Two-Species Ion Arrays for Quantum Logic Spectroscopy

and Entanglement Generation

by

David B. Hume

B.S., University of Kentucky, 2002

A thesis submitted to the

Faculty of the Graduate School of the

University of Colorado in partial fulfillment

of the requirements for the degree of

Doctor of Philosophy

Department of Physics

2010
This thesis entitled:

Two-Species Ion Arrays for Quantum Logic Spectroscopy and Entanglement Generation

written by David B. Hume

has been approved for the Department of Physics

David J. Wineland

Dr. Neil Ashby

Date

The final copy of this thesis has been examined by the signatories, and we find that both the content and

the form meet acceptable presentation standards of scholarly work in the above mentioned discipline.
iii

Hume, David B. (Ph.D., Physics)

Two-Species Ion Arrays for Quantum Logic Spectroscopy and Entanglement Generation

Thesis directed by Dr. David J. Wineland

The quantum states of trapped atomic ions can be highly isolated from external perturbation, and

precisely manipulated with applied laser fields. This makes them an excellent medium for quantum-limited

experiments such as quantum information processing and precision spectroscopy. A relatively small number

of ion species have been used for these types of experiments because most species are difficult to laser cool

and detect directly. This thesis demonstrates a way to overcome this limitation by use of sympathetic cooling

and state detection based on quantum logic. We apply these techniques to mixed-species arrays of 27Al+ and

9Be+ ions. A mathematical model for the motion of such a two-species array is presented in order to explore

some features of the ion dynamics that are relevant for the experiments. Repetitive quantum nondemolition

measurements of the electronic states of a 27Al+ ion show detection fidelities as high as 99.94 %. We also

demonstrate the simultaneous detection of two 27Al+ ions and observe a detection fidelity of 99.8 %. The

basic ideas behind the detection strategy are extended to potentially enable similar experiments on a more

general class of atomic and molecular ions.

We have also investigated, theoretically and experimentally, a method for preparing entangled Dicke

states in trapped atomic ions. We consider a linear chain of N ion qubits that is prepared in a particular

Fock state of motion, |m⟩. The m phonons are removed by applying a laser pulse globally to the N qubits,

and converting the motional excitation to m flipped spins. The global nature of this pulse ensures that the

m flipped spins are shared by all the target ions in a state that is a close approximation to the Dicke state

|D

(m)

N ⟩. We calculate numerically the fidelity limits of the protocol and find small deviations from the ideal

state for m = 1 and m = 2. We have demonstrated the basic features of this protocol by preparing the Bell

state |D(1)

2

⟩ in two 25Mg+ target ions trapped simultaneously with an 27Al+ ancillary ion.
Dedication

For Addie.
v

Acknowledgements

The work described in this thesis comes from a collaboration of many excellent scientists. In particular

I owe a huge amount of gratitude for the hard work and patience of my closest advisors, Dave Wineland

and Till Rosenband, without whom nothing in the experiment would have ever worked. Their technical

expertise, creativity and nuanced knowledge of physics have guided this project from the start. I am also

greatly indebted to Chin-Wen “James” Chou, Piet Schmidt and Jeroen Koelemeij, who worked skillfully,

tirelessly (and even cheerfully!) to make the aluminum clocks tick. I would like to give a special thanks to Jim

Bergquist who has generously shared his knowledge, hard work and encouragement to help me at every step

along the way. Many more people have contributed to designing, building, operating and troubleshooting

the experiment. Listing all of their names and contributions would really boost my page count, but I don’t

want to risk leaving anyone out, so instead I will just acknowledge all of the past and current members of

the ion storage group collectively. Over the years, the people in the group have been a consistent source of

ideas and inspiration as well as experimental support. I also want to acknowledge the work of the members

of the Optical Frequencies Measurement Group at NIST, which has been essential for much of the work in

this thesis. In addition, Neil Ashby who has served as my titular advisor, has been a valuable source of

scientific guidance.

Just as important has been the support of my friends and family who saw me through graduate school

with an incredible amount of love, understanding and grace. In particular I want to thank Lenore Sparks

who (1) physically sat me down and forced me to dictate the really onerous parts of the dissertation while

she dutifully typed them, (2) proof-read the whole mess of it when it came down to do-or-die time and

(3) dragged me multiple times from my disintegrating shack in Eldorado Springs where I would hide from
vi

anything thesis-like. My parents and brother performed similarly heroic tasks. I really hope someday I can

return the favors. Thanks!
vii

Contents

Chapter

1 Introduction 1

1.1 Single-Ion Optical Clocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Ion Trap Quantum Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Laser Ion Interactions 10

2.1 Ion Energy Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Laser Ion Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Morris-Shore Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3.1 Two-Level Morris-Shore Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3.2 Multilevel Morris-Shore Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Apparatus 24

3.1 Ion Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.1.1 Ion Trap Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.1.2 Experimental Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Laser Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2.1 Be+ lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2.2 Al+ lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2.3 Mg+ Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2.4 Experimental Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
viii

3.3 Experimental Primitives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3.1 Loading ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3.2 Laser Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.3.3 Al+ State Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3.4 Monitoring and Locking the Trap Conditions . . . . . . . . . . . . . . . . . . . . . . . 45

4 Two-Species Ion Dynamics 48

4.1 Mathematical Model for Multiple Species Ion arrays . . . . . . . . . . . . . . . . . . . . . . . 49

4.2 Ion Reordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.2.1 Two Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2.2 Three Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2.3 Higher Ion Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.3 Micromotion Measurement and Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.4 Sympathetic Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5 Quantum Nondemolition Measurements of Trapped Ions 66

5.1 General QND measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.2 Experimental QND Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.2.1 Bayesian Analysis of Photon Histograms . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.2.2 Be+ Resonance Fluorescence as a QND measurement . . . . . . . . . . . . . . . . . . 71

5.2.3 Qubit Detection through Repetitive QND Measurements . . . . . . . . . . . . . . . . 74

5.3 Generalizing Indirect QND Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.3.1 QND Detection through Arbitrary States of Motion . . . . . . . . . . . . . . . . . . . 88

5.3.2 QND Measurements for Entanglement Generation . . . . . . . . . . . . . . . . . . . . 92

5.3.3 Detection by Modulated Fluorescence . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
ix

6 Preparation of Dicke States in an Ion Chain 103

Bibliography 112
x

Figures

Figure

2.1 Relevant Be+ levels and lasers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Relevant Al+ levels and lasers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Conceptual diagram for the two-level Morris-Shore tranformation . . . . . . . . . . . . . . . . 17

2.4 Conceptual diagram for a multilevel Morris-Shore transformation . . . . . . . . . . . . . . . . 21

3.1 Ideal linear Paul trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Trap dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3 Physical traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.4 Blue Doppler and red Doppler AOM schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.5 Raman AOM scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.6 Diagram of the fiber-noise cancelation setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.7 3P1 AOM setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.1 Two ion reordering thresholds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.2 Two-ion “twist” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.3 Three-ion reordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.4 Time dilation from micromotion for mixed species two-ion arrays . . . . . . . . . . . . . . . . 60

4.5 Doppler cooling rates for a mixed species two-ion array . . . . . . . . . . . . . . . . . . . . . . 63

5.1 Be+ resonance fluorescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.2 Relevant energy levels in 27Al+ and 9Be+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
xi

5.3 Fidelity for single Al+ measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.4 Red sideband Rabi flopping for 2 Al+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.5 Spectroscopy with 2 Al+ and 1 Be+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.6 Detection data with 2 Al+ and 2 Be+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.7 Comparison of detection performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.8 Conceptual diagram for measurement induced entanglement . . . . . . . . . . . . . . . . . . . 94

5.9 Experimental diagram for fluorescence modulation . . . . . . . . . . . . . . . . . . . . . . . . 100

5.10 Fluorescence modulation data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.1 Simulated Dicke state fidelities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.2 Detection data from entangled Mg+ states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
Chapter 1

Introduction

Quantum theory describes the behavior of the physical world at a fundamental level. Despite being

supported by precise measurements on a vast range of physical systems, it continues to confound human

imagination with seemingly bizarre implications. For example, the Copenhagen interpretation of quantum

mechanics seems to imply the existence of nonlocal effects [Bell 64] - a condition challenged historically as

unphysical [Einstein 35]. However, this property of quantum mechanics has been confirmed repeatedly in

measurements with increasing accuracy [Aspect 82, Rowe 01], making it an accepted aspect of the theory if

not completely understood. At the same time, quantum mechanical effects play an increasingly central role

in the technology that drives the modern world. The band structure of semiconducting materials arises from

a quantum mechanical description of the electrons in crystalline materials and is central to the operation of

semiconducting devices such as the transistors in computer chips. These aspects of quantum theory makes

it simultaneously one of the most confirmed, debated, applied and misconstrued discoveries of science.

From the original conception of quantum theory, atomic spectroscopy has played a central role in motivating its development and confirming its predictions. Measurements in atomic spectroscopy, particularly

observations of the spectra of atomic hydrogen, guided the formulation of the Bohr atom in 1913 - a direct

precursor to the formal quantum theory developed in subsequent years. Likewise, precise observations of the

hyperfine splitting in hydrogen [Lamb 47] presented a puzzle to physicists in the 1940’s that was explained

with the development of quantum electrodynamics (QED). The predictions of QED continue to be tested

with improved precision in atomic physics labs.

Conversely, theoretical developments in quantum theory have given rise to ground breaking progress
2

in atomic spectroscopy. In 1917 Albert Einstein wrote a paper, “On the quantum theory of radiation”

[Einstein 17], which showed theoretically the necessity for stimulated emission of radiation. This concept

laid the foundation for the eventual amplification of radiation through stimulated emission, first with the

invention of the ammonia maser [Gordon 54] then with its optical counter part, the laser [Maiman 60]. Since

that time, the laser has become a bedrock tool for the study of atomic systems. The spatial and temporal

coherence of laser light give rise to an array of techniques that together have improved numerous aspects of

atomic spectroscopy.

In this way, quantum theory and atomic spectroscopy have often played a complementary role in

pushing forward our understanding of the physical world and improving our ability to interact with it. The

interplay between these two scientific programs has been invaluable to both, and the symbiotic relationship

is likely to continue far into the future. Currently, the breadth of applications and the achievable performance in atomic spectroscopy is increasing rapidly. A relevant example is the recent development of optical

atomic clocks, which combine advances in laser stability and laser frequency measurements with newly developed techniques in the trapping and cooling of atoms and ions to produce clocks that have demonstrated

performance exceeding that of existing microwave frequency standards.

On the other side, quantum mechanics has undergone a revolution in the past couple of decades with

the advent of quantum information processing (QIP) theory. This is the study of how information behaves

and can be manipulated when encoded in coherent quantum states. It has implications for encryption,

computation, metrology and quantum simulation. Again here, atomic spectroscopic techniques are central

to some efforts to realize the protocols in QIP. Likewise, the theoretical developments in QIP motivate

novel and challenging experiments in atomic physics, sometimes providing researchers with new avenues for

experiments or a new perspective on the physical systems they investigate.

In this thesis I describe a set of experiments that, along with a diversity of other experimental efforts

underway, lie at the intersection of QIP and precision spectroscopy. I believe the work reflects on the

continued interaction between developments in quantum mechanics and experimental atomic physics. In

particular, we demonstrate how techniques developed in the context of QIP can also be applied to enable

precision spectroscopy on an interesting atomic system. We apply QIP techniques in an ion trap to enable
3

the detection of electronic states 27Al+ (Ch. 5), enabling precision spectroscopy of the 1S0 →3P0 “clock”

transition. Reapplying the same capabilities in a different experiment, we use 27Al+ as an ancillary ion to

entangle the internal states of two 25Mg+ (Ch. 6). To motivate these experiments I will first introduce the

ideas behind single-ion optical clocks and ion trap quantum computing.

1.1 Single-Ion Optical Clocks

The Bureau International des Poids et Mesures (BIPM) is responsible for defining the SI system of

units. Since 1967 the second has been defined based on a resonance frequency in atomic cesium1:

The second is the duration of 9192631770 periods of the radiation corresponding to the

transition between the two hyperfine levels of the ground state of the caesium 133 atom.

By defining the speed of light in vacuum as c = 299792458 m/s, atomic clocks also provide the basis for

the definition of the meter. Since these base units, the meter and the second, both appear in many derived

units (i.e. Hz = 1/s) thinking of atomic clocks as timekeeping devices is an oversimplification. Much of the

motivation for improving the stability and accuracy of atomic clocks is not directly related to timekeeping,

since the signals from state-of-the art clocks are not maintained continually and the distribution of the

signals globally is subject to various inaccuracies. However, stable atomic clocks have been applied to a

variety of important technological problems such as network synchronization and precise satellite navigation

as required, for instance, in GPS satellites. They have been central to tests of Einstein’s relativity theory

[Hafele 72, Vessot 80]. Since the development of the first atomic clocks they have been continually improved,

both in terms of their frequency stability and their absolute accuracy2. State-of-the-art cesium fountain

clocks use laser-cooled atoms and exhibit fractional frequency inaccuracies below 5 × 10−16 [Parker 10]

dominated by frequency shifting effects such as Stark shifts due to background blackbody radiation [Itano 82].

Frequency stability of Cesium fountain clocks has been reported in the range of 4 × 10−14 at a one second

integration time [Santarelli 99], averaging in time according to the Allan deviation as σy(τ ) ∝ 1/

√

τ , where

τ is the total measurement duration.

1 This is the official English translation of the French text published by BIPM. It specifically refers to a cesium atom at rest

and and at a temperature of 0 K.

2 For readers interested in an accessible history of atomic clocks and atomic timekeeping I recommend Splitting the second:

the story of atomic time by Tony Jones [Jones 00].
4

Recently, a new generation of atomic clocks based on optical transitions in trapped atoms or ions

(optical clocks) have been developed that demonstrate higher accuracy and stability than the Cesium fountains. Although the potential for high accuracy optical clocks was recognized in the pioneering efforts of

Hans Dehmelt as early as 1981 [Dehmelt 81], the experimental demonstrations have been enabled by recent

advancements of several techniques in atomic physics. For example, laser cooling of trapped atoms to low

temperatures allows for recoil-free optical spectroscopy. This suppresses inhomogeneous Doppler broadening of the optical resonance. In addition, it reduces small shifts of the atomic absorption spectrum due to

relativistic time dilation. As another example, the improvement of laser stabilization [Young 99] by use of

isolated, ultra-high finesse optical cavities has been key for the measurement of narrow optical resonances.

Laser stabilization remains an important area of research for further improving optical clock performance.

Finally, the invention of the self-referenced femtosecond frequency comb [Jones 00] has simplified the measurement of optical frequencies. In a relatively compact and reliable package these lasers transfer the stability

of an optical signal into the electronically countable radio frequency (RF) domain3.

Clock performance is quantified primarily on the basis of accuracy and stability, both being normalized

to the center frequency of the atomic resonance. The inaccuracy of a clock quantifies how well systematic

shifts to the clock frequency are characterized. Generally, these frequency shifts are calibrated through separate measurements, theoretical considerations, and sometimes by periodically monitoring the shifts during

clock operation. The clock stability, on the other hand, quantifies the magnitude of random statistical noise

in the clock frequency. It is typically characterized in terms of the Allan deviation. Ultimately the stability

of a clock is limited by quantum projection noise, and the Allan deviation in this case is given by

σy(τ ) ≈

∆ν

πν0

√

Tc

τN

(1.1)

where ∆ν is the linewidth of the transition at frequency ν0, Tc is the experimental cycle time, τ is the

averaging time and N is the number of atoms probed simultaneously. The higher frequency, ν0, of an optical

atomic clock, being many orders of magnitude greater than a microwave clock, is the central feature that

leads to higher stability and accuracy.

3 Before the invention of the femtosecond laser frequency comb optical frequencies could only be counted through a complicated mechanism by which signals between the optical and RF domains where phase-locked in stages.
5

Most optical clock experiments currently underway take one of two primary approaches. One is based

on individual atomic ions trapped in a RF Paul trap, like the experiments of this thesis, while the other is

based on an atomic ensemble trapped in an optical lattice. Each of these approaches offers a number of basic

advantages and disadvantages that affect various performance characteristics and have a bearing on other

practical considerations. One attractive feature of an ion optical clock, is that the ion’s charge provides a

means for trapping it and precisely controlling its position. With an RF Paul trap, trap depths on the order

of ∼ 1 eV and secular mode frequencies above 1 MHz can be achieved. The trap depth ensures that long

interaction times can be achieved between an ion and an applied electromagnetic pulse during spectroscopy.

For example, the typical lifetime of a laser-cooled ion in a room-temperature trap is on the order of an hour

(often limited by chemical reactions with residual hydrogen in vacuum) while the interrogation pulse time

is currently limited by laser instability to ∼ 300 ms. This interrogation time leads to Fourier transform

limited linewidths on the order of 1 Hz. The ion can remain trapped through many experimental cycles so

that Tc does not include the extra time that would be required for reloading the ion. On the other hand,

the Coulomb repulsion between ions limits the ion density that can be achieved. In fact, frequency shifts

due to the presence of the RF trapping fields have forced the high-accuracy ion clocks developed so far to

operate with a single ion (or a 1 dimensional array of a small number of ions) at the null of the trapping

fields. With N = 1, the clock stability (Eq. 1.1) suffers compared to neutral atom clocks that typically have

many thousands of atoms. Optical transitions in ions are typically at a higher frequency compared to neutral

atoms (ultraviolet rather than visible frequencies) due to the higher nuclear charge.

Several ion systems have been investigated as a frequency reference for an optical atomic clock,

including Hg+, In+, Sr+, and Yb+. One motivation behind the experiments in this thesis is an optical

clock based on 27Al+, which is unique among ion clocks studied experimentally in that it does not have a

suitable electronic structure to perform direct laser cooling and state detection. Instead laser cooling of Al+

is performed sympathetically via the Coulomb interaction with a simultaneously trapped ion, which can be

directly laser cooled. Although, we specifically study Al+ for precision spectroscopy, similar techniques may

be applied to other atomic and molecular ions. These ion species may be attractive for use as a frequency

reference in an atomic clock or other precision spectroscopy experiments such as a search for the possible
6

time-variation of the fundamental constants. The basis for the techniques I describe is a procedure for

mapping information in the electronic state of one ion, through a collective mode of motion and into the

electronic state of the sympathetic cooling ion. These ideas were inspired by progress in the field of ion trap

quantum computing, which I will introduce now.

1.2 Ion Trap Quantum Computing

The basic unit of information in a classical computer is the bit, which can take one of two values: 0

or 1. In analogy the basic unit of quantum information is the quantum bit or qubit. The qubit, like the

classical bit, is a system containing two states: |0⟩ and |1⟩. However, unlike the classical bit, a qubit can

exist in a superposition state |ψ⟩ = c0|0⟩ + c1|1⟩, where the complex amplitudes obey |c0|

2 + |c1|2 = 1. The

Hilbert space of a qubit is conveniently represented by the Bloch sphere. In this representation c0 = sin(θ/2)

and c1 = cos(θ/2)e

iφt, such that ψ is specified by two parameters θ and φ corresponding to a point on a

sphere of unit radius. From this representation it is clear that, whereas a classical bit occupies a discrete

value, a qubit occupies a point in a continuous space.

Another key difference between quantum information and classical information relates to observation

or measurement of a quantum state. In the Copenhagen interpretation, a superposition state “collapses” to

a particular basis state when measured. For instance, a measurement on |ψ⟩ yields the result |0⟩ (|1⟩) with

probability |c0|

2

(|c1|

2

). Since quantum measurement is a central theme in this thesis (Ch. 5) I will introduce

it more formally here, following the treatment in [Nielsen 00]. Extending the Hilbert space of state |ψ⟩ to

arbitrary dimensions we can write,

|ψ⟩ =

∑

i

ci|ψi⟩, (1.2)

where we assume an orthonormal basis such that ⟨ψi|ψi

′ ⟩ = δii′ and |⟨ψ|ψ⟩|2 =

∑

i

|ci|

2 = 1. A projective

measurement of ψ can be described in terms of a set of projection operators {Pˆm} with eigenvalues {m}.

I assume that the decomposition of state ψ in Eq. 1.2 is done in terms of the measurement basis such

that Pˆm = |ψm⟩⟨ψm| and PˆmPˆ′

m = δmm′Pˆm. Then any measurement on |ψ⟩ is described by the observable

Oˆ =

∑

m mPm and has an expectation value of ⟨ψ|Oˆ|ψ⟩. Any individual measurement of |ψ⟩, however, can

give only a single result corresponding to a particular eigenvalue. Since I have defined the decomposition of
7

|ψ⟩ in terms of the measurement basis, the outcome of the measurement will be m with probability |cm|

2

and the quantum state after measurement will be |ψm⟩. In this formal outline of a projective measurement

it is typically assumed that the result of the measurement is unchanged by further application of Oˆ since the

eigenvalue m will be observed with unit probability. In practice, this is a difficult experimental proposition,

since quantum states are often perturbed by any interaction with the environment (or equivalently the

experimental apparatus) as required by measurement. A measurement that does not perturb the projected

quantum state has been termed a quantum nondemolition (QND) measurement, which I will examine in the

context of trapped ion experiments in Ch. 5.

The physical interpretation of state projection during measurement has been the subject of significant

debate, but here I consider it as a fundamental postulate of quantum theory. Independent of any interpretation, quantum state projection has profound implications in QIP, the recognition of which led to some of the

first theoretical work in quantum information theory. A well-known example is the no-cloning theorem, first

reported in 1982 [Wootters 82], which states that an arbitrary unknown quantum state cannot be copied. In

quantum cryptography, information transmitted via coherent quantum states enables secure communication.

Here, the no-cloning theorem forbids a potential eavesdropper from intercepting a quantum message, copying

it, and sending one copy forward. Any attempt to gain information in this way can be detected due to the

inevitable state projection. The idea of secure communication by use of a quantum information channel was

developed soon after the no-cloning theorem [Bennett 84].

Given the early results in quantum cryptography, it is natural to ask what other applications exist for

quantum information. In particular, could a computer that processes quantum information give an advantage

in efficiency over a classical computer? In 1985 David Deutsch studied this problem and discovered a simple

example in which a quantum algorithm could outperform the best classical counterpart [Deutsch 85]. In

the intervening years several important quantum algorithms have been developed. 1n 1994 Peter Shor

discovered algorithms for prime factorization and computing the discrete logarithm [Shor 94], which are of

great importance because they provide an exponential speed-up compared to the best classical algorithms

for problems that are central in classical cryptography. These key results and others have motivated the

continued theoretical investigation of QIP.
8

In parallel, experimental groups around the world, working with a diverse array of physical systems

have begun serious efforts to implement the basic building blocks of a quantum computer. Any physical

system capable of QIP must meet a certain set of requirements related to qubit initialization, measurement,

coherence, and interaction. One famous formulation of these requirements is called the Divincenzo Criteria

[DiVincenzo 01]. An experimental effort to meet these requirements presents numerous challenges, not the

least of which is the notoriously fragile nature of a quantum state. Any interaction between a qubit and the

environment can collapse the qubit state, just like a measurement, causing decoherence. For this reason, the

qubits in a quantum computer must be highly insulated from their environment, while allowing for strong

interactions with the experimental systems that control the computation. This juxtaposition and other

technical challenges have made progress in experimental quantum computing both difficult and interesting.

Despite the difficulties, basic demonstrations of QIP protocols have been made in several physical

systems. Perhaps the most advanced of these efforts uses individual trapped ions to store information. This

idea, originally proposed by Cirac and Zoller [Cirac 95], takes two spectroscopically isolated states in the

atomic system as a qubit. Typically these states are initialized with optical pumping and measured with

fluorescence detection. The qubit state of an individual ion can be manipulated by applying properly tuned

laser pulses that rotate the qubit state on the Bloch sphere. The breakthrough in this proposal was a protocol

for coupling the state of two ions stored in the same trap through a collective state of motion, to perform a

two-qubit controlled-not (CNOT) gate. The CNOT gate is one member of a finite set of operations necessary

for universal quantum computation, and, like the classical controlled-not gate can be summarized by the

following truth table,

|00⟩ → |00⟩

|01⟩ → |01⟩

|10⟩ → |11⟩

|11⟩ → |10⟩.

Here the state of the two qubits is compressed into a single ket (i.e. |0⟩1 ⊗ |1⟩2 ≡ |01⟩). The first qubit is

the “control” qubit that modulates a bit flip on the second, “target” qubit. The quantum CNOT operation
9

can be used to entangle two quantum bits. If the initial state of the two qubits is |ψ⟩ = 1/

√

2(|0⟩ +

|1⟩)|0⟩, after a CNOT gate, state will be |ψ

′

⟩ = 1/

√

2(|00⟩ + |11⟩). This is called an entangled state

because it cannot be separated into the form |ψ⟩1 ⊗ |ψ⟩2. In addition to being a basic building block for

quantum computing experiments, the CNOT gate is the central operation that enables spectroscopy of

Al+ in this thesis. The current state of the art for ion trap QIP involves the entanglement of up to eight

trapped-ion qubits [Leibfried 05, H ̈affner 05], and two-qubit CNOT gates with fidelities as high as 99.3 %

[Leibfried 03, Benhelm 08]. Both of these are significant steps towards realizing practical applications of

QIP. By contrast, the work I discuss involves operations that entangle only two qubits and require relatively

low fidelity (in our case ∼85 %) to demonstrate precision spectroscopy on an otherwise inaccessible system.

We have operated an atomic clock based on Al+ in this way and have achieved inaccuracy as low as 8×10−18

and clock stability of 2.8×10−15 at a one second averaging time. More broadly, this shows, in principle, that

the controlled interaction of two quantum systems can open new doors in the field of precision metrology.

Whereas large-scale QIP requires the precise interaction of many quantum bits, applications in metrology

are well within reach with the current state of the art.
Chapter 2

Laser Ion Interactions

2.1 Ion Energy Levels

The singly-ionized elements of group II in the periodic table (alkaline earth metals) have one electron

in their outer shell, and thus exhibit a relatively simple atomic structure. In a general sense, this is the

feature that allows them to be efficiently laser cooled and detected in single ion experiments and for that

reason they are the ions most commonly used. 9Be+ has a nuclear spin of I = 3/2 so its 2S1/2 ground

state splits into two hyperfine levels with F = 2 and F = 1 respectively. The relevant levels are shown in

Fig. 2.1. With them I show the main atomic transitions used in our experiments and label the lasers used in

addressing them as described in Sec. 3.2. We use two particular levels in the ground state (F = 1, mF = −1)

and (F = 2, mF = −2) as a qubit. In the experiments the ions are subjected to a quantization magnetic

field of 1 - 4 G, which causes Zeeman splitting of the mF sublevels of the hyperfine manifolds so that they

are spectroscopically resolved. At these magnetic fields the linear Zeeman effect is a good approximation to

the energy shifts, which is given by ∆E = μBgF mF B, where B is the magnetic field magnitude, μB is the

Bohr magneton, gF is the g-factor for the hyperfine level (+1/2 or -1/2 for F = 2 and F = 1 respectively),

and mF is the Zeeman sublevel.

Aluminum has one naturally-occurring isotope with nuclear spin I = 5/2. The singly-ionized atom has

two outer-shell electrons and thus exhibits a helium-like structure, with a 1S0 ground state. It was chosen

for our experiments because the 1S0 → 3P0 transition is particularly well suited as a stable reference for

an optical atomic clock [Rosenband 07, Rosenband 08, Chou 10]. The 3P0 state has a lifetime of 20.6(1.4)
11

Figure 2.1: Relevant Be+ levels and lasers.
12

Figure 2.2: Relevant Al+ levels and lasers.

s corresponding to a natural linewidth of 8 mHz, and has a measured frequency of 1121015393207857.4(7)

Hz. Details of the Al+ optical clock have been omitted from this dissertation to focus instead on the state

detection techniques. For this, the relevant levels are sketched in Fig. 2.2. We generally use the 1S0, mF = 5/2

and 3P0, mF = 5/2 levels as a qubit. Both the ground state and 3P0 energy levels, with angular momentum

J = 0, exhibit only a weak dependence on the magnetic fields through the Zeeman effect. The g-factor for

the 1S0 arises primarily from the nuclear g-factor and has been measured as g

1S0) = −0.00079248(14). The

g-factor for the 3P0 arises from a combination of the nuclear g-factor and mixing with states have J ̸= 0,

and has been measured as g

3P0) = −0.00197686(21). The difference in the g-factors for these two manifolds

is experimentally accessible and has been measured to be g[

1S0] − g[3P0] = 0.00118437(8) [Rosenband 07].

The 3P1 state with J = 1 has gJ ∼ 1.5.

The energy level diagrams also show the electronic transitions typically addressed by laser beams in
13

the experiments. These laser-ion interactions form the core of our experimental process.

2.2 Laser Ion Interactions

The evolution of a quantum mechanical system obeys the Schr ̈odinger equation,

−i~

∂ψ

∂t = Hψ, ˆ (2.1)

where Hˆ is the total Hamiltonian for the system. Analytical solutions to Eq. 2.1 exist for a relatively

small number of simple quantum systems. However, given Hˆ , the quantum dynamics of the system can,

in principle, be simulated numerically. The challenge in generating an accurate numerical model for an

experimental system is in producing a Hamiltonian that incorporates all of the significant effects present.

Conversely, it can be difficult experimentally to generate a physical system that approximates a relatively

simple Hamiltonian. Fortunately, trapped ions do provide an experimental realization of an interesting, and

relatively simple, quantum system. In this section I introduce the model we use to describe the behavior of

trapped ions under the influence of applied fields. This will be a brief outline that closely follows previous

treatments of the subject [Wineland 98], but I will point out particular details that relate to mixed-species

ion arrays, and which may differ slightly from earlier work.

We model the behavior of a trapped ion as spin-1/2 particles coupled to a quantum harmonic oscillator.

This model takes into account only a small subset of the internal and external degrees of freedom, and is

justified only to the extent that we can insulate the system from other effects. A real ion moves in three

dimensions, and contains many electronic eigenstates. The effects of these additional states in the system can

often be ignored because they do not participate in the system dynamics. For example, since the motional

sidebands are resolved, a pulse tuned to one resonance but we can laser cool all modes, and spectroscopically

isolate one mode, so that the “spectator” modes have little effect on the system dynamics. The Hamiltonian

for free evolution (without the application of external fields) of the system consists of terms representing the

ion motion, Hˆm, and the electronic states, Hˆ

e, such that,

Hˆ

0 = Hˆm + Hˆe. (2.2)
14

For a single mode of motion,

Hˆm = ~ωm(ˆnm +

1

2

), (2.3)

where ωm/2π is the frequency and ˆnm is the number operator for mode m. To include more modes of motion,

Eq. 2.3 would need to include a sum over the relevant degrees of freedom. Through optical pumping we can

spectroscopically isolate a two-level subspace of electronic states, so that we treat the electronic degrees of

freedom as to a spin-1/2 particle. In an N-ion array,

Hˆ

e = ~

∑

N

i=1

ωiSˆ

z,i, (2.4)

where ωiis the electronic resonance frequency for the ith ion and Sz,i is the z-component of the fictional spin

for the ith ion. Usually, we make the assumption that all ions of the same species have the same resonance

frequency. In practice, this means that any perturbation to the frequency does not exhibit a significant

gradient along the length of the ion array, which is the case in our experiments.

The Hilbert space of the system spans the eigenspace of Hˆ

0, consisting of N + 1 degrees of freedom (one for each ion and one for the motion). A particular eigenstate of the system is represented by,

|sl⟩ = |Mz⟩1|Mz⟩2 . . . |Mz⟩N |n⟩, with |Mz⟩i ∈ {|↓⟩i, |↑⟩i}. To simplify notation, we can drop the ion subscripts and assume that the ion spin states appear in order (i.e. |↓⟩1|↓⟩2 → |↓↓⟩). A general pure state in the

system can be written as

2

∑N

l=1

∑∞

n=0

cl,n|sl⟩|n⟩ (2.5)

where each |sl⟩ is one of the set of 2N spin states. In the interaction picture we consider how the state

amplitudes in Eq. 2.5 evolve under the application of applied laser fields.

The electric field, E(x, t), of an applied laser beam interacts with the electronic state of an ion through

the electric dipole operator, μd, producing the Hamiltonian,

Hˆ

I =

∑

N

i=1

−μd,i· E(x, t). (2.6)

For a plane wave E(x, t) = E0 cos(k · x − ωLt + φL)

ˆξ, and we identify ~Ω0 ≡ −μd · E0ˆξ. We can rewrite

Eq. 2.6 as

Hˆ

I = ~Ω0

∑

N

i=1

(S+,i + S−,i)(e

i(k·x−ωLt+φ) + e−i(k·x−ωLt+φ)

), (2.7)
15

where S+,i(S−,i) is the angular momentum addition (subtraction) operator for ion i. Switching to an

interaction picture,

H′

I ≡ e

iH0t/~HI e−iH0t/~

H′

I = ~Ω0Sˆ+,i exp(i[ηi(ˆae−iωmt + a

†

e

iωmt

) − ∆t + φ]) + h.c. (2.8)

where ηi ≡ k · xiis the Lamb-Dicke parameter for the ith ion, and ˆa (

ˆa

†) is the raising (lowering) ladder

operator for the relevant motional mode. To arrive at Eq. 2.8 we make the rotating wave approximation,

so that terms oscillating at a frequency ω0 + ωL are ignored. The evolution of the state amplitudes, cl,n,

is determined by the matrix elements ⟨sl

′ |⟨n

′

|Hˆ ′

I

|sl⟩|n⟩. In experiments we use laser pulses that are nearresonant either with the carrier transition at ω0 or a motional sideband at ω0 ± (n

′ − n)ωm. Thus, the only

non-zero matrix elements are those for which a single ion undergoes a spinflip. Furthermore, in the resolved

sideband regime we can safely take only the matrix elements where |ωL − ω0| ≈ |(n

′ − n)ωm|. That is, we

ignore the off-resonant coupling to other motional sidebands, which is justified because Ω0/ωm ≪ 1. With

these assumptions the coupling strength between two states can be written as [Wineland 98],

Ωi = Ω0 exp[−η

2

i /2](n<!/n>!)1/2

η

|n

′−n|

i L

|n

′−n|

n<

(η

2

i

). (2.9)

This matrix element involves a spinflip on only the ith ion and connects the Fock state n to the state n

′

, with

n< and n> the lesser and greater of these respectively. The generalized Laguerre polynomial, L

|n

′−n|

n< (η

2

i

)

has the ith ions Lamb-Dicke parameter, ηi, as its argument.

The discussion above applies to single photon transitions like those employed in interactions with

Al+. In the case of Be+ we drive transitions between Zeeman sublevels in the ground state hyperfine

manifolds using two-photon Raman transitions. In this case the 2P1/2 and 2P3/2 optically excited states act

as intermediary states that are never populated to a significant degree. In general, if the Raman detuning,

∆R, from the optically excited state is large compared to the on-resonance Rabi rate, Ω0, this state can

be “adiabatically eliminated” [Wineland 98], and we recover an interaction Hamiltonian that has the same

form as Eq. 2.8. However Ω0 is replaced with Ω′

0 = Ω2

0/∆R, ∆ = ω0 − (ω1 − ω2) and ηi ≡ (k1 − k2) · xi

,

where ω1, ω2, k1 and k2 are the frequencies and k-vectors of the the two Raman beams. From this we see

that, in order to have reasonably fast Rabi rates for sideband transitions, the Raman beams must not be
16

copropagating. In most cases, for Be+ Raman beams we choose k1 − k2 to lie along the trap axis so that

we address only axial modes of motion.

In our experiments we globally address all ions simultaneously with laser beams having a waist much

larger than the interion spacing. If N ions of the same species are simultaneously addressed with a sideband

pulse, there will, in general be N degenerate states that are simultaneously resonant, and the state evolution

becomes a complicated process involving coupling with all of these levels. In this case there will not generally

be analytical solutions for Schr ̈odingers equation and we resort to numerical integration. However, in some

cases, I have found the state evolution can be conceptually simplified by a basis transformation called the

Morris Shore transformation.

2.3 Morris-Shore Transformation

The Morris Shore transformation [Morris 83] simplifies the evolution of a degenerate two-level system

by factoring it into some number of independent two-level systems and isolated “dark” states. This would

be the situation, for example, of resonantly driving the F = 2 to F = 1 hyperfine transition of 9Be+, where

F = 2 is five-fold degenerate and F = 1 is three-fold degenerate at B = 0 T. It applies to an arbitrary

initial state with rf fields of arbitrary polarization. A conceptual schematic of this idea is given in Fig. 2.3.

With certain conditions, the Morris-Shore transformation can be extended to the case of systems with a

greater number of energy levels [Rangelov 06]. Both of these mathematical techniques apply to situations

encountered in the ion trap experiments described here. Since I have found this technique useful I will review

some relevant results and point out how they apply to ion experiments.

2.3.1 Two-Level Morris-Shore Transformation

Following the discussion in [Morris 83] and [Rangelov 06], we consider a quantum system Fig. 2.3

with two energy levels, Ea and Eb, with a number of degeneracies da and db respectively. The interaction

Hamiltonian describing this system’s evolution under the application of external fields may contain matrix

elements connecting each of the states at Ea to each of the states at Eb, making a total of da × db linkages.

However, we assume the off diagonal terms connecting states in the same energy level are zero. This situation
17

úb÷

ú ̄ ̄ ̄ ïñ 1ñ

ú ̄ ̄­ ïñ 0 0 ñ ú ̄­ ̄ ïñ 0ñ ú­ ̄ ̄ ïñ 0ñ

ú ̄ ̄ ̄ ïñ 1ñ

Ea

Eb

V

ñ

E =b Ñ( - ) w w 0 m

E =a 0

W1 W2 W3 W = ( ÷ 1 2 3 + + )1/2 W W W

úb÷ñ1 ñ2 úb÷ñ3

V÷

ñ

ñ1

úb ñ1 úb ñ2

úa ñ1 úa ñ2 úa ñ3 úa ñ4

úb÷ñ1 úb÷ñ2

úa÷ñ1 úa÷ñ2 úa÷3ñ úa÷ñ4

(a)

(b)

D

Figure 2.3: Conceptual diagram for the two-level Morris-Shore tranformation
18

can be represented by a Hamiltonian in the rotating-wave approximation of the form,

Hˆ =









ˆ0a Vˆ

Vˆ † ∆ × ˆIb









, (2.10)

where ˆ0a the null matrix in the subspace a (the result of choosing Ea = 0), Vˆ is a da ×db rectangular matrix

where Vij is the coupling strength between state i and state j. The lower right block matrix is the identity

matrix in the subspace b multiplied by the constant detuning ∆.

If ⃗c(t) is the state vector corresponding to the state of the system at some time, t, then the task

at hand is to solve simultaneously the set of da + db linear differential equations given by the Schr ̈odinger

equation,

i

d

dt⃗c(t) = Hˆ (t)⃗c(t). (2.11)

We can apply a unitary transformation Uˆ to the basis states such that ⃗c ′ = U⃗c ˆ and Aˆ′ = UˆAˆUˆ †, for

any operator A acting on the system. This preserves the values of all observables since Uˆ †Uˆ = ˆI. The

Morris-Shore unitary transformation mixes only the degenerate levels, so we consider unitary operators of

the form

Uˆ =









Uˆ

a 0

0 Uˆ

b









. (2.12)

The transformed Hamiltonian, UˆHˆUˆ † has off-diagonal blocks Vˆ ′ ≡ Uˆ

aVˆUˆ

b and its Hermitian conjugate. To

reduce this interaction to a basis with at most two-state couplings, we require that the rectanguluar matrix Vˆ ′

contain at most one non-zero element per row and per column. Notice that, under this condition, the square

matrices Vˆ ′Vˆ ′† = Uˆ

aVˆ Vˆ †Uˆ †

a and Vˆ ′†Vˆ ′ = Uˆ

bVˆ †VˆUˆ †

b

are both diagonal. This identifies the condition that

Ua and Ub diagonalize Vˆ Vˆ † and Vˆ †Vˆ respectively as a necessary condition for the desired transformation.

In [Morris 83] it is demonstrated that Vˆ Vˆ † and Vˆ †Vˆ have the same eigenvalues ignoring the |da − db| null

eigenvalues of the larger matrix, and that the previous condition on Uˆ

a and Uˆb always exists, providing the

desired transformation.

The basis states after the transformation, expressed in terms of the original basis, for states at Ea

and Eb are the eigenvectors of Vˆ Vˆ † and Vˆ †Vˆ respectively. If λi

is a particular non-zero eigenvalue, those

eigenvectors corresponding to λi form a two-state system with the effective Rabi rate √

λi.
19

In Fig. 2.3 (b) we sketch one application of the Morris Shore transformation for state evolution in

ion qubits, with qubit states denoted |↓⟩ and |↑⟩ respectively. We consider a string of three ions initialized

in the electronic state |↓⟩ and prepared in the Fock State |1⟩ for a particular mode of motion. A laser pulse

tuned to resonance for the first red sideband of the qubit transition is applied to the ions, connecting the

initial state to the three possible excited states, which are in the ground state of motion and have one qubit

excitation. The coupling strength between the ground state and excited states, represented by Ωi for ith

ion, might be unequal due to imbalanced laser illumination or unequal motional amplitudes. These coupling

strengths affect the eigenstates and the Rabi rates for the transformed system.

For zero detuning, the Hamiltonian matrix is

Hˆ =

























0 Ω1 Ω2 Ω3

Ω1 0 0 0

Ω2 0 0 0

Ω3 0 0 0

























, (2.13)

so that

Vˆ =

[

Ω1 Ω2 Ω3

]

. (2.14)

Since there is only one state in the ground-state manifold there will be only one non-zero eigenvalue

of Vˆ †Vˆ , given by Ω′2 = Ω2

1 + Ω22 + Ω23

. This eigenvalue corresponds to the normalized eigenvector

|b

′

1

⟩ = (Ω2

1 + Ω2

2 + Ω2

3

)

−1/2

















Ω1

Ω2

Ω3

















(2.15)

The factorization also identifies the “dark” states of the system (state vectors that do not evolve under

application of Hˆ ) as vectors in the null space of Vˆ †Vˆ . In the present example the state (η

2

1 +η

2

2

)

−1/2

(−Ω2|↑↓↓

⟩ + Ω1|↓↑↓⟩) corresponds to a zero eigenvalue.

The result above can be generalized to an arbitrary number of ion qubits undergoing red sideband

excitation from the Fock state |1⟩. The Rabi rate for the transformed system is given by

Ω

′2 =

∑

N

i=1

Ω

2

i

(2.16)
20

and the excited state can be written as

|b

′

1

⟩ =

1

Ω′

(

Ω1| ↑↓ . . . ↓⟩ + Ω2| ↓↑ . . . ↓⟩ + . . . + ΩN | ↓↓ . . . ↑⟩). (2.17)

This transformation is applied to the simulations of some experiments in Sec. 6 as well as Sec. 5.2.3.

2.3.2 Multilevel Morris-Shore Transformation

This convenient basis change has also been applied to systems with more than two degenerate manifolds [Rangelov 06]. If the system has NE energy levels, each level will have some degree of degeneracy, dj

with j ∈ [1, NE]. The total number of states is ∑

j

dj , but we consider, as before, a Hamiltonian, Hˆ that

only connects states in adjacent energy levels. The maximum number of linkages in the Hamiltonian for the

system is then ∑

j<NE

djdj+1. The desired change of basis transforms the system into a set of isolated systems

with, at most, NE levels each. This can significantly simplify the evolution of the system because now the

number of non-zero off-diagonal matrix elements Hˆ is ∑

j<NE

dj,min where dj,min is the lesser of dj and dj+1.

The Hamiltonian for a degenerate NE-level system can be expressed in block-matrix form as

Hˆ =

























ˆ0 Vˆ

1

ˆ0 · · ·

Vˆ †

1 ∆2 Vˆ

2

ˆ0 Vˆ †

2 ∆3

.

.

.

.

.

.

























, (2.18)

where each matrix ˆ0 is the null matrix with the appropriate dimensionality dj × dj

′ , the matrix Vˆ

j contains

all the elements connecting states in system j to system j + 1, and ∆ˆ

j is the scalar detuning of the applied

fields from the state j multiplied by the identity. In analogy to the two-level transformation described above,

this Hamiltonian can be transformed by a block-diagonal unitary matrix of the form

Uˆ =

























Uˆ

1

ˆ0 ˆ0 · · ·

ˆ0 Uˆ

2

ˆ0

ˆ0 ˆ0 Uˆ

3

.

.

.

.

.

.

























. (2.19)
21

E1

E2

ú ̄ ̄ ̄ñ

ú ̄ ̄­ñ ú ̄­ ̄ñ ú­ ̄ ̄ñ

ú ̄­­ñ ú­ ̄­ñ ú ­ ̄ ­ ñ

ú0ñ

ú1ñ

ú2ñ

ú ̄ ̄ ̄ñ

ú2ñ

W1

W2

3W1

"####

2W2 W2 W2

E3

ú1ñ

ú0ñ

úb÷ñ1 úb÷ñ2 úb÷ñ3

úc÷ñ1 úc÷ñ2 úc÷ñ3

úb÷ñ1

ú­ ̄ ̄ñ ú ̄  ̄­ ñ "### ( + + ú ̄ ̄­ñ)

3

1

=

úc÷ñ1

ú  ̄ ­­ ñ ú­ ̄­ñ "### ( + + ú ­ ̄ ­ñ)

3

1

=

Figure 2.4: Conceptual diagram for a multilevel Morris-Shore transformation

The transformed Hamiltonian,

Hˆ ′ = UˆHˆUˆ † =

























ˆ0 Uˆ

1Vˆ1Uˆ †

2

ˆ0 · · ·

Uˆ

2Vˆ †

1 Uˆ †1 ∆1 × ˆI Uˆ

2Vˆ

2Uˆ †

3

ˆ0 Uˆ

3Vˆ †

2 Uˆ †2 ∆2 × ˆI

.

.

.

.

.

.

























, (2.20)

has upper off-diagonal blocks Vˆ ′

j = Uˆ

jVˆ

jUˆ †

j+1 representing the interactions in the transformed Hamiltonian, which must be diagonal after removing null rows and columns. As a consequence, the matrices

Vˆ ′

j Vˆ ′†

j = Uˆ

jVˆjVˆ †

j Uˆ †j

and Vˆ ′†

j Vˆ ′

j = Uˆ

j+1Vˆ †

j Vˆ

jUˆ †

j+1 must be diagonal. If all Vˆ ′

j Vˆ ′†

j

are to be diagonal under the transformation Uˆ, then Uj+1 must simultaneously diagonalize Vˆ †

j Vˆ

j and Vˆj+1Vˆ †

j+1 for j ∈ [1 NE −1].

Thus, we arrive at the stringent constraint that

[

Vˆ †

j Vˆ

j , Vˆj+1Vˆ †

j+1]

= 0, (2.21)

where brackets indicate the commutator. If this condition is satisfied, any eigenstate |ψ

(μ)

j

⟩, is connected via

Hˆ ′to at most two other eigenstates |ψ

(μ)

j−1

⟩ and |ψ

(μ)

j−1

⟩ where we have identified the eigenvectors in any two

adjacent energies with respect to a common eigenvalue λ

(μ)

j

. The eigenvalues are, again, the square of the

coupling strength between the two levels.

Following the example of Fig. 2.3 (b) we present a concrete example of the application of the generalized Morris-Shore transformation applied to trapped ions. Similar calculations have been used in Sec. 6 in
22

analyzing the state evolution of multiple ions under global sideband pulses. The energy levels and couplings

are sketched in Fig. 2.4. As in the previous example, we consider three ions each prepared in the state | ↓⟩

and addressed by a red sideband laser pulse. This time they begin in the motional Fock state |2⟩M.

There are three energy levels to consider. The ground state of the system is non-degenerate and the

red sideband interaction connects to all three states at energy E2 in the untransformed basis. Likewise each

of these states connects to two states at energy E3, making six total non-zero matrix elements between the

upper two energy levels. The off-diagonal blocks in the Hamiltonian can be written as

V1 =

[

iΩ1 iΩ1 iΩ1

]

(2.22)

V2 =

















iΩ2 iΩ2 0

iΩ2 0 iΩ2

0 iΩ2 iΩ2

















. (2.23)

We can confirm that [

Vˆ †

1 Vˆ

1, Vˆ2Vˆ †

2

]

= 0. Note that we have assumed equal motional amplitudes and equal

illumination for all ions, which is necessary for satisfying the condition of Eq. 2.21. In the Lamb-Dicke limit,

the only other conditions under which Eq. 2.21 holds is the trivial case of with zero motional amplitude for

two of the ions, however, outside of the Lamb Dicke limit other non-trivial solutions to Eq. 2.21 may exist.

The product Vˆ

1Vˆ †

1

is a scalar since the ground state is non-degenerate, and the coupling strength for

connecting the ground state to the next level is given by √

Vˆ

1Vˆ †

1 =

√

3Ω1. The complementary product, Vˆ †

1 Vˆ

1, is a 3 × 3 matrix with a single non-null eigenvector corresponding to the entangled state

|b

′

1

⟩ = 1/

√

3(| ↓↓↑⟩ + | ↓↑↓⟩ + | ↑↓↓⟩)|1⟩M (see Sec. 6), which connects to | ↓↓↓⟩|2⟩M. The evaluation of eigenvectors and eigenvalues for the matrices, Vˆ †

2 Vˆ

2, Vˆ2Vˆ †

2

gives the states involved in coupling levels E2 and E3

and their respective strengths. Since Eq. 2.21 is satisfied, we know |b

′

1

⟩ is an eigenvector also of Vˆ

2Vˆ †

2

. It

has eigenvalue 4Ω2

2 and connects to the state |c

′

1

⟩ = 1/

√

3(| ↑↑↓⟩ + | ↑↓↑⟩ + | ↓↑↑⟩)|0⟩M in the manifold of E3.

There are two other eigenvectors and eigenvalues for both of the matrices Vˆ †

2 Vˆ

2 and Vˆ2Vˆ †

2

, and each of the

sets defines an isolated two-level system under the red sideband interaction.

In this section we have reviewed the mathematical formalism behind a useful basis change called the

Morris-Shore transformation. In certain trapped ion experiments this allows us to express complicated state
23

dynamics in terms of the evolution of isolated systems each with a small number of equally spaced states.

In particular, when considering sideband pulses globally addressing multiple ions, as in the experiments of

Ch. 6 and Ch. 5 this transformation represents a conceptual and numerical simplification.
Chapter 3

Apparatus

3.1 Ion Trap

3.1.1 Ion Trap Theory

In this section I review some of the basic features of ion traps and point out how some results relate

to practical experimental considerations. The fundamental property of an ion that gives us an experimental

handle for controlling its external degrees of freedom is its charge. The Lorentz force law relates the force on

a particle from local electric and magnetic fields, E and B respectively, to a particle’s charge, Q and velocity

v by

F = Q(E + v × B). (3.1)

These fields provide an experimenter with the means to influence the motion of any charged particle. The

goal may be to accelerate a proton to near the speed of light, or to hold an atomic ion nearly motionless as

in the experiments of this thesis.

Various configurations of electric and magnetic fields have been devised to trap ions. Many aspects

of these methods have been compiled in recent books on the subject [Ghosh 95, Major 05], which are good

overviews of the science and history of ion trapping and are useful references toward primary sources.

Two major categories of ion traps are the Penning trap and the Paul trap. The Penning trap combines

a strong magnetic field with electric fields to provide three-dimensional harmonic confinement. The Paul

traps discussed here use time-varying electric fields (sometimes in combination with static electric fields) to

achieve the same goal.
25

x

`

y

`

V0cosHWT tL

V0cosHWT tL

0

0

(a)

U , rf gnd 0

V cos( t) 0 T W

z

y

x

0 V DC, rf gnd

(b)

Figure 3.1: (a) A cross section of the ideal linear Paul trap. Radio frequency voltages applied to the electrodes

(gray corner regions) set up an oscillating quadrupolar potential (contour lines). The electric field (arrows)

ideally vanishes at the trap center. Any displacement of the ion (green circle) from the trap center leads

to driven motion (micromotion) perpendicular to the direction of displacement at the frequency ΩT . (b)

A model for a simple linear Paul trap. In addition to the radial confinement provided by rf fields, a static

potential U0 applied to the endcaps (yellow) provides axial confinement. The center electrodes (white) are

held at DC ground.

For a particle of mass m and charge Q confined harmonically in three dimensions, the external potential

energy can be described generally by

QΦ = 1

2

∑

3

i=1

kiu

2

i

, (3.2)

where i ∈ {x, y, z} and kiis the oscillator spring constant in coordinate ui. The Laplace equation, ∇2Φ = 0

provides the constraint

∑

i

ki = 0, (3.3)

such that either ki = 0 for all i (the trivial case of a free particle) or ki < 0 for some i. Since a negative spring

constant is anticonfining this means that no configuration of static electric fields can be used to harmonically

trap charged particles - one case of a general result called Earnshaw’s theorem. However, Earnshaw’s theorem

does not preclude the use of time-varying electric fields to confine ions.

The linear Paul traps in this thesis use RF electric fields to constrain a chain of positively charged

atomic ions to one axis, and DC electric fields to provide stable confinement in the third dimension. The

development of this hybrid rf-DC trap was preceded by the development of spherical Paul traps that used

RF fields for confinement in all three dimensions [Schuessler 69, Motz 67]. In either case a single ion is

pushed toward a null in the RF electric fields by their time averaged effects (called the ponderomotive force)

[Motz 67]. If there are multiple ions in a trap they will experience mutual repulsion according to Coulomb’s
26

law. In the case of a spherical Paul trap they will tend to displace each other from the null of the RF field,

and will experience driven motion. This RF ion motion can cause several deleterious effects in the context of

precision spectroscopy and quantum information processing [Berkeland 98]. The development of the linear

Paul trap was motivated by the idea of reducing these effects while trapping multiple ions in a single trap

[Prestage 89, Raizen 92]. The linear Paul trap sets up a nodal line, along which the RF electric fields vanish,

constraining the ions to one dimension through the ponderomotive force then “capping” the ends of the

nodal line with DC fields.

The ideal linear Paul trap consists of four parallel, rod-like electrodes. I follow the convention that

the rods extend in the z direction, which defines the axis of the ion chain to be at the trap centroid in the

x-y plane. I define the x direction to connect two diagonally opposing electrodes (x electrodes) and the y

direction to connect the remaining two electrodes (y electrodes). With these definitions, a radio frequency

voltage V0cos(ΩT t) applied to the x electrodes, while the y electrodes are held at ground, sets up an oscillating

quadrupolar potential given by

Φrf =

V0cos(ΩT t)

2

(

1 +

x

2 − y2

R2

)

, (3.4)

where R denotes the shortest distance from the trap axis to an electrode. A snapshot in time of such a

potential is depicted in Fig. 3.1(a). The contour lines represent equipotentials and define the borders of

the four electrodes (gray corner regions). Any deviation from the hyperbolic electrode border will lead to

higher-order terms in Φrf , as well as a weakening of the quadrupolar potential. For a radial ion position,

r =

√

x

2 + y2 ≪ R these effects are small and can be taken into account by multiplying Eq. 3.4 by a factor

on the order of unity [Wineland 98]. The arrows in Fig. 3.1(a) show the relative magnitude and direction of

the electric field

Erf = −∇Φrf =

V0cos(ΩT t)

R2

(−xxˆ + yyˆ), (3.5)

at a particular RF phase. This oscillating field vanishes at the trap axis. As the ion is displaced radially from

the center of the trap it gains kinetic energy due to its driven motion at frequency ΩT , which is in a direction

perpendicular to the direction of displacement. This geometry becomes important when considering the

measurement and compensation of excess micromotion.
27

I sketch a model for the three dimensional geometry of a linear Paul trap in Fig. 3.1(b). Two

diagonally opposing cylindrical electrodes carry RF potential, while the remaining two segmented electrodes

are kept at RF ground. We apply a DC voltage U0 to the outer segments of these electrodes (called endcaps)

to provide confinement of the ions in the axial direction. To second order this produces a harmonic trapping

potential in the z direction and an “antitrapping” potential in the radial direction that is given by

QΦs =

1

2

kz

(

z

2 −

[

εx2 + (1 − ε)y

2

]

)

, (3.6)

where kz ≡ 2κU0Q is the axial spring constant for an ion of charge Q, and both κ and ε are geometrical

factors. For the harmonic approximation in Eq. 3.6 to be valid we must have κu2 << 1 for u = x, y, z.

Consider a single ion of mass m and charge q trapped in a potential given by Φ = Φs + Φrf . In the

axial direction, only DC terms appear in the potential energy V = −QΦ. Treating the problem classically,

the equation of motion

z ̈ = −

kz

m

z, (3.7)

is the equation for a simple harmonic oscillator, which is solved by

z = Re[z0e

i(ωzt+φz)

], (3.8)

where ωz =

√

kz

m . The amplitude, z0, and phase φz are given by initial conditions.

In the radial directions, the equations of motion involve contributions from the DC potential (Eq. 3.6)

and the oscillating RF potential (Eq. 3.4). They can be cast into a form identical to the canonical Mathieu

equation [McLachlan 47],

d

2ui

dζi

= [a + 2qicos (2ζ)]ui = 0, (3.9)

with a = −4QκU0/mΩ

2

T = −2kz/mΩ

2

T

(for the symmetric case with ε =

1

2

) and qx = −qy = 2QV0/ΩT mR2.

The solution to this equation to lowest order in qiis

ui(t) = Aicos(ωit + φi)[1 + qi

2

cos(ΩT t)], (3.10)

where Ai and φi are determined by initial conditions. In the preceding analysis we have assumed the trap

strength is equal in the x and y coordinates and aside from the RF phase in Eq. 3.10 the solutions for

i = x, y are the same. In the following discussion we use i = r to refer to either coordinate.
28

The motion at frequency

ωr = (a + q

2

r /2)1/2ΩT /2 (3.11)

is called secular motion and is the higher amplitude motion in the experiments discussed here since a ≪ qr ≪ 1.

We refer to the driven motion at frequency ΩT as micromotion. This motion is important because, in the

frame of the ion, it can divert optical power of an applied laser pulse into micromotion sidebands, depleting

the carrier, and potentially affecting the efficiency of laser cooling. Typically in our experiments the ions

are cooled to low temperatures (∼ 0.1 mK) so the motional amplitude, Aiqi, is small and these effects of

micromotion are negligible. In the case of ion frequency standards, where the relativistic time-dilation shift

of δν/ν = ⟨v

2

⟩/2c

2 may cause appreciable shifts at relatively low velocities (⟨v2

⟩ = 1 m2/s

2

), micromotion

amplitudes at these low temperatures cannot be ignored.

Considering only the secular motion (that is averaging over the micromotion), the ion behaves as if

it is confined in a harmonic potential well in the radial direction. The spring constant for this harmonic

oscillator is given by kr = mω2

r = 4Q2V

2

0 /mR4 − kz/2. The first term arises from the radial RF fields, while

the second term accounts for the antitrapping effect of the axial trapping fields. If we write

ar ≡ 4Q

2V2

0 /mR4

, (3.12)

then we can define an effective pseudopotential, ΦP as

QΦP =

1

2

arr

2

/m. (3.13)

This identifies the radial secular motion as harmonic motion and is called the pseudopotential approximation.

Note that the harmonic pseudopotential is different from the axial harmonic potential in that it is

inversely proportional to the ion mass. Thus, heavier ions are bound less tightly to the axis of the trap.

The solution to the Mathieu equation given in Eq. 3.10 is valid only for ai ≪ qi ≪ 1. In the case that qi

approaches one, ΩT ≃ ωi and the ion motion can become unstable. This is the key feature that enables the

use of radiofrequency electric fields for mass spec applications. Since ωx ∝ 1/m lighter mass ions have a

larger q-parameter, and enter the region of instability at lower V0.

In setting up the single ion experiments discussed in this thesis, we sometimes accidentally load

unwanted ions of different mass from the experimental ions. If VO can be raised to a sufficient level we
29

can, in principle, selectively eject unwanted lighter-mass ions. This can be difficult in practice because, in

designing the ion traps, we prefer radial confinement as high as possible to maintain linear crystals (ωr ≫ ωz)

while being able to resolve the motional sideband spectrum. Without being able to significantly raise V0 or

lower ΩT we cannot significantly affect trap stability in this way. For unwanted heavier-mass ions, however,

we can use a different technique. By raising U0 (or equivalently lowering V0) we decrease ki = ar/m − kz/2.

A heavier mass ion will reach an unstable condition where ki = 0 at lower U0 (or higher V0), and will be

ejected from the trap. We have found this technique to be reliable for eliminating ions with mass ratios as

low as μ = 1.11 (i.e. separating BeH+ from Be+).

3.1.2 Experimental Trap

One of the major design goals for the ion traps used in our experiments is a high radial confinement

frequency. This allows for a well-resolved motional mode spectrum while maintaining fast sideband pulses.

With the constraint that ωr > ωz for linear chains of ions, the scale of the trapping frequencies is set by the

radial confinement strength. Maximizing this amounts to maximizing the parameter ar of Eq. 3.12, which

scales as 1/R4(ωr ∝ 1/R2), so, for high confinement frequencies, we desire a small electrode-ion distance.

However, practical considerations such as machining tolerances often limit how small trap features can be

made. In addition, experiments on individual ions often suffer from anomalous heating of the ion secular

motion [Turchette 00, Wineland 98]. This can reduce the fidelity of quantum protocols and cause timedilation shifts in ion-based atomic clocks [Rosenband 08]. Although the exact mechanism for this heating is

an active question [Dubessy 09], controlled observations [Deslauriers 06] of the heating rate for a particular

mode show that n ̄ ̇ ∝ Rp where p ∼ 4.

In the experiments described in this thesis, we used two different experimental systems. Most of the

work was done in a system designed for experiments on Be+ and Al+, while the experiments in the final

chapter involve a separate system designed for Mg+ and Al+. Aside from the two laser systems that interact

with Al+, the experiments, including the linear Paul traps are completely separate. To compare the relevant

parameters of these two traps, we sketch a general picture of the electrodes surrounding the trap region in

Fig. 3.2(a), and tabulate values for the two traps in Fig. 3.2(b). The radial confinement frequency is also
30

R

wg

wc

wv

V cos( t) 0 T W

U U0

0

wv

wh

Perpendicular to z-axis Along z-axis

V cos( t) 0 T W

V cos( t) U 0 T W

0

U0

(a)

(b)

Trap wv [μ m] wh [μ m] wg [μ m] wc [μ m] R [μ m] V0 [V] ΩT [MHz] U0 [V]

Wafer 200 450 20 200 250 ∼ 400 103 12

Knifeblade 600 600 200 1600 425 1000 59 100

Figure 3.2: (a) Diagram of a generic trap structure showing the relevant physical dimensions. These measurements, together with the applied electric fields, and geometrical factors on the order of unity, determine

the trap frequencies. (b) The dimensions for the two traps with the Be-Al trap labeled “wafer” and the

Mg-Al trap labeled “knifeblade”.

proportional to the applied RF voltage, V0. To obtain higher ωr we mount the ion trap at the end of a

coaxial, quarter-wave RF resonator and typically obtain a quality factor on the order of 200. Coupling 5-20

W of RF power into the resonator achieves the voltages tabulated in Fig. 3.2(b).

A picture of the wafer trap used in Ch. 5 is shown in Fig. 3.3(a). This trap was originally used

in a sympathetic cooling experiment [Barrett 03], then adapted for work on quantum logic spectroscopy

[Schmidt 05] with 9Be+ and 27Al+. With a single 9Be+ ion we generally work with conditions such that

{ωx, ωy, ωz} ≃ {14, 13, 4} MHz. The design is based on a gold coated, laser-machined, alumina substrate

to define both the RF and DC electrodes [Rowe 02]. Two identical wafers (one flipped relative to the other

by 180 degrees about the long axis) separated by an insulating spacer produce a close approximation to the

model of a linear Paul trap in Fig. 3.1(b). In addition, the machining tolerances for the microfabricated

electrode structures can be very low, leading to well characterized trap conditions.

The second trap was designed specifically for the Al+ optical clock, where factors limiting the clock

accuracy, such as micromotion compensation and ion heating rate, are a more severe limitation. Thus the
31

Figure 3.3: (a) A photograph of the gold-plated alumina wafer trap. (b) A drawing of the knife-blade style

trap used in the experiments of Ch. 6 with the end of a copper RF resonator also shown.

trap is larger and constructed from solid metal electrodes with the nearest insulating surface relatively far

from the ion, reducing the effects of stray charges. This trap was used in the experiments of Ch. 6 where we

use 25Mg+ as a sympathetic cooling and quantum logic ion. For a single 25Mg+ ion and the parameters in the

table of Fig. 3.2(b), we obtain frequencies of {ωx, ωy, ωz} ≃ {6.7, 4.9, 3.0} MHz. In this case, the difference

in the x and y mode frequencies comes primarily from applying DC voltages to both center electrodes.

Each ion trap is mounted in an ultrahigh vacuum system at the end of a quarter-wave coaxial resonator.

The basic features of both vacuum systems are the same. The traps are surrounded by a cylindrical glass

envelope featuring several flat windows that allow optical access for both the laser beams and an ∼F/1

imaging system. This glass envelope connects to a stainless steel vacuum chamber via a glass-metal seal.

The vacuum chamber consists of several arms that house an ion pump, a titanium sublimation pump and

an ion gauge for monitoring pressure. We baked out the assembled systems at a temperature near 200 C 1

while pumping on it with a turbopump, which was later disconnected. The final pressure we achieve for

both systems is near 1 × 10−11 Torr. The quarter wave resonator is partly inside and partly outside the

vacuum chamber, with the inner conductor transmitted through a vacuum feed-through. The RF trap drive

is coupled using an antenna into the resonator mode and the coupling is optimized by monitoring the reflected

power through a directional coupler. The RF power incident on the resonator can be switched between two

levels using a TTL switch and an attenuator. In order to maintain ions in the trap while switching between

these two arms of the circuit we found it was necessary to adjust the cable lengths properly so that the RF

1 The exact value differed between the two systems due to different temperature sensitivity of the components.
32

phase was matched at the resonator. The voltages applied to the endcap electrodes and the separate bias

electrodes are fed into the vacuum system via a feed through and filtered using RC filters both internal and

external to the vacuum chamber providing corner frequencies near 10 kHz.

3.2 Laser Systems

The lasers required for interacting with our ions are all at frequencies in the ultraviolet spectral range.

The laser sources involve frequency stabilized visible or infrared lasers that are then frequency doubled or

quadrupled using nonlinear optical crystals in a buildup cavity. I briefly describe the laser systems used to

interact with Be+, Al+ and Mg+ separately here.

3.2.1 Be+ lasers

To interact with the Be+ atomic system, we require lasers at 313 nm. We derive these beams from

three separate frequency-doubled dye lasers at 626 nm. For convenience, we call the laser systems: 1) Blue

Doppler (BD), which is used for Doppler cooling and state detection, 2) Red Doppler (RD), which is used

for repumping and state preparation, and 3) Raman, which is used for driving stimulated Raman transitions

between the qubit levels. All of these laser systems are home-built, verdi-pumped, ring dye lasers in a bowtie

configuration. The basic design was developed in house over several years [Bergquist 10], and I will give only

a brief overview of the system here.

A thin jet of Rhodamine dye (dissolved in ethylene glycol) passes through the strong focus of the laser

cavity with the liquid-air interfaces oriented at Brewster’s angle with respect to the cavity mode. The 532

nm Verdi laser at a power between 2 W and 6 W is focused onto a point in the dye stream that is coincident

with the focus of the cavity mode. Although the emission spectrum of the Rhodamine dye is many tens

of nanometers wide, several intracavity elements limit the laser gain to a narrower frequency region around

626 nm. From coarsest to finest they are: 1) a birefringent filter, for which the angle is adjusted with a

micrometer screw, 2) a current controlled etalon (thin etalon), and 3) a piezo-mounted etalon (thick etalon).

The thick etalon is actively stabilized with a slow servo loop to the maximum laser output using a lock-in

amplifier. One mirror of the bowtie cavity is mounted on a piezoelectric transducer. This provides a means
33

to stabilize the overall cavity length to an external reference cavity (finesse approximately 300). To lock

to the reference cavity, we use a Hansch-Couillaud (H.-C.) [Hansch 80] locking scheme based on the phase

shift between beams transmitted and reflected from the reference cavity. The difference signal from 2 photo

diodes is amplified and fed back to the dye laser cavity piezo. The combination of these frequency selective

elements produces a laser linewidth of approximately 100 kHz, which is sufficient for precisely interrogating

the ∼20 MHz optical resonance in Be+. In addition, we lock the laser to a molecular absorption feature in

I2 by performing saturated absorption spectroscopy in a gas cell [Preston 96]. Here, two weak laser beams

(probe beam and reference beam) traverse the length of the I2cell. The probe beam is overlapped using a

polarizing beam splitter (PBS) with a strong, counter-propagating saturation beam. The transmitted probe

and reference beams are focused onto an autobalancing difference detector. The electronic signal from the

difference detector is demodulated and amplified to feed back on a ring piezo in the reference cavity. This

feedback loop stabilizes the reference cavity, and in turn the dye laser, to the molecular iodine signal.

Depending on the Verdi pump power, these dye lasers produce between a few hundred mWs and 1

W of laser power at 626 nm. The bulk of this light is coupled into an optical fiber and transmitted to a

separate laser table. Here, the visible light is frequency doubled in a build up cavity. The design of the

frequency doubling cavity is nearly the same for all three Be+ laser systems. A type-II, critically phase

matched, Brewster-cut barium triborate (BBO) crystal of 1 cm length is mounted at the strong focus of a

bow tie cavity at Brewster’s angle. We typically couple approximately 90% of the incident light into the

cavity and achieve a build up between 50 and 100. We compensate astigmatism introduced by the BBO

crystal by using an appropriate angle of reflectance on the curved mirrors of the build up cavity. We lock the

length of the cavity to the input light using an H.-C. locking scheme with feedback on a small, piezo-mounted

mirror. This produces a beam of 313 nm light of 10-25 mW in the case of the RD and BD lasers. In the

case of the Raman laser, with significantly higher visible laser power, we obtain approximately 100 mW. The

astigmatism of these UV beams is corrected using the combination of a cylindrical lens and a spherical lens.

From these UV laser sources, we derive pulses of the correct frequency and duration for experimental

operation using a series of AOMs. I will describe the AOM system independently for the three laser systems.

For the RD laser, we need separate beams that connect the two ground state hyperfine manifolds to the P1/2
34

resonance. We lock the visible laser to a molecular iodine feature such that the second harmonic is centered

between the resonances of the two hyperfine ground state manifolds and the P1/2. In order to produce the

two frequencies required for repumping, we span the hyperfine splitting using two AOM’s tuned to produce

beams shifted by ±636 MHz. Both beams are switched using a 200 MHz AOM in double pass configuration

(Fig. 3.2.1).

vert. pol.

BD Split

+200

BD

Detuned

Sw

+200

BD SW

x2

+200

50/50

RD

+200

+600

BD

+200 BD detuned

+600 BD

BD doubler

BBO

Blue Doppler (BD) and 

Red Doppler (RD)

Beam lines

RD doubler

BBO

vert. pol. 50/50

-636

+636

RD 

split1

RD 

split2

RD SW

x2

+200

BD

Figure 3.4: Blue Doppler and red Doppler AOM schemes.

For Doppler cooling and detection with the BD laser, we require one beam that is tunable across

the |↓⟩Be → |P⟩Be optical resonance (BD), and a separate beam detuned 200 MHz lower in frequency (blue
35

Doppler detuned or BDD). Near the output of the doubling cavity, a 200 MHz AOM splits the near resonant

beam toward a double pass AOM with a tuning range of approximately 40 MHz (Fig. 3.2.1). A third 200

MHz AOM is used to switch the detuned beam and to overlap it with the near resonant beam. The two

beams from the BD laser are overlapped with the two beams of the RD laser using a 50/50 beam splitter.

Together, they are spatially filtered using a 25 um pinhole at the focus of a telescope and directed towards

the trapping region. All of the beams need to be σ

− polarized. The polarization is filtered using a Wollaston

prism and adjusted with a quarter wave plate mounted on a mirror mount so that the angle can be precisely

adjusted.

The Raman laser is tens of GHz detuned from either of the 2S1/2 →2 P1/2 or 2S1/2 →2 P3/2 resonances.

Due to this detuning, the linewidth and the absolute frequency of the Raman laser is not as critical as for

BD or RD. For this laser system we neither stabilize to an external reference cavity nor to molecular iodine.

We require two beams separated in frequency by the hyperfine splitting (approximately 1.2 GHz) with the

difference frequency tunable in order to scan across the ion motional spectrum. We accomplish this with a

double pass AOM in both beams as well as a single pass switch AOM in both beams, all centered near 200

MHz. One set of AOMs is aligned to diffract the beam into the -1 order producing “red Raman” (RR) while

the other set of AOMs is aligned to diffract the beam into the +1 order producing “blue Raman” (BR). The

double pass AOM in the RR beamline has a tuning range of 40 MHz to provide the necessary adjustments in

the difference frequency (Fig 3.2.1). Both beams are spatially filtered and focused into the trap with beam

waists of approximately 20-40 μm. The BR beam is set to π-polarized and the RR beam is σ

−-polarized,

both using a quarter-wave plate in combination with a half-wave plate, to drive transitions between the two

qubit levels.

3.2.2 Al+ lasers

To interact with the Al+ atomic system we require two laser sources near 267 nm. I will call them

respectively the 3P0 laser and the 3P1 laser referring to the atomic resonance each one drives. The basic

design of both of these systems is the same. The laser source is a commercial fiber laser near 1070 nm with a

approximately 50 mW in a linewidth of a few kHz. This light is amplified to a power between 1 W and 2 W
36

To Trap

RR 

200 X2

BR 

200 X2

RR Switch 

200 

BR Switch 

200

Raman Doubler

BBO

Figure 3.5: Raman AOM scheme.
37

Figure 3.6: Diagram of the fiber-noise cancelation setup.

with a home-built Yb-doped fiber amplifier pumped with laser diodes (4 W) near 976 nm. The 976 nm pump

light is combined with the 1070 nm signal light into a dual-clad fiber using a wavelength-division multiplexer

and they co-propagate in the gain fiber. To limit the effect of optical feedback to the laser source we use two

optical isolators - one between the laser source and the amplifier and another at the output of the amplifier.

The 1070 nm light is frequency doubled using non-critical phasematching (T ∼ 140 C) in a lithium triborate

(LBO) crystal (2 cm in length) surrounded by a build-up cavity. Both laser systems achieve several hundred

mW of visible power, most of which is directed through an optical fiber to the optical table that supports

the ion trap. A small part of the visible light is sent via optical fiber towards an ultra high-finesse cavity.

The laser is locked to the cavity using a Pound-Drever-Hall (PDH) locking scheme that feeds back on an

AOM at the fiber laser output. The visible light is frequency doubled in a bowtie cavity using BBO in the

case of the 3P0 laser and potassium dideuterium phosphate (KD*P) in the case of the 3P1 laser. The 3P1

laser typically produces 5 - 25 mW of UV power while the 3P0 laser typically produces 1 mW. For most

experiments only a small fraction of this laser power is needed.

The 3P0 laser needs to have an ultranarrow linewidth on the order of 1 Hz for addressing the 1S0 → 3P0

resonance. For this reason the reference cavity (finesse ∼ 50000) used to stabilize it is housed in a vacuum
38

chamber that is carefully isolated from vibrations and thermal fluctuations [Young 99]. In addition, the

laser beam can acquire noise due to vibrations in the optical fibers that transmit visible light between

optical tables. A beam coupled into the fiber with a linewidth of 1 Hz might be broadened to the kHz level

at the output facet. This noise is actively canceled by monitoring the phase of a beatnote between a reference

beam near the input to the optical fiber and a partial reflection from the fiber endface [Bergquist 92, Ma 94].

A diagram of the basic setup for canceling fiber noise is given in Fig. 3.2.2. The stable input light is diffracted

by an AOM with frequency F0 (generally 40 - 80 MHz), and the +1 order diffracted beam is launched into

the fiber. The end face of the fiber is cleaved flat so that a 4 % reflection travels back through the fiber

and overlaps with the first order diffraction from the AOM. The reference beam is derived by retroreflecting

the zero-order beam from the AOM. The geometry of this setup ensures that the retroreflected beam is

aligned to be diffracted into the -1 order of the AOM, and is overlapped with the undiffracted return beam

from the fiber. These two beams impinge on a fast photodiode which picks up their beatnote at frequency

2 × F0. The phase of this beatnote is compared to an RF reference signal and the phase difference is fed

back to a voltage-controlled oscillator (VCO), which drives the AOM. Separate, but nearly identical systems

compensate fiber noise for the light directed through optical fiber toward several different components of

the experimental system including the ion trap and the reference cavity. In contrast, the 3P1 laser system

requires a linewidth on the level of 1 - 10 kHz, so the reference cavity used to stabilize it is not housed in a

highly isolated environment and it is not necessary to compensate for noise introduced in the optical fibers.

The 267 nm laser beams from both the Al+ laser systems is switched toward the ion trap using AOMs.

In the case of the 3P0 laser two AOMs centered at 200 MHz switch the light through two different laser

access ports towards the ions. In the case of the 3P1 laser system, a greater tuning range (10s of MHz) is

required to tune across the ion Zeeman structure and motional spectrum. We use a double-pass AOM to

provide a tuning range of 40 MHz. Two single-pass AOMs near 200 MHz, can separately switch the beam

through either of two laser access ports towards the trap (Fig. 3.2.2). One of these (3P1-σ) is used to provide

a pure σ

+ or pure σ− polarization depending on the side of the Al+ Zeeman structure being probed. This

polarization depends on the rotation of a half-wave plate, which is mounted on a servo-motor system and

controlled digitally by adjusting the duty cycle of a TTL signal. The polarization of the second beam 3P1-π
39

can be adjusted to be pure π polarization or σ

+/σ−.

To Trap To Trap 3P1 Doubler

KD*P

3P1 Tune

2x 200

3P1 pi 

switch

200

3P1 Sigma 

switch

200

Figure 3.7: 3P1 AOM setup.

3.2.3 Mg+ Lasers

To interact with Mg+ we require lasers at 280 nm. The capabilities necessary for controlling the Mg+

atomic system (i.e. cooling, state preparation, detection and stimulated raman transitions) are similar to

those described for Be+. In the case of Mg+, a different scheme for repumping and state preparation has

been developed to eliminate the need for a laser equivalent to the RD laser. Instead of two beams resonant

with the S1/2 →P1/2 transition we use a combination of pulses connecting S1/2, F= 2 to P3/2, as well as

stimulated Raman pulses connecting the two hyperfine levels. The end result is the same, with the pulses

driving population from one qubit level to another with a minimal number of spontaneous scattering events

that could heat the ions. The two laser sources, Mg-Doppler and Mg-Raman, are respectively derived from

1) a dye laser at 560 nm similar to the Be+ laser systems and 2) a fiber laser at 1120 nm similar to the

Al+ laser systems. Mg-Doppler is frequency stabilized by dither locking to a reference cavity external to the

dye laser cavity. The visible light is frequency doubled to 280 nm with a BBO doubling cavity. The 1120

nm fiber laser has a natural stability that is sufficient for stimulated Raman transitions in Mg+ and is not

stabilized to any external reference. Like the Al+ laser system it is frequency doubled first to 560 nm using

an LBO doubling cavity, then to 280 nm using a BBO doubling cavity. The 1.8 GHz hyperfine splitting in
40

Mg+ is spanned for both Mg-Doppler and Mg-Raman with a series of AOMs similar to the basic design of

the Be+ laser systems, but in slightly different configurations.

3.2.4 Experimental Control

In a general sense, the input to each ion experiment is a series of laser pulses at a particular frequency,

duration and phase. The output is a number of photon counts observed during a detection pulse at the end

of the experiment. Both the application of laser pulses and the acquisition of ion fluorescence is controlled

by a field-programable gate array (FPGA) system designed specifically our ion trap experiments [Langer 06].

At a higher level, we use a C++ program written in-house by Till Rosenband that controls and schedules

sets of experiments and analyzes their outcome. Although it is outside the scope of this thesis (and my

knowledge) to discuss technical details of the system, I will give a basic overview of it and describe how the

architecture is necessary for our experiments.

The FPGA is clocked at 62.5 MHz and is programmed to control the frequency and phase of a set of

direct digital synthesizers (DDSs), which have a clock speed of 1 GHz. The FPGA also applies TTL pulses

of the appropriate duration to switch the RF signals toward various AOMs. The photon counts during

fluorescence detection are imaged onto a photomultiplier tube (PMT) and the resulting electronic signals are

passed through a thresholding circuit that converts them to a TTL signal, which is input to the FPGA. The

FPGA records the number of photon counts for a particular experiment and stores them where they can be

read out digitally. A particular experiment is defined by the sequence of RF pulses, sometimes overlapping

in time, and each consisting of (at most2) a DDS frequency, a DDS phase and a TTL pulse duration. The

set of instructions that we use to define this pulse sequence is called a digital control file (dc-file) and it is

translated into FPGA code by a compiler designed in-house [Langer 06]. This system allows for precisely

defined pulse frequencies, phases and times all switchable at a fast time-scale. The FPGA system can scan

variables so that the ion response as a function of any of the pulse characteristics can be observed. This

affords a great degree of flexibility in quickly designing and redesigning an experiment.

The C++ control program performs the basic tasks of writing a series of dc-files, and interpreting the

2 Many pulses are applied to an AOM at a fixed frequency, so they consist of only the TTL pulse duration.
41

results of experiments. These capabilities combined allow for a high degree of automation in the experimental

control. For example, it enables automatic calibration of atomic resonance frequencies and optimum pulse

durations. It also enables interleaved experiments so that certain aspects of the experiments can be actively

monitored and adjusted to protect the experimental results from slow uncontrolled drifts in such parameters

as bias fields near the ion or laser frequencies.

As an example of the way this automation is applied, consider the precise calibration of an atomic

resonance frequency. The experiment consists, of laser cooling, an interrogation pulse near the atomic

resonance frequency, and detection of the ion state. The FPGA is instructed to scan the interrogation pulse

across a certain range near the frequency where the atomic resonance should be, and the results are read out

as a series of photon counts each referring to a particular frequency. The C++ program can then perform

such tasks as binning and averaging the photon counts and plotting the average fluorescence as a function of

frequency. It can also fit the plot to a known functional form, thereby extracting a calibration of the atomic

resonance frequency. Furthermore, the program is set up to sequentially calibrate many parameters, such

that, for example, all of the resonances in the motional spectrum and their Rabi rates can be calibrated at

once. This is convenient because many tasks must be performed periodically to maintain optimum calibration

and this reduces the work to a single button press.

A related feature that has been crucial to performing the experiments described in this thesis is

the ability to continuously servo such parameters as resonance frequencies or trap voltages. To do this

the parameter can be modulated between two states from one set of ion experiments to the next, and the

difference in the fluorescence signal from the ion serves as an error signal. If a particular “primary” experiment

requires the constant recalibration of a parameter, the control program interleaves these servo experiments

with the primary experiments to maintain a continuous calibration. As an example, our experiments require

that position of the ions be maintained on the trap axis to avoid micromotion. The position often changes due

to stray electric field from photoelectrons that build up on insulating surfaces. To servo this, we modulate a

control electrode and perform an experiment that looks for the presence or absence of micromotion. Several

experiments can be designed to make ion fluorescence sensitive to micromotion, but in all cases the servo

attempts to adjust the center voltage of the control electrode such that the fluorescence signal is balanced
42

and the ion rests at a micromotion minimum.

3.3 Experimental Primitives

Several operations are necessary for carrying out the experiments of this thesis, and I will describe

the basics for the Be-Al experiment in particular here. The protocols for the Mg-Al experiment are similar,

but will not be discussed.

3.3.1 Loading ions

The loading of Be+ is performed by electron-impact ionization of neutral beryllium atoms from a

thermal beam. The thermal beam comes from a beryllium oven, which consists of a thin beryllium wire

coiled around a tungsten “heater” and is mounted near the trap volume. We run a current of ∼0.9 A through

the tungsten wire, heating the beryllium near its melting point of 1278 C. A cone of metal surrounding the

oven shields most of the electrode surfaces from the beryllium atoms and transmits only a small solid angle

of the beam through an aperture toward the trapping region. Simultaneously, an electron gun filament is

heated by running ∼1 A of current through it, emitting electrons, which are accelerated across a voltage of

50-80 mV and bombard atoms in the trapping region. Typically, after 20-40 seconds, an atom will be ionized

in the trap with a sufficiently low energy for it to be laser cooled by the detuned BD beam. This ion will

be imaged on the CCD camera and appear as a bright dot, which allows us to manually stop the loading

process before more ions appear.

We can only load Al+ when a Be+ is already trapped, since we can not detect the Al+ directly through

fluorescence. However, the presence of a second “dark” ion can be detected by monitoring the position of

the Be+, which moves by a couple microns due to Coulomb repulsion. The process is similar to loading Be+.

The aluminum oven is composed of an aluminum reservoir heated by a tungsten wire, both embedded in

an alumina tube. Neutral aluminum atoms are either ionized by electron impact or by resonance-enhanced

two-photon ionization (photoionization). For photoionization, we use a commercial external cavity diode

laser with a wavelength of 396 nm. Aluminum has a 3s

23p ground state with two fine-structure components,

2P3/2 and 2P1/2, split by a wavenumber of approximately 100 cm−1

. Both of the ground state fine-structure
43

components are significantly occupied at the oven temperature and we address the 2P3/2 →2 S1/2 transition

with the photoionization laser. From the 2S1/2 state, another 396 nm photon is sufficient to reach the

ionization threshold at  ̃6 eV.

Since the presence of an Al+ ion can only be inferred by its effect on Be+, the successful loading of

a Be-Al pair must be confirmed by measuring the motional frequencies of the two-ion array. Similarly, for

loading a larger number of ions the number of dark ions can often be determined by examining the position

of the Be+ ion(s), but the species of the ions can only be verified through spectroscopy of the motional

spectrum.

3.3.2 Laser Cooling

Laser cooling is the first step in most experiments. This process first involves Doppler cooling all

normal modes of motion by applying a series of BD laser pulses [Wineland 79, Wineland 78]. First a BDdetuned pulse of 200 μs serves to cool ions that may have too high an energy to be efficiently cooled by the

near resonant beam. Next we apply a series of near resonant pulses (detuned red by γ/2) lasting 200-1000

μs depending on the number of ions and their configuration. For a single Be+ ion there are only 3 normal

modes which are all cooled efficiently, so a single pulse for 200 μs is sufficient. For the ion array Al-Be-Al,

we need two pulses at different trap conditions, lasting close to 1 ms in order to approach the Doppler limit

(see Sec. 4.4 and Sec. 5.2.3.2). In general, due to differences in motional coupling strength, each particular

configuration of ions will require different trap conditions and Doppler cooling times to reach the Doppler

limit.

After Doppler cooling, for many experiments, we also cool the axial modes of motion close to the

ground state [Diedrich 89] by resolved-sideband Raman cooling. Raman cooling involves a series of pulses

with the Raman lasers addressing the first red sideband for each axial mode separated by repumping pulses

using the RD laser. We choose the range of Fock-states to cover in the cooling process which improves

the cooling efficiency by adjusting the sideband π-time based on the Fock state according to Eq. 2.9. For

example, consider the case of cooling one Be+ and one Al+ using sideband pulses on Be+. We cool both the

in-phase and out-of-phase axial modes beginning from a π-time appropriate for n = 15 using 2 repetitions
44

at each π-time. The π-times are calculated based on the calibrated π-time near the ground state of motion,

the known Lamb-Dicke parameter and Eq. 2.9. We interleave cooling cycles on the two modes so that first

we apply a sideband pulse at the in-phase mode frequency, for the appropriate time, repump using RD, then

repeat the process for the out-of-phase mode frequency. Interleaving cooling cycles reduces the probability

that one mode heats while the other is being cooled. The interleaved cycles are repeated at the same πtimes, then we move to the next lower Fock-state π-times and perform another set of cooling cycles. In this

example, we step down the ladder from n = 15 to n = 1 performing interleaved cooling cycles for each Fock

state on the two axial modes twice each. The entire process takes approximately 1 ms.

3.3.3 Al+ State Preparation

At the beginning of each experiment involving the Al+ system, the state of Al+ must be prepared in

either 1S0, mF = ±5/2. This is accomplished by applying a series of lasers pulses on transitions from 1S0

to 3P1 using the 3P1-σ beam with polarization σ

+ or σ− matched to the sign of the target mF state. We

attempt to maximize the efficiency of this process by first optimizing the polarization of the 3P1-σ beam

and second calibrating the frequencies and π-times of all of the transitions leading to the target state.

To optimize the polarization of 3P1-σ we perform a depumping experiment. If the polarization is

pure σ then, in applying a pulse to 1S0, mF = ±5/2 → 3P1, mF = ±7/2, the transition probability will be

maximized. However, if the polarization is imperfect, there will be a finite probability of making a transition

to 3P1, mF = ±5/2 or 3P1, mF = ±3/2, both of which can decay out of the target 1S0 state. This will

show up in the experiment as a drop in the transition probability. To optimize the polarization we run

this experiment continuously and adjust the angle and rotation of both a HWP and QWP to maximize the

transition probability.

We also calibrate the frequency and π-times of the optical pumping pulses in order to pump Al+ more

efficiently to the target state. To do this, we first calibrate the frequency and π-times of the two carrier

transitions at opposite sides of the Zeeman spectrum, (1S0, mF = ±5/2) → 3P1, (mF = ±7/2). Although

during the calibration the polarization is not perfected, we still can get a reasonable signal assuming the

polarization is not grossly misaligned. Based on the difference frequency between the two transitions, the
45

magnetic field can be determined. With this, we use the Breit-Rabi formula, to determine the frequencies

of all of the transitions3. In addition, we scale the Rabi rates based on the Clebsch-Gordon coefficients for

each of the transitions. Based on these calibrations we apply a series of pulses at a particular frequency,

duration and polarization appropriate for the target state desired. The lifetime of 3P1 is approximately 300

μs, which is the limiting timescale for optical pumping. At any step in the process we do not know which

state the Al+ system occupies, so we apply a series of pulses sweeping towards the target state, then wait

for a time that allows the 3P1 state to decay. We repeat this through a specified number of repetitions. To

confirm that we have reached the target state, we apply pulses to the target state from (1S0, mF = ±5/2)

to 3P1, (mF = ±7/2) and see that we do indeed make transitions.

3.3.4 Monitoring and Locking the Trap Conditions

Due to the effects of stray charges on insulating surfaces in the trap several trap conditions can drift

during the course of a set of experiments. As a result we typically use periodic, interleaved experiments of

the type described in Sec. 3.2.4 to fix the ion positions in three dimensions and to lock the axial confinement

frequency.

3.3.4.1 Micromotion

Radial displacement of the ions due to uncompensated radial bias fields cause excess micromotion.

All of the experiments we perform require the micromotion to be minimized, because its amplitude can

affect the laser cooling efficiency and cause time-dilation shifts to the ion resonance frequency. To actively

compensate micromotion, we use the strategy described in Sec. 4.3 to servo the DC bias fields (see Sec. 3.2.4).

To detect uncompensated bias fields, we look for radial mode amplitude along the trap axis by applying a

sideband pulse with the Raman beams. To the extent that the Raman beams are well-aligned ∆k will

be directed along the trap axis and should not couple at all to the radial modes. However, if there is a

radial bias field, the mode amplitudes change direction, and the radial modes will, in general, have some

projection along ˆz. This can be observed by looking for Rabi flopping on the radial sidebands. For the

3 An evenly spaced set of frequencies based on the linear Zeeman effect would be sufficient at the B-fields we typically apply.
46

different ion configurations used in this thesis (Be-Al, Al-Be-Al, and Be-Al-Al-Be) we choose an appropriate

radial mode for which the axial mode amplitude on Be+ increases significantly with a radial bias field. We

modulate the voltage on compensation electrodes symmetrically about the nominally compensated value.

An uncompensated field shows up as a difference in the transition probability between the two modulation

states, which we compensate by adjusting the nominal center. In addition to changing the mode amplitudes,

a radial bias field will change the mode frequencies. For this reason, in conjunction with the bias fields, we

monitor the mode frequencies by modulating the sideband pulse frequency about the nominal center and

adjusting based on an imbalance in the transition probability. To fully compensate the bias fields we servo

two orthogonal radial directions (along the ˆx and ˆy directions). We generally achieve at least a 50:1 ratio

between the transition strengths of the carrier and the micromotion sideband.

3.3.4.2 Trap Frequency

Stray charge buildup on the trap can also cause a change in the axial trap frequency, which affects

the fidelity of sideband pulses. This is actively compensated using a similar servo strategy to that described

above. In this case we apply an RF signal to a trap endcap to excite motion at the ion in-phase axial

frequency. If the RF signal is at the motional resonance it will drive energy into the motion. We can

detect the motion because it depletes the strength of a qubit carrier transition into the motional sidebands.

We use a servo to fix the motional resonance to a particular value, by modulating the frequency of the

RF signal between two symmetric values about resonance. We raise or lower the voltage to all endcap

electrodes simultaneously to balance transition probabilities between the two modulated values. In this way

the resonance frequency is fixed to be the desired value.

3.3.4.3 Axial Position

Due to imperfections in the trap geometry there is residual micromotion directed along the axis of

the trap, for which the amplitude depends on the ion position in relation to the trap electrodes. As a

result, this micromotion amplitude changes with stray charge buildup. We actively compensate this during

the course of experiments by relying the the alignment of the laser beams with the minimum of the axial
47

micromotion. If we align the beams at conditions where we know axial micromotion is compensated, we

can then adjust the position of the ions to be centered in the laser beams to restore this compensation.

The parameter modulated in this case is a differential endcap voltage, which adjusts the ion position axially

without changing the motional frequencies. We balance the qubit carrier Rabi rate in the two modulated

states to identify the compensated differential voltage.
Chapter 4

Two-Species Ion Dynamics

Two fundamental requirements for both precision spectroscopy and quantum information processing

in an ion trap are the ability to laser cool and detect a single ion. This places a stringent constraint on the ion

electronic structure, and only a small number of ion species, including the singly ionized atoms of group IIa

and IIb, can be directly investigated in this way [Metcalf 99]. In particular, Doppler cooling [Wineland 79]

and resonance fluorescence detection [Bergquist 86] both benefit from the presence of a strongly-allowed

cycling transition. For most ions, the electronic level structure is complicated enough that isolating a cycling

transition is experimentally infeasible. Other ions, such as Al+, have appropriate transitions but only at laser

wavelengths that are currently inaccessible [Wineland 02]. Molecular ions are also interesting candidates for

spectroscopy and quantum information processing [DeMille 02, Schmidt 06], but optical pumping to many

states of the rovibrational degrees of freedom generally prohibit cycling transitions. Another situation where

Doppler cooling can be problematic is in quantum information processing (QIP) experiments. Here, direct

cooling may be possible, but doing so would disturb the information stored in the qubit [Kielpinski 02].

All of these considerations motivate the use of two ion species trapped in a single trap and coupled via

Coulomb repulsion to enable sympathetic laser cooling. In this technique the primary ion species is cooled

indirectly through its Coulomb interaction with the cooling ion species. Sympathetic cooling has been

demonstrated in several systems [Barrett 03, Blinov 02, Schmidt 05]. In recent experiments, sympathetic

recooling of ion qubits has enabled the entanglement of harmonic oscillator states in an ion trap array [Jost 09]

as well as the realization of repeated logic operations without loss of fidelity [Home 09]. The exchange of

energy between two ion species also provides the means for indirect state detection, whereby information in
49

an ion’s electronic structure is transferred to the shared motion and read out by the sympathetic cooling ion.

A particular scheme for indirect detection has been applied to precision measurements of 27Al+[Schmidt 05,

Rosenband 07].

In this chapter we investigate the properties of two-species ion arrays in a linear rf Paul trap. First,

in Sec. 4.1, we present a mathematical model for dynamics of the two-species arrays. In Sec. 4.2 we

describe a method for fixing the arrangement of the ions, which takes advantage of the mass dependence

of the trap confinement. We show specific examples of this for small numbers of ions. In Sec. 4.3 we

consider the implications of these ideas for the suppression of motion at the frequency of the rf potential, or

“micromotion”. We discuss how properties of a two-species ion array can enhance or diminish micromotion

caused by uncontrolled electric fields and we briefly review how micromotion can be actively compensated

by monitoring the motional spectrum. In the final section, Sec. 4.4, we consider Doppler cooling in the

context of multiple species ion arrays and examine ways of optimizing the cooling efficiency.

4.1 Mathematical Model for Multiple Species Ion arrays

The basic geometry of a linear rf Paul trap [Paul 90] has been described in numerous publications [Wineland 98]. In this section, we denote the ion positions as the components of u, such that

{u3i+1, u3i+2, u3i+3} respectively specify the x, y and z coordinates of the ith ion. We consider an array of N ions such that and i ∈ [0, N − 1]. In some cases it will be convenient to use the equivalent notation

ri = {xi, yi, zi}. We take the z coordinate to be parallel to the axis of the trap, along which the rf electric

fields vanish. Typically in experiments, the saddle point in the static potential coincides with a point on

the rf nodal line. This is the condition that minimizes the effects of micromotion, because the static fields

do not displace the ions into the rf pseudopotential. However, in what follows, it will be important to also

consider the effects of a bias field, Er = Exxˆ + Eyyˆ.

The equilibrium positions of N trapped ions in three dimensions are determined by minimizing the

potential energy of the system with respect to the 3N degrees of freedom. The total potential energy includes

terms from the external electric fields as well as terms from the Coulomb interaction between ions. We write
50

the potential energy from static external fields as

Vdc =

1

2

N

∑−1

i=0

∑

3

j=1

κju

2

3i+j + Q

N

∑−1

i=0

(Exu3i+1 + Eyu3i+2), (4.1)

where the first term sums components from the static trapping fields, specified by the parameters κ1, κ2

and κ3, which represent the curvatures associated with the static potential in the x, y and z directions

respectively. To satisfy Laplace’s equation, we must have ∑

j

κj = 0. Since we rely on the static potential

to provide confinement in the z direction, κ3 > 0 and κ1 + κ2 < 0. The particular values of these curvatures

are determined by the magnitudes and geometries of the applied fields. The second term in Eq. 4.1 takes

into account a bias field in the x and y directions. A bias field in the z direction amounts to a shift of the

coordinate origin, and has been omitted.

In the pseudopotential approximation we write the potential energy from the radial rf fields as

Vrf =

1

2

ar

N

∑−1

i=0

1

mi

(u

2

3i+1 + u

2

3i+2), (4.2)

where miis the mass of the ith ion and ar characterizes the strength of the radial confinement. The potential

energy of a single ion of mass m0 is Vdc +Vrf. In this case we can easily relate {κj} and ar to experimentally

accessible quantities. Since the axial confinement depends only on the static trapping potential, κ3 = m0ω

2

z

,

where ωz is the observed resonance frequency for ion motion along the trap axis. In the radial direction we

have ar/m0 + κ1 = m0ω

2

x

, and ar/m0 + κ2 = m0ω

2

y

, which together with κ1 + κ2 = −κ3 fully characterize

the trap conditions for a single ion. Note that these measurable frequencies for a single ion are independent

of ER, a condition which is not true for multiple ions of different species.

For N > 1, the ions interact via their mutual Coulomb repulsion. This contributes to the total

potential energy with

Vion =

Q2

8πε0

N

∑−1

i,i′=0

i̸=i

′

1

|ri,i′ |

(4.3)

where |ri,i′ | is the distance between ions i and i

′ and is given by

|ri,i′ | =

[∑

3

j=1

(u3i+j − u3i

′+j )

2

] 1

2

. (4.4)

The total potential energy, then, is

Vtot = Vdc + Vrf + Vion. (4.5)
51

The equilibrium ion positions, u

′

, which minimize this potential have been found analytically for N = 2

and N = 3 in the special case of an axial array [James 98]. In all the results that follow u

′

is found by

numerically minimizing Vtot.

In analyzing the motion of the ions about their equilibrium positions, we write the time dependent

position as u(t) = u

′ + q(t). For laser cooled ions (T ≈ 0.1 mK) the motional amplitudes are on the order

of 10 nm, while the inter-ion spacing is on the order of 3 μm. This allows us to take the limit of small

oscillations and approximate the potential energy to second order in q(t). If we define coordinate indices for

the 3N dimensional space as l ≡ 3i+j and l

′ ≡ 3i′ +j′

, we can identify the components of a mass matrix M ̃

as Ml,l′ = δl,l′mi. Similarly the components of a spring constant matrix K ̃ are given by Kl,l′ =

∂

2Vtot

∂ql∂ql

′

|{ql}=0.

Now we can write the Lagrangian as

L =

1

2

 ̇q

TM ̃  ̇q −

1

2

q

T K ̃ q, (4.6)

where superscript T denotes the vector transpose. This Lagrangian gives the 3N equations of motion

M ̃  ̈q + K ̃ q = 0. (4.7)

All forces in Eq. 4.7 are linear, leading to oscillatory solutions for q(t) of the form

q(t) = Re[Cve

i(ωt+φ)

], (4.8)

where the scaling factor C and the oscillator phase φ are determined by initial conditions. Substituting Eq.

4.8 into Eq. 4.7 leads to the following eigenvalue equation

M ̃ −1K ̃ v = ω

2v. (4.9)

In solving the eigenvalue equation we get the 3N oscillator frequencies {ω

(α)} and their corresponding

normalized eigenvectors {v

(α)}, for which the component v(α)

3i+j

corresponds to the ith ion’s relative amplitude

in the direction specified by j (x, y, or z) for the mode labeled α. To simplify the notation in what follows

we drop the label α and refer to a particular mode implicitly.

The scaling factor C depends on the ion motional state. The state of each mode can be expanded in

terms of Fock number states, |n⟩, for which the characteristic amplitude is

C(n) = (2n + 1)1/2C0, (4.10)
52

where C0 is the amplitude for the ground state of motion. The amplitudes for the ground state of motion

can be found using the total kinetic energy,

1

4

~ω =

ω

2

2

v

TM ̃ vC2

0

, (4.11)

which identifies the effective mass as meff = v

TM ̃ v leading to the usual expression for the extent of the

harmonic oscillator ground state wavefunction,

C0 =

(

~

2meffω

)1

2

. (4.12)

Putting this expression with Eq. 4.10 in Eq. 4.8 we can characterize the motion of the ions for any particular

distribution of Fock states. The motional amplitudes and frequencies are central in evaluating the interaction

between a array of ions and applied laser beams.

The model discussed here represents an ideal linear Paul trap, with the addition of uniform bias

fields. Any real trap will exhibit deviations from this ideal model. Although these deviations will usually be

sufficiently small to keep the ion dynamics qualitatively unchanged, we mention some common imperfections

here. First, the model ignores misalignment between the rf nodal line and the static potential axes. One way

a misalignment is manifest is by the existence of a gradient of the x-y bias electric field as a function of the z

coordinate. This particular effect can be accounted for in Eq. 4.1 by applying the appropriate functions Ex(z)

and Ey(z). Another effect of trap asymmetry is axial micromotion, which refers to rf fields directed along the

potential axis. The corresponding rf pseudopotential in the ˆz direction can be accounted for by a modification

of Eq. 4.2 to include a term that depends on the coordinates u3i+3. A third complication which can occur

is higher-order terms in both the static and rf potentials. Traps generally exhibit small anharmonicities

but these effects can be significant in certain situations, particularly with multizone microfabricated traps

[Rowe 02] and surface-electrode traps [Seidelin 06].

4.2 Ion Reordering

The transport of atomic ions has been studied in the context of single-zone linear traps [Splatt 09],

multi-zone linear traps [Rowe 02], and two-dimensional trap structures [Hensinger 06, Blakestad 09]. The

emphasis in these experiments was on the requirements for large-scale quantum information processing. A
53

sequence of quantum gates between selected groups of ions, may require different ions to interact at different

steps in a single experiment. The problem discussed here, by contrast, concerns fixing an array of differentmass ions into a particular order independent of the initial order. These techniques are relevant for many

experiments involving a small number of ions of unequal mass, including precision spectroscopy and quantum

information experiments.

We consider a two-species ion array that has N0 ions of mass m0 and N1 ions of mass m1, (N0, N1 ∈ [1,

N]). There are N − 1 possible mass combinations, and for each of those there are N!/N0!N1! ion orders.

Those ion orders that are mirror images of each other will exhibit the same motional spectrum making them

indistinguishable in our experiments, where we address all ions simultaneously with a uniform laser field.

In the following discussion we group the mirror-image orders together and refer to them jointly as an ion

configuration.

Uncontrollable collisions with background gas in the ion trap (typically with a timescale of minutes)

will tend to heat the ions into a uncrystallized thermal state. Subsequent cooling of the ions into a linear

array will put the ions in a random order. We desire, then, a method to rearrange the ions into a particular

configuration independent of the initial order. If the radial confinement is sufficiently stronger than the axial

confinement the ions will form an axial array with xi, yi = 0 for all i. Under these conditions, Vrf = 0 so the

total potential energy is independent of the mass and all N!/N0!N1! ion orders will be stable. One way to

change the ion configuration, is to increase the ratio ωz/ωx or ωz/ωy to a point where the ion array becomes

purely radial. We use ωx, ωy and ωz to refer to the secular frequencies for a single trapped ion. Another way

to change the ion configuration is to introduce a radial bias field such that the ion pseudopotential energy is

nonzero. By adjusting the trap voltages, we can find some conditions that have only one stable equilibrium

configuration. This will not be a linear array of ions, but instead a two-dimensional ion geometry with some

ions displaced radially into the pseudopotential.

4.2.1 Two Ions

We begin by considering the simplest case of two ions, one of mass m0 and one of mass m1 held

together in a linear Paul trap. There is only one possible on-axis ion configuration since the two ion orders
54

2 4 6 8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

μ

Threshold R

ω

0.4 0.5 0.6 0.7 0.8

−4

−2

0

2

Rω

x, z [

μ m]

μ = 3 x

0

z

0

z

1

x

1

Figure 4.1: Threshold frequency ratio (Rω = ωz/ωx) as a function of mass ratio μ = m0/m1 for two ions.

Here Rω refers the trap conditions for a single ion of mass m1. (Inset) A particular example of the ion

positions in two dimensions for μ = 3 as a function of Rω. The sharp change in ion position where zi → 0

indicates the threshold used to generate the larger plot.

are mirror images. To the extent that the laser intensity is uniform across the ions, the absorption spectrum

of the two ion orders are equivalent, but it may be desirable to maintain a particular order to reduce the

effect of laser intensity gradients, and it is instructive to examine how this case behaves when adjusting trap

parameters.

For simplicity we consider the cylindrically symmetric case where κ1 = κ2 = −κ3/2 and restrict the

discussion to two dimensions, x and z, which captures the relevant features of the more general case. We

specify the trap confinement in terms of the motional frequencies of a single ion of mass m1 and define

μ ≡ m0/m1. We study first the conditions under which the axial ion array becomes radial as a function of

the frequency ratio Rω = ωz/ωx. For μ = 1, the ion array is radial if Rω > 1. As μ decreases, the threshold

ratio increases because the lighter-mass ion is more tightly bound by the pseudopotential. Likewise, with

higher μ, the heavier ion is less tightly bound radially and the threshold reduces. In Fig. 4.2.1 we plot the

threshold Rω at which the ions go from a radial to axial array as a function of μ.
55

Experimentally, Rω can be raised by increasing endcap voltages, or, equivalently, by lowering the rf

voltage. If we begin with an axial array and raise the endcap voltages, the ions reorient into a radial array.

At the threshold conditions between a radial and axial array, one of the radial mode frequencies reduces to

zero. Since the voltages are adjusted during a finite period of time, the transition is non-adiabatic and the

ions will gain kinetic energy during the process, which generally causes them to decrystallize. If there is

motional damping, for example due to laser cooling, the ions recrystallize into the new equilibrium positions.

The threshold conditions correspond to parameters at which a radial mode frequency of the two-ion array

reaches zero.1 A vanishing mode frequency is a general feature of trap conditions at the boundary between

ion configurations.

The inset of Fig. 4.2.1 shows a particular example (μ = 3) of the equilibrium ion configurations as

a function of the trap frequency ratio. Note that, in this example, a particular radial orientation has been

chosen for the two ions, although the reverse arrangement (i.e. xi → −xi for i = 1, 2) is equivalent. There

is a probability of 1/2 that either arrangement is observed. Similarly, if the trap conditions are relaxed back

through the threshold to make an axial array, either ion order is equally probable. This means that changing

Rω alone is not useful for choosing an orientation of two ions.

The ion positions can also be manipulated by the application of a radial electric field, Ex. For Ex ̸= 0

the heavier ion will occupy a position further from the axis due to the mass dependence of the ponderomotive

potential. In Fig. 4.2.1 we give a particular example of the ion positions as a function of the radial bias

field for μ = 3 and {ωx, ωy, ωz} = 2π × {13, 13, 4} MHz. Here we note the qualitative differences between

this case and the case of changing Rω. First, there is no sudden change in position as the ions go from an

axial to a radial array. However, as with the case of raising Rω, a radial mode frequency drops to zero at the

point where the ions form a radial array. Here, there is a particular orientation defined by the sign of Ex,

which is important in fixing the ion order. With a radial array of known orientation, one can apply a electric

field gradient, Ez = dEz/dx × x, to turn the ion array [Splatt 09]. If the applied voltages are subsequently

reduced to zero to form an axial array, the ions will have a known order. This technique has been applied

1 For two ions there are four radial modes, two in radial direction. In one mode, the ions move in-phase, while in the other,

the ions move out of phase. The out-of-phase mode is always lower in frequency and this is the mode that reaches zero frequency

at threshold. Conceptually this makes sense because the ions must move in opposite directions to form a radial array
56

0 250 500 750 1,000

−10

−8

−6

−4

−2

0

2

4

x,z [

μ m]

E

x

 [V/m]

x

0

z

0

x

1

z

1

Figure 4.2: Plot of ion positions as a function of a bias field Ex in two dimensions for μ = 3 and {ωx, ωy,

ωz} = 2π × {13, 13, 4} MHz.

experimentally to arrange a particular order in an array of ions containing Mg+ and Al+ [Chou 10, Hume 09].

4.2.2 Three Ions

With three ions of masses m1, m0 and m0 respectively, there are two possible ion configurations.

These are distinguished by a unique mode spectrum. Specifically, the ion of mass m1 can be in the center of

the array (symmetric configuration) or on the outside (asymmetric configuration). In most experiments on

such an ion array, we want to maintain a particular configuration so that we have fixed motional sideband

frequencies and amplitudes. Here, again, we consider the ion configuration for a particular mass ratio μ as

a function of Rω.

We imagine beginning with an ion array in a particular axial configuration, then ramping Rω adiabatically to a higher value and back down to the original value while laser cooling. As the trap conditions

change, the ions take on particular arrangements. We characterize these arrangements based on which axial
57

configurations can be generated when Rω is returned to the original value. An arrangement is asymmetric

or symmetric if it can only return to that particular configuration when relaxed to the axial state. The ions

can also form a radial array such that zi = 0 for all i. In this case, neither configuration is deterministically

generated as the trap relaxes. Finally, the trap conditions can be such that the ions are not confined.

For the case where the ions start as a symmetric array, the configuration regions are plotted in

Fig. 4.2.2. There are several features to note from this diagram. First, it is impossible to make an asymmetric

configuration if μ < 1 by only adjusting the frequency ratio. The ion arrangement either transitions directly

from symmetric to radial (at Rω = 1) or from symmetric to not confined, depending on μ. In the latter case,

the boundary occurs at ω

′

x/ω′z = 1 where ω

′

x and ω

′

z

refer to the mode frequencies for a single ion of mass

m0. For μ > 1 there is an asymmetric region near Rω = 1 where only a triangular arrangement of ions is

stable, with the heavier ions to one side. This region can be useful for deterministically reordering the ions.

If Rω is ramped adiabatically to a point in this region then adiabatically relaxed to the original conditions,

the ions will assume an asymmetric configuration independent of the initial configuration. We note also that

Symmetric

Not Confined Radial

Asymmetric

1/R

ω

Figure 4.3: Diagram of ion configurations for three ions of masses {m1, m0, m1} as a function of μ and

1/Rω, with Ex = 0
58

the lower bound of the graph, at the value 1/Rω = 0, marks the transition from a radial ion array to one

that is not confined for μ > 1.

In a similar way, the symmetric 3-ion array can be deterministically produced by adjusting Ex for

any value of μ ̸= 1. Imagine beginning from an asymmetric configuration with μ < 1 (two heavier ions).

As Ex is ramped up, the heavier ions are pushed deeper into the pseudopotential relative to the lighter

ones. In the z direction, the barrier that keeps the heavier ions from assuming a position centered about

the minimum of the external static potential is the Coulomb repulsion from the lighter ion. As the radial

distance between the lighter and heavier ions grows with increasing Ex this potential barrier is reduced until

the axial confining fields overcomes it and the ions move towards z = 0. With laser cooling, the ions will

crystallize into a triangular array that can be adiabatically relaxed to an axial symmetric configuration. The

same process will reconfigure three ions with μ > 1.

4.2.3 Higher Ion Numbers

As the number of ions in an array increases the number of ion configurations also increases. For 2 ions of

mass m0 and 2 ions of mass m1, there are four different ion configurations, for 2 ions of mass m0 and 3 ions of

mass m1 there are six. In general, it will not be possible to deterministically produce all of these configurations

using the techniques we have described. However, some subset of the configurations may be accessible. For

example, for the four-ion case with N0 = N1 = 2, and μ = 3 we find that the configuration m1−m0−m0−m1

can be produced deterministically by raising Rω. In addition, the configuration m0 − m1 − m0 − m1 can be

produced by raising Ex. On the other hand, we have not found a way to produce the configuration m0 −

m1 −m1 −m0 deterministically using these techniques. Other techniques, such as introducing field gradients

[Splatt 09] may be useful. We note also that any configuration that cannot be produced deterministically

can, in principle, be produced probabilistically. Since all configurations are distinguishable by their sideband

motional spectrum, any one of them can be produced by randomly rearranging the ions (i.e. by decrystallizing

then recooling), checking the sideband spectrum, then repeating the process until the desired configuration

is observed.
59

4.3 Micromotion Measurement and Compensation

Micromotion or motion resulting from the rf trap fields at frequency ωrf and is an unavoidable effect

of the ponderomotive trapping potential. In ideal trapping conditions where ωx, ωy ≫ ωz, such that the

ions form a linear string along the rf nodal line, there is still some micromotion due to the finite extent of

the ion wavepackets even in the ground state of motion. If, additionally, there are uncompensated radial

electric fields, the ions will experience excess micromotion. Relatively high micromotion amplitudes can

decrease the efficiency of laser cooling due to the depletion of the cooling resonance into the micromotion

FM sidebands. Even if the micromotion amplitudes are very small they can limit the accuracy of precise

frequency measurements due to relativistic time dilation [Rosenband 08, Chou 10].

In this section we focus on the effects of excess micromotion and the corresponding time-dilation

on the measured frequency in precision spectroscopy. The fractional time dilation shift of a transition at

frequency, ν0, for an ion of mass, mi, is given by

∆ν

ν0

≃ −

⟨KE⟩

mic

2

, (4.13)

where ⟨KE⟩ is the time-averaged kinetic energy of the ion. The kinetic energy due to excess micromotion is

equal to the pseudopotential energy of the ion at the equilibrium position. Using Eq. 4.2 we can write

⟨KE⟩ =

ar

2mi

(x

2

i + y

2

i

). (4.14)

Suppose there are two ions of mass m0 and m1 (μ = m0/m1) held in a trap, and consider the time

dilation shift on the ion of mass m1. For a given value for the uncompensated radial bias field, Ex, the

equilibrium position will depend on ωx, ωz and μ. In Fig. 4.3 we plot the frequency shift from Eq. 4.13 and

Eq. 4.14 as a function of 1/Rω for several values of μ. In this example, we have chosen Ex = 10 V/m and

ωx/2π = 6 MHz. As long as we are considering small displacements such that xi/zi ≪ 1 for both ions, the

results are proportional to E2

x

for a particular ωx.

One conclusion we draw from Fig. 4.3 is that the sensitivity of a precision frequency measurement

to uncontrolled radial electric fields depends on the mass ratio in a two-species ion array. The external

electric field causes a tilt of the ion array with respect to the trap axis, which results in an additional radial
60

0.3 0.4 0.5 0.6

0

2

4

6

8

x 10−18

1/R

ω

∆

ν / 

ν

0

μ = 0.3

μ = 3

Figure 4.4: Time dilation shifts due to excess micromotion for a two ion array as a function of 1/Rω = ωx/ωz,

for several values of μ = m0/m1

component of the Coulomb repulsion between the two ions. Depending on the mass ratio, this can enhance

or diminish the effect of the external field on the position of the primary ion. In general, for the purpose of

minimizing micromotion due to stray electric fields, it is preferable to use an ancillary ion that is heavier.

For certain conditions, the micromotion amplitude is nulled for the lighter-mass ion independent of Ex.

This is the condition where Ex is equal to the x-component of the field from the heavier ion at the position

{x1, z1} and it occurs precisely when ωx = ωz for the single, heavier ion. This electric-field-insensitive point

may allow for passive nulling of excess micromotion for high accuracy measurements. Note, however, that

these conditions cannot be simultaneously obtained for both the x and y directions because this implies

a degeneracy in motional frequencies, which is incompatible with efficient cooling. A small difference in

the radial motional frequencies can allow for near-perfect nulling of excess micromotion in the two radial

directions while maintaining cooling efficiency.

Excess micromotion can also be actively compensated by detecting it and adjusting voltages on the
61

trap electrodes to suppress it. A direct method for doing this is by applying a laser beam at a frequency

ω0 ± ωRF , where ω0 is the frequency of an optical resonance, and looking for fluorescence, or coherent Rabi

flopping. Excess micromotion is minimized in the direction of the laser beam k-vector when the sideband at

ωRF is minimized. To do this in both radial directions, two nearly orthogonal laser beams in the x-y plane are

required. However, stray electric fields causing excess micromotion also alter the motional mode amplitudes

and directions in two-species ion arrays. The use of these effects for micromotion compensation has been

explored previously [Barrett 03]. As a simple example, consider two ions of different mass aligned perfectly

on the axis of a linear trap and probed with a laser beam directed along the trap axis. In this condition, the

radial motion sidebands at frequencies ω0 ± ω

(α)

, with α referring to any mode in the x or y direction, will

be absent. However, a bias electric field will displace the two different-mass ions into the pseudopotential,

causing excess micromotion and altering the normal mode directions. As a result, the secular modes that

are purely radial in the ideal configuration will have some axial component, which can be detected with the

axial beams. To the extent that the probe beam is purely axial, the excess micromotion can be nulled by

nulling these sidebands, which aligns the ion array along the trap axis. In the next section we will consider

further the effect of bias fields on the amplitude and direction of secular motion in the context of sympathetic

cooling.

4.4 Sympathetic Cooling

The precise manipulation of ion internal and external states requires low temperature. High motional

temperature can cause broadening of or FM sidebands on atomic resonance, as well as shifts of the resonance

frequency due to time dilation. In addition, the ion temperature affects the rate and coherence of Rabi

flopping due to the Debye-Waller effect [Wineland 98]. For these reasons, in most experiments on individual

trapped ions, the first step in the experimental process is Doppler laser cooling. This section addresses

Doppler cooling efficiency in two-species ion arrays.

Sympathetic laser cooling using two ion species has been explored in the context of quantum information processing [Kielpinski 00, Barrett 03]. In this technique one (cooling) ion species acts as a refrigerant for

the other (primary) species, with both species stored in the same trap and coupling through collective modes
62

of motion. There are several advantages of sympathetic cooling. First it allows laser cooling of ion species

that cannot be directly cooled, which includes most atomic ions, molecular ions, and ions with no atomic

structure such as a proton. Second, the spectroscopic isolation of the primary system from the cooling laser

means that the laser cooling leaves the primary system largely undisturbed. Experiments where the ions

must be cooled between or during coherent operations become possible. One potential problem with this

idea is that there are N0 ions scattering photons rather than N (N = N0 + N1). This means that there is

less cooling power for the 3N motional modes compared to a single-species ion array.

The rate that energy is extracted from the ion system during laser cooling can limit both the experimental cycle time and the minimum temperature. For example, Doppler cooling times are often on

the order of 1 ms, comparable with other rate-limiting steps such as fluorescence detection. The minimum

temperature achievable is affected by the cooling rate compared to the external heating rate. Here, we ignore

recoil heating and parameterize the laser cooling rate for a particular mode in terms of the damping time

τ

(α)

d

. The cooling rate is given by  ̇n

(α) = −n(α)/τ (α)

d +  ̇next where we assume a rate of external heating of  ̇next.

The minimum temperature occurs when the heating rate and cooling rate are equal, yielding n

(α)

min = τ

(α)

d

n ̇ ext.

The assumption that we can ignore the effects of recoil heating is justified to the extent that n

α

min is greater

than the limit due to recoil heating, nD. In ideal circumstances, nD ≃ Γ/2, where Γ is the linewidth of the

resonance used for cooling.

Damping times for Doppler cooling depend on a number of experimental parameters including the

intensity, geometry and detuning of the cooling laser beams. Theoretical cooling rates have been derived for

a range of conditions [Wineland 79, Itano 82]. In the following analysis we focus on the effect of ion mode

amplitudes. Suppose a single cooling ion is trapped with another ion of a different species. Only the cooling

ion motional amplitudes are relevant for the cooling rate. If the temperature is sufficiently low that the

Doppler broadening of the cooling resonance is insignificant, τ

(α)

d ∝ (1/η(α)

i

)

2 where η

(α)

i

is the Lamb-Dicke

parameter of ion i and mode α. If r

(α)

i

is the ground-state motional amplitudes for the cooling ion, and k is

the cooling laser k-vector then η

(α)

i = k · r

(α)

i

.

The mode amplitudes represented by r

(α)

i

are determined based on Eq. 4.9, and Eq. 4.12. A pair

of ions of unequal mass held in the same linear trap have unequal motional amplitudes for each mode. In
63

particular, the radial modes, affected by the mass-dependent pseudopotential, often exhibit large imbalances

in mode amplitudes. As an example, we consider a 9Be+-

27Al+ array (μ = 3). We take trap conditions

such that a single Be+ has normal mode frequencies {ωx, ωy, ωz} = 2π × {13, 14, 4}. The minimum mode

amplitude for a single trapped Be+ is 6.3 nm. With the Be+ and Al+ in those trap conditions, the radial

out-of-phase modes have amplitudes 0.3 and 0.4 nm for the x and y directions respectively. For typical laser

geometries, the cooling rate of the lowest amplitude mode is more than 300 times lower than for a single Be+.

A mode with a small amplitude for the cooling ion will have a relatively large amplitude for the primary ion

as shown in Eq. 4.11. The small amplitude for cooling can limit the minimum temperature for that mode

and the large amplitude for the primary ion can make the elevated temperature a significant experimental

issue. Furthermore, while external heating of an equal-amplitude, out-of-phase mode (i.e. with two equal

mass ions) will be strongly suppressed [Kielpinski 00], modes with unbalanced amplitudes will heat under

spatially uniform fluctuating electric fields. These factors together make unbalanced motional amplitudes

problematic for effective sympathetic cooling.

The problematic mode amplitudes can often be enhanced significantly by adjusting trap parameters.

Figure 4.5: Relative cooling rates for a two ion array with μ = 3 as a function of Rω and Ex when: (a)

Directly cooling the lighter ion. (b) Directly cooling the heavier ion.
64

We illustrate this in Fig. 4.4 (a) by plotting |r

(αmin)

1

|

2 vs. Rω and Ex for the μ = 3 example, where αmin is the

mode with the lowest amplitude at each set of trap parameters. In most situations, this is the mode that will

limit the rate of Doppler cooling, and, in the presence of external heating, limit the minimum temperature.

Note that Rω in these plots refers the ωx/ωz for the heavier ion (i = 0), and |r

(αmin)

i

|

2

is normalized to the

highest value in the respective plots. In Fig. 4.4 (b) we plot |r

(αmin)

0

|

2

for the same parameters. The blue

bounding line represents the threshold between an axial and radial array. For the case of directly cooling

the lighter ion, the limiting rate can be significantly enhanced by adjusting the confinement parameters to a

specific region. In an experiment, higher cooling efficiency and lower temperature can be reached by applying

the proper endcap and radial bias voltages, then adiabatically relaxing the trap back to an axial array. An

enhancement in mode amplitudes can also be made in the case of directly cooling the heavier ion, although

the effect is not as pronounced and only occurs in one region close to the threshold for a radial array.

The qualitative features of Fig. 4.4 are very similar for other values of μ. In addition, the cooling rate

for higher numbers of ions can be enhanced with this technique. As with two ions, the specific values for

Rω and Ex that are optimal depend on μ, and the details of the ion configuration. Other factors that affect

the rate of cooling have been omitted from this discussion. Since η

(α)

i = k · r

(α)

i

, it is not only the absolute

value |r

(α)

i

| that is important, but also its direction with respect to the cooling lasers. As the ions move off

axis under the application of Ex, the mode vectors change direction. If k · ri = 0 for a particular mode, that

mode will not be cooled and instead will heat due to recoil. Another important consideration is the excess

micromotion the ions experience as they are pushed off axis. If the micromotion has a component in the

direction of k it can reduce cooling efficiency and, in some cases, cause heating. These issues can generally

be managed with the proper choice of trap conditions and cooling laser beam geometry.

4.5 Conclusion

We have examined several ways in which the dynamics of two-species ion arrays can affect experiments

on trapped ions. In several aspects, including equilibrium positions, micromotion compensation, and laser

cooling, these systems behave in a qualitatively different way than a single-species ion array. This both

presents experimental challenges and provides methods for overcoming them. In addition to the benefits
65

of sympathetic cooling and indirect state detection, the behavior of a two-species ion array allows for new

methods to measure and compensate excess micromotion. The examples presented mainly focused on two

ions of different species, but using the same model discussed in Sec. 4.1, the results can be extended to

larger ion numbers. The techniques and considerations discussed here apply to a number of experiments

already performed, including the experiments of this thesis. Given the interest in two-species ion arrays

for spectroscopy and quantum information processing, these considerations may also be helpful in future

experiments.
Chapter 5

Quantum Nondemolition Measurements of Trapped Ions

Reliable state detection plays a central role in many quantum physics experiments including quantumlimited metrology and quantum information processing. In practice, detection fidelity is limited by state

perturbations and noise during the measurement process. For example, for a single trapped ion, the discrimination between internal states presents a number of technical challenges. The excitation of a single atom

by an optical or rf photon adds only a small amount of energy to the system (∆E ≃ 10−25 to 10−19 J),

and achieving high enough signal-to-noise ratio (SNR) requires a huge amplification. At the same time, it is

important to reduce the time interval required for detection both for atomic clock experiments (to maximize

clock stability) and for quantum computation (to reduce error correction overhead [DiVincenzo 01]).

These constraints together put a high performance requirement on any measurement scheme1. A

couple of methods have been proposed and developed to detect the state of individual ions. For example,

the Fock-state of cyclotron motion for a single electron in a Penning trap has been detected electronically by

observing the spin-dependent current induced on a trap electrode [Peil 99]. However, the charge-to-mass ratio

of the electron is thousands of times larger than for atomic ions. The higher mass and consequently lower

motional frequencies lead to smaller currents that are difficult to detect in a reasonable integration time with

sensitive electronics. The standard method for detecting the internal state of a single trapped ion, proposed

by Hans Dehmelt [Dehmelt 75] and first implemented concurrently in three labs [Nagourney 86, Bergquist 86,

Sauter 86], is state-dependent resonance fluorescence. One feature that ties this method together with the

1 As a practical matter, it is also desirable to keep the ion trapped for subsequent experiments and keep its ionization state

unchanged. This requirement eliminates possibilities such as using state-dependent photoionization and detecting the emitted

electron with a microchannel plate.
67

observation of spin-dependent current is that they are both examples of quantum nondemolition (QND)

measurements. The QND nature of these schemes allows for the huge amplification factors necessary to

detect a single atom’s excitation.

The outline of this chapter is as follows: Section 5.1 introduces the idea of a QND measurement,

which will be the basis for all of the state detection protocols discussed. In our experiments, information

is ultimately transferred from the ions to the measurement apparatus via resonantly scattered photons that

impinge on a photon counter. Section 5.2.1 considers the Bayesian analysis of photon counts to determine

the result of a measurement. This analysis is applied in Sec. 5.2.2 to measurements of Be+ via resonance

fluorescence. In Sec. 5.2.3, I present a method for indirectly performing adaptive QND measurements

on a trapped ion by storing it with another ion of a different species. The measurement performance is

characterized experimentally and the ideas are extended to the simultaneous, indirect state detection of

multiple ion qubits. Finally, I consider alternative measurement schemes (Sec. 5.3.1) that may be more

suitable for different atomic systems. These ideas are relevant to measurements that generate entanglement

(Sec. 5.3.2) as well as a measurement protocol based on time-resolved photon counting (Sec. 5.3.3).

5.1 General QND measurements

The term nondemolition, loosely speaking, refers to the preservation of a projected eigenstate during

measurement of a quantum system. To be clear, it does not concern the projection of the quantum system

into an eigenstate of the measurement operator, which is an unavoidable consequence of the measurement

postulate in quantum mechanics (Ch. 1). The idea of a QND measurement was originally developed in the

context of gravitational wave detection [Caves 80], in which the measurement must distinguish harmonic

motion at the quantum scale. Standard measurement schemes tend to perturb the system being measured,

erasing the delicate information contained in the quantum state. The motivation behind a QND measurement

is to construct a measurement scheme that does not perturb the observable being measured, so that the

scheme can be repeated with the same outcome. As an imperfect QND measurement is repeated or applied

continuously, the measurement error probability due to classical noise decreases, while the probability of state

perturbation increases. This implies a minimum error probability at a particular measurement duration. The
68

measurement duration is a separate parameter to characterize the measurement efficiency.

The principles of a QND measurement can be stated succinctly in terms of general quantum mechanical

operators. We consider a quantum system, S, measured by another quantum system (meter), M. We would

like to measure an observable Oˆ

S by coupling system S to system M. The interaction between the systems is

described by the operator Hˆ

SM, and we assume there is an observable of M, OˆM, that is directly measurable.

The first condition,

[OˆM, Hˆ

SM] ̸= 0, (5.1)

ensures that the interaction Hamiltonian Hˆ

SM indeed affects the state of the meter. However, this is not

enough because the meter must evolve dependent on the state of the measured system. Therefore,

Hˆ

SM ̸= HˆS ⊗ HˆM, (5.2)

which is the condition that the interaction Hamiltonian is not separable into independent actions on S and

M. The final condition,

[Oˆ

S, Hˆ ] = 0, (5.3)

where Hˆ = Hˆ

SM + Hˆ0 and Hˆ0 describes the free evolution of the system, ensures that the projected state

of system S remains stable in time. The details of these conditions and examples of their application can be

found in two reviews of the subject [Caves 80, Braginsky 96].

No measurement of a real quantum mechanical system perfectly meets the condition of Eq. 5.3. For

example, when detecting the state of a quantum harmonic oscillator, the surrounding environment will have

finite temperature leading to a finite probability of thermal excitation during measurement. Similarly, the

electronic states of atomic systems can decay, whether due to their finite natural lifetime or to weak offresonant coupling to applied fields. The extent to which a QND measurement meets the criteria of Eq. 5.1

and 5.3 has been quantified formally in different contexts [Ralph 06, Holland 90]. Here, the performance of

an imperfect quantum nondemolition measurement is quantified primarily in terms of a single parameter

- the minimum error probability that can be achieved through repetitive measurements. If the conditions

stated above are met in a meaningful way, an imperfect QND measurement can be repeated to decrease

the probability of measurement error. As the repeated measurement cycles continue, however, the proba-
69

bility of perturbing the state increases. At a certain average measurement duration (or a average number

of measurement cycles) these two counteracting effects give rise to a certain minimum error probability.

The measurement duration itself is a separate, but often related, parameter describing the measurement

performance. I explore both of these performance measures in the next sections.

5.2 Experimental QND Measurements

All of the detection protocols used in our experiments can be considered as a QND measurement, and

in this section I discuss the measurement strategies in these terms. All of the information we extract from

system S about the ion electronic states, Si, comes in the form of photon counts from a photomultiplier tube

(PMT). This is true whether we are directly measuring the ions via resonance fluorescence as in Sec. 5.2.2,

or are indirectly inferring their state after interacting with other ions held in the same trap as in Sec. 5.2.3.

To provide a basis for these particular types of measurement I first describe the statistical analysis of photon

histograms.

5.2.1 Bayesian Analysis of Photon Histograms

The accurate evaluation of measurement outcomes depends on our previous knowledge of the photon

histograms corresponding to the NS possible ion states, {S1, S2, . . . , SNS }. The observable measured is

Oˆ

S =

∑

NS

i=1

λi|Si⟩⟨Si|. (5.4)

In most experiments we think of the ion states in terms of the electronic basis, with each of the N ions

occupying either qubit state | ↓⟩ or | ↑⟩ so that i ∈ [1, 2N ]. However, in the case of direct fluorescence

measurements we assume the ions are equally illuminated by the detection beam and equally imaged onto

the PMT so that they cannot be distinguished. The observable in this case is the number of fluorescing ions,

which is represented in Oˆ

S by assigning all states with equal number of ions in |↓⟩ the same eigenvalue.

If P(p|Si) is the probability of observing p photons given the state Si, then Bayes’ rule,

P(Si|p) = P(p|Si)

∑

k P(p|Sk)

, (5.5)
70

gives the probability of the ion state being Si given p photon counts. The measurement outcome is Si

′ , for

which P(p|Si

′ ) is greatest, and the probability of error is given by

Perr(p) = 1 − P(Si

′ |p). (5.6)

The mean error probability can be written as

Perr =

∑∞

p=0

Perr(p)

[∑

NS

k=1

P0(Sk)P(p|Sk)

]

, (5.7)

where P0(Sk) is the prior probability of state Sk and the factor in brackets gives the overall probability of

observing p photons. The prior probabilities are usually assumed to be 1/NS for all states. Substituting this

and Eq. 5.6 and 5.5 into Eq. 5.7 we can write

Perr =

1

Ns

∑∞

p=0

∑

k̸=i

′

P(p|Sk), (5.8)

remembering that the value of i

′ depends on the value of p. Although this expression involves an infinite

sum, as a practical matter it can be truncated at a reasonable maximum number of photons, pmax

2

.

The analysis above can be extended to indirect measurements of ion states. In indirect state measurements, information is transferred from the ions of system S to the ions of system M via a collective

mode of motion, then system M is measured directly. With repeated cycles of this process, the outcome of

a measurement is based on a series of photon counts, rather than the single number p. If C represents the

number of measurement cycles, then the series of photon counts is represented by {p} = {p1, p2, . . . , pC }.

Each number of photon counts carries information about the same state. Given a particular state, the

probability of observing the whole set {p} is

P({p}|Si) = ∏

C

j=1

P(pj |Si). (5.9)

Bayes’ rule (Eq. 5.5) applied to the probability distributions of Eq. 5.9 provides the measurement outcome,

Si

′ and the error probability is given by,

Perr({p}) = 1 − P(Si

′ |{p}). (5.10)

2

In the experiments of this chapter a single fluorescing ion might produce an average of p = 10 photons with a 1-σ spread

of √p in the measurement duration so pmax ∼ 30 is a safe assumption. Depending on the desired accuracy of the calculation,

one way to scale the cutoff with higher ion numbers N is pmax = Np + X

√

Np, where X is the number of standard deviations

required for the desired accuracy
71

All of the expressions above (Eq. 5.5- Eq. 5.7) can be applied equivalently to multiple measurements with

the substitution p → {p}. The mean error probability for a particular number of cycles is given by averaging

over all possible sets {p}. As with the sum over p, above, there are an infinite number of terms in the series,

but the series can be terminated using a reasonable maximum photon number. In this case, the number of

terms summed is p

C

max

, leading to,

Perr =

1

Ns

p

C∑max

l=1

∑

k̸=i

′

P({p}l|Sk). (5.11)

Evaluating Eq. 5.11 can become computationally intensive for a relatively small number of ions and

measurement cycles. An approximation to the mean error probability can be determined by a Monte-Carlo

simulation. In this method, the set of conditional probabilities, {P(p|Si)|i ∈ [1, NS]} are taken as input

based on separate calculations (see, for example, Sec. 5.3.1). Each state in the set {S1, S2, ..., SNS } acts as the

input state to a certain number of trials of the Monte Carlo simulation. If S1 is the input state then a series

of C photon counts are randomly chosen based on the known distribution P(p|S1). The error probability is

calculated based on Eq. 5.9 and Eq. 5.10 with Si, Si

′ → S1, and the mean error rate {Perr(Si)} is the average

of these individual error probabilities. Alternatively (and less efficiently) the most likely state Si

′ can be

determined based on the set of probabilities P({p}|Si). If Si

′ ̸= S1 an error has occurred and the estimated

mean error rate is the ratio of errors to trials. Depending on the accuracy desired, a certain number of

trials are performed with each of the possible input states, yielding the set of mean error rates, {Perr(Si)}.

The overall mean error rate is given by Perr =

∑

i P0(Si)Perr(Si). Known rates of spontaneous decay (or

off-resonant repumping) can be applied to system S in the simulation to obtain a reasonable estimate for

the minimum error probability. This numerical method has been used in several simulations for the different

scenarios described below.

5.2.2 Be+ Resonance Fluorescence as a QND measurement

In the experiments described in this chapter we use 9Be+ as a sympathetic cooling ion, and, in the

language QND measurements, as a quantum meter. Before discussing those experiments it is useful to

consider the direct measurement of Be+ through resonance fluorescence as a QND measurement. Here,
72

the primary system, S, is the Be+ atomic system with each Be+ ion considered as a qubit. The meter

system, M, is not as well characterized, but consists of the set of electromagnetic modes connecting the

ion to the PMT. The Be+ qubit is comprised of two hyperfine Zeeman sublevels of the 2S1/2 ground state,

|↓⟩Be ≡ |F = 2, mF = −2⟩ and |↑⟩Be ≡ |F = 1, mF = −1⟩. If we define |P⟩Be ≡ |2P 3

2

; F = 3, mF = −3⟩, the

states | ↓⟩Be and | ↑⟩Be are distinguished by applying a σ

− polarized laser pulse of duration tD tuned to

resonance for the |↓⟩Be → |P⟩Be cycling transition [Monroe 95]. For a given photon collection efficiency, ε,

and a given background count rate, γBG, the measurement fidelity for this system is limited by off-resonant

repumping of |↑⟩Be to |↓⟩Be. A detailed analysis of this process that considers electron shelving in different

Zeeman sublevels is presented in [Langer 06], following [Acton 06]. For the repumping of |↑⟩Be to |↓⟩Be here,

a simple exponentially decaying model is sufficient, for which the probability of having been repumped after

detection interval, tD is given by 1 − e

−γrtD . In terms of the saturation parameter, s0, the laser detuning

δ = ωHF , and the 2P3/2 linewidth, Γ, the repump rate is given by

γr = C↓

Γ

2

s0

1 + s0 + (2δ/Γ)2

, (5.12)

where C↓ is the coupling coefficient between the |

2P 3

2

; F = 3, mF = −2⟩ and the hyperfine level F = 2. We

make the realistic assumption that scattering from |F = 2, mF = −1⟩ to |↓⟩Be is fast compared to 1/γr and

tD.

The parameters, γr, γBG and γc ≡ εΓ, are sufficient for calculating photon histograms for any detection

time. Fluorescence histograms for 0 to 4 fluorescing Be+ ions based on realistic experimental parameters,

are displayed in Fig. 5.1(a). For the case of all ions in state |↑⟩Be the histogram represents the low level of

background counts as well as the long tail corresponding to ions fluorescing after being repumped. With

larger numbers of ions the overlap between adjacent ion states becomes more significant, both because the

repump probability increases and because the neighboring Poisson distributions have greater overlap. In

general, this favors longer detection times in order to further disperse the Poisson means (at the expense of

larger repump probability). Here, we assume that all of the ions are equally illuminated by the detection

laser and are imaged with equal efficiency so that, if there are NBe ions, there are NBe+1 states to distinguish.

The sets of photon histograms give the average measurement fidelity for distinguishing the states of using
73

Eq. 5.7. The simulated measurement error for NBe ∈ [1, 4] as a function of tD is shown in Fig. 5.1(b), with

the minimum error for 1 Be+ occurring at 200 μs while the minimum error probability for multiple Be+ is

approximately 500 μs.

0 10 20 30 40 50

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Photon Counts

Probability

0 200 400 600 800

10−2

10−1

100

tD [μ s]

P

err

1

2

3

4

1 Be

NBe

2 Be

3 Be 4 Be

0 Be

tD

 = 200 μ s

(b) (a)

Figure 5.1: (a) Histograms of photon counts calculated from realistic experimental parameters including

background counts and repumping from |↑⟩Be to |↓⟩Be. (b) Mean measurement infidelity as a function of

detection time tD for different numbers of ions. The small wiggles are due to finite sampling.

Strictly speaking, this measurement does not satisfy the condition 5.3. To begin with, the interaction

of the atom with the light field promotes |↓⟩Be to |P⟩Be so that Hˆ

SM contains terms proportional to |P⟩Be⟨↓|Be,

which does not commute with Oˆ

S = |↓⟩Be⟨↓|Be. This can be dealt with formally by identifying a scattering

event as a two photon process (as in [Wineland 79]), which, to the extent that the polarization is pure

σˆ

−, leaves the internal state unchanged. Equivalently, we can identify the observable as Oˆ

S = |↓⟩Be⟨↓|Be +

|P⟩Be⟨P|Be so that 5.3 is satisfied. An imperfection that cannot be formally avoided is off-resonant repumping
74

of |↑⟩Be, which introduces to Hˆ

SM terms proportional to |↓⟩Be⟨↑|Be. However, every photon scattered is a

complete measurement of the electronic state in that it results in the projection of the state into |↓⟩Be or

|↑⟩Be, and in principle can be counted. Since γr ≪ Γ, Eq. 5.3 is a good approximation and since γr/γc < 1,

Eq. 5.1 is satisfied in a meaningful way.

5.2.3 Qubit Detection through Repetitive QND Measurements

Direct detection of an ion qubit through resonance fluorescence requires a relatively strong interaction

of the ion to the environment, since only a small fraction of the photons scattered are detected. One way to

isolate a qubit further from the environment is to couple the primary quantum system, S, to a controlled

quantum meter, M, which is then directly detected [DiVincenzo 01, Haroche 06, Schaetz 05, Schmidt 05,

Gleyzes 07]. The measurement interaction, HSM, is now a controlled coherent process between two quantum

systems, which is inherently well insulated from the environment. Furthermore, the interaction between the

meter system and the measuring apparatus need not involve the primary system at all. These conditions

can help in satisfying Eq. 5.3. An important feature of a QND measurement is its repeatability [Meunier 06,

Gleyzes 07, Lupascu 07], which allows for high fidelity state detection in the presence of noise. The repetitive

transfer of information from S to M followed by detection of the meter state provides a natural mechanism

for real-time measurement feedback, which can further enhance detection efficiency [Armen 02, Cook 07].

We now apply the ideas of QND measurements to high-fidelity indirect measurements of one and twoqubit systems in an ion trap. In these experiments, one or two 27Al+ ions form the primary quantum system

S and one or two Be+ ions make up M. The ions are stored together in a linear, rf Paul trap and they are

interact through their mutual Coulomb repulsion. Both ion species can be considered as qubits, each having

an auxillary internal state used in the detection procedure (Fig. 5.2). In Al+, the qubit states consist of the

|

1S0; F = 5

2

, mF =

5

2

⟩ ≡ |↓⟩Al ground state and the metastable |

3P0; F = 5

2

, mF =

5

2

⟩ ≡ |↑⟩Al optically excited

state. Motional sideband transitions through the |

3P1; F = 7

2

, mF =

7

2

⟩ ≡ |3P1⟩Al auxillary state mediate the

measurement interaction, transferring information from S to M as described below.
75

Figure 5.2: Relevant energy levels in 27Al+ and 9Be+. The states | ↓⟩Al and | ↑⟩Al form the qubit to be

measured. The mF =

7

2

Zeeman sublevel of the 3P1 state forms a closed transition with |↓⟩Al. The widely

separated excited state lifetimes in Al+, 21 s and 300 μs, allow for many repetitions of the detection procedure

in a single experiment. The qubit states in Be+ are distinguished by state-dependent resonance fluorescence

[Monroe 95].
76

5.2.3.1 Be-Al:

The simplest ion system to consider for repetitive QND measurements consists of two ions, one ion

as the primary system and one as the meter. With Al+ in an unknown qubit state, the measurement

procedure begins by laser-cooling the two ions to the motional ground state [Barrett 03] and initializing Be+

to |↓⟩Be [Monroe 95]. Then, a series of laser pulses transfers the information in the Al+ system first to a

shared motional state then to the Be+ internal state. The Be+ ion is detected by resonance fluorescence as

described in Sec. 5.2.2. The individual steps and their durations are:

(1) Doppler cooling all modes (≃ 600 μs)

(2) Raman cooling axial modes to ground state (1 ms)

(3) Be+ preparation to the state |↓⟩Be (1 μs)

(4) State transfer between Al+ and Be+ (≃ 25 μs)

(5) Be+ state detection (200 μs).

In step (4) we couple the ions through the axial in-phase mode (ωm = 2π × 2.62 MHz) [Schmidt 05].

Here, we denote Fock states of motion as |n⟩m. First, a π-pulse on the |↓⟩Al|0⟩m → |3P1⟩Al|1⟩m sideband

transition inserts a quantum of motion into the mode dependent on the ion being in the |↓⟩Al state. The

information in the collective motion is then transferred to the internal state of Be+ using a π-pulse on the

|↓⟩Be|1⟩m → |↑⟩Be|0⟩m transition. This sequence implements an entangling operation,

(

α|↓⟩Al + β|↑⟩Al)|↓⟩Be → α|

3P1⟩Al|↑⟩Be + β|↑⟩Al|↓⟩Be. (5.13)

After measurement, the Al+ ion is projected into |↑⟩Al with probability |β|

2

. Because the 1S0 to 3P1 transition

is closed, the Al+ ion is projected into the manifold of the |↓⟩Al and |

3P1⟩Al states with probability |α|2

.

Although temporary optical excitation into |

3P1⟩Al represents a departure from the strict definition of a

QND measurement, we can formally address this (as in Sec. 5.2.2) by defining the state |↓⟩Al to include

the eigenspace spanned by the 1S0 and 3P1 states. Spontaneous emission from the 3P1 state (τ1 ≃ 300 μs)
77

effectively re-prepares Al+ back to |↓⟩Al with probability greater than 99 % before another detection cycle is

implemented.

Imperfect cooling and transfer pulses give rise to a single-cycle detection error of approximately 15

%. However, the fidelity can be improved by repeating the procedure. For the jth cycle of the measurement

procedure, a number of photons pj is scattered from the Be+ ion and collected in a photomultiplier tube.

The entire measurement yields a series of photon counts, {p}, that are used to determine the Al+ state

(see Sec. 5.2.1). Here, we analyze the photon counts in real-time, providing the means to actively control

the measurement process. Before the first cycle, we assume equal prior likelihoods for Al+ qubit states.

The probability, P(p|Si), of observing p photons given state Si of the Al+ system is determined based on

histograms continuously updated from previous measurements 3. The probability, P({p}|i), of Si producing

the observed series of photon counts is P({p}|Si) = ∏

j P(pj |Si). Applying Bayes’ rule,

P(Si|{p}) = P({p}|Si)

∑

k P({p}|Sk)

, (5.14)

yields the probability of a particular state Si given the observed series of photon counts. Here, k spans all

states in the Al+ system, in this case |↓⟩Al and |↑⟩Al. This procedure provides both the most likely state of

Al+, Si

′ , and also the probability of measurement error, 1−P(Si′ |{p}). The first assigns detection outcomes,

while the second is used to optimize the measurement process. Specifically, we repeat the detection cycles

only until the aggregate detection reaches a desired (threshold) error probability.

To experimentally determine the error rate for state discrimination, we compare two consecutive

detection sequences, each of which separately determines Si

′ and reaches a specified minimum error probability. If the two results agree, both detections are counted as correct, while disagreement signifies an error.

This analysis allows us to compare the actual error rate with the real-time prediction. Errors quantified

in this way also represent the fidelity of quantum state preparation [Lupascu 06]. The results for a single Al+ ion are plotted in Fig. 5.3 as a function of the specified threshold likelihood ratio (the greater of

P({p}||↑⟩Al)/P({p}||↓⟩Al) and P({p}||↓⟩Al)/P({p}||↑⟩Al)). The observed errors agree well with the predicted

error rate for fidelities up to 99.94 %. In the case of detecting |↓⟩Al, the observed error rate is as low as

3 Updates are made by applying an exponential filter to previous measurement results (1/e at 100 to 1000 measurements),

providing accurate histograms even if experimental parameters slowly drift.
78

101 102 103 104

10−4

10−3

10−2

10−1

Threshold Likelihood Ratio

Error Probability

0 5 10 15 20

Mean Detection Cycles

Error Rate 10−2

10−4

Figure 5.3: Observed (symbols) and calculated (solid line) error rates for repeated detections plotted as

a function of the threshold likelihood ratio, which is defined as the greater of P({p}||↑⟩Al)/P({p}||↓⟩Al)

and P({p}||↓⟩Al)/P({p}||↑⟩Al). The 1S0 detections (triangles) reach a repeatability of 99.99 % while 3P0

detections (squares) are limited by the state lifetime to 99.94 %. This is achieved for a desired likelihood

ratio of 103, requiring a mean number of detection cycles equal to 6.54. Inset: Simulation of qubit detection,

using experimental histograms, comparing the case in which the number of detection cycles is fixed (filled

circles) to that in which it is adaptive (open circles). The ability to estimate errors in real time significantly

increases measurement efficiency.

9 × 10−5consistent with the threshold chosen. However, for |↑⟩Al detections, the state lifetime τ0 = 21 s

limits the number of times the detection cycle can be repeated and still yield an accurate prediction. Here,

the observed error rate reaches a minimum at 6 × 10−4, then increases to above 1 × 10−3 as we demand

higher measurement confidence through more repetitions. This error rate agrees with that predicted from

the decay rate and the interval between detection cycles. The |↑⟩Al state lifetime and measurement cycle

duration, tc, set an upper bound on the attainable detection fidelity; that is, no measurement can achieve

an error rate lower than the probability of decay before the first detection, tc/τ0 ≃ 10−4in the experiment

here.

To investigate the gain in sensitivity achieved with adaptive measurements, we perform a MonteCarlo simulation of the detection procedure based on experimentally observed histograms. We compare

the adaptive scheme, which uses the minimum number of cycles necessary to achieve a given fidelity, to
79

a detection scheme where the number of detection cycles is set to a particular value [Fig. 5.3(inset)]. The

simulation assumes uncorrelated errors and includes spontaneous decay based on the known 3P0 state lifetime

and an experimental cycle time. In both the adaptive and fixed cases the final state determination results

from Bayesian analysis, which gives optimal results based on the detection record. As a function of mean

detection duration, the adaptive detection gives an exponentially lower error rate until reaching the minimum

error imposed by spontaneous decay. Although the minimum error achieved is nearly equal for both cases,

the adaptive scheme decreases the average time required to reach the minimum by more than a factor of

two. In the absence of effects such as spontaneous decay, which disturb state populations, this gain in

measurement efficiency would continue independent of the detection time.

5.2.3.2 Al-Be-Al:

Adaptive detection also provides a means to measure and prepare the state of a multiple qubit system

without the need for individual qubit addressing. To demonstrate the detection of two Al+ ions, we investigate two different schemes. First, we describe detection with a three ion array (one Be+ centered between the

two Al+), then show how we improve the measurement performance by using a four ions (two Al+ centered

between two Be+). The single meter qubit in the three ion array can yield at best one bit of information

in a measurement cycle. However, repeating the measurement process and varying the parameters of the

interaction yields more information [Haroche 06].

The first step in the three ion experiments is to ensure that the ions occupy the correct configuration

(Al-Be-Al). This is accomplished by applying a bias field perpendicular to the trap axis, ER ≃ 500 V/m,

which is strong enough to put the ions in a triangular configuration with the two Al+ ions on the outside

because of the mass dependence of the pseudopotential. Relaxing to ER = 0 V/m prepares the symmetric

axial ion configuration. In each measurement cycle all nine normal modes are Doppler cooled. The antisymmetric modes, which normally do not couple to Be+, are Doppler cooled sympathetically by again distorting

the ion array with a bias field, ER ≃ 300 V/m, then adiabatically relaxing the system back to the aligned

configuration (see Sec. 4.4). The two axial modes that couple to Be+ are cooled to the ground state and

then Be+ is prepared in |↓⟩Be.
80

0 50 100 150 1

2

3

4

5

6

7

Photon Counts

Time (μs)

T1

T2

Figure 5.4: Red sideband Rabi flopping on the |↓⟩Al − |3P1⟩Al transition beginning with one quantum of

motion. The fluorescence signal is obtained after transferring the motional information to the qubit state of

Be+. The three possible numbers of ground state Al+ ions, N↓ = 2 (solid circles), N↓ = 1 (empty squares),

and N↓ = 0 (empty triangles), are distinguished by their sideband Rabi rate. For zero ground state ions,

the known signal admixture from one ground state ion due to detection errors was removed. We determine

the state of excitation in the aluminum ion system after making multiple mapping sequences with pulses of

duration T1 or T2.
81

While the primary steps of the 3-ion procedure are identical to the 2-ion procedure above, the weaker

mode couplings for Be+ lead to a longer Doppler cooling duration (∼ 4 ms) and Raman cooling duration (∼

2 ms). To couple the Al+ system to the Be+ system (step 4) we use a slightly different procedure:

(1) |↓⟩Be|0⟩m → |↑⟩Be|1⟩m π-pulse

(2) |↓⟩Al|1⟩m ↔ |3P1⟩Al|0⟩m (variable duration)

(3) |↑⟩Be|1⟩m → |↓⟩Be|0⟩m π-pulse,

where the total duration is approximately 50 - 100 μs. The first pulse inserts a quantum of motion and

the second pulse removes it dependent on the Al+ state. If we apply the Morris-Shore transformation (see

Sec. 2.3) to the |↓↓⟩Al state evolution in step (2) we can describe the rsb Rabi flopping for this case in terms

of an effective two-state system. The Al+ states, then, are distinguished by their relative Rabi flopping

rates between |1⟩m and |0⟩m, which depend only on the number of ions, N↓ ∈ {0, 1, 2}, in the state |↓⟩Al.

Step (3) translates a motional excitation into a flipped spin in the Be+ qubit. Experimental excitation

curves for these three cases are plotted in Fig. 5.4 as a function of pulse duration. With N↓ = 0 no Rabi

flopping occurs. With N↓ = 1, in the Lamb-Dicke limit, Rabi flopping between | ↓⟩Al|1⟩m and |

3P1⟩|0m⟩

will proceed with a Rabi rate given by ΩR,1 ≃ Ωcη, where Ωc is the carrier Rabi frequency and η is the

Lamb-Dicke parameter for each of the aluminum ions [Monroe 95]. With N↓ = 2, both ions contribute

coherently to the sideband excitation. Rabi flopping carries the aluminum system to an entangled state,

|↓↓⟩Al|1⟩m → √

1

2

(

|

3P1 ↑⟩Al + |↓ 3P1⟩Al)

|0⟩m with characteristic Rabi rate ΩR,2 ≃

√

2Ωcη [King 98]. We find

good agreement between the fitted Rabi flopping rates and the rates expected based on separate calibrations

of Ωc and η.

Two particular transfer pulse durations, T1 = 30 μs and T2 = 80 μs, exhibit good discrimination

between one particular state and the other two. A combination of detection cycles using these two pulse

durations distinguishes the three states. In this scheme, the real-time Bayesian analysis plays the additional

role of determining the pulse duration that maximizes contrast between the two most-likely states based

on previous measurements. As before, we measure the error rate for state discrimination by comparing

consecutive detection sequences, and find a detection fidelity of 98.3%.
82

−300 −200 −100 0 100 200 300 400 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Detuning (Hz)

Transition Probability

0 10 20 30 0 

0.4

0.8

Pulse Time (ms)

Trans. Prob.

Figure 5.5: Signal from 3P0 spectroscopy using two Al+ and one Be+. Obtaining this signal depends on the

ability to prepare the |↓↓⟩Al or |↑↑⟩Al state of the Al+ system because transitions between |↓↑⟩Al and |↑↓⟩Al

will not be detected. Inset: Rabi flopping on the 3P0 transition. Signal contrast is limited by fluctuations

in the 3P0 excitation, rather than detection efficiency.
83

We have performed spectroscopy of the |↓⟩Al → |↑⟩Al transition on two Al+ ions using this scheme.

Note that the observation of a full-contrast |↓⟩Al → |↑⟩Al resonance depends on our ability to prepare the state

|↓↓⟩Al or |↑↑⟩Al before the spectroscopy pulse since transitions from |↓↑⟩Al to |↑↓⟩Al will not be distinguished.

Figure 5.5 shows a Fourier-limited lineshape and Rabi flopping (inset), for which the experiments beginning

in the states |↓↑⟩Al and |↑↓⟩Al have been filtered out in post-processing. Deterministic preparation of states

| ↑↑⟩Al or | ↓↓⟩Al could be performed efficiently by use of sideband pulses mapping the Al+ state to Be+

[Schmidt 05]. Although we have high detection fidelity (98.3 %), the temperature of some radial secular

modes limits signal contrast. Also, the measurement is slow, taking more than 0.5 s to reach the minimum

error probability. This is due to the poor signal contrast seen in Fig. 5.4 as well as the relatively long cooling

times necessary in each measurement cycle.

5.2.3.3 Be-Al-Al-Be:

To explore the improvement of measurement fidelity and efficiency by using more Be+ ions, we have

carried out experiments with two Al+ ions positioned in the center of two Be+ ions. This can improve the

cooling rate and increase the fluorescence signal. We choose trap conditions such that {ωx, ωy, ωz} = {13.5,

14.1, 3.0} MHz for a single Be+, which leads to good separation in frequency for all 12 normal modes of the

Be-Al-Al-Be ion array. The ions are ordered by reducing the rf trap power by 5 dB to form a diamond-shaped

configuration with Al+ ions displaced radially from the axis, then returning the rf power to the experimental

settings.

The steps in the detection procedure are identical to the steps for 1 Al+ and 1 Be+, although the laser

cooling durations are somewhat greater. This results in a total cycle time of ∼ 4.5 ms. As with 1 Al+ and

1 Be+, the transfer of state information (step (4)) consists of a blue sideband pulse on the |↓⟩Al → |3P1⟩Al

transition in Al+ followed by a red sideband pulse on the Be+ qubits. In this case, the first pulse will

lead to motional state |0⟩m, |1⟩m, or |2⟩m depending on the number of Al+ in |↓⟩Al and on the duration

of the sideband pulse. For the state |↑↑⟩Al, no motional excitation will occur, while for |↑↓⟩Al and |↓↑⟩Al

there will be at most one excitation. For | ↓↓⟩Al, a distribution of |0⟩m, |1⟩m, and |2⟩m will occur. We

desire a pulse duration that differentiates the three Al+ states well. To find this we note that the π-pulse
84

0 5 10 15 20 25 30

0

0.05

0.1

0.15

0.2

0.25

Photon Number

Probability

(b)

Figure 5.6: Data from the measurement of 2 Al+ with 2 Be+. (a) Experimental photon histograms displayed

as stepped lines for the three relevant Al+ states: |↓↓⟩Al (black), |↓↑⟩Al and |↑↓⟩Al (blue), and |↑↑⟩Al (red).

(b) Demonstration of quantum jumps between the three Al+ states. The frequency of the laser probing the

transition |↓⟩Al → |↑⟩Al (x axis) was scanned during the experiment. Each point represents the average of

190 detection cycles.

duration, t↓, that achieves full transfer from |0⟩m to |1⟩m for |↓↑⟩Al and |↑↓⟩Al is close to the pulse duration,

t↓↓, that maximizes the probability of two motional excitations for |↓↓⟩Al. Specifically, in the Lamb-Dicke

limit, t↓↓ = (2/3)1/2t↓ = π/√

6Ωcη [King 98]. We choose an intermediate pulse duration (the average of t↓↓

and t↓) that is near optimum for both cases simultaneously. Similarly, when choosing a pulse duration for

transferring the motional excitation to the Be+ qubits, we want to maximize the distinction between the

three motional states. Once again, the pulse durations t1 and t2 that achieve maximum transfer for |1⟩m

and |2⟩m respectively are similar (t1 = (3/4)1/2t2 = π/√

8Ωcη) a duration equal to the mean of t1 and t2

was chosen.

As opposed to the experiment with 1 Be+ and 2 Al+, this single set of pulse durations was used for all

detection cycles. The resulting experimental fluorescence histograms are plotted in Fig. 5.6 (a). In Fig. 5.6

(b), we plot the first quantum jump signal observed with this scheme. Three distinct fluorescence levels are

observed for the cases of N↓ = 0, 1, and 2. In this experiment, we used 190 detection cycles for each data

point. The points with an intermediate fluorescence level, as the fluorescence drops from one value to a lower

value, are most likely due to a spontaneous decay that occurred during the detection cycles. 4

4 Note that the mean fluorescence gathered at any one point leaves out all higher-order moments of the photon distribution.

To see this, as a hypothetical example, consider a point where the mean fluorescence signal is half that for two fluorescing Be+.

The distribution might be evenly divided between signals from 0 and 2 fluorescing Be+, or purely due to 1 fluorescing Be+.
85

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

10−4

10−3

10−2

10−1

100

Time (s)

Error Probability

2 Be 2 Al

1 Be 1 Al

1 Be 2 Al

Figure 5.7: Comparison of the detection error as a function of time for the three protocols studied, 1 Al+ 1

Be+ (black), 2 Al+ 1 Be+ (blue), and 2 Al+ 2 Be+ (red). The curves come from a Monte-Carlo simulation

of the detection process based on experimental histograms and cycle times for each case(See Sec. 5.2.1). The

fidelity limit in the simulation comes from the |↑⟩Al lifetime of τ0 = 21.4 s limits.
86

To quantify the fidelity using the maximum likelihood method, we perform an experiment with two

independent measurement sets, separately reaching a maximum error probability. The histograms used for

the likelihood determination come from experimental histograms for the three states that are continuously

updated. We record a 99.8 % fidelity in a measurement duration of ∼ 130 ms, which represents a substantial

increase in measurement efficiency compared the previous case with 1 Be+ and 2 Al+.

To summarize the measurement efficiency results, we compare the efficiency of the three detection

schemes by performing a Monte-Carlo simulation using experimental photon histograms and cycle times

together with the known lifetime of state |↑⟩Al. The results are plotted in Fig. 5.7. We note that when

the minimum measurement error is limited by the spontaneous decay, as we have assumed here, a good

approximation for the error is I = 1−e

−topt/NAlτ0

, where topt is the optimum average measurement duration,

and NAl is the number of Al+ ions. Thus, for a particular NAl, a higher optimum fidelity will always be

achieved in a shorter measurement duration, although the source of the improvement may be either shorter

measurement cycles or more accurate single-shot measurements. The minimum infidelities observed by the

simulation for 1 Be+ and 1 Al+, as well as 2 Be+ and 2 Al+ are slightly different from the measured minimum

infidelities. This is partly due to statistical fluctuations in both the simulation and the experiment. It might

also indicate slightly correlated errors between measurement cycles. The histograms tend to change slowly

over time because of laser frequency drifts, magnetic field fluctuations and drifts in laser alignment. If these

occur on a timescale longer than tc, but short compared to the histogram update rate it would result in

correlated errors, which are not present in the simulation. Unaccounted drifts in photon histograms could

also affect the results by causing a deviation of the photon histograms used in the simulation from those

sampled in the experiment.

As a concluding remark, although both experiments involving 2 Al+ were performed with an axially

symmetric ion configuration, the use of an asymmetric configuration has advantages. In some cases, the

Doppler cooling rate is improved because the 3N modes of motion couple more strongly to the cooling ion.

For example, in the case of Al-Be-Al, 3 different modes (one mode in each direction x, y and z) do not

Using only the mean fluorescence measurement these two possibilities would appear identical, but the maximum likelihood

method described above would clearly distinguish them.
87

couple to Be+. In these, the mode amplitudes for the two Al+ are equal and opposite while the amplitude

for Be+ is zero. On the other hand, all nine modes of the asymmetric configuration, Be-Al-Al and its

mirror image, couple to Be+ with finite amplitude. Furthermore, numerical simulations have shown that the

cooling rates achievable by distorting the asymmetric configuration with a bias field are higher than those

for the symmetric case. Another potentially beneficial feature of any asymmetric configuration, independent

of the total ion number, is that each ion generally exhibits a unique set of mode amplitudes. In contrast

to the experiments above, this permits discrimination between states like | ↓↑⟩Al and | ↑↓⟩Al. The unique

sideband coupling strength for the different eigenstates results in a unique distribution of Fock states after a

sideband pulse. As opposed to the other detection protocols described here, including resonance fluorescence

detection, this enables discrimination of the 2N eigenstates of N ion qubits without the need for individual

ion addressing or spatially resolved photon counting, which might be useful in both ion spectroscopy and

quantum information processing. For example at the end of a quantum computation, in general, all 2N

states must be distinguished. In spectroscopy this capability would avoid the necessity of preparing S in a

particular state before the spectroscopy pulse.

5.3 Generalizing Indirect QND Measurements

In the preceding sections we have shown how the transfer of atomic state information can be accomplished by use of resolved sideband pulses on both the M and S. This protocol uses several special features

of the Al+ system to enable both efficient transfer of state information and repetitive measurements. In

particular, the scheme relies on the condition that 1/ωm ≪ τ1 ≪ τ0, which allows resolved sideband pulses

and efficient repreparation of the state |↓⟩Al. To limit the effect of τ1 on measurement efficiency, we desire the

more stringent constraint that τ1 ≪ tc, so that the duty cycle for measurement is not dominated by the decay

time of the 3P1 excited state. This inequality is satisfied to varying degrees in the experiments of Sec. 5.2.3.

Another simplifying feature of the Al+ system is that a closed transition can be isolated between the |↓⟩Al

and |

3P1⟩ states by using pure ˆσ+ or ˆσ− polarized light, which ensures that Eq. 5.3 is satisfied. Interesting

atomic and molecular systems rarely exhibit this combination of conditions. Here I briefly consider other

methods for state-dependent motional excitation that may relax the requirements on the primary atomic
88

system.

5.3.1 QND Detection through Arbitrary States of Motion

A general formulation of the indirect detection of ion states makes no assumption about the method

used to map electronic state information from S into the motion, nor the method used to transfer motional

excitation into the electronic state of M. However, I still assume that resonance fluorescence detection

ultimately transfers information in M to the measuring apparatus. The efficiency for any such measurement

depends only on a single set of conditional probabilities {P(p|Si)}, relating photon probabilities to the initial

state. In principle, the process that leads to these conditional probabilities can be completely incoherent

and might not involve the electronic state of M at all, e.g. a state-dependent heating of the ion array leads

to a loss in fluorescence from |↓⟩Be (see Sec. 5.3.3). In this section I restrict my discussion to processes that

involve the electronic state of M, like the detection protocols described above. If we expand an arbitrary

motional state in terms of the Fock states, we can write,

P(p|Si) = ∑

j,n

P(p|Mj )P(Mj |n)P(n|Si). (5.15)

This divides the problem of calculating P(p|Si) into three distinct steps. The factor P(p|Mj ) has already been

calculated numerically in Sec. 5.2.2 for Be+ from considerations of the photon counting rate and repumping

rate. A similar calculation can be applied to other atomic ions. In what follows we individually consider the

factors P(Mj |n) and P(n|Si).

A first step in expanding the applicability of indirect state detection to a larger group of atomic and

molecular ions is to consider producing state-dependent motion without driving resolved sideband transitions

on system S. The synthesis and analysis of various ion motional states have been described previously

[Meekhof 96, Leibfried 96]. In that work they show that general Fock state distributions can be sensitively

reconstructed by analyzing state populations for a single ion qubit after applying a sideband pulse of variable

duration. We consider here the use of sideband pulses on system M to distinguish arbitrary states of motion

for the purpose of detecting the electronic state of S. We also point out where these ideas could be useful

for spectroscopy of atomic or molecular systems with certain features. As before, the quantum coherence
89

between individual Fock state components does not affect the measurement statistics, and the probabilities

represented by P(n|Si) are just the diagonal elements of the reduced density matrix after tracing over

electronic degrees of freedom.

As an example, with a single ion in system M and for a general distribution of Fock states, Pn, we

consider applying a laser pulse tuned to the red sideband of the |↓⟩M → |↑⟩M resonance . The ion state

probability evolves as,

P(|↓⟩M) = ∑∞

n=0

Pn cos2(Ωnt) (5.16)

P(|↑⟩M) = 1 − P(|↓⟩M), (5.17)

where Ωn =

√

nΩcη in terms of the meter ion’s Lamb-Dicke parameter, η ≪ 1. The amplitude of spectral

components in the red sideband state evolution at frequencies {Ωn| n ∈ [0∞]} can be used to reconstruct

Fock state distributions Pn. To efficiently distinguish only a limited set of known distributions, a small

number of measurements at one or more particular pulse durations are sufficient. The pulse durations can

be optimized (as in Sec. 5.2.3) by minimizing Eq. 5.8 as a function of t. With more than one ion in system

M, the state evolution during a sideband pulse becomes more complicated. For any set of states in M the

coupling strengths between states can be calculated as in Eq. 2.9, and the state evolution can be determined

numerically by integrating Schrodinger’s equation.

5.3.1.1 Thermal States:

Two important classes of ion motional states are thermal states and coherent states. Thermal states

occur as the result of stochastic processes such as laser cooling or heating due to electric field fluctuations.

From the ground state, an array of ions can be heated to a thermal state characterized by the mean motional

occupation, nth, which is an ensemble average over many preparations of the same state. The individual

Fock state probabilities are given by,

Pth(n) = n

n

th

(nth + 1)n+1 . (5.18)

By substituting Eq. 5.18 into Eq. 5.16 and evaluating the sum, we obtain the |↓⟩M probability as a function

of the pulse duration, which can be optimized for minimum detection error. As an illustration, consider a
90

system S comprised of a single ion with the same basic features in its atomic structure as Be+ (Fig. 5.2). The

relevant features are qubit levels |↓⟩ and |↑⟩, with |↓⟩ connected to an auxiliary excited state with resonant

linewidth ΓS . We assume that Γ ≫ ωm so that the motional sidebands are not resolved. Beginning with S

in an unknown state, the system is prepared by laser cooling to the ground state using system M. Then a

laser pulse detuned by ΓS /2 blue of the strong resonance in S is applied, which heats the ions depending on

the qubit state. If the ion is in |↓⟩M the ion array will heat, while if it is in |↑⟩M they will remain close to

the ground state. These two states of motion can be distinguished as described above.

In the absence of resolved sidebands, the use of thermal states for transferring information would be

advantageous in any scenario where it is less efficient or impractical to directly collect the photons from

the auxiliary resonance in S. For example, the resonance in S might be at a laser wavelength beyond

the spectral response of efficient optics and detectors. Alternatively, the rate of photon scattering from S

might be too low for efficient optical detection. This can occur if the linewidth ΓS is relatively small, or if

there exists a slow decay channel branching from the auxiliary excited state. In either case more efficient

detection could be achieved by analyzing ion motion with M. Finally, we note that the fidelity of fluorescence

detection can be limited (as with Be+) by off-resonant repumping of |↑⟩M. Since optical collection efficiency

is generally on the order of 1 × 10−3, many photons are scattered for every one collected. With detection by

ion heating, relatively few scattered photons will create a measurable change in ion motion, and in some cases

the maximum detection fidelity with thermal state detection could be better than with direct fluorescence

detection.

5.3.1.2 Coherent States:

Another important class of motional states are the coherent states (a general discussion of coherent

state properties is presented in [Haroche 06] and [Walls 08]). These states can be generated by applying a

classical driving force to the ions, i.e. by applying an oscillating electric field at frequency ωm. A coherent

state |α⟩ is decomposed in the Fock state basis as,

|α⟩ = e

−|α|

2

2

∑∞

n=0

α

n

√

n!

|n⟩, (5.19)
91

where α is a complex number. The probabilities of the individual Fock states are

|⟨n|α⟩|2 =

e

−|α|

2

|α|

2n

n!

. (5.20)

A coherent state can be generated dependent on the electronic state by applying Raman beams with frequency

ωL and ωL − ωm respectively, detuned from an optical resonance of one qubit state in S by a frequency

∆R ≫ ωm [Wineland 98].

For a relevant example consider a single Al+ as system S. Suppose the Raman lasers have k-vectors

k1 and k2 with k-vector difference ∆k ≡ k1 − k2 = ∆kzˆ. The σ

+ components of the lasers set up an

oscillating dipole force at frequency ωm that couples only to the state |↓⟩Al

5

. A resonant classical force

applied to an ion for some time tF acts as a displacement operator D(α) in phase space. For the purposes of

this section we are concerned only with the absolute value, |α(t)|, since the phase of the coherent state does

not affect the measurement outcome. We assume that the electric field amplitude for the σ

+ components of

both Raman beams are equal so that we can quantify their coupling strength to the carrier transition |↓⟩Al →

|

3P1⟩Al with a single Rabi frequency Ω0. Then the magnitude of the coherent state is |α(t)| = ηΩ2

0

tF /∆R.

State dependent coherent states can be produced up to an amplitude near the Lamb-Dicke regime η ∼ 1,

√

2|α(t)|

2 + 1∆kzz0 ≃ 1, where extent of the motional wavepacket is on the same order as the effective

wavelength of the driving fields. Note also, that in this scheme, with a component of σ

− polarization, that it

is possible to drive two-photon Raman transitions between the groundstate mF = 5/2 and mF = 1/2. This

process can be avoided by going to a sufficient magnetic field such that the Zeeman transition is sufficiently

detuned, or by going to a counter-propagating beam configuration with only σ

+ polarization.

The main benefit of transferring information between ion systems through coherent states of motion

is that excitation to the intermediate electronic state, in this case |

3P1⟩Al, can be strongly suppressed. If

negligible state population is transferred to the intermediate state, Eq. 5.3 can be satisfied without the

presence of a cycling transition. This capability could be applied to atomic and molecular systems with

complicated structures, to allow repeated detection of a long-lived state. The use of Raman beams also

suppresses the probability of spontaneous emission, PSE which is essential for maintaining coherence between

5

In general, the magnitude of the dipole force depends on the polarizabilities of the electronic states of S at the frequency

ωL, and a variety of laser configurations can be utilized dependent on the details of the atomic system.
92

electronic eigenstates (See Sec. 5.3.2). In the Al+ example,

PSE =

tF

τ1

(

2Ω2

0

∆2

R

+

Ω

2

3/2

∆2

3/2

)

, (5.21)

where Ω3/2 and ∆3/2 are the coupling strength and detuning respectively of the ˆσ

− polarization component

coupling to the state |

3P1, mF = 3/2⟩. With typical experimental parameters PSE < 1 × 10−2

, |α| ∼ 1 and

tF < 100 μs.

5.3.2 QND Measurements for Entanglement Generation

Resonance fluorescence detection projects the ion into an eigenstate of the atomic system on a timescale

of nanoseconds, extinguishing any coherence that existed in the atomic state before measurement. Likewise,

the techniques for indirect detection presented thus far project the system S into an atomic eigenstate with

near unit probability after a single measurement cycle. This happens primarily as a result of photon scattering

from system M, but also as a result of spontaneous emission from |

3P1⟩Al. Measurements that result in only

partial quantum state collapse are an interesting way of studying the progressive loss of coherence during

measurement [Katz 06]. This could be achieved in our experiments by transferring information through

coherent states, as described above, or by other mechanisms that avoid spontaneous decay from |

3P1⟩Al. It

could also be beneficial to make measurements that project the ion state into a basis other than the basis

of atomic eigenstates. For both fluorescence detection and indirect detection it is possible to effectively

make a measurement in any basis by first mapping the desired measurement basis into atomic eigenstates,

measuring as usual, then mapping the atomic eigenstates back into the measurement basis. However the

measurement fidelity and the fidelity of the final state in the measurement basis are both limited by the

fidelity of the mapping operation. To state it a different way, this is a QND measurement only to the extent

that the mapping operations are perfect. In the case of effectively projecting a state into an entangled basis,

for example, the process requires two entangling operations, which typically have low fidelity.

Entanglement has been generated deterministically and observed in trapped ions in a number of different ways. For example, the original Cirac-Zoller proposal has been implemented [Schmidt-Kaler 03], and

several subsequent proposals that address sources of infidelity in the Cirac-Zoller proposal have been ex-
93

plored experimentally [Leibfried 03, Benhelm 08]. Entanglement has also been generated probabilistically

by observing coincidence counts in scattered photons from two ions [Matsukevich 08]. In this section, motivated by the high fidelities achieved in measuring atomic eigenstates, we explore a different method for

probabilistically generating entangled states through indirect measurements. As in the previous probabilistic

scheme, the entanglement is heralded. This means that, when the entangling operation has been successful,

there is an observable effect that does not disturb the entanglement, i.e. fluorescence from the Be+ system.

In principle, the measurement can be repeated to achieve higher fidelity, since it is a QND measurement.

Below we show how these features provide a means to reduce the effects of most sources of infidelity, such

as residual Stark shifts and photon scattering, at the expense of a lower rate for generating entanglement.

The basis for our entanglement scheme is a two-ion parity measurement. The parity operator for two

qubits is given by,

Π =ˆ | ↑↑⟩⟨↑↑| + |↓↓⟩⟨↓↓| − (| ↑↓⟩⟨↑↓| + | ↓↑⟩⟨↓↑|). (5.22)

We refer to the states with eigenvalue 1 as even-parity states and states with eigenvalue -1 as odd-parity

states. Suppose we measure Π for a general 2-qubit pure state, ˆ |ψ⟩ = c0|↓↓⟩ + c1| ↓↑⟩ + c2| ↑↓⟩ + c3| ↑↑⟩,

with ∑

3

i=0

|ci|

2 = 1. We will observe an even-parity state with probability P1 = |c0|

2 + |c3|2 and an odd parity

state with probability P−1 = |c1|

2 + |c2|2

. The state projection during such a measurement produces the

final states,

|ψ1⟩ =

c0|↓↓⟩ + c3|↓↓⟩

√

P1

(5.23)

|ψ−1⟩ =

c1| ↓↑⟩ + c2| ↑↓⟩

√

P−1

. (5.24)

In general, these are entangled states and the degree of entanglement can be controlled by the preparation

of |ψ⟩. Maximal entanglement can be generated from a separable state if, for instance, ci = 1/2 for all

i. On the other hand if the state before measurement is the separable state |↓↓⟩, the projected state after

measurement will still be |↓↓⟩. Therefore, parity measurements are not measurements in an entangled basis,

but, with the proper preparation of the input state, the measurement can project a separable state into an

entangled state. To accomplish this, there is an additional requirement beyond the QND requirements of

Eq. 5.1-Eq. 5.3. Specifically, coherence between any states of the same parity must be maintained. Thus,
94

 = 0

1

2

0

1

2

5/2 7/2

Be Al Al Be

Mode

Vectors

Force 

Vectors

for 

σ + Interference pattern

= 3/2

= 5/2

(a) (b)

(c)

(d)

Figure 5.8: A set of drawings that portray several aspects of the scheme for projecting two Al+ into an

entangled state by QND measurements. (a) States of Al+ and laser configuration (b) Laser beam geometry

and ion configuration. (c) Mode vectors for the mode at ωm and dipole force vectors for the state |↓↓⟩Al.

The states |↑↓⟩Al and |↓↑⟩Al will experience this force for only the ion in state |↓⟩Al, while the state |↑↑⟩Al

experiences negligible forces. (d) The forces and mode vectors in (c) result in the excitation of coherent

states of motion depending on the ion states. For |↓↓⟩Al and |↑↑⟩Al no motional excitation occurs, whereas

|↓↑⟩Al and |↑↓⟩Al are excited to coherent states of magnitude |α| but out of phase by π rad.

the effects that can limit the fidelity in deterministic entanglement protocols, such as spontaneous emission

and fluctuating Stark shifts, can also limit the fidelity for this probabilistic protocol.

In measurements of atomic eigenstates, neither spontaneous emission nor Stark shifts, affect the measurement fidelity. The relatively straightforward protocols of Sec. 5.2.3 have a high probability of spontaneous

emission from |

3P1⟩Al and Stark shifts on |↓⟩Al of ∼ 10 rad in a single measurement cycle. To maintain coherence between states of equal parity, we need a protocol that reduces both of these effects. I have already

described how to avoid spontaneous emission by mapping information through coherent states of motion,

and will apply those ideas to a scheme for entangling two Al+ ions. A pair of off-resonant Raman beams,

with frequencies ωL and ωL − ωm respectively, detuned from the |↓⟩Al → |3P1⟩Al resonance by ∆R produce

a coherent motional state with small probability of excitation to |

3P1⟩Al. In Fig. 5.8 (a), we show the

directions of the laser beams and the external magnetic bias field relative to the ion array. As in Sec. 5.2.3

(c) we choose a symmetric configuration with 2 Al+ centered between 2 Be+ because we can fix the ion array

in this order by adjusting trap parameters and the symmetry ensures equal mode amplitudes for both Al+
95

ions for all modes. The two laser beam k-vectors, k1 and k2, are oriented such that ∆k = k1 − k2 is directed

along the trap axis. The frequencies and polarizations chosen set up a sinusoidal interference pattern in the

σ

+ polarized light that causes a z-dependent Stark shift, ES . The corresponding dipole force, F = −∂ES /∂z

oscillating at frequency ωm acts as a resonant, classical driving force on an in state |↓⟩Al. In Fig. 5.8 (b) we

present a conceptual picture of the intensity modulation and the resulting forces with respect to the mode

vectors for stretch mode (∆φm = π).

We choose the Al+ ion spacing, d, such that ∆kd = 2πlλ, where lλ is a whole number. For Be-Al-AlBe this occurs, for instance, when the stretch mode frequency is ωm = 4.778 MHz, for which d = 3.02μm

and lλ = 16. Since any Al+ in state |↑⟩Al experiences negligible force, and the force vectors for |↓↓⟩Al do not

overlap with the mode vectors, ideally, no motion will be produced for the even parity states. In particular,

F↓↓ = F↑↑ = 0, likewise F↓↑ = −F↑↓ = F0 sin (ωmt), where F0 is the peak force magnitude for a single

ion. A resonant classical force acts like a displacement operator, D(α) = e

αa†+α∗a

in phase space. The

displacements for each state are depicted in Fig. 5.8 (c). The magnitude of the displacement for |↑↓⟩Al and

|↑↓⟩Al is given by |α| = F0z0tF /2~, where tF is the duration of the driving force, and z0 is the ground state

amplitude of motion for the Al+ ions. For the trap conditions described above z0 = 3.57 nm. The presence or

absence of motion can be used to distinguish the parity of the two Al+ state using a red sideband transition

on the Be+ system.

The Stark shifts that drive the coherent motion also introduce phase shifts to the atomic states. The

phase shift is the accumulated Stark shift over the duration of the drive pulse and is given by,

∆Φ = N↓Emax

S

2

∫ tF

t=0

cos2

(

−

ωmt

2

)

dt =

N↓Emax

S

tF

4

, (5.25)

where N↓ is the number of Al+ in state |↓⟩Al. To the extent that the laser beam intensity is equal across

the two ions, the differential Stark shift between |↓↑⟩Al and |↑↓⟩Al will be zero. However for the even parity

states, the overall differential Stark shift is E

max

S tF

2

. With reasonable parameters, i.e. those for which α > 1,

The differential Stark shift between |↓↓⟩Al and |↑↑⟩Al is on the order of ∼ 20 rad. With slow intensity drifts,

which can arise from fluctuations in the laser beam position, this can cause significant decoherence between

the even parity states. One way to reduce the effects of these fluctuations is to switch the detuning of the
96

laser beams between ∆R and −∆R. This changes the sign of the Stark shift on |↓⟩Al. If the detuning is

switched on a timescale fast compared to the timescale of intensity fluctuations it will suppress slow drifts

in the Stark shift. However, the sign of the force on the two ions also reverses. To compensate for this, the

differential phase between the two Raman beams can be switched between ∆φ = 0 and ∆φ = π.

With the techniques described, we can produce coherent motion dependent on the parity of the twoion state, with a small probability of spontaneous emission and suppressed Stark shifts. Now we describe

two particular schemes to exploit this capability to produce heralded entangled states. The process for both

follows these steps:

(1) Doppler cool using Be+ (600 - 2000 μs)

(2) Cool axial states near the ground state of motion (∼ 1500 μs)

(3) Apply a π/2 pulse on the |↓⟩Al → |↑⟩Al transition (∼ 500 μs)

(4) Apply coherent displacement (α = 0.1 − 2, tF = 10 − 200 μs)

(5) Be+ rsb (∼ 10 μs)

(6) Be+ fluorescence detection ∼ 400 μs

Step (3) produces the equal superposition |ψ⟩Al =

1

2

(|↓↓⟩Al + |↓↑⟩Al + |↑↓⟩Al + |↑↑⟩Al) to which a parity measurement is applied. The duration of step (4) is an estimate based on experimental observations and is

adjusted based on the intensity of the laser beams and whether the intent is to produce √

1

2

(|↓↓⟩Al + |↑↑⟩Al)

or √

1

2

(|↓↑⟩Al + |↑↓⟩Al) as described below.

To produce the even parity entangled state |↑↓⟩Al and |↓↑⟩Al are driven to a coherent state of amplitude,

α = 1 − 2, so that |⟨0|α⟩|2 = e

−|α|

2

is relatively low, and a high measurement fidelity can be obtained with

a small number of measurements. To maximize this fidelity at the expense of entanglement success rate,

we can select only those experiments where all Be+ measurements of step (6) indicate the state |↓↓⟩Be, i.e.

no motion was excited by the displacement drive. One difficulty with this protocol is that the long drive

times lead to larger Stark shifts, which may not be perfectly canceled by switching ∆R. Another option is

to produce √

1

2

(|↓↑⟩Al + |↑↓⟩Al), which will experience no differential Stark shifts to the extent that the laser
97

beam intensity is balanced between the two ions. However, because the sign of α is opposite for the two

odd-parity states, the phase difference between |↓↑⟩Al and |↑↓⟩Al in the projected state,

|ψ⟩⟨n|α⟩ =

e

1

2

|α|

2

|α|

n

√

2n!

(|↓↑⟩Al + (−1)n|↑↓⟩Al), (5.26)

depends on whether the state of motion projects into an even or odd Fock state. The Fock state number can

generally not be determined based on the Be+ fluorescence measurement. One way to control the Fock-state

number is to use only a small displacement α ∼ 0.2, for which the probability of projecting the odd parity

states projecting into a Fock state higher than |1⟩ is small. In particular, |⟨2|α⟩|2/|⟨1|α⟩|2 = |α|

2/2, so that,

for example, the value α = .2 gives a probability of n = 1 that is 50 times that of n = 2. However, if

only those detection results that show n > 0 are taken as a success, the probability of success in a single

experiment is |⟨1|α⟩|2 = 0.04. One issue in this protocol, is that the ground state cooling must be nearly

perfect to ensure that Pth(|1⟩) ≪ |⟨1|α⟩|2. This concern and others make the realistic prospect of reaching

F > 0.9 experimentally challenging. Independent of entanglement fidelity it may provide and interesting

path to study quantum coherence by partial-collapse measurements.

In the above, I have presented some details of a new protocol for generating entangled states based on

QND measurements of the parity of an unentangled states. I specifically explored the experimental issues for

carrying out the protocol on a Be-Al-Al-Be ion array. Two significant challenges are Stark-Shift cancelation

for Al+ and the relatively low fidelity of fluorescence measurements on Be+, both of which may be improved

with other atomic systems, or improved experimental protocols. At the time of writing this thesis we have

demonstrated most individual steps in this protocol experimentally. We have shown ground state cooling

of Be-Al-Al-Be. We have produced coherent states dependent on the Al+ states with these four ions and

have shown elimination of the motion for the case |↓↓⟩Al by adjustment of the Al+ ion spacing. In addition

we measured Stark shifts from the coherent drive with a single Al+ and demonstrated the suppression of

the Stark shifts during the coherent drive by switching ∆R and ∆φ simultaneously. Unfortunately the

experiment was not completed because of by experimental difficulties, the failure of a laser system and other

experimental priorities. It is possible also that a two-photon Zeeman state-changing process was affecting

the experiment, which is a fact that we had not appreciated at the time. Although the experiments are
98

challenging, they present a novel approach to generating entangled two-ion states and provide a means to

address some of the infidelities present in deterministic protocols. In addition, these ideas can be used to

study the loss of coherence due to partial quantum state collapse in ions, which has previously not been

studied.

5.3.3 Detection by Modulated Fluorescence

The detection schemes discussed so far require several separate laser beams for system M to perform

Doppler cooling, detection, repumping and Raman pulses. In this section I discuss the possibility of eliminating the need for resolved-sideband Raman pulses on system M and performing indirect state detection with

only a single laser beam tuned near resonance of a strongly allowed transition in the measurement system. I

consider again detecting the state of one ion by coupling it electrostatically to a meter ion in the same trap.

The basic idea behind the scheme is to use the meter ion to detect the presence of a state-dependent classical

driving force near resonance with a secular mode frequency. This can be created by applying Raman beams

that have a difference frequency tuned to the motional frequency for an axial mode. The effect we observe is

a modulation of the fluorescence rate as a function of the phase of the classical driving force. In particular,

we consider applying a Doppler cooling laser, propagating along direction ˆkL, to the meter ion tuned to

frequency ωL such that ωL −ω0 = −Γ/2, where ω0 is the resonance frequency for the cooling transition, and

Γ is its linewidth.

The ion system simultaneously subjected to laser cooling and a harmonic driving force can be accurately modeled as a classical damped, driven harmonic oscillator. The one-dimensional equation of motion

for such a system is

z ̈ + 2γdz ̇ + ω

2

Mz = F0 cos(ωF t + θF ), (5.27)

where we have denoted the relevant motional degree of freedom as z. Here γd is the classical amplitude

damping rate, and F0 is the amplitude of the driving force at frequency ωF and phase θF . This differential

equation can be solved by assuming oscillatory solutions of the form

z = z0e

iωF t

. (5.28)
99

We substitute z,  ̇z and  ̈z in Eq. 5.27 and equate real and imaginary components. The steady-state motional

amplitude, zS, is given by,

zS =

F0

m

1

√

(ω

2

M − ω

2

F

)

2 + (2γdωF )2

. (5.29)

which is independent of initial conditions.

The ion velocity is just the real part of the derivative of Eq. 5.28. If we assume that the ion is driven

on resonance, then the peak velocity is given by

V0 =

F0

2mγd

. (5.30)

If we denote the peak ion velocity in the direction of the laser beam used for detection as Vk = V0zˆ ·

ˆkD,

in the rest frame of the ion the laser frequency observed will be modulated at the frequency of the secular

motion such that

ω

′

L = ωL[1 + Vk cos(ωMt + θM)]. (5.31)

The scattering rate as a function of time can now be written as

R(t) = R0

1 + [

2ωLVk cos(ωMt+θM)

cΓ − 1

]2

, (5.32)

where R0 is the on-resonance scattering rate assuming the saturation parameter, S0 ≪ 1. If we also assume

that the maximum Doppler shift is small compared to the atomic linewidth, or 2ωLV0/cΓ ≪ 1 (a condition

which is justified with experimental parameters later), we can expand the scattering rate to first order in

this parameter and arrive at the expression,

R(t) = R0

2

[1 + 2ωLVk

cΓ

cos(ωMt + θM)] (5.33)

Thus, we can detect the presence of a classical driving force by looking for a sinusoidal modulation of the

fluorescence rate at frequency ωM.

A figure of merit for any implementation of this detection scheme is the modulation depth defined as

M ≡ ∆R/R0, where ∆R ≡ R+ −R− is the difference between the maximum and minimum fluorescence rate

of Eq. 5.32. For M ≪ 1, this becomes

M =

2ωLVk

cΓ

. (5.34)
100

PMT

Discriminator

FPGA

DDS

V cos( t + ) 0 M M w q

z

Doppler

cooling

laser

Bin 1 2 3

. . .

0

p

6

2p

6

3p

6

. . .

. . .

DDS

Phase

Photons

kL

°

Figure 5.9: A simple schematic of the experimental setup for detecting modulation of the fluorescence from

25Mg+ at the axial motion frequency. The ion is driven near a motional resonance by a DDS applying an rf

signal to one endcap. Meanwhile a Doppler cooling laser is applied to the (2S1/2, F = 3) → 2P3/2 transition

and the fluorescence counts from a PMT are recorded by an FPGA. The counts are binned in 12 different

bins according to the rf phase at the time they were recorded, so that differences in the fluorescence rate in

an rf cycle can be distinguished.

One approximation to the maximum modulation achievable would be the situation where R+ = R0. Using

Eq. 5.32, this requires that 2ωLVk/cΓ = 1. Under this condition R− = R0/5. Because ∆R ∼ R0 the

minimum time scale to distinguish a driven ion from an undriven ion is approximately the same as the time

scale for fluorescence detection.

We demonstrate some aspects of the detection of a classical driving force using a single 25Mg+. The

experimental setup is sketched in (Fig. 5.9). We perform laser cooling and detection on the (2S1/2, F = 3)

→ 2P3/2 cycling transition with Γ = 2π × 41.4 MHz. We apply a classical driving force by modulating

the voltage on one endcap at a frequency close to the axial mode frequency, ωM = 2π × 3.06 MHz. The

modulation signal is also fed into an FPGA board that records photon counts from the PMT. Photon counts

are binned based on the phase of the modulation signal at the time they occurred [Berkeland 98].

We can observe a motional resonance by scanning the frequency of the endcap modulation and analyzing the photon count rate as a function of modulation phase. At each drive frequency, ωF , we compute a
101

0

0.04

0.08

0.12

M

2

3.05 3.06 3.07 3.08

0

Modulation Frequency [MHz]

Mod. Phase [rad]

−π/2

−π/4

π/4

π/2

Figure 5.10: Modulation of ion fluorescence at the motional resonance frequency. (a) The square of the

modulation depth (Eq. 5.34) as a function of drive frequency frequency (blue circles). (b) Modulation phase

(θm in the text) as a function of the drive frequency. The error bars in (a) and (b) represent a 68% confidence

interval in the fit of observed fluorescence to to a sine function.

weighted least-squares fit to the function

R(t) = R0

2

[1 + M cos(ωF t + θM)], (5.35)

which comes from the substitution of Eq. 5.34 into Eq. 5.33. This provides the modulation depth, M, as well

as the modulation phase θM. In Fig. 5.10 (a) we plot M2 vs. ωF for a particular magnitude of the driving

field. We observe a Lorentzian lineshape with a linewidth of 5.3 kHz. This corresponds to a damping time

of the axial mode due to Doppler laser cooling of 30 μs, which agrees well with the prediction from laser

cooling theory of 34 μs. In Fig. 5.10 (b) we plot θM as a function of the drive frequency and observe a π

phase shift as the frequency is scanned through resonance.

To use this type of signal for state detection we trap the sympathetic cooling ion together with the
102

ion to be studied. The classical driving force is created by two Raman beams with difference frequency ωm.

Generally the propagation direction of the two beams is set such that ∆k = k1 − k2 is parallel to the z-axis

of the ion trap, but here we assume an arbitrary direction for ∆k. In this scheme the Lamb-Dicke limit

imposes a bound on the achievable fluorescence modulation. The motional amplitude at the boundary of

the Lamb-Dicke regime, zLD is defined by ∆k · zzˆ S = |∆k|zS cos(θk) = 1 where θk is the angle between ∆k

and the z axis. The peak velocity of the cooling ion in the direction of the detection laser is then

Vk =

ωM

|∆k| cos(θk)

z2

z1

zˆ ·

ˆkD, (5.36)

where z2 and z1 are the relative amplitudes of motion for the meter ion and primary ion respectively for

the particular secular mode chosen. It is important to note that although the Lamb-Dicke limit becomes

less restrictive for smaller |∆k|, the driving force is also reduced, requiring more laser power to achieve high

motional amplitudes for a given motional damping rate.

The primary benefit of the scheme discussed here is the elimination of several separate laser beams

used for Raman transitions and ground state cooling. This might significantly simplify the experimental

requirements for spectroscopy of Al+, by reducing the complexity of the laser setup and reducing the laser

power requirement. Although a state-dependent measurement has not yet been perfomed on Al+ in this way,

we have demonstrated the principle with motion driven by an rf field, and further experiments are currently

underway.

5.4 Conclusion

In this chapter I have explored several techniques for indirect state detection of Al+ ions and demonstrated high-fidelity state detection on systems with one and two Al+ ions. I also explored ideas for performing indirect state detection by use of coherent states of motion. Many of the protocols described can be

adapted to other ion systems, and in some cases may perform better with alternate atomic structures. The

fidelity of detection demonstrated and the flexibility of detection mediated by ion motion may lead to new

experiments making measurements of otherwise unexplored atomic and molecular systems. These techniques

may enable a broad array interesting experiments in spectroscopy and quantum information science.
Chapter 6

Preparation of Dicke States in an Ion Chain

Entanglement is a fundamentally nonclassical feature of quantum mechanics and has been recognized

as an important resource in quantum information science [Nielsen 00]. The coherent manipulation of entangled quantum states can be useful for important tasks such as computation, cryptography, and metrology. In

addition, the insights gained from research with entangled states may contribute to a better understanding

of the foundations of quantum theory and shed light on physics at the boundary between the quantum and

classical worlds.

Various entangled states have been experimentally demonstrated in a wide variety of physical systems

including photons, condensed matter systems, atoms in optical lattices, and trapped ions [Insight 08]. In the

case of trapped ions, Bell states have been generated with high fidelities [Leibfried 03, Benhelm 08]. For larger

numbers of ion qubits, GHZ states [Leibfried 05, Sackett 00] and W states [H ̈affner 05] have been realized.

As the number of physical parts in a quantum system grows, we can find more classes of entangled states

[D ̈ur 00] and potentially more interesting applications, but the state preparation and verification becomes

more challenging. Here we explore numerically an approach to preparing multiple trapped ions in a general

class of entangled states called Dicke states. We then demonstrate the basic protocol on a mixed-species ion

chain composed of two Mg+ ions coupled to a single Al+ ion.

The Dicke state |D

(m)

N ⟩ is the equal superposition of all basis states of N qubits having exactly m

excitations [Dicke 54]. If we denote the qubit states as |↓⟩ and |↑⟩, we can write an arbitrary Dicke state in

the following form

|D(m)

N ⟩ =

(

N

m

)− 1

2 ∑

k

Pk

(

| ↓⊗(N−m)↑

⊗m⟩

)

. (6.1)
104

The sum is over all (N

m

)

permutations (produced by the permutation operator Pk) with m qubits in the state

|↑⟩. The W states are a special case of |D

(m)

N ⟩ with m = 1. Entanglement in Dicke states is highly resilient

against external perturbations and measurements on individual qubits [Stockton 03, D ̈ur 01]. Through

projective measurements on some of the qubits in the system, we can obtain states of different entanglement

classes. Thus, the Dicke states can serve as a versatile resource for the preparation of multipartite entangled

states.

In addition to the technique used in [H ̈affner 05], other methods for the generation of Dicke states

with trapped-ion qubits have been proposed [Retzker 07, Linington 08a, L ́opez 07]. The basic features of

our approach follow those proposals. We consider a chain of N ion qubits each initialized in the state | ↓⟩

and collectively cooled to the ground state of motion for a particular mode. By addressing a single ion in the

chain with a laser pulse tuned to the mth higher-frequency motional sideband (blue sideband) [Meekhof 96]

we produce the state

|ψ⟩ = | ↓1↓2 . . . ↓N⟩ ⊗ |m⟩M, (6.2)

where we have labeled the motional Fock state with a subscript M. From |ψ⟩, a laser pulse of appropriate

duration addressing all N qubits and tuned to the first lower-frequency motional sideband (red sideband),

creates a state that is a close approximation of |D

(m)

N ⟩.

We define the fidelity for an arbitrary final state, ρ, as F ≡ ⟨D

(m)

N |ρ|D

(m)

N ⟩. In the case with m = 1

(W states), the procedure outlined above can achieve arbitrarily high fidelity. For higher-order Dicke states

the attainable fidelity is reduced because the red sideband pulse does not transfer population completely

from |ψ⟩ to |D

(m)

N ⟩. The previous proposals showed that this imperfection can be mitigated by postselection

[Retzker 07], or by a generalization of adiabatic rapid passage [Linington 08b, Linington 08a]. However, the

simplified method presented here is sufficient to achieve fidelities limited by other experimental imperfection

for a range of Dicke states.

One experimental challenge in the above scheme is to individually address a single ion to produce the

state |ψ⟩. High confinement frequencies, and consequently small inter-ion spacing, are desirable to resolve

the motional sideband spectrum. This makes individual spatial addressing difficult. We avoid this problem

by introducing to the N-qubit ion chain a single ancillary ion of a different atomic species. Simultaneous
105

0.990

0.995

1.00

D

(1)

2

 S

D

(1)

3

D

(1)

4

D

(1)

2

 A

0.1 1 10

0.99

0.96

Fidelity

μ

D

(2)

4

D

(2)

5

D

(2)

6

a

b

D

(3)

6

D

(1)

6

D

(1)

5

Figure 6.1: (a) Fidelity of |D

(1)

N ⟩ state generation (by use of the in-phase axial mode of motion) as a function

of mass ratio, μ, for N = 2 . . . 6. In the case of N = 2 we consider both the symmetric case labeled S, with

the ancillary ion in the center, and the asymmetric case labeled A, with the ancillary ion on the outside.

The latter case is the one chosen in the experiment. For N = 3 to N = 6 we assume that the ancillary ion

occupies a central position in the ion chain. (b) Fidelity of |D

(2)

N ⟩ and |D

(3)

N ⟩ state generation as a function

of mass ratio for N = 4 . . . 6. Here the ancilla ion occupies a central position for all cases.

trapping of individual ions of two species has been used for sympathetic cooling [Barrett 03] and indirect

state detection [Hume 07]. Spectroscopic resolution of the two atomic species ensures that any laser pulse

applied to one species will leave the internal states of the other ion species unchanged. The state |ψ⟩ can

be created by applying a global laser pulse tuned to the mth sideband (or equivalently m pulses sequentially

tuned to the first sideband) of an accessible transition in the ancilla.

Two-species ion chains typically contain ions of unequal mass, which leads to unequal motional mode

couplings. To see how this affects state fidelity, we consider the generation of W states (|D

(1)

N ⟩), for which

the fidelity can be perfect in the case of equal masses. For the i

th ion in the chain, we denote the groundstate motional amplitude as zi

. The Lamb-Dicke parameter is defined by ηi ≡ kzi, where k is the laser

beam k-vector projection along the ˆz direction. In the Lamb-Dicke limit (ηi ≪ 1), the strength of the ion’s

coupling to the red sideband pulse is Ωi = Ω0ηi, where Ω0 is the carrier (|↓⟩|m⟩M → |↑⟩|m⟩M) Rabi rate.
106

Beginning from the state |ψ⟩ with m = 1, under the red sideband interaction, the N-ion state acts like a

two-level system as described by the Morris-Shore transformation [Morris 83]. The motion undergoes Rabi

oscillations between the two Fock states |1⟩ and |0⟩ at a frequency, Ω′that obeys the relation

Ω

′2 =

∑

N

i=1

Ω

2

i

(6.3)

with all Ωi real. Meanwhile the qubits evolve from |ψ⟩ to the state

|ψ

′

⟩ =

1

Ω′

(

Ω1| ↑↓ . . . ↓⟩ + Ω2| ↓↑ . . . ↓⟩ + . . . + ΩN | ↓↓ . . . ↑⟩), (6.4)

in which the terms in the superposition are the same as those in |D

(1)

N ⟩ but the state amplitudes are proportional to the individual ion coupling strengths, Ωi

. The obtainable W-state fidelity, F = |⟨D

(1)

N |ψ

′

⟩|2, can be

written in terms of the coupling strengths as

F =

1

NΩ′ 2

(∑

N

i=1

Ωi

)2

. (6.5)

To quantify the effect on fidelity of different mass ratios and different ion configurations, we first

calculate the motional amplitudes [Kielpinski 00]. Each configuration is specified by the number of qubit

ions, N, and the mass ratio μ = Mancilla/Mqubit. With one exception noted below we assume that the

position of the ancilla is at the center of the chain (for N even) or adjacent to the center of the chain

(for N odd). These configurations can be prepared deterministically by adjusting trap parameters. We

calculate equilibrium positions for the N + 1 ions, then determine the amplitude of small oscillations about

equilibrium. The in-phase, axial mode of motion exhibits the smallest deviations between individual ion

motional amplitudes and is used for all fidelity calculations. These amplitudes give us the coupling strengths,

Ωi, and F follows from Eq. (6.5). The results are presented in Fig. 6.1(a), where we have assumed the

Lamb-Dicke limit for simplicity, although fidelities for {ηi} ≈ 1 are similar. The case labeled D(1)

2 S is the

only one that yields a theoretically perfect fidelity for any μ ̸= 1 because that symmetric configuration, with

the ancilla in the center of two qubits, gives equal motional amplitudes for the two outer ions. The cases

with N = 3 . . . 6, as well as that labeled D(1)

2 A, with the ancilla at the outside position, allow high fidelity

over a wide range of mass ratios.

In Fig. 6.1 (b) we present the result of a similar calculation for the states |D

(2)

N ⟩ and |D

(3)

N ⟩ with
107

N = 4 . . . 6. During the application of the red sideband pulse the ions state evolves in a complicated manner.

To determine a practical upper limit on the fidelity, we numerically find the first maximum of the quantity

F = ⟨D

(2)

N |ρ(t)|D

(2)

N ⟩ as a function of the red sideband pulse duration. Here ρ(t) is the reduced density matrix

of the qubit system at pulse duration t after tracing over the motional degrees of freedom. We find fidelities

for the case m = 2 as high as 0.99 and for m = 3 above 0.96. The effect of unequal masses is found to be

similar to the W state case with a mass ratio of 10 reducing the optimum fidelity at the 1 % to 2 % level.

These numbers refer to the fidelities at the first maximum during the red sideband evolution, but, in the

absence of decoherence, F could in general be made higher by evolving for a longer duration and reaching a

later maximum.

Note that the fidelities for μ = 1 in Fig. 6.1 are equal to the fidelities obtainable in the case where

there is no ancilla and |ψ⟩ is generated by spatially addressing a single ion with a strongly focussed laser

beam. Here, there are only N ions in the chain, and they share equal motional amplitudes for the in-phase

axial mode. Under the red sideband interaction beginning from |ψ⟩ the system can be transformed to a basis

where it evolves as a ladder of m + 1 states spaced equally in energy [Rangelov 06]. Here the red sideband

Hamiltonian is symmetric with respect to ion exchange, so all ions participate equally in the entangled state.

We have demonstrated the basic features of the protocol in an experiment with one 27Al+ ancillary ion

and two 25Mg+ qubit ions. The ions are trapped in a linear RF Paul trap [Rowe 02], with trap frequencies

for a single 25Mg+ ion {ωx, ωy, ωz} = 2π × {5.54, 6.46, 2.55} MHz where z denotes the axis of the ion chain.

In 25Mg+ we use two Zeeman sub-levels of the ground-state hyperfine manifold as a qubit. We define

|↓⟩ ≡ |2S 1

2

, F = 3, mF = −3⟩ and |↑⟩ ≡ |2S 1

2

, F = 2, mF = −2⟩. In 27Al+ the relevant qubit levels are

defined as |

1S0⟩ ≡ |1S0, F =

5

2

, mF = −

5

2

⟩ and |

3P1⟩ ≡ |3P1, F =

7

2

, mF = −

7

2

⟩. The ions are loaded into

the trap via photo-ionization.

The order of the ions is maintained as Mg-Mg-Al by monitoring the motional spectrum and adjusting

DC voltages of the trap electrodes to regain the correct order when necessary. Specifically, we raise the DC

endcap voltages and apply a radial bias field to configure a radially oriented, linear ion chain with Al at

one end. Then we apply differential endcap voltages to twist the radial chain to the desired orientation and

finally relax the voltages back to the experimental parameters. In the order Mg-Mg-Al, the amplitudes of
108

0 5 10 15 20 25 30

0

0.02

0.04

0.06

0.08

0.1

Photon Counts

Probability

−1

−0.5

0

0.5

1

Ramsey Phase, φ (rad)

Parity

a

b

π 2π 3π

Figure 6.2: (a) Example experimental histogram of photon counts from fluorescence detection of entangled

ions (bars) and its fit (dot-dashed line). The stepped lines represent calibrated histograms P(n|0), P(n|1),

and P(n|2), scaled respectively by the fit parameters {c0, c1, c2} = {0.08, 0.80, 0.12}. This contributes to

the overall fidelity with ρ↓↑,↓↑ + ρ↑↓,↑↓ = 0.80(1). (b) Parity of qubit state after applying a rotation R(

π

2

, φ)

of varying phase (squares) and the parity after first applying the rotation R(

π

2

,

π

2

), then applying a second

rotation, R(

π

2

, φ), of varying phase (circles). The residual oscillation in the former case (squares) arises from

small even-parity populations in ρ. The measurements give a an overall fidelity F = 0.77(2).

motion for the Mg+ ion in the in-phase mode are equal to within 1%, having a negligible impact on fidelity,

while the Mg+ ion spacing is small (3 μm) to facilitate equally strong interaction with the laser beams.

Initially, the three axial motional modes of the ions are cooled close to the ground state by resolved

sideband Raman cooling of Mg+ [Monroe 95]. This process fills the largest part of our experimental duty

cycle, about 2 ms. We observe residual phonon numbers n < 0.1 for all axial modes. Optical pumping

ideally prepares the system in |↓↓⟩|1S0⟩|0⟩M. The preparation of |D(1)

2

⟩ starts with a laser pulse that adds

one phonon to the in-phase axial mode by driving the blue sideband transition |

1S0⟩|0⟩M → |3P1⟩|1⟩M in

the Al+ ion, changing the ion state to | ↓↓⟩|3P1⟩|1⟩M. Then a laser pulse removes one phonon from the

in-phase mode by driving the red sideband transition in the Mg+ ions. This ideally produces the state

√

1

2

(| ↓↑⟩ + | ↑↓⟩)|

3P1⟩|0⟩M [King 98].
109

In terms of the individual density matrix elements, the fidelity of the final Mg+ state, ρ, is

F =

1

2

(ρ↓↑,↓↑ + ρ↑↓,↑↓ + ρ↓↑,↑↓ + ρ↑↓,↓↑). (6.6)

We measure the fidelity with the same technique used in previous experiments [Sackett 00]. The oddparity population, ρ↓↑,↓↑ + ρ↑↓,↑↓, is measured using resonance fluorescence histograms from ρ. The offdiagonal term, ρ↓↑,↑↓ + ρ↑↓,↓↑, is inferred from the parities of the states obtained by applying a qubit

rotation R(

π

2

, φ) of variable phase. Here we use the convention that R(θ, φ) implements the transformation,

|↓⟩ → cos(

θ

2

)

|↓⟩ − ie−iφsin(

θ

2

)

|↑⟩ and |↑⟩ → −ie+iφsin(

θ

2

)

|↓⟩ + cos(

θ

2

)

|↑⟩. In terms of the parity operator

for two qubits,

Π = |↓↓⟩⟨↓↓| + | ↑↑⟩⟨↑↑| − (| ↓↑⟩⟨↓↑| + | ↑↓⟩⟨↑↓|), (6.7)

we define Π(θ, φ) ≡ tr (R†(θ, φ)ρR(θ, φ)Π). We can then write ρ↓↑,↑↓ + ρ↑↓,↓↑ =

1

2

[Π( π

2

, 0) + Π( π

2

,

π

2

)]. A

signature of the state |D(1)

2

⟩ is that it produces an even parity state independent of the π/2 analysis pulse

phase.

We detect the state of the Mg+ qubit by applying σ−polarized laser pulse of duration 200 μs resonant

with the |↓⟩ → |2P3

2

, F = 4, mF = −4⟩ cycling transition and counting photons. We fit observed photon

histograms to a weighted sum of the distributions P(n|0), P(n|1) and P(n|2), which correspond to the

probability of observing n photons given 0, 1 or 2 ions in the state |↓⟩. Assuming equal illumination and

equal photon collection efficiency for the two ions, we can write the distributions above in terms of single-ion

count distributions as

P(n|0) = PBG(n) ∗ P↑(n) ∗ P↑(n) (6.8)

P(n|1) = PBG(n) ∗ P↑(n) ∗ P↓(n) (6.9)

P(n|2) = PBG(n) ∗ P↓(n) ∗ P↓(n), (6.10)

where G(n) ∗ H(n) ≡

∑

m≤n G(n − m)H(m) is the discrete convolution of the distributions G and H. Here

the convolved distributions refer to photon count probabilities from background scattering (BG), as well as

the two qubit states |↓⟩ and |↑⟩. In the case of P↑(n) we account for off-resonant repumping from |↑⟩ to |↓⟩

by assuming an exponential decay rate, γ, such that if the ion began in |↑⟩, the state probabilities at time t
110

are given by c↑ = e

−γt and c↓ = 1−c↑ [Langer 06]. The photon count rates and repump rate that determine

the distributions result from a simultaneous fit to two reference histograms taken just before or after the

experiment.

In any particular measurement, we fit the unknown distribution

Pρ(n) = ∑

2

i=0

ciP(n|i) (6.11)

with ∑

i

ci = 1 to the observed series of Np samples of photon counts, {n1, n2 . . . nNp }. We use a maximum

likelihood method where we maximize the quantity

L =

∏

Np

j=1

Pρ(nj ) (6.12)

by adjusting the parameters ci.

An example of the results of measuring photon counts from the entangled state as well as a fit to the

distribution are shown in Fig. 6.2(a). We measure odd-parity populations of the initial entangled state of

c1 = −0.80(1). In Fig. 6.2(b) we display the measured parity of our state as a function of the phase of the

analysis pulse (squares) as well as a sinusoidal fit. The mean value of Π( π

2

, 0) and Π( π

2

,

π

2

) is 0.74(2). These

numbers together yield an overall fidelity of F = 0.77(2). To further confirm the presence of entanglement,

we applied the pulse R(

π

2

, 0) to ρ followed by a second analysis pulse R(

π

2

, φ) of varying phase (circles), again

measuring the parity of the final state. Here we observe quantum state interference in the sinusoidal parity

oscillation with a period of π and an amplitude of 0.70(3).

Several experimental imperfections contribute to the overall infidelity of 1 − F = 0.23(2). The largest

error stems from random failures to optically pump to the Al+ |

1S0, mF = −

5

2

⟩ ground state at the 10

% level, which has been tested in separate experiments. The next largest error is due to beam pointing

fluctuations. If the two Mg+ ions are unequally illuminated by the Raman beams during the entangling

pulse, two errors arise. The state amplitudes will accumulate a differential phase due to unequal Stark shifts

and the state probabilities will be different due to unequal coupling strengths. We balance the Stark shifts by

suppressing the beating of fringes in a Ramsey experiment, in which each Raman beam is applied separately

during the wait period. Even at the optimum beam position, beam pointing fluctuations can cause errors,

which we estimate to be approximately 5 %. Additional errors are caused by imperfect ground-state cooling
111

of the in-phase mode of motion (0.05 quanta remaining, 3 % infidelity), Al+ decoherence from laser and

magnetic field noise (3 % infidelity), and Al+ spontaneous emission (300 us upper state lifetime) during the

|

1S0⟩|0⟩M → |3P1⟩|1⟩M pulse (15 μs pulse duration, 1 % infidelity). These estimated errors add to 0.22 and

agree with the experimental fidelity.

In summary, we have explored numerically and experimentally a protocol for the creation of Dicke

states in a trapped ion chain. The pulse sequence we use involves just two consecutive laser pulses, tuned to

the resonance of a motional sideband and addressing all ions simultaneously. Infidelities due to unequal ion

masses and inequivalent positions in the chain can be small (≪ 1%) for creating W states, and the infidelity

for creating Dicke states of two and three excitations can be sufficiently low to enable future interesting

experiments. We have demonstrated the basic features of the process on an ion chain composed of two Mg+

ions coupled to a single Al+. If the technical errors observed in this demonstration are reduced, scaling the

experiment to a larger number of ions would require the same number of steps, which makes it an attractive

method for enabling the study of multipartite entangled states.
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