Quantum dynamics of single trapped ions

D. Leibfried

University of Colorado and National Institute of Standards and Technology, Boulder,

Colorado 80305-3328

R. Blatt

Institut fu ̈r Experimentalphysik, Universita ̈t Innsbruck, A-6020 Innsbruck, Austria

C. Monroe

FOCUS Center and Department of Physics, University of Michigan, Ann Arbor,

Michigan 48109-1120

D. Wineland

National Institute of Standards and Technology, Boulder, Colorado 80305-3328

(Published 10 March 2003)

Single trapped ions represent elementary quantum systems that are well isolated from the

environment. They can be brought nearly to rest by laser cooling, and both their internal electronic

states and external motion can be coupled to and manipulated by light fields. This makes them ideally

suited for quantum-optical and quantum-dynamical studies under well-controlled conditions.

Theoretical and experimental work on these topics is reviewed in the paper, with a focus on ions

trapped in radio-frequency (Paul) traps.

CONTENTS

I. Introduction 282

II. Radio-Frequency Traps for Single Charged Particles 283

A. Classical motion of charged particles in rf traps 283

1. Classical equations of motion 283

2. Lowest-order approximation 284

3. Typical realizations 285

B. Quantum-mechanical motion of charged

particles in rf traps 285

1. Quantum-mechanical equations of motion 286

2. Lowest-order quantum approximation 287

C. Special quantum states of motion in ion traps 287

1. The number operator and its eigenstates 287

2. Coherent states 288

3. Squeezed vacuum states 288

4. Thermal distribution 289

III. Trapped Two-Level Atoms Coupled to Light Fields 289

A. The two-level approximation 290

B. Theoretical description of the coupling 290

1. Total Hamiltonian and interaction

Hamiltonian 290

2. Rabi frequencies 291

3. Lamb-Dicke regime 292

4. Resolved sidebands 292

5. Unresolved sidebands 293

6. Spectrum of resonance fluorescence 293

C. Detection of internal states 294

1. The electron shelving method 294

2. Experimental observations of quantum

jumps 295

D. Detection of motional-state populations 295

IV. Laser Cooling of Ions 296

A. Doppler cooling 296

B. Resolved-sideband cooling 298

1. Theory 298

2. Experimental results 299

C. Electromagnetically induced transparency

cooling 300

1. Cooling in the Lamb-Dicke regime 301

2. Scattering rates in EIT cooling 302

3. Experimental results 303

V. Resonance Fluorescence of Single Ions 303

A. Excitation spectroscopy, line shapes 303

B. Nonclassical statistics, antibunching, and

squeezing 304

C. Spectrum of resonance fluorescence, homodyne

detection of fluorescence 305

VI. Engineering and Reconstruction of Quantum States

of Motion 307

A. Creation of special states of motion and

internal-state/motional-state entanglement 307

1. Creation of number states 307

2. Creation of coherent states 308

3. Creation of squeezed states 310

4. ‘‘Schro ̈dinger-cat’’ states of motion 310

5. Arbitrary states of motion 312

B. Full determination of the quantum state of

motion 312

1. Reconstruction of the number-state density

matrix 313

2. Reconstruction of s-parametrized

quasiprobability distributions 313

3. Experimental state reconstruction 314

VII. Quantum Decoherence in the Motion of a Single

Atom 315

A. Decoherence background 316

B. Decoherence reservoirs 316

1. High-temperature amplitude reservoir 316

2. Zero-temperature amplitude reservoir 318

3. High-temperature phase reservoir 319

C. Ambient decoherence in ion traps 320

VIII. Conclusions 320

Acknowledgments 320

REVIEWS OF MODERN PHYSICS, VOLUME 75, JANUARY 2003

0034-6861/2003/75(1)/281(44)/$35.00 281 ©2003 The American Physical Society
Appendix: Couplings of Light Fields to The Internal Electronic

State 320

1. Dipole coupling 321

2. Quadrupole coupling 321

3. Raman coupling 321

References 322

‘‘ . . . we never experiment with just one electron or atom

or (small) molecule.’’—E. Schro ̈dinger, 1952

I. INTRODUCTION

In the last 30 years, experiments with single trapped

ions or charged fundamental particles have provided key

contributions to many fields in physics. This review does

not attempt to cover all these fields and will therefore

concentrate on the use of trapped ions in quantum optics and the closely related field of coherent control of

the internal state and the motional state of ions in the

trap potential. With the availability of spectrally narrow

light sources based on lasers, these fields have prospered

in the last 15 years and are still in rapid development,

gaining momentum as a result of the new challenges encountered in quantum information processing. While

there are several review papers and books covering the

more traditional work in ion traps [see, for example,

Brown and Gabrielse (1986); Paul (1990); Gosh (1995)],

those on ion-trap-based frequency standards (Diddams

et al., 2001, and references therein), and those that concentrate on quantum information processing (Steane,

1997; Wineland et al., 1998), there is no comprehensive

work covering quantum optics, coherent control of the

motion, and motional-state reconstruction with trapped

ions. The purpose of this review is to fill this gap.

One seemingly obvious approach to understanding

the interaction of atoms and light is to isolate and confine a single atomic system, put it to rest or at least into

a well-characterized state of motion, and then direct

light fields onto that isolated system in a precisely controlled manner. This idea seems quite straightforward,

but can be difficult to convert into a feasible experiment.

Traps for neutral atoms often have a rather shallow trapping potential that can also depend on the electronic

state of the atom, thus perturbing the internal states and

entangling them with the motion. With ion traps, which

couple to the excess charge of the trapped particle, potential wells that are up to several electron volts deep

and do not depend on the internal electronic state of the

ion can be realized. The most popular forms of ion traps

are the Penning trap (Penning, 1936), in which the

charged particles are held in a combination of electrostatic and magnetic fields, and the traps developed by

Wolfgang Paul (Paul, 1990), in which a spatially varying

time-dependent field, typically in the radio-frequency

(rf) domain, confines the charged particles in space. In

this review only the latter type will be considered.

The intrinsically low signal levels one faces in the detection of single particles can be overcome by laserinduced fluorescence. On a dipole allowed transition a

single ion can scatter several million photons per second

and a sufficient fraction may be detected even with detectors covering a small solid angle and having low

quantum efficiencies. Moreover, several variations of the

electron shelving technique suggested by Dehmelt

(1975) can distinguish internal electronic states of the

trapped ion with a detection efficiency close to unity.

The first single-particle trapping experiments in the

present context were those on electrons confined in a

Penning trap at the University of Washington by Wineland, Ekstrom, and Dehmelt (1973). The same paper

contained a proposal to store a single atomic ion, an idea

further discussed by Dehmelt (1973) in the same year. It

took another seven years until Neuhauser et al. (1980)

reported the first experiment with a single barium ion in

an rf trap. This experiment was followed by one of

Wineland and Itano (1981), who used a single magnesium ion in a Penning trap. To date there are about 20

groups all over the world that work with single ions in rf

traps.

An important ingredient for work with atomic systems

was the advent of laser cooling. It was independently

proposed by Ha ̈nsch and Schawlow (1975) for free particles and by Wineland and Dehmelt (1975a) for trapped

particles. The first atomic laser cooling experiments

were reported independently by Wineland, Drullinger,

and Walls (1978) on Mg1 ions and by Neuhauser et al.

(1978) on Ba1 ions. In the following years laser cooling

of collections of ions or single ions found widespread use

in many different research groups around the world. Laser cooling to the ground state of the trapping potential

was first achieved on a single Hg1 ion by the National

Institute of Standards and Technology (NIST) group

(Diedrich et al., 1989).

Many experiments with single ions have the goal of

using a narrow optical or microwave transition between

electronic states of the ion(s) for frequency standards

(Diddams et al., 2001, and references therein). Others

deal with the quantum-mechanical aspects of the light

emitted by a single ion and the quantum dynamics of the

ions’ motion in the trapping potential. The latter field

gained considerable interest after it was realized that

these dynamics closely resemble the Jaynes-Cummings

model that was well known from cavity QED (Blockley,

Walls, and Risken, 1992). These are the experiments

that will be preferentially covered in this review. In recent years a number of groups have also started working

with single ions in an effort to implement quantum information processing following the proposal by Cirac

and Zoller (1995), a field that is very closely related to

our subject (Steane, 1997; Wineland et al., 1998).

Following this Introduction, the second section consists of a discussion of the classical and quantum motion

of single ions in rf traps, including the driven motion due

to the trapping rf field (micromotion). Section III introduces two-level atoms and describes the coupling of

trapped ions to the light field. With these tools at hand,

we study laser cooling in Sec. IV and resonance fluorescence in Sec. V. In Sec. VI we review experimentally

realized methods for engineering and reconstructing

282 Leibfried et al.: Quantum dynamics of single trapped ions

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
quantum states of motion. Section VII summarizes experiments on decoherence and reservoir engineering

with single trapped ions.

II. RADIO-FREQUENCY TRAPS FOR SINGLE CHARGED

PARTICLES

In this section the equations of motion of a charged

particle in different common types of rf traps are discussed. Only trap types that lead to an electric potential

F(x,y,z,t) of approximately quadrupolar spatial shape

in the center of the trapping region are considered here.

It is further assumed that the potential can be decomposed into a time-dependent part that varies sinusoidally

at the rf drive frequency vrf and a time-independent

static part:

F~x,y,z,t!5U

1

2 ~ax21by21gz2!

1U ̃ cos~vrft!

1

2 ~a8x21b8y21g8z2!.

(1)

The condition that this potential has to fulfill the

Laplace equation DF50 at every instant in time leads

to restrictions in the geometric factors, namely,

a1b1g50, (2)

a81b81g850.

From these restrictions it is obvious that no local threedimensional minimum in free space can be generated, so

the potential can only trap charges in a dynamical way.

As we shall see below, the drive frequency and voltages

can be chosen in such a way that the time-dependent

potential will give rise to stable, approximately harmonic motion of the trapped particles in all directions.

One choice for the geometric factors is

a5b5g50, (3)

a81b852g8,

leading to three-dimensional confinement in a pure oscillating field. A second choice is

2~a1b!5g.0, (4)

a852b8,

leading to dynamical confinement in the x-y plane and

static potential confinement for positively charged particles in the z direction as used in linear traps (Paul,

1990).

First we give an overview of the classical equations of

motion and their solutions, and approximations to these

solutions are studied. A quantum-mechanical picture of

ions trapped in rf fields following the approach of

Glauber (1992) is then derived and it is shown that in

the range of trapping parameters used in the experiments discussed here, the quantized motion of trapped

ions can be modeled by static potential harmonic oscillators to a very good approximation.

A. Classical motion of charged particles in rf traps

1. Classical equations of motion

The classical equations of motion of a particle with

mass m and charge Zueu in a potential of the form given

by Eq. (1) were first studied by Paul, Osberghaus, and

Fischer (1958). They are decoupled in the spatial coordinates. Only the motion in the x direction will be discussed below; the other directions can be treated analogously. The equation of motion is

x ̈ 52 Zueu

m

]F

]x 52 Zueum @Ua1U ̃ cos~vrft!a8#x (5)

and can be transformed to the standard form of the

Mathieu differential equation

d2x

dj2 1@ax22qx cos~2j!#x50 (6)

by the substitutions

j5 vrft

2 , ax54ZueuUa

mvrf

2 , qx52ZueuU ̃ a8

mvrf

2 . (7)

The Mathieu equation belongs to the general class of

differential equations with periodic coefficients. The

general form of the stable solutions follows from the

Floquet theorem (McLachlan, 1947; Abramowitz and

Stegun, 1972),

x~j!5Aeib xj (n52`

`

C2nei2nj

1Be2ib xj (n52`

`

C2ne2i2nj , (8)

where the real-valued characteristic exponent bx and

the coefficients C2n are functions of ax and qx only and

do not depend on initial conditions. A and B are arbitrary constants that may be used to satisfy boundary

conditions or normalize a particular solution. By inserting Eq. (8) into Eq. (6) one obtains a recursion relation,

C2n122D2nC2n1C2n2250,

D2n5@ax2~2n1bx!

2#/qx , (9)

that connects the coefficients and bx to ax and qx .

Simple rearrangements and recursive use of Eq. (9)

yield continued fraction expressions for the C2n ,

C2n125 C2n

D2n2 1

D2n122 1

...

, (10)

C2n5 C2n22

D2n2 1

D2n222 1

...

,

and for bx

2

,

Leibfried et al.: Quantum dynamics of single trapped ions 283

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
bx

2

5ax2qx

S 1

D02 1

D22 1

...

1

1

D02 1

D222 1

... D

.

(11)

Numerical values for bx and the coefficients can be extracted by truncating the continued fractions after the

desired accuracy is reached. The contributions of higher

orders in the continued fraction rapidly drop for typical

values of ax and qx used in the experiments described

here.

The region of stability in the ai-qi plane (i

P$x,y,z%) is bounded by pairs of ai and qi that yield

either bi50 or bi51 (Paul, Osberghaus, and Fischer,

1958; Gosh, 1995). The stable region that contains the

points (ai ,qi)5(0,0) for all iP$x,y,z% is often called

the lowest stability region. The traps relevant for the

experiments discussed here work inside this lowest

stable region with ai'0.

The exact shape of the stability regions depends on

the actual parameters in Eq. (1). For three-dimensional

(3D) rf confinement, for example, with the parameters

as in Eq. (3), the trap electrodes are often cylindrically

symmetric around one axis (usually labeled the z axis),

leading to the parameter relations a85b852g8/2 and

a5b52g/2. The parameters ai and qi in the Mathieu

equations along the different axes will then obey

az522ax522ay , qz522qx522qy . (12)

A trapped particle will be stable in all three dimensions

if

0<bi<1, for all iP$x,y,z%. (13)

Figure 1(a) shows a plot of the lowest stability region for

the cylindrically symmetric Paul trap. The axes displayed are az and qz , and the corresponding values for

the x and y dimensions can be found from Eq. (12). The

borderlines of stability in the x and y direction are identical. For the linear trap, the parameters have the relations

qy52qx , qz50. (14)

In this case the first stability region is symmetric around

the qx axis, since the borderlines of stability in the two

directions are mirror images of each other [see Fig.

1(b)]. General traps with no intrinsic symmetry can have

even more complicated stability diagrams, because the

borderlines of stability might not be connected by

simple relations such as in the two cases discussed here

[see Eq. (4)]. The radially defocusing effects of the static

potential along the z axis in a linear Paul trap can also

lead to modifications in the stability diagram, especially

if the confinement along this axis becomes comparable

in strength to the radial confinement (Drewsen and

Brøner, 2000).

2. Lowest-order approximation

The lowest-order approximation to the ion trajectory

x(t) in the case (uaxu,qx

2

)!1 can be found by assuming

C64.0. Then, together with the initial condition B

5A, Eq. (9) yields

bx'Aax1qx

2

/2,

x~t!'2AC0 cosS bx

vrf

2 tDF 12 qx2 cos~vrft!G , (15)

FIG. 1. Trap stability: (a) Stability diagram for a cylindrically symmetric trap with rf confinement in all three axes (a5b

52g/2,a85b852g8/2); (b) stability diagram for a linear trap (a1b52g,a852b8,g850).

284 Leibfried et al.: Quantum dynamics of single trapped ions

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
identical to the solution found by the pseudopotential

approximation (Dehmelt, 1967; Gosh, 1995). The trajectory consists of harmonic oscillations at frequency n

5bxvrf/2!vrf , the secular motion, superposed with

driven excursions at the rf frequency vrf . The driven

excursions are 180° out of phase with the driving field

and a factor qx/2 smaller than the amplitude of the secular motion. These fast, small oscillations are therefore

dubbed micromotion. If micromotion is neglected, the

secular motion can be approximated by that of a harmonic oscillator with frequency n. Most theoretical papers covering the subject of this review assume this approximation. In the course of this paper we shall see that

it is justified in most cases if the ions are at reasonably

low kinetic energy, even if we treat the center-of-mass

motion of the ion quantum mechanically.

3. Typical realizations

One of the most popular trap configurations is the

cylindrically symmetric 3D rf trap (az522ax

522ay ; qz522qx522qy). It can be realized with the

electrode configuration shown in Fig. 2 where a5a8

5b5b8522g522g8 and where, to a good approximation, a52/(r0

2

12z0

2

). This last expression holds exactly

when the electrode surfaces coincide with equipotentials

of Eq. (1) and holds reasonably well with truncated electrodes as shown in Fig. 2. Typically, the traps are operated in the left-hand portion of the stability diagram

[Fig. 1(a)], where qz,0.5; however, the entire stability

diagram has been experimentally explored, including

parametric instabilities, in an impressive series of experiments by the group of Werth (Alheit et al., 1996).

A second very useful trap electrode configuration is

that for the linear rf trap (qz5g850; qy52qx) shown

schematically in Fig. 3. This trap is essentially a linear

quadrupole mass filter (Paul, 1990) that has been

plugged on the ends with a static axial z potential. If the

axial potential is made fairly weak compared to the x,y

potentials, two or more trapped ions will line up along

the trap axis. This can be useful for addressing individual ions with laser beams. Typically these traps are

also operated in the left-hand portion of the stability

diagram [Fig. 1(b)] with qx,0.5.

B. Quantum-mechanical motion of charged particles in rf

traps

Even a simple account of the cooling process in ion

traps, as well as the description of nonclassical states,

relies on a quantum-mechanical picture of the motion.

Since the trapping potential is not a static, but rather a

time-dependent potential, it cannot be taken for granted

that quantization of the motion in the effective timeaveraged potential already gives an adequate picture. In

the first quantum-mechanical treatment of the timedependent potential by Cook, Shankland, and Wells

(1985), the authors derived an approximate solution of

the Schro ̈dinger equation and concluded that the stability regions of classical and quantum-mechanical motion

are identical. They also found that the dominant effect

of the time-dependent potential is to multiply the wave

function of the static pseudopotential by a timedependent phase factor. The essence of these findings

was confirmed and further elaborated upon by Combescure (1986), who first derived an exact solution, and

later by Brown (1991) and Glauber (1992). All these

treatments are semiclassical in the sense that the trapping rf field is not quantized, but rather represented as a

classical electromagnetic potential of the form of Eq.

(1). The treatment given here follows the elegant approach of Glauber (1992).

FIG. 2. Schematic drawing of the electrodes for a cylindrically

symmetric 3D rf trap. Typical dimensions are r0.&z0

.100 mm– 1 cm with U ̃ .100– 500 V, uUu.0 – 50 V, and

vrf/2p.100 kHz– 100 MHz. FIG. 3. Schematic drawing of the electrodes for a linear rf

trap. A common rf potential U ̃ cos(vrft) is applied to the dark

electrodes; the other electrodes are held at rf ground through

capacitors (not shown) connected to ground. The lower right

portion of the figure shows the x-y electric fields from the

applied rf potential at an instant when the rf potential is positive relative to the ground. A static electric potential well is

created (for positive ions) along the z axis by applying a positive potential to the outer segments (gray) relative to the center segments (white). Typical dimensions are r0

.100 mm– 1 cm with U ̃ .100– 500 V, uUu.0 – 50 V, and

vrf/2p.100 kHz– 100 MHz.

Leibfried et al.: Quantum dynamics of single trapped ions 285

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
1. Quantum-mechanical equations of motion

For the quantum-mechanical treatment of the motion

we assume that the time-dependent potential is quadratic in each of the three Cartesian coordinates of the

center of mass of the trapped particle. Then, as in the

classical motion, the problem is separable into three

one-dimensional problems. In one dimension and replacing the coordinate x by the respective operator xˆ,

we can write the time-dependent potential V(t) as

V~t!5 m

2 W~t!xˆ 2, (16)

where

W~t!5 vrf

2

4 @ax12qx cos~vrft!# (17)

can be thought of as a time-varying spring constant that

will play a role similar to v2 in the static potential harmonic oscillator. With these definitions, H(m)

, the

Hamiltonian of the motion, takes a form very similar to

the familiar Hamiltonian of a static potential harmonic

oscillator:

Hˆ (m)

5 pˆ 2

2m

1

m

2 W~t!xˆ 2, (18)

and we can immediately write down the equations of

motion of these operators in the Heisenberg picture:

xˆ

 ̇ 5 1

i\ @xˆ,Hˆ (m)

#5 pˆ

m ,

pˆ

 ̇ 5 1

i\ @pˆ ,Hˆ (m)

#52mW~t!xˆ, (19)

which can be combined into

xˆ

 ̈ 1W~t!xˆ 50. (20)

It is easy to verify that this equation is equivalent to the

Mathieu equation (6) if one replaces the operator xˆ with

a function u(t). This fact can be used to find solutions to

Eq. (20) by utilizing a special solution of the Mathieu

equation subject to the boundary conditions

u~0!51, u ̇ ~0!5in. (21)

This solution can be constructed from Eq. (8) with A

51, B50,

u~t!5eib xvrft/2 (n52`

`

C2neinvrft[eib xvrft/2F~t!, (22)

where F(t) is a periodic function with period T

52p/vrf . In terms of the coefficients of this solution,

Eq. (21) takes the form

u~0!5 (n52`

`

C2n51, n5vrf (n52`

`

C2n~bx/21n!.

(23)

This solution and its complex conjugate are linearly independent; they therefore obey the Wronskian identity

u*~t!u ̇ ~t!2u~t!u ̇ *~t!5u*~0!u ̇ ~0!2u~0!u ̇ *~0!

52in. (24)

The unknown coordinates xˆ(t) and u(t) satisfy the same

differential equation, so the complex linear combination

Cˆ ~t!5A m

2\n i$u~t!xˆ

 ̇~t!2u ̇ ~t!xˆ~t!% (25)

is proportional to their Wronskian identity and also constant in time:

Cˆ ~t!5Cˆ ~0!5 1

A2m\n

@mnxˆ~0!1ipˆ ~0!#. (26)

Moreover the right-hand side is exactly the annihilation

operator of a static potential harmonic oscillator of mass

m and frequency n,

Cˆ ~t!5Cˆ ~0!5aˆ, (27)

which immediately implies the commutation relation

@Cˆ ,Cˆ †#5@aˆ,aˆ †#51. (28)

This static potential oscillator will be called the reference

oscillator in the remainder of this section.

The Heisenberg operators xˆ(t) and pˆ(t) can be reexpressed in terms of u(t) and the operators of the reference oscillator using Eq. (25):

xˆ~t!5A \

2mn $aˆ u*~t!1aˆ †u~t!%,

pˆ ~t!5A\m

2n $aˆ u ̇ *~t!1aˆ †u ̇ ~t!%, (29)

so their entire time dependence is given by the special

solution u(t) and its complex conjugate.

For later calculations it is convenient to have expressions for a basis of time-dependent wave functions in the

Schro ̈dinger picture. Again the reference oscillator used

above is very helpful in this task. In analogy to the static

potential case we shall consider a set of basis states un,t&

in which n51,2, ...,`. These states are the dynamic

counterpart of the harmonic-oscillator number (Fock)

states. The ground state of the reference oscillator

un50&n obeys the condition

aˆun50&n5Cˆ ~t!un50&n50, (30)

but since the Heisenberg operator Cˆ is connected to the

Schro ̈dinger picture counterpart Cˆ S by Cˆ (t)

5Uˆ †(t)Cˆ SUˆ (t), with Uˆ (t)5exp@2(i/\)Hˆ (m)

#, we immediately get

Cˆ S~t!Uˆ ~t!un50&n5Cˆ S~t!un50,t&50 (31)

by multiplying Eq. (30) with Uˆ (t) from the left and noting that Uˆ (t)un50&n is the Schro ̈dinger state of the

time-dependent oscillator that evolves from the ground

state of the static potential reference oscillator. Since the

time dependence of the Schro ̈dinger operator CS(t) is

due entirely to the explicit time dependence of u(t), Eq.

(31) is equivalent to

286 Leibfried et al.: Quantum dynamics of single trapped ions

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
@u~t!pˆ 2mu ̇ xˆ #un50,t&50, (32)

or reexpressed in coordinate space

H u~t!

\

i

]

]x8

2mu ̇ ~t!x8J ^x8un50,t&50. (33)

The normalized solution is

^x8un50,t&5S mn

p\ D

1/4 1

$u~t!%

1/2 expF

im

2\

u ̇ ~t!

u~t!

x8

2

G .

(34)

In complete analogy to the static potential harmonic oscillator, all other states of a complete orthonormal base

can be created by repeated operation on the ground

state with the creation operator Cˆ S

† (t):

un,t&5 @Cˆ S

†

~t!#

n

An! un50,t& (35)

expressed in coordinate space, and by rewriting u(t)

such as in Eq. (22), these states are

^x8un,t&5expF 2iS n1

1

2 D ntGxn~t!, (36)

with

xn~t!5 1

A2nn!

S mn

p\ D

1/4 exp$2in arg@F~t!#%

$F~t!%

1/2

3HnHF mn

\uF~t!u

2G

1/2

x8J

3expH mn

2\ F 12 iF ̇ ~t!nF~t!

G x8

2

J ,

where Hn is the Hermitian polynomial of order n. The

classical micromotion appears in the wave functions as a

pulsation with the period of the rf driving field. For a

static potential harmonic oscillator the evolution of the

energy eigenstates only multiplies the wave function by

a phase factor (which is why they are called stationary

states). In the time-dependent potential studied here,

the same is true, but only for times that are integer multiples of the rf period Trf52p/vrf . The states given by

Eq. (36) are not energy eigenstates (they periodically

exchange energy with the driving field in analogy to the

classical micromotion), but they are the closest approximation to stationary states possible in the timedependent potential. Therefore they are often called

quasistationary states.

These features will be illustrated in the next few sections, where we shall find the lowest-order corrections to

the static potential oscillator picture in close analogy to

the classical pseudopotential solution presented in Sec.

II.A.2. We also discuss the analogous operator to the

number operator for the static potential harmonic oscillator and some special classes of motional states in the

ion trap.

2. Lowest-order quantum approximation

The lowest-order approximation for the quantummechanical states will be studied by first deriving an approximate expression for the special solution u(t).

Again uaxu,qx

2

!1 and C6450 is assumed. Together with

the initial conditions of Eq. (21) one finds

bx'Aax1qx

2

/2, n'bxvrf/2,

u~t!'exp~int!

11~qx/2!cos~vrft!

11qx/2 , (37)

essentially the lowest-order classical solution found earlier in Eq. (15). It still must be stressed that the frequency of the reference oscillator n is equal to the characteristic exponent bxvrf/2 only in this lowest-order

approximation. The periodic breathing of xn(t) with period Trf is now obvious, as one can see in the approximate expression x0(t) for the ground-state wave function:

x0~t!5S mn

p\ D

1/4A 11qx/2

11~qx/2!cos~vrft!

3expS H i

mvrfsin~vrft!

2\@2/qx1cos~vrft!#

2 mn

2\ J x8

2

D ,

(38)

while the phase factor in Eq. (36) is governed by the

ground-state pseudoenergy \n/2. This expression is

identical to the static harmonic potential ground-state

wave function if one sets vrf50.

C. Special quantum states of motion in ion traps

In this section various classes of motional states in ion

traps will be discussed, some nonclassical in nature and

some more reminiscent of classical motion. For each of

the classes theoretical proposals on how to create them

in an ion trap have been brought forward and all have

been created and observed experimentally.

1. The number operator and its eigenstates

To exploit the close analogy between the confinement

in an rf ion trap and that in a static harmonic potential,

it is advantageous to express the motional states in the

basis of the eigenstates of the reference oscillator number operator. We shall first do this in the Heisenberg

picture. Since Cˆ (t) is time independent [see Eq. (27)],

the operator

Nˆ 5Cˆ †

~t!Cˆ ~t!5aˆ †aˆ (39)

is also time independent and the eigenstates are just the

familiar number or Fock states of the static potential

harmonic reference oscillator with the usual ladder algebra,

aˆun&n5Anun21&n , aˆ †

un&n5An11un11&n ,

Nˆ un&n5nun&n . (40)

Transforming to the Schro ̈dinger picture we get

Leibfried et al.: Quantum dynamics of single trapped ions 287

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
Uˆ †

~t!Nˆ Uˆ ~t!5Uˆ †~t!Cˆ †~t!Uˆ ~t!Uˆ †~t!Cˆ ~t!Uˆ ~t!

5Cˆ S

†

~t!Cˆ S~t!. (41)

The eigenstates and eigenvalues of these operators

are easily inferred from Eq. (35) in the last section:

Cˆ S~t!un,t&5Anun21,t&,

Cˆ S

†

~t!un,t&5An11un11,t&, (42)

implying

Nˆ S~t!un,t&5nun,t&. (43)

These Schro ̈dinger-picture eigenstates can therefore be

used in complete analogy to the static potential harmonic oscillator, and all algebraic properties of the static

potential ladder operators carry over to Cˆ S(t) and

Cˆ S

† (t). The only difference is that these states are not

energy eigenstates of the system, since the micromotion

periodically changes the total kinetic energy of the ion.

Nevertheless, due to the periodicity of the micromotion,

it makes sense to connect the quantum number n to the

energy of the ion averaged over a period Trf52p/vrf of

the drive frequency. This connection will be further explored in Sec. IV on laser cooling.

Any motional state can be expressed as a superposition of the number states

C5(

0

`

cnun,t&, (44)

and a number of these expansions will be used in what

follows in this paper. For convenience we shall set un,t&

5un& and only write the time dependence explicitly if it

helps to clarify matters.

2. Coherent states

In a static potential harmonic oscillator, a coherent

state of motion ua& of the ion corresponds to a Gaussian

minimum-uncertainty wave packet in the position representation whose center oscillates classically in the harmonic well and retains its shape. The wave packet has

the same shape as the ground-state wave function.

Glauber has shown that the states that evolve out of an

initial coherent state in the dynamic trapping potential

are also displaced forms of the Gaussian ground-state

Eq. (34) (Glauber, 1992). The displaced Gaussian does

the same breathing as the ground state, but does not

spread, and its center of gravity follows the classical trajectory of an ion in the trap (now secular motion and

micromotion). States of this type were first considered

by Schro ̈dinger (1926) when he tried to construct wave

packets that reflected the classical motion of a harmonic

oscillator.1

The term ‘‘coherent state’’ was first used by Glauber

(1963, 1964) in connection with quantum states of a light

field. There are different ways to define a coherent state

(see Klauder and Skagerstam, 1985, for a review); for

example, they are the eigenstates of the annihilation operator with complex eigenvalue a:

Cˆ S~t!ua&5aua&. (45)

It is easy to prove that states with coefficients in the

number-state basis expansion Eq. (44),

cn5 an

An!

exp~2uau

2/2!, (46)

are eigenstates of this operator, and the probability distribution among number states is Poissonian,

Pn5ucnu

25u^nua&u25~ ̄nne2 ̄n!/n! with  ̄n5uau2.

(47)

Another popular choice is to represent coherent states

as the action of a displacement operator,

Dˆ ~a!5exp@aCˆ S

†

~t!2a*Cˆ S~t!#, (48)

on the vacuum state, namely,

Dˆ ~a!u0&5ua&. (49)

The action of successively applied displacement operators is also additive up to phase factors,

Dˆ ~a!Dˆ ~b!5Dˆ ~a1b!e1/2(ab*2a*b)

, (50)

so the displacements form a group with Dˆ (0)5I

ˆ as a

neutral element. Note that the extra phase on the righthand side makes the displacement operations noncommutative in general.

3. Squeezed vacuum states

In any quantum state the product of the variance in

position and momentum has a lower bound of \2/4,

given by the Heisenberg uncertainty relation. The

ground state of a static potential harmonic oscillator and

all other coherent states are minimum-uncertainty states

in which the variance in position is (Dx)25^x2

&2^x&

2

51/(mn)\/2 and the variance in momentum is (Dp)2

5(mn)\/2. If one now ‘‘squeezes’’ the position variance

the momentum variance must become wider, so the

Heisenberg uncertainty relation is still fulfilled. In the

course of the time evolution the squeezed position wave

packet will not retain its shape, but will become wider

for half an oscillation period before it contracts back to

the original width after a full period. The momentum

wave packet contracts and expands accordingly so that

at any time the uncertainty is minimal (Walls 1986; Walls

and Milburn, 1995). The coefficients in expansion (44)

for the so-called squeezed vacuum state are

1

One curiosity in his paper is that he considered the real part

of the constructed wave packet to reflect the physical state,

since the probability interpretation of quantum mechanics was

not yet established then.

288 Leibfried et al.: Quantum dynamics of single trapped ions

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
cn5H S 2Abs

bs11 D

1/2S bs21

bs11 D

n/2

~21/2!

n/2 An!

~n/2!!

einf , n even

0, n odd.

(51)

The parameter bs describes the squeezing of the state,

namely, the position variance of the squeezed state is

reduced at certain times by

Dxs5Dx0 /bs , (52)

where Dx0 is the variance of the ground state. The

ground state is recovered for bs51 (therefore the name

‘‘squeezed vacuum state’’). For bs.1 the position wave

function is narrower than that of the coherent state,

while for 0,bs,1 the momentum wave function has

this property. The angle f describes the alignment of the

squeezed state with respect to the position and momentum directions. This can best be visualized in phase

space. The Wigner function of squeezed states has elliptical equicontour lines (Walls and Milburn, 1995). If one

of the major axes of these ellipses is aligned with the

position coordinate axis, f is equal to zero. The center

of mass of the Wigner function of a squeezed vacuum

state coincides with the origin of phase space.

The probability distribution Pn for a squeezed

vacuum state is independent of f and again restricted to

the even states,

Pn5 2Abs

bs11 S bs21bs11 D

n

22n n!

@~n/2!!#

2 , n even. (53)

For strong squeezing this distribution has a tail that

reaches to very high n; for example, with bs540, 16%

of the population of the squeezed vacuum is in states

above n520.

Squeezed vacuum states, like coherent states, have a

very compact operator representation. They are generated from the ground state by the operator

Sˆ ~«!5expH

1

2 «*Cˆ S~t!

22 1

2 «@Cˆ S~t!

†#2

J , (54)

where «5reif and r is related to bs by bs5e2r

.

4. Thermal distribution

If the ion is in thermal equilibrium with an external

reservoir at temperature T the average weight of the

excitation of the state un& will be proportional to the

Boltzmann factor exp@2n\n/(kBT)#, where kB is the

Boltzmann constant. Of course it does not make sense

to assign a temperature to a single realization of a

cooled ion. However, if the ion is coupled to that reservoir and the number operator Nˆ is measured many

times (making sure that the ion reequilibrates after each

measurement), one can extract a temperature from the

average result  ̄n from this ensemble of many different

realizations according to

T5 \n

kB lnS

 ̄n11

 ̄n D

. (55)

In considering an ensemble it is appropriate to characterize the state by a density matrix. Moreover, in the

spirit of choosing the density matrix with the maximum

ignorance (and therefore maximum entropy), the offdiagonal elements have to be zero. This makes it impossible to write the thermal distribution in the form of Eq.

(44) that would correspond to a density matrix with nonzero off-diagonal elements for T.0. So, even if the term

‘‘thermal state’’ is often used in the literature, ‘‘thermal

distribution’’ seems to be a more appropriate reminder

of the ensemble nature of this entity.

After some minor algebra to normalize the trace of

the states weighted by the Boltzmann factors, the density matrix may be written as

%ˆ th5 1

 ̄n11 (n50

`

S  ̄n

 ̄n11 D

n

un&^nu, (56)

with level population probability

Pn5  ̄nn

~ ̄n11!

n11 . (57)

III. TRAPPED TWO-LEVEL ATOMS COUPLED TO LIGHT

FIELDS

With the help of suitable electromagnetic fields the

internal levels of trapped ions can be coherently coupled

to each other and the external motional degrees of freedom of the ions. For strongly confined ions and a suitable tuning the coupling is formally equivalent to the

Jaynes-Cummings Hamiltonian (Jaynes and Cummings,

1963). Consequently much of the work devoted to coherent interaction of trapped ions has been inspired by

the important role this coupling plays in quantum optics.

Beyond this special case there are many possibilities

connected to the interchange of multiple motional

quanta, in close analogy to multiphoton transitions in

quantum optics. Moreover, the light field inducing the

coupling can act as a source of energy, so that energy

conservation implicit in atom-photon couplings does not

have to be fulfilled in the interaction of internal states

and the motion of trapped ions, allowing interactions in

which both the internal state of the atom and its motion

undergo a transition to a higher-energy level. Finally, if

the full quantum-mechanical picture of the motion, including corrections due to micromotion, is considered,

another class of transitions becomes possible that involves exchange of motional quanta at integer multiples

Leibfried et al.: Quantum dynamics of single trapped ions 289

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
of the rf driving field or combinations of integer multiples of the driving field and the secular motion (micromotion sidebands).

A. The two-level approximation

In most of this review the internal electronic structure

of the ion will be approximated by a two-level system

with levels ug& and ue& of energy difference \v5\(ve

2vg). This is justified for real ions if the frequencies of

the electromagnetic fields that induce the coupling are

only close to resonance for two internal levels and if the

Rabi frequencies describing the coupling strength are

always much smaller than the detuning relative to offresonant transitions. Such a reduction is appropriate for

most of the experimental situations described in this paper.

The corresponding two-level Hamiltonian Hˆ (e) is

Hˆ (e)

5\~vgug&^gu1veue&^eu!

5\

ve1vg

2 ~ug&^gu1ue&^eu!

1\

v

2 ~ue&^eu2ug&^gu!. (58)

Since any operator connected to a two-level system can

be mapped onto the spin-1/2 operator basis, Hˆ (e) and

related operators can be conveniently expressed using

the spin-1/2 algebra that is represented by I

ˆ, the 232

unity matrix, and the three Pauli matrices. In the particular case at hand the mapping is

ug&^gu1ue&^eu°I

ˆ, ug&^eu1ue&^gu°sˆ x ,

i~ug&^eu2ue&^gu!°sˆ y, ue&^eu2ug&^gu°sˆ z. (59)

With this mapping Hˆ (e) is reexpressed as

Hˆ (e)

5\

v

2 sz , (60)

where the energy is rescaled by 2\(ve1vg)/2 to suppress the state-independent energy contribution in Eq.

(58).

B. Theoretical description of the coupling

To describe the interaction of the trapped atom with

light fields in a simple but sufficient way, it is assumed,

such as in the preceding section, that the motion of the

atom bound in the trap is harmonic in all three dimensions. The descriptions presented below will include the

explicit time dependence of the trapping potential, but

in many cases it is sufficient to model the motion of the

ion as a three-dimensional static potential harmonic oscillator, because the general theory introduces only very

minor changes if the modulus of the dimensionless Paultrap parameters ax and qx

2 related to the static and rf

potential (see Sec. II.A) is much smaller than 1. This is

true for the traps used in the experiments discussed in

this review. The generalized description of the coupling

of internal states and motion follows the approaches of

Cirac, Garay, et al. (1994) and Bardroff et al. (1996).

It is also assumed that it is sufficient to treat the light

field in the lowest order in its multipole expansion that

yields a nonvanishing matrix element between the nearresonant electronic states in question. This assumption is

justified by the fact that the extension of the electronic

wave function is much less than the wavelength of the

coupling field. For dipole allowed transitions the field

will be treated in the familiar dipole approximation,

while for dipole forbidden transitions only the quadrupole component of the field is considered. For Raman

transitions, the near-resonant intermediate level will be

adiabatically eliminated, making these transitions formally equivalent to the other transition types (see below).

1. Total Hamiltonian and interaction Hamiltonian

The total Hamiltonian Hˆ of the systems considered

here can be written as

Hˆ 5Hˆ (m)

1Hˆ (e)1Hˆ (i), (61)

where Hˆ (m) is the motional Hamiltonian along one trap

axis, Eq. (18), as discussed in Sec. II. Hˆ (e) describes the

internal electronic level structure of the ion (see Sec.

III.A), and Hˆ (i) is the Hamiltonian of the interactions

mediated by the applied light fields.

As summarized in the Appendix, electric dipole allowed transitions, electric quadrupole allowed transitions, and stimulated Raman transitions can be described in a unified framework that associates a certain

on-resonance Rabi frequency V, effective light frequency v, and effective wave vector k with each of these

transition types. The effective light frequencies and

wave vectors are identical to the frequency and wave

vector of the coupling light field for dipole and quadrupole transitions, but equal to the frequency difference

v5v12v2 and wave vector difference k5k12k2 of the

two light fields driving the stimulated Raman transitions.

For running wave light fields all three transition types

can be described by a coupling Hamiltonian of the form

Hˆ (i)

5~\/2!V~ug&^eu1ue&^gu!

3@ei(kxˆ S2vt1f)1e2i(kxˆ S2vt1f)

#. (62)

In the spin-1/2 algebra we can reexpress

ue&^gu°sˆ 151/2~sˆ x1isˆ y!,

ug&^eu°sˆ 251/2~sˆ x2isˆ y!. (63)

For simplicity the discussion is again restricted to one

dimension, and the effective wave vector k is chosen to

lie along the x axis of the trap. The generalization to

more dimensions is straightforward.

The simplest picture of the dynamics induced by the

light field arises after transformation into the interaction

290 Leibfried et al.: Quantum dynamics of single trapped ions

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
picture with the free Hamiltonian Hˆ 05Hˆ (m)

1Hˆ (e) and

the interaction Vˆ 5Hˆ (i)

.

With Uˆ 05exp@2 (i/\)Hˆ 0t# the transformed interaction

Hamiltonian is

Hˆ int5Uˆ 0

†

Hˆ (i)Uˆ 0

5~\/2!Ve~i/\!H

ˆ (e)

t

~s 11s 2!

3e2 ~i/\!H

ˆ (e)

t

e~i/\!H

ˆ (m)

t

@ei(kxˆ 2vt1f)

1e2i(kxˆ 2vt1f)

#e2 ~i/\!H

ˆ (m)

t

5~\/2!V~s 1eiv0t

1s 2e2iv0t

!e~i/\!H

ˆ (m)

t

@ei(kxˆ 2vt1f)

1e2i(kxˆ 2vt1f)

#e2 ~i/\!H

ˆ (m)

t

. (64)

Multiplying the time-dependent factors in the above expressions leads to exp@6i(v6v0)t#. Two terms are rapidly oscillating because v and v0 add up, while the two

other terms oscillate with frequency d5v2v0!v0 .

Since the contribution of the rapidly oscillating terms

hardly affects the time evolution, they can be neglected.

Doing so is called the rotating-wave approximation for

historical reasons.

The transformation of the motional part of the Hamiltonion into the interaction picture is equivalent to a

transformation of this part from the Schro ̈dinger to the

Heisenberg picture. The position operator xˆ S will be replaced by its Heisenberg-picture version xˆ(t) as given in

Eq. (29). Introducing the Lamb-Dicke parameter h

5kx0 , where x05A\/(2mn) is the extension along the

x axis of the ground-state wave function of the reference

oscillator mentioned in Sec. II.B, yields

kxˆ~t!5h$aˆ u*~t!1aˆ †u~t!%, (65)

and the interaction Hamiltonian in the rotating-wave approximation takes its final form as

Hˆ int~t!5~\/2!Vsˆ 1 exp„i$f1h@aˆ u*~t!1aˆ †u~t!#2dt%...

1H.c. (66)

The time dependence of the exponent is governed by

the frequency difference d and the time dependence of

u(t). Recalling the form of the solution Eq. (22) and

expanding part of the exponent in the Lamb-Dicke parameter

exp„i$f1h@aˆ u*~t!1aˆ †u~t!#2dt%...

5ei(f2dt)

(

m50

` ~ih!

m

m! H aˆ e2ib xvrft (n52`

`

C2n*

3e2invrft1H.c.J

m

(67)

it is easily verified that anytime the detuning satisfies

~l81lbx!vrf5d, (68)

with l and l8 as integers and lÞ0 if l8Þ0, two of the

terms in the Hamiltonian will be slowly varying. The

dominant contribution in the time evolution of the

trapped and illuminated ion will come from this term,

while the others can be neglected in a second application

of the rotating-wave approximation. For a given l and l8

one would speak of a detuning to the lth secular sideband of the l8th micromotion sideband, a terminology

coming from the classical picture of the ion vibrating in

the trap’s well with secular and micromotion frequencies. In the frame of reference of the ion the monochromatic light field is therefore phase modulated at these

two frequencies. For example, if one of the modulation

sidebands coincides with the transition frequency v0 of

the ion at rest, this sideband can induce internal state

transitions.

The exact general form of the resonant term can be

calculated, in principle, by performing a polynomial expansion of the above expression, but this is unnecessary

for most practical cases since often h!1, (uaxu,qx

2

)!1,

so the coupling strength of higher orders in l and l8

vanishes quickly. The coupling strength for some special

cases will be calculated in Sec. III.B.2. In all experiments

covered in this review only sidebands with l850 were

driven. Terms with ul8u>1 could then be neglected. Further, it is assumed that (uaxu,qx

2

)!1, so bxvrf'n and

C0'(11qx/2)21, as in Eq. (37). Then the interaction

Hamiltonian simplifies to

Hˆ int~t!5~\/2!V0s 1 exp$ih~aˆ e2int

1aˆ †eint

!%ei(f2dt)

1H.c., (69)

with the scaled interaction strength V05V/(11qx/2).

This scaling reflects the reduction in coupling due to the

wave packet’s breathing at the rf drive frequency.

2. Rabi frequencies

Depending on the detuning d, the interaction Hamiltonian (69) will couple certain internal and motional

states. If the exponent containing the ladder operators in

Eq. (69) is expanded in h this will result in terms containing a combination of s 6 , with l aˆ-operators and m

aˆ †-operators rotating with a frequency of (l2m)n5sn.

If d'sn these combinations will be resonant, coupling

the manifold of states ug&un& with ue&un1s&. The coupling strength, often called the usuth blue (red) sideband

Rabi frequency for s.0 (s,0), is then (Cahill and

Glauber, 1969; Wineland and Itano, 1979)

Vn,n1s5Vn1s,n5V0u^n1sueih(a1a†)

un&u

5V0e2h2/2husu

An,!

n.!

Ln,

usu

~h2!, (70)

where n, (n.) is the lesser (greater) of n1s and n, and

Ln

a (X) is the generalized Laguerre polynomial

Ln

a ~X!5 (

m50

n

~21!

m~ n2m n1a !

Xm

m! . (71)

Leibfried et al.: Quantum dynamics of single trapped ions 291

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
3. Lamb-Dicke regime

The interaction Hamiltonian [Eq. (69)] and the Rabi

frequencies [Eq. (70)] are further simplified if the ion is

confined to the Lamb-Dicke regime where the extension

of the ion’s wave function is much smaller than 1/k. In

this regime the inequality hA^(a1a†)2

&!1 must hold

for all times. The exponent in Eq. (69) can then be expanded to the lowest order in h,

Hˆ LD~t!5~\/2!V0s 1$11ih~aˆ e2int

1aˆ †eint

!%ei(f2dt)

1H.c., (72)

and will contain only three resonances. The first resonance for d50 is called the carrier resonance and has the

form

Hˆ car5~\/2!V0~s 1eif 1s 2e2if !. (73)

This Hamiltonian will give rise to transitions of the type

un&ug&↔un&ue& with Rabi frequency V0 . These transitions will not affect the motional state.

The resonant part for d52n is called the first red

sideband and has the form

Hˆ rsb5~\/2!V0h~aˆ s 1eif 1aˆ †s 2e2if !. (74)

This Hamiltonian will give rise to transitions of the type

un&ug&↔un21&ue& with Rabi frequency

Vn,n215V0Anh (75)

that will entangle the motional state with the internal

state of the ion. Indeed this Hamiltonian is formally

equivalent to the Jaynes-Cummings Hamiltonian, the

workhorse of quantum optics, and it is this analogy that

inspired many workers originally coming from quantum

optics to do investigations in the field of trapped ions. It

is also responsible for the remarkable similarity of investigations done in cavity QED (Englert et al., 1998;

Varcoe et al., 2000; Raimond et al., 2001) to some of the

experiments done in ion traps presented below. This interaction removes one quantum (phonon) of the secular

motion while the ion goes to the excited state, similarly

to the absorption of a light quantum in cavity QED.

The counterpart of this interaction is the first blue

sideband, resonant for d51n:

Hˆ bsb5~\/2!V0h~aˆ †s 1eif 1aˆ s 2e2if !. (76)

This Hamiltonian will give rise to transitions of the type

un&ug&↔un11&ue& with Rabi frequency

Vn,n115V0An11h. (77)

It has no direct counterpart in the atom-photon realm

because such a process would violate energy conservation and is sometimes called anti-James-Cummings coupling. Because of this interaction and the additional possibility of driving higher-order sidebands (when h is not

too small), trapped-ion experiments can yield inherently

richer dynamics than cavity QED experiments. With

suitable light pulses on the carrier, red and blue sidebands, a classical driving force to create coherent or

squeezed states, and the ability to cool to the ground

state, one can engineer and analyze a variety of states of

motion as discussed in more detail in Sec. VI.

By choosing larger detunings, d5ln with ulu.1, in

principle, one can obtain a number of nonlinear couplings, for example, a ‘‘two-phonon’’ coupling,

Hˆ tp5~\/2!V0~h2/2!@a2s 1eif 1~a†!

2s 2e2if # (78)

for l522. We note, however, that this interaction and

those with ulu.2 are not easy to realize in the laboratory

since efficient ground-state cooling has been achieved

only in the Lamb-Dicke regime. There the coupling

strength will be significantly reduced compared to carrier and first sidebands, because h!1.

4. Resolved sidebands

So far we have assumed that the two internal levels of

the ions have an infinite lifetime, leading to arbitrarily

narrow carrier and sideband resonances. In practice this

is only approximately true.2 If the frequency of a sufficiently stable laser beam has intensity such that uVn,mu

!n for all n,m it is possible to observe well-resolved

carrier and sideband resonances. For a detuning d5ln

1d8 with d8!n the ion’s dynamics is governed by the

few resonant terms in Eq. (69) to which the laser happens to be tuned. This allows for clean manipulation of

the internal and motional states and also for cooling to

the ground state, as will be described in Sec. IV.B. If we

neglect dissipative terms, the time evolution of the general state

uC~t!&5 (n50

`

cn,g~t!un,g&1cn,e~t!un,e& (79)

is governed by the Schro ̈dinger equation

i\]tuC~t!&5Hˆ intuC~t!&, (80)

which is equivalent to the set of coupled equations

c ̇ n,g52i

(12ulu)

ei(d 8t2f)

~Vn1l,n/2!cn1l,e , (81)

c ̇ n1l,e52i

(11ulu)

e2i(d8t2f)

~Vn1l,n/2!cn,g . (82)

This set of equations may be solved by the method of

Laplace transforms, yielding the solution

F

cn1l,e~t!

cn,g~t! G5Tn

l

F

cn1l,e~0!

cn,g~0! G , (83)

with

2

For example, for quadrupole transitions with a lifetime on

the order of 1 s, linewidths would be limited to about 1 Hz.

292 Leibfried et al.: Quantum dynamics of single trapped ions

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
Tn

l 5

5

e2i(d8/2)t

F cos~fn

l t/2!1i

d8

fn

l sin~fn

l t/2!G 2i

Vn1l,n

fn

l ei(f1ulup/22d8t/2) sin~fn

l t/2!

2i

Vn1l,n

fn

l e2i(f1ulup/22d 8t/2) sin~fn

l t/2! ei(d 8/2)t

F cos~fn

l t/2!2i

d8

fn

l sin~fn

l t/2!G 6 (84)

and fn

l 5Ad8

21Vn1l,n 2 . The solution describes a generalized form of sinusoidial Rabi flopping between the states

un,g& and ue,n1l& and is essential for the quantum-state

preparation and analysis experiments described later.

5. Unresolved sidebands

Allowed optical electric dipole transitions for ions

have a linewidth of around several tens of MHz, typically beyond the highest secular motional frequencies

observed in ion traps. In this case spontaneous emission

cannot be neglected when considering the dynamics of

the internal and motional states of an ion driven by the

light field. Spontaneous emission will have two consequences: the coherent evolution of the internal two-level

system is interrupted, and the recoil of the emitted photon leads to randomized momentum kicks in the external motion.

A convenient picture of the dynamics is provided by

the master-equation formalism (Walls and Milburn, 1995;

Schleich, 2001). The trapped ion is modeled as being

coupled to a zero-temperature reservoir of optical

modes in the vacuum state. This is a fair assumption

since the average occupation number of optical modes

at room temperature is extremely small. The master

equation is an equation of motion for the reduced density matrix r that describes the time evolution of the

internal and motional states of the ion only, the reduction of the total problem brought about by tracing over

the many degrees of freedom of the environment or reservoir modes. The exact derivation is very technical, so

the reader is referred to Stenholm (1986) and Cirac et al.

(1992) for details. In brief, the reservoir modes are assumed to be coupled to the ion by an interaction of the

Jaynes-Cummings form,

Hrc5~\/2!(

l

gl~x!~saˆl

†

eiDlt1s1aˆle2iDlt

!, (85)

where aˆl , gl(x), and Dl are the destruction operator,

position-dependent coupling strength, and detuning of

the lth reservoir mode, respectively. If we expand the

equation of motion for r to second order in this reservoir interaction and assume the reservoir modes that are

resonant with the two-level atom to be in zerotemperature distributions, we arrive at the master equation

dr

dt 52 i\ @Hˆ (m)

1Hˆ (e)1Hˆ (i),r#1Ldr. (86)

Damping by spontaneous emission is contained in the

Liouvillain

Ldr5 G

2 ~2s2 ̃rs12s1s2r2rs1s2!, (87)

where G is the spontaneous emission rate. To account for

the recoil of spontaneously emitted photons the first

term of the Liouvillian contains

 ̃r5 1

2 E

21

1

dzY~z!eikxˆ zre2ikxˆ z, (88)

with Y(z)53(11z2)/4 the angular distribution pattern

of spontaneous emission for a dipole transition.

In most cases of interest the master equation has been

solved numerically, often assuming the Lamb-Dicke

limit as an additional restriction. One example of this

work will be discussed in the context of laser cooling of

a trapped ion.

6. Spectrum of resonance fluorescence

The master equation (86) can also be used to derive a

formal expression of the light emitted by a trapped ion

in the Lamb-Dicke limit. Lindberg (1986) has studied

the spectrum of light emitted by a harmonically trapped

two-level atom and its interaction with a traveling-wave

laser field. Later, Cirac, Parkins, et al. (1993) derived

general relations applicable to multilevel systems, general trapping potentials, and traveling-wave and

standing-wave configurations. The expressions for the

motional sidebands appearing in the spectrum can be

written in terms of correlation functions of internal operators and of steady-state expectation values of external operators. The internal operators can be evaluated

with the optical Bloch equations for an atom at rest,

while the external operators are derived from the applied trapping forces using the formalism introduced by

Cirac et al. (1992).

Specifically, for a single harmonically trapped ion in a

traveling-wave field configuration, and for the lowintensity limit at which V0!G, one obtains for the motional sidebands the following spectral contributions:

Sbsb~v!5 Q~c!g ms

g ms

2 1~v2n!

h2

S V0

2 D

2

^a†a&ss

3U

cos c

d2ig

1

1

~d1n!2ig

U

2

, (89)

Srsb~v!5 Q~c!g ms

g ms

2 1~v1n!

h2

S V0

2 D

2

~^a†a&ss11!

3U

cos c

d2ig

1

1

~d2n!2ig

U

2

. (90)

Leibfried et al.: Quantum dynamics of single trapped ions 293

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
Here, Q(c), V0 , and n denote the angular distribution

for the dipole radiation, the Rabi frequency, and the motional frequency, respectively. The width g ms of the motional sidebands is given by

g ms5h2

S V0

2 D

2

@P~n1d!2P~n2d!# (91)

and the final (steady-state) excitation energy is

^a†a&ss5 P~n2d!1aP~d!

P~n1d!2P~n2d!

, (92)

with

P~d!5 g

g21d2 (93)

and a52/5 for a dipole transition and the spontaneous

decay rate G5g/2.

Thus, aside from the coherent peak, the spectrum of

resonance fluorescence consists of two Lorentzian functions centered at v.6n with different heights which

depend on the observation angle c. The width of the

motional sidebands is given by half the cooling rate,

which in the Lamb-Dicke limit is narrower than V0.G

by a factor of the order h2!1 (Lindberg, 1986). The

asymmetry of the motional sidebands reflects the population of the trap levels un& and thus mirrors the ion’s

residual excitation energy in the trap potential.

For large intensities and on resonance the spectrum

exhibits ac Stark splitting with a center line at the laser

frequency, two symmetric sidebands at the Rabi frequency V0 , and width proportional to G (the Mollow

triplet; Mollow, 1969). The ion motion then leads to additional (narrow) sidebands such as in the low intensity

case.

For an ion in a standing-wave field configuration, expressions similar to Eq. (89) are found with an explicit

dependence on the ion position with respect to the

standing-wave phase. For details we refer the reader to

the work by Cirac et al. (1992).

C. Detection of internal states

The ions used in the experiments described here can

be detected by laser-induced fluorescence on an electric

dipole allowed transition, usually identical to the transition used for Doppler cooling. On such a transition a

single ion can scatter several millions of photons per

second and a sufficient fraction may be detected even

with detectors covering a small solid angle and having

5–20 % quantum efficiencies. The fluorescence is either

detected in a spatially resolved manner on chargecoupled-device (CCD) cameras or imager tubes, or on

photomultipliers. Cameras and imager tubes have been

used in a number of experiments to provide pictures of

ion clouds and crystals (see, for example, Diedrich et al.,

1987; Wineland et al., 1987; Drewsen et al., 1998; Mitchell et al., 1998; Na ̈gerl et al., 1998). Since the spatial extension of a cold ion’s wave function is typically smaller

than the wavelength of the fluorescence light, single ions

in a crystal show up as bright dots with a size determined

by the resolution limit of the imaging optics, that is,

around 1 mm with a good imaging lens.

Photomultipliers usually do not offer spatial information directly, but rather pick up the collected fluorescence with better quantum efficiency than CCD cameras

or imagers. With a total detection efficiency of 1023 (including solid angle and quantum efficiency) of the fluorescence rate, even a single ion will lead to about 50 000

counts/s on a dipole allowed transition with a 10-ns upper state lifetime.

This signal can be used not only to detect the ion

itself, but also to distinguish between internal states of

the ion with extremely high detection efficiency. This

technique was suggested by Dehmelt (1975) as a tool for

spectroscopy and was dubbed electron shelving because

the ion’s outer electron can be ‘‘shelved’’ into a state in

which it does not fluoresce.

1. The electron shelving method

Electron shelving makes use of an internal level structure that is well described by a V-type three-level system. In addition to ug& and ue& there is a third level ur&,

and it is assumed that the transitions ug&↔ue& and

ug&↔ur& can be independently driven by laser fields (see

Fig. 4). If the lifetimes of ug& and ue& are much longer

than that of ur& the transition ug&↔ur& may be strongly

driven, resulting in many scattered photons if the ion is

projected into ug& by the first scattering event. On the

other hand, if no photons are scattered, the ion was projected into state ue& by the interaction with the driving

field.

Depending on the background light scattered into the

photomultiplier and its dark count rate, ug& and ue& can

be distinguished with high confidence in rather short detection periods (typically on the order of a few tens of

ms). Apart from this convenient way of measuring the

quantum state, Dehmelt’s proposal triggered a number

of theoretical papers on ‘‘quantum jumps’’ between ug&

and ue& that would manifest themselves by extended periods of time with few or many photons detected, respectively, if both transitions are driven simultaneously.

The fact that one describes the behavior in the time of a

FIG. 4. V-type three-level system for electron shelving. In addition to transitions ug&↔ue&, transitions to a third level ur& can

be independently driven by laser fields. The lifetimes of ug&

and ue& are usually much longer than that of ur&.

294 Leibfried et al.: Quantum dynamics of single trapped ions

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
single atom, undergoing random transitions,3 rendered

the usual density-matrix approach inadequate, because

it yields only ensemble averages, not individual trajectories. A number of different correlation functions were

finally used by different workers to attack these problems. They are all related to g(2)(t), the probability of

detecting another photon, originating from the ur&→ug&

spontaneous emission, at time t5t, if one was detected

at t50. A review of both theoretical and experimental

work on this problem is presented by Blatt and Zoller

(1988).

2. Experimental observations of quantum jumps

Experimental observations of quantum jumps of

single trapped ions were first reported at about the same

time in three different laboratories (Bergquist et al.,

1986; Nagourney et al., 1986; Sauter et al., 1986).

In all three experiments the ug&↔ur& transition was a

dipole allowed transition also used for Doppler cooling,

while the weak transition was a dipole forbidden quadrupole transition that was excited by an incoherent hollow cathode lamp (Nagourney et al., 1986), was far offresonance scattering and collisional excitation (Sauter

et al., 1986), or was excited by a coherent laser source

(Bergquist et al., 1986). Indeed the observed fluorescence showed the random intensity jumps (see Fig. 5)

and all statistical properties to be in accord with theoretical predictions (see Blatt and Zoller, 1988, and references therein).

D. Detection of motional-state populations

A finite Lamb-Dicke parameter h implies that

ug&↔ue& transitions depend on the motional state of the

ion [see Eqs. (83) and (84)]. In general the internal state

may get entangled with the motional state for interactions on the carrier or any sideband. The dependence

can also be used to map the motional state of the ion

onto the internal state, which can subsequently be measured with high efficiency as described above. This mapping is straightforward in the resolved sideband regime,

where dissipation plays no role.

Practically, since h!1 and the ions are in the LambDicke regime, the most interesting transitions are the

first red and blue sidebands, since their coupling to lowest order is linear in h [see Eq. (70)]. The first blue

sideband proved to be very useful in characterizing the

number-state distribution of various states of motion. By

resonantly driving an ion in the starting state

uC~0!&5ug&(n50

`

cnun& (94)

on the blue sideband for various times t and measuring

the probability Pg(t)5^C(t)u(ug&^gu ^ I

ˆ m)uC(t)& to find

the ion in the ground state after the interaction, where

I

ˆ m is the identity operator in the motional-state space,

one obtains the following signal, which can be easily derived from Eq. (83) for the case l511:

Pg~t!5 1

2 F 11 (n50

`

Pn cos~Vn,n11t!G . (95)

Here Pn5ucnu

2 is the probability of finding the atom in

the nth motional number state. As long as Vn,n11 are

distinct frequencies, the occupation of all number states

Pn can be found by Fourier transforming the signal

equation (95). The Rabi frequencies Vn,n11 are distinct

inside the Lamb-Dicke regime, where the blue sideband

frequencies scale as An11 to lowest order in h [see Eqs.

(70) and (76)]. Outside this regime the frequencies will

not rise monotonically as a function of n, so distinguishing them is more complicated.

3

Indeed it was proposed to use the length of light and dark

periods for the generation of perfect random numbers (Erber

and Puttermann, 1985).

FIG. 5. Quantum jumps of a single

138Ba1 ion. If the ion makes the transition to

ue&[D5/2 , the fluorescence drops. After a

mean time equal to the excited-state lifetime

(ca. 32 s in this example), a spontaneous transition returns the ion to ug&[S1/2 and the fluorescence on the ug&↔ur&[P1/2 transition returns to a higher level.

Leibfried et al.: Quantum dynamics of single trapped ions 295

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
The Pn correspond to the diagonal elements of the

density matrix r nm of the motional state. One can also

extend the method described above, using additional

manipulations of the motional state, to yield the offdiagonal elements of r nm and characterize the complete

quantum state of motion. Theoretical methods and experimental implementation of two such methods will be

described in Sec. VI.B.

IV. LASER COOLING OF IONS

The general goal of cooling is to reduce the kinetic

energy of an ion after it was loaded into the trap, ideally

to a point at which the ion is in the ground state of the

trapping potential with very high probability. Ion traps

typically confine ions up to a temperature that corresponds to about one-tenth of their well depth, about 1

eV or 10 000 K. Cooling from these starting temperatures requires a high scattering rate of the cooling light,

so it is advantageous to use a dipole transition to a fairly

short-lived upper level for this stage. For most traps and

ions commonly used the first cooling stage will therefore

occur in the unresolved sideband regime, since the lifetime of the upper state is considerably shorter than one

period of oscillation in the trap. In this case, cooling in a

trap or cooling free atoms is essentially the same (Wineland and Itano, 1979). For example, the limiting kinetic

energy under this type of cooling turns out to be the

same as the limit of Doppler cooling for free atoms.

To reach the motional ground state with high probability a second stage of cooling is necessary, typically

with a lower scattering rate, but now in the resolvedsideband regime. For cooling to the ground state, three

different methods have been used to date, namely, cooling on a dipole forbidden quadrupole transition, cooling

by stimulated Raman transitions, and cooling utilizing

electromagnetically induced transparency (EIT cooling).

All these methods will be briefly discussed in this section. For a more comprehensive treatment the reader is

referred to the literature cited below and a review of

cooling techniques in ion traps by Itano et al. (1992).

A. Doppler cooling

Cooling in a trap was examined in conjunction with

the first Doppler cooling experiments (Neuhauser et al.,

1978; Wineland et al., 1978; Wineland and Itano, 1979;

Itano and Wineland, 1981). A semiclassical picture

based on the master equation was developed by Stenholm and co-workers, as reviewed by Stenholm (1986).

In these treatments a purely harmonic secular motion

with no micromotion of the ion was assumed. This was

somewhat unsatisfactory from the point of view of experiments in which additional cooling and heating effects related to micromotion were observed (Blu ̈ mel

et al., 1989; DeVoe et al., 1989). Finally, the most complete picture of cooling to date, including micromotion,

was derived by Cirac, Garay, et al. (1994). We first

present a very simple, more qualitative picture of Doppler cooling, before introducing the basic building

blocks and results of the general approach, including micromotion.

In the simple picture micromotion is neglected and

the trapping potential is approximated by the timeindependent pseudopotential

Vp~x!5 1

2 mn2x2. (96)

This description applies to axial motion in a linear trap

[see Eq. (4)], where micromotion is absent. If the motion of the trapped ion is taken to be classical, its velocity obeys

v~t!5v0 cos~nt!. (97)

If the radiative decay time is much shorter than one

oscillation period, n!G, one cycle of absorption and

spontaneous emission occur in a time span in which the

ion does not appreciably change its velocity. In this case

the averaged radiation pressure can be modeled as a

continuous force that depends on the ion’s velocity. If

the cooling laser field is a single traveling wave along the

ion’s direction of motion, every absorption will give the

ion a momentum kick Dp5\k in the wave-vector direction of the light field, while the emission will generally

be symmetric about some point. Emission will then lead

to zero-momentum transfer on average, but to a random

walk in momentum space, similar to Brownian motion.

The rate of absorption-emission cycles is given by the

decay rate G times the probability of being in the excited

state ree5^eurˆue&. Therefore the average force is

S dp

dt D a

'Fa5\kGree , (98)

and the excited-state probability is given by (Loudon,

1973)

ree5 s/2

11s1~2d eff /G!

2 , (99)

where s52uVu

2/G2 is the saturation parameter proportional to the square of the on-resonance Rabi frequency

V. The detuning is composed of the detuning D5v

2v0 of the light wave with respect to the resonance

frequency of the atom at rest and the Doppler shift:

d eff5D2k•v. For small velocities, close to the final temperature reached by laser cooling, where the Doppler

broadening is small compared to G, Fa can be linearized

in v:

Fa'F0~11kv!, (100)

with

F05\kG

s/2

11s1~2D/G!

2 (101)

the averaged radiation pressure that displaces the ion

slightly from the trap center and

k5 8kD/G2

11s1~2D/G!

2 (102)

296 Leibfried et al.: Quantum dynamics of single trapped ions

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
the ‘‘friction coefficient’’ of the cooling force that will

provide viscous drag if D is negative. The cooling rate,

averaged over many oscillation periods, is then

E ̇ c5^Fav&5F0~^v&1k^v2

&!5F0k^v2&, (103)

since ^v&50 for a trapped particle. Thus, without taking

the random nature of the light-scattering events into account, the ion would cool to zero energy. In practice this

cannot happen since even if the ion has zero velocity it

will continue to absorb and emit photons. The emission

rate for v50 is Gree(v50), with the recoil taking directions dictated by the emission pattern of the transition

used (typically an electric dipole transition). Since the

emission pattern is symmetric the average momentum

change is ^Dp&50, but the momentum undergoes diffusion, ^Dp2

&Þ0. As usual in random-walk processes the

average distance covered by the diffusion is proportional

to the square root of the number of recoil kicks or, in

other words, the second moment of the distribution of

random processes is proportional to the number N of

recoils: ^Dp2

&}(\k)2N. Not only the emission, but also

the random times of absorption of photons lead to momentum kicks, but this time only along the axis defined

by the wave vector of the cooling beam. This still gives

rise to diffusion due to the discreteness of the absorption

processes. Again the diffusion will be proportional to

the square root of the number of absorptions. Furthermore, unless the cooling transition is driven weakly, s

!1, absorption and emission will be correlated, leading

to an altered diffusion. While all these effects are included in the more general approach discussed later and

also discussed in the literature (Lindberg, 1984; Stenholm, 1986, and references therein), this correlation will

be neglected for the simple picture here. The momentum kicks due to absorption and emission will then have

the same rate but different directionality. This can be

taken into account by scaling the emission term with a

geometry factor j that reflects the average component of

the emission recoil kick along the x axis and takes the

value j52/5 for dipole radiation (Stenholm, 1986). For

the final stage of cooling v will be close to zero, so the

heating rate is approximately

E ̇ h5 1

2m

d

dt ^p2

&5E ̇ abs1E ̇ em

5E ̇ abs~11j!

. 1

2m ~\k!

2Gree~v50!~11j!.

(104)

Equilibrium will be reached if heating and cooling proceed at an equal rate, so one can infer the final temperature from equating Eqs. (103) and (104):

m^v2

&5kBT5 \G

8 ~11j!F~11s!

G

2D 1

2D

G G , (105)

with kB the Boltzmann constant. For this simple model,

which neglects the correlation between absorption and

emission, the minimum temperature will be

Tmin5\GA11s

4kB

~11j!, (106)

reached for a laser detuning D5GA11s/2.

The simple picture presented so far provides some insight into the cooling mechanism but neglects several

aspects of cooling in rf traps that can be important in

practice. In many experiments using Doppler cooling,

sidebands due to micromotion are observed in fluorescence spectra. Especially when the rf drive frequency is

comparable to or larger than the natural linewidth of the

cooling transition, one might wonder what effects this

has on the cooling process.

For example, when including micromotion, the kinetic

energy of the trapped particle has to be reconsidered.

The forced oscillations at the rf drive frequency vrf add

kinetic energy in excess of the energy in the secular motion. However, for the cooling dynamics, which evolve

on a much slower time scale than this fast oscillation, it

is appropriate to look at the kinetic energy averaged

over one period 2p/vrf of the micromotion (denoted by

the overbar):

Ekin5^pˆ ~t!

2

&

2m . (107)

The process of cooling can then be defined as an approach to minimizing this quantity.

For Doppler cooling, the spatial extension of the final

motional state is usually small compared to the cooling

light wavelength, so it is appropriate to limit a study of

cooling dynamics to the Lamb-Dicke regime. It is also

assumed that the trap operates in the (uaxu,qx

2

)!1 regime of micromotion. Assuming that the cooling light is

a traveling wave and following Cirac, Garay, et al.

(1994), one may expand the interaction Hamiltonian

(66) to first order in h (since the time origin for cooling

is unimportant, the phase f is irrelevant and set to zero

in the following):

Hˆ int

LD~t!5~\/2!V@sˆ 1exp~2idt!1H.c.#

1~\/2!VH (n52`

`

ihC2nsˆ 1e2idt

3@aˆ e2i(n1nvrft)1aˆ †ei(n1nvrft)

#1H.c.J .

(108)

The interpretation is straightforward. The first term represents the strong carrier excitation with Rabi frequency

V, while the other terms represent pairs of combined

secular and micromotion sideband excitations at detunings 6(n1nvrf) (n50,61,62, . . . ) with weaker Rabi

frequencies hVuC2nu. For the assumed conditions the

magnitude uC2nu rapidly drops with increasing unu. If all

sidebands are resolved, uvrfu@n@G, one may choose the

detuning in such a way that only one term in Eq. (108) is

resonant. This case will be described in the next section.

In the case in which G>vrf ,n, the dynamics are more

involved and for quantitative insight the master equation connected to Hamiltonian (108) has to be solved.

The sidebands and their relative strength are sketched

in Fig. 6. The extra sidebands lead to additional channels

Leibfried et al.: Quantum dynamics of single trapped ions 297

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
of absorption and emission. Absorption on the red secular sidebands will lead to a decrease in kinetic energy,

while absorption on the blue secular sidebands will lead

to an increase. Note that all sidebands at nvrf1n will

lead to heating even if n,0. This means that although

the laser might be red detuned from the resonance of

the atom at rest (carrier), the ion can be heated, especially if the excess micromotion due to unwanted static

potentials in the trap is not compensated (Blu ̈ mel et al.,

1989; DeVoe et al., 1989; Berkeland et al., 1998; Peik

et al., 1999). For good compensation the coupling

strength quickly drops with unu and the effect of these

extra resonances is not very dramatic. The ‘‘simple’’

Doppler limit stated in connection with Eq. (106) might

not be reached in all experiments due to these additional

heating contributions, but it still is a good rule of thumb.

In the NIST 9

Be1 experiments the rf drive frequency

was quite high, vrf /(2p)'230 MHz, compared to a 20-

MHz natural linewidth of the S1/2→P3/2 transition. The

cooling laser was detuned by about half a linewidth, so

in this experiment micromotion sidebands did not play a

role. Indeed, at a trap frequency of 11.2 MHz, a Doppler

cooling limit of  ̄nexp50.47(5) was reached, consistent

with the theoretical limit of  ̄n50.484 (Monroe et al.,

1995).

B. Resolved-sideband cooling

In a regime where the effective linewidth due to decay

from the excited state with rate  ̃G is lower than the motional frequency n, the individual motional sidebands

become resolved.4 One can then tune the cooling laser

to the first red sideband and cool the ion close to the

motional ground state if no heating mechanisms other

than the recoils of the cooling transition are present.

To reach this regime on a dipole allowed transition

one can use either a very stiff trap with high motional

frequencies or a weakly allowed dipole transition. Both

approaches have been pursued (Jefferts et al., 1995; Peik

et al., 1999), but so far cooling to the ground state in the

sense that the state ug,n50& is produced with high probability has involved either dipole forbidden transitions

or Raman transitions.

1. Theory

The cooling process in the resolved-sideband limit has

previously been described (Neuhauser et al., 1978; Wineland and Itano, 1979; Stenholm, 1986; Cirac et al., 1992;

Cirac, Garay, et al., 1994). The essential physics can be

understood with a simple model. To avoid unwanted

motional-state diffusion due to transitions of high order

in h it is very advantageous to start resolved-sideband

cooling inside the Lamb-Dicke regime. In fact all successful experiments on ground-state cooling have featured an initial Doppler cooling stage that achieved the

Lamb-Dicke regime. For a typical linewidth of several

tens of MHz on the Doppler cooling transition a sufficiently high motional frequency greater than about 1

MHz is necessary.

In the Lamb-Dicke regime and with no other heating

mechanisms present, the cooling laser is detuned to d

52n, the first red sideband. To first order in h, Eq.

(108) is reduced to the resonant red sideband (n50,

since the micromotion sidebands are far off resonance

and can be neglected), the carrier detuned by n, and the

blue sideband detuned by 2n:

Hˆ int

LD~t!5~\/2!V@sˆ 1eint

1sˆ 2e2int1ih~sˆ 1aˆ 1sˆ 2aˆ †!

1ih~sˆ 1aˆ †ei2nt1sˆ 2aˆ e2i2nt

!#. (109)

One way to find the final motional state would be to

insert this interaction Hamiltonian into the master equation (86) and solve it numerically, as was done by Cirac,

Garay, et al. (1994), but since every cooling cycle involves spontaneous emission, coherences never play a

strong role and the problem can be approximately

4

The effective rate  ̃G can be modified to be different from the

spontaneous decay rate of the excited state Gnat ; see, for example, Eq. (116) below.

FIG. 6. Spectrum of the secular and micromotion sidebands involved in the cooling process. The height of the bars represents the

coupling strength.

298 Leibfried et al.: Quantum dynamics of single trapped ions

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
solved with rate equations. Every cooling cycle (absorption followed by spontaneous emission) involves a transition from the ground state ug& to the excited state ue&

on the red sideband and a subsequent decay on the carrier (in the Lamb-Dicke regime the ion will predominantly decay on the carrier; therefore other decay channels can be neglected to lowest order in h). The cooling

rate is then given by the product of the excited-state

occupation probability pe(n) of a motional state un& and

its decay rate  ̃G. As in Eq. (99) a Lorentzian line shape

is assumed for this excitation, but now with detuning

2n:

Rn5 ̃

GPe~n!5 ̃G ~hAnV!

2

2~hAnV!

21 ̃

G2

. (110)

This rate depends on n and vanishes once the ground

state is reached. The ground state is therefore a dark

state of the red sideband excitation and the ion would be

pumped into that state and reside there without any

competing mechanisms. In the absence of any other

heating sources the dominant channel out of the ground

state is off-resonant excitation of the carrier and the first

blue sideband. Actually both these processes contribute

to the heating on the same order. The carrier is excited

with a probability of @V/(2n)#

2 [see Eq. (99) with  ̃

G

5G,V!d eff5n] but will mostly decay back on the carrier

transition. Decay on the blue sideband after carrier excitation that leads to heating only occurs with a rate of

@V/(2n)#

2 ̃h2 ̃

G. Note that the Lamb-Dicke factor  ̃h for

this decay is not equal to the one of the excitation, because the emitted photon can go in any direction, not

only along the wave vector of the cooling beam, and

some experimental arrangements use a three-level system in which the emitted photon does not have the same

wavelength as the cooling light (see below). The second

dominant heating process is excitation on the first blue

sideband with probability $hV/@2(2n)#%

2 (see above,

but now d eff52n) followed by decay on the carrier with a

rate of $hV/@2(2n)#%

2 ̃

G. For the final stage of the cooling the problem may be restricted to the ground state

and the first excited state with rate equations

p ̇ 05p1

~hV!

2

 ̃

G

2p0FS V

2n

D

2

 ̃h2 ̃G1S hV

4n D

2

 ̃

GG , (111)

p ̇ 152p ̇ 0

for the probabilities p0 ,p1 to be in the respective states.

In the steady state, p ̇ i50,p1512p0 , this yields

 ̄n'p1'S  ̃

G

2n

D

2

FS  ̃h

hD

2

1

1

4G . (112)

The factor in square brackets is of order one and n

@ ̃G, so the particle is cooled to the ground state with

probability p0'12@ ̃

G/(2n)#

2 very close to one.

Several methods have been used in experiments to

determine the final mean excitation number  ̄n after

resolved-sideband cooling. This review will be restricted

to the simplest and most robust method used so far, the

comparison of the probability Pe(t) to end up in the

excited electronic state ue& after excitation of the ion on

the red and blue sidebands. From Eq. (95) for the first

blue sideband and the analogous expression for the red

sideband [derived from Eq. (83) for l521], using

Pe(t)512Pg(t) and the assumption that the final motional states after cooling have a thermal distribution

(see Sec. II.C.4), one gets

Pe

rsb~t!5 (

m51

`

S  ̄n

 ̄n11 D

m

sin2

~Vm,m21t!

5  ̄n

 ̄n11 (

m50

`

S  ̄n

 ̄n11 D

m

sin2

~Vm11,mt!

5  ̄n

 ̄n11 Pe

bsb~t! (113)

using Vm11,m5Vm,m11 . This means that the ratio of

these probabilities is

R5 Pe

rsb

Pe

bsb 5  ̄n

 ̄n11 (114)

independent of drive time t, carrier Rabi frequency V,

or Lamb-Dicke parameter h (Turchette, Kielpinski,

et al., 2000). The ratio R can be inferred from a frequency scan over both sidebands while keeping the light

intensity and excitation time constant and will directly

yield the mean occupation

 ̄n5 R

12R (115)

of the thermal motional state.

2. Experimental results

Several groups have reported experiments that cool

up to four ions close to the ground state. The first report

of ground-state cooling was by the Boulder group, cooling 198Hg1 on a quadrupole allowed transition. The trap

was adjusted to be nearly spherical, with a secular frequency of 2.96 MHz using an appropriate positive dc

bias on the ring electrode. After 20 ms of Doppler cooling on the strong 2S1/2→2P1/2 transition this laser was

shut off and another laser on the 2S1/2→2D5/2 first red

sideband transition was turned on for 200–500 ms. Since

the natural lifetime of the 2D5/25ue& state limits the

scatter rate to about (1/2)Gnat56 photons/s, the upper

level was intentionally broadened by coupling it to the

short-lived 2P3/25uaux& level with an additional laser

source (repumper). This effectively broadens the linewidth to (Marzoli et al., 1994)

 ̃

G5 Vaux

2

~Gaux1Gnat!

214Daux

2 (116)

if the Rabi frequency on the ue&→uaux& transition is

Vaux and the auxiliary laser is detuned from resonance

by Daux . The choice of this Rabi frequency and detuning

will determine the effective linewidth  ̃G and in turn the

Leibfried et al.: Quantum dynamics of single trapped ions 299

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
cooling rate and final thermal distribution. After shutting off the sideband cooling laser, the repumper was left

on for another 5 ms to ensure that the ion returned to

the ground state ug&. Following the cooling cycle another laser pulse with roughly saturation intensity was

swept over the red and blue sidebands of the 2S1/2

→2D5/2 transition to record the excitation probability on

the sidebands. The probability was found with the electron shelving method by observing the 2S1/2↔2P1/2 fluorescence (see Sec. III.C) and averaging over about 40

sweeps consisting of one measurement for each frequency setting. From the strength of the sidebands the

final mean occupation number of the motional state was

determined as outlined in Sec. IV.B.1 to be  ̄n50.05

60.012. In this experiment the final temperature could

be determined in only two dimensions.

Ground-state cooling in all three dimensions was first

achieved by the NIST group (Monroe et al., 1995). The

trap used in this experiment had motional frequencies of

(nx ,ny ,nz)52p(11.2,18.2,29.8) MHz. The 9

Be1 ion

was first Doppler cooled well into the Lamb-Dicke regime in all three dimensions on the 2S1/2→2P3/2 transition. After Doppler cooling alone the motional states

had ( ̄nx , ̄ny , ̄nz)5(0.47,0.30,0.18), measured with the

sideband ratio method outlined above. Then a total of

15 cycles of interspersed resolved-sideband cooling

pulses (order xyz xyz ...) were applied, five cycles in

each of the three directions. Each cycle consisted of a

pulse on the red sideband of the stimulated Raman transition from the (2S1/2 ,F52,mF522)5ug& to the

(2S1/2 ,F51,mF521)5ue& state using the 2P1/2 state as

a virtual intermediate state [see also the Appendix, Eq.

(A3)]. The pulse time was adjusted to make a p pulse on

the ug&un51&→ue&un50& transition, typically taking 1–3

ms. Following this a resonant ue&→2P3/2s2 repump pulse

of about 7-ms length optically pumped the ion to ug& via

the 2P3/2 level. After the complete set of cooling pulses a

probe pulse interrogated the transition probability to

ue&. Cooling and probe pulses made up one experiment,

typically repeated 1000 times with the same settings. The

probe pulse was swept over all six red and blue sidebands, thus mapping out all relevant transition probabilities. From the ratio of corresponding sidebands the

final average motional quantum numbers were inferred

to be ( ̄nx , ̄ny , ̄nz)5(0.033,0.022,0.029).

Peik et al. (1999) report cooling a single 115In1 ion to

 ̄n50.7(3) corresponding to 58% ground-state occupation. The cooling transition was the weakly allowed

5s2 1S0→5s5p 3P1 intercombination line with an experimentally observed linewidth of G5(2p)650 kHz.

The trap frequency n was about (2p)1 MHz.

At the University of Innsbruck, ground-state cooling

was achieved on a single 40Ca1 ion for various motional

frequencies from n5(2p)2 MHz up to 4 MHz (Roos

et al., 1999). The method used was very similar to the

experiment in 198Hg1. The cooling transition was the

well-resolved red sideband on the ug&5S1/2(m521/2)

→ue&5D5/2(m525/2) quadrupole transition. The 1.045

s lifetime of the upper level was shortened by an additional laser that coupled ue& to the quickly decaying P3/2

level. The most efficient cooling occurred when the red

sideband Rabi frequency V1,0 on the quadrupole transition was roughly equal to the effective linewidth  ̃

G [see

Eq. (116)].

The ion was Doppler cooled into the Lamb-Dicke regime with a 2.6-ms pulse of light on the S1/2→P1/2 transition. A second laser resonant with the D3/2→P1/2 transition prevented optical pumping to the metastable D3/2

level. A short pulse of s1 polarized light on the S1/2

→P1/2 transition optically pumped the ion to ug&. Following this the two lasers for sideband cooling were

switched on for varying times between 3 and 7 ms. Another s1 pulse ensured that the ion was prepared in ug&

after the cooling, counteracting the possibility of the

ion’s being pumped into the S1/2(m511/2) state by the

cooling. For the polarizations and branching ratios

present in the experiment this would happen on average

after about 90 (ug&→ue&→P3/2) cooling cycles (Roos

et al., 1999).

Finally the cooling result was detected by comparing

the red and blue sideband transition probabilities on the

ug&→ue& transition as described above. Figure 7 shows

the results for a single ion cooled at a 4.51-MHz motional frequency. From the residual noise around the red

sideband transition frequency an upper limit of 99.9%

ground-state occupation was inferred. We note that

ground-state cooling using resolved-sideband transitions

was also experimentally demonstrated for up to four

ions (King et al., 1998; Sackett et al., 2000; Rohde et al.,

2001) and for neutral atoms in optical lattices (Hamann

et al., 1998; Perrin et al., 1998; Vuletic et al., 1998).

C. Electromagnetically induced transparency cooling

From the previous sections it might have become intuitively clear that any type of laser cooling in a trap

depends on the balance of absorption and emission of

photons on the red and blue sidebands. Every

absorption-emission cycle may be viewed as a scattering

FIG. 7. Sideband absorption spectrum on the S1/2→D5/2 transition of a single calcium ion (Roos et al., 1999) in a trap with

4.51-MHz motional frequency along the cooled axis: s, after

Doppler cooling; d, after resolved-sideband cooling; (a) red

sideband; (b) blue sideband. Each data point represents 400

experiments.

300 Leibfried et al.: Quantum dynamics of single trapped ions

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
event, with some events leading to cooling, some to

heating, and some to no change in the motional state.

The ion will be cooled on average if the events that

dissipate motional energy are more probable than the

heating events. Typically the likelihood of cooling events

decreases as the kinetic energy decreases, while the

heating events settle at a fixed (possibly very low) rate.

Equilibrium is reached when cooling and heating events

on average balance each other. The derivation of the

Doppler cooling limit, Eq. (105), in Sec. IV.A is an example of that mechanism in which the velocityindependent heating rate Eq. (104) is balanced with the

velocity- (and therefore kinetic-energy-) dependent

cooling rate Eq. (103).

Inspection of Eq. (106) and its derivation reveals that

the minimum Doppler temperature is dictated by the

line shape of the cooling transition [see Eq. (99)]. This

implies that one might influence the cooling process by

tailoring the line shape of the cooling transition. In the

preceding sections our discussion was limited to effective two-level systems and laser intensities around or below saturation with not too strong coupling of the atom

to the light field. The basic idea of electromagnetically

induced transparency (EIT) cooling is to go beyond this

scenario and utilize the strong coupling of one laser to

the atom in a three-level L-type scheme to create an

absorption profile for the second (weaker) laser that is

advantageous for cooling (Morigi et al., 2000).

We shall first derive a generalized treatment of the

cooling that works for an arbitrary scattering rate on the

cooling transition. The cornerstones of this treatment

are outlined in Stenholm (1986). We shall derive some

general statements about what kind of dependence of

the scattering rate on the relative detuning of atom and

laser is most useful. We shall then derive the scattering

rate in the L system for an atom at rest and set the

parameters according to the design principles found earlier.

1. Cooling in the Lamb-Dicke regime

We shall now calculate the cooling and heating rates

for a laser-driven, trapped ion with arbitrary scattering

rate W(D). A general scattering (absorption-emission)

cycle will proceed from ug,n& over ue,n8& to ug,n9&, and,

in principle, one could find the scattering rates for all

possible combinations (n,n8,n9) and derive rate equations for the probability P(n) to be in a certain state n

based on all these scattering rates. Here we shall limit

possible scattering paths by assuming that some sort of

precooling (for example, Doppler cooling) has left the

ion in a state with thermal distribution in or close to the

Lamb-Dicke regime with h2

 ̄n<1. In this approximation

the treatment cannot describe the complete cooling dynamics, but as long as the cooling method in question

reaches the Lamb-Dicke regime, it will yield useful expressions for the cooling limit and the cooling rate towards the final state. We assume that we know the scattering rate W(D) at laser detuning D of the atom at rest.

This quantity can usually be found by solving Bloch

equations for the atom and calculating the steady-state

population rex in the excited state. The scattering rate is

then this population times the total decay rate G of the

excited state, W(D)5Grex . In the Lamb-Dicke regime

absorption and emission will be dominated by carrier

and first-order red (blue) sidebands with transition probabilities proportional to V2 and h2V2n @h2V2(n11)#,

respectively, and we can neglect all processes of higher

order in h. Scattering on the carrier (ug,n&→ug,n&) will

be the most frequent process but will not change the

probabilities Pn to be in a certain n state. Motionalstate-changing events are limited to ug,n&→ug,n

6(0,1)& via the intermediate states ue,n& or ue,n61&. By

comparing the Rabi frequencies on carrier and sideband

we can see that the scattering rate on a path involving

one red (blue) sideband transition (for example, ug,n&

→ue,n&→ug,n11&) will be suppressed by the factor h2n

@h2(n11)# compared to the carrier transition. In more

detail, the rates Rn61 n will go as

Rn11 n 5W~D!h2

~n11!1W~D2n!h2~n11!,

Rn21 n 5W~D!h2n1W~D1n!h2n, (117)

where the first contribution comes from the scattering

path through ue,n& and the second from the path ue,n

61&. For example, in the second path on Rn11 n the atom

absorbs at detuning D2n since the remaining energy \n

goes into the motion. Knowing these rates we can immediately write down rate equations for the motional-level

populations

d

dt P~n!5Rn

n11

Pn111Rn

n21

Pn212~Rn21 n 1Rn11 n !Pn

5A2@Pn11~n11!2Pn~n!#

1A1@Pn21n2Pn~n11!#, (118)

with the n-independent coefficients

A65h2

@W~D!1W~D7n!#. (119)

The rate equation (118) can be converted into an equation of motion for the average motional quantum number

d

dt ̄n5 (n51

`

n

d

dt Pn52~A22A1! ̄n1A1 (120)

by appropriate pairing and resumming of the respective

populations in connection with the A6 coefficients. As

long as the cooling rate 2(A22A1),0,  ̄n will evolve

towards the final state of cooling, the steady state of Eq.

(120) given by

 ̄nf5 A1

A22A1

5 W~D!1W~D2n!

W~D1n!2W~D2n!

. (121)

It is instructive to plug the scattering rates WL(D) of a

Lorentzian line into this formalism. Then the Doppler

limit  ̄nfn.G/2 is recovered for G.n while  ̄n!1 for G

!n. For best cooling results with a general scattering

rate we want to minimize  ̄nf , which happens naturally if

the scattering rate on the red sideband W(D1n) is

Leibfried et al.: Quantum dynamics of single trapped ions 301

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
much bigger than the carrier and blue sideband rates,

W(D) and W(D2n), respectively.

2. Scattering rates in EIT cooling

The general idea of EIT cooling in a L system is to

use a dark resonance to completely suppress the carrier

scattering (Morigi et al., 2000). With a prudent choice of

parameters of the two light fields driving the system we

shall then also be able to fulfill W(D2n)!W(D1n).

We denote the levels, detunings, and Rabi frequencies of

the two transitions as indicated in Fig. 8. The equations

of motion for the density matrix (Bloch equations) can,

in principle, be derived from Eq. (86) or by adding phenomenological damping terms to the unitary evolution

that correctly reflect the decay of the ue& state. The

Bloch equations in our system are

drrr

dt 5i

Vr

2 ~rre2rer!1Grree ,

drgg

dt 5i

Vg

2 ~rge2reg!1Ggree ,

drrg

dt 5iF~Dg2Dr!rrg1

Vg

2 rre2 Vr2 regG ,

drre

dt 5iF

Vr

2 ~rrr2ree !1

Vg

2 rrg2DrrreG 2 G2 rre ,

drge

dt 5iF

Vg

2 ~rgg2ree !1

Vr

2 rgr2DgrgeG 2 G2 rge ,

(122)

with G5Gg1Gr . We shall assume that reequilibration of

the internal-state dynamics is much faster than the external motional dynamics, so we can solve for the steady

state of the Bloch equations to describe the scattering

events at all times. Doing so and using the conservation

of probability rrr1rgg1ree51 one can (after some algebra) derive the steady-state solution for ree (Janik

et al., 1985):

ree~D!5 4D2Vg

2

Vr

2

G

D , (123)

where D5Dg2Dr and

D58D2Vr

2

Vg

2

G14D2G2

~Vr

2

Gg1Vg

2

Gr!

116D2

@Dr

2

Vg

2

Gr1Dg

2

Vr

2

Gg#18DrDVg

4

Gr

28DgDVr

4

Gg1~Vr

2

1Vg

2

!

2

~Vr

2

Gg1Vg

2

Gr!. (124)

This rather complicated expression quickly simplifies if

we set Gg5aG and assume Dr'Dg and Vg!(Vr ,Dr),

following the idea that the strong laser field on the

ur&↔ue& transition optically pumps the internal state to

ug& and also modifies the scattering rate of a comparably

weaker beam on the ug&↔ue& transition into W(D)

5Gree(D), which is advantageous for cooling. This

yields

W~D!' D2Vg

2

G

a@D2G214~Vr

2

/42DDg!

2#

. (125)

Figure 9 shows the qualitative behavior of the scattering

rate vs detuning D for (Dr ,Dg).0. Indeed W(D) vanishes at D50, so the carrier is completely suppressed.

The position of the two maxima is given by

D651

2 ~6ADr

2

1Vr

2

2Dr!. (126)

We want the narrow bright resonance at positive detuning to coincide with the red sideband, D15n:

n51/2~ADr

2

1Vr

2

2Dr!. (127)

The reader may recognize that this condition is equivalent to saying that the Stark shift of the ur&→ug& resonance has to be equal to the motional frequency. For this

choice of parameters, W(1n) takes the largest value

possible, while W(2n) then assumes a comparatively

small value from the wing of the broad bright resonance

to the left of the origin. To find the cooling limit quantitatively, we can start from Eq. (121) with W(0)50.

Using Eq. (127) we get after some algebra

 ̄ns5 W~2n!

W~n!2W~2n!

5S G

4Dr

D

2

. (128)

FIG. 8. Notation of levels, detunings, Rabi frequencies, and

decay rates used in the calculation of an electromagnetically

induced transparency (EIT) line shape. FIG. 9. Qualitative scattering rate on the ug&↔ue& transition as

Dg is varied for Dr.0. In addition to the broad resonance

around Dg'0, a narrow bright resonance appears to the right

of the dark resonance at Dg5Dr . The distance between the

dark and the bright resonance is equivalent to the ac Stark

shift of ue& caused by the strong beam.

302 Leibfried et al.: Quantum dynamics of single trapped ions

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
To ensure that  ̄ns!1 we have to choose Dg5Dr@G,

while Vr has to be set to accommodate Eq. (127).

Finally we should note that our treatment of EIT

cooling is only an approximation, since we neglected all

recoils happening when the system is relaxing back into

the steady state after scattering a photon on the sideband transitions. Fortunately, since we start in the

Lamb-Dicke regime, the relaxation proceeds predominantly through carrier scattering, thus not altering the

average motional quantum number. Morigi et al. (2000)

have done a Monte Carlo simulation that also takes

these effects into account. Indeed the cooling dynamics

are slightly slower in this simulation, but very close to

the approximate treatment.

3. Experimental results

The Innsbruck group has demonstrated EIT cooling

with a single calcium ion (Roos et al., 2000). The L system was implemented within the S1/2→P1/2 transition,

whose Zeeman sublevels m511/2 and m521/2 constitute a four-level system. By applying the strong laser on

the s1 transition (ur&[uS1/2 ,m521/2&,ue&[uP1/2 ,m

51/2&) and the weaker cooling laser on a p transition

(from ug&[uS1/2 ,m511/2&), they ensured that very

little population would ever be in the extra uP1/2 ,m

521/2& state, so this was effectively a three-level system. An additional laser is used to repump the ion from

the D3/2 level, but the branching ratio to that level was

so small that the above conclusion was not seriously

compromised.

The two beams were generated by splitting frequencydoubled light from a Ti:sapphire laser near 397 nm into

two suitable beams with the help of two acousto-optic

modulators (the same beam was also used for Doppler

precooling of the ion). The second-order Bragg reflexes

of two acousto-optical modulators driven at around 90

MHz had a blue detuning of about Ds5Dp.75 MHz

relative to the S1/2↔P1/2 line center (natural linewidth of

the transition G'20 MHz). The beams (typically about

50 mW in the strong and 0.5 mW in the weak beam) were

then focused into a '60-mm waist onto the single ion

in a trap with oscillation frequencies (nx ,ny ,nz)

5(1.69,1.62,3.32) MHz. The k vectors of the two beams

enclosed an angle of 125° and their k-vector difference

Dk had a component along all three trap axes. The beam

intensity was controlled with the power of the acoustooptical modulator rf drive.

The ion was first Doppler precooled. The Doppler

cooling limit on this was ( ̄nx , ̄ny , ̄nz)'(6,6,3) in the trap

used. The EIT cooling beams were applied for periods

between 0 and 7.9 ms, with  ̄n reaching its asymptotic

limit after about 1.8 ms. The final mean occupation number was probed on the resolved sidebands of the

S1/2↔D5/2 transition, either using the technique described in Sec. IV.B.1 or fitting experimental Rabi oscillations to those expected for a small state with thermal

distribution (Meekhof et al., 1996; Roos et al., 1999).

In this manner EIT cooling of the y and the z oscillations at 1.62 and 3.32 MHz was investigated. The intensity of the strong beam was fine-tuned by observing the

resolved-sideband excitation on the red sideband of either mode and minimizing it. The lowest observed mean

vibrational quantum number was  ̄ny50.18, corresponding to 84% ground-state probability. On the z mode at

nz53.3 MHz, after the intensity of the s1 beam was

increased to adjust the ac Stark shift, a minimum mean

vibrational number of  ̄nz50.1 was observed, corresponding to a 90% ground-state probability. The cooling

results were largely independent of the intensity of the p

beam as long as it was kept much smaller than the s1

intensity. The intensity ratio Is /Ip.100 was varied by a

factor of 4, with no observable effect on the final  ̄n.

From the dependence of the mean vibrational quantum

number on the EIT pulse length, an initial cooling rate

of 1 quantum per 250 ms was found for the y direction.

In addition, both modes were simultaneously cooled

by setting the intensity of the strong beam for an ac

Stark shift roughly halfway between the two mode frequencies. Both modes were cooled simultaneously with

(p0)y558% and (p0)z574% ground-state probability.

V. RESONANCE FLUORESCENCE OF SINGLE IONS

The observation and analysis of resonance fluorescence emitted from atomic systems provide an important tool for investigation of the interaction between

matter and radiation. Essentially there are three

complementary strategies available for these measurements:

(i) Excitation spectroscopy comprises the observation of resonance fluorescence as a function of the

detuning of the exciting electromagnetic radiation;

(ii) the spectrum of resonance fluorescence is obtained by measuring the spectral distribution of

the emitted resonance fluorescence at a fixed detuning of the exciting radiation;

(iii) quantum-dynamical information can be obtained

by measuring the time interval statistics of the

photon-counting events from resonance fluorescence, usually in the form of correlation functions.

All three techniques have been used to investigate

single-ion resonance fluorescence.

A. Excitation spectroscopy, line shapes

A single two-level atom at rest in free space (Dehmelt, 1973) is the ideal object for the observation of

resonance fluorescence. The outcome of corresponding

experiments can be completely described in terms of the

optical Bloch equations, as given by Eq. (86), where the

excited-state population ree as a function of the detuning D of the exciting radiation describes the observed

line shapes. However, since the detuning also sensitively

influences the cooling and heating dynamics, the observed line shape in excitation spectroscopy usually exhibits this influence as well. More precisely, for weak

confinement (i.e., trap frequency n,G, the natural lineLeibfried et al.: Quantum dynamics of single trapped ions 303

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
width of the excited state) and a laser detuning far below resonance (i.e., D,0 or ‘‘red detuning’’), the

Doppler-broadened line shape resembles that of a thermal ensemble of atoms; however, the Doppler broadening results from a convolution of overlapping sidebands.

For a red detuning close to resonance, cooling sets in

and the line shape can be described very well by the

Lorentzian of an atom at rest. For a positive (or ‘‘blue’’)

detuning D.0, heating dominates and the large number

of resulting sidebands broadens the transition to such an

extent that the absorption sharply drops near resonance

D50. Therefore excitation spectra of a single two-level

ion exhibit an asymmetric (‘‘half-Lorentzian’’) shape

due to the cooling and heating dynamics (Nagourney

et al., 1983). For three- and more-level systems with several lasers contributing to the cooling and heating dynamics the observed line shapes can become quite involved and difficult to interpret (Reiß et al., 1996).

Especially well-investigated cases are the three-level

(eight-sublevel) systems of single trapped Ba1 and Ca1

ions (Janik et al., 1985; Siemers et al., 1992). These

ions must be modeled by a L-shaped three-level

S1/2↔P1/2↔D3/2 (or, accounting for all Zeeman sublevels, eight-level) system. The excitation usually involves

two dipole transitions, sharing the fluorescing level P1/2 ,

and excitation spectroscopy is obtained by tuning only

one exciting laser across resonance while the second one

is kept fixed below resonance. Thus net cooling usually

can be observed even for a certain range of blue detunings. Three-level (eight-sublevel) systems can exhibit

line shapes that agree very well with the theoretical descriptions given by the optical Bloch equations for an

atom at rest. In fact, the richness of these systems allows

one to investigate in quantitative detail the effects of

optical pumping, the appearance of dark resonances,

and Raman processes (Schubert et al., 1992, 1995; Siemers et al., 1992). The excellent quantitative agreement

between theory and experiment allows one to precisely

derive all spectroscopic parameters, such as the detunings, the Rabi frequencies, and the orientation of the

quantization axis required for more detailed quantumoptical calculations, e.g., correlation functions, as discussed below.

B. Nonclassical statistics, antibunching, and squeezing

A powerful way to acquire detailed information about

the quantum dynamics of an atom is the statistical analysis of the measured stream of photon counts. In particular, nonclassical features of the resonance fluorescence

are observed in the second-order correlation functions

(intensity correlations)

g(2)~t!5 ^E2~t!•E2~t1t!•E1~t1t!•E1~t!&

@^E2~t!•E1~t!&#

2 . (129)

Here the scattered light field of the atomic resonance

fluorescence is described by the electric-field operators

E1 and E2 (Glauber, 1963, 1964) and for t>0. This

function is proportional to the probability of detecting a

second photon at time t after a first one has been observed at t50. Whereas classical fields show correlation

functions g(2)(0)>1 and g(2)(t)<g(2)(0), the nonclassical resonance fluorescence of a single atom exhibits the

so-called antibunching behavior, i.e., g(2)(0)50 and

g(2)(t).g(2)(0). This condition entails sub-Poissonian

emission probability for t50. Both antibunching and

sub-Poissonian statistics have been observed in

quantum-optical experiments with atoms (Kimble, Dagenais, and Mandel, 1977; Short and Mandel, 1983);

however, corrections for the fluctuating atom numbers in

the atomic beam had to be applied.

For a single two-level atom the intensity correlation

function is

g(2)~t!5 ree~t!

ree~`!

, (130)

where ree represents the population of the excited state

of the atom. Since g(2)(t) is a conditional probability as

a function of time, it allows one to observe directly the

quantum-dynamical behavior of the matter-radiation interaction. Likewise, any motional effects affecting the

resonance fluorescence can be characterized from an

analysis of g(2)(t).

With a single trapped Mg1 ion Diedrich and Walther

(1987) observed antibunching and sub-Poissonian statistics of an individual two-level atom. In that experiment

antibunching in the resonance fluorescence of one, two,

and three trapped ions was observed and the antibunching property decreased as predicted for increasing ion

numbers, since for an increasing number of independent

atoms the photon counts become more and more uncorrelated. The dynamical interaction between matter and

radiation leads to Rabi oscillations and for a single ion

the observed intensity correlation agrees very well with

the theoretically predicted function (Carmichael and

Walls, 1976),

g(2)~t!512e23Gt/2F cos Vt1

3G

2V sin Vt G , (131)

where V25V0

2

1D22(g/4)2, with the Rabi frequency

V0 at resonance, the natural linewidth G, and the detuning D. The single-atom resonance fluorescence clearly

exhibited sub-Poissonian statistics. The deviation of the

distribution from Poisson statistics is usually described

in terms of Mandel’s Q parameter,

Q5 ^~Dn!

2

&T2^n&T

^n&T

, (132)

where ^n&T5(n50 ` npn(T) denotes the mean number of

detected photons, with pn(T) the probability for detecting n photons within the time T. This parameter is related to the photon correlation function (Mandel, 1979;

Merz and Schenzle, 1990) by

Q~T!5 2Z

T E

0

T

dt~T2t!@g(2)~t!21#, (133)

where Z5^n&T /T is the mean count rate. While for any

classical radiation Q(T)>0 at all T, the measurement

of Diedrich and Walther revealed Q52731025. In that

304 Leibfried et al.: Quantum dynamics of single trapped ions

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
experiment the classical residual (driven) micromotion

of the trapped ion was observed as a modulation in the

intensity correlation; however, secular motion could not

be detected.

In an experiment with one and two trapped Hg1 ions

Itano et al. (1988) were able to observe antibunching

and sub-Poissonian statistics on the P1/2↔D3/2 transition. Spontaneous emission on this transition is very

weak (G552 s21); however, each decay can be observed

since it interrupts the otherwise steady photon flow on

the strong (G543108 s

21) P1/2↔S1/2 transition that is

used for excitation and detection. Thus each spontaneous decay on the weak transition is detected by the absence of many photons due to electron shelving (see Sec.

III.C). Using this technique, antibunching on the weak

transition was observed and, due to the greater detection efficiency of the electron shelving technique (as

compared to usual photon counting in the experiment

by Diedrich and Walther), a Q value of Q520.25 for

one and two ions could be measured.

In a subsequent experiment at the University of Hamburg, Schubert et al. (1992, 1995) observed antibunching

and sub-Poissonian statistics (Q52731024 due to limited efficiency of photon counting) in the resonance

fluorescence of a single Ba1 ion, which cannot be modeled as a two-level system. Aside from antibunching and

sub-Poissonian statistics, the observed intensity correlations in this case revealed maximum photon correlations

much larger than what is possible with two-level atoms,

as well as photon antibunching with much larger time

constants of the initial photon anticorrelation. Thus a

detailed study of the internal dynamics due to optical

pumping and the preparation of Zeeman coherences became possible. Furthermore, the exact form of the observed g(2) functions allowed the direct observation and

quantitative description of the preparation of superposition states vs mixed states of a single trapped particle

(Schubert et al., 1995).

Aside from the antibunching property, which directly

reflects the photon nature of light, other nonclassical effects are observable in resonance fluorescence. An example is the squeezing property, which is usually observed in a homodyne detection scheme (Slusher et al.,

1985; Wu et al., 1986) and which describes an asymmetric noise behavior of the quadrature components of the

electromagnetic radiation. Squeezing has also been predicted to appear in the resonance fluorescence of single

two-level atoms (Walls and Zoller, 1981) and three-level

atoms (Vogel and Blatt, 1992). Due to limited detection

efficiency, Mandel (1982) has shown that the observable

effect from squeezing in the photon statistics of the resonance fluorescence of a single atom is extremely weak.

Alternatively, the detection of homodyne intensity correlations is expected to offer an easier way to observe

squeezing in the resonance fluorescence of a single

trapped ion, as was proposed by Vogel (1991) and Blatt

et al. (1993). However, no experimental results are currently available.

C. Spectrum of resonance fluorescence, homodyne

detection of fluorescence

Information about the dynamics of the interaction between matter and radiation can also be gathered by

spectral analysis of the emitted resonance fluorescence.

The spectrum of resonance fluorescence emitted by a

free two-level atom interacting with a traveling-wave laser field was studied by Mollow (1969). For low excitation intensities the fluorescence spectrum exhibits an

elastic peak centered at the incident-laser frequency v L ,

while for higher intensities an inelastic component becomes dominant, with contributions centered at the frequencies v L and v L6V0 (for resonant excitation),

where V0 denotes the Rabi frequency. This so-called

‘‘Mollow triplet’’ arises from the dynamic (or ac) Stark

splitting of the two-level transition. As outlined in Sec.

III.B.6 above, the observation of the fluorescence spectrum provides an alternative method for detecting the

internal quantum dynamics as well as the (quantum)

motion in the trap.

With a single trapped and laser-cooled Ba1 ion, Stalgies et al. (1996) analyzed the spontaneously emitted

light on the P1/2↔S1/2 transition at 493 nm with a FabryPerot interferometer and recorded emission spectra of

the single-ion fluorescence. The piezo-tuned confocal filter resonator provided a spectral resolution of about

16.7 MHz. This was sufficient to resolve the spectral

components of the transition, which has a natural width

of about 20 MHz. The observed spectra show the expected dynamical Stark effect and are in agreement with

theoretical predictions based on calculations using optical Bloch equations and parameters that were derived

from excitation spectra as outlined above in Sec. V.A.

Refined spectroscopic information on the fluorescence

spectrum can be gained by beating the fluorescent light

with a local oscillator and subsequent homodyne or heterodyne analysis. The elastic component scattered by an

ensemble of cold-trapped atoms was first detected by

Westbrook et al. (1990) with atoms confined in an optical lattice. With a single trapped and laser-cooled Mg1

ion, Rayleigh scattering on the S1/2↔P3/2 transition was

observed by Ho ̈ffges, Baldauf, Eichler, et al. (1997) and

Ho ̈ffges, Baldauf, Lange, and Walther (1997). Heterodyne detection allowed for a spectral resolution of the

elastic component with a linewidth of 0.7 Hz.

As shown above in Sec. V.A, the motion of trapped

particles in an external potential shows up as sidebands

to the elastically scattered component. The height of the

sidebands then presents a measure of the motional amplitude, and their width reflects the cooling rate. Such

motional sidebands were first detected in the fluorescence spectrum of ensembles of cold atoms in optical

lattices (Jessen et al., 1992) and they were only recently

observed in the homodyne spectra of single trapped

ions. With a single trapped and laser-cooled Yb1 ion,

Bu ̈hner and Tamm observed motional sidebands in the

resonance fluorescence of the P1/2↔S1/2 transition at

369 nm of 172Yb1 at a radial secular motion of 785 kHz

(Bu ̈hner and Tamm, 2000; Bu ̈hner, 2001). The measured

Leibfried et al.: Quantum dynamics of single trapped ions 305

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
maximum width of the sidebands was about 1.5 kHz at

saturation intensity and for an optimal detuning of the

cooling laser of D.G. Taking into account residual

broadening by the detection technique (i.e., detection

bandwidth of the spectrum analyzer and fluctuating trap

voltages), the authors concluded that the observed cooling rate agreed well with predictions based on the twolevel cooling calculations.

With a single trapped 171Yb1 ion that has a hyperfine

splitting, a weak spontaneous decay to a hyperfine component of the excited D3/2 state leads to intermittent

resonance fluorescence on the P1/2↔S1/2 transition due

to electron shelving. This results in an additional inelastic component of the observed fluorescence spectrum

(Hegerfeldt and Plenio, 1995), and its height and width

are uniquely determined by the shelving conditions. This

leads to a Lorentzian component centered on the elastic

scattering and is interpreted as a broadening due to a

‘‘stochastic intensity modulation.’’ In the experiment this

feature was observed and found to be in agreement with

the theoretical prediction (Bu ̈hner and Tamm, 2000).

With a single trapped Ba1 ion, heterodyne measurements were also performed by the Innsbruck group. The

measured width of the elastic component was limited by

the resolution bandwidth of 64 MHz of the spectrum

analyzer. Since the phase of the driven micromotion at

vrf52p 18.53 MHz is well defined, the corresponding

micromotion sidebands were observed with the same

linewidth. From the height of the sidebands a modulation index corresponding to a residual micromotion amplitude of 26 nm was derived. This measurement allows

one to detect and compensate for residual micromotion,

and it is sensitive to micromotion amplitudes of about 1

nm (Raab et al., 2000). Furthermore, sidebands due

to secular motion at frequencies (vx ,vy ,vz)/2p

5(0.62,0.65,1.3) MHz were observed using an excitation

technique: By exciting the ion motion with a weak rf

field (applied to nearby electrodes) one can detect coherent sidebands with a high signal-to-noise ratio, since

this results in a driven motion and thus is similar to the

detection of micromotion sidebands. Scanning the excitation across the sideband frequency results in a resonant enhancement of the ion motion, making the sidebands easily observed. Extrapolating the observed

linewidth to zero amplitude of the exciting rf field eventually reveals the cooling rate limiting the width of the

motional sidebands. The experimentally observed linewidths of about 750 Hz agree within the error limits with

the calculated cooling rates (Raab et al., 2000). These

findings were corroborated by direct observation of the

sidebands as shown in Fig. 10 (Raab et al., 2001). No

asymmetric sidebands were observed in either the Ba1

experiment (Innsbruck) or the Yb1 experiment (Braunschweig), since in both cases only Doppler cooling was

available and the residual quantum numbers of the harmonic motion were between 10 and 30 (depending on

the ion and the respective secular frequencies), leading

to almost equal sideband amplitudes.

Using a new technique, the Innsbruck group recently

studied the interaction of a single trapped Ba1 ion with

a part of the emitted resonance fluorescence (Eschner

et al., 2001). For this, part of the emitted light (about 4%

of the solid angle) is collimated with a lens to a plane

mirror about 25 cm away from the ion and then retroreflected (via the same lens) onto the ion. The fluorescence and its mirror image are then observed with a

photomultiplier opposite the mirror collecting both the

directly emitted and the back-reflected part. Moving the

mirror using a piezoelectric transducer and overlapping

the direct and reflected parts allows one to observe interference fringes that result from the interfering beam

paths (direct and reflected paths) and the visibility of the

interference fringes can be calculated from the firstorder correlation function

g(1)~t!5 ^E2~t2t!E1~t!&

^E2~t!E1~t!& . (134)

The emitted field E6(t) can be expressed in terms of the

atomic polarization, which in turn can be calculated

from the optical Bloch equations. As expected, the visibility decreases with increasing laser intensity due to

the increasing ratio of inelastic to elastic scattering; that

is, the internal dynamics of the interaction can be investigated through a study of the interference fringes. In

addition, external atomic motion influences the interference since the ion oscillation results in a phase modulation of the field E6(t). This effect allows one to measure

the amplitude of any residual ion motion via an observation of the interference fringes. Moreover, the ion position with respect to the mirror can be determined on

the scale of the extension of the wave function, which is

several nm.

Including the back-reflected part of the emitted fluorescence in the optical Bloch equations reveals that the

reflected part actually acts back on the ion and changes

the excited-state populations and thus the overall resonance fluorescence. This back action has been detected

by an independent observation of the excited-state

population as a function of the mirror position (i.e.,

FIG. 10. Heterodyne signal on a secular sideband of the undriven motion of a single Ba1 ion (Raab et al., 2001). The fitted width corresponds to a cooling rate of 2p 900 s21.

306 Leibfried et al.: Quantum dynamics of single trapped ions

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
whether there is constructive or destructive interference). This can be considered inhibition and enhancement of the resonance fluorescence by the reflected (and

phase-sensitive) presence of the self-emitted photon.

This back action mediated by the external mirror has

also been observed with the radiation of two ions; that

is, two ions interfering with the mirror image of each

other show the same effect (Eschner et al., 2001). This

technique opens the way for quantum feedback scenarios in which single emitted fluorescence quanta can

be used to influence and eventually control the internal

and external quantum dynamics of trapped ions.

VI. ENGINEERING AND RECONSTRUCTION OF

QUANTUM STATES OF MOTION

A. Creation of special states of motion and internal-state/

motional-state entanglement

Fueled by the strong analogy between cavity QED

and the trapped-ion system, various theoretical proposals have been made on how to create nonclassical and

arbitrary states of motion of trapped ions or even how to

create states in which the internal degree of freedom is

entangled with the external motion in a highly nonclassical way (Schro ̈dinger-cat states). The theoretical efforts

were complemented by the first experiments on nonclassical states of motion by the groups at NIST and the

University of Innsbruck. In the NIST experiments, coherent, quadrature squeezed number states and superpositions of number states of trapped Be1 were prepared (Meekhof et al., 1996). The preparation of

number states was also achieved for trapped Ca1 at the

University of Innsbruck (Roos et al., 1999). The NIST

group succeeded in entangling coherent motional states

with opposite phases to the two internal states

(Schro ̈dinger-cat state; Monroe et al., 1996).

The common starting point for all motional-state engineering experiments so far has been the ground state

of motion, reached by resolved-sideband cooling methods as described in Sec. IV.B. Since resolved-sideband

cooling has only been successfully applied inside the

Lamb-Dicke regime, most of the state preparation also

took place in this regime. The description of interactions

will be restricted to this limit in the following and only

generalized where appropriate.

1. Creation of number states

Several techniques for the creation of number states

of motion have been proposed, using quantum jumps

(Cirac, Blatt, et al., 1993; Eschner et al., 1995), adiabatic

passage (Cirac, Blatt, and Zoller, 1994), or trapping

states (Blatt et al., 1995). Despite the possibilities described in these papers, a simple technique involving

multiple p pulses is the only one to have been used in

experiments so far. The ion is initially cooled to the ug,0&

number state. Higher-n number states are created by

applying a sequence of resonant p pulses of laser radiation on the blue sideband, red sideband, or carrier. A p

pulse corresponds to a time tp,n so that the argument in

Eq. (84) is fn

l tp,n5uVn1l,nutp,n5p. Here, resonant

means d850. The phase factors 2i and eif are physically irrelevant in the present context since they factor

out of the wave function; they are just kept for mathematical consistency. For these conditions the solution

matrices Tn

l take the simple forms

Tn

05S 0 2ieif

2ie2if 0 D (135)

for the carrier and

Tn

61

5S 0 eif

2e2if 0 D (136)

for the red (l521) and blue (l511) sidebands. For

creating pure number states, the phase factors of successive pulses will just add up to an overall phase factor

that is irrelevant, therefore one can set these phase factors equal to one for all these pulses without losing generality. When generating superpositions of number

states these phases will determine the relative phase of

the number-state components and cannot be disregarded. Starting from the state ug,0&, with a blue sideband pulse, T0

1

ug,0&5ue,1&, the first excited motional

state connected with the ue& internal state is created.

This may either be converted to 2iug,1&5T1

0

ue,1& or

walked higher up the number-state ladder with a red

sideband pulse T1

21

ue,1&5ug,2&. In this manner one can

step through to high number states, as was done in an

experiment of the NIST group using Raman transitions

between two hyperfine levels of Be1 (Meekhof et al.,

1996). Once the number state is created, the signature of

the state can be found by driving transitions on the first

blue sideband as described in Sec. III.D. The rate of the

Rabi flopping, Vn,n11 in Eq. (70), depends on the value

of n of the number state occupied. According to Eq.

(95) the expected signal is proportional to 1

1cos(Vn,n11t). The experimentally observed signal also

turned out to be damped, probably due to a combination of uncontrolled noise sources (see below). This was

accounted for by introducing n-dependent exponential

damping with constants gn in the fits of the blue sideband excitation curves,

Pg

fit~t!5 1

2 @11cos~Vn,n11t!exp~2gnt!#. (137)

The observed flopping curves are approximately described by Eq. (137), as can be seen in Fig. 11(a) for an

initial ug,n50& number state created in this experiment.

The fit to Eq. (137) is drawn as a solid line and yielded

V0,1 /(2p)5188(2) kHz and g0511.9(4) s21.

In this experiment Pg(t) was recorded for a series of

number states ug,n& up to n516 and the Rabi-frequency

ratios Vn,n11 /V0,1 were extracted. They are plotted in

Fig. 11(b), showing very good agreement with the theoretical frequencies, which include the trap’s finite LambDicke parameter h50.202 [solid line in Fig. 11(b)].

The damping constant gn was also extracted and observed to increase with n [the data are consistent with

gn'g0(n11)0.7]. The source of the damping was beLeibfried et al.: Quantum dynamics of single trapped ions 307

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
lieved to be in part due to uncontrolled magnetic-field

fluctuations at the position of the ion and frequency and

intensity fluctuations of the two Raman light fields used

in the experiment. One example of such noise sources,

the magnetic fields due to currents switching at the

60-Hz line frequency and its harmonics, was successfully

suppressed by producing a compensation field of the

right amplitude and phase with extra coils close to the

trap.

Another source of dissipation was an unexpectedly

high heating rate out of the motional ground state in

these experiments. The observed heating rate was about

one quantum per millisecond, about three orders of

magnitude higher than expected from thermal electronic

noise (Wineland et al., 1998; Turchette, Kielpinski, et al.,

2000).

The n0.7 scaling of the damping constants with number

state was extracted from fits to the data in this experiment. Theoretical work has incorporated spontaneous

emission and heating to explain the observed scaling

(Plenio and Knight, 1996, 1997; Bonifacio et al., 2000; Di

Fidio and Vogel, 2000; Budini, de Matos Filho, and

Zagury, 2002).

The second experiment on number states was carried

out at the University of Innsbruck (Roos et al., 1999).

There a single 40Ca1 ion was cooled to the ground state

on the S1/2→D5/2 quadrupole transition (see also Sec.

IV.B), leaving the ion in the n50 number state 99.9% of

the time. The n51 number state was created by applying a p pulse on the blue sideband and then incoherently

repumping the excited electronic state ue& (in this experiment corresponding to the mf525/2 sublevel of the

D5/2 state) via the P3/2 level with light at 854 nm. Since

the repumping was done inside the Lamb-Dicke

regime,5 it did not change the motional state for most of

the experiments, thus leaving the ion in the ug&un51&

number state with high probability. Figure 12 shows the

Rabi flopping dynamics on the blue sideband for the two

prepared number states [(a) n50 and (b) n51]. The

Rabi frequency V0,1 is 21(1) kHz and the frequency ratio

is V1,2 /V0,1'1.43, close to &, with the asymptotic value

of Eq. (77) for the Lamb-Dicke parameter approaching

zero. In this experiment a contrast over 0.5 was maintained for about 20 periods. Since a heating rate of

about one quantum in 190 ms (1/70 ms) was measured

along the axial (radial) direction for the Innsbruck trap,

heating was not believed to play a leading role during

the '1-ms duration of individual experiments. The decay in contrast was mainly attributed to magnetic-field

fluctuations in the laboratory and intensity fluctuations

in the laser beams.

2. Creation of coherent states

Coherent states of motion can be produced from the

un50& state by a spatially uniform classical driving field

(Carruthers and Nieto, 1965), by a ‘‘moving standing

wave’’ (Wineland et al., 1992), by pairs of standing waves

(Cirac, Blatt, and Zoller, 1994), or by a nonadiabatic

shift of the trap center (Janszky and Yushin, 1986; Yi

and Zaidi, 1988; Heinzen and Wineland, 1990). In experiments of the NIST group, the first two approaches

were taken. For the classical drive, a sinusoidally varying

potential at the trap oscillation frequency was applied to

one of the trap compensation electrodes for a fixed time

of typically 10 ms with varying amplitudes. This gave rise

to an approximately spatially homogeneous force on the

ion of the form

F~t!5eE0 sin~vdrivet2w!, (138)

which can be associated with an interaction potential

HI52xˆ F~t!52A \

2mn $aˆ u*~t!1aˆ †u~t!%F~t!.

(139)

5

h 854'0.03 for the D5/2→P3/2 transition and h 393'0.09 for the

spontaneous decay, so the probability for two carrier transitions is (12h 854

2 )(12h 3932 )50.99.

FIG. 11. Number states in the NIST experiment (Meekhof

et al., 1996): (a) Rabi oscillations on the blue sideband for the

ground state in the trap (n50); (b) d, measured ratio of Rabi

frequencies for different number states; solid lines, theoretical

predictions for several Lamb-Dicke parameters.

FIG. 12. Number states in the Innsbruck experiment (Roos

et al., 1999): (a) Rabi oscillations on the blue sideband for the

ground state (n50); (b) Rabi oscillations as in (a) but now for

the number state un51&.

308 Leibfried et al.: Quantum dynamics of single trapped ions

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
Putting in the lowest-order approximation, Eq. (15),

u(t)'exp(int)$@11(qx/2)cos(vrft)#/(11qx/2)%, already

displays the general behavior. As was pointed out by

Carruthers and Nieto (1965), this interaction will change

the quantum state of the oscillator only if the drive is

resonant with its motion. In the lowest-order approximation this would mean a resonance at vdrive'n and

two weaker resonances at vdrive'n6vrf . To lowest order, the presence of micromotion will then just rescale

the coupling strength by a factor 1/(11qx/2) for a drive

frequency close to n. For vdrive5n there are two stationary terms in Eq. (139) and two rotating at 2n. If we

neglect the nonstationary terms (rotating-wave approximation) HI becomes time independent and can easily be

integrated, yielding the evolution operator

U~t!5exp@~Vdt!aˆ †2~Vd

*t!aˆ #5D~Vdt!, (140)

with

Vd5 1

11qx/2

2eE0

2\ A \2mn

eiw . (141)

The drive coherently displaces an initial motional state

uC& to

uC8&5Dˆ ~Vdt!uC&. (142)

The coherent displacement is proportional to the time

the drive is applied and the amplitude of the driving

field. Experimentally the drive was calibrated by applying it to the motional ground state with a certain voltage

amplitude and fitting the internal-state evolution on the

blue sideband Pg(t) to the expected signal of a coherent

state ua& (see below) with the magnitude uau

25 ̄n as the

only free parameter.

In the ‘‘moving standing-wave’’ approach, two laser

beams with a frequency difference of Dv and detuned

by about Ddrive'212 GHz from the S1/2→P1/2 transition

in Be1 were used. The detuning was much bigger than

the linewidth G of the P1/2 state, so this state was populated with only an extremely small probability,

Pp<4S V0,s

DdriveD

2

, (143)

where V0,s52z^sue2(r•E0)uP1/2&zÕ\ is the on-resonance

Rabi frequency of an ion in the hyperfine state us& and is

driven by one of the equally strong beams E1,2

5E0 cos(k1,2x2v1,2t1w1,2), in complete analogy to the

Appendix. At the position of the ion the interference of

the two off-resonant light fields resulted in a timevarying light-field intensity I5^(E11E2)2

& with a cross

term modulated at Dv5v12v2 (here the averaging

brackets ^•••& denote averaging over a time T with uv1

2v2u!1/T!v1,2). The other components are time independent and will not lead to any consequences for the

ion’s motion. The difference frequency part leads to a

spatially and time-varying intensity with a corresponding

ac Stark shift of level us& that can resonantly drive the

motion if Dv5v12v25n. The dipole force on the ion

is proportional to the spatial derivative of this ac Stark

shift,

Fdip5\

V0,s

2

Ddrive

Dk sin~Dkx1Dvt1Dw! (144)

for (Dv,V0,s)!Ddrive . If the oscillation amplitude of the

ion remains in a small enough range, Dk^x&!1, we can

approximate sin(Dk^x&1Dvt1Dw)'sin(Dvt1 ̃w) for a

suitable  ̃w. Then the dipole force is of the form discussed in the context of Eq. (139) and will result in a

coherent displacement of the motional wave function

that grows linearly with the coupling time. Note that the

range where this assumption is valid coincides with the

definition of the Lamb-Dicke regime, so this drive will

closely approximate a coherent drive while h2

 ̄n!1.

By an appropriate choice of the internal levels and

beam polarization, the dipole force can even create a

state-dependent mechanical drive that allows one to entangle the internal states of the ion with the motion. As

an example we shall discuss the level scheme used in

Be1 in the NIST experiment. Here the ug& state was

chosen to be the (F52,mF522) hyperfine substate and

ue& was (F51,mF521) of the 2S1/2 manifold. The light

fields were polarized s2 and detuned 212 GHz from

the 2P1/2 level. This resulted in a driving dipole force for

ue& with the (1,21) state coupling to the (2,22) of the

2P1/2 manifold, but there exists no (3,23) substate in

this manifold that could couple to ug&. This level therefore remains unaffected by the drive. The dependence of

the drive on the internal state was crucial for entangling

internal and motional states as described in Sec. VI.A.4.

As in the predictions of the Jaynes-Cummings model,

the internal-state evolution Pg(t) driven on the blue

sideband, Eq. (95), will undergo collapses and revivals

(von Foerster, 1975; Eberly et al., 1980), a purely quantum effect (Shore and Knight, 1993). Figure 13 shows an

example of Pg(t) after the creation of a coherent state

of motion. Similar behavior has been observed in the

context of cavity QED (Brune, Schmidt-Kaler, et al.,

1996). The inset of Fig. 13 shows the probabilities of the

FIG. 13. Rabi oscillations for a coherent state (Meekhof et al.,

1996). The data (dots) are displayed together with a fit to a

sum of number states having a coherent-state population distribution (line). The fitted value for the mean quantum number is  ̄n53.160.1. The inset shows the amplitudes of the

number-state components (bars) with a fit to a Poisson distribution, corresponding to  ̄n52.960.1 (line).

Leibfried et al.: Quantum dynamics of single trapped ions 309

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
number components, extracted by Fourier transformation of the signal. The probabilities of different motional

levels display the expected Poissonian distribution over

n (see Sec. II.C.2). In the experiment the observed revival for higher- ̄n coherent states was attenuated due to

progressively faster decay rates of the higher-n number

states (see Sec. VI.A.1); for states with  ̄n*7 no revival

was observed.

3. Creation of squeezed states

A ‘‘vacuum squeezed state’’ of motion can be created

by a parametric drive at 2n (Janszky and Yushin, 1986;

Yi and Zaidi, 1988; Heinzen and Wineland, 1990), by a

combination of standing- and traveling-wave laser fields

(Cirac, Blatt, and Zoller, 1994), or by a nonadiabatic

change in the trap spring constant (Janszky and Yushin,

1986; Yi and Zaidi, 1988; Heinzen and Wineland, 1990).

In the experiment of the NIST group the ion was cooled

to the ground state and then irradiated with two Raman

beams that differed in frequency by 2n, driving Raman

transitions between the even-n levels within the same

hyperfine state. In analogy to the coherent drive described in Sec. VI.A.2 the interaction can be thought of

as a parametric drive induced by an optical dipole force

modulated at 2n. The squeeze parameter bs (defined as

the factor by which the variance of the squeezed quadrature is decreased) grows exponentially with the driving

time.

Figure 14 shows Pg(t) for a squeezed state prepared

by the NIST group in this way. The data were fitted to a

vacuum squeezed-state distribution, allowing only bs to

vary. The fit of the data in Fig. 14 is consistent with a

squeezed state with a squeeze parameter bs540610

(corresponding to a noise level 16 dB below the zeropoint variance in the squeezed quadrature component).

For this squeezing parameter, the average motional

quantum number is  ̄n'7.1.

The probability distribution for a vacuum squeezed

state is restricted to the even states, as can be seen from

Eq. (53). This distribution is very flat; with bs540, 16%

of the population is in states above n5 20 (see Sec.

II.C.3). The NIST experiment was conducted with a

Lamb-Dicke parameter of 0.202. The Rabi-frequency

differences of these high-n levels were small [see Fig.

11(b)] and began to decrease with n after n520. The

levels could therefore no longer be well distinguished in

frequency, so it was no longer possible to extract the

level populations by Fourier transform of Pg(t).

4. ‘‘Schro ̈dinger-cat’’ states of motion

In Schro ̈dinger’s original thought experiment (Schro ̈-

dinger, 1935) he describes how one could, in principle,

transform a superposition inside an atom to a large-scale

superposition of a live and dead cat by coupling cat and

atom with the help of a ‘‘diabolical mechanism.’’ Subsequently the term ‘‘Schro ̈dinger’s cat’’ developed a more

general meaning, first referring to superposition states of

macroscopic systems (the poor cat would be in such a

state in the original paper) and later, especially in the

context of quantum optics, referring to a superposition

of two coherent states ua& and ub& (see Sec. II.C.2) with a

separation in phase space ua2bu much larger than the

variance of a coherent state. Such a state would allow

one to distinguish the positions of the two coherent contributions by a position measurement with very high reliability, and they are macroscopic in the sense that their

maximum spatial separation is much larger than the

single-component wave-packet extension.

In an experiment of the NIST group, an analogous

state for a single trapped ion was engineered (Monroe

et al., 1996). Out of an equal superposition of internal

states (ue& and ug&) a combined state of the motional

and internal degrees of freedom of the form ug&ua&

1ue&uaeif & was created in which the two coherent motional components had the same amplitude but different

phases. The motional components of the superposition

were indeed separated in space by distances much

greater than their respective wave-packet spread for

suitable phases f.

This situation is interesting from the point of view of

the quantum measurement problem associated with

wave-function collapse, which was historically debated

by Einstein and Bohr, among others. One practical approach toward resolving this controversy is the introduction of quantum decoherence, or the environmentally

induced reduction of quantum superpositions into statistical mixtures and classical behavior (Zurek, 1991, 2001).

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 ‘‘Schro ̈dinger-cat’’ states, except at extremely short-time scales. The creation of mesoscopic

Scho ̈dinger-cat states in controlled environments allows

one to do controlled studies of quantum decoherence

and the quantum/classical boundary (Brune, Hagley,

et al., 1996; Myatt et al., 2000; Turchette, Myatt, et al.,

2000). Some of this work is described in Sec. VII.

The creation of Schro ̈dinger-cat states of a single ion

in the experiments cited above relied on a sequence of

laser pulses. The coherent states were excited with the

use of a pair of Raman laser beams. The key to the

experiment was that the displacement beams were both

polarized s1, so that they did not affect the ue& internal

state (see Sec. VI.A.2). It is this selectivity that allowed

FIG. 14. Rabi oscillations for a squeezed state (Meekhof et al.,

1996). The data (dots) are displayed together with a fit to a

sum of number states having a squeezed-state population distribution with bs540610, which corresponds to  ̄n'7.1.

310 Leibfried et al.: Quantum dynamics of single trapped ions

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
the components of a superposition of internal states to

be associated to different motional states.

The evolving state of the system during the sequence

is summarized in Fig. 15. Following (a) laser cooling to

the ug&un50& state, the Schro ̈dinger-cat state was created by applying several sequential pulses of the Raman

beams: (b) A p/2 pulse on the carrier transition split the

wave function into an equal superposition of states

ug&u0& and ue&u0&. (c) The selective dipole force of the

coherent displacement beams excited the motion correlated with the ue& component to a state ua&. (d) A p pulse

on the carrier transition then swapped the internal states

of the superposition. (e) Next, the displacement beams

excited the motion correlated with the new ue& component to a second coherent state uaeif &. (f) A final p/2

pulse on the carrier combined the two coherent states.

The relative phases [f and the phases of steps (b), (d),

and (f)] of the steps above were controlled by phase

locking the rf sources that created the frequency splitting of the Raman or displacement beams, respectively.

The state created after step (e) is a superposition of

two independent coherent states, each correlated with

an internal state of the ion (for f5p),

uC&5 ua&ue&1u2a&ug&

& . (145)

In this state, the widely separated coherent states replace the classical notions of ‘‘dead’’ and ‘‘alive’’ in

Schro ̈dinger’s original thought experiment. The coherence of this mesoscopic superposition was verified by

recombining the coherent wave-packet components in

the final step (f). This resulted in different degrees of

interference of the two wave packets as the relative

phase f of the displacement forces [steps (c) and (e)]

was varied. The nature of the interference depended on

the phases of steps (b), (d), and (f) and was set to cause

destructive interference of the wave packets in the ug&

state. The interference was directly measured by detecting the probability Pg(f) that the ion was in the ug&

internal state for a given value of f. The signal for particular choices of the phases in (b), (d), and (f) is

Pg~f!5 1

2 @12Ce2a2(12cos f) cos~a2 sin f!#, (146)

where a is the magnitude of the coherent states and C

51 is the expected visibility of the fringes in the absence

of any mechanisms decreasing the contrast, such as, for

example, imperfect state preparation, motional heating,

or decoherence. The experiment was continuously

repeated—cooling, state preparation, detection—while

slowly sweeping the relative motional phase f of the

coherent states.

Figure 16 shows the measured Pg(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 (c) and (e) from above]. The unit

visibility (C.1) of the interference feature near f50

verifies that superposition states were produced instead

of statistical mixtures, and the feature clearly narrows as

a increases. The amplitude of the Schro ̈dinger-cat state

was extracted by fitting the interference data to the expected form of the interference fringe. The extracted

values of a agreed with an independent calibration of

the displacement forces. Coherent-state amplitudes as

high as a.2.97(6) were measured, corresponding to an

average of  ̄n.9 vibrational quanta in the state of motion. This indicates a maximum spatial separation of

4ax0583(3) nm, which was significantly larger than the

single wave-packet size characterized by x057.1(1) nm

as well as a typical atomic dimension (.0.1 nm). The

individual wave packets were thus clearly spatially sepaFIG. 15. Steps for creation of a Schro ̈dinger-cat state (Monroe

et al., 1996). For detailed explanation, see text.

FIG. 16. Phase signal of a Schro ̈dinger-cat state for different

magnitudes of the coherent displacement (Monroe et al.,

1996).

Leibfried et al.: Quantum dynamics of single trapped ions 311

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
rated and also 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 accelerates. In the experiment, decoherence due to coupling to a thermal reservoir is expected to result in the loss of visibility in the

interference pattern of C5e2a2lt

, where l is the coupling constant and t the coupling time. This exponential

reduction of coherence with the square of the separation

(a2 term) is thought to be the basic reason that bigger

‘‘cats’’ decay faster (Zurek, 1991). In Fig. 16(d), the observed loss of contrast at the largest observed separation

already indicates the onset of decoherence.

The precise control of quantum wave packets in the

ion-trap version of Schro ̈dinger’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

(Ekert and Josza, 1996) and quantum cryptography

(Bennett, 1996). It was employed in a series of experiments that are described in Sec. VII.

5. Arbitrary states of motion

The methods presented above for creating a

Schro ̈dinger-cat state can be generalized to arbitrary

states of motion. The general strategy for creating such

states was first described by Law and Eberly (1996) in

the context of cavity QED and later generalized to superpositions of internal states and arbitrary motional

states of a trapped ion by Gardiner et al. (1997) and

Kneer and Law (1998). No experiments to realize these

proposals have been reported so far.

B. Full determination of the quantum state of motion

The controlled interaction of light and rf electromagnetic fields with trapped ions allows one not only to prepare very general states of motion, but also to completely determine these quantum-mechanical states

using novel techniques. The ability to prepare a variety

of nonclassical input states in ion traps that can, for example, exhibit negative values of the Wigner function

makes state reconstruction in ion traps an attractive

goal. In addition, comparing the results of the state determination with the intended state allows one to quanitfy the accuracy of preparation and detrimental effects

such as, for example, decoherence and motional heating.

The first proposal for motional-state reconstruction in

ion traps was put forward by Wallentowitz and Vogel

(1995). Their method encodes the information on the

motional state in the ground-state occupation of an

atom driven on the first red and blue sidebands simultaneously. A Fourier integral transform of the measured

data then yields the motional density matrix in a generalized position representation. Poyatos, Walser, et al.

(1996) proposed to measure either the Husimi Q function or the quadrature distribution function P(x,u) by

applying a suitable sequence of phase shifts, coherent

displacements, and squeezing operations to the motional

state and subsequently measuring the occupation of the

motional ground state with a method involving adiabatic

passage (Cirac, Blatt, and Zoller, 1994). D’Helon and

Milburn (1996) proposed to tomographically reconstruct

the Wigner function of the vibrational state by measuring the electronic state population inversion after applying a standing-wave pulse to the ion that must reside at

a node of the standing wave. Bardroff et al. (1996) proposed using a technique they called quantum-state endoscopy, exploiting the explicit time dependence of the

rf trapping potential. The technique relies on driving the

ion on combined secular and micromotion sidebands

and monitoring the internal-state evolution. Leibfried

et al. (1996) proposed and experimentally realized two

techniques that reconstruct either the density matrix in

the number-state basis or the Wigner function with the

help of coherent displacements and monitoring of the

internal-state evolution on the blue sideband, similar to

the number-state population measurements described

above. These techniques and the experimental results

will be discussed in more detail below. Lutterbach and

Davidovich (1997) proposed a very elegant method to

directly measure the Wigner function from the inversion

of the electronic states after a coherent displacement of

the original motional state and a driving pulse of appropriate length on the blue sideband. Freyberger (1997)

describes a scheme for measuring the motional density

matrix in the number-state basis, which, like the proposal of Poyatos, Walser, et al. (1996), relies on coherent

displacement and a filtering measurement to determine

the population in the motional ground state. Finally,

Bardroff et al. (1999) proposed a scheme to measure the

characteristic function of the motional state by applying

internal-state changing and coherent driving light pulses,

very similar to the procedure for preparing a Schro ̈-

dinger cat described in Sec. VI.A.4.

The two reconstruction methods described by Leibfried et al. (1996) were specially developed to fit the

technical possibilities and constraints of the NIST apparatus and could therefore be experimentally realized.

We shall restrict the more detailed discussion to these

methods. The first method reconstructs the density matrix in the number-state basis, while the second yields

the Wigner function of the motional state of a single

trapped atom.

Both measurement techniques rely on the ability to

displace the input state to several different locations in

phase space. Specifically, a coherent displacement (see

Sec. VI.A.2) of the form U(2a)5U†(a)5exp(a*a

2aa†

) (2a simplifies the notation below) controlled in

phase and amplitude is used in those schemes. In a manner similar to the characterization of motional-state

populations described in the sections above, radiation

on the blue sideband is then applied to the atom to extract the signal Pg(t,a) (notice that this signal will vary

for different displacements). The internal state at t50 is

always prepared to be ug& for the various motional input

states, so, according to Eq. (95), modified for the rates of

loss of contrast for different number states (see above),

the signal averaged over many measurements is

312 Leibfried et al.: Quantum dynamics of single trapped ions

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
Pg~t,a!5 1

2 H 11 (

k50

`

Qk~a!cos~Vk,k11t!e2g kt

J ,

(147)

where Qk(a) denotes the population of the coherently

displaced number state U(a)uk&. For no coherent displacement the signal, Eq. (95), is recovered and contains

information about the populations of the number states

only. But since these measurements are repeated for several magnitudes and phases of the coherent displacement, information can be extracted about the coherence

properties of the state, especially the off-diagonal elements of the density matrix—or the Wigner function can

be reconstructed from the measured displaced populations Qk(a).

1. Reconstruction of the number-state density matrix

As outlined above, the readily measurable quantities

in an experiment are the number-state populations

Qk(a) of the coherently displaced unknown initial state

characterized by the density matrix r. The quantities

Qk(a) are related to r by

Qk~a!5^kuU†

~a!rU~a!uk&5^a,kurua,k&, (148)

where ua,k& is a coherently displaced number state

(Moya-Cessa and Knight 1993). Hence every Qk(a) is

the population of the displaced number state ua,k& for

an ensemble characterized by the input density matrix r.

Rewriting Eq. (148) in terms of number-state densitymatrix elements r n,m yields

Qk~a!5 1

k!

^0uakU†

~a!rU~a!~a†!

ku0&

5 1

k!

^au~a2a!

kr~a†2a*!kua&

5 e2uau

2

uau

2k

k! (n,m50

`

(

j,j850

k ~a*!

n2j

am2j8

n!m!

3~21!

2j2j8

S k

j D

3S k

j8

D A~m1j!!~n1j8!!r n1j8,m1j . (149)

To separate the contributions of different matrix elements r n,m , one can choose the coherent displacements

to lie on a circle in phase space,

ap5uauexp@i~p/N!p#, (150)

where pP$2N,...,N21%. The number of angles 2N

on that circle determines the maximum number of states

nmax5N21 that can be included in the reconstruction.

With a full set of measured populations Qk(ap) of the

state displaced along 2N points on a circle, a discrete

Fourier transform of Eq. (149), evaluated at the values

ap , results in the following matrix equations:

Qk

(l)

[ 1

2N (

p52N

N21

Qk~ap!e2il(p/N)p

5 ( n5max(0,2l)

`

g kn

(l)

r n,n1l , (151)

with matrix elements

g kn

(l)

5e2uau

2

uau

2k

k! (

j850

min(k,n)

(

j50

min(k,l1n)

uau

2(n2j2j8)1l

3~21!

2j2j8

S k

j D S kj8

D A~l1n!!n!

~l1n2j!!~n2j8!! (152)

for every diagonal r n,n1l of the density matrix. To keep

the matrix dimension finite, a cutoff for the maximum n

in Eq. (151) is introduced, based on the magnitude of

the input state. For an unknown input state, an upper

bound on n could be extracted from the populations

Qk(a). If these are negligible for k higher than a certain

kmax and all displacements a, they are negligible in the

input state as well, and it is convenient to truncate Eq.

(151) at nmax5kmax . The resulting matrix equation is

overcomplete for some l, but the diagonals r n,n1l can

still be reconstructed by a general linear least-squares

method (Press et al., 1992).

2. Reconstruction of s-parametrized quasiprobability

distributions

As pointed out by several authors, all s-parametrized

quasiprobability distributions F(a,s) have a particularly

simple representation when expressed in populations of

displaced number states Qk(a) (Royer, 1984; MoyaCessa and Knight, 1993; Banaszek and Wodkiewicz,

1996; Wallentowitz and Vogel, 1996):

F~a,s!5 1

p (n50

`

@~s11!/2#

n

(

k50

n

~21!

k

S n

kD Qk~a!.

(153)

For s521 the sum breaks down to one term and

F(a,21)5Q0(a)/p gives the value of the Husimi Q

quasiprobability distribution at the complex coordinate

a. The reconstruction scheme of Poyatos, Walser, et al.

(1996) is based on the Husimi Q distribution. For s50

the Wigner function F(a,0)5W(a) for every point a in

the complex plane can be determined by the single sum

W~a!5 2

p (n50

`

~21!

nQn~a!. (154)

In the reconstruction performed by the NIST group, the

sum was carried out only to a finite nmax , as described

above. Since truncation of the sum leads to artifacts in

the quasiprobability distributions (Collett, 1996), the experimental data were averaged over different nmax . This

smoothes out the artifacts to a high degree.

In contrast to the density-matrix method described in

Sec. VI.B.1, summing the displaced probabilities with

their weighting factors provides a direct and local

Leibfried et al.: Quantum dynamics of single trapped ions 313

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
method of obtaining the quasiprobability distribution at

the point a in phase space, without the need to measure

at other values of a. This also distinguishes the method

from experiments that determines the Wigner function

by inversion of integral equations (tomography) (see,

for example, Smithey et al., 1993 and Dunn et al., 1995).

Finally, for a known density matrix the Wigner function can be derived by expanding Eq. (154) in the

number-state basis,

W~a!5 2

p (n50

`

~21!

n (

k,l50

`

^a,nuk&r kl^lua,n&, (155)

with the matrix elements given by (l>n) (Cahill and

Glauber, 1969)

^lua,n&5An!/l!al2ne21/2uau

2

Ln

(l2n)

~uau

2!, (156)

where Ln

(l2n) is a generalized Laguerre polynomial. Using this approach a plot of the Wigner function using

reconstructed density-matrix data may be created (Leibfried, Meekhof, et al., 1998).

3. Experimental state reconstruction

In the experiments of the NIST group (Leibfried

et al., 1996; Leibfried, Monroe, and Pfau, 1998), the coherent displacement needed for the reconstruction mapping was provided by a spatially uniform classical driving field at the trap oscillation frequency n (see Sec.

VI.A.2). The field was applied on one of the trap compensation electrodes for a time of about 10 ms. The rf

oscillators that created and displaced the states were

phase locked to control their relative phase. Different

displacements were realized by varying the amplitude

and the phase of the displacement oscillator. For every

displacement a, Pg(t,a) was recorded. Qn(a) was then

extracted from the measured traces by a numerical

singular-valued decomposition method (Press et al.,

1992). To determine the amplitude uau of each displacement, the same driving field was applied to the un50&

ground state and the resulting collapse and revival trace

(see Sec. VI.A.2) were fitted to that of a coherent state

with the amplitude a as the only variable.

The accuracy of the reconstruction was limited by the

uncertainty in the applied displacements, the errors in

the determination of the displaced populations, motional heating, and decoherence during the measurement. The value of the Wigner function was found by

the sum, Eq. (154), with simple error propagation rules.

The density matrix was reconstructed by a linear leastsquares method, and the error was estimated with the

help of a covariance matrix (Press et al., 1992). As the

size of the input state increases, state preparation and

the relative accuracy of the displacements become more

critical, thereby increasing their uncertainties.

Surface and contour plots of the Wigner function of

an approximate un51& number state are shown in Fig.

17. The plotted surface is the result of fitting a linear

interpolation between the actual data points to a 0.1

30.1 grid. The octagonal shape is an artifact of the eight

measured phases per radius. The white contour represents W(a)50. The negative values around the origin

highlight the nonclassical character of this state. The

measured Wigner function W(a) was rotationally symmetric within experimental errors. The minimum of the

large negative part of the Wigner function around the

origin has a value of 20.25, close to the largest negative

value possible for a Wigner function in the chosen

phase-space coordinates of 21/p. This highlights the

fact that the prepared input state is very nonclassical.

In contrast to the number state, the state closest to a

classical state of motion in a harmonic oscillator is a

coherent state. The reconstruction of the number-state

density matrix of a small coherent state with amplitude

ubu'0.67 is depicted in Fig. 18. The reconstructed offdiagonal elements are generally slightly smaller than

those expected from the theory of a pure coherent state.

In part this is attributed to decoherence during the measurement, so the reconstruction shows a mixed-state

character rather than a pure coherent-state signature.

The reconstructed Wigner function of a second coherent state with amplitude ubu'1.5 is shown in Fig. 19. The

FIG. 17. Surface and contour plots of the reconstructed

Wigner function W(a) of an approximate n51 number state

(Leibfried et al., 1996). The negative values of W(a) around

the origin highlight the nonclassical nature of this state.

FIG. 18. Experimental amplitudes r nm and phases Qnm of the

number-state density-matrix elements of an uau'0.67 coherent

state (Leibfried et al., 1996).

314 Leibfried et al.: Quantum dynamics of single trapped ions

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
plotted surface is the result of fitting a linear interpolation between the actual data points to a 0.1330.13 grid.

The approximately Gaussian minimum-uncertainty

wave packet is centered 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 minimumuncertainty half width of Aln(2)/2'0.59 in the chosen

phase-space coordinates. To suppress truncation artifacts in the Wigner function summation (154), the data

were averaged over nmax55 and nmax56.

The NIST group also created and reconstructed a coherent superposition of un50& and un52& number

states. This state is ideally suited to demonstrate the sensitivity of the reconstruction to coherences. The only

nonzero off-diagonal elements should be r02 and r20 ,

with a magnitude of ur02u5ur20u5Ar00r22'0.5 for a superposition with about equal probability of being measured in the un50& or un52& state. In the reconstruction

shown in Fig. 20 the populations r00 and r22 are somewhat smaller, due to imperfections in the preparation,

but the coherence has the expected value of ur20u5ur02u

'Ar00r22.

Finally, a thermal distribution was generated by only

Doppler cooling the ion. The reconstruction of the resulting state is depicted in Fig. 21. As expected, there are

no coherences, and the diagonal, which gives the

number-state occupation, shows an exponential behavior within the experimental errors, indicating a mean occupation number  ̄n'1.3.

VII. QUANTUM DECOHERENCE IN THE MOTION OF A

SINGLE ATOM

Decoherence in quantum systems (for an overview

see Zurek, 2001, and references therein) has been a subject of enduring interest because it relates to the fundamental distinction between quantum and classical behavior. More recently, quantum decoherence has come

to the forefront because it is the most significant obstacle for applications in quantum information science

(DiVincenzo, 2001). We shall review the decoherence

studies of motional quantum states of a single harmonically bound atom. Since this problem is closely related

to decoherence of superposition states of a single mode

FIG. 20. Reconstructed density-matrix amplitudes of an approximate (1/&)(u0&2iu2&) number-state superposition. The

amplitudes of the coherences indicate that the reconstructed

density matrix is close to that of a pure state (Leibfried et al.,

1996).

FIG. 19. Surface and contour plots of the reconstructed

Wigner function W(a) of an approximate coherent state

(Leibfried et al., 1996). The approximately Gaussian

minimum-uncertainty wave packet is centered 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 minimumuncertainty half width of A(1/2)ln(2)'0.59.

FIG. 21. Reconstructed density-matrix amplitudes of a thermal distribution of states after imperfect Doppler cooling ( ̄n

'1.3). As one would expect for a thermal distribution, no

coherences are present within experimental uncertainties and

the populations drop exponentially with n (Leibfried et al.,

1996).

Leibfried et al.: Quantum dynamics of single trapped ions 315

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
of the electromagnetic field, a large body of theoretical

research in quantum optics can be applied.6

As was briefly discussed in Sec. VI.A.5, arbitrary

states of motion of a single atom can be prepared (in

principle) using a variety of techniques. Following the

discussion in Sec. VI.A.4, the motional and internal

states can be manipulated with coherent radiation, generating operations of the form

ug&un&→cos uug&un&1eif sin uue&un8&. (157)

This operation faithfully maps the motional state under

study to the internal state, which can subsequently be

measured with high accuracy. Before this mapping/

measurement step, superpositions of motional states can

decay naturally due to ambient fluctuating fields. Applying larger fields that dominate over the ambient level

improves experimental control and increases the data

rate. More importantly, applying different types of decohering fields before mapping/measurement allows the

study of a more general class of reservoir couplings.

A. Decoherence background

Consider the decoherence of a superposition state

c1uc1&1c2uc2&, where uc1& and uc2& are states of a single

mode of motion. This mode of motion will be taken as

the quantum ‘‘system’’ under investigation. In a basic

model of decoherence (Zurek, 1991, 2001), it is assumed

that the mode couples to an external environment having initial state ufe&. The total initial state of the motion

and environment is thus c05(c1uc1&1c2uc2&) ^ ufe&.

Due to the coupling between the system and the environment, the joint state evolves to

c05~c1uc1&1c2uc2&) ^ ufe&→cfinal

5c1uc1

8&ufe1&1c2uc28&ufe2&. (158)

To illustrate how this causes decoherence, we assume

the coupling is such that uc1

8&5eiz 1uc1& and uc28&

5eiz 2uc2&; then, after the interaction, cfinal

5c1uc1&ufe1&1ei(z 22z 1)c2uc2&ufe2&, with the initial

states of the system correlated with different states of

the environment. For strong coupling, the environmental states will typically be distinct, with ^fe2ufe1&.0.

Now if the final environmental states are uncontrolled

and unmeasured, these degrees of freedom must be

ignored or mathematically traced over. In this case,

the pure state cfinal becomes a statistical mixture

expressed by the density matrix r final5uc1u

2

uc1&^c1u

1uc2u

2

uc2&^c2u. Hence the off-diagonal or coherence

terms of the pure-state density matrix (c1uc1&

1c2uc2&)(c1

*^c1u1c2*^c2u) are lost to the environment.

It is sometimes useful to incorporate a quantum measuring device or quantum ‘‘meter’’ into the scheme

above using a von Neumann chain (Zurek, 1991). Here

it is assumed that the quantum system, initially given by

c1uc1&1c2uc2&, is first coupled to a quantum meter and

the combination is then coupled to the environment. In

the first stage of coupling,

~c1uc1&1c2uc2&) ^ uc M& ^ ufe&

→~c1uc1

8&uc M1&1c2uc28&uc M2&) ^ ufe&. (159)

Upon coupling to the environment, the evolution is

~c1uc1

8&uc M1&1c2uc28&uc M2&) ^ ufe&

→c1uc1

9&uc M1 8 &ufe1&1c2uc29&uc M2 8 &ufe2&, (160)

and, as before, if ^fe2ufe1&.0 and the final environmental states are not measured, then the final state of the

system and meter is expressed by the density matrix

uc1u

2

uc1

9&^c19u ^ uc M1 8 &^c M1 8 u1uc2u

2

uc2

9&^c29u ^ uc M2 8 &^c M2 8 u.

(161)

Thus the correlation between the system and meter

states is established, yielding the expected classical result, but the quantum coherence is lost. Including the

quantum meter more closely describes the ion experiments discussed below because the system (the ion’s motion) is not directly measured but is instead coupled to

the ion’s internal state, which acts as the meter.

B. Decoherence reservoirs

Below, several simple decoherence reservoirs are discussed, and experimental demonstrations of decoherence in these reservoirs are summarized. Decoherence

of harmonic-oscillator superposition states into a variety

of reservoir environments has been investigated extensively in theory; see, for example, Caldeira and Leggett

(1985), Walls and Milburn (1985, 1995), Collett (1988),

Zurek (1991), Buzˇek and Knight (1995), and Poyatos,

Cirac, and Zoller (1996). Typically a system harmonic

oscillator is coupled to a bath of environmental quantum

oscillators, with particular couplings determined by the

type of reservoir.

1. High-temperature amplitude reservoir

The interaction Hamiltonian for amplitude damping is

HI5~\/2!(

i50

`

Giabi

†

1H.c., (162)

where a is the lowering operator for the system oscillator, bi

† is the raising operator for the ith environmental

oscillator, and Gi gives the strength of coupling between

the system oscillator and the ith environmental oscillator. The case in which the system oscillator is a single

mode of the electromagnetic field has been investigated

in the experiments of Brune, Hagley, et al. (1996). This

model and similar ones specific to trapped-ion experiments are discussed theoretically (Murao and Knight,

6

See, for example, Caldeira and Leggett, 1985; Walls and Milburn, 1985; Collett, 1988; Zurek, 1991; Vogel and Welsch, 1994;

Buzˇek and Knight, 1995; Poyatos, Cirac, and Zoller, 1996;

Schleich, 2001.

316 Leibfried et al.: Quantum dynamics of single trapped ions

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
1998; Schneider and Milburn, 1998, 1999; Wineland

et al., 1998; Bonifacio et al., 2000). One model assumes

that the motion of a trapped ion couples to the

(noisy) uniform electric field E caused by the environmental oscillators through the potential U

52Zueux"E, where Zueu is the ion’s charge and x its

displacement from the equilibrium position. Such a

model corresponds to the case of a noisy electric field

due to a resistor coupled between the trap electrodes

(Wineland and Dehmelt, 1975b; Wineland et al., 1998)

and is described by the Hamiltonian in Eq. (162).

In the experiments, a hot resistor is simulated by applying a random uniform electric field between the electrodes that has some spectral amplitude at the ion’s motional frequency n (Myatt et al., 2000; Turchette, Myatt,

et al., 2000). This is achieved using a commercial function generator producing pseudorandom voltages that is

connected through a bandpass filter to one of the trap

electrodes. The initial coherence and subsequent decoherence of the motional-state superpositions were measured using single-atom interferometry, as described in

detail by Myatt et al. (2000) and Turchette, Myatt, et al.

(2000).

Briefly, an initial state of the type shown in the lefthand side of Eq. (160) was first created, with uc1

8&

5ua&, uc M1&5ue&, uc2

8&5ua8&, uc M2&5ug&, and c15c2

51/&. This is illustrated schematically in part (2) of Fig.

22. Here ua& and ua8& are coherent states of amplitudes a

and a8.

After preparation, the superposition was exposed to

the fluctuating field for a fixed time [Fig. 22, part (3)].

The components of the motional-state superposition

were then recombined by reversing the steps that created it [Fig. 22, part (4)] and a final p/2 pulse was applied to the internal states. Finally, the internal state was

measured. This sequence was repeated many times for

various values of the relative phase f R between the creation and reversal steps. The contrast of the resulting

interference fringe characterizes the amount of coherence remaining in the final state. The results are displayed in Fig. 23. The contrast of the fringe decays as

exp(2kua2a8u

2

^En

2

&), where A^En

2

& is the root-meansquare value of the applied noise and k is a constant.

Similar scaling was also observed for the ambient fluctuating fields after exposure for varying times (Myatt

et al., 2000; Turchette, Myatt, et al., 2000). The exponential dependence of the decoherence rate on the size of

the cat state Da[ua2a8u agrees with theoretical predictions and indicates why it is difficult to preserve superpositions on a macroscopic scale.

The experimental situation here is slightly different

from the theoretical models outlined above. First, the

environment acted only on the system (motional state)

and not the meter (internal state), so that uc M1 8 &

5uc M1& and uc M2 8 &5uc M2& in Eq. (160). In addition, in

this case the action of the system on the environment is

nearly negligible, so that ufe1&'ufe2&'ufe&. Because

the noise voltage En in this experiment could, in principle, be measured classically without disturbing ufe& appreciably one could subsequently apply an operation to

the ion that reverses the environmental effect. In practice this could be readily done for the engineered noise

FIG. 22. Phase-space representation of interference experiment (Ramsey, 1963) for coherent-state superpositions

coupled to an amplitude reservoir (from Myatt et al., 2000 and

Turchette, Myatt, et al., 2000). Starting from the initial system/

meter state ua50&ug&, the Schro ̈dinger-cat state shown in

panel (2) is created by applying a p/2 pulse on the internal

state followed by a state-dependent optical dipole force. Noise

is then applied to the motion simulating coupling to a hot resistor. This causes a random displacement b as shown in panel

(3). The steps used to create the cat state are then reversed,

but with a phase shift f R on the final p/2 pulse. Finally, the

probability Pg of finding the ion in the ug& internal state is

measured. Ideally, this results in Pg5 1

2 $12cos@fR

12 Im(b*Da)#%, having the sinusoidal oscillations with f R

characteristic of Ramsey interferometry. Noise reduces the

Ramsey fringe contrast because b must be averaged over a

distribution of values. For Gaussian noise with variance sb ,

this yields a contrast of exp(24sb

2

uDau

2

).

FIG. 23. Loss of coherence of a Schro ̈dinger-cat state caused

by coupling to an amplitude reservoir (from Myatt et al., 2000).

Amplitude noise was applied for a fixed duration of 3 ms but

with varying amplitude. The fringe contrast for all experiments

is normalized to that observed in the absence of applied noise.

The scaling shows that the decoherence rate is proportional to

the square of the phase-space separation Da of the superposition components.

Leibfried et al.: Quantum dynamics of single trapped ions 317

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
but not for the ambient noise. Without performing such

a measurement, however, it is necessary to average over

the possible environmental states $ufe&%, and this leads

to the decoherence exhibited in Fig. 23.

2. Zero-temperature amplitude reservoir

In the cavity QED decoherence experiments of

Brune, Hagley, et al. (1996), the system oscillator frequency was around 50 GHz and the ambient temperature was T'0.6 K, so the quantum number of the equilibrium oscillator was around 0.05 and the reservoir

temperature was effectively near zero. In the ion experiments, the oscillator frequency was around 5 MHz and

the ambient temperature was 300 K or greater, implying

a very large equilibrium oscillator number (Turchette,

Kielpinski, et al., 2000). Nevertheless, it is possible to

simulate a T50 reservoir for the ions, following a suggestion by Poyatos, Cirac, and Zoller (1996). The technique relies on laser cooling, as illustrated in Fig. 24.

Coherent Raman beams drive the ug&un&↔ue&un21&

transition with Rabi rate V. At the same time, an optical

pumping beam causes spontaneous Raman transitions

from ue&un& to ug&un& at rate g. From the diagram in Fig.

24, it is clear that all populations tend towards the state

ug&u0&, which is characteristic of a T50 reservoir. By

varying the intensities of the lasers, the reservoir parameters can be experimentally controlled.

In these experiments, the evolution of the initial state

c5 1/& (u0&1u2&)ug& was observed for varying lengths

of reservoir interaction time. The coherence was measured using an interference experiment similar to that of

the previous section. However, here the internal state

serves several functions: it allows the preparation of the

motional superposition and is the quantity ultimately

measured, but also, in the middle of any particular experiment, it acts as part of the environment due to the

coupling of Fig. 24. The resulting data are shown in Fig.

25, where each point is the contrast of the interference

fringes after interacting with the reservoir for the given

time. Two cases are shown, g.V and g,V. In the first

case, the coherence simply decays due to coupling to the

reservoir. Here it is natural to take the reservoir to be

the internal state plus the rest of the environment, which

includes the spontaneously scattered photons. The initial

nonexponential decay is a manifestation of the quantum

Zeno effect (Myatt et al., 2000; Turchette, Myatt, et al.,

2000) and arises because the condition g@V is not rigorously satisfied. In contrast, when g,V the coherence

between the u0& and u2& states disappears and reappears

over time, with an overall decay of the fringe contrast.

The underlying effect is population transfer back and

forth (Rabi oscillation) between the states ug&u2& and

ue&u1&. Indeed, for g→0, near-perfect revival of the

FIG. 24. Technique for implementing a zero-temperature environment (Myatt et al., 2000). A pair of laser beams drives the

ug&un&↔ue&un21& transition at rate V, while spontaneous Raman scattering is used to make transitions ue&un& to ug&un& at

rate g. When g@V, the internal states together with the spontaneous events act as a T50 reservoir for motional states.

FIG. 25. Decoherence of the motional superposition state 1/& (u0&1u2&) coupled to the

engineered zero-temperature reservoir (from

Myatt et al., 2000 and Turchette, Myatt, et al.,

2000). The superposition is prepared with a

sequence of laser pulses and the reservoir applied for a variable time. The preparation

pulses are then reversed with a variable phase

shift, and the final state is recorded as a function of this phase. The data show the contrast

of the resulting interference pattern for two

cases of the relative values of V and g. The

initial contrast is not unity due to imperfections in the state preparation and reversal

pulses.

318 Leibfried et al.: Quantum dynamics of single trapped ions

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
fringe contrast is obtained (Turchette, Myatt, et al.,

2000), since in this case the environment is restricted to

just the internal states and the apparent decoherence is

easily reversed. A scheme for observing a similar reversal in the context of cavity QED is discussed in Raimond

et al. (1997).

In the zero-temperature experiment with g.0, once

the atom scatters a photon through the spontaneous Raman process, no measurement can be made even in principle to restore the initial superposition. That is, the

emission (and subsequent absorption by a measuring apparatus or the environment) of a spontaneous photon

projects the atom into a definite state (ug&u1& in the context of Fig. 24) and phase information is irreversibly lost.

This situation is identical to that of the cavity experiments of Brune, Hagley, et al. (1996). The decoherence

in this second ion experiment can be contrasted with the

first, in which the environment could, in principle, be

measured and its effects reversed. The data for g,V do

illustrate how coherence lost to the environment can be

recovered, and an alternative explanation states that

transferring the ug&u2& component of the superposition

to the ue&u1& state provides ‘‘which-path’’ information in

the interferometer—the paths being the ug&u0& and

ug&u2& parts of the superposition. The oscillation in

which-path information is analogous to that seen in the

experiments of Chapman et al. (1995), Du ̈rr et al. (1998),

Bertet et al. (2001), and Kokorowski et al. (2001).

The zero-temperature experiment also illustrates a

fundamental dilemma in explaining decoherence. If the

environment is restricted to be the internal state by taking g→0, then the coherence information can be recovered. Even in the case of spontaneous emission, coherence need not be lost if the photon is emitted into a

high-quality cavity from which it can later be recovered,

as has been observed in the cavity QED experiments

(Maıˆtre et al., 1997; Varcoe et al., 2000). Therefore it

seems that decoherence is needed only to describe situations in which, for practical or technical reasons, information pertaining to the overall system is lost. These

limitations seem only to be practical and not fundamental unless some mechanism that is so far missing in quantum mechanics is found to cause intrinsic decoherence.

For a summary of such alternatives, see Leggett (1999).

3. High-temperature phase reservoir

The interaction Hamiltonian for a phase damping is

HI5~\/2!aa†

(

i50

`

Gibi

†

1H.c. (163)

This interaction does not change the energy of the system oscillator and can be considered a model for

quantum-nondemolition measurements (Walls and Milburn, 1985).

In the experiments, a phase reservoir is realized by

modulating the trap frequency, thus advancing (or retarding) the phase of harmonic motion in the trap.

Gaussian noise is symmetrically applied to the trap electrodes to produce a noisy electric-field gradient, and the

noise is uniform up to a cutoff frequency well below the

trap frequency so that no energy is transferred to the ion

motion.

Motional decoherence caused by this phase noise is

most clearly demonstrated in a superposition of Fock

states of the form c1un&1c2un8&. Similar techniques to

that described above were used to characterize phase

decoherence in a variety of Fock-state superpositions

(Myatt et al., 2000; Turchette, Myatt, et al., 2000), and

the results are plotted in Fig. 26. In analogy to the case

of amplitude damping, we find that the decoherence rate

scales with the square of the distance between the superposition’s constituents, here meaning the squared difference in Fock-state indices.

FIG. 26. Loss of coherence of various Fockstate superpositions caused by coupling to a

phase reservoir (from Myatt et al., 2000).

Phase noise was applied for a fixed duration

of 20 ms but with varying amplitude. The

fringe contrast for all experiments is normalized to that observed in the absence of applied noise. The scaling shows that the decoherence rate is proportional to the square of

the Fock-state number Dn of the superposition components.

Leibfried et al.: Quantum dynamics of single trapped ions 319

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
C. Ambient decoherence in ion traps

The ambient motional decoherence observed in all experiments can be characterized by high-temperature amplitude damping: its characteristics are the same as those

caused by thermal electronic noise in the resistance of

the electrodes or resistance coupled to the electrodes.

At the relatively low ion oscillation frequencies, where

the characteristic wavelength is much larger than the

electrode spacing, the ambient decoherence is adequately described by thermal (Johnson) noise associated with lumped circuit elements attached to the electrodes, or equivalently, thermally fluctuating dipole

oscillators in the electrode bulk (Turchette, Kielpinski,

et al., 2000).

Typical heating rates (expressed as quanta per second

from the motional ground state) are observed to be

around ^dn/dt&.1032104 s

21 for nz.10 MHz, and the

distance from the ion to the nearest electrode surface is

around 150 mm (Turchette, Kielpinski, et al., 2000).

However, given the estimated electrode resistance

(Wineland et al., 1998) and attached circuit elements, an

electrode temperature of 106 K or greater is needed to

explain most of the heating results (Turchette, Kielpinski, et al., 2000). The principle cause of the anomalously

large heating is not understood at this time, but some of

its characteristics have been determined. The fluctuating

field has no sharp spectral features in the range from 2

to 20 MHz, and it seems to be emanating from the electrodes themselves. More local-field sources such as collisions with background gas or free electrons are ruled

out because these collisional sources would heat the internal modes of two ions at nearly the same rate as the

center-of-mass modes. However, the internal mode heating is observed to be negligible compared to the heating

of the center-of-mass modes (King et al., 1998), indicating that the fields at the site of the ions are approximately uniform spatially. [We note that the experiments

of Rohde et al. (2001) disagree with this finding.] In contrast, a more distant field source, such as electrical noise

from laboratory equipment, is contradicted by observations that the heating rate depends sensitively on the

trap size R, scaling typically as R2k for k on the order of

5. Thermal noise arising from circuit resistance should

scale as k52 (Wineland et al., 1998), but noise from

fluctuating patch potentials on the electrode surfaces is

predicted to scale as k'4 (Turchette, Kielpinski, et al.,

2000). The scatter of the data for 9

Be1 ions is rather

large but is consistent with the patch potential model.

Data on other traps (Diedrich et al., 1989; Rohde et al.,

2001) also appear to be consistent with this value of k.

Previous experiments (Turchette, Kielpinski, et al.,

2000) have given some indication that beryllium deposition onto the electrodes causes a higher heating rate.

Such deposition occurs because these ions are created

by ionization neutral beryllium atoms that pass near the

center of the trap after being emitted from a wide-angle

source. In this process, some of the atoms from the

source miss the trap and are deposited on the electrodes.

Preliminary evidence indicates that, by physically masking the electrodes from direct deposition, one can

achieve a decrease in heating rate by a factor on the

order of 100 (Rowe et al., 2002). In any case, we conjecture that for clean metal electrode surfaces, free from

oxides or adsorbed gases that could support mobile electrons, the heating should approach that predicted by

thermal electronic noise ^dn/dt&.1 s21.

VIII. CONCLUSIONS

Single ions confined in rf traps offer two nearly ideal

basic quantum systems, a two-level system, represented

by two of the internal electronic states, and an approximate harmonic oscillator, represented by the motion.

With appropriate light fields these two subsystems can

be coupled in a number of ways, leading to a wealth of

possible studies. In this review the underlying theory

and a number of experiments utilizing this system have

been discussed, with special emphasis on laser cooling,

resonance fluorescence, quantum-state engineering,

quantum-state reconstruction, and motional decoherence of single ions.

This system will continue to be studied and, very

likely in the future, much more complicated superposition states will be realized. Although an important goal

of ion-trap experiments is to realize arbitrary entangled

states for many ions, as required in quantum computing,

for example, the single-ion experiments discussed here

will continue to be the test bed for studies of operation

fidelity and decoherence.

ACKNOWLEDGMENTS

We want to thank James Bergquist, John Bollinger,

Ignacio Cirac, Ju ̈rgen Eschner, Wayne Itano, Steven Jefferts, David Kielpinski, Brian King, Christopher Langer,

Dawn Meekhof, Volker Meyer, Giovanna Morigi, Christopher Myatt, Hanns-Christoph Na ̈gerl, Christian Roos,

Mary Rowe, Cass Sackett, Ferdinand Schmidt-Kaler,

Quentin Turchette, Christopher Wood, and Peter Zoller

for their cooperation and many stimulating discussions.

Work at the University of Innsbruck was supported by

the Austrian Fonds zur Fo ̈rderung der wissenschaftlichen Forschung (Grant No. SFB15 and START Grant

No. Y147-PHY), by the European Commission [TMR

networks ‘‘Quantum Information’’ (Grant No. ERBFRMX-CT96-0087) and ‘‘Quantum Structures’’ (Grant

No. ERB-FMRX-CT96-0077)], and by the Institut fu ̈r

Quanteninformation GmbH. Work at NIST was supported by NSA/ARDA, ONR, ARO, and NRO. Work

at Michigan was supported by NSA/ARDA, ARO, and

NSF.

APPENDIX: COUPLINGS OF LIGHT FIELDS TO THE

INTERNAL ELECTRONIC STATE

The coupling of electromagnetic fields to charges is a

complex subject. It has been extensively studied, for example, by Cohen-Tannoudji et al. (1989). Deriving the

relevant interaction Hamiltonians HI from first prin320 Leibfried et al.: Quantum dynamics of single trapped ions

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
ciples is beyond the scope of this review, so we shall

restrict ourselves to stating these Hamiltonians and listing their properties as far as they are necessary to describe the atom-field couplings studied here. In all cases

the electromagnetic field(s) will not be quantized, but

treated as a classical plane-wave field of the form

E~x,t!5E0@ei(kx2vt)

1c.c.#, (A1)

with the real field amplitude E0 . We shall assume (a)

that \v'Ee2Eg so all electronic states except ug& and

ue& can be neglected and (b) that all ac Stark shifts, represented by the diagonal elements ^juHIuj&, j5$g,e%, are

lumped into the definitions of Ej , namely, Ej5Ej0

1^juHIuj&, where Ej0 is the energy of level j in absence

of the coupling. We can then expand HI in the remaining off-diagonal terms,

HI5~ug&^eu1ue&^gu!^guHIue&, (A2)

where we have chosen a convention in which the matrix

element ^guHIue& is real.

In the remainder of this appendix we shall study these

matrix elements for the types of transitions used in the

described experiments. Doppler cooling and fluorescence experiments relied on dipole transitions. Experiments with resolved sidebands relied on the excitation

of quadrupole transitions or stimulated Raman transitions between two long-lived states.

1. Dipole coupling

For dipole coupling to a single outer-shell electron the

interaction Hamiltonian is

HD5e2x•E~x,t!, (A3)

with the electron charge e2 . For a plane-wave electric

field of the form of Eq. (A1) this becomes

HD5e2x•E0@ei(kx2vt)

1c.c.# (A4)

and the matrix element is

^guHDue&5e2^gu~E0•x!ue&@ei(kx2vt)

1c.c.#. (A5)

Comparison of this equation with Eq. (62) yields

~\/2!V5e2^gu~E0•x!ue& (A6)

for dipole transitions. Since the dipole interaction is an

odd function of position, only matrix elements between

states of opposite parity differ from zero. The actual numerical value of the matrix element depends on the angular momentum values of ug& and ue& and the field polarization. Details on this can be found, for example, in

James (1998).

2. Quadrupole coupling

For quadrupole coupling to a single outer-shell electron the interaction Hamiltonian is

HQ5(

i,j

e2qij

]

]xi

Ej , (A7)

with the quadrupole tensor

qij51

2 S xixj2 13 d ijx2

D . (A8)

For a plane-wave electric field like Eq. (A1) the Hamiltonian simplifies to

HQ51

2 e2kx~E0•x!@iei(kx2vt)

1c.c.# (A9)

and the matrix element is

^guHQue&5 1

2 e2k^gux~E0•x!ue&

3@ei(kx2vt1p/2)1c.c.#. (A10)

Comparison of this equation with Eq. (62) yields

~\/2!V5e2~k/2!^gux~E0•x!ue& (A11)

for quadrupole transitions, where the phase factor p/2 is

lumped into f in Eq. (62). Since the quadrupole interaction is an even function of position, only matrix elements between states of the same parity differ from

zero. Again the actual numerical value of the matrix element depends on the angular momentum values of ug&

and ue& and the field polarization [see, for example,

James (1998)]. To relate the order of magnitude of quadrupole transition matrix elements to those of the more

familiar dipole transitions, one can deduce from Eqs.

(A10) and (A5) that their approximate ratio is a0k

.1023 – 1024, where a0 is the Bohr radius, so quadrupole transitions have a much weaker decay and higher

saturation intensity when driven by a laser source.

3. Raman coupling

An alternative way to create an effective two-level

system is to couple two ground-state levels by twophoton stimulated Raman transitions (Heinzen and

Wineland, 1990; Monroe et al., 1995). The Raman transitions are induced by two light fields whose frequency

difference matches the separation of the two groundstate levels (plus the relatively small detunings to, for

example, a sideband). Each beam is close to resonance

with an allowed dipole transition to a short-lived excited

electronic state u3& but sufficiently detuned to make

population of that state negligible. While the coupling is

enhanced, the near-resonant excited state can be adiabatically eliminated in the theoretical treatment (Wineland et al., 1998), leaving an effective two-level coupling

between the two ground states. The coupling is formally

equivalent to a narrow single-photon transition if one

makes the following identifications:

v↔v12v2 ,

k↔Dk5k12k2 . (A12)

Here v1 ,k1 (v2 ,k2) are the frequency and wave vector

of the light fields coupling ue& (ug&) to u3&. If both fields

are detuned from resonance by DR , the coupling

strength is given by

Leibfried et al.: Quantum dynamics of single trapped ions 321

Rev. Mod. Phys., Vol. 75, No. 1, January 2003
~\/2!V52\

uVg3Ve3u

DR

eiDf , (A13)

where \Vg3 and \Ve3 are the dipole matrix elements of

ug& and ue& to u3& as discussed above and Df is the phase

difference of the two light fields. This phase difference

can be lumped into f in Eq. (62).

The experimental advantages of stimulated Raman

transitions lie in the fact that they combine strong optical electric-field gradients with good stability of the crucial frequency difference. The frequency difference usually is in the GHz range and can therefore be

synthesized by rf sources with very long (. hours) coherence time. The linewidth of the laser plays a negligible role as long as it is small compared to the detuning

DR . The coupling strength can be considerably higher

than that for narrow one-photon transitions to metastable levels (e.g., quadrupole transitions) for the same

laser power. However, the coupling can lead to ac Stark

shifts on the order of the Rabi frequency that have to be

controlled and accounted for or compensated by judicious choice of polarizations, beam intensities, and detunings. The fact that the effective k is the vector difference of the two light-field wave vectors allows for

variation of k by changing the relative angle of the two

propagation directions. As a consequence the LambDicke factor can be tuned from almost zero [copropagating beams, h5(uk1u2uk2u)x0] to h5(uk1u1uk2u)x0

(counterpropagating beams). The ability to make transitions motionally insensitive (h'0) can be very helpful

in certain experimental situations.

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