity had a mode-matched power enhancement factor of about 110, for light at 515-nm. Thermal lensing in our crystal limited the overall conversion efficiency to about 30%. Even though we achieved slightly higher

conversion efficiencies in BBO with cylindrical focusing, the thermal lensing effects limited the useful fun- damental power in the external cavity to a level where

a simpler spherical cavity would offer no risk of crystal damage. From our results, we conclude that the small increase in second harmonic power due to elliptical focusing probably does not warrant the expense

and complication of cylindrical mirrors or lenses.

^ We note that Kubota et al. [16] were able to achieve a stable output power of about 800 mW at 266 nm by doubling 532 nm

radiation in BBO in an external ring cavity. This would either imply a better BBO crystal with extremely low absorption in the

ultraviolet, or that the absorption at 266 nm is lower than at 257

nm.

TN-181
214 A. Steinbach et al. / Optics Communications 123 (1996) 207-214

Acknowledgements

We thank H. Patrick and M. Young for careful read- ing of the manuscript. F.C. Cruz is pleased to acknowledge the financial support of CAPES (Brazil).

References

1 1

I G.D. Boyd. DA. Kleinman, J. Appl. Phys. 39 (1968) 3597.

|2| F.M. Librecht and J. A. Simons, IEEE J. Quantum Electron.

11 (1975) 850.

[3| D.J. Kuizenga, Appl. Phy,s. Lett. 21 (1972) 570.

|41 Y. Taira, Jpn. J. Appl. Phys. 31 (1992) L682.

|51 Y. Taira, CLEO'93 proceedings, pp. 634-637.

|6| A. Yariv, Quantum Electronics (Holt, Rinehart and Winston, New York, 1976) p. 51.

[7] C. Zimmermann, R. Kallenbach, T.W. Han.sch and J. Sandberg, Optics Comm. 71 (1989) 229.

18] Z.Y. Ou and H.J. Kimble, Optics Lett. 13 ( 1993) 1053.

[9] K. Fiedler,S. Schiller, R. Paschotta, R Kurz and J. MIynek,

Optics Lett. 18 (1993) 1786.

[10] M. Boshier, Ph.D. Thesis, University of Oxford (United Kingdom), 1988; available from UMl in association with

the British Library. [11] T.W. Hansch and B. Couillaud, Optics Comm. 35 (1980)

441.

[12] J.C. Beigquist, H. Hemati and W.M.

43 (1982) 437. I13]W.J. Kolowsky, CD. Nabors and R.L. Quantum Electron. 24 (1988) 913

14] E.S. Polzik and H.J. Kimble, Optics Lett. 16 ( 1991 ) 1400.

15] Coherent, private communication;

Y. Taira, private communication.

16] Kubota el al.. Optics Lett. 19 ( 1994) 189.

Itano. Optics Comm.

Byer. IEEE J.

TN-182
Sum-frequency generation

of continuous-wave light at 194 nm

Dana J. Berkeland, Flavio C. Cruz, and James C. Bergquist

Over 2 mW of continuous-wave tunable 194-nm light is produced by sum-frequency mixing approximately 500 mW of 792-nm and 500 mW of 257-nm radiation in beta-barium borate (BBO). The powers in both fundamental beams are enhanced in separate ring cavities whose optical paths overlap in the Brewstercut BBO crystal. Due to the higher circulating fundamental powers, the sum-frequency-generated power is nearly 2 orders of magnitude greater than previously reported values. © 1997 Optical Society

of America

Key words: Sum-frequency generation, second-harmonic generation, ultraviolet light, lasers, diode

lasers, optical cavities.

Tunable narrowband sources of ultraviolet radiation

have applications in laser cooling and in spectroscopy

of atoms and molecules. In particular, light at 194.1

nm drives the 5cf^°6s ^Si/2-5d^°6p ^Pi/2 resonance

in singly ionized mercury. This allows laser-cooling,

Raman-cooling, double-resonance experiments and

other experiments on strings of electromagnet!cally

trapped Hg"^ ions. ^ 2 Because available nonlinear

materials cannot be phase matched for secondharmonic generation (SHG) to 194 nm,^ light at this

wavelength is produced by sum-frequency generation

(SFG). Also, because most materials absorb in the

ultraviolet, only a few crystals are suitable for generation of light in this region. Previously, several

microwatts of continuous-wave 194-nm radiation

have been produced by SFG in potassium pentaborate (KBS)* and 31 |jlW in beta-barium borate (BB0).5

Here we report the generation of over 2 mW of coherent cw radiation at 194 nm.

Figure 1 shows an overview of the apparatus. As

in Refs. 4 and 5, the wavelengths of the fundamental

beams are 792 and 257 nm, where the 257-nm light is produced by frequency doubling the 515-nm light

from a single-frequency argon-ion laser. The fundamental beams are enhanced in separate resonant

cavities whose smallest waists overlap. At this in- tersection, the beams propagate coUinearly in a

The authors are with the Time and Frequency Division, National

Institute of Standards and Technology, 325 Broadway, Boulder, Colorado 80303. Received 12 September 1996; revised manuscript received 27 February 1997.

Brewster-cut, angle-tuned BBO crystal to produce

the sum-frequency-generated 194-nm radiation.

Light at 515 nm is provided by an argon-ion laser

made to run at a single frequency by a temperaturetimed intracavity etalon. An active servo adjusts

the temperature of the etalon to maintain maximum

power, which eliminates mode hops. Feedback to the position of a piezo-mounted cavity mirror stabilizes the laser frequency relative to a low-finesse ref- erence cavity, reducing short-term fluctuations to

less than 1 MHz. Long-term drift is removed by

feedback to the reference cavity length so that the

laser frequency maintains resonance with a hyperfine feature in molecular iodine.^

The 515-nm beam is mode matched into a power

enhancement cavity in which a 3 mm x 3 mm x

5 mm, angle-tuned, Brewster-cut BBO crystal is placed. The cavity consists of two 10-cm radius of

curvature mirrors, a 30-cm radius of curvature mirror, and a flat input coupler that transmits 1.8% of

the input power. The round-trip length of the cavity

is 1.35 m, and the 28-|jLm minimum beam waist is between the two 10-cm mirrors. The crystal is heat

sunk when pressed into a slot in an aluminum block.

Thin indium foil is placed between the crystal and the

aluminum to ensure good thermal contact. The

crystal is placed at the 28-|jLm waist so that the fundamental beam propagates as an wave for type I phase matching. A dichroic beam splitter reflects 97% of the harmonically generated 257-nm light

while transmitting the 515-nm light. The cavity is locked to resonance with the incident radiation by use

of the Hansch-Couillaud method,'^ as are all other

cavities in this experiment. With the crystal in

20 June 1997 / Vol. 36, No. 18 / APPLIED OPTICS 4159

TN-183
To stabilizing and

<^ analyzing optics

4r-0-

To stabilizing and

analyzing optics

Power

Amplifier Diode

Laser

Fig. 1. Optical layout. DBS, dichroic beam splitter; CL, cylin- drical lens; PD, photodiode; PZT, piezoelectric transducer. Not shown are optics and electronics for stabilizing and analyzing laser frequencies or for stabilizing enhancement cavities.

place and at low circulating powers, the power en- hancement factor is 120.

However, with an incident power of 2 W, the en- hancement is reduced to roughly 90 because of nonlinear conversion loss. As the circulating power

increases, fractionally more of the fundamental

power is transferred to the second-harmonic beam.

This increases the cavity losses, lowering the power

enhancement factor. Consequently, the overall con- version efficiency £ = P257/-P515. where P515 is the

input power at 515 nm and P257 ^^ the output power

at 257 nm, does not increase linearly with the input

power. Rather, it is determined by^'^

^

= 4r^

[2- Vl-T(2-L- Ve-nPsis)]'

(1)

where T is the transmission of the input coupler, L is the round-trip loss not including conversion losses,

and T\ is the single-pass efficiency of the crystal measured at low powers. Figure 2 shows the theoretically expected values of e as a function of input power,

along with our measured values of e, corrected by the

Fresnel loss of the second-harmonic beam at the exit

face of the crystal. For 5.6 W of input power we

-——

-"

y

/^^

/ . • /

•

•

I

f

1

Input Power (W)

Fig. 2. Theoretically expected values of SHG efficiency from Eq.

(1) (solid curve) and measured values (filled circles). Calculated

values use ti = 6.0 x 10"^ W-\ L = 0.007 m, and T = 0.015.

obtain 2.0 W of second-harmonic radiation. The

measured efficiencies, however, are significantly

lower than the theoretical values for input powers of

2 W and greater.

This discrepancy is explained by the small, but

finite, absorption of the harmonically generated UV

light. As more 257-nm radiation is generated, the

crystal develops an anisotropic temperature gradient

that is due to the nonuniform intensity distribution of

the harmonic power. Because the indices of refrac- tion are temperature dependent, a thermal lens is created in the crystal. This changes the modematching efficiency between the input bearn and the

cavity, which decreases the circulating power. ^° The resulting feedback between the intracavity

power and the thermal lens can in turn destabilize

the cavity resonance.^ The heat sink controls the

crystal temperature sufficiently so that the cavity can

be tightly locked with at least 6 W of input power.

However, because the heat sink does not eliminate

the thermal lens, our optimum conversion efficiency

is limited to 40%.

Approximately 500 mW of 257-nm radiation is mode matched into a second power enhancement cavity containing a second, angle-tuned BBO crystal

used for SFG to 194 nm. Because the 257-nm radiation is generated in angle-tuned BBO, its beam profile at the crystal surface is approximately

rectangular along the walk-off direction.^-^i The

beam's width is the walk-off distance pZ, where p is the walk-off angle and I is the length of the crystal.

Because the overlap of this beam profile with a Gaussian distribution of width Wq = l/2pZ is^^

0.89, (2)

the second-harmonic beam can be mode matched

with as much as ^ = 89% efficiency into a cavity with

spherically symmetric Gaussian modes. After passing through suitably chosen spherical and cylindrical

mode-matching lenses, approximately 85% of the in- put beam is coupled into the enhancement cavitys

TEMoo mode.

The 257-nm power enhancement cavity is formed

by two 30-cm radius of curvature mirrors and two fiat mirrors, which includes a 9% transmitting input coupler. The round-trip length is 1.79 m and the minimum waist, located midway between the two curved

mirrors, is 39 |xm. The power enhancement factor of

the empty cavity is approximately 27. When the

Brewster-cut BBO crystal is positioned at the cavity

waist so that it is near Brewster's angle, the build-up

factor can be as great as 21. Due to gradual UVinduced degradation of the cavity surfaces, the

build-up factor at the time of this measurement was

roughly 12. With 500-mW input, then, the circulating 257-nm power is 6 W.

As in the SHG cavity, the high power and intensity

of the circulating 257-nm radiation (>240 kW cm~^

for 6-W circulating power) has several adverse effects

4160 APPLIED OPTICS / Vol. 36, No. 18 / 20 June 1997 TN-184
on the 257-nm enhancement cavity. Thermal lens- ing in the SFG BBO crystal destabilizes the intracavity power as explained above. The baseline of the

Hansch-Couillaud dispersion signal also drifts significantly. This may be due to the increased intensity of reflected higher-order cavity modes combined

with slight beam misalignments. It is also possible

that changes in the crystal birefringence slightly al- ter the polarization of the light reflected from the

cavity. Changes in the crystal's indices of refraction

also alter the phase-matching conditions for SFG.

Thus the difference in the 194-nm beam position

when the cavity is locked into resonance versus when

it is swept through resonance is roughly 3 mrad.

The beam also moves as the crystal temperature fluc- tuates over the course of the day. These problems

are reduced greatly when the SFG crystal is heat

sunk in the same way as the SHG crystal, although

the 257-nm intracavity power can still vary by more

than 10% over tens of minutes.

Approximately 500 mW of narrowband singlefrequency radiation at 792 nm is produced by a master oscillator power amplifier system. ^^ The master

laser is an external-cavity diode laser in the Liftman

configuration, whose free-running line width is less than 0.5 MHz. The frequency of the master laser is stabilized to an external cavity whose length is ser- voed to light from the iodine-stabilized argon-ion la- ser. Approximately 18 mW from the master laser are mode matched into a tapered amplifier^^ acting as a slave laser. With no more than 1 . 1 A into the slave

laser, as much as 600 mW can be generated safely.

We verified that the frequency of the slave laser is phase locked to the master laser with as little as 3

mW of injected power. The shape of the slave laser

beam is transformed so that it is nearly spherically

Gaussian, after which the beam is mode matched into a power enhancement cavity with greater than 90%

efficiency.

The 792-nm enhancement cavity is formed by two

30-cm radius of curvature mirrors and two fiat mirrors. The round-trip length of the cavity is 1.83 m,

and the minimum cavity waist size is approximately

68 |jLm. The transmission of the input coupler is 1.5% and the build-up factor of the empty cavity is approximately 80. With the crystal placed near the

cavity waist so that its surfaces are at Brewster's

angle, the power enhancement is reduced only

slightly. Thus for 600-mW input power, the circulating power is 48 W. Aside from a 10% decrease in

intracavity power when the 257-nm cavity is locked

to resonance, the 792-nm cavity is not affected by

thermal lensing in the crystal at these powers.

The 792-nm and 257-nm cavities are situated so that the intracavity beams coincide at the crystal

position near the minimum beam waists. The

Brewster-cut SFG BBO crystal is 4 mm X 4 mm X

5.75 mm, and cut at 71.6° to the optic axis. Both

fundamental beams propagate as o waves inside the

crystal, while the SFG beam propagates as an e wave. To make the beams collinear inside the crystal, the

incidence angles of the fundamental beams at the

crystal are adjusted to account for the difference in

refraction angles. Only the 792-nm beam is strictly at Brewster's angle. Because of the differences in

refraction angles of the SFG and fundamental beams,

the 194-nm beam is separated sufficiently from the

792-nm and 257-nm beams to pass the cavity optics

easily. For the conditions described here, a National

Institute of Standards and Technology calibrated

photodiode measures 1.4 mW of 194-nm radiation

2.0 m from the crystal. If absorption in air and the

Fresnel loss at the exit surface of the crystal are accounted for, approximately 2.2 mW of radiation is generated.

To conclude, we have produced more than 2 mW of

194-nm radiation, which is 2 orders of magnitude

greater than previously reported. This is due partly

to production of 500 mW of 257-nm radiation by SHG

in BBO, whereas previous experiments produced only

20-30 mW by SHG in ammonium dihydrogen phosphate (ADP).4.5 Furthermore, although the quality

of the 257-nm power enhancement cavity is not improved over that reported in Ref. 5, the build-up fac- tor of the 792-nm power enhancement cavity is seven

times that of Refs. 4 and 5. The 194-nm power may

still be increased by approximately a factor of 3 when

the intracavity beam waist sizes are decreased closer

to the optimum size of approximately 15 ixm^^ and

when the build-up factor of the 257-nm power en- hancement cavity is increased. However, the higher

intensities (especially at 257 nm) might cause crystal

damage or severe thermal problems.

We thank S. Mioc, D. Meekhof, M. Young, and D.

Sullivan for careful reading of this manuscript.

F. C. Cruz acknowledges the support of Coordenadoria de Aperfeifoamento de Pessoal de Nivel Superior (CAPES) (Brazil).

References and Notes

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

D. J. Wineland, "Ionic crystals in a linear Paul trap," Phys.

Rev. A 45, 6493-6501 (1992).

2. J. Miller, D. Berkeland, F. Cruz, J. Bergquist, W. Itano, and D. Wineland, "Cryogenic linear trap for accxirate spectroscopy," in Proceedings of the 1996 IEEE International Frequency Control Symposium (Institute of Electrical and Electronics Engineers,

New York, 1996), pp. 1086-1088.

3. The crystal SraBegBaOy (SBBO) has produced wavelengths as

short as 177.3 nm by SHG. C. Chen, B. Wu, W. Zeng, Y. Wang, N. Ye, and L. Yu, "New nonlinear optical crystal Sr2Be2B207: growth and properties," in Quantum Electronics & Laser Science Conference, Vol. 10 of 1996 OSA Technical

Digest Series (Optical Society of America, Washington, D.C.), pp. 229-230. The crystal KBeaBOgFzdCBBF) has produced 184.7-nm SHG output. C. Chen, Z. Xu, D. Deng, J. Zhang,

G. K. L. Wong, B. Wu, N. Ye, and D. Tang, "The vacuum

ultraviolet phase-matching characteristics of nonlinear optical

KBeaBOgFz crystal," Appl. Phys. Lett. 68, 2930-2932 (1996).

4. H. Hemmati, J. C. Bergquist, and W. M. Itano, "Generation of continuous-wave 194-run radiation by sum-frequency mixing

in an external ring cavity," Opt. Lett. 8, 73-75 (1983).

5. M. Watanabe, K. Hayasaka, H. Imago, and S. Urabe,

"Continuous-wave sum-frequency generation near 194 nm

with a collinear double enhancement cavity," Opt. Commun.

97, 225-227 (1993).

20 June 1997 / Vol. 36, No. 18 / APPLIED OPTICS 4161

TN-185
6. J. C. Bergquist, H. Hemmati, and W. M. Itano, "High power

second harmonic generation of 257 nm radiation in an external

ring cavity," Opt. Commun. 43, 437-442 (1982).

7. T. W. Hansch and B. Coioillaud, "Laser frequency stabilization by polarization spectroscopy of a reflecting reference cavity," Opt. Commun. 35, 441-444 (1980).

8. A. Steinbach, M. Rauner, F. C. Cruz, and J. C. Bergquist, "CW

second harmonic generation with elhptical Gaussian beams,"

Opt. Commun. 123, 207-214 (1996).

9. W. J. Kozlovsky, C. D. Nabors, and R. L. Byer, "Efficient second harmonic generation of a diode-laser-pumped CW Nd:YAG la- ser using monolithic MgOtLiNbOg external resonant cavities," IEEE J. Quantum Electron. 24, 913-919 (1988).

10. E. S. Polzik and H. J. Kimble, "Frequency doubling with

KNbOg in an external cavity," Opt. Lett. 16, 1400-1402 (1991).

11. G. D. Boyd and D. A. Kleinman, "Parametric interaction of focused Gaussian light beams," J. Appl. Phys. 39, 3597-3639

(1968).

12. M. G. Boshier, "Precise laser spectroscopy of the hydrogen

1S-2S transition," Ph.D. dissertation. (University of Oxford,

Oxford, England, 1988).

13. F. C. Cruz, M. Rauner, J. H. Marquardt, L. Hollberg, and J. C. Bergquist, "An all solid-state Hg"^ optical frequency standard,"

in Proceedings of the Fifth Symposium on Frequency Stan- dards and Metrology, J. C. Bergquist, ed. (World Scientific, Singapore, 1996), pp. 511-513.

14. SDL-8630; the mention of brand names in this paper is for information purposes only and does not constitute an

endorsement of the product by the authors or their institutions.

4162 APPLIED OPTICS / Vol. 36, No. 18 / 20 June 1997 TN-186
JOURNAL OF APPLIED PHYSICS VOLUME 83, NUMBER 10 15 MAY 1998

Minimization of ion micromotion in a Paul trap

D. J. Berkeland,^^ J. D. Miller,'" J. C. Bergquist, W. M. Itano, and D. J. WIneland

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

(Received 26 November 1997; accepted for publication 12 February 1998)

Micromotion of ions in Paul traps has several adverse effects, including alterations of atomic

transition line shapes, significant second-order Doppler shifts in high-accuracy studies, and limited confinement time in the absence of cooling. The ac electric field that causes the micromotion may

also induce significant Stark shifts in atomic transitions. We describe three methods of detecting

micromotion. The first relies on the change of the average ion position as the trap potentials are changed. The second monitors the amplitude of the sidebands of a narrow atomic transition, caused

by the first-order Doppler shift due to the micromotion. The last technique detects the Doppler shift induced modulation of the fluorescence rate of a broad atomic transition. We discuss the detection

sensitivity of each method to Doppler and Stark shifts, and show experimental results using the last technique. [50021-8979(98)05610-2]

I. INTRODUCTION

Because of their low velocities, cooled and confined ions can provide the basis for accurate and stable frequency stan- dards and atomic clocks. For example, for '^^Hg"^ ions trapped in an rf Paul trap and laser cooled to the Doppler

limit, the magnitude of the fractional second-order Doppler

(time dilation) shift of transition frequencies can be as low as 2X10~'^. ' However, due to the ion motion synchronous

with the trap ac field (the "micromotion"), this shift can be

orders of magnitude larger if the average ion position is not

at the nodal position of the trap's ac electric field. To realize the high accuracy of a trapped-ion frequency standard, the ion micromotion must be minimized. In this article, we dis- cuss ion micromotion in a Paul trap and its associated effects on stored ions and their transition frequencies. We also describe methods to detect and minimize micromotion, and

present experimental data using one of these methods.

II. MICROMOTION IN A PAUL TRAP

For brevity, we characterize motion of a single ion in one type of Paul trap that may be particularly useful for high-accuracy spectroscopy. The results in this section can be generalized to other types of Paul traps. If several ions are stored in the same trap, the equations of motion must be

modified to include modes of collective motion. However,

the conclusions about micromotion and its effects are still valid.Figure 1 shows a schematic diagram of a linear Paul

trap.'"^ Electrodes 1 and 3 are held at ground potential, while

the potential of electrodes 2 and 4 is Vq cos(nO. Typically,

for atomic ions, 0/27r> 100 kHz and |Vol<1000V. Near

the axis of the trap the potential due to the electrodes is

V(x,y,t)=^ 1 + cos(n?). (1)

"'Electronic mail: dana.berkeland@boulder.nist.gov

•"Present address; KLA, Austin, TX.

R is the perpendicular distance from the trap axis to the trap electrodes [shown in Fig. 1(b)], and R'=R (/?'=/? if the

trap electrodes are hyperbolic cylinders of infinite length).^''' The gradient of the corresponding electric field confines the ion radially in a harmonic pseudopotential.^ To confine the ion axially, two endcaps held at potential Uq create a static potential Uix,y,z). Near the center of the trap, U{x,y,z) can

be approximated by

U{x,y,z)- ^[.^ {x'+y^)l (2)

where k( < 1 ) is a geometrical factor and Zq is shown in Fig.

1(a). Here, for simplicity, we have neglected the (small)

component of alternating electric field along the z axis caused by the electrode configuration shown in Fig. 1. Linear

trap electrode geometries which eliminate this field are dis- cussed in Refs. 1 and 2 (see also Sec. V). From Eqs. (1) and

(2), the total electric field is

xx — yy\ E{x,y,z,t)=-Vo\—^T2-\<^os{at)

R' xUo . . . -^=2-[2zz-xx-yy'].

•^0 (3)

The equations of motion for a single ion of mass m and

charge Q in the above field are given by the Mathieu equation

ft2

M, + [a, + 2g, cos(flf)] — M, = 0, (4)

where u=u.^x + u^y + u^z is the position of the ion using the coordinate system shown in Fig. 1(b), and from Eq. (3),

£ __4QkUo

and

2QVo

n^R^HT^' ^^-°-

(5)

(6)

5025 TN-187

Downloaded 25 Sep 2001 to 132.163.135.43. Redistribution subject to AlP license or copyright, see http://ojps.aip. org/japo/iapcr.jsp
5026 J. Appl. Phys., Vol. 83, No. 10, 15 May 1998 Berkeland et al.

Trap.

Axis

endcaps

r^® (D^

Vo cos(Qt) ^_

>® ®^ '^

X X / y

(b)

FIG. 1. Linear Paul trap (a) side view and (b) axial view. A string of trapped

ions is shown schematically in (a). For clarity, the endcaps are not shown in

(b). The trap electrodes are labeled 1, 2, 3, and 4. The trap axis defines the

z-axis, and the origin of the z axis is centered between the two endcaps.

where kg is the Boltzmann constant, and the kinetic energy is characterized by a temperature T^ . Typically, in the radial direction, \ai\<q'f{i = x,y). The energy of the radial secular motion is then approximately equal to that of the radial micromotion. In this case,

£'/^,^^b7', = jwm,,.co,(j = x,j). (12)

The energy of the secular motion can be reduced by

cooling."^ As the amplitude «], of the secular motion is re- duced, the micromotion and its corresponding energy are also reduced according to Eqs. (8) and (10). The Dopplercooling limit of the ion temperature due to secular motion in one direction is'°

Tn =

hy

2r„

(13)

where y is the linewidth of the cooling transition. As an example, for the '^^Hg+ 5d^%s ^Sm^Sd^^p ^P^a transi- tion used for laser cooling, y=2'7r-70MHz. The Dopplerlimited cooling temperature is Tq=1.1 mK.

If, in addition to the trap fields described above, the ion

is also subjected to a uniform static electric field E^^ , Eq. (4) becomes

Ui + [aj + 2q^ cos(nr)] —— m, = , (14)

For convenience, we also define the unit vectors

u^ = x, Uy = y, and u^ = z- (7)

In the typical case where |g,|'=^l and |a,|<^l, the first-order solution to Eq. (4) is^

M,(0==="l, COS(fa),f-f(p5,) 1 + y COS(nf)

where

,^inV«,+k-

(8)

(9)

and <p5, is a phase determined by the initial conditions of the ion position and velocity. The "secular" motion of the ion is the harmonic oscillation at frequency o), and amplitude W],

The motion corresponding to the cos(n?) term is driven by

the applied ac field, and is called "micromotion."

From Eq. (8), the kinetic energy of the ion averaged over

a period of the secular motion is

,2 ~ 1 2 21,, ^

g- + 2a,

(10)

where the first term in the last two expressions is the kinetic energy due to the secular motion, and the second term is the

kinetic energy due to the micromotion. For motion parallel to the trap axis, qZ = 0, so the average kinetic energy is due

entirely to secular motion. Because the secular motion is typically thermal, incoherent motion, the kinetic energy due

to motion in the c direction is

£'/c.-=2^er- =

1 2 2 (ii;

To lowest order in a, and q, , the solution to Eq. (14) is

",(0 = [wo, + "i, cos(co,r-h(p5,)] 1 + y COS(nO

where

«o,= 4eEd,-M, QEdc-M,

m(a,+ j^f)n^ mo),

, (15)

(16)

The field E^ displaces the average position of the ion to "o^^Ojc^ + woyy + woz^. but does not directly change m,,. The ac electric field at position Uq causes micromotion of amplitude 2'"0i^( along m, . We will call this "excess micromotion," to distinguish it from the unavoidable micromotion

that occurs when the secular motion carries the ion back and

forth through the nodal line of the ac field. Unlike secular motion, excess micromotion cannot be significantly reduced

by cooling methods because it is driven motion. Excess micromotion can also be caused by a phase dif- ference (Pac between the ac potentials applied to electrodes 2 and 4. For example, in the trap shown in Fig. 1, the potential

applied to the electrode 4 may be -I- Vq cos(nr-l- j<Pac)> and to electrode 2, Vq cosCHr— j^^c)- If fac"^ 1' these potentials are approximately equal to Vg cosCfl?) — jVotpac sinCflr) and

Vq cos(no+2V^o'Pac sin(nr), respectively. Near the trap axis, the additional field due to the ± 2 Vg^ac sin(fl?) terms is ap- proximately that of two parallel plates held at potentials — 2^0'Pac sin(nr) and separated by 2^/a." The value of a

depends on the geometry of the trap. We use the method of van Wijngaarden and Drake'" to calculate the dipole moment

for our trap (/? = 0.81mm, electrode radius / = 0.38mm),

and find a = 0.75. If we include a uniform static field, the

total electric field near the center of the trap is TN-188
J. Appl. Phys., Vol. 83, No. 10, 15 May 1998 Berkeland et al. 5027

E{x,y,zj)=- -^72 ixx-yy)cosiat)- —2-

X{2zz-xx-yy) + Eit:

+

2R

sin(n?)x. (17)

With the additional oscillating electric field due to cp^,- , the equations of motion in the y and z directions remain un- changed from Eq. (15). However, if we solve the equation of motion in the x direction to lowest order in a^ , q^ and cp^,.

,

and use R' = R in Eq. (6), then

";c(0 = ["o;c + wix cos{(o„+(ps^)][l+ \q, cosCHr)] - i q^Ratp^^ sin(nO. (18)

Equation (18) shows that unless cp^^—Q, the excess micromotion in the x direction will not vanish for any average ion

position Uq . From Eqs. (15), (16), and (18), the average kinetic en- ergy due to motion along m, is L^.2,2^l2r.2, 4 / (29,Edc-«,

£^,= J '""w(<+ ¥?/"') + m \(2a, + g,')n^ + ^m(g,/?a<P3,n)-<5,.^. (19)

In order to compare the size of the last two terms relative to the first, it is useful to write them as k^T^i/l, where T^, is the equivalent (pseudo) temperature for the kinetic energy due to the excess micromotion along «, . A uniform static

field along the axial direction does not change E^. , since it only shifts the position of the minimum of the static potential

Uix,y,z)- For a '^^Hg"*" ion in a trap with \a,\<qj<l and

a)jf = 27r- 100 kHz, a 1 V mm^' uniform field along the x

direction increases T^^ by 1.4 X 10"^ K. For R= l.O mm and a = 0.75, a phase shift of <Pac= 1 ° between the trap electrode

potentials increases T^^^ by 0.41 K. These effective temperatures are orders of magnitude greater than the 1.7 mK temperature associated with the secular motion at the Dopplercooling limit. These phase shifts and electric fields may be reasonably

expected. A phase shift can be caused by asymmetries in the

electrical impedances of the electrodes. For example, a phase

shift will occur if the leads to the trap electrodes have dif- ferent inductances due to different lengths or geometrical

arrangements. A uniform electric field of magnitude

1 V mm"' may develop in a millimeter-sized trap in several ways. Often, an effusive oven located on one side of the trap

is used with an electron-emitting filament to produce ions

inside the trap. In this case, the trap electrodes may become

unevenly coated with the oven contents, which could cause

contact potentials of a fraction of a volt. Additionally, the

trap electrodes may become unevenly charged when this coating or other dielectric or oxide layer is charged by the emitted electrons. Finally, patch effects due to different crystal planes at the surface of the electrodes also can produce

surface potential variations of roughly 100 mV. Although the magnitude of stray fields caused by patch effects and charging of the trap electrodes can be reduced by heating the trap electrodes in situ,^^ no technique can eliminate these fields. Below, we will give general expressions for the effects of excess micromotion and several methods to detect it. To

provide examples, we consider the Doppler shifts and the ac Stark shift of the '^''Hg^ 5d^%s ^Sy2-^5d^6s^ ^D^a electric quadrupole transition at 282 nm, and the sensitivity of

the various methods used to detect these shifts. Because ef- fects from excess micromotion are negligible in the limit that \a\>q], in the following sections all examples assume that l^/l^^f^l {i = x,y). We take the physical trap parameters

as «'«=/?=I.Omm, a = 0.75, n = 27rlOMHz, cu^^w^, = 27rI.0MHz, and g;,'^^^,^ 0.28.

III. EFFECTS OF EXCESS ION MICROMOTION

The first-order Doppler shift due to excess micromotion

can significantly alter the excitation spectrum of an atomic

transition. The spectrum can even change so that a laser heats the ions at frequencies where laser cooling is normally

expected.'*'^ Assume that the electric field of the laser used

to excite the ion has amplitude Eq, frequency wiaser. phase

<Piaser' ^"^ wave vector k. From Eqs. (15) and (18), in the

rest frame of an ion undergoing excess micromotion, this laser field becomes

E(f) = Re{Eo exp[/k-U-/aj,aser?+(p,a5er]} = Re{Eo exp[/k-(Uo+u')-/W|aserf+'Plaser]}> (20)

where u' is the amplitude of the excess micromotion. To

isolate the effect of excess micromotion, we have assumed

that |moiI^I"iiI ^nd |^a^ac|^|Mi,|. From Eqs. (15) and

(18),

k-u'(r) = jen cos(nr+<5),

where

/3= VU" 2 ^,"0.9/ + T^;t9;c^«^ac . 2 ,^,

sin S- k^q^Racp^

and

cos S=

V(22,=;,,yA:,Mo,g,)' + (M;c^«<Pac)^'

yJ{21,=^ykiUo,q,f + {k^q^Ra(p^^)^

(21)

(22)

(23)

(24)

With the Bessel function expansion

oc

exp[/yS cos(nr+<5)]= 2 Jni/3)exp[in{nt+ 5+ tt/2)],

(25)

Eq. (20) can be written as 100

Eoexp[/k-Uo] ^_^Jni/3)

Xexp[ - / a>|^ser'+ <Plaser+ '"(^f + '5+ 'n"/2)]

TN-189 (2^^
5028 J. Appl. Phys., Vol. 83, No. 10, 15 May 1998 Berkeland et at.

P.'

1

- A

0.6- \ —P-0

0.8 • _\

-• P-5

[1=10

0.4

J/J Vv

•p.20

0.2'

0- \" 1 "l^ 1 H 1 1

Ion motion also produces a second-order Doppler (timedilation) shift of atomic transition frequencies

V, (29)

p;

0.8

0.6 ••

0.4

0.2 •

-1 1

(a)

I

where v is the atomic transition frequency and V is the ion

velocity. From Eqs. (12) and (19), the fractional shift due to motion along «, can be written as

Ai^Dl 1 mc

fcBTiiai + qf) 4

—p.o

0-2 + '

QQi^dc-Ui

m \(2a, + ^/)n

64

Sl. (30)

-A. kJLk JL

-25 -20 -15 -10 -5 5 10 15 20 25

(b)

FIG. 2. Effect of micromotion on the spectrum of Pg (excited state population). We plot P'g = Pglhy/(2^\Eol)f for various values of /3. For both

graphs, we assume that the ion is driven below the saturation limit, (a) Cl/y^OA. For ^=10, heating occurs in the regions <0.6<(a;|jse, -'^a.omVr<0 and (o>,^,-<')^„J/y>0.6. (b) n/y= 10. For /3>0, heating can occur when the laser frequency is tuned near, but above the center of, any of the sideband frequencies.

We define ^ |E| as the amplitude of the interaction matrix

element between atomic levels \e) and |^) coupled by an

electric field Re{Ee """*""''}. Here, w is the field frequency and

(p is the field phase. If the field of Eq. (26) interacts with the atom in the low intensity limit

If Ejjc = 0, <Pac~0 3nd T^=1J mK, motion in the x direction

for a '^Hg"^ ion contributes -8X10~'^ to the fractional second-order Doppler shift. However, if <Pac~l°' the fractional shift becomes — 9X10~'^. A 1 Vmm~' field along

the X axis further increases the magnitude of the fractional

shift by 3X10" '1 The ac field that causes micromotion can also cause sig- nificant ac Stark shifts. The Stark shift due to the field the ion experiences is approximately

^Vs=as{E{u,t)^), (31)

where cr^ is the static Stark shift constant and (£'(u,r)^) is the time-averaged square of the electric field at the ion position. To lowest order in g, and a, , substituting Eq. (18) into Eq. (17) gives

(£?(u,r))^

mD.^kBTi a^+2qj ^IqiE^-Uj

2Q^ 2ai + q\

1 lmq^Ra(p^^Cl'

2a,+ q]

^|Eol

<^y. (27)

32

2\ 2 (32)

then the steady-state solution to the optical Bloch equation

for the upper level population P^ is''*'^

:^|En JlW)

(^a •^a

(28)

where a>atom 's the resonance fi-equency of the atomic transition. Figure 2(a) shows the excitation spectrum calculated from Eq. (28) for various magnitudes of micromotion, for iKy and in the low intensity limit. As fi increases from 0, the frequency modulation from the excess micromotion first broadens the transition. This decreases the rate at which a

laser can cool the ion. For larger values of fi, the line shape can develop structure that causes the laser to heat, rather than cool the ion, even when wiaser^^atom^O. Figure 2(b) shows

the effect of micromotion when fi^y. As yS increases, the excitation spectrum develops sidebands at ±nVl (n = 1,2,3,...), and the strength of the carrier transition decreases. Heating now occurs when the laser frequency is tuned near, but above the center frequency of any of the sidebands.

The second term is much greater than the square of Ejjj.w,

when \ai\<q]<\, so & small uniform static field can induce

a large Stark shift. Stark shifts have been measured for the three Zeeman

components of the 1.76 /xm 6^S^i2—*S^D^i2 transition in a ~ „ 2 , ci 71' single '^^Ba'"' ion.'^ The ion, cooled to about 1 mK, was

confined in a spherical Paul trap in which ft = 27r-26MHz

and Wj. = 27r-2.6 MHz. A static electric field was applied

along the z direction, and the shift of the transition frequency was measured as a function of the field strength. The measured values of 0-5 were on the order of + 10~^ Hz/(V/m)"

for each Zeeman component. With these values of cr^ , the fractional Stark shift of the transition frequency in the ab- sence of uniform static electric fields was calculated to be

10~'^. A 5 mV potential across the 0.3 mm diameter trap was predicted to cause a fractional shift of + 1.2X 10~'^. We have estimated the static Stark shift constant 0-5 of

the 282 nm '^'Hg+ 5t^'°65 ^5,/2^5c?^6.y^ -D5/2 electric quadrupole transition. First, we calculate the matrix elements between the Sd^^bs 'Sy2 ground state and the closest few

states with the 5d^^6p configuration. This gives a5 = 2.1 X 10" ^"^ cm-' for the polarizability of the ground state. To

estimate the polarizability of the 5 ^^6^" ^D^^ state, we con- TN-190
J. Appl. Phys., Vol. 83. No. 10, 15 May 1998 Berkeland et al. 5029

sider transitions from the 5d^6s^ ^D^n state to the various

states with the configuration 5d^6s6p. In the 5d^6s6p

states, one of the two 65 electrons of the '^D^u state has been

excited to a 6p state. That transition is qualitatively similar

to exciting the single 6s electron of the 5d^^6s ^5^2 state to one of the 5d^^6p states. Hence we assume that the matrix elements between the 5d^6s^ states and the 5^5/2656^ states are similar to those between the 5d^%s and 5d^%p states. The center of gravity of the 5d^6s^—*5dl/2^s6p transitions

is about 208 nm. This wavelength is not much longer than

that of the first 5d^%s^5d^%p transition at 194 nm. Thus we expect that the polarizability a^ of the 'D5/2 state is on

the order of that of the ground state. Both states will be

shifted down in energy by a static electric field, since all electric dipole transitions connect them only to higher-lying

states. For the electric quadrupole transition, then, we esti- mate that Io-jI^I.OX 10"^ Hz/(V/m)l

This number lets us estimate the magnitude of the Stark

shift for the 282 nm quadrupole transition, using the parameters stated earlier. lfE^^—0, ^ac~0' ^nd the ion is cooled to the Doppler limit in the x direction, then the Stark shift of the 282 nm quadrupole transition due to the electric field in the x

direction is lAi^s/j^l^l.lX 10"'^. A 1 Vmm"' static field along the x direction causes a fractional Stark shift of

I A 1^5/1^1^9x10"''*. If ^ac=l°. the magnitude of this shift increases by about 3X 10"''*. Under favorable circumstances, the second-order Doppler and ac Stark shifts from excess micromotion can be

made to cancel. If we consider only the effects of excess micromotion, we have

QE{u,t) = m

d'u'

dt' m

dW,

dt

(33)

where V^ is the velocity of the excess micromotion. From

Eq. (31), the ac Stark shift can be written as

A 1^5= 0-5 mCt

(vl). (34)

Using Eq. (29), we can write the sum of the fractional second-order Doppler and ac Stark shifts as Av5 A 1^20 05

V ~Q

1 2? {vD- (35)

For 0-5 >0, it might be possible to make the factor in brackets equal to 0. As an example, for the 282 nm quadrupole

transition in '^^g+, if 0-5= 1 X 10"^ Hz/(V/m)-, this factor

is for = 277-8.4 MHz, close to the condition of the ex- periment reported in Sec. V.

Finally, if several ions are stored in the same trap, excess micromotion can also increase the magnitude of the secular motion. The micromotion and secular motion of a single ion

in a Paul trap are highly decoupled, so excess micromotion

will typically not increase the secular motion. However, if two or more ions are in the trap, the energy of the excess micromotion of any ion can be parametricaily coupled into the energy of the secular motion of the other ions.^'*'^'^ Since the micromotion is driven by the ac field, this heating

is continuous and can limit the lowest temperature attainable

by cooling methods. In the absence of cooling mechanisms,

the ions can gain enough energy to leave the trap.

IV. DETECTION OF EXCESS MICROMOTION

Different techniques can be used to detect excess micromotion caused by a uniform static field E^^ or phase differ- ence <Pac between the trap electrode potentials. In the first of

these methods, which is sensitive to excess micromotion

caused by static fields, the time-averaged ion position is monitored as the pseudopotential is raised and lowered.'* If an imaging system is used to view the ion as it is translated, then the ion position in the plane of observation can be de- termined to the resolution limit of the optics. Translations can also be detected in any direction by monitoring the dis- tance that a focused laser beam must be translated to maintain the maximum photon scattering rate from the ions. Let Amo, be the measured translation along m, when the secular frequency is reduced from cu,i to w,2. From Eqs. (16) and

(30) (taking \ai\<q'f<tl), when the secular frequency is

(Oji , the fractional second-order Doppler shift due to excess micromotion along m, is

Ai/Q- w,2 AMo,wn

'i2 (36)

From Eqs. (16) and (32), the Stark shift due to the ac field along M, is

( A 1/5), = 0-5 ^ 2

O),, - W,2

(t)j2 mAuQiOJiifl]

(37)

As an example of the sizes of the detectable shifts, we as- sume the same parameters for the trap and ion ('^^Hg"*") as above. We assume also that the ion position changes by

|AmoxI~25 /um in the x direction when the pseudopotential

is lowered to 0)^ = 217-0.5 MHz. Then the second-order Doppler shift when W;f = 277- 1.0 MHz is Ai'd2/*'= ~ 1-5 X 10"'"*, and the Stark shift of the electric quadrupole transition is |Av5/j^|^4XlO"''*.

This technique can also be used by modulating the pseudopotential (by modulating Vq at frequency Wnjoj-^n)

while the ion is located in the waist of a laser beam tuned to a cycling transition (for example, the Doppler-cooling transition). We assume that |a,|^g,<gl and that the modulation

is adiabatic, so while the magnitude of the excess micromotion changes, the magnitude of the thermal motion is ap- proximately constant Suppose that the laser beam has a transverse Gaussian intensity profile

-,.2, /(r) = /o exp(-2rVwo), (38)

and that the ion lies on the half-intensity radius of the beam

(39)

The secular frequency is given by

W,' = W, -1- A O), COS( (O^odl + <Pmod)

.

where

(40)

TN-191
5030 J. Appl. Phys., Vol. 83, No. 10, 15 May 1998 Berkeland et al.

Afa)^ Aw„

(41)

^mod is the phase of the modulation, and A Vq is the modulation of the trap rf amplitude. Here, for simplicity, we as- sume that Aa),/w,<tl. The ion position averaged over a cycle of the rf potential varies as

"0, = «0,~A«0, COS(Wn,odf+(Pn,od),

where

2mo,AVo,.

Amo, = —7} ('=Jc>>')-

(42)

(43) We define r= rr as the vector from the laser beam axis to the ion position (such that r-k—0). If Auo-r<^M'o, then the la- ser intensity in the rest frame of the ion is

/=/r

1 V21n2 AUo-f COS(Wn,od^+(Pn,od)

Ho -A/ COs(Wn,odr+(p„od). (44)

In the low intensity limit, the detected fluorescence signal is ^rf=2^max-A/?dCos(Wn,odr+9n,od)- Hcrc, A /?rf is the amplitude of the signal synchronous with the pseudopotential

modulation, and R^ax 'S the signal when the ion is at the center of the laser beam profile. We can write

A/frf A/

R. —= ^J2\n2

'0

AUn

Wo

I |AmoI

V2 In 2 cos 6i^

(45) where 6^^ is the angle between Auo and r. From the measured value of Ai?rf//?n,ax and for a known value of AVq/ ^o- we can determine Mqi from Eqs. (43) and (45). From Eqs.

(16), (30), and (32), we can then determine the corresponding values of At-fl/v and (E'f), analogous to Eqs. (36) and

(37). Generally, cos 6^ is not known, but it can be maximized and the direction of Au can be determined by moving

the laser beam appropriately.

The main disadvantage to the above techniques is that they are not sensitive to excess micromotion caused by a phase shift ^^c between the potentials applied to the trap electrodes. If mox~0 but (p^^^O, the average ion position

will not change as the pseudopotential is raised and lowered,

as indicated in Eq. (18). Techniques that sense the magnitude

of the first-order Doppler shift caused by the excess micromotion eliminate this problem.

We will assume that we measure the effects of the first- order Doppler shift on an optical transition with natural width y. Previously, first-order Doppler shifts of microwave

spectra have been used to determine the temperature of the secular motion of trapped ions.'^'^*^ We first take the case in which Cl>y. The micromotion can be monitored by measuring the scattering rate Rq when the laser is tuned to the

carrier (w|aser~ w^toni— 0) and R^ when tuned to the first sideband (a)|aser~'^atom= -^)^' [see Fig. 2(b)]. From Eq.

(28), in the low intensity linnit.

/?! -/?(/?) ^0 Jl(^)

' (46)

where /3 is defined in Eq. (22). For fi<l,

R\ , T /^(i;8)^

^0

and since

y8=|k-u'| =

k V,

n

(47)

(48)

the fractional second-order Doppler shift can be written as Avd2 CI ck cos 6'^J Rq

' Ry

(49)

where 6^i^ is the angle between k and the direction of the excess micromotion. From Eqs. (34), (47), and (48), the cor- responding Stark shift can be written

^Vs=2(Ts

Icl' Qk cos d^k ^0

(50)

As an example, we assume that we probe the sidebands on

the 282 nm transition in '^^Hg"^ in a trap with the parameters

listed previously. If ^^^ = and Ri/Rq = 0.1, then the second-order Doppler shift is A 1^02/ 1^= — 9 X 10"'^. The

corresponding Stark shift is |Ai^5/i'|«2.5X 10~'^

In the limit fi<^y, a sensitive method to detect excess micromotion monitors the modulation of the ion's fluores- cence signal due to the first-order Doppler shift. ^^~^ We will

call this the "cross-correlation" technique because the modulation is correlated to the ac potentials applied to the

trap electrodes. For simplicity, we assume that the amplitude

of the first-order Doppler shift is much less than the line- width y. From Eqs. (15) and (18), the velocity due to excess micromotion is given by

1 V^(0 = -r2 "0,9/^ sin(nf)M,

—q^Ra(pj£l cos,{Cit)x. (51)

In the frame of an ion undergoing excess micromotion, the frequency of the laser is Doppler shifted by -kV^

= ySn sin(nr+^, where jS and d are defined in Eqs. (22), (23), and (24). In the low intensity limit, the detected fluo- rescence rate is thus

-.,\2 R,=R

(ir)

d "^ max I 2 (iy)' + ['^a.om-Wlaser->en ^m{Clt + S)f

(52)

We take Watom~<^iaser= T^^, which is a natural choice since

this minimizes the temperature of the Doppler-cooled ions and because it maximizes the cross-correlation signal. If l3VL<y, then

Ra 1 /Sn sin(n/+5)_ 1 A/?rf sin(fi/ + <5) R 2 y

p

(53)

Using Eq. (48), we can write the fractional second-order Doppler shift as

TN-192
J. Appl. Phys., Vol. 83, No. 10, 15 May 1998 Berkeland et al. 5031

AJ^D2 A/?,

4 \ck cos 9^1, R^

(54)

Biasing Rod 3

where, again, 6^i^ is the angle between k and the direction of

the excess micromotion. With Eqs. (34) and (48), the corre- sponding Stark shift can be written as

^Vs=(Ts

1 / my a AR,

2 \ Qk cos d^k Rm^:

(55)

As an example, we consider '^^Hg"*" ions, using the previously stated trap parameters, and y=27r-70MHz for the 194 nm Doppler-cooling transition. If ^^<: = and

A/?^//?max~0.1, then the second-order Doppler shift is Avj}2lv=-5X\Q~^^, and the Stark shift of the electric quadrupole transition is lAt-j/i^l^l.SX 10"'^. As opposed to the first method for sensing micromotion,

the cross-correlation technique can determine whether the ac

potential applied to electrode 2 is out of phase with that applied to electrode 4 (see Fig. 1). If a deliberately applied

static electric field moves the ion to different positions in the

trap, the phase of the fluorescence modulation at frequency

n depends on q>^^ . The atomic velocity is 90° out of phase

with the force due to the ac electric field. Thus if <Pac=0, the phase of the cross-correlation signal jumps by 180° as the average position of the ion crosses the nodal line of the ac

field. Also, when the ion is on the nodal line, the signal at frequency D, vanishes. However, if (p^c^O, from Eq. (18), the phase of micromotion in the x direction continuously

varies as the varies as the average ion position is changed.

Furthermore, the amplitude of the micromotion is never zero. This behavior can be used to determine the relative contri- butions of stray static electric fields and electrode potential

phase shifts to the excess micromotion. In general, the ef- fects of Ejc can be eliminated by purposely applying a static field Eappiied= ~E(jc.; <Pac ^au be made zero by loading elec- trodes 2 and 4 with the appropriate reactances.

Still, avoidable effects may confuse the interpretation of

the cross-correlation signal. For example, as the ion moves

back and forth across a nonuniform laser beam intensity profile, the fluorescence is modulated at frequency Cl, even if k-V^ = 0. This modulation is minimized when the ion is at the center of the (symmetric) laser beam, regardless of its average distance from the ac field's nodal position. However,

this condition can be detected—the phase of this fluores- cence modulation is sensitive to lateral translations of the

laser beam, which is not true if the fluorescence modulation

is due to the first-order Doppler shift.

It is also important to avoid tuning the laser frequency

too close to the atomic frequency. In this case, the fluores- cence modulation due to the first-order Doppler shift [Eq.

(52)] is deceptively small. This situation, though, is easily

checked by detuning the laser frequency farther from atomic

resonance to see if the fluorescence modulation amplitude

increases.

Finally, to determine that the micromotion is zero in all three dimensions, three laser beams must interact with the

ion. These beams must not be coplanar, to ensure sensitivity

Trap Electrodes Beam 1

Biasing Rod 3

(a)

O

Beams ©

1 and 2

Biasing Rod

rap Rod

Beam 3

(b)

FIG. 3. Experimental setup to observe and minimize micromotion using the

cross-correlation technique. The ions are at the intersection of the three laser beams, (a) Top view of the section of the trap in which the ions are located. Beam 3 is not shown, (b) View along the trap axis. For clarity, the origin of

the coordinate system has been translated.

to micromotion in every direction. Unless the three beams

are orthogonal, this technique is not equally sensitive to ion motion in all directions, as illustrated below.

V. EXPERIMENTAL DEMONSTRATION OF THE

FLUORESCENCE MODULATION TECHNIQUE

Figure 3 illustrates the experimental configuration we

use to detect and minimize micromotion of a string of ions in our linear Paul trap, using the cross-correlation technique. In

this trap, <Pac = within the experimental resolution. Typically, about ten ions, whose extent is small compared to their distance from the electrodes, are stored in the trap. In this case the fluorescence modulation signals from each ion add

in phase. Laser beams I and 2 propagate along sin d[ix

—y)/-\/l]^cos 6z, where 6=20°. Beam 3 propagates along

(x + y)/V2. The three beams intersect at the ions' position.

Static electric potentials are applied to four biasing rods run- ning parallel to the trap electrodes, creating an additional

field that is nearly uniform at the site of the ions. When the

potentials on the four rods are appropriately summed, the

electric fields along the (x±y)/V2 directions can be separately controlled. We detect the fluorescence modulation with a STARTSTOP time-to-amplitude converter (TAC).^^ The TAC generates an analog pulse having a height proportional to the time delay between a START and a STOP pulse. A fluores- cence photon, detected by a photomultiplier tube, generates

the START pulse. An amplifier discriminator generates a STOP pulse for each negative-going zero crossing of the trap ac potential. The counting rate of fluorescence photons is TN-193
5032 J. Appl. Phys., Vol. 83, No. 10, 15 May 1998 Berkeland et al.

typically much less than the frequency of the ac field. Also,

the time between photon detections is much greater than the time the TAC takes to reset for the next START pulse. Thus,

each detected photon results in an output from the TAC,

proportional to the time to the next STOP pulse. This process would be inefficient if the START and STOP trigger sources were reversed, because not every START pulse would be

followed by a STOP pulse within a period of the ac electric

field. Finally, the height of the output pulse from the TAC is measured by a triggered analog-to-digital converter and

binned according to height by a computer, which acts as a multichannel analyzer.^^ A spectrum of the fluorescence in- tensity as a function of the phase of the ac electric field is typically built up within a few seconds. The fluorescence modulation signals due to beams 1,2,

and 3 are separately measured, then the static fields are ad- justed to minimize the fluorescence modulation for each

beam. Since the micromotion is directed along the ac electric

field, in general, the direction of the micromotion is not the

direction of the ion displacement from the trap axis. For

example, in the trap of Figs. 1 and 3, micromotion along

(x + y)/'\/l indicates that the ions are displaced along (i — y)/Vl, and vice versa. To begin, we compare the cross- correlation signal A/?rf/^n,a^ with only beam 1 present to ^^rf/^max with only beam 2 present. The signals due to the two beams will differ if the ions experience micromotion

along £. Such axial micromotion is due to the endcaps,

which produce a (small) component of the ac electric field along £. From symmetry, this axial micromotion should be

minimized when the ions are equidistant from the endcaps. A

differential potential is applied between the two endcaps to translate the ions along the trap axis until the signals from

beams 1 and 2 are nearly equal. Next, the static field along

(x + y)/'V^ is adjusted to move the ions to a position at which

A/?^/i?n,ax from beams 1 and 2 are each minimized. Typically, we must iterate these adjustments before ^Rj/R^^

«=0 for both beams 1 and 2. Finally, a static field along {x — y)/Vl is applied to null the amplitude of the signal from

beam 3. After this we check that the signals from beams 1 and 2 have remained negligible. If they have increased, we

repeat the entire process until the micromotion is eliminated

in all three dimensions.

Figure 4 shows some fluorescence modulation signals collected with the setup shown in Fig. 3, when only beam 1

100

60

O 40

20

n -T * f^ *^ *

V ;^-^ »- * -I *t

0.1 0.2

(a)

0.3 0.4 0.5 time (2n/n)

0.6 0.7 0.8

150

0.2 (b)

0.3 0.4 0.5 0.6 time (2jt/n)

0,7 0.8

150

120

cS 90 c

o 60 o

30

* *T»i*

••

vv* - ^•v; • • * ^"/^ ""^ ***

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 (c) time (2n/a) FIG. 4. Experimental fluorescence modulation signals for beam 1 of Fig. 3, using eight ions in the linear trap (points) and fit (solid line). Displacement

of the ions from the trap axis along {x + y)/V2 is (a) 0.9±0.3 fj.m, (b) 6.7 ±0.4 yum, and (c) — 6.7±0.4 yitm.

is present. The laser is tuned near the 194 nm 5d^^6s ^5,/2 -*5d^%p ^Py2 transition. Here, n = 27r- 8.6 MHz, o)^ = w^=«27r- 65 kHz, and coi^^^— Watom— ~ 7^^- The micromotion has been nulled in all three dimensions as just described.

TABLE I. Summary of the sensitivities of various techniques to the fractional second-order Doppler shift and the Stark shift. Formulas are approximations

assuming |a,|<(y,^<? 1 {i = x.y). Here, (Ai^2D^'')i 'S the fractional second-order Doppler shift due to motion along direction m, , (Ai^^); the Stark shift due to the electric field along direction li, , k the wave vector, and y the width of the excited transition, m the ion mass, Q the ion charge, CI the trap drive frequency,

<jt)j the secular frequency along direction «, , and 0^^ the angle between the direction of the micromotion and k.

Method Afin/f A 1^5

Directly monitor ion displacement Amq, along

direction u, as co, is changed from co,^ to &Ji2

Measure ratio /?, /Rq of scattering rates of carrier and

first sideband {il>y)

Monitor cross-correlation signal t^RjIR^^^ due to

first-order Doppler shift {.Q.<y)

^Vdi\ ' / '^i2 ^"Oi'^/l|

n /?, \ck cos ^^/ Rq

1 / y A/?, ^^

4 \r* cos 0^ R^

{Avs)i=a-s\—i r

a),2 mAMo,a),ifi\

2cr, 2m[V

\Qk cos 0^1 Rq

2 '\Qk cos 0^R„

TN-194
J. Appl. Phys., Vol. 83, No. 10, 15 May 1998 Berkeland et al. 5033

Next, only beam 1 is used as the ions are translated along

(i + >')/V2 to induce excess micromotion along {x — y)ly/2.

Figure 4(a) shows the cross-correlation signal when the ions

are located near the trap axis. The fit to the data gives

A7?^//?n^=0.043± 0.014, corresponding to a fractional second-order Doppler shift of Aj^o2/v=-(9±3)X 10"'^ due to motion along the propagation direction of beam 1. Figures 4(b) and 4(c) are the fluorescence modulation signals

when the ions are deliberately shifted by approximately

±6.7 /im along {x + y)lVl. For these signals, l^RdlRiaiA

=0.3, corresponding to a second-order Doppler shift of

Ai'oi/^'— ~5X 10~'^ due to motion along the propagation

direction of beam 1 . The Stark shift due to the ac field along

the propagation direction of beam 1 is lA^-^/^l^l.O

XIO"'^

The fluorescence modulation signals from all three laser beams can be reduced to the level shown in Fig. 4(a). We

consider the case in which the signals from beams 1 and 2 have the same sign. Then from Eq. (54), the fractional second-order Doppler shifts due to excess micromotion is -(0±0.2)X10~'* along £,-(8±2)X 10"'^ along {x

-y)l^, and -(0.9±0.3)X 10"'^ along (x-Fy)/v5. These

values add to give a total shift of AvD2/t'= — (9±2)

X10~'^. Similarly, from Eq. (55), these signals indicate a

total Stark shift of |Aj/5/i'|=e(1.9±0.4)X 10"'"'. These

small shifts illustrate this method's effectiveness in reducing

micromotion.

To conclude, the micromotion of ions in a Paul trap has

several related adverse effects. In high-resolution spectros- copy, the most significant are the second-order Doppler shift and a possible Stark shift due to the ac electric fields. Because these shifts can be substantial, it is critical that micromotion be eliminated in all three dimensions. Table I lists the methods discussed in this article, and the corresponding for- mulas for determining the second-order Doppler shift and the Stark shift from the relevant signals. The first method monitors the spatial motion of the ions as the pseudopotential is varied, whereas the last two methods monitor the effects of

first-order Doppler shift on the atomic line shape. The

spatial-monitoring techniques are insensitive to micromotion

caused by a phase shift between the ac potentials applied to the trap electrodes. Apart from this, which technique is most

sensitive to micromotion depends on the parameters of the

trap, laser beams, and atomic transition.

ACKNOWLEDGMENTS

This work was supported by ONR and ARO. The authors thank Dietrich Leibfried, Don Sullivan, Brent Young,

and Matt Young for helpful comments on the manuscript.

' D. J. Wineland, J. C. Bergquist, J. J. Bollinger, W. M. llano, D. J. Heinzen, S. L. Gilbert, C. H. Manney, and M. G. Raizen, IEEE Trans. Ultrason. FeiToelectr. Freq. Control 37, 515 (1990).

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'*C. Tamm (unpublished). Talk presented in Proceedings of the Fifth Symposium on Frequency Standards and Metrology, edited by C. Bergquist (World Scientific, Singapore, 1996). "H. a. Schuessler and H. S. Lakkaraju, in Precision Measurement and

Fundamental Constants II, edited by B. N. Taylor and W. D. Phillips, Natl. Bur. Stand. (U.S.). Spec. Publ. 617 (1984), p. 103. -"M. Jardino, F. Plumelle, M. Desaintfuscien, and J. L. Duchene, in Proceedings of 38th Annual Symposium on Frequency Control, IEEE

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*U.S. GPO 2002-0-773-018/40206

TN-195

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