High Fidelity Quantum Information Processing

with Trapped Ions

by

Christopher E. Langer

B.S., Seattle Pacific University, 1999

M.S., University of Colorado at Boulder, 2005

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

2006
This thesis entitled:

High Fidelity Quantum Information Processing with Trapped Ions

written by Christopher E. Langer

has been approved for the Department of Physics

David Wineland

Eric Cornell

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.
Langer, Christopher E. (Ph.D., Physics)

High Fidelity Quantum Information Processing with Trapped Ions

Thesis directed by Dr. David Wineland

Quantum computing (QC) offers the ability to efficiently solve certain computational problems which are intractable classically. Recently, most of the requirements for

QC have been demonstrated with ions, and the current focus in ion-trap QC is tackling

problems of scalability. Large-scale QC, requiring error correction, is possible only if

the error rates in the system are suppressed to very low levels. The focus of my thesis

work has been to characterize the errors in our ion-trap system and reduce them to levels where fault tolerance may be achieved. Two areas were of primary focus: memory

errors and errors due to the presence of laser light.

A dominant source of memory error is decoherence induced by fluctuating magnetic fields. We addressed this problem and created long-lived qubit memories using

a first-order magnetic-field-independent hyperfine transition. Our results with 9Be+

qubits showed a coherence time of approximately 15 seconds, an improvement of over

five orders of magnitude from previous experiments. Using pessimistic models for memory decoherence over time, the memory error for the duration of error correction is

∼ 10−5, below known fault-tolerance thresholds.

Errors during quantum gate operations must also be maintained to low levels

to enable efficient error correction. In many atomic-ion based QC architectures, offresonant laser light is used to perform quantum gate operations with stimulated-Raman

transitions. In such schemes, spontaneous photon scattering is a fundamental source

of decoherence. We experimentally studied decoherence of coherent superpositions of

hyperfine states of 9Be+ in the presence of off-resonant laser light. Our results indicated

that decoherence is dominated by inelastic Raman photon scattering which, for suffi-
iv

cient detunings from the excited states, occurs at a rate much smaller than the elastic

Rayleigh scattering rate. For certain detunings, the measured decoherence rate is a

factor of 19 below the calculated total scattering rate indicating that qubit coherence is

maintained in the presence of photon scattering. Using the measured decoherence rate

and experimental parameters from this experiment, the calculated error due to spontaneous scattering during a 2-qubit gate is also below known fault-tolerance thresholds.
Dedication

To my parents H. Arthur Langer and Gloria E. Langer

who inspired me to study the physical world,

to my daughter Maren E. Langer

who makes me laugh and loves to ride bicycles,

and most importantly to my wife Megan M. Langer

whose love and encouragement make life, even in graduate school, desirable.
vi

Acknowledgements

Special thanks goes first and foremost to Dave Wineland, my research advisor,

who not only taught me to perform experiments and understand the interaction of

trapped ions with lasers, but who also rode his bike at a relaxed pace so that I could

tag along. Thanks to Roee Ozeri with whom I spent many sleepless nights in attempts

at collecting good data. Roee, thank you also for stimulating discussions which made

the subject matter of this thesis understandable to me. Thanks to Didi Leibfried and

Manny Knill for explaining the intricacies of quantum theory and to John Jost for his

excellent contributions to the experiments. Special thanks goes to the many scientists

with whom I have had the pleasure of working and who all contributed much to this

work, namely: Till Rosenband, Piet Schmidt, John Chiaverini, Tobias Schaetz, Murray

Barrett, Brian DeMarco, David Hume, Jim Bergquist, John Bollinger, Wayne Itano,

Joe Britton, Brad Blakestad, Rainer Reichle, Janus Wesenberg, Seigne Seidelin, Ryan

Epstein, David Keilpinski, Cass Sackett, Volker Meyer, Mary Rowe, Quentint Turchette,

Amit Ben-Kish, Brana Jelenkovic, David Lucas, and Chris Monroe.
Contents

Chapter

1 Introduction 1

2 The 9Be+ Qubit 7

2.1 Atomic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Hyperfine Structure and Magnetic Field—Breit-Rabi Solution . . . . . . 10

2.3 Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4.1 Dark to Bright State Optical Pumping . . . . . . . . . . . . . . . 22

2.4.2 Improving Dark to Bright State Leakage Through Shelving . . . 25

2.4.3 Optical Pumping Rates—Kramers-Heisenberg Formula . . . . . . 31

2.4.4 Optimizing Measurement Fidelity . . . . . . . . . . . . . . . . . . 33

2.4.5 Bright to Dark State Optical Pumping . . . . . . . . . . . . . . . 35

2.4.6 Measurement via Photon Arrival Times . . . . . . . . . . . . . . 37

2.5 Quantum Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.5.1 Single-Qubit Quantum Gates . . . . . . . . . . . . . . . . . . . . 49

2.5.2 Stimulated Raman Transitions . . . . . . . . . . . . . . . . . . . 49

2.5.3 Coupling Electronic States to Motional States . . . . . . . . . . . 53

2.5.4 Ground State Cooling . . . . . . . . . . . . . . . . . . . . . . . . 56

2.6 Two-Qubit Quantum Gates . . . . . . . . . . . . . . . . . . . . . . . . . 60
viii

2.6.1 Phase Gate in xy-Basis . . . . . . . . . . . . . . . . . . . . . . . 61

2.6.2 Phase Gate in z-Basis . . . . . . . . . . . . . . . . . . . . . . . . 69

2.6.3 Entanglement Verification . . . . . . . . . . . . . . . . . . . . . . 75

3 Apparatus 79

3.1 Ion Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.1.1 Ion Trap Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.1.2 Gold Plated Alumina Wafer Trap . . . . . . . . . . . . . . . . . . 86

3.2 Magnetic Field Coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.3 Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.3.1 Imaging System . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.3.2 Blue/Red Doppler Beam Lines . . . . . . . . . . . . . . . . . . . 93

3.3.3 Raman Beam Line . . . . . . . . . . . . . . . . . . . . . . . . . . 97

3.3.4 Electro-Optic Raman Beam Line . . . . . . . . . . . . . . . . . . 103

3.4 Experiment Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

3.4.1 Introduction to Field Programmable Gate Arrays . . . . . . . . . 106

3.4.2 Introduction to Direct Digital Synthesis . . . . . . . . . . . . . . 108

3.4.3 Experiment Controller Design . . . . . . . . . . . . . . . . . . . . 111

4 Robust Quantum Memory 133

4.1 Field-Independent Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . 134

4.2 Magnetic Field Noise Model . . . . . . . . . . . . . . . . . . . . . . . . . 141

4.3 Magnetic Field Drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

4.4 Heating Induced Measurement Degradation . . . . . . . . . . . . . . . . 152

4.5 Systematics at the Field-Independent Point . . . . . . . . . . . . . . . . 159

5 Off-Resonant Spontaneous Photon Scattering 163

5.1 Raman Inelastic vs. Rayleigh Elastic Photon Scattering . . . . . . . . . 165
ix

5.2 Experimental Verification . . . . . . . . . . . . . . . . . . . . . . . . . . 180

5.2.1 Measurement of the Raman Scattering Rate . . . . . . . . . . . . 181

5.2.2 Coherence in the Presence of Spontaneous Photon Scattering . . 187

5.3 Field-Independent Point Shift due to the Light Shift . . . . . . . . . . . 194

Bibliography 198

Appendix

A Glossary of Acronyms 209

B Daughter Board Schematics 212

B.1 FPGA Daughter Board . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

B.2 DDS Daughter Board . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

C Histogram Data Analysis using Maximum Likelihood 220

C.1 The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

C.2 Background—Least Squares Fit as a Maximum Likelihood Estimator . . 220

C.3 Method of Maximum Likelihood . . . . . . . . . . . . . . . . . . . . . . 222

C.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

C.4.1 Single Poisson Distribution . . . . . . . . . . . . . . . . . . . . . 224

C.4.2 Weighted Poisson Distribution . . . . . . . . . . . . . . . . . . . 224

C.5 Experimental Data Fitting in Practice . . . . . . . . . . . . . . . . . . . 225
x

Tables

Table

2.1 Field-independent points in 9Be+ . . . . . . . . . . . . . . . . . . . . . . 16

2.2 Measurement errors for cut-off method vs. photon arrival method . . . . 46

3.1 FPGA opcodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

3.2 DDS OPCODEs minimum durations . . . . . . . . . . . . . . . . . . . . 121

3.3 PMT memory bit map . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

3.5 DDS Bus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

3.7 Logic Bus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.1 Laser power requirements to achieve 10−4spontaneous scattering error

during a π pulse for different ion species. . . . . . . . . . . . . . . . . . 179
Figures

Figure

2.1 Energy level diagram of 9Be+ . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Breit-Rabi solution for 9Be+ . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Laser frequencies and 9Be+ levels . . . . . . . . . . . . . . . . . . . . . . 18

2.4 Qubit measurement histograms . . . . . . . . . . . . . . . . . . . . . . . 22

2.5 Shelving the dark state . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.6 Photon count distribution for the dark state . . . . . . . . . . . . . . . . 30

2.7 Simulated measurement error using photon arrival times . . . . . . . . . 45

2.8 Stimulated Raman transitions . . . . . . . . . . . . . . . . . . . . . . . . 50

2.9 Raman sideband cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.10 Sideband cooling temperature . . . . . . . . . . . . . . . . . . . . . . . . 59

2.11 Phase space displacement trajectory . . . . . . . . . . . . . . . . . . . . 63

2.12 Laser configuration for Z-phase gate . . . . . . . . . . . . . . . . . . . . 73

2.13 Parity oscillation for “cat” states up to six ions . . . . . . . . . . . . . . 78

3.1 Schematic Paul trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.2 Alumina wafer trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3.3 Optical system near the ion trap . . . . . . . . . . . . . . . . . . . . . . 90

3.4 Laser frequencies and 9Be+ levels . . . . . . . . . . . . . . . . . . . . . . 92

3.5 Blue Doppler (BD) beam line . . . . . . . . . . . . . . . . . . . . . . . . 95
xii

3.6 Red Doppler (RD) beam line . . . . . . . . . . . . . . . . . . . . . . . . 98

3.7 Raman beam line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

3.8 Raman beam with EOM . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

3.9 FPGA design flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

3.10 DDS block diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

3.11 Experiment controller block diagram . . . . . . . . . . . . . . . . . . . . 112

3.12 FPGA design block diagram 1/2 . . . . . . . . . . . . . . . . . . . . . . 114

3.13 FPGA design block diagram 2/2 . . . . . . . . . . . . . . . . . . . . . . 115

3.14 Histogram memory state machine . . . . . . . . . . . . . . . . . . . . . . 116

3.15 LVDS transmission line with multiple transceiver “drops” . . . . . . . . 126

3.16 Picture of DDS Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

3.17 Sample pulse sequence script . . . . . . . . . . . . . . . . . . . . . . . . 129

4.1 Field-independent qubit levels . . . . . . . . . . . . . . . . . . . . . . . . 135

4.2 Frequency scans at different TR . . . . . . . . . . . . . . . . . . . . . . . 140

4.3 Field-independent transition frequency parabola . . . . . . . . . . . . . 140

4.4 Ramsey phase scan contrast vs. time . . . . . . . . . . . . . . . . . . . . 142

4.5 Sample magnetic field change intervals . . . . . . . . . . . . . . . . . . . 147

4.6 Theoretical contrast decay vs. Ramsey interval . . . . . . . . . . . . . . 150

4.7 Magnetic field drift and servo . . . . . . . . . . . . . . . . . . . . . . . . 153

4.8 Detection loss induced by ion heating—experiment . . . . . . . . . . . . 155

4.9 Detection loss induced by ion heating—theory . . . . . . . . . . . . . . . 158

4.10 AC Zeeman shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

5.1 Rayleigh vs. Raman scattering . . . . . . . . . . . . . . . . . . . . . . . 166

5.2 Relevant scattering energy levels. . . . . . . . . . . . . . . . . . . . . . . 170

5.3 Probability of photon scattering during a π pulse . . . . . . . . . . . . . 176
xiii

5.4 Experimental sequence for population relaxation and coherence relaxation rate measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

5.5 Coherence and population relaxation for a particular detuning . . . . . 184

5.6 Decoherence and relaxation rates vs. detuning . . . . . . . . . . . . . . 189

5.7 Coherence decay rate vs. population relaxation rate . . . . . . . . . . . 191
Chapter 1

Introduction

In his famous 1982 paper, Richard Feynman pointed out that classical computer

simulation of quantum mechanical systems was very difficult. [Feynman 82]. In the same

work, however, he mentioned that it may be possible to simulate quantum systems with

other quantum systems, namely a quantum computer, and writes, “It’s not a Turing

machine, but a machine of a different kind.” Interest in the field of quantum information

processing became very large when Peter Shor wrote his 1994 paper on the efficient

factorization of large numbers [Shor 94] using a quantum computer, and since then,

there has been an explosion of activity in the field. The efficient factorization of large

numbers, if possible, would compromise public key cryptography.

An obstacle to implementation of Shor’s algorithm and hence factoring, of course,

was the lack of a quantum computer. Although some of the earliest demonstrations of

quantum algorithms were simulated using ensembles of nuclear spins in liquid state nuclear magnetic resonance [Jones 01] including factoring [Vandersypen 01], a deterministic quantum computer based on prepared pure quantum states remained a challenge.

In 1995, Ignacio Cirac and Peter Zoller proposed an architecture of a quantum computer based on trapped ions [Cirac 95] which is the basis for most of the ion-trap based

quantum computing experimental research efforts including this current work.

Cirac and Zoller’s proposal consisted of a linear array of trapped ions cooled to

their ground state of motion. Using tightly focused laser beams, the states of indi-
2

vidual ion qubits in the string could be rotated, and interactions between two ions in

the string could be implemented by coupling the internal states of the individual ions

with the collective motion of a particular mode. This satisfied the requirement of a

universal gate set for universal quantum computation [Barenco 95], namely that any

quantum algorithm could be constructed from a primitive set of gates. In particular,

if an experimentalist had at his or her disposal arbitrary single qubit rotations and a

single universal entangling gate between any two qubits, then an arbitrary quantum

circuit could be constructed from these primitives. This is analogous to constructing

classical logic circuits from a network of classical NAND (NOT AND) gates. In this

sense, the set of single qubit rotations and the two-qubit entangling gate is “universal.”

In addition to the universal gate set, other requirements exist for quantum computation

[DiVincenzo 98], and most of these had already been developed and demonstrated in

ions. Enumerating, the five DiVincenzo criteria are [DiVincenzo 98]:

(1) A scalable physical system with well characterized qubits.

(2) The ability to initialize the state of the qubits to a simple fiducial state, such

as |00 · · · 0i.

(3) Long qubit coherence times, much longer than the gate operation time.

(4) A universal set of quantum gates.

(5) A qubit specific measurement capability.

Item (2) was accomplished via optical pumping and laser cooling [Wineland 79]; item (3)

had been demonstrated in experiments devoted to making highly stable atomic clocks

[Bollinger 91]; item (5) was realized in resonance fluorescence and electron shelving

experiments [Wineland 80, Nagourney 86, Bergquist 86]. Cirac and Zoller’s proposal

filled the gaps of items (1) and (3). Many of the intermediate steps required by the CiracZoller proposal were quickly demonstrated, namely ground state cooling [Monroe 95b]
3

and conditioned two-qubit quantum gates between the internal state of a single ion and

its motion [Monroe 95a]. It was quickly realized that the Cirac-Zoller proposal had some

technical drawbacks, and it wasn’t until 2003 that the original Cirac-Zoller two-qubit

entangling gate was realized [Schmidt-Kaler 03].

Meanwhile, two-qubit entangling phase gates based on conditional displacements

in the motional mode’s phase space were proposed [Mølmer 99, Sørensen 99, Sørensen 00,

Milburn 01] and implemented [Sackett 00, Leibfried 03]. These gates had the technical

advantages that individual addressing was not required and that ground state cooling

was not required so long as the ions were sufficiently cold to be in the Lamb-Dicke

regime. Furthermore, the ion’s motion was only excited in a transient fashion such that

after the application of the gate, the ion’s motional state returned to its original state.

These gates appeared to be more robust than the Cirac-Zoller gate, and the highest

fidelity deterministically generated two-qubit entangled states to date were generated

by these gates [Leibfried 03].

As of a few years ago, all of the DiVincenzo requirements enumerated above

have been demonstrated with ions by multiple groups, and some simple quantum algorithms have been demonstrated including: dense coding [Schaetz 04], quantum teleportation [Barrett 04, Riebe 04], quantum error correction [Chiaverini 04], quantum

semiclassical Fourier transform [Chiaverini 05], Grover search [Brickman 05], and entanglement purification [Reichle 06]. In addition, entanglement of six [Leibfried 05] and

eight [Haffner 05] ions has been demonstrated. As with all possible implementations of

QIP, the current challenge in ion-trap quantum information processing was (and still

is) contained in the second word of the first item in the above list, namely, “how do

you build a scalable physical system which can perform quantum information processing?” Part of the technical challenge of the Cirac-Zoller proposal was just this; it was

not scalable. As the number of ions in a single linear trap increased, the number of

motional modes increased, and it would be very difficult to spectrally resolve a single
4

mode of motion on which to perform quantum gates. Furthermore, for the number of

ions needed to perform Shor’s factorizing algorithm, a trap length of many tens of meters would be required with impractically high voltages. The current architecture which

most of the ion-trap quantum information processing community has embraced is that

of a multiplexed array of traps where ions are shuttled back and forth between different

interconnected trapping regions and can interact in small numbers (say 2) in isolated

traps such that the large number of motional modes is not problematic, and high trap

frequencies can be used to implement fast quantum gates [Wineland 98, Kielpinski 02].

During transport, the ions would inevitably acquire kinetic energy which must be removed prior to performing two-qubit gates; this can be accomplished via sympathetic

cooling with a different ion species. The current progress in this area is that ions have

been shuttled and split in a linear trap array [Rowe 02, Barrett 04], sympathetic cooling

has been demonstrated [Rohde 01, Blinov 02, Barrett 03], ion transport around a corner

in a two-dimensional trap array has been realized [Hensinger 06], and new trap fabrication techniques have been developed [Stick 06, Seidelin 06, Britton 06]. In addition,

two trap foundries, Lucent Bell Labs and Sandia National Labs, have been established

to build even more complicated trap structures to support the ion-trap quantum information processing effort.

The idea of scalability in a physical system encompasses more than the architecture alone. For example, the number of qubits and the number of quantum gate

operations required to factorize a number which would break public key cryptography

is in the thousands of qubits and in the millions of operations. What about the errors?

If the probability of error during a single operation is 2, then the probability that N

operations will succeed without error is (1 − 2)

N , an exponentially small number. This

is not scalable in any sense of the word. Fortunately, the theory of quantum error correction has been developed, and by encoding the quantum information into codes with

redundancy, arbitrarily long computations can be performed with a constant predeter-
5

mined probability of error for the entire computation so long as the errors per operation

are below what is termed the “fault-tolerance threshold” (for a review, see [Preskill 97]).

The fault-tolerance thresholds depend on the architecture, noise model, and the particular error-correcting code used. The current estimates for the thresholds are somewhere

in the range of 10−4to 10−2. These numbers should be taken with caution. In some

sense, they are too pessimistic. With a probability of error of 10−2, a computation

utilizing ∼ 102 gates could be performed without error correction at all, and upon repetition of the experiment and averaging the results, we could obtain some confidence

in the answer. However, we are concerned with scalable quantum computation, and

for that, we require gate errors to be below threshold. In fact, when the errors are

only slightly below threshold, the resource requirements for the error-correcting codes

become very large; therefore, the errors should be well below the threshold, probably

by an order of magnitude. The smaller are the errors, the simpler is the code. If a

quantum computer is to be scalable, then its errors per operation at the most basic

level must be suppressed below the fault-tolerance threshold. This is the focus of this

thesis—to investigate ways to reduce the errors to the smallest possible levels in order

to make fault-tolerant scalable quantum information processing with ions possible. We

tackled this problem in two areas: (1) reducing memory errors below threshold and (2)

demonstrating that spontaneous scattering of photons during quantum gate operations

induces a probability of error less than the fault-tolerance threshold. Hence the title of

this thesis is “High fidelity quantum information processing with ions.”

I begin this thesis with a discussion of the 9Be+ qubit where I explain how to

implement all of the DiVincenzo criteria for quantum computation with ions (Ch. 2).

I then describe the experimental apparatus used (Ch. 3). In Ch. 4, the experimental

demonstration of long coherence times (DiVincenzo criteria #3) is discussed, and the

quantum memory is shown to have a probability of error below threshold. Finally,

I discuss our results on the measurement of decoherence induced by the spontaneous
6

scattering of photons. As all of the quantum gate operations involve the interaction of

ions with off-resonant laser light, this is an important topic of study as it appears to

be the most important fundamental source of decoherence. We demonstrate that the

probability of decoherence induced by spontaneous scattering can be suppressed to very

low levels, and at the levels demonstrated in Ch. 5, the probability of error during a

two-qubit gate operation induced by spontaneous scattering is below the fault-tolerance

threshold.
Chapter 2

The 9Be+ Qubit

Quantum information processing requires information to be stored and manipulated in quantum systems. In general, multilevel quantum systems can be used for

processing; however, the simplest quantum system is the two-level system, namely the

qubit. 9Be+ ions have multiple internal electronic levels, and we spectroscopically isolate two levels to form a qubit. In this chapter, I describe the relevant atomic structure

of 9Be+. I derive the Breit-Rabi formula for J =

1

2

hyperfine structure and show

which transitions are suitable as “field-independent” qubits in the S1/2 manifold. I then

explain how we initialize the system to a well defined state and how we perform statedependent measurements. A detailed analysis of measurement error is given. We find

that 9Be+ qubit measurements can be fault-tolerant in principle. Finally, I discuss how

we implement a universal gate set for ions, namely single qubit rotations and a single

entangling two-qubit gate. Combined with the demonstration of long coherence times in

Chapter 4, this list meets most of the DiVincenzo criteria for building a quantum computer [DiVincenzo 98] with the exception of a scalable architecture. As mentioned in the

introduction (Ch. 1), work is in progress on building scalable architectures for ion traps

[Kielpinski 02, Rowe 02, Barrett 04, Hensinger 06, Stick 06, Seidelin 06, Britton 06].
8

2.1 Atomic Structure

The 9Be+ ion, like the Alkali neutral atoms, has a single valence electron. Its

gross structure has a ground S orbital with a P orbital as its lowest lying excited state.

The S ↔ P transition is accessible via electric dipole radiation at 313 nm, and the

excited state has a radiative lifetime of 8.2 ns (19.4 MHz excited state line width). The

spin-orbit interaction gives rise to the fine structure of the excited P orbital splitting

this level into two manifolds of states with well defined electron angular momentum. In

spectroscopic notation, these manifolds of states are labeled 2p

2P3/2 and 2p2P1/2 or

P3/2 and P1/2for short. An energy level diagram is shown in Fig. 2.1. The fine structure

splitting of the P orbital is 197.2 GHz [Bollinger 85]. There is no fine structure in the

S orbital. Its spectroscopic notation is 2s

2S1/2

, or S1/2for short.

The nuclear spin of 9Be+ is I = 3/2 and couples to the electron angular momentum generating the hyperfine interaction. This interaction gives rise to a splitting of

the levels into states with well defined total angular momentum described by the quantum number F at low magnetic field. In the ground S orbital, the hyperfine splitting is

1.25 GHz [Wineland 83]; in the excited P1/2 manifold, the hyperfine splitting is 237 MHz

[Bollinger 85]. The P3/2 hyperfine splitting is less than 1 MHz [Poulsen 75]. The states

in each hyperfine manifold are degenerate at zero magnetic field, and in the presence

of a small magnetic field, this degeneracy is lifted by the linear Zeeman effect. The

Zeeman energy shift is −μBgF mF B where the g-factor gF is −1/2 and +1/2 for F = 2

and F = 1 respectively for S1/2. The projection of the ion’s total angular momentum on

the magnetic field axis is ~mF , μB is the Bohr magneton, ~ is Planck’s constant divided

by 2π, and B is the strength of the magnetic field. We use two hyperfine ground states

in the S1/2 manifold as the qubit. Any two states will do; however, as we will see in

Section 2.2, certain states are better suited for quantum information processing than

others. The F = 1 hyperfine manifold lies higher in energy than the F = 2 manifold,
9

9Be+ Level Diagram

2p 2P1/2

2p 2P3/2

2s 2S1/2

313 nm

F = 2

F = 1

F = 1

F = 2

mF = –2

mF = –1

mF = 0

mF = 1

mF = 2

mF = –2 mF

 = –1 mF

 = 0 mF

 = 1 mF

 = 2

mJ = 3/2

mJ = 1/2

mJ = –1/2

mJ = –3/2

Hyperfine < 1 MHz

197.2 GHz

237 MHz

1.25 GHz

Figure 2.1: Energy level diagram of 9Be+. The fine structure splitting of the excited P

orbital is 197.2 GHz. The hyperfine splitting is 237 MHz for the P1/2 manifold and less

than 1 MHz for the P3/2 manifold. The hyperfine splitting of the ground S1/2states is

1.25 GHz. The S ↔ P transition is electric dipole allowed at 313 nm.
10

and its radiative lifetime is estimated to be on the order of 1015 seconds [Wineland 98].

2.2 Hyperfine Structure and Magnetic Field—Breit-Rabi Solution

The Hamiltonian for the combined system including the hyperfine and Zeeman

interactions is

H = hAI · J − μ · B

= hAI · J − μB(gJJ + gI I) · B

(2.1)

where A is the hyperfine constant in units of Hz, h is Planck’s constant, ~J and ~I

are the electron and nuclear angular momentum operators respectively, and gJ/I is the

corresponding Land ́e g-factor. The operators J and I are dimensionless. In the following

discussion, we take the magnetic field to be along the z-axis. The quantum numbers of

interest are F, J, and I and their corresponding z-projections mF , mJ , and mI . The

quantum number F represents the total angular momentum of the atom F ≡ J+I. For

J = 1/2, as is the case for alkali-like atoms in the ground S1/2state (including 9Be+),

the hyperfine splitting (hfs) is ∆Ehfs = hAF+, where F+ ≡ I + 1/2; F+ = 2 for 9Be+.

For magnetic fields where the Zeeman interaction is small compared to the hyperfine interaction, the eigenstates of the Hamiltonian are states with well defined total

angular momentum. In this case, the Zeeman interaction can be treated as a perturbation in the |F, mF i basis, and we obtain a shift linear in B with slope proportional to mF

as discussed in Section 2.1. This is the low-field regime. In the opposite high-field regime

where the magnetic field is so large that the Zeeman interaction dominates, the eigenstates are described by the |I, J, mI , mJ i basis (or |mI , mJ i for short since I and J are

constants in the S1/2 manifold). In the high field regime, the hyperfine interaction can

be treated as a perturbation in the |mI , mJ i basis. The result is ∆EmI ,mJ = hAmImJ .

The stretched states (mF = ±F+ where F+ ≡ I +J) are eigenstates of the Hamiltonian
11

for all magnetic field strengths. The energy levels for these states are

hmF = ±F+|H|mF = ±F+i = hAJI ∓ μBB(gJ J + gI I). (2.2)

In the intermediate field regime, the non-stretched eigenstates are neither |F, mF i

nor |mI , mJ i, but rather superpositions of these states. We must solve for the eigenstates

by diagonalizing the Hamiltonian in Eq. (2.1). For J = 1/2, this system can be solved

analytically, and its solution is termed the Breit-Rabi formula.

To solve the system described by Eq. (2.1), we take advantage of the fact that the

total angular momentum projection mF ≡ mJ + mI is a conserved quantity in either

basis at all magnetic fields. As such, the problem is reduced to diagonalizing a family

of two-dimensional matrices parameterized by mF . We define the states

|±i ≡ |mJ = ±1/2, mI = mF ∓ 1/2i (2.3)

which conserve mF . For a general angular momentum operator ~L with quantum

numbers L and ML describing its magnitude and z-projection, we define the ladder

operators,

L± ≡ Lx ± iLy. (2.4)

They have the properties

L±|L, MLi =

p

(L ∓ ML)(L ± ML + 1)|L, ML ± 1i (2.5)

so long as M ± 1 lies within the range {−L, −L + 1, .., L}. Using J± and I± based on

Eq. (2.4), we can write the Hamiltonian [Eq. (2.1)] as

H = hAIzJz +

hA

2

(J+I− + J−I+) − μBB(gJ Jz + gI Iz). (2.6)

Using Eqs. (2.6) and (2.5), we calculate the matrix elements,

h±|H|±i = −

1

4

hA − μBBgImF ±

1

2

(hAmF − μBB(gJ − gI ))

h±|H|∓i =

1

2

hAq

F

2

+ − m2

F

. (2.7)
12

Diagonalizing H in the |±i basis yields the eigenvalues

E(F, mF ) = hA μ−

1

4

+

g

0

I

1 − g

0

I

mF x ±

q

F

2

+ + 2mF x + x

2

¶

(2.8)

and (non-normalized) eigenvectors





mF + x ±

q

F

2

+ + 2mF x + x

2

q

F

2

+ − m2

F



 |+i + |−i. (2.9)

where x ≡ −

μBBgJ (1−g

0

I

)

hA is a normalized dimensionless measure of the strength of the

magnetic field, and g

0

I ≡

gI

gJ

is the ratio of nuclear to electron g-factors. The ± sign in

Eqs. (2.8) and (2.9) correspond to the F = I ± 1/2 states at low magnetic field. The

energy spectrum in Eq. (2.8) is the well-known Breit-Rabi formula [Woodgate 92, § 9.6].

A nice feature of the physics of Breit-Rabi solution for quantum information

processing is that in the intermediate field regime, the spectral lines exhibit curvature.

The curvature of these lines allows us to identify transitions where the differential energy

shift vs. magnetic field vanishes to first order. Using such a transition as a qubit is

advantageous for quantum information processing because fluctuations in magnetic field

couple negligibly to the qubit frequency. The use of field-independent qubits as robust

quantum memories is discussed in Chapter 4.

The energy levels as a function of magnetic field for 9Be+ are shown in Fig. 2.2a.

Three of the “field-independent” transitions at non-zero field are indicated by arrows

in Fig. 2.2a. The |F = 2, mF = 0i ↔ |F

0 = 1, m0

F = 0i transition is field-independent

at zero magnetic field. However, due to the degeneracy of the magnetic sub-levels

at zero field, a small finite field would be required to spectrally resolve these levels.

This would induce a linear field dependence to the transition frequency and hence increase the coupling of magnetic field noise to the qubit frequency. The lowest non-zero

field-independent point at B = 2.54 × 10−5 T between states |F = 2, mF = 1i ↔

|F

0 = 1, m0

F = −1i

1

is also less practical because performing rotations via optical

1 This transition is similar to the |F = 1, mF = −1i ↔ |F0 = 2, m0

F = 1i transition in rubidium at

3.23 × 10−4 T [Harber 02, Treutlein 04].
13

stimulated Raman transitions (see § 2.5.2) would require four laser beams2. The two

field-independent transitions near 0.0119 T occur at slightly different magnetic fields

because of the g

0

I

1−g

0

I

mF x term in Eq. (2.8). This term is on the order of ∼ 10−4x,

thus the magnetic field at which these two field-independent points occur differ by

roughly ∼ 10−4×0.0119 T. Data used for calculating Fig. 2.2 are the hyperfine constant

A = −625.008837048(10) MHz, electron g-factor gJ = −2.00226206(42), and nuclear to

electron g-factor ratio g

0

I ≡

gI

gJ

= 2.134779853(2) × 10−4[Wineland 83].

The energy difference between different F-levels E(F = I +

1

2

, mF ) − E(F

0 =

I −

1

2

, m0

F

) is given by

∆E

hA =

g

0

I

1 − g

0

I

∆mF x +

q

F

2

+ + 2mF x + x

2 +

q

F

2

+ + 2m0

F

x + x

2 (2.10)

where ∆mF ≡ mF − m0

F

. Practical transitions utilizing optical stimulated Raman

transitions for single qubit rotations require ∆mF = 0, ±1. For ∆mF = 0, differentiating

Eq. (2.10) with respect to x and setting to zero yields a field independent point at

x = −mF . This is the condition for the |F = 2, mF = 0i ↔ |F

0 = 1, m0

F = 0i transition

at B = 0 and the |F = 2, mF = 1i ↔ |F

0 = 1, m0

F = 1i transition at B ' 0.02231 T.

The stretched states do not have a ∆mF = 0 transition. The most straight forward way

to determine the locations of the ∆mF = ±1 field-independent points is to differentiate

Eq. (2.10) with respect to x, set this to zero, and numerically solve for x. Examining

Eq. (2.10), we notice that the g

0

I

1−g

0

I

is small; therefore, these transitions exist in pairs

with different x locations differing on the order of ∼ g

0

I

x in the neighborhood of

x0 = −

F

2

+ + m + m2 −

p

F4 + m2(m + 1)2 − F2(1 + 2m + 2m2)

2m + 1

, (2.11)

the solution to ∂∆E

∂x |g

0

I=0 = 0 where m = min(mF , m0

F

). The transitions |F = 2, mF =

1i ↔ |F

0 = 1, m0

F = 0i at B = 0.01196 T and |F = 2, mF = 0i ↔ |F

0 = 1, m0

F = 1i at

B = 0.01194 T are examples of this duplicity.

2 Because of nuclear spin selection rules, stimulated Raman transitions involving two laser beams

can only change mF by at most 1. Therefore, ∆mF = 2 transitions would require four laser beams.
14

1000

500

0

–500

0 5 10 15 20 25 30

E/h [MHz]

B [mT]

(F, mF)

(1, –1)

(1, 0)

(1, 1)

(2, 2)

(2, 1)

(2, 0)

(2, –1)

(2, –2)

F = 2

F = 1

mF = –2

mF = –1

102.8

86.7 76.4 69.1

86.5

76.2

1207.495,843

(b)

(a)

|↓〉

|↑〉

Figure 2.2: (a) Energy levels of the ground S1/2state in 9Be+ as a function of

magnetic field calculated via the Breit-Rabi formula. The hyperfine constant A =

−625.008837048(10) MHz, electron g-factor gJ = −2.00226206(42), and ratio of nuclear

to electron g-factor g

0

I ≡

gI

gJ

= 2.134779853(2) × 10−4 used in this calculation are taken

from Ref. [Wineland 83]. The arrows indicate first-order magnetic field-independent

transitions at 11.94 mT, 11.96 mT, and 22.31 mT. (b) Detailed transition frequencies

of S1/2 at 11.94 mT used in the experiments of Chapters 4 and 5. Frequencies are in

MHz. The field-independent transition is indicated by the arrow. The states of the

qubit are labeled as | ↓i and | ↑i.
15

At very high magnetic fields, field-independent transitions exist between different

nuclear hyperfine states (mI ) within an electron spin manifold (fixed mJ ) due to the

nuclear spin Zeeman effect. One such transition exists between states |mJ =

1

2

, mI =

−

3

2

i and |mJ =

1

2

, mI = −

1

2

i at x ' −36.73 (B ' 0.8194 T). This particular transition

was used as a 303 MHz frequency standard, and coherence times exceeding ten minutes

were observed [Bollinger 91]. Other high-field field-independent transitions exist in 9Be+

and are listed in Table 2.1.

When the coupling of magnetic field fluctuations to the transition frequency vanishes to first order, the second order coefficient determines the transition frequency’s

dependence on the field. At such extremum points, the transition frequency can be

approximated by the following

ν(B) ' ν(B0) + c2(B − B0)

2

(2.12)

where ν is the transition frequency in Hz, B0 is the magnetic field at which the

field-independent transition occurs, and c2 ≡

1

2h

∂

2∆E

∂B2 |B=B0

. c2 for the various fieldindependent points is tabulated in Table 2.1. Small values for c2 are advantageous

to reduce phase noise induced by ambient magnetic field fluctuations. As such, the

303 MHz frequency standard in Ref. [Bollinger 91] used the field-independent transition at B0 ' 0.8194 T which has one of the smallest second order coefficient c2 in

9Be+. Unfortunately, optical stimulated Raman transitions (useful for individual qubit

addressing in quantum information processors) between these states are inefficient because the nuclear spin is largely decoupled from the electron spin at such high magnetic

fields. This, in addition to the relative ease it is to produce smaller magnetic field

strengths, spurred us to pursue field-independent transitions near 0.0119 T for quantum information processing (see Chapter 4). The experiments in Chapters 4 and 5 used

the |F = 2, mF = 0i ↔ |F

0 = 1, m0

F = 1i transition as the qubit. A detailed energy

diagram of the S1/2 manifold at 0.00194 T required for this qubit is shown in Fig. 2.2b.
16

Transition ν [MHz] B0 [T] c2 [Hz/μT2]

|F, mF i ↔ |F

0

, m0

F

i

|2, 0i ↔ |1, 0i 1250.01767410 0 +3.140 × 10−1

|2, 1i ↔ |1, −1i 1250.01752212 2.5403 × 10−5 +2.355 × 10−1

|2, 0i ↔ |1, 1i 1207.49584322 1.1945 × 10−2 +3.049 × 10−1

|2, 1i ↔ |1, 0i 1207.35280753 1.1964 × 10−2 +3.049 × 10−1

|2, 1i ↔ |1, 1i 1082.54706095 2.2307 × 10−2 +3.626 × 10−1

|1, 1i ↔ |1, 0i 322.551896190 1.6018 × 10−1 −5.293 × 10−4

|2, 0i ↔ |2, 1i 324.547848987 1.7472 × 10−1 −3.123 × 10−4

|2, 0i ↔ |2, 1i 321.168429685 6.7740 × 10−1 +7.562 × 10−6

|2, 0i ↔ |1, 0i 303.016377260 8.1944 × 10−1 −7.749 × 10−6

Table 2.1: Field-independent points in 9Be+

There are three parameters in the Breit-Rabi formula which determine the transition frequencies: A, g

0

I

, and x. The hyperfine constant A sets the y-scale. The xlocations of field-independent points in the intermediate field regime are determined by

quantum numbers (F+, mF ) with g

0

I

causing small variations in the x-positions. Fieldindependent x-locations in the high field regime (eigenstates approximated by |mI , mJ i)

are determined solely by g

0

I

. In the limit g

0

I → 0, the high-field field-independent points

tend toward x → ∞. The magnetic field B can be determined from x given A and gJ .

However, in practice, x is determined by measuring a transition frequency with non-zero

first order dependence on changes in the magnetic field using knowledge of A and g

0

I

.

Once x is measured, the energy levels of all other hyperfine states can be determined.

In particular, the frequency of a field-independent transition can be calculated as a

function of a field-dependent transition frequency. This calculation and corresponding

measurement data are shown in Fig. 4.3.

The eigenvectors of the hyperfine Hamiltonian are described by Eq. (2.9). Writing

the eigenvectors as |0i = α0|−i + β0|i and |1i = α1|−i + β1|+i for the two states of

the field independent transition, we find that at any field-independent point, α0 = −α1

and β0 = β1 in the limit g

0

I → 0. Including g

0

I

causes a small correction on the order

of 10−4. This can be seen by optimizing Eq. (2.10) for x with g

0

I → 0 and substituting
17

x into Eq. (2.9). This has implications for quantum information processing because

two-qubit gates relying on differential Stark shifts are less efficient (see § 2.6.2). The

Stark shift of each qubit level is a function of |α|

2/∆ where ∆ is the detuning of the

Raman laser from one of the exited P1/2 or P3/2states. Therefore, the differential Stark

shift for a field-independent qubit will have factors of the form 1

∆ −

1

∆+ω0

'

ω0

∆

1

∆ where

ω0 is the qubit transition frequency. To reduce the effects of spontaneous scattering, we

desire large detunings ∆, and the factor ω0/∆ is typically 10−2 or less. A more detailed

discussion of two-qubit gates based on differential Stark shifts can be found in § 2.6.2.

2.3 Initialization

One of the DiVincenzo criteria for quantum computation is the ability to initialize

the system to a well defined state. We accomplish this via optical pumping. In the

experiments, we use circularly polarized ˆσ

+ laser light propagating along the direction

of the magnetic field to optically pump the state of the ions to the stretched |F =

2, mF = 2i state. These laser beams consisted of the near resonant “Blue Doppler”

(BD) beam tuned 1

2

γ (γ = 2π × 19.4 MHz, the natural linewidth) to the red of the

cycling transition |S1/2, F = 2, mF = 2i ↔ |P3/2, F0 = 3, m0

F = 3i, the “BD Detuned”

beam tuned 400 MHz to the red of the cycling transition, the “Red Doppler” (RD)

beam tuned slightly to the red of the |S1/2, F = 1, mF = 1i ↔ |P1/2, F0 = 2, m0

F = 2i

transition, and the “Repumper” beam tuned on resonance with the |S1/2, F = 2, mF =

1i ↔ |P1/2, F0 = 2, m0

F = 2i transition (see Fig. 2.3). The BD Detuned beam performs

pre-cooling for very hot ions where the Doppler-broadened resonance makes the nearresonant BD cooling less efficient. This beam has approximately 500–1000 μW of laser

power in a ∼30 μm waist. The near-resonant BD beam is responsible for fast Doppler

cooling after the pre-cooling stage, and its intensity is set to approximately half of

the saturation intensity (∼0.5 μW in 25 μm waist). The RD beam assists in optically

pumping the F=1 manifold when the polarization of BD is imperfect. However, for field-
18

F = 2

F = 1

mF = –2

mF = –1

102.8

86.7 76.4 69.1

86.5

76.2

1207.495,843

F = 2

F = 1

mF = –2

mF = –1

39.9

28.5 23.2 20.1

28.2

23.1

261.8

2P3/2

197 GHz

2P1/2

313.20 nm

2S1/2

mJ = 3/2

mJ = 1/2

mJ = –1/2

mJ = –3/2

mF = –3

mF = +3

223.4

BD d

etun

e

BD

RD

repumper

~80 GHz

Figure 2.3: Laser frequencies and 9Be+ levels
19

independent qubits at 0.0119 T, imperfect BD polarization is less problematic than in

the low field case (∼1 mT) because the depumping transition |S1/2, F = 2, mF = 2i ↔

|P3/2, mJ = 1/2, mI = 3/2i is 223 MHz (11.5 γ) off-resonant. Furthermore, the highpower BD Detuned beam, even though it is ∼600 MHz off-resonant from the |S1/2

, F =

1i ↔ |P3/2, mJ = 3/2i transition, power broadens this transition to a ∼600 MHz width;

therefore, BD Detuned is also efficient at optically pumping the F = 1 manifold. The

Repumper beam optically pumps the |F = 2, mF = 1i state during Raman side-band

cooling (§ 2.5.4). Empirically we found that pre-cooling for a few hundred microseconds

increased the ion lifetime in the trap. We suspect that ions became very hot occasionally

after collisions with thermal background gas, and pre-cooling with an off-resonant beam

would re-cool these hot ions after such an event.

After Doppler cooling and preparing the electronic state of the ion to the stretched

state, the ion was further cooled to the motional ground state via resolved side-band

cooling for some experiments (see § 2.5.4). Side-band cooling was performed on the

|F = 2, mF = 2i ↔ |F = 2, mF = 1i transition at 103 MHz. The final stage of

side-band cooling involved repumping the ion back to the stretched |F = 2, mF = 2i

state, now in its motional ground state |n = 0i. (|ni are Fock states of a harmonic

oscillator potential, and n represents the phonon occupation number). Side-band cooling can prepare the ion in the motional ground state with 99% fidelity [Wineland 98].

However, the ion’s motional state is only of importance prior to performing two-qubit

logic gates. For the schemes outlined in this thesis (§ 2.6), the Lamb-Dicke criteria

only need be satisfied. Therefore, a ground state preparation fidelity of 99% is sufficient

for fault-tolerant quantum computation (see § 2.6 and Refs. [Sørensen 00, Steane 03]).

To prepare the ion in the | ↑i in the qubit subspace, we performed a π-rotation on

the |F = 2, mF = 2i ↔ | ↑i transition (see § 2.5.2). This is the starting point for

computation.

The initialization infidelity of the ion’s electronic state depends on the impurity
20

of the RD laser-beam polarization since this is the last optical pumping operation prior

to the start of the computation. This error, in principle, can be eliminated completely.

However, it is instructive to calculate the preparation error given some impurity in

the polarization. We consider the fidelity of preparing the stretched |F = 2, mF =

2i state after optical pumping with RD and the Repumper, both ˆσ

+ polarized. We

now consider the optical pumping rates away from |F = 2, mF = 2i due to imperfect

Repumper polarization since it is closest to resonance with the |S1/2, F = 2, mF = 2i ↔

|P1/2i transition. We assume for the present analysis that the Repumper beam has 2σ+

admixture of ˆσ

+ polarization and 2π admixture of ˆπ polarization.

The optical pumping rate from state i to state f through a single excited state is

given by:

Γi→f =

γ

2

s0

1 + s0 + 4 δ

2

γ

2

ci→f (2.13)

where γ is the natural line width, s0 ≡ I/Is is the on-resonance saturation parameter, δ

is the detuning of the laser from the excited state through which the scattering occurs,

and ci→j is a coupling coefficient3. I is the laser intensity, and Is is the saturation

intensity [Metcalf 99, § 2.4]. For scattering through either |P1/2, F = 2, mF = 2i via

πˆ polarized light or through |P1/2, F = 2, mF = 1i via ˆσ

− polarized light, summing

over coupling coefficients yields P

j

ci→j = 2/9 (independent of the magnetic field).

The optical pumping rate into the stretched state also has a coupling coefficient of

2/9. Therefore, the ratio of rates gives the steady state fraction and hence the error in

initialization:

einit =

2π

1 + δ

2

π/γ02

+

2σ−

1 + δ

2

σ− /γ02

(2.14)

where γ

0 ≡

1

2

√

1 + s0 is half of the power broadened line width. For 9Be+ at 0.0119 T,

the field independent point, δπ = 103 MHz and δσ+ = 143 MHz. With 0.1% admixture of

both ˆπ and ˆσ

+ polarizations and an on-resonance saturation parameter of s0 = 1/2, the

3 The calculation of the coupling coefficient ci→j is very similar to the calculation of b

(k)

ij in § 2.4.3.
21

error in initialization is 2.0 × 10−5, below the fault-tolerant limit [Steane 03, Knill 05].

2.4 Measurement

Qubit measurements are performed via state-dependent resonance fluorescence.

Originally proposed by Dehmelt et al. [Dehmelt 75] and later demonstrated by Wineland

et al. [Wineland 80], Nagourney et al. [Nagourney 86], Sauter et al. [Sauter 86], and

Bergquist et al. [Bergquist 86], this technique takes advantage of a closed transition’s

ability to scatter many photons without optically pumping out of the closed transition.

A resonant laser on a strong closed transition is applied to the ion causing a projective

measurement of the atomic state collapsing the wave function of the ion into one of the

ion’s atomic eigenstates. If the collapsed ion state is found to be in one of the states

participating in the closed transition, then strong fluorescence occurs. Conversely, if

the ion’s state collapses into a state that does not participate in the closed transition,

then a very small amount of fluorescence occurs. For the 9Be+ qubit, we use the cycling transition |S1/2

, F = 2, mF = 2i ↔ |P3/2, F0 = 3, m0

F = 3i excited by ˆσ+ light as

the strong closed transition. We tune the BD laser on resonance with approximately

half of the saturation intensity. If the ion collapses into the |F = 2, mF = 2i state,

the ion scatters approximately 20 million photons per second. The qubit states | ↑i

and | ↓i (see Fig. 2.2b) are transferred to the states |F = 2, mF = 2i (“bright”) and

| ↑i (“dark”) respectively by a sequence of two π-pulses on their respective transitions

prior to applying the resonant BD laser for fluorescence measurement. With a detector

efficiency of 3 × 10−3, we collected on average 12 photons on a photomultiplier tube

(PMT) in a 200 μs detection interval if the ion was projected into the “bright” state.

(The photon collection apparatus is described in detail in § 3.3.1). The “dark” state

| ↑i scatters negligible photons since the detection laser is 1.2 GHz off resonance. The

typical background photon count for the dark state was ∼0.1 photon in the 200 μs

detection interval.
22

5 10 15 20 25 30

0

0.02

0.04

0.06

0.08

0.1

5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

5 10 15 20 25 30

0

0.1

0.2

0.3

0.4 (a) (b) (c)

Figure 2.4: Typical histogram of qubit measurements for the qubit state prepared in

(a) | ↑i, (b) | ↓i, and (c) equal superposition of | ↑i and | ↓i. The histograms are

normalized by the number of experiments, N = 1000 in this case.

The distribution of collected photons from the bright state ideally follows Poisson

statistics with a mean given by the scattering rate, detection efficiency, and detection

interval—12 photons in our case. If the ion does not scatter any photons, background

light collected on the detection system generates a histogram of photon counts that also

obeys Poisson statistics. The mean of the “dark” distribution was 0.1. We performed

multiple repetitions of the experiments and generated a histogram of photon counts.

Typical histograms for qubit states in | ↑i, | ↓i, and superpositions of these two are

shown in Fig. 2.4. To determine the probability that the qubit state was in either | ↑i or

| ↓i, we fit the measurement histogram to a weighted sum of two Poisson distributions.

The bright mean of the Poisson distribution used for fitting is extracted by fitting

measurement data for a prepared |F = 2, mF = 2i state. The dark mean is determined

by fitting the background counts collected to a Poisson distribution when the detection

laser was tuned far off resonance (∼1 GHz). The fitting procedure used was the method

of maximum likelihood (see Appendix C and [Freund 92]).

2.4.1 Dark to Bright State Optical Pumping

One fundamental source of measurement error occurs due to off-resonant optical

pumping of the dark state into the bright state. When this occurs, the dark state

begins to participate in the cycling transition, and the collected photons obey Poisson
23

statistics with a reduced mean. For the dark state | ↑i, the probability of scattering into

the bright state follows an exponential distribution assuming a two level system. We

expect the distribution of collected photons then to be the convolution of an exponential

distribution and the Poisson distribution [King 99, Acton 05]. We give the analysis

following that of Acton et al. [Acton 05].

For an ion originally in the dark state at time t = 0, the probability that the ion

will be found in the bright state at time t is given by

f(t) = 1 − e

−ωt (2.15)

where ω is the scattering rate into the bright state and is calculated from atomic parameters and the laser intensity at the ion. To a first approximation,

ω =

γ

2

s0

1 + s0 +

4δ

2

γ

2

cBR (2.16)

where γ is the natural line width, s0 ≡ I/Is is the on-resonance saturation parameter,

δ is the detuning of the detection laser from the dark state to |P3/2, mJ = 1/2i state

transition, and cBR is the coupling coefficient for scattering through this state into

the bright stretched state. Here, we have neglected scattering into |F = 2, mF = 1i

followed by scattering into the bright state. This is a second order off-resonant scattering

process, and the above analysis still gives a conceptual view of the repumping error

process. Later I will use a rate-equation treatment to show the repumping error in

the case where the dark state is transferred (or “shelved”) to an auxiliary state where

multiple scattering events must occur in order for the dark state to participate in the

cycling transition. In this more general treatment, all allowed scattering paths are taken

into account; therefore, I will delay its discussion until then. Using field-independent

qubits, cBR is no longer a simple product of Clebsch-Gordon coefficients because the

eigenstates are no longer states with well defined total angular momentum (see § 2.2).

The eigenstates are more generally described by superpositions in the |mI , mJ i basis:
24

|darki = α|mI = mF + 1/2, mJ = −1/2i + β|mI = mF − 1/2, mJ = +1/2i. Using

this definition, cBR =

2

9

α

2

. For 9Be+ at 0.0119 T, γ = 2π × 19.4 μs

−1

,

2δ

γ = 128, and

α = −0.793. At s0 =

1

2

, the scattering rate from the dark state into the bright state is

ω = 2.6 × 10−4 μs

−1

.

Eq. (2.15) describes the probability that the dark state will be found in the bright

state at time t. The probability that the dark state will scatter into the bright state

between t and t + dt is given by its derivative,

 ̇f(t)dt = ωe−ωtdt. (2.17)

If the dark ion enters the bright state at time t, the collected photons will exhibit Poisson

statistics with a mean given by

λ(t) = rbgγcτD + γc(τD − t) (2.18)

where τD is the detection interval, γc = η

γ

2

s0

1+s0

is the rate of collected photons from the

cycling transition, η is the photon collection efficiency, and rbg is the rate of background

collected photons normalized by γc. The mean of the background distribution is λbg =

rbgγcτD, and the mean of the bright distribution (in the absence of background) is

λ0 = γcτD. Including background, the bright distribution has a mean of λbg + λ0 =

(1 + rbg)γcτD. λ(t) in Eq. (2.18) is a function of the random variable t. We desire

the distribution of λ, and we are given λ(t) and  ̇f(t)dt, the probability of the dark

ion entering the bright state between t and t + dt. The fundamental transformation

law of probabilities states that for a probability distribution p(x)dx of the random

variable x, the distribution for a function y(x) of the random variable x follows the rule

|p(x)dx| = |p(y)dy|. Using this transformation law for the function λ(t), we find

g(λ)dλ =







ω

γc

e

− ω

γc

[λbg+λ0−λ]

dλ λbg < λ ≤ λbg + λ0

e

−ωτD λ = λbg

, (2.19)
25

the probability that the collected photons from a dark ion will exhibit Poisson statistics

with mean λ. When the dark ion never optically pumps to the bright state, the mean

of the photon distribution is simply the background mean λbg. The probability that the

dark ion will exhibit Poisson statistics with mean λbg is e

−ωτD which is the probability

1 − f(τD) that the ion never optically pumps to the bright state.

The calculation of the dark state photon count distribution is now straight forward. Given the density of means g(λ) and conditional probability of detecting n

photons given λ (the Poisson distribution P(n|λ) = e−λλ

n

n!

) the distribution of photon

counts for an ion originally in the dark state is simply the convolution:

pdark(n) = e

−ωτD

Ã

P(n|λbg) + ωe

− ω

γc

λbg

γcn!

Z λbg+λ0

λbg

e

ωλ/γc e−λλndλ

!

= e

−ωτD

·

P(n|rbgγcτD) + ωγn

c

e

−ωτDrbg

(γc − ω)

n+1

©

P2(n + 1,(γc − ω)(1 + rbg)τD)

− P2(n + 1,(γc − ω)rbgτD)

a

 ̧

(2.20)

where P2(a, x) ≡

1

(a−1)!

R x

0

e

−yya−1dy is the incomplete Gamma function normalized

such that P2(a, ∞) = 1. We have explicitly shown the τD dependence on the second

line in Eq. (2.20).

2.4.2 Improving Dark to Bright State Leakage Through Shelving

Off-resonant repumping of the dark state can further be suppressed by transferring

the dark state to a hyperfine state where multiple scattering events would be required

before the dark state would be pumped into the bright state. Fig. 2.5 illustrates this

basic idea. If the dark state is |F = 1, mF = 1i, even though scattering is suppressed

because the detection beam is off resonance, only a single scattering event is required

to pump this state into the bright |F = 2, mF = 2i state. Conversely, if the dark

state is |F = 2, mF = −2i, a minimum of four scattering events would be required for

this state to pump to the bright state4. Because scattering into the bright state is a

4 The quantum number mF can only change by 1 per scattered photon at most because the electron

spin is largely decoupled from the nuclear spin in the P3/2 state. This occurs because the hyperfine
26

multi-order off-resonant photon scattering process, we expect Eq. (2.15) to no longer be

valid. Eq. (2.15) describes the probability of the electron to be in the |F = 2, mF = 2i

state at time t. Once we determine the correct f(t), we can proceed as in the previous

section to calculate the distribution of photon counts for the dark state.

The dynamics of the probability of the electron to be in state |F = 2, mF = 2i can

be calculated using a rate-equation treatment. Let the vector v represent the population

in each of the hyperfine states. In 9Be+ the length of this vector is 8. We define the rate

matrix Γ as the matrix whose elements Γij are the optical pumping rates from state

j → i. Because the detection beam is ˆσ

+ polarized and population can only scatter into

states with ∆mF ∈ {0, +1}, the columns of Γ have at most three entries. We ignore

Rayleigh elastic scattering by letting Γii = 0 as it does not affect the optical pumping

process. The rate of population transfer is given by:

∂vi

∂t =

X

j

Γijvj − vi

X

j

Γji. (2.21)

In matrix notation, Eq. (2.21) can be written as

∂v

∂t = M · v (2.22)

where M ≡ Γ − D(1

T

v

· Γ), 1v is a column vector of ones, and the operator D(x) is

defined as a diagonal matrix whose diagonal elements are the elements of the vector x.

The matrix M has at most four entries per column or row with the diagonal elements

dependent on the other three non-zero entries in the same column (the sum of each

column is zero to conserve total population). Taking the Laplace transform of Eq. (2.22)

yields:

sV(s) − v(0) = M · V(s). (2.23)

where V(s) is the Laplace transform of v(t), v(0) is the initial condition, and s is the

Laplace transform variable. Eq. (2.23) is an algebraic linear system of equations which

splitting in the P3/2 manifold (< 1 MHz) is small compared to the Zeeman splitting (223 MHz). The

nuclear spin quantum number mI is conserved through emission of a photon from the exited P state to

the ground state.
27

F = 2

F = 1

mF = –2 mF

 = –1

|bright〉

|dark 1〉

|dark 2〉

mJ = 3/2

mJ = 1/2

2P3/2

detection

laser σ

+

∆3/2

∆3/2

∆z

Figure 2.5: Shelving the dark state for increased detection efficiency. During the detection interval, the |dark 2i state has a smaller probability of off-resonantly scattering into

the |brighti state than the |dark 1i state. In the first case, a minimum of four scattering

events must occur, whereas in the latter case, only 1 scattering event must occur. In the

figure, ∆3/2is the detuning of the detection laser from the |P3/2, mJ = 3/2i manifold

for different S1/2 hyperfine states, and ∆z = 233 MHz is the P3/2 excited state Zeeman

splitting between the mJ = 3/2 and mJ = 1/2 manifolds at 0.0119 T. Not shown are

four hyperfine sub-levels in each mJ manifold in the P3/2level all within 1 MHz.
28

is easily solved, and its solution is:

V(s) = −(M − s1)

−1v(0) (2.24)

where 1 is the identity matrix. The inverse Laplace transform of Eq. (2.24) gives the

time domain solution for the state vector v(t) given the initial condition v(0).

Some comments on the solution Eq. (2.24) are in order. Firstly, we note that the

columns of the matrix −(M − s1)

−1 are the solutions for the initial conditions where

100% of the population starts in a particular hyperfine state. Secondly, the inverse of

a matrix is proportional to the inverse of its determinant; therefore, each element is

proportional to the inverse of the characteristic polynomial of M. In terms of control

theory, we can interpret −(M−s1)

−1 as a transfer function, and the poles of the transfer

function represent the characteristic frequencies of the system. These frequencies are

independent of the initial conditions. Looking at a single term further, we can rearrange

the term in to a sum of partial fractions:

[(M − s1)

−1

]ij =

pij (s)

q(s)

=

X

k

a

(k)

ij

s + ωk

(2.25)

where q(s) is the characteristic polynomial common to all terms, pij (s) is a polynomial

in s, −ωk are the roots of q(s) [q(s) = Q

k

(s + ωk)], and

a

(k)

ij = lim s→−ωk

(s + ωk)pij (s)

q(s)

=

pij (ωk)

Q

l6=k

(ωl − ωk)

(2.26)

are constants assuming q(s) has no double roots. This is the case for our detection

system. Thirdly, we know physically that ˆσ

+ polarized light optically pumps any initial

state to the |F = 2, mF = 2i stretched state, so we know the solution v(t → ∞)

approaches |F = 2, mF = 2i. As such, we can use the final value theorem for Laplace

transforms, lims→0 sV(s) = limt→∞ v(t), to state that the characteristic polynomial
29

q(s) must have a single zero at s = 0. Furthermore, the 1/s term in the partial fraction

expansion of (M − s1)

−1 has a coefficient of −1 for the |F = 2, mF = 2i row and 0 for

all other rows (pij (s) has a zero at s = 0 for all rows other than |F = 2, mF = 2i). Since

we are concerned with the probability of detecting the electron in state |F = 2, mF = 2i

with the electron starting in some other hyperfine state at t = 0, we are only concerned

with the |F = 2, mF = 2i row of −(M−s1)

−1

. Its elements represent different hyperfine

states as initial conditions.

We are now prepared to give some insight into the time-domain solution. Taking

the inverse Laplace transform of −[(M − s1)

−1

]i=(2,2),j gives the probability that the

state |ji will have optically pumped into the bright state |F = 2, mF = 2i by time t:

P(bright|j) = 1 −

X

k

a

(k)

i=(2,2),je

−ωkt

. (2.27)

The frequencies −ωk are the non-zero roots of the characteristic polynomial; the j index

specifies the initial condition, and a

(k)

i=(2,2),j are determined from Eq. (2.26). Comparing

to f(t) in Eq. (2.15), we can simply replace e

−ωt with the sum P

k

a

(k)

i=(2,2),je

−ωkt and

follow the analysis of the previous section to obtain the distribution of collected photons

as:

pdark(n|j) = X

k

a

(k)

i=(2,2),je

−ωkτD

·

P(n|rbgγcτD) + ωkγ

n

c

e

−ωkτDrbg

(γc − ωk)

n+1

×

©

P2(n + 1,(γc − ωk)(1 + rbg)τD)

− P2(n + 1,(γc − ωk)rbgτD)

a

 ̧

.

(2.28)

Plots of the dark distribution Eq. (2.28) for different initial conditions are shown in

Fig. 2.6. Parameters used in the calculation of the rate matrix are the following. The onresonance saturation parameter for the cycling transition is s0 = 0.5, and the frequency

of the detection laser is on resonance with the cycling transition. The detunings and

hyperfine state superposition amplitudes α and β are calculated from the Breit-Rabi

formula (see § 2.2) at 0.0119 T, the field-independent point. For initial conditions where
30

(a) (b)

(c) (d)

0 1000 2000 3000 4000

0.2

0.4

0.6

0.8

1

H1,1L

H2,1L

H1,0L

H2,0L

H1,−1L

H2,−1L

H2,−2L

IC HF,mFL

0 50 100 150 200

1

2

3

× 01

−4

H1,1L

H1,−1L

H2,−2L

IC HF,mFL

0 5 10 15 20

1

2

3

4

× 01

−2

H1,1L

H2,1L

H1,0L

H2,0L

H1,−1L

H2,−1L

H2,−2L

IC HF,mFL

0 5 10 15 20

1

2

3

4

5

× 01

−5

t [μs] t [μs]

number of collected photons number of collected photons

Figure 2.6: (a) Probability of a 9Be+ ion to be in state |F = 2, mF = 2i at time t

given different initial conditions (IC) at t = 0 calculated using Eq. (2.27). (b) Close

up of probability of electron to be in state |F = 2, mf = 2i for the two best states

|F = 2, mF = −2i and |F = 1, mF = −1i and the |F = 1, mF = 1i state for comparison.

(c) Photon count distribution for the dark state for different initial conditions (IC). (d)

Close up view of the photon count distribution for the state |F = 2, mF = −2i. In both

(c) and (d) the probability of detecting 0 photons is off the y-scale and near 1. The

mean of the bright distribution is λ0 = 10 in τD = 200 μs (γc = 5 × 10−2

). Background

is zero (rbg = 0).
31

|ji = |F = 1, mF = 1i or |F = 2, mF = 1i such that a single scattered photon could

optically pump the state |ji into the bright state, Eq. (2.28) has a single dominant term

and approximates the simpler solution in Eq. (2.20). In this case, the dark distribution

exhibits a bimodal feature (the first mode is at n = 0 and off the y-scale near 1; the

second mode is near n = 4). The multi-order scattering processes do not exhibit this

property.

2.4.3 Optical Pumping Rates—Kramers-Heisenberg Formula

The non-zero elements Γij of the rate matrix in Eq. (2.21) can be calculated using

the Kramers-Heisenberg formula [Ozeri 05, Loudon 00, §8.7]

Γij = g

2

γ

 ̄

 ̄

 ̄

 ̄

 ̄

X

k

b

(k)

ij

∆k

 ̄

 ̄

 ̄

 ̄

 ̄

2

(2.29)

where g =

Eμ

2~ =

γ

√s0

2

√

2

, E =

p

2I/c20 is the laser-beam electric field amplitude, c is

the speed of light, 20 is the vacuum dielectric constant, and μ = |hP3/2, F = 3, mF =

3|d · σˆ

+|S1/2

, F = 2, mF = 2i| is the magnitude of the cycling transition electric-dipole

moment. Here, d is the electric dipole operator, I is the laser-beam intensity, γ is

the natural line-width, and s0 ≡ I/Is is the on-resonance saturation parameter. The

amplitude b

(k)

ij =

P

q

hi|d · σˆ

q

|kihk|d · σˆ

+|ji/μ2

is the normalized second-order coupling

from state |ji to state |ii through the excited state |ki, and ∆k is the detuning of the

detection laser from the |ji ↔ |ki transition frequency. The sum over k in Eq. (2.29) is

over all intermediate excited states |ki. For this case, we will only be concerned with

mJ ∈ {1/2, 3/2} manifolds in the P3/2level since the detection laser is ˆσ

+ polarized and

near resonance with only these levels. Furthermore, we only consider Raman inelastic

scattering where i 6= j since only these events cause optical pumping.

States |ii and |ji can be written as |li ≡ αl|mI = mF +1/2, mJ = −1/2i+βl|mI =

mF − 1/2, mJ = +1/2i for l ∈ {i, j}. We distinguish between two types of Raman

scattering, namely ∆mF = 0 transitions where the scattered photon does not change
32

the z-projection of total angular momentum, and ∆mF = 1 where the final total zprojection is increased by 1. For ∆mF = 0 transitions, Raman photon scattering occurs

through two excited states, one in each of the mJ = 3/2 and mJ = 1/2 manifolds in

the P3/2level. The result is:

Γ

∆m=0

ij = g

2

γ

 ̄

 ̄

 ̄

 ̄

1

3

αiαj

∆3/2 + ∆z

+

βiβj

∆3/2

 ̄

 ̄

 ̄

 ̄

2

= g

2

γ

4

9

|αjβj |

2





1 + 3

2

∆z

∆3/2

∆3/2 + ∆z





2 (2.30)

where we have used the orthonormality of the hyperfine states and the definitions of

∆3/2 and ∆z in Fig. 2.5. Raman photon scattering for the ∆mF = 1 case only scatters

through the |P3/2, mI = mF + 1/2, mJ = 1/2i states. Its rate is:

Γ

∆m=1

ij = g

2

γ

2

9

|βiαj |

2

(∆3/2 + ∆z)

2

. (2.31)

Eqs. (2.30) and (2.31) break down when the detuning ∆3/2is comparable to half

of the power broadened line width. We can include radiative damping in Eqs. (2.30)

and (2.31) by including an imaginary term in the detuning [Loudon 00, §8.8–8.9]. Letting ∆3/2 → ∆3/2 + iγ0

, we obtain the following result:

Γ

∆m=0

ij = g

2

γ

4

9

|αjβj |

2

1

∆2

3/2

3

1 + 3

2

∆z

∆3/2

 ́2

+

3

γ

0

∆3/2

 ́2

·

1 + ∆z

∆3/2

−

3

γ

0

∆3/2

 ́2

 ̧2

+

3

γ

0

∆3/2

 ́2 3

2 + ∆z

∆3/2

 ́2

Γ

∆m=1

ij = g

2

γ

2

9

1

¡

∆3/2 + ∆z

¢2 + γ

02

(2.32)

where γ

0 ≡

γ

2

√

1 + s0 is half of the power broadened line width. This method of adding

iγ0to the detuning in the Kramers-Heisenberg formula on a 2-level atom reproduces the

well known saturated scattering rate in Eq. (2.16). Letting γ

0 → 0, Eq. (2.32) reduces

to Eqs. (2.30) and (2.31).

The parameters α, β and ∆3/2 are calculated using the Breit-Rabi formula (see

§ 2.2). We use g

2 = γ2

s0/8 with s0 = 1/2 to generate the Γ matrix used in the

calculation of Fig. 2.6.
33

2.4.4 Optimizing Measurement Fidelity

The goal of a single qubit measurement is to determine whether the ion was

projected into the bright |F = 2, mF = 2i state or a dark state. As discussed in § 2.4,

this is determined by observing the fluorescence of the ion. The bright state scatters

many photons, whereas the dark state scatters a much smaller number of photons.

Experimentally, we collect fluorescence for a fixed detection interval τD, and we establish

a cut-off for the number of collected photons nc such that if we collect more than nc

photons, we determine that the ion was projected into the bright state. Likewise, if

we collect at most nc photons, we claim that the ion was projected into a dark state.

The optimum cut-off nc is a function of τD, the background photon collection rate,

the distribution of collected photons in the dark state pdark(n) [see Eq. (2.28)], and to

which hyperfine state we choose to map the original | ↓i state. From § 2.4.2, we see that

|F = 2, mF = −2i is the optimum dark state. We now focus attention on optimizing

τD and nc.

The probability of falsely determining that a bright ion was dark is given by

Pnc

n=0 P(n|γcτD(1 + rbg)) where P(n|λ) ≡ e

−λλn/n! is the Poisson distribution with

mean λ (the bright-ion distribution). Likewise, the probability of falsely determining

that an ion prepared in the dark state |ji was bright is given by P∞

n=nc+1 pdark(n|j) =

1−

Pnc

n=0 pdark(n|j) [see Eq. (2.28)]. The total error in measuring a qubit is the average

of these two (assuming a uniform prior distribution of states). We wish to minimize the

total error by varying nc and τD. It so happens that nc = 0 is optimal independent of

which dark is used if the background is zero (rbg = 0). This can be physically understood

by considering the knowledge acquired about the state once a single photon is detected.

If there is no background and the photon collection efficiency is small such that we can

neglect the probability of collecting one of the photons scattered by states other than

the bright state, then the collected photon is generated by scattering on the cycling
34

transition. That is, whether the original state was dark or bright, the detection of a

photon implies that the state of the ion directly after the photon is collected is bright.

This is because the cycling transition is closed, and once the ion is optically pumped to

the bright state, it stays there. In the presence of background, this is not necessarily

the case. The collected photon could have been generated by the cycling transition, or

it could have been generated by the background. Depending on the background level,

different nc are optimal. We can include the probability of collecting any of the photons

scattered by the dark states as they optically pump to the bright state similarly to

including background. For collection efficiencies on the order of η ∼ 10−3, nc = 0 is

optimal. A straight forward way to determine nc is to numerically calculate the optimal

error by varying τD for different nc.

Assuming zero background, the optimal cut-off is nc = 0, and we can optimize

the average error by varying τD. With a photon collection efficiency of 2.5 × 10−3

(this corresponds to collecting 10 photons in 200 μs), we find the optimal measurement

interval to be τD = 196 μs yielding an average measurement error of 8.6 × 10−5. With

an f/1 optical collection system and a PMT quantum efficiency of 20%, a detection

efficiency of 5 × 10−3is possible. Using such a system, we can collect 20 photons in

200 μs, and the optimal measurement interval is 121 μs in which we collect on average

12.1 photons for the bright state. In this case, the measurement error is 1.0 × 10−5.

High-finesse cavities can increase total photon collection efficiencies to as high as ∼30%

[Keller 04, Acton 05]; however, using simple measurement optics and transferring the

dark state to a “far-away” hyperfine state where multiple scattering events are required

before the dark state can optically pump to the bright state, we obtain measurement

errors below the fault-tolerance threshold [Steane 03, Knill 05].

It is interesting to compare the far-away hyperfine dark state measurement fidelity

to an ion with a large hyperfine splitting and a dark state only ∆mF = 1 away from the

bright state. In Ref. [Acton 05], the authors calculate the ideal measurement fidelity
35

for such a case in 111Cd+. With a photon collection efficiency of η = 1 × 10−3 and

a hyperfine splitting of 14.5 GHz, the minimum infidelity in 111Cd+ is 5 × 10−3. The

smallness of the infidelity is due solely to the relatively large hyperfine splitting in

111Cd+ decreasing the optical pumping rate. For comparison, in 9Be+ with a 1.2 GHz

hyperfine splitting, photon collection efficiency of η = 1 × 10−3, and a far-away dark

state |F = 2, mF = −2i, the minimum infidelity is 1.1×10−3. However, the suppression

of dark to bright optical pumping in the far-away dark state case shows its true power in

the scaling of error rate with photon collection efficiency. At η = 5×10−3, the infidelity

in 111Cd+ is 1.2 × 10−3, whereas in 9Be+, it is 1.0 × 10−5. Measurement errors could

be suppressed even further using a different ion with larger nuclear spin such as 25Mg+.

In addition, quantum logic operations may be used to increase measurement fidelity

[Schaetz 05]. In Ref. [Acton 05] the authors optimize the worst case error calculated

by taking the maximum of the errors in falsely inferring the bright or dark state. We

calculate the average error (for reasons, see § 2.4.6) which are ∼ 2 − 6% lower than the

worst case error.

We note that these theoretical low error probabilities for measurement using faraway dark states have not been verified experimentally. Background light and imperfect

transfer pulses limit the measurement error. However, without trying too hard, we can

achieve measurement errors less than 1% day to day using the |F = 2, mF = −2i dark

state.

2.4.5 Bright to Dark State Optical Pumping

A similar source of measurement error is bright ions being optically pumped to

a dark state during the detection interval due to imperfect BD polarization. If the BD

polarization has a small admixture of ˆπ (ˆσ

−) polarization, the |S1/2

, F = 2, mF = 2i

state can couple to the |P3/2, mJ = 1/2i (|P3/2, mJ = −1/2i) state and decay to a dark

state. We reduced this source of error by tilting and rotating a λ/4 wave plate while
36

monitoring the fraction of dark experiments for a prepared bright ion. We also apply

smaller shimming magnetic fields orthogonal to the primary field to overlap the magnetic

field axis with the BD beam direction. In practice, however, this was unnecessary since

altering the λ/4 wave plate was sufficient for reducing this source of error to below

10−3. In previous experiments with 9Be+ at magnetic fields near 1 mT, shimming fields

were required [Barrett 03]. The moderate 0.0119 T magnetic field offers an advantage

in this area in that the depumping ˆπ (

ˆσ−) transitions are off-resonant by 223 (446)

MHz due to the Zeeman shift of the excited P3/2states. The alignment procedure

described above also improves state preparation because the RD and Repumper beams

are co-propagating with BD. This source of error can be eliminated in principle.

The bright to dark optical pumping error during measurement can be quantified

using analysis similar to § 2.4.1. We replace ω in Eq. (2.16) with 2πω where 2π is

the fraction of the BD laser intensity in the ˆπ polarization component. Because the ˆπ

polarized depumping transition through the |P3/2, mJ = 1/2i state is closer to resonance

than the ˆσ

− polarized transition, we calculate the worst case error assuming all of the

impure polarization is ˆπ. 2πω is the rate of depumping out of the bright state. Ideally the

measurement laser propagates parallel to the magnetic field vector. The ˆπ polarization

component results from misalignment of the measurement beam direction with respect

to the magnetic field vector. The probability of collecting n photons for an ion initially

in the bright state is the convolution of the Poisson distribution with the exponential

distribution (see § 2.4.1 and Ref. [Acton 05]):

pbright(n) = e

−2πωτD P(n|(1 + rbg)γcτD) + 2πωγn

c

e

2πωτDrbg

(γc + 2πω)

n+1

× {P2(n + 1,(γc + 2πω)(1 + rbg)τD) − P2(n + 1,(γc + 2πω)rbgτD)} . (2.33)

With 2π = 10−3 and using the coupling coefficient cBR = 2/9 for the total optical

pumping to both |F = 1, mF = 1i and |F = 2, mF = 1i states, we calculate the

probability of detecting zero photons in the 200 μs detection interval to be 3.0 × 10−4.
37

With a collection efficiency of η = 5 × 10−3, the optimal infidelity is 1.3 × 10−4 at τD =

192 μs collecting on average 19.2 photons in the bright state. In principle, polarization

errors can be as low as 10−5. In this case, the error in falsely determining the bright

state as dark is dominated by the Poisson distribution at n = 0, and the optimal error

rates for measurement are those given in § 2.4.4.

2.4.6 Measurement via Photon Arrival Times

In the measurement scheme described earlier, one collects a number of photons for

a fixed detection interval, and if the number of collected photons n is greater than the

cut-off photon number nc, then we infer that the qubit was projected into a particular

state, | ↑i for example. We found that for zero background, nc = 0 is optimal, and

there exists an optimal detection interval τD which minimizes the probability of falsely

inferring the quantum state. Alternatively, rather than count the number of photons

collected in a fixed detection interval, we may record the arrival times of the photons

{tk} into our measurement apparatus. Certainly there is more information in the set

{tk} than in the number of photons collected, and we explore a method of qubit state

inference based on {tk} in this section.

The problem is to determine whether the ion was projected into the bright state

|bi or dark state |di given the arrival times of N photons {tk} (k ∈ {1, 2, · · · , N})

during the detection interval τD > tN . We calculate the likelihood that the set {tk} was

generated by the state |bi or the state |di, and we infer the qubit state based on the

method of maximum likelihood. The likelihood function L({tk}; α) is the conditional

probability of generating the data set {tk} given the vector α. In our case, the vector

α simply describes whether the ion was projected into |bi or |di and is one dimensional

and binary. Practically speaking, we calculate the likelihood function for |bi and |di

and choose the state which has the higher likelihood. We first consider the likelihood

of the first photon arriving at time t1 for the |bi and |di states. In the analysis that
38

follows, we assume zero background photons and no depumping from |bi. The case

where background photons exist will be handled later.

Once the first photon arrives at time t1, independent of whether the ion was in the

|bi or |di state, the ion is in the fluorescing state |F = 2, mF = 2i for the remainder of

the measurement interval (t > t1). As such, the probability of detecting photons for the

remaining collection times {t2, t3, · · · , tN } is determined by the bright state distribution.

Therefore, the likelihood of the remaining photon arrival times {t2, t3, · · · , tN } is the

same for both the |bi and |di states. As such, we only need to consider the likelihood

of the first photon arrival time t1.

The conditional probability of collecting a photon between t and t + dt given the

bright or dark state is given by the following:

Pb(t)dt = γce

−γctdt

Pd(t)dt =

μZ t

0

dt

0

γce

−γc(t−t

0

)  ̇f(t0

)

¶

dt

= M(t)γce

−γctdt

= M(t)Pb(t)dt

(2.34)

where M(t) ≡

R t

0

dt

0

e

γct

0

 ̇f(t

0

). In the calculation of Pd(t) in Eq. (2.34), the dark to bright

state pumping event can occur any time between 0 and t1 with probability  ̇f(t). Once

the pumping event occurs, the ion can scatter photons at the rate γc into the collection

system for the remainder of the interval. Pb(t) and Pd(t) are the probability densities

of detecting the first photon at time t conditioned on the ion being in state |bi or |di

respectively. The likelihood that the bright state produced a photon between t1 and

t1 + dt is Pb(t1)dt, and the likelihood that the dark state produced a photon between t1

and t1 + dt is Pd(t1)dt. Using the method of maximum likelihood, we conclude that the

ion was in state |di if Pd(t)/Pb(t) = M(t) > 1. Because M(t) is an increasing function,

this method of inference is equivalent to inferring the qubit state is |di if and only if

t > tc where tc is defined by the relation M(tc) = 1.
39

The method of state inference based on maximum likelihood optimizes the average error in false inference assuming a uniform prior distribution. The uniform prior

distribution in this case is equal probability of |bi or |di preparations. The average

error, not the worst case error, is the relevant parameter for quantum error correction.

Furthermore, the assumption of a uniform prior distribution actually closely resembles

many of the measurements that occur in a quantum information processor5. This is

the case for measurements during quantum teleportation and teleporting gate protocols.

For the case of state preparation and verification where the actual prior distribution is

very close to a single state, using a uniform prior distribution when calculating the

average error is pessimistic. For this case, one may develop an inference method which

takes the prior distribution into account in order to minimize the expected error when

integrating the prior distribution of states over the posterior distribution of errors. Here

we do not concern ourselves with prior distributions other than the uniform distribution.

As an example, we consider the system where only one scattering event is required

to optically pump |di into |bi. In this case, f(t) = 1 − e

−ωt as in Eq. (2.15), which gives

M(t) = ω

γc−ω

£

e

(γc−ω)t − 1

¤

. In the general case where |di is some other dark hyperfine

state, then f(t) = 1 −

P

k

ake

−ωkt as in Eq. (2.27), and M(t) has the form:

M(t) = X

k

akωk

γc − ωk

h

e

(γc−ωk)t − 1

i

(2.35)

Solving for tc using Eq. (2.35) is difficult analytically.

The measurement error for a particular t1 is the probability of false inference

for this point. In our inference method, we choose between two values (i.e. |di or |bi)

based on whether Pd(t1) > Pb(t1) or not6. Normalizing by the total probability that

either |bi or |di produced at photon at t1, the probability that |di is more likely is

Pd(t1)/(Pb(t1) + Pd(t1)). Our inference method then is to infer |di if Pd(t1)/(Pb(t1) +

Pd(t1)) > 0.5. Using this definition, we can calculate the conditional probability of false

5 E. Knill, private communication.

6 This is equivalent to the condition M(t1) > 1 which is equivalent to the condition t1 > tc.
40

inference given we were actually in state |di or |bi:

P(falsely infer b at t|d) =







Pd(t)

Pb(t)+Pd(t)

t ≤ tc

0 t > tc

P(falsely infer d at t|b) =







0 t < tc

Pb(t)

Pb(t)+Pd(t)

t ≥ tc

. (2.36)

The worst case error of 50% occurs if we collect the first photon at t1 = tc. At this point,

we cannot identify which state produced t1 as both states are equally likely. Given t1,

not only do we infer which state produced it based on the query t1 > tc, we know

the error in false inference directly from Eq. (2.36). Therefore, we can reject certain

measurements when the individual error becomes too large. The posterior distribution

of errors [Eq. (2.36)] may be useful to certain error correcting protocols.

The expected error in falsely inferring |bi given |di is:

2b =

Z tc

0

Pd(t)dt

= f(tc) − e

−γctc

(2.37)

where we have used Eq. (2.34) and the fact that M(tc) = 1 and f(0) = 0. Similarly,

the expected error in falsely inferring |di given |bi is 2d = 1 −

R tc

0

Pb(t)dt = e

−γctc

.

Therefore, the average expected measurement error is the average of 2d and 2b, namely

2ave =

1

2

f(tc).

It is worth checking that the average error is optimal. We can calculate the probability of false inference for the bright and dark states as a function of a different cut-off

time t

0

c

. In doing so, we obtain 2ave(t

0

c

) = 1

2

h

f(t

0

c

) + e

−γct

0

c (1 − M(t

0

c

))i. Differentiating 2ave(t

0

c

) with respect to t

0

c and setting to zero yields M(t

0

c

) = 1, implying t

0

c = tc.

Therefore, the maximum likelihood inference method gives the optimal average error.

For the case where f(t) = 1 − e

−ωt, the maximum likelihood estimate for tc is

tc =

1

γc−ω

log ¡

γc

ω

¢

, and the average error is 2ave =

1

2

·

1 −

3

ω

γc

 ́ ω

γc−ω

 ̧

. With |di = |F =
41

1, mF = 1i, s0 = 0.5, and η = 2.5 × 10−3, then ω = 2.6 × 10−4, γc = 5.0 × 10−2, and we

find tc = 106 μs yielding an average error of 2ave = 1.4 × 10−2. For the far-away dark

state |di = |F = 2, mF = −2i and the same parameters, the cut-off time is tc = 196 μs

with an optimal average error of 2ave = 8.6×10−5. With a collection efficiency of 5×10−3

as in Fig. 2.6, 2ave = 1.0 × 10−5. These average errors are the same as those calculated

in § 2.4.4, and the cut-off time tc is the same as the optimal τD. This is expected

because the inference methods are essentially the same. In the absence of background,

the optimal cut-off photon number nc is zero; therefore, inferring the state based on

whether t1 is greater or lesser than tc is the same as detecting zero or more photons

in τD. Knowing the time the first photon arrived t1 does provide us with the posterior

distribution of errors which may be useful to error correcting protocols. For example, in

the three qubit redundancy code, three ancilla qubits entangled with the logical qubit

are measured and are expected to yield the same result. The error correcting protocol

takes a majority vote for these three ancilla measurements. However, if we know the

time of the first photon arrivals, we can weight the measurements by the posterior

probability of false inference and obtain more confidence in the ancilla measurements.

2.4.6.1 Background Photons

We saw in the previous section that in the absence of background, comparing the

time of arrival of the first photon t1 to a cut-off time tc is equivalent to the optimal

measurement scheme for counting photons in a fixed detection interval τD. We will

show in this section that in the presence of background, using the full set of photon

arrival times {tk} we can achieve lower average measurement errors than in the case of

counting photons in a fixed detection interval. As an example of this point, if we detect

photons for an interval longer than τD, the error in dark to bright repumping dominates

and the average error increases. However, we loose nothing by detecting longer using

the method of photon arrival times. These extra photons may hold no more useful
42

information, but they can also easily be ignored.

We begin by calculating the likelihood that the bright or dark state produced

the set {tk} in the interval [0, τD). We define the background photon collection rate

as γbg ≡ rbgγc, and the conditional probability that the background will produce a

photon between t and t + dt after a time t of no photons is Pbg(t)dt = γbge

−γbgtdt. In

general, the conditional probability that a single event X will occur given that either

distribution a or distribution b generated the event X is P(X|a or b) = P(X|a)P(X ̄|b)+

P(X ̄|a)P(X|b) where P(X ̄|c) is the conditional probability that event X will not occur

given distribution c. This rule can be understood by realizing that we are examining the

probability of a single event X. It can only be generated by a or b exclusively. Therefore,

we must sum the probabilities that distribution a generated X AND distribution b did

not OR distribution b generated X AND distribution a did not. Following this rule, the

conditional probability that a photon will arrive at time t given the ion is in the bright

state and there exists background at rate γbg is Pb+bg(t)dt = (γc + γbg)e

−(γc+γbg)tdt

which is expected. The probability distribution in time for two independent scattering

mechanisms each with exponential distributions is an exponential distribution with rate

equal to the sum of the individual rates. In more detail, the probability that a photon

will not arrive by time t for an exponential distribution with rate γ is 1−

R t

0

γe−γt0dt

0 =

e

−γt

. The probability that a photon will arrive at time t for an exponential distribution

with rate γ is γe−γtdt. Using the rule above, we obtain our expression for Pb+bg(t)dt.

Generalizing to N photon arrival times, the likelihood function assuming the qubit is

in the bright state is

Lb({tk}, τD)dt

N = (γc + γbg)N e−(γc+γbg)

PN

k=1 ∆tk dt

N e−(γc+γbg)(τD−tN )

= (γc + γbg)

N e−(γc+γbg)τD dtN

(2.38)

where ∆tk ≡ tk − tk−1, t0 ≡ 0, and N is the number of photons collected. The

e

−(γc+γbg)(τD−tN )

term is the probability that neither the bright state nor the background produced a photon in the interval from tN to τD. The likelihood function for
43

the dark state is

Ld({tk}, τD)dt

N = Pd(t1)dt

μ

1 −

Z ∆t1

0

Pbg(t

0

)dt

0

¶

× (γc + γbg)

N−1

e

−(γc+γbg)(τD−t1)

e

−γbg∆t1 dtN−1

+

μ

1 −

Z t1

t0

Pd(t

0

)dt

0

¶

Pbg(∆t1)dt

·

Pd(t2)dt

μ

1 −

Z ∆t2

0

Pbg(t

0

)dt

0

¶

× (γc + γbg)

N−2

e

−(γc+γbg)(τD−t2)

e

−γbg∆t2 dtN−2

+

μ

1 −

Z t2

t1

Pd(t

0

)dt

0

¶

Pbg(∆t2)dt

·

Pd(t3)dt

μ

1 −

Z ∆t3

0

Pbg(t

0

)dt

0

¶

× (γc + γbg)

N−3

e

−(γc+γbg)(τD−t3)

e

−γbg∆t3 dtN−3

+ · · ·

+

Ã

1 −

Z tN−1

tN−2

Pd(t

0

)dt

0

!

Pbg(∆tN−1)dt

×

·

Pd(tN )dt

μ

1 −

Z τD−tN−1

0

Pbg(t

0

)dt

0

¶

e

−γc(τD−tN )

+

Ã

1 −

Z τD

tN−1

Pd(t

0

)dt

0

!

Pbg(τD − tN−1)dt

 ̧

· · ·  ̧ ̧.

(2.39)

Our inference method is to infer |di if the likelihood ratio Lr ≡ Ld/Lb is greater than

1, otherwise infer |bi. After some simplification, this ratio is:

Lr = (1 − γr)M1

+ γrf1(1 − γr)M2

+ γ

2

r

f1f2(1 − γr)M3

+ · · · + γ

N−1

r

f1f2 · · · fN−1(1 − γr)MN

+ γ

N

r

f1f2 · · · fN fN+1

= (1 − γr)M1 + (1 − γr)

X

N

k=2

γ

k−1

r Mk

k

Y−1

l=1

fl + γ

N

r

N

Y

+1

l=1

fl

(2.40)
44

where

Mk ≡ M(tk)e

−γctk−1 e−γbg∆tk

for k < N, (2.41)

MN ≡ M(tN )e

−γctN−1

, (2.42)

fk ≡ e

γc∆tk

[1 − f(tk) + f(tk−1)]

+ e

−γctk−1

[M(tk) − e

γc∆tkM(tk−1)] for k ≤ N, (2.43)

fN+1 ≡ e

γc(τD−tN )

[1 − f(τD) + f(tN )]

+ e

−γctN [M(τD) − eγc(τD−tN )M(tN )], and (2.44)

γr ≡

γbg

γc + γbg

. (2.45)

Notice that in the limit of no background γr → 0, Eq. (2.40) reduces to M(t1) which is

expected. The likelihood ratio Lr for the empty set of zero photons in the measurement

interval τD is Lr({}, τD) = f1 = M(τD) + e

γcτD (1 − f(τD)). We note that Lr({}, τD) is

greater than 1 for systems where the repump rates are less than γc. Said another way,

in the absence of photons, the maximum likelihood estimator for the state is the dark

state with confidence exponentially approaching 1. Once a photon arrives, however, the

state estimate may change depending on when the photon arrives.

We can calculate the measurement error for our inference method in a fashion

similar to the previous section. We first calculate the probability of false inference given

we perfectly prepared states |bi and |di and take the average. The probability of falsely

inferring |bi and |di respectively is:

2b =

Z

Lr<1

Ld({tk}, τD)dt

N

2d =

Z

Lr≥1

Lb({tk}, τD)dt

N . (2.46)

Eq. (2.46) is difficult to calculate analytically due to the complex N-dimensional

boundary defined by Lr({tk}, τD) = 1. To determine the errors in Eq. (2.46), we

simulated 107 data sets of photon arrival times {tk} of length at least N = 35 taken
45

(a)

(b)

0.00

0.01

0.03

0.05

0.10

0.30

γ

bg/γc

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

0 5 10 15 20 25 30 35

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

0 100 200 300 400 500 600 700 800 900 1000

photon number

measurement interval [μs]

error error

Average error vs. measurement interval

Average error vs. photon number

0.00

0.01

0.03

0.05

0.10

0.30

γ

bg/γc

Figure 2.7: Simulated measurement error using photon arrival times. (a) Average

measurement error as a function of the number of photons used for state inference. (b)

Average measurement error as a function of the detection interval. The simulation used

107 data sets {tk} of length at least N = 35 photon arrival times of at least 1 ms total

duration taken from the bright and dark distributions for each background level. The

bright state scattering rate used was γc = 5 × 10−2 μs

−1

. The ratio of the background

scattering rate to the bright state scattering rate γbg/γc is shown in the legend. The

theoretical minimum error of 8.58 × 10−5 agrees with the zero background simulation

within the statistical uncertainty. The dark state was |di = |F = 2, mF = −2i.
46

(1) photon arrival method (2) photon counting method

γbg/γc 2ave n10% t10% [μs] tcnt [μs] 2ave nc τD [μs] λ0 λbg

0 8.58 × 10−5 1 191 196 8.58 × 10−5 0 196 9.80 0

0.01 1.17(3) × 10−4 5 383 278 2.82 × 10−4 3 303 15.1 0.15

0.03 1.51(4) × 10−4 7 442 309 4.18 × 10−4 4 323 16.2 0.48

0.05 1.91(4) × 10−4 9 483 336 5.19 × 10−4 6 376 18.8 0.94

0.10 2.95(5) × 10−4 12 559 381 7.79 × 10−4 9 434 21.7 2.2

0.30 8.08(9) × 10−4 25 719 456 2.45 × 10−3 16 470 23.5 7.1

Table 2.2: Optimal measurement errors for the two methods: (1) maximum likelihood

using the set of photon arrival times {tk} and (2) total photon counts in a fixed measurement period τD. The bright state scattering rate is 5 × 104

s

−1

. The nc cut-off photon

number in method (2) is the number of photons required in the interval τD for the bright

state to be inferred. In method (1), n10% is the number of photons we should collect

to achieve a measurement error within 10% of the optimal measurement error. Similarly, t10% is the minimum measurement interval required to obtain measurement error

within 10% of the optimal measurement error. For comparison, tcnt is the measurement

interval in method (1) where the measurement error equals the optimal measurement

error in method (2). λ0 is the mean of the bright distribution when collecting photons

for τD in the absence of background. λbg is the mean of the background distribution.
47

from the bright and dark distributions with different background levels for a detection

interval of at least τD = 1 ms. For data sets where less than N photons arrived before

τD, we extended the set to size N. We calculated the likelihood ratio Lr and the

probability of falsely inferring dark given bright and vice versa using the first n of N

photon arrival times {tk}. In this fixed n method, we let the measurement interval

τD equal the arrival time of the last photon tn. We also inferred the state based on

the likelihood ratio for variable sized sets {tk} with τD fixed and examined the average

error as a function of τD. The results are shown in Fig. 2.7. The simulation used

γc = 5 × 104s

−1 and the shelved dark state |di = |F = 2, mF = −2i with s0 = 0.5.

The fractional background scattering rates rbg were between 0 and 0.3. We observe that

the average measurement error drops exponentially with increasing photon number n

until the error reaches a threshold above that of zero background. Similarly, the average

error drops exponentially as a function of τD until it saturates at the same lower bound.

The threshold for saturation is background dependent. In addition, the error for the

zero background simulation agrees with the theoretical estimate within the statistical

uncertainty after the first photon, and additional photons do not lower the error. This

agrees with our theoretical hypothesis that in the absence of background, only the

arrival time of the first photon carries useful information for state inference. More

generally, there appears to be a background dependent critical photon number for which

additional photons add negligible additional information useful for state inference. We

quantify the critical photon number n10% as the number of photons required to achieve a

probability of measurement error within 10% of the optimal measurement error. For zero

background, this critical photon number is 1; for 1% background, this critical photon

number is 5. Practically, we expect that by collecting the arrival times of the first five

photons, we can achieve average measurement errors in the presence of 1% background

(typical in our experiments) very close to the zero background case. A similar critical

measurement interval t10% exists such that detecting photons for an interval longer than
48

t10% yields a probability of measurement error within 10% of the optimal. Detecting

longer adds negligible (only 10%) additional information useful for state inference. For

the 1% background case, t10% = 383 μs.

We compare the simulated average measurement errors in Fig. 2.7 to the calculated optimal average measurement errors using the cut-off photon number nc method

in the presence of background. The results are tabulated in Table 2.2. We observe that

we can achieve approximately a factor of 3 improvement in the average error using the

photon arrival times vs. counting the number of photons arriving in a fixed detection

interval τD. The parameters t10% and n10% are the minimal measurement time and

photon number respectively required to achieve a measurement error probability within

10% of the optimal measurement error probability. For comparison, we also tabulate

tcnt, the measurement interval in the photon counting method where the probability of

measurement error equals that of the optimal measurement error in the photon counting

method. We observe that for the same probability of measurement error, the photon

counting method using maximum likelihood can be performed in approximately 10%

less time.

2.5 Quantum Gates

As discussed in the introduction, a sufficient gate set for universal quantum computation consists of arbitrary single qubit rotations and a single entangling quantum

gate [Barenco 95]. All gates in this thesis involve the interaction of one or more ions

with laser beams. We acknowledge that there exist alternate methods for performing

gates [Mintert 01, Schrader 04] including spin rotations using resonant magnetic fields.

However, laser-ion interactions have the advantage that closely spaced ion qubits can

be individually addressed with laser beams [Wineland 98, Riebe 04], and strong magnetic field gradients required for some two-qubit entangling gates are easily generated

[Cirac 95, Wineland 98].
49

2.5.1 Single-Qubit Quantum Gates

The single qubit gates are implemented using stimulated Raman transition with

co-propagating laser beams (see § 2.5.2 below). Co-propagating laser beams have the

advantage that the phase as a function of space of the effective field driving the qubit

transition is given by ∆~k · ~x where |∆~k| ∼ 2π

30cm

is the difference in the two k-vectors

of the two co-propagating laser beams and ~x is the ion position (there is no need to

quantize the ion position in this treatment since the coupling to the motional states

is negligible). Therefore, interferometric stability of the laser fields is not required for

single-qubit rotations. However, for high-fidelity single-qubit rotations where the phase

error should be kept below 10−4, the total optical path from the location where the

two Raman beams are generated to the ion should not fluctuate by more than 5 μm.

To reach this level of phase stability, the modulator frequency may need to be phaselocked to the optical heterodyne beat frequency measured close to the trap. We did

not implement such a feedback loop in the experiments of this thesis. In addition,

electronic state changing transitions with co-propagating Raman laser beams can occur

independent of the motional state (the Lamb-Dicke parameter is η ∼ 10−7for 9Be+).

In the section that follows, I show how the phase of the single qubit rotation can be

controlled by the differential phase between the two Raman lasers generated by stable

microwave sources. Given rotations about any axis in the x − y plane of the Bloch

sphere, we can generate arbitrary rotations about any axis on the Bloch sphere.

2.5.2 Stimulated Raman Transitions

The basic building blocks of our gates are two-photon stimulated Raman transitions. Two states which do not couple directly through an applied radiation field

can obtain an effective coupling mediated via the radiation field coupling these states

to an auxiliary level. A diagram of such a situation is shown in Fig. 2.8. The qubit
50

δ

∆

|1〉

|2〉

|3〉

b

r

Figure 2.8: Energy level diagram for two-photon stimulated Raman transitions. The

symbols b and r represent the “blue” and “red” lasers respectively. The electric field of

laser beam i has the form E~

i = ˆ2iEi cos(~ki

· ~x − ωit + φi) where i ∈ {b, r}. The detuning

of the blue laser from the |1i ↔ |3i transition frequency is ∆, and the detuning of the

difference frequency ωb − ωr from the |1i ↔ |2i transition frequency is δ.
51

states are levels |1i and |2i, and the auxiliary state is level |3i. These states have

energies ~ω1, ~ω2, and ~ω3 respectively corresponding to a base Hamiltonian H0 =

~ω1|1ih1| + ~ω2|2ih2| + ~ω3|3ih3|. For ions commonly used in QIP, level |3i is typically

one of the lowest P states, and the |1i and |2i levels are hyperfine ground states. We

apply two lasers labeled by the indices b and r for “blue” and “red” respectively each

having an oscillating electric field of the form E~

i = ˆ2iEi cos(~ki

·~x−ωit+φi) for i ∈ {b, r}.

Here, Eiis the amplitude of the electric field, ˆ2iis its polarization, ~kiis the wave vector,

~x is the position of the ion, ωiis the frequency of the laser, and φiis the phase of

the laser. The interaction Hamiltonian for this system is given by the electric-dipole

interaction HI = −~μ · (E~

b + E~

r) where ~μ is the electric-dipole operator. Writing an arbitrary state as |Ψi =

P3

k=1 cke

−iωkt

|ki and applying Schr ̈odinger’s equation, we obtain

a system of differential equations for the coefficients ck:

ic ̇1 = g

∗

b

e

i∆t

c3

ic ̇2 = g

∗

r

e

i(∆−δ)t

c3

ic ̇3 = gbe

−i∆t

c1 + gre

−i(∆−δ)t

c2 (2.47)

where gb ≡ −Ebe

i(~kb·~x+φb)

2~

h3|~μ·2ˆb|1i and gr ≡ −Ere

i(~kr·~x+φr)

2~

h3|~μ·2ˆr|2i are the on-resonance

Rabi frequencies between states |1i ↔ |3i and |2i ↔ |3i, and we have ignored terms

oscillating near 2ωb,r by the rotating wave approximation. There is no direct coupling

between c1 and c2 in Eq. (2.47); however, as we will see below, the coupling of each of

these coefficients to c3 mediates an effective coupling between c1 and c2.

c3 can be removed from Eq. (2.47) by adiabatic elimination. We separate the fast

phase evolution in c3 from the slower envelop by letting c

0

3 = e

i∆t

c3. Upon taking its

derivative and substituting into the last equation in Eq. (2.47), we find

∆c

0

3 + ic ̇

0

3 = gbc1 + gre

iδtc2. (2.48)

We now make the assumption that the majority of c3’s time dependence comes from the

fast phase evolution induced by the large detuning ∆, namely that |∆c

0

3

| À |c ̇

0

3

|. This is
52

valid as long as the two-photon Rabi frequency, which we are in the process of deriving,

is small compared to ∆, which can be verified at the end of the calculation. Using this

assumption to ignore the ic ̇

0

3

term in favor of ∆c

0

3

in Eq. (2.48) and substituting it into

Eq. (2.47) yields

ic ̇1 = δs1c1 + Ω∗e

iδtc2

ic ̇2 = Ωe

−iδtc1 + δs2c2 (2.49)

where δs1 ≡ |gb|

2/∆ and δs2 ≡ |gr|2/∆ are the AC Stark shifts of levels |1i and |2i

respectively and Ω ≡ gbg

∗

r/∆ is the two-photon Rabi frequency between states |1i and

|2i. We note that the two-photon Rabi frequency is proportional to exp[i(

~kb −~kr) · ~x +

(φb −φr)]; therefore, we can control the phase of rotations by controlling the differential

optical phase between the two laser beams. The Stark shifts can be removed from

the problem by a redefinition of the |1i ↔ |2i transition frequency, or alternatively

absorbing the differential Stark shift into the definition of the Raman beam difference

frequency detuning δ. We move to a frame rotating with the Stark shifts by letting

ck ≡ c

0

k

e

iδsk for k ∈ {1, 2}. Eq. (2.49) is transformed then to the following:

ic ̇

0

1 = Ω∗

e

iδ0t

c

0

2

ic ̇

0

2 = Ωe

−iδ0t

c

0

1

(2.50)

where δ

0 ≡ δ + δs1 − δs2.

Eq. (2.50) are the equations for the standard 2-level problem. Its solution and

many other aspects are discussed at length in a variety of works, not the least of

which include [Allen 87, Sakurai 94, Metcalf 99]. The time dependence can be removed from Eq. (2.50) by moving to another rotating frame with the transformation

c

0

1 ≡ c

00

1

e

iδ0t/2 and c0

2 ≡ c

00

2

e

−iδ0t/2

. The new coefficients obey the equation

ic ̇ = M · c (2.51)
53

where

M ≡





−δ

0/2 Ω

Ω

∗

δ

0/2





and c ≡





c

00

2

c

00

1



 .

Taking the Laplace transform of Eq. (2.51) and solving for the Laplace transformed

vector C(s) yields:

C(s) = (iM + s1)

−1

· c(0)

=

1

s

2 + Ω02





s + i

δ

2 −iΩ

−iΩ

∗

s − i

δ

2



 · c(0) (2.52)

where C(s) is the Laplace transform of the time-domain vector c(t), c(0) is the initial condition, 1 is the identity matrix, s is the Laplace transform variable, and Ω0 ≡

p

|Ω|

2 + δ2/4. The time-domain solution is obtained by taking the inverse Laplace

transform of Eq. (2.52):





c

00

2

(t)

c

00

1

(t)



 =





cos(Ω0t) + i

δ

2Ω0 sin(Ω0

t) −i

Ω

Ω0 sin(Ω0

t)

−i

Ω∗

Ω0 sin(Ω0

t) cos(Ω0t) − i

δ

2Ω0 sin(Ω0

t)



 ·





c

00

2

(0)

c

00

1

(0)



 . (2.53)

The time-domain solution Eq. (2.53) is the general solution to the two-level problem. On resonance, the population of the two levels flops back and forth at a rate given

by the on-resonance Rabi frequency |Ω|, and the axis about which the Bloch vector

rotates is given by arg(Ω).

2.5.3 Coupling Electronic States to Motional States

In the previous section we showed how we can obtain an effective coupling between

two ground internal states of an ion (qubit states) by using two laser beams detuned

from an excited state with a difference frequency near resonance with the qubit transition frequency. The evolution of the state amplitudes [Eq. (2.50)] can be equivalently

described by a single two-level quantum system driven with a near resonant field. We

make the identification |1i → | ↓i and |2i → | ↑i with the qubit transition frequency
54

defined as ω0. The base Hamiltonian is H0 = Hint where Hint = ~

ω0

2

σz and σz is the

Pauli-z operator. The interaction Hamiltonian in the interaction picture with respect

to Hint can be written as HI = Ω0e

i∆~k·~xe−iδtσ+ + h.c. [see Eq. (2.50)] where we have

explicitly showed the e

i∆~k·~x factor imbedded in the Rabi frequency definition in the previous section. Here, Ω0 ≡

EbEr

4~2∆

h↑ |~μ · 2ˆr|3ih3|~μ · 2ˆb| ↓ie

i∆φ

, ∆~k ≡ ~kb −~kr, ∆φ ≡ φb − φr,

and σ± =

1

2

(σx ±iσy) are raising (| ↑ih↓ |) and lowering (| ↓ih↑ |) operators for the effective 2-level system (σx,y are Pauli matrices). Writing a general state in the interaction

picture as |Ψi = c↑| ↑i + c↓| ↓i and applying Schr ̈odinger’s equation i~

∂

∂t|Ψi = HI |Ψi,

one can readily verify the evolution of the amplitudes is exactly that of Eq. (2.50).

The ion’s quantized motion can be taken into account by including the harmonic

oscillator Hamiltonian Hosc = ~ωza

†a (neglecting the zero-point energy 1

2

~ωz) in the

base Hamiltonian H0 = Hint+Hosc where ωz is the oscillation frequency. The interaction

Hamiltonian is modified by replacing the x vector with its operator equivalent: ∆~k·~x →

η(a

† + a) where η ≡ ∆kzz0 is the Lamb-Dicke parameter. The phase due to the mean

position of the ion ∆~k · ~x (~x a real vector) can be included in the definition of ∆φ

contained in Ω0. Here, z0 ≡

p

~/2mωz is the spread of the motional ground-state wave

function, ∆kz is the projection of ∆~k onto the mode direction z, and m is the ion mass.

Writing the general state of the ion including its motion as

|Ψi =

X

s∈{↓,↑}

X∞

n=0

cs,n(t)|si|ni, (2.54)

the interaction Hamiltonian in the interaction picture can be obtained by the transformation H0

I = e

iHosct/~HIe−iHosct/~

:

H0

I = ~Ω0σ+ exp(i[η(a

†

e

iωzt + ae−iωzt

) − δt]) + h.c.. (2.55)

Schr ̈odinger’s equation i~

∂

∂t|Ψi = H0

I

|Ψi then gives the evolution of the state |Ψi.

We are primarily concerned with near resonant transitions where the detuning is

near an integer multiple of the motional mode frequency, namely δ = (n

0 − n)ωz + ∆
55

where n

0 and n are integers and ∆ ¿ ωz, Ω0. When this is the case, the two states

| ↑, n0i and | ↓, ni form a two-level system and evolve according to:

c ̇↑,n0 = −i

1+|n

0−n|

e

−i∆tΩn0

,nc↓,n

c ̇↓,n = −i

1−|n

0−n|

e

i∆tΩn0

,nc↑,n0 (2.56)

where Ωn0

,n is given by [Wineland 98, Cahill 69, Wineland 79]

Ωn0

,n ≡ Ω0|hn

0

|e

iη(a

†+a)

|ni|

= Ω0e

−η

2/2

μ

n<!

n>!

¶1/2

η

|n

0−n|L

|n

0−n|

n<

(η

2

) (2.57)

and n< (n>) is the lesser (greater) of n

0 and n; Lα

n

is the generalized Laguerre polynomial

L

α

n

(x) = Xn

m=0

(−1)m

μ

n + α

n − m

¶

x

m

m!

. (2.58)

For tightly confining traps and relatively cold ions, the ion’s amplitude of motion is small compared to the wavelength of radiation, and the Lamb-Dicke criteria

hΨmotion|∆k

2

z

z

2

|Ψmotioni

1/2 ¿ 1 is satisfied. The Lamb-Dicke criteria should not be

confused with the less restrictive condition η ¿ 1 [Wineland 98] as ion motion with

large amplitude can sample different portions of the wavefront even though η may be

small. For Fock states with motional occupation number n, the Lamb-Dicke criteria is

satisfied when η

√

2n + 1 ¿ 1. In this limit, Eq. (2.57) can be expanded to the lowest

order in η, and the Rabi frequency has the simpler form

Ωn0

,n ' Ω0η

|n

0−n|

μ

n>!

n<!

¶1/2

1

|n0 − n|!

. (2.59)

In the Lamb-Dicke limit, the interaction Hamiltonian also takes on a simpler form. We

can expand the exponential in Eq. (2.55) to lowest order in η to obtain:

H0

I ' ~Ω0σ+e

−iδt[1 + iη(ae−iωzt + a†

e

iωzt

)] + h.c.. (2.60)

With this approximation, spin flip transitions simultaneous with the creation or annihilation of motional quanta is apparent. If the laser field is resonant with the spin
56

|↓〉|n = 0〉

|↓〉|n = 1〉

|↓〉|n = 2〉

|↓〉|n = 3〉

|↑〉|n = 0〉

|↑〉|n = 1〉

|↑〉|n = 2〉

|↑〉|n = 3〉

ωz

. . . .

. . . .

|↓〉|n = 0〉

|↓〉|n = 1〉

|↓〉|n = 2〉

|↓〉|n = 3〉

|↑〉|n = 0〉

|↑〉|n = 1〉

|↑〉|n = 2〉

|↑〉|n = 3〉

ωz

. . . .

. . . .

(a) stimulated Raman (∆n = −1) (b) spontaneous Raman (∆n = 0)

Figure 2.9: Raman sideband cooling is a two-step process. (a) Step 1: starting in

the higher-energy | ↑i state, perform a blue sideband π-pulse to remove one phonon of

motional energy. (b) Step 2: optically pump back to the | ↑i via spontaneous Raman

scattering. Repeat this processes until the ground motional state |n = 0i is reached.

flip transition frequency plus one motional quanta (δ = ωz, the “blue” sideband) and

we neglect oscillating terms, the Hamiltonian reduces to H0

I = η~(iΩ0σ+a

† − iΩ∗

0σ−a).

Similarly, if the laser difference frequency is tuned to the qubit frequency minus the

oscillation frequency (δ = −ωz, the “red” sideband), the Hamiltonian has the Jaynes

Cummings form [Jaynes 63] H0

I = η~(iΩ0σ+a − iΩ

∗

0σ−a

†

). The expansion of the Hamiltonian in Eq. (2.60) does not allow for the annihilation or creation of more than one

phonon simultaneous with a spin flip transition, nor does it include the Debye-Waller

factor e

−η

2/2

[Wineland 98, Wineland 79]. The multiple phonon creation/annihilation

terms can be obtained by examining higher order terms in Eq. (2.55) or using the Rabi

frequencies calculated in Eqs. (2.57) or (2.59).

2.5.4 Ground State Cooling

After initialization, the ions are Doppler cooled to their Doppler temperature. For

typical trap frequencies in the few MHz region, the ions’ motion is in a thermal state

with average occupation number hni ' 2 − 3. The ions can be further cooled to their
57

ground motional state |n = 0i via Raman sideband cooling [Wineland 79, Diedrich 89,

Monroe 95b]. Cooling to the ground state and hence preparing a pure motional state

is a requirement for some two-qubit gates [Cirac 95] and increases the fidelity of other

two-qubit gates which are less restrictive in this regard [Leibfried 03, Sørensen 00]. A

diagram of the stimulated Raman sideband cooling process is shown in Fig. 2.9. For

an ion in the | ↑i|n = 2i state, we can apply a π-pulse on the blue sideband to remove

one motional quanta from the ion simultaneous with a spin-flip transition. The ion

spin state can be reset to | ↑i via optical pumping. In the Lamb-Dicke limit, recoil

heating is suppressed during the optical pumping process [Dicke 53, Bergquist 87]. The

process to remove one motional quanta is repeated until the ion reaches the ground state.

Even though the heating of the system on average is equal to the recoil energy after

scattering of a photon, the increase in the motional occupation number is suppressed

in the Lamb-Dicke limit. The probability of increasing the phonon number by one

through the repump process is η

2

(n + 1). There is still a small probability of increasing

the phonon number through repumping. However, because the cooling process repeats,

this additional heat can be removed by succeeding cooling cycles. Once in the ground

state, additional cooling cycles become inactive. Because the optical pumping process

resets the ion to the | ↑i state, if the ion is in the motional ground state | ↑i|n = 0i, the

blue sideband drive is off resonance with any transition to the | ↓i state as there is no

|n = −1i motional state. Furthermore, the | ↑i is a dark state for the optical pumping

laser; therefore the ion will not scatter any photons once in the | ↑i state and hence does

not recoil heat back to the Doppler temperature. The fact that additional cooling cycles

do not affect ions in the ground state implies some robustness to the cooling process.

Because the ion’s motion is in a thermal distribution after Doppler cooling, there

is a significant fraction of experiments where the ion’s motional occupation number is

greater than that corresponding to the Doppler temperature. Furthermore, the pulse

duration required to perform a π-pulse on the blue sideband differs based on the starting
58

motional state |ni (see Eq. (2.57) in § 2.5.3). We can cool all experiments where the ion

has at most L phonons by the following procedure. We first apply a blue sideband pulse

for the duration required to induce the transition | ↑i|n = Li → | ↓i|n = L−1i followed

by optical pumping to the | ↑i|n = L − 1i state. This is accomplished by choosing

the duration of the pulse τL to be such that ΩL,L−1τL = π/2. The population of the

|n = Li state is now completely transferred to the |n = L − 1i state. The population

in the other states make only partial transitions reducing their phonon number by one

because the Rabi frequencies for these transitions are different. However, next we apply

a blue sideband pulse for τL−1 = π/(2ΩL−1,L−2) followed by the optical pumping step

completely clearing out the |n = L − 1i motional state. This processes is repeated

clearing out motional states |n = L − 2i, |n = L − 3i, · · · , |n = 1i until the final pulse

transfers perfectly all of the remaining population in the |n = 1i state to the |n = 0i

state. For a thermal (Boltzmann) distribution with hni = 3, 15 cooling cycles of this

form is sufficient to successfully cool to the ground |n = 0i state 99.5% of the time.

Additional cooling cycles will increase this percentage.

Using a sequence of π-pulses for perfectly transferring population from motional

states |ni → |n − 1i for n from L to 1 is analogous to frequency chirping a red-detuned

Doppler cooling laser for weakly bound neutral atoms. As the frequency of the Doppler

laser is chirped from the red of the Doppler broadened atomic linewidth closer to resonance, different velocity classes of atoms are cooled and redistributed closer to zero

velocity. In this manner, the population of atoms in different velocity classes piles up

near zero velocity. Similarly in Raman sideband cooling, as the cooling process progresses, the population of different motional states of the ion is cleared out and piles up

near |n = 0i.

The temperature of a cold harmonically-bound ion near the ground state can

be measured by examining its motional spectra [Diedrich 89, Monroe 95b]. For an ion

in electronic state | ↑i and a thermal state with average motional occupation number
59

0

2

4

6

8

10

12

-3.82 -3.77 -3.72 -3.67 -3.62 -3.57 -3.52

0 ms delay

3 ms delay

Raman difference frequency − ω0 [MHz]

Average counts in 200 μs

Red sideband

0

2

4

6

8

10

12

3.5 3.55 3.6 3.65 3.7 3.75 3.8

0 ms delay

3 ms delay

Raman difference frequency − ω0 [MHz]

Average counts in 200 μs

Blue sideband

Figure 2.10: The temperature of an ion after Raman sideband cooling can be determined by measuring the transition probabilities on the red and blue sidebands. The

y-axis measures the average number of photons collected in a 200 μs measurement interval λ. An ion in the | ↑i = |F = 2, mF = 2i gives the baseline fluorescence (λ↑ ∼11

photons/200 μs). The transition probability to the | ↓i = |F = 2, mF = 1i is proportional to λ↑ − λ. The filled diamonds are data for a temperature measurement

taken directly after Raman sideband cooling and give hni = 0.06. The open circles are

data for a temperature measurement taken 3 ms after Raman sideband cooling giving

hni = 0.88. Data was taken on 2005-04-25 in John Jost’s trap. Similar measurements

of the ion temperature for various delays gives a heating rate of approximately 0.19

quanta/ms.
60

hni, the ratio of transition probabilities to | ↓i for the blue and red sidebands is hni

hni+1

independent of the laser intensity, pulse duration, and trap frequency in the Lamb-Dicke

limit [Turchette 00]. Therefore, we can determine hni by making measurements of the

ratio of the transition probabilities on the red and blue sidebands. A simple example

of this measurement is when the ion is in the ground state. Here, the blue sideband

transition probability vanishes indicating that hni = 0. Fig. 2.10. shows temperature

measurement data immediately after cooling to the ground state and 3 ms there after.

The blue sideband vanishes for the filled diamonds indicating that ground state cooling

is achieved. After cooling to the ground state and waiting for a delay (3 ms in Fig. 2.10),

the temperature can be probed again to measure the heating rate of the ion. Repeating

the temperature measurement for multiple delays gave a heating rate of 0.19 quanta/ms

in John Jost’s trap on 2005-04-25.

2.6 Two-Qubit Quantum Gates

To complete the universal gate set for ions, we desire an entangling gate between the spin states (more specifically, the subsystem of two hyperfine ground states

of 9Be+) of two ions. The direct interaction between two ions’ internal states is negligible in our systems; however, an effective interaction is obtained by coupling the

internal states of the ions to their collective motion. The original proposal to generate a universal two-qubit entangling gate in this manner was given by Cirac and Zoller

[Cirac 95], and its experimental realization was recently demonstrated by Schmidt-Kaler

et al. [Schmidt-Kaler 03]. Since Cirac’s original proposal, schemes for generating entangling two-qubit gates for ions based on spin dependent motional displacements have

been proposed [Mølmer 99, Solano 99, Milburn 01, Garc ́ıa-Ripoll 03] and implemented

[Sackett 00, Leibfried 03, Haljan 05b]7. Such gates have the advantage that the ions

7 A. M. Steane et al. have reported demonstration of two-qubit z-basis gates as well in private

communication.
61

do not need to be cooled to the ground state so long as they remain in the Lamb-Dicke

limit, and individual ion addressing is not required. These technical advantages can

make these type of gates desirable over the original Cirac and Zoller gate, and entangled states created with these methods have been realized with fidelities as high as 97%

[Leibfried 03].

Two gates are discussed in this section, and both operate in essentially the same

manner. A mode of a two-ion crystal is displaced around a closed path in phase space

conditioned on its two-qubit state and obtains a phase proportional to the area enclosed

by the path taken in phase space. The differences between the two gates are the basis

on which the phases are conditioned and the methods for generating the conditional

displacements. I begin this section by explaining the theory of operation of conditional

phase gates in the xy-basis, namely the Mølmer-Sørensen (MS) gate. I then discuss

phase gates in the z-basis, namely the “Z-phase gate”. We have experimentally realized

both of these gates [Sackett 00, Leibfried 03]. Our published fidelities are higher for

the Z-phase gate; however, direct application of the Z-phase gate on field-independent

qubits is very inefficient. The work arounds to this problem are discussed.

2.6.1 Phase Gate in xy-Basis

The Mølmer-Sørensen gate generates conditional displacements in phase space by

application of a bichromatic field to the ions. We recall from § 2.5.3 Eq. (2.55) that the

interaction Hamiltonian of a single effective field which couples the spin state of a single

ion to its motional state (non-negligible Lamb-Dicke parameter η) in the interaction

picture is given by:

H0

I = ~Ωσ+ exp(i[η(a

†

e

iωzt + ae−iωzt

) − δt + φ]) + h.c. (2.61)

where δ ≡ ω−ω0 is the detuning of the effective field from the qubit transition frequency

(generated by two lasers with a difference frequency ω, see § 2.5.2), σ+ ≡ | ↑ih↓ |
62

(σ− ≡ | ↓ih↑ |) is the qubit raising (lowering) operator, and we have explicitly shown

the phase φ of the effective field previously contained in the definition of Ω (see § 2.5.2

and 2.5.3). We assume Ω is real and that all of its complex phase is contained in

φ. In the Lamb-Dicke limit (η

p

hni + 1 ¿ 1, hni the average phonon number), the

exponential in Eq. (2.61) can be expanded to lowest order in η to obtain:

H0

I ' ~Ωσ+e

−i(δt−φ)

[1 + iη(a

†

e

iωzt + ae−iωzt

)] + h.c.. (2.62)

The gate scheme by Mølmer and Sørensen is to apply two effective fields at equal

and opposite detunings ±δ of equal strengths near the blue and red sidebands [Ωb =

Ωr = Ω; subscript ’b’ and ’r’ represent “blue” (+) and “red” (−) detunings and phases

φb,r respectively]. For a single ion, the application of a bichromatic field results in

the transformation e

−i(δt−φ) → e−i(δt−φb) + ei(δt+φr) = eiφs (e−i(δt−φm) + ei(δt−φm)

) in

Eq. (2.62) where we have replaced φb and φr with the more symmetric definitions

φs ≡

1

2

(φb + φr) and φm ≡

1

2

(φb − φr). The resulting Hamiltonian is:

H0

I = 2~Ω cos(δt − φm)σ−φs

− ~ηΩσ π

2 −φs

(a

†

e

i[(ωz−δ)t+φm] + ae−i[(ωz−δ)t+φm]

)

− ~ηΩσ π

2 −φs

(a

†

e

i[(ωz+δ)t−φm] + ae−i[(ωz+δ)t−φm]

)

(2.63)

where σφ ≡ cos(φ)σx + sin(φ)σy is a rotated Pauli matrix in the xy-plane [σ± =

1

2

(σx ±

iσy)], and η is the Lamb-Dicke parameter for the selected mode of the two-ion crystal

(the center of mass mode of two ions has a Lamb-Dicke parameter 1/

√

2 times the LambDicke parameter for a single ion). The first term and last two terms are off-resonance,

and we neglect them in the remainder of this treatment. Their contribution to the gate

error is discussed in [Sørensen 00]. The first term, called the “direct” term by Sørensen,

is an off-resonant drive of the carrier transition and is absent in the analogous Z-phase

gate. However, with special control of the field intensities, its contribution can be made

to vanish exactly [Sørensen 00]. Its presence should not be taken lightly because its

resonant coupling strength is 1/η stronger than the gate drive strength.
63

x

p

φm

|↑〉φs

|↓〉φs

klaser kr

kb

∆kr = klaser− kr

∆kb = kb− klaser

(a)

(b)

(c)

laser power spectrum

frequency

ωlaser

ωlaser + ω0 ω +δ laser− ω0+δ

φm(z) = φm

(0) + 1/2 (∆kb

− ∆kr)z

sin φm(z)

∆z ~ 2−3 μm

λeff~ 220 nm

red sideband blue sideband

Figure 2.11: (a) Phase space displacement trajectory. The different states in the σπ/2−φs

basis follow different paths in phase space. Both paths are circular, and the motional

state returns to its starting location after the period 2π/(ωz − δ) independent of the

spin state. Both paths circumscribe the same area with the same sense of rotation. The

starting direction is given by the phase φm, and as long as it remains stable, the area

circumscribed is independent of this phase. (b) Laser configuration for the MølmerSørensen gate. By arranging the ∆kb,r vectors in opposite directions, the φs phase

dependence becomes negligible (see text and Ref. [Haljan 05a]). (c) The φm phase

front oscillates in space with an effective wavelength λeff = λ/√

2 in the optical regime.

By controlling the trap strength, two ions can be spaced an integer number of effective

wavelengths (∼ 10) causing both ions to see the same φm phase.
64

We can gain an intuitive picture of state evolution under the Hamiltonian in

Eq. (2.63) by examining the propagator for infinitesimal durations dt. At time t,

the infinitesimal propagator has the form e

−iH0

I

(t)dt/~ = exp(dα(t)a† − dα∗

(t)a) =

D(dα(t)) where D(α) is the displacement operator of complex displacement α, and

dα(t) = 2iηΩσ π

2 −φs

e

i[(ωz−δ)t+φm]dt is the infinitesimal displacement for the MS gate.

The strength of the infinitesimal displacement is proportional to the |n = 0i ↔ |n = 1i

sideband Rabi frequency ηΩ and is conditioned on the qubit state in the σπ/2−φsbasis.

The direction of the displacement at time t is given by arg(dα(t)) = ±[π/2 + (ωz −δ)t+

φm] where the ± signs are taken for the ± qubit states in the σπ/2−φsbasis. We observe

that the direction of the displacement in xp phase space in the interaction picture8

rotates linearly in time with a rate given by the detuning of the effective field from the

first sideband frequency. Assuming the gate drive strength ηΩ is constant during the

gate, the resulting path taken in phase space is a circle, and the motional state of the

ion returns to its starting location after the period 2π/(ωz − δ) (see Fig. 2.11).

Application of the MS gate drive to two or more ions can be examined by summing

over the individual Hamiltonians for each ion [Eq. (2.63)] and noting that the motional

mode is common to all ions9. When taking this sum, it is important to remember

that the ions have different phases φs and φm due to their different locations [φ

(i)

r,b =

φ

(0)

r,b + (∆~kr,b · zˆ)zi where zi

is the location on the axis of the i

th ion]. By arranging the

effective k-vectors ∆~kr and ∆~kb in opposite directions (see Fig. 2.11b), the difference in

φs for the two ions φ

(1)

s − φ

(2)

s '

1

2

ω0

c ∆z ∼ 10−4

is essentially zero10 [Haljan 05a], and

for convenience, we choose φs = 0. The small difference in basis between the two ions

can be compensated by conjugating the gate by a small z-rotation on one of the qubits.

More importantly, fluctuations in ion position or equivalent fluctuations in the laser

8 The interaction picture phase space has xp coordinates which rotate at the trap frequency with

respect to the lab frame xp coordinates.

9 We neglect coupling to the other axial modes as they are off resonant. Coupling to the other modes

is further suppressed by poor mode matching of the driving effective field to the auxiliary modes.

10 We have ∆φs =

1

2

(∆φb + ∆φr) = 1

2

[∆kb∆z + ∆kr∆z] = 1

2

(∆kb + ∆kr)∆z, and if ∆kb and ∆kr

are in opposite directions, ∆kb + ∆kr → ω0/c, and ∆φs '

1

2

ω0

c ∆z.
65

beam optical path will cause minimal fluctuations in the differential φs phase seen by

the ions. Therefore, the Pauli-matrix σπ/2−φs → σy in Eq. (2.63) is common to all ions

when we take the sum. The optical arrangement which makes φs common to all ions is

also responsible for making φm position dependent (φ

(i)

m = φ

(0)

m +

1

2

[(∆~kb−∆~kr)·zˆ]zi, see

Fig. 2.11c). We see from Fig. 2.11c that the φm phase front oscillates with a length scale

given by the effective wavelength λeff of the laser field (2π/λeff = |∆~kb−∆~kr|). By tuning

the trap strength appropriately, we can space the two ions by an integer of effective

wavelengths of the φm phase front such that each ion sees the same φm. For more

than three ions, the ions are not equally spaced; however, an ion spacing arrangement

can be found for which the ions see the same φm to within a reasonable approximation

[Leibfried 05]. It is notable that the laser configuration in Fig. 2.11b provides stability

for the φs for long periods whereas the φm phase may change between applications of the

gate within a single quantum algorithm due to lack of interferometric stability. However,

the area circumscribed by the displacements of a closed trajectory is independent of this

phase. What is important is that the relative φm phase between two ions does not change

and that φm does not change during the application of the gate. This puts constraints on

the spectrum of noise of relative beam positions and on DC electrode voltage stability.

We don’t expect fluctuations in φm to cause problems at the 1% error level. However,

these error sources should be examined more closely when attempting gates at the 10−4

error level or below. With these considerations, the interaction picture Hamiltonian for

N ions can be constructed by making the replacement σπ/2−φs → 2Jy/~ in Eq. (2.63)

where Jξ = ~

PN

i

σξi/2 is the ξ

th component of the collective spin operator for the N-ion

string (ξ ∈ {x, y, z}). The N ion interaction picture interaction Hamiltonian becomes:

H0

I ' −2ηΩJy(a

†

e

i[(ωz−δ)t+φm] + ae−i[(ωz−δ)t+φm]

) (2.64)

We note that the Hamiltonian [Eq. (2.64)] has the form

H0

I/~ = iα ̇(t)a

† − iα ̇∗

(t)a (2.65)
66

where  ̇α(t) = 2iηΩ(Jy/~)e

iφmei(ωz−δ)t

. The propagator for Eq. (2.64) can be constructed

by repeated application of the infinitesimal propagator dU(t) = e

−i(H0

I

/~)dt = D(  ̇α(t)dt)

and the relation D(α)D(β) = D(α+β)e

iIm(αβ∗)

[Walls 94] such that the full propagator

has the form

U(t) = Y

M

i=1

D(  ̇α(tM+1−i)dt)

= D(

X

M

i=1

α ̇(ti)dt)

Y

M

i=1

e

iIm(  ̇α(ti)dtPi−1

j=1 α ̇

∗(t0

j

)dt0)

= D(α(t)) exp μi

Z t

0

Im[α

∗

(t

0

)  ̇α(t

0

)]dt0

¶

(2.66)

where

α(t) = Z t

0

α ̇(t

0

)dt0 =

2ηΩJye

iφm

~(ωz − δ)

3

e

i(ωz−δ)t − 1

 ́

(2.67)

is the total displacement at time t. Here, we have broken up the time evolution into M

infinitesimal displacements of size  ̇α(ti)dt for i ∈ {1, · · · , M}. We convert back to the

continuum with the introduction of the integrals in Eqs. (2.66) and (2.67). The integral

in Eq. (2.66) is straight forward, and the resulting propagator is

U(t) = D(α(t)) exp "i

4η

2Ω2J2

y

~

2(ωz − δ)

μ

t −

sin[(ωz − δ)t]

ωz − δ

¶#

. (2.68)

We can verify that the propagator U(t) in Eq. (2.68) satisfies Schr ̈odinger’s equation i~

∂

∂tU(t) = H0

IU(t) by using the alternate definition D(α) = e

−|α|

2/2

e

αa†

e

−α

∗a

[Walls 94] and assuming the form U(t) = D(α(t))e

if(t)

. As an intermediate step, we

calculate

D ̇(α)D

†

(α) = −

1

2

(  ̇αα∗ + αα ̇

∗

) +  ̇αa† − α ̇

∗D(α)aD†

(α)

= −

1

2

(  ̇αα∗ − α ̇

∗α) +  ̇αa† − α ̇∗

a

= −iIm(α

∗α ̇) − iH0

I/~

(2.69)

where we have used D(α)aD†(α) = a−α [Walls 94] and the definition of the Hamiltonian

in Eq. (2.65). From Schr ̈odinger’s equation, we have −iH0

I

/~ = UU ̇ † = DD ̇ † + i

 ̇f.

Substituting Eq. (2.69) and integrating yields f(t) = R t

0

Im[α

∗

(t

0

)  ̇α(t

0

)]dt0. Substituting
67

f(t) into the assumed propagator D(α(t))e

if(t) gives Eq. (2.66), and hence U(t) in

Eqs. (2.66) and (2.68) satisfies Schr ̈odinger’s equation.

The phase factor e

if(t) has a nice geometric interpretation. At time t, we can

close the path taken in phase space by displacing the motional state back to the

starting location without altering the phase of the motional state. The total unitary

transform is then D†(α(t))U(t) = e

if(t)

. We define the massless position and momentum operators x = (a

† + a)/

√

2 and p = i(a

† − a)/

√

2 to convert the phase integral

f(t) = R

P

Im(α

∗dα) = 1/~

R

A

dxdp = A/~ [Leibfried 03]. The phase acquired by the

motional state as it traverses the path P is equal to the area A circumvented in phase

space divided by ~.

For τ = 2π/(ωz−δ), the motional mode in phase space has completed a full circle,

α(τ ) = 0, and U(τ ) = exp(i4η

2Ω2

(Jy/~)

2

τ /(ωz −δ)). We note that U(τ ) is independent

of φm as expected. The gate time τ and the corresponding detuning ωz − δ are chosen

such that 4η

2Ω2

τ /(ωz − δ) = π/2, and the gate operator becomes UMS = e

i(Jy/~)

2π/2

.

We note that the rate at which the motional state is driven around a circle in phase

space (2ηΩ)2/(ωz − δ) is the two-photon Rabi frequency for making transitions from

| ↓↓i|n = 0i → | ↑↑i|n = 0i through the two degenerate states | ↓↑i|n = 1i and

| ↑↓i|n = 1i (see Ref. [Sørensen 99]).

The gate operator UMS = e

i(Jy/~)

2π/2 derived above is valid when driving the gate

on the center of mass (COM) mode because each ion in the chain is driven equally and

in the same direction. However, axial modes other than the COM mode and all axial

modes for ion strings containing different masses do not exhibit this property. For two

ions, the stretch mode where the ions oscillate out of phase, heats more slowly than the

COM mode where the ions oscillate in phase [King 98]. As such, the stretch mode is

preferable for driving two qubit gates. Therefore, we must modify the previous analysis

slightly to take this into account.

The motion of the ions in the Lamb-Dicke limit is best described using the method
68

of normal modes [Goldstein 80]. The amplitude and phase of each ion’s motion in the

crystal for a particular mode is proportional to the eigenvector for that mode. The

mode frequencies and mode vectors have been calculated for small numbers of ions

[James 98, Kielpinski 00] including different masses [Morigi 01]. The eigenvectors for

the normal modes form a complete orthogonal set. Therefore, we can decompose the

displacement vector (formed out of the displacement of each ion at the different ion

positions) driving the ions into components driving each of the modes. Since we isolate

a particular mode on which to drive the gate (by tuning the difference frequencies of

the Raman lasers close to a particular mode sideband), we are concerned with this

particular component of the displacement vector, and we ignore the off-resonant drive

of the other modes. For the mode in question, we are given the normal mode vector

~v normalized such that ||~v|| = N where N is the number of logic ions in the string.

We are also given the equilibrium position of the ions in the string which results in

different phases φ

(i)

m for each i

th ion. For two logic ions, φm can be made the same

for each ion by tuning the ion spacing (see Fig. 2.11). In the previous analysis, we

made the transformation P

i

σ

(i)

y e

iφ(i)

m → 2(Jy/~)e

iφm. The direction of the force driving

the harmonic motion on each ion is proportional to σ

(i)

y e

iφ(i)

m . Here, we modify this

transformation by taking the projection of the driving force vector for the ions onto

the mode vector and let P

i

σ

(i)

y e

iφ(i)

m →

P

i

σ

(i)

y e

iφ(i)

m vi

. For the two-ion stretch mode

with mode vector ~v = (1, −1) and the ions spaced such that φ

(i)

m = φm is the same

for both ions, P

i

σ

(i)

y e

iφ(i)

m vi = e

iφm(σ

(1)

y − σ

(2)

y ), and we can make the replacement

4(Jy/~)

2 → (σ

(1)

y −σ

(2)

y )

2

such that the gate operator becomes UMS = e

i(π/2)(σ

(1)

y −σ

(2)

y )

2

.

This has the consequence that when the ions are spaced by an integer number of effective

wavelengths of the φm phase front (see Fig. 2.11c), the stretch mode is driven when the

two ions are in different states in the σy-basis (i.e. | ↑↓iy or | ↓↑iy). Opposing driving

forces on the two ions couples to the stretch mode whereas common-mode driving forces

do not. The resulting transformation of the UMS gate on the stretch mode in the y-basis
69

is

UMS| ↑↑iy = | ↑↑iy

UMS| ↑↓iy = | ↑↓iye

iπ/2

UMS| ↓↑iy = | ↓↑iye

iπ/2

UMS| ↓↓iy = | ↓↓iy. (2.70)

2.6.2 Phase Gate in z-Basis

The Z-phase gate closely resembles the xy-basis Mølmer-Sørensen gate except for

a change of basis. The frequencies of the two Raman lasers are tuned such that the

difference frequency between the two beams ω ≡ ωb − ωr is near a particular motional

mode frequency ωz. We can safely ignore any qubit state couplings as the effective

field frequency ω is off resonance from the qubit transition frequency ω0. We can

follow an analysis analogous to that in § 2.5.2 to obtain an effective field which couples

different motional states within the same qubit-state manifold. The two Raman laser

fields couple the states |mSi|ni and |mSi|n

0

i to a set of excited states which mediate

an effective coupling between |mSi|ni and |mSi|n

0

i resulting in the matrix elements

[Wineland 98, Wineland 03]:

hmS, n0|HI |mS, ni = ~Ω(mS)e

iφe−iωthn0

|e

iη(a

†+a)

|ni + δS(mS) (2.71)

where

Ω(mS) = EbEr

4~

2

X

i

hmS|~μ · 2ˆr|iihi|~μ · 2ˆb|mSi

∆i

, (2.72)

and

δS(mS) = X

j∈{b,r}

E2

j

4~

2

X

i

|hmS|~μ · 2ˆj |ii|2

∆i

(2.73)

is the Stark shift of qubit state |mSi (mS ∈ {↑, ↓}). The sums in Eqs. (2.72) and (2.73)

are over all intermediate excited states i, and ∆iis the detuning of the Raman laser
70

from the excited state i (∆ À ω, ωz). The phase of the effective field φ = φb − φr is the

differential phase between the two Raman beams.

We desire a strong coupling Ω(mS) and a small differential Stark shift δS(↑) −

δS(↓). Using orthogonal linearly polarized light for the two Raman lasers, the differential Stark shift is of order (ω0/∆)Ω(mS) [Wineland 03], and with a small elliptical

polarization on one of the Raman beams, depending on the choice of the qubit states,

the differential Stark shift can be made to vanish exactly [Wineland 03]. Alternatively,

the differential Stark shift can be absorbed into the definition of the qubit transition

frequency ω0, and we therefore ignore it in the remainder of this treatment.

The excited states are adiabatically eliminated to obtain the effective Hamiltonian

(in the Schr ̈odinger picture) for the motional states in the |mSi subspace

HI (mS) = ~Ω(mS)e

iφe−iωteiη(a

†+a) + h.c.. (2.74)

Transforming to the interaction picture H0

I = e

iHosct/~HIe−iHosct/~ where Hosc = ~ωza†a

yields:

H0

I

(mS) = ~Ω(mS)e

iφe−iωt exp h

iη(a

†

e

iωzt + ae−iωzt

)

i

+ h.c., (2.75)

and upon expanding Eq. (2.75) to lowest order in η, we obtain a Hamiltonian similar

to Eq. (2.63):

H0

I

(mS) ' ~Ω(mS)e

iφe−iωt h

1 + iη(a

†

e

iωzt + ae−iωzt

)

i

+ h.c.

= 2~Ω(mS) cos(ωt − φ)

+ η~Ω(mS)

3

ia†e

i[(ωz−ω)t+φ] − iae−i[(ωz−ω)t+φ]

 ́

+ η~Ω(mS)

3

ia†e

i[(ωz+ω)t−φ] − iae−i[(ωz+ω)t−φ]

 ́

.

(2.76)

There are three differences between the Z-phase gate Hamiltonian [Eq. (2.76)] and the

MS Hamiltonian [Eq. (2.63)]. The first difference lies in the first term. In the MølmerSørensen gate, the first term is an off-resonant drive of the carrier transition; in the

Z-phase gate, the first term acts as a Stark shift modulated at the frequency of the
71

effective field. In both cases, the duration of the gate can be chosen such that this term

has no effect [for the Z-phase (MS) gate, this corresponds to ωt = 2πn (δt = 2πn),

n an integer]. The resonant strength of this error term is ∼ 1/η times the gate drive

strength in both cases. The second difference is that for the MS gate, the gate drive

strength is proportional to ±Ω for states in the ±σy basis (φs = 0 in Eq. (2.63)) and is

proportional to Ω(mS) for the different |mSi states in the Z-phase gate. We see that in

the MS gate, the strength of the gate drive is equal and in opposite directions for the

two qubit states in the ±σy basis. This is not necessarily the case for the Z-phase gate.

The strength Ω(mS) is a function of the qubit states and laser polarization. We desire a

large state dependence to the gate drive strength; that is, we would like |Ω(↑)−Ω(↓)| as

large as possible. The third difference is that there is no φs phase in the Z-phase gate.

That is, the basis on which the displacements in phase space are conditioned is always

the σz basis. Similarly, the φm phase in the MS gate is duplicated by the phase φ +π/2

in the Z-phase gate. In both cases, this phase oscillates in space with a length scale

set by the effective wavelength of the effective field along the trap axis (see Fig. 2.11c).

The last terms in these Hamiltonians are ignored in both cases by the rotating wave

approximation.

Similar to the end of the previous section, we must project the displacement drive

strength vector onto the mode vector ~v. Summing over all ions in the string, we can

write the Hamiltonian [Eq. (2.76)] in the form of Eq. (2.65) (ignoring the first and last

error terms) where

α ̇(m

{i}

S

, t) = ηei(ωz−ω)tX

i

Ω(m

(i)

S

)e

iφivi (2.77)

and viis the component of the mode vector for the i

th ion. The notation m

{i}

S

indicates

the qubit configuration in the σz basis for the set of ions. For the qubit states |↓i = |F =

2, mF = −2i and |↑i = |F = 1, mF = −1i used in the Leibfried et al. [Leibfried 03]

experiment, Ω(↓) = −2Ω(↑) [Wineland 03]. As indicated in the previous section, the
72

mode vector for the two-ion stretch mode is ~v = (1, −1); upon spacing the ions such

that the phase φ is the same for two ions, the strength of the displacement drive for

the different two-qubit configurations becomes  ̇α(↓↑, t) = −α ̇(↑↓, t) = 3ηei(ωz−ω)tΩ(↑)e

iφ

and  ̇α(↓↓, t) =  ̇α(↑↑, t) = 0. Following the analysis of the previous section, we can choose

the detuning of the effective field from the mode frequency ω − ωz such that after the

period τ = 2π/(ωz − ω) we obtain the gate operator UL = e

i(π/2)(σ

(1)

z −σ

(2)

z )

2

. The period

τ and the detuning ω − ωz are calculated using the relation 9η

2Ω(↑)2

τ /(ωz − ω) = π/2.

The duration τ for the stretch mode two-ion Z-phase gate satisfies the relation

η

2

|Ω(↑) − Ω(↓)|

2

τ /(ωz − ω) = π/2 where τ = 2π/(ωz − ω). Substituting for ωz − ω in

favor of τ yields the relation

η|Ω(↑) − Ω(↓)|τ = π. (2.78)

Therefore, the speed of the gate is directly proportional to |Ω(↑) − Ω(↓)|, and as such,

we would like to maximize this rate. Ω(mS) is calculated using Eq. (2.72). We write

the qubit eigenstates as (mS ∈ {↑, ↓})

|mSi = αmS|mJ = −1/2, mI = mF (mS) + 1/2i

+ βmS|mJ = +1/2, mI = mF (mS) − 1/2i

(2.79)

where αmSand βmSare calculated from the Breit-Rabi formula (see § 2.2), and at low

magnetic fields, αmSand βmSare Clebsch-Gordan coefficients. Because the detuning

of the Raman lasers is large compared to the hyperfine splitting of the P1/2 and P3/2

manifolds, we can sum over all excited states in the P1/2 manifold separately from those

in the P3/2 manifold when calculating Ω(mS) in Eq. (2.72). We define the polarizations

according to their left circular (ˆσ

−), right circular (ˆσ+), and linear (ˆπ) components such

that ˆ2l = l+σˆ

+ + l−σˆ− + l0πˆ and l2

+ + l

2

− + l

2

0 = 1 for l ∈ {b, r}. For the P1/2 manifold,
73

ε

r

 = (σ

+ − σ−

ε )/√2 b = (σ

+

 + σ

−

)/√2

B

Figure 2.12: Raman laser configuration for the Z-phase gate. The blue and red Raman

lasers propagate such that the difference in their k-vectors lies along the trap z-axis.

The polarizations are both linear and orthogonal to each other and to the magnetic field

B~ .

we obtain

X

i∈P1/2

hmS|~μ · 2ˆr|iihi|~μ · 2ˆb|mSi

∆i

=

|μ|

2

∆1/2

1⁄2

|αmS|

2

μ

2

3

b+r

∗

+ +

1

3

b0r

∗

0

¶

+ |βmS|

2

μ

2

3

b−r

∗

− +

1

3

b0r

∗

0

¶ 3⁄4

=

|μ|

2

3∆1/2

£

(b0r

∗

0 + b+r

∗

+ + b−r

∗

−) + ¡

|αmS|

2 − |βmS

|

2

¢

(b+r

∗

+ − b−r

∗

−)

¤

, (2.80)

and similarly for the P3/2 manifold,

X

i∈P3/2

hmS~μ · 2ˆr|iihi|~μ · 2ˆb|mSi

∆i

=

|μ|

2

3∆3/2

1⁄2

2(b0r

∗

0 + b+r

∗

+ + b−r

∗

−)

−

¡

|αmS|

2 − |βmS

|

2

¢

(b+r

∗

+ − b−r

∗

−)

3⁄4

(2.81)

where μ = hP3/2, F = 3, mF = 3|~μ · σˆ

+|S1/2

, F = 2, mF = 2i is the cycling transition

electric dipole moment, and ∆1/2(∆3/2) is the detuning of the Raman lasers from the

P1/2(P3/2) state.

A few observations on Eqs. (2.80) and (2.81) are in order. Firstly, the state

dependence enters two places: (1) the term |αmS

|

2 − |βmS

|

2 and (2) the detunings ∆1/2

and ∆3/2. The relative detuning for the two qubit states is ω0/∆j for j ∈ { 1

2

,

3

2

} where

the qubit frequency is ∼1 GHz and the detunings are of order ∼100 GHz. Therefore, the
74

different detunings for the different qubit eigenstates only modify the gate speed slightly

if |αmS|

2 − |βmS

|

2 differs by order unity for the qubit eigenstates. Secondly, the gate

speed is optimized when b+r

∗

+ − b−r

∗

− is maximized. We are constrained to the use of

linearly polarized light to minimize the differential Stark shift [Wineland 03]. However,

we can maximize b+r

∗

+ − b−r

∗

− while maintaining linearly polarized light by letting

b− = b+ = 1/

√

2 and r− = −r+ = 1/

√

2. This polarization configuration is obtained

with the laser beam geometry in Fig. 2.12. The third observation is that for fast gates,

we require qubit eigenstates such that

 ̄

 ̄

|α↑|

2 − |α↓|2

 ̄

 ̄

is large11 . Unfortunately, field

independent qubits have the property that α↑ ' −α↓ + O(10−4) (see § 2.2); therefore,

direct application of the Z-phase gate to field independent qubits is inefficient. In order

to efficiently use the Z-phase gate on field independent qubits, we must temporarily

transfer out of the field independent subspace, perform the gate, and transfer back to

the field independent subspace. If errors induced by magnetic field noise are small over

the duration of these steps, this may be a viable option. Alternatively, we could perform

the Mølmer-Sørensen gate directly. Because |α↑|

2 − |α↓|2 ∼ 10−4

for field independent

qubits, we might consider using the difference in detuning to drive the gate, particularly

for ions where the hyperfine splitting is large (such as Cd+ where the hyperfine splitting

is 14.7 GHz). However, it is desirable to have large detunings from the excited states

to reduce the error due to Raman inelastic spontaneous scattering (see [Ozeri 05] and

Ch. 5). Therefore, given the availability of other equivalent gates (namely the MølmerSørensen gate), the Z-phase gate is not the gate of choice for field independent qubits.

Using the approximation ω0/∆j ¿ 1 for j ∈ { 1

2

,

3

2

}, the differential displacement

drive between the qubit eigenstates is

Ω(↑) − Ω(↓) = −

EbEr|μ|

2

6~

2

∆F

∆1/2(∆1/2 − ∆F )

(b+r

∗

+ − b−r

∗

−)(|α↑|

2 − |α↓|2

) (2.82)

where ∆F is the fine structure splitting between P3/2 and P1/2. The duration of the

11 By normalization, |αmS

|

2 − |βmS

|

2 = 2|αmS

|

2 − 1; therefore, the speed of the gate is proportional

to

 ̨

 ̨|α↑|

2 − |α↓|2

 ̨

 ̨.
75

stretch mode two-ion Z-phase gate satisfies Eq. (2.78).

2.6.3 Entanglement Verification

Both the Z-phase gate and the Mølmer-Søresen gate have been used to generate

maximally entangled Schr ̈odinger “cat” states of N ions of the form |Ψi = | ↑ · · · ↑

i+e

iφ0

| ↓ · · · ↓i [Sackett 00, Leibfried 03, Leibfried 05]. In general, multi-qubit quantum

states are characterized by quantum state tomography [Haffner 05]. However, for the

case of “cat” states, the target density matrix ρcat = |ΨihΨ| has only four non-zero

elements, and the fidelity F ≡ hΨ|ρ|Ψi =

1

2

(P(↑) + P(↓)) + ρ(↑↓) of an arbitrary density

matrix ρ is characterized by the three experimentally measurable quantities: P(↑) —

the population of the | ↑ · · · ↑i state, P(↓) — the population of the | ↓ · · · ↓i state,

and the far off-diagonal element of the density matrix ρ(↑↓) ≡ ρ↑···↑,↓···↓ [Sackett 00]. In

this section, I discuss how these three parameters are measured following the method of

Sackett et al. [Sackett 00]. Fidelities greater than 0.5 are sufficient to prove N-particle

entanglement [Sackett 00].

The populations P(↑) and P(↓) are determined by state dependent resonance fluorescence (see § 2.4 and Appendix C). After preparing the “cat” state and measuring all

qubits in the z-basis, we obtain a histogram of collected photon counts upon repeated

preparations and measurements. This histogram is fit to a sum of N + 1 reference histograms for 0-ions bright, 1-ion bright, up to N ions bright to obtain the probability that

N and 0 ions are bright and hence the populations P(↑) and P(↓)respectively. Measurement of the ρ(↑↓)

requires the probabilities Pj that j ions are bright for j ∈ {0, 1, · · · , N},

and these are extracted using the same histogram analysis.

The coherence term ρ(↑↓)is determined by performing an additional π/2 interrogation pulse with a variable phase φ to all the qubits prior to measuring Pj . That is,

we perform the unitary rotation U(φ) = √

1

2

PN

i=1 h

1 − i

3

σ

(i)

x cos φ + σ

(i)

y sin φ

 ́i trans-
76

forming the state into

|Ψ1i = U(φ)|Ψi = 2−(N+1)/2

£

(|0i − ieiφ|1i)· · ·(|0i − ieiφ|1i)

+ e

iφ0

(−ie−iφ)

N (|0i + ieiφ|1i)· · ·(|0i + ieiφ|1i)

¤

=

X

2N

k=0

ck|ki

(2.83)

where

ck = 2−(N+1)/2(−ieiφ)

n1(k)

h

1 + (−1)n1(k)e

i(φ0−Nφ−Nπ/2)i

(2.84)

|ck|

2 = 2−N [1 + (−1)n1(k)

cos(φ0 − Nφ − Nπ/2)]. (2.85)

We have made the associations | ↑i → |0i, | ↓i → |1i, and the binary number k labels

the state |ki. The function n1(k) is the number of 1’s in the binary number k. From

histogram analysis (see Appendix C), we extract Pj , the probability of measuring j ions

in the |1i state. Using Eq. (2.85), Pj for the N-ion “cat” state is

Pj =

X

{k:n1(k)=j}

|ck|

2 = 2−N

μ

N

j

¶

£

1 + (−1)jcos(φ0 − Nφ − Nπ/2)¤. (2.86)

From Pj we can construct the parity12

Π(φ) ≡

X

N

j=0

(−1)jPj = cos(φ0 − Nφ − Nπ/2). (2.87)

We measure the parity as a function of φ which oscillates N times in 2π, and the

amplitude of this oscillation is in fact 2ρ(↑↓)[Bollinger 96].

An alternative method for measuring the coherence term ρ(↑↓) other than parity

oscillations is to transform the cat state into a superposition of | ↑ · · · ↑i and | ↓ · · · ↓i

with respective populations that are a function of the phase φ of the transformation

pulse [Leibfried 04, Leibfried 05]. The populations of | ↑ · · · ↑i and | ↓ · · · ↓i oscillate

back and forth N times in 2π as a function of φ. Because the fluorescence of the ions

is proportional to the number of ions in the ↑ state, the fluorescence data can be used

12 The factors 2−N PN

j=0 `N

j

 ́

(−1)j = 0 and 2−N PN

j=0 `N

j

 ́

= 1 by the binomial theorem (1 + x)

N = PN

j=0 `N

j

 ́

x

j

for x = −1 and x = 1.
77

directly without the need to fit the histograms to obtain the populations of 0 up to N

ions bright. This method has been used to verify entanglement for N-ion Schr ̈odinger

cat states up to N = 6 ions [Leibfried 05]. Fluorescence oscillations as a function of

the phase φ of the decoding pulse are shown in Fig. 2.13 [Leibfried 05]. We observe

that fidelity drops considerably as the number of ions is increased verifying the need

for higher-fidelity operations and/or quantum error correction and hence very low error

rates for quantum memory (see Ch. 4) and quantum gates (see Ch. 5).
78

N = 1

(reference)

N = 3

F = 0.89(3)

N = 4

F = 0.76(1)

N = 5

F = 0.60(2)

N = 3

F = 0.509(4)

Figure 2.13: Fluorescence oscillation for “cat” states up to six ions [Leibfried 05]. In

the figure, N is the number of ions entangled, and F is the fidelity (see text).
Chapter 3

Apparatus

3.1 Ion Traps

In order to perform quantum information processing experiments, our qubits

must be well isolated from the environment. This is accomplished by trapping ion

qubits in ultra high vacuum using linear Paul traps. In this section, I will describe

the theory of linear Paul traps followed by descriptions of the trap apparatuses used in

the experiments. Because the trapping electric fields couple strongly to the ion’s net

charge and only weakly to the internal states, the ion’s internal states are left nearly

unperturbed and well isolated from the environment leaving them available for use as

robust quantum information carriers.

3.1.1 Ion Trap Theory

Due to their net charge, ions can be confined in free space with an appropriate

arrangement of electric fields. It is interesting to note that static electric fields cannot

confine charged particles by Earnshaw’s theorem. A simple explanation of why this is the

case is contained in Gauss’s law which states that the divergence of the electric field at

a point in space is proportional to the charge density at that point. Trapping of positive

(negative) charge from static external fields requires negative (positive) divergence of

the electric field, and in empty space, the charge density for the external field is zero.

From a slightly different perspective, we can consider the potential energy of a positive
80

charge in the static quadratic electric potential Φ(x, y, z) = Φ0(Ax2 + By2 + Cz2). By

Laplace’s equation ∇2Φ = 0, we must have A + B + C = 0; therefore, although it may

be possible to trap in two directions (say A, B > 0 for Q > 0), the third direction will

be anti-trapping (C = −A − B < 0).

Although static electric fields cannot confine charge, time-varying fields can.

There are a number of works describing the theory of charged particle traps including two books [Major 05, Ghosh 95]. I refer the reader interested in primary sources to

the references contained in [Major 05]. The traps used in the research of this thesis are

linear RF Paul traps. The linear Paul trap is a Paul mass filter plugged at the ends

with static potentials. A schematic diagram of a linear Paul trap [Raizen 93] similar to

those used in this thesis is shown in Fig. 3.1. A potential of the form V0 cos ΩT t + Ur is

applied to the white diagonally opposing RF electrodes placed a distance R away from

the trap z axis. The segmented control electrodes, also placed a distance R from the

trap axis, are held at RF ground. This produces a potential near the axis of the form

ΦRF '

1

2

κRF (V0 cos ΩT t + Ur)

μ

1 +

x

2 − y2

R2

¶

(3.1)

where κRF is a geometry factor on the order of unity resulting from the fact that the

rods are not equipotential surfaces for the electric quadrupole field. As I will show later,

this potential provides radial confinement of the ions. The ions are confined along the

axis by applying a static potential U0 to the end cap electrodes while holding the middle

segmented electrodes at DC ground. Taylor expanding the static potential along the

axis results in a static potential of the form:

Φs '

κDCU0

Z

2

0

·

z

2 −

1

2

2DCx

2 −

1

2

(2 − 2DC)y

2

 ̧

(3.2)

where Z0 is the length of the middle segmented electrode and κDC and 2DC are geometry

factors on the order of unity similar to κRF . The potentials in Eqs. (3.1) and (3.2) are

the lowest order terms in the multipole expansion of the potentials about the axis where
81

RF electrode

High DC potential

control electrode

Low DC voltage

control electrode

Positive ion

End view

y x

z

Figure 3.1: Schematic diagram of a linear RF Paul trap. An oscillating potential is

applied to the white RF electrodes producing an approximate quadrupole electric field

pattern in the xy plane (see end view). This provides radial confinement of the ions

to the RF electric field null along the z axis. Axial confinement is provided by the

segmented control electrodes. Applying higher DC potential to the end caps relative to

the middle DC electrode provides a potential minimum along the axis for the ions. If

axial confinement is weaker than radial confinement, multiple ions will configure into a

linear chain along the axis.
82

the ions will be trapped. For laser cooled ions in typical traps with strengths of a few

MHz, the extent of the ions’ motion about their equilibrium positions is on the order of

a few tens of nanometers. Given trap dimensions on the order of R ∼ 100 μm, higher

order terms in the multipole expansion of the potential are negligible.

The equations of motion for an ion with mass m and charge q are given by

x ̈j = −

q

m

∂

∂xj

(ΦRF + Φs) for j ∈ {x, y, z}. Motion in the z-direction is simple harmonic

motion with frequency ωz =

p

2qκDCU0/(mZ2

0

). In the x and y directions, the equations

of motion are homogenous differential equations of the Mathieu type (i ∈ {x, y}):

d

2xi

dτ 2

+ (ai + 2qi cos 2τ )xi = 0 (3.3)

where τ = ΩT t/2 is a dimensionless time, ax = −

4q

mΩ2

T

3

κDC U0

Z2

0

2DC −

κRF Ur

R2

 ́

, ay =

−

4q

mΩ2

T

3

κDC U0

Z2

0

(2 − 2DC) + κRF Ur

R2

 ́

quantify the static potentials, and qx = −qy =

2qκRF V0

mΩ2

T R2

is a measure of the strength of the RF potential. The Mathieu equation can be solved

using Floquet solutions, and for an in-depth treatment in the context of ion traps, I

refer the reader to [Major 05]. However, in practice, we work in the regime where

|ai| < q2

i ¿ 1, and we can invoke the pseudopotential approximation.

Under the conditions |ai| < q2

i ¿ 1, we can separate the motion of a single

coordinate xiinto two parts xi = u + 2 where u À 2 is a slowly varying function

representing the slow time scale position of the ion and 2 is a small amplitude high

frequency motion. We further assume that d

2

2

dτ 2 À d

2u

dτ 2 which we will verify later. (In

what follows, we drop the subscript i on the parameters ai, qi, and xi to simplify the

notation). Using these approximations, we can rewrite the Mathieu equation as

d

2

2

dτ 2

+ 2q cos(2τ )u = 0

⇒ 2 =

q

2

cos(2τ )u. (3.4)

We now back substitute 2 in Eq. (3.4) into the Mathieu equation (3.3) to obtain the

equation

d

2u

dτ 2

+ (a + q

2

cos22τ )u +

1

2

aq cos(2τ )u = 0. (3.5)
83

Averaging Eq. (3.5) over one period of the RF cycle yields harmonic motion in u with

frequency β =

p

a + q

2/2. The resulting xi coordinate of the ion’s position has then

the approximate form:

xi = u + 2 = Ai cos(βi

ΩT

2

t + φi)(1 + qi

2

cos ΩT t) (3.6)

where Ai and φi are determined by initial conditions and βi ≡

q

ai + q

2

i

/2 (i ∈ {x, y}).

We observe that under the conditions a < q2 ¿ 1, the frequency βΩT /2 ¿ ΩT and

hence d

2

2

dτ 2 À d

2u

dτ 2 is satisfied. Furthermore, because q ¿ 1, the assumption u À 2 is also

verified. The large amplitude slower motion of the ion at frequency βiΩT /2 is termed the

“secular” motion, and the smaller amplitude faster motion at the RF drive frequency

ΩT is termed the “micromotion”. The micromotion amplitude is proportional to the

instantaneous secular position.

Ignoring the micromotion, we see that the ion’s secular motion in the xy directions

can be equivalently determined by a harmonic pseudopotential Φp of the form

qΦp =

1

2

m(ω

2

xx

2 + ω2

y

y

2

) (3.7)

where ωi =

ΩT

2

q

ai +

q

2

i

2

for i ∈ {x, y}. When the radial confinement is much stronger

than the axial confinement, the secular frequency is approximated by ωi '

|qi|

2

√

2

ΩT =

√

qκRF V0

2mΩT R2

.

The approximate solutions (3.6) are the lowest order terms of the more general

Floquet solution (see [Major 05]). The pseudopotential approximation gives insight

into the frequency of secular motion, and the motion is stable so long as 0 < βi < 1 for

i ∈ {x, y} [Major 05]. In the pseudopotential approximation, βi =

q

ai + q

2

i

/2; however,

this is an approximation valid only in the small a, q2limit. Therefore, higher order terms

in the expansion of βi become important when examining the stability diagram near
84

βi . 1. Taken from [Major 05], βiis given by the continued fraction expression:

β

2

i = ai + fi(βi) + fi(−βi)

fi(βi) = q

2

i

(2 + βi)

2 − ai −

q

2

i

(4+βi)

2−ai−···

. (3.8)

One can plot stability diagrams in the a−q plane for which motion in both the x and y

directions is stable simultaneously. In practice, however, having the axial trap frequency

ωz less than the radial trap frequencies ωx and ωy and having the radial trap frequencies

much less than ΩT /2 results in stable ion motion. Furthermore, under these conditions,

the micromotion amplitude is small (small qi), and chains of ions will align along the

axis of the trap (ωz ¿ ωx,y).

Examining Eq. (3.6), the radial motion (say x motion) of the ion is slow secular

motion at a frequency ωx modulated at the RF drive frequency ΩT . Consequently, the

motional frequency spectrum contains frequencies ωx and ΩT ± ωx. The higher order

terms in the Floquet solution have frequencies at nΩT ± ωx, n an integer. Therefore,

noise at the RF drive frequency will not drive the ions motion. However, electrical

noise at ΩT ±ωx will parametrically drive the ions motion. The situation changes when

a stray electric field in the radial direction exists. The ion finds a new equilibrium

position where the force from the stray field is balanced by the force induced by the

pseudopotential. The motion of the ion in the x direction is modified to the following:

x = [Ax cos(ωxt + φx) + ∆xstray](1 + qi

2

cos ΩT t) (3.9)

where ∆xstray =

qEstray

x

mω2

x

is the displacement induced by the stray field. Here, we see that

the ion has motion at the RF drive frequency. Fortunately, this motion is separable

from the secular motion, and coupling of the RF drive energy into the secular motion

should not occur except when mediated by collisions.

Micromotion reduces the efficiency of Doppler cooling and state dependent resonance fluorescence. Consider resonance fluorescence of an ion with micromotion momentarily. In the ion’s reference frame, the resonant laser will appear to be modulated at
85

the RF drive frequency ΩT and will consequently have sidebands at ΩT due the Doppler

effect. For typical RF drive frequencies near 100 MHz, these sidebands are outside of

the natural line width of the cycling transition (19.4 MHz) and negligibly contribute to

the scattering rate. Therefore, the intensity on resonance in the ion’s reference frame

is reduced limiting the efficiency of resonance fluorescence. Furthermore, when using

state-dependent resonance fluorescence for qubit measurement (a transition involving

one qubit eigenstate is on-resonance hence the state is “bright”, whereas transitions

involving the other qubit eigenstate are off-resonance hence the state is “dark”), some

of the sidebands are closer to resonance to optical pumping transitions out of the dark

state increasing the probability of repump error (see § 2.4). Micromotion can be minimized by applying external static fields to compensate for stray fields. Because the

resonant scattering rate decreases with micromotion, we can use measurements of the

scattering rate to minimize micromotion.

Another novel technique for minimizing micromotion exists when trapping two

ions of different mass [Barrett 03]. Because the pseudopotential is mass dependent, the

displacement from the RF null along the axis is also mass dependent. In the presence

of a stray electric fields, a two-ion crystal of different masses will tilt with respect to the

trap z-axis. This couples the radial and axial modes, and the axial mode spectrum is

modified. By observing the axial mode spectrum while applying compensating electric

fields, the stray fields and the micromotion can be eliminated.

The analysis of the ion motion so far has been for a classical charged particle.

The quantum dynamics can be included in the pseudopotential approximation by simply

transforming to the quantum harmonic oscillator with Hamiltonian Hosc = (a

†a+

1

2

)~ωi

for i ∈ {x, y, z} where a and a

† are the quantum harmonic oscillator annihilation and

creation operators respectively. The ion’s position and momentum operators for a particular mode i with frequency ω are expressed as x = x0(a

† + a) and p = p0(a† − a)

where x0 ≡

p

~/(2mω) and p0 ≡ i

p

~mω/2. In the z-direction, the potential is har-
86

monic to a very good approximation, and transforming the quantum harmonic oscillator

is well substantiated. However, transforming to the quantum harmonic oscillator for the

ion’s radial motion which classically is described by Floquet solutions is less clear. This

problem has been extensively studied theoretically [Major 05, Leibfried 03, Ghosh 95,

Brown 91, Stenholm 92, Gheorghe 92, Schrade 95, Nieto 00, Gheorghe 00, Bardroff 96].

In the small a and q limit (a here is the parameter in the Mathieu equation and not

the harmonic oscillator annihilation operator), Bardroff [Bardroff 96] and Kielpinski

[Kielpinski 01] show that the ion’s motion can be described by a quantum harmonic

oscillator with oscillating squeezing at the RF drive frequency. In practice, this “breathing” motion changes somewhat an ion’s interaction with laser beam electric fields; this

effect can be compensated in the determination of Rabi rates, etc.

In the context of quantum information processing, we isolate a single axial mode

along the z-axis which is well described by the quantum harmonic oscillator. The motion

is only involved during two-qubit gates (see § 2.6) which involves coherently displacing

the ion wave packet around in phase space conditioned on the ions’ internal states. After

the completion of the gate, the motional mode’s wave packet is restored to its original

location in phase space. Because the z-axis motion is not affected by the RF motion,

we neglect the full quantum dynamics of the Mathieu equation. The interested reader

is referred to the references.

3.1.2 Gold Plated Alumina Wafer Trap

The ion trap used for the experiments of this thesis is shown in Fig. 3.2. It was

constructed from two identical gold plated alumina wafers. The lower wafer is rotated

about the trap axis by 180◦such that the RF electrodes reside at two opposite corners

of the four-rod quadrupole configuration in Fig. 3.1. The wafers are spaced by 200 μm

thick alumina wafers. A “bias” board with a 0.6 × 4.4 mm slot to allow laser beam

access is placed 400 μm from the back trap wafer using alumina spacers. Application of
87

a DC potential to the bias board gives the experimenter control over the principle axes

of the radial modes. To eliminate Be plating of the experimental trap zones, a “mask”

board with two holes cut in it to allow laser beam access to all trapping regions and

neutral Be access to the load trap is placed 1.5 mm from the bias board using alumina

spacers. The eight control electrodes provide six different trapping regions, and ions

can be shuttled between these regions by application of time-varying potentials to the

control electrodes. The larger volume trap residing over electrode 2 (trap 2) is used

for loading and hence is termed the load trap. Experiments where shuttling was not

required typically used the trap residing over electrode 5. Separation of ions occurs over

electrode 6 as its small width of 100 μm provided the narrowest potential wedge for

separation, and thereby minimized heating of the ions during the separation process.

The RF filter board provides low pass filtering for the control electrodes from

their source voltages using a first order RC circuit. The 820 pF capacitors also serve as

RF shorting capacitors to ground, and consequently, the control electrodes serve as an

RF ground.

The RF potential is provided by a quarter wave cylindrical unterminated transmission line which acts as a step-up transformer for the RF input. The RF input is

coupled into the resonator inductively [Jefferts 95]. With a resonator Q in the few hundreds, we obtain RF potentials of a few hundred volts with input RF powers of a few

watts. This provides radial confinement of the ions with trap frequencies ∼ 15 MHz.

The control electrode potentials are provided by digital to analog converters in the few

volts range which provides axial confinement of the ions with trap frequencies in the

3—5 MHz range.

The trap is loaded from a neutral 9Be source via electron-bombardment. Neutral

9Be wire is wrapped around a tungsten filament which when heated sublimates neutral

9Be. The 9Be source, termed the “Be oven”, is resistively heated with ∼ 1 A of electrical

current. The neutral Be is ionized in the trapping region by electron bombardment with
88

~ 1 cm

RF filter board

segmented

linear trapping

region

8 76 5 4 3 2 1

RF

top trap wafer bottom trap wafer

bias board

mask board

sideview

(bottom wafer view)

Figure 3.2: Gold plated alumina wafer trap. The trap has 8 control electrodes and 1

RF electrode per wafer. The control electrodes are numbered from left to right from 1

to 8. Trapping over electrode 2 (trap 2) is termed the load trap, and trap 5 is typically

termed the experiment trap. The width of the (j

th) control electrode is: (2) 500 μm,

(3) 500 μm, (4) 360 μm, (5) 200 μm, (6) 100 μm, (7) 200 μm. The spacing between the

nearby edges of adjacent electrodes is 20 μm.
89

the “electron gun.” The electron gun is constructed out of a thorated tungsten filament

which is biased at −80 V with respect to ground. Applying ∼ 1 A electrical current to

the electron gun resistively heats the filament until the thermal energy of the electrons

surpasses the work function and ejects electrons from the surface. Once free from the

filament, the electrons are accelerated through the 80 V potential towards the trapping

region bombarding the neutral Be atoms creating ions. When ions are created within

the capture range of the trap, they become trapped. Upon Doppler cooling, the ions

form cold crystals which are easily seen with the imaging system.

3.2 Magnetic Field Coils

A schematic of the trap apparatus, nearby optics, and the magnetic field coils

are shown in Fig. 3.3. The magnetic field coils shown in Fig. 3.3 provide the 0.01194 T

magnetic field required for field-independent qubits (see § 2.2 and Ch. 4). Nine turns of

hollow 1/8 inch outer diameter copper refrigerator tubing are wound around a cylindrical

aluminum coil form of 3.725 inch radius. The planes of two coils of this type are placed

0.5 inches away from the ion position and placed symmetrically about the ion. Because

the coils require ∼ 100 A of current to generate the 0.01194 T magnetic field generating

∼ 500 W of heat, the coils were water cooled. The turns of the coils were insulated from

each other by wrapping the copper tubing in heat shrink tubing prior to winding it on

the mount.

3.3 Optics

There are three laser systems used for quantum information processing with 9Be+.

These generate light for the three beam lines: (1) the “blue Doppler (BD) line”, (2)

the “red Doppler (RD) line”, and (3) the “Raman line”. The wavelength of these lasers

is 313 nm. We generate the 313 nm radiation using three ring dye lasers at 626 nm

and frequency double the light to 313 nm using resonant BBO doubling cavities. The
90

RF resonator

quartz envelope

ion

ion trap

magnetic field coils

f/1 imaging lense

ion pump,

titanium sublimation

 pump,

ion gauge

f = 15 cm

λ

Gla

/4

pol

n-laser

arizer

BD/RD experiment beams

f = 15 cm

λ/4

f = 15 cm

λ/4

RCO

λ/2

R90

f = 15 cm

λ/4

BD/RD load beams

B

trap axis

(σ+ − σ−

)/√2

π

(σ+ + σ−)/√2

polarization

definitions

Figure 3.3: Optical system near the ion trap. Ion fluorescence is collected by the

f /1 imaging lens. Magnetic field coils provide the 0.01194 T field required for fieldindependent qubits. The Raman beams RCO and R90 propagate perpendicular and

anti-parallel to the magnetic field such that the difference ~k-vector ∆~k lies along the trap

axis. The BD and RD beam lines propagate along the magnetic field and are polarized

σˆ+. RCO is circularly polarized such that its polarization is ˆ2CO =

1

2

(ˆσ+ − σˆ−) + √

i

2

πˆ.

The R90 beam is horizontally linearly polarized, namely ˆ290 = √

1

2

(ˆσ+ + ˆσ−). The BD

load beam is circularly polarized.
91

frequencies of the three lasers are depicted in Fig. 3.4. The “BD line” consists of the

“BD” and “BD detune” beams which are near resonant with the S1/2 ↔ P3/2transition

frequency (D2 line, wavelength 313.13 nm). The “RD line” consists of the “RD” and

“repumper” beams which are near resonant with the S1/2 ↔ P1/2transition frequency

(D1 line, wavelength 313.20 nm). The “Raman line” beams are off-resonant from both

the S1/2 ↔ P3/2 and S1/2 ↔ P1/2transition frequencies and typically are detuned

∼ 80 GHz blue of the D1 line. The detuning of the Raman laser from the P1/2 and

P3/2levels affects the speed of gate operations as well as the probability of spontaneous

scattering (see Ch. 5). Because the BD and RD lines are resonant with the D2 and D1

lines in 9Be+, their absolute frequencies are stabilized to molecular absorption lines in I2

using saturated absorption spectroscopy [Preston 96]. The Raman laser, however, is not

stabilized to an atomic reference as its relatively small slow frequency drift1 only affects

the detuning of the laser and not the difference frequency between the Raman beam

pairs. All of the dye lasers are frequency stabilized to external optical reference cavities

by measuring polarization changes of the laser reflected from the cavity [Hansch 80].

In addition, the resonant BBO doubling cavities are locked to resonance with the laser

frequencies using the same technique [Hansch 80].

3.3.1 Imaging System

A schematic of the trap apparatus and nearby optics is shown in Fig. 3.3. During

qubit measurement (see § 2.4), ion fluorescence is collected through an f /1 imaging

lens and projected onto either a photon counting camera or a photo-multiplier tube

as selected by an electronic flipper mirror. When imaged on the camera, the ion fluorescence is shown as a picture on an oscilloscope. When imaged on the PMT, the

photons are counted by the FPGA-based experiment controller (see § 3.4). The optics

of the imaging system consists of an f /1 objective lens placed approximately 40 mm

1 The Raman laser frequency is stable to within a few hundred MHz over many hours.
92

F = 2

F = 1

mF = –2

mF = –1

102.8

86.7 76.4 69.1

86.5

76.2

1207.495,843

F = 2

F = 1

mF = –2

mF = –1

39.9

28.5 23.2 20.1

28.2

23.1

261.8

2P3/2

197 GHz

2P1/2

313.20 nm

2S1/2

mJ = 3/2

mJ = 1/2

mJ = –1/2

mJ = –3/2

mF = –3

mF = +3

223.4

BD d

etun

e

BD

RD

repumper

blue raman (

π)

red raman (

π)

blue raman (σ+)

red raman (σ−)

~80 GHz

400 MHz

Figure 3.4: Laser frequencies and 9Be+ levels. Numbers are in units of MHz except

where marked.
93

from the ion which provides ∼ 5× magnification. The objective images the ion at the

location of a 600 μm diameter aperture after which a second-stage lens provides another

∼ 25× magnification of the first image. The total magnification of the imaging system

is ∼ 125. The 600 μm aperture at the location of the first image reduces stray light

since, to a high degree, only light originating from the object plane passes through the

aperture. In addition, another iris is used as a second aperture at the second image

near the PMT/camera to mask stray light other than the ion signal, particularly light

scattered from nearby ion trap electrode surfaces. With this system, we typically could

achieve ion fluorescence signal to background ratios of over 100.

3.3.2 Blue/Red Doppler Beam Lines

The beam lines enter the ion trapping region through four optical ports in the

trap apparatus (see Fig. 3.3). The BD/RD experiment beams propagate along the

magnetic field direction and are circularly polarized ˆσ+. This allows BD to perform

Doppler cooling and state-dependent resonance fluorescence on the cycling transition

|

2S1/2

, F = 2, mF = 2i ↔ |2P3/2, F = 3, mF = 3i. The BD laser power is set such

that the intensity at the ion is half of the saturation intensity. With a beam waist of

∼ 30 μm, the BD laser power is ∼ 0.5 μW. The BD detune beam is a higher power

laser beam (∼ 500 μW) tuned to the red of the BD beam by ∼ 400 MHz. This provides

efficient Doppler cooling of very hot ions whereas the relatively small detuning and

intensity of the BD beam provides little cooling for hot ions. The BD detune beam

is also used for loading for the same reason. During the loading process, the ions are

created via electron bombardment of hot neutral atoms, and a red-detuned high power

laser beam assists in cooling these ions considerably. For this reason, the BD detuned

beam is detuned by another 500–1000 MHz to the red during loading. Because the trap’s

loading region and experimental region reside in different locations, we found it useful

to send the BD detuned beam through another optical port into the loading region such
94

that the experiment beam and loading beam did not have to be moved every time we

needed to load another ion (see “BD/RD load beams” in Fig. 3.3). The load beam is

circularly polarized. During the loading process, the magnetic field is turned off such

that the load beam’s ~k-vector provides the quantization axis. This establishes a cycling

transition when the load beam is circularly polarized.

The BD beam line is shown in Fig. 3.5. The BD dye laser at 626 nm is doubled in

a resonant BBO doubling cavity where the UV light exits vertically polarized. The beam

is split using the “BD split” AOM establishing the separate “BD” and “BD detune”

beams of the BD beam line. The +1st order of the BD split AOM is sent to the “BD

SW” (BD switch) double-pass AOM after which it is recombined with the BD detuned

beam at the “BD detuned SW” (BD detuned switch) AOM. The BD and RD lines

are then combined on a 50/50 beam splitter prior to the combined BD/RD beam lines

being split into the “BD/RD load beams” and the “BD/RD experiment beams” which

enter the trapping regions from the lower two optical ports in Fig. 3.3. This optical

configuration allows for a frequency scannable resonant BD beam and a non-frequency

scannable 400 MHz red detuned BD detune beam. The double-pass BD SW AOM

allows for the BD beam to have its frequency changed during a single experiment from

on resonance for state-dependent resonance fluorescence (see § 2.4) to red detuned by

half of the excited state line width for optimum Doppler cooling. Furthermore, the

optical system allows for the BD and BD detuned beams to be applied individually or

together independently. Alternate optical configurations exist. For example, the BD

detuned SW could be rotated such that the −1

st order was used rather than the +1st

order. This would create an 800 MHz frequency difference between the resonant BD

beam and the BD detuned beam which may help even more in cooling very hot ions

after collisions with thermal background gas. Furthermore, because the optical power

of the BD beam is a few orders of magnitude lower than the BD detuned beam, the

BD split AOM could be replaced with a 90/10 beam splitter or a piece of glass with 4%
95

vert. pol.

BD split

+200

BD detune

SW

+200

BD SW

x2

+200

50/50

RD, Repumper

0

+200

+600

BD

+20

+60

0 BD detune

0 BD

λ/2

PBS

BD, RD

load beam line

BD, RD

experiment beam line

BD doubler

BBO

Figure 3.5: Blue Doppler (BD) beam line. The two beams in the BD beam line (“BD”

and “BD detune”) are generated with three AOMs. The “BD split” AOM sends light to

the “BD SW” double pass AOM setup. The BD beam is combined with the BD detune

beam on the “BD detune SW” AOM forming the “BD beam line”. The beam line is

combined with the Red Doppler (RD) beam line prior to being sent to the trap at two

different optical ports (see Fig. 3.3).
96

reflectivity. Combining these two alterations would establish a BD detuned beam with

a frequency 600 MHz red of the BD beam frequency. Lastly, we found in the robust

quantum memory experiment (Ch. 4) that BD detuned leakage light caused a systematic

Stark shift of the qubit transition. Optical shutters were used for the long-coherence

time measurement; however, the BD detuned SW AOM could be double-passed to

reduce scattered light. For long experiments such as measuring a 15 second coherence

time, optical shutters are appropriate. However, for quantum information processing

experiments where the entire experiment may last only a few milliseconds, the long

delays and slow repetition rates of optical shutters are a limiting factor, and configuring

the BD detune SW as a double-pass AOM may be advantageous. Removing the BD

split AOM and converting the BD detune SW into a double-pass using the −1

st order

creates two beams separated in frequency by 800 MHz.

The RD beam line optical setup is shown in Fig. 3.6. The two beams of the

RD beam line are the “RD” beam and the “repumper” beam. These two beams are

separated in frequency by the |

2S1/2

, F = 2, mF = 1i ↔ |2S1/2, F = 1, mF = 1i

transition frequency of 1120 MHz. The RD beam is resonant with the |

2S1/2

, F =

1, mF = 1i ↔ |2P1/2, F = 2, mF = 2i transition frequency, and the repumper beam

is resonant with the |

2S1/2

, F = 2, mF = 1i ↔ |2P1/2, F = 2, mF = 2i transition

frequency. The frequency different of 1120 MHz is generated with two 636 MHz AOMs

and another 152 MHz AOM. The reason the 1120 MHz frequency difference is not

generated by two AOMs at 560 MHz is for compatibility of the optical system with the

27Al+ clock experiment sharing the same laser systems. The 27Al+ clock experiment

uses a 9Be+ ion as a refrigerant at near zero magnetic field and requires the RD and

repumper beams to be separated by 1272 MHz, corresponding to the difference frequency

between the |

2S1/2

, F = 2, mF = −1i and |

2S1/2

, F = 1, mF = −1i states of 9Be+ at

low magnetic field. The dashed lines in Fig. 3.6 show the 27Al+ clock experiment RD

beam line; removable mirrors are inserted between the two 636 MHz AOMs to route the
97

beam through the 152 MHz AOM such that our experiment has the correct difference

frequency. The repumper and RD beams are combined on the +636 MHz AOM. The

“RD SW” double-pass AOM provides a global frequency scannable offset and acts as an

optical switch. The RD and repumper beams are combined with the BD beam line on a

50/50 beam splitter prior to being sent to the BD/RD experiment beams and BD/RD

load beams (see also Figs. 3.5 and 3.3).

The RD beam line polarization at the position of the ions is ˆσ+ similar to the BD

line (see Fig. 3.3). This polarization allows the RD and repumper beams to perform

near recoilless optical pumping of the ion to the stretched |

2S1/2

, F = 2, mF = 2i state

through the 2P1/2 manifold. The optical pumping is recoilless (in the Lamb-Dicke limit)

in the sense that the stretched |

2S1/2

, F = 2, mF = 2i state is a dark state for the optical

pumping process and the probability of scattering from |F, m = 1i with motional Fock

state |n = 0i to |F = 2, mF = 2i|n = 1i is η

2 where η is the Lamb-Dicke parameter (see

§ 2.5.4). In contrast, if BD was used for optical pumping out of the |F, mF = 1i states,

after scattering into the stretched |F = 2, mF = 2i, the ion would scatter many photons

on the cycling transition and equilibrate back to the Doppler temperature (hni ' 3).

3.3.3 Raman Beam Line

The Raman beam line is shown in Fig. 3.7. After exiting the doubler, the beam

passes through a spatial filter of diameter 75 μm which is used more as an alignment

tool rather than an optical mode filter due to the presence of the optical fibers later

in the beam line. At the magnetic field-independent point (0.01194 T), the transition

frequencies between different mF states within the same F manifold are not degenerate

and span the range of 69—103 MHz. We generate two beams with difference frequencies

in this range by using two high-bandwidth deflector AOMs in a double-pass configuration which operate in the frequency range of 160—240 MHz (“R-90” and “R-CO”

in Fig. 3.7). The difference frequency between these two beams can be set to any fre-
98

RD doubler

BBO

vert. pol. 50/50

BD, BD detune

-636

+636

+152

+636 RD

 -484 repumper

RD SW

x2

+200

to trap load

and experiment

zones

Figure 3.6: Red Doppler (RD) beam line. The RD beam line consists of the “RD” beam

and the “repumper” beam separated by 1120 MHz. The dashed line shows the beam

path used by the 27Al+ clock experiment. Removable mirrors direct the beam through

the 152 MHz AOM such that we have the frequency difference needed at 0.01194 T.

The RD SW AOM is frequency scannable. The beam line is combined with the Blue

Doppler (BD) beam line prior to being sent to the trap at two different optical ports

(see Figs. 3.3 and 3.3).
99

quency in the range from 0 to 160 MHz. The outputs of these two AOMs form the RCO

and the R90 beam lines entering the trap from the upper two optical ports in Fig. 3.3.

Using just these two AOMs, we can perform co-carrier rotations, 90◦-carrier rotations,

sideband cooling, and Z-phase two-qubit gates between pairs of states within the same

F manifold (see § 2.5). Co-propagating carrier (co-carrier) rotations are performed by

sending two frequency components to the R-CO AOM. For example, sending 175 MHz

and 226.4 MHz to the R-CO AOM will generate two laser beams with a difference frequency of 102.8 MHz co-propagating at 90◦

to the magnetic field vector and can drive

the |F = 2, mF = 2i ↔ |F = 2, mF = 1i transition.

The two AOMs R-CO and R-90 are not sufficient to drive F-changing transitions,

however. In particular, the field-independent qubit transition |F = 2, mF = 0i ↔ |F =

1, mF = 1i at 1207 MHz is of this type. To generate frequency differences spanning the

F-changing transition frequency range of 1018—1516 MHz, we use the ∆F AOM which

operates over the range of 500—740 MHz2. The ∆F AOM configuration double-passes

both the 0th order and the +1st order such that after the double-pass, there exist two copropagating laser beams with a difference frequency twice that of the AOM frequency.

The lens makes both beam paths after the first pass through the AOM parallel such that

they can reflect off a common retro reflecting two-mirror apparatus. The two-mirror

vertical retro reflector serves to provide a vertical displacement of the retro reflected

beam relative to the incoming beam. After the retro reflected beam passes through

the lens, the vertical displacement is converted into an angular change such that the

returning beam intersects the original beam directly at the acoustic wave focus of the

AOM maximizing the diffraction efficiency through the second pass of the AOM. This is

a common optical arrangement for a double-pass AOM. All of the 200 MHz double-pass

2 The largest F-changing transition |F = 2, mF = −2i ↔ |F = 1, mF = −1i near 1516 MHz falls

outside the double-passed range of the ∆F AOM; however, we may be able to drive this transition

at the expense of less AOM efficiency. We have not tried this in the lab, and in practice, there is no

foreseeable need.
100

Raman doubler

BBO

vert. pol.

R-CO

x2

+(160 - 240)

R-90

x2

+(160 - 240)

+(500 - 740) beam cleaning 75 μm

fiber

λ/4

Glan-laser

polarizer

λ/2

fiber

λ/4

Glan-laser

polarizer

λ/2

∆F DEFL-CO

+(160 - 240)

DEFL-90

+(160 - 240)

trap

axis

ion positions

in different traps

B

Deflector AOMs (DEFL-XX) scan

the laser beams over a range of1 mm at the trap. The beams at the 

deflection points of the AOMs are 

imaged onto the final lens such thatdeflections at the AOM become angular

changes at the final lens. 

telescopenot to scale

telescopenot to scale

photodetector

photodetector

Figure 3.7: Raman beam line. Both the 0th and the +1st orders of the ∆F AOM are

double-passed generating two laser frequencies differing by 1000 — 1480 MHz spanning

the large range of F-changing transition frequencies in 9Be+ at 0.00194 T. The deflector

AOMs (DEFL-CO and DEFL-90) move the beams across a range of 1 mm at the trap.

The wide bandwidth of the R-CO and R-90 AOMs allow a difference frequency between

the two beams from 0 up to 160 MHz spanning all mF changing transition frequencies

within the same F manifold.
101

AOM configurations use right angle prisms to provide vertical retro reflection. Because

we require very good overlap of the second pass 0th order with the second pass +1st

order in the ∆F AOM, we used two separate mirrors mounted at ±45◦ with respect

to the horizontal with their normal vectors in the plane of laser beam travel. This

provided the additional degrees of freedom to simultaneously maximize the second pass

diffraction efficiency and 0th order/+1st order beam overlap. The lens on the incoming

beam to the ∆F AOM focuses the laser to ∼ 80 μm at the AOM such that the beam

fits within the AOM’s acoustic focus of 100 μm. When performing rotations or twoqubit z-basis gates, the ∆F AOM is off, and the double-passed 0th order propagates

to the R-90 and R-CO AOMs. The insertion loss of the double-passed 0th order beam

when the ∆F AOM is off is less than 20%. When performing F-changing rotations or

two-qubit Mølmer-Sørensen gates, the ∆F AOM is on.

Fibers are used on both RCO and R90 beam lines to clean the optical spatial mode

as well as provide some passive beam pointing stability. Because the polarization of the

light exiting the fiber drifts over time, the polarization optics λ/2, λ/4 and a Glanlaser polarizer are placed at the exit of the fiber. Small bending of the fibers increases

the optical loss, hence paddles could not be used to adjust the polarization. Instead,

the λ/4 and λ/2 plates are rotated to optimize the laser power transmitted through

the Glan-laser polarizer. Both polarization drift and beam pointing fluctuations at

the entrance of the fiber translate into intensity fluctuations after the polarizer at the

exit of the fiber. The intensities of the beams are sampled on photo detectors and fed

back on the RF power of the R-90 and R-CO beams to stabilize the intensity after the

fiber. The broadband deflector AOMs DEFL-CO and DEFL-90 provide beam steering

capability. The deflection point of each of the deflector AOMs is imaged onto the

center of the corresponding final lens using a telescope (not to scale in Fig. 3.7) such

that beam steering operations with the AOMs translates into angular deviations at the

corresponding final lens. The beams can span over 1 mm distance covering multiple
102

trapping regions in the trap.

The polarizations of the RCO and R90 beams is shown in Fig. 3.3. The copropagating Raman beam line (RCO) enters the trap from the upper-left optical port

in Fig. 3.3, and its polarization is ˆ2CO =

1

2

(ˆσ+ − σˆ−) + √

i

2

πˆ. Because the two copropagating Raman beams have the same polarization, the two beams are equivalent to

a single laser beam with amplitude modulation. We require both ˆπ and ˆσ+/σˆ− polarization components to drive transitions with ∆mF = 1. For linear polarized amplitude

modulated light with both ˆπ and ˆσ+/σˆ− components (generated, for example, by a

linear polarization at 45◦ with respect to the vertical), the Rabi frequency contains two

terms which destructively interfere suppressing the Rabi frequency. However, by adding

a phase shift of π/2 to the ˆπ-polarized beam with respect to the (ˆσ+ − σˆ−)/

√

2 component, the two terms in the calculation of the Rabi frequency constructively interfere.

We establish this polarization scheme by introducing the λ/4 plate in the RCO beam

path. The polarization of the R90 beam is nominally horizontally linearly polarized;

the λ/2 plate is used to rotate the vertical polarized beam to this configuration. The

differential Stark shift of non-field-independent transitions induced by linear polarized

light, although suppressed, is not zero. We can tune the differential Stark shift to zero

by adding a small elipticity to the beam. The λ/4 plate is used for this purpose.

Because the phase of qubit superpositions is measured relative to the difference

phase between the two Raman beams, it is important to have good relative phase stability between the two beams in the co-propagating Raman pair. In the Raman beam

line depicted in Fig. 3.7, the Raman beam pair for the field-independent qubit is generated at the ∆F AOM. After the ∆F AOM, the beams are co-propagating; therefore,

any phase noise such as index of refraction changes in air or optical path fluctuations,

is common to both beams, and the differential phase is unaffected to a high degree.

However, the path of the two beams is not co-propagating between the ∆F AOM and

the retro reflecting mirror setup. To reduce differential phase fluctuations, the beam
103

paths where the two beams are physically separated is kept as small as possible. In our

setup, the beams traverse different paths for only 20 cm. Throughout these paths, the

two beams are separated horizontally by approximately 1 mm and travel through the

same optical elements in the hopes that vibrations in the optical elements and index

changes in the air will be mostly common mode. In addition, a box is placed over the

AOM/lens/mirror setup to reduce air currents and differential beam pointing fluctuations. We measured the phase noise between the two beams by optical heterodyne

detection followed by rf homodyne measurement to obtain a signal proportional to the

phase difference between the two beams. The phase was passively stable to within 0.05

radians rms for minutes with zero detectable drift. In principle, the optical heterodyne/rf homodyne measurement scheme could be used to feedback on the phase of the

∆F AOM and achieve even lower phase noise.

3.3.4 Electro-Optic Raman Beam Line

The long coherence time measurements in Ch. 4 used a different Raman beam

line. There was no need for sideband cooling or coupling the internal states of the ion

to its motion. Therefore, a single laser beam was used entering the trap through the

RCO port (see Fig. 3.3). As discussed in the previous paragraph, the two beams of

a co-propagating Raman beam pair can cause phase noise on qubit superpositions if

the beams are physically separated over a finite distance. For the long coherence-time

experiment, we required very high phase stability between the Raman beam pairs, and

we accomplished this by using the electro-optic modulator (EOM) based Raman beam

line shown in Fig. 3.8. The two radio-frequency (RF) EOMs at 1.018 GHz and 1.207

GHz respectively phase modulate the horizontal polarization component while leaving

the vertical polarization component free of modulation. These frequencies drive the

|F = 2, mF = 2i ↔ |F = 1, mF = 1i and |F = 2, mF = 0i ↔ |F = 1, mF = 1i

transitions in 9Be+. The 45◦ Z-cut KD*P crystal orientation which allows this type
104

of modulation also has static birefringence which gives a relative phase retardation

between the horizontal and vertical polarization components of a few thousand π. The

Rabi frequency with this scheme is proportional to sin φ where φ is the retardation

between the two polarization components. We compensated the static birefringence by

measuring the retardation φ with a polarization analyzer similar to that in [Hansch 80]

and fed back on the temperature of the birefringent KD*P crystal in the DC EOM (see

Fig. 3.8). This stabilized the retardation φ to its optimal value of π/2 mod 2π.

Even though the EOM scheme in Fig. 3.8 has the advantage that the two Raman

beams (generated by the two polarization components of a single laser beam) are copropagating and hence have a very small susceptibility to differential phase noise, we

eventually switched to the AOM based scheme in Fig. 3.7 due to a number of problems

with the EOM scheme. Because of the large capacitance of the KD*P crystal in the RF

EOM (the dielectric constant of KD*P is ∼ 5020), we could only achieve an oscillator

Q of ∼ 50 at the same resonance frequency. Consequently, over 10 W of RF power was

required to achieve a phase modulation index high enough maximize the optical power

in the first sideband. Our RF amplifiers which had a saturated output power of 7 W

could not achieve this. Nevertheless, with 7 W RF power, we could not operate the

EOMs at a duty cycle larger than 1.5% without suffering thermal lensing of the beam as

it passed through the KD*P crystal. Because temperature changes in the KD*P crystal

modified the birefringence of the crystal, such large impulses of energy deposited on

the crystal made locking the retardation using temperature controllers very difficult.

The only way we could operate these EOMs was to maintain a constant duty cycle

throughout the experiment. We accomplished this with sophisticated software which

would automatically insert extra EOM pulses during wait times and extra delays to

maintain the constant duty cycle requirement. In practice, this was quite problematic,

and after the long coherence time experiment was finished, we switched the optical setup

to that in Fig. 3.7.
105

Raman doubler

BBO

vert. pol.

beam cleaning

75 μm

Polarization

analyzer

Intensity

stabilizer

RF EOM

1.2 GHz

RF EOM

1.0 GHz

SW AOM DC EOM

200 MHz

To ion

Figure 3.8: Raman beam line used in long coherence time measurement in Ch. 4. Two

radio-frequency (RF) electro-optic modulators (EOMs) at 1.018 GHz and 1.207 GHz

respectively phase modulate the horizontal polarization component while leaving the

vertical polarization component free of modulation. The static birefringence of the

EOMs is compensated with a polarization detector and feedback on the temperature of

an additional birefringent KD*P crystal (DC EOM).
106

3.4 Experiment Control

Quantum information processing with trapped ions as described in this thesis

requires the controllable interaction of modulated laser sources with the ions. The experimenter must control the frequency and phase of the modulation source as well as

the duration of the laser pulse. In addition, a minimum of one measurement is performed every experiment as described in section 2.4. Some experiments require the

application of classically conditioned quantum gates based on measurement outcomes

[Barrett 04, Riebe 04, Chiaverini 04, Chiaverini 05]. This is particularly true for quantum error correction schemes where measurements of ancilla qubits yields information

about what error, if any, occurred on the computational logic qubits. Such schemes are

required for large-scale quantum information processing. To accommodate all of these

necessary capabilities, I designed and built a field programmable gate array (FPGA)

based experiment controller for use in quantum information processing experiments with

trapped ions.

Throughout this section, many acronyms are used. The acronyms are defined at

first use in the text as well as in Appendix A.

3.4.1 Introduction to Field Programmable Gate Arrays

Field programmable gate arrays (FPGAs) are reconfigurable digital logic devices.

They consist of primitive components capable of generating arbitrary logic functions.

The primitive components vary from device to device and may include logic blocks such

as registers, look-up tables, shift registers, memory, buffer drivers, adders, multipliers,

counters, multiplexers, and in some cases processors. It is possible to create an entire

system on a chip including processor, memory, and input/output functionality. Because

the designer has access to hardware-level components, the designs can be very fast

(few nanoseconds delay) as compared to software running on a computer. FPGAs are
107

completely reconfigurable. If the experimenter needs more functionality, he or she can

implement the functionality by reprogramming the FPGA. At the time of this thesis,

the two largest manufacturers of FPGAs are Xilinx and Altera.

Each of these manufacturers sell software tools to assist in the design and configuration of their FPGAs. For simpler designs, the designer can use schematic capture

tools; however, for larger designs, this method becomes very cumbersome. Schematic

capture tools allow the designer to draw the schematic in block diagram format interconnecting different modules with wires. More complicated designs are written using

a Hardware Description Language (HDL). Using an HDL, the designer can write the

design in a top-down modular format making the management of larger designs less

complicated. The two most common HDLs are Verilog and VHDL3. I focus on VHDL

in this work. The software tools sold by the FPGA manufacturers typically have both

schematic capture and HDL design tools built into one integrated development environment.

VHDL is a high level text-based hardware description language. It was originally

developed for hardware description and simulation and was later used for synthesis

[Perry 02]. VHDL is a modular language allowing for top-down design with components.

There are two types of statements in VHDL—concurrent statements and sequential

statements. The designer uses concurrent statements to describe combinatorial logic (i.e.

AND gates, multiplexers, etc.) and uses sequential statements to describe synchronous

logic (i.e. counters, state-machines). I refer the reader to [Perry 02] for a VHDL tutorial

and reference.

FPGA manufacturers and third party companies sell development and evaluation

boards for FPGAs at a very reasonable price (few hundreds of dollars to few thousands

of dollars). Typically included on these boards is external memory, input/output ports

(i.e. RS-232, Ethernet, USB), and plug-in connectors for user daughter boards with

3 VHSIC (Very High Speed Integrated Circuit) Hardware Description Language
108

general purpose I/O. Some boards have multiple fast high resolution analog-to-digital

and digital-to-analog converters (ADCs, DACs) for use in digital feedback controllers.

Such boards are in use in the experiments of Hideo Mabuchi at Caltech [Geremia 04,

Stockton 02].

The design process consists of writing a hardware description in VHDL and verifying the design. This process is illustrated in Fig. 3.9. All of these steps can be

performed using the software tools provided by the FPGA manufacturer. After describing the design in VHDL, the designer performs a behavioral simulation to verify the

design’s functionality. This is where the designer spends most of his or her effort. The

behavioral simulator allows the designer to view all intermediate logic signals given some

input logic stimulus. Separate components can be simulated independently allowing the

designer to build large logic systems in a modular fashion. After the functionality is

verified, the design is synthesized and implemented. During this phase, the hardware

description written in VHDL is compiled into an optimized gate-level description using

the primitive components of the particular device (multiplexers, look-up tables, etc.).

The synthesis tool provides timing information and constraints such as gate delays and

input setup time for the primitive components to be used during timing simulation. The

designer, after timing simulation, may choose to rewrite parts of the hardware description in an attempt to optimize the speed and/or area of the design. All of these processes

can occur without access to the physical hardware. After the timing simulation results

are satisfactory, the final step is testing the design on actual hardware.

3.4.2 Introduction to Direct Digital Synthesis

Direct Digital Synthesis (DDS) is a technique for generating an analog signal

source derived from a stable reference clock given a digital frequency input. Schematically, the DDS is broken up into two sections—the digital part and the analog part

(see Fig. 3.10). The digital section consists of a digital phase accumulator and a sine
109

Figure 3.9: FPGA Design Flow.
110

Ref

Clock

Phase

Accumulator

Amplitude/Sine

Conv. Algorithm

D/A

Converter

Tuning word specifies 

output frequency as a 

fraction of Ref Clock 

frequency

DDS Circuitry

In Digital Domain

Sin(x)/x

Freq Tuning Word

Phase Offset Word

In Analog Domain

Figure 3.10: Block Diagram of a DDS. Figure taken from Analog Devices, A Technical

Tutorial on Digital Synthesis, (1999).

look-up table. The phase accumulator simply ramps a digital phase word modulo its

range (interpreted as 2π) at a rate proportional to the frequency tuning word input.

The digital phase word is the phase written in binary, and the digital phase word range

is 0 to 2n − 1 where n is the number of bits in the digital phase word. The phase offset

word input is summed into the phase accumulator as well allowing the user to make fast

phase changes. The output of the phase accumulator feeds the sine look-up table where

the phase is converted into a digital amplitude. The digital amplitude is converted into

an analog signal with a fast digital-to-analog converter (DAC) in the analog section.

The output of the DDS is a sampled sinusoid, and with proper low-pass filtering, frequencies up to the Nyquist frequency (1/2 reference clock frequency) can be generated.

DDS synthesizers are stand-alone integrated circuits and can easily be controlled by an

FPGA.

DDS technology offers many advantages over more traditional frequency synthesis

techniques such as phase locked loops (PLL). The DDS frequency is digitally tuned;

therefore, the DDS has high accuracy, low frequency drift, low phase drift, and large
111

frequency dynamic range. 32 bits of frequency resolution is common, and some devices

have up to 48 bits. Furthermore, phase continuous frequency switching can be performed

in a few reference clock cycles (tens of nanoseconds). Disadvantages of DDS technology

are limited spectral purity, quantized DAC noise (equal to one least-significant bit),

and the existence of image frequencies. (It is notable, however, that it is possible to

band-pass filter the frequency images for higher frequency applications). At the time

of this writing, devices with reference clock frequencies up to 1 GHz exist4, and agile

frequency synthesis at higher frequencies can be achieved with the integration of PLL

translation loop or divider loop circuits. However, frequency switching time for PLL

circuits will be limited to the rise time of the loop filter. For fast frequency switching

at high frequencies, single side band (SSB) modulation can be used.

3.4.3 Experiment Controller Design

The FPGA based experiment controller was designed to have variable duration

digital output pulses, fast (sub-microsecond) frequency/phase switching, integrated photon counting input from a photo-multiplier tube (PMT), and the ability to branch the

pulse sequence conditioned on measurement. To reduce memory requirements, support was added for subroutines and loops. The fast frequency/phase switching was

accomplished by the FPGA controlling a shared communications bus (DDS Bus) used

to program individual DDS modules. Each DDS module contained one Analog Devices

AD9858 DDS.

The FPGA card used was the Xilinx XtremeDSP Kit. It contained one user

programmable Virtex IV FPGA, 42 general purpose I/O pins, 2 14-bit 65 MSPS5

ADCs, 2 14-bit 160 MSPS DACs, 512 kB6 of off-chip memory, and an external clock

4 Analog Devices AD9858. This device also has a 2 GHz analog mixer and 400 MHz phase-frequency

detector with charge pump functional blocks for versatile PLL integration.

5 Mega-samples per second

6

kilobyte
112

PCI Interface

Memory

System 

Controller

DDS Controller

Pulse Controller

Data Collector

25

19

Digital I/O Bus

DDS Bus

PMT Input (2)

External Trigger

FPGA

Computer

Digital I/O Box

DDS Box (8)

Detection System

PMT

Figure 3.11: Block diagram of the experiment controller.
113

input. This card interfaces to a computer through the PCI7 bus. The FPGA is

programmed to interface to the Digital I/O Box and the DDS Box through the Digital

I/O Bus and the DDS Bus respectively (see Section. 3.4.3.2). The Digital I/O Box has

16 digital output channels, 2 digital counter input channels, and 1 external triggering

digital input. The DDS Box has 8 DDS modules, each of which can be individually

controlled by the FPGA through the DDS Bus using module select addressing logic

circuitry. Although the DDS Box contained only 8 modules, the DDS Bus was designed

to control up to 32 DDS modules.

3.4.3.1 FPGA Design

A block diagram of the FPGA design is illustrated in Figs. 3.12, 3.13, and 3.14.

The PCI IFACE block coordinates communication between the computer and the FPGA.

The FPGA card ships with software drivers (Nallatech FUSE library) which allows the

programmer to perform simple operations on the FPGA such as resetting and reconfiguring the FPGA and performing read/write operations over the PCI bus. The FPGA

card provides other digital circuitry (a smaller dedicated FPGA) to handle the PCI

communications according to the PCI specifications and transfer the read/write information to the user FPGA in a well defined format (see [Nallatech 03] for details). The

FPGA design must contain logic to handle this communication.

The FPGA is configured once writing the hardware description into the FPGA.

The hardware description is written in such a way so as to allow different experimental

sequences or “program sequences” to be downloaded into the FPGA’s internal memory.

When the experimenter desires to run a different program sequence, it is downloaded

to the FPGA memory without reconfiguring the FPGA.

After the FPGA is configured, all computer interactions with the FPGA are

implemented with read/write cycles. For example, after the program sequence is down7 Peripheral Component Interconnect
114

FPGA 1/2

Memory blocks are dual port Block RAM withport 1 R/W (connected to PCI_IFACE)port 2 read only (connected to ADDR_CNTR)PCI_IFACE ADDR(13..11) are MEM_SEL bitsfor the different Block RAMs. ADDR(9..0) areconnected to 18 x 1k Block RAMS, andADDR(10..0) is connected to DDSMEM 9 x 2k.

REGISTERSLOGICMEM1LOGICMEM2TIMINGMEM1TIMINGMEM2CTRLMEMDDSMEMHISTOMEMPMT_MEMPMT_REG

0x00000x08000x10000x18000x20000x28000x30000x38000x40000x4800

Memory Block Base ADDR

REGISTERS

REG_CTRLREG_LASTADDRREG_NUMEXPREG_HIST_SIZEREG_COMMANDREG_DDS_READ_BYTEREG_LOGIC_ANDREG_LOGIC_ORREG_NUMCOUNTS

012345678

ADDR(13..11) are MEM_SEL bits;must be ‘000’ for register read/write

register addr

0 = CTRL_RUN1 = CTRL_RUNEXPIRE2 = CTRL_NEWDATA_READY3 = CTRL_SETDDS4 = CTRL_XTRIG5 = CTRL_WAND6 = CTRL_WOR

set to start, unset to stop; if RUNEXPIRE=’1’, this is unset internally after NUMEXPif set, then stop after NUMEXP, else run continuously until CTRL_RUN is unset from PCset internally when new data is available, to reset from PC, use CTRL_WAND w/ 0xBDset to run a DDS command, CTRL_RUN should be ‘0’ to run a DDS commandset to enable external triggeringset to AND the contents of the dataword to REG_CTRL; AND takes precident over ORset to OR the contents of the dataword to REG_CTRL; AND takes precident over OR

bit REG_CTRL

REG_COMMAND 18..14 = opcode 13..0 = dataword same format as CTRLMEMCTRL_SETDDS must be setCTRL_RUN must be unset for thisto do anything. This bit is reset afterthe command is finished.

LOGICMEM1

18 x 1k

LOGICMEM2

18 x 1k

TIMEMEM1

18 x 1k

DDSMEM

9 x 2k

CTRLMEM

18 x 1k

TIMEMEM2

18 x 1k

PCI_IFACE_BUS

PULSER_MEM

LOGICOUT(17..0)

ADDR_CNTR

ADDR_CNTR

DETECT_GATE(3..0) <== (3..0)

TIMING_WORD(49..36) <== (17..4)

TIMING_WORD(17..0)

(17..14)==>OPCODE(13..0)==>DATTIMING_WORD(35..18)

AWORD

DDSMEM_ADDR

DDSMEM_OUT

DDSMEM_OUT(7..0)DDSMEM_ADDR(10..0)DATAWORD(13..0)OPCODE(3..0)TIMING_WORD(49..0)DETECT_GATE(3..0)LOGICOUT(17..0)

ADDR_CNTR(9..0)

ADDR(13..11) are MEM_SEL bits

COMM_CLK for PCI_IFACETIMER_CLK for port 2 w/ADDR_CNTR and DDSMEM_ADDR

ADIO

RENl_WENlRDl_WRlINTlEMPTYBUSYRSTl

PCI_IFACE

ADDR

NEXT_ADDR

DATA_OUT

READ_STROBE

WRITE_STROBE

DATA_IN

PCI_IFACE_BUS

ADIO_BUS

COMM_CLK

TIMER_CLK

REG_BLOCK_ADDDRREG_REPEAT_CNT

DATAWORD(13..0)

OPCODE(3..0)

ADC0(13..0)ADC1(13..0)ADDR_CNTR(9..0)

TIMER_CLK

EXPIRE XTRIG_RUN

TIMING_WORD

LASTADDR

Blockstack

PULSER_PROC

XTRIG_CTRL

XTRIG

CTRL_RUN

EXPIRE

DDS_RUN_COMMAND Figure 3.12: Block diagram of the FPGA design part 1/2. Thick arrows are multi-bit

logic signals (or buses); thin arrows are single-bit signals. Direction of arrow indicates

direction of data flow.
115

RSTTIMER_CLKDDS_DATAWORD(13..0)DDS_OPCODE(4..0)DDSMEM_OUT(7..0)DDSMEM_ADDR(10..0)

DDS_DATA(7..0)

DDS_ADDR(5..0)

DDS_RDl

DDS_WRl

DDS_IOUPDATE

DDS_MRESET DDS_RUN_COMMAND

DDS_SEL(2..0)

DDSCTRL

INT

REG_CTRL

TIMER_CLK

REG_CTRL_WRITEREG_CTRL_WSTROBE

CTRL_STATEMACHINE

When EXP_CNTR_EXPIRE triggers -set CTRL_RUN to ‘0’ if CTRL_RUNEXPIRE = ‘1’ -set CTRL_NEWDATA_READY -trigger INT

EXPIRE

REG_NUMEXP EXP_CNTR_EXPIRE

EXP_CNTR

COLLECT_DATA

CNTRHISTO_MEMSTATEMACHINE

PMTDATAMEM

PMT_IN (0,1)

DETECT_GATE(3..0)

PCI_IFACE (read only)

TIMER_CLK

CTRL_RUN

PI_PMT_ADDRPI_PMT_COUNTPI_PMT_DATA_OUTPMT_ADDRPMT_COUNT

VAL

COLLECT_DATAstate machine

idle

countingdonecounting

updatehistogram

if DG ≠ 0

if DG = 0

(DG = DETECT_GATE)

RST

-wait for at least one of DG(3..0) to go high-clear CNTR-store DG in REG_DG-wait for DG(3..0) to change-count on rising edge of PMT-get histo_mem @ addr=REG_DG & CNTR_VAL add +1 and store in  REG_NEWVAL-write REG_NEWVAL to histo_mem @ addr = REG_DG & VAL

FPGA 2/2

Figure 3.13: Block diagram of the FPGA design part 2/2.
116

clear all

wait no

new data

no new

data

wait new

data

new data

clear old

data

finish

clear old

data

wait exp

new data

wait

pc read

finish

EXP_CNTR_EXPIRE

EXP_CNTR_EXPIRE

EXP_CNTR_EXPIRE

EXP_CNTR_EXPIRE

NEWDATAREADY = 1

NEWDATAREADY = 0

CNTR_EXPIRE

histo_mem.vhd

state machine for clearing data and write protecting

data for the collect_data.vhd module

2003-10-07 CEL

WEN_ACTIVE = 0

all memory clear,

wait for EXP_CNTR_EXPIRE

WEN_ACTIVE = 0

collect data in one memory

the other is clear

wait for EXP_CNTR_EXPIRE

to start filling other memory

WEN_ACTIVE = 1

datamem si full, wait for NEWDATAREADY

to go to 1 before going to new data

WEN_ACTIVE = 1

wait for NEWDATAREADY to go low

this indicated PC has read the memory,

then toggle READ_FROM

WEN_ACTIVE = 1

clear data not being written to or

read from. Still writing and reading

WEN_ACTIVE = 1

finish clearing

old data

WEN_ACTIVE = 0

EXP_CNTR_EXPIRE

EXP_CNTR_EXPIRE EXP_CNTR_EXPIRE and CNTR_EXPIRE EXP_CNTR_EXPIRE

NEWDATAREADY = 0

WAIT FOR exp_cntr_expire,

then toggle WRITE_TO

WEN_ACTIVE = 0

both data memories

are full, wait for pc read,

then clear all

Figure 3.14: Histogram memory state machine for COLLECT DATA module in

Fig. 3.13.
117

loaded to the FPGA through a sequence of write cycles, the pulse sequence is “started”

by writing the value ‘1’ to the control register (REG CTRL). Similarly, the FPGA can

be stopped by writing the value ‘0’ to the control register. Data (fluorescence counts)

is retrieved through a sequence of read cycles. Consequently, the PCI IFACE module

was the first block that I designed, and its success was paramount to getting the rest

of the design working. It transforms the ADIO bus signals (ADIO(31..0), RENl WENl,

RDl WRl, INTl, EMPTY, BUSY, RSTl, see [Nallatech 03]) into the PCI IFACE BUS

signals used to read/write the PULSER MEM block and the registers (see Fig. 3.12).

The PULSER MEM module contains the memory for the pulse sequencer. There

are four blocks of memory in the PULSER MEM module: (1) LOGICMEM, (2) TIMEMEM, (3) CTRLMEM, and (4) DDSMEM. The LOGICMEM stores the digital logic

channels which are output for variable lengths of time. The duration of these pulses

is stored in TIMEMEM. The design allows for a limited set of control flow, and the

commands and data associated with this are stored in CTRLMEM. Finally, the DDSMEM stores frequency and phase data used to program the DDS modules over the

DDS Bus. The PULSER MEM LOGICOUT bus is routed off the chip to the Digital

I/O Box (see Fig. 3.11). The memory modules in PULSER MEM are dual port Xilinx

Block RAMs. Port 1 is read/write and connected to the PCI IFACE bus, and port 2 is

read only and addressed by ADDR CNTR. Port 1 allows the computer to write to and

read from the contents of memory to load new pulse sequences and read them back for

verification. Port 2 is used by the remainder of the FPGA design for operation. The

Block RAMs output the data at address ADDR CNTR within one TIMER CLK cycle

after ADDR CNTR changes. ADDR CNTR is driven by the PULSER PROC module.

The PULSER PROC module controls the address counter ADDR CNTR and

the address into the DDS memory, DDSMEM ADDR. When the CTRL RUN bit of

the REG CTRL register is ‘0’, PULSER PROC sets ADDR CNTR to zero, and the

first pulse is output on the LOGICOUT bus. This is the non-running state. When
118

CTRL RUN is ‘1’, PULSER PROC is running. The simplest operation is as follows.

When running, PULSER PROC loads the TIMING WORD located at ADDR CNTR

(starting at zero, the first pulse) into a decrementing counter. Upon counter expiration, ADDR CNTR is incremented, the next LOGICOUT pulse is output, the

next TIMING WORD is loaded into the counter, and the process repeats. When

ADDR CNTR reaches REG LASTADDR, PULSER PROC triggers the EXPIRE signal, sets ADDR CNTR to zero, and the next experiment begins. The EXPIRE pulses

are counted by the EXP CNTR module, and when the count of experiments matches

REG NUMEXP, the EXP CNTR EXPIRE signal triggers an interrupt for the host computer. At this point, if the CTRL RUNEXPIRE bit is set to ‘1’ in the REG CTRL register, the CTRL RUN bit is reset to ‘0’, and the PULSER PROC enters the non-running

state. If the CTRL RUNEXPIRE bit is ‘0’, the PULSER PROC module remains in the

running state and continues to run the next experiment starting at ADDR CNTR = 0.

In addition to the linearly incrementing ADDR CNTR run mode described above,

the PULSER PROC module also supports branching the ADDR CNTR. It does so by

processing a limited set of OPCODEs (see Table 3.1). The CTRLMEM block of the

PULSER MEM module outputs the OPCODE and the DATAWORD addressed by

ADDR CNTR for processing. The DATAWORD takes on different meanings based on

the OPCODE being processed. For example, when the PULSER PROC encounters

the OPCODE OP ENTER BLOCK, it interprets the DATAWORD to be the number of times the following block should be repeated. In contrast, if the OPCODE is

OP CALL SR, the DATAWORD represents the address to which to branch for a subroutine call. The OPCODEs and the meaning of the corresponding DATAWORDs are

shown in Table 3.1.

The OPCODEs listed in Table 3.1 represent two types: (1) ADDR CNTR control flow and (2) DDS control. ADDR CNTR control flow OPCODEs are processed

by PULSER PROC, whereas the DDS control OPCODEs (value 0xB and above) are
119

OPCODE Value DATAWORD Meaning

OP NOP 0x0 Bit 0 = ADC accumulate mode(1), normal

mode (0), bit 8 = ADC channel in ADC accumulate mode

OP ENTER BLOCK 0x1 Block repeat count

OP EXIT BLOCK 0x2 N/A

OP BRANCH 0x3 Branch address

OP BRANCH GREATER 0x4

OP BRANCH LESS 0x5

OP BRANCH EQUAL 0x6

OP BRANCH NOT EQUAL 0x7

Bits 9..0 branch address, bits 13..0 register address to compare; Register data: bits

13..10 detect gate, bits 9..0 value to compare.

OP CALL SR 0x8 Subroutine branch address

OP RETURN SR 0x9 N/A

OP SET REG 0xA Bits 13..10 register address, bits 9..0 register

value to write

DDS Opcodes

OP SETFREQ 0x0B Address of 4 byte frequency in DDSMEM

OP SETPHASE 0x0C 14 bit phase

OP SETRF 0x0D Address in DDSMEM of 4 byte frequency followed by 2 byte phase (low byte first on both

frequency and phase)

OP SET SEL 0x0E DDS selection value

OP DDS WRITE BYTE 0x11 Bits 13..8 address of DDS register, bits 7..0

data to write

OP DDS READ BYTE 0x12 DDS register address to read

OP DDS MRESET 0x13 N/A

Table 3.1: FPGA opcodes.
120

processed by the DDSCTRL module (see Fig. 3.13). The DDSCTRL module is responsible for communicating with the DDS modules on the DDS Bus, and it uses DDSMEM

as a resource. For example, if a certain pulse has the OPCODE OP SET FREQ, the

DDSCTRL module interprets the DATAWORD as the address in DDSMEM where the

frequency is stored. It then sets the DDSMEM ADDR to this value, reads the first byte

of the frequency tuning word from DDSMEM OUT and routes this to the DDS DATA

bus while setting the DDS ADDR to 0xA8. A few clock cycles later9, DDSCTRL increments DDSMEM ADDR, retrieves the next byte of the frequency tuning word from

DDSMEM OUT and sends this out the DDS DATA bus at DDS ADDR 0xB. This process is repeated until all of the bytes of the frequency tuning word are written to the

DDS Bus. Next, the frequency update signal is pulsed on the DDS Bus to write the

new frequency tuning word to the internal registers of the DDS. Consequently, there

is a minimum pulse duration for each DDS OPCODE. The minimum pulse duration

varies as a function of the DDS OPCODE because certain OPCODEs must transfer

more information to the DDS than others. In the case of OP SETFREQ, the minimum

time is 23 TIMER CLK cycles10 and transfers four bytes of information to the DDS

(32 bits of the frequency tuning word). In contrast, the minimum pulse duration for

OP SETPHASE is shorter because only two bytes of information is transferred to the

DDS (14 bits of the phase offset word). The minimum pulse duration for all of the DDS

OPCODEs is shown in Table 3.2.

The DDS Bus supports individual addressing of different DDS modules. This is

accomplished via the DDS selection address (or DDS board address). Each DDS module

has a 5-bit DIP11 switch which is set to a unique binary value. This identifies each DDS

module from the others. The FPGA interfaces to an individual DDS module by first

8 This is the register address on the Analog Devices AD9858 DDS of bits 7..0 of the 32-bit frequency

tuning word. See the AD9858 datasheet for further details.

9 The communications clock for the DDS is intentionally divided down by a factor of three to make

DDS Bus communications more robust.

10 At the TIMER CLK frequency of 62.5 MHz, this corresponds to 368 ns.

11 Dual inline package
121

DDS OPCODE Minimum Minimum Duration

TIMER CLK at 62.5 MHz (ns)

Cycles

OP SETFREQ 23 368

OP SETPHASE 10 160

OP SETRF 29 464

OP SET SEL 3 48

Table 3.2: DDS OPCODEs minimum durations.

setting the DDS selection bits prior to executing a DDS OPCODE. This is accomplished

in two steps: (1) execute OP SETSEL to set the DDS selection bits to the DDS module

you wish to address, and (2) execute the DDS OPCODE of interest (OP SETFREQ,

OP SETPHASE, or OP SETRF).

The OPCODEs for both control flow and DDS operations are stored in CTRLMEM to allow these OPCODEs to be processed synchronously with the experiment.

However, there are other operations that we desired which did not require intra- experiment operation. These include DDS reset and the ability to write and read a single

byte to any address on the DDS. CTRLMEM is only 18 bits wide and stores a 14

bit DATAWORD and a 4 bit OPCODE. The extra OPCODEs are 5 bits wide and

processed via the register REG COMMAND. The FPGA must be in the non-running

state (CTRL RUN set to ‘0’) for these commands to be processed. The user first

writes a command to REG COMMAND including a 5 bit opcode and 14 bit dataword.

Next, the user sets the CTRL SETDDS bit in the REG CTRL register to initiate the

command. The OPCODE and DATAWORD are routed to the DDSCTRL module,

and the DDS command is executed. In the case of OP DDS READ BYTE, the byte

read from the DDS is stored in REG DDS READ BYTE. The user can then read the

REG DDS READ BYTE from the computer.

Experimental measurements are performed by counting photons on a photomultiplier tube, and the experiment controller has 2 counter inputs to accommodate
122

this: PMT1 and PMT2. Management of PMT counts occurs in the COLLECT DATA

module in Fig. 3.13. Even though there are two only PMT inputs, counts are stored

in 14 different sections of memory based on the DETECT GATE signal bus. The DETECT GATE signal bus is 4 bits wide, and the most significant bit indicates the PMT

input. If the 3 least significant bits of DETECT GATE are all zero, then the COLLECT DATA state machine is in the non-counting state. Therefore, each PMT input

has 7 different DETECT GATE sections of memory where the counts are collected into

a histogram and stored.

At the beginning of an experiment, the COLLECT DATA state machine is in the

idle state waiting for DETECT GATE(2..0) to be non-zero. When DETECT GATE

changes, the COLLECT DATA module begins counting rising edges from the PMT

input. Counting continues until DETECT GATE changes. If DETECT GATE(2..0)

changes to a non-zero value, counting continues for the new DETECT GATE. After

DETECT GATE returns to zero, the count is used to update the histogram at the

location in memory associated with both the DETECT GATE(2..0) and the count.

In addition to maintaining a histogram of photon counts, the COLLECT DATA

block also stores the raw counts in the PMT data memory in succession. Because

the actual pulse sequence can change from experiment to experiment based on intraexperiment measurements and conditional branch instructions, then number of measurements during each experiment is not known a priori. However, this information

can be extracted from the raw counts (see Table 3.3). The most significant bit, termed

the “successive detect bit,” is ‘0’ for the first count in an experiment and ‘1’ for each

additional count. This labels the first count in an experiment so that the user can reconstruct which pulse sequence occurred. The DETECT GATE is stored in bits 16–13,

and the actual count value is stored in bits 12–0. The maximum count value which can

be stored is 8192.

During an experiment, the most recent count for each DETECT GATE is stored
123

PMT Memory Bit Map

Bit 17 16..13 12..0

Meaning Successive detect bit DETECT GATE Count value

Table 3.3: PMT memory bit map.

in a register such that the PULSER PROC module can compare the PMT count to

threshold values when processing conditional branch OPCODEs such as

OP BRANCH LESS.

3.4.3.2 DDS Bus and Digital I/O Bus

The FPGA communicates to the DDS modules and the Digital I/O box via 2

buses—the DDS Bus and the Digital I/O Bus. The bus signals are listed in Table 3.5

and Table 3.7. Each bus uses low voltage differential signaling (LVDS) over 100 Ω

twisted pair transmission lines. Because the FPGA only provides 42 general purpose

logic I/O pins, I designed a daughter interface board to the FPGA board which converts

the LVTTL12 logic levels of the FPGA to LVDS signals. The daughter board schematic

and board layout can be found in Appendix B.

LVDS adds many advantages to digital communications over LVTTL. Since it is

a differential signal standard, it is less susceptible to interference. This allows it to use

lower voltages and hence lower currents. It uses terminated transmission lines which

reduces transients. This all sums up to faster communications speeds. My first attempt

to communicate with DDS modules using fanned-out LVTTL had many problems including cross-talk induced DDS resets even at low communication speeds.

Each DDS module uses an unterminated LVDS transceiver as a “drop” on the

transmission line (see Fig. 3.15). The transmission line is terminated into 100 Ω on

each end with one end residing at the FPGA. The “drop” is an unterminated LVDS

transceiver residing on the transmission line with short electrical traces to reduce the

alterations to the transmission line’s characteristic impedance. When the FPGA drives

the bus, the signal propagates down the transmission line until it is terminated into

100 Ω. Because the load impedance is matched to the transmission line, there is minimal

reflection. When one of the DDS modules drive the bus, it drives two 100 Ω transmission

12 Low voltage transistor-transistor logic
124

DDS Bus 50-Pin Header

Name Pin Pin Name

DDS PROF SEL H0 50 49 DDS PROF SEL L0

DDS PROF SEL H1 48 47 DDS PROF SEL L1

DDS MRESET H 46 45 DDS MRESET L

DDS RDL H 44 43 DDS RDL L

DDS SEL H0 42 41 DDS SEL L0

DDS SEL H1 40 39 DDS SEL L1

DDS SEL H2 38 37 DDS SEL L2

DDS SEL H3 36 35 DDS SEL L3

DDS SEL H4 34 33 DDS SEL L4

DDS WRL H 32 31 DDS WRL L

DDS IOUPDATEL H 30 29 DDS IOUPDATEL L

DDS ADDR H0 28 27 DDS ADDR L0

DDS ADDR H1 26 25 DDS ADDR L1

DDS ADDR H2 24 23 DDS ADDR L2

DDS ADDR H3 22 21 DDS ADDR L3

DDS ADDR H4 20 19 DDS ADDR L4

DDS ADDR H5 18 17 DDS ADDR L5

DDS DATA H0 16 15 DDS DATA L0

DDS DATA H1 14 13 DDS DATA L1

DDS DATA H2 12 11 DDS DATA L2

DDS DATA H3 10 9 DDS DATA L3

DDS DATA H4 8 7 DDS DATA L4

DDS DATA H5 6 5 DDS DATA L5

DDS DATA H6 4 3 DDS DATA L6

DDS DATA H7 2 1 DDS DATA L7

Table 3.5: DDS Bus
125

Logic Bus 40-Pin Header

Name Pin Pin Name

NC 40 39 NC

LOGIC OUT H15 38 37 LOGIC OUT L15

LOGIC OUT H14 36 35 LOGIC OUT L14

LOGIC OUT H13 34 33 LOGIC OUT L13

LOGIC OUT H12 32 31 LOGIC OUT L12

LOGIC OUT H11 30 29 LOGIC OUT L11

LOGIC OUT H10 28 27 LOGIC OUT L10

LOGIC OUT H9 26 25 LOGIC OUT L9

LOGIC OUT H8 24 23 LOGIC OUT L8

LOGIC OUT H7 22 21 LOGIC OUT L7

LOGIC OUT H6 20 19 LOGIC OUT L6

LOGIC OUT H5 18 17 LOGIC OUT L5

LOGIC OUT H4 16 15 LOGIC OUT L4

LOGIC OUT H3 14 13 LOGIC OUT L3

LOGIC OUT H2 12 11 LOGIC OUT L2

LOGIC OUT H1 10 9 LOGIC OUT L1

LOGIC OUT H0 8 7 LOGIC OUT L0

XTRIG H 6 5 XTRIG L

PMT IN H0 4 3 PMT IN L0

PMT IN H1 2 1 PMT IN L1

Table 3.7: Logic Bus
126

FPGA

LVTTL

DDS1

LVTTL

LVDS

100 Ω

100 Ω 100 Ω twisted pair

DDS3

LVTTL

DDS2

LVTTL

Figure 3.15: LVDS transmission line with multiple transceiver “drops”

lines in parallel—one transmission line terminating at the FPGA and the other at the

opposing end. We use the Texas Instruments LVDM series LVDS transceivers for this

task as they can source twice the normal current required to drive 100 Ω loads; these

chips are designed specifically for this purpose.

3.4.3.3 DDS Module

The DDS module consists of an Analog Devices AD9858/PCB evaluation board

containing the AD9858 DDS chip and an interface daughter board. The evaluation

board provides an LVTTL header to program the DDS. Our daughter board interfaces

to this header by converting the LVDS signals to LVTTL in the appropriate fashion.

The daughter board also provides board addressing with a 5-pin DIP switch. When the

FPGA desires to communicate with a particular DDS, it first sets the DDS selection

bits to the desired DDS board address. The DDS daughter board compares the DDS

selection bits to the board address set by the DIP switch. If they match, the bus

signals are routed to the DDS; otherwise, the signals are ignored. This allows multiple

DDS modules to share the same bus. The schematics for the DDS daughter board are

shown in Appendix B. A picture of the DDS Box containing 8 DDS modules is shown

in Fig. 3.16. The clock source for each DDS module is a 1 GHz synthesized source

referenced to the NIST Hydrogen maser.
127

Daughter board

DDS Bus Cable

DDS Evaluation Board

Figure 3.16: Picture of DDS Box containing 8 DDS modules. Each module consists of an

Analog Devices AD9858/PCB evaluation board with the AD9858 DDS and a daughter

interface board. The daughter boards are “drops” on the LVDS transmission line (flat

cable) with the last daughter board on the cable terminated (top left).
128

3.4.3.4 Software Interface

The FPGA experiment controller as described in the previous sections, though

very functional, provides a very low-level interface with which to program experiments.

The Nallatech FUSE software library exports all of the necessary functions to download

and run experiments on the controller. As we discussed above, all computer interactions

with the FPGA are either read or write statements. However, writing an experimental

pulse sequence requires converting the duration of pulses into a number of clock cycles,

setting the control memory to handle appropriate commands to set DDS frequencies

and phases, and setting the control memory to handle program execution flow. This

process can become very complicated even for simple experiments.

To facilitate simpler pulse sequence design, I developed a pulse-programming

language and a corresponding parser for the language. The experimenter defines names

of pulses corresponding to a set of logic channels which should be asserted high when

the pulse is active. Variables can also be defined. The pulse sequence is described by

a list of “pulse” statements containing the name of the pulse followed by its duration

(see Fig. 3.17; language keywords are highlighted in blue). The pulse statement can

have an optional DDS command such as “setfreq” or “setphase.” Pulses also have

optional labels marked by the keywords “tscan” and “fscan.” These labels alert the

parser to remember the addresses where these pulses reside in the FPGA memory such

that changes to the duration and/or frequency/phase of the individual pulses can be

made quickly. The language provides some simple control flow statements such as the

“if” statement, “block repeat” statement (simpler version of the “for” loop in most

programming languages), and subroutine calls. The language takes advantage of the

branching capabilities of the FPGA design to accomplish this.

The parser is only one part of the interface library (fpgart.dll) to the experiment

controller. The software library maintains a memory map of the parsed file, maps of
129

//experiment to flop on the carrier

name bd = 3,10;

name rd = 4;

name cool = bd,rd;

name wait = none;

name detect = bd,pmt(1);

name raman = 7,8;

var ddsRaman = 2;

var tpi = 2.34; //durations are in microseconds

var fCar = 201.65; //frequencies are in MHz

pulse cool 200.0; //durations are in microseconds

pulse wait 1 setsel(ddsRaman);

pulse wait 1 setfreq(fCar) fscan(fsRaman);

pulse raman tpi tscan(tsRaman);

pulse wait 1;

pulse detect;

pulse wait 1;

if(pmt(1) < 4) {

pulse cool 1000;

}

pulse wait 1;

Figure 3.17: Sample pulse sequence script
130

the pulse names, variables, and tscan/fscan identifiers. It exports functions which allow

the user to make changes to the pulse information associated with tscan and fscan

identifiers. It allows the user to set the number of experiments and histogram size, and

it provides functionality to download the measurement data stored on the FPGA.

I also developed a graphical user interface (pulser ui.exe) which consumes the

fpgart.dll library. This interface gives the experimenter tools to perform scans of the

time, frequency, and phase. The data for each scan parameter is stored in a file, and

the results are displayed on plot. This software is particularly useful for quick test

experiments that scan over experimental parameters. For example, a simple Rabi flopping experiment requires scanning the duration of a laser pulse at a fixed frequency

after the ion has been initialized. The experimenter would write a pulse sequence script

file, parse and download it to the FPGA, select the tscan identifier associated with the

flopping duration, and scan it over the desired range. The pulser ui would then display

the measurement results as a function of the duration of the pulse on a plot, and if the

experiment was successful, sinusoidal flopping data would result.

3.4.3.5 Concluding Remarks

In the previous sections I tried to give an overview of the design of the FPGA

based experiment controller we used in the experiments described in this thesis. After

its first few years of use, many of the bugs have been worked out, and the controller

has become quite reliable. However, looking back, there are a few things which I would

improve in a second generation of such a device. I will outline these thoughts here.

The first limitation we have encountered are the small number of pins. The

XtremeDSP Kit only provides 42 general purpose I/O pins for the user’s design. We

use 23 pins for the DDS Bus and 19 for the Digital I/O Box. This can be further

broken down. Of the 19 pin group, 16 are TTL output, 2 counter inputs, and 1 external

triggering input. For small experiments, 16 TTL channels is sufficient, but as the
131

experiments get more complicated, this number is too small. The obvious solution to

this problem is to use a different FPGA card which has more pins. Many commercial

FPGA cards have as many as 100 general purpose pins, so this would certainly solve

this problem. These boards have other features as well. For example, some boards

give access to the Virtex IV’s high speed serial ports which can communicate at over

1 gigabit per second. A new design could use such a serial port for the DDS bus by

modifying the DDS daughter board to interface to a serial communications protocol.

This would free up pins and reduce cross talk between signals on the parallel bus.

Choosing a new FPGA card should not be taken lightly. The current design takes

advantage of PCI interrupts that the XtremeDSP kit provides. After the set number of

experiments expires, the computer is interrupted allowing it to modify scan parameters

and restart the experiment. Many of the alternate cards do not generate interrupts

for the computer; therefore, an alternate scheme for alerting the computer that data is

available must be used. Changing the DDS Bus to a serial bus rather than a parallel

bus would free up some pins, so changing the FPGA card may not be necessary. Using

an alternate FPGA card should be considered, however.

The current design also uses on-chip memory to store the pulse sequence, DDS

data, and measurement data. Because of the size of the original FPGA on the first

version of the XtremeDSP kit13 , I only allocated enough memory for 1024 pulses.

Granted, the experimenter’s requirements are heavily reduced because of variable length

pulses, subroutines, and loops; however, we are currently running into the limits of such

a small amount of program memory. The newer Virtex IV FPGAs have more on-chip

memory, so extending this amount should be trivial. This has not been done yet,

however. In addition to expanding on-chip memory, the design could utilize the 512

kB of off-chip memory. This is an excellent location to store the measurement data.

Alternate FPGA cards may have a larger size of external memory as well.

13 Virtex II XC2V2000
132

The opcode processing also suffers from a flawed design. The CTRLMEM block

is 18 bits wide due to the size of the original Block RAMS. Because I wanted to make

phase changes as fast as possible, I made the DATAWORD 14 bits such that the entire

phase word could fit in the DATAWORD. Consequently, the DDS CTRL block did

not have to query the DDSMEM for the phase word reducing the latency between the

DDS CTRL block and the DDS module in question. This forced the OPCODE size

to be 4 bits. Unfortunately, we desired more OPCODEs than 24, so I developed the

following work around. Certain DDS commands (reset, read byte, and write byte) did

not need to be executed during an experiment. I made the DDS opcode 5 bits wide

and substituted a ‘0’ for the most significant bit when the card was running. When the

card was stopped, however, the full 5-bit OPCODE from the REG DDS COMMAND

register was routed to the DDS CTRL block. This is obviously a bad design, and a

future design should plan for a larger OPCODE space from the beginning, even if it

means using a smaller DATAWORD and addressing the phase word in the DDSMEM.

The last modification I would make in a future version is to convert the DDS Bus

into a more general purpose bus. This would allow different peripheral modules like fast

DACs to interface to the same bus. It is important to note that the bus definition is

one of the most important aspects of the design. Once the bus is defined, the peripheral

modules can be designed and redesigned at will as long as the bus specifications do

not change. Similarly, the FPGA design can change as long as it communicates with

the bus in the same fashion. Good bus design allows for modularity and expansibility.

One of the successes of the current design is that the DDS Bus has board addressing

functionality that allowed it to be expanded to address up to 32 DDS modules. Future

redesigns of the bus should be carefully thought out. In a future version, I would give

a strong thought to using a fast serial bus with shift registers on the daughter boards

converting the signals back into parallel words.
Chapter 4

Robust Quantum Memory

In section § 2.2 we calculated the energy spectrum of the ground S1/2 hyperfine

states of 9Be+. We found that at certain magnetic fields there exists transitions whose

frequency has no first order dependence on the magnetic field. In this chapter, we use

the particular field-independent transition |F = 2, mF = 0i ≡ | ↓i ↔ |F = 1, mF =

1i ≡ | ↑i as our qubit and investigate its properties. In particular, we measure the

location where the transition is field-independent and measure the coherence time of

the qubit. Our results indicate that the coherence time of this qubit is approximately

15 seconds. I develop a simple model of memory decoherence based on random magnetic

field changes every fixed sample interval. Exponential memory coherence decay models

indicate that the memory error per measurement interval is ∼ 10−5. More sophisticated

models indicate that the memory error per measurement interval may be ∼ 10−7. With

such a long coherence time, this qubit is suitable for use in a fault-tolerant quantum

computing architecture [Knill 05, Steane 03, Kielpinski 02].

In the later half of the chapter, I discuss some experimental challenges to measuring long coherence times. Some of these include magnetic field drift, detection loss due

to ion heating, and systematic frequency shifts. In general, many of the characteristics

of a good qubit memory are common to atomic clocks, such as transition frequency

stability. In particular, the 9Be+ field independent qubit is a relatively stable radio

frequency clock near 1.2 GHz. We do note, however, that the requirements for good
134

atomic clocks, and their performance characteristics, far exceed the 9Be+ qubit discussed in this thesis. Nevertheless, we performed a systematic study of frequency shifts

so that we could be reasonably confident that long coherence times were attainable.

4.1 Field-Independent Qubits

Atomic systems have proven themselves as good candidates for quantum information storage through their use in highly stable atomic clocks [Diddams 04]. Here,

the principle of using first-order magnetic-field-independent transitions is well established. A typical clock transition |F, mF = 0i ↔ |F

0

, mF0 = 0i between hyperfine states

of angular momentum F and F

0

in alkali atoms has no linear Zeeman shift at zero

magnetic field, and coherence times exceeding 10 minutes have been observed [Fisk 95].

Unfortunately, the degeneracy of magnetic sublevels at zero magnetic field makes it

more advantageous to operate at a nonzero field in order to spectrally resolve the levels,

thereby inducing a linear field dependence of the transition frequency. However, fieldindependent transitions between hyperfine states also exist at nonzero magnetic field.

For example, coherence times exceeding 10 minutes have been observed in 9Be+ ions at

a magnetic field B = 0.8194 T [Bollinger 91].

In neutral-atom systems suitable for quantum information processing (QIP),

field-independent transitions at nonzero magnetic field have been investigated in rubidium [Harber 02, Treutlein 04]. The radio-frequency (RF)/microwave two-photon hyperfine transition |F = 1, mF = −1i ↔ |F

0 = 2, mF0 = 1i is field-independent at approximately 3.23 × 10−4 T , and coherence times of 2.8 s have been observed [Treutlein 04].

In these and the clock experiments, transitions were driven by microwave fields on large

numbers of atoms. Using microwaves, it may be difficult to localize the fields well enough

to drive individual qubits unless a means (e.g., a magnetic-field gradient or Stark-shift

gradient) is employed to provide spectral selection [Mintert 01, Schrader 04], a technique that has the additional overhead of keeping track of the phases induced by these
135

F = 2

F = 1

mF = –2

mF = –1

102.8

86.7 76.4 69.1

86.5

76.2

1207.495,843

|↓〉

|↑〉

|Α〉

1018.0074

Figure 4.1: Frequencies of transitions relevant to experiments in the text. The qubit

states are indicated by | ↓i and | ↑i. The state |Ai is the starting state for the experiments as it is easily initialized via optical pumping (see § 2.3). |Ai is also used for

detection. The transition |Ai ↔ | ↑i is linearly dependent on the magnetic field. We

use frequency measurements of this transition as a probe of the magnetic field. Units

are in MHz.

shifts. With transitions induced by laser beams, the addressing can be accomplished by

strong focusing [Riebe 04] or by weaker focusing and inducing transitions in separate

trap zones [Barrett 04]. In contrast to microwave fields, optical fields (using appropriate

geometry [Cirac 95, Monroe 95b]) provide stronger field gradients that are desirable for

coupling ion motional states with internal states, a requirement for certain universal

multi-qubit logic gates [Cirac 95, Wineland 98]. Here, we explore the coherence time of

a single atomic ion qubit in a scalable QIP architecture using laser beam addressing.

In recent 9Be+ QIP experiments utilizing 2s 2S1/2 hyperfine states: |F = 2, mF =

−2i and |F = 1, mF = −1i as qubit levels, fluctuating ambient magnetic fields caused

significant decoherence [Barrett 04, Chiaverini 04]. There, the qubit transition depended linearly on the magnetic field with a coefficient of approximately 21 kHz/μT

(see Fig. 2.2 in § 2.2). Thus, random magnetic field changes of 0.1 μT (typical in our

laboratories) would dephase qubit superpositions (to a phase uncertainty of 1 rad) in

80 μs. To mitigate this decoherence, refocussing spin-echo π-pulses were inserted in

the experimental sequences [Barrett 04, Chiaverini 04] to limit the bandwidth of noise

to which the qubits were susceptible. However, these effects could not be eliminated

completely, and fluctuating fields remained a major source of error in these experiments.
136

The energy spectrum of the ground hyperfine states of 9Be+ as a function of

magnetic field is calculated in § 2.2 and shown in Fig. 2.2 and repeated in Fig. 4.1.

At B0 ' 0.01194 T, the transition |F = 2, mF = 0i ≡ |↓i ↔ |F = 1, mF = 1i ≡ |↑i

(frequency ν↑↓ ' 1.2 GHz) is first-order field-independent with second-order dependence

given by (0.305 Hz/μT

2

)(B−B0)

2

(see Table 2.1). Given random magnetic field changes

of 0.1 μT, we expect superpositions of |↓i and |↑i to dephase in approximately 50 s.

The transition |F = 2, mF = 2i ≡ |Ai ↔ |↑i (frequency ν↑A ' 1.0 GHz) is first-order

field sensitive with linear dependence of 17.6 kHz/μT for B = B0. We use frequency

measurements of this transition as a probe of the magnetic field.

In the experiment, a single 9Be+ ion is confined to a zone of the trap described

in § 3.1. The ion is optically pumped to the state |Ai, and its motion is Doppler cooled

by use of the cycling transition |Ai ↔ |2p

2P3/2

, F0 = 3, mF0 = 3i (see § 2.3). We detect

the state of the 9Be+ ion through state-dependent resonance fluorescence on the cycling

transition (see § 2.4). Using coherent rotations described below, we measure the |↓i, |↑i

“qubit” level populations by mapping the states |↑i and |↓i to |Ai and |↑i respectively

and measuring the state |Ai.

Coherent rotations between states |Ai ↔ |↑i and |↑i ↔ |↓i can be represented by

(in the Bloch sphere representation)

R(θ, φ) = cos θ

2

I − isin

θ

2

cos φ σx − isin

θ

2

sin φ σy, (4.1)

where I is the identity matrix, σi are Pauli operators, θ is the rotation angle, and φ is

the angle from the x-axis to the rotation axis (here, assumed to be in the x-y plane).

These rotations are driven by two-photon stimulated Raman transitions using focused

laser beams (see § 2.5.2). We modulate one polarization component of a single laser

beam with an electro-optic modulator. This technique simplifies the stabilization of

differential optical path length fluctuations between the two Raman beams (generated

by the two polarizations). The difference in optical path, due to the static birefringence
137

of the modulator, is stabilized to its optimal value of λ/4 by measuring the retardation

with an optical phase detector and feeding back on the temperature of an additional

birefringent crystal in the beam path (see § 3.3.4).

The use of co-propagating Raman beams for rotations was important for these

experiments because any differential path-dependent phase fluctuations between the

Raman beam pair would manifest itself as phase error on the qubit superposition. We

took great care to make sure that the Raman beam pair was co-propagating for the

entire beam path. This ensured that any phase fluctuations in the laser beams would

be common mode, and the differential phase fluctuations would be negligible. The

electro-optic phase modulation scheme for Raman transitions proved very useful for the

long coherence time measurement even though there were many technical drawbacks to

its implementation (see § 3.3.4).

We perform two different types of Ramsey experiments in this chapter, the Ramsey frequency scan and the Ramsey phase scan. The Ramsey frequency scan is more

useful for frequency measurements as in Fig. 4.3, whereas the phase scan is useful for

determining the contrast of Ramsey fringes alone as in Fig. 4.4. The Ramsey frequency

scan description is as follows. To measure the frequency of a two-level system with

states labeled |0i and |1i

1

, the Ramsey frequency scan experiment begins by preparing the system in a superposition state |Ψ1i = √

1

2

(|1i − i|0i) by first preparing |1i

followed by application of the rotation R(

π

2

, 0). After a delay TR, the system evolves

to |Ψ2i = √

1

2

(e

iδTR |1i − i|0i) where δ is the detuning of the local oscillator frequency

driving the transition from the qubit frequency (we assume TR is much greater than

the duration of the π/2 pulses). In our case, the local oscillator frequency is the Raman laser beam pair difference frequency. Here, we assume the detuning δ is fixed. In

general, the phase factor δTR is the integrated detuning over the Ramsey interval TR.

Application of a second rotation R(

π

2

, 0) and making multiple projective measurements

1 We assume state |1i lies higher in energy than state |0i.
138

of identically prepared systems yields the probability of detecting the |1i state:

P1-fscan =

1

2

[1 − cos(δTR)]. (4.2)

Repeating the experiment for different δ and fitting to the form a −

b

2

cos(δTR) allows

us to extract the frequency of the qubit transition. The fit parameter a allows for

a bias in the measurement signal which may occur from background light during the

measurement operation for example. The fit parameter b is responsible for the contrast

of the scan. Any frequency fluctuations in the detuning (either generated by an unstable

local oscillator or an unstable qubit frequency) will reduce the contrast b. For frequency

measurements of transitions which are first-order sensitive to the magnetic field such as

the |Ai ↔ | ↑i transition in Fig. 4.1, the Ramsey interval TR is limited to values on the

order of 100 μs due to the drop in contrast b from magnetic field noise in the laboratory.

However, frequency measurements of the field-independent qubit can be much longer as

indicated in Fig. 4.4.

The Ramsey frequency scan experiment is used to measure the qubit transition

frequency to high accuracy. It is also used to measure the differential Stark shift. By

applying the Stark shifting laser during the Ramsey interval TR, the phase of the qubit

superposition will evolve at a rate given by the difference between the local oscillator

frequency and the Stark shifted qubit transition frequency. Performing Ramsey frequency scan experiments with and without the Stark shifting beam yields frequency

measurements with a difference frequency equal to the differential Stark shift.

In units of frequency, Eq. (4.2) is periodic in 2π/TR. Therefore, care must be

taken to make sure we are measuring the detuning about the central Ramsey fringe.

For example, measurements about ω0 + 2πn/TR will yield the same results where n is

an integer and ω0 is the qubit frequency. We ensure we are examining detunings about

the central fringe (n = 0) by performing frequency scans at different TR. Data for

frequency scans of the field-independent qubit transition at different Ramsey intervals
139

TR is shown in Fig. 4.2. The fringe shown in Fig. 4.2 is the central fringe because the

minima of multiple frequency scans at different TR overlap. We observe that the minima

in Fig. 4.2 do not overlap perfectly. The small shift of the minimum is due to the AC

stark shift of the Raman laser during the π/2 pulses. The central frequency is shifted

by ∆ωss = ωss(1 + π

4

TR

τπ/2

)

−1 where ωss is the differential AC stark shift of the qubit

transition frequency induced by the Raman laser, τπ/2is the duration of the π/2 pulse,

and TR is the Ramsey interval measured between the end of the first π/2 pulse to the

beginning of the last π/2 pulse. In the calculation for ∆ωss, we assume ωss is much

smaller than the Rabi frequency with no assumption on the size of TR as compared to

τπ/2.

To characterize the field-independent transition, we perform Ramsey spectroscopy

[Ramsey 63] on the two transitions |Ai ↔ |↑i and |↓i ↔ |↑i for different magnetic fields

using the Ramsey frequency scan method. The data is shown in Fig. 4.3. The magnetic

field is determined from the ν↑A measurement. By measuring ν↑↓ at B = B0 for different

RF trapping strengths and extrapolating to zero, we can determine the corresponding

AC Zeeman shift produced by the trap’s RF currents. This shift [1.81(2) Hz] was

removed from the data in Fig. 4.3. The solid curve in Fig. 4.3 is calculated using the

Breit-Rabi formula (see § 2.2).

We measure the qubit coherence time by adjusting the magnetic field to the

minimum of Fig. 4.3 and performing Ramsey phase scans on the |↓i ↔ |↑i transition

for different Ramsey intervals TR. The 9Be+ ion is first Doppler cooled and prepared

in the state |↑i. We then apply the rotation R(

π

2

, 0), creating the superposition state

|Ψ1i = √

1

2

(|↑i − i|↓i) and wait for the Ramsey interval TR during which the state

evolves to |Ψ2i = √

1

2

(e

iφD |↑i − i|↓i). The phase φD is given by the integrated detuning

of the Raman beams’ frequency difference from the qubit transition frequency over the

Ramsey interval TR. The Raman beam’s difference frequency is derived from a stable

hydrogen maser based local oscillator which is itself periodically calibrated to the NIST
140

Overlapped Ramsey Fringes

0

1

2

3

4

5

6

7

8

-50 -40 -30 -20 -10 0 10 20 30 40 50

Detuning [Hz]

Figure 4.2: Frequency scans of the field-independent transition | ↓i ↔ | ↑i for different

Ramsey intervals TR. The filled diamonds have TR ' 16 ms; the open squares have

TR ' 32 ms; the filled triangles have TR ' 64 ms.

-8 -4 0 4 8

0

5

10

15

1,207,495,843 [Hz]

B 11,944.6 [ T]

Figure 4.3: Field-independent transition frequency vs. magnetic field. The fieldindependent transition frequency (y-axis) is for the |F = 2, mF = 0i (| ↓i) ↔ |F =

1, mF = 1i (| ↑i transition). The x-axis is determined via frequency measurements of

the field-dependent transition |F = 2, mF = 2i ↔ |F = 1, mF = 1i near 1018 MHz.

Circles are measured data; the solid curve is a theoretical prediction from the BreitRabi formula (see § 2.2). The statistical uncertainty in each datum is ∆B . 3 nT and

∆ν↑↓ . 0.3 Hz. Figure taken from Ref. [Langer 05].
141

F1 cesium fountain clock. A second rotation R(

π

2

, φ) is then applied with φ variable.

Repeating the experiment many times and performing a projective measurement of the

state |↑i as described in § 2.4 ideally yields

P↑ =

1

2

(1 − cos (φD + φ)), (4.3)

the probability of measuring the state |↑i. The measurement sequence is repeated

for different phases φ, and the detected probability P↑ is fit to the function f = a −

b

2

cos (dφ + φD). The fit parameter d allows for magnetic-field drift in time as successive

phase points are recorded sequentially; d is close to unity for all scans in this data set.

Phase scans for TR = 4 ms and 4 s are shown in Fig. 4.4a. Any fluctuation in φD during

the Ramsey interval TR will reduce the contrast b. The form of contrast decay vs. TR is

dependent on the spectrum of magnetic field noise which has components corresponding

to times both long and short compared to TR. The coherence time is limited in part by

slow drift of the magnetic field over the measurement time scale of a single point. For

the TR = 4 s data in Fig. 4.4a, this time scale is 400 s. Moreover, since the measurement

of the contrast can take many hours for the longer Ramsey intervals, the magnetic noise

environment can vary over different points in Fig. 4.4b. As a benchmark, we fit the

contrast b for different TR to the exponential b(TR) = b0e

−TR/τ (Fig. 4.4b) and find

τ = 14.7 ± 1.6 s. In principle, if the magnetic-field drift is small for the period of a

single measurement, we can interrupt data collection to measure (via ν↑A) and correct

for magnetic-field deviations from B0.

4.2 Magnetic Field Noise Model

Fitting the contrast to exponential decay, although useful as a bench marking

tool, is somewhat naive. We can calculate the expected contrast as a function of time

using the following simple model. Suppose we perform a Ramsey phase scan experiment

with Ramsey interval TR where the measurement after the final phase-scanned π/2
142

0 1 2 3 4 5 6 7

0.0

0.2

0.4

0.6

0.8

1.0

contrast

Ramsey interval TR [s]

(b)

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

0.8

1.0

P

phase 

(a)

Figure 4.4: (a) Ramsey data at TR = 4 ms (squares) and 4 s (triangles). The y-axis

represents the probability of measuring the state |↑i. The contrast b for the 4 ms data

is 0.933 ± 0.014 and for the 4 s data is 0.742 ± 0.043. The φD '1 rad phase shift

in the 4 ms data is due to detuning the local oscillator by the differential Stark shift

(∼4.2 kHz) such that the Ramsey π/2-pulses are resonant. (b) Contrast vs. Ramsey

interval TR. Each datum represents the fitted contrast b for a phase scan with Ramsey

interval TR. The solid curve is a weighted least-squares fit to the data with reduced

χ

2 ' 1.16. Figure taken from Ref. [Langer 05].
143

pulse, averaged over many experiments, yields A cos(φ + φ0) = 1 − 2P↑ where P↑ is the

probability of measuring | ↑i (see e.g. Eq. 4.3). Here, φ is the phase scan variable,

and φ0 is some constant offset phase arising from skewness the phase noise distribution,

for example. We further assume that the local oscillator is tuned exactly to resonance

and any bias in the measurement signal has been removed. Perfect contrast is defined

by A = 1. Suppose that the accumulated phase noise is distributed according to some

probability density function p(x) which is responsible for the decrease in contrast A

from the ideal A = 1. In the absence of phase noise (and hence perfect contrast), the

final phase-scanned signal is given by cos φ. Given phase noise distributed according to

p(x), the actual measured signal is the expectation value of cos(φ+x−μx) with respect

to the density function p(x) where μx ≡ hxi is the mean of the phase noise distribution.

The phase noise with respect to its mean x−μx rather than x is added to φ because we

assume that the local oscillator is tuned on resonance with the qubit transition frequency

including the systematic frequency shift μx/TR. Therefore, the actual measured signal

is given by A cos(φ + φ0) = Rp(x) cos(φ + x − μx)dx from which we can solve for A.

Using the trigonometric identity cos(α+β) = cos(α) cos(β)−sin(α) sin(β) and equating

the coefficients of cos φ and sin φ, we obtain the system of equations

A cos(φ0) = hcos(x − μx)i

A sin(φ0) = hsin(x − μx)i (4.4)

where hf(x)i is the expectation value of f(x) according to the density function p(x).

The contrast can be calculated from A2 = hcos(x − μx)i

2 + hsin(x − μx)i2

. For density

functions symmetric about the mean where p(μx+x) = p(μx−x), then hsin(x−μx)i = 0,

and A = hcos(x−μx)i. For asymmetric density functions about the mean, hcos(x−μx)i

gives a lower bound for the contrast. Therefore, the study of coherence decay induced

by symmetric density functions is useful as it gives a lower bound to the contrast. In

addition, in many practical situations such as transition frequencies which have a linear
144

dependence on changes in the magnetic field, the phase probability density function is

symmetric about the mean, and the contrast is given by hcos(x − μx)i. As mentioned

above, asymmetric density functions will have a non-zero phase offset φ0.

For the remainder of this section, we assume, without loss of generality, that the

density function p(x) has mean 0. In practice, if the mean of the phase noise density

function is not zero, we can simply redefine the qubit frequency to take this systematic

phase shift over the Ramsey interval into account. As a simple example, suppose the

phase noise is distributed by the gaussian distribution: p(x) = exp(−x

2/2σ2

x

)/

p

(2π)σx

where σ

2

x

is the variance in the phase probability density. Because p(−x) = p(x), the

equation for the contrast simplifies to:

A =

Z ∞

−∞

p(x) cos(x)dx

= e

−σ

2

x/2

.

(4.5)

For symmetric phase probability density functions, the contrast is A = hcos(x)i =

h1 − x

2/2! + x4/4! − · · · i ' exp(−σ2

x/2) for any symmetric probability density function

with negligible probability density for x > 1. For gaussian distributed phase noise,

Eq. (4.5) is exact. For other density functions, as long as the standard deviation σx is

small compared to 1 and higher moments are also small, Eq. (4.5) is a valid approximation. In particular, for asymmetric distributions, the contrast is given as in Eq. (4.5)

with the variance σ

2

x modified by a factor 1 −

1

36

μ

2

3

σ2

x

where μ3 ≡ h(x − μx)

3

i is the third

moment about the mean. To determine how the contrast A decays with time, we must

determine how the variance σ

2

x

changes with Ramsey interval TR and substitute it into

Eq. (4.5).

We assume the magnetic field deviation ∆B is distributed according to the probability density function pB(∆B) and the qubit transition frequency has zero first order

dependence on changes in the magnetic field. This in turn will distribute the frequency shifts from resonance ω = c2∆B2 according to p(ω) = pB(∆B(ω))/

dω

d∆B ×
145

(multiplicity of ∆B(ω)) = pB(

p

(ω/c2))/

p

(c2ω) by the fundamental transformation

law of probabilities [Freund 92, Press 03]. This density function has a non-zero mean,

and we tune the local oscillator to account for this such that we have an effective asymmetric density function p(ω) centered at ω = 0. The constant c2 ' 2π×0.3 rad s−1 μT

−1

is taken from Table 2.1.

As a model, we let the frequency shifts from resonance ω induced by fluctuating

magnetic fields randomly change every sample interval Ts according to the distribution

p(ω). The k

th sample taken from p(ω) is ωk, and the frequency ωk is held constant over

the Ts interval. Here, the amplitude of phase fluctuations is described by p(x), and the

spectrum is described by Ts (i.e. the frequency of noise in this model is 1/Ts). For Ts

much less than the Ramsey interval TR, the phase x is approximated by

x = Ts

X

N

k=1

ωk (4.6)

where N ≡ floor(TR/Ts),2 and we have neglected the remaining time interval TR −TsN.

(Because TR

Ts

− 1 < N ≤

TR

Ts

, the remaining interval TR − TsN is less than Ts which we

assume is much less than the Ramsey interval TR.) The phase x as a random variable is

well described by the random walk problem with variable step size [Reif 65, §1.9] where

the step size ωk is taken from the distribution of frequency shifts from resonance p(ω).

Frequency noise modeled in this manner is termed “random walk frequency modulation”

(RWFM) in the time metrology literature. Phase noise generated by RWFM is an

integrated Wiener noise. Solutions to this problem in terms of stochastic differential

equations are given in [Galleani 03].

Independent of the density function p(ω), the variance in the phase x is

σ

2

x = NT2s σ

2

ω ' TRTSσ

2

ω

(4.7)

where σ

2

ω

is the variance of p(ω). To see this, we simply take the expectation value of

the square of Eq. (4.6) and note that factors of the form hωjωki = hωj ihωki = 0 for j 6= k

2 The function floor(x) is defined as the largest integer less than or equal to the real number x.
146

because different random frequencies drawn from p(ω) are independent. Therefore, the

only non-zero terms in hx

2

i are the N terms of the form T

2

s

hω

2

k

i, and hence we arrive

at Eq. (4.7) (remember, hxi = 0). For Ts ¿ TR, the variance in the phase is linear

in the Ramsey interval TR, and the contrast decays exponentially with time constant

τ = 2/(Tsσ

2

ω

). We note that for field independent transitions, the frequency probability

density function is not symmetric, and the approximation of Eq. (4.5) is a lower bound

valid only for small σ

2

x

.

We now consider the general case with no restrictions on the size of Ts. We first

consider the case where TR ≥ Ts (N ≥ 1). The time of the first frequency change t1

occurs randomly within the interval [0, Ts)

3 after which successive frequency changes

occur every Ts interval after t1. The last interval is in general less than Ts. We let the

t1 random variable be uniformly distributed over the first Ts interval. There are two

cases to consider (see Fig. 4.5). First, if t1 < TR − NTs, then there are N intervals of

length Ts and two intervals shorter than Ts, namely t1 and the last interval tL which

make up the entire Ramsey interval TR. The last interval is never independent. In this

first case, it is tL = TR − NTs − t1. The second case occurs when t1 ≥ TR − NTs. Here

there are N −1 intervals of length Ts, and the final interval is tL = TR −(N −1)Ts −t1.

The Ramsey interval TR only has enough duration to allow N steps of size Ts if the first

interval is less than the remainder TR − NTs; therefore, if t1 is greater than this value,

the last Ts interval vanishes.

We wish to calculate the variance in the phase σ

2

x = hx

2

i. For TR ≥ Ts, the phase

random variable is given by:

x =







ω1t1 + ωL(TR − NTs − t1) + PN+1

k=2 ωk for t1 < TR − NTs

ω1t1 + ωL[TR − (N − 1)Ts − t1] + PN

k=2 ωk for t1 ≥ TR − NTs

, (4.8)

where ωk for k ∈ {1, 2, .., N + 1, L} are all independent. We can simplify this equation

3 The notation [a, c) defines an interval such that b ∈ [a, c) is equivalent to (⇔) a ≤ b < c. Similarly,

b ∈ (a, c) ⇔ a < b < c and b ∈ (a, c] ⇔ a < b ≤ c.
147

0 TR

0 TR

TS

TSTS

TSTSt

1

t

1

t

L

t

L

(a)

(b)

TR− NTS

TR− (N − 1)TS− t1

TR− NTS− t1

N − 1 intervals of length TS

N intervals of length TS

Figure 4.5: Sample magnetic field change intervals for N ≡ floor(TR/Ts) = 3. The first

interval t1 is uniformly distributed in the interval [0, Ts). (a) t1 < TR − NTs case: there

are N intervals of size Ts. (b) t1 ≥ TR − NTs case: there are N − 1 intervals of size Ts.

In all cases, the last interval tL is dependent on t1.
148

by introducing g1(t1) ≡ 1 if t1 ∈ [0, TR − NTs), otherwise g1(t1) = 0. Eq. (4.8) can now

be rewritten as:

x = t1ω1 + [TR − (N − 1)Ts − (t1 + Tsg1)]ωL + g1TsωN+1 + Ts

X

N

k=2

ωk. (4.9)

The phase variance is calculated by evaluating hx

2

i using Eq. (4.9) and the fact that

different ωk are independent. As such, the phase variance is the sum of the variances of

each term with a different ωk, and all terms are proportional to σ

2

ω

. The result is:

hx

2

i

σ

2

ω

= ht

2

1

i + h[TR − (N − 1)Ts − (t1 + Tsg1)]2i + T

2

s

hg

2

i + (N − 1)T

2

s

. (4.10)

We observe that we must evaluate expectation values of g1, g

2

1

, t1, t

2

1

, and g1t1. Since

t1 is uniformly distributed over the [0, Ts) interval, these are easily calculated. After

evaluation, substitution, and simplification, the phase variance simplifies a great deal

to:

hx

2

i =

μ

TR

Ts

−

1

3

¶

T

2

s σ

2

ω

(4.11)

for TR ≥ Ts.

For TR < Ts (N = 0), the Ramsey interval TR has either two intervals both

less than TR or a single interval equal to TR. The probability of having two intervals

with different frequency deviations is TR/Ts. This can be understood by thinking about

many repetitions of the experiment performed sequentially. If the frequency deviation

changes every Ts, then floor(Ts/TR) experiments will all have the same frequency, and

the frequency will change in the middle of the next experiment. Therefore, the probability of having a frequency change in a single experiment is TR/Ts. We model this by

letting the random variable t1 be uniformly distributed over the [0, Ts) interval, and if

a particular sample t1 happens to lie in the interval [0, TR), we let the frequency change

at t1. In this case, the phase is x = t1ω1 + (TR − t1)ω2. If t1 happens to be within the

interval [0, TR), the Ramsey interval experiences only a single frequency, and x = TRω2.

We can introduce the function g2(t1) = t1 if t1 ∈ [0, TR), otherwise g2(t1) = 0, simi-
149

lar to before, to rewrite the phase as x = g2ω1 + (TR − g2)ω2. After calculating the

intermediate expectation values hg2i and hg

2

2

i, the phase variance simplifies to

hx

2

i =

μ

1 −

1

3

TR

Ts

¶

T

2

Rσ

2

ω

(4.12)

for TR < Ts. Summarizing, the phase variance as a function of the Ramsey interval TR

is:

σ

2

x =







3

1 −

1

3

TR

Ts

 ́

T

2

R

σ

2

ω

for TR < Ts

3

TR

Ts

−

1

3

 ́

T

2

s σ

2

ω

for TR ≥ Ts

. (4.13)

A few remarks on Eq. (4.13) are in order. First, σ

2

x and its first two derivatives

with respect to TR are continuous across the TR = Ts boundary. Furthermore, the

discrete function N = floor(TR/Ts) has dropped out of the equation, and σ

2

x

is a smooth

function of TR. Finally, as before, for long Ramsey intervals TR À Ts, the decay is

exponential, and for short Ramsey intervals TR ¿ Ts, the decay is gaussian. This

is the best of both worlds. The long time behavior has decaying contrast which is

slower than gaussian, namely exponential, and the short time behavior has decaying

contrast which is slower than exponential, namely gaussian. We might suspect that the

14.7 s coherence time measured in the previous section might have short term errors

during the measurement interval quadratically smaller than the 10−5level proclaimed

in [Langer 05].

The coherence time is defined as the Ramsey interval TR where the contrast

decays to the e

−1

level. In this model, this occurs when σ

2

x = 2. Defining y = TR/Ts

and a = Tsσω, we must solve (1 −

y

3

)y

2a2

for y < 1 and (y −

1

3

)a

2

for y ≥ 1. For

y ≥ 1, the solution is y0 =

2

a

2 +

1

3

requiring a

2 ≤ 3. That is, for low noise levels and

fast modulation frequencies, the coherence time lies in the exponential decay section.

Solving σ

2

x = 2 for y < 1 gives the equation y

3 − 3y2 + b = 0 where b =

6

a

2 ∈ (0, 2)

because a

2 > 3. This cubic equation only has one solution which is physically viable,

namely y0 = 1 + 2 cos( θ−2π

3

) where θ = tan−1(

√

3(2a

2−3)

a

2−3

). Summarizing, the coherence
150

0 2 4 6 8 10

0

0.2

0.4

0.6

0.8

1

Time [s]

Contrast

Figure 4.6: Theoretical contrast decay vs. Ramsey interval. The curves are the contrast

vs. TR for different sample intervals Ts. Ts in the figure (top down) is: 1 s, 3 s, 10 s,

and 30 s. The magnetic field uncertainty is σB = 0.4 μT.

time τ in this model is given by:

τ =







1

3 +

2

T2

s σ2

ω

T

2

s σ

2

ω ≤ 3

1 + 2 cos ¡

θ−2π

3

¢

T

2

s σ

2

ω > 3

(4.14)

where θ ≡ tan−1

μ√

3(2T2

s σ2

ω−3)

T2

s σ2

ω−3

¶

. We note that there exists continuity of the coherence

time across the T

2

s σ

2

ω = 3 boundary.

Sample contrast decay curves are shown in Fig. 4.6. The quadratic decay at short

times is indicative of the gaussian decay in this section. In the laboratory, a dominant

source of magnetic field noise occurs at 60 Hz due to noise in AC line currents in the

walls and electronic instruments. Using the model with Ts = 0.0167 s (the period

of 60 Hz) and the measured τ = 14.7 s coherence time, we find the uncertainty in the

magnetic field at 60 Hz to be σB = 1.0 μT. This is approximately an order of magnitude

higher than we measure in the laboratory indicating that 60 Hz noise is not the limiting

factor in the 14.7 s coherence time. Rather, slow magnetic field drift with Ts À TR is

more likely the cause. Nevertheless, assuming that all of noise is in the 60 Hz spectral

line, we can calculate the memory errors during 1-qubit and 2-qubit gates as well as

the measurement interval. Remember, the contrast is a function of the Tsσω product;
151

therefore, even though the 1 μT noise field at 60 Hz is artificially high, the product

Tsσω is in agreement with the measured 14.7 s coherence time. Using the model, the

error during a 1-qubit gate (1 μs) is 4 × 10−12, 2-qubit gate (10 μs) is 4 × 10−10, and

measurement interval (100 μs) is 4×10−8, all much below the fault-tolerance threshold of

10−4for the Steane code [Steane 03]. We note that the region of validity where Eq. (4.5)

represents a lower bound to the contrast is when σ

2

x

is small, which is definitely the case

here.

In summary, for short times, memory errors are quadratic in the time interval, and

at all times, memory errors scale as the magnetic noise amplitude to the fourth power.

Therefore, with minimal speed and minimal shielding, the memory errors induced by

magnetic field noise can be suppressed to very low levels. We note that such small

memory errors are extremely difficult to measure and these arguments are estimates

only. We also point out that slow magnetic field drift (Ts À TR) plays a strong role in

the decoherence. However, slow drift is relatively easy to correct using slow feedback

control. The next section addresses this issue.

4.3 Magnetic Field Drift

Slow magnetic field drift, as seen in the previous section, can cause significant

memory decoherence. Fortunately, slow time scale drift can be controlled via feedback.

For the measurements in § 4.1, no feedback was implemented. As seen in Fig. 4.7a,

the magnetic field drift can vary from day to day. The drift on Jan. 21, 2005 in

Fig. 4.7a is tolerable, whereas the drift on Jan. 28, 2005 is not. We remove drift by

periodically interrupting the experimental sequence to measure the magnetic field via

Ramsey spectroscopy on an auxiliary transition which depends linearly on the magnetic

field deviation. The measurement is then used to null the field deviation with separate

feedback coils. The stabilized magnetic field is shown in Fig. 4.7b. The transition used

to interrogate the magnetic field was |F = 2, mF = 2i ↔ |F = 2, mF = 1i at 0.01196 T
152

(transition frequency is 103 MHz).

The inset in Fig. 4.7b indicates that our magnetic field servo system stabilizes the

slow magnetic field drift to zero offset from the set point. The scatter in the data is due

to statistical uncertainty in the frequency measurements of the |F = 2, mF = 2i ↔ |F =

2, mF = 1i transition frequency. For the measurements in Fig. 4.7b, Ramsey frequency

measurements were performed with a Ramsey interval of TR = 100 μs and averaging

for N = 100 experiments on each side of the Ramsey fringe. With perfect contrast, we

would expect statistical uncertainty in the measured magnetic field to be 0.015 μT. The

sample standard deviation in the data of the inset in Fig. 4.7b is 0.017 μT, consistent

with statistical measurement uncertainty assuming Ramsey fringe contrast of 88%.

4.4 Heating Induced Measurement Degradation

One challenge we faced while taking the measurements of § 4.1 was detection

degradation induced by ion heating. The experiments inherently required leaving the ion

in the dark for periods on the order of seconds, and consequently, the ion would acquire

kinetic energy. The mechanisms responsible for ion heating are not well understood,

and this is still an active area of research [Deslauriers 06, Wineland 98, Turchette 00].

The most likely candidate is fluctuating patch potentials on electrode surfaces. Measurements of the heating rate for short intervals are described in § 2.5.4.

The measurement prescription described in § 2.4 is immune to small amounts of

heating. However, when left in the dark for many seconds, the ion’s kinetic energy is

large enough to reduce the scattering of the detection beam due to Doppler broadening.

We performed the following experiment to measure the detection loss due to heating.

A single ion was Doppler cooled and prepared in the bright |F = 2, mF = 2i state.

The ion was then left in the dark in the absence of cooling for a variable duration after

which the ion was detected. Data are shown in shown in Fig. 4.8. Detection loss for

John Jost’s trap is better than that of Brian DeMarco’s trap by approximately an order
153

Time [103 s]

∆B [

μT]

0

1

2

0 2 4 6 8

01/28/05

01/21/05

Magnetic field drift (a)

(b)

-100

-50

0

50

100

150

200

250

300

350

400

0 50 100 150 200 250 300 350 400

set point

measured

Time [s]

B

− 118003 [10

−1 μT]

Magnetic field servo 10/27/05

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

300 320 340 360 380 400

Time [s]

B − 118003 [10

−1 μT]

Figure 4.7: Magnetic field drift and servo. (a) Typical magnetic field drift for two

different days without feedback. (b) Stabilized magnetic field drift via slow feedback.

The dashed line is the magnetic field set point. The open circles are measurements of the

magnetic field via frequency measurements of the |F = 2, mF = 2i ↔ |F = 2, mF = 1i

transition. The inset in (b) is a close up view of the measured magnetic field data

for times 300 – 400 s after the start of the graph. The magnetic field is stabilized at

this point, and the scatter in the data is due to statistical uncertainty in the frequency

measurements.
154

of magnitude. In John Jost’s trap, the amount of Doppler cooling during initialization

did not appear to play a role.

We model the detection loss induced by heating using the following simple model.

First, assume the momentum p of the ion is gaussian distributed with mean zero and

width p

(hp

2

i). If the mean were not zero, the ion would fail to be trapped. Here, hp

2

i is

expectation value of p

2 and is equal to the variance of p given that the mean is zero. We

further assume that the trap frequency is small compared to the natural line width of the

excited 2P3/2state. The photon scattering rate of the detection laser detuned by δ from

the cycling transition resonance (|

2S1/2

, F = 2, mF = 2i ↔ |2P3/2, F = 3, mF = 3i) for

an ion originally prepared in the bright state |F = 2, mF = 2i is given by

f(δ) = γ

2

s0

1 + s0 +

4δ

2

γ

2

(4.15)

where γ is the natural line width of the |

2P3/2

, F = 3, mF = 3i state, and s0 is the

on-resonance saturation parameter. With the detection laser tuned on resonance, an

ion with momentum p along the trap axis ˆz will see the detection laser Doppler shifted

from resonance by an amount δ = keffp/m where keff = ~k · zˆ is the projection of the

detection laser’s wave vector along the trap axis. We can now write the on-resonance

photon scattering rate as a function of the ion’s momentum as

f(p) = b

1 + 4p

2

∆p

2

(4.16)

where b ≡

γ

2

s0

1+s0

, and ∆p ≡

mγ0

keff

is the Lorentzian width of the scattering rate measured

in units of momentum. Here, m is the mass of the ion, and γ

0 ≡ γ

p

(1 + s0) is the

power broadened line width. We let

g(p|hp

2

i) = 1

p

2πhp

2i

e

− 1

2

p

2

hp2i (4.17)

be the conditional probability density for the ion’s momentum given the variance in the

momentum distribution is hp

2

i. For an ion in the ground state of a harmonic oscillator,
155

0

2

4

6

8

10

12

0 0 .5 1 1.5 2 2 .5 3 3 .5 4

1 ms c o o ling

10 ms c o o ling

1s c o o ling

Detection loss vs. time J. Jost’s trap 2004-11-30

Time [s]

Average counts

(a)

(b)

Time [s]

Detection loss vs. time B. DeMarco’s trap 2004-03-24

0

2

4

6

8

10

12

0 0 .2 0 .4 0 .6 0 .8 1 1.2 1.4 1.6 1.8 2

Average counts

Figure 4.8: Detection loss induced by ion heating. (a) Measurements in John Jost’s

trap on Nov. 30, 2004. (b) Filled squared are measurements in Brian DeMarco’s trap

on Mar. 24, 2004. The open circles with the dashed trend line are the fitted means of

the bright distribution from Ramsey phase scan data as a function of Ramsey interval

TR. Data was taken on August 10, 2004 on a single 9Be+ ion with sympathetic Doppler

cooling being performed on a simultaneously stored 24Mg+ ion in the same trap. We

observe that the detection loss induced by ion heating is reduced in this case.
156

Eq. (4.17) is an exact description of the probability of measuring the ion’s momentum

with the variance given by hp

2

i =

1

2

p

2

0 =

1

2

~mω, half of the square of the momentum

wave function spread p0. Here, ω is the harmonic oscillation frequency. At low thermal

energies where the ion’s momentum state is a statistical mixture of the lowest momentum

Fock states, Eq. (4.17) is a rather poor description. However, as we will see below, in

terms of calculating the detection laser scattering rate, it is sufficient. In addition, for

large energies on the order of a few hundred or more, the ion’s momentum distribution

in thermal equilibrium is gaussian, and Eq. (4.17) is a good approximation.

The variance in the momentum can be calculated using the Virial theorem which

states that for a system of harmonic oscillators in thermal equilibrium, the total energy

of the system is equally divided between the kinetic energy and the potential energy

[Pathria 96]. For the quantum harmonic oscillator, we find h

p

2

2m

i =

1

2

hEi =

1

2

(hni+

1

2

)~ω.

Therefore, the momentum variance can be expressed in terms of the average harmonic

oscillator occupation number hni as hp

2

i = (hni +

1

2

)p

2

0

.

The thermal average scattering rate for an ion in a thermal average occupation

number hni can be expressed as

h(hni) = Z ∞

−∞

f(p)g(p|hp

2

i)dp

=

b

q

2π(hni +

1

2

)p0

Z ∞

−∞

1

1 + 4 p

2

∆p

2

e

− 1

2

p

2

(hni+ 1

2

)p

2

0 dp

=

b

√

π

e

∆y

2

π∆y erfc(∆y)

(4.18)

where ∆y =

∆p

2p0

√

2hni+1

=

γ

0

2keffq m

~ω(2hni+1) , and erfc(x) ≡ 1 − √

2

π

R x

0

e

−t

2

dt is the

complimentary error function of x. The integral in Eq. (4.18) is similar to the integral

in the Voigt profile, a spectral line shape common in many branches of spectroscopy

which arises from Doppler broadening of a Lorentzian profile. The Voigt profile is the

convolution of a gaussian distribution with the Lorentzian distribution. In Eq. (4.18) we

integrated over these two distributions and did not take the convolution resulting in the
157

Voigt profile evaluated at line center. This is a consequence of the fact that we tuned

the detection laser on resonance. If we would have left the detuning of the detection

laser as a free parameter, h would have been a Voigt profile function of the detuning.

A plot of Eq. (4.18) is shown in Fig. 4.9a. We observe that for 9Be+ ions in a

ω = 2π × 4 MHz trap with the detection laser beam direction oriented 45◦to the trap

axis with half of the saturation intensity, the on-resonance scattering rate drops to half

of its original value after the ion heats up to a motional occupation number of hni ' 420.

The dashed curve in Fig. 4.9a was obtained by calculating the expectation value of the

scattering rate for a statistical mixture of momentum space wave functions based on a

thermal distribution with mean phonon occupation number hni. We truncated the sum

of expectation values for Fock state momentum space wave functions at n = 500, hence

the two plots disagree for hni & 200. We observe that the two approaches agree well

for hni . 200 which was the region where we were originally unsure that the gaussian

distribution of momentum was valid. Recall from § 2.5.1 that the measured heating rate

for John Jost’s trap was hn ̇i ' 0.25 ms−1. Naively assuming the heating rate is constant

over this temperature range, we expect the ion to obtain a temperature hni ' 500 after

2 seconds in the dark causing the on-resonance scattering rate to drop to approximately

half. This appears consistent with the data in Fig. 4.8.

The appropriate method for dealing with detection loss is sympathetic cooling

(see [Barrett 03]). However, for the long coherence time measurements, we handled this

problem using data analysis. For each Ramsey interval TR, we collected histograms

of photon counts for each phase point. In general, these histograms were bimodal

with one mode near zero (the dark distribution mean) and the other near 10 (the bright

distribution mean). Because of detection loss induced by ion heating during the Ramsey

interval, the bright mean would drop for the longer Ramsey intervals. We desired to

extract the means of the bright and dark distributions so that we could fit the histograms

to a sum of two poisson reference histograms, one for the bright distribution and one
158

200 400 600 800 1000

0.2

0.4

0.6

0.8

1

Average phonon occupation number 〈n〉

Normalized scattering rate vs. 〈n〉

Ramsey interval TR

Mean of bright poisson

distribution vs. Ramsey interval

 0

 2

 4

 6

 8

 10

 12

 14

 0 1 2 3 4 5 6 7

(a)

(b)

Figure 4.9: (a) Detection loss induced by ion heating—theory. Solid curve is calculated

using Eq. (4.18). Dashed curve is calculated by taking the expectation value of the

scattering rate for a statistical mixture of momentum space wave functions based on

a thermal distribution with mean phonon occupation number hni. The sums in the

wave function calculation are truncated at n = 500, hence the departure of the two

curves near hni ' 200. (b) Mean of the poisson distribution for the bright state as a

function of Ramsey interval TR. The mean is determined by fitting a sum of two poisson

distributions with variable means to the sum of all histograms with same TR for the

data in Fig. 4.4a.
159

for the dark distribution (see method of maximum likelihood in Appendix C). Since the

means of the reference histograms would vary as a function of Ramsey interval, using the

same means for the reference poisson distributions would yield bad fit results. Therefore,

we determined the means for the bright and dark reference poisson distributions by

summing the histograms for all phase points for an individual Ramsey interval TR and

fit this histogram to a sum of two poisson distributions with variable means using the

method of maximum likelihood (see Appendix C). Then, the histograms for each phase

point were fit to a weighted sum of two poisson distributions with fixed mean given by

the previous step. The weight parameter determined the fraction of experiments for

each phase point where the bright state was inferred. We then plotted the probability

of measuring the bright state as a function of phase and fit it to the phase scan function

as described in § 4.1 to obtain the contrast. The mean of the bright distribution as a

function of Ramsey interval extracted in this manner was similar to the data in Fig. 4.8

and is shown in Fig. 4.9b.

4.5 Systematics at the Field-Independent Point

Fluctuating systematic frequency shifts of the qubit transition can be a source

of phase noise in QIP experiments. Therefore, we performed a simple study of the

systematic frequency shifts in our system. In regards to long coherence times, we only

concerned ourselves with systematic frequency shifts on the order of 1 Hz or larger.

Therefore, I do not do a full systematics study in this section. Rather, I cover only

those shifts which were detectable on the 1 Hz level or higher and those which caused

problems on the way to the 15 seconds coherence time.

The second largest frequency shift was due to the second order AC Zeeman effect. Recall that field independent transitions have a second order dependence on the

magnetic field and can be expressed by a Taylor series expansion about the magnetic
160

field extremum:

ω(B) = ω0 + c2∆B

2

(4.19)

where ∆B is the deviation of the magnetic field from the field-independent point, ω0

is the qubit transition frequency at ∆B = 0, and c2 is the second-order coefficient (see

Table 2.1 in § 2.2). From Eq. (4.19) we find that an oscillating magnetic field will

induce a frequency shift in the amount of c2h∆B2i = c2hB2i. We note that this shift

is proportional to the square of the amplitude of an oscillating magnetic field; it is

independent of the oscillation frequency, and its sign is given by the sign of c2.

The largest source of oscillating magnetic field was RF currents in the trap’s RF

electrodes. We characterized the frequency shift induced by these currents by measuring

the qubit transition frequency for different RF input power levels to the resonator.

Because the RF power stored in the resonator is proportional to both the input power

and the average of the square of the magnetic field, the systematic frequency shift is

linear in the input RF power. For Brian DeMarco’s trap, the qubit frequency shift vs.

RF power is shown in Fig. 4.10. We find the slope of the shift vs. RF input power to

be 2.27(2) Hz/W. For the data in Fig. 4.3, the trap was operated at 0.80 W inducing

an AC Zeeman shift of 1.81 Hz. This shift was removed from the data in Fig. 4.3.

Because strong RF electric fields are required for trapping, we could not eliminate

AC Zeeman shift induced by the trap completely. However, we were able to operate the

trap at the weakest strength allowable to minimize the AC Zeeman shift. Fortunately,

only fluctuations in frequency shifts cause decoherence. At the trap’s weakest point,

the AC Zeeman shift was approximately 0.5 Hz. We estimated the RF power stored in

the resonator to be more stable than 1%. Therefore, the estimated uncertainty in the

qubit frequency due to the trap’s RF currents was roughly 5 mHz.

The third largest systematic frequency shift was the AC Stark shift of near resonant leakage light from the detuned BD laser beam. The reason for BD detuned leakage
161

AC Zeeman Effect induced by RF resonator currents

qubit frequency shift [Hz]

RF resonator input power [W]

0

10

20

30

40

50

60

0 5 10 15 20 25 30

Figure 4.10: Systematic AC Zeeman shift in Brian DeMarco’s trap. Data was taken on

June 29, 2004. Open circles are measurement data; solid line is a linear best fit. The

slope of the shift is 2.27(2) Hz/W.
162

light was the existence of a single-pass AOM switch used to block the beam (see § 3.3.2

Fig. 3.5). Using a double-pass AOM configuration as a switch may have mediated this

problem somewhat. The detuned BD laser beam had a few hundred μW of optical

power focused to a spot size of roughly 30 μm when turned on and was tuned approximately 400 MHz to the red of the detection cycling transition (see § 3.3.2). The shift

induced by leakage light from this beam was measured to be as high as 30 Hz. We

reduced this shift by attenuating the laser beam by 26 dB with a neutral density filter.

The differential AC Stark shift of the qubit transition is linear in the optical power.

Therefore, we estimated this shift to be approximately 80 mHz. In principle, optical

shutters can be used to reduce this shift even further.

In addition to leakage light shifts and second order Zeeman shifts, we checked for

Stark shifts of the 24Mg+ laser, spatial dependence of the second order Zeeman effect

along the trap axis, and shifts induced by the room lights. None of these had any

measurable shift detectable at the 1 Hz level.
Chapter 5

Off-Resonant Spontaneous Photon Scattering

In Ch. 2 I showed how to implement the requirements for quantum computation

with ions. In particular, the universal quantum gate set (single qubit rotations and a

two-qubit entangling gate) is implemented by interacting the ions with off-resonant laser

beams. Therefore, it is necessary to consider the decoherence of quantum information

stored in ions due to the presence of off-resonant laser light.

For laser sources, the quantum description of the light is closely approximated by

a coherent state of the quantum electro-magnetic field [Loudon 00, § 7.4 pp. 304–310].

Furthermore, a coherent state of light is equivalent to a classical field plus a quantum

field originally in the vacuum state [Mollow 75, Cohen-Tannoudji 92, pp.597–601]. The

decohering mechanisms then can be broken up into two parts: (1) decoherence induced

by the classical field and (2) decoherence induced by the quantum vacuum field. The

decoherence of the classical field is easily described by classical noise. Common classical

laser noise sources are due to fluctuations of the intensity, frequency, and phase. Faulttolerant quantum computation will require classical noise sources to be small enough

such that the errors imparted to the quantum information carriers are well below the

fault tolerance threshold. However, I will show in this chapter that we were able to

control in certain experiments the errors induced by classical noise sources to levels

below the fault-tolerance threshold. This involved a handful of experimental techniques

such as refocusing slow-timescale coherent errors (i.e. over-rotations or accumulated
164

phase). In principle, classical noise is classically controllable (via feedback) and can be

eliminated. In practice, all experimental techniques for eliminating classical noise must

be used to achieve fault-tolerance including classical controllers and other techniques

such as refocusing and composite pulse techniques [Vandersypen 04].

The interaction of the ion with the quantum electro-magnetic vacuum, however,

is a fundamental source of decoherence. Physically, the quantum vacuum is responsible

for excited state decay, i.e. spontaneous emission, and in our experimental setup, the

vacuum is responsible for the spontaneous scattering of photons from an off-resonant

laser beam. The question then arises, “what are the decohering effects of spontaneous

photon scattering?” This question we experimentally answer in this chapter [Ozeri 05].

In the context of quantum information, we are interested in decoherence of quantum

information, i.e. qubit coherence, and in our experimental system, the qubit eigenstates

are two hyperfine ground states of 9Be+. Therefore, we experimentally study the decohering effects of off-resonant laser light on coherent superpositions of hyperfine ground

states [Ozeri 05].

Previous experiments with neutral atoms have studied the coherence of hyperfine superpositions in the presence of laser light [Andersen 03, Kuhr 03]. In these experiments, the dominant source of decoherence was classical noise such as laser intensity noise or ambient magnetic fields. The coherence of Zeeman ground states in

the presence of resonant spontaneous scattering was investigated by Cohen-Tannoudji

[Cohen-Tannoudji 62]. In these experiments, coherence was shown to be preserved when

the Zeeman splitting of the ground states was small compared to the excited state line

width. Here, we study the coherence of hyperfine states separated in energy much more

than the excited state line-width in the presence of off-resonant spontaneous photon

scattering [Ozeri 05].
165

5.1 Raman Inelastic vs. Rayleigh Elastic Photon Scattering

The photon scattering process for an atomic ion (or neutral atom) can be broken

into two separate types: (1) Rayleigh elastic scattering and (2) Raman inelastic scattering. These two different types of scattering are depicted in Fig. 5.1. Rayleigh elastic

scattering describes events where the post-scattering state of the atom is unchanged.

Consequently, the scattered photon has the same frequency and polarization of the incoming laser [Mollow 69, H ̈offges 97]. No information about the state of the atom is

carried away by the scattered photon. Directly after the scattering process and prior

to measurement, the atom and the scattered photon are in a product state. Therefore,

any measurement of the scattered photon will not collapse the state of the atom. Theoretically, if we trace over the degrees of freedom of the scattered photon, the reduced

density matrix of the atom is unaffected and remains in a pure state. In particular,

if the atom was in a coherent superposition prior to scattering, it remains so after the

scattering process.

The situation is quite different for the case of Raman inelastic scattering. Here,

the atom changes state after the scattering process. If the two ground states are separated in energy as they are in Fig. 5.1, then the scattered photon can have a different

frequency than the incoming laser. Hence, the frequency of the scattered photon is

entangled with the state of the atom. Consider an atom prepared in an equal superposition of the two ground states in Fig. 5.1 prior to scattering. After scattering, if

the scattered photon frequency is larger than that of the laser, then the post-scattering

state of the atom is the lower state (by energy conservation), no longer a superposition

state (see right-most diagram in Fig. 5.1b). Similarly, if the scattered photon frequency

is smaller than that of the laser, then the post-scattering state of the atom is the higher

ground state, also not a superposition (see left-most diagram in Fig 5.1b). In both

cases, Raman scattering projects the state of the atom. In terms of quantum informa-
166

Rayeigh elastic

scattering

(a) (b) Raman inelastic scattering

Figure 5.1: Rayleigh and Raman scattering of a coherent superposition by an offresonant laser (straight vertical arrows). Curvy arrows represent scattered photons.

Rayleigh scattered photons have the same frequency and polarization of the incoming

laser and do not decohere superposition states. A Raman scattered photon’s frequency

and/or polarization is entangled with the state of the atom and measures the atom’s

state collapsing the superposition to one of the atom’s eigenstates.
167

tion theory, the scattered photon carries away information (i.e. in its frequency) about

the quantum state of the atom and hence acts like a projective measurement operator.

In terms of quantum theory, the state of the atom is entangled with the state of the

scattered photon, and upon tracing over the degrees of freedom of the scattered photon,

the reduced density matrix for the atom is in a mixed state. The same arguments apply

to the polarization state of the scattered photon. If the two ground states of the atom

are eigenstates with different z-projection of angular momentum, then the scattered

photon’s polarization is also entangled with the state of the atom [Blinov 04].

Generally, if the scattered photon(s) reveal information as to the quantum state

of the atom, then the scattering process causes decoherence. For example, the Raman

scattering process measures the state of the atom after a single scattering event. In the

case of Rayleigh scattering of a single photon as in Fig. 5.1a, there is no way for the

experimenter to infer the state of the atom based on a measurement of the scattered

photon. However, it may be possible to collect many Rayleigh scattered photons and

infer the rate of Rayleigh scattering within some uncertainty based on counting statistics.

If the two ground states have different Rayleigh scattering rates, then the collection of

Rayleigh scattered photons acts as a measurement, and the quantum state of the atom

will collapse if the Rayleigh scattered photons provide enough information to distinguish

between the two ground states. This process is a measurement which occurs over an

interval [Geremia 04] where the atomic state collapses over time as the uncertainty in

the measurement of the scattering rates reduces over time. For the experiments in this

chapter, we used magnetic field insensitive hyperfine states which have the same total

scattering rates within 1% for the range of detunings used.1 Therefore, in order to

have the atomic state collapse due to the information leaked from the atom by the

difference in scattering rates, the atom would have to scatter more than 104 photons.2

1 The reason for the difference in scattering rates is dominated by the different detunings for the two

states which differ by ∼1 GHz out of ∼100 GHz.
168

The extreme example of different scattering rates performing measurement and hence

collapsing the atomic state is in the case of state dependent resonance fluorescence (see

§ 2.4). Here, the laser is tuned on resonance such that one ground state has a large

scattering rate (tens of MHz), and the other has a very small scattering rate (few kHz).

In this case, the state of the atom is projected in a few 10’s of nanoseconds as these

scattering rates are very distinguishable.

We consider scattering from an off-resonant laser of intensity I and polarization

2ˆ ≡ 2+σˆ+ + 20πˆ + 2−σˆ− where ˆπ is parallel to the quantizing magnetic field and ˆσ+

(ˆσ−) is right (left) circularly polarized with respect to the magnetic field direction. The

polarization is normalized such that P

k∈{+,−,0}

2

2

k = 1. The rate of photon scattering

for an atom originally in state |ii scattering into state |fi is given by the HeisenbergKramers relation (see [Loudon 00] and § 2.4.3)

Γi→f = |g|

2

γ

 ̄

 ̄

 ̄

 ̄

 ̄

 ̄

X

k

a

(k)

i→f

∆k

 ̄

 ̄

 ̄

 ̄

 ̄

 ̄

2

(5.1)

a

(k)

i→f =

1

μ2

X

q

hf|d · σˆq|kihk|d · 2ˆ|ii (5.2)

where g =

Eμ

2~

, E =

p

2I/c20 is the laser-beam electric field amplitude, c is the speed

of light, 20 is the vacuum dielectric constant (in the expression for the electric field

amplitude, everywhere else, 20 is the ˆπ polarization component), and μ = |hP3/2, F =

3, mF = 3|d · σˆ+|S1/2, F = 2, mF = 2i| is the magnitude of the cycling transition

electric-dipole moment. Here, d is the electric dipole operator, and γ is the natural

line-width. The amplitude a

(k)

i→f

is the normalized second-order coupling from state |ii

to state |fi through the excited state |ki, and ∆k is the detuning of the laser from the

|ii ↔ |ki transition frequency. The sum over k in Eq. (5.1) is over all intermediate

excited states |ki. Raman inelastic scattering describes state-changing events where

i 6= f, and Rayleigh elastic scattering describes events where i = f leaving the state

2 The uncertainty in a measurement of the scattering rate is proportional to the square root of the

number of photons collected. Therefore, to distinguish between two scattering rates which fractionally

differ by 10−2, a minimum of 104 photons would need to be scattered.
169

unchanged. The Raman scattering rate out of state |ii is ΓRaman

i =

P

f6=i Γi→f , and

the Rayleigh scattering rate is ΓRayleigh

i = Γi→i

. It should be noted that the final state

|fi is the final state of the atom-scattered-photon system, and scattered photons with

different modes (including different polarizations) are distinguishable. Therefore, the

d · 2ˆ operator in Eq. (5.2) must be constrained so that non-zero matrix elements only

occur for scattered photon polarizations consistent with the final state |fi.

We will consider detunings near the first excited state fine structure levels 2PJ

accessible via the electric-dipole interaction where J ∈ { 1

2

,

3

2

} for alkali-like atoms such

as 9Be+. It is convenient to introduce the effective amplitude a

(J)

i→f =

P

k∈J

a

(k)

i→f

for

scattering through all levels in the J fine-structure manifold. For detunings ∆k large

compared to hyperfine and Zeeman splittings in the 2PJ manifolds3, the scattering rate

from atomic state |ii to state |fi simplifies to

Γi→f = |g|

2

γ

 ̄

 ̄

 ̄

 ̄

 ̄

 ̄

a

(J=1/2)

i→f

∆

+

a

(J=3/2)

i→f

∆ − ∆F

 ̄

 ̄

 ̄

 ̄

 ̄

 ̄

2

(5.3)

where ∆ is the detuning of the laser from the |ii ↔ |2P1/2i transition frequency, and

∆F is the fine structure splitting between 2PJ manifolds (J ∈ { 1

2

,

3

2

}) (see Fig. 5.2). In

Eq. (5.3), the final state of the system |fi represents the final atomic state as well as the

final polarization state of the scattered photon. The integration over the final density

of states of the scattered photon is contained in γ in Eq. (5.3).

For ground hyperfine states of the form (see § 2.2)

|li ≡ αl|mJ = −

1

2

, mI = m

(l)

F +

1

2

i + βl|mJ = +

1

2

, mI = m

(l)

F −

1

2

i, (5.4)

3 The hyperfine splitting of the 2P3/2 manifold at zero magnetic field is less than 1 MHz; the hyperfine

splitting of the 2P1/2 manifold at zero magnetic field is 237 MHz. At B = 0.0119 T, the field-independent

point, the Zeeman splitting of the 2PJ levels is of order a few hundred MHz.
170

F = 2

F = 1

mF = –2

mF = –1

1207.495,843

|↓〉

|↑〉

|Α〉

2P1/2

2P3/2

2S1/2

313 nm

∆

∆F

ωL

Figure 5.2: The relevant energy levels of 9Be+ for the spontaneous scattering experiment. ωL is the frequency of the decohering laser beam and is detuned ∆ from the

| ↓i ↔ |2P1/2i transition frequency.
171

the scattering amplitudes can be written as

a

(J=1/2)

i→f = 2+

"

2

3

αiα

∗

f

δm

(i)

F

,m

(f)

F

δ2ˆs,σˆ+ −

√

2

3

αiβ

∗

f

δm

(i)

F +1,m

(f)

F

δ2ˆs,πˆ

#

+20

"

1

3

δi,f δ2ˆs,πˆ −

√

2

3

(αiβ

∗

f

δm

(i)

F +1,m

(f)

F

δ2ˆs,σˆ− + βiα

∗

f

δm

(i)

F −1,m

(f)

F

δ2ˆs,σˆ+)

#

+2−

"

2

3

βiβ

∗

f

δm

(i)

F

,m

(f)

F

δ2ˆs,σˆ− −

√

2

3

βiα

∗

f

δm

(i)

F −1,m

(f)

F

δ2ˆs,πˆ

#

(5.5)

a

(J=3/2)

i→f = 2+

"μ

1

3

δi,f +

2

3

βiβ

∗

f

δm

(i)

F

,m

(f)

F

¶

δ2ˆs,σˆ+ +

√

2

3

αiβ

∗

f

δm

(i)

F +1,m

(f)

F

δ2ˆs,πˆ

#

+20

"

2

3

δi,f δ2ˆs,πˆ +

√

2

3

(αiβ

∗

f

δm

(i)

F +1,m

(f)

F

δ2ˆs,σˆ− + βiα

∗

f

δm

(i)

F −1,m

(f)

F

δ2ˆs,σˆ+)

#

+2−

"μ

1

3

δi,f +

2

3

αiα

∗

f

δm

(i)

F

,m

(f)

F

¶

δ2ˆs,σˆ− +

√

2

3

βiα

∗

f

δm

(i)

F −1,m

(f)

F

δ2ˆs,πˆ

#

(5.6)

where we have used the orthonormality condition αiα

∗

f + βiβ

∗

f = δi,f for states with the

same mF throughout Eqs. (5.5) and (5.6); δn,m is the Kronecker delta function defined

as δn,m = 1 for n = m, 0 otherwise. The scattered photon polarization is represented

by ˆ2s, and the Kronecker delta function on the polarization δ2ˆs,σˆk, k ∈ {+, −, 0} ensures

that the final state of the system is consistent with conservation of angular momentum

(ˆσ0 ≡ πˆ). We note that every individual term is distinguishable in the amplitudes of

Eqs. (5.5) and (5.6). This is because each term represents a different distinguishable

final state whether it is a different atomic state or a different polarization state of the

scattered photon. In practice, for a particular final state, the amplitude a

(J)

i→f will only

have one non-zero term. It is convenient to define the scattering amplitude for a single

term given the polarization of the laser and the polarization of the scattered photon. We

define a

(J)

ik→fq

as the scattering amplitude from the initial atomic state |ii scattering into

the final atomic state |fi via photon scattering from a laser with polarization ˆσk where

the scattered photon has polarization ˆσq. The scattering amplitudes then simplify to

a

(J)

i→f =

X

k,q

2ka

(J)

ik→fq

δ2ˆs,σˆq. (5.7)
172

For example, a

(J=1/2)

i+→f+

=

2

3

αiα

∗

f

δm

(i)

F

,m

(f)

F

. Examining Eqs. (5.5) and (5.6) we observe

that a

(J=3/2)

ik→fq

+ a

(J=1/2)

ik→fq

= δi,f ; that is, for each polarization component of the laser

and the scattered photon, the Raman scattering amplitudes through the 2P1/2 and the

2P3/2 are equal in magnitude and opposite in sign. Consequently, the Raman scattering

rate [see Eq. (5.3)] is proportional to

 ̄

 ̄

 ̄

1

∆ −

1

∆−∆F

 ̄

 ̄

 ̄

2

, and for detunings much larger than

the fine structure (∆ À ∆F ), the Raman scattering amplitudes destructively interfere.

For large detunings, the Raman scattering rate scales as ∼ ∆−4, whereas the Rayleigh

scattering rate scales as ∼ ∆−2.

The Raman scattering rate for an initial state |ii is calculated by summing over

all final states to which the initial state can scatter. Because the scattering amplitude

through the 2P3/2level is equal in magnitude and opposite in sign to that through the

2P1/2 manifold, the Raman scattering rate is straight forward to calculate. Using the

orthonormality of hyperfine states with the same mF , we obtain

Γ

Raman

i = |g|

2

γ

 ̄

 ̄

 ̄

 ̄

1

∆

−

1

∆ − ∆F

 ̄

 ̄

 ̄

 ̄

2

X

k

2

2

k

X

f6=i,q

|a

(J=1/2)

ik→fq

|

2

=

2

9

|g|

2

γ

 ̄

 ̄

 ̄

 ̄

1

∆

−

1

∆ − ∆F

 ̄

 ̄

 ̄

 ̄

2

£

2

2

+α

2

i

(1 + 2β

2

i

) + 2

2

0 + 2

2

−β

2

i

(1 + 2α

2

i

)

¤

.

(5.8)

Similarly, the Rayleigh scattering rate is given by

Γ

Rayleigh

i =

1

9

|g|

2

γ

·

2

2

+

 ̄

 ̄

 ̄

 ̄

2α

2

i

∆

+

1 + 2β

2

i

∆ − ∆F

 ̄

 ̄

 ̄

 ̄

2

+ 2

2

0

 ̄

 ̄

 ̄

 ̄

1

∆

+

2

∆ − ∆F

 ̄

 ̄

 ̄

 ̄

2

+ 2

2

−

 ̄

 ̄

 ̄

 ̄

2β

2

i

∆

+

1 + 2α

2

i

∆ − ∆F

 ̄

 ̄

 ̄

 ̄

2  ̧

,

(5.9)

and the total scattering rate is given by the sum Γtotal

i ≡ Γ

Raman

i + γ

Rayleigh

i

. These

expressions are valid when the detunings ∆ and ∆ − ∆F are large compared to the

hyperfine and zeeman splittings in the excited 2P1/2 and 2P3/2 manifolds.

The suppression of Raman scattering with large detuning can be physically understood by considering the fine structure interaction. The fine structure couples the

electron spin to the electron’s orbital angular momentum; the electron spin does not
173

couple to the electric field of the light directly. The electric field does couple to the

electron’s orbital angular momentum, however, and the spin-orbit coupling then gives

an effective coupling between electron spin and the light field. When the detuning is

large compared to the fine structure splitting, the 2PJ levels are less distinguishable to

the scattering laser, and the laser scatters from an atom with an effective P level without fine structure. Therefore, Raman scattering events, which involve the reorientation

of the electron spin, are suppressed for detunings large compared to the fine structure

splitting. The suppression of Raman scattering events relative to Rayleigh scattering

for large detunings has been observed by Cline et al. [Cline 94] by observing population

changes due to scattering. The same reasons apply to the suppression of stimulated

Raman transitions for detunings large compared to the fine structure splitting. The

two-photon Rabi frequency scales as ∼ ∆−2for ∆ À ∆F compared to ∼ ∆−1 when the

laser is tuned near a single fine-structure transition.

In the context of quantum information processing, we are interested in the spontaneous scattering error during quantum gate operations. We can quantify this error by

considering the probability of Raman scattering during a π pulse, for example. Both the

two-photon Rabi frequency (see § 2.5.2) and the spontaneous scattering rate are functions of the laser intensity and the detuning. The duration of a π pulse τπ is defined by

the relation |Ω|τπ =

π

2 where Ω is the two-photon Rabi frequency. For small probabilities of photon scattering, the probability of scattering a single Raman photon during a

π pulse is given by 1

2

3

Γ

Raman

↓ + ΓRaman↑

 ́

τπ =

π

2

1

2

3

Γ

Raman

↓ + ΓRaman↑

 ́

/|Ω|. During a π

pulse, the qubit spends an equal amount of time in both the | ↑i and | ↓i states, hence

we average the scattering rates of the two qubit eigenstates. To simplify the analysis,

we consider forming a qubit out of a ∆mF = 0 hyperfine transition. Results for different

pairs of hyperfine states which are suitable for use as qubits will differ by factors close

to unity. However, the functional form of the Raman scattering error vs. detuning will

be similar.
174

In § 2.5.2 we calculated the stimulated Raman two-photon Rabi frequency between a pair of ground state levels which were coupled by two lasers to a single excited

state. Generalizing this result, the Rabi frequency for two lasers b and r with polarizations 2l = l+σˆ+ + l0πˆ + l−σˆ− with l ∈ {b, r} for stimulated Raman transitions through

the multiple excited states is given by

Ω = μ

2EbE∗

r

4~

2

X

k

h↑ |d · 2ˆr|kihk|d · 2ˆb| ↓i

∆kμ2

(5.10)

where Elis the electric field amplitude for laser l ∈ {b, r} and the sum in Eq. (5.10) is

over the intermediate excited states |ki. Using the excited states in the 2P1/2 and 2P3/2

manifolds, the Rabi frequency becomes

Ω = μ

2EbE∗

r

4~

2

1⁄2

1

∆

·

2

3

α↓α

∗

↑

b+r

∗

+ +

1

3

(α↓α

∗

↑ + β↓β

∗

↑

)b0r

∗

0 +

2

3

β↓β

∗

↑

b−r

∗

−

 ̧

+

1

∆ − ∆F

·

(

1

3

α↓α

∗

↑ + β↓β

∗

↑

)b+r

∗

+

+

2

3

(α↓α

∗

↑ + β↓β

∗

↑

)b0r

∗

0 + (α↓α

∗

↑ +

1

3

β↓β

∗

↑

)b−r

∗

−

 ̧3⁄4

=

μ

2EbE∗

r

4~

2

2

3

·

1

∆

−

1

∆ − ∆F

 ̧

α↓α

∗

↑

(b+r

∗

+ − b−r

∗

−)

(5.11)

where we have used the orthogonality condition α↑α

∗

↓+β↑β

∗

↓ = 0 for states with the same

mF to simplify the second line of Eq. (5.11). We observe that ˆπ polarized light does not

contribute to the Rabi frequency. Furthermore, using linearly polarized light requires

that the polarizations of the b and r beams be orthogonal to maximize b+r

∗

+ − b−r

∗

−.

Linearly polarized light has the additional advantage that differential Stark shifts between non-field independent transitions is suppressed. For field independent transitions, the differential Stark shift is suppressed independent of polarization. We let

b+ = b− = r+ = −r− = √

1

2

resulting in b+r

∗

+ − b−r

∗

− = 1. Using these polarizations for

the b and r beams, the Raman spontaneous scattering rate (Eq. (5.8)) for atomic state

|ii from both beams is

Γ

Raman

i =

(E2

b + E2

r

)μ

2

4~

2

2

9

γ

 ̄

 ̄

 ̄

 ̄

1

∆

−

1

∆ − ∆F

 ̄

 ̄

 ̄

 ̄

2

(1 + 4α

2

i β

2

i

) (5.12)
175

Simplifying the analysis, we specialize to the case of the |F, mF = 0i ↔ |F

0

, m0

F = 0i

clock transition at low magnetic fields where α

2

i = β

2

i =

1

2

for i ∈ {↑, ↓}, and we let

the electric field amplitude for the two Raman laser beams be equal Eb = Er = E.

Under these conditions, the Raman scattering rate (Eq. (5.12)) is the same for the two

qubit states | ↓i and | ↑i; that is ΓRaman

↓ = ΓRaman↑ ≡ Γ

Raman. Comparing Ω to the

spontaneous scattering rate (Eq. (5.8)), we find the probability of Raman spontaneous

scattering during a π pulse to be

PRaman−π =

π

2

Γ

Raman

|Ω|

=

2πγ

3

 ̄

 ̄

 ̄

 ̄

∆F

∆(∆ − ∆F )

 ̄

 ̄

 ̄

 ̄

. (5.13)

Similarly, the probability of Rayleigh scattering during a π pulse on the clock transition

is given by (ΓRayleigh

↓ = ΓRayleigh↑ ≡ Γ

Rayleigh)

PRayleigh−π =

π

2

Γ

Rayleigh

|Ω|

=

πγ

3∆F

 ̄

 ̄

 ̄

 ̄

(3∆ − ∆F )

2

∆(∆ − ∆F )

 ̄

 ̄

 ̄

 ̄

. (5.14)

Finally, the probability of scattering any photon, Rayleigh or Raman, is given by the

sum of Eqs. (5.13) and (5.14) (Γtotal

↓ = Γtotal↑ ≡ Γ

total):

Ptotal−π =

π

2

Γ

total

|Ω|

=

πγ

∆F |∆(∆ − ∆F )|

h

(∆ − ∆F )

2 + 2∆2

i

. (5.15)

We note that in deriving the relations in Eqs. (5.13), (5.14), and (5.15) we used the

approximation that the detunings ∆, ∆ − ∆F are much greater than the hyperfine

splitting, zeeman splittings, and the power-broadened line width γ

√

s0 + 1 of the excited

2PJ manifolds (J ∈ { 1

2

,

3

2

}) where s0 is the on-resonance saturation parameter.

A plot of the Raman spontaneous scattering error probability vs. detuning

(Eq. (5.13)) is shown in Fig 5.3 along with the total scattering probability (Eq. (5.15))

for comparison. We find that the Raman spontaneous scattering probability during

a π pulse drops quadratically to zero when the detuning ∆ is much larger than the

fine structure splitting ∆F . In contrast, the total scattering probability asymptotically

approaches the finite non-zero value of 3πγ

∆F

. The global minimum in the total spontaneous scattering rate of Ptotal−π =

4πγ

∆F

3

3−2

√

2

3

√

2−4

 ́

' (2.83) πγ

∆F

occurring at a detuning of
176

-3 -2 -1 0 1 2 3 4

1

2

3

4

5

∆/∆F

Pscat/(3πγ/∆F

)

Figure 5.3: Solid curve is the calculated probability of inelastic Raman scattering during

a π pulse. The dashed curve is the probability of total scattering during a π pulse. The

detuning (x-axis) is in units of the fine structure splitting ∆F . The probability (y-axis) is

in units of the limiting probability of scattering at large detunings 3πγ

∆F

;

3πγ

∆F

= 9.3×10−4

for 9Be+.
177

∆ = ¡√2 − 1

¢

∆F between the two fine-structure manifolds is slightly lower than the

asymptotic value. If it were the case that all spontaneously scattered photons would

cause decoherence, then it would appear that an ion with a large fine structure splitting

∆F would be advantageous to reduce the probability of photon scattering. For 9Be+,

the saturated probability of photon scattering is 9.3 × 10−4, considerably high when

compared to Steane’s fault-tolerance threshold [Steane 03]. We note that according to

Knill [Knill 05], having errors below fault-tolerance thresholds is not necessary to perform useful quantum computation. However, it is desirable to push the errors in our

system to very low levels, and we use the 10−4threshold of Steane as a benchmark to

which we can compare all of our errors. Even though it may be possible to perform

useful quantum computing with errors above this threshold [Knill 05], certainly if we

can push our errors well below this threshold, then we are at a good starting point for

building a large scale quantum computer.

By increasing the detuning ∆, we can suppress Raman spontaneous scattering

and hence reduce the error during quantum gates. However, as we see from Eq. (5.11),

as the detuning becomes large compared to the fine structure splitting ∆F , both the

probability of spontaneous Raman scattering during a π pulse and the Rabi frequency

decrease quadratically. That is to say, reducing the error comes at a cost. If we are to

maintain the same Rabi frequency and reduce the error due to spontaneous scattering

by an order of magnitude, we would require an order of magnitude more laser power.

From a different perspective, if we keep the laser power constant, then reducing the

spontaneous scattering by an order of magnitude will slow the gate speed by an order

of magnitude.

We can quantify the laser power requirements for a given probability of spontaneous scattering error as follows [Ozeri 06]. Examining Eqs. (5.11) and (5.13), we can
178

write the probability of Raman scattering during a π pulse as

PRaman−π =

8~

2π|Ω|

E2

γ

μ2

. (5.16)

The natural line width γ and the dipole moment squared μ

2 are proportional, and the

ratio is a function of atomic constants, namely

γ

μ2

=

ω

3

3/2

3~π20c

3

(5.17)

where ω3/2is the |

2S1/2

i ↔ |2P3/2i transition frequency, 20 is the vacuum dielectric

constant, and c is the speed of light. Furthermore, for a gaussian laser spatial mode

with waist w0 at the ion, the electric field amplitude can be expressed in terms of the

waist and the laser power Plaser:

E

2 =

4Plaser

πw2

0

c20

. (5.18)

Using these expressions, the probability of Raman photon scattering during a π pulse

can be written in terms of the Rabi frequency, waist, laser power, and atomic constants:

PRaman−π =

2

3

|Ω|

Plaser

~πw2

0ω

3

3/2

c

2

. (5.19)

We find that the probability of error induced by spontaneous scattering during a π

pulse is proportional to the Rabi frequency and inversely proportional to the laser power.

As was mentioned previously, speed and laser power are trade-offs if the probability of

error is kept constant. From a different perspective, given a required gate speed, the

laser power must be greater than

Plaser ≥

2

3

|Ω|

PRaman−π

~πw2

0ω

3

3/2

c

2

. (5.20)

Furthermore, ions with longer wavelengths have lower spontaneous scattering error for

the same laser power. This results from the fact that the natural line width is proportional to ω

3

3/2

resulting from the fact that higher energy states have shorter lifetimes
179

Ion λ [nm] Plaser [mW]

9Be+ 313 3.36

43Ca+ 393 1.69

113Cd+ 215 10.5

199Hg+ 165 23.0

25Mg+ 280 4.72

87Sr+ 408 1.52

137Ba+ 456 1.09

Table 5.1: Laser power requirements to achieve 10−4spontaneous scattering error during

a π pulse for different ion species.

because of the larger density of states for the electromagnetic vacuum at higher frequencies.

In principle, the waist w0 when diffraction limited can be as small as the wavelength. However, the waist size in practice is set by other experimental parameters such

as the inter-ion spacing when performing two-qubit gates. In the former, the Raman

spontaneous scattering error probability would scale linearly with the frequency ω3/2

whereas in the latter, the error probability scales with the frequency cubed ω

3

3/2

.

The probability of spontaneous scattering error can be compared across different

ion species using Eq. (5.20). In Tbl. 5.1 we show the laser power required in each Raman

beam for different ion species to achieve an error during a π pulse of 10−4. We assume

a waist of w0 = 20 μm and a Rabi frequency of Ω = 2π × 250 kHz corresponding to a

π pulse duration of 1 μs. For 9Be+, a minimum laser power of 3.4 mW is required to

reach a probability of error below 10−4. For the set of ions under consideration for use

in quantum information processing, the range of ω3/2frequencies spans approximately

an octave making the laser power requirements for the different ions fall within an order

of magnitude. This makes the decision of which ion to use for QIP more a function

of technical parameters such as which ion has a wavelength where high power readily

available laser sources exist rather than physical reasons such as which ion has the largest

fine structure splitting. When considering the laser power requirements for fault-tolerant
180

two-qubit gates, the ion mass in addition to the frequency ω3/2is another physical

parameter which plays a role into which ion has the lowest laser power requirements

[Ozeri 06]. The reason for this is that the two-qubit gate speed is proportional to the

Lamb-Dicke parameter η which is inversely proportional to the square root of the mass

(see § 2.6). The largest contribution to the two-qubit gate error is Raman spontaneous

scattering [Ozeri 06], and the probability of error is ∼ 2/η larger than the one-qubit

gate. The ability of two ions to scatter Raman photons gives the factor of 2, and the

gate speed is 1/η slower. A more detailed discussion of the spontaneous scattering error

during two-qubit gates can be found in [Ozeri 06].

5.2 Experimental Verification

In the previous section I gave arguments why elastic Rayleigh spontaneous scattering does not cause decoherence of superposition states. In this section, I describe

the experiment we performed to verify these claims [Ozeri 05]. The Raman spontaneous scattering rate is measured and compared to the measured decoherence rate of

prepared superposition states. It is shown that Raman spontaneous photon scattering

is responsible for coherence decay, and that Rayleigh scattering does not cause decoherence.

In the experiment, a single 9Be+ ion is confined in a single zone of the linear Paul

trap (see § 3.1). The ion is Doppler cooled and prepared in the stretched |Ai ≡ |F =

2, mF = 2i state. Coherent superpositions are then prepared in the | ↑i ≡ |F = 1, mF =

1i and | ↓i ≡ |F = 2, mF = 0i hyperfine ground states (see Fig. 5.2). At a magnetic

field of 0.01194 T, these two states form a qubit with a transition frequency which has

zero first-order dependence on small changes in the magnetic field, and long coherence

times have been observed (see Ch. 4 and [Langer 05]). Both z-basis eigenstates (| ↑i and

| ↓i) and coherent superpositions of the qubit states are prepared by first performing

a π rotation on the |Ai ↔ | ↑i transition using a stimulated Raman pulse (see § 2.5.2)
181

followed by a qubit rotation of the form

R(θ, φ) = cos(θ/2)I − isin(θ/2)(cos φσx + sin φσy) (5.21)

where θ is the rotation angle about an axis in the xy-plane at an angle φ from the

x-axis. Here, σx and σy are Pauli matrices, and I is the identity matrix. The | ↓i state

is prepared by first preparing | ↑i followed by a π qubit rotation. Qubit measurements

are performed via state-dependent resonance fluorescence (see § 2.4). A ˆσ+ polarized

laser resonant with the |Ai ↔ |2P3/2, F = 3, mF = 3i transition frequency applied

to the ion causes the state |Ai fluoresce strongly whereas the other hyperfine states

do not. To measure the probability of the ion to be in a particular qubit eigenstate,

the particular qubit eigenstate is transferred to the state |Ai prior to measurement

through a sequence of π rotations. For example, to measure the state | ↓i, we apply

a π rotation on the qubit transition transferring | ↓i → | ↑i followed by a π rotation

on the | ↑i ↔ |Ai transition transferring the population now in | ↑i to |Ai. The |Ai

state is measured, which effectively measures the population in | ↓i prior to the transfer

sequence. Similarly, to measure | ↑i, we first transfer | ↑i to |Ai through a π rotation

on the | ↑i ↔ |Ai transition prior to |Ai measurement.

5.2.1 Measurement of the Raman Scattering Rate

The Raman scattering rate is measured by performing the experimental sequence

in Fig. 5.4a. The eigenstate |ii ∈ {| ↑i, | ↓i} is prepared as described above. The

ion is then illuminated with the decohering laser beam detuned from the |

2S1/2

, F =

2, mF = 0i ↔ |2P1/2i transition frequency by ∆ with polarization ˆσ+. The population

remaining in state |ii is then measured. The application of the decohering laser beam

causes Raman spontaneous scattering which, by definition, causes the population in the

state |ii to optically pump to other states. By measuring the population remaining

in state |ii as a function of the duration of the decohering beam, we can extract the
182

prep |↑〉, |↓〉 meas |↑〉, |↓〉

decohering (a) beam

(b) prep |↑〉 meas |↑〉

θ = π/2

φ = 0

θ = π

φ = π

θ = π

φ = 0

θ = π

φ = π

θ = π/2

φR = 0, π

τ

echo 2 τecho

N τecho

decohering

beam

Figure 5.4: (a) Experimental sequence for population relaxation measurement. The

eigenstate |ii (i ∈ {↑, ↓}) is prepared followed by application of the decohering laser

beam (thick arrow). The population of state |ii is measured as a function of decohering

beam duration Nτecho. (b) Experimental sequence for coherence decay measurement.

An equal superposition of | ↑i and | ↓i is prepared with the first θ = π/2 pulse. A spinecho sequence follows where θ = π pulses are applied with alternating phase φ = 0, π

such that slow rotation angle errors are refocused as well as slow Stark shifts and Zeeman

shifts. The final θ = π/2 pulse at phases φ = 0, π followed by measurement interrogates

the superposition. The decohering beam is applied in the Ramsey intervals. Coherence

vs. τecho is measured. The number of intervals τecho is N for simple comparison with

the population relaxation rate measurement.
183

rate of population decay. Rayleigh elastic scattering also occurs with application of

the decohering laser beam; however, because Rayleigh scattering, by definition, does

not change the atomic state of the ion, the population remaining in the |ii state is

unaffected. Typical data for population decay is shown in Fig. 5.5. The grey-filled

squares are proportional to the population remaining in the state | ↑i after application

of the decohering laser beam to the prepared | ↑i state. For the data in Fig. 5.5, the

detuning was ∆ = 2π × 227.5 GHz.

The dynamics of population decay can be described using an analysis similar to

that in § 2.4.2. Recall from § 2.4.2 that a ˆσ+ polarized near-resonant laser beam caused

optical pumping of the dark state to the bright state during state-dependent resonance

fluorescence measurements. The dynamics of optical pumping was solved using Laplace

transforms. For application to measurement, we were concerned with the population

optically pumped into the bright state |Ai, whereas here, we are concerned with the

population remaining in the prepared state |ii ∈ {| ↑i, | ↓i}. In both cases, the optical

pumping rates from state |ii to state |fi are given by the Heisenberg-Kramers relation

with a slight modification in the near-resonant case (see § 2.4.3). We were able to

obtain a closed-form solution because ˆσ+ polarized light optically pumps only in one

direction of angular momentum, namely increasing mF . Since we are concerned only

with the population remaining in the prepared state |ii, we can simplify the analysis.

We let |ii ≡ |F, mF i and define F

0

to be the total angular momentum quantum number

other than F such that F

0 6= F. The possible states to which |ii can optically pump

under the application of ˆσ+ polarized light are |F, mF + 1i, |F

0

, mF + 1i, and |F

0

, mF i.

Because |F, mF + 1i and |F

0

, mF + 1i cannot optically pump back to |F, mF i due to the

polarization of the decohering laser beam, population optically pumped to these states

is lost from |F, mF i forever. However, population optically pumped to |F

0

, mF i may

repump back to |F, mF i.

We define vF (t) and vF0(t) to be the population in the states |F, mF i and |F

0

, mF i
184

0 1 2 3 0

4

8

12

Photon counts

τ [ms]

Figure 5.5: Coherence and population relaxation data for a decohering beam detuning

of ∆ = 2π ×227.5 GHz. The difference between the open (closed) circles is proportional

to the coherence of a prepared hyperfine superposition state in the absence (presence)

of the decohering laser for duration τ (x-axis) in each arm of the Ramsey spin-echo

sequence. The gray-filled squares are proportional to the population remaining in the

prepared | ↑i state after the decohering laser has been applied for the duration τ in each

arm of the Ramsey spin-echo sequence. It is clear from the data that the timescales for

coherence relaxation and population relaxation are similar.
185

respectively at time t, and we define v ≡





vF

vF0





to be the two-dimensional vector

of these populations. Given the arguments above concerning the angular momentum

directionality of optical pumping with ˆσ+ polarized light, the rate equation for the

optical pumping process can be written as follows:

 ̇v = Γ · v (5.22)

where

Γ ≡





−ΓF→F0 − ΓF→(mF +1) ΓF→F0

ΓF0→F −ΓF0→F − ΓF0→(mF +1)



 . (5.23)

The terms in Eq. (5.23) are defined as follows: ΓF→F0 is the optical pumping rate from

state |F, mF i to state |F

0

, mF i and ΓF→(mF +1) is the optical pumping rate from |F, mF i

to both |F, mF + 1i and |F

0

, mF + 1i states. Similarly, the rates ΓF0→k are defined as

the optical pumping rates from |F

0

, mF i to states k ∈ {F,(mF + 1)} in an analogous

fashion. The Raman scattering rate out of state |F, mF i is the upper-left term in the

matrix (5.23). We define the hyperfine states in terms of their mJ and mI components

as in Eq. (5.4). Due to the orthonormality condition αiα

∗

j + βiβ

∗

j = δi,j for hyperfine

states |ii and |ji with the same mF , the terms in the matrix (5.23) are constrained,

namely: ΓF→F0 = ΓF0→F and ΓF→(mF +1) + ΓF0→(mF +1) = 1 (δi,j is the Kronecker delta

function).

Using the Heisenberg-Kramers relation [Eq. (5.3)], the matrix Γ can be written

as Γ ≡ Γ0M where

Γ0 ≡

2

9

g

2

γ

 ̄

 ̄

 ̄

 ̄

1

∆

−

1

∆ − ∆F

 ̄

 ̄

 ̄

 ̄

2

, (5.24)

M ≡





−α

2

(1 + 2β

2

) 2α

2β2

2α

2β2 −β2

(1 + 2α

2

)



 , (5.25)

and the symbols in the definition of Γ0 are defined in the previous section (§ 5.1); α

and β are the coefficients of the |mJ , mI i components for the |F, mF i state as defined
186

in Eq. (5.4). It is evident from Eq. (5.25) that ΓF→F0 = ΓF0→F . Similarly, because

ΓF→(mF +1) (ΓF0→(mF +1)) is proportional to α

2

(β

2

), it is also clear that ΓF→(mF +1) +

ΓF0→(mF +1) = 1. With these definitions, the Raman scattering rate out of state |F, mF i

is given by

Γ

Raman

F = Γ0[α

2

(1 + 2β

2

)]. (5.26)

We desire to extract the Raman scattering rate of state |F, mF i from the data in

Fig. 5.5. The coefficients α and β are known from the Breit-Rabi solution (see § 2.2),

and the only free parameter which we would like to extract from the data in Fig. 5.5

is Γ0. Once Γ0 is known, we can calculate the Raman scattering rate using Eq. (5.26).

The functional form to which we fit the data to extract Γ0 is given by the dynamics of

optical pumping which we now derive.

The dynamics of population relaxation can be solved similar to § 2.4.2 using

Laplace transforms. Taking the Laplace transform of Eq. (5.22) and solving for V(s) ≡

L(v(t)) where L(f(t)) is the Laplace transform of f(t) and s is the Laplace transform

variable, we obtain the solution (in s) V(s) = −(Γ0M−sI)

−1v(0) where I is the identity

matrix. After some simplification, the matrix −(Γ0M − sI)

−1

can be written as

−(Γ0M − sI)

−1 =

1

q(s)





s + Γ0(a + β

2

) −aΓ0

−aΓ0 s + Γ0(a + α

2

)



 (5.27)

where a ≡ 2α

2β2 ∈ [0, 1], q(s) = (s + ω+)(s + ω−), and

ω± =

Γ0

2

3

1 + 2a ±

p

4a

2 − 2a + 1 ́

. (5.28)

The frequencies ω± are non-negative real numbers. We are concerned with the dynamics

of vF (t) with the initial conditions vF (0) = 1 and vF0(0) = 0. Therefore, we only consider

the upper-left term in Eq. (5.27). This term can be written as a sum of partial fractions:

VF (s) = s + Γ0(a + β

2

)

Γ0

√

4a

2 − 2a + a

μ

1

s + ω−

−

1

s + ω+

¶

, (5.29)
187

and the time-domain solution is given by the inverse Laplace transform of Eq. (5.29):

vF (t) = (4a

2 − 2a + 1)−1/2

·

e

−ω−t

μ

a + β

2 −

ω−

Γ0

¶

− e

−ω+t

μ

a + β

2 −

ω+

Γ0

¶ ̧ . (5.30)

The terms ω±/Γ0 are independent of Γ0; Γ0 only enters in the exponential terms e

−ω±t

in Eq. (5.30) as a linear scaling factor of the frequencies.

The population decay data in Fig. 5.5 (grey-filled squares) is fit to Eq. (5.30) by

varying Γ0, and the Raman scattering rate is determined by Eq. (5.26).

5.2.2 Coherence in the Presence of Spontaneous Photon Scattering

The coherence remaining in a prepared superposition state after application of

the decohering beam is measured in a similar fashion to the population decay. The

experimental sequence is shown in Fig. 5.4b. A coherent superposition of qubit states

|Ψ1i = √

1

2

(| ↑i − i| ↓i) is prepared by first preparing | ↑i followed by the rotation R(θ =

π/2, φ = 0) [see Eq. (5.21)]. After a spin echo sequence with delay, the superposition is

interrogated with a final R(θ = π/2, φ = 0) pulse followed by measurement of the state

| ↑i. Because the spin echo sequence refocuses any detuning of the local oscillator with

respect to the qubit transition frequency, the measurement of the | ↑i state always yields

either the bright or dark extremum of the equivalent Ramsey phase scan experiment

depending on whether or not there are an odd or even number of spin echo pulses

respectively (see Fig. 4.4a in Ch. 4). The experiment is repeated with the final π/2

pulse having a phase φ = π yielding the opposite extremum. In the absence of the

decohering beam, the measurement record for this experiment yields the open circles

data in Fig. 5.5. The difference between the bright data and the dark data is directly

proportional to the coherence of the superposition state at the time of measurement. We

see from Fig. 5.5 that in the absence of the decohering laser, the coherence is constant

as a function of time indicating we have a good quantum memory, consistent with the

results of Ch. 4.
188

The application of the decohering laser in the intervals between the spin echo

pulses did cause decoherence of the prepared superposition, and data for these measurements are shown as filled circles in Fig. 5.5. The difference between the upper

and lower filled circles is a measure of the coherence, and fitting the difference between

these two curves as a function time to an exponential yields the decoherence rate in the

presence of light. It is notable that the population relaxation measurement (filled-grey

squares) decays on the same time-scale as the coherence. This implies that the rate of

coherence decay is similar to the rate of population decay at the particular detuning

where this data was taken (∆ = 2π × 227.5 GHz in Fig. 5.5).

In order to prove that the decoherence rate is the same as the Raman scattering

rate, we repeated the coherence and population decay measurements for different detunings ∆ of the decohering laser. The differential Stark shift of the qubit transition

frequency was also measured at each detuning. The measured decoherence rate and the

measured Raman scattering rate were normalized by the measured differential Stark

shift to remove any dependence on the laser intensity. The differential Stark shift is

measured rather than the total scattering rate because the total scattering rate is too

small to measure directly. In Fig. 5.6 we plot the normalized measured Raman scattering rate (open circles) and the normalized measured decoherence rate (filled circles) as

a function of laser detuning. Also plotted in Fig. 5.6 are the calculated total scattering

rate (dashed curve) and calculated Raman scattering rate (solid curve) both normalized

by the calculated differential Stark shift for comparison. There are no fit parameters in

the calculated total scattering rate or the calculated Raman scattering rate normalized

by the calculated differential Stark shift. Both these rates are a function of atomic

parameters, namely the fine-structure splitting ∆F = 197.2 GHz, the excited state linewidth γ = 19.4 MHz, and the hyperfine state coefficients α and β derived from the

Breit-Rabi formula (see § 2.2). It is clear from Fig. 5.6 that both the measured Raman scattering rate and the measured decoherence rate are in good agreement with the
189

Figure 5.6: The measured decoherence rates (filled circles) and population relaxation

rates (open circles) normalized by the measured differential Stark shift are plotted vs.

the detuning of the decohering laser beam. The dashed curve is the calculated total

scattering rate normalized by the calculated differential Stark shift. The solid curve

is the calculated Raman scattering rate normalized by the calculated differential Stark

shift. Similar to Fig. 5.3, the total scattering rate normalized by the differential Stark

shift asymptotically approaches a finite non-zero value of ' 0.0154 whereas the Raman

scattering rate normalized by the differential Stark shift quadratically approaches zero

for detunings large compared to the fine structure splitting. Both the measured normalized decoherence rates (filled circles) and population decay rates (open circles) are

in good agreement with the calculated normalized Raman scattering rate curve (solid

line) indicating that Raman spontaneous scattering is responsible for both population

relaxation and decoherence of superposition states and that elastic Rayleigh scattering

does not cause decoherence.
190

calculated Raman scattering rate curve implying that Raman spontaneous scattering is

responsible for coherence decay. In addition, Rayleigh elastic scattering does not cause

decoherence of superposition states. In particular, at a detuning of ∆ = −331.8 GHz,

the measured decoherence rate is more than a factor of 19 below the calculated total

scattering rate implying that coherence is maintained in the presence of spontaneous

photon scattering.

The measured Raman scattering rate and the measured decoherence rate are

compared directly in Fig. 5.7. Fig. 5.7 plots the measured decoherence rate (y-axis)

against the measured Raman scattering rate (x-axis), both which are normalized by

the measured Stark shift. The solid curve is a weighted least squares fit to the power

law y = x

α yielding the best-fit parameter α = 0.997(4). This comparison implies that

the Raman scattering rate and the decoherence rate are the same. From a different

perspective, the scattering of one Raman photon is sufficient to destroy coherence. We

recall from the study of two-level atoms that the decoherence rate is bounded below by

half of the excited state decay rate [Metcalf 99] which is a population relaxation process.

Here, we find that the decoherence rate is bounded below by the Raman scattering rate,

also a population relaxation process. The reason that the two-level atom has a lower

bound to the decoherence rate half of the excited state decay rate and not the excited

state decay rate itself is because only one of the two states of the two-level atom decays,

namely the excited state. The ground-state does not decay in the two-level atom model.

However, in the presence of an off-resonant decohering laser, both hyperfine states decay

due to spontaneous scattering which makes the lower bound to the decoherence rate the

Raman scattering rate itself and not half. If in some model of the two-level atom the

ground state also decayed to other levels at some rate, then the decoherence would be

bounded below by the average of the two rates—the excited state decay rate and the

ground state decay rate. If the ground state decay rate is the same as the excited state

decay rate, then the decoherence rate is the same as the population decay rate.
191

0.00 0.01 0.02

0.00

0.01

0.02

1/(

τdec

∆St

)

ΓRaman/∆St

y = x

α

 α = 0.997(4)

Figure 5.7: The measured coherence decay rate (y-axis) is plotted against the measured

population relaxation rate (x-axis), both normalized by the measured differential Stark

shift. The filled circles are measured data; the solid line is a weighted least squares fit

to the power law y = x

α yielding the best-fit parameter α = 0.997(4) indicating that 1

Raman scattered photon is sufficient to destroy the coherence of a superposition state.
192

The decoherence rates we measured were very low for the larger detunings, and

successful measurement of the decoherence rate limited by spontaneous scattering required suppression of all other decohering mechanisms below that of Raman spontaneous

scattering. In particular, laser intensity noise induces phase noise due to the differential

Stark shift of the qubit transition frequency. The total scattering rate normalized by the

differential Stark shift asymptotically approaches the value of 0.0154. Consequently, in

order to measure decoherence rates dominated by Raman spontaneous scattering, the

phase noise induced by the differential Stark shift had to be suppressed to well below

this value. For the particular detuning of ∆ = −331.8 GHz where we measured a decoherence rate over a factor of 19 below the total scattering rate, the phase noise induced

by the differential Stark shift was below 10−3.

Multiple experimental techniques were used to suppress Stark shift phase noise.

First the laser beam propagated through a short 30 cm section of optical fiber close

to the trap to reduce beam pointing fluctuations after the output of the fiber. Beam

pointing fluctuations into the fiber remained a problem, and fluctuating beam position

on the entrance to the fiber translated into laser power noise at the fiber exit. This

was overcome by sampling the laser power after the fiber and feeding back on the RF

power level of an acousto-optic modulator before the fiber. The bandwidth of the laser

power servo was limited to less than 100 kHz; therefore, fast noise was not controlled.

However, the phase noise induced by the differential Stark shift is the integrated Stark

shift frequency noise, and high frequency noise averages to very low phase noise levels

upon integration. The limited servo bandwidth was more of a problem when switching the decohering laser on as the control system would require ∼ 10 μs to lock after

turning on the laser. The spin-echo arm length τecho (see Fig. 5.4b) was typically in the

few milliseconds range before spontaneous scattering decohering effects would manifest

themselves, and the ∼ 10 μs Stark shift noise, after integration on the ion, was undetectable. The final experimental technique we used to suppress Stark shift phase noise
193

was the spin-echo sequence in Fig. 5.4b. Stark shift noise with frequencies much lower

than 1/τecho would refocus under the spin-echo sequence as would any slowly changing

systematic frequency shift, occurring from magnetic field drift, for example. We found

that we could not tolerate spin-echo intervals τecho longer than a few milliseconds. At

the time of the experiment, we speculated that the laser must have some 60 Hz intensity

noise which limited the spin-echo technique to durations shorter than the 17 ms period

of the 60 Hz cycle even though we could not detect laser power noise at 60 Hz. Our

speculations were that the photo detector used to servo the laser power may have had

some 60 Hz noise which it injected into the laser via the feedback circuit. In hindsight

we realized that the ˆσ+ polarized decohering laser beam shifted the location of the magnetic field independent point, and that magnetic field noise at 60 Hz was limiting the

spin-echo arm duration τecho to much less than 17 ms (see § 5.3).

Because the spin-echo duration τecho was limited to on the order of a few milliseconds, a large number of spin-echo pulses were added to semi-continually refocus

systematic frequency shifts induced by Stark shifts and Zeeman shifts. For the larger

detunings, we used 18 spin-echo pulses. This made the measurement more susceptible

to rotation angle errors induced by intensity noise on the Raman lasers. We circumvented this problem by alternating the phase of subsequent spin-echo pulses. This had

the effect that if the Rabi frequency drifted at a rate much slower than 1/τecho, then

rotation angle errors on a spin-echo pulse would be automatically corrected on the next

pulse. For example, if the first spin-echo pulse over-rotated by an angle 2 about the

x-axis, because the phase φ of rotation on the next spin-echo pulse was π, the rotation

of the subsequent spin-echo pulse, about the −x-axis would undo the over-rotation so

long as the Rabi frequency did not change dramatically between application of the two

pulses. This is effectively a refocussing scheme for the rotation about the x-axis whereas

the usual spin-echo sequence refocuses phase rotation about the z-axis.

The techniques outlined in the previous two paragraphs were necessary to sup-
194

press phase noise induced by differential Stark shifts to levels below that of Raman

spontaneous scattering. In principle, the same techniques can be used when performing quantum gates or quantum algorithms. In conclusion, by detuning the laser far

from the fine structure, we should be able to perform quantum gates below the faulttolerance threshold. In particular, at the detuning ∆ = −331.8 GHz used in Fig. 5.6,

the probability of Raman scattering during a π pulse is ∼ 1 × 10−5(see Eq. (5.13)).

Two-ion gates require application of the laser for a duration 1/η longer than the π-pulse

one-qubit gate where η is the Lamb-Dicke parameter, and the presence of a second ion

increases the probability of Raman scattering during a two-qubit gate to be 2/η times

larger than the one-qubit gate [Ozeri 06]. For the typical Lamb-Dicke parameter used

in the laboratory of η ' 0.3, the probability of Raman scattering during a two-qubit

gate is approximately ∼ 6 × 10−5for ∆ = −331.8 GHz. Both the one-qubit gate error

probability and the two-qubit gate error probability due to spontaneous photon scattering are below Steane’s fault-tolerance threshold [Steane 03], and we have experimentally

measured decoherence induced by spontaneous Raman scattering at this level.

5.3 Field-Independent Point Shift due to the Light Shift

For circularly polarized light propagating along the magnetic field direction, the

differential Stark shift of the qubit transition is dependent on the magnetic field strength.

The Stark shift turn modifies the magnetic field at which first-order sensitivity of the

qubit transition frequency vanishes. The experiments described in this chapter used a

σˆ+ polarized laser beam as the decohering beam, and consequently we could not simultaneously work at the field-independent point for both situations where the decohering

beam was on and when the decohering beam was off. We did not realize this at the time,

and we tuned the magnetic field such that the ion was at the field-independent point in

the absence of light. When the decohering beam was on, the qubit transition frequency

picked up an linear dependence on the magnetic field, and the qubit became more sus-
195

ceptible to magnetic field noise. In hind-sight, we believe that the reason we could not

use spin-echo arms longer than a few milliseconds was because 60 Hz magnetic field

noise was causing significant decoherence. In this section, I derive the field-independent

point shift induced by the differential Stark shift dependence on the magnetic field.

Hyperfine states labeled |li = αl|mJ = −1/2, mI = mF + 1/2i + βl|mJ =

+1/2, mI = mF − 1/2i in the presence of an off-resonant laser beam with polarization ˆ2 = 2ππˆ + 2−σˆ− + 2+σˆ+ will have an AC Stark shift given by:

δs =

1

3

g

2

2

2

π

μ

1

∆

+

2

∆ − ∆F

¶

+

2

3

g

2

∆

¡

α

2

2

2

+ + β

2

2

2

−

¢

+

g

2

∆ − ∆F

·

2

2

+

μ

1

3

α

2 + β2

¶

+ 2

2

−

μ

α

2 +

1

3

β

2

¶ ̧

=

1⁄2

g

2

3∆

£

2

2

π + 22

2

−

¤

+

g

2

3(∆ − ∆F )

£

22

2

π + 2

2

−

¤

3⁄4

+

2

3

g

2α2

¡

2

2

+ − 2

2

−

¢

·

1

∆

−

1

∆ − ∆F

 ̧

(5.31)

where the bracketed {} term in Eq. (5.31) is not a function of the magnetic field. The

magnetic field dependence enters Eq. (5.31) through the coefficient α (see Eq. (2.9) in

§ 2.2). We have replaced β

2 with 1−α2

in the last line of Eq. (5.31) by the normalization

condition. We note that the Stark shift induced by linearly polarized light (2

2

+ = 2

2

−) has

no magnetic field dependence. Therefore, we need not worry about memory decoherence

induced by magnetic fields when performing quantum gates with laser beams as the laser

beams are linearly polarized.

The differential Stark shift of the qubit transition can be calculated using Eq. (5.31)

and the fact that α

2

is nearly the same4for both states of a field-independent qubit (see

§ 2.2). We label the states of the qubit as |0i and |1i with corresponding coefficients α0

4 The coefficients actually differ by roughly 10−4 due to the nuclear Zeeman effect. See § 2.2.
196

and α1 respectively. The differential Stark shift between these states is:

∆δs = const. −

2

3

g

2

¡

2

2

+ − 2

2

−

¢

·

α

2

0ω0

μ

1

∆2

−

1

(∆ − ∆F )

2

¶

− ∆α

2

1

μ

1

∆

−

1

∆ − ∆F

¶  ̧

' const. + cs∆α

2

1

(5.32)

where ∆α

2

1 ≡ α

2

1 −α

2

0

, cs ≡

2

3

g

2

¡

2

2

+ − 2

2

−

¢

3

1

∆ −

1

∆−∆F

 ́

, and we have used the fact that

the qubit transition frequency ω0 ¿ ∆1/2, ∆3/2. We absorbed the α

2

0ω0

3

1

∆2 −

1

(∆−∆F )

2

 ́

term into the constant term because its dependence on the magnetic field is suppressed

compared to cs∆α

2

1 by a factor of ∼ ω0/∆, ω0/(∆−∆F ). The qubit transition frequency

in the presence of a Stark shifting laser beam can now be described as

ω(B) = ω0 + c2∆B

2 + c(1)

s ∆B + c

(2)

s ∆B

2

(5.33)

where the constant term in Eq. (5.32) has been absorbed into ω0. The constant c2 is

the second order coefficient calculated using the Breit-Rabi formula (see § 2.2), and

the coefficients c

(1)

s and c

(2)

s are the first and second order coefficients from a Taylor

expansion of ∆δs, namely: c

(1)

s ≡ cs

∂∆α

2

1

∂∆B

|B0and c

(2)

s ≡

1

2

cs

∂

2∆α2

1

∂∆B2 |B0

. We recall from

§ 2.2 that ∆α

2

1

(B0) = 0, and the magnetic field dependence of ∆α

2

1

can be determined

from Eq. (2.9). We minimize Eq. (5.33) by differentiating, setting to zero, and solving

for ∆B to find the new location of the magnetic field independent point:

∆B0 = −

1

2

c

(1)

s

c2 + c

(2)

s

. (5.34)

For the |F = 2, mF = 0i ↔ |F = 1, mF = 1i qubit used in this and the previous

chapters, c

(1)

s = 0.485 cs

∂x

∂B and c

(2)

s = −0.0141 cs

¡

∂x

∂B ¢2

where ∂x

∂B = −

μBgJ (1−g

0

I

)

A~2 =

−4.483 × 10−5 μT

−1

is obtained from the x definition in § 2.2 following Eq. (2.9). The

magnetic field independent point offset can be re-expressed using these numbers as:

∆B0 = −

1

2

s0

(−8.53 × 10−3)

h

(2

2

+ − 2

2

−)

3

γ

∆ −

γ

∆−∆F

 ́i−1

+ (1.3 × 10−6)s0

[μT] (5.35)
197

We find from Eq. (5.35) that the deviation of the magnetic field independent point

saturates near 0.5 T. However, for typical detunings in the tens of GHz range, saturation

does not occur until the on-resonance saturation parameter is on the order of 106(∼1 W

laser power in ∼30 μm waist). The spontaneous scattering experiments in this chapter

used detunings in the hundreds of GHz range with an on-resonance saturation parameter

of s0 ∼ 105causing a deviation of the magnetic field independent point by 1 mT. This is

quite far away from the optimal point. The qubit transition acquires a linear magnetic

field dependence with slope of 600 Hz/μT. With a 0.1 μT step in the magnetic field, we

would expect the qubit to dephase by 1 radian in 17 ms.
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Appendix A

Glossary of Acronyms

AC: Alternating current—generic term for describing electronic signals which vary in

time.

ADC: Analog to digital converter—integrated circuit which converts analog signals to

digital signals.

DAC: Digital to analog converter—integrated circuit which converts digital signals to

analog signals.

DC: Direct current—generic term for describing electronic signals which are constant

in time.

DDS: Direct digital synthesis—frequency synthesis technique where output is a sampled sinusoid generated by a digital frequency.

DIP: Dual-inline package—a type of electronics package where there are two parallel

rows of pins.

FPGA: Field programmable gate array—reconfigurable programmable logic device.

HDL: Hardware description language—language used to describe digital logic circuits.

Most common HDLs are VHDL and Verilog.

I/O: Input/output.
210

kB: kilobyte—1024 bytes.

LSB: Least significant bit.

LVTTL: Low voltage TTL (transistor-transistor logic)—logic family where logic ’1’ is

represented by 3.3 V and logic ’0’ is represented by 0 V.

LVDS: Low voltage differential signal—differential digital communications specification for 100 Ω transmission lines. Communications speeds can reach hundreds

of MHz. Differential voltage swing is 350 mV.

MSB: Most significant bit.

MSPS: Mega-samples per second.

NIST: National Institute off Standards and Technology

PCI: Peripheral component interconnect—local bus standard for connecting peripherals to computers. Clock speeds are 33 or 66 MHz, and the data word is 32 bits

wide.

PLL: Phase-locked loop—feedback control loop used to lock the phase of one oscillator

to another.

PMT: Photo-multiplier tube—photon counting detector used for detecting single photons in low-light conditions.

QIP: Quantum information processing.

RF: Radio frequency.

SSB: Single side band—frequency modulation technique which suppresses one sideband

in favor of another.

TTL: Transistor-transistor logic—logic family where logic ’1’ is represented by 5 V and

logic ’0’ is represented by 0 V.
211

USB: Universal serial bus—serial communications standard common in most modern

computers.

VHDL: VHSIC (very high speed integrated circuit) hardware description language.

VHSIC: Very high speed integrated circuit.
Appendix B

Daughter Board Schematics

B.1 FPGA Daughter Board

FPGA daughter board schematic page 1/3.
213

FPGA daughter board schematic page 2/3.
214

FPGA daughter board schematic page 3/3.
215

FPGA daughter board layout.
216

B.2 DDS Daughter Board

DDS daughter board schematic page 1/3.
217

DDS daughter board schematic page 2/3.
218

DDS daughter board schematic page 3/3.
219

DDS daughter board layout.
Appendix C

Histogram Data Analysis using Maximum Likelihood

The measurement process for ion-trap quantum information processing (QIP)

experiments typically involves collecting a number of photons on a detector for a fixed

detection interval. Many experiments are performed for the same set of experimental

parameters resulting in a histogram of photon counts (see § 2.4). From this histogram,

we would like to know what fraction of the total experiments are M out of N ions

fluorescing. I demonstrate the maximum likelihood method to determine this fraction.

C.1 The Problem

Given: (1) a histogram h(n) of photon counts in a fixed detection interval where

n is the number of photon counts and N ≡

P

n

h(n) is the total number of experiments

and (2) a parameterized model probability distribution P(n; a) from which these events

were selected where a is a vector of parameters for the distribution, find the parameters

a which maximize the likelihood function L(a). In practice, the parameters a are the

statistical weights and means of bright and dark photon count distributions.

C.2 Background—Least Squares Fit as a Maximum Likelihood

Estimator

For data (xi, yi), i ∈ {1, 2, · · · , N} which are assumed to fit a parameterized model

y(x; a), the method of weighted least squares is commonly used to determine the “best
221

fit” parameters a. In this method, the χ

2

statistic,

χ

2 =

X

N

i=1

μ

yi − y(xi; a)

∆yi

¶2

, (C.1)

is minimized where ∆yiis the measurement error on yi.

Minimizing χ

2

is equivalent to maximizing the likelihood function where the data

(xi, yi) are normally distributed about the model y(x; a) with variance for point (xi, yi)

given by ∆y

2

i

[Press 03]. The likelihood function L(a) is defined as the probability of

measuring data (xi, yi) given parameters a, namely:

L(a) ∝

Y

N

i=1

exp μ−

1

2

μ

yi − y(xi; a)

∆yi

¶2¶

. (C.2)

Maximizing L is equivalent to minimizing its negative logarithm, which is equivalent to

minimizing χ

2

in Eq. (C.1).

The problem we are trying to solve is not one of finding the best fit parameters

a of a model y(x; a) where x is an independent variable and y is a dependent variable.

Rather, we have a set of photon counts {xi} all of which are independent; the data are

not paired data (xi, yi) as is the case with least squares fitting, and we want to determine

from which probability distribution (parameterized by a) the data {xi} came. Thus, it

would be incorrect to attempt to fit the histogram of counts h(n) to a parameterized

distribution using minimum least squares. As an example, suppose N = 105experiments

were performed measuring the photon counts from a single ion that is initially prepared

in a superposition of bright and dark states. The data is a histogram of counts with

maximum photon number no more than 40. If we blindly use least squares fitting we

would make the association xi = n and yi = h(n) where i ∈ {0, 1, · · · , 40} effectively

give the fitting routine ∼40 data points even though we performed 105experiments.

Some questions naturally arise which we must answer if we attempt to use least squares

fitting: (1) What is the error ∆yi? (2) How do you handle zero counts [h(n) = 0 for

some n]? (3) What is the error for zero counts?
222

A natural guess for the error ∆yiis 1 for all i because if we happened to perform

N − 1 experiments instead, one of the h(n) would have 1 less experiment. However, we

know h(n) exactly, thus we should expect to have zero error. We might think about

estimating the error in xi or n, but this is also discrete which makes it difficult to

estimate. I propose there is a better way to determine the parameters a, namely the

method of maximum likelihood [Freund 92].

C.3 Method of Maximum Likelihood

Suppose we have a probability distribution P(n; a) parameterized by the vector

a of measuring n photons, and we have a measured histogram of counts h(n). The

likelihood function is:

L(a) ∝

Y

N

i=1

P(xi; a) (C.3)

where xiis the number of photons counted for experiment i ∈ {1, 2, · · · , N}. The

histogram h(n) is defined as the multiplicity of events where n photons were measured,

thus the likelihood function can also written as

L(a) = Y∞

n=0

C(n, h(n))P(n; a)

h(n)

, (C.4)

where C(n, h(n)) is a combinatorial factor and N =

P

n

h(n). The maximum likelihood

estimate a0 is calculated by maximizing the likelihood (or equivalently and more simply

lnL):

lnL = const. +X∞

n=0

h(n) ln P(n; a) (C.5)

which is equivalent to solving the system of equations:

∂ lnL

∂ak

=

X∞

n=0

h(n)

P(n; a)

∂P(n; a)

∂ak

= 0. (C.6)

In practice, h(n) = 0 for n > nmax for some nmax relatively small (for one ion, nmax ∼

50), and we can safely truncate the sum in Eq. (C.6) at n = nmax as each term contains

a factor of h(n).
223

Even for linear models P(n; a) = P

k

akfk(n), L and lnL are non-linear, thus

non-linear root finding or non-linear minimization is required to find a0. In general,

non-linear minimization is numerically easier than non-linear root finding [Press 03].

Once a0 is determined, error estimates on a0 are straight forward. For the number of

experiments N relatively large, the likelihood function is strongly peaked about a0. We

can approximate the likelihood function as a multivariate gaussian:

L(a) ' exp μconst. +

1

2

δa

T Aδa + · · · ¶

(C.7)

where δa ≡ a − a0 and

Aij ≡

∂

2

∂ai∂aj

lnL(a)

 ̄

 ̄

 ̄

 ̄

a0

= −

X

n

h(n)

·

∂P(n; a)

∂ai

∂P(n; a)

∂aj

1

P(n; a)

2

−

∂

2P(n; a)

∂ai∂aj

1

P(n; a)

 ̧ ̄

 ̄

 ̄

 ̄

a0

(C.8)

are the negative inverse elements of the covariance matrix.

This approximation is valid for large N; however, a more robust method for

determining the error is to construct synthetic data sets of the same size (N) from the

newly found fitted parameters, perform the fitting procedure for these synthetic data

sets, and calculate the sample variance on the best fit parameters for the synthetic data

sets. This method of resampling is robust and can be used to analyze any bias in the

fitted parameters [Press 03].

For linear models,

P(n; a) ≡

X

k

fk(n)ak, (C.9)

the equation for maximum likelihood [Eq. (C.6)] and the A matrix have a simpler form:

∂ lnL

∂ak

∝

X

n

h(n)

fk(n)

P(n; a)

(C.10)

Aij = −

X

n

h(n)

fi(n)fj (n)

P(n; a0)

2

(C.11)
224

Notice that Aii is negative indicating that the maximum likelihood occurs at a0 rather

than the minimum likelihood.

C.4 Examples

C.4.1 Single Poisson Distribution

Suppose we assume the photon counts are distributed according to the Poisson

distribution. Then, a = ν and P(n; a) = P(n; ν) where

P(n; ν) ≡

ν

n

e

−ν

n!

(C.12)

From Eq. (C.6) using Eq. (C.12), we must solve:

0 = X

n

h(n)

∂

∂ν ln P(n; ν)

=

X

n

h(n)

∂

∂ν μ

n ln ν − ν − ln n!

¶

=

1

ν

X

n

h(n)n −

X

n

h(n)

= N

μ

n ̄

ν

− 1

¶

,

(C.13)

indicating ν =  ̄n, the sample mean [ ̄n ≡

1

N

P

n

h(n)n].

The error σν = 1/

√

−Aνν is evaluated as follows.

−Aνν = −

∂

2

∂ν2

lnL = −

∂

∂ν

X

n

h(n)

μ

n

ν

− 1

¶

=

X

n

h(n)

n

ν

2

 ̄

 ̄

 ̄

 ̄

ν0

=

N

n ̄

(C.14)

We find σν =

p

n/N  ̄ is the distribution error (√n ̄) averaged down by 1/√N as expected.

C.4.2 Weighted Poisson Distribution

When performing repeated measurements of identically prepared states of M ions,

the expected histogram of measured photon counts is a weighted sum of the photon count

distributions of having k ∈ {0, 1, · · · , M} ions bright. The photon count distribution
225

of k ions bright is a Poisson distribution with mean νk. We let P(n; a) be defined as

follows:

P(n; a) ≡

μ

1 −

X

M

k=1

bk

¶

g(n; ν0) +X

M

k=1

bkg(n; νk) (C.15)

where a = a(b1, b2, · · · , bM), νk are defined positive constants, and g(n; νk) is a poisson

distribution with mean νk. Here, bk ≥ 0 is the weight of the k-ions bright poisson

distribution with mean νk, and PM

k=1 bk ≤ 1. Alternatively, bk is the probability that

k ions are bright. We would like to maximize the likelihood by varying bk for k ∈

{1, 2, · · · , M} and hence determine from the data the probability that k ions are bright.

We can use non-linear minimization routines to minimize:

− lnL = −

X

n

h(n) ln P(n; a) (C.16)

and the errors can be calculated as

σbk =

μX

n

h(n)

gk(n)

2

P(n; a0)

2

¶−1/2

(C.17)

This treatment assumes νk are fixed. However, νk may as well be varied, and

a would be a function of these as well. In this case, P(n; a) is no longer linear in ak;

however, it is still linear in the weights bk. In this case, Eq. (C.17) is valid for the errors

in the bk, but it is not valid for errors in the νk. For errors in νk, Eq. (C.8) must be

used instead.

C.5 Experimental Data Fitting in Practice

In practice, we collect a large number of histograms h(n) for different experimental

parameters. If the detection interval is the same for these histograms, then the different

histograms can be summed together to generate a single large histogram hr(n). This

histogram can be fit letting both the weights bk and means νk vary to determine the

reference poisson distributions defined by the set {νk}. The means {νk} may have

constraints such as νk(ν0, ν1) = ν0+kν1. Once {νk} are determined, we can fit individual
226

histograms h(n) varying only the weights bk. The errors in bk would then be given by

Eq. (C.17).