Nobel Lecture: Superposition, entanglement, and raising Schro ̈dinger’s cat*

David J. Wineland

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

Colorado 80305, USA

(published 12 July 2013)

DOI: 10.1103/RevModPhys.85.1103

CONTENTS

I. Introduction 1103

II. Some Early Steps Toward Quantum Control 1104

III. Controlling the Quantum Levels of Individual

Trapped Ions 1105

IV. Manipulating Ion Motion at the Quantum Level 1107

V. SchrO ̈ dinger’s Cat 1108

VI. Enter Quantum Information 1109

A. Quantum simulation 1110

B. Spectroscopy and quantum metrology 1110

VII. Summary 1111

Acknowledgments 1111

References 1111

I. INTRODUCTION

Experimental control of quantum systems has been pursued

widely since the invention of quantum mechanics. In the first

part of the 20th century, atomic physics helped provide a test

bed for quantum mechanics through studies of atoms’ internal

energy differences and their interaction with radiation. The

advent of spectrally pure, tunable radiation sources such as

microwave oscillators and lasers dramatically improved these

studies by enabling the coherent control of atoms’ internal

states to deterministically prepare superposition states, as, for

example, in the Ramsey method (Ramsey, 1990). More recently this control has been extended to the external (motional)

states of atoms. Laser cooling and other refrigeration techniques have provided the initial states for a number of interesting studies, such as Bose-Einstein condensation. Similarly,

control of the quantum states of artificial atoms in the context

of condensed-matter systems is achieved in many laboratories

throughout the world. To give proper recognition to all of these

works would be a daunting task; therefore, I will restrict these

notes to experiments on quantum control of internal and

external states of trapped atomic ions.

The precise manipulation of any system requires lownoise controls and isolation of the system from its environment. Of course the controls can be regarded as part of the

environment, so we mean that the system must be isolated

from the uncontrolled or noisy parts of the environment. A

simple example of quantum control comes from nuclear

magnetic resonance, where the spins of a macroscopic ensemble of protons in the state j #i (spin antiparallel to an

applied magnetic field) can be deterministically placed in a

superposition state j #i þ j "i (jj

2 þ jj2 1⁄4 1) by application of a resonant rf field for a specified duration.

Although the ensemble is macroscopic, in this example

each spin is independent of the others and behaves as an

individual quantum system.

But already in 1935, Erwin Schro ̈dinger (Schro ̈dinger,

1935) realized that, in principle, quantum mechanics should

apply to a macroscopic system in a more complex way, which

could then lead to bizarre consequences. In his specific

example, the system is composed of a single radioactive

particle and a cat placed together with a mechanism such

that if the particle decays, poison is released, which kills the

cat. Quantum mechanically we represent the quantum

states of the radioactive particle as undecayed 1⁄4 j "i or

decayed 1⁄4 j #i, and live and dead states of the cat as jLi

and jDi. If the system is initialized in the state represented

by the wave function j "ijLi, then after a duration equal to the

half life of the particle, quantum mechanics says the

system evolves to a superposition state where the cat is alive

and dead simultaneously, expressed by the superposition

wave function

 1⁄4 1

ffiffiffi

2

p 1⁄2j "ijLi þ j #ijDi: (1)

Schro ̈dinger dubbed this an entangled state because the state of

the particle is correlated with the state of the cat. That is, upon

measurement, if the particle is observed to be undecayed, one

can say with certainty that the cat is alive, and visa versa. But

before measurement, the particle and cat exist in both states.

This extrapolation of quantum mechanics from individual

quantum systems to the macroscopic world bothered

Schro ̈dinger (and a lot of other people). As one way out of

the dilemma, in 1952, Schro ̈dinger (Schro ̈dinger, 1952b) wrote

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

atom or (small) molecule. In thought experiments,

we sometimes assume that we do; this invariably

entails ridiculous consequences ... .’’

But of course these days, this argument doesn’t hold and

we can in fact experiment with individual or small numbers of

quantum systems, deterministically preparing superpositions

and entangled superpositions. Our control is best when we

deal with very small numbers of particles, which enables us to

realize many of the gedanken experiments that provided the

basis for discussions between Schro ̈dinger and the other

founders of quantum mechanics. And, we can also make

small analogs of Schro ̈dinger’s cat, which are by no means

*The 2012 Nobel Prize for Physics was shared by Serge

Haroche and David J. Wineland. These papers are the text of

the address given in conjunction with the award.

REVIEWS OF MODERN PHYSICS, VOLUME 85, JULY–SEPTEMBER 2013

0034-6861= 2013=85(3)=1103(12) 1103  2013 Nobel Foundation, Published by The American Physical Society
macroscopic but have the same basic attributes. So far, it

appears that our inability to make macroscopic ‘‘cats’’ is due

just to technical, not fundamental, limitations. Admittedly,

these technical limitations are formidable, but one can be

optimistic about increasing the size of these states as technology continues to improve.

This contribution is based on the lecture I gave at the Nobel

ceremonies in 2012. It is mostly a story about our group at the

National Institute of Standards and Technology (NIST) in

Boulder, Colorado, whose combined efforts were responsible

for some of the contributions to the field of trapped-ion

quantum control. It will be a somewhat personal tour, giving

my perspective of the development of the field, while trying

to acknowledge some of the important contributions of

others. For me, the story started when I was a graduate

student.

II. SOME EARLY STEPS TOWARD QUANTUM CONTROL

From 1965 to 1970, I was a graduate student in Norman

Ramsey’s group at Harvard. Norman, with his close colleague

Dan Kleppner and student Mark Goldenberg, had recently

invented and demonstrated the first hydrogen masers

(Goldenberg, Kleppner, and Ramsey, 1960; Kleppner,

Goldenberg, and Ramsey, 1962). As part of this program,

Norman wanted to make precise measurements of the hyperfine frequencies of all three isotopes of hydrogen, so I chose to

work on deuterium. The experiment was relatively straightforward, complicated a bit by the relatively long wavelength

(92 cm) of deuterium’s hyperfine transition relative to

that of hydrogen (21 cm) (Wineland and Ramsey, 1972).

Most importantly, this experiment taught me to pay close

attention to, and control as best as possible, all environmental

effects that would shift the measured transition frequency from

that found for an isolated atom. In addition to enjoying the

detective work involved in this, I also became hooked

on the aesthetics of long coherence times of superposition

states (1 s in the masers), and their importance in atomic

clocks. Norman received the 1989 Nobel Prize in physics

for his invention of the separated-fields method in spectroscopy and development of the hydrogen maser (Ramsey, 1990).

During my time as a graduate student, I also read about and

was intrigued by the experiments of Hans Dehmelt and

his colleagues Norval Fortson, Fouad Major, and Hans

Schuessler at the University of Washington. The trapping of

ions at high vacuum presented some nice advantages for

precision spectroscopy, including the elimination of the

first-order Doppler shifts and relatively small collision shifts.

The Washington group made high-resolution measurements

of the 3Heþ hyperfine transition, which has internal structure

analogous to hydrogen, by storing the ions in an rf (Paul) trap.

One challenge was that detection by optical pumping was

(and still is) not feasible because of the short wavelengths

required. Therefore, in a heroic set of experiments, state

preparation was accomplished through charge exchange

with a polarized Cs beam that passed through the ions.

Detection was accomplished through a charge-transfer process (3Heþ þ Cs ! 3He þ Csþ) that depended on the internal state of 3Heþ, followed by detection of the depleted 3Heþ

ion number by observing the ions’ induced currents in the trap

electrodes (Fortson, Major, and Dehmelt, 1966; Schuessler,

Fortson, and Dehmelt, 1969).

Although these experiments were what first attracted me to

ion trapping, my postdoctoral research with Dehmelt, starting

in the fall of 1970, was focused on experiments where

collections of electrons were confined in a Penning trap for

a precise measurement of the electron’s magnetic moment or

g factor. These experiments were started by Dehmelt’s graduate student, Fred Walls, who later became a colleague at the

National Bureau of Standards. After a while, it became clear

that systematic effects would be much better controlled if the

experiment could be performed on single electrons.

Therefore, the first task was to isolate a single trapped

electron. This was accomplished by first loading a small

number of electrons into the trap and driving their nearly

harmonic motion (60 MHz) along the magnetic field direction. This motion could be detected by observing the

currents induced in the electrodes (proportional to the number

of electrons). By adjusting the strength of the drive to a

critical level, occasionally one of the electrons would gain

enough energy to strike a trap electrode and be lost. Steps in

the induced current level could then be used to determine

when one electron was confined in the trap (Wineland,

Ekstrom, and Dehmelt, 1973). Subsequent experiments on

single electrons by Robert Van Dyck, Paul Schwinberg, and

Dehmelt were used to make precision measurements of the

electron’s g factor (Van Dyck, Schwinberg, and Dehmelt,

1977; Dehmelt, 1990). For this and the development of the

ion-trapping technique, Dehmelt and Wolfgang Paul shared

the Nobel Prize in 1989, along with Ramsey.

The modes of motion for a single charged particle in a

Penning trap include one circular mode about the trap axis

called the magnetron mode. For the electron g-factor experiments, it was desirable to locate the electron as close to the trap

axis as possible by reducing the amplitude of this mode. This

could be accomplished with a form of ‘‘sideband cooling’’

(Wineland and Dehmelt, 1975a, 1976) as demonstrated by Van

Dyck, Schwinberg, and Dehmelt (1978). Around this time, I

was also stimulated by the papers of Arthur Ashkin (Ashkin,

1970a, 1970b) on the possibilities of radiation pressure from

lasers affecting the motion of atoms. In analogy with the

electron sideband cooling, Dehmelt and I came up with a

scheme for cooling trapped-ion motion with laser beams

(Wineland and Dehmelt, 1975b) (see below). The cooling

could also be explained in terms of velocity-dependent radiation pressure as in a concurrent proposal by Ted Ha ̈nsch and

Art Schawlow (Ha ̈nsch and Schawlow, 1975). We didn’t anticipate all of the uses of laser cooling at the time, but it was

clear that it would be important for high-resolution spectroscopy of trapped ions. For example, the largest systematic

uncertainty in the 3Heþ experiment (Schuessler, Fortson,

and Dehmelt, 1969) was the uncertainty in the time dilation

shift, which would be reduced with cooling.

In the summer of 1975, I took a position in the Time and

Frequency Division of NIST (then NBS, the National Bureau

of Standards). My first task was to help make a measurement

of the cesium hyperfine frequency, the frequency reference

that defines the second. The apparatus, NBS-6, had been built

by David Glaze of the Division. It was a traditional atomic

beam apparatus but had a relatively long distance between

1104 David J. Wineland: Nobel Lecture: Superposition, entanglement, ...

Rev. Mod. Phys., Vol. 85, No. 3, July–September 2013
Ramsey zones of 3.75 m. With it, we realized a fractional

accuracy of 0:9  1013 (Wineland et al., 1976). At that time,

the Division was more service oriented, with very little basic

research. Fortunately my group leader, Helmut Hellwig, had a

progressive view of the Division’s future and was able to

obtain NBS support to initiate laser-cooling experiments.

That support, along with some seed money from the Office

of Naval Research (ONR), enabled us to start a project on

laser cooling in the fall of 1977. With Robert Drullinger (a

local laser expert) and Fred Walls, we chose to use 24Mgþ

because of its simple electronic structure and Penning traps,

because of our prior experience with them. This was a very

exciting time, being able to work on a project of our choosing,

and by the spring of 1978, we had obtained our first cooling

results (Wineland, Drullinger, and Walls, 1978). In our experiments we observed currents in the trap electrodes induced

by the ions’ thermal motion and hence had a direct measurement of the ions’ temperature. Meanwhile, Peter Toschek’s

group in Heidelberg (joined by Dehmelt, who was on sabbatical) was working toward the same goal, using Baþ ions

confined in an rf-Paul trap. They, with colleagues Werner

Neuhauser and Martin Hohenstatt, also observed the cooling

at about the same time (Neuhauser et al., 1978), through the

increased trapping lifetime of ions. In a near coincidence,

although there was no contact between the groups, the manuscripts were received by Physical Review Letters within one

day of each other (Peter Toschek’s group ‘‘won’’ by one

day!). The cooling observed in both experiments is typically

called Doppler cooling, where the oscillation frequency of the

ions’ motion is less than the linewidth of the cooling transition. Theoretical groups were becoming interested in the

cooling, and some of the earlier work is discussed in

Letokhov, Minogin, and Pavlik (1977), Kazantsev (1978),

and Stenholm (1986).

To us, the cooling of course provided a start toward

improving clocks and in 1985, working with John

Bollinger, John Prestage, and Wayne Itano, we demonstrated

the first clock that utilized laser cooling (Bollinger et al.,

1985). But as physicists, we were excited by just the cooling

process itself. So, in addition to clock applications, it would

eventually lead to reaching and controlling the lowest quantized levels of motion for a trapped particle (below).

III. CONTROLLING THE QUANTUM LEVELS OF

INDIVIDUAL TRAPPED IONS

One of the obvious next steps was to isolate single ions. In

addition to the aesthetic appeal of this, as for single electrons,

the systematic errors in spectroscopy would be smallest in

this case (Dehmelt, 1982). By observing steps in the ion laser

fluorescence, the Heidelberg group was able to isolate Baþ

single ions (Neuhauser et al., 1980). With Wayne Itano, we

subsequently used this fluorescence ‘‘steps’’ method to observe single 24Mgþ ions (Wineland and Itano, 1981). The

Heidelberg group also made photographs of a single ion, and

because of its relatively long fluorescence wavelength

(493 nm), with a magnifier, a single Baþ ion can be observed

with the human eye!

In NIST single-ion experiments we chose to focus on Hgþ

because for frequency-standard applications, 199Hgþ has a

relatively high ground-state hyperfine clock transition

frequency of 40.5 GHz (Major and Werth, 1973; Cutler,

Giffard, and McGuire, 1982; Prestage, Dick, and Maleki,

1991) and also a narrow 2S1=2-

2D5=2 optical transition

[ð

2D5=2Þ ’ 86 ms], which could potentially be used as an

optical frequency standard (Bender et al., 1976). Although

optical pumping of 199Hgþ could be achieved with radiation

from isotopically selected Hgþ fluorescence lamps (Major

and Werth, 1973; Cutler, Giffard, and McGuire, 1982;

Prestage, Dick, and Maleki, 1991), laser excitation was

made difficult because of the short (194 nm) wavelength

required. Jim Bergquist in our group, with colleagues

Hamid Hemmati and Wayne Itano, first developed the required source by sum-frequency mixing a doubled Arþ laser

at 515 nm with 792 nm from a dye laser in a potassium

pentaborate crystal (Hemmati, Bergquist, and Itano, 1983).

We used an rf trap with a simple ring-and-end-cap structure

shown in Fig. 1, similar to that used by the Heidelberg group.

By the mid-1980s ion trappers were able to directly

address one of Schro ̈dinger’s questions, which formed the

title for his publication ‘‘Are there quantum jumps?’’

(Schro ̈dinger, 1952a, 1952b). Three similar demonstrations

were made in 1986 (Bergquist et al., 1986; Nagourney,

Sandberg, and Dehmelt, 1986; Sauter et al., 1986; Blatt and

Zoller, 1988); for brevity, we describe the experiment of

Bergquist et al. Referring to Fig. 1, a nearly harmonic binding

potential called a pseudopotential (Paul, 1990) is formed by

applying an rf potential between the ring electrode and the

end-cap electrodes (held in common). The relevant optical

energy levels of a Hgþ ion are indicated in the upper left-hand

part of the figure. The 2S1=2 ! 2P1=2 electric-dipole transition

[ 1⁄4 194 nm, ð

2P1=2Þ ’ 2:9 ns] was used for Doppler laser

cooling. If continuously applied, a steady fluorescence from

the ion would be observed and could be used to produce

images of the ion. If 2S1=2 ! 2D5=2 resonance radiation were

single Hg 1 mm +

2S1/2

2D5/2

2P1/2

194 nm 

cooling 

and detection 

“optical clock” transition 

Hg+

endcap 

endcap 

ring 

counts 

t (s) 1 2 3 0 

70 

282 nm 

FIG. 1 (color). Schematic of the trap for single Hgþ ion studies. An

rf potential is applied between the ring electrode and end-cap electrodes (which are in common), forming an rf ‘‘pseudopotential’’ for

the ion. The relevant Hgþ energy levels are indicated, including the

narrow 2S1=2 ! 2D5=2 ‘‘optical clock’’ transition. The data in the

upper right-hand part of the figure show the number of 194 nm

fluorescence photons detected in 10 ms detection bins vs time when

both transitions are excited simultaneously (Bergquist et al., 1986).

Absence of detected counts indicates that the ion is in the 2D5=2 state.

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Rev. Mod. Phys., Vol. 85, No. 3, July–September 2013
applied simultaneously, one would expect the 194 nm fluorescence to decrease because of excitation to the 2D5=2 state.

A density-matrix description, valid for an ensemble of

atoms, would predict a reduced but steady fluorescence

rate. But what would be observed for a single ion?

(Cook and Kimble, 1985; Erber and Putterman, 1985;

Cohen-Tannoudji and Dalibard, 1986; Javanainen, 1986;

Kimble, Cook, and Wells, 1986; Pegg, Loudon, and Knight,

1986; Schenzle, DeVoe, and Brewer, 1986). In fact the ion’s

fluorescence does not steadily decrease, but switches between

the full value and no fluorescence, effectively indicating quantum jumps between the 2S1=2 and 2D5=2 states. For the data

shown in the upper right-hand corner of Fig. 1, the 194 nm

fluorescence photon counts registered by a photomultiplier

tube were accumulated in 10 ms time bins and plotted as a

function of elapsed time to show the jumps. In a more general

context, a measurement of the quantum system composed of

the 2S1=2 and 2D5=2 states can be made by applying the 194 nm

‘‘measurement’’ beam for 10 ms and observing the presence or

absence of fluorescence. The 2S1=2 ! 2P1=2 transition is sometimes called a ‘‘cycling transition’’ because when the 2S1=2

state is excited to the 2P1=2 state, the ion decays back to the

2S1=2 state, emitting a photon, and the excitation/decay process

is then repeated. Neglecting the occasional decays of the 2P1=2

state to the 2D3=2 state (Itano et al., 1987), this procedure

approximates an ideal measurement in quantum mechanics

because the detection of the state is nearly 100% efficient and

because the state of the Hgþ ion, either the 2S1=2 or 2D5=2 state,

remains in its original condition after the measurement.

Dehmelt dubbed this ‘‘electron shelving’’ detection

(Dehmelt, 1982), where in this example the ion is shelved to

the 2D5=2 state. Such measurements are also called quantum

nondemolition (QND) measurements (Braginsky and Khalili,

1996; Haroche and Raimond, 2006). The method of detection

by state-dependent fluorescence has now become rather ubiquitous in atomic physics.

To perform spectroscopy on the 2S1=2 ! 2D5=2 transition

( ’ 282 nm), radiation was first applied near the transition

frequency in the absence of the 194 nm beam; this avoids

perturbations of the energy levels from the 194 nm beam. The

282 nm beam was then switched off, followed by measurement of the ion’s state with the 194 nm beam. This process

was repeated many times, and by stepping the frequency of

the 282 nm beam, spectra like that shown in Fig. 2 are

obtained (Bergquist, Itano, and Wineland, 1987). To interpret

this spectrum, we must consider the motion of the ion. Along

any mode axis the motion is nearly harmonic, so in the frame

of the ion, the laser beam appears to be sinusoidally frequency modulated due to the first-order Doppler shift. Thus

the central feature or ‘‘carrier,’’ which corresponds to the

transition frequency, is surrounded by frequency-modulation

sidebands spaced by the motional frequency of the ion

(Dicke, 1953). An equivalent picture is that the ion can absorb

radiation while simultaneously gaining or losing one quantum of motion, which leads to absorption features spaced by

the frequency of motion around the carrier.

As in many atomic physics experiments, by using highly

coherent radiation, we can initialize an ion in an eigenstate and

deterministically prepare superpositions; e.g., j #i!j #iþj "i.

To extract the values of jj and jj, we detect as described

above. A single measurement indicates either the j #i orj "istate

with respective probabilities P 1⁄4 jj

2 and 1  jj2. Quantum

fluctuations or ‘‘projection noise’’ in the measurements are

characterized with a variance ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPð1  PÞ=M p , where M is the

number of measurements on identically prepared atoms (Itano

et al., 1993). Therefore, accurate measurements of P generally

require many repeated experiments. Similarly, Ramsey-type

experiments where the two pulses are separated in time can

measure the relative phase between  and . From these types

of measurements, many ion trap groups now routinely produce

and verify superposition states of single ions that have coherence times exceeding 1 s. [For ion ensembles, coherence times

exceeding 10 min have been demonstrated (Bollinger et al.,

1991; Fisk et al., 1995).]

The Hgþ clock project at NIST, led by Jim Bergquist, has

been a long but very successful story. First, an accurate clock

based on the 40.5 GHz hyperfine transition of a few 199Hgþ

ions confined in a linear Paul trap achieved systematic errors

of about 4  1014 (Berkeland et al., 1998). Although we felt

these errors could be substantially reduced, we also realized

that the future of high-performance clocks was in the optical

domain, so we focused on the 2S1=2 ! 2D5=2 optical clock

transition. For many years it had been appreciated that

higher frequency was advantageous in terms of measurement

FIG. 2 (color). Spectroscopy of the 2S1=2 ! 2D5=2 transition on a

single 198Hgþ ion. Referring to Fig. 1, for each measurement cycle,

the ion is prepared in 2S1=2  j #i state by allowing it to decay to that

level. Then, application of a 282 nm ‘‘probe’’ laser beam is alternated

with a 194 nm measurement beam. The j #i and 2D5=2  j "istates are

detected with nearly 100% efficiency by observing the presence or

absence of 194 nm scattered light. By stepping the frequency of the

probe beam and averaging over many measurements, we obtain the

spectrum shown where we plot the probability of the ion remaining in

the 2S1=2 state Pð

2S1=2Þ vs the 282 nm laser beam frequency. In a

quantum picture of the motion, the central feature or ‘‘carrier’’

denotes transitions of the form j #ijni ! j "ijni, where n denotes

the motional Fock state quantum number. ‘‘Red’’ and ‘‘blue’’ sidebands correspond to j #ijni ! j "ijn þ nitransitions with n 1⁄4 1

or þ1, respectively. The central feature or carrier is essentially

unshifted by photon recoil, since the recoil is absorbed by the entire

trap apparatus as in the Mo ̈ssbauer effect; see, e.g., Dicke (1953),

Lipkin (1973), and Wineland et al. (1998).

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Rev. Mod. Phys., Vol. 85, No. 3, July–September 2013
precision; basically the higher oscillation frequencies allows

one to divide a time interval into finer units. But two things

were needed: a laser with high enough spectral purity to take

advantage of narrow optical transitions, and a practical means

to count cycles of the ‘‘local oscillator,’’ in this case the laser

that would excite the clock transition. In our lab, Brent

Young, Bergquist, and colleagues were able to make a

cavity-stabilized laser at 563 nm, which was doubled to

produce the clock radiation. The 563 nm source had a linewidth of less than 0.2 Hz for an averaging time of 20 s (Young

et al., 1999). It is now understood that the linewidth was

limited by thermal fluctuations in the mirror surface, currently still the limit for the most stable lasers. The solution to

the second problem is by now well known. The relatively

rapid development of optical combs by Jan Hall (Hall, 2006),

Ted Ha ̈nsch (Ha ̈nsch, 2006), their colleagues, and other

researchers meant that it was now possible to effectively

count optical cycles. Including these developments, in

2006, Bergquist and colleagues demonstrated a 199Hgþ optical clock with a systematic uncertainty of 7:2  1017, the

first clock since the inception of atomic clocks that had

smaller systematic errors than a cesium clock (Oskay et al.,

2006).

IV. MANIPULATING ION MOTION AT THE QUANTUM

LEVEL

An interesting next step would beto control anion’s motion at

the quantum level. Since a cold trapped ion’s motion along any

mode axis is harmonic to a very good approximation, in a

quantum description (Neuhauser et al., 1978; Wineland and

Itano, 1979; Stenholm, 1986), we express its Hamiltonian in

the usual way as ħ!zaya with !z the oscillation frequency

(along the z axis here) and a and ay the lowering and raising

operators for the ion motion. The operator for the ion’s position

about its mean value is z 1⁄4 z0ða þ ayÞ, where z0 1⁄4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ħ=2m!z

p

is the spread of the ground-state wave function, with m the ion’s

mass. In principle, we could detect the ion’s motion through the

current it induces in the trap electrodes, as was done for electrons. In practice, however, a far more sensitive methodisto map

information about the motional states onto internal states of the

ion and read those out as described above. For this, we need to

efficiently couple an ion’s internal states to its motion. To see

how this works, consider a single trapped ion that has a singleelectron electric-dipole transition with resonance frequency !0.

If this transition is excited by a laser beam of frequency !L

propagating along the z axis, the interaction is given by

HI 1⁄4 er~  E^ 0 cosðkz  !Lt þ Þ

1⁄4 ħðþ þ Þðeiðkz!LtþÞ þ eiðkz!LtþÞ

Þ; (2)

where r~ is the electron coordinate relative to the ion’s core, e is

the electron charge, ^, E0, and k are, respectively, the laser

beam’s electric-field polarization, amplitude, and wave vector,

and  is the electric-field phase at the mean position of

the ion. The operators þð1⁄4 j "ih# jÞ and ð1⁄4 j #ih" jÞ are

the internal-state raising and lowering operators, and  

E0h" jr~  ^j #i=2ħ, with j #i and j "i denoting the ion’s ground

and optically excited states as above. If we transform to an

interaction picture for the ion’s internal states (þ ! þei!0t

)

and motion states (ay ! ayei!zt

) and assume !L ’ !0, then

neglecting terms that oscillate near 2!0 (rotating wave approximation), Eq. (2) becomes

HI ’ ħþei1⁄2kzð!L!0Þtþ þ H:c:

’ ħþei1⁄2ð!L!0Þt

1⁄21 þ iðaei!zt þ ayei!ztÞ

þ H:c: (3)

Here, H.c. stands for Hermitian conjugate and   kz0 1⁄4

2	z0= is the Lamb-Dicke parameter, which we assume here

to be much less than 1. For an ion of mass 40 u (e.g., 40Caþ) in a

well with !z=2	 1⁄4 3 MHz and  1⁄4 729 nm, we have z0 1⁄4

6:5 nmand 1⁄4 0:056. For!L 1⁄4 !0 and  !z, to a good

approximation we can neglect the nonresonant term in Eq. (3)

and obtain HI ’ ħeiSþ þ H:c: This is the Hamiltonian for

carrier transitions or, equivalently, spin-vector rotations about

an axis in the x-y plane of the Bloch sphere. If we assume !L 1⁄4

!0  !z (laser tuned to the ‘‘red sideband’’), and absorb phase

factors in the definition of , the resonant term gives

HI ’ ħðþa þ ayÞ: (4)

This Hamiltonian describes the situation where a quantum of

motion is exchanged with a quantum of excitation of the ion’s

internal state. It is most commonly known as the JaynesCummings Hamiltonian from cavity QED, which expresses

the exchange of energy between the internal states of an atom

in a cavity and the photons confined by the cavity (Jaynes and

Cummings, 1963; Haroche and Raimond, 2006). In the cavityQED experiments of Serge Haroche, Jean-Michel Raimond,

Michel Brune, and their colleagues in Paris, the atoms play

much the same role as they do in the ion experiments; however,

in the cavity-QED experiments, the relevant harmonic oscillator

is that which describes a field mode of the cavity, whereas in the

ion case, the relevant harmonic oscillator is that associated with

the ion’s motion (Sauter et al., 1988; Blockley, Walls, and

Risken, 1992). Over the years, this connection has led to some

interesting and complementary experiments between the two

types of experiments (Haroche and Raimond, 2006).

In the trapped-ion world, this type of exchange at the quantum level was first used in the electron g-factor experiments of

Dehmelt and colleagues, where a change of the electron’s

cyclotron quantum number was accompanied by spin flip of

the electron, which could be detected indirectly (Dehmelt,

1990). If we apply HI of Eq. (4) to an atomic ion in the state

j #ijni, where n denotes the harmonic oscillator’s quantum state

(Fock state), we induce the transition j #ijni!j "ijn1i. This

corresponds to the absorption feature labeled n 1⁄4 1 in

Fig. 2, and reduces the energy of motion by ħ!z. When the

ion decays, on average, the motion energy increases by the

recoil energy R 1⁄4 ðħkÞ

2=ð2mÞ, where k 1⁄4 2	=. Typically,

we can achieve the condition R  ħ!z, so that in the overall

scattering process the motional energy is reduced. In Fig. 2,

the carrier absorption feature is labeled n 1⁄4 0, indicating

photon absorption without changing the motional state.

This is a manifestation of the ‘‘recoilless’’ absorption of the

Mo ̈ssbauer effect [see, e.g., Dicke (1953), Lipkin (1973), and

Wineland et al. (1998)], but in the visible wavelength region.

Continuous application of the red-sideband transition provides a relatively straightforward way to laser cool the ion to

near the ground state of motion. After many scattering events,

the ion reaches the j #ijn 1⁄4 0i state, a ‘‘dark state’’ in which

scattering stops, since the j "ijn 1⁄4 1i state does not exist.

David J. Wineland: Nobel Lecture: Superposition, entanglement, ... 1107

Rev. Mod. Phys., Vol. 85, No. 3, July–September 2013
The process is not perfect, since scattering in the wings of

n 1⁄4 0, þ1 transitions leads to some residual recoil heating,

but the condition hni  1 can be achieved. This is easily

verified because absorption on the n 1⁄4 1 red sideband

nearly disappears, but the n 1⁄4 þ1 blue-sideband absorption remains. In 1989, with Frank Diedrich, who was a

postdoc in our lab, we achieved near-ground-state laser cooling in two dimensions, in essentially the way described

here (Diedrich et al., 1989). Later in an experiment led by

Chris Monroe, we achieved near-ground-state cooling in 3D

using two-photon stimulated-Raman transitions (Monroe,

Meekhof, King, Jefferts et al., 1995).

In addition to suppressing Doppler shifts in spectroscopy to

the highest degree possible (Wineland et al., 1987), one

motivation for sideband cooling was the intrinsic appeal of

(actively) placing a bound particle in its ground state of

motion, the lowest energy possible within the limitations

imposed by quantum mechanics. Here, the ground state is

a Gaussian-shaped wave packet with spread ffiffiffiffiffiffiffiffihz2i p 1⁄4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ħ=2m!z

p  z0 and energy ħ!z=2. We were also interested

in generating nonclassical states of motion (Heinzen and

Wineland, 1990; Cirac, Blatt et al., 1993; Cirac, Parkins

et al., 1993; Cirac et al., 1996) or entangled states of spins

(Wineland et al., 1992; Bollinger et al., 1996). For these

experiments, cooling to the ground state of motion provides

a clean starting point for motional state manipulation. [In the

Paris experiments, the ground state of the cavity mode can be

achieved either by thermally cooling to hni  1 by operating

at low temperature or by extracting photons with atoms sent

through the cavity in a process analogous to ion sideband

cooling (Haroche and Raimond, 2006).]

The red-sideband interaction of Eq. (4) and the ‘‘bluesideband’’ interaction (HI ’ ħþay þ H:c:, for !L 1⁄4

!0 þ !z) that induces j #ijni ! j "ijn þ 1i transitions, provide simple tools for the manipulation of an ion’s motional

states. For example, starting from j #ijn 1⁄4 0i, and applying a

series of blue-sideband, red-sideband, and carrier 	 pulses,

Fock states for a selected value of n can be deterministically

prepared (Meekhof et al., 1996). From j #ijn 1⁄4 0i, we can

also make coherent states ion motion by forcing the ion at its

motion frequency with an oscillating classical uniform field

(Carruthers and Nieto, 1965) or by applying an oscillating

optical-dipole force (Meekhof et al., 1996), which results

from spatial gradients of laser-beam-induced ac Stark shifts.

A coherent state of a quantum particle is very much like an

oscillating classical particle but, as opposed to a classical

particle that can be pointlike, the shape of the quantum

particle’s wave packet is the same as it is in the ground state.

In a clever but straightforward scheme suggested by Chi

Kwong Law and Joe Eberly (Law and Eberly, 1996) arbitrary

motional state superpositions can be prepared (Ben-Kish

et al., 2003). As a final example, the red-sideband interaction

applied for a ‘‘	-pulse’’ duration t 1⁄4 	=ð2Þ provides

internal-state to motion-state transfer

ðj #i þ j "iÞj0i ! j #iðj0i þ j1iÞ: (5)

V. SCHRO ̈ DINGER’S CAT

The optical-dipole force is interesting because the

strength of the force can depend on the ion’s internal state.

In 1996 (Monroe et al., 1996), using state-dependent

optical-dipole forces, we were able to produce an analog to

the Schro ̈dinger’s cat state in Eq. (1), which had the form

 1⁄4 1

ffiffiffi

2

p 1⁄2j "iji þ j #ij  i; (6)

where ji denotes a coherent state. The amplitude of the

particle’s oscillatory motion is equal to 2z0. The spatial part

of the state in Eq. (6) represents two wave packets that

oscillate back and forth but are 180	 out of phase with each

other and therefore pass through each other at the center of

the trap every half cycle of oscillation. Here, the analogy to

Schro ̈dinger’s cat is that the spin states of the ion are like the

states of the single radioactive particle and the coherent states

of the ion, which follow more macroscopic classical trajectories, are like the state of the cat; e.g., the ion at its left

extremum point ’ live cat, ion at its right extremum ’ dead

cat. Figure 3 describes how this state was produced.

To analyze the experiment, in Fig. 3(e), we can control the

phase of the amplitude such that the coherent state is ei

rather than . Near the condition  1⁄4 0, the probability Pð#Þ

of the ion to be in state j #i oscillates as a function of  due to

interference of the two wave packets. This verifies the coherence between the two components of the cat superposition

state. These interference oscillations are very analogous to the

fringe oscillations observed in Young’s-slit-type experiments

performed on individual photons, electrons, neutrons, or

atoms, but in those experiments the particle wave packets

disperse in time, whereas the wave packets in a harmonic

oscillator do not, and in principle last arbitrarily long.

In Monroe et al. (1996), for the condition described by

Eq. (6), the maximum separation of the wave packets was

⎪↑〉i

π/2 π π

F

(a) (b) (c) (d) (e) (f)

⎪↓〉i

π/2

-F

FIG. 3 (color). Depiction of the harmonic oscillator potential and

the wave packets for each component of the ion’s internal states,

denoted j "i and j #i. The images are snapshots in time; for images

(c) through (f ) the wave packets are shown at the extremes of their

motion. The areas of the wave packets correspond to the probability

of finding the atom in the given internal state. (a) The initial wave

packet corresponds to the ground state of motion after laser cooling

and preparation of the j #i internal state. (b) A 	=2 carrier pulse

creates the internal-state superposition 1ffiffi

2

p ðj #i þ j "iÞ. (c) An oscillating optical-dipole force is applied that excites only the j "i

component of the superposition to a coherent state of amplitude

, creating the state 1ffiffi

2

p ðj #ijn 1⁄4 0i þ j "ijiÞ. (d) The spin states are

flipped by applying a carrier 	 pulse. (e) The wave packet associated with the j "i state is excited by the optical-dipole force to an

amplitude of , that is, out of phase with respect to the first

excitation. This is the state of Eq. (6). (f ) To analyze the state

produced in step (e) and verify phase coherence between the

components of the cat wave function, we apply a final 	=2 carrier

pulse and then measure the probability Pð#Þ of the ion to be in

state j #i (see text). From Monroe et al., 1996.

1108 David J. Wineland: Nobel Lecture: Superposition, entanglement, ...

Rev. Mod. Phys., Vol. 85, No. 3, July–September 2013
4z0 ’ 83 nm, while the size of the wave packets z0 was

7.1 nm [see also McDonnell et al. (2007) and Poschinger

et al. (2010)]. Of course, one might object to dignifying the

state produced by calling it a Schro ̈dinger cat since it is so

small. In fact as we tried to make jj larger, the quality of the

superposition became more susceptible to decoherence

caused by noisy ambient electric fields (Myatt et al., 2000a,

2000b; Turchette et al., 2000), limiting the size that was

obtained. However, as far as we know, this is just a technical,

not fundamental limitation and we should eventually be able

to make a cat with jj large enough that the wave packets are

separated by macroscopic distances.

VI. ENTER QUANTUM INFORMATION

Following Peter Shor’s development of a quantummechanical algorithm for efficient number factoring

(Shor, 1994), there was a dramatic increase of activity in

the field of quantum information science. The potential

realization of general-purpose quantum information processing (QIP) is now explored in many settings, including atomic,

condensed-matter, and optical systems.

At the 1994 International Conference on Atomic Physics

held in Boulder, Colorado, Artur Ekert presented a lecture

outlining the ideas of quantum computation (Ekert, 1995), a

subject new to most of the audience. This inspired Ignacio

Cirac and Peter Zoller, who attended the conference and were

very familiar with the capabilities (and limitations) of trappedion experiments, to propose a basic layout for a quantum

computer utilizing trapped ions (Cirac and Zoller, 1995).

This seminal paper was the first comprehensive proposal for

how a quantum information processor might be realized. In

their scheme, quantum bits or ‘‘qubits’’ are realized with two

internal states of the ion, e.g., the j #i and j "i states above. The

ion qubits are held in a trap shown schematically in Fig. 4.

The motion of the ions is strongly coupled by the Coulomb

interaction and is best described by the normal modes of a kind

of pseudomolecule. Typically, the motion of each mode is

shared among all the ions and can act as a data bus for transferring information between ions. A single-qubit gate or rotation (the relatively easy part) is implemented by applying a

focused laser beam or beams onto that ion and coherently

driving a carrier transition as described above. The harder part

is to perform a logic gate between two selected ions. This can

be accomplished by first laser cooling all modes to the ground

state. The internal qubit state of one ion is then transferred onto

the qubit formed from the ground and first excited state of a

particular mode of motion (laser beam 1 in Fig. 4), as indicated

in Eq. (5). Laser beam 2 then performs a logic gate between the

(shared) motion qubit state and a second selected ion. Since the

second ion is generally in a superposition state, before the gate

operation is performed, the wave function for the spin and

motional state of the second qubit can be written as j #ij0i þ

j #ij1i þ 
j "ij0i þ j "ij1i. One type of logic gate imparts a

minus sign to the j "ij1i component of the wave function

by coherently driving a 2	 transition j "ij1i!jauxij0i !

j "ij1i, where jauxi is a third ‘‘auxiliary’’ internal state of

the ion (Cirac and Zoller, 1995). Flipping the sign of the j "ij1i

component of the wave function realizes an entangling twoqubit ‘‘	-phase’’ gate and is universal for computation.

Finally, the initial transfer step on the first ion is reversed,

restoring the motion to the ground state and effectively having

performed the logic gate between the internal qubit states of

the two laser-beam-selected ions. At NIST, since we had

recently achieved ground-state cooling with stimulatedRaman transitions on hyperfine qubit states, we were able to

quickly demonstrate a universal gate between a hyperfine

qubit and a motional mode qubit (Monroe, Meekhof, King,

Itano, and Wineland, 1995). The complete Cirac-Zoller gate

between two selected qubits was subsequently demonstrated

by the Innsbruck group, led by Rainer Blatt (Schmidt-Kaler

et al., 2003).

More streamlined gates were subsequently devised in

which multiple ions are addressed simultaneously by the

same laser beams (Sørensen and Mølmer, 1999, 2000;

Solano, de Matos Filho, and Zagury, 1999; Milburn,

Schneider, and James, 2000; Wang, Sørensen, and Mølmer,

2001). These gates also have the advantage that it is not

necessary to prepare all modes in the ground state; it is only

necessary that each ion is maintained well within the LambDicke regime [hz2ið=2	Þ

2]. These ‘‘geometric’’ gates

can be viewed as arising from quantum phases that are

acquired when a mode of the ions’ motion is displaced in

phase space around a closed path; the phases accumulated are

proportional to the enclosed area in phase space. The different

gates can be viewed in a common framework, the main

difference being whether or not the forces act on the spin

states in the z basis (eigenstates j #i, j "i) or in the x; y basis

[eigenstates of the form 1ffiffi

2

p ðj #i þ ei
j "iÞ, 1ffiffi

2

p ðj #i  ei
j "iÞ]

(Lee et al., 2005). The forces required for the displacements

are usually implemented with optical-dipole forces as in the

Schro ̈dinger cat example. Since the forces are state dependent,

the differential geometric phases generate entangling gates.

Two-qubit phase gates have been implemented in the z basis

(Leibfried et al., 2003; Home et al., 2006) and in the x; y basis

(Sackett et al., 2000; Haljan et al., 2005; Benhelm et al., 2008;

FIG. 4 (color online). Scheme for quantum computation proposed

by Cirac and Zoller (Cirac and Zoller, 1995). Quadrupolar electrodes are configured to produce a linear array of trapped-ion qubits

(filled black circles). Two diagonally opposite rods support an rf

potential to realize a ponderomotive pseudopotential transverse to

the trap’s (horizontal) axis. Static potentials applied to the end

segments of the electrodes confine ions along the axis. Ideally, all

motional modes are laser cooled to the ground state before logic

operations. The quantized modes of motion can be used as a data

bus to share information between the internal-state qubits of ions

that are selected by focused laser beams (see text).

David J. Wineland: Nobel Lecture: Superposition, entanglement, ... 1109

Rev. Mod. Phys., Vol. 85, No. 3, July–September 2013
Kim et al., 2009). In the Innsbruck experiment of Benhelm

et al. (2008), a Bell state with fidelity 0.993(1) was produced,

setting a standard for all QIP experiments. The use of singleand multiqubit gates has enabled the demonstration of several

ion-based QIP algorithms; see, for example, Blatt and

Wineland (2008) and Blatt and Roos (2012). At NIST most

such demonstrations were led by Didi Leibfried. Chris

Monroe’s group at the University of Maryland is leading

efforts on an entirely different scheme for ion entanglement

generation based on performing joint measurements on photons that are first entangled with ion qubits (Moehring et al.,

2007; Olmschenk et al., 2010; Monroe et al., 2012). This

scheme has the advantage that the ions don’t have to be in

the Lamb-Dicke regime, and it also enables entanglement of

widely separated qubits because of the relative ease of transferring photons over large distances.

The basic elements of the Cirac-Zoller proposal are carried

forward in the different variations of trapped-ion QIP. This

proposal rejuvenated the field of trapped ions and today there

are over 30 groups in the world working on various aspects

of quantum information processing. These include groups at

the University of Aarhus; Amherst College; University of

California, Berkeley; University of California, Los Angles;

Duke University; ETH Zu ̈rich; University of Freiburg;

Georgia Tech; Griffiths University; Imperial College;

University of Innsbruck; Lincoln Laboratories; Mainz

University; University of Hannover and PTB (Germany);

MIT; NIST (USA); NPL (UK); Osaka University; Oxford

University; Joint Quantum Institute at the University of

Maryland; Universite ́ de Paris; Saarland University

(Saarbru ̈cken); Sandia National Laboratory (USA); Siegen

University; Simon Fraser University; National University of

Singapore; Sussex University; University of Sydney;

Tsinghua University; University of Ulm; University of

Washington; Wabash College; and the Weizmann Institute.

A. Quantum simulation

In the early 1980s, Richard Feynman proposed that one

quantum system might be used to efficiently simulate the

dynamics of other quantum systems of interest (Feynman,

1982; Lloyd, 1996). This is now a highly anticipated

application of QIP and will likely occur well before useful

factorization is performed. Of course, the universality of a

large-scale quantum computer will allow it to simulate any

quantum system of interest. However, it is also possible to use

the built-in available interactions in a quantum processor to

simulate certain classes of physical problems. For trapped

ions, it has been possible to use the interactions employed in

the various gates to simulate other systems of interest,

for example, nonlinear optical systems (Leibfried et al.,

2002), motional quantum dynamics as in an electron’s

Zitterbewegung (Gerritsma et al., 2010), or the properties of

a ‘‘quantum walk’’ (Schmitz, Matjeschk et al., 2009;

Za ̈hringer et al., 2010). Currently, efforts are underway in

several laboratories to use QIP interactions to simulate various

dynamics including those of condensed-matter systems. Some

of the basic ideas for how this might work with ions have been

outlined in Wunderlich and Balzer (2003), Porras and Cirac

(2004, 2006), Deng, Porras, and Cirac (2005), Pons et al.

(2007), Scha ̈tz et al. (2007), Chiaverini and Lybarger

(2008), Taylor and Calarco (2008), Clark et al. (2009),

Johanning, Varon, and Wunderlich (2009), Schmitz,

Friedenauer et al. (2009), Schmied, Wesenberg, and

Leibfried (2011), Blatt and Roos (2012), Britton et al.

(2012), Korenblit et al. (2012), and Schneider, Porras, and

Schaetz (2012). Here, logic gate interactions between ions i

and j invoke a spin-spin–like interaction of the form ui^ uj^ ,

where u^ 2 fx;^ y;^ z^g. Spin rotations about a direction u^ act like

magnetic fields along u^. These basic interactions have been

implemented on up to 16 ions in an rf trap (Scha ̈tz et al., 2007;

Friedenauer et al., 2008; Kim et al., 2009, 2010; Edwards

et al., 2010; Islam et al., 2012; Korenblit et al., 2012). One

interesting aspect of this work is the study of quantum phase

transitions by varying the relative strengths of the (simulated)

spin-spin and magnetic field interactions. Under appropriate

conditions, the effects of spin ‘‘frustration’’ are now becoming

apparent. The basic interactions have also been implemented

on over 100 spins in a Penning trap experiment led by John

Bollinger at NIST (Britton et al., 2012), where the ions

naturally form into a triangular array. In the Innsbruck group,

simulations including engineered dissipation have also been

implemented (Barreiro et al., 2011; Blatt and Roos, 2012), and

a striking demonstration of a digital quantum simulator has

been made (Lanyon et al., 2011; Blatt and Roos, 2012), in

essence the first universal quantum computer.

B. Spectroscopy and quantum metrology

Some potential applications of quantum control and QIP

are motivated by the idea of using entangled states to improve

spectroscopic sensitivity (Wineland et al., 1992, 1994;

Bollinger et al., 1996; Leibfried et al., 2004; Roos et al.,

2006; Goldstein et al., 2009) and demonstrations of this

increased sensitivity have been made (Meyer et al., 2001;

Leibfried et al., 2004, 2005; Roos et al., 2006; Leroux,

Schleier-Smith, and Vuletic ́, 2010; Monz et al., 2011).

These demonstrations were made in the limit that noise was

dominated by ‘‘projection noise,’’ the fundamental noise

arising from the fluctuations in which state the system is

projected into upon measurement (Wineland et al., 1982;

Itano et al., 1993). This might be the case in a spectroscopy

experiment where the interrogation time is limited by a

particular experimental constraint, like the duration of flight

of atoms in a cesium fountain clock or by the desire to hold

the temperature of ions below a certain value if they are

heated during interrogation. However, if significant phase

noise is present in either the atoms themselves (Huelga

et al., 1997) or the interrogating radiation (Wineland et al.,

1998; Buzek, Derka, and Massar, 1999; Andre ́, Sørensen, and

Lukin, 2004; Rosenband, 2012), the gain from entanglement

can be lost. This puts a premium on finding probe oscillators

that are stable enough that the projection noise dominates for

the desired probe duration.

Some ions of spectroscopic interest may be difficult to

detect because they either don’t have a cycling transition or

lack a cycling transition at a convenient wavelength. In

some cases, this limitation can be overcome by simultaneously storing the ion(s) of spectroscopic interest with

a ‘‘logic’’ ion or ions whose states can be more easily

1110 David J. Wineland: Nobel Lecture: Superposition, entanglement, ...

Rev. Mod. Phys., Vol. 85, No. 3, July–September 2013
detected. Following the Cirac and Zoller scheme, we can

use the internal-to-motion-state-transfer process described

above. Here, the idea is to first transfer the two states of

interest in the spectroscopy ion to the ground and first

excited states of a mode of the ions’ coupled motion. This

is then followed by mapping the motional states onto the

logic ion, which is subsequently measured (Wineland et al.,

2002). In a project led by Till Rosenband at NIST, this

technique has been used to detect optical transitions in

27Alþ ions by transferring the relevant 27Alþ states to a

9Beþ or 25Mgþ logic ion, which is then measured (Schmidt

et al., 2005). It is now used routinely in an accurate optical

clock based on 27Alþ (Rosenband et al., 2008; Chou, Hume,

Koelemeij et al., 2010) and might also be extended to

molecular ions. Currently, the 27Alþ single-ion optical clock

has the smallest systematic error of any clock at somewhat

below 1 part in 1017 (Chou, Hume, Koelemeij et al., 2010).

This level of precision has enabled observations of the

predictions of Einstein’s general theory of relativity on a

human scale, such as time dilation for bicycling speeds and

the gravitational redshift for height changes of around

30 cm (Chou, Hume, Rosenband, and Wineland, 2010).

Such clocks may become useful tools in geodesy.

The information transfer and readout process employed

in the 27Alþ=9Beþ clock experiments typically had a fidelity of about 0.85, limited by errors caused by the ions’

thermal motion in modes not used for information transfer

[so-called ‘‘Debye-Waller’’ factors from Mo ̈ssbauer spectroscopy (Lipkin, 1973; Wineland et al., 1998)]. However,

the quantum logic detection process is a QND type of

measurement in that it doesn’t disturb the detected populations of the 27Alþ ion. It can therefore be repeated to gain

better information on the 27Alþ ion’s (projected) state. By

use of real-time Bayesian analysis on successive detection

cycles, the readout fidelity was improved from 0.85 to

0.9994 (Hume, Rosenband, and Wineland, 2007). This experiment shares similarities with those of the Paris cavityQED group, where successive probe atoms are used to

perform QND measurements of the photon number in a

cavity (Dele ́glise et al., 2008). In Hume, Rosenband, and

Wineland (2007), the same atom (9Beþ) is reset after each

detection cycle and used again. Also, because the detection

was accomplished in real time, the procedure was adaptive,

requiring on each run a minimum number of detection

cycles to reach a certain measurement fidelity.

VII. SUMMARY

I have tried to give a brief account of some of the developments that have taken place in the area of quantum state

manipulation of small numbers of trapped atomic ions. With

apologies, I have omitted several aspects of this subject and

for the topics discussed here, I primarily used examples from

the NIST, Boulder group. Much of the other work has been

discussed in various comprehensive articles and reviews; see,

for example, Cirac et al. (1996), Wineland et al. (1998),

Sˇasˇura and Buzˇek (2002), Leibfried, Blatt, Monroe, and

Wineland (2003), Lee et al. (2005), Blatt and Wineland

(2008), Duan and Monroe (2008, 2010), ; Ha ̈ffner, Roos,

and Blatt (2008), Kielpinski (2008), Monroe and Lukin

(2008), Blatt and Roos (2012), Korenblit et al. (2012),

Monroe et al. (2012), and Schneider, Porras, and Schaetz

(2012). Reviews on advanced clocks including those based on

ions are contained in Gill (2005, 2011), Maleki (2008), and

Margolis (2009) [see also Madej et al. (2012) and references

therein].

ACKNOWLEDGMENTS

Certainly my role in this work is very small when compared

to that of my colleagues both at NIST and around the world,

who have made so many important contributions. Having been

recognized by the Royal Swedish Academy of Sciences is

really more recognition of our field rather than individual

accomplishment; many others are at least as deserving. Just

the work of the NIST group was due to the combined efforts of

a very large number of people. I have been lucky to work with

NIST permanent staff members Jim Bergquist, John Bollinger,

Bob Drullinger, and Wayne Itano for my entire career, and we

have been fortunate to be joined by Didi Leibfried and Till

Rosenband in the last decade. Chris Monroe was a very

important part of our group from 1992 to 2000 and now has

his own group at the University of Maryland. Of course our

successes would not have happened if not for the dedication of

many students, postdocs, and visiting scientists to our group,

numbering over 100 people. Having a group working directly

together or on related problems has been a source of strength

for us, and the congenial atmosphere over the years has made

our efforts so enjoyable. Throughout my career, our group has

enjoyed the support and encouragement of NBS/NIST management. My direct supervisors over the years, Helmut

Hellwig, Sam Stein, Don Sullivan, and Tom O’Brian, have

always supported our goals and desires as much as possible.

More recently, we have also enjoyed the support of Carl

Williams, who heads NIST’s quantum information program.

We are all indebted to our laboratory director, Katharine

Gebbie, for her support and encouragement. Perhaps one

measure of her success is that I am the fourth person, after

Bill Phillips, Eric Cornell, and Jan Hall, to receive a Nobel

prize during her tenure as lab director. We are also grateful for

the support of agencies outside of NIST, such as AFOSR,

ARO, DARPA, ONR, and various intelligence agencies who

have supported our work on quantum information. I have great

respect for the leaders of some of our group’s strongest competition such as Rainer Blatt (Innsbruck) and Chris Monroe

(University of Maryland) and have enjoyed their friendship for

many years. It was also a great pleasure to share this recognition with Serge Haroche. I have known Serge for about

25 years and have enjoyed both his group’s elegant science

and also the mutual friendship that my wife and I have shared

with him and his wife, Claudine. Most importantly, I have been

very fortunate to have the support, understanding, and patience

of my wife Sedna and sons Charles and Michael. I thank John

Bollinger, Wayne Itano, Didi Leibfried, and Till Rosenband

for helpful suggestions on the manuscript.

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