Light-Assisted Collisions and Quantum State Tomographyof Single-Atom Motion in Optical Tweezers
byMark O. Brown
B.S., Virginia Polytechnic Institute and State University, 2015
M.S., University of Colorado, Boulder, 2017

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2022
Committee Members:
Cindy Regal, Chair
Jose D’Incao
Ana Maria Rey
Adam Kaufman
Kelvin Wagner

ii
Brown, Mark O. (Ph.D., Physics)
Light-Assisted Collisions and Quantum State Tomography of Single-Atom Motion in Optical Tweezers
Thesis directed by Prof. Cindy Regal

This thesis presents experiments with single 87 Rb atoms trapped in arrays of tightly confining
optical tweezers. Optical tweezer arrays are a pioneering new platform for studying ultracold atomic
physics and quantum computation. I begin by presenting our recent work in the preparation oflarge atom arrays and the quantum motional control of single atoms. Then, I present a new opticaltweezer loading technique and our studies of the processes by which single atoms can be loadednear-deterministically into the microtraps. These processes depend sensitively on the long-range
molecular physics of pairs of atoms, so I additionally discuss our theoretical calculations of themolecular potential landscapes relevant for such collisions. Finally, I present our all-mechanicaltime-of-flight-based quantum tomography of atoms in carefully prepared excited motional states ofthe optical tweezers.

iii

Acknowledgements

I must begin by thanking my advisor, Cindy. Even as a young graduate student, Cindy
placed a great deal of trust in me and gave me a great deal of freedom and responsibility in runningthe lab in B232. Combining that with the right amount of guidance and mentorship resulted in anexperience I do not think I could have gotten anywhere else. I am leaving with more confidence,experience, and knowledge than I could have possibly imagined at the start of my PhD. For that I
will be forever grateful.
I must of course also thank those who built the B232 lab for me to work in: Adam and Brian.
My first day in the lab was Adam’s last day, but I have referenced his thesis more times than I
can count. I was lucky enough to work with and learn from Brian for the first year and a halfof graduate school. When Brian finished his PhD, the lab postdoc Yiheng Lin carried me frombumbling ignorance to a place where I could really run the lab myself. After Yiheng left, Tobias
Thiele would become a great friend, always being up for getting coffee or lunch in just 5 minutes.
Tobias was always willing to discuss any problems we encountered and was a great mentor to me
throughout the middle of my PhD. Additionally I must thank Ana Maria Rey and her theory group,
especially Sean and Robert, for all of the fruitful discussions we have had over the years.
A special thanks go out to the Randall, Max, Daniel, Thanmay, Ting You, Chris, Zhenpu,
Ting-Wei, Willa, Steven, and the optomechanics crew. Thanks for your friendship and all of theirwork and scientific contributions over the years. Willa and Steven will continue to do great work in
the B232 I’m sure. I look forward to seeing what new developments come out of the lab!

Contents

Chapter
1 Introduction

1

1.1

Optical Tweezer Arrays: Historical Background . . . . . . . . . . . . . . . . . . . . .

1

1.2

This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.3

Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.4

List of Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52 Apparatus for Ground-State Optical Tweezer Experiments

6

2.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2

An Updated Physical Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3

2.4

6

2.2.1

Laser Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.2

Mechanical Shutters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.3

Sill Objective Lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Computer Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.1

Chimera Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3.2

Climate Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Other Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4.1

Intraaction AODs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.4.2

NI 5451 Arbitrary Waveform Generator . . . . . . . . . . . . . . . . . . . . . 30

2.4.3

Other Devices of Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

v
3 Raman Sideband Cooling, Tunneling, Rearrangement, and Other Techniques

34

3.1

Larger Tweezer Arrays

3.2

The Evening of Optical Tweezers / Intermodulation Compensation

3.3

Flashing Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.4

Raman Sideband Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.5

3.6

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
. . . . . . . . . 36

3.4.1

Use of In-Trap ΛGM for Better Initial Cooling . . . . . . . . . . . . . . . . . 39

3.4.2

Raman Sideband Cooling of Larger Arrays . . . . . . . . . . . . . . . . . . . 43

Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.5.1

Dumping Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.5.2

Considerations on Tunneling Fidelity . . . . . . . . . . . . . . . . . . . . . . . 47

3.5.3

Recent Two-Tweezer Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.5.4

Three-Tweezer Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.5.5

Predicting Two-Particle Tunneling with Interactions . . . . . . . . . . . . . . 56

3.5.6

Adiabatic Ramsey Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Rearrangement Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.6.1

Flashing Rearrangement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.6.2

Jerk Rearrangement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.6.3

Stiff Rearrangement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4 Gray-Molasses Based Optical-Tweezer Loading: Controlling Collisions for Scaling AtomArray Assembly

70

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.2

Loading Studies and Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.3

4.2.1

Loading Experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.2.2

Loading in Large Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.2.3

Loading Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Recent Further Explorations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

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4.4

Role of Λ-enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.5

Wider 87 Rb D1 F ′ = 2 Parameter Scans . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.6

Loading on the 87 Rb D1 F ′ = 1 Transition . . . . . . . . . . . . . . . . . . . . . . . . 81

4.7

Technical Details for Grey Molasses Optical Tweezer Loading and Rearranging . . . 83

4.8

4.7.1

Optical Tweezers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.7.2

Laser Cooling and Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.7.3

Imaging, Data, and Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.7.4

Scaling Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5 Molecular Potentials for Ultra-Long Range 87 Rb Dimers

88

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.2

Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.3

Definitions, Notation, and Key Relations . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.3.1

5.4

5.5

Angular Momentum Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

Born-Oppenheimer Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.4.2

Λ Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.4.3

Spin Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.4.4

ÎBO Inversion Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.4.5

κ̂BO Kronig Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.4.6

Born-Oppenheimer Symmetry Labels

5.4.7

Resonant Dipole Born-Oppenheimer Interactions . . . . . . . . . . . . . . . . 100. . . . . . . . . . . . . . . . . . . . . . 100

Fine Structure Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.5.1Ω Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.5.2

ÎFS Inversion Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.5.3

κ̂FS Kronig Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

vii

5.6

5.5.4

Fine Structure Symmetry Labels . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.5.5

Fine Structure Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.5.6

HF S Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Hyperfine Structure Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.6.1|Φ| Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.6.2

ÎHFS Inversion Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.6.3

κ̂HFS Kronig Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.6.4

Hyperfine Structure Symmetry Labels . . . . . . . . . . . . . . . . . . . . . . 111

5.6.5

Hyperfine Structure Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.6.6

HF S Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.6.7

HHF S Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.7

Hyperfine Potentials Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . 115

5.8

Discussion of Rotation Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.9

Fine Structure and Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.10 Hyperfine Structure With Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.11 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6 Time-of-Flight Quantum Tomography of Single Atom Motion
6.1

6.2

131

Optical tweezers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.1.1

Tweezer generation and control via acousto-optic deflectors . . . . . . . . . . 139

6.1.2

Tweezer loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

6.1.3

Trap depth and frequency calibrations . . . . . . . . . . . . . . . . . . . . . . 142

6.1.4

Excited Fock state preparation via tunneling . . . . . . . . . . . . . . . . . . 144

Imaging methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.2.1

Imaging setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6.2.2

In-tweezer RPGC imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6.2.3

Time-of-flight imaging characterization . . . . . . . . . . . . . . . . . . . . . . 148

viii
6.2.4
6.3

6.4

6.5

Time-of-flight image analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Single-atom temperature characterization . . . . . . . . . . . . . . . . . . . . . . . . 155
6.3.1

Raman sideband spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

6.3.2

Ballistic expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Quantum state tomography and related characterizations . . . . . . . . . . . . . . . 156
6.4.1

Single Fock-state momentum distribution analysis . . . . . . . . . . . . . . . 156

6.4.2

Maximum likelihood estimation algorithm . . . . . . . . . . . . . . . . . . . . 158

6.4.3

Alternative tomography methods . . . . . . . . . . . . . . . . . . . . . . . . . 161

Error estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.5.1

Estimating statistical error through bootstrapping . . . . . . . . . . . . . . . 165

6.5.2

Noise and imaging systematic effects . . . . . . . . . . . . . . . . . . . . . . . 169

6.5.3

Trap anharmonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Bibliography

176

ix

TablesTable
2.1

Table of Selected 87 Rb Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2Sill Objective Lens Aberration Coeﬀicients . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1

Tunnel-Coupled Tweezers Analytic Calculations . . . . . . . . . . . . . . . . . . . . . 57

5.1

Angular Momentum Definitions and Other Notation. . . . . . . . . . . . . . . . . . . 129

5.2

Hund’s Case Energy Hierarchies

6.1

Camera and imaging settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6.2

Factors contributing to imaging blur . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

6.3

Noise sources in our images. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

6.4

State populations predicted through different types of analysis . . . . . . . . . . . . 166

7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Figures

Figure
2.1

The Main Experiment Apparatus Layout . . . . . . . . . . . . . . . . . . . . . . . .

8

2.2

The Main Experiment Apparatus Layout, Viewed from the Side . . . . . . . . . . . .

9

2.3

A 87 Rb Level Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4

The D1 Laser Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5

The 850 nm Tweezer TA Laser Layout . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.6

Photos of the Sill Objective Lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.7

Photos of the Tweezer Rail System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.8

Screenshots of Chimera Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.9

Tweezer Array Frequency Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.1

Atom Array Images. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2

Direct Flashing Imaging Vs. Normal Imaging Comparison . . . . . . . . . . . . . . . 40

3.3

Harmonic Oscillator Excitation Populations. . . . . . . . . . . . . . . . . . . . . . . . 41

3.4

Harmonic Oscillator Statistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.5

The Relative Performance of In-Trap ΛGM. . . . . . . . . . . . . . . . . . . . . . . . 42

3.6

Simultaneous Raman Spectroscopy of 36 Atoms. . . . . . . . . . . . . . . . . . . . . 44

3.7Visualizations of Tunnel-Coupled Optical Tweezers. . . . . . . . . . . . . . . . . . . . 49

3.8

Tunneling of Single Atoms Between Two Tweezers. . . . . . . . . . . . . . . . . . . . 53

3.9

Tunneling of a Single Atom Between Three Tweezers.

. . . . . . . . . . . . . . . . . 55

xi
3.10 Two-Tweezer Variational Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.11 Two-Atom Tunneling Simulation Energies . . . . . . . . . . . . . . . . . . . . . . . . 59
3.12 Adiabatic Ramsey Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.13 Rearrangement into a Line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.14 Atoms Spelling Out JILA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.15 Stiff Rearrangement in 2D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.1

Enhanced Loading and Rearrangement in Large Arrays. . . . . . . . . . . . . . . . . 72

4.2

ΛGM Comparative Loading Studies. . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.3

Scan of Repump Power and Frequency Parameters. . . . . . . . . . . . . . . . . . . . 79

4.4

A Wider F ′ = 2 ΛGM Scan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.5

A D1 F ′ = 1 ΛGM Scan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.1

Molecular Potentials with Only Fine Structure Interactions . . . . . . . . . . . . . . 109

5.2

Molecular Potentials with Hyperfine Structure Interactions . . . . . . . . . . . . . . 117

5.3

Molecular Potentials with Hyperfine Interactions Zoomed-In . . . . . . . . . . . . . . 118

5.4

Molecular Potentials with Rotation and Fine Structure . . . . . . . . . . . . . . . . . 124

6.1

Tomography sequence and notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

6.2

Single-atom Fock state preparation and imaging. . . . . . . . . . . . . . . . . . . . . 136

6.3

Motional quantum state tomography via time-of-flight imaging and MLE . . . . . . 137

6.4

Schematic Fock-state momentum distribution experiment timing diagrams . . . . . . 140

6.5

Schematic tomography experiment timing diagrams . . . . . . . . . . . . . . . . . . . 141

6.6

Center of mass oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.7

Excited state tunneling resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6.8

The point spread function of our imaging system . . . . . . . . . . . . . . . . . . . . 147

6.9

Background analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

6.10 Ground state expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

xii
6.11 Inverse Abel Transform Wigner Function. . . . . . . . . . . . . . . . . . . . . . . . 162

6.12 Inverse Radon Transform Wigner Function . . . . . . . . . . . . . . . . . . . . . . . 164
6.13 State Populations Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
6.14 Simulating the Effects of Gaussian Noise on Maximum Likelihood Estimation . . . . 169
6.15 Fits of an anharmonic oscillator to center of mass oscillations . . . . . . . . . . . . . 171
6.16 Simulated Tomography Fidelity and Wigner Negativity . . . . . . . . . . . . . . . . 175

Chapter 1
Introduction

1.1

Optical Tweezer Arrays: Historical Background
The field of neutral atom atomic physics has undergone a number of evolutions, but it received

a particularly great boost with the invention of the laser. Coming soon after the maser, the laser
was first publicly conceptualized in the late 1950’s and first implemented in 1960 [1, 2]. The opticaltweezer would be invented a decade later where it was first used to trap microparticles [3]. After acouple years in 1975, laser cooling was proposed and implemented in 1978 [4, 5, 6]. The magneto
optical trap (MOT), capable of trapping a large cloud of neutral atoms, was developed a decade
afterwards [7]. Then, single atoms would be isolated and studied in a MOT [8]. There are many
useful reviews of the physics and history of these foundational developments to which I would direct
the interested reader [9, 10, 11, 12].
In the ’90s and early 2000s, the great progress was made in the study of degenerate quantum
gases of bosonic atoms [13, 14], fermionic atoms [15], and molecules [16, 17]. These degenerategases continue to be fruitful areas of study to this day and are the starting points for many newexperiments [18, 19, 20]. For example, quantum gas microscope experiments often start by using
condensation and related phase transitions to prepare low-entropy states, and follow with leveraging
single atom imaging and increasingly fine control over potentials and ultracold dynamics to conduct
novel quantum simulation experiments [21, 22, 23, 24, 25, 26, 27].
In parallel, developments in the loading of single atoms and molecules into optical tweezers
have enabled a more bottom-up approach to studying atomic systems. Beginning in 2001, this2
work continued for about ten years at a smaller scale, where research focused on establishing thecore functionality and characterizations of such systems [28, 29, 30, 31, 32] and then harnessing
strong and long-range Rydberg interactions [33, 34, 35, 36]. In the past ten years or so, there has
been a small explosion of optical-tweezer based single-atom experiments and increasing interest in
the platform. Now, there exist many more projects utilizing similar systems for studying fermions,molecules, alkali-earth atoms, and even some commercial enterprises hoping to use the platform
for quantum computing [37, 38, 39, 40, 41, 42, 43, 44, 45, 46].

1.2

This Thesis
In general, improvements in quantum state readout and state preparation herald progress inthe field of atomic physics. New methods for optical trapping predicted a new generation of trappedatom experiments. Motional ground-state preparation via Raman sideband cooling preceded awave of quantum information experiments with trapped ions. The preparation of Bose-Einsteincondensates anticipated the current plethora of quantum simulation experiments.
With the optical tweezer platform, we push the limits of single neutral atom control into new
regimes. Here, we are equipped with single-atom imaging and spin and motional control opticalpumping, microwaves, and Raman transitions, and the mechanical manipulation of our tweezers.
More uniquely, we also have control over the exact number and position of the atoms we work with.
One of the major scientific themes of this thesis is the use of these tools to study lightassisted multi-atom collisions in new regimes for the development of novel techniques for the loadingof single particles into tightly confining optical potentials. Light-assisted collisions are routinely
used in the field, but under their technical application lies rich molecular physics waiting to bestudied [47, 48, 49, 39]. We apply the tools enabled by our single-atom control and the interplayof laser-cooling techniques to study this problem. In our experiments, we find that because of
the combination of the ultracold temperatures of our atoms and the tight but overall low-energy
confinement of the optical potential, that tweezer platforms are uniquely situated to study very
long-range and low-energy light-assisted molecular collisions.

3
Another major scientific theme of this thesis is the quantum control of single particles near
their motional ground state. While the optical tweezer platform is particularly attractive for Rydberg atom-based experiments and quantum computing, our focus has often been on near-groundstate physics [50, 51, 52]. Quantum motional control, as distinct from control of quantum spins,comes with unique opportunities for unique measurement techniques, such as time-of-flight imaging, and applications in the quantum control of macroscopic objects [43, 53, 54]. Working withground-state atoms entails its own set of challenges and rewards. Optical tweezers are in many
ways more individually controllable than alternative traps such as optical lattices, and so tweezers
promise many unique ways to prepare states and dynamics potentials that would be more challenging in other systems. These capabilities include dynamic movement and reconfiguration of thearray of atoms, including into structures which are closely-spaced on the order of the wavelength of
atomic transitions, capabilities which can be used for whole other category of experiment involving
atom-light interactions [55, 56].
One of the core components of this control in our experiments is quantum tunneling fromtweezer to tweezer. An optical lattice, achieving its potential structure through the interferenceof only two beams, it is a very clean and suitable environment for realizing tunneling [57]. Ina tweezer system, however, the potential must be constructed piece by piece, which makes it atechnically challenging endeavor [51, 58, 42]. In this thesis, I characterize and push our motional
control to further limits, enabling the production and tomography of non-classical motional states.
We study these states using free-space imaging in the first such measurement of single ground-statebosons. We also apply methods similar to those used in optical homodyne tomography to map
out the quantum state of our single particles. Our demonstration will enable future experimentswith single atoms or larger particles to explore quantum effects in more complex and macroscopic
systems [59, 60, 61, 62].

4

1.3

Outline
The scientific content of this thesis focuses primarily on the developments we made in en-

hanced loading, molecular physics, and quantum tomography. While I contributed to experiments
in vector magnetometry, metalens optics, and in a more distant sense, Rydberg physics, I would
point readers to the future but relatively forthcoming theses from our group of Christopher Kiehl
and Ting-Wei Hsu for more information about this work.
Chapter 2 is titled “Apparatus for Ground-State Optical Tweezer Experiments”. Here, I will
discuss the experimental apparatus upgrades I made, mostly in the earlier stages of my PhD. This
includes upgrades to laser and optics systems, mechanical innovations, our control system, andother important physical developments made during my PhD.
In Chapter 3, I will discuss a number of atom-based experiments, techniques, and upgrades
we developed. This includes new, previously unpublished iterations on Raman sideband coolingand quantum tunneling experiments conducted in recent years, as well as novel rearrangement
techniques among other new technical developments.
Chapter 4 is all about our new technique for the enhanced loading of single atoms into opticaltweezers. It reviews our published work [63] and explores further unpublished experiments we haverecently conducted to learn more about this mysterious process, including details on the role of theΛ-enhancement and gray-molasses level structure on the loading process.
In Chapter 5, I will present theoretical calculations of the ultra-long range molecular potentials that we believe are important for the enhanced loading technique discussed in Chapter
4. Previous works have modeled the collisional process but tend to approximate the effect of theexcited state molecular potential on the loading dynamics. By calculating the potentials fully, we
find that the true nature of these potentials significantly alters our conception of how the loading
process works. These calculations pave the way for future experiments and understanding of thesecomplicated dynamics.
Lastly, Chapter 6 will focus on our all-mechanical time-of-flight-based quantum tomography

5experiment. This experiment makes use of the rapid cycle time of the experimental apparatus, ourground state cooling, quantum tunneling, and new free-space imaging techniques. The measurements give direct access to new observables and unprecendented information about single particle
quantum states in our apparatus. We leverage all these tools to do full quantum tomography on
motionally excited states of our optical tweezer potentials.

1.4

List of Publications
For reference, I am including here a list of papers which I have contributed to.
1. B. J. Lester, Y. Lin, M. O. Brown, A. M. Kaufman, R. J. Ball, E. Knill, A. M. Rey,

and C. A. Regal, Measurement-Based Entanglement of Noninteracting Bosonic Atoms, Phys. Rev.
Lett. 120, 193602 (2018).
2. T. Thiele, Y. Lin, M. O. Brown, and C. A. Regal, Self-Calibrating Vector Atomic
Magnetometry through Microwave Polarization Reconstruction, Phys. Rev. Lett. 121, 153202(2018).
3. M. O. Brown, T. Thiele, C. Kiehl, T.-W. Hsu, and C. A. Regal, Gray-Molasses OpticalTweezer Loading: Controlling Collisions for Scaling Atom-Array Assembly, Phys. Rev. X 9, 011057(2019).
4. T.-W. Hsu, W. Zhu, T. Thiele, M. O. Brown, S. B. Papp, A. Agrawal, and C. A.
Regal, Single Atom Trapping in a Metasurface Lens Optical Tweezer, ArXiv:2110.11559 [Physics,
Physics:Quant-Ph] (2021).
5. M. O. Brown, S. R. Muleady, W. J. Dworschack, R. J. Lewis-Swan, A. M. Rey, O.
Romero-Isart, and C. A. Regal, Time-of-Flight Quantum Tomography of Single Atom Motion,
arXiv:2203.03053 [quant-ph] (2022).
6. M. O. Brown, et al., Molecular Potentials for Ultra-Long Range 87 Rb Dimers, (unpublished).

Chapter 2
Apparatus for Ground-State Optical Tweezer Experiments

2.1

Overview
In this section I introduce the physical apparatus used to conduct the experiments discussed

elsewhere in this thesis. The experimental apparatus originated through the hard work of Adam
Kaufman and Brian Lester among others, and as such, many details about the apparatus at earlier
times are best found in Adam’s thesis [64] and Brian’s thesis [65]. However, here I will provide abrief overview for completeness, while more discussion of new and modified systems will begin in
Sec. 2.2.
An experiment for cooling and trapping single atoms must take place in ultra-high vacuum,or else collisions between the atoms of interest and other atoms and molecules will contribute toheating and loss from traps. The vacuum apparatus we use is very simple, consisting of an ion
pump, evaporative rubidium getters, and a single glass cell in the shape of an octagonal prism where
the cooling and trapping occurs. The vacuum system we use is itself unchanged from the vacuum
system developed by Adam Kaufman and described in his thesis [64]. We anticipate changing
the vacuum system at some point in the future in order to introduce fresh 87 Rb getters, upgrade
the glass cell to a version with flatter windows and more specialized optical coatings, and makeuse of a 2D+ magneto optical trap (MOT) cell we purchased from Cold Quanta. Many otherspecies of atoms require more specialized vacuum apparatus such as Zeeman slowers, source ovens,
or make use of multi-chamber systems in other ways. Thanks to the simple electronic structureand favorable physical parameters of 87 Rbsuch as having a large mass, such complications are not7necessary here (Table 2.1).
Parameter
Value
Mass
1.44316060(11) × 10−25
Relative Natural Abundance
27.83(2)F = 2 ↔ F = 2 Scattering lengths
95.44
F = 1 ↔ F = 1 Scattering Length
100.4
F = 1 ↔ F = 2 Scattering Length
98.006
D2 Transition (5S1/2 → 5P3/2 )Decay Rate
38.11(6) × 106
Natural Linewidth (Γ)
2π × 6.065(9)Recoil Velocity
5.8845
Recoil Temperature
361.96
Doppler Temperature
146′
Isat D2 Resonant Isotropic Polarization 2 → 3 3.576(4)Isat D2 Resonant Cycling |2, 2⟩ → |3, 3⟩
1.669(2)Isat D2 Far-Detuned
2.503(3)D1 Transition (5S1/2 → 5P1/2 )Decay Rate
36.10(5) × 106
Natural Linewidth (Γ)
2π × 5.746(8)Recoil Velocity
5.7754
Recoil Temperature
348.66
Doppler Temperature
138
Isat D1 Far-Detuned
4.484(5)

Units
kg
%
a0
a0
a0

Source
[66]
[66]
[67]
[67]
[67]

s−1
MHz
mm/s
nK
µK
mw/cm2mw/cm2mw/cm2

[66]
[66]
[66]
[66]
[66]
[66]
[66]
[66]

s−1
MHz
mm/s
nK
µK
mw/cm2

[66]
[66]
[66]
[66]
[66]
[66]

Table 2.1: Table of Selected 87 Rb Constants. A table displaying a variety of 87 Rb constants
that I have found useful over the years. Also see Fig. 2.3.

In order to cool and trap atoms in this cell, we must make use of a plethora of lasers andlaser systems. These systems have changed non-trivially and are therefore documented in detail in
Sec. 2.2.1. The physical arrangement of the laser systems around the vacuum chamber is viewable
in Fig. 2.1 and Fig. 2.2.
At the heart of the electrical control system for our experiment is our digital output (DO)system, which outputs a series of precisely timed digital pulses. These pulses trigger other devices,such as cameras and radio frequency (RF) waveform generators at the appropriate time, they
open and close electrical gates such as those for magnetic field coils, and they open and close RF
switches which has the effect turning on or off laser beams. This system originated as a DIO-64
event analyzer/control module (long since discontinued) system from Viewpoint Systems, and more

Figure 2.1: The Main Experiment Apparatus Layout. This figure is not precisely to scale but care has been taken to make it arelatively accurate representation of the experiment layout. 850 nm light is shown as red light, 795 nm light is orange, and 780 nm lightis yellow. Where two beams overlap, the two colors are represented with colored hatches in the beam. Platforms are indicated by shadedboxes, where darker shades indicate higher elevations. Most common optics are identified in the legend in the top left corner. The tweezers
diverge after the objective and are not blocked, but this divergence and other similar unused beams are not shown for clarity. The vacuum
chamber is roughly outlined in green. The bottom Raman path goes underneath the vacuum chamber, as is more evident by looking at
Fig. 2.2 which is a companion to this figure.

8

9Figure 2.2: The Main Experiment Apparatus Layout, Viewed from the Side. This figure
is viewed from the side of the objective lens, rather than the side of the axial beam path. It is moreevident from this angle that the bottom Raman beam goes under the vacuum chamber. See legend
in Fig. 2.1.

recently has been implemented through a custom FPGA system designed to emulate the DIO-64
system.
Also crucial are analog input and output systems. Analog outputs (voltage range -10 V to
+10 V) are mostly used for controlling relative laser powers or magnetic field currents. As other10
systems in the experiment are increasingly digital, we find fewer uses for analog outputs now thanwe used to. Analog inputs are primarily used for experiment diagnostics and automatic laser power
calibrations. The analog inputs and outputs we use are National Instruments cards PCI-6221 andPXI-6722 respectively.
Laser cooling sequences and spin state preparation typically require careful control overmagnetic fields in the vicinity of the atoms. To control the magnetic fields, we use 8 electriccoils, 2 of which are configured in an anti-Helmholtz configuration for the creation of a MOT, and
6 of which are in a Helmholtz configuration for zeroing background magnetic fields or setting a
constant magnetic field. As we only need to drive currents less than 10 amps, these coils are simple
copper coils and are driven by custom but simple current servo systems based off of Hall detectors.
We use a number of Keysight (formerly Agilent) arbitrary waveform generators, mostly models 33522A/B for a number of purposes:• Time-varying intensity servo set points for our tweezer servo system• Square waves with specific relative phases and duty cycles for flashing imaging techniques• Calibrated pulses for pulsed Raman sideband cooling techniques.
We currently use voltage-controlled oscillators (VCOs) for a number of purposes, including
determining lock offsets and driving acousto-optical deflectors and modulators (AODs and AOMs).
However, we are slowly gravitating toward the use of a home-built direct digital synthesis (DDS) RF
generator based on Analog Devices AD9959 evaluation boards. These have numerous advantages
in that they provide cleaner, programmable outputs, have wider frequency output ranges, and are
more stable than VCOs, which are useful features for most of our applications. Our system for
driving the optical tweezer deflection AODs is more non-trivial and discussed in Sec. 2.4.2.
Lastly, we require a flexible microwave source for driving electro-optical modulators and asawed-off microwave horn. In particular, even for a single experiment we generally need to beable to change the microwave frequency during different experiment stages, for example from thesettings optimized for Λ-enhanced grey molasses (ΛGM) loading to those optimized for in-trap

11
ΛGM cooling. This is typically accomplished using a “list”-mode, where the generator switchesfrom one list entry (containing a frequency and power to output) to another on being triggered by
the DO system.
In the past, we have used a Rhode-Schwarz microwave generator (RSG) to generate thesignals we need for these devices. Recently, especially after a power supply failure in the RSG, wehave gravitated toward using devices from Windfreak Technologies. The devices we use are the“SynthHD PRO (10MHz – 24 GHz) Dual Channel Microwave Generator”, and the “SynthHD (54MHz – 13.6 GHz) Dual Channel Microwave Generator”.

2.2

An Updated Physical Apparatus
In the time since Brian’s departure, there have been a number of changes to our physicalsetup with important optics and electronics were introduced among other things which I will discussin detail.

2.2.1

Laser Systems

87 Rb is the only atom species we use in the lab. We have numerous laser systems that we use

to address the various 87 Rb transitions (Fig. 2.3). At the start of my PhD, most of the lasers in
the lab were external cavity diode lasers (ECDLs), typically constructed with anti-reflection coated
diodes from Eagleyard Photonics and a laser grating in a Littrow configuration. These lasers were
not initially very stable and would come unlocked several times a day. I eventually tracked thisdown to be a fixable electronics issue in the servo systems. Afterwards, these lasers were very
functional, however they were still relatively sensitive to vibrations, temperature, and humidity
drifts due to the macroscopic cavity that exists outside of the laser diode. As such, it was still
diﬀicult to keep them locked for more than a day or so. Meanwhile, the company Photodigm began
refining their distributed Bragg reflector (DBR) lasers to have a low enough linewidth to be useable
for our 87 Rb systems. As a result, we slowly started switching from ECDLs to DBR lasers until at
this point we only use DBRs.

12

Figure 2.3: A 87 Rb Level Diagram. Not to scale. Values are primarily from [66]. The dipson the right-hand-side of the levels are reflective of the relative light shifts of the 850 nm opticaltweezer traps. Shown on the left hand side are lines indicating the major laser systems we use in
our experiment, including the tweezer lasers, 780 nm lasers, and both configurations of 795 nm
laser systems that have been used.

The DBR lasers have only a few downsides, including that they are relatively expensive (∼$5k
per module), and some models, in particular our initial high power 795 nm models, broke faster13
than expected (∼ 6 months) under normal use. In the past several years, however, we have not
had such troubles. None of our 780 nm modules have died, including one that has been in use for∼ 10 years.
One of our strategies, both for dealing with these occasional issues and for characterizing
our lasers, is to separate the “head” of the laser system (i.e. the laser module itself followed byany required beam shaping and optical isolation) from the rest of the laser system by an opticalfiber connection. This approach has several advantages: if a laser dies, it is easy to immediately
replace the laser head with a portable “platform laser” which can substitute for the table-mounted
laser until a replacement arrives and is installed without needing to realign the rest of the laser
system. As well, the early fiber launcher provides an easy way to test the laser’s wavelength witha wavemeter, and the fiber spatially filters the laser for the rest of the system’s alignment. This
split setup is visible in the 795 laser schematic (Fig. 2.4).

2.2.1.1

780 nm Laser System

We use four main 780 nm lasers in our experiment. One is the Raman laser, which is ∼50
GHz red-detuned and typically unlocked. This laser is split into the four Raman beams that go to
the experiment.
I will call the other lasers the “D2 master” laser, the “D2 f = 2” laser, and the “D2 f = 1”
laser, where f is the total angular momentum including electronic orbital, electronic spin, andnuclear spin angular momentum. Historically, the D2 f = 2 laser has also been called the “MOT”
laser despite it being used for more than just MOT light, and the D2 f = 1 laser has been called
the “repump” laser. The term “master” here is for historical continuity purposes. If renaming
it today I would call it the “reference” laser, since the laser is not the “master” in either the“mastery”, “master/slave” or “master/replica” senses. The D2 master laser is static throughout all
experiments and is locked to the 87 Rb D2 f = 2 → f ′′ = 3 line (Fig. 2.3) (the f ′ symbol will bereserved for the 87 Rb D1 line). We use a standard saturation spectroscopy setup in order to create
the relevant line structure, and use a frequency modulation spectroscopy setup to convert these14transition lines into usable error signals [68, 69, 70]. This is as opposed to modulation transfer
spectroscopy setups which are also common in the field [71]. The frequencies of the other twolasers are changed dynamically throughout a single experiment repetition, and they are lockedwith reference to the master laser via an offset lock system. As the linewidth of 87 Rb transitions
is relatively large and the level structure forgiving, the locking electronics we use are custom but
simple proportional-integration (PI) systems.
The D2 f = 1 laser is split into two separate beams, referred to as the “repump” beam andthe “σ+ repump” beam, both tuned near the f = 1 → f ′′ = 2 transition. The repump beam
serves to repump the atoms during the standard MOT, red-detuned polarization gradient cooling(RPGC), and imaging phases of a typical experiment. The σ+ repump is specifically used during
any optical pumping procedures to minimize the number of photons needed to optically pump into
the |f, mf ⟩ = |2, 2⟩ state. This is especially important for Raman sideband cooling procedures.
The D2 f = 2 laser is split into many more beams. As such, we need slightly more power fromthis laser, so the DBR laser seeds a 1 W 780 nm tapered amplifier from Eagleyard. The beam thensplits into what are typically referred to as the “diag. MOT”, “side MOT”, “σ+ optical pumping(OP)”, and “probe” beams. Most of these are tuned to the D2 f = 2 → f ′′ = 3 transition, withthe exception of the optical pumping beam, which is given a slight shift via AODs to be resonantwith the D2 f = 2 → f ′′ = 2 transition.
2.2.1.2

795 nm Laser System

ΛGM Paper Configuration

Originally, the 795 nm laser system used for our Λ-enhanced

gray molasses (ΛGM) scheme (discussed in Chapter 4) consisted of a single laser. Early in the
beam handling, the laser was split along two paths, with one path going into an AOD and thena standard saturation spectroscopy setup which locked the laser frequency on this path to the
87 Rb D

1 f =2→f′ = 2 transition. Then, when changing the frequency that this AOD operated

at, the lock would compensate and shift the laser current to maintain the lock, thus shifting theother path’s frequency relative to that of the locked path. The other path would go through a

15fiber-based electro-optical modulator (EOM) and then a tapered amplifier. This setup worked wellenough for that paper, but pretty significantly limited the effective frequency range that we could
scan over during that experiment. It also wasn’t that stable.
Modern Configuration

Before too long therefore, we upgraded this system to have two

separate lasers, which I will refer to as the “D1 master laser” and the “D1 f = 2” laser. (Note,the D1 f = 2 laser has had a number of informal names in the lab over the years, including the“worker” laser and the “main D1 ” laser, before I settled on this more operational name).
The D1 master laser in this new configuration is locked (with no AOD this time) directly to
the 85 Rb f = 2 → f ′ = 1 transition (note the different species!) (Fig. 2.3). The lock is a relatively
standard lock, using saturation spectroscopy to get the desired transition line signal, and then againfrequency modulation spectroscopy to generate an error signal. This configuration is very similar
to that of the 780 nm master lock.
The D1 f = 2 laser is then offset locked by a variable detuning red-shifted from the master
laser (Fig. 2.3). Being red-shifted by a small amount from the 85 Rb D1 f = 2 → f ′ = 1 transition
puts you blue-shifted from the 87 Rb D1 f = 2 → f ′ = 2 transition, as is desired for the Λ-enhanced
grey molasses (ΛGM) scheme. Our offset lock configuration is much more widely tunable, with atuning range of approximately 1 GHz, far superior to our previous tuning range of ∼ 40 MHz.
The D1 f = 2 laser then enters a fiber EOM which is driven with an approximately 6.834 GHz
signal in order to give a tunable sideband on the D1 f = 2 light that are close to the 87 Rb D1
f = 1 → f ′ = 2 transition which can act as coherent repumping light, as is desired for ΛGM(Fig. 2.3). The fiber-EOM is not particularly eﬀicient or stable, and so the resulting light is passed
through a 3 W Tapered Amplifier (TA) in order to get enough light for conducting ΛGM. We don’t
need anywhere near 3 W, this was just the only tapered amplifier available at the time.
This arrangement works well, although it has a couple downsides. The EOM coupling is notparticularly stable, which mostly only makes it a nuisance to maintaining laser powers. Nowadays
these powers are part of an automatic calibration system, so there are no problems as long as theoverall power doesn’t drop too low. One might also generally worry that the extra sidebands of

16
the EOM could complicate the effect of the ΛGM, although we have seen no direct symptoms ofthis. An alternative would be adding a separate, 3rd D1 f = 1 laser for repumping, mimicking
our 780 nm setup, and upgrading our offset locking system to be capable of a robust phase lockbetween these two lasers. Quite possibly the resulting light would still need to seed a TA in orderto get suﬀicient power, but this different configuration might help with some of the other issues.

Figure 2.4: The D1 Laser Layout. Visible are the D1 master laser, D1 f = 2 laser, saturation
spectroscopy and offset lock beating setup on the left-hand side. The tapered amplifier and diagonal/ side path splitting setup is on the right-hand side. See legend in Fig. 2.1.

2.2.1.3

850 nm Laser System (Tweezers)

The tweezer light source has changed many times over the years, but most recently comesfrom a temporally incoherent light source. The incoherent light source that we use is a superluminescent LED (SLED) module from a company called Superlum, with part number BLMSmini-381-HP3-PM-PD-OI. The module outputs about 25 mW of light with a bandwidth of 15 nm,
which we immediately filter down to a full-width half-maximum of 3.2 nm using a band pass filter
from Semrock, part number LL01-852-12.5. The resulting light has a fairly short coherence lengthof ∼80 µm. As a result of this short coherence length, reflections and back reflections of thetweezer light in our tweezer rail system should generally not coherently interfere with the main
tweezer beams. This type of interference was generally a concern for us, as we are very sensitive
to temporal fluctuations of our tweezer potentials, slow or fast. When we implemented this, we

17
saw some improvement in the relative bias fluctuations of our tweezers, but we did not see any
serious improvement of the overall performance of the system as measured by the atom’s responseto the potentials, in the form of the measured trap shape or tunneling properties. However, we still
use this source, mostly as there’s little reason to use anything else and this might still help on themargins.
In the past, we have used a number of TA setups, including a single-pass 1 W TA, a doublepass 2 W TA, and currently a 1 W TA followed by a 3 W TA (Fig. 2.5). The double-pass 2 W
TA setup worked somewhat well but did not seem very stable and was tricky to align properly.
Therefore, when the TA eventually died, we decided to change our setup to the two-TA system we
use now.
After filtering down the SLED output, we only have a couple mW of incoherent light, which
is not enough to properly seed a TA, as TAs typically require ∼30-50 mW to get their full power
output. Therefore, we use the incoherent light to first seed a 1 W TA, which is under seeded but canstill easily output 100’s of mW of power. The excess light has occasionally been used for additional
beams or tests in our system, but usually we just use this to have comfortably enough power toproperly seed the 3 W TA with the required amount of power.
In general, we find that these TA modules die rather quickly (with a lifetime of 1-2 years)
if driven at their full power consistently, while they last much longer if driven at lower powers.
Therefore, while we have in the past operated our 3 W TA at full power (in particular to getenough power to load 100 traps during the ΛGM paper), recently we drive it at about half of it’s
maximum output power and work more conservatively with smaller arrays of traps, which are allthat is necessary for our current work regardless.

2.2.1.4

Ti:Sapphire Laser

We own a Ti:Sapphire laser which has been used for a number of purposes over the course of
my PhD. It is a very flexible laser, capable of outputting on the order of 1 W of optical power at
all wavelengths that we use in the lab. It is seeded by a 10 W green pump laser from Lighthouse

18

Figure 2.5: The 850 nm tweezer TA laser layout. Visible is the 1 W TA setup above, whichseeds the 3 W TA setup below. Beam shaping after the TA chips is tricky. Sometimes an extra
pickoff would be added after the two isolators in the 1 W section in order to get some extra 850 nm
light. The multiple filters may be unnecessary but are attractive in reducing any long tail ofamplified spontaneous emission (ASE) that the TA chips may output. See legend in Fig. 2.1.

Photonics, a “Sprout” model. Higher power models are available. As such, it has at various times
been used as our Raman laser, tweezer laser, and even for some 795 nm testing purposes. Soon, weplan on using it to create an optical lattice to compliment the tweezer potentials.

19
2.2.2

Mechanical Shutters

In an experiment such as this it is often required to completely extinguish light entering theexperiment apparatus from a given laser path. This is especially true for lasers that are resonantwith an atomic transition and which if left unattended, would result in slow heating of the atoms
after they have been ground-state cooled. It is also crucial to block the input to our EMCCD
camera at various times, especially in the tomography experiment discussed in chapter 6. At theheight of data taking for this experiment, because of the large number of images necessary, we weredestroying the imaging shutter every couple weeks.
In general, we use acousto-optical deflectors with varying RF input powers to variably attenuate laser beams for dynamic control during an experiment. Even after extinguishing the RF powercompletely, however, one is still limited by scattering off of the AOD crystal which can scatter into
an optical fiber at small but significant rates. It is possible, and possibly optimal, to use chained
acoustic optical modulators in order to attenuate by a significant enough amount for our purposes.
However this is rather expensive and creates longer and less stable laser paths. It is worth noting
as well that double-pass AOD configurations do not generally solve this scattering effect, as one
may be limited by backwards-scattering from the first pass through the AOD unless the crystal islarge enough to support two physically separated deflection regions for each pass.
We therefore often turn to mechanical shutters which physically obstruct the laser path inorder to completely extinguish laser beams when necessary. Originally we primarily used expensive
shutters from the company Uniblitz, typically model LS6 6 mm. These shutters are fast andhave convenient mounting configurations, especially for connecting to our EMCCD where we need
to be able to block the entire opening to the camera sensor. However they are at this moment
still relatively expensive. We also experimented with Thorlabs shutters (part number SHB025T),
however these have similar drawbacks to the Uniblitz shutters.
Other experiments deal with similar challenges. However, the relatively fast repetition rateof our experiments compared to other cold-neutral-atom experiments, combined with the relatively

20
slow rate of the actual science experiment compared to more strongly interacting systems like ions,create a perfect storm. We really need shutters to get suﬀicient extinction during the longer sciencestages of the experiment, but we actuate them frequently enough that they break on relatively
short time-scales.
As a result, over the course of my PhD, we have used a number of different designs forcustom mechanical shutters. We tried designs based on speaker actuators as outlined by [72], andby rotating flags driven by simple motors as outlined by [73]. Both of these designs are cheaper thancommercial options, but in our experiment both can also break frequently and perform irregularly
in devious ways. A shutter that occasionally sticks or bounces open is worse than no shutter at all.
We had relatively more success with the rotary design [73], and a lucky few shutters have lasted
a number of years and are still in use. However, our continued struggles lead us to consider otheralternatives.
Our most recent innovation is the use of free-space micro-electromechanical system (MEMS)
technology from the company Agiltron, specifically their 700 µm FSVOA devices. As the namesuggests these devices can be used as variable optical attenuators, but if carefully configured, thesedevices can be completely attenuating, and we believe that their reliance on MEMS-technology,
as well demonstrated by the telecom industry, will lead to significantly longer lifetimes than otherdesigns which involve mechanical collisions as the shutters close. Indeed, none of the 4 devices weoriginally made two years ago have failed or had any issues, a much better track record than ourother custom solutions. The small aperture size means one must focus through the device. We are
currently working on implementing one of these shutters for use in our imaging system as well. This
was an original idea and is unpublished. Ideally one might use fiber-based MEMS shutters, but
unfortunately we struggled to find fiber-based MEMS shutters which were polarization-maintaining
and single-mode at the wavelengths of interest to us (780 nm - 850 nm). Perhaps in the future suchdevices will be available.

21
2.2.3

Sill Objective Lens

At the start of my PhD, the lab was using a high numerical aperture (NA) objective lensfrom the company ASE Optics. The lens performed at around an effective NA of ∼ 0.5, and while
this was good enough for a wide variety of experiments, it is generally desirable to have as tight andwell-behaved optical tweezers as possible. This is especially true for experiments with side-band
cooling, where our ground-state fidelity is limited primarily by having relatively low axial trap
frequencies, and for experiments with tunneling, which are thought to be sensitive to aberrationsin the trap in a number of ways. As such, before my thesis started, the lab purchased two high-NA
objectives from the company Sill Optics. The initial motivation for purchasing two lenses was fortheir potential for double-objective setups with one objective on either side of the vacuum chamber,
although we have never attempted this in practice.

Figure 2.6: Photos of the Sill Objective Lens. (Left) The Sill lens viewed from the front.
Visible is a ring assembly which has been epoxied to the front of the lens using Hysol, which holds
two mirrors in place. The mirrors are angled to make it possible to create an axial lattice with asingle beam, although this has never been utilized. (Right) The sill lens viewed in situ. Visibleon the bottom side of the image is the phase plate, which has been attached to the objective via J.
B. Weld epoxy.

22
2.2.3.1

Sill Lens Objective Characterization

It was found after the objective lenses arrived that the two objectives were defective andcould not meet their quoted performance. As such, near the beginning of my PhD, we embarked
on a long sequence of tests and modifications in order to try to improve the performance of thelenses. This report is a collection of work done by various students in the lab, including myself,
Junling Long, and Ludovic Brossard.
The first task was to measure the aberrations of the objective lens. This was done using what
was effectively a Michelson interferometer setup. A large input beam was split along two paths,
with one path hitting a flat mirror, and the other traveling through the objective and hitting areference sphere held inside an unevacuated vacuum chamber cell. The reference sphere was held
inside the unevacuated vacuum chamber cell primarily in order to best mimic the effect of the cell
window on the performance of the objective. The reference sphere was a small silicon-nitride ballbearing.
The primary results of the measurement are listed in Table 2.2.3.1. As can be seen here, themain measured aberration is coma, which has an egregious magnitude above twice the wavelengthin phase, a clear explanation of the problems we encountered early on with the lens. The RMS
wavefront error of the uncorrected lens was 0.92 λ at 850 nm.
After this measurement, we commissioned a phase plate, a very slightly shaped piece ofglass, that was designed in order to correct these aberrations. The piece was ordered from Light
Machinery. Although originally spec’d to correct the Sill lens to a very high accuracy, the company
also had trouble machining the specific profile we ordered, and the result was imperfect aberration
correction. The measured aberrations of the corrected Sill lens are also denoted in table 2.2.3.1. The
results, while not as good as hoped, reduced the remaining aberrations to approximately the level
which we expect from out “rail” system that we use to generate the tweezer array: an RMS errorof approximately λ/5. We realized that for further improvement, we would have to try something
more active.

Name
Abberation Coordinates
Unmodified
With Phase Plate

H. Astig.
(2,-2)-0.08
0.10

V. Astig.
(2,2)
0.00
0.16

H. Coma(3,-1)
1.44
0.02

V. Coma
(3,1)
2.05
-0.18

Spherical
(4,0)
0.08
-0.01

RMS

23

0.92
0.147

Table 2.2: Sill Objective Lens Aberration Coeﬀicients. Listed are the coeﬀicients before andafter various corrections. Numbers are in units of λ, the wavelength, at 850 nm. Even after adding
the phase plate, there were non-trivial aberrations remaining, which we eventually corrected witha deformable mirror.

2.2.3.2

Deformable Mirror

As a result of the limitations of the phase-plate, we purchased a deformable mirror from
Boston Micromachines for further objective correction. The deformable mirror is an active optic
which uses electrostatic actuators to continuously deform a continuous mirror surface. Options areavailable with segmented surfaces for less interactuator coupling but high-frequency artifacts fromthe actuator edges. We purchased a multi-C model, which features 137 actuators capable of 1.5 µm
stroke in an approximately circular configuration.
The deformable mirror model has numerous advantages over other adaptive optics. Unlike
digital mirror devices commonly used in quantum gas microscope experiments, the mirror is nearly
perfectly eﬀicient [74]. Unlike liquid-crystal based spatial light modulators, the device is driven witha static voltage, reducing noise concerns, and it is fast, featuring up to 100 kHz update speeds [75].
It is important to realize, however, that the deformable mirror is in this case primarily to be used
for static, or very slowly (over days) changing aberrations, as the aberration calibration processitself is slow.
The deformable mirror is placed before either tweezer AOD in our rail system, so that thesame aberration configuration is applied to all tweezers uniformly. The configuration is visible inschematic form in Fig. 2.1 and is visible in Fig. 2.7. The downside of this configuration is that thebeam must travel through a long optical path from the Deformable Mirror (DM) before creating
the tweezers, and because of limitations imposed by the pre-existing rail setup, the DM is not in a
precise 4-f configuration with the tweezers. Therefore, applying coma for example on the DM will

24
not result in exactly coma being applied to the tweezers. A future configuration might place theDM in a 4-f configuration with the tweezers, with the DM being placed in-between the AODs andthe large tweezer expansion telescope. This might make the operations of the DM more predictable
and result in better optimizations.

Figure 2.7: Photos of the Tweezer Rail System. (Left) The horizontal Intraaction AOD,
nestled within a rather bulky custom heat sink to help maintain low thermal fluctuations in the
AOD. (Right) The deformable mirror configuration right out of the launcher into the rail setup.
The deformable mirror is visible near the top with a long ribbon cable coming out of it.

Our original calibration of our deformable mirror was slow and churning, as we had to manually change the deformable mirror pattern between each measurement of the trap frequency or
depth, which we were attempting to optimize. In the end, we were able to improve the performanceof the Sill lens creating a single tweezer potential, achieving a single tweezer NA of approximately
0.64, where the Sill lens with the phase plate but with no DM corrections performed with an NAof approximately 0.55.
Unfortunately, however, we ran into two major problems with this plan. The most importantof which is that while a single tweezer experienced this nice optimized NA, when we tested the same
DM configuration while producing many tweezers, the result was much poorer, and we did not see

25significant improvement from the DM. It may be possible to specifically optimize the performanceof an array of traps using these mechanisms, but we could no longer expect that there would be
one good configuration for all of our tweezer experiments. The most likely culprit for this behavior
is that the measured imaging point spread function (Fig. 6.8), suggests that the tweezers may havea somewhat long tail, causing multiple tweezers to distort each other, even when they are different
frequencies and far separated.
The second issue was that the DM’s aberration correction calibration seemed to drift relatively
quickly. As mentioned, the original calibration was somewhat manual and lengthy, which decreased
our appetite for recalibrating the DM frequently. Recently, however, control over the deformablemirror has been incorporated into our experiment control system, Chimera, and so automatedcalibrations should be possible in the future which should greatly alleviate this issue.
Despite its limitations, the deformable mirror remains a powerful optic for use in our lab.
Future uses of this may include better trap optimization, or optimization in a different configuration,
or of optimization of different optics. As well, the DM can change it’s profile very fast (∼100 kHz),and so it is possible that we could use the DM to programmatically modulate the trap shape, which
could be useful for creating non-classical superposition states. Similarly, the DM could be used toprogrammatically change the trap shape to, for example, make a nearly quartic potential rather
than a nearly harmonic potential.

2.3

Computer Control Systems

2.3.1

Chimera Control

At the beginning of my PhD, the control system was split between two separate computers.
One contained the camera control system, which was a stock program (Andor Solstis) for controllingthe camera on its own computer, and the second was a computer running Windows XP that ran
the old Visual Basic 6 code, which was responsible for programming the DO system and AO
system. Many of the additional systems already discussed did not exist at the time. The setup was

26functional, and served previous generations well. However, there were a number of limitations that
resulted from this setup:• Communication between the different systems in this configuration is diﬀicult. In particular,
we anticipated eventually wanting to do rearrangement of atoms within tweezers. Thisrequires communication between the camera system, which detects atoms, and the NIAWG
system, which is capable of moving atoms.
• With three computers, there are more potential points of failure of the system. If any
computer crashed or died, the experiment would stop.
• With three control computers, running the experiment is significantly more complicated.
Even starting a run required a non-trivial sequence of commands which had to be done in
the right order on the different computers.
• Visual Basic 6 is an old language no longer supported by modern operating systems. As
such, the computer which ran this needed to run Windows XP, which at the time of my
PhD was no longer supported by Microsoft, and so had to exist off of the JILA internet
network in order to prevent infection by malware, which is rather inconvenient.
• Being restricted to Visual Basic 6 means that it is diﬀicult to upgrade the control systemwith new modern graphical user interface (GUI) features.
I could go on, but this short list indicates the drawbacks of the previous configuration, and our
motivation for an upgrade. At the start of my PhD, I started developing an independent NIAWG
control system, and after that experience, I embarked on a complete overhaul of the rest of thesystem. As the system was initially mostly a combination of these three independent systems, I
called it Chimera, a fitting reference to the mythological animal composed of many different animal
parts. To understand the system now at the end of my PhD, it is useful to understand a bit of thehistory of development.

27
Initially, I was ignorant of modern GUI design principles and tools. The first implementationof the combined control program was implemented in the “raw” Win32 API in C++. As a result,most of the original implementation was basically C-code with minimal class structure or modern
C++ design principles. At some point, I realized that there was a better API to use, called
the “Microsoft foundation classes” (MFC) application programming interface (API). This was astep forward, but the MFC API was still Windows-exclusive and quite old, not making good use ofmodern C++ design or tools. Recently, furthered by having extra remote time during the pandemic,
I performed a last upgrade, upgrading the system to using Qt 5 instead of MFC. Qt is a modern,well-maintained, well-documented, cross-platform API for creating GUIs in a variety of languages.
It provides many powerful language features which make adding advanced GUI functionalities mucheasier than with MFC. For example, adding tool-tips to GUI elements is quite complicated in MFC
and took me several months to implement properly, while in Qt it is as simple and natural as a
one-line call to a built-in function.
In the end, Chimera is mostly similar to any other system that controls experimental apparatus in a lab, although I’d like to think it’s a very well-designed and modern such system. Several
illustrative screenshots are visible in Fig. 2.8. Chimera has several relatively unique features thatare worth emphasizing:• Real-time data analysis: it is possible to select specific locations on the incoming images,
and monitor incoming data for programmable patterns such as loading, survival, tunneling,
and more.
• Automated laser calibrations: Most of our laser powers are controlled by sending a variable
DC voltage into a mixer, which variably attenuates an RF signal going to an AOD. Chimera
can programmatically vary this voltage and measure the resulting laser powers throughphotodetectors which are connected to Chimera’s analog input system. It can thus createa “voltage versus laser power” calibration curve, which can be re-run at regular intervals
to accommodate for drifting laser powers.

28• Automated experiment calibrations: There are many regular experiments that should be
run in order to make sure all systems in the experiment are working correctly. These include
MOT loading tests, MOT temperature tests, ΛGM cooling tests, single atom loading tests,
single atom imaging tests, Raman frequency calibrations, and Raman sideband coolingtests among others. Chimera is capable of running a set of pre-designated calibrations attimes when the experiment is not typically in use, such as the early morning, creating arecord of the system’s performance as well as keeping systems well tuned.

Figure 2.8: Screenshots of Chimera Control. (Left) The Andor camera window of Chimera.
On the right side of this window, one can see the display showing real-time images collected fromour camera, and on the left side of this window one can see real-time data analysis plots, among
other camera settings. From top to bottom, the real-time analysis plots show the (1) first picture
count histograms (different colors correspond to the different atom locations being analyzed), (2)second picture count histograms, (3) The loading fraction as a function of the parameter being
scanned (in this case a Raman transition frequency), and (4) the survival as a function of theparameter being scanned. (Right) The “main” window of Chimera, which shows the main statuslogs and the automated laser calibration system. The listview on the bottom right shows settingsfor different laser calibrations, and the plot above and on the right shows the calibration curve inreal-time as the data comes in. There are four other windows for controlling other various othersystems in the experiment not shown.

These features reflect the fact that as experiments like this one get more complicated, there
is an increased need for automation and streamlining of the data collection process. Chimera is aheavily multithreaded program, as most GUI programs are. While the plotted real-time analysis
data above is not time-sensitive, Chimera is also capable of performing fast real-time calculations
such as those necessary for doing real-time rearrangement of single atoms in the middle of anexperiment. This will be discussed in depth in Sec. 3.6.

29
2.3.2

Climate Control

We monitor the temperature and humidity in the lab with a home-made temperature monitoring system. The sensors are sensitive to temperatures to about 0.1 degree C, and some thoughthas been put into attempting to purchase or make more sensitive sensors in the future.
In general, we find that the temperature is stable within the lab to approximately 1 degree C
over the course of the day, with the temperature fluctuating less around the somewhat isolated andairflow-controlled main experiment table. The temperature tends to be more stable in the winter,and less stable in the summer, presumably due to differences in the way JILA’s central air handlingsystem handles the two cases. Midway through my PhD, we moved the thermostat sensor frombeing stationed near the door to the room, where there is presumably more noise and fluctuations
coming from the hallway and door, to the middle of the lab. We found that this improved thestability to its current level.

2.4

Other Systems

2.4.1

Intraaction AODs

At the beginning of my PhD, we used Gooch and Housego acousto-optic devices in orderto generate the tweezers. We use a number of AOMs and AODs in the experiment, mostly from
Gooch and Housego, which at various times has labeled their products either as AOMs or AODs.
While perfectly serviceable, these devices had a relatively low bandwidth for our tweezer deflection
application, which limited the number of tweezers we could make simultaneously. So as we scaledup the capabilities of the platform, we desired more suitable deflectors.
We eventually purchased a pair of AODs from the company Intraaction which designs AODs
with a phased-array RF-to-acoustic transducer design which allows deflections over a much wider
frequency. Most AODs are driven roughly by carefully attaching a piezo-electric transducer to one
side of the AOD crystal which converts the RF wave on the input to a mechanical wave in the
crystal. The limiting factor of the bandwidth is that the deflection angle, and therefore the correct

30
matching input angle, changes as a function of the drive frequency. Therefore while the transducer
might be able to drive the crystal over a large bandwidth, the laser will only be deflected over a
relatively narrow frequency range without realignment.
By using a phased-array transducer design, it is possible for the angle of the acoustic wave tochange as a function of frequency in order to compensate this effect. This is the basic operational
principle behind the Intraaction AODs, and why they outperform our previous system.
The model number of the AODs we purchased is ATD-1803DA2.850. The center frequency of
the AODs is spec’d to be 180 MHz, with a quoted bandwidth of about 90 MHz and a corresponding
deflection range of 36 milli-radians. The AODs have an active aperture of 3x4 mm. The laser beam
inside the AOD has a 1/e2 waist of 0.8 mm. The combination of the active aperture size andthe total RF bandwidth in principle limits how many resolved tweezers it would be possible tomake, but we are limited by other factors in the experiment first. In practice, we also find that
we can use the AODs over a broader bandwidth of approximately 120 MHz. We find the driving
performance of the AODs to be slightly irregular, and find that we need between 2 W and 4 W of
RF power in order to eﬀiciently drive the AODs. This drive leads to some non-trivial heating of
the AOD devices, and so in order to maintain thermal stability as much as possible, throughout
the experiment sequence these AODs are driven with a constant total RF power, and the AODs
are fitted with some aggressive heat sinks and fins (Fig. 2.7).
Especially when driving the AOD with many tones, intermodulation effects are significant.
Compensation for these effects is discussed at length in Sec. 3.2. In general we are able to maintainoverall 2-axis deflection eﬀiciencies above 50% even when driving with as many as 11 RF tones,
although these eﬀiciencies vary non-trivially with the overall RF power and details of the drive
configuration.

2.4.2

NI 5451 Arbitrary Waveform Generator

In 2015 we acquired a National Instruments arbitrary waveform generator (NIAWG colloquially), model 5451. This PXIe card is in principle capable of outputting arbitrary, individually31
programmed voltage samples at 400 million samples per second (MS/s) on each of two output
channels. Designed partially with IQ mixing setups in mind, this device was purchased for the
purpose of driving our two tweezer-deflection generating AODs, where we would use the arbitrary
functionality to program different relative tone powers, phases, and frequencies, while doing more
complicated operations like precise frequency ramps for moving the tweezers and carefully designed
flashing patterns, and real time programming for active rearrangement experiments.
Although expensive, the NIAWG can accomplish most of these things eﬀiciently. In particular, our use of it in conducting rearrangement procedures will be discussed in Sec. 3.6. In working
with the device we encountered one major flaw, which is that in operating the device in our desiredmode, “scripting mode” with frequent triggering, the NIAWG is incapable of operating consistently
at its maximum sample rate of 400 MS/s and we therefore we have generally operated the device
at 320 MS/s. This means that the maximum frequency and total bandwidth we can output isapproximately 160 MHz as limited by the Nyquist limit.
Newer designs in the lab and around JILA make use of custom-programmed FPGA devices toachieve similar functionality. While not-truly arbitrary, most of the functionality described above,with the possible exception of some of the flashing techniques, can be accomplished with an arrayof DDS cores on a suﬀiciently fast and large FPGA device.
Because we create tweezers with different AOD deflections, different tweezers within a given
row or column will always have slightly different frequencies and therefore not interfere, except on
fast time-scales which the atoms are not generally sensitive to. However, in a 2D array, diagonalgroupings of tweezers within the array will have the same frequency, and can interfere (Fig. 2.9).

2.4.2.1

Mix-up Setup

An astute reader will have noticed that there is a mismatch between the center frequency of
the AODs and the available frequency range of the NIAWG system. This is real, and our solution
is to mix up the NIAWG output. In this way, we get all the flexibility of the NIAWG over it’s
relatively large 160 MHz bandwidth, but at a higher frequency. We mix up the NIAWG output

32

Figure 2.9: Tweezer Array Frequency Distribution. A depiction of a 10x10 tweezer array,with the different frequencies of each tweezer represented by the color of the tweezer. Tweezers indiagonal slices of the 2d array have the same frequency, despite shifts from the AODs. A typicalspacing for loading, imaging, and cooling is approximately 10 MHz, while a typical spacing fortunneling is accomplished with 4-5 MHz tones.

by combining it with a 255 MHz tone from a Marconi microwave generator in a high quality mixer
from Marki Microwave, part number T3-12LQP. The output contains upper and lower sidebands,
and we use a nice custom low pass RF filter from the company K&L Microwave in order to filter
out the unneeded higher tone and carrier leakage, part number 7L340-229.5/T500-O/O{135}. The
output then goes through a pre-amplifier, and then a standard 5 W amplifier from Mini-circuits,part number ZHL-5W-1+. In general, we find that intermodulation in the AODs themselves are
the leading contribution to non-linear effects in the RF chain, so we do not concern ourselves withpurchasing a higher power or particularly low noise amplifier.

33
2.4.3
2.4.3.1

Other Devices of Note
Electro-Optical Modulators

We use fiber EOMs from the company EOSpace, part numbers PM-0S5-10-PFA-PFA-780SUC315-UL and PM-0K5-10-PFA-PFA-800-UL for creating sidebands for Raman transitions andto generate a coherent repumper for the ΛGM schemes. These EOMS work well at the high 6.834
GHz frequencies required for both of these schemes, although their transmission is not very stable.
Putting more than a couple mW of optical power through these devices requires heating up thedevices in order to avoid photo-bleaching effects.

Chapter 3
Raman Sideband Cooling, Tunneling, Rearrangement, and Other Techniques

In this chapter I describe various refinements of Raman sideband cooling, quantum tunneling,
and various other experiments developed over the course of my PhD. As opposed to Chapter 2,
which deals with experiment hardware, these are all developments that use single atoms.

3.1

Larger Tweezer Arrays
At the beginning of my PhD, the largest array of atoms that we had loaded was a 2 × 2 array.

Larger arrays of atoms are useful for a variety of our experiments and futures plans, and so early
on we decided to embark on a number of projects designed to improve the size of the array we use.
These projects were:• Upgrading our tweezer laser power source to a 2 W and eventually a 3 W tapered amplifier(Sec. 2.2.1.3), in order to have suﬀicient optical power to generate the tweezer array.
• Switching from the ASE lens to the Sill lens (Sec. 2.2.3), hoping that the Sill lens would
have a larger field of view and produce tighter traps than the ASE lens and therefore beable to support more tweezers among other benefits.
• Upgrading from generic waveform generators to the NIAWG 5451 (Sec. 2.4.2), which was
required to do intermodulation compensation and generating large numbers of RF tones tocreate many AOD deflections.

35• Upgrading to the IntraAction AODs (Sec. 2.4.1), which have a larger bandwidth and cantherefore support more deflections than our original AODs.
As a result of these upgrades, at our peak we were able to produce and load a 10x10 arrayof atoms with reasonably good performance (Fig. 3.1). Although the array could probably be
expanded a little to 11x11 or 12x12, at this size, we are running into a number of hard limitations:• Needing more laser power. This is probably the easiest problem to fix with suﬀicientfunding, as I believe one could get more power than we have now with better optimized
tapered amplifier systems, or because Ti:Saph laser systems with larger seeds than our
10 W seed are commercially available.
• The bandwidth of our AODs. It is hard to get a wider bandwidth of deflections without
negatively changing the spacing of our optical tweezers per unit of RF frequency difference
between tones. It is important that this ratio remain small in order to perform quantum
tunneling. Other technologies, such as liquid crystal spatial light modulators, deformablemirror devices, and microlens arrays, can help address this problem as the array of tweezers these technologies generate. However all of these are either static or relatively slow
compared to AOD-based systems and so dynamical tunneling may be challenging usingthem.
• The finite field of view and spot size of our objective. Improving this would require a new
objective system with a large field of view and similar trap performance. These objectives
are quite expensive and take a lot of time and resources to design and test.
In the meantime, a number of groups have expanded their atom systems significantly larger,
and to my knowledge the largest such system is 361 usable traps [76].

36

Figure 3.1: Atom Array Images. (a) A binned, averaged picture of a 10x10 array of atoms.
Differences in intensity reflect a combination of field of view effects and background noise. (b)An unbinned, averaged picture of a 10x10 array of atoms. Notable is a significant “pedestal” ofbackground noise that the atoms sit on, suggesting a long tail of the imaging point spread function,and some aberrations affecting the atom images in the bottom left corner of the image.

3.2

The Evening of Optical Tweezers / Intermodulation Compensation
In order to create an array of tweezers, we drive our tweezer AODs with a sum of up to 10

RF tones simultaneously. In general, the maximum voltages achieved by a sum of 10 RF tones will
be much larger than the voltage achieved by a single tone, even if the same amount of total power
is in both sets of drive tones. This happens when there is a significant amount of constructive
interference between the different tones at a specific time in the waveform. As such, a sum of tones
will generally be more sensitive to nonlinearities in the AOD’s response, which will both reduce theoverall deflection eﬀiciency, add higher order intermodulation effects outside of the original tones,
and cause relative un-evenness between the different deflections. As such, we wish to minimize
these negative effects.
To first order, the effects are very predictable by simply examining the total many-tone

37
waveform. After calculating the waveform given a number of RF frequencies, amplitudes, andphases, one can estimate how bad the intermodulation is likely to be based off of metrics such as
the peak voltage in the waveform or high order moments of the voltage of the waveform. Typically,
we will set up a function which calculates such a metric as a function of the many individual
phases, amplitudes, and frequencies of the RF tones, and then plug the function into a minimization
algorithm which will find a set of phases which minimizes this metric. These parameter spaces tend
have many local minima, so we use a basin-hopping algorithm to find a good global minimum. We
then fix the phases of our tones. One could check the resulting intermodulation on an RF spectrum
analyzer, but this routine is robust enough that we typically do not bother, and instead turn to
optimizing the evenness of the traps.
In general, after the phase-optimization algorithm described above, the traps are even enoughto at least load into all traps. However, for larger arrays, it can be helpful at this point to image thetweezer array directly on a camera, where one can quickly see “by eye” how evened the trap is. We
can then manually even the powers of the RF tones in order to even the array by eye. In general,due to aberrations in the objective lens, evening an array on such a camera does not result in an
evened array inside our vacuum chamber, and so at this point we must turn to optimizing based on
tests that probe the potentials that the atoms actually experience. We can measure the evennessof the traps using a couple metrics, such as the trap depth or the trap frequency. Typically thesewill not be simultaneously optimized due to fluctuating trap shapes across the field of view of theobjective. Most commonly we optimize the depth, which we measure by measuring the light-shiftof the tweezers on the D2 f = 2 → f ′′ = 3 transition [64].
This depth optimization procedure is an iterative one. We measure the depths by analyzing
the light shift they induce on an accessible transition. Then, we assume that the depths areproportional to the powers in each of the RF tones (effectively ignoring intermodulation effects) [64].
With this assumption and a given set of powers and depths, we can predict a new set of powers
which will minimize the variation among the depths. This is our next iteration’s powers. We
again measure the depths using these new RF powers, and repeat the procedure until the depths

38
converge to the desired level of precision. For a 1D array, we are limited by our ability to precisely
and quickly measure the trap depth and temporal fluctuations, but typically achieve depths thatare even to within approximately 1%. For a 2D array of traps, we are more fundamentally limitedby spatially dependent aberrations in our tweezer generation system that make it impossible tocreate an evened set of traps given our limited set of controls over the tweezers. As such, we canonly achieve depths that vary with a standard deviation of approximately 5% of the average depth.

3.3

Flashing Imaging
For large atom arrays, effective imaging can be affected significantly by the uniformity of thetweezer array. In particular, variations in the trap depths result in variable light shifts on the atom,which can dramatically change the scattering rate of the imaging light (Fig. 3.2). Additionally, theoptical tweezer is anti-trapping for excited state 87 Rb atoms, which may lead to some additional
heating in typical configurations.
Inspired by the work done at Harvard, as we ramped up the size of the arrays we were using,
we started using what we refer to as a “flashing imaging” technique [77]. The technique consists of
rapidly alternating the optical tweezer with the imaging light. The flashing is fast enough that theatom’s motion is primarily influenced by the time averaged tweezer potential, but the configuration
is such that the light only scatters off of the atom when the tweezers are off and therefore when
the atom experiences no tweezer light shift and no anti-trapping. This avoids the issues with thetweezer uniformity, and so is especially useful in the initial setup and calibration of the tweezer
array before the depths are evened.
In our case we typically flash the traps and imaging light with period of 0.5 µs. We cantypically gain enough fluorescence to identify an atom after 5-15 ms of imaging time. Notably, theflashing is done with the tweezer’s intensity servo AOD, which is separate from and before eitherof the tweezer deflection AODs that we flashed for rearranging procedures (Sec. 3.6). The imaging
light is in a common σ+, σ− red-detuned PGC configuration with zero’d background fields. The
imaging light is typically detuned from the D2 f = 2 → f ′′ = 3 transition by about 30 MHz or about

39
5Γ, where Γ is the natural linewidth of the transition. This balances the need to scatter photons at
a suﬀicient rate to be detectable above background light with the need to keep atoms cold enoughto remain trapped. The imaging light is flashed with a 42% duty cycle, while the trapping lightis flashed with a 50% duty cycle. In our system, the waveforms are typically applied with a phaseoffset of 165 degrees in order place the imaging light to be centered in the time when the tweezers
are off, however this is in general sensitive a variety of delays determined by things such as the
position of the laser light in the AODs which are used to flash the light. Since these parameters are
all programmatically controlled via Chimera, they can easily be optimized by scanning them andsimultaneously maximizing the imaging survival and the mean count value seen when an atom isloaded.
Our analog servo is typically active and set to a constant set point during this flashingprocedure, but the servo bandwidth, which is typically on the order of 100 kHz, is slow enough that
we do not run into significant issues with this. We have experimented with using “sample and hold”
servo techniques to mitigate some servo fluctuations that happen during and immediately after theprocedure, but we found that keeping this functionality well-tuned in our system was challenging
and generally not worth the additional effort.

3.4

Raman Sideband Cooling
Resolved Raman sideband cooling is a crucial technique in preparing ground-state atoms forour experiments [78, 58]. Throughout this thesis, we use a pulsed method, details of which arerecorded in Adam’s thesis [64]. However, we experimented with a number of small variations while
attempting to cool larger arrays of atoms, and techniques that make use of the ΛGM we developed.

3.4.1

Use of In-Trap ΛGM for Better Initial Cooling

One of the most important parameters for Raman sideband cooling is the initial temperatureof the atoms. In order to be deep in the Lamb-Dicke regime where sideband cooling is effective, theinitial size of the atomic wave packet needs to be small, or all else being equal the atom needs to40

Figure 3.2: Direct Flashing Imaging Vs. Normal Imaging Comparison. (a) Normal
imaging, where some atom locations have dramatically reduced imaging signal compared to theothers, and (b) flashing imaging, which despite uneven and poor loading in the given data set,can still produce reasonable count histograms. Insets: averages of the images used to produce thehistograms. Much of the variation still seen is due to the finite field of view of our objective andother light collection issues. Notably, this data set is not particularly representative of the best
imaging signal which is possible, but it is a good direct comparison between flashing imaging andregular imaging. Red outlines in the inset indicate tweezer locations.

already be relatively cold in the trap. We typically accomplish this with an initial stage of coolingin between the initial atom imaging stage and the Raman sideband cooling stage. Traditionally,a stage of in-trap optimized red-detuned polarization gradient cooling (RPGC) was used for thisstage. However, with the development of the ΛGM we use for loading techniques (Chapter 4), we
had an additional tool at our disposal here.
There are many ways to characterize the temperature of the atoms in a trap, in order to judge
how well different cooling techniques work. Near the ground state of the trap, Raman sideband
spectroscopy can be used to great effect. However for hotter temperatures such as our pre-Raman
sideband cooling temperature, we use another technique. We start by loading and imaging ouratoms and then either apply in-trap optimized ΛGM cooling, RPGC cooling, or no cooling at all,

41

Figure 3.3: Harmonic Oscillator Excitation Populations. A visualization of the populationof a given quantum state n depending on the temperature of the single atom. At low populations,the atom is increasingly concentrated in the lowest harmonic oscillator level, but as expected athigh temperatures states are increasingly evenly populated. Contours are placed at populationsof 0.1, 0.01, 0.001, and 0.0001. (a) Plot for typical radial frequency of 150 kHz, and (b) plot for
typical axial trap frequency of 35 kHz.

leaving the atom at its relatively hot post-imaging temperature. We then reduce the trap depth tothe scale of 1 µK before raising the trap back to its relatively deep imaging depth and conducting
a final image. We then analyze the pair of images to determine how likely the atom is to survive
this lowering process as a function of the reduced depth. This data can be analyzed to determine
the atom’s temperature, but it also immediately indicates that the ΛGM cooling works marginallybetter than the RPGC cooling (Fig. 3.5) [31]. As a result, we have used the in-trap optimized
ΛGM cooling before Raman sideband cooling ever since. The main difference of the optimizationof the ΛGM for in-trap cooling rather than loading is the detuning. While the detuning for theloading process may be on the order of 20 MHz from the free-space resonance, the detuning of thein-trap optimized ΛGM may be as much as 100 MHz in order to cool most effectively. Otherwise,
the procedures are identical.

42

Figure 3.4: Harmonic Oscillator Statistics. A variety of useful plots showing the conversion
between various useful measures of an atom’s temperature in a harmonic potential. (a) Probabilityof being in a given harmonic oscillator n level as a function of the mean occupation number n̄which is the metric typically reported based on sideband spectroscopy. (b) Probability of being ina given harmonic oscillator n level as a function of the atom’s temperature, using a typical radial
trap frequency of 150 kHz and a typical axial trap frequency of 35 kHz. (c) Mean occupation leveln̄ as a function of the atom’s temperature, using the same trap frequencies as (b).

Figure 3.5: The Relative Performance of In-Trap ΛGM. After applying in-trap optimized
ΛGM (blue data), a larger fraction of atoms survive at a given dropping depth than either using
RPGC (orange data) or using no additional cooling (green data). Note that the system at thistime was calibrated for large traps, and there may be a non-trivial offset to our calibration of thetrap depth here. Cooling is done at a typical trap depth of 1.0 µK

43
3.4.2

Raman Sideband Cooling of Larger Arrays

One topic that is of perennial interest to the group is the construction of many-atom degenerate gases starting from individually cooled single atoms. While we did not publish any work on
this front during my PhD, we did do various exploratory experiments in this direction (Sec. 3.5).
The pre-requisite for this work however is starting with a larger array of ground-state cooled atoms,
which is challenging in a number of ways.
The main challenge we experienced with sideband cooling of larger arrays is that of trap
uniformity. Our evening procedure (Sec. 3.2) is not capable of creating a perfectly even set of traps,
but in fact the bigger issue we faced was differences in the trap shapes that became noticeable near
the edges of a 10-by-10 array. This manifested itself most noticeably in the splitting of Raman
transitions into multiple frequencies. Similar splitting has been noticed in other experiments [79],and is presumably due to high order aberrations in the optical tweezer potentials, making them
relatively anharmonic. In particular, simple astigmatism effects should not cause such splittings,
as the atom should always sit in the circle of least confusion of an astigmatic potential.
Still, relatively eﬀicient cooling could be achieved of the large array of traps, at least in theradial direction of the traps. The radial n̄ value of the data in Fig. 3.6 is 0.10(22), where thenumber in parenthesis is the standard deviation of the results, obtained by fitting individual data
sets with a sum of three Gaussian distributions. This result was achieved also by modulating theRaman cooling frequency at 10 kHz in order to effectively broaden the cooling drive. Many typesof experiments would still be sensitive to the remaining thermal population in the tweezers, butthese results are still promising for a number of types of experiments that we dream of. We believethat the best approach for improving these results is improving the traps themselves, which could
be done with a combination of new objective systems and lattice systems.

44

Figure 3.6: Simultaneous Raman Spectroscopy of 36 Atoms. Different colors represent
different atom locations, the black dataset is the average of all individual atoms and the black error
bars represent the standard deviation of the single atom values.

3.5

Tunneling

3.5.1

Dumping Techniques

A general problem in early tunneling experiments was getting the tweezers to be shallow
enough in order to have strong tunneling at large tweezer spacings. Our typical approach in thepast was to simply change the set point of our servo system in order to servo at low powers. However,this means that for low power servoing the photodetector is detecting small optical signals which
are susceptible to noise, and that the servo system is working on very low voltage signals which
are diﬀicult to get working properly in our analog servo. The main tweezer light at the point ofservoing typically has maximum powers ranging from 3 mW to 200 mW depending on the numberof tweezers used in a given experiment, (although only a small fraction is used to servo) which
is attenuated to give a readout of approximately 10 V on our photodetector, which is a Thorlabs

45
PDA10A photodetector which has been modified with some extra filtering in order to have improved
noise performance. Without extra techniques, the minimum voltage during a tunneling experimentmight then be approximately 1 mV.
Other attempts included using a digital servo which had been utilized by some ion trappinggroups at NIST [80], and using a logarithmic photodetector which had been in use by some groups
at Harvard. These both basically worked but were not much improvement over our original system,
and came with their own technical problems.
Eventually, I found a better solution: instead of lowering the set point, we dumped extra
power in “auxiliary tweezers” that would be placed far away from the tweezers of interest. Thus,
the servo would be servoing the same overall power (counting the power in the auxiliary tweezers
in addition to the tweezers of interest), but there would be less power in the tweezers of interest
reducing the overall depth that is relevant for the atoms. This works very effectively, but requires
the ability to dump extra power into tweezers. In our system this is done by having an extra set oftweezers a large distance (and RF frequencies) away from the main tweezes, and similar approaches
might be diﬀicult for doing similar techniques with larger arrays where it’s diﬀicult to put extra
tweezers far away from the tweezers of interest.
This technique allowed us to get to extra small depths spaced extra wide in order to attempt
tunneling in this regime. However, it is also generally useful in other circumstances, as in various
configurations it allows the intensity servo to be run at the same set point settings while having
the absolute depth of the tweezers change. This is particularly important for configurations thatuse different numbers of atoms, as it may be desirable to, in multiple configurations, have similarly
large ranges of servoing while having very different total powers. Without this technique, one
would either have to manually alter the servo parameters between configurations or attempt tooperate the servo over an even larger range than an individual experiment requires. The latter isundesirable because the intensity servo can be somewhat finnicky to keep working well at a variety
of different depth parameters for many experiments. It is easy for the servo to drift and start
ringing at extreme parameters.

46This trick works well enough that we can make our traps shallow enough such that even after
sideband cooling, the traps are so shallow that they only support ground-state atoms. This can bea useful technique as it allows us to release any excited state atoms from the trap before conducting
an interesting experiment. This allows us to make cooling or temperature issues manifest as loss,
which it is generally possible to post-select away, rather than have these thermal atoms contaminate
other signals. Notably, one expects rather discretized loss from our optical tweezers as individuallevels, including eventually the ground-state, should be either supported or not supported at various
depths. We measure however that the atoms are lost gradually as we reduce the trap depth. Thisis suggestive that at these extremely low depths, the traps may be affected by fluctuations in
depth or shape during the ramp process which cause some variable fraction of the atoms to be lost
during the ramp. There are no indications that atoms are gradually lost at the shallow depth as afunction of time, which might be the expected manifestation of consistent noise at the final depth
or gravitationally assisted tunneling out of the trap.
Notably, we find that we must put the dump tweezers unexpectedly far away from the main
tweezers. For example, for tunneling experiments we will typically set the NIAWG to output 75 MHz
and 87 MHz tones to the horizontal AOD, creating a 12 MHz spacing which will shrink to a 5 MHz
spacing during the tunneling portion of the experiment. We find however that we must use verticalfrequencies of 81 MHz and 117 MHz, a spacing of 36 MHz (24 MHz being insuﬀicient), where the
81 MHz tone creates the “main” tweezers which we optimize and analyze for tunneling, while extra
power is dumped into the 117 MHz tweezers. We find that if we put the dump tweezers closer, that
there is a large amount of loss in the main tweezers when they move closer together to initialize thetunnel coupling. This is surprising. The nature of intermodulation, which is normally the culprit
for effects like this, is such that each axis should independently effectively have extra (usuallyweak) tweezers that are offset from the main tweezers by multiples of the frequency differences inthe driving RF tones, which should not allow for any sort of cross-coupling effects. Therefore thenature of this issue is suggestive of a combination of cross-coupling between the two RF systems
and intermodulation. Another possibility is a combination of frequency-reducing very-high-order47
intermodulation and a second-order AOD deflection, which could conspire to produce very weakextra light near the moving traps which could cause heating. This is one of several possibly related
indications of strange couplings within our tweezer generation system, the other main one including
issues we face with the “stiff” rearrangement scheme 3.6.3.

3.5.2

Considerations on Tunneling Fidelity

In general, we expect that quantum tunneling procedures are particularly sensitive to theshape of our optical traps (Fig. 3.7). In order to understand this and other parts of this section, I
will briefly overview the idea behind quantum tunneling in a double-well potential.
We start by assuming that the tweezer potential is symmetric and the tweezers are far separated from each other, in which case the ground state of the system has a degeneracy of 2, andcan be constructed as any two orthogonal superpositions of the ground states of the two individual
tweezers. The natural two states of the system when the tweezers never come close are |L⟩ and|R⟩, the two single-tweezer ground states. In the case that the tweezers couple, the ground statedegeneracy is broken, but the potential is still symmetric under the parity operator along the twotrap axis. As a result, the parity eigenstates |ψ + ⟩ = |L⟩ + |R⟩ and |ψ − ⟩ = |L⟩ − |R⟩ with parity
eigenvalues +1 and −1 respectively will still be good eigensolutions of the problem. Our initialstate preparation is of atoms in the first basis, which can be represented in the second basis, forexample as |ψ⟩(0) = |L⟩ = |ψ + ⟩ + |ψ − ⟩. Then, when the tunnel coupling breaks the degeneracy,
the state will evolve as |ψ⟩(t) = |ψ + ⟩ + |ψ − ⟩e−iJt , resulting in the state at a later evolution timet = π/J being |ψ⟩(π/J) = |ψ + ⟩ − |ψ − ⟩ = |R⟩, and hence we have quantum tunneling. Hong-OuMandel (HOM) interference occurs when two bosonic atoms tunnel and interfere with each otherin coupled tweezers, and is an important demonstration of the type of physics we can explore with
our system [64, 81, 51, 82, 57, 52].
With this foundation established, it is not too diﬀicult to see in concrete ways how the detailed
nature of the tweezer potential can affect these dynamics. If the tweezer potential is not symmetric,
then the sum of single tweezer ground states is no longer an eigenstate of the parity operator, and48
the state will not transfer perfectly. The tweezer potential can be asymmetric for a number ofreasons, including that the bias or relative depth of the two tweezers can be different. Asymmetricaberrations from our objective system may make each tweezer individually asymmetric, which
would also make the sum asymmetric. Spatially dependent aberrations may make each tweezer
different from each other, which would break this asymmetry as well.
The above analysis treats the problem in one-dimension, but the impact of the other dimensions can be significant as well. In general, the different coordinates of a gaussian potential are
separable nearest to the bottom of the trap, and thus we hope that since the tunnel coupling should
ideally be a perturbation only in the radial direction, that the tunnel coupling can not couple stateswith different axial excitations. However, considering that the axial excited states are energeticallyclose, if there is a small axial coupling, it might be easy for either off-resonant couplings or bias
fluctuations to result in transfer to excited states which would negatively affect our tunneling.
All of these issues are diﬀicult to probe directly but guide our attempts to improve thetunneling performance of our system.

3.5.3

Recent Two-Tweezer Tunneling

The above considerations (Sec. 3.5.2) were a major motivating factor for our work in developing the Sill lens as a replacement for our previous objective. If the tunneling is very sensitive tothe nature of the traps, then we’d like the traps to be as tight and well-behaved as possible, whichmight be achievable with a new custom objective lens. It should not come as too large of a surprise
however, considering the struggles we had with the Sill lens (Sec. 2.2.3), that at this point thetunneling with the new objective is no better than before. Key metrics, such as the single particle
tunneling contrast and the Hong-Ou-Mandel (HOM) dip were not improved by the introduction of
the new objective.
In particular, other than simply having the different objective, our improvements to ourcontrol over the trap depth (Sec. 3.5.1) allowed us to test tunneling at even further spacingsand shallower depths, which had seemed to work well previously. Unfortunately, we did not find49

Figure 3.7: Visualizations of Tunnel-Coupled Optical Tweezers. All Gaussians have atypical waist of 700 nm, although these plots could be also be displayed in a dimensionless fashion as
a function of the trap spacing over the trap waist. (a) Definitions of various parameters for tunnelcoupled optical tweezers. (b) Simulated image depicting typical spacing and relative powers of main
tweezers, which hold atoms, and dump tweezers, which generally don’t and are never analyzed or
optimized, for a two main tweezer configuration during, for example, the loading stage. With only
two main tweezers, most of the power is in the dump tweezers, as the configuration with zero
dumping is typically set to support for a larger array. (c) Numerical calculation of the spacing
between the two minima of a double-gaussian potential as a function of the gaussian spacing. (d)Visualization of variations of the bias between tweezers. (e) Visualizations of variations of thespacing of the tweezers. When the spacing is reduced to a single trap waist, the bump between thetwo potentials disappears. (f) Visualizations of variations of the single tweezer depth.

improved performance here, but rather we believe we run into more fundamental limitations ofshallow tweezer tunneling. At typical depths, the tunnel coupling between the ground and thenearby excited nx = 1 motional state is expected to be larger than the ground-to-ground coupling
due to the tweezer geometry, but should still be finite and small enough to not significantly affectground-to-ground tunneling. At particularly shallow depths and larger spacings, as one approaches

50
the regime where there is only one bound state per tweezer, we expect that the ground-to-excited
coupling increases dramatically before eventually becoming ill-defined as the bound-states are lost.
In this regime, the ground-to-excited state coupling can become quite large and potentially interferewith the ground-to-ground tunnelling on the relevant side of the tunneling resonance. This is highly
visible in Fig. 3.8(d), where the tunneling resonance in bias-space is highly asymmetric. Eventually,
we additionally suffer significant single-particle loss when working with shallower and further spaced
traps. We may also be increasingly affected by issues with the trap shape as at reduced depths
the wavefunction is larger and samples more of the flawed potential. Additionally, it is possiblethat we suffer issues in coupling to axial excited states as the axial trap frequency approaches thetunneling frequency. There are some hints of asymmetry and extra resonances in bias scans suchas Fig. 3.8(d), however it is in general diﬀicult to tell whether these issues come from issues withstate preparation, tunneling initialization, or such extra resonances.
The tunnel coupling is generally increased by working at a closer spacing, and decreased
by working in deeper tweezers. We can therefore achieve similar tunneling coupling strengths byboth decreasing the spacing and increasing the tunneling depth, and by doing this we can attempt
to avoid the issues discussed above. In general we find that tunneling works best at tunnel rates
greater than 100 Hz, and presumably at weaker couplings we are sensitive to other types of noisesuch as intensity noise in our traps. At short distances and deep traps, however, we suspect that
we are increasingly sensitive to shot-to-shot bias fluctuations. This is supported by the fact that atdeep depths, it is diﬀicult to get coherent tunneling, where we have to consistently sit on-resonance,
but that adiabatic rapid passage works pretty well (Chapter 6, Fig. 6.7).
Even at large depths, we still see hints of unexpected, non-trivial structure in our tunnel
couplings. These are visible in scans of the bias over large ranges, which were completed in order
to find ground-to-n = 1 and ground-to-n = 2 states as was required for our quantum tomography
work (Fig. 3.8(c)). In these scans, we can see significant unexpected structure in the tunnelingtransfer, which may indicate couplings to unexpected states, and is worth future investigation.
This structure may result from aberrations and structure in the tweezer potential, couplings to51
axial or orthogonal radial excitations, or combinations thereof. One should also not completely
rule out more mundane issues with state preparation and tunneling initialization, as it can be
non-trivial to cleanly create the desired initial state at the beginning of the tunneling procedure,
and we have less experience doing this in the highly asymmetric traps required for excited statetransfer.
Many of these comments regarding possible explanations for tunneling issues are at some level
speculative. The direct things that we can measure while tunneling are the tunneling contrast, HOM
contrast, and tunneling decay rates, which are all relatively indirectly related to these potential
causes, making it in general diﬀicult to precisely identify the causes of reductions in the tunnelingperformance.
An example set of coherent tunneling data with HOM interference is shown in Fig. 3.8(b).
The HOM dip is seen in data where an atom was loaded in each tweezer and manifests as a reductionin the probability of measuring one atom in each tweezer (P11 ) at times when each atom has had achance to delocalize over the tweezer pair and interfere with the other. HOM interference causes thebosons to bunch up and prefer to lie in the same tweezer at these times, in which case our imaging
system will tend to kick out the pair of atoms and we will observe no signal, i.e. a reduction in
P11 . Ideally, after loading two atoms, we measure either one atom in each tweezer or zero atoms
at all. However, we occasionally observe a single atom in the final image, which we refer to as a
2 → 1 event (referring to two atoms in the initial image, and a single atom in the final image). Thiscan occur because a single atom was lost due to finite temperature effects or background collisions,cases which should be discarded in the HOM data, or cases where at the end of the experiment,
two atoms were in the same tweezer but the imaging only kicked out one of the pair of atoms. In
this latter case, the single atom remaining is a sign of HOM bunching, and this should be includedin the data set and contribute to the P11 drop.
The data in Fig. 3.8(b) is conservative, and throws away all such single atom measurements,
resulting in a minimum HOM P11 signal of approximately 0.30. This can be viewed as a upperbound estimate of the HOM dip, which is significantly lower than classical expectations already.

52
Aggressively including all such single atom measurements in the analysis, as if they all resulted fromimperfect imaging instead of single particle loss, yields an optimistic minimum P11 signal of approximately 0.19, which can be viewed as a lower bound. Both of these limits are discussed in Adam
Kaufman’s thesis [64], and the latter is typically plotted. Independent imaging measurements, conducted well after the tunneling measurements, indicate that at typical imaging parameters, whenattempting to image two atoms loaded into one tweezer, a single atom remains after the image
approximately 25% of the time, and the rest of the time zero atoms remain. This imaging imperfection is consistent with numbers reported by Adam. There are then several simple ways of
estimating the true P11 fraction before the imaging takes place.
(1) Knowing the expected number of single-particle loss-based 2 → 1 events based on independent single particle loss measurements, one can attribute the remaining 2 → 1 events toimperfect-imaging.
(2) Knowing the imaging imperfection rate (approximately 25%) and knowing the number of
2 → 0 events observed, one can calculate the number of imaging-based 2 → 1 events we
expect to have occured and include this many in the calculation of the P11 dip. This
analysis was completed and suggests a true P11 minimum of approximately 0.24, which sits
nicely in between these bounds. Confidence in this estimate could be reasonably improvedby conducting such imaging tests at the exact same parameters as were used in tunnelingmeasurements, and in more reasonable temporal proximity.
(3) Lastly, one can look at fluctuations in the observed 2 → 1 fraction as a function of tunnelingtime, and therefore effectively the true pre-imaging P11 fraction, in order to estimate how
many of the 2 → 1 events are due to this imaging issue. This was effectively done in Adam’s
thesis [64].
This area in general is fruitful ground for interesting tests of our system, and in general wethink that there are many ways to continue to create and study interesting non-classical motional
states. However, such testing is very time-consuming, especially considering the need to keep the

53

Figure 3.8: Tunneling of Single Atoms Between Two Tweezers. Shown in all plots are
the probabilities of an atom tunneling from the left tweezer to the right tweezer (red data), andthe probability of tunneling from the right tweezer to the left tweezer (blue data). (a) A typicaltunneling bias scan taken at a single-tweezer depth of 1.23 µK and a spacing of 0.84 µm. One canrefer to Fig. 3.7 for illustration of the bias parameter. Notably, the tunneling does not perfectly go
to zero outside of the bias resonance here, and the tunneling resonance does not rise all the way
to 1, which are a consistent problems in optimizing the tunneling. (b) A reasonably good timetunneling scan taken at the shallow depth of 0.25 µK and a gaussian spacing of 0.84 µm. Shown are
the tunneling probabilities as a function of the amount of time during which the tunnel couplingis on. Also shown are the probabilities of measuring one particle in each tweezer post-selected oneither zero or two atoms surviving, typically referred to as P11 (black data). Some typical problems
are seen here in that the tunneling atom does not transfer all the way to the opposing tweezer,
but returns to the original tweezer nicely. Typically, at these shallow depths, the tunneling time
traces remain well-behaved longer than the bias scans do. (c) A wide bias scan at the relatively
deep depth of 1.89 µK and relatively wide spacing of 0.94 µm which was close to what was used for
preparation of 1st and 2nd excited states for the tomography work discussed in Chapter 6. Shown
are many unexpected resonances surrounding the expected n = 0 → n = 2 resonance. Notably, thereverse transfer is very small, a signal which is useful for diagnosing temperature issues in the state
preparation. On the far left side of the resonance, we expect increasingly incoherent dumping of
the atom from one tweezer into the other. (d) A tunneling bias scan at a relatively wide spacingof 0.94 µm and shallow depth of 0.32 µK. In this regime, on the relevant side of the bias resonance,
the transfer population does not nicely go towards zero, but rather it seems that the atom is almost
immediately delocalized between the tweezers on the relevant side of the resonance.

54
Raman sideband cooling highly optimized throughout testing. As a result, as it became clear that
our tunneling performance was not significantly improved over previous results, we did not delay
too much in sustained tests at this point in my PhD, and instead moved on to new experiments invarious attempts to circumvent these issues.

3.5.4

Three-Tweezer Tunneling

One of the categories of things that we were interested in was expanding the tunnelingexperiments to more tweezers, and potentially involving more atoms. To that end, we wished
to analyze a single atom tunneling between 3 optical tweezer potentials. This problem can beapproximated as an implementation of the following simple Hamiltonian: 0 J3 0  J3 0 J3 
0 J3 0(3.1)

Where each atom is directly coupled to the motional eigenstates of its nearest neighbors. We
neglect couplings to tweezers further away or any relative biases between tweezers, although this
could be easily added. Diagonalizing this Hamiltonian gives three non-degenerate eigenstates:|L⟩ − |R⟩√2√√
|L⟩ + 2|C⟩ + |R⟩ i 2J3 t
e
ψ2 =
√ 2
|L⟩ − 2|C⟩ + |R⟩ −i√2J3 t
ψ3 =
e
2
ψ1 =

(3.2)
(3.3)
(3.4)
(3.5)

For single-tweezer eigenstates centered on the left, center, or right tweezer being labeled|L⟩, |C⟩, and |R⟩ respectively. Then, given that you start in the leftmost tweezer (similar to startingin the right), the probability of being in any tweezer evolves in time as:55|⟨L|ψ⟩| =2( cos(√2J t) + 1 )2
3

2
( cos(√2J t) − 1 )2
3|⟨R|ψ⟩|2 =2 √2J3 t)1−
cos(2|⟨C|ψ⟩|2 =
4

(3.6)
(3.7)
(3.8)

While if you start in the center tweezer, the probabilities evolve with:√
1 − cos(2 2J3 t)|⟨R|ψ⟩| = |⟨L|ψ⟩| =4√
1 + cos(2 2J3 t)|⟨C|ψ⟩|2 =
4
2

2

(3.9)
(3.10)

These calculations form the basis of our expectations for a single particle tunneling between
three wells. In adding more wells, it becomes somewhat more tedious to set the biases of thethree tweezers to enable resonant tunneling, but I believe it is not particularly complicated. Afteraligning this bias, we experimentally measure the probability of an atom being in a given tweezer
as a function of the tunneling time during which the tunnel coupling is active (Fig. 3.9).

Figure 3.9: Tunneling of a Single Atom Between Three Tweezers. Shown are the probability of an atom finishing in the center tweezer given that it starts in the left tweezer (red), thecenter tweezer (green) or the right tweezer (blue). Ideally the red and blue should be identical, andthe green curve should drop completely to zero and rise completely to one.

56
These results show nice, relatively coherent tunneling of a single atom between three opticaltweezers, and suggest that our system could likely be expanded to support tunneling in more
tweezers. We did not seriously examine multiparticle dynamics in three tweezers, as we expected
it to be diﬀicult to see appreciable interference affects without significant work. However, since
our goal was to examine the interactions between many particles, we soon proceeded to otherexperiments in order to explore these dynamics, and did not return to single particle tunnelingdynamics afterwards given the challenges we encountered there.

3.5.5

Predicting Two-Particle Tunneling with Interactions

In general calculating the eigenstates of double-well Gaussian tweezers is done using relatively
involved discrete variable representation (DVR) techniques. This has been done in the past in
specific circumstances with the help of theorist Michael Wall [83]. Additionally, Sean Muleadyhas also assisted us in understanding how various factors such as additional couplings and thermal
population could be affecting our tunneling performance.
A simpler approach is to assume that the ground states will be approximately a sum ofGaussians, and to use the variational principle in order to create an estimate for the parameters of
the sum of Gaussians as a function of the spacing of the two tweezers (Fig. 3.7). The variational
principle is specifically used to calculate the parameters of a single particle in a symmetric sum of
two gaussian wavefunctions, and does not involve interactions. This variational calculation could
be expanded to include two particles or asymmetric sums of the wavefunctions for more accuracy(Fig. 3.10).
With this solution, we can then estimate the interaction strengths, kinetic energy, potential
energy and total energies as a function of the tweezer spacing. For gaussian wavefunctions witha given spacing, these energies can be determined analytically and calculated quickly. To facilitate understanding of the ground-state dynamics, we calculate these parameters for four potential
combinations of the ground state, which we refer to as the “Mott-insulator”-like state, ΨM I , the“double-filled” state ΨD , the “superfluid-like” state Ψsup , and the “anti-superfluid”-like state, or57
what would extend to a superfluid with a single excitation, Ψasup . The states, interaction energies,
and kinetic energies are recorded for the x-dimension of the problem in Table 3.1, and the actual
values as a function of distance, using the variationally calculated parameters for two 3D gaussian
potentials, are plotted in Fig. 3.11. These parameters were mostly analyzed as a function of thespacing of the two tweezers. Typically, we initialize the tunneling with either an intensity ramp or abias ramp, so these solutions might be better calculated as a function of either of those parameters.
State Definition∆
ΨD = ND ψg (x1 + ∆2 )ψg (x2 + 2 )(∆ΨMI = NMI ψg (x1 − ∆2 )ψg (x2 +) 2 )∆+ψg (x1 + ∆2 )ψg (x2 − 2 )
(
)
(
∆
)
+
ψ
(x
+
)
Ψsup = Nsup ψg (x1 − ∆
g
1
2
2
)
(
)
∆
× ψg (x2 − ∆
)
+
ψ
(x
+
)
g
2
2
2
(
(
)
∆
Ψasup = Nasup ψg (x1 − ∆
)
−
ψ
(x
+
)
g
1
2
2
)
(
)
∆
× ψg (x2 − ∆
)
−
ψ
(x
+
)
g 2
2
2

Interaction Energy√
ID = π2 wa0
2
ID exp(2χ2 )+1Kinetic Energy22~
KD = mw
2
0

K
)
( D×
2χ2
1 − exp(2χ
2 )+1

ID ×

K
)
( D×
2χ2
1 − exp(χ
2 )+1

ID ×

K
( D×
)
2χ2
1 + exp(χ2 )+1

exp(2χ2 )+4 exp(χ2 /2)+3
exp(2χ2 )+2 exp(χ2 )+1exp(2χ2 )−4 exp(χ2 /2)+3
exp(2χ2 )−2 exp(χ2 )+1

Table 3.1: Tunnel-Coupled Tweezers Analytic Calculations. Various energies of interest offour two-particle one-dimensional wavefunctions involving various sums of Gaussians. χ = ∆/w0 ,where ∆ is the spacing of the two wavefunction centers and w0 is the wavefunction waist (not theprobability distribution waist, which is used elsewhere). ψg = exp(−2x2 /w02 ). The parameters of
the Gaussian are to be determined using the variational method as shown in Fig. 3.10. Adding
the energy contributions of the other dimensions is straightforward (if tedious) if one assumes thatthe potential is separable and that the solutions in those dimensions are simple Gaussians.

3.5.6

Adiabatic Ramsey Experiments

In general, we have had relatively good success with adiabatic tunneling procedures, wherean adiabatic transfer between the ground-state of one tweezer and the first excited state of anothertweezer can be done with fidelities as high as 92.4+1.9−2.5 %. These procedures also leverage that thecoherence time of our tunneling procedure is relatively long. Although, there may be more nuance

58

Figure 3.10: Two-Tweezer Variational Solutions. Variational Calculation of various groundstate solutions to the double-well problem. The axial and y (perpendicular to the multiple tweezeraxis) trap frequencies monotonically increase as the separation decreases and the tweezer overlap
increases, increasing the effective trap depth. The frequency along the multiple tweezer axis decreases as the tweezers approach each other and the barrier between the two tweezers lowers andreduces the effective trap frequency. Lastly, the location of the wavefunctions more or less follows
the bimodal minimums of the potential rather than the actual locations of the gaussians, which
diverge at close spacings. At spacings closer than the waist of the tweezer (2σ ≈ 700 nm), data
for which is not shown, there is no barrier between the tweezers. The variational solution in thisregion initially suggests a finite spacing between the multiple gaussians to effectively make a lowertrap-frequency solution, before eventually collapsing to having no spacing.

to the state we prepare using these procedures, as is explored in Chapter 6. While the HOM contrast
with diabatically initialized tunneling was always limited, we hoped to be able to circumvent someof these issues with an adiabatically prepared state.
Adiabatic state preparation naturally occurs very slowly, and as such it makes itself a natural
method to explore the detailed influence of atom-atom interactions on tunneling procedures. By
adiabatically preparing a multi-particle state in regimes where the interactions are significantly
stronger or weaker than the tunneling, we would hope to be able to see different tunneling behavior.
This would be an important stepping stone to observing similar regimes with more particles, andwith enough atoms we might see the transition between these regimes manifest as a quantum phasetransition between a mott insulator and a super-fluid.
The main method we used to explore this is effectively an adiabatic prepared Ramsey experiment. First, we attempt to adiabatically bring a pair of traps to be resonant with each other. We
experimented in doing this either by lowering the trap depth, which slowly increases the tunnel cou-

59

Figure 3.11: Two-Atom Tunneling Simulation Energies. Calculation of the various energies
in the 3D double well problem as a function of the tweezer spacing at a fixed tweezer depth. States
are defined and 1D energies are calculated in Table 3.5.5, parameters of the solutions there aretaken from the variational calculations (Fig. 3.10). (a) The total energy minus the mean potential
energy averaged between the different states. The purpose of the subtraction is simply to makethe difference between states more visible, as the potential energy is significantly larger than theother energy scales of the problem. At large distances, the Mott-insulator-like state, which is what
we create when we do standard ground-state preparation of two atoms on the same tweezer which
contributes a positive interaction energy. (b) The interaction energy. The interaction energies
decrease slightly as the tweezers start to approach each other as the x-axis trap frequency decreases,
but eventually increases due to the increased confinement from the other axes. Superfluid and antisuperfluid states have nonzero interactions throughout which gives them higher energy than the
ground state at large tweezer distances. (c) The kinetic energy. All kinetic energies are identical atlarge distances but split as wavefunction overlap between the two tweezers becomes non-negligible.
The superfluid state minimizes the kinetic energy. (d) The potential energy of the system, whichmonotonically decreases at smaller spacings as the tweezer overlap, and the effective overall depth,
increases.

pling, or by slowly changing the bias between tweezers. In the case of a single atom, this procedure
should coherently delocalize the atom between the pair of tweezer potentials. We then abruptly
change the bias of the tweezers, causing the two parts of the wavefunction to evolve depending on
how large a bias is applied. The bias is then abruptly returned to zero, and the tunnel couplingis adiabatically reduced. Mapping out the final location of the atom as a function of the time thewavefunction evolves in uneven traps results in measurement of characteristic Ramsey oscillations

60
between the two traps. In configurations with strong interactions, we expect no delocalization andno Ramsey oscillations, whereas with weak interactions and relatively strong tunneling, we expect
delocalization during the adiabatic passage followed by full Ramsey oscillations.
In the end, however, we were not able to observe the regime where the interactions werestronger than the tunneling and see the effect of the interactions in an unambiguous way on thetunneling. This may be due to a number of factors. The interaction strength in our system may
have been smaller than we expected for traps of this relatively shallow depth. Also, noise issues
may have prevented us from being adiabatic with respect to the interaction strength anyways. In
general, the interaction strength can be relatively large in deep traps. However, we need to beadiabatic in shallower traps to still be able to tunnel effectively.

Figure 3.12: Adiabatic Ramsey Oscillations. Included are the probabilities for single particles
to be in the left well, whether they started in the left (green data) or in the right (red data).
Additionally included is the probability to measure a particle in each tweezer (black data), giventhat two particles were loaded and that either two or zero survived, post-selecting out cases where
only one particle survived as this is possibly a result of single particle loss.

3.6

Rearrangement Techniques
As long as it is not possible to deterministically load an optical tweezer, rearrangement of

already loaded atoms is the best way to reduce the positional entropy associated with unloaded
tweezers, and thus is crucial for quantum simulation and quantum computation experiments with

61
larger arrays of atoms. Rearrangement techniques have proven to provide fruitful ground for many
publications, starting with the first demonstrations simultaneously published [84, 85], followed bya series of various expansions, optimizations, and implementations in different systems, including
our own. Rearrangement has been demonstrated in a number of groups working with either opticallattices or optical tweezer arrays [86, 87, 88, 89, 90, 76, 91, 92].
The tweezer-generating system we use, which uses two orthogonal acousto-optical deflectors,
can only create grids of traps, although the grid can potentially be unevenly spaced. Additionally,
tweezers within the grid generally have different frequencies. In particular, if we attempted to use
our ordinary tweezer system to create an extra row of traps which traveled between nearby rows
in order to transport traps from one row to another, intermodulation effects within the unevenly
spaced array of tweezers would cause additional, unwanted tweezers to appear with unpredictable
amplitudes throughout the array, which would perturb other trap locations and cause loss. Practically speaking then, it is not possible to do rearrangement in this configuration and we are limited
to producing only a regularly spaced array of traps.
Unlike many groups that use slow, shear-wave acousto-optical deflectors to create tweezer
arrays, in our group we use fast longitudinal-wave devices. This opens up several possibilities
which we were eager to try to take advantage of, and which I describe below. All of these methods
worked at a reasonable level, and were suﬀicient to allow us to generate some nice pictures andresults that we included in the ΛGM paper. However, in order to be competitive in large arrays,
rearrangement techniques need to be able to move a particle from site to site with > 90% fidelity,with many applications requiring > 99% in order to consistently assemble perfect large arrays. As
such, we determined that the methods we developed here were insuﬀicient at the best fidelities
we could achieve, and that pardon any breakthroughs, in our system it would be best to simply
implement a second tweezer system which could be moved independently with respect to the array,similar to the techniques that other groups use. However, I record the results of the techniques wetried here in order to illustrate some of the challenges and pitfalls that others should watch out for.

62
3.6.1

Flashing Rearrangement

In this first technique, we attempted to move tweezers by rapidly alternating our RF generation system between two RF configurations. One configuration would be the RF to generate astatic grid of traps, in which most atoms would sit in. The second configuration would be a single
moving trap, or occasionally several moving traps when we were feeling ambitious. If this alternating is done rapid enough, atoms should only see the average of the two optical potentials. However,since the optical potentials are separated in time from each other, there is no interference of thetwo, avoiding problem (1) above, and additionally it is possible to break the rectangular symmetry
posed in problem (2) above. So this method seems promising in principle. We use a Hungarianminimum cost-matching algorithm to calculate the sequence of moves required to rearrange theatoms into a given configuration.
Timing Details

In these experiments, since the procedure to rearrange atoms must be

done in real time after getting information from the camera, a number of things must happen forrearrangement to occur.
• The first image is completed and received by Chimera from the camera, which involves low
level processing by the computer which I have no control over.
• The image must be processed in order to determine where atoms have been loaded.
• The sequence of desired moves, in abstract form (this location to that location) must be
calculated• The RF voltages (each sample must be individually calculated, as this is how the NIAWG
is programmed) corresponding to the sequence of moves must be calculated or assembled
into the total waveform if precalculated• The total rearrangement RF waveform must be streamed to the NIAWG. In general, when
the NIAWG is in scripting mode and looping on a waveform while waiting for a trigger,
it loads only the immediate next waveform into its output buffers. During this looping

63
however we are free to actively write and overwrite waveforms that are to be outputtedlater in the experiment, such as a large waveform set aside for rearrangement.
Notably, the each individual move’s RF can be pre-calculated and held in memory for reasonable array sizes, and so this step which might otherwise be somewhat taxing for this form ofarbitrary waveform generator is circumvented, and the stage of combining the RF for the individual
moves only takes ∼3 ms. The image processing and sequence of move calculations are both quite
fast on these time-scales and take only ∼1 ms. The time-limiting steps are the time it takes for theimage to be received by the Andor camera, which takes about 25 ms for typical image sizes, andthe time it takes to stream the waveform to the NIAWG, which is about 50 ms for waveforms that
contain about 100 individual moves, or 6 ms worth of moving time. It is likely possible to increase
the streaming speed by using a faster PCIe MXI card, which is the card used to communicate with
the NIAWG’s PXIe crate. We are currently using the slowest such card.
Various Rearrangement Parameters

In general the depth of the tweezers being moved

is much larger than that of the static array, and the optimal depth varies depending on how fast
the move is being conducted. We found that it worked best to use relatively fast movement timesof 60 µs. In particular, we found that moving slower made various issues worse rather than better.
This seems likely because there is a period of time at the start and end of a rearrangement movewhere the atom sees both the static tweezer and the moving tweezer, and that this back and forth
pattern is a large perturbation that is probably capable of causing heating at frequencies that asingle move is not capable of. Generally, intensity fluctuations are symmetric and only couple
states of the same parity, causing them to drive transitions when they are near multiples of twice
the trap frequency, while pointing fluctuations such as this referenced issue are capable of driving
transitions at all multiples of the trap frequency.
Flashing Speed

The biggest limitation of this technique is the speed at which tweezers

can be effectively flashed. While the AOD responds at very high frequencies, it takes a non-trivial
amount of time for an acoustic wave to travel across the body of the AOD crystal. Our AOD

64
crystals are telerium-dioxide, which has a acoustic wave velocity of 4.2 mm µs−1 , meaning thatfor our 3 mm wide crystals, it takes 700 ns for the acoustic wave to travel across the crystal (thebeam size inside the crystal however only has a waist of 0.8 mm). If we naively switch the RF
configurations abruptly, inside the devices the tweezer light will hit both acoustic wave fronts atthe same time. Specifically, if switching the horizontal (vertical) AODs between RF patterns fha
and fhb (fva and fvb ), while we will get the desired deflections of fha × fva and fhb × fvb , we also
get undesired tones at fha × fvb and fhb × fva . As a result, we typically operated near a flashingrate of 1 MHz, and with a dead time of ∼100 ns, considering that our tweezers do not entirely fill
the AOD devices.
Unfortunately, these speed-based limitations are hard to improve on for AOD-generated
tweezers. In general, it is desirable to make AOD crystals and the inputted laser size large, as
this increases the number of resolvable deflections that the AOD can generate. This downside isaddressable by using a smaller wavelength trap light which can, at the same size, make more spots,
but we are limited in how short of a wavelength we can use without causing significant scattering
off of the 87 Rb atoms. We are therefore not able to simply focus into the AOD or use a smaller
crystal without sacrificing the size of the array. We are also limited in what materials it is feasible
to make an AOD crystal out of. Some materials, such as fused silica, have higher speeds, up to
about 6 mm µs−1 , but it is unclear to what extent an increase of only 50% solves this problem,and considering the significant time investment in testing a new AOD system compared to theeffort of implementing a second independent tweezer system, and we decided the former was not
worth the potential upside. However a combination of improvements along these lines along with
better-performing objective optics might improve the performance of this method of rearrangement.
Fidelities

We found that for a single, deterministic transfer between two tweezers, it waspossible to optimize the rearrangement to be highly eﬀicient, near 100% fidelity and not easily
distinguishable from background loss in our system. However, for a system of 3 × 6 tweezers, whichis the size we were using at the time, there are 54 possible moves between neighbors that need
to work very eﬀiciently. We found that it was quite challenging if not impossible to optimize our

65system to be eﬀicient for so many different patterns at the same time. We found that our best
results, after a large amount of individual move optimization, we could prepare a single loaded
optical tweezer in a single experimental cycle with only about 90% fidelity (note that some groups
report fidelities after many repeated cycles of rearranging and re-imaging to improve results). Thiswas significantly higher than the loading eﬀiciency of about 50% that we had at the time, but notsignificantly better than previous results we had obtained with enhanced loading [93] (the ΛGM
procedure had not been developed yet). With an eﬀiciency of 90%, it would be diﬀicult to prepare
any macroscopic arrangement of atoms with a reasonable repetition rate. However, it was still
possible at these rates to create pictures of letters and spell out words (Fig. 3.14).
Flashing Rearrangement Conclusions

In the end, after many optimizations, we were

faced with a number of confounding issues in the flashing rearrangement that made the technique
as a whole seem undesirable.
• Unexpected loss resonances requiring traps to be at very specific depths during the movement.
• Some movements would cause loss in nearby tweezers which in principle should be unaffected.
• Some unexplained issues prevented some moves from ever reaching a high move fidelity at
any parameters• Many of the optimized parameters did not seem very stable and required re-optimization
on a weekly scale.
Many of these affects may be explained by a combination of factors:• Nontrivial variations in trap shape across the field of view of our objective• Our tweezers, like our PSF, may have a long tail which might be able to non-trivially affecttweezers which we would ordinarily think are far enough away.

66
It should be emphasized that it is still possible that these issues could be addressed and thatthe method could work. Indeed, future systems that have more well-behaved objectives may wish
to test these ideas, as there are significant advantages in having a self-contained rearrangement
system, including the need for consistently registering the two sets of potentials. However, it isimportant to remember that there is already a well-tested system for doing rearrangement, namely
the introduction of a completely separate tweezer system in order to do the rearrangement.

Figure 3.13: Rearrangement into a Line. (a) The initial loading pattern of a 3x6 array oftweezers. Lines show an example of how atoms could be rearranged to make a one-dimensional lineof atoms. (b) A line of atoms, after being rearranged.

3.6.2

Jerk Rearrangement

Another idea which avoids some of the issues of flashing rearrangement, but introduces others,
is what I call “Jerk” rearrangement. In this method, we turn off the main array, then use a single
very deep tweezer to move the target atom from one site to another very fast (on the order of 1
µs per move), and then turn the original array back on, before the original atoms had time to fly
away. This idea is essentially flashing rearrangement in the speed limit of a single flash. Somewhat
surprisingly, it worked with a similar fidelity as the original flashing rearrangement. It sufferedfrom similar issues in that the exact depth (which was generally much deeper) required careful,site-selective calibration, and the movement of the particularly deep tweezer tended to cause loss

67

Figure 3.14: Atoms Spelling Out JILA. Four separate images of atoms after being arranged
into letters to spell out “JILA”.

in nearby tweezers.

3.6.3

Stiff Rearrangement

Another technique which seems simpler to implement than the others, but that is less powerful, is what I call the stiff rearrangement technique. This is the technique which went into theenhanced loading paper discussed in Chapter 4 [63]. In this technique, the initial loading configuration is analyzed to determine if one can make a target final array by removing entire rows andcolumns of the initial array, and the shrinking the rest (Fig. 3.15). Although all rearrangement
techniques benefit from enhanced loading techniques, this is especially true with stiff rearrangement.
After densely loading an array, we first obtain the location of each atom using a single image(left panel Fig. 3.15(a)). Even with dense loading, the probability of loading a specific set of 6 × 6
traps is exponentially small (dashed lines in Fig. 3.15(b)). However, there are many potential setsof (sometimes disjoint) 6 × 6 traps embedded in the 10 × 10 array. We then search for such aconfiguration of completely loaded 6 × 6 traps. If successful, we turn off the extra traps to remove
the excess atoms, and then contract and shift the identified disjoint array in a single move (rightpanel) [84]. At the time of this work, successful rearrangement to a square n = 36 (6 by 6) array

68

Figure 3.15: Stiff Rearrangement in 2D. (a) Single image of 80 atoms in an array of 100
traps with separation 2 µm (left). Binarized loading (center) indicating empty traps (red squares),occupied traps (gray pixels), and loaded traps selected for parallel rearranging (yellow pixels)
to a defect-free 6 × 6 array. For the single-shot images, in each trap pixel (background pixel)
the atom count threshold (average threshold of nearest neighbors) is subtracted for clarity fromthe count number recorded by the camera (see Appendix). (b) Monte-Carlo-simulated successprobabilities (SP ) of finding a filled disjoint array of atoms within 10 × 10 traps as a function ofloading eﬀiciencies (solid lines), and corresponding loading probability of a specific array without
rearrangement (dashed lines). Colors indicate target array sizes: 3 × 3 (red), 4 × 4 (green), 5 × 5(blue), 6 × 6 (orange), and 7 × 7 (black). The light gray area is only accessible with average loading
greater than 80%, and the dark gray area for greater than 90%.

only worked in 0.1% of cases due to unexpected loss observed when turning off rows and columns, inwhich an atom is effectively lost with a 17%-chance. This loss is technical in nature, and potential
causes include intermodulation in the tweezer-generating RF tones or collisions between the trappedand dropped atoms. It is potentially related to other issues seen in the dumping of extra power
into distant optical tweezers (Sec. 3.5.1) and the non-trivial imaging point-spread function of oursystem (Fig. 6.8).
However, as illustrated in Fig. 3.15(b), observing this array with this parallel technique would
have been impossible without enhanced loading. In increasing the loading probability P from 60%
to 80%, the percentage of experiment runs in which one could possibly extract a defect-free 6 × 6

69
array goes from 0.02% to 37%. Notably, this entire procedure is completed using only a pair of
acousto-optic deflectors to control the optical tweezers.

Chapter 4
Gray-Molasses Based Optical-Tweezer Loading: Controlling Collisions for
Scaling Atom-Array Assembly

In this chapter, I discuss our work on developing a new optical tweezer technique using Λenhanced gray molasses. It contains a mix of work that was done for our published result, and avariety of updated work that was done in recent months of 2021 after we could utilize a variety ofrecent experimental upgrades [63].
To isolate individual neutral atoms in microtraps, experimenters have long harnessed molecular photoassociation to make atom loading distributions sub-Poissonian. While a variety of approaches have used a combination of attractive (red-detuned) and repulsive (blue-detuned) molecular states, to-date all loading experiments have been predicated on red-detuned cooling. In ourwork, we present a shifted perspective – namely, the eﬀicient way to capture single atoms is toeliminate red-detuned light in the loading stage, and use blue-detuned light that both cools theatoms and precisely controls trap loss through the amount of energy released during atom-atomcollisions in the photoassociation process. Subsequent application of red-detuned light then assures
the preparation of maximally one atom in the trap. Using Λ-enhanced gray molasses for loading,
we study and model the molecular processes and find we can trap single atoms with 90% probability even in a very shallow optical tweezer. Using 100 traps loaded with 80% probability, wedemonstrate one example of the power of enhanced loading by assembling a grid of 36 atoms using
only a single move of rows and columns in 2D. Our insight will be key in scaling the number ofparticles in bottom-up quantum simulation and computation with atoms, or even molecules.

71

4.1

Introduction
In quantum simulation and computing, the assembly of large arrays of individually control-

lable particles is a frontier challenge. Ultracold gases of neutral atoms have long simulated quantum
physics on a macroscopic scale, and quantum gas microscopes are now a window to microscopic
dynamics [22, 94]. However, the desire for control of individual atoms, in particular for quantum
computing, motivates pursuing bottom-up engineering of neutral atom arrays [36, 35, 51, 58]. In
a Maxwell’s demon approach, experimenters image single atoms and subsequently rearrange theminto a desired pattern. The resulting ordered arrays have presented new opportunities in studies
of multi-particle quantum dynamics [95, 96, 84, 97, 98, 99, 91, 86, 87]. Yet, compared to trapped
ions, single neutral atoms are still diﬀicult to trap and assemble.
In our work, we form ordered atom arrays by combining dense loading of large optical tweezer
arrays with atom imaging and rearrangement [Fig. 4.1]. Using Λ-enhanced gray molasses (ΛGM)
on the D1 line of 87 Rb [100, 101], we can load single atoms with high eﬀiciency in a trap shallower
than required for standard sub-Poissonian loading [28] and nearly an order of magnitude shallower
than required for previous enhanced loading [48]. While we demonstrate the idea with an arrayof optical tweezers in 2D, dense loading could also be used in optical lattices or in microtraps in3D [86, 87]. We predict our technique will scale-up neutral-atom array assembly by expanding
rearrangement algorithms and by enabling considerably larger ordered arrays.
To isolate single atoms in optical tweezers or lattices, one typically drives light-assisted collisions in the collisional blockade regime using red-detuned light [28, 29]. In this case, atoms arephotoassociated to attractive molecular states in which they accelerate towards each other and gain
kinetic energy that predominantly expels both from the trap [Fig. 4.1(a)]. If the collisions occur
quickly enough to dominate the dynamics, as is the case in microtraps, a single atom is left about
half the time. In the pioneering work of Ref. [48], after adding a blue-detuned laser to drive atoms
into repulsive molecular states, the energy gained in the collision was tuned to induce single atom
loss [48, 102, 93, 103]. Loading eﬀiciency was enhanced to 90%, but at the cost of requiring large

72

Figure 4.1: Enhanced Loading and Rearrangement in Large Arrays. (a) Sketch of laser
configuration, and molecular energies versus interatomic separation. Solid (dashed) arrows showcooling (repump) lasers with indicated detunings. (b) Atoms loaded into an array of 100 traps withdepth U formed by optical tweezers undergo blue-detuned light-assisted collisions. (c) Schematic ofparallel rearrangement to form defect-free array with target atoms (white) after removing a subset
of loaded atoms (yellow).

trap depths (U/kB ∼ 3 mK compared to 1 mK for red-detuned loading) and hence making use of
the technique untenable in large arrays.
Here we resolve the conflict that has existed in previous work with enhanced loading – namelythat red-detuned cooling drives lossy collisions and competes with desired blue-detuned collisions.
By using ΛGM, we have the ability to cool into the trap and photoassociate with the same bluedetuned laser [Fig. 4.1(a)], and we can control the energy atoms are given in the collision byvarying the laser’s detuning. Further, we can make use of red-detuned and blue-detuned molecular
photoassociation processes at will. In particular, we first modify the atom number distribution in
the microtrap with blue-detuned cooling (ΛGM). We then apply red-detuned light, which bothassures that not more than a single atom remains and, if it remains, images it. The loading

73
behavior studied in a single trap agrees with a model of consecutive light-assisted collisions torepulsive molecular states. Our model further allows us to identify paths to even more eﬀicient
single-atom loading.
We find we can load a single optical tweezer with a trap depth of U/kB = 0.63(6) mK with
89(1)% eﬀiciency, and a 10 × 10 array with 80.49(6)% eﬀiciency [Fig. 4.1(b)]. We also demonstrate a proof-of-principle rearrangement technique that relies on the enhanced loading to createa 6 × 6 defect-free array using a simplified sequence of parallel moves of entire rows and columns[Fig. 4.1(c)] [84]. Lastly, we discuss how the eﬀiciency of both this simplified rearrangement, as
well as atom-by-atom assembly, scale exponentially with initial filling of the array.

4.2

Loading Studies and Modeling

4.2.1

Loading Experiments

Generally, in Λ-enhanced gray molasses (ΛGM) [101, 100] the cooling laser is set blue-detunedof a type-II (F ′ ≤ F ) transition and in a Λ configuration with a coherent repump laser [Fig. 4.1(a)].
Because of its greater isolation from nearby hyperfine manifolds, we chose to operate the ΛGM
detuned from the 5P1/2 |F ′ = 2⟩ state [in contrast to, e.g. 5P3/2 |F ′′ = 2⟩]. Note, we were motivatedto use ΛGM mainly as a natural way to blue-detune both cooling and repump lasers, which isa somewhat different motivation than in recent quantum degenerate gas experiments with lightatoms and molecules – namely that gray molasses works on open transitions, and Λ-enhancement
results in lower temperatures [104, 105, 106, 107, 101, 100, 108, 109, 110, 111, 112, 113, 114].
We first present results from loading a single optical tweezer using ΛGM, and compare to
standard loading using red-detuned polarization gradient cooling (RPGC) [Fig. 4.2(b,c)]. We capture 87 Rb atoms in a magneto-optical trap (MOT) and then cool them into a spatially-overlapped
optical tweezer with depth U with either ΛGM or RPGC. After the cooling and loading stage,
we apply RPGC with parameters optimized for fluorescence imaging of the atoms. Initially, thisquickly removes remaining atom pairs, and then images whether a single atom or no atom remains

74

Figure 4.2: ΛGM Comparative Loading Studies. (a) Flow-diagram of experiments. (b,c)
Average loading eﬀiciency in to a single optical tweezer from 150 experiment repetitions as a functionof bare laser detuning from the free-space resonance and trap-depth, for RPGC (b) and ΛGM (c).
(d) Monte-Carlo simulated mean trap occupation with bi-linear vertical scale (black and grayaxes). In the ΛGM-step, the initial trap occupation (dashed, red line) is reduced (cyan line) by
blue-detuned collisions (blue area). In the RPGC imaging step, red collisions (red area) further
reduce the trap occupation (red line). The resultant is compared with data (gray points) from(c) averaged for U/kB > 0.65 mK (see text). (e) Histograms (cyan) of trap occupancy from theMonte-Carlo after the ΛGM-step for detunings indicated by the black circles in (d), compared toa Poisson-distribution (black) with the same mean trap occupation, and atom distribution after
the RPGC-step (red in i). (f) Loading eﬀiciency as a function of ΛGM-laser detuning ∆LGM fora single tweezer (black) [see also cut along the black dashed line in (c)] and a regular array of 100
tweezers (purple) at U/kB ≈ 0.55(5) mK. Error bars indicate statistical 1σ-confidence interval (seeSec. 4.7.3). Throughout panels, green lines are AC-Stark-shifted |F = 2⟩ − |F ′′ = 3⟩ transition (forRPGC) and AC-Stark-shifted |F = 2⟩ − |F ′ = 2⟩ transition δtrap (for ΛGM), and blue lines areδtrap + 2U/h.

in the trap [Fig. 4.2(a)]. The procedure is repeated to determine average single-atom loading efficiencies, i.e., the fraction with which a single atom is found after both the loading and imaging
stages. See Fig. 4.2(a) and Sec.4.7.3 for experimental-sequence timing and details of the imaging
analysis. Also, see Fig. 4.1(a) and Sec. 4.7.2 for detailed laser configurations.

75
Fig. 4.2(b,c) shows the loading probability P as a function of both laser detuning from theclosest atomic free-space resonance and trap depth, for both RPGC and ΛGM. With ΛGM weobserve 89(1)% loading eﬀiciency at (∆ΛGM , U/kB ) = (45 MHz, 0.55(5) mK), and we can stillload with ∼ 80% eﬀiciency at trap depths of U/kB ≈ 0.27(3) mK. These findings are remarkableas with the same optical power we can load tweezer arrays that are more densely filled and twoto three times larger compared to RPGC loading. The maximum RPGC loading of 64(1)% for(∆RPGC , U/kB ) ≈ (−14 MHz, 1.1(1) mK) is among the highest reported for RPGC [93, 103, 97,84]. In the simplest picture of RPGC, one expects 50% loading, but, in agreement with otherstudies [103], additional processes result in ∼ 35% of the collisions causing only one atom to leave
the trap.
A physically rich picture can be gained from studying the detuning dependence of ΛGMUloading [Fig. 4.2(c)]. First, note that the trap light results in an AC Stark shift δtrap = 32.8 MHzmK kB

of the atomic transition in the center of the trap [green lines in Fig. 4.2(c,d,f)]. The blue line in
Fig. 4.2(c,d,f), which marks a shift of 2U/h from the trap-shifted resonance, is a key energy scale
for the physics of the enhanced loading. At shifts smaller than 2U/h, the collision does not give apair of zero-temperature atoms sitting at the bottom of the trap enough energy for either to escape,while at larger detunings both atoms will be expelled. A finite temperature, and hence an initialcenter of mass motion, will blur the transition, and indeed is necessary for inducing the desired
single-atom loss. Although our data are roughly consistent with this picture, we look more closely
by plotting the data of Fig. 4.2(c) against a dimensionless detuning h(∆ΛGM − δtrap )/U . We do
this for all data traces U/kB ≥ 0.65 mK [Fig. 4.2(d)], and observe a number of interesting features.
For example, we observe a ∼ 60% loading probability for small detunings and that the maximum
loading peaks below the 2U/h shift (blue line).

4.2.2

Loading in Large Arrays

We have also performed a loading study for an array of 10×10 optical tweezers spaced by 2 µm.
We display the measurement at U/kB = 0.55(5) mK as the purple line in Fig. 4.2(f). Compared to

76
the single-trap data at similar U (black), the data are shifted to smaller detunings, and we observea maximum loading of 80.49(6)% in a single run averaged over the 10×10 array. These effects could
be due to a variety of consequences of the larger array: variations in trap shape and depth or overall
degradation of the optical spot sizes (see Sec. 4.7.1). Note that our experimental apparatus was
designed and optimized to entangle closely-spaced atoms in ground states, in contrast to systems
that interface (farther-spaced) Rydberg atoms. This places constraints on the tweezer array, suchas acousto-optic device mode, trap-light detuning, optical power, and high-NA field-of-view, thatmean with typical loading we are limited to working with array sizes less than 10 × 10. However,ΛGM-loading allows us to scale up both the size of our arrays as well as the total number of atoms
we can trap, and is a unique realization for enhanced loading in optical tweezers.

4.2.3

Loading Model

To elucidate trends in Fig. 4.2(c), we have carried out a Monte-Carlo calculation of thecollision dynamics. At the time of the original experiment, we had not yet dived deep into themolecular physics of the two-atom systems, as is done in Chapter 5. Therefore, our original model
did not take into account the detailed nature of these potentials and the effect that they may haveon the loading process. Those considerations will be discussed in chapter 5. Here, I discuss ouroriginal model.
Generally, we expect loading to be affected by both collisions and the ΛGM cooling performance, and both may be influenced by the non-trivial light shifts and polarization gradients in thetweezer traps. Modeling the interplay of these effects is beyond the scope of this work, but we canunderstand the collisional process quantitatively if we assume the continuous ΛGM cooling canload at least a few atoms per trap, and re-thermalizes any atoms remaining after a collision. The
simulation starts by preparing a Poisson-distributed number of atoms Natom with a mean numberN̄atom = 5 and temperature T , where N̄atom was chosen > 2.5 to avoid loading zero atoms initially.
To simulate the finite experiment cooling time, we calculate a finite number of 5000 time steps eachhaving two atoms collide once if they are closer than 100 nm. A collision might eject none, one, or77both atoms out of the trap depending on the final energy of each atom, which is determined by theirpre-collision energy and the collisional energy gain E = h [∆ΛGM − δtrap ]. This simple assumption
makes this simulation easier, but future more accurate simulations could take as inputs results fromthe molecular physics calculations in Chapter 5. This process continually reduces Natom in eachtime step. At the end, the RPGC imaging is simulated by assuming that it entails a fast collisional
process at the start of the image where red-detuned collisions reduce atom numbers in a mannerconsistent with our red loading - namely we reduce any remaining Natom > 1 by 2 with a chanceof 65% and by 1 with a chance of 35% until Natom ≤ 1.
Fig. 4.2(d) shows the result of the Monte-Carlo simulation by indicating the mean trap
occupation N̄atom as a function of the normalized collisional energy gain. During ΛGM-loading, theinitial atom number (red dashed line) is reduced (cyan line). During RPGC imaging N̄atom is further
reduced [red line in Fig. 4.2(d)]. Fig. 4.2(e) shows the simulated atom-number distribution (Natom )in the trap and how atom loss in the ΛGM and RPGC-phase modifies the Poisson distribution.
We observe three physical regimes: For E ≪ 2U [see panel (i) in Fig.

2(e)], the ΛGM-

phase has little effect as almost no atom loss occurs, hence the initial Poisson-distribution (black
distribution) is not modified (blue distribution). The initial phase of the RPGC-imaging step,
however, reduces the number of atoms to 0 or 1, yielding a RPGC-like 65% mean trap occupation(red distribution). In contrast, for E ≫ 2U (panel iv), two-body losses dominate in the ΛGMphase as every collision expels both atoms from the trap resulting in Natom = 0 (Natom = 1)
in ∼ 50% of the cases if the loaded Natom was even (odd). Hence after ΛGM, Natom < 2 andthe red-detuned imaging phase does not modify the atom-number distribution anymore. At thetransition E ≈ 2U (panel iii), both single atom and two-body losses occur in the ΛGM-phase withroughly equal probability because of the finite temperature. Since two-body losses tend toward an
equal distribution of Natom = 0 and 1, and single-atom loss toward Natom = 1, we load a single
atom (Natom = 1) in 75% of the cases. Again, RPGC-imaging does not modify the distribution
as Natom < 2 after ΛGM. Maximal loading probability is found at E < 2U (panel ii) where
only single-atom loss occurs. Here, any occurrence of Natom = 0 is a result of either no atoms

78
having been loaded initially, or the ΛGM-step has not finished (finite Natom > 1 after ΛGM), andRPGC-imaging then ejects pairs atoms.
This simple model indicates no fundamental limitations to the loading eﬀiciency and thatby optimizing the trap size, atom temperature, and related parameters, it may be possible toreach higher loading fractions. Note that the only free parameter that affects the prediction of
the simulation is the atom temperature T in the trap. The simulation describes our data well for
T =120(10) µK, which needs to be understood as an average value for the different trap-depths
U that were investigated. This value is close to the free-space ΛGM temperature we measureof T ≈50 µK, which is higher than typical values, likely due to non-ideal beam geometries (seeSec. 4.7.2).
This model could be refined with a more detailed understanding of the molecular physics
involved, which we investigate in Chapter 5.

4.3

Recent Further Explorations

4.4

Role of Λ-enhancement
In gray molasses, the atom is at in general in a coherent superposition of states in a given

manifold that experiences coherent population trapping and are therefore somewhat dark (“gray”)
to the cooling light. In ΛGM configurations, the coherent superposition is extended to involve statesin multiple ground-state manifolds, which generally has the effect of improving the cooling limit.
The ΛGM configuration was originally chosen simply because it is a convenient cooling configuration
which used entirely blue-detuned light. It was not initially clear if the loading would benefit
specifically from the Λ-enhancement, and as Λ-enhancement can be technically more challenging
than regular gray molasses, we thought it would be useful to explore more thoroughly.
Unenhanced gray-molasses-like cooling is easily available for us by detuning our repump
frequency to be near-resonant with the atomic transition instead of with the Raman resonance where
there is strong Λ-enhancement. Notably, here the repump is operating on the type-I F = 1 → F ′ = 2

79transition (F ′ > F ). There exist cycling transitions here, and we therefore do not expect that any
population in this manifold can be dark to the repumping light. Rather, the dark state is entirelyin the F = 2 manifold, and the repumper only serves to pump into this state. This contrasts withthe behavior on the Raman resonance, where the dark state is in general expected to coherently
include population in both ground-state manifolds, with the F = 1 population prevented fromrotating through Raman transition into the F = 2 manifold via interference with the F = 2 state
populations. It should be noted however that, in general, predicting the dark state itself is nontrivial as it depends carefully on the polarizations of the light fields being applied, and in thepresence of the tweezer potential would spatially vary.
Recently, we’ve taken more data indicating that we can get some enhancement with this “regular” gray-molasses, but that we get much better enhancement with the Λ configuration (Fig. 4.3).
In Fig. 4.3(a) we scan the frequency around this free-space resonance, and we see that on the(+27)

blue-detuned side of it we get some enhancement, with a maximum loading eﬀiciency of 0.717(−25) .
However, there is significantly higher loading eﬀiciency on the Raman resonance, where in this(+15)

configuration with 7 atoms, we get a maximum loading eﬀiciency of 0.810(−17) .
It is as of yet not entirely clear why the loading works better with the Λ-enhancement. Several
theories exist, including that with the Raman resonance it is possible to get significant repumping
action with the repumper far blue-detuned without unnecessarily broadening the transition. Acting
on a broadened transition might prevent us from selecting specifically repulsive potentials. It isalso true that we expect the ΛGM configuration to result in a different equilibrium population
distribution among the single atom F and mF levels. This might also have impacts on the symmetryof the colliding atoms, which could affect which transitions are excited to. More research is required
to further inform the mechanism at play here.
In addition, we note in passing that in the scan we can see the broadening of the transition
with increasing repump power, and that we strangely see the loading suddenly drop at high power.
It is not yet clear why this is and may be an uninteresting technical issue with our setup. We create
the repump light using a fiber-based EOM, and it is possible we are simply running into technical80

Figure 4.3: Scan of Repump Power and Frequency Parameters. a A wide scan, where
we see loading where the repump light is near-resonant with the atomic transition. Slight loading
enhancement without coherent Lambda-enhanced cooling is visible on the blue side of the transition.
The Raman resonance is not visible because of it’s narrowness, but exists on the right-hand sideof this data. The zero is the approximate location of the free-space resonance. b A zoom in on
the Raman resonance, where we benifit from Lambda-enhanced cooling. Note the different x-axis
scale.

issues when we drive it suﬀiciently hard.

4.5

Wider 87 Rb D1 F ′ = 2 Parameter Scans
With a new laser system and a new laser locking configuration as discussed in Sec. 2.2.1.2, it

was later possible to take larger frequency scans than in our original work. This resut is shown in
Fig. 4.4. The broadened scan shows a number of interesting features.
We see poor loading at high detunings and low trap depth, possibly due to ineﬀicient freespace cooling in this regime. The loading reverts to ≤ 50%, at larger depths. Perhaps in the larger
depth the trapping volume is effectively bigger so that collisions are rarer, leading to poor loadingin this region. Perhaps the variations in the light shifts that the atom sees creates a situation where
it is diﬀicult to tune the detuning of the loading to be relevant for all shifts. It is not yet clear.
Finally, notably the loading is non-zero at large depths and shallow detunings, where theLGM light is red-detuned (and heating) in the deepest part of the trap. This may be due to amechanism like that the ΛGM is still cooling at the edges of the trap, where the light shift is81

Figure 4.4: A Wider F ′ = 2 ΛGM Scan. A scan of the loading on the F ′ = 2 ΛGM systemthat is wider than original scans could be. We see a number of new interesting additional features
here at large depths and larger detunings, which are discussed in the main text.

smaller, and that it’s possible to retain an atom in this region.

Loading on the 87 Rb D1 F ′ = 1 Transition

4.6

An interesting further test case that was readily available to us was attempting to load in adifferent loading configuration, where the main (repump) laser is tuned to the D1 F = 2 → F ′ = 1(F = 1 → F ′ = 1) transition instead of the D1 F = 2 → F ′ = 2 (F = 1 → F ′ = 2) D1 transition(Fig. 2.3). The transition is such that it should support dark states similar to our traditionalloading configuration, so we can expect that it is possible to do ΛGM on this transition similarly.
In the new configuration, the lasers are all red-detuned of the F ′ = 2 transitions, but it is a-priori
not clear that this is important since these transitions are 800 MHz away from the transition ofinterest.
Practically, we do this by locking the D1 Master to the 87 Rb D1 F = 2 → F ′ = 2 transitionand offsetting by ∼−1000 MHz (with an extra 230 MHz included to account for an AOD shift),whereas normally we lock to the 85 Rb F = 2 → F ′ = 1 transition and offset by ∼−900 MHz.
Future laser locking configurations may be able to scan smoothly between these transitions without
relocking the laser.
We begin the test by checking the free-space temperature of this different ΛGM configuration,

82
and find a temperature of 12 µK, which is consistent with the temperatures we get from the F ′ =2 ΛGM configuration in our system, so we can conclude that the free-space single-atom ΛGM
performance is similar to that in the F ′ = 2 configuration and should not cause significant issues.
We then examine loading in this configuration in Fig.

4.5. Qualitatively, the dependenceof the loading on the laser frequency and trap depth follows the exact same patterns as that on
the F ′ = 2 transition, with the exception that the peak loading eﬀiciency is significantly lower,(+35)

maximizing at 0.601(−34) . This is similar to the RPGC configuration, which was also capable ofloading slightly greater than 50%.

Figure 4.5: A D1 F ′ = 1 ΛGM Scan. Scan of the loading performance as a function of trap depth
and loading frequency with the f = 2 → F ′ = 1, f = 1 → F ′ = 1 ΛGM loading configuration.
Features of the loading are almost identical to the F ′ = 2-based configuration, except that the
maximum loading eﬀiciency here is significantly lower, and the loading feature is generally moreflat-topped than before.

This is a very valuable result, as it is illustrative of the complexities of the ΛGM loadingmechanism. With traditional pictures of the loading, it is not at all obvious why the new configuration does not enhance the loading eﬀiciency. Our best explanation for this behavior is actually
the nature of the molecular states is significantly different on these transitions than on the F ′ = 2
based ones. In particular, in Chapter 5 we show that many of the blue-detuned molecular potentials that are relevant for this transition do not have significant repulsive barriers and are actuallymostly attractive in nature (Sec. 5.7). This would be diﬀicult to predict without the full calculation

83
discussed there and goes a long way towards an explanation of this behavior.

4.7

Technical Details for Grey Molasses Optical Tweezer Loading and Rearranging

4.7.1

Optical Tweezers

We generate an array of optical-tweezer traps spaced by 2 µm in the xy-plane by passing asingle 850 nm laser beam through two orthogonal longitudinal-wave TeO2 acousto-optical deflectors(AODs) with center frequencies (bandwidths) of 180 MHz (90 MHz). Each modulator is driven
with a sum of radio-frequency (RF) tones with frequency (amplitude) that can be individuallyand dynamically adjusted to control the position (intensity) of different tweezer-rows and columns.
The relative phases of the tones are set to minimize intermodulation in the RF setup. The arrayof deflections created by the AODs is then imaged by a 0.6-NA-objective lens into a glass cell.
This creates a trap with a 0.68 µm waist for a single tweezer, and traps with an average waistof 0.75 µm for a 10 × 10 array. The standard deviation of the trap depths was minimized to 8%
by optimizing the RF amplitudes. Trap depths are calibrated by measuring light-shifts of in-trap
atomic transitions as a function of trap power and applying a linear fit; the slope gives a calibrationof the intensity of trap light the atom experiences, which can be used to directly calculate the trap
depth [115]. Errors on trap depths are 1-σ errors extrapolated from the errors on the slope of thelinear fit. The lifetime of atoms in the traps is limited to 5 sec by the background pressure.

4.7.2

Laser Cooling and Loading

In all experiments, three beam paths are used to address the atoms. Two (diagonal) paths
are along the diagonals of the xy-plane, and a third (acute) path in the xz-plane is at an angle of
55° from the z-axis to avoid the objective [64]. All lasers along these paths are retro-reflected andin a σ + σ − polarization configuration.
Our magneto-optical trap (MOT) is spatially overlapped with the trap array and cools atoms
for 500 ms to a temperature of ∼ 100 µK, measured by imaging its ballistic expansion. The cooling84(repump) laser is red-detuned from the D2 |F = 2⟩ → |F ′′ = 3⟩ (|F = 1⟩ → |F ′′ = 2⟩) transition,
and applied on all three beam paths (on only the diagonal paths). In the case of the 20-ms-long
RPGC stage we cool the atoms to 10 µK. For this, we detune the cooling (repump) laser by ∆RPGC(20 MHz), set the intensities at 1.3 Isat (0.1 Isat ) on the diagonal paths and 4.5 Isat (0 Isat ) on theacute path, and zero the magnetic fields.
In the case of the 200-ms-long Λ-enhanced gray molasses (ΛGM) stage, we apply a coolinglaser that is detuned by ∆ΛGM from the D1 |F = 2⟩ → |F ′ = 2⟩ transition at 2.5 Isat (0.4 Isat )on the acute (diagonal) paths. We create the coherent repump beam from the cooling laser on theacute path using an electro-optic modulator. The repump beam is detuned by ∆ΛGM + 0.14 MHz
from the D1 |F = 1⟩ → |F ′ = 2⟩ transition and at 1.5 Isat . Note that the optimal ΛGM freespace temperature of 50 µK is reached for ∆ΛGM ≈ 15 MHz and is likely limited by the beam pathgeometry and repump light configuration.

4.7.3

Imaging, Data, and Statistics

Regardless of the loading configuration, we image the atoms using another RPGC stage
with the cooling beam (∆RPGC = −19 MHz at 3 Isat ) only on the acute path. We alternate thetweezer-light with the imaging light at 2 MHz to scatter light when atoms are experiencing nolight shifts. This configuration is maintained for 20 ms during which we collect scattered photons
on an EMCCD camera, super binned to 4 × 4 pixels to reduce readout noise. As we now discuss
experimental evidence for, this red-detuned imaging process quickly kicks out any pairs of atoms
that might exist, for example, in the case of a gray molasses loading stage with a small detuning[Fig. 4.2(d) and (e)]. Accordingly, it does not resolve a tweezer’s occupation number following thegray molasses stage [illustrated by the cyan line in Fig. 4.2(d)], but rather maps the atom number
onto 0 or 1. If this loss did not occur quickly compared to the imaging time, we would sometimes
collect numbers of photons significantly larger than our calibrated single-atom scattering rate. We
do not observe this signature despite high experimental statistics, suggesting a sub-percent impactof these effects on the imaging.

85
At every atom location individually, to determine a count threshold that indicates the presence of an atom in the trap, we create a histogram of all counts during an experiment and fit it witha sum of two Gaussians. The threshold with maximal fidelity F is found, where F = 1−(Ef p +Ef n ),with Ef p (Ef n ) being the expected rate of false positives (false negatives) from the fits. This converts a sequence of counts to a sequence of Booleans which is averaged to determine the loading
probability. By finding thresholds for each trap individually and subtracting them from the images
in Fig. 3.15(a), we compensate for a spatially varying background noise and, due to the limited
field of view of our high-NA lens, the different numbers of photons we collect for each trap.
All errors reported indicate 1-σ equal-tailed Jeffrey’s prior confidence intervals [116]. The
loading eﬀiciencies reported in the main text for RPGC (64(1)%), ΛGM (89(1)%), and 10×10-ΛGM(80.49(6)%) were obtained by analyzing 2000, 1000, and 5000-per-atom repetitions with thresholdfidelities 0.987, 0.998, and 0.993 respectively.

4.7.4

Scaling Arguments

The full potential of dense loading using ΛGM will come when combined with the mostadvanced atom-by-atom rearrangement algorithms in 2D or 3D [97, 86, 87]. In principle, if theMaxwell’s demon atom rearrangement operation is perfect, and if a large number of traps are used,
one would expect very large arrays could be created through a bottom-up approach regardless of
the loading. Practically, however, there are many factors that steeply limit creating large-scale
systems with optical tweezer assembly. In particular, a finite atom lifetime compared to the timerequired for rearrangement, and also simply the number of traps that can be created and loaded.
Our approach addresses these problems because it is eﬀicient in shallow traps. Hence, with
ΛGM more loadable traps can be created with the same amount of optical power. Additionally,
2D algorithms fill defects in a target array of size n with a sequence of m ∝ (1 − P )n1.4 single
moves, which we verified with Monte-Carlo simulations [97]. In scaling up array sizes, the time andnumber of moves required become lengthy, lowering the probability of successful rearrangement(SP ) as errors ϵ due to finite move fidelities and background collisions suppress this success rate as

86e−mϵ [97]. Increasing the loading probability P from P = 60% to P = 90% would decrease m bya factor of 4, making larger array sizes more obtainable and exponentially improving the successprobability SP .

4.8

Conclusion
In conclusion, by gaining control over photoassociation to molecular states we have demon-

strated enhanced loading of arrays of shallow optical tweezers. As described in Secs III A and B, weachieved a strong relative improvement on our trapped atom-numbers, but experimental platforms
designed to host more optical traps than our system will stand to benefit even more. For example,
Ref. [97] loads approximately 50 atoms with a 2D-array of 100 traps of 1 mK depth. With the same
optical power and ΛGM, one would expect to utilize 370 traps of 0.27 mK depth and, based on
the shallow-depth loading of single atoms at P = 80% of Fig. 2(c), load approximately 300 atoms– a six-fold increase. Further, the density of the filling will affect the number of moves requiredin rearrangement [84]. Using a technique that moves atoms individually [97], our Monte-Carlosimulations indicate that rearranging 300 atoms at P = 50% requires approximately 900 moves
on average, whereas at P = 80% it requires 320 moves. As a result, the probability to retain all
300 atoms in the rearrangement protocol increases roughly from 0.1% to 10% when going fromP = 50% to P = 80%, assuming a 420 second atom lifetime [117], 1 ms per move, and a 99.3%
move fidelity [97].
We have now more thoroughly studied ΛGM on the 87 Rb D1 5S1/2 − 5P1/2 transition. It
will be interesting to continue to explore a variety of other related cooling techniques in future
experiments. This could include transitions with narrower linewidths, such as the 5S1/2 → 6P1/2
transition. It is also known that gray molasses is effective on the 5S1/2 − 5P3/2 transition [113],
and future studies could compare the salient molecular physics in each manifold [118], especiallyconsidering details of the molecular potentials which we explore in Chapter 5. Further, there ismuch room for explorations of the interplay of collisions and cooling in microtraps for a host of bluedetuned cooling mechanisms with alkali atoms, other atomic species, and even molecules [119, 110].

87
In general we find that these loading studies are full of rich dynamics and unexplained behavior. It is both interesting from a physical perspective and important for practical applications,so we anticipate conducting future research on this topic in order to more fully understand thecomplicated multi-particle dynamics of these loading configurations.

Chapter 5
Molecular Potentials for Ultra-Long Range 87 Rb Dimers

5.1

Introduction
In this chapter, we present early work on the calculation of long-range molecular potentials

for a pair of 87 Rb atoms in the regime of interest for our recent recent work involving light-assisted
collisions between atoms in optical tweezers which are used, among other purposes, to load single
atoms into optical tweezers. Compared to common photo-association experiments, these collisions
are uniquely very long range (& 10 nm), and sensitive to low-energy scales on the scale of the trap
depth (∼ 20 MHz) and ultracold collisional energies (∼ 200 kHz). As such, while in general thetheory of ultracold photoassociation is well established, especially in the context of photoassociation
experiments, feschbach resonances, and Rydberg interactions, there remain many questions about
our unique regime that are yet to be studied [120, 121, 122, 123, 124, 118].
The potentials calculated include asymptotic Born-Oppenheimer (BO) interactions, finestructure (FS) and hyperfine structure (HFS) interactions. While related potentials and calculations appear in the literature, there is relatively little explicit discussion of how to properly
symmetrize all the interactions and calculate in these potentials [125, 126]. Doing this requires
assimilating literature over a very broad range of historical times and notational standards. Speaking personally, as a humble experimentalist this was a uniquely challenging endeavor. It should
be understood that developing thorough understanding of these calculations is a very subtle task.
This chapter of my thesis should be understood as my best effort to explain these calculations butthat there are numerous aspects of the molecular physics here which still elude me, and there is

89
still much work that to expand upon. However, we still think it is useful to summarize my notes
here for future work and for others who may find potentials such as these useful. We find thathaving explicit calculations of the potentials has significantly refined our picture of how the collisions result in loading processes, and yields plausible explanations for many phenomena observed
experimentally.
Most of the work in our lab is with individual 87 Rb atoms, or with many weakly-interacting
87 Rb atoms in their electronic ground-states. Ground state atom-atom interactions can be approx-

imated as S-wave collisions that are parameterized by a single constant: the scattering length [127].
For 87 Rb this is asc ≈ 97a0 where a0 is the Bohr radius, although the value varies by 1% dependingon the hyperfine states of the interacting atoms [127].
In the case of loading single atoms into a tweezer using light-assisted collisions, collisions
are additionally complicated as a result of the interaction of the atoms with the collisional light.
Several other groups have previously made progress with relevant characterizations of single-atomloading and light-assisted collisions [29, 128, 47]. To first order, this technique, which utilizes molecular potentials which are blue-detuned from the separated-atom resonances, can be understood as
involving optical shielding effects which have been observed and studied in a number of other contexts [129, 130, 131]. In this case, atoms are excited by the loading light into an “S+P” molecularpotential, where the valence electron of one atom is in the ground state l = 0 “S” potential, wherel is the orbital angular momentum quantum number, and the other is in the excited l = 1 “P”
potential. This generally results in a significant amount of heating of pairs of atoms, causing pairsof atoms to be lost from the tweezer. The actual dynamics of the loading procedure are presumably
then very dependent on the details of the molecular structure of the atoms. We became especiallyinterested in this subject matter upon discovering our gray-molasses-based loading technique, andsubsequently began to study these structures in greater detail in order to elucidate the exact mechanisms of the loading technique and hopefully to be able to predict and improve loading rates [63].
Starting in the spring of 2020, I began constructing these calculations with the help of regular
discussions with Jose D’Incao and some helpful notes from Paul Julienne [132].

90
While 87 Rb is a “simple” alkali atom, accurately calculating the molecular potentials experienced by two such atoms is technically rather challenging. Despite the prevalance of 87 Rb studies
in the literature, these calculations are not generally available in the region of interest for the experiment. Most previous studies originate from photoassociation spectroscopy work and deal withenergy scales many orders of magnitude larger than those of interest to us. As a result, most of
the calculations in the field don’t go so far as to include the hyperfine interactions, for instance.
However, several photoassociation review articles[120, 118] are still especially useful here. Several
books are also useful references, especially for some of the theory regarding the symmetries inmolecules [133, 134].

5.2

Problem Statement
It is easy to get lost in the results of these calculations, and lose track of exactly what purpose

they serve and how they should be interpreted. As such, we start with a pedagogical explanationof our problem, the BO approximation which is essential to our approach, our methods, and how
the results should be interpreted.
In general we are interested in characterizing the way in which two atoms collide with eachother. In particular, for single-atom loading processes, we are interested in characterizing theamount of energy that the pair of atoms might gain in a collision, and what influences theseenergies. As such, the approach is to write down the full Hamiltonian of the system and use it toderive eigenstates, eigenenergies, and analyze scattering processes. This is challenging, however,since the Hamiltonian representing the interaction of two neutral atoms consists of the interactions
between two nuclei and two electrons, and generally more considering the complicated electronelectron interactions between heavy atoms that are very relevant at short distances. As such, a
simplification should be made, and the standard approach is the use of the BO approximation.
Anticipating this, in the center-of-mass frame, we can write down our Hamiltonian in asuggestive manner:

91

H = TR + HEE (R, ⃗e)(5.1)

Where R is the internuclear distance, TR = −(~2 /2µ)∂ 2 /∂R2 is the relevant kinetic energy
operator with µ the reduced nuclear mass, and HEE (R, ⃗e) includes “everything else” in the Hamiltonian, including electron kinetic energy operators, electron-electron interactions, electron-nucleus
interactions, and any other corrections one wishes to include. ⃗e represents all other coordinates,
such as electron coordinates, that would be needed to describe the Hamiltonian.
The Hamiltonian has no explicit time-dependence, so we are quickly dealing with the timeindependent Schrodinger’s equation. What we’d like to do, is to use the separability ansatz
that ψ(R, ⃗e) = ψR (R)ψ∗e (⃗e), but one quickly arrives at a problem: E − TR ψR (R)/ψR (R) =HEE (R, ⃗e)ψ∗e (⃗e)/ψ∗e (⃗e). At this point, continuing with standard separation of variables strategies, we would normally declare that each side is a function of different variables and therefore
that in order to be equal, they must both be constant. This strategy however pales in face of the
R-dependence of the ĤEE . Therefore, we now introduce the BO approximation, that R and thecenter of mass kinetic energy are effectively constant, and that the problem is effectively separable
and can be stated as solving the pair of equations:⃗ e (R, ⃗e) = VEE (R)ψe (R, ⃗e)
HEE (R, E)ψ

(5.2)

EψR (R) = [TR + VEE (R)]ψR (R)(5.3)

Where we have reluctantly re-introduced the R-dependence back into ψe (and removed theasterisk). This approximation greatly simplifies the analysis. This assumption can also be statedas the assumption that the nuclear motion is very slow compared to all other dynamics in thesystem. This should generally be true, as any nucleus is many orders of magnitude heavier than it’s
electrons. This assumption might not hold for suﬀiciently high energy collisions, but the collisions
we are interested in are especially cold, so we can make the assumption with some comfort.
Our problem then, can be stated as calculating VEE (R), the eigenvalues of HEE (R). Since

92
each eigenvalue can be expected to have a dependence on R, the result which we will show are thena set of R dependent potential energy curves. For a given curve, we would then get a standard
one-dimensional Schrodinger’s equation for the nuclear separation wavefunction, which could be
solved or analyzed to determine the positions of bound-states, scattering amplitudes, and othermetrics of interest.
In this paper, we will primarily work with the Hamiltonian
HEE (R) = HBO (R) + HF S + HHF S(5.4)

where HBO is the long internuclear distance asymptotic order Born-Oppenheimer Hamiltonian, HF S is the fine structure Hamiltonian, and HHF S is the hyperfine structure (HFS) Hamiltonian. The rotational Hamiltonian HR = L̂2R /2mµ R2 where mµ is the reduced mass, could be
included as well, but as discussed in Sec. 5.8, we believe a great amount of insight can be gained
without including the rotational couplings. We still cover some of the considerations for including
rotation in Sec. 5.9 and Sec. 5.10. In general, the Born-Oppenheimer Hamiltonian includes all of
the Coulomb interactions between particles in the system among other terms, and are a functionof all of the other coordinates in the system as well. In this case, since we are only interested in
the long-range behavior of S+P molecules, we take only the asymptotic longest-range terms fromit which are dipole-dipole interactions for these molecules and which are only a function of R.
The different sub-Hamiltonians in HEE are easily stated in the bases in which they are
diagonal. However, the challenge here is that these are different bases, and none of these individual
bases will have good symmetry properties for the total Hamiltonian. We must therefore carefully
study the symmetry properties of each Hamiltonian in order to construct a final basis which has
good symmetry, and then relate the bases relevant for the above Hamiltonian to our final basis.

5.3

Definitions, Notation, and Key Relations
Now, we must introduce some complicated notation for describing the molecular states. In

general, for work such as this, it’s a good idea to reference textbooks dedicated to angular momen-

93
tum algebra. There are many options, several that I’ve used are cited here [135, 136].
In the single-atom case we had 5 important angular momenta that are generally considered:ˆla , ŝa , îa , ĵa , and fˆa . Now, we have at least all of these angular momenta for both atoms, plus themolecule’s rotational angular momentum, and many more combinations thereof. I report all theangular momenta in Table 5.1.
I am following several patterns in the definitions of angular momentum in this chapter, which
will hopefully make it easier to follow. Single particle angular momentum, such as la and fb , arereferred to with lower case labels and a subscript of “a” or “b”. Multi-particle angular momentum
such as J = ja + jb are referred to using upper case labels. In the previous chapters we refer to thetotal single atom angular momenta as F in order to remain consistent with our previous work andAngular momentum projections of angular momentum along a space-fixed axis are referred
to using the common mX notation, where X can be replaced with any angular momentum operator. Projections along specifically the internuclear axis are referred to using Greek letters(Λ, λα , Σ, σα , Ω, ωα , Φ, ϕα , I, ια ) that still follow the capitalization scheme used for operators. The
rotational angular momentum and total angular momentum including rotation are referred to using
special symbols N , and LJR or LF R depending on whether HFS is included. Note that in somereferences, J and F are used for the total angular momentum including rotation, while j and f
are the total angular momentums excluding rotation. Hopefully the Table 5.1 helps clarify any
confusion on this.
There are a couple pattern breakings, including the fact that capital “Iota” is typically
identical to the letter “I”, and so instead we use I in place of the capital “Iota” symbol. Note aswell that the symbols for the projections of various on the angular momentum (Λ, Σ, Ω) are not tobe confused with the labels of different values of Λ (Λ = 0 gets labeled Σ, Λ = 1 gets labeled Π,
etc. when referring to specific BO potentials later in this chapter).
When a combined angular momentum, such as ja = la + sa , is a good quantum number fora state, the constituent quantum numbers are also good. The question then arises as to whichquantum numbers are included in the state descriptions. When included, they will be included in94parentheses near the relevant quantum number such as |LΛ(la lb )⟩. There are many cases in which
there is only one reasonable way to combine angular momentum, and in these cases the extra
angular momentum are omitted. As such, I will simply declare here:|LΛ(la lb )⟩ ≡ |LΛ⟩|SΣ(sa sb )⟩ ≡ |SΣ⟩|II(ia ib )⟩ ≡ |II⟩(5.5)

|jα ωα (lα sα )⟩ ≡ |jα ωα ⟩|fα ϕα (jα (lα sα )iα )⟩ ≡ |fα ϕα ⟩There are notable cases with two molecules where there is at least the potential for ambiguity.
These include J which can be and will be constructed either as |JΩ(ja jb )⟩ or |JΩ(LS)⟩. Notably,
these are very different states, and in general ⟨JΩ(ja jb )|JΩ(LS)⟩ ̸= 1. Similarly, for F , one will needto construct states either through |F Φ(fa fb )⟩ or |F Φ(JI)⟩ states. For these cases, the constituent
quantum numbers will always be included.

5.3.1

Angular Momentum Symbols

Clebsch-Gordon Symbols:

While the Clebsch-Gordon symbols are typically taught at

the undergraduate physics level, a firm understanding of them is key for later discussions and there
are even some unique nuances to their use here. We consider for example that you have a systemwith any two arbitrary angular momentum, j1 and j2 . You might define a basis with well-definedsuch angular momentum and projections, |j1 mj1 ⟩|j2 mj2 ⟩. However, suppose you have an interactionthat couples the two angular momentum, an interaction with significant j1 ·j2 content. Then, statesof the form |j1 mj1 ⟩|j2 mj2 ⟩ are not eigenstates fo this interaction, and so these generally stop being
good quantum numbers. However, the eigenstates of the total momentum operator, j := j1 +j2 , areeigenvalues of the above interaction, and so we can expect all of |jmj (j1 j2 )⟩ to be good quantum
numbers. Here, as elsewhere, I use the parenthesis to group together j1 and j2 in order to indicate

95that j is composed of these two angular momentum. In other words, while the individual projectionsof individual spins are no longer conserved, the projection of the total spin is still conserved, as
is the total angular momentum. It is important to note that the individual spins j1 and j2 arestill conserved for every hamiltonian we deal with in this chapter, although these extra quantum
numbers are often dropped as it is typically obvious from the context what they should be (e.g.
for two electrons with total spin S, these sub-momenta would be the individual electron spins).
However I will tend to be very explicit here as there are so many to keep track of and occasionallythese details are important. As a result, depending on the presence of interaction terms, we willneed to perform basis transformations between these two bases, and to do those we need the matrix
jm j jelements ⟨jmj (j1 j2 )|j1 mj1 j2 mj2 ⟩ := Cj1 mjj 1j22mj . These constants are known as the Clebsch-Gordon
1

2

symbols. Note again that while this ’C’ symbol is common, the extra angular momentum in the
superscript are often omitted for brevity. It is useful to note that one can consider symbols where thejm j jvalues of j1 and j2 don’t match, but they are zero: ⟨jmj j3 j4 |j1 mj1 j2 mj2 ⟩ = δj1 ,j3 δj2 ,j4 Cj1 mjj 1j22mj
1

2

for some extra potentially independent momentum j3 and j4 . We can then write the decomposition:|jmj (j1 j2 )⟩ =∑mj1 ,mj2

9j Symbols

jm (j j )2
Cj1 mjj j12 m|j1 mj1 ⟩|j2 mj2 ⟩
j
1

(5.6)

2

There exist in general what are called “3j” and “6j” symbols, which are used

to describe the coupling of 2 and 3 angular momentum, but most useful in this chapter are the “9j”
symbols [135, 136]. In general, suppose you have 4 angular momentum ji for i ∈ (1, 2, 3, 4). Then,denote ĵij = ĵi + ĵj , and j1234 ≡ j∗ for brevity. There is a common matrix element to consider
between the coupling of these four angular momentum:







j1 j2 j12 




√


⟨j∗ mj∗ (j12 j34 )|j∗ mj∗ (j13 j24 )⟩ ≡ j̆12 j̆34 j̆13 j̆24 j3 j4 j34









j13 j24 j∗ 


(5.7)

The matrix element almost looks like a Clebsch-Gordon symbol, but is dramatically more
complex. We use here a common abbreviation x̆ = 2x + 1 for any angular momentum quantum

96
number x. The symbol is called a “breve” symbol, and this is only to shorten long equations we will
see later. The big matrix-looking term in curly brackets is not a matrix, but rather the 9j symbol.
The 9j symbol should be thought of as a function of 9 arguments which is only defined non-zero forphysically sensible angular momentum numbers. The matrix element ⟨j∗ mj∗ (j12 j34 )|j∗ mj∗ (j13 j24 )⟩can also be written as an appropriate sum of Clebsch-Gordon coeﬀicients.

5.4

Born-Oppenheimer Interactions

5.4.1

Introduction

In general, the problem of finding eigenenergies and eigenstates of a neutral dimer molecule
involves at least four particles, two electrons and two nuclei, each of which generally has botha motional wavefunction and a spin wavefunction. The BO approximation (Sec. 5.2), the mostcommon approximation in molecular physics, is the approximation that the relative nuclear spatial
wavefunctions are separable from the rest of the system’s wavefunction. In this approximation, therest of the Hamiltonian is diagonalized with a constant value of R in order to get the potential
energies of the different possible collisional states at the internuclear distances R. This procedurecan be done at a variety of values of R in order to turn these eigenenergies into energy curves
V (R) which tell you what the potential energy of the system is at a particular internuclear distanceR. Then, these potential energy curves could be plugged back into Schrodinger’s equation with
the nuclear kinetic energy operator in order to find vibrational and rotational eigenstates of themolecule, do scattering analysis, and other procedures. In this chapter, we focus only on calculating
the asymptotic potential energy curves and understanding the associated symmetries and otherphysics behind them.
In order to start our analysis, it is essential to identify the relevant quantum numbers of
the states we are dealing with. Just as the relevant quantum numbers determine, for example, FSinteraction energy offsets, the quantum numbers for BO states will determine BO energies. Here,we will start by identifying the symmetries and quantum numbers relevant when considering only

97
the Born Oppenheimer potentials, and disregarding FS, HFS, and rotation. In the coming sections,
we will carefully discuss how the new interactions break previous symmetries and introduce newsymmetries. A thorough understanding of these symmetries is both key for calculating the results
as well as interpreting the resulting curves. Although these symmetries have been discussed at
length over many years, it is diﬀicult to find text with the symmetries of all of the interactions wewill consider all discussed in the context of each other, so here we will do a thorough review.
Consider two alkali atoms that are very far separated, such that they are without any BOinteractions at all. For a hydrogen-like single atom, the potential is spherically symmetric, leading
to the nice spherical harmonics as solutions to the angular part of the equation. As such, there isa good basis to describe the electronic states of this system, namely the separated atom basis:|na la mla ; nb lb mlb ⟩ = |na la mla ⟩|nb lb mlb ⟩(5.8)

n, l, and ml are the principle quantum number, orbital angular momentum quantum number, andorbital angular momentum projection quantum number respectively. The subscripts a and b here
refer to which nuclear center the quantum numbers belong to, as they will for the rest of this paper.
When we wish to refer to an arbitrary single nuclear center, we will use α. Here, the quantum
number mlα points along an arbitrary (so called “space-fixed”) axis. As we build up our set ofquantum numbers and notation for these problems, we refer the reader to Table 5.1 where we havecompiled all the quantum numbers for reference.
When the atoms are close enough to experience the non-radially symmetric BO interactions,the spherical symmetry is broken. In general, as we get closer, eigenfunctions of the new interacting
Hamiltonian will be complex superpositions of states different of different n, l, and ml , making theseno longer good quantum numbers. We can consider a total angular momentum L̂ = ˆla + ˆlb , but forthe same reason this too is not a good quantum number. It is important to realize an importantnuance here though. While the B.O. solutions themselves do not have good values of la and lb , they
asymptotically connect to the separated atom states which do. Because we are only working withthe asymptotic order B.O. interactions, the states we are working with oftentimes will be assumed

98
to have good la and lb quantum numbers, and it takes care to realize when it is appropriate to
introduce these.

5.4.2

Λ Symmetry

While the molecular system is not spherically symmetric, it is instead cylindrically symmetric
about the internuclear axis. Therefore, if we pick our angular momentum’s quantization axis along
the internuclear axis, the BO interactions will not mix states of different total angular momentum
projections along this axis. We call the projection of L̂ = ˆla +ˆlb , the total orbital angular momentum,on the internuclear axis “Λ”. We can refer to the projections of the single angular momentum lα(when relevant) along this specific axis λα , although these individual projections are not goodquantum numbers of BO eigenstates. It turns out as well, that unlike in the single atom casewhere states of different mla were all degenerate without an external electric or magnetic field,
Born Oppenheimer states with different magnitudes of Λ, i.e. different values of |Λ|, are notdegenerate. Therefore, states are typically represented by their value of |Λ|, and indeed sometimes
in the literature the absolute value is dropped and understood to be implicit. We will discuss thelabels used for these states below.

5.4.3

Spin Symmetry

The BO potential includes primarily electrostatic interactions and does not include electronic
spin-dependent interactions. Therefore the total spin states Ŝ = ŝ1 + ŝ2 of the molecule are notmixed, and S is a second good quantum number for B.O. states. Furthermore, different spinstates (singlet vs triplet) will have different energies due to the impact the spin state has on thesymmetrization of the electronic wavefunction. The spin S’s projection along the internuclear axis,
called Σ, is also unbroken by BO interactions, but states with the same S and different Σ aredegenerate until we introduce fine structure interactions in the next section.

99
5.4.4

ÎBO Inversion Symmetry

For homonuclear molecules, the system is also symmetric on the inversion of all electron
coordinates through the center of charge of the molecule. For homonuclear systems this is also thecenter of mass of the molecule. We label the operator which does this inversion ÎBO , and give its
eigenvalues the symbol IBO = ±1. The states of positive parity with IBO = 1 are typically called“gerade” states (g), and states with negative parity “ungerade” (u). Oftentimes in the literature
the symbol used for its eigenvalues is given as σ = 0, 1, and then when this symmetry impacts
a phase somewhere terms are multiplied by (−1)σ instead of just IBO . This notation could be
confused with the spin projection quantum numbers, so we instead insist on using IBO .
5.4.5

κ̂BO Kronig Symmetry

Lastly, states can also be eigenstates of the operator that flips the wavefunction about a plane
containing the internuclear axis. This is sometimes called a “Kronig” symmetry. We therefore
call the operator that does this flip κ̂BO , and since it is a parity operator it has eigenvalues again
κBO = ±1. In general, I recommend carefully considering the states at hand before considering
the operation of this Kronig symmetry, as there are numerous subtleties in the action of κBO on
a given state. For example, in the long-distance limit discussed here, the result can be derived byanalyzing the single-atom basis states that the B.O. molecular states dissociate to. For a longer
discussion, we reference [134]. For the work in this paper, the following rule suﬀices:κ̂BO |LΛ⟩ = (−1)−Λ |L, −Λ⟩(5.9)

Thus states of Λ = 0 (which are called Σ states, not to be confused with Σ as the projectionof S on the internuclear axis) have positive parity. Note that states with |Λ| = 1 (which are calledΠ states) at the BO level do not have well defined parity as they are usually defined, but one canconstruct states of good κ̂BO parity using various Π states, for example |LΛ⟩±(−1)−Λ |L, −Λ⟩ [120].
As the B.O. hamiltonian does not contain spin-dependent interactions, we need not consider the

100
action of operators like this on the spin of the state yet.
This symmetry is slightly surprising from an atomic physics perspective since states from ournormal hydrogen atom basis |na la mla ⟩ are not eigenstates of this operator. However, if one wished,
one could easily use a different basis for the Hydogen atom which respected this symmetry, as e.g.
the superpositions (using |na la mla ⟩ notation) |111⟩ ± |11, −1⟩ are eigenstates of this operator.
It is not hard to see from this discussion that there is some ambiguity in the definition ofκ̂BO , in particular by choosing a different plane to invert through one can reverse the phase of thesestates [133]. We use the typical phase standard which was set by Condon and Shortley [137].

5.4.6

Born-Oppenheimer Symmetry Labels

With these three main good quantum numbers, BO eigenstates typically get labeled by theBOfollowing symbols: 2S+1 Λκg/u. For example, for alkali atoms the ground S + S states split into only

1 +two states: 3 Σ+
g and Σu . In fact, it is a common exercise to do a simple BO calculation for theH2 molecule. It is much more complicated to do excited state calculations as the excited S + P
3 + 1 + 1 + 3
3
1
1
states split into 3 Σ+
g , Σu , Σg , Σu , Πg , Πu , Πg , and Πu . Note that as mentioned prior theΠ states are not simultaneous eigenfunctions of κ̂BO and thus have no ± superscript. For moreon these calculations, see e.g. [138]. These states can also be labeled in ket form with a more fullset of quantum numbers |LΛIBO κBO S⟩ (with la , lb , sa , and sb implicitly good quantum numbers aswell). We will introduce the nuclear spin later. In the asymptotic limit one could add la and lb tothis ket. More quantum numbers which are degenerate at this level but important later could also
be added, such as the single-atom electron and nuclear spins sα , iα , and the various projections ofthese and their combined angular momenta on the internuclear axis. The correct way to construct
the bases for future calculations which require identifying these quantum numbers will be discussed
in Sec. 5.5.5.

101
5.4.7

Resonant Dipole Born-Oppenheimer Interactions

BO states always have good values of S, |Λ|, IBO , κBO . At close distances, they do not havegood L values. In order to evaluate fine structure corrections at all distances it is necessary toproperly relate the BO eigenstates to the separated atom basis, which would require e.g. being
able to identify BO states in terms of a superposition of atomic states with different orbital angularmomentum. However, all BO states dissociate to states with good L at long distances. Since weare only interested in the long-distance limit of the BO states, our BO bases will always have goodL, and more specifically L = 1, simplifying this calculation greatly. Since for homonuclear atoms
the asymptotic order BO interaction is the resonant dipole-dipole interaction, BO eigenstates aresuperpositions of single-atom excitation states of the form:|ΛIBO κBO S⟩ → |L = la , Λ = λa , IBO κBO ⟩|SΣ⟩√= (|00la λa ⟩ + p|la λa 00⟩)|SΣ⟩/ 2(5.10)

where the kets in the latter equality are of the form |la λa lb λb ⟩ = |la λa 00⟩ for example. pshould be chosen to make the entire wavefunction of the two electrons antisymmetric with respect
to particle exchange. In the case that la = lb for potentials which are not “S+P” in nature, or for
heteronuclear molecules, we can take p = 0. The other parts of the wavefunction that factor into
this are the spin part and the spatial part, so we can therefore relate p to the total spin and spatial
inversion quantum number of the state:p = (−1)S IBO(5.11)

This relation ensures that the wavefunction of the two electrons antisymmetric with respect
to particle exchange. This parity quantum number is an alternative representation of the otherquantum numbers used. It is evidently convenient here, but in general we will continue using IBO
and S for stylistic continuity with other resources.
With the symmetries of the BO potentials established, we can move on to considering the

102
magnitudes of these potentials and how to include them in our eventual calculation. The BO
potentials are extremely deep compared to the energy scales we are interested in. The magnitudesof these potentials are on the order of the energy spacing between different angular momentum
quantum numbers, so e.g. on the order of the D1 and D2 lines for alkali earths, or hundreds ofTHz. As well, the length scale at which these full calculations are important is the length at which
the overlap of the atomic electron wavefunctions about the two nuclei is significant, or typically< 1 nm.
In general, calculating these curves exactly is quite complex. For our purposes, however,
we are interested in low-energy, very long range interactions, and so it suﬀices to take only theasymptotic terms from the BO potentials, which can be approximated as being of the form Cx /Rxfor various values of x. The coeﬀicients Cx are known as “dispersion coeﬀicients”. For the caseof a homonuclear S + P dipole, the asymptotic order interactions are well-known to be 1/R3
resonant dipole-dipole interactions. These can be thought of as arising from a combination of
exchange symmetrization requirements and the resonant exchange of the single excitation between
the two atoms. The exact values can be calculated through relatively involved perturbation theory
calculations where the perturbation is the Coulomb interactions between atoms far apart from eachother [139]. The result to first order in perturbation theory is()(−1)le +mle +1 [(−1)le +S IBO ]2le
V = − 2le +1
2le + 1l e + m leR
1

(5.12)

×⟨ne le |rle |ng 0⟩2
For excited states with quantum numbers ne , le , mle and ground state quantum numbers
ng , lg = 0, mg = 0, and excited state electron cooredinate r. The term in parenthesis is a binomial
coeﬀicient. The equation is valid for states with le ≥ 1. The first order perturbation theory to
S + S atoms with le = 0 is zero, and one must go to second order perturbation theory to get theleading contribution, which turns out to be 1/R6 as one might have guessed.
Note that these calculations are independent of which nuclear center is excited and so we use

103
the “e” subscript to reflect the excited quantum number, deviating from our “a/b” notation.
One can see in Eqn. 5.12 that for S + P molecules the asymptotically go as 1/R3 , and that
because of the dipole matrix element ⟨ne le |rle |ng 0⟩ the coeﬀicient will only be non-zero for statesthat are connected by electric dipole transitions. This calculation does not include fine or HFS in
the states involved, but a straightforward generalization would presumably show the same thing,which will become important later. Similarly, the asymptotic potentials for S + D molecules with
le = 2 are driven by electric quadrupole 1/R5 interactions.
The fact that the form of the dispersion coeﬀicients for S+P potentials is relatively simple
leads it to commonly be described in terms of only a few parameters for the different BO states:

C6
2C3+ 6 + O(R−8 )
3
R
R
C6
2C3
V1 Σ+
+ 6 + O(R−8 )3 + =3
g , ΣuRR
C3C6
V3 Πg ,1 Πu = 3 + 6 + O(R−8 )
R
R
C3
C6
V1 Πg ,3 Πu = − 3 + 6 + O(R−8 )RR

V 3 Σ+
1 + = −g , Σg(5.13)

In general these constants, C3 and others, and those higher order corrections have beenmeasured or calculated (especially for common atoms like alkali metals) and can be looked up (forexample, [139] is a great reference here). These are the energies which I will plug into HBO .

5.5

Fine Structure Interactions
The inclusion of FS interactions was originally done by Movre and Pischler [140], and so theresulting potentials are sometimes called Movre-Pischler potentials. The FS interactions turned
out to be quite important historically as they predicted the presence of the relatively long-range0−
g and 1u bound states in alkali atoms. These states which are relatively long range and looselybound compared to BO molecules, but still experimentally detectable [141]. [142] is a useful goodreference for this calculation.
The atomic fine structure interaction for atom a is given by AFS (ˆla · ŝa ) where AFS is a

104
typically experimentally determined constant. This energy tends to become important at large
distances where the electron wavefunction overlap becomes small. This is the regime where the twoatom’s wavefunctions overlap very little and therefore need not consider corrections such sa · lb or
la · lb . Therefore, here we consider the total FS Hamiltonian is

HFS = AFS (ˆla · ŝa + ˆlb · ŝb )(5.14)

Generally the fine structure interaction will result in a spatially varying mixing of the BO
states, with the mixing being relatively weaker at short distances where the terms from the BO
interactions dominate, then becoming stronger and eventually becoming the normal single-atom
mixed states in the long distance limit. The basis in which this Hamiltonian is diagonal is theseparated atom basis with coupled lα and sα :|ja ωa ; jb ωb ⟩ = |ja ωa ⟩|ja ωa ⟩(5.15)

with ĵα = ˆlα + ŝα and ωα = λα + σα , which satisfies

AFS

HFS |ja ωa ; jb ωb ⟩ =)∑ (jα (jα + 1) − lα (lα + 1) − sα (sα + 1) |ja ωa jb ωb ⟩(5.16)

α∈(a,b)

Where we recall that states with good jα also have good lα and sα , but that we need notconsider the nuclear spin iα yet. To calculate the fine structure interaction matrix elements, onemust convert each two-atom BO state to the appropriate sum of single-atom fine structure states.

5.5.1Ω Symmetry

Similar to L and lα , J and jα , (with Jˆ = ĵa + ĵb ) will not be good quantum numbers dueto the HBO interactions, but the projection of J on the internuclear axis, which is called “Ω” will
be. The FS operator won’t mix states of different Ω. Thus the total matrix will be block-diagonalin these states. Specifically, similar to |Λ|, with only FS interactions, states of different Ω but thesame |Ω| are degenerate, and so typically only positive values are reported.

105
5.5.2

ÎFS Inversion Symmetry

Despite adding the electron spin into the total Hamiltonian, the problem’s two nuclei are
still identical at this stage, and so the Hamiltonian and therefore the eigenfunctions still respect
the IBO symmetry. In other words, IBO is still a good quantum number, g and u are still goodsymmetry labels, and ÎBO = ÎFS .
5.5.3

κ̂FS Kronig Symmetry

It is not hard to see that the Kronig symmetry κ̂BO as defined in Eqn. 5.9 does not commute
with ˆlα · ŝα terms in HFS , but that we can define a similar operator κ̂FS which flips the spin operator
as well which does commute with this term. In this case:yκFS |LΛSΣ; Ω⟩ = (−1)L−Λ+S−Σ |L, −Λ, S, −Σ, −Ω⟩(5.17)

Similar to before, the phase here is tricky and has some amount of arbitrariness to it. Notethat κFS is not explicitly related to κBO . As well only states with Ω = 0 have well-defined nondegenerate κ̂FS symmetries, but the κ̂FS symmetries will be important for constructing states ofκ̂HFS symmetry in the next sections [120].
5.5.4

Fine Structure Symmetry LabelsFS
As such, these states are typically labeled by symbols of the form ΩκIFS, where IFS values areusually replaced by “g/u” for IFS = ±1 respectively, although I will typically still use ±1 inside
bra/ket symbols. For example, 87 Rb S+P molecules support 8 different state symmetries at this+ − −level: 0+
g , 0u , 0g , 0u , 1g , 1u , 2g , and 2u .

5.5.5

Fine Structure Basis

We aim to construct states with good Ω, IFS , and κFS quantum numbers. The first task here
is to determine how to compose states of these good quantum numbers out of BO states, which

106
don’t have a well defined Σ or Ω. If you recall, the BO eigenstates have good ΛS IBO and κBOquantum numbers, and have good L, la , and lb only in the long-distance limit. Only by correctly
constructing the new basis can we successfully add the BO energies to the FS energies. The
degeneracy responsible for this disconnect is the Σ quantum number. By enumerating the statesof different Σ, we can expand our BO basis and construct states of good BO and FS symmetryout of the expanded basis. However, it is not enough to merely add the different Σ states to the
BO basis, as states of good Σ do not generally have good κFS . Instead we can iterate through the
new values of Σ, but construct the new states with good κFS instead. This can be achieved withsomething like the following lines of pseudo-code:for state in BO_Basis:for Sigma in [-state['S'], ..., state['S']]:
state['Omega'] = Sigma+state['|Lambda|']if state['|Lambda|'] != 0 and state['Omega']==0:
for kappa_FS in [-1,1]:
newState['K_FS'] = kappa_FS
expandedBasis.append(newState)else if Sigma>=0:
state['k_FS']=(-1)^(state['L']-state['|Lambda|']+state['S']-Sigma)
if newState not in BoFsBasis:
BoFsBasis.append(newState)
The ÎFS symmetry does not need to be handled explicitly here because this quantum number
is contained in the original BO basis states. After constructing this basis, we can construct subbases by simply separating out states of particular |Ω|, κFS , and IFS . Each of the above states
as constructed is already an eigenstate of the BO Hamiltonian, so the only remaining step is theconversion of FS eigenstates to this basis. The states here can then be separated into the relevant
sub-bases based on their |Ω|, κFS , and IFS quantum numbers and handled separately.

107
5.5.6

HF S Transformation

The FS Hamiltonian is diagonal in the two-atom separated jα basis|ja ωa ; jb ωb ⟩(5.18)

Therefore in order to add this asymptotic BO Hamiltonian, we must do a frame transformation based on the following matrix elements:⟨|Ω|, L|Λ|; S|Σ|IBO IFS κBO κFS |ja ωa ; jb ωb ⟩(5.19)

We will do the frame transformation in three steps. This can be viewed as first transformingfrom the good FS basis to the uncoupled two-atom l-s basis, followed by transforming into the BO
basis with good Σ values, followed by transforming into the BO basis with good |Ω| and κFS :⟨|Ω|, L|Λ|; S|Σ|; IBO κBO κFS |ja ωa ; jb ωb ⟩ =⟨|Ω|, L|Λ|; S|Σ|IBO κBO κFS |ΩLΛ; SΣ⟩(5.20)

×⟨ΩLΛ; SΣ|la λa sa σa ; lb λb sb σb ⟩×⟨la λa sa σa ; lb λb sb σb |ja ωa ; jb ωb ⟩Or more specifically one will want to do this transformation for each of the sub-bases discussed
in the Sec. 5.5.4. The first element deals with complications from the κFS symmetry.

⟨|Ω|, L|Λ|; S|Σ|; IBO κBO κFS |ΩLΛ; SΣ⟩ = κFS(5.21)

for states with negative Ω and 1 for states with positive Ω, by convention. To be clear, it
is also required that the absolute values and signed angular momentum values on both sides are
consistent, this is not an accident of notation. For example the matrix element with |Λ| = 2 on theleft hand side and Λ = −1 on the right hand side would be zero. I could also express this with a lot

108
of delta functions, but this would be tedious. This relationship can be understood by applicationof the statement above about how κFS acts on states with the given quantum numbers.
Here is the second element:a lb⟨ΩLΛ; SΣ; |la λa sa σa ; lb λb sb σb ⟩ = pla ClLΛl
C SΣsa sb
a λ a lb λ b s a σ a s b σ b(5.22)The last one is the easiest to deal with as it is only based on normal angular momentum
combination rules:⟨la λa sa σa ; lb λb sb σb |ja ωa ; jb ωb ⟩ = Cljaaλωaasa σa Cljbbλωbbsb σb(5.23)

I list the resulting total Hamiltonians HBO + HF S below. These can be viewed as the blockdiagonal components of the total Hamiltonian. The symbols to the left of the matrices indicate thequantum numbers corresponding to the state. For example, it should be understood that the firstelement of the first matrix is, in terms of the basis |0, 10; 10; 1, −11⟩, ⟨|(HBO +HF S )|0, 10; 10; 1, −11⟩.
While there are still a fair number of states here, before adding hyperfine structure it is stillreasonable to write this all down.

109

2S+1 |Λ|κBO
IBO

3Π

0−
u :

g

3 Σ+
u
3Π



C3
2∆
3 −
R √ 3
2∆
3

3 Σ+
g

0−
g :




u

C3
2∆
−R
3 − 3

−



C3
∆
3 Σ+
g  R3 − 3
1 Π+  − ∆
g 
1g :
3

∆
3Π
g
3

√

2∆
3

√



2∆
3

3
− 2C
−∆
3
R3

−

√

3Π



2∆
3



3
− 2C
−∆
3
R3

C3
∆
−R
3 − 3

∆
3

∆
3

3
− 2C
−∆
3
R3

[

0+
u :



∆
3

3 Π+
g

1 Σ+
u
3Π

C3
2∆
−R
3 − 3
√

u

2∆
3






 1u :
1 Π+ 
u 

3Π
u
[

2u :



g

C3
∆
3 Σ+
u − R 3 − 3

]
C3
R3





3 Π+
u

C3
−R
3

√

C3
2∆
3 −
R √ 3
2∆
3

1 Σ+
g

+
 0g :−∆
32g :2S+1 |Λ|κBO
IBO



2∆
3



2C3
−∆
3
R3
√

2∆
3




3
− 2C
−∆
3
R3



−∆
3

∆
3

−∆
3

C3
−∆
3
R3

∆
3

∆
3

∆
3

2C3
−∆
3
R3







]

(5.24)
Diagonalizing all of these matrices realizes at every position R yields the curves in Fig. 5.1.
This discussion may seem heavy-handed for the fine structure interactions where in the end
there are only 16 different states in 8 different symmetry classes, making the problem possible totackle “by hand” and without need for heavy systemization or rigor. However, the problem will
suddenly become much more complex when we add the HFS interactions, after which we will haveto keep track of 384 states.

5.6

Hyperfine Structure Interactions
Adding hyperfine coupling now breaks the ÎBO symmetry because the molecule is generally

not symmetric about the internuclear plane when the two nuclei can have different spins (an exception are the Φ = 5 states which both involve nuclear spins that are perfectly aligned with eachother and the electronic spins no matter which nuclear center holds the excitation). This leaves
only the total parity, the total angular momentum projection on the internuclear axis Φ, and thesymmetry about a plane that contains the internuclear axis for Φ = 0 states [125]. Notably, even at

110

Figure 5.1: Molecular Potentials with Only Fine Structure Interactions. Note carefully
the different vertical an horizontal scales. Potentials are given different colors and line styles
according to their symmetry properties. a All molecular potentials at large energy scales andrelatively short distance scales. b Molecular states dissociating to the 5P3/2 + 5S1/2 atomic states,
at smaller energy scales and longer distance scales. Clearly visible are the 1u and 0+
g long range
bound-state-supporting potentials predicted by this calculation. c Molecular states dissociating to
the 5P1/2 + 5S1/2 atomic states, at smaller energy scales and longer distance scales. d Molecularstates dissociating to the 5P3/2 + 5S1/2 atomic states at very small energies and distances relevant
for loading processes. The red lines, indicating the free-space hyperfine energies, are displayedin order to give a visual indication of at what energy and distance scale we expect the hyperfine
potentials to start to become significant at.

these smaller energy scales, the asymptotic Born-Oppenheimer interactions are still the only thingwhich is mixing the various single-atom eigenstates and making this problem interesting.
Evaluating the coupling constant is slightly more tricky than the fine structure constantbecause there are more manifolds for the hyperfine constant whose splittings are generally very
different from one another. For example, in 87 Rb, the ground state splitting is approximately 6.834
GHz, while the splitting of the D1 line is about 800 MHz. As such, to evaluate the hyperfine
coupling, I simply forced the diagonal elements of the hyperfine coupling matrix in the F basis to111
match the experimental values. This includes explicitly plugging in the values for the 4 different
hyperfine manifolds on the D2 transition.

5.6.1|Φ| Symmetry

Similar to previous interactions, we now must consider that the total angular momentumis F̂ = fˆa + fˆb , which is not conserved due to the lack of spherical symmetry. However, thatagain because of the cylindrical symmetry of the problem, the projection of F on the internuclear
axis, called Φ, is still conserved by all interactions in the problem. Again, until we add rotation,
magnetic fields, or other hamiltonians, the states with different-signed Φ values but the same |Φ|are degenerate.

5.6.2

ÎHFS Inversion Symmetry

With the inclusion of the HFS interactions, the inversion symmetry becomes much more
complicated. Because the spins of the two nuclear centers in general have different states, theproblem is in general no longer symmetric under the exchange of only electron coordinates. We
can however define a new total inversion symmetry operator ÎHFS which exchanges the coordinatesof both of the electrons and both of the nuclear spins. As this is a parity operator, it again has
eigenvalues IHFS = ±1. Note that the inclusion of this symmetry is expected to make the eventual
calculation of selection rules more non-trivial, as the electromagnetic field primarily interacts withthe light electrons rather than the nucleus, and so is easier to define selection rules for states thathave good ÎBO or ÎFS symmetry.
5.6.3

κ̂HFS Kronig Symmetry

The HFS interaction has the character of a spatially-dependent îα · ĵα interaction. Our
previous Kronig symmetry operator κ̂FS does not commute with these interaction because it only
flips the electron spin and not the nuclear spin, and so we must again revise the operator to include
both a flip of the electrons spatial wavefunction about a plane, flipping the electron spin state, as112well as the nuclear spin state. This is defined as κ̂HFS , and it more or less replaces κ̂FS .
5.6.4

Hyperfine Structure Symmetry Labels

As a result of the above considerations we label these stats of good symmetry with symbols
HFSof the form |Φ|κIHFS, where IHFS values are now “+/-” for IHFS = ±1 (no longer “g/u”). There are

14 different non-degenerate 87 Rb S+P molecular state symmetries at this level:

+ − −
HFS
|Φ|κIHFS∈ {0++ , 0− , 0+ , 0− , 1+ , 1− , 2+ , 2− , 3+ , 3− , 4+ , 4− , 5+ , 5− }5.6.5

(5.25)

Hyperfine Structure BasisWith the symmetry properties of the different Hamiltonians established, we can now identify
a basis of good symmetry for the total Hamiltonian and begin the process of transforming ourindividual Hamiltonians into this basis. In this section we will do this as discussed above without
yet including rotation. Notably, this basis reduces to the FS basis described above if one sets
ia = ib = 0. Adding rotation is rather significantly more complicated and will be discussed inSec. 5.10. Eigenstates of the new total Hamiltonian again fall into two categories. All states havea good quantum number Φ, the projection of F̂ = fˆa + fˆb on the intermolecular axis. States with
Φ = 0 also have good IHFS and κHFS quantum numbers, while states with Φ ̸= 0 only have a good
IHFS quantum number. In principle one could construct states of good κHFS with Φ ̸= 0 as well

but they are degenerate until we add rotation.
There are 384 total states but states with opposite sign Φ and identical other quantum
numbers are degenerate. So there are 230 states remaining to keep track of. We can expand
the Born-Oppenheimer with fine structure to include hyperfine structure with something like thefollowing pseudo code:

113
for state in boFsBasis:for I_ in [|i_a-i_b|, ..., i_a + i_b]:
for Iota in [-I_,..., I_]:
for Omega in [-state['|Omega|'], state['|Omega|']]:
state['Phi'] = Iota+Omega
if Omega != 0 and Phi == 0:
for k_HFS in [-1,1]:
state['k_HFS'] = k_HFS
if newState not in boHfsBasis:boHfsBasis.append(newState)
else:state['k_HFS'] = state['k_FS']*(-1)^(I_-Iota)
if newState not in boHfsBasis:boHfsBasis.append(newState)
One can then construct the basis of good total symmetries out of these states after identifying
the total inversion symmetry of each state.

5.6.6

HF S Transformation

The FS basis is often referred to with only the quantum numbers relevant for the FS interaction, i.e. |ja ωa ⟩.
Here we are interested in the transformation⟨|Φ||Ω|; L|Λ|; S|Σ|; I|I|IBO IHFS κBO κFS κHFS |ja ωa ia ιa ; jb ωb ib ιb ⟩This is more or less the same as the transformation given in Sec. 5.5.5 just with the extra
nuclear spin quantum numbers which must match, so I omit any further discussion of this here.

114
5.6.7

HHF S Transformation

The HFS Hamiltonian is diagonal in the two-atom separated basis |fa ϕa ; fb ϕb ⟩. And therefore
we must form the basis transformation between this and the expanded BO basis out of the following
matrix elements:⟨|Φ||Ω|; L|Λ|; S|Σ|; I|I|IBO IHFS κBO κFS κHFS |fa ϕa ; fb ϕb ⟩(5.26)

We will do this frame transformation in three steps again. This can be viewed as first transforming from the good HFS basis to the uncoupled two-atom l-s-i basis, followed by transforming
the l-s-i states into the BO basis with good Σ and I values, followed by finally transforming into
the BO basis with good |Σ|, |I| and κHFS :⟨|Φ||Ω|; L|Λ|; S|Σ|; I|I|IBO IHFS κBO κFS κHFS |fa ϕa ; fb ϕb ⟩ =∑⟨|Φ|, |Ω|, L|Λ|; S|Σ|; I|I|IBO IHFS κBO κFS κHFS |ΦΩ; LΛ; SΣ; II⟩(5.27)

×⟨ΦΩLΛ; SΣ; II|la λa sa σa ia ιa ; lb λb sb σb ib ιb ⟩×⟨la λa sa σa ia ιa ; lb λb sb σb ib ιb |fa ϕa ; fb ϕb ⟩The first element deals with complications from the κHFS symmetry. It turns out that withthe way I have constructed my bases, this is a little more complicated than before. In general, astate with good |Φ| and κHFS is composed of the two states with equal and opposite Φ with a phasedetermined by κHFS :⟨|Ω|, L|Λ|; S|Σ|; I|I|IBO κBO κFS |ΩLΛSΣ; II⟩= κFS κHFS (−1)(5.28)I−|I|

For elements with negative Φ and 1 for elements with positive Φ. To be clear, it is also
required that the absolute values and angular momentum values on both sides match, this is not
an accident of notation. For example the matrix element with |Λ| = 2 on the left hand side and115
Λ = −1 on the right hand side would be zero. I could also express this with a lot of delta functions,but this would be tedious. Note that both κ symbols are ±1 so overall this is just a phase. My
bases could likely be defined in such a way to make this a little bit more sensible, but this is theway it works in my calculations.
The second element is relatively straightforward:⟨ΦΩLΛ; SΣ; II|la λa sa σa ; lb λb sb σb ⟩ =∑a lb
pla +S IBO ClLΛl
C SΣsa sb C IIia iba λ a lb λ b s a σ a s b σ b i a ι a i b ι b(5.29)The last one is the easiest to deal with as it is only based on normal angular momentum
combination rules:⟨la λa sa σa ia ιa ; lb λb sb σb ib ιb |fa ϕa ; fb ϕb ⟩ =∑(5.30)

Cljaaλωaasa σa Cljbbλωbbsb σb Cjfaaωϕaaia ιa Cjfbbωϕbbib ιbThen, to evaluate the hyperfine coupling, we simply force the eigenvalues of the hyperfine
coupling matrix to match the experimental values.
It is notable especially in this case that completely calculating the above transformation
matrices may not be the most eﬀicient way of doing the calculation, as even after subdividing thebasis into the block-diagonal symmetry classes, we are still working with relatively large bases whichmakes calculating the full transformation matrices rather numerically taxing, especially consideringthat one generally has to transform into the complete l-s-i basis for part of this transformation whichis large even for 87 Rb. As a result, practically speaking I generally find that it is faster to simply
transform individual good symmetry states into the relevant basis for calculating energies one at atime rather than constructing the entire matrix above.

5.7

Hyperfine Potentials Results and Discussion
The potentials including hyperfine interactions at a zoomed out level are visible in Fig. 5.2.

A number of potentials at a more zoomed in range are visible in Fig. 5.3. Note that all the116states with Φ ̸= 0 are doubly degenerate corresponding to the positive and negative phi states.
The legends of the figures here report the value of Φ and the number of non-degenerate states inparentheses. These potentials reveal a number of interesting features which are worth taking noteof. In order to analyze these in the context of the loading, we note that historically it has expectedthat specifically the repulsive nature of blue-detuned molecular potentials has been thought to beimportant for enhanced loading processes. Additionally, beyond the linewidth of the transition
which is approximately 6 MHz, the most important energy scale of the loading problem is the trap
depth. This is up to approximately 20 MHz, which is quite small on the scale of these potentials.
On the D1 f = 1, f ′ = 2 and D1 f = 2, f ′ = 2 transitions (Fig. 5.3(c,d)), which are the setof excitations used for the loading which works well, there is a nice repulsive potential for nearly
every symmetry which is accessible on each transition. There are only a few exceptions such as
the highest Φ symmetry states available on each transition, which only have one attractive andone repulsive potential available to split between the two inversion symmetry classes. Specifically,
the 3+ and 4− states on the f = 1 and f = 2 transitions respectively only have a single attractive
potential. But the relative plethora of repulsive states otherwise should make it relatively easy to
primarily excite to repulsive states.
On the D1 f = 1, f ′ = 1 and D1 f = 2, f ′ = 1 transitions (Fig. 5.3(e,f)), where Λ-enhanced
gray molasses is still possible but the loading of single atoms does not work as well, we see quite
different behavior. For many available symmetry classes, especially on the f = 1 transition, there
are no available repulsive transitions, or the repulsive transitions have only very shallow repulsive
bumps. As a result, we expect it to be relatively diﬀicult to selectively excite to repulsive potentials
and control the amount of energy gained in a collision.
Lastly, on the D2 f = 1, f ′′ = 2 and D2 f = 2, f ′′ = 2 transitions where again Λ-enhanced
gray molasses is possible, (Fig. 5.3(a,b)), we see in general a less clear picture, especially on thef = 2 transition. However, there are a number of avoided crossings at relatively large distances
which cause many repulsive potentials to again only have relatively shallow barriers, which we
expect to be important in the loading dynamics. It is therefore not so surprising that minimal

117
enhanced loading is achieved in this configuration.

Figure 5.2: Molecular Potential curves calculated with hyperfine structure included.
Different line styles and colors indicate the different symmetries of the states. Legend symbols are
HFS
of the form |Φ|κIHFS. Zoomed out figures at the scale of Fig. 5.1 are omitted as at those large energy
scales the HFS is not visible. (a) A relatively high-energy, short distance view of potentials that
dissociate to the free space D2 line. (b) A relatively low-energy, long-distance view of potentials
that dissociate to the free space D2 ground state f = 1 states. (c) Similar, but dissociating to
the free space D2 ground state f = 2 states. (d) A relatively high-energy view of potentials that
dissociate to the free space D1 line. (e) A relatively low-energy, long-distance view of potentials
that dissociate to the free space D1 ground state f = 1 states. (f) Similar, but dissociating to thefree space D1 ground state f = 2 states.

5.8

Discussion of Rotation Effects
The Hamiltonians above are diagonal in bases of non-rotating states, such that all nuclear

motion is along the inter-nuclear axis. These rotating states are eigenstates of the HamiltonianL̂2R /2mµ R2 with eigenvalues ~2 ℓ(ℓ + 1)/(2mµ R), for inter-nuclear angular momentum L̂R and reduced mass mµ . Relating the eigenbasis of this rotation Hamiltonian to the stationary bases of theother Hamiltonians is possible but non-trivial and beyond the scope of our present analysis, which
we believe captures most of the dynamics that we are interested in.

118

Figure 5.3: Molecular Potentials with Hyperfine Interactions Zoomed-In. Zoomed inviews at the hyperfine dissociation limits of some of the potentials seen in Fig. 5.2(b,c,e,f). (a)The states which dissociate to the large separated atom D2 f = 1, f ′′ = 2 manifold. (b) The states
which dissociate to the large separated atom D2 f = 2, f ′′ = 2 manifold. Near R = 40 nm, there
are a number of avoided crossings with potentials that eventually dissociate to the higher energy
f = 2, f ′′ = 3 manifold. (c) The states which dissociate to the separated atom D1 f = 1f ′ = 2
manifold. This set of states, combined with the states in (d), are where the enhanced loading
technique generally works the best. (d) The states which dissociate to the separated atom D1
f = 2f ′ = 2 manifold. (e) The states which dissociate to the separated atom D1 f = 1f ′ = 1
manifold. This manifold in particular is lacking in good repulsive states. (f) The states which
dissociate to the separated atom D1 f = 2f ′ = 1 manifold.

119
The maximum rotational kinetic energy scale of the collision is set by considering a collision
between two atoms whose kinetic energy is just below twice the trap depth and is entirely rotational,
which would be approximately 40 MHz. This energy scale is well below the HFS splitting of 87 Rb.
As a result, even in this statistically improbable limit, the rotational kinetic energy should not mix
states of good symmetry as described above very significantly.
Transitions between states with different rotational kinetic energy eigenvalues are limited byan angular momentum selection rule ∆ℓ = ±1. Therefore, the first effects one might see if including
rotation become important when neighboring rotational states have differences in energy that arelarger than the linewidth of the transition. Otherwise, the extra rotational states simply slightly
broaden existing transitions. This occurs as atoms in optical tweezers are typically loaded after subdoppler free-space cooling, they have relatively little kinetic energy and can only populate relatively
low energy rotational kinetic energy states. For example, typical red-detuned polarization gradientcooling and Λ-enhanced gray molasses techniques make atoms reach free-space temperatures ofapproximately 10 µK or approximately 200 kHz. In the most extreme case of a glancing collision,
this energy could be entirely rotational. Our analysis of the potentials without rotation indicatethat collisions should generally happen at distances greater than 25 nm. At this distance and withthese energies, the highest rotational state which might be reasonably populated would only haveℓ = 1. At this distance the difference in energy between ℓ = 2 and ℓ = 1 states is 0.75 MHz,
which is well below the 87 Rb D1 transition linewidth of ∼ 6 MHz. Therefore in the worst case,the interaction between different rotational states would contribute only a small broadening of thenormal transition, which is a small enough effect to neglect in our case. At larger distances, it ispossible to populate higher rotational kinetic energy states, but the difference in energy between
states drops still.
At the more extreme energy scale, two atoms that are very hot inside an optical tweezer
might have a rotational kinetic energy as high as nearly twice the trap depth, or on the order of
40 MHz, at which point a similar calculation gives a maximal energy splitting of approximately 30
MHz, which is noticeably above the linewidth of the transition. This means that at this extreme120
some collisions which have somewhat significantly higher centrifugal repulsive barriers. However,since for these collisions all of the kinetic energy is rotational, little distance is traveled along theinter-nuclear axis, and so little kinetic energy, other than from photon recoil, should be gained fromthis type of collision.

5.9

Fine Structure and Rotation
Adding rotation to the problem complicates matters greatly. The original reference for thisderivation is [143]. Notably, this resource is more general than my notes here and includes extra
general quantum numbers to allow one to fully define the electronic configuration for more complicated atoms than alkali atoms. If you’ve made it this far, this is where the calculation gets much
more diﬀicult. Therefore I begin by omitting the HFS which will make the angular momentum
algebra significantly easier. The rotational Hamiltonian here is

HR =

L̂2R
2mµ R2(5.31)

where R is again the distance between atomic nuclei. The operator has standard eigenvalues~2 ℓ(ℓ+1)/2mµ R2 , where ℓ is the rotational excitation quantum number. HR doesn’t commute witheither HBO or HF S , so at this point we have three Hamiltonians in our main total Hamiltonian.
Depending on R and the rotational state, any of these Hamiltonians can dominate, and so different
quantum numbers are good, and different state bases are best to describe the states. The different
cases of which terms are strongest are called Hund’s cases in the literature, and there are five ofthem labeled (a)-(e). A description of Hund’s cases is available in Table 5.2. Notably, Hund’s cases
don’t consider the strength of the hyperfine interactions at all, so they will be limited to being
useful in this section. HBO is diagonal in case (a) and case (b) bases, HF S is diagonal in the case(c) states, and HR is diagonal in cases (d) and case (e). Therefore, since we have been working in
the BO basis for all this work, the challenge of adding this to the Hamiltonian to the calculation
amounts to finding the transformation matrices which transform states from case (e), where HR is

121
diagonal and easy to evaluate, to case (a), which should be the best basis to describe the states inour case.
With rotation included in the Hamiltonian, the case (a) states do not have good rotational
angular momentum but they have good total angular momentum LJR and it’s space-fixed projection mJR , and so should be written as |LJR mJR ΛSP⟩, where P is the space-fixed total parity
eigenvalue, either of ±1. Notably, many references will commonly use the letter J for the total
angular momentum including rotation in the FS case and ĵ = ĵa + ĵb for the electronic angularmomentum omitting rotation. This breaks the pattern in definitions I’ve established (and which
other references mostly follow) however. In any case there is the potential for confusion. So for thisthesis I follow my own preference and use the unique operator LJR for the total angular momentum and Jˆ = ĵa + ĵb for the two-atom electronic angular momentum, maintaining the pattern thatlower-case quantum numbers with subscripts are the individual atom angular momentum, while
their capitalized counterparts are two-atom combined angular momentum. Since the projection of
the rotational angular momentum along the internuclear axis is zero by construction, the projectionof LJR along the internuclear axis is Ω, the same as J. Refer to Table 5.1 for all angular momentum
definitions.
States in case (e) are then described by quantum numbers LJR , mJR , ℓ, J, ja , and jb , and they
are related to the case (a) basis as|LJR mJR (ℓJ(ja jb ))⟩ =∑|LJR mJR ΛSΣp⟩⟨ℓJ(ja jb )|ΛSΣP⟩LJR(5.32)

ΛSΣ

Where it should be understood that the matrix element ⟨ℓJ(ja jb )|ΛSΣp⟩LJR between the
eigenstates of the rotational Hamiltonian and the eigenstates of the BO Hamiltonian varies depending on the value of the total angular momentum LJR . This equation can be thought of as
the definition of this matrix element. Therefore the matrix elements we need to calculate are⟨ℓJ(ja jb )|ΛσΣ⟩. How can one calculate this? The answer is to, in a somewhat orthogonal way,
explicitly calculate the relationship between these states at large internuclear distance R, at which

122
point the matrix element can be recovered. This calculation involves a good amount of non-trivial
angular momentum algebra and very careful consideration of various phase factors which I will
not attempt to reproduce here, and will instead only outline the approach. For more details andreferences I encourage the reader to study the original work [143].
I start by generally constructing the states |LJR mJR ΛSΣP⟩, which include rotation throughthe total angular momentum quantum number LJR and it’s space-fixed projection mJR , out of the
non-rotating BO eigenstates |ΛSΣ⟩. The rotational wavefunction can be added by making use of
the Wigner D matrices which represent solutions to the rigid rotor problem in quantum mechanics
and which are related to the spherical harmonic functions. Careful consideration of symmetrizationand normalization, especially of the Wigner D matrices leads to the conclusion that this must be|LJR mJR ΛSΣP⟩ =√[1
2LJR + 1 ∗LJR√
DmJR Ω (αβγ)|Λ; SΣ⟩
8π 3
2 − δΛ0 δΣ0(5.33)∗LJR+ P(−1)la +lb +LJR −S Dm(αβγ)| − Λ; S, −Σ⟩
JR −Ω

]

This equation is general and should hold for all R, which is possible because of the hidden
R dependence of the BO Eigensetates |ΛSΣ⟩. Through a non-trivial amount of nuanced angularmomentum algebra, one can expand this solution in the limit of large R to express it in terms ofthe case (e) states which have good j, ℓ, ja , and jb to get the following result:R→∞|LJR mJR ΛSΣP⟩ ====⇒∑
J,ℓ,ja ,jb√|LJR mJR (ℓJ(ja jb ))⟩2 − δΛ,0 δΣ,0(1 − P(−1)la +lb +ℓ )(−1)ℓ−Ω−LJR
2πℓ0
×CJ,−ΩL⟨JΩ(ja jb )|LΛ; SΣ⟩JR Ω(5.34)
This can now be used to directly extract the transformation matrix element:

123⟨ℓJ(ja jb )|ΛSΣP⟩LJR = (−1)ℓ−Ω−LJR

1 + (−1)lb +ℓ+p (1 − δΛ,0 δΣ,0 )√2 − δΛ,0 δΣ,0(5.35)ℓ0
×CJ,−ΩL⟨JΩ(ja jb )|LΛ; SΣ⟩JR Ω

Now we can make use of the 9j relation discussed before (Eqn. 5.7), to quickly arrive at theresult for the matrix element above:⟨JΩ(ja jb )|LΛ; SΣ⟩ = ⟨JΩ(ja jb )|∑′J ΩLS ′CLΛSΣ|J Ω(LS)⟩

J′JΩLS= CLΛSΣ⟨JΩ(ja jb )|JΩ(LS)⟩





 la s a j a 





√


JΩLS
= S̆ j˘a j˘b L̆CLSΛΣ lb sb jb









L S J (5.36)

With the recognition that j1 = la , j2 = lb , j3 = sa , j4 = sb , j12 = L, j34 = S, j13 = ja andj24 = jb .
This relation allows you to calculation transformation matrices to convert all the case (e)basis states to case (a) in order to evaluate the Hamiltonian for each value of ℓ and as a functionof R. Specifically calculating (⟨ℓJ(ja jb )|ΛSΣp⟩LJR )† HR (R)⟨ℓJ(ja jb )|ΛSΣp⟩LJR for a the relevantbases states for a given value of LJR . With some herculean mastery of angular momentum algebra, a previous author has calculated analytical forms of Eqn. 5.35 and written them down in
one reference [142]. These results are quite useful, although, we believe there is some mistakein this reference’s equation 11, which gives a non-unitary transformation matrix which does notnumerically match my results in numerically calculating the matrix element above.
For a given value of LJR , states can have either positive or negative total parity. It turns outthat the results of these calculations result in pairs of closely spaced rotational energy levels whichhave different parity, and so each state gets an additional label. States with parities (−1)LJR aresaid to have “e” parity, and states with parities −(−1)LJR are said to have “f” parity. For example,
the state with LJR = 5 and negative total parity is also of “e” parity and the state with positive

124
total parity is of “f” parity. This scheme ensures that each state in the doublet has different suchparity. The “e” and “f” labels are independent of the Hund’s case basis used. In other words, thematrix elements above do not mix states of different “e/f” parity. This parity label has historicallyfluctuated and been the source of confusion in the field, but was resolved in the 70’s. [144]. In theend, these results are used to calculate the potentials seen in Fig. 5.4. Similar analysis can be done
for the hyperfine potentials discussed in the next section.

Figure 5.4: Molecular Potentials with Rotation and Fine Structure. Lines of various
red colors represent states with different total angular momentum J and gerade symmetry. Linesof various blue colors represent states with different total angular momentum J and ungerade
symmetry. (a) Large energy scale, short distance view of the 5P3/2 potential. (b) Large energy
scale, short distance view of the 5P1/2 potential. (c) Lower energy scale, long distance view of the
5P3/2 potential. (d) Lower energy scale, long distance view of the 5P1/2 potential.

5.10

Hyperfine Structure With Rotation
This section is less well tested than the previous and should be viewed as notes on progressto the full calculation of potentials which include HFS and rotation simultaneously. These notes
are partially aided based off of some unfinished notes which were received from Paul Julienne in

125
the course of my studies [132]. The calculation should proceed similarly to the calculation above
but with more angular momentum algebra to keep track of. With the inclusion of the hyperfine
interactions, our total angular momentum is now AF and it’s space-fixed projection is mF R . In thiscase with the HFS included, there are sparingly few resources to establish a common standard forthe total angular momentum quantum numbers here, so there is less room for confusion, althoughsome other resources may wish to define the total angular momentum as F and use f = fa + fb forthe total atomic angular momentum without rotation. But we will stick with LF R and mF R .
Working with case (a) is slightly more tricky because we want to have states of good total parity. Similarly to the case without rotation, we can start by adding rotational states to the BO states√∗LF R
F R +1Dm(αβγ)|ΛSΣII⟩.
with HFS by using the Wigner D-matrices which gives terms like 2L8π
2F R ,Φ∗LF R
Here Dm(αβγ) is the complex conjugate of the Wigner rotation matrix for angles α, β, and γ,
F R ,Φ

which are arbitrary angles in the present context. In general these angles are unimportant and so
will often be omitted from the matrix symbol. However, this is surprisingly not a state of good
total parity, so instead we form a linear combination of these states with appropriate parity, andwe get our case a basis:|LF R mF R ΛSΣIIP⟩ = √1
2 − δΛ0 δΣ0 δI0[√

2LF R + 1
8π 2

∗LF R
× Dm
(αβγ)|Λ; SΣ; II⟩F R ,Φ

(5.37)]∗LF R+ p(−1)la +lb +LF R −I−S Dm(αβγ)| − Λ; S, −Σ; I, −I⟩
F R ,Φ

It’s only at large R that the BO states gain good values for the quantum numbers la , lb . As
such, it’s only as R → ∞ that

126

|Λ; SΣ; II⟩ =∑|la λa ⟩|lb λb ⟩|SΣ⟩|II⟩⟨la λa lb λb |LΛ⟩

λa ,λb|la λa ⟩|lb λb ⟩|SΣ⟩|II⟩ =∑|F Φ(fa fb )⟩⟨F Φ(fa fb )|la λa lb λb ; SΣ; II⟩(5.38)

F,fa ,fb|F Φ(fa fb )⟩ =∑F
Dm
|F mF (fa fb )⟩FΦmF

It should be possible to use these relations and some non-trivial angular momentum algebra
to show that√R→∞

|LF R mF R Λ; SΣ; IIP⟩ ====⇒
×∑2 − δΛ0 δΣ0 δI02π|LF R mF R (ℓF (fa fb ))⟩(1 − P(−1)la +lb +ℓ )(−1)ℓ−Φ−LF R(5.39)

F ℓfa fbℓ0
×CF,−ΦL⟨F Φ(fa fb )|λa λb SΣII⟩F RΦ

Where now we have a much harder additional matrix element to calculate.
And this should give the result:⟨F ℓ(fa fb )|ΛSΣIIP⟩LF R = (−1)ℓ−Φ−LF R

1 + (−1)lb +ℓ+p (1 − δΛ0 δΣ0 δI0 )√2 − δΛ0 δΣ0 δI0(5.40)ℓ0
CF,−ΦL
⟨F Φ(fa fb )|LΛ; SΣ; II⟩F RΦThere are 6 basic angular momentum and multiple ways to couple them to construct statesof total angular momentum F . In the case here, there are two ways to construct the total electronic
angular momentum J. We have Jˆ = ˆla + ŝa + ˆlb + ŝb , and we can first couple the two orbital angularmomentums and spins to create states |JmJ (LS)⟩, or we can construct the individual electronic
angular momentum first and create states |JmJ (ja jb )⟩. Similarly, we have four angular momentum
ja , jb , ia , and ib to construct states of good F in two critical ways to create two bases: |F mF (JI)⟩or |F mF (fa fb )⟩. This is going to lead to two Wigner 9j symbols. And this is suggestive that weare looking to construct the matrix elements ⟨F mF (fa fb )|F mF (JI)⟩ and ⟨JmJ (ja jb )|JmJ (LS)⟩out of the given matrix element, both of which will reduce to Clebsch Gordon coeﬀicients. With

127
these patterns identified, it’s just angular momentum algebra to get to the result. In the following
I explicitly label fa (ja ia ) and similar terms until the middle of the derivation to emphasize thecancellation with the ja′ terms in the sum. It is important to keep in mind thought that any timein these derivations fa is a good quantum number, ja (and la and sa ) and ia are as well, and I justtypically omit these because there is only one standard way to combine these angular momentum.

⟨F Φ(fa (ja ia )fb (jb ib ))|la λa lb λb ; SΣ; II⟩
(
)
∑
ClLΛ
= ⟨F Φ(fa (ja ia )fb (jb ib ))||LΛ⟩ |SΣ; II⟩a λa ,lb λb
=

∑

L

ClLΛ
⟨F Φ(fa (ja ia )fb (jb ib ))|a λa ,lb λb

L

=

∑

(

ClLΛ
C JΩ ⟨F Φ(fa (ja ia )fb (jb ib ))|a λa ,lb λb LΛSΣ

×
=

∑

)

|JΩja′ jb′ ⟩⟨JΩja′ jb′ |JΩ(LS)⟩ |II⟩

=
(∑

)
JΩLS
CLΛSΣ
|JΩ(LS)⟩ |II⟩J∑ja′ jb′

L,J

(∑

∑

C JΩ ⟨F Φ(fa fb )|ClLΛa λa ,lb λb LΛSΣL,J)
F ′ ΦJICJΩII|F ′ Φ(JI)⟩ ⟨JΩ(ja jb )|JΩ(LS)⟩F′(5.41)C JΩ C F ΦJI ⟨F Φ(fa fb )|F Φ(JI)⟩⟨JΩ(ja jb )|JΩ(LS)⟩ClLΛ
a λa ,lb λb LΛSΣ JΩII

L,J





la


(
)
√

∑
a lb
=
ClLΛl
C JΩLS C F Φj̆a j˘b f˘a f˘b L̆S̆ j̆ I˘ lba λa ,lb λb LΛSΣ JΩII


L,J



L


0



(
)
√

∑
JΩ1S F ΦJI
˘
˘
˘
˘
˘
3j̆a jb fa fb S̆ J I 1=
C1ΛSΣ CJΩII


J



1

sa
sb
S
sa
sb
S






ja 

ja



jb   jb









J J



ja

ja 





jb   jb









J J

ia
ib
I
ia
ib
I




fa 



fb 




F



fa 



fb 




F

In the last step I substitute L = 1, la = 0, lb = 1 as this is always true for alkali metals in
S+P potentials. I also remove the sum and Clebsch-Gordon coeﬀicient and sum, as there is always
exactly one non-zero Clebsch-Gordon coeﬀicient of value 1 in this sum. Where in second-to-last
I use the Wigner 9j relations (Eqn. 5.7) with the appropriate substitutions. If this is diﬀicult to128follow, consider reviewing the fine structure calculation or the notation section. Now we can reallyappreciate the breve. It is useful to realize that a number of sums in this derivation, the F ′ , ja′ ,and jb′ sums, immediately reduce to single values because of the matrix left hand side of the innerproduct will select out the appropriate numbers. The Clebsch-Gordon coeﬀicient with only orbital
angular momentum is typically 1.

5.11

Future Work
We believe that there are many avenues for future work in this area. Among the extensions

to these calculations, the primary remaining addition is the inclusion of rotation into the potentials
which have HFS. However, other additional corrections such as relativistic retardation effects havebeen significant in other contexts [145], and may be interesting to explore here as well.
There are many additional properties of the potentials which are of interest. These include
the transition line strengths, excitation rates, state lifetimes, transition selection rules, and otherproperties of the states relevant for loading processes [146]. Many more radical extensions of thiswork are also possible, such as the studying of three-body potentials and collisions, or studying
different species with different level structures in this context.
These potentials could be used to model and study loading processes in more advanced ways.
Previous experiments with single atoms in optical tweezers have attempted to characterize andmodel the nature of related collisions using a variety of methods, but without the insight thatthese potentials provide. Similar methods could be used to experimentally validate some of thesecalculations [47, 128].

129

Description
ˆlαL̂ = ˆla + ˆlbŝαŜ = ŝa + ŝbĵα = ˆlα + ŝαJˆ = L̂ + ŜL̂JR = L̂ + Ŝ + L̂R
L̂ = L̂ − L̂ − ŜR
JRîαIˆ = îa + îbfˆα = ĵα + îαF̂ = Jˆ + IˆL̂F R = Jˆ + Iˆ + L̂R

AngularMomentumOperator Ẑ

lα
L
sα
S
jα
J
L
JR
ℓ
iα
I
fα
F
LF R

Ẑ 2 Eigenvalues(~2 z(z + 1))λαΛ = λa + λbσα
Σ = σa + σbωα = λa + σaΩ=Λ+ΣThe same Ω
0 (Always)
ια
I = ιa + ιb
ϕ α = ωα + ι α
Φ=Ω+IThe same ΦBody-FixedẐz Eigenvalues(~mz )mlαm = mla + mlb
L
ms α
mS = msa + msb
mjα = mlα + msα
mJ = mL + mS
mJR = mL + mS + µµ = m − mL − mS
JRmiα
mI = mia + mib
mfα = mjα + miα
mF = mJ + mI
mF R = mJ + mI + µSpace-FixedẐz Eigenvalues(~mz )

Table 5.1: Angular Momentum Definitions and Other Notation.

Single Electron Orbital A.M.
Total Electron Orbital A.M.
Single Electron Spin
Total Electron Spin
Single Electron Total A.M.
Total Electron A.M.
Total A.M. Except Nuclear Spin
Nuclear Rotational A.M.
Single Nuclear Spin
Total Nuclear Spin
Single Atom Total A.M.
Total Non-Rotational A.M.
Total A.M.

130

Hund’s Case
(a)
(b)
(c)
(d)
(e)

Energy Hierarchy|EBO | ≫ |EF S | ≫ ER
|EBO | ≫ ER ≫ |EF S ||EF S | ≫ |EBO | ≫ ER
ER ≫ |EBO | ≫ |EF S ||EF S | ≫ ER ≫ |EBO |Table 5.2: Hund’s Case Energy Hierarchies. The various Hund’s cases and their energy
hierarchies. |EBO | denotes the magnitude of the electrostatic energy or the BO energy, |HF S |denotes the magnitude of the FS energy, and ER , which is strictly positive, denotes the magnitudeof the rotational energy.

Chapter 6
Time-of-Flight Quantum Tomography of Single Atom Motion

Time of flight is an intuitive way to determine the velocity of particles and lies at the heartof many capabilities ranging from mass spectrometry to fluid flow measurements. Here we show
time-of-flight imaging can realize tomography of a quantum state of motion of a single trapped
atom. Tomography of motion requires studying the phase space spanned by both position andmomentum. By combining time-of-flight imaging with coherent evolution of the atom in an opticaltweezer trap, we are able to access arbitrary quadratures in phase space without relying on coupling
to a spin degree of freedom. To create non-classical motional states, we harness quantum tunnelingin the versatile potential landscape of optical tweezers, and our tomography both demonstrates
Wigner function negativity and assesses coherence of non-stationary states. Our demonstrated
tomography concept has wide applicability to a range of particles and will enable characterizationof non-classical states of more complex systems or massive dielectric particles [147].
The creation and full reconstruction of quantum states featuring genuine non-classical behavior has played a key role in the development of quantum systems. Such reconstructions areperhaps most familiar in quantum optics, where preparing and measuring modes of the electromagnetic field in non-classical states were striking demonstrations of the quantum nature of light.
In these experiments, state characterization has been accomplished both with homodyne tomography [148, 149, 150, 151] and by coupling photons to a spin degree of freedom in cavity or circuit
quantum electrodynamics (QED) [152, 153]. The associated quasiprobability distributions that are
obtained, such as the Wigner function, are useful tools in analyzing non-classical behavior. While

132
quasi-classical coherent states have strictly positive Wigner functions, other states, such as excited
Fock states and Schrödinger cat states, can exhibit regions of negative phase-space density thathave no classical analog.
For particles with mass, the observation of non-classical states of motion is equally intriguing.
Early experiments with trapped ions created non-classical states using trap displacements andRaman sideband transitions, the spin-phonon analog to cavity QED. They verified the generationof squeezed states, Fock states, and cat states among others [154], and have continued to explore
a rich space of tomography methods [155]. Meanwhile, the control of quantum motion of objects
has expanded greatly in recent years, for instance with the ability to couple artificial spins tomechanical solid-state acoustical excitations [156, 157]. Another example is particle levitation,
where one can achieve environmentally-isolated masses whose wavefunction can be expanded overlarge scales for fundamental studies with massive particles [158, 159, 160]. It is now possible tocool the center-of-mass motion of a dielectric particle to its quantum ground state [53, 161, 162],
and it has been proposed that quantum state creation and characterization for these masses canbe explored using nonlinear potential landscapes and time-of-flight tomography [163, 164]. But todate directly measuring rotated quadratures in position and momentum [165], the natural analogy
to optical homodyne tomography, has not been harnessed to characterize a non-classical state of asingle trapped particle.
In this work, we demonstrate tomography of a single neutral atom prepared in non-classicalmotional states using time-of-flight imaging (Fig. 6.1). Time-of-flight samples a particle’s momentum distribution (Fig. 6.1a), and has been used in optical tweezers to measure thermal single
atoms [166] and to probe spin correlations in few-fermion systems [167]. Time-of-flight imaging and detection has also been used to great effect in neutral atomic gases and optical lattices [168, 169, 170, 171, 172], and has enabled momentum distribution measurements of atomic
ensembles prepared in squeezed, Fock, and superposition states [173], and tomography of the dynamics of a Bose-Einstein condensate [174]. In our work, using detection with single atom sensitivity, and measurements at multiple quadrature angles, we carry out full tomography and reveal

133

Figure 6.1: Tomography sequence and notation. (a) Many time-of-flight images of a single atom
released from the optical tweezer are averaged to measure the momentum distribution of a quantum
state of motion. (b) An initial state (green) is prepared in an optical tweezer. The state evolves
over a time te . The tweezer is turned off from a depth of Vf , and the atom expands in free space(blue arrows) for a fixed flight time tf . (c) The distribution measured after a given evolution time
te in the trap, p̃(θ) = p̃(ωx te ), is a generalized quadrature measurement of the initial state. At
specific te , this generalized quadrature can be equivalent to the momentum quadrature p = p̃(0) or
the position quadrature x = (x0 /p0 )p̃(π/2).

134
negative valued Wigner functions with a single particle [163].
As illustrated in Fig. 6.1, to obtain a time-of-flight image we suddenly turn off the opticaltweezer and allow the atom to fly in free space for a fixed time tf . We then average many such
images to determine the momentum distribution at the time of release. We extract arbitrary
quadrature distributions by combining time-of-flight imaging with in-trap harmonic evolution for atime te (Fig. 6.1b,c). We start with a state ψx (te = 0) that we want to characterize, and measuring
this initial state via time of flight gives the momentum quadrature p̃(te = 0) = p. If we allow theatom with mass m to evolve in an ideal harmonic trap, the in-trap momentum after a time te is the√rotated quadrature p̃(θ) = p cos θ+(p0 /x0 )x sin θ (Fig. 6.1c), where θ = ωx te , and x0 = ~/(2mωx )√and p0 = m~ωx /2 are the characteristic length and momentum of the harmonic oscillator withangular frequency ωx . By varying the evolution time te we can extract an ensemble of quadrature
distributions p̃(θ) for θ ∈ [0, 2π], analogous to what is done in optical homodyne tomography. The
quadratures can be used to reconstruct the complete quantum state of the particle or equivalently
the Wigner function W (x, p).
In our experiments we test our protocol with multiple motional states, such as Fock statesand displaced Fock states. To create the near-ground state of a single neutral 87 Rb atom, we use
Raman sideband cooling [175]. To then create non-classical motional states, we use versatile controlof quantum tunneling in the optical tweezer [58]. This capability does not rely upon internal statesand spin-motion coupling as in standard trapped ion settings [154, 176], and hence, can be extended
to polarizable particles with no controlled internal degrees of freedom. With large mass dielectric
particles, while quantum tunneling is not experimentally feasible, it has been proposed that othernon-harmonic potentials created by optical tweezers can enable quantum state synthesis [177, 178].
Our experiments start by stochastically loading single 87 Rb atoms into optical tweezer traps
using grey optical molasses and ascertaining the presence or absence of an atom through an initial intrap fluorescence image. We then use optical molasses cooling followed by three-dimensional Raman
sideband cooling in a trap of depth 1.0 mK to prepare the atom close to the three-dimensionalmotional ground state |nx , ny , nz ⟩ = |0, 0, 0⟩ (Sec. 6.1) [175]. The trap is then adiabatically ramped

135
down to a depth of 0.33 µK where remaining thermal population is allowed to escape to further
purify the initial state.
In the first set of experiments, we measure the momentum distribution of motional statesat te = 0 (Fig. 6.2). After preparing the state of interest, we adiabatically ramp the trap toa final depth Vf /kB = 2.4 µK and then abruptly turn off the trap (Sec. 6.1). We then wait aflight time tf before applying resonant light for τ = 10 µs and collecting fluorescence through thehigh numerical aperture (NA) lens that creates the optical tweezers on an electron-multiplying
CCD (EMCCD) camera. We repeat this procedure to realize multiple instances of single atom
momentum measurements, and collect enough data to create an averaged momentum distributionthat is observable above the camera noise. Experimental runs where an atom is not detected in theinitial in-trap fluorescence image are used to characterize our imaging background, which is thensubtracted from our captured momentum distribution (Sec. 6.2.3).
We first characterize the expansion dynamics of an atom prepared close to the ground stateof the optical tweezer with angular trap frequencies ωx,y,z (Fig. 6.2a,d). The initial ground-stateroot mean square (RMS) size σx,y in position space is estimated as x0 = 86 nm, which is well
below the resolution of our imaging system (Fig. 6.1a). At an expansion time of tf = 0.5 ms, theatomic probability distribution has expanded to an RMS size of 2.4(1) µm in the atom plane, whichis resolved by our imaging system (Fig. 6.2d). By studying the expansion as a function of flighttime tf , we can ascertain that the expansion’s kinetic energy observed in the radial directions of(kB × 0.256(16) µK)/2 is partly driven by the expected zero-point kinetic energy of the harmonic
oscillator, Ezp /2 = ~ωx,y /4 = (kB × 0.188(1) µK)/2 (Sec. 6.3.2). The difference in energy is due tothe finite temperature of the atoms, which can also be seen in Raman spectroscopy (Sec. 6.3.1).
Starting with a ground-state atom, we create nx = 1 or nx = 2 motional Fock states using
one-dimensional tunneling in a double well. The tweezer is moved to 0.88 µm from a second empty
optical tweezer on the right, and with both tweezers near a depth of 1.8 µK we bring the groundstate of the left tweezer nearly energetically resonant with higher-nx states of the right tweezer.
Then, with an adiabatic sweep of the relative tweezer depths, the atom is transferred into the target

136
excited state of the right tweezer (Fig. 6.2b,c) (Sec. 6.1.4) [58]. The two tweezers are then slowly
separated and the intensity of the left tweezer is ramped to zero, releasing any atom that did not
successfully transfer to avoid polluting the final image (Sec. 6.2.4). We abruptly turn the remaining
right tweezer off from 2.4 µK and proceed with the same imaging procedure as for the nx = 0 state.
The resulting nx = 1 and nx = 2 momentum distributions in Fig. 6.2e,f show characteristic fringing
that is expected of the excited motional states.
ad

2.4 µm86 nm

p̃(0) = pbe0.88 µmcfFigure 6.2: Single-atom Fock state preparation and imaging. (a),(b),(c) Illustration of
motional state preparation of n ∈ {0, 1, 2} states, respectively. (d),(e),(f) Time-of-flight momentum distributions at te = 0 and for tf = 0.5 ms of nx = (0, 1, 2) states for which (64008, 48309,
58899) images were averaged, respectively.

We now proceed to the full tomographic characterization of motional states. In these experiments, we study multiple quadrature distributions by waiting a variable amount of evolution
time te before releasing the atom and imaging the result. We can visualize the quadrature data
as time-sequence waterfall plots (Fig. 6.3a,d,g), which are derived from our raw distributions bydeconvolving with the imaging point spread function (PSF) and integrating out the vertical axis
(Sec. 6.2.4). We study states with a goal of testing the capacity of time-of-flight tomography to

137

0.11

x/x0

7

0

-7

0

-0.11
e

−π Phase (Rad)
i
|0i

0.1

7

W(x, p)

0.11

-7
7

|3i

0

|6i

-7
0

π
ωx te (Rad)-0.098h

W(x, p)

7

0

|9i

2π

c

0
p/p0

7

-7
f
π

|3i
|6i

0
p/p0

7

|3i

|6i

|9i

−π

|9i

0.05
0

-0.05
0

-5

0
x/x0

5

k
2.5

-7
|0i

|6i

0.1

Phase (Rad)

|0i

|3i

7

-7
-7

|0i

π

j

0.098

W(x, 0)W(x, p)te (ms)
0.05

0

g

x/x0

p̃(ωx te )/p0

-0.11
b

x/x0

te (ms)
0.05
0.1

7
0
-7
7
0
-7

7
0
-7
7
0
-7

|9i

0

d

|0i

|3i

|6i

|9i

hp̃(ωx te )/p0 i

te (ms)
0.05
0.1

0

a

0

-2.5

-7

0
p/p0

7

0

0.5
te (ms)1

Figure 6.3: Motional quantum state tomography via time-of-flight imaging and maximum likelihood estimation. Measured quadrature data, Wigner function, and density matrix
Hinton plot for the displaced nx = ny = 0 state (a,b,c); nx = 1 state (d,e,f) and displaced nx = 1
state (g,h,i) both with a slight squeezing operator also applied. Quadrature data: Waterfall
plots (a,d,g). Each vertical slice corresponds to a raw quadrature distribution, such as in Fig. 6.2,
after deconvolving with the imaging PSF and integration along the vertical axis. The measured
data (upper waterfall plot) is compared to the expectation for ideal preparation and harmonic time
evolution based upon our protocol (lower waterfall plot). Wigner functions: Wigner functions(b,e,h) show classical positive values as red and non-classical negative values as blue. We normalize
the Wigner function such that the ideal negativity of a pure nx = 1 state is −1/π. Density matrices: Hinton plots (c,f,i) for density matrices as reconstructed via MLE. The area of each square is
proportional to the magnitude of the corresponding element’s complex value; the color of the square
represents the element’s phase. (j) p = 0 slice of reconstructed Wigner functions from (h) (red).
Equivalent slices as reconstructed through a bootstrapping method (black) that characterize thestatistical uncertainty of our reconstruction algorithm. (k) Evolution of the measured center of thecoherent state (green circles) and damped sinusoidal fit (black line), which is used to characterize
the trap frequency and anharmonicity.

138
identify non-classicality and phase preservation, as well as subtle non-stationary features. First, wecreate a coherent state by starting with an nx = 0 state in the Vf /kB = 2.4 µK trap and abruptly
displacing the optical tweezer by 180 nm. We find the state oscillates back and forth in the trap as
expected (Fig. 6.3a). Next, we produce an nx = 1 Fock state using the same protocol as the data
presented in Fig. 6.1b. In addition, after the tunneling and optical tweezer separation, the depth issuddenly doubled, resulting in application of a squeezing operation. The atom is then released froma trap of Vf /kB = 3.6 µK. We observe that the state is mostly stationary, as expected, with theaddition of a slight breathing from the squeezing induced by the depth jump (Fig. 6.3d). Lastly,
we combine multiple techniques by starting with an nx = 1 state, applying the sudden doubling of
the trap depth, and in addition displacing the optical tweezer by 140 nm. As shown in Fig. 6.3g,
we are able to observe the expected oscillation dynamics of the state in the trap.
To reconstruct the quantum state from the quadrature data, we choose to use maximum
likelihood estimation (MLE) [179] (Sec. 6.4.2). An appropriately designed MLE algorithm takes
the quadrature data and returns the density matrix that is most likely to reproduce this data [180].
We implement an iterative MLE protocol based on the standard optical homodyne tomography
literature [181], and from the density matrix, the Wigner function is directly recovered (Fig. 6.3).
The results of applying the MLE algorithm to the quadrature data are then presented inFig. 6.3b,c,e,f,h,i. The coherent state displays significant off-diagonal coherences but, as expected,
a positive Wigner function (Fig. 6.3b,c). The non-Gaussian state preparation associated with
Fig. 6.3d,g results in the density matrices and Wigner functions with negative values displayed inFig. 6.3e,h,f,i. The value and statistical error on the density matrix and Wigner function negativity
is estimated by a bootstrapping technique [181], in which we randomly sample predicted quadratureand noise distributions based on the MLE result and our camera noise characterization respectively.
We then extract the density matrices of these data sets to create a statistical ensemble of density
matrices and Wigner functions (Sec. 6.5). The nearly-stationary nx = 1 state displays a dominantnx = 1 component and the Wigner minimum is found to be −0.060(6) (Fig. 6.3e). Adding adisplacement demonstrates non-trivial off-diagonal coherences and a negative Wigner function value

139
at the displaced center of −0.064(6) (Fig. 6.3h,j).
A full assessment of the reconstructed wavefunction must also consider systematic errors(Sec. 6.5). Trap anharmonicity, for example, will result in measured quadratures that do not
simply follow from the rotated quadratures of the ideal protocol. We determine the impact ofanharmonicity by theoretically assessing the tomography protocol based on a model using measured
trap parameters. We estimate our trap anharmonicity by studying the coherent state oscillationsof Fig. 6.3a over a longer time (Fig. 6.3k). The center of the Gaussian oscillates at 7.84(5) kHz, anddecays with a time constant of 0.63(14) ms. A model of the trap containing an anharmonic term isfitted to match the observed damping. For a displaced nx = 1 state in this model, we compare thereconstructed states obtained from MLE after evolution in a trap with and without our modelledanharmonic terms. We observe only a small infidelity of < 5% between the resulting reconstructedstates, and the Wigner function minimum for anharmonic evolution is smaller in magnitude by< 0.01 compared to harmonic evolution, and remains negative (Sec. 6.5.3). In the future, thelarge dynamic range and control afforded by optical traps can be used to control the harmonicity.
Specifically, a shallow double-well or other anharmonic traps could be used for state creation, andthe tomography could be carried out after ramping to a much deeper and less anharmonic trap.
We have demonstrated quantum tomography of non-classical single atom motional states.
By using time-of-flight imaging we have measured negative-valued Wigner functions with nontrivial phase-space structure. This work lays the foundation for tomography and characterizationof massive levitated particles without exploitable spin structure [160, 177, 164]. Further, time-offlight imaging of single atoms will enable study of high-n motional superposition states [182], highly
squeezed states, and interference [183] of complex delocalized states.
Acknowledgements: We thank Tobias Thiele, Steven Pampel, and Ting-Wei Hsu for valuable insights and technical assistance, and Konrad Lehnert and Adam Kaufman for input on the
manuscript. We acknowledge funding from NSF Grant PHYS 1734006, ONR Grant N00014-17-12245 and Grant N00014-21-1-2594, NSF QLCI Award OMA 2016244, and the U.S. Dept. of Energy,
Oﬀice of Science, National Quantum Information Science Research Centers, Quantum Systems Ac-

140celerator, and the Baur-SPIE Endowed Professor at JILA. W. J. D acknowledges support from an
NSF Graduate Fellowship.

6.1

Optical tweezers

IaPosition

Atom in tweezer

I,II,III

Tweezers off

II,III
Empty tweezer

Left tweezer
bias

II,IIIb
0.5

Load left tweezer

I,II,III

I

II

I
II
III

I,II,IIInx = 1 tunnelingnx = 2 tunnelingTweezer
depth

c

III

1.0 mK
0.58 mK
I,II,III

1.8 µK

0.33 µK
0
Loading

In-trap
image

Raman
cooling

PurifyBring
together

Adiabatic
passage
50 ms

Drop
atom
I,II,III Vf = 2.4 µK
Separateand ordrop
Wait

Evolution
time
teTime offlighttf

Free-space
image
0.01 ms

Figure 6.4:
Schematic timing diagram for Fock-state momentum distribution experiment (Fig. 6.2) Not-to-scale experiment diagrams, showing (a) the position of the atoms andtweezers, (b) the bias of the tweezers, and (c) the depth of the atom-holding tweezer. Stationary
Fock-state momentum distributions in which we prepared nx = 0, 1, 2 states are labeled I, II, andIII respectively. Round markers indicate adiabatic passage into an excited state is not used inexperiment I that works with the nx = 0.

6.1.1

Tweezer generation and control via acousto-optic deflectors

Tweezer generation:

The optical tweezers are generated by sending light at a wavelengthof 850 nm through two orthogonal acousto-optic deflectors (AODs) that are driven simultaneouslywith multiple RF tones to create multiple deflections. In the case of this experiment, the horizontal
AOD is used to generate the two deflections used for tunneling, in order to avoid complications
arising from gravity.
Depth and bias control:

The vertical AOD is used to direct extra laser power far away

141
aPosition

Atom in tweezer

Tweezers off
IV
Displacement

IV,V,VI
V,VI

VI

Empty tweezer

Left tweezer
bias

V
b
0.5

Load left tweezer

IV,V,VI

V,VI

nx = 1 tunnelingIV

Tweezer
depth

c

IV
V
VI

V,VI Drop empty
right tweezer

1.0 mK
Drop
V,VI Vf = 3.6 µK atom

0.58 mK
IV,V,VI

1.8 µK

0.33 µK
0
Loading

In-trap
image

Raman
cooling

PurifyBring
together

Adiabatic
passage
50 ms

IV
Separateand ordrop
Wait

2.4 µK
Evolution
time
teTime offlighttf

Free-space
image
0.01 ms

Figure 6.5: Schematic timing diagram for tomography experiment (Fig. 6.3) Not-toscale experiment diagrams, showing (a) the position of the atoms and tweezers, (b) the bias of thetweezers, and (c) the depth of the atom-holding tweezer. Displaced coherent state data shown in
Fig. 6.3a-c is labeled IV. Non-displaced nx = 1 data shown in Fig. 6.3d-f is labeled V. Displacednx = 1 data shown in Fig. 6.3g-j is labeled VI. Round markers indicate adiabatic passage into an
excited state is not used in experiment I that works with the nx = 0.

from the main tweezers that hold atoms, which allows us to reduce the depth of the main tweezers
by many orders of magnitude even given the limited dynamic range of our intensity servo. The
relative depths of the traps are modified by dynamically adjusting the amount of RF power in eachtone driving the horizontal AOD.
Tweezer position and movement:

The tweezers are moved by changing the frequenciesof the RF tones that drive the AODs. The speed of this movement is then limited by the size ofthe laser beam inside the AOD crystal and the time it takes the acoustic wavefront to cross thisdistance (∼ 100 ns). This is very fast compared to the tweezers’ radial trap frequencies. However,there is 10 µs of electronic jitter in the time between the trigger to change the RF frequency andwhen the frequency jumps. This could be easily improved in future experiments, but is the likely
explanation of small phase offsets noticeable in the center-of-mass oscillation data (Fig. 6.6).

142
6.1.2

Tweezer loading

During the initial loading stage, V /kB = 0.58 mK. The loading procedure is stochastic, andthe ΛGM loading technique we use is capable of up to ∼90% loading eﬀiciency [184]. However,
because interleaving background images without atoms provides useful information, we use suboptimal loading (50% to 80% eﬀiciency) (Fig. 6.4 and Fig. 6.5).

6.1.3

Trap depth and frequency calibrations

Trap depth calibration:

We calibrate our trap depth V by measuring the light shift of

the trap on the D2 87 Rb F = 2 → F ′ = 3 transition at relatively high depths (∼ 0.1 mK to 1 mK).
We extrapolate this calibration to low depths where the shift is smaller than the linewidth of thetransition and therefore diﬀicult to measure directly.
Trap frequency calibration methods:

We can measure trap frequencies ωx,y,z in deep

traps through Raman sideband spectroscopy. We can then estimate the trap frequencies at smaller√
depths according to ωx,y,z ∝ V .
Independently, displacing a state in a harmonic oscillator ideally causes the state’s centerof-mass momentum to oscillate at the trap frequency. These oscillations can then be measured inorder to independently determine the trap frequency at the small depths used for motional state
preparation and tomography (Fig. 6.6). The oscillations can be analyzed either by calculating⟨p̃(te )⟩ from the deconvolved quadrature data (Fig. 6.3a,d,g), or they can be analyzed by fitting
the quadrature data and tracking the location of the fit. Specifically for the displaced nx = 0
state, which is approximately Gaussian, the fits more reliably characterize the state as the fit isless susceptible to far-off-axis noise.
Comparing frequency calibrations:

The nx = 0 state was released from a depth of

Vf = 2.4 µK. Extrapolating the Raman sideband spectroscopy trap frequency calibrations tothis depth predicts ωx,y /2π = 6.8 kHz and ωz /2π = 1.4 kHz. We measure via the center-of-mass
oscillations of the displaced nx = 0 state 7.84(5) kHz (Fig. 6.6).

143

2

hp̃(te )/p0 i

1
0

−1
−2

0
0

0.5

1
0.04

0.08

0.12

te (ms)Figure 6.6: Center of mass oscillations of data from Fig. 6.3. Displaced nx = 0 centers ofGaussian fits (green circles), and for comparison ⟨p̃⟩ (open green circles) [in a 2.4 µK trap]. Nondisplaced nx = 1 ⟨p̃⟩ (purple squares), and displaced nx = 1 ⟨p̃⟩ (blue diamonds) [in a 3.6 µK trap].
The nx = 0 (nx = 1) is fit with a decaying sinusoid to find an nx = 0 (nx = 1) oscillation frequencyof 7.84(5) kHz (9.05(11) kHz) (black and blue curves). Inset: Re-print of Fig. 6.3k for reference,
which displays the same fitted centers of the nx = 0 data (green circles) and their fit (black curve)
over a longer time.

144
The nx = 1 displaced and non-displaced states were released from a larger depth of 3.6 µK,
where we measure via the center-of-mass oscillations 9.05(11) kHz for nx = 1 (Fig. 6.6). Extrapolating the measured nx = 0 trap frequency to the larger depth predicts 9.60(6) kHz. The difference
between the value 9.05(11) kHz and 9.60(6) kHz is most likely due anharmonicity in these relatively
shallow traps. In Sec. 6.5.3, we discuss a theoretical model for our trap that accounts for theseobservations.

6.1.4

Excited Fock state preparation via tunneling

Tunneling parameters and calibrations:

In order to complete the adiabatic ground-

to-excited state transfer (Fig. 6.2b,c), we must find and characterize the appropriate tunnelingresonance and adiabatic transfer process. Tunneling is always done at a depth of V = 1.8 µK andwith the tweezers spaced at a set Gaussian function spacing of 0.88 µm. Assuming the tweezers
are Gaussian, the spacing between the double-well minima is expected to be 0.78 µm based uponthis setting. For only the characterization of the resonances, we load atoms into both tweezers
individually in order to measure both the intended transfer from ground to excited state andthe unintended reverse transfer from excited state to ground state, which is a result of imperfectground-state preparation.
We calibrate the relative tunneling depth (∆V ) by comparing the width of a ground-toground tunneling resonance to its oscillation frequency. This is done at shallower depths where
the ground-to-ground tunneling resonance is measurable. At the relatively deep depths used forground-to-excited-state tunneling, the nx = 0 → nx = 0 tunnel coupling is too weak to be easily
measured. We calculate where we expect it to be located based on the splitting between the nx = 1
and nx = 2 resonances and the assumption that the trap is harmonic, and set this location as∆V = 0.
nx = 0 → nx = 1 characterization:

We characterize the nx = 0 → nx = 1 resonant

tunneling transfer eﬀiciency as a function of ∆V . We fit the transfer eﬀiciency with a Gaussianfunction and find that the resonance occurs at ∆V = 294.5(2) nK and has a RMS size of 3.6(3) nK

145(Fig. 6.7a). Ramping the relative depths across a width of 69.3 nK centered on this resonanceover 50 ms achieves adiabatic rapid passage to the excited state at an eﬀiciency of 92.4+1.9−2 %.
Meanwhile, we find transfer in the reverse direction, nx = 1 → nx = 0, to be 11+3
−2 %, reflecting that
there is a small population in excited states of the tweezer, which is capable of transferring thereverse direction. This is consistent with our expectations from characterization of our single-atom
temperatures via other methods where we estimate 90% radial ground-state fraction (Sec. 6.3.1).

Figure 6.7:
Excited state tunneling resonances. (a) The nx = 1 tunneling resonance.
Probability for an atom to tunnel from the ground state to nx = 1 (teal squares) and from nx = 1
to the ground state (gold circles). (b) The nx = 2 tunneling resonance. The probability for an
atom to tunnel from the ground state to nx = 2 (blue squares) and from nx = 2 to the groundstate (red circles). Due to the ground-state cooling there are nearly no atoms in the excited stateof the second tweezer to transfer backwards.

nx = 0 → nx = 2 characterization:

We similarly find an nx = 0 → nx = 2 tunnelingresonance located at ∆V = 589.0(11) nK which has a RMS size of 12.1(13) nK (Fig. 6.7b). Ramping
the relative depths across a width of 111 nK centered on this resonance over 50 ms achieves adiabatic
rapid passage to the excited state at an eﬀiciency of 86.0+2−3 %. We find transfer in the reverse
direction, nx = 2 → nx = 0, to be 3.2+1.7−1.1 %, which is significantly smaller than the reverse transfer
measured on the nx = 0 → nx = 1 resonance. This reflects that there is very little population innx = 2 that is capable of transferring the wrong direction, as is expected after Raman sideband
cooling.

Parameter
Vertical shift speed
Horizontal shift speed
EM gain setting
Frame transfer mode
Pixel size
Pixel binning
Camera temperature
Ideal imaging NA
Time-of-flight imaging time
RPGC imaging time

Value
2 MHz
1 MHz
×300
Off
16 µm
1x1
−60 ◦ C
0.55
10 µs
5 ms

146

Table 6.1: Camera and imaging settings. (upper section) Camera settings directly programmed or reported by the manufacturer . Here, we report the vertical and horizontal shift speed
as they are programmed. However, our camera is mounted sideways (Fig. 6.9). (lower section)
Imaging parameters.

6.2

Imaging methods

6.2.1

Imaging setup

The camera used is an Andor IXON-EM+ back-illuminated electron-multiplying CCD (EMCCD) camera, model number DU-897E-C00-#BV-9GT. Camera settings for the experiment arelisted in Table 6.1. The imaging light at 780 nm is collected through the same high-NA objective
lens used to create our tweezer array, split from the optical tweezer light using a dichroic mirror,
and focused using a 1 m focal-length achromatic doublet.

6.2.2

In-tweezer RPGC imaging

RPGC imaging configuration:

Red-detuned polarization gradient cooling (RPGC) andassociated light scattering is used to determine if an atom is loaded at the start of the experiment.
Time-of-flight images are post-selected based the presence of an atom in the RPGC image. Sometime-of-flight images that record cosmic ray events are additionally removed in post-selection. Timeof-flight images where atoms are not loaded are used for background analysis. The RPGC coolingis in a balanced, σ + -σ − configuration with zeroed background magnetic field. During the imaging,
we alternate trap light and imaging light at a 2 MHz flashing rate [77, 185]. This eliminates light

147
shifts and anti-trapping effects during the scattering of the imaging light to create more uniform
images.
RPGC image point spread function:

One measure of the point spread function (PSF)

of our imaging system is the average of in-trap RPGC images of single atoms (Sec. 6.2.4). Thisaveraged image represents the effect of the lens NA, aberrations, and the finite spatial extent of theatom in the trap during imaging. We note that this PSF may vary spatially within the field of viewof the imaging system, but we utilize the central point as a representative value. The measured
PSF (Fig. 6.8) is roughly an astigmatic Gaussian with a long asymmetric tail. We subtract the
averaged image’s background and fit the result with a 2D Gaussian to extract effective RMS PSF
sizes of 0.445(2) µm (horizontal) and 0.328(2) µm (vertical) (Table 6.2). This result is the PSF used
for deconvolution in the tomography analysis. Considerations of systematic error based on thischoice are discussed in Sec. 6.5.

Figure 6.8: Imaging system point spread function. Averaged image intensity (color bars) is
normalized so that the integration of the averaged image is 1. The image shown is the average of
131,000 RPGC images with a single atom present. (a) The PSF on a linear color scale. (b) A sliceof the PSF and a Gaussian fit to the PSF displayed on a linear y-axis. (c) The PSF displayed ona logarithmic color scale, emphasizing the observed long tail and structure due to aberrations. (d)A slice of the PSF and Gaussian fit to the PSF displayed on a logarithmic y-axis.

148
6.2.3

Time-of-flight imaging characterization

Time-of-flight imaging configuration:

The time-of-flight imaging is near-resonant andoperates with an intensity I ≫ Isat . The light is applied in a power-balanced σ+ , σ− polarization
configuration on the D2 f = 2 to f ′ = 3 87 Rb transition with zero background magnetic field.
Collected photon number:

In a 10 µs image, as we nearly saturate the atomic transition,

we expect to scatter 180 photons from a single atom. Based on the final quadrature distributions
and characterizations of our camera system’s count-to-photon conversion ratio at our EM gain
setting (0.0124 photons per count), we estimate that we collect ∼ 5 − 10 photons, depending on
which data set is analyzed, suggesting an overall collection eﬀiciency of 3%-6%.
Time-of-flight imaging blur considerations:
In addition to the imaging resolution PSF discussed above, in time of flight there are additional blurring effects [166, 186], and in this section we estimate these effects theoretically (Table 6.2).
For Vf /kB = 2.4 µK, the confinement of the atom prior to release, combined with the measured atom temperature, we predict an initial atomic probability distribution RMS size of 100 nm.
During the τ = 10 µs imaging time, atoms move due to their initial velocity a RMS distance of
57 nm. Atoms additionally undergo random-walk motion due to the scattering of the imaging
photons. We estimate during τ the atom moves an additional RMS distance of 0.83 nm [166].
The released atomic wavefunction expands in 3D before being imaged onto the 2D image
plane of our camera. The extent of the atom probability distribution in the z direction combinedwith a finite depth of focus is another potential blurring effect. We apply geometric optics and the
expected impact of diffraction to an impulse response of the form δx, y × exp(−z 2 /(2σz2 )), whereσz here is the predicted z-size of the atomic wavefunction in our experiments after 0.5 ms of flighttime. Diffraction alone would result in a PSF size of σx = 300 nm, and our calculation indicatesthat the DOF increases this to σx = 320 nm, which is 110 nm as a value added in quadrature.
First 10 µs time-of-flight distribution measurement:

A useful experimental compar-

149

In-trap measurement
Measured PSF σx
Measured PSF σyExpectations
Expected diffraction limit σ
In-trap thermal distribution σT (0)Atomic flight measurement
Measured PSF first 10 µs σx
Measured PSF first 10 µs σyAtomic flight calculations
DOF effect σDOF
Thermal displacement during τ , στ
Random walk during τ , σrecoil445(2) nm
328(2) nm
300 nm
100 nm461(5) nm
366(4) nm
110 nm57 nm
0.83 nmTable 6.2: Factors contributing to imaging blur. The upper table reflects in-trap measurements and the expected diffraction limit calculated using the NA of the ideal tweezer lens. The
lower table reflects blurring from effects unique to the time-of-flight imaging and that can be sensitive to the length of a single image τ .

150ison to the theoretical estimate of time-of-flight blur is the size of a time-of-flight distribution of anear ground state atom in its first 10 µs of flight, the smallest time-of-flight distribution possible
without shortening the imaging time (Fig. 6.10a). This distribution is affected by the photonscattering random walk during a single image and some of the thermal displacement effects, although not DOF effects. Note, however, that it is a noisier PSF estimate than the in-trap RPGC
based PSF estimate (Sec. 6.2) due to the significantly shorter single image time. We find this timeof-flight distribution has Gaussian RMS size of 0.461(5) µm and 0.366(4) µm in the horizontal andvertical directions respectively, which is only slightly larger than the in-trap measurement and issimilarly astigmatic, indicating that the dominant source of blur is the imaging optics (Table 6.2).
Error based on our ability to account all blurring effects is discussed in Sec. 6.5.
Clock-induced charge noise:

We are limited by our camera’s so-called clock induced

charge (CIC) [186]. These are events in which a pixel on the camera gains a “clock-induced” chargeduring the process by which photoelectrons are shifted into gain and readout registers. The CIC
is amplified the same as any charge, making it indistinguishable from the photoelectrons we wish
to measure. The number of such charges scales with the number of pixels in a single image. As
such, there is a tradeoff between improved imaging resolution and camera noise. CIC is reducedby shifting electrons across the camera sensor and gain registers at the maximum rate. Even at
the fastest shift speed of 0.3 µs per shift, Andor reports that CIC occurs at a rate of 1.8 × 10−3
events/pixel. Due to technical issues our Andor camera is only currently capable of operating at
0.5 µs per shift, which will increase CIC.
Background characterization:

An undesirable effect of averaging together many images

is a sensitivity to background patterns on the camera, including “hot” pixels that have consistently
higher counts, a small amount of background light, patterns in the camera’s sensitivity, large profiles
in counts which are a result of the camera’s cleaning and shifting processes, and CIC (Fig. 6.9).
Most of the spatial variation in these noise sources is only horizontal, depending on the proximityof a column of pixels to the camera’s readout sensor. Some of these sources vary over time. To
combat these issues, backgrounds subtracted from our data are taken from the same experimental

151
sequence as the data of interest, and are therefore temporally interleaved with the desired data.
Most of these sources, such as the overall profile left by the cleaning process, are large comparedto the averaged imaging signal, but very consistent.
c

1
b

5 µm

5 µm

100.0

10−2
Probability

102.5

97.5
95.0

10−4

92.5
90.0

10−6

Averaged camera counts

a

10−8
0

200

400
600
Camera counts

800

1000

Figure 6.9: Background and noise analysis. (a) The measured probability of observing agiven count value in a single background image averaged over all columns in the images, whichcharacterizes our camera noise. (b) Inset: a representative, single, background image. High points
in the image are primarily due to CIC. (c) An averaged background, the average of many single
background images. The CIC has mostly averaged out, and a non-negligible background profile
has appeared, along with several hot spots. Vertical and horizontal integrations of the backgroundare also displayed, emphasizing that most of the profile is in the horizontal direction (direction ofAndor vertical shift, see Table 6.1), increasing toward the readout register on the left.

Gaussian camera noise characterization:

After subtracting the background from anaveraged image but before further post-processing, there remains noise in the averaged image which
is visible, for example, near the edges of the images in Fig. 6.2, where we expect negligible real
signal. The noise on a single pixel approximately samples a Gaussian distribution and is primarily aresult of camera CIC, so we refer to this as Gaussian camera noise. Our analysis takes into accountpotential spatial dependence of the noise across the camera.
We characterize this noise by first carefully measuring, from a very large data set of background images, the single-image count-probability distribution for every column of pixels (Fig. 6.9a).
If we analyze this noise probability distribution, we find an average CIC rate of 7.0×10−2 events/pix,
which is larger than ideal rates, largely due to our slower shift speed (Table 6.3) [186]. With thesecolumn-wise-measured count-probability distributions, we can simulate an ideal uncorrelated back-152
ground image by sampling from the measured single-image count-probability distributions. We
can then create an averaged background by creating an arbitrary number N of sampled single
background images and averaging them together. We finally simulate many averaged backgrounds
in order to reconstruct the probability distribution of a count value in an averaged background forevery column of pixels. The resulting probability distribution approximately follows a Gaussianfunction with a standard deviation σnoise , which decreases as N increases. We simulate σnoise as a√function of N , and fit the result with a function of the form σnoise = A/ N . The resulting valuesof A vary slightly by column, as expected, with a mean value averaged over all columns of A = 26.5
counts and a standard deviation among the columns of 1.4.
In practice, we find that the observed noise in our tomography data is larger than thissampling analysis would predict, as determined by analyzing the edges of background-subtracted
quadrature distributions. Specifically, we observe RMS noise (1.6, 1.05, 1.16) times larger for thenx = 0 displaced, nx = 1 non-displaced, nx = 1 displaced states, respectively, than the predicted
noise from images with no atoms discussed above. The discrepancy is likely due to a combinationof effects, such as differences in the background between different data sets and small correlationsin the background data which average out in the larger background data set used for the above
analysis. Non-trivial correlations exist in the background between data sets which are taken inclose temporal proximity to each other, suggesting that some of the background signal may be
due to fluctuating weak light signals, for example coming from light elsewhere in the apparatus.
Some signal and noise may also come from atom-scattered photons as a result of our PSF having along tail (Fig. 6.8). To be consistent our observations, we always use the observed approximately
Gaussian camera RMS noise from the tomography images when bootstrapping datasets. We discuss
error based on the amount of Gaussian camera noise in Sec. 6.5.
Magnification:

We first measure the magnification of our imaging system using gravity

in an atom-drop experiment. We prepare an atom in a tweezer, release it, and then take a picture
a variable amount of time later to reconstruct the gravitationally assisted descent of the atom’s
center-of-mass position. We fit the signal with a Gaussian and plot the center as a function ofSpecified camera noise
CIC specified rate (ideal) 1.8 × 10−3 events/pix
Readout noise< 1e−Dark current
1.5 × 10−4 e− /(pix · sec)
Measured noise characteristics
CIC measured rate
7.0 × 10−2 events/pix
Measured readout noise σ 5.4 counts
EM gain signal γ73.1 counts
Count offset
88.4 counts

153

Table 6.3: Noise sources in our images. Note that we expect our CIC rate to be larger than thespecified value both due to our inability to run the camera at its fastest shift speed and because
our measured value may be contaminated by a small amount of scattered light.

flight time and fit to y(tf ) = y0 + (1/2)at2f to extract the acceleration a of the magnified signal on
the camera. The magnification is then a/(9.8 m/s2 ), which we find to be ×64(1). This is the value
used for calculations throughout the text.
Alternatively, we can measure the magnification based on measuring the nx = 1 Fock state.
The distinctly non-Gaussian nature of the momentum distribution is a useful calibration because
the fringe pattern is measurable in the image plane and independently predictable in the atom planegiven m, ωx , and tf . In contrast to the Gaussian nx = ny = 0 state, it is discernible from thermal
population, which manifests as spatially-Gaussian-distributed signal with an RMS size dependent
on the population’s temperature. Specifically, we know how large the atom-plane σx should begiven the functional form of the nx = 1 state (Eq. 6.3,6.4,6.5), and we know from analyzing our
momentum distributions (Sec. 6.4.1) how large the image-plane σx is. Taking the ratio of these twosizes gives us a magnification of ×66, which is consistent with our previous measurement.
6.2.4

Time-of-flight image analysis

Imaging signal loss:

In some cases, whether due to background collisions or thermal exci-

tations, atoms that are imaged in the first RPGC image of the experiment are lost from the opticaltraps before the time-of-flight imaging procedure. In the case of optical homodyne tomography,
where one typically measures the occupation of an optical mode, the loss of a photon registers as a

154
measurement of the n = 0 state. In our case, the loss of an atom does not result in the measurementof an atom in the n = 0 state; rather, it simply results in a reduction of the signal to noise of theexperiment, so no additional analysis is necessary to account for this.
Richardson-Lucy deconvolution:

The point spread function (PSF) of our imaging sys-

tem is non-zero in size and astigmatic, resulting in blurring of the measured quadrature distribution.
In order to compensate for these effects, we deconvolve our measured quadrature with an estimateof our PSF (Sec. 6.2). While convolving two-dimensional imaging data is trivial, deconvolution is
diﬀicult numerically, being is very sensitive to any imaging noise.
We use the Richardson-Lucy deconvolution algorithm as implemented in the scikit-image
python package. This iterative algorithm consists of iterating the relation

D(k+1) = D(k) ×(I
D(k) ~ P)~ P∗ ,(6.1)

where D(k) is the quadrature distribution after k iterations of the algorithm, I is the original distribution, P is the PSF, P ∗ is the PSF flipped along all axes, and ~ is the convolution operator.
Multiplication and division of all terms is done element-wise. D(0) is initialized as an array uniformly filled with 1. This algorithm iteratively approaches the quadruatre distribution that, when
convolved with our measured PSF, maximizes the likelihood of reproducing the measured quadruatre distribution, subject to Poissonian noise in our photon signal. This iterative method is similar
to one that we use for density matrix reconstruction (Sec. 6.4.2).
In the equation above, a crucial intermediary step involves division, which in the presenceof suﬀicient noise can produce issues related to the division by small numbers issues and floating
point arithmetic. Therefore, the algorithm implements a filter step, whereby if an element in theterm I (k) ~ P is smaller than the filter value ζ0 , the division result is floored to zero. While thisretains the spirit of the algorithm, it also introduces added complexity into choosing the number of
iterations to perform. As such, given the PSF there are two free parameters in the deconvolution
algorithm: the number of iterations of the algorithm and the filter value ζ0 . We discuss how we

155minimize error due to this in Sec. 6.5.

6.3

Single-atom temperature characterization

6.3.1

Raman sideband spectra

A well-established method of characterizing the temperature of a single atom in a harmonic
trap is Raman sideband spectroscopy that evaluates the imbalance between the red and blue Raman
sidebands [175]. We do Raman sideband cooling at a trap depth of 1 µK where we measure a radial
trap frequency of 139(4) kHz and an axial trap frequency of 28(6) kHz. Our cooling is capable ofproducing 3D ground-state fractions greater than 90%. We estimate that over the course of several
weeks of data taking required for this experiment n̄x = 0.10(4) in the radial dimension based on
Raman sideband spectroscopy. This corresponds to a temperature of (kB T /2)/(Ezp /2) = 0.83(7),where Ezp = ~ωx /2 is the zero point energy. Our Raman sideband spectroscopy addresses both thex and y-dimensions so we do not isolate the x-axis temperature. However, expansion momentum
distributions indicate that there is no significant difference between radial axes (Sec. 6.3.2). The
Raman spectroscopy temperature is consistent with our observation of 11+3
−2 % reverse transfer
during g-e adiabatic tunneling transfer (Sec. 6.1.4). In the axial dimension, which we are relatively
insensitive to, we measure n̄z = 0.08(10).

6.3.2

Ballistic expansion

When the trap is shut off the atom’s wavefunction will expand according to the standard
ballistic expansion formula√σx (t) =2EKE 2t + σx (0)2 ,
m f

(6.2)

for a given expansion kinetic energy EKE , which is the observable we directly measure via time-offlight imaging. At high temperatures, EKE → kB T /2. However, as T → 0, the expansion energy
diverges from the thermal energy as EKE is asymptotically dominated by the kinetic part of the

156
harmonic-oscillator ground-state zero-point energy Ezp /2 = ~ωx /4.
To measure this expansion kinetic energy, we conduct a similar experiment as that of Fig. 6.2,
but we vary the expansion time tf before imaging (Fig. 6.10a). We fit the averaged momentum
distributions with a 2D Gaussian function, and then fit the Gaussian RMS values as a function of
expansion time with Eq. 6.2 in order to extract an expansion energy from this data.
The experimentally measured results show a ballistic expansion kinetic energy of EKE =(kB × 0.256(16) µK)/2, which is close to the theoretical minimum at these trap parameters of(kB × 0.188(1) µK)/2 (Fig. 6.10). It makes no statistically significant difference whether we use
σx , σy , (σx + σy )/2, or if we deconvolve the time-of-flight distributions with the in-trap PSF firstwith reasonable deconvolution parameters, therefore we quote the temperature obtained from analyzing the mean σ of the non-deconvolved data σfit = (σx + σy )/2. This corresponds to a temperature of 0.205(12) µK (Fig. 6.10c), which is significantly different than the ballistic expansion
energy. The temperature determined from ballistic expansion corresponds to a normalized energyof (kB T /2)/(Ezp /2) = 1.08+11−10 , which is similar to the value measured via Raman spectroscopy of(kB T /2)/(Ezp /2) = 0.83(7) (Sec. 6.3.1). The slight discrepancy may suggest that the trap depth
ramps that occurs between Raman cooling and the release of the atom for ballistic imaging are not
perfectly adiabatic, or it may arise from fluctuations in the Raman cooling eﬀiciency during thelong experiments required to measure the atoms in free-space.

6.4

Quantum state tomography and related characterizations

6.4.1

Single Fock-state momentum distribution analysis

In general, the momentum space wavefunction is related to the position space wavefunction
via a Fourier transform. The momentum space distribution of the nx = 0, nx = 1, and nx = 2√√
states with ground-state RMS momentums p0,x = m~ωx /2 and p0,y = m~ωy /2 are|ϕ0 (px , py )|2 =( ( p2p2y ))xexp −+,
2πp0,x p0,y2p20,x 2p20,y
1

(6.3)

157

b
σfit (µm)

a0 ms

0.1 ms0.2 ms2.65 µm

2
1
0
0.0
c
6

0.1

0.2
0.3
tf (ms)

0.4

2

4

0.5

EK E
Ezp /24
2

0.3 ms

0.4 ms

0.5 ms00

kB T /2
Ezp /2

Figure 6.10: Ground state expansion. (a) The averaged time-of-flight distributions used in
the temperature measurement, labeled by the starting time of the 10 µs time-of-flight image. Color
scales for different distributions are set individually. (b) The average RMS size σfit = (σx +σy )/2 ofthe 2D fits of the data in part (a) (black circles) and their ballistic expansion fit (green line) versus
expansion time tf . Error bars are too small to be visible. (c) Relation between the measurable
expansion kinetic energy EKE and the underlying thermal kinetic energy kB T /2 (blue line), a slope1 reference (black line) and our measured data point (red point) which is significantly offset fromthe slope-1 reference, indicating that the zero-point energy is significant in this expansion.

158|ϕ1 (px , py )|2 =|ϕ2 (px , py )|2 =( ( p2p2y ))
x
2
p
exp
−
+
,
2πp30,x p0,y x
2p20,x 2p20,y114πp0,x p0,y

( p2

)2

x
−1
p20,x

( ( p2p2y ))
xexp −+.
2p20,x 2p20,y

(6.4)
(6.5)

As a baseline estimate of the stationary state, we can fit the raw momentum distributions
from Fig. 6.2d-f with a variable-weighted sum of the above momentum-space distributions convolved
with our imaging PSF to estimate the populations in the different states we prepare. We scale theRMS x and y sizes of the above nx = 0 (nx = 1 and nx = 2) states based on our displacement-based
trap-frequency measurement of the nx = 0 (nx = 1) states, tf , and the independently measured
imaging magnification of ×64(1). It is important to fix the RMS size for characterization of theground-state, as a ground-state Gaussian of one trap frequency can be easily confused with athermal state with a smaller trap frequency. The excited state momentum distributions, which
do not suffer from this issue, can be used as independent calibrations of the magnification system(Sec. 6.2.4), but for this analysis we fix the size of these functions as well. This analysis is a
useful preliminary diagnostic of our state preparation and imaging system. The nx = 0 state wasmeasured to have nx = (0, 1, 2) populations of (0.93(3), 0.07(2), 0.00(2)) which is consistent with our
expectations based on Raman sideband spectroscopy (Sec. 6.3.1), free-space expansion (Sec. 6.3.2),
and excited state preparation (Sec. 6.1.4). The nx = 1 state was measured to have diagonal
populations of (0.260(10), 0.651(14), 0.089(15)), and the nx = 2 state had diagonal populationsof (0.395(14), 0.125(18), 0.480(16)). Deviations of these excited state results from the ideals of(0, 1, 0) and (0, 0, 1) likely result from a combination of imperfect state preparation (Sec. 6.1.4) andblurring effects in the imaging which are not accounted for by the PSF deconvolution (Sec. 6.2).
Most such blurring effects that reduce the fringe contrast naturally manifest as population in lower
states, so these numbers can be reasonably understood as lower-bound estimates on our Fock state
preparation procedure. These considerations are treated in more detail in the context of the full
tomography in Sec. 6.5.

159
6.4.2

Maximum likelihood estimation algorithm

We now turn to analysis of the full tomographic quadrature data of Fig. 6.3.
Quadrature data:

The data in Fig. 6.3a,d,g for displaced nx = 0, non-displaced nx =1, and displaced nx = 1 states consist of data sets of 9, 64, and 64 quadrature measurements
respectively. Each quadrature measurement is the average of a variable number of pictures. The
median numbers of pictures averaged for these individual quadrature measurements were 10852,
6191, and 11320 pictures for the data in Fig. 6.3a,d,g. The non-displaced nx = 1 data was takenwith notably fewer pictures than the other data sets.
We note that N quadrature measurements are needed to distinguish phase oscillations in
the quadrature distributions differing in angular frequency by N ωx , assuming the quadrature measurements are equally spaced by phase angle 2π/N . Hence, N such measurements are needed toreconstruct a density matrix occupying Fock states up to nx = N/2 [179]. However, even in the casewhere we have occupation of states with nx ≥ N/2, we expect to accurately capture the density
matrix up to the N/2 off-diagonal. Given the expected form of the density matrix, considering
independent measures of the atom temperature and the size of the trap displacements, we estimate
the number of quadrature measurements used is suﬀicient for capturing all non-trivial elements ofthe density matrix within error bounds.
MLE algorithm:

Maximum likelihood estimation (MLE) is a common statistical tech-nique for estimating the free parameters of a model based on how likely they are to reproduce agiven data set. In the case of quantum tomography, we can take the quantum state as a model
for producing measured observables, where the free parameters are typically the complex-valued
matrix elements of the density matrix in some basis. MLE will then return the density matrix most
likely to result in the observed data. This approach is attractive as we may a priori impose physicalassumptions about our state, as opposed to other methods like the inverse Radon transform thatmay predict nonphysical states [181].
To determine the density matrix, we utilize an iterative MLE algorithm based off of the dis-

160
cussion in Ref. [181], which we briefly review here. Suppose we have a set of projective measurement
outcomes, {(p̃j , θj )}, with shorthand p̃ = p̃(θ), labelled by index j and occurring with frequency fj
in our data set. Here, p̃j is the result of a measurement of p̃(θj ) = p cos θj +mωx x sin θj , corresponding to the quadrature distribution at phase angle θj . For any given pair (p̃, θ), we may form the
projection operator Π (p̃, θ) = |p̃, θ⟩⟨p̃, θ| so that the probability of obtaining such a measurementin the state ρ is given by
P (p̃, θ) = Tr (ρ Π (p̃, θ)) .

(6.6)

The problem of MLE then corresponds to finding the quantum state ρ that maximizes the likelihoodof obtaining the data set {(p̃j , θj )},L (ρ) =∏P (p̃j , θj )fj ,

(6.7)

j

while also subjecting the resulting state to various physical constraints. Namely, ρ must be a tracenormalized, Hermitian, positive semi-definite matrix in a convenient, physically-motivated basis of
our choosing.
An iterative algorithm to maximize Eq. (6.7) is motivated by the observation that, for thestate ρ0 maximizing L, we must have fj ∝ P (p̃j , θj ) in the large sample limit. Introducing theoperator
R (ρ̂) =and noting that

∑

∑
j

fj
Π (p̃j , θj ) ,P (p̃j , θj )(6.8)

j Π (p̃j , θj ) ∝ I for identify matrix I, this translates to the condition

ρ0 ∝ R (ρ0 ) ρ0 R (ρ0 ) .

(6.9)

We may then leverage this to form the basis of an iterative algorithm,()()
ρ(k+1) ∝ R ρ(k) ρ(k) R ρ(k) ,(6.10)

where we start with the initial trial state ρ(0) ∝ J, where J is a matrix of ones in our truncatedFock basis, which has a nonzero probability for each possible outcome. We iterate this procedure

161
√
(
)
until the error, as defined via T ρ(k) , ρ(k+1) for trace distance T (ρ, ρ′ ) = 21 Tr( (ρ − ρ′ )2 ) is lessthan 10−4 or 500 iterations have been completed.
In practice, it is necessary to choose an appropriate reduced basis in which to restrict the setof possible states. For our purposes, we restrict ourselves to motional Fock states with occupationn ≤ nmax for a maximum occupation nmax = 25. The projection operators can then be defined via
Πmn (p̃, θ) = ⟨m|Π (p̃, θ) |n⟩(6.11)= ⟨m|p̃, θ⟩⟨p̃, θ|n⟩with the standard harmonic oscillator matrix element⟨n|p̃, θ⟩ = ie−inθ

( 1 )1/4 H (p̃/p √2)2n0√e−(p̃/p0 ) /4
2
2πp0
2n n!

(6.12)

for Hermite polynomials Hn .
After reconstructing the density matrix, the state’s Wigner function may be obtained via
√
nmax
1 ∑
k!
2
k −iϕ(m−n) −|α|2 /2
W (x, p) =(6.13)
ρmn (−1) ee|α|∆ L∆k (|α| )π(k + ∆)!
m,n=0

for α = x/x0 + ip/p0 , ϕ = arg(α), k = min(m, n), and ∆ = |m − n|, where L∆
k are the generalized
Laguerre polynomials.

6.4.3

Alternative tomography methods

There are many alternatives to our use of maximum likelihood estimation. We explored
several in order to better understand and support our state preparation and tomography. Eachmethod has different drawbacks, making different assumptions about our states. We discuss several
alternative methods below for thoroughness.
Inverse Abel transform.

It is possible to derive the Wigner function based on a single

time-of-flight image based on the assumption that the state’s Wigner function should be radiallysymmetric [187], such that the quadrature distribution is time-invariant. These assumptions should
hold for Fock states and phase-randomized states.

162
The Wigner functions of harmonic oscillator eigenstates are radially symmetric, meaning that
W (x, px ) → W (p̃). In particular,1x2p2exp(− 2 − x2 )2πx0 p0
2x0 2p0
(
)
(
2
2
1 2p
σx2 p22xx2 )Wnx =1 (x, px ) =
−
1
exp
−
+
−
π ~2
σx2
~2
σx2
Wnx =0 (x, px ) =(6.14)(6.15)The prepared states displayed in Fig. 6.2 should be primarily of fock state character. To theextent that they are not pure Fock states, it is reasonable to suspect that they are an incoherent
or phase-randomized mixture of Fock states, as this infidelity would likely be driven by thermal
population, laser intensity and bias noise issues, and other issues which we do not coherently control.
Such incoherent or phase-randomized states would have Wigner functions which would be radiallysymmetric. It has been shown and demonstrated that for radially symmetric Wigner functions, theWigner function is related to it’s quadrature distribution via the inverse Abel transform [187, 150]:

1
W (p̃) = −
2π

[∫

∫ ∞]

−∞

+
−p̃

p̃

Φ′ (p̃)√[dr]
r2 + p̃2(6.16)We use this result to reconstruct the approximate Wigner function of our single atom states
based on the assumptions discussed above. Specifically, we integrate the deconvolved profiles of
the raw data in Fig. 6.2 vertically, average the quadruatre distribution between the positive andnegative values of p̃, and numerically evaluate the above integral to obtain the Wigner function(Fig. 6.11).
Inverse Radon transform.

Another alternative method for the construction of a states

Wigner function is the use of the inverse Radon transform [188], which is given by the following
relation:1
W (x, p) = 24π[dθ]Φ(p̃, θ)

[dη]

[dp̃]
−∞

∫ π

∫ ∞

∫ ∞

−∞

0

× |η| exp(iη(p̃ − x cos(θ) − p sin(θ)))

163

Figure 6.11: Inverse Abel Transform Wigner Function a The Wigner function the n = 1 statedata from Fig. 6.2 as reconstructed through the inverse Abel transform. Here, the one-dimensional
result of the transform is symmetrically wrapped around the origin in order to compare to otherestimation of the Wigner function. b The raw one-dimensional Wigner function used to create parta of this figure. To obtain this result, we assume our data in Fig. 6.2 are stationary. c The raw
image reprinted from Fig. 6.2 which this Wigner function is derived from.

164
Where Φ(p̃, θ) is the measured quadrature distribution. The inverse Radon transform is
widely used in the field of medical topographical imaging. Both the inverse Radon transform andmaximum likelihood estimation make assumptions about the underlying system. For the integral
transform such assumptions are necessary to evaluate the divergent kernel in the above relation,and in MLE one must pick a finite basis with which to use in the algorithm. Maximum likelihood
estimation is widely preferred in quantum state tomography, however, because it is easier to enforce
reasonable physical assumptions about the state in this method than with the integral transform.
In particular, the inverse Radon transform can result in obviously non-physical states which havenon-Hermitian density matrices which have non-unity trace and negative diagonal values. When
we evaluate it for our tomography data presented in Fig. 6.3, it suggests, as an explanation for
camera noise in our data, non-trivial populations in unreasonably highly excited states. However,it can still be illustrative to view the results of the inverse Radon transform, as it still consistently
reconstructs the main features of the data, including a strongly negative and displaced center forour displaced nx = 1 atom data. Therefore, we include the result of taking the inverse Radontransform of our deconvolved quadrature data in Fig. 6.12.
Pattern function overlap.

Another method for reconstructing the density matrix is theuse of the so-called ”pattern functions”. These can be used to reconstruct a density matrix directlyfrom the measured quadrature distributions without the use of maximum likelihood estimation.
The pattern functions f˜k,ℓ (fp̃ ) are not unique, but one convenient form is given by [189]:

√

2k k!
|fp̃ |fp̃ℓ−k2ℓ ℓ!

(6.17)

2× exp(−fp̃2 /4)Lℓ−kk (fp̃ /2)(6.18)

f˜k,ℓ (fp̃ ) = f˜ℓ,k (fp̃ ) = π(−i)ℓ−kThese can be used to construct a density matrix via the following relation:

165

0.06

7

0.04

0

0.00

W (x , p )x /x 0

0.02

−0.02

−0.04
−7

−0.06
−7

0

7

p /p 0Figure 6.12: Inverse Radon transform Wigner function. The measured Wigner function of adisplaced nx = 1 state. The inverse Radon transform is more sensitive in many ways to camera noise
than the maximum likelihood estimation is, although both methods result in a Wigner function
with the same primary features.

166

1
ρk,ℓ = 2
4π

∫ 2π

∫ +∞

[dθ]w̃(fp̃ , θ)(6.19)

× exp(i(k − ℓ)θ)f˜k,l (−fp̃ )(6.20)−∞[dfx ]
0

Where w̃(fp̃ , θ) is the 1-dimensional Fourier-transform of the quadrature data w(x, θ), andfp̃ is the spatial frequency coordinate, and Lnm (z) is the generalized Laguerre polynomial.
The main downside to this method is that like the inverse-Radon transform, when applied
to noisy data, this method produces unphysical density matrices with negative diagonal elements.
Normalization and comparison to other methods when these effects are significant are not trivial.
Error from this method can be estimated with a similar bootstrapping routine.

6.5

Error estimation

6.5.1

Estimating statistical error through bootstrapping

To present the Wigner functions distributions in Fig. 6.3 we apply MLE directly to theobserved quadrature data. When this is done, we obtain Wigner function maximum negativitiesof -0.052 (-0.043) for the non-displaced (displaced) nx = 1 states discussed in Fig. 6.3. However,this calculation does not naturally lend itself to estimation of the error on these values or on thedensity matrices.
The calculation of confidence intervals for the results of the MLE algorithm is a non-trivial
task. We take the bootstrapping approach outlined in Ref. [181]. We begin by calculating the
expected quadrature distributions as a function of θ and p̃ for the non-bootstrapped MLE density
matrix, which is obtained from our measured quadrature data. For every value of θ, we thensimulate projective measurements of p̃ by sampling from these expected quadrature distributions,and use this to generate an ensemble of bootstrapped images.
Separately, based on our Gaussian camera noise analysis, we reconstruct noise that is added toeach bootstrapped quadrature distribution. Specifically, we use Gaussian-distributed noise with aStateStat. nx = 0
Stat. nx = 1
Stat. nx = 2
Disp. nx = 0
Sq. nx = 1
Sq. Disp. nx = 1
Disp. nx = 0
Sq. nx = 1
Sq. Disp nx = 1
Stat. nx = 0
Stat. nx = 1
Sq. nx = 1
Stat. nx = 2
Disp. nx = 0
Sq. Disp. nx = 1

0
1
0.958
0
0.366
0.012

Ideal States
0
0
0
0
0
0.041
1
0
0.204
0.0759
0.135
0.175

0
0
0.001
0
0.027
0.262

P (n = 0) P (n = 1) P (nx = 2) P (nx = 3) P (nx > 3)xx
Least-Squares Single Image Fit
0.93(3)0.07(2)0.00(2)
X
X
0.260(10)0.651(14)0.089(15)X
X
0.395(14)0.125(18)0.480(16)X
X
Maximum Likelihood
0.338(14)0.214(15)0.148(9)0.095(10)0.205(16)0.227(8)0.477(9)0.092(8)0.039(5)0.166(7)0.455(7)0.133(8)0.113(6)0.116(7)0.184(7)Aggressively Deconvolved Maximum Likelihood
0.336(10)0.259(15)0.184(12)0.104(9)0.117(12)0.324(7)0.514(7)0.024(3)0.029(3)0.109(5)0.642(6)0.080(3)0.077(3)0.082(3)0.119(4)
1
0
0
0
0.327
0.417

0
0
0
0
0
0.001

0.034(8)0.081(5)0.021(2)0.067(12)0.102(5)0.038(3)X
X
X

P (nx > 9)0−0.318−0.318−0.1320−0.318-0.002(5)-0.070(5)-0.124(4)-0.001(4)-0.060(6)-0.064(6)

X
X
X

Min W

0.318
0.142
0.142
0.318
0.318
0.142

0.141(13)0.136(5)0.231(3)0.110(11)0.113(6)0.133(6)

X
X
X

Max W

1
1
1
1
1
1

0.470(16)0.448(6)0.749(8)0.361(15)0.351(7)0.443(9)X
X
X

PurityTable 6.4: State populations predicted through different types of analysis. Least-Squares Single Image Fit: The fractionof each of the stationary (stat.) nx = 0, nx = 1, and nx = 2 states. Results come from fitting the data with a sum of the three relevant
quantum states which have been convolved with the fitted PSF of our imaging system. The standard deviation of the nx = 0 (nx = 1
and nx = 2) fit is fixed by our measurement of the displaced oscillation period of the nx = 0 (nx = 1) state. All errors are least squared
errors from the fitting routine, notable as most other errors reported come from MLE bootstrapping simulations. Maximum Likelihood:
The results of the full maximum likelihood analysis and bootstrapping to our various displaced (disp.) and squeezed (sq.) quadrature
measurements, as used in Fig. 6.3. Notably, only the on-diagonal values and uncertainties are reported, while off-diagonal coherences
contribute significantly to the Wigner function negativity. Aggressively Deconvolved Maximum Likelihood: The results of setting
the deconvolution filter value to ζ0 = 1.9 counts in all analysis. Filtering aggressively will reduce the effects of noise, as is evidenced by the
reduced fraction of each state with n > 10, which we attribute to noise. However, aggressively filtering may overestimate the negativity of
the Wigner function.

167

168
standard deviation consistent with what we observe in the tomography quadrature data (Sec. 6.2.3).
Note, for the highly-averaged statistics of our experiment, the quadrature distributions are wellsampled, and the projection noise is small compared to the Gaussian camera noise.
At this point, we have an ensemble of bootstrapped quadrature distributions, which we canrun through our deconvolution algorithm and MLE, as done for our raw experimental data. This
yields an ensemble of density matrices, whose variation reflects the Gaussian camera noise in thequadrature data.
The diagonal density matrix elements derived from the direct MLE and the bootstrapped
ensemble are displayed in Fig. 6.13a,b,c. While off-diagonal components are also important forthe state inference, this picture provides a simple method to compare different techniques andexpectations. From this comparison we can see the bootstrapped ensemble results in slightly lesspopulation in higher nx states. From these density matrices we calculate the Wigner negativities,
as presented in the main text. The minimum value of the Wigner function for the nearly-stationarynx = 1 state is found to be −0.060(6) and the displaced nx = 1 state of −0.064(6) Recall, for the
direct MLE analysis we obtain Wigner function maximum negativities of -0.052 (-0.043) for the
non-displaced (displaced) nx = 1 states discussed in Fig. 6.3. While all analyses clearly point tonegative Wigner values, we note the difference between the non-bootstrapped and bootstrapped
analyses is likely indicative of subtle effects such as non-ideal noise and the non-invertibility of ournumerical deconvolution routine (Sec. 6.2.3).
Lastly, we note that our reconstructed Wigner function for the displaced and non-displacednx = 1 states are qualitatively similar despite having independent noise patterns (Fig. 6.3d,g),
which indicates the robustness of the MLE reconstruction protocol.

6.5.2

Noise and imaging systematic effects

Systematic error due to variable Gaussian noise:

We can simulate the systematiceffect of varying amounts of Gaussian noise on our MLE state reconstruction by again using ourbootstrapping procedure. For this simulation, we construct mixed states with only diagonal ele-

169

Figure 6.13: State populations visualization via diagonal populations (top row) Populations of harmonic oscillator states measured in our tomography as calculated through the error
bootstrapping (blue bars) or the direct MLE result (black bars) for (a) the displaced nx = 0 state,(b) the non-displaced nx = 1 state, and (c) the displaced nx = 1 state. (c) inset: populationsof the experimentally displaced nx = 1 data set after being displaced back to the origin. (bottomrow) For comparison, we display the populations of theoretically ideal states, which do not include
experimental imperfections in state preparation, with (red bars) and without (grey bars) the applied squeezing operator for (d) the displaced nx = 0 state, (f) the non-displaced nx = 1 state,and (e) the displaced nx = 1 state.

170
ments based on the least squares fits of the heavily averaged momentum distributions in Fig. 6.2, as
this data has the least relative noise. By adding variable amounts of Gaussian noise to this initialstate and conducting MLE on the result, we find that increasing amounts of Gaussian noise biases
the reconstruction to overestimate the population in higher-n states, and underestimates the true
negativity of the Wigner function (Fig. 6.14).

State population or min(W (xm , pm ))nx = 01.0

nx = 1anx = 2nx = 3nx = 4bmin(W (xm , pm ))

c

0.8
0.6
0.4
0.2
0.0
−0.2
0.0

0.2

0.4

0.6

0.8

1.0 0.0

0.2

0.4

0.6

0.8

1.0 0.0

0.2

0.4

0.6

0.8

1.0

RMS Gaussian camera noise (counts)

Figure 6.14: Simulating the effects of Gaussian noise on maximum likelihood estimation (a) Simulation of noise on a numerically displaced nx = 0 state with undisplaced diagonalnx = (0, 1, 2) populations of (0.93(3), 0.07(2), 0.00(2)) and no off-diagonal populations, the values
suggested by least squares fitting of the highly-averaged momentum distributions in Fig. 6.2, fordemonstration. Shown are the mean values (solid lines) of populations nx = 0 through nx = 4,
averaged over 50 simulations at each RMS noise value. 95% of the single-simulation observed values
fall within the color bands to demonstrate variation of these parameters between simulations. (b)Simulation of noise on an undisplaced but numerically squeezed nx = 1 state with undisplaced andunsqueezed nx = (0, 1, 2) populations of (0.260(10), 0.651(14), 0.089(15)) picked similarly from theleast squares fitting routine for demonstration. Additionally shown are the mean Wigner values atthe location of the true minimum (xm , pm ) (red line), and 95% of single-simulation Wigner function
values at this location lie within the red color band. (c) Similar to (b) with the same initial populations, but for a displaced nx = 1 state. A key reference point is the measured Gaussian camera
noise, as measured in our real quadrature data in Fig. 6.3 for each state (blue dotted lines).

Point-spread function and blurring effects:

As discussed in Sec. 6.2, in order toaccurately reconstruct our state, we must take into account aberrations and blurring effects in thetime-of-flight images. Ideally, one could measure and characterize all such effects to reconstruct
the true point-spread function, which can then be used to deconvolve the measured quadrature
distribution. Because characterizing all blur is challenging for certain types of effects, we use

171
a conservative value in our analysis, namely the in-trap size measured with RPGC, which is oursmallest PSF estimate (Sec. 6.2). A less conservative estimate is the larger time-of-flight distributionof the atoms immediately after being released.
We verify that using larger PSF functions for deconvolution in the analysis of displaced andnon-displaced nx = 1 Fock state data (Fig. 6.3d-j), up to a factor of 1.5 larger in both dimensions,
results in uniformly more negative Wigner functions. For example, using the first-10 µs time-offlight distribution (Table 6.2), which also has slightly less astigmatism, has a negligible impact on
our Wigner negativity — the reconstructed non-displaced nx = 1 Wigner function would have aminimum value from the non-bootstrapped MLE analysis of -0.055 (instead of -0.052).
Other deconvolution parameters:

In general, we expect the optimal deconvolution

parameters to depend on the nature of the state being reproduced and the amount of noise present.
To choose suitable values for our analysis, we perform bootstrapping similar to that done forstatistical error estimation, except that we allow the deconvolution parameters in the bootstrapped
ensemble to vary from the direct MLE reconstruction parameters. We then compare the resulting
bootstrapped density matrix to the direct MLE result to find optimal parameters that reproduce
the non-bootstrapped density matrix with the highest possible fidelity, where the fidelity is defined
as

(
F=

(√
)) 2
√
√
Tr
ρ1 ρ2 ρ1.

(6.21)

Let ζ0 and ζb denote the deconvolution filter values for our direct MLE and boostrapped density
matrices, respectively. We choose ζ0 such that the boostrapped density matrix with filter value ζbyields the highest fidelity when ζb = ζ0 . For the displaced nx = 1 data set, we find that a filter
value of ζb = 0.69 counts and between 2 and 10 iterations of the deconvolution algorithm reliably
reproduces the density matrix for a wide range of ζ0 . We thus choose to use this value, ζ0 = 0.69,
in our main analysis. To minimize additional artifacting, we also apply only 2 iterations of thedeconvolution algorithm, for which our bootstrapping routine reproduces the non-bootstrapped
density matrix with a fidelity of 0.934(2).

172
We apply a similar analysis to find that the highest-fidelity parameters for the displacednx = 0 analysis also corresponds to a filter value of ζ0 = 0.69, and that the best for the nondisplaced nx = 1 is ζ0 = 1.03, both similarly with 2 iterations of the algorithm. The non-displacednx = 1 requires slightly larger filtering to compensate for having slightly increased noise due fewer
points in this data set.
Noise and imaging analysis summary:
In conclusion, noise and imaging blur processes can influence the MLE algorithm when carrying out tomography using imperfect camera imaging. We have shown through a survey of systematiceffects that their impact is either small or they result in our state reconstruction being a conservative underestimate of parameters such as the magnitude of the Wigner function negativity andstate preparation fidelity.

6.5.3

Trap anharmonicity3𝑝(𝜔" " 𝑡! ) /𝑝#

2
1
0
-1
-2
0.0

0.2

0.4

0.6

0.8

1.0

𝑡! (ms)Figure 6.15: Best-fit dynamics for ⟨p̃(ωx te )⟩/p0 generated by Eq. (6.22) for an initial displaced
ground state (blue, dotted), compared to experimental results from Fig. 6.6 for an initial displacednx = 0 state (green).

Characterizing trap anharmonicity

To analyze the effect of trap anharmonicity on

state reconstruction, we consider the dynamics of a single atom in a one-dimensional anharmonic
oscillator, which serves as a phenomenological model for the experimental optical tweezer potential.

173
Specifically, we consider dynamics generated by the Hamiltonianmωx2 x2
p2
+
+ Λx4
2m [
2
]
1
2
2
4
=~ωx (px0 /~) + (x/x0 ) + λ(x/x0 ) ,4

H=(6.22)

which describes an atom of mass m in a harmonic trap with frequency ωx , plus an additional quartic
term of characteristic strength Λ. In the second line, we have rewritten this potential in terms of√
the harmonic length x0 = ~/(2mωx ), and defined the dimensionless parameter λ = Λx40 /(~ωx )as the relative strength of the anharmonic correction.
We can obtain estimates of ωx and λ by comparing the predictions of our phenomenologicalmodel with experimental observations. Specifically, we compare experimental data for the longtime center-of-mass dynamics of an initially-displaced nx = 0 state (see Fig. 6.2k) to the average
momentum ⟨p̃(ωx te )⟩ obtained from Eq. (6.22) for an initially-displaced nx = 0 state. In thecase that λ < 0, we add a minimal sextic term ~ωx (2λ2 /3)(x/x0 )6 that ensures our potential
remains stable and retains only a single local minimum, and thus induces only local deformationsof our ground state; this term plays a negligible role in the resulting dynamics for the consideredparameters.
Treating ωx , λ, and the initial state displacement xi as free parameters, we perform a leastsquares fit of ⟨p̃(ωx te )⟩ to the experimental center-of-mass oscillations. We obtain best-fit valuesof λ = −0.0037(4), ωx = 2π × 8.50(5) kHz (corresponding to x0 = 83 nm) and xi = 166(5) nm. As
shown in Fig. 6.15, we observe that the fitted model robustly captures the observed damping. As acrude estimate of the relative significance of the harmonic and quartic potential terms in the Hamiltonian when considering the dynamics of this displaced state, we can compare their expected values
at te = 0. By approximating moments by their harmonic values, we find 4|λ|⟨(x/x0 )4 ⟩/⟨(x/x0 )2 ⟩ ≈0.13. While this indicates that the harmonic term remains dominant for displacements of this size,
the anharmonic term is not so small as to be easily discounted without further analysis. Separately,
we note the the best-fit estimate for the trap frequency in this model (8.50(5) kHz, with x0 = 83 nm)
is comparable to the observed oscillation frequency in the experiment, 7.84(5) kHz with x0 = 86 nm.

174
The deviation between these values is consistent with the negative λ in our model, which generally
leads to a slower observed oscillation frequency for a displaced nx = 0 state than what is expected
for the corresponding ideal harmonic oscillator.
Maximum likelihood estimation in an anharmonic potential

We now examine howanharmonicity affects state reconstruction, utilizing the Hamiltonian in Eq. (6.22) as a model forthe tweezer potential. Following the protocol in the main text, we spatially displace an initialstate ρi by a fixed amount xi , which in the Schrödinger picture can be described by acting thedisplacement operator D(xi ) = e−ipxi /~ on the initial state. We then evolve for a time te , describedby the unitary operator U (t) = e−ite H/~ . The resulting dynamical state is expressed as,ρ (te ) = U (te ) D(xi )ρi D† (xi )U † (te ) ,(6.23)

from which we extract the time-evolved momentum distribution P (p, te ) = Tr (ρ (te ) |p⟩⟨p|) for
momentum eigenstate |p⟩. This distribution is used as the input for the iterative MLE algorithm(Sec. 6.4.2).
To make connections with the experiment, we would like to analyze the effect of an anharmonic trap on reconstruction of a displaced nx = 1 state. Because the prepared state contains
contributions from other Fock states, we assume the initial state ρi can be well modelled by an
incoherent mixture of the low-energy eigenstates of the Hamiltonian,ρi = P0 |0⟩⟨0| + P1 |1⟩⟨1| + P2 |2⟩⟨2|.

(6.24)

Here, Pnx denotes the probability of staring in the eigenstate |nx ⟩; we denote this state by the
triplet P = (P0 , P1 , P2 ). By choosing Pnx that generally resembles the makeup of the state in
the experiment we are able to analyze the effects of initial mixed states in both the dynamics
and ensuing state reconstruction, and demonstrate robustness of MLE tomography over a range ofpossible initial states.
As the state reconstruction data for the displaced nx = 1 state, featured in Fig. 6.3g-k, isobtained using a deeper trap depth and smaller displacement than that used in our initial analysis

175
of the anharmonic oscillator model Eq. (6.22) for the nx = 0 center-of-mass oscillations, we modify
the previously obtained best-fit parameters by an appropriate scaling. Assuming that H is linear√√
in the trap depth V , we have that λ ∼ 1/ V and ωx ∼ V . Hence we can rescale based on therelative ratios of both the depth and displacement.
In Fig. 6.16a, we plot the fidelity of our reconstructed state, ρMLE , to the actual underlying
state, ρ(0) = D(xi )ρi D† (xi ), for a variety of compositions P. In the figure, we indicate P that
produces a state before displacement with a similar spatial extent in the trap as the nx = 1 statesin Fig. 6.3e, h. This point, indicated by the purple star, corresponds to P = (0.28, 0.57, 0.15), andcorresponds to estimates obtained in Fig. 6.13 for the diagonal composition of the state preparedin the trap.
Within our model and for the indicated state, we find that the reduction in the fidelity
owing to anharmonic effects does not exceed 5%. However, for higher populations of the nx = 2
state, a larger reduction in fidelity is expected from anharmonicity, and a deeper trap or smaller
displacement would be needed to reduce the effect of anharmonicity and faithfully characterize suchstates. Moreover, we find that the diagonal elements of the density matrix indicated by the purple
star remain very similar (Fig. 6.16b).
While the fidelity provides some measure of similarity between our states, of specific interestis the robustness of the nonclassical nature of the prepared states, which may be identified throughthe Wigner negativity γ ≤ 0, where γ = minx,p W (x, p). Therefore, we also plot the differencein negativity between ρ(0) and ρMLE in Fig. 6.16c. Over the entire range of P considered, thedifference in obtained negativity never exceeds 0.06, and for the indicated estimates of our nx = 1
initial state composition, for which γ ≈ −0.04, the corresponding MLE state remains negative witha difference in negativity of < 0.01.

176

a

b

ℱ

c

0.6

𝛿𝛾

𝜌(0)
𝜌!"#

𝑃%

𝑃%

𝑃&!

0.4

0.2

0.0
𝑃$

0

5
𝑛'

10
𝑃$

Figure 6.16: (a) Fidelity F (Eq. (6.21)) between the state ρ (0) and the reconstructed state ρMLE
for initial states ρi characterized by a range of P = (Pn=0 , Pn=1 , Pn=2 ), after evolution with ouranharmonic model Eq. (6.22). (b) Comparison of the diagonal elements for nx ≤ 10 of ρ(0) andρMLE after a displacement back to the origin, corresponding to P = (0.28, 0.57, 0.15) (purple starin a). This point corresponds to a state with similar spatial extent as the nx = 1 state shown in
Fig. 6.3e, h. Here, nx labels the Fock states of the oscillator in Eq. (6.22). (c) For each P, wethen compute the Wigner negativity, γ = minx,p W (x, p), for our initial state ρ(0), as well as the
Wigner negativity γMLE of the reconstructed state. We plot the difference δγ = γ − γMLE ; positive
regions indicate parameters for which ρMLE estimates more nonclassicality than the underlying
state, whereas negative regions indicate parameters for which ρMLE is an upper bound on thisnegativity. The dashed black dashed line separates nonclassical states with γ < 0 from those with
γ = 0.

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