A high optical access cryogenic optical tweezer arraybyTing-Wei Hsu
B.S., Purdue University, 2016
M.S., University of Colorado Boulder, 2022

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
2024

Committee Members:
Cindy Regal, Chair
Adam Kaufman
Daniel Slichter
Ana Maria Rey
Longji Cuiii
Hsu, Ting-Wei (Ph.D., Physics)
A high optical access cryogenic optical tweezer array
Thesis directed by Prof. Cindy Regal

Arrays of single trapped neutral atoms are an emerging platform for quantum metrology,
simulation, and information processing. They have seen the fastest qubit number scaling in thepast few years. In this thesis, I present a set of new technologies, a cryogenic optical tweezer
array and trapped single atoms using a metasurface focusing lens, that we developed for the next
generation of optical tweezer platforms.
In the first part of the thesis, I describe a new apparatus that combines a cryogenic environment with high-NA optical access. We demonstrate a trapped array of single atoms with avacuum-limited lifetime of up to 3000 seconds, which is difficult to achieve in room temperature
tweezer experiments. Additionally, we characterize the heating of atoms in our apparatus and findno adverse effects from residual vibration introduced by the cryocooler. Interferometric measurements reveal residual vibration to be 2 nm RMS. With the long vacuum-limited lifetime, we measure
imaging and cooling losses of trapped single atoms that can often be overwhelmed by vacuum loss
in typical apparati. We demonstrate an imaging loss rate of 3.8(4) × 10−4 per 14 ms image, with
an imaging assignment infidelity of 10−6 and a cooling loss rate of 5(4) × 10−5 per 25 ms coolingpulse, among the lowest reported in alkali atoms. We also demonstrate preliminary single qubit
rotations with a microwave antenna within the cryogenic housing and characterize magnetic field
fluctuations to 0.5 mG over a period of half an hour, comparable to other tweezer experiments.
In the second part of my thesis, I describe a new approach to manipulate and control trapped
single atoms in optical tweezers through the use of optical metasurfaces. Optical metasurfaces areplanar photonic elements composed of a periodic array of subwavelength dielectric nanostructures.
These nanostructures couple, either resonantly or off-resonantly, and re-radiate the incoming lightwith a transformed phase, polarization, and amplitude determined by the nanostructure’s shape,

iii
size, and material composition. In this thesis, we create a 0.55 NA metasurface lens with a focusing
efficiency of up to 58% to trap single atoms to sub-micron positional confinement, and detect
single 87 Rb atoms though the same lens. We also perform field-of-view measurements and find adiffraction-limited field-of-view of ±10 µm, which is in good agreement with numerical simulation.
Finally, I present potential future metalens designs with qualities beneficial for creating large neutral
atom tweezer arrays, including a doublet lens with a large field-of-view of up to 2 mm, a lens without
chromatic aberration, and polarization multiplexed lenses for dynamically changing the trapping
potential or for achromatic imaging of the trapped atoms.

Dedication

This work is dedicated to my family and to the friends who have made an impact on my
life.

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Acknowledgements

First and foremost, I would like to thank my advisor, Cindy Regal. The guidance andmentorship she provided during my PhD journey were exceptional. Working and learning in theRegal lab has been an amazing experience that I could not have imagined getting elsewhere. I am
grateful for the freedom she gave me to pursue topics that interested me during the design andconstruction of the experiments.
Furthermore, I would like to thank the collaborators on the cryogenic experiments, Adam
Kaufman and Daniel Slichter, as well as collaborators on the metalens project, Wenqi Zhu, Amit
Agrawal, and Scott Papp. They offered invaluable scientific insight and technical expertise acrossdifferent disciplines within AMO physics. I highly value the guidance they provided, and it wasalways refreshing to hear perspectives from other areas of the AMO community.
Next, I want to thank my lab mates for always being amazing through thick and thin and for
putting up with my occasional minor (or often major) OCD tendencies when it comes to building
the experiment (I often say, ’If it doesn’t work, at least it looks nice and shiny.’ If it looks nice,the chances of it working are probably pretty high.). It has been a pleasure to work with such afantastic team in B240 (Zhenpu, Ting You, and Matteo). Each of you has expertise crucial to thesuccess of the experiment, and I have learned a great deal from all of you.
Zhenpu, with whom I have had the pleasure of working with for the past 4 years, has extensive knowledge in science and experimental techniques. I can’t imagine how much longer thisexperiment would take without him. We’ve learned a great deal from each other through the countless discussions we have had together. I have greatly enjoyed working with Ting You as well, andvi
I appreciate the many contributions he has made to the experiment. Matteo, with his backgroundin trapped ions, brings technical expertise and experience that complement our skill set as neutral
atom trappers. His insights from an ion trapper’s perspective have proven very useful on several
occasions. Additionally, I want to thank the undergraduate student Max, who has done excellent
work building electronics and optics that we still use today.
I would also like to acknowledge the support from the JILA staff shop, including both theelectronics and instrument shops. I have learned a great deal about electronics and machining fromthe experts who make JILA’s research possible. I am particularly grateful to James Uhrich, Calvin
Schwadron, and Adam Ellzey for their support, not only in the construction of the experiment but
also as friends.
Finally, I would like to thank Amie for her friendship and support during my thesis writing
and throughout the final year of my PhD. She taught me a great deal about maintaining a balanced
life and emotional stability. I am very grateful for her companionship during this time.
One thing I will definitely miss are the late nights with Zhenpu, whether in the machine
shop or in the lab. Neither of us can sleep if there’s an unsolved problem, and we both enjoy
turning chunks of metal into chips. I will also miss going to the gym with Zhenpu at night andthen returning to work until midnight. I couldn’t ask for better lab mates than Zhenpu, Ting You,and Matteo. Now that the experiment is in Zhenpu’s and Ting You’s hands, I’m excited to see
what new developments will emerge in the future.

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Contents

Chapter
1 Introduction
1.1

1Quantum information processing and simulation . . . . . . . . . . . . . . . . . . . .

2

1.1.1

Neutral atom QIP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2

Cryogenic optical tweezer array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.3

Survey of cryogenic used in other quantum system . . . . . . . . . . . . . . . . . . .

8

1.3.1

9

Horizon of impact on neutral atoms and molecules . . . . . . . . . . . . . . .

1.4

Metasurface lens optical tweezer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.5

Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.6

List of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Cryogenic tweezer array apparatus

16

2.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2

Design of the cryogenic tweezer array apparatus . . . . . . . . . . . . . . . . . . . . . 18
2.2.1

Major parts of the experimental apparatus . . . . . . . . . . . . . . . . . . . 18

2.2.2

Science MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.3

DC electric field control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.4

Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2.5

Coldbox and cryopumping finger design concept . . . . . . . . . . . . . . . . 35

2.2.6

2D MOT assembly and 2D to 3D MOT transfer . . . . . . . . . . . . . . . . 37

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2.2.7

Magnetic field in 3D MOT (science) chamber . . . . . . . . . . . . . . . . . . 41

2.2.8

BBR microwave

2.2.9

Thermal design and analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.2.10 Optical access . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.3

2.4

Microscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.3.1

High NA objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.3.2

Tweezer light source and optical rail . . . . . . . . . . . . . . . . . . . . . . . 55

2.3.3

Pinhole testing of the objective . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Cryostat testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.4.1

AR coating testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.4.2

Vibration measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

2.4.3

Heat load measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

2.4.4

Tweezer cold window alignment measurement . . . . . . . . . . . . . . . . . . 69

2.4.5

Cold filter testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3 Cryogenic tweezer atom characterization

75

3.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.2

MOT lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.3

3.4

3.2.1

Residual flux of 2D MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.2.2

Dipole-push beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Single atom lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.3.1

Heating in optical tweezer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.3.2

Cooling loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.3.3

Imaging loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.3.4

Pulsed cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

3.3.5

Vacuum-limited single atom lifetime . . . . . . . . . . . . . . . . . . . . . . . 98

Tweezer FoV atom charactrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

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3.5

3.4.1

NA measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

3.4.2

LGM loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Impact of vacuum, loading rate and imaging loss on achieving large atom array . . . 107
3.5.1

Imaging loss vs detuning and intensity . . . . . . . . . . . . . . . . . . . . . . 107

3.5.2

Effect of imaging loss on defect-free array . . . . . . . . . . . . . . . . . . . . 109

3.5.3

Effect of vacuum loss on defect-free array . . . . . . . . . . . . . . . . . . . . 111

4 Cryogenic tweezer qubit manipulation

113

4.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

4.2

Microwave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.2.1

B-field calibration and noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

4.2.2

Rabi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4.2.3

Ramsey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

4.3

Eddy current measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

4.4

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.4.1

Atom Array, SLM and optical lattice . . . . . . . . . . . . . . . . . . . . . . . 125

4.4.2

Cryogenic enhancement: cryopmping . . . . . . . . . . . . . . . . . . . . . . . 126

4.4.3

ITO coating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5 Metasurface lens experiment

128

5.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.2

UHV system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.3

5.2.1

ColdQuanta source cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.2.2

Science glass cell AR coating . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.2.3

Science glass cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.2.4

Metalens sample mounting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

Metasurface Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.3.1

Optical metasurface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

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5.3.2

Anisotropic plasmonic nano-antennas. . . . . . . . . . . . . . . . . . . . . . 141

5.3.3

Dielectric metasurface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

5.4

Design principle of metasurface and metalens . . . . . . . . . . . . . . . . . . . . . . 147

5.5

Metalens sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

5.6

5.7

5.8

5.5.1

Non-polarization multiplexed lens

. . . . . . . . . . . . . . . . . . . . . . . . 150

5.5.2

Metasurface fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

5.5.3

Optical testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

5.5.4

Metalens PSF with subwavelength pinhole . . . . . . . . . . . . . . . . . . . . 155

5.5.5

Atom image system test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

Testing with atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
5.6.1

Single atom loading and detection . . . . . . . . . . . . . . . . . . . . . . . . 161

5.6.2

Trap depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

5.6.3

Trap frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

5.6.4

Tweezer array setup and FoV . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

5.6.5

Focusing efficiency and FoV simulation . . . . . . . . . . . . . . . . . . . . . . 168

5.6.6

Discussion and challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

Future sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
5.7.1

Polarization multiplexed lens . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

5.7.2

Doublet design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

5.7.3

dual wavelength lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
5.8.1

Polarization multiplex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

5.8.2

Large FoV doublet metalens . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

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186

Bibliography

Appendix
A Montana cold window test202

A.1 Double window thermal test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
A.2 Indium seal dimensional stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
B SAES Rb Dispenser Activation

205

B.1 The Alkali Metal Dispenser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
B.2 Dispenser Activation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
C Anodic Bonding

208

C.1 Anodic Bonding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

C.2 Bonding Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

C.3 Metalens Cell Bonding Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
D Basic Principle of cryopumping

212

D.1 Cryopumping Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
D.1.1 Cryocondensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
D.1.2 Cryosorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
D.1.3 Cryotrapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
D.2 Outgassing at low temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
E Stinger Cryostat

218

E.1 Stinger cryocooler operating principle . . . . . . . . . . . . . . . . . . . . . . . . . . 218
E.2 The UHV cryostat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
E.3 Compressor helium contamination issue . . . . . . . . . . . . . . . . . . . . . . . . . 222

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F Laser phase noise from cryostat vibration

225

F.1 Converting a vibration spectrum to phase noise . . . . . . . . . . . . . . . . . . . . . 225
G Release-recapture Gaussian vs harmonic trap

228

G.1 Harmonic trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
G.2 Gaussian and Airy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
G.3 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

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TablesTable
2.1

List of window substrate used in cryogenic Rydberg tweezer experiment . . . . . . . 52

3.1

Single atom cooling and detection parameters . . . . . . . . . . . . . . . . . . . . . . 100

5.1

Metalens Focusing Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

5.2

Metalens Doublet Coefficients, NA 0.7, FoV= ± 1.1 mm, WD= 7 mm . . . . . . . . 177

B.1 Typical activation behavior of SAES alkali metal dispenser . . . . . . . . . . . . . . 207

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FiguresFigure1.1

Rydberg atom quantum computing . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.2

Cryogenic tweezer arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3

Metalens single atom trapping

2.1

Cryo overall view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2

MOT Coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3

87 Rb Rydberg DC Stark polarizability

2.4

Example electrode configurations for DC electric field control . . . . . . . . . . . . . 26

2.5

Cryogenic tweezer array DC electrode . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.6

Cryogenic tweezer array active E-field control simulation . . . . . . . . . . . . . . . . 28

2.7

Cryogenic tweezer array vacuum design schematic . . . . . . . . . . . . . . . . . . . . 31

2.8

Cryogenic tweezer array vacuum system . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.9

Cryogenic tweezer 2D MOT vacuum assembly . . . . . . . . . . . . . . . . . . . . . . 38

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

. . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.10 Cryogenic tweezer 2D MOT Optics package . . . . . . . . . . . . . . . . . . . . . . . 40
2.11 Cryogenic tweezer bias coil assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.12 Microwave spectrum inside of coldbox . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.13 BBR photoionization and Rydberg state decay . . . . . . . . . . . . . . . . . . . . . 45
2.14 Compact double pane cold window design . . . . . . . . . . . . . . . . . . . . . . . . 47
2.15 Coldbox titanium thermal break . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

xv
2.16 Indium seals and cutter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.17 Coldbox optical access . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.18 Tweezer light source and optical rail . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.19 Special Optics objective pinhole test . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.20 AR coating liquid nitrigen stress test . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.21 Coldbox vibration measurement setup . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.22 Coldbox vibration measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.23 Temperature vs. heat load (bare cryostat) . . . . . . . . . . . . . . . . . . . . . . . . 67
2.24 Copper braid and shield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.25 Temperature vs. added heat load on the coldbox . . . . . . . . . . . . . . . . . . . . 68
2.26 Coldbox tweezer window alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.27 Special Optics objective angular tolerance ZEMAX . . . . . . . . . . . . . . . . . . . 71
2.28 DC electrode lowpass filter characterization . . . . . . . . . . . . . . . . . . . . . . . 72
2.29 Microwave bandpass filter characterization . . . . . . . . . . . . . . . . . . . . . . . . 74
3.1

Cryogenic tweezer MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.2

Cryogenic tweezer MOT background loading

3.3

Single atom lifetime timing sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.4

Dark lifetime without cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.5

Release and recapture temperature measurement method . . . . . . . . . . . . . . . 86

3.6

Single atom heating rate of various configurations . . . . . . . . . . . . . . . . . . . . 87

3.7

Tweezer RIN and heating rate

3.8

Cooling loss measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.9

Pulse cooling temperature measurement . . . . . . . . . . . . . . . . . . . . . . . . . 92

. . . . . . . . . . . . . . . . . . . . . . 80

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.10 Single atom imaging histogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.11 Single atom imaging loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.12 Single atom pulse cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

xvi
3.13 Single atom vacuum-limited lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.14 Cryogenic tweezer trap depth and trap frequency . . . . . . . . . . . . . . . . . . . . 102
3.15 Cryogenic tweezer NA over FoV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.16 LGM Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
3.17 Cryogenic tweezer LGM B-field vs time . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.18 Cryogenic tweezer LGM vs FoV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
3.19 Single atom imaging loss detuning dependence . . . . . . . . . . . . . . . . . . . . . 108
3.20 Sequence used for the rearrangement Monte-Carlo simulation . . . . . . . . . . . . . 110
3.21 Effect of imaging and cooling loss on defect-free array probability . . . . . . . . . . . 111
3.22 Effect of vacuum loss on defect-free array probability . . . . . . . . . . . . . . . . . . 112
4.1

Cryogenic tweezer qubit level diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.2

Cryogenic tweezer qubit optical pumping

4.3

Cryogenic tweezer bias magnetic field calibration and noise . . . . . . . . . . . . . . 118

4.4

Cryogenic tweezer array microwave Rabi . . . . . . . . . . . . . . . . . . . . . . . . . 120

4.5

Cryogenic tweezer array microwave Remsey . . . . . . . . . . . . . . . . . . . . . . . 121

4.6

Cryogenic tweezer array eddy current characterization . . . . . . . . . . . . . . . . . 123

5.1

Metalens UHV Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5.2

ColdQuanta Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.3

Metalens source MOT atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.4

Metalens science cell AR Coating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.5

Metalens science cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

5.6

Metalens sample mounting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

5.7

Microwave Metamaterial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

5.8

Metasurface Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

5.9

Plasmonic metasurface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142. . . . . . . . . . . . . . . . . . . . . . . . 116

5.10 Dielectric Pancharatnam-Berry phase metasurface . . . . . . . . . . . . . . . . . . . 144

xvii
5.11 Dielectric HCA metasurface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.12 Metasurface design principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.13 Non-polarization multiplexed metalens . . . . . . . . . . . . . . . . . . . . . . . . . . 152
5.14 0.55NA metalens zemax phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

5.15 Metalens Fabrication Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
5.16 Metalens Focus Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
5.17 Metalens Pinhole Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
5.18 Metalens Pinhole Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
5.19 Metalens MOT Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
5.20 Metalens Atom Trapping Histogram . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
5.21 Metalens Trap Depth Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
5.22 Metalens Trap Frequency Measurement . . . . . . . . . . . . . . . . . . . . . . . . . 166
5.23 Metalens FoV Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
5.24 Atom array and metalens field of view . . . . . . . . . . . . . . . . . . . . . . . . . . 169
5.25 Metalens efficiency and FoV simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 170
5.26 Polarization Multiplexed Metalens Principle . . . . . . . . . . . . . . . . . . . . . . . 174
5.27 Polarization Multiplexed Metalens Phase. . . . . . . . . . . . . . . . . . . . . . . . 175

5.28 Polarization Multiplexed Metalens PSF . . . . . . . . . . . . . . . . . . . . . . . . . 176
5.29 Metalens Doublet Schmidt Corrector . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
5.30 Metalens Doublet Zemax Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
5.31 Metalens Doublet Surface Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
5.32 Hybrid Metalens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
5.33 Polarization dependent trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
5.34 Metalens doublet compare to state-of-the-art objective . . . . . . . . . . . . . . . . . 185
A.1 Cryostat double window test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
C.1 Anodic bonding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

xviiiD.1 Different cryopumping regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
D.2 Cryopumping saturation curve of common gases. . . . . . . . . . . . . . . . . . . . 214

D.3 Xenon isotherm of some technical surface at 77 K . . . . . . . . . . . . . . . . . . . . 216
E.1 ColdEdge technologies Stinger Cryocooler block diagram . . . . . . . . . . . . . . . . 220
E.2 ColdEdge technologies Stinger Cryocooler block diagram . . . . . . . . . . . . . . . . 221
E.3 Helium flow of typical clogging event and slow degradation over long operating time 224
F.1 Vibration induced laser phase noise and vibration comparison to Jonathan Home
group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
G.1 Airy, Gaussian and harmonic trap shape . . . . . . . . . . . . . . . . . . . . . . . . . 229
G.2 Airy, Gaussian and harmonic trap Release and Recapture . . . . . . . . . . . . . . . 231

Chapter 1

Introduction

Scientists have been studying quantum mechanical phenomena for over a century, but it’s only
recently that we have begun to have the ability to create and manipulate quantum systems. Overthe past few decades, the field of atomic physics has undergone numerous advances. The inventionof lasers and improvements in optics and electronics provide new ways of creating quantum atomic
systems with very low entropy. With laser cooling of atomic vapors, the first degenerate quantum
gas with bosonic atoms was created and their quantum mechanical properties studied [1, 2].
While degenerate quantum gases are thriving, progress is also being made in cooling andcontrolling the quantum state of single neutral particles at a microscopic level, for example, opticaltrapping of single neutral atoms [3–5]. In this bottom-up approach, similar to the trapped ions [6,7],
individual neutral quantum particles can be assembled into large systems, and their quantum
mechanical properties can be harnessed. However, there are still challenges in fully controlling theseneutral particles as a quantum resource, and extending control in bottom-up systems remains amajor research interest.
Quantum systems are inherently fragile, as they are sensitive to perturbations from theenvironment, but also making them excellent candidates for sensing and metrology applications.
For example, with Rydberg atoms, due to their large dipole moments, it is possible to detect electric√fields down to 5 µVcm−1 / Hz [8]. Correspondingly, there are strong interactions between nearby
Rydberg atoms, which provides a way to achieve controlled interactions for quantum simulationand quantum information processing.

2
On the other hand, due to the quantum mechanical nature of atoms, properties that satisfy
certain quantum selection rules are very robust against environmental perturbations, leading to longquantum coherence. For example, the 1 S0 → 3 P0 transition of strontium atoms can be utilized tocreate the most accurate atomic clocks, with a uncertainty of 2 × 10−18 [9]. These long-coherence
quantum states can be used for information storage.
These properties open new avenues, including the simulation of quantum systems in solidstate physics, chemistry, and materials science [10, 11]. Additionally, large quantum systems withsingle-particle control can be used to realize quantum computers, devices capable of solving problems such as factorization of large integers [12], solving linear systems of equations [13], and searching unsorted databases [14].

1.1

Quantum information processing and simulation
Consider a system with N quantum particles that has states |0⟩ and |1⟩. One would need 2N

complex amplitudes to describe the collective quantum state. This number grows exponentially withthe size of the system. With a moderate-sized quantum system of N = 100, the computational
resources needed to describe it already exceed the capacity of all classical computing hardware
currently available in the world. In 1982, Feynman suggested [15] that in order to study a quantum
mechanical system, it might be more feasible to simulate it in a more controlled analogous quantum
system. This allows us to study the behavior of larger quantum systems by exploring sub-portionsof the system in a more controlled manner. Some examples of quantum simulation include phasetransitions in the Hubbard model [16, 17], dynamics of interacting spin chains [18, 19], and theantiferromagnetic ordering in optical lattices [20, 21].
In a universal quantum computer, qubits are used to store and carry out computational tasks.
These qubits are quantum two-level systems, well isolated from the environment and with good
coherence. Contrary to classical computers, a universal quantum computer will only need N qubits
to represent the N -particle quantum state completely. Through a combination of one and two-qubit
operations, any unitary transformation between many qubits can be realized [22]. This allows the

3universal quantum computer to simulate the time evolution of any quantum systems with only a setof fundamental quantum logic operations. Apart from simulating quantum mechanical dynamics,
the universal quantum computer can also be programmed such that the time evolution gives answers
to computational problems that are resource-intensive to solve on a classical computer. Somenotable examples include finding the prime factors of large integers through Shor’s algorithm [12]and speeding up unsorted database searches with Grover’s algorithm [14].

1.1.1

Neutral atom QIP

The concept of using neutral atoms as a quantum computing platform emerged relatively
recently, about 15 years after ion traps and superconducting circuits. It was proposed in the early
2000s [23–25] that the long-range dipole-dipole interaction of Rydberg atoms could be tailoredfor entanglement and quantum computation. Rydberg atoms are atoms with electrons excited toa large principal quantum numbers (n = 50 − 100), resulting in a large induced dipole momentand long-range interaction spanning distances up to several micrometers, with natural decay times
on the order of tens to hundreds of microseconds. Typically, Rydberg transitions can be excitedwith a single high-energy UV photon or one blue and one NIR photon via an intermediate state.
The qubit states require long coherence; thus, they are normally encoded using weakly interacting
ground states. However, depending on the application, qubits encoded in the Rydberg state arealso common in quantum simulation, where resonant dipole-dipole interaction is used to implement
spin-exchange interaction [26].
Reaching a regime where Rydberg interactions can be harnessed for quantum computing
requires tools for trapping single atoms. One of the tools that provides single-particle-level trappingand control of neutral atoms is optical tweezers. Optical tweezers are microscopic optical dipole
traps capable of confining single atoms at the sub-micron spatial scale. They achieve this by firsttrapping many atoms, and through the process of light-assisted collisions, only single atoms remain
in each trap [3]. Many traps can be generated using diffraction with acousto-optic modulators(AOMs) or spatial light modulators (SLMs), providing a highly scalable and configurable platform

4
without the need for hardware changes [27, 28].
With this bottom-up approach, the first Rydberg atom two-qubit gate was demonstrated in2010 [29,30]. Since then, optical tweezers single-atom arrays have become one of the fastest-growing
quantum computing platforms in terms of fidelity and scalability [31, 32].

1.2

Cryogenic optical tweezer array
Currently, in neutral atom quantum systems, qubit fidelity is reaching the point where quan-tum error correction is possible [31, 33]. To realize a useful fault-tolerant universal quantum computer, a large number of qubits is required. For example, with the currently achievable error rate,
2-qubit gate fidelity, and error-correcting scheme, factoring a 2048-bit RSA integer requires approximately 107 qubits [34]. Several approaches are possible to achieve qubit number scaling anderror correction. For instance, coherent transport of atoms between different functional zones [35],
mixed-species operation with spectator qubits [36, 37], and a network of small modular quantum
processors connected via coherent optical links [38,39]. As the size of the quantum system scales tohundreds or even thousands of qubits for quantum error correction, extremely weak or infrequent
single-particle effects become increasingly relevant and problematic.
One challenge is to achieve better isolation of the atoms from the vacuum environment. With
the shallow trap depth of optical tweezers compared to the trapped ions, the probability of losing an
trapped atom after collision with residual background gas is almost unity. Considering a vacuumlimited lifetime τvac for a trapped single atom, the probability of losing an atom over certain time t ise−t/τvac . For typical state-of-the-art experiments at room temperature, τvac typically ranges from a
few minutesminutes [40–42] to recent advancements extending up to around 20 minutes with multiround bake-out, chamber inside coated with fresh titanium, and no extra components enablingattainment of XHV pressure [43]. This time is enhanced as the number of qubits increases. Given
N qubits, the probability of losing one atom after some time t now becomes e−N t/τvac [Fig. 1.1(a)].
Furthermore, as computation protocols become more complex and approaches rely on coherent
transport and rearranging of the atoms, atom arrangement and quantum error correction sequences

5

a

b

Figure 1.1: Examples of impact of a cryogenic environment in our apparatus (a) Probability ofdefect-free array versus qubit number in optical tweezer array under different vacuum lifetimes.
Assuming no imperfection, and the array assembly time to be tassembly = 50 + 0.85N (ms) basedon Ref. [27]. Orange is vacuum-limited lifetime of typical room temperature experiment, green is
cryogenic experiment. Showing an order of magnitude improvement in defect-free probability for
an array size of 1000 atoms. (b) 87 Rb Rydberg atom total decay lifetime (Γtotal = Γnatural + ΓBBR )versus BBR environment temperature. A factor of 3 increase in lifetime is possible by reducing theenvironment temperature from 300 K to 30 K for the 70S state. Black symbol is natural lifetime(Γnatural ) of the Rydberg state versus principle quantum number (left axis). The color symbols areratio of BBR induced total Rydberg lifetime (Γtotal ) to the Rydberg stat natural lifetime (Γnatural )(right axis). The Rydberg state natural lifetime (Γnatural ) is the intrinsic lifetime of the state in
the absence of any external decay mechanisms.

can take up to several hundred milliseconds to seconds, which is many orders of magnitude longer
than the simple two-qubit operation [29]. For quantum error correction and quantum simulation,
the entangled system can scale to hundreds of atoms [35, 44, 45]. The loss of trapped atoms due tobackground gas collision during the sequence time is no longer negligible. Therefore, achieving along single-atom lifetime is of paramount importance to achieve large-scale quantum computation
with tweezer-trapped atoms.
Another challenge is the reduction of Rydberg state lifetime due to room temperature blackbody radiation (BBR). There are two types of transitions that can be driven by the BBR, one
is microwave transition to the adjacent Rydberg state and another one is infrared transitions tothe states near ground state. In general the decay out of the particular Rydberg state is mainly
through the microwave transition to the nearby Rydberg state. We will discuss this in more de-6tail in Sec. 2.2.8, showing the Rydberg decay contributions from different states at various BBR
temperatures.
This effect is particularly significant in circular Rydberg states where the natural lifetime,in the absence of BBR, can reach seconds but is reduced by orders of magnitude at room temperature [46–48]. This is because each photon only carries one angular momentum, therefore, decay
to the ground state is prohibited and the only decay channel out of these high angular momentum
state is to other circular states of nearby Rydberg levels that is driven by the microwave BBR [49].
In fact, there are experiments that successfully trapped circular Rydberg state atoms in a cryostat
and observe the effect of BBR on the circular state lifetime [50]. To fully leverage the lifetimeof these circular states, a cryogenic environment with extensive microwave background control istypically required. This is beyond the design capability of the cryogenic tweezer array platform we
described in this thesis.
Even in states typically used in current high-fidelity Rydberg gates, which we utilize in thisthesis, thermal photons from BBR shorten the Rydberg state lifetime by a few factors. For example
a factor of 2-3 (from 400 to 150 µs) in the 60S-80S states of 87 Rb [51], impairing Rydberg gate
fidelity during qubit operation [52]. We see that around 2/3 of the decay is induced by the room
temperature BBR and the rest from natural decay to the ground state [Fig. 1.1(b)].
Furthermore, for quantum simulation with large Rydberg ensembles, dephasing caused bydecay to nearby Rydberg states is detrimental [53, 54]. These unwanted Rydberg states introduce
erroneous dipole-dipole interaction, causing avalanching decoherence triggered by the microwave
BBR [55,56]. In quantum computation, decay to nearby Rydberg states is also problematic because
it is generally difficult to detect these errors as these states remain dark during detection similar
to the target Rydberg state, but with incorrect interaction [57]. For these cases, trapping at a 30
K environment is sufficient to suppress these decay to a negligible level (5 times shorter than thenatural lifetime of the state).
Combining optical tweezers and a cryogenic environment can solve these aforementioned
problems. Cryogenic shielding limits the BBR background and low outgassing. In addition, cryop-

7umping at 4 K can freeze out almost all residual gas other than helium. However, until recently,
there were no cryogenic adaptations of optical tweezers and quantum gas microscope-type experiments due to the stringent optical alignment and flatness required. A notable example from CNRS
leverages cryogenic temperatures to achieve a 6,000-second lifetime in optical tweezers [58, 59],
albeit with limited optical access and optical lens performance.
In this thesis, we introduce a new apparatus capable of high-NA optical access in a low
vibration cryogenic environment. Our cryogenic Rb optical tweezer array apparatus incorporates
an extensive set of components for quantum state manipulation in a configuration with high opticalaccess for many atomic addressing and trapping beams.
For ground state hyperfine qubit manipulation, we integrate a microwave antenna inside thecryogenic shield to drive microwave qubit transitions at rates up to tens of kHz. Additionally, ourapparatus includes a Raman laser to drive qubit transitions at rates of several hundreds of kHz forfaster operation and to minimize decoherence.
Moreover, in terms of high optical access, this apparatus distinguishes itself from the setup in
Ref. [58,59] by employing an in-vacuum objective mounted at room temperature, focusing throughthe cold window on the custom cryogenic shields. This configuration creates a diffraction-limited
field of view (FoV) of ±225 µm at 0.55 NA, allowing for the creation of large arrays with uniform
loading and detection.
The cryogenic shield, maintained at temperatures below 45, K, provides some BBR shielding
and an environment with low outgassing. This temperature is primarily chosen for non-circularstate Rydberg atoms, where the natural lifetime is short and does not require such a low BBR
environment for decay suppression. For circular Rydberg states, a lower temperature BBR environment is needed to fully leverage the long natural lifetime of such states [50]. In addition to a cold
temperature, to fully attenuate microwave radiation relevant to Rydberg transitions, ITO coatingis also required. Although this is not utilized in the presented first iteration of our experiment, ourdesign is fully compatible with having ITO coating on the windows.
With simply a 4 K cryopumping coldfinger, we have been able to measure a vacuum lifetime8of 3000 seconds, one of the longest observed in optical tweezer systems [40–42]. Furthermore, thelarge optical access of our apparatus makes it compatible with a 3D cross-beam bi-chromatic opticallattices that can simultaneously trap both ground and Rydberg state atoms. Looking ahead, thisis a viable option for generating several thousand tightly confined trapping potentials in a laserpower-efficient manner.

1.3

Survey of cryogenic used in other quantum system
Cryogenic apparati have a long standing history in atomic, molecular and optical (AMO)

community and in superconducting circuits. Here I survey a few systems for comparative information.
Superconducting qubits are nonlinear quantum oscillators that use circuit elements madeof superconducting materials to minimize dissipation. To introduce nonlinearity and address individual energy levels, Josephson junctions are used. The inductance of the Josephson junction is
proportional to the current in the resonator [60]. The energy levels of these oscillators typicallylie in the GHz range. To access the ground state, they must be cooled to temperatures of tensof millikelvin. Therefore, it is necessary to place them in a dilution refrigerator to suppress any
thermal photons that could easily excite the energy levels of these resonators and destroy theircoherence. For comparison, thermal photons from a 4 K environment peak at 400 GHz, which is
sufficient to break Cooper pairs, whereas thermal photons at 50 mK, although peaking at 5 GHz
which is close to the qubit operation frequency, are 1010 times smaller in intensity. Placing thesequbits at the lowest available cryogenic temperature has enabled long qubit coherence of up to 1
ms [61] and high-fidelity two-qubit gate operations with errors as low as 6 × 10−3 [62].
Ion trap use individually trapped ions as qubits. Ions possess many favorable qualities,
including optical and microwave transitions with long coherence times. They can be laser cooled,trapped in large quantities, and individually addressed and configured. They were among the earliest systems to demonstrate quantum state control and entanglement [6]. Recently, ion traps havebeen designed in chip form as a means to shuttle individual ions around and scale up the number of

9qubits. However, problems arise when ions are close to the chip surface (tens to hundreds of µm).
Impurities on the surface of the trap, thermal voltage and current noise (Johnson-Nyquist noise),and patch charges can generate electric field noise that modifies the trapping potential and causes
motional state heating. Because ion trap quantum computing often uses these motional states forentanglement or requires the ions to be near the motional ground state during gate operations [63],heating reduces fidelity. It has been shown that placing the ion traps in a cryogenic environment
can decrease motional heating by several orders of magnitude [63–65]. In fact, some of the best
single qubit and Bell state fidelity has been achieved in cryogenic ion traps [66, 67]. Cryogenic
environments enable a longer period of time before ions are hit by a background gas molecule,
reduce thermally activated surface electric field noise that causes motional state heating [68], andalso allow for rapid trap sample exchange.

1.3.1

Horizon of impact on neutral atoms and molecules

The technique developed in this thesis opens up a pathway to achieve high-quality opticalaccess and robust optical alignment of beams required to trap and cool neutral particles whilecycling from 400 K down to cryogenic temperatures. Moreover, the closed-cycle flow cryostat,vibration decoupling, novel thermal isolation design, and interferometric testing conducted on oursystem demonstrate its compatibility with optical lattice and cavity-type experiments, such as thosein quantum gas microscopes and cavity optical lattice clocks. This sets our technique apart fromthe apparatus demonstrated in Ref. [58]. Adapting the technique we developed in this thesis will
enable the construction of a new generation of systems capable of leveraging BBR suppression,
reduced surface electric field noise (thermally induced) [63], and enhancing vacuum lifetime offered
by a cryogenic environment without significant compromises in optical performance (Fig. 1.2).
This new generation of cryogenic systems can find applications in many neutral atom experiments, such as quantum metrology and trapped molecules. In quantum metrology, BBR-induced
frequency shifts have long been recognized as a substantial hurdle to overcome in the pursuit ofhigh-accuracy atomic clocks [69]. A few minutes of room temperature vacuum lifetime can result in10
substantial atom loss during the interrogation of typical optical clock transition lifetimes exceeding
100 seconds (e.g., 160 seconds for 87 Sr) [41], leading to a reduced signal-to-noise ratio and requiring
more cycles to achieve the required precision. Further, in recent molecular quantum computing
proposals [70], room temperature BBR is relevant for driving transitions among ro-vibrational qubit
subspaces.

Figure 1.2: Cryogenic tweezer arrays provide a way to reduce noise and loss associated with thesurrounding environment. The benefits of a high optical access system may find applications inmetrology (image from Ref. [41]), hybrid quantum systems, quantum information processing (image
from Ref. [71]) and quantum simulations (image from Ref. [45]).

1.4

Metasurface lens optical tweezer
In the second part of the thesis, I describe a new approach to manipulate and control trapped

single atoms in optical tweezers through the use of optical metasurfaces. In the previous section,
we introduced the importance of scaling the quantum system for quantum information processing
applications. As more quantum systems are implemented with cryogenic technologies, maintaining

11high-resolution optical control through conventional multi-element objective lenses may become
problematic if these lenses need to be placed at cryogenic temperatures. One upcoming challenge
in the neutral atom array community is developing scalable and multi-functional optical componentsthat enable site-selective manipulation of hyperfine states and Rydberg excitations while operatingin constrained environments with low scattering and cross-talk.
In ion trap experiments, limited optical access in cryogenic chip-scale ion traps has propelled along-standing efforts in developing integrated optical components. In these cryogenic chip ion traps,
fidelity is limited by optical alignment and vibration due to the presence of multi-layer radiation
shields. It has been demonstrated that scalable planar photonics integrated with the ion trap
chip can efficiently deliver light in a cryogenic environment. Eliminating the need to align beams
into the cryostat has enabled improved robustness, parallelism, and addressing capabilities [72–74]. Neutral atoms can take on a similar trajectory and have many unique requirements. For
example, control of single neutral atoms relies heavily on optical potentials for trapping, either inlattices or arrays of optical tweezers. Development of active components, from acousto-optic devices
to spatial light modulators, are important for moving and addressing individual atoms [75, 76].
Static components that reduce reliance on large conventional optics for trapping and focusing will
also alleviate constraints in increasingly complex vacuum chambers, such as the cryogenic system
mentioned above, and improve scalability.. Advancing these specialized optical systems will benefit
from connection to the forefront of integrated photonics development.
Metasurfaces are planar photonic elements composed of a periodic array of subwavelength
dielectric or metallic nanostructures that have made significant impact on photonic systems inrecent years. Contrary to traditional optical elements that rely on refraction for phase shift, thenanostructures constituting a metasurface couple, resonantly or off-resonantly, and re-radiate theincoming light with a transformed phase, polarization, and amplitude determined by the nanostructure shape, size, and material composition [77]. Figure 1.3(a) shows incident wavefront modified by
the metasurface and focus to a point. Electromagnetic modeling, device optimization and fabrication of nanostructures with unprecedented complexity and resolution have enabled multi-functional

12

Figure 1.3: Main: Illustration of optical tweezer potential (orange) focused by the metasurface
lens (blue). The concentric pattern is created by the nanopillars arranged in a phase pattern similar
to a Fresnel lens. (a) A planar wavefront incident onto the metasurface lens. The light re-emitted
by the individual nanopilar, with different phase, interfere at a distance and create a focused point.
The nanopillars are made of silicon less than a micrometer tall, on top of a fused silica substrate.
(c) Single Rb87 trapped by the optical tweezer potential created by the metasurface lens at its
focus. Credit: Sean Kelley/NIST

control of the optical wavefront [77, 78]. By spatially varying the constituent nanopillar width in apattern similar to a Fresnel lens the re-radiated light can converge at the far field to form a focal
point, i. e. create a metasurface lens, shown in Fig. 1.3.
One major distinction between metasurface and conventional computer controlled holograms,
for example, liquid-crystal spatial light modulator (LC-SLM) or digital micromirror devices (DMDs)
is the gradient of wavefront control. LC-SLMs and DMDs have been used in combination with conventional high-numerical-aperture (high-NA) lenses in atom-array experiments to create arbitraryreconfigurable potentials through spatial modification of the optical wavefront using pixels larger
than the optical wavelength. Metasurfaces in contrast consist of deep-subwavelength pillars andoperate based upon a waveguide effect that provides large bend angles that can be used for highNA optics and aggressive wavefront shaping. To tailor the wavefront, the pillars have a controlled
size, spacing and shape, which further enables capabilities such as polarization [79], wavelength
multiplexing [80] and large FoV focusing [81].
In atomic physics, metasurface optics are intriguing to explore given their combination of high13
performance, multifunctional response, and low form-factor. Initial explorations in metasurfaces
for atomic experiments have been used to create beamshaping and splitting elements in magnetooptical traps [82, 83].
In this thesis, we open up the use of metasurfaces to optical dipole traps, in the form of tightlyfocused optical tweezers. We successfully trap single atoms with sub-wavelength confinement andsimultaneously image the atoms using a single high-NA (0.55) metasurface lens. The FoV of thelens is also investigated, and the measurements are compared to numerical simulations with good
agreement. To achieve the sub-wavelength trapping, a much more stringent focusing efficiencies
and optical phase control is required. The demonstration of successful trapping an array of single
atoms is an indication that potential deleterious effects of metasurface optics, for example, scattered
light, the presence of undeflected zero-order light, or deformations due to absorption and heating of
the lens makes negligible contributions to the trapping performance of large-spacing tweezers. This
exemplary demonstration in this thesis, the first in single neutral atom array community, opens up
new avenues of control and creating large mm-scale array of trapped single atoms.

1.5

Thesis outline
Chapter 2 we discusses the design details and considerations for constructing the high opticalaccess radiation shield, which serves as the core of the cryogenic optical tweezer apparatus. We thenpresent the mechanical and thermal testing of the cryostat and the high optical access cryogenic
shield, including vibration measurements, optical alignment over temperature, and thermal load
characterization.
In Chapter 3, we present atomic measurements of single atoms within the apparatus, demonstrating the capabilities of creating long-lived atom arrays. We also test the optical performanceby measuring the FoV and NA of the tweezer with atoms. Furthermore, we show that with thelong vacuum lifetime, one can measure and characterize cooling and imaging loss of single atoms
to a level that was previously inaccessible due to vacuum loss. We demonstrate an imaging loss
rate of 3.8(4) × 10−4 per 14 ms image with an imaging infidelity of 10−6 . Additionally, we report a14cooling loss of 5(4) × 10−5 per 25 ms cooling pulse, among one of the lowest ever reported in alkali
atoms [43]. Finally, we show that our apparatus is compatible with the LGM enhanced loading
technique previously developed in our group [84], achieving array loading efficiency over 80% acrossthe FoV of the objective.
Chapter 4 presents single qubit rotation using the microwave antenna and an understandingof the Rabi frequency from the measurement based upon microwave field simulation. We alsomeasure magnetic field stability and eddy current decay using microwave spectroscopy.
In Chapter 5, we describe the trapping of single atom array by combining a high-NA metasurface lens with tunable acousto-optic deflectors and characterize the tweezer foci with the trapped
atoms. We measure the trap size (NA) of the tweezer generated by the metalens and compare it
to the theoretical value. Additionally, we optically characterize the metalens with the pinhole andcompare the results to the atom measurements, finding good agreement. Finally, I present somepotential future metalens designs with qualities beneficial for creating large neutral atom tweezer
arrays, including a large FoV of up to 2 mm, a lens without chromatic aberration, and polarization
multiplexed lenses for dynamically changing the trapping potential or achromatic imaging of thetrapped atoms.

15

1.6

List of publications
For reference, I am including the list of publications which I have contributed to.

(1) 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.” Physical Review
X 9(1), 011057 (2019).
(2) 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.” PRX Quantum 3(3), 030316(2022).
(3) C. Kiehl, D. Wagner, T-W. Hsu, S. Knappe, C. A. Regal, and T. Thiele. “Coherence ofRabi oscillations with spin exchange.” Physical Review Research 5(1), L012002 (2023).
(4) A high optical access cryogenic tweezer array with a 3000 second lifetime (in preparation)

Chapter 2

Cryogenic tweezer array apparatus

2.1

Overview
In recent years, techniques developed have allowed us to trap atoms with light for long periodsof time, and the ability to drive coherent excitation to the Rydberg state has enabled on-demandinteractions. The rapid progression of our ability to control and manipulate individual atoms hasreached a point where the environmental isolation achievable at room temperature may become alimiting factor. Looking ahead, several challenges in neutral atom quantum systems could benefit
from placing atoms in a cryogenic environment. The main challenges include achieving better
vacuum isolation and shielding from room temperature BBR.
To achieve a large-scale quantum computer with hundreds or even thousands of atomic qubits,a long-lived array is required. Typically, state-of-the-art room temperature optical tweezers experiments have vacuum lifetimes on the order of 100-300 seconds [41, 42]. This is insufficient when
considering that rearranging a large atom array can take hundreds of milliseconds. For instance, tomaintain an array of 1000 atoms for 300 ms with 90% survival, a vacuum lifetime of 3000 seconds
is necessary.
To achieve sufficient isolation, a vacuum environment in the extremely high vacuum (XHV)
regime around 10−12 torr is necessary (we will discuss this in more detail in Sec. 2.2.4). Recently, an
array with a vacuum lifetime of approximately 1200 seconds has been demonstrated using a room
temperature setup [43]. However, the materials allowed in the vacuum chamber are very restricted.
While achieving XHV at room temperature is challenging, it is routinely accomplished in cryo-17
genic experiments, such as those in high-energy physics [85] and ion trapping [86]. Consequently,
placing the atom array into a cryogenic environment is beneficial to further increase isolation frombackground gas collisions.
Another important challenge for neutral atom array experiments is shielding from the room
temperature BBR environment. In quantum metrology, it is well-known that BBR backgroundfrom room temperature can affect optical clock transitions, thereby introducing systematic errors
in measurements [69]. In applications of quantum simulation and information utilizing Rydberg
atoms, controlling the BBR environment is crucial. The typical transition energy between one Rydberg state to another falls within the range of 10-100 GHz, and room temperature BBR contains
significant energies in this spectral range. This can cause decoherence by driving atoms to other
Rydberg states and creating unwanted interactions. In quantum information, such decoherence introduces errors in gate operations [57]. In quantum simulation, for example, studying many-bodydynamics with Rydberg dressed ensembles [53,54] and with resonantly driven Rydberg systems [87],researchers have observed that room temperature BBR microwaves can induce dephasing of individual Rydberg atoms and spontaneous avalanche dephasing of the entire ensemble, triggered by
microwave BBR [55, 56].
Therefore, the natural progression to achieve better environment isolation is to place theatoms in a cryogenic environment. In this chapter, I describe the design and testing of a newcryogenic optical tweezer array apparatus. First, I discuss the design considerations of various
aspects of the apparatus that are essential for the Rydberg atom quantum computing platform,
including magnetic field control, vacuum environment considerations, and atom delivery. Then, I
will show the design and simulations for the DC electric field and microwave environment essential
for Rydberg and ground-state qubit control. Next, I will present the testing of the microscope
objective and the design of the optical tweezer system. Finally, I will present performance testsof various parts of the apparatus, including cryostat vibrations, thermal loads and cooling power,
tweezer optical alignments, and cryogenic filters for the DC electric field and microwave control.

18

2.2

Design of the cryogenic tweezer array apparatus
The use of cryogenics has a long-standing history in the atomic, molecular, and optical (AMO)

experimental community. Some early demonstrations in neutral atoms [88] and ion trapping [86]have shown improvements in vacuum lifetime in a cryogenic environment compared to room temperature. In more recent years, there have been various efforts to implement cryogenic apparatus
in ion trapping [68], circular Rydberg atoms [50,89], and Bose-Einstein condensation [90] to reduce
two-level noise, couple atoms to superconducting cavities/resonators, and achieve better vacuum.
However, due to the stringent optical alignment and flatness required, there were no cryogenic
adaptations of optical tweezers and quantum gas microscope-type experiments until recently. The
most notable example of combining high-NA optical tweezers with cryogenics is an experiment at
CNRS leveraging 4 K cryogenic shielding to achieve a 6000-second lifetime in an array of single
trapped atoms [58], albeit with limited optical access due to layered radiation shields similar to aconventional cryostat design.
In this work, we present a cryogenic solution that takes a different approach compared tothe traditional layered radiation shield design. Here, we construct a custom radiation shield thatmaintains the required optical alignment and flatness while still being able to attach to a standard
cryostat interface. This approach is fully compatible with many technologies already developed
for room temperature AMO experiments, such as in-vacuum objectives, and can be adapted todifferent types of AMO experiments that require a large amount of high-quality optical access with
only slight modifications, such as quantum gas microscopes, optical tweezers, optical lattices, andhigh-finesse cavities.

2.2.1

Major parts of the experimental apparatus

The cryogenic tweezer apparatus can be separated into four major sections: atom source,science chamber, vacuum pumps, and the cryostat interface. The overall schematic CAD model ofthe experimental apparatus is shown in Fig. 2.1. The major components and a brief description

19
are listed as follows.
Source MOT chamber: The source MOT chamber, discussed in Sec. 2.2.6, contains the Rb
atom source from an alkali metal dispenser in a glass cell [yellow part in Fig. 2.1], pre-cooledin the transverse direction to form a collimated beam. The glass cell is connected to anintermediate pumping stage through a pinhole to limit the vacuum conductance. The entire
assembly is connected to the main chamber with a gate valve so that it can be isolated
when the main chamber is vented. The source MOT assembly includes an optics package
to provide all the necessary cooling lasers and can be moved to align the atom beam withthe 3D MOT region inside the coldbox. The details of 3D MOT is in Sec. 2.2.2.
Science chamber: The main science chamber [depicted in gray in Fig. 2.1] CAD model, is astandard Kimball Physics chamber (MCF800-SphSq-G2E4C4A16) with viewports locatedin appropriate locations for the MOT beams, enabling high-NA optical access for the opticaltweezer. The design for the window mounting for high-NA optical access is in Sec. 2.2.10.
The main chamber contains the custom high optical access radiation shield (the coldbox),
in-vacuum objectives (details in Sec. 2.3.1), and various electrical connections for thermometry, microwave (discussed in Sec. 2.2.8), and DC electric field electrodes (discussed
in Sec. 2.2.3).
Ion pumps and NEG: The science chamber is pumped by three pumps: a 25 l/s ion pump, a
SAES NexTorr Z200 200 l/s NEG + 5 l/s ion pump, and a SAES CapaciTorr Z200 200 l/s
NEG. The pumps are green in Fig. 2.1 CAD model. The NEGs efficiently pump reactive
gases, such as water and hydrogen, which are dominant in an unbaked UHV system. The
non-reactive gases, such as CO and noble gases (e.g., helium, argon, etc.), are pumped by
the ion pump. The details of the vacuum design is discussed in Sec. 2.2.4
Cryostat interface: The UHV cryostat comes with a 4.5” CF flange, which provides the mechanical mounting, stinger helium flow interface, and electrical connection for the heaterand sensor in the cryostat. The cryostat connects to the science chamber through a flexible

20
vacuum bellow for small alignment adjustments and a reducer to interface with the 2.75”
CF port on the science chamber. The orange-colored part in Fig. 2.1 is the Stinger UHV
cryostat assembly. The details for the Stinger cryostat is described in Appendix E.

2D Source MOT

MOT coil assembly

Gate valve

3 l/s ionpump

25 l/s ionpump
Science Chamber

Flexible coupling

MOT beams

SAES
NexTorr Z200

Stinger UHV Cryostat

MOT beams
Objective(in vacuum)

SAES CapaciTorr
Z200

0

10

20 cm

Figure 2.1: Overall view of the cryogenic Rydberg tweezer array apperatus. Yellow: 2D source
MOT providing a pre-cooled source of Rb atoms. This part connects to the main chamber with agate valve (MDC part number: 302000). Orange: Part of the chamber that contains the cryostat
and interface with the Stinger cryocooler. Green: pumps for the main chamber, include the 25
l/s ion pump, SAES NexTorr Z200 combo pump and SAES CapaciTorr Z200. Gray: Main sciencechamber housing the custom high optical access cryogenic shield called ”the coldbox” and thein-vacuum objectives.

21
2.2.2

Science MOT

For optical tweezer experiments, optical and magnetic field access for creating a MOT isessential. A MOT is the starting stage for all neutral atom trapping experiments, as it pre-cools
and traps the atomic vapor from room temperature to a more manageable temperature around100-200 µK. A MOT requires a quadrupole magnetic field with a gradient on the order of 10 G/cm.
One of the most common approaches to obtaining such a gradient in a cryogenic environment is
to use superconducting coils, as described in Ref. [50, 58, 90]. This is because the layered radiation
shields commonly used in cryostats prevent the coil from being placed close to the atoms outside of
vacuum. With the finite cooling power of the cryostat, joule heating from placing regular conductors
to generate such a magnetic field gradient inside the cryostat is not permitted. However, withour custom large optical access radiation shield, the distance between the radiation shield andthe atmosphere can be as short as 35 mm. This allows us to place coils in an anti-Helmholtz
configuration, 75 mm apart, which enables us to generate a 10 G/cm gradient with a modest 2.5 A
current through the coils. Another important aspect is the location of the MOT beams. Generally,
three pairs are required, and they need to have projections onto three orthogonal axes, which puts
constraints on the numerical aperture (NA) of the optical tweezer, as shown in [Fig. 2.2(b)]. The
location of the MOT beams is largely determined by the small viewports on the science vacuum
chamber.
The cross-section of the science chamber reveals the cold box and the MOT coils positioned
from the bottom of the recessed viewport within the groove surrounding the glass window, placed
as close to the cold box as possible [Fig. 2.2(a)]. The MOT coil is affixed to aluminum heatsinks,
which also serve as support structures for mounting the coil onto the viewport. The magnetic fieldcalculation generated from the MOT coils in the anti-Helmholtz configuration, with a total of 92
turns per coil, yields a fairly uniform 4 G/cm/A gradient [Fig. 2.2(c)]. In situations requiring astrong bias field in the vertical direction, the polarity of one of the coils can be reversed to operate
in Helmholz configuration, working alongside the external Z-bias coils (along the gravity direction)

22
to generate an additional magnetic field of around 10 G/A, as calculated in Fig. 2.2(d).

ac

MOT coil heatsink

75 mm

70 mm

d
b
Tweezer
0.55 NA

Figure 2.2: (a) Cross section showing the location of MOT coils with respect to the cold box. (b)Horizontal cut through the center of the tweezer and coldbox showing MOT beam positions withrespect to the optical tweezer. (c) The calculated magnetic field gradient generated from the MOT
coils in anti-Helmholtz configuration, giving a 4 G/cm/A gradient. (d) Magnetic field created by
the MOT coil in Helmholtz configuration showing a 10 G/A bias field in vertical direction. Thiscan be use in conjunction with the bias coil to create a larger field.

23
2.2.3

DC electric field control

Rydberg-mediated two-qubit gates and quantum simulations utilize long-range dipole-dipole
interactions between adjacent Rydberg atoms. Depending on the principal quantum number ofthe Rydberg state, the interaction between Rydberg atoms can extend up to tens of micrometers.
However, the large electric-dipole moments of the Rydberg states also make these atomic statessusceptible to external perturbations. Depending on the exact state prepared and the angularmomentum of such states, they can be sensitive to DC electric or magnetic fields. For example,
in the case of rubidium 70S1/2 state, the polarizability is 0.6 GHz/(V/cm)2 , while 70P3/2 and70D5/2 have polarizabilities more than 4 times larger than the 70S1/2 state. The polarizability alsostrongly depends on the principal quantum number; the larger the principal quantum number, thegreater the DC Stark polarizability. Plotting the DC Stark polarizability as a function of principal
quantum number (n) reveals that it approximately follows an n7 power law, as shown in Fig. 2.3.
This means that controlling the local electric field at the atom location is necessary as we go tohigher n for larger interaction.

Figure 2.3: 87 Rb Rydberg DC Stark polarizability vs. principle quantum number showing thepolarizability approximately follows a n7 power law with respect to the principle quantum number.
This polarizability allows us to estimate the required electric field stability in our setup.

24
Because the DC Stark polarizability in Rydberg atoms is induced by the external electricfield, it is proportional to the strength of the external field. Thus, the energy shift is second-orderproportional to the electric field, as can be seen in the unit of the y-axis in Fig. 2.3. To determine
the level of electric field control required for our experiment, we can perform a simple estimation
based on the Rabi frequency of the Rydberg transition that we wish to drive. Considering a Rabi
oscillation with a period of 1 µs, the Rabi frequency would be Ω = 2π × 1 MHz. While the intrinsic
lifetime-limited linewidth of the Rydberg transition can be on the kHz scale, typical Rabi frequencies
in the MHz range result in power-broadened spectral widths. We can compare the induced energy
shift by the external electric field (E) and the Rabi frequency using the polarizability α = 0.6
GHz/(V/cm)2 for the 70S1/2 state with the following equation:∆UStark1Ω≈= αE 2
ℏ
2

r
=⇒

E≈

2Ω
=
α

r

2π
≈ 0.1 V/cm
600

(2.1)

From the equation above, we can observe that an external electric field drift on the order of 0.1
V/cm will shift the resonance by 1 MHz, which is sufficient to prevent excitation to the Rydberg
level.
There have also been observations where the stray electric field from patch charges accumulated on dielectric surfaces near the Rydberg atom causes drifts in the transition frequency [91,92].
This effects the fidelity of ground-to-Rydberg excitation because over the course of few minutes
the laser becomes detuned and require frequent calibration. In Refs. [91, 92], the closest dielectricsurface is on the order of 4 mm from the atoms, and the effect of patch charges became a problem
without external compensation. In our design, we choose to keep the closest surface, typically forthe tweezer optical access, 8 mm away from the atoms. The longer distance is intended to minimize
the effect of patch charges, especially because these patch charges may be more difficult to eliminate
in a cryogenic environment. With active electric field control in our platform, one of the goal is
to compare the slow drift of the electric field at the atom location using the Rydberg transition
when the coldbox is at room temperature versus at 45 K. With this comparison, we will be able

25
to answer some open questions regarding the impact of charge accumulation on low-temperature
dielectric surfaces.
We can estimate how much patch charge (qp ) will create an electric field (Ep ) with the same
magnitude from Eq. 2.1 using simple Coulomb’s Law with r = 8 mm.

Ep =

qp
= 0.1 V/cm
4πε0 r2=⇒

qp ≈ 106 e(2.2)

Furthermore, due to the second-order sensitivity, the fluctuation of ∆UStark causing by the
E-field fluctuations (dE) approaches zero when the DC E-field (Edc ) is zero. We can see this bytaking the derivative of Ω.
dΩ= αEdcdE(2.3)

Thus, by minimizing Edc , the effect of dE (fast fluctuating E-field noise from the environment) in
Rabi frequency is minimized, and the coherence is improved given the same rms E-field fluctuation.
To compensate for the slowly drifting E-field from patch charges, as well as fields created fromstray voltage sources, to near zero requires active DC electric field control near the trapped atoms.
There have been several approaches tested by other groups [92–95] that incorporate in-vacuum
electrodes in Rydberg atom experiments to compensate residual electric fields. These approaches,using octopole designs aimed at achieving individual electric field control in the x, y, and z axes
with sub-percent inter-axis coupling [92], enable compensation of residual fields to be less than 1
mV/cm [94]. However, they require bulky metal ring structures that are situated further awayfrom the atoms [Fig. 2.4(a,b)].
To achieve active E-field control in our apparatus, we have chosen an approach with metal
traces on alumina PCBs. Alumina is a well-characterized substrate that is UHV compatible. We
get the PCBs from CERcuits, a Belgian company specialized in ceramic PCBs. The metal traces are
directly deposited onto the alumina substrate with virtually no outgassing. This method allows us
to produce different electrode designs relatively easily, with excellent dimensional accuracy, without
altering the way they are mounted on the apparatus.

26

a

b

Figure 2.4: (a) Electrode example from Ref. [92, 93] using small sections of concentric metal structure to achieve active electric field control with % level inter-axis coupling. The different voltage
configuration at the bottom shows how an electric field in a particular direction can be generated by
combining voltages in different segments. (b) Electrode examples from Ref. [94], to compensate theelectric field with good homogeneity and very low residual level to less than 1 mV/cm for precision
measurement of the Rydberg excitation spectrum.

Initially, the pattern took on a design similar to Fig. 2.4(b), with sectioned rings centered on
the tweezer cold windows to create an octopole design. Although this design gives good crosstalk
performance between axes, we found that it is hard for us to achieve a homogeneous E-field over
the whole field of view (FoV) of the tweezer, most likely due to the large FoV (approximately ±0.25mm) compared to the size and separation of the ring sections (approximately 8 mm). After someiteration, we found that it is better to have straight electrodes forming the same octopole geometry
rather than curved electrodes, the design shown in Fig. 2.5(a). The electrodes are silver conductive
traces on 0.5 mm thick alumina substrate, the length of the electrode pattern is 13 mm, and they
are separated 15 mm apart arranged as a square. The substrate has a 9 mm hole cut at the centerfor the tweezer light to pass through. The relative position of the two alumina substrates andelectrodes with respect to the atoms and tweezers is shown in Fig. 2.5(b). The substrates are 19

27
mm apart from each other, with the trapped atoms located in between (8.5 mm from the atoms).
This geometry ensures that the alumina substrate, which is dielectric, is located further away than
the tweezer windows, thus not introducing stray charges closer to the atoms.

a

b

13 mm

13 mm

y

15 mm
z

8.5 mm19 mm

Figure 2.5: (a) DC electric field electrode design based on bar electrode octo-pole to independently
control the electric field in x,y and z direction based on providing voltages on different pairs of electrodes. (b) Cross sectional cut of the coldbox, showing the relative positions of the two electrodes
substrates surrounding the atoms. The electrode substrates are 8.5 mm from the atoms, 19 mmapart from each other, attached to the side of the coldbox with metal springs (not showing).

To verify the design of the bar-octopole electrode, we set up an simulation with the 3Dmodel of the coldbox and electrodes into COMSOL and applied up to ±100 volts on the electrodes
with configuration shown in Fig. 2.6(a). Then, a series of cuts on the tweezer plane was made,and the homogeneity as well as the cross-coupling between the x, y, and z axes were analyzed.
The resulting electric field vector on the tweezer plane (y-z plane) shows a large region with good
uniformity [Fig. 2.6(b)]. To more precisely determine the uniformity and cross coupling of the fields
a series of vertical (x=0, y, z=0) and horizontal (x=0, y=0, z) cuts on the x=0 plane [Fig. 2.6(b)]are made, shown in Fig. 2.6(c). Zoom in on portions of Fig. 2.6(c) shows that the bar electrodedesign can achieve cross-coupling between axes of less than 0.2% [Fig.2.6(d)], and electric field
uniformity of better than 0.5% [Fig.2.6(e)].

28

a

0V
0V

b

+100 Vy+100 Vzx-100 V-100 V

0V

0V
Tweezer plane = y-z planecdE-Field along tweezer (y-z) plane cut (+/- 100 V on electrode)eFigure 2.6: (a) Electrode positions relative to the atoms. The voltages set up for the simulation
is marked next to the individual electrode. A voltage of ±100 V is applied to the two opposing
electrodes to create an electric field in the z-direction. (b) Plotting the generated electric field in zdirection across the tweezer plane (y-z plane), using the voltages applied in (a). (c) Line cut across
tweezer plane (y-z plane) along y and z axis, showing the cross section profile of the electric fields
in vertical and horizontal direction. (d,e) zoom in portion of (c) showing the E-field uniformity (e)and cross coupling to other axis (d).

29
2.2.4

VacuumVacuum is one of the most important aspects of this apparatus, as one of the goals is toachieve very low atom loss due to background gas collisions. In this apparatus, the vacuum levelacross different parts of the system spans ranges from high-vacuum (< 10−6 torr) to extreme-ultrahigh vacuum (XHV) (< 10−12 torr). It is not a trivial task to maintain vacuum levels across a rangeof 106 . To further complicate matters, the outgassing sources at different stages of vacuum alsoneed to be considered. Having a good understanding of the material properties and characteristics
under different temperatures is crucial to achieving this level of vacuum at the location of theatoms.
In order to determine the required vacuum environment for tweezers with lifetime of thousands of seconds, we need some estimations based on previously published results on atom losses due
to background gas collisions. A typical rule of thumb from MOT measurements at room temperature is a lifetime of 1 second corresponds 10−8 torr, and it’s inversely proportional to the pressure.
A pressure to MOT loss rate coefficient of P = 2 × 10−8 Torr s/τ is measured with a Rb MOT at
a trap depth of 1 K [96]. With this conversion, we estimate a pressure of 5 × 10−12 Torr for a loss
rate of 1 hour−1 , which was observed in a cryogenic MOT with a 4 K environment [88]. However,there are two caveats: typical MOTs exhibit lower loss rates than tweezers due to larger trapping
volume, capture velocity, and Doppler cooling from the MOT beams. Also, due to the large portionof residual gas being hydrogen, the ion pump current reading is not an accurate indication of thereal pressure in the chamber and can vary by a factor of 2 to 5 [97]. This inaccuracy arises from thedifferent mass-to-charge ratio of hydrogen compared to nitrogen, which is typically what is used
for ion pump pressure conversion.
To illustrate the effect of residual gas colliding with the Rb atoms we consider the hydrogenmolecules, which are the dominant residual gas in typical UHV systems, colliding with Rb atoms [58,98, 99], the equipartition theorem gives the following velocity:

30

1
3
mH2 VH22 = kB T
2
2

s
=⇒

VH2 =

3kB T
≃ 1900 m/s @ T = 300 K
mH2

(2.4)

Now if the Rb atoms are initially trapped in a MOT, effectively at rest, the resulting velocity ofhead-on collision is
V87 Rb =2mH2 VH2≃ 80 m/s @ T = 300 K
mH2 + m87 Rb(2.5)

This is only a factor of 3 larger than the typical MOT capturing velocity, taking into considerationthat most collisions occur at grazing angles and MOT trap depths are generally deeper (∼ 1 K)
compared to the tweezer (∼ 1 mK) by a factor of 103 . In addition, the local pressure at the atoms
is strongly dependent on the geometry of the vacuum chamber compared to the pressure measured
at the pump.
Therefore, it is common to observe tweezer lifetimes that are shorter than MOT lifetimes,
and both can be shorter than the lifetime corresponding to the pressure measured at the pumps[92, 100, 101]. To achieve a vacuum lifetime similar to that in Ref. [88], our goal is to create avacuum environment with a pressure below 5 × 10−12 Torr. This pressure is also consistent with
the inferred pressure from another cryogenic tweezer apparatus in Ref. [58] and P = 6.2 × 10−9Torr s/τ determined with Li MOT cold-atom vacuum standard described in Refs. [102, 103] withhydrogen as the predominant background pressure at room temperature.
Now that we have established the pressure on the source side of the chamber and the tweezer
side of the chamber, we can design the vacuum system accordingly. First we establish the schematic
design of the vacuum system including various vacuum conductance Ci , pressures of different regions
Pi , outgassing rates Qi , pumping speeds Si , and temperatures (Fig. 2.7). This allows us to better
determine the required pressure in the intermediate vacuum chambers.
Once we have identified the intermediate vacuum chamber pressure and expected outgassing
rate, illustrated in Fig. 2.7, we can start to determine the conductances C1 , C2 , and C3 . To
maximize differential pumping between the stages, we design these conductances to be as small as
possible. Using equations for conductance in the molecular flow regime, where the mean free path31

Tweezer
RegionSciencechamberIntermediate
source chamber

Atom
SourceT ~ 45 K
P1 = 10-12 torr
Q1 = small

T = 300 K
P2 ~ 10-10 torr
Q2 = unknown

T = 300 K
P3 ~ 10-8 torr
Q3 = unknown

T = 300 K
P4 = 10-7 torr
Q4 = large

C1

Cp1

C2

Cp2

Cp3

C3

Cp4

Sp1

Sp2

Sp3

Sp4

4K finger
Cryo pumping

NEG + ion

NEG + ion

Conductance

Figure 2.7: Schematic of cryogenic tweezer array vacuum systems including various outgassing rates
Qi , pumping speeds Si and expected pressure Pi of different region of the vacuum systems as wellas the conductance Ci between them. This provides a good guide for how different stages of systemshould be designed and the parameters required.

is much larger than the dimensions of the vacuum chamber, we arrive at the following numbers.
The equations for different geometries can be found in Refs. [98,99]. The units for lengths in Eq. 2.6
is in centimeters.
d3
0.23= 12.1= 0.02 ls−1 @ T = 300K −→ 0.006 ls−1 @ T = 30K
l
5
0.153
C2 = 12.1= 0.024 ls−1 @ T = 300K
1.5

C1 = 12.1(2.6)

C3 = 11.6 A = 11.6 π 0.032 = 0.05 ls−1 @ T = 300K
Now we can determine the required pumping speed (Spi ) and the pump conductance (Cpi ) in
conjunction with the expected outgassing rate (Qi ) within the given region in the system using
methods from Ref. [98, 104, 105].
We will now discuss the requirements and pumping design at different regions of the system,
starting from parts with the worst vacuum to the best vacuum. The atom source chamber is thepart that has the worst vacuum. Due to the continuous operation of the rubidium dispenser fromheating, there is significant outgassing from the decomposition of the chromate salts that generate
the rubidium vapor. The source glass cell acts as a rubidium vapor cell, with the typical rubidium

32ad = 2 mm, l = 50 mm

Gatevalve

3 l/s ion pump
Atom Source
glass cell

Intermediatechamber, P = 10-10 torr

Flexible coupling

Intermediatechamber, P = 10-8 torrcbTweezer region
P = 10-12 torr

45 K coldboxdd = 1.5 mm, l = 15 mm

d = 0.6 mm, l = 1 mm

SAES NexTorr 200 l/s NEG
45 K cold shield
4 K cold tip
Cryo pumpcryo pump appendage
for outer chamber

Figure 2.8: Cryogenic tweezer apparatus vacuum system showing the source chamber, intermediatevacuum chambers (science and differential pumping chamber after the source cell) and the coldbox.
(a) The zoom in portion of 45 K differential pumping tube on the coldbox with the conductance
C1 < 0.02 l/s. (b) The differential pumping tube between two intermediate chamber with theconductance C2 = 0.024 l/s. (c) The differential pumping pinhole separating the rubidium 2D
MOT source vapor cell from the rest of the chamber with conductance C3 = 0.05 l/s. (d) The
cross section of the science chamber showing the 4 K cryopumping cold finger and appendages.
As well as the NEG pumps (in green) partially protrude into the chamber to maximize pumping
capacity. In vacuum objective and the coldbox are also visible.

vapor pressure near room temperature being on the order of 10−7 torr [106]. There are no active
pumps inside the source glass cell other than the pinhole with conductance C3 ; since the majority

33
of the partial pressure is from rubidium, it does not affect the operation of the source cell. The
dimensions for C3 are 0.6 mm in diameter and 1 mm in length [Fig. 2.8(c)].
The next stage is the intermediate pumping chamber, which is pumped by a 3 l/s ion pump
and a 25 l/s NEG. In order to maintain a pressure in the order of 10−8 to 10−9 torr, a differential
pumping ratio of 100 is needed. With the value of C3 being 0.05 l/s and outgassing Q3 in theintermediate chamber being minimal, the total pumping speed Sp3 = 28 l/s easily satisfies the ratioof 100.
The differential pumping in the final chamber just before the tweezer region is more complicated due to the large number of items inside this volume. This chamber contains several plastic
parts, including PEEK thermal stand-offs, Kapton-coated wires, stainless rigid coaxial cables, microwave connectors, and Rogers PCB (Teflon). Fully characterizing the outgassing properties ofthis chamber is very difficult; only estimations can be made. Since most of the components contributing to outgassing are UHV plastics (data from Refs. [107–109]) we can estimate the types ofresidual gases to be expected from unbaked plastics in an UHV system, which are mostly H2 , H2 O,
CH4 , CO, and CO2 .
NEG pumps offer several advantages over other types of pumps, the most important ones
being high pumping speed in a small form factor and no magnets, so stray magnetic fields are
eliminated. Therefore, we choose to use multiple NEGs that partially protrude inside the chamber
to achieve high conductance and pump reactive gases efficiently. However, ion pumps are stillneeded for pumping non-reactive gases such as helium, argon, and CH4 . Three main pumps (green
in Fig. 2.1) connecting to the main chamber through the 2.75” CF ports. A small tube with
conductance C2 [Fig.2.8(b)] separates the two intermediate chambers. The dimensions for C2 are
1.5 mm in diameter and 15 mm long, with a conductance of 0.024 l/s. The NEGs are mounted insuch that the pumping elements partially protrude into the main chamber to maximize pumping
capacity [Fig. 2.8(d)]. The total pumping speed Sp2 = 200 × 2 + 25 = 425 l/s. With this pumpconfiguration, we measure a pressure in the low 10−10 torr at room temperature and low 10−11 torr
when the apparatus is cold.

34
Finally, we determine the requirements for the tweezer region. We opt for an aggressive
differential pumping tube on the cold box, as shown in Fig. 2.8(a). The dimensions for C1 are 2
mm in diameter and 50 mm long, with a conductance of 0.02 l/s at 300 K. However, since the tube√
is at 45 K, the conductance scales as T, reducing to 0.006 l/s at 45 K.
Next, we estimate the effective cryopumping from the cold surfaces inside the cold box. While
the precise pumping speed is challenging to calculate, we estimate the conductance from the trapped
atoms to the 4 K cold finger to be in the order of 25 l/s. Typical cryopumping capacity on metal
surfaces is in the order of 50 l/s/cm2 [110–113]. Since we want the conductance to be the limiting
factor, we design our cold finger with slits so the surface area is in the order of 50 cm2 , resulting
in an effective pumping speed of hundreds of l/s, as depicted by the gray appendages surroundedby the 45 K shield in Fig. 2.8(d). For a more detailed discussion about the effect of temperature
on pressure of different gasses, temperature vs gasses desorb from surfaces and cryopumping see
Appendix D.

35
2.2.5

Coldbox and cryopumping finger design concept

The logic of design of the coldbox and the cryopumping coldfinger is the following. Since
we know the outgassing rate of a surface is significatly reduced at low temperature, we aim tocreate a low temperature vacuum shield (coldbox) that has limited ability to cryopump but does
not create additional adverse effect and is very clean in terms of outgassing properties. For thiscold vacuum shield to be effective, the conductance from inside to the outside of the shield islimited. Then we place a 4 K cryopumping cold finger inside the coldbox with optimized surface
area and conductance to cryopump on residual gasses. The primary gas load in a system that isalready UHV at room temperature is mainly hydrogen which we rely mostly on cryosorption at 4
K to pump out. We choose metal for the cryopumping cold finger for several reasons, one is thevolume that this cryopump act on is very small so we do not expect large gas load to saturate thesurface quickly, assuming nightly regeneration of the cryopumping surface by warming up to 20
K. Additionally, the limited conductance between the coldfinger to the auxiliary ion pumps makes
regeneration longer. Having porous materials such as activated charcoal will generate significant
gas load and long regeneration time. With this configuration, we estimate a factor of 200 differential
pumping ratio and ultimate pressure on the order of 10−12 torr. This estimation is based on the
conductance (0.05 l/s) to the outside of the coldbox, cryopumping pressure curve from (Fig. D.2)
and the surface sorption ability given by (Fig. D.3) from Appendix D.
One observation we have made is that despite the coldbox being held at 45 K, which should
be sufficient to freeze out most of the gases other than hydrogen, helium ,and neon, the outgassing
behavior seems to be different after the coldbox is baked (using the build in heater) at 100 ◦ C
for a few days. We see almost and order or magnitude increase of single atom lifetime from∼ 400 s to ∼ 3000 s. One hypotheses is that, the gases absorbed on the surface of the coldbox
has an activation energy required for their release. Below this energy threshold, the surface act
as a reservoir, continuously releasing background gas even at low temperatures. After baking,these reservoirs are depleted and the outgassing is further reduced. However, the effect of this36
outgassing is not evident on the ionpump pressure reading on the outer vacuum chamber due tosmall conductance between inside of the coldbox and outer vacuum. After baking the coldbox wehave virtually no outgassing from the cryogenic shield at 45 K, with the differential pumping ratio
to be on the order of a few hundreds we expect the pressure in the tweezer region to be at least in
the range of 10−12 torr. This will provide a lifetime in the order of thousands of seconds for atoms
trapped in the tweezers.
One major difference between our experiment and many ion-trap experiments where theions are enclosed in a 4 K radiation shield is that our experiment has a simpler design and larger
optical access, but sacrificing the advantage of having large amount of 4 K cryopumping surface inline of sight from the atoms. Due to the low temperature of 4 K surfaces, residual gas moleculesthat collides with the wall are order of magnitude more probable of being trapped by the surface
comparing to the 45 K coldbox due to Van-der-Waal’s force and surface molecular potentials [110,113]. Even for the gases not being absorbed on the first collision, the temperature of gas moleculesthat bounces off are significantly reduced and have lower probability of ejecting the atoms/ionsfrom the trap. Therefore the experiments where the ions are trapped inside 4 K enclosures/boxes
typically have better vacuum 10−13 to 10−15 torr (inferred by using the ion in the trap), comparing
to the few 10−12 torr we achieve [68, 114].

37
2.2.6

2D MOT assembly and 2D to 3D MOT transfer

From Refs. [58, 59], we know that the oven + Zeeman slower approach as the atom source
is problematic due to line-of-sight collisions from hot rubidium vapor. To exacerbate matters,
the length of the Zeeman slower creates a rather collimated atom beam from the thermal source.
Without a gate valve, collisions will result in an effective lifetime on the order of 300 seconds evenwhen the vacuum environment is ideal [59].
To circumvent this issue, we decided to use a 2D MOT as our atom source to better control
the atom flux entering the science chamber. The advantage of using a 2D MOT with an alkali metal
dispenser instead of an oven is that there will be no line-of-sight between the rubidium source andthe tweezer array. In this section, we describe the construction of the 2D MOT, including thevacuum package, atom source, and optics assembly. This approach allows us to create a compact,pre-cooled source of rubidium that can be easily integrated into the main experiment and aligned
to the small differential pumping stages on the coldbox.
The vacuum package consists of a custom glass cell from ColdQuanta, with electrical feedthroughs on the side to supply current for the AMD (see Appendix B for dispenser details). On
the glass cell, there is a pinhole, as shown in Fig. 2.8(c), with a conductance of 0.05 l/s to limit
the rubidium vapor exiting the glass cell. After the pinhole, there is a 1” thick 2.75” CF adapter
with side ports, creating a small pumping volume for the NEGs and ion pump to be attached,as shown in Fig. 2.9(a). The custom NEG assembly contains two 12 l/s NEG pumping elements
welded onto the 1.33” CF electrical feedthrough [Fig. 2.9(b)]. The atomic beam passes through thespace between the two NEG pallets.
A set of 4 permanent magnet bar assemblies creates a 25 G/cm quadrupole magnetic field in
the transverse direction for the transverse cooling of the atomic vapor. The center of the field canbe translated by changing the position of the magnet assembly through a translation stage. Thisaligns the 2D MOT with the first differential pumping pinhole on the 2D MOT cell [Fig. 2.10(a)].
Another critical component of the 2D MOT assembly is the optics package. Unlike con-

38

aglass cellgate valveRb

3 l/s ion pumppinholeRb dispenserb

NEG 25 l/s
2D MOT Magnet

2D MOT optics package

flexible bellow

Figure 2.9: Cryogenic tweezer apparatus 2D source MOT vacuum assembly. The 2D MOT optics
package is offset from the vacuum chamber for clarity. The Assembly includes a set of 4 permanent
magnet rods generating a 25 G/cm 2D magnetic field gradient. A translation stage allows theposiiton of the zero field to be aligned to the differential pumping pinhole. A 3 l/s ion pump isconnected to the intermediate pumping chamber and a gate valve is used to isolate the assembly
when the science chamber vents to air. (a) Shows the zoom in portion of glass cell and intermediatechamber for differential pumping. Including the ion pump and custom NEG assembly with 25 l/s
pumping speed. (b) The picture of the custom NEG assembly consists of two 12 l/s NEG elements
welded on a 1.33” CF UHV electrical feedthough.

ventional approaches that utilize a large or elliptical laser beam [95, 115], which necessitate bulky
lenses for beam generation, we opted to split a small beam into several paths to create an elongated trapping region, as depicted in Fig. 2.10(b). This multistage method has been studied inRefs. [116, 117], showing similar atom flux compared to the use of single large beam. Since thebeams are wide and the alignment requirements are relatively relaxed, we chose to 3D print someof the optomechanics for beam distribution. The grey parts in Fig. 2.10(b) are 3D printed using
HP Multi Jet Fusion in Nylon PA 12, and the beam splitting optics are glued to the 3D printed

39
parts. The completed optics package is shown in Fig. 2.10(d). The only requirement for operationis two fiber-coupled light sources for the MOT beams and push beam. After cooling the Rb atoms
from the background vapor in the transverse direction (bright dot in the center of Fig. 2.10(a)), theatoms are propelled by a push beam with a velocity of 15-20 m/s into the science chamber throughall the differential pumping tubes.
To characterize the performance of the 2D MOT, we measure the atom number loaded inthe 3D MOT versus time, showing approximately 107 trapped atoms during typical operation, as
depicted in Fig. 2.10(c) orange trace. The atom flux is extracted through fitting the slope of theatom number verses time at t = 0. The dashed line represents the fitted atom flux inside the
coldbox at the tweezer region. With the push beam, we observe 4 × 106 atoms/s, and 3 × 105
atoms/s without the push beam but with the transverse cooling 2D MOT beams, which is morethan 10 times smaller. Essentially no atoms are observed if both beams are switched off.

40

a

30/70

b

50/50

mirrormirror

Rb dispenser
MOT fiber launcher

50/50 BS2D MOTcpush beam launcher

mirror + quarter waveplatedFigure 2.10: (a) Viewing from the push beam direction of the 2D MOT showing the pinhole and the
Rb dispensers. The 2D MOT, aligned to the pinhole, is the bright dot at the center of the image. (b)Cryogenic tweezer apparatus 2D source MOT optics package. Grey parts are 3D printed with HP
Multi Jet Fusion in Nylon PA 12. The MOT beams (red and purple) are split into 3 equal intensity
paths and retro-reflected. Push beam (yellow) launched from the back pushes the transversely
cooled atom out of the pinhole. (c) 3D MOT loading as a function of time showing 107 trapped
atoms. Dashed line are the loading rate with and without the push beam showing 10× differencein flux. (d) Photo of the completed 2D MOT optics package.

41
2.2.7

Magnetic field in 3D MOT (science) chamber

The DC magnetic field coils are necessary to cancel stray magnetic fields such as the Earth’s
magnetic field and fields created by magnetic objects nearby the vacuum chamber, such as opticalbreadboards and screws. The location of the coil is rather constrained, so there is really only
one configuration to place three pairs of coils for magnetic field control in all three dimensions[highlighted in blue in Fig. 2.11(a)]. However, the coil is still designed with the possibility ofgenerating up to a 100 Gauss bias field in mind. This is for a potential plan to prepare circular
Rydberg states in the cryostat, which require an electric to magnetic field crossover. Due to thespace constraint, the bias coils are not in a Helmholtz configuration, but the size of the coils islarge enough, therefore within the Field of View (FoV) of the objective, a reasonable magnetic field
homogeneity is maintained.
Due to the complicated mounting geometry of the nested coils, the coil mounts are 3D printed
with HP Multi Jet Fusion in glass-reinforced nylon for high-temperature resistance in case the coils
warm up during experimental operation. Because of the odd shape of the coil, we used COMSOL
to see if the coil produces a uniform field and also to obtain the coil current to field conversion. The
result shows that within a ±1 cm region near the center, the magnetic field uniformity is around
0.02%. Figure 2.11(b) is the simulated field of the y-axis for illustration, but the simulation was
done in all axes and showed similar uniformity. For both the x and y axes, we have 5.5 G/A. The
z-axis is designed to be used as a circular state bias coil in the future, so it has 13 G/A.

42

b

a
bias coilassembly

tweezeryxz

OP + Rydberg

Figure 2.11: (a) Blue highlight is the bias coil assembly. Containing rectangular coil with roundcorners for x-axis, elliptical shape coil for y-axis and circular coils for z-axis. (b) is the COMSOL
simulation for y axis bias coil at 1A current. Inset is zoom in of ±1 cm region near the center of thecoil showing 0.02% B-field uniformity. Simulations are also done on the x and z direction showing
similar results in terms of uniformity.

2.2.8

BBR microwave

One of the requirements for the qubit state is a long coherence time. In neutral atom
quantum computing platforms, it is common to use the mf = 0 ground hyperfine state (firstorder not sensitive to B-fields) as qubit states [35, 118]. These states typically exhibit very long
T1 (relaxation) times on the order of a few seconds or more, so the coherence is typically limited
by environmental stability such as magnetic fields and AC-Stark shifts created by other light.
With clever dynamical decoupling and magnetic field noise compensation, T2 coherence times ofseveral seconds are achievable, but typically the bare coherence (T2∗ ) is on the order of a few
milliseconds [32, 35].
The hyperfine level typically has energy splittings on the order of a few gigahertz. In the caseof 87 Rb, the clock transition between F=1 and F=2 is 6.835 GHz. It is difficult to efficiently couple
microwaves into the metal chamber and metal coldbox from outside of the vacuum; therefore, we
decided to put a resonant loop antenna inside the coldbox to drive this transition as a diagnostic
tool. Alternatively, a pair of Raman lasers can be used to drive this transition at a higher Rabi

43frequency. Having an antenna inside the coldbox allows us to place it only 15 mm away from theatom; however, this also creates a potential path for noisy room temperature BBR microwaves toenter the chamber.
Typically, a microwave source has broadband noise that is significantly higher than the room
temperature BBR. So connecting it directly to the antenna inside the coldbox can introduce photons
in the range of several tens of GHz. At n = 70, the adjacent Rydberg level spacing is 20 GHz;
the room temperature BBR can easily couple these levels and create decay channels to unwanted
quantum states.
To prevent unwanted room temperature BBR microwaves from entering the coldbox, a PCB
with a microwave bandpass filter is attached to the coldbox and thermalized to the coldbox temperature of 45 K. This rejects the room temperature BBR outside of the passband and replaces the
spectrum with 45 K microwave BBR.
To estimate the difference between room temperature and 45 K microwave spectra up to 100
GHz, which is within the transmission range of the coaxial cable, we calculate the BBR spectra ofperfect blackbody at these two temperatures [orange and blue in Fig. 2.12(a)]. Within this range,
we see a ratio of around 7 in power density [black trace in Fig. 2.12(a)]. To calculate the effect of
microwave bandpass filter we multiply the room temperature BBR spectrum with the filter transfer
function and add the 45 K BBR spectrum on top of it [black dash in Fig. 2.12(b)] is the resulting
spectrum. Comparing the residual transmitted BBR through the filter with 45 K BBR spectrum[orange trace in Fig. 2.12(b)] we can see that beyond the passband of 6-8 GHz of the filter, theroom temperature microwave BBR is attenuated to less than the BBR from the 45 K environment.
This ensures that the microwave the atoms see is limited by the 45 K BBR instead of the 300 K
microwave BBR coupled in from the coaxial cable.
BBR microwave impact on Rydberg atom lifetime
The reason the microwave environment needs careful consideration in our coldbox is that the coupling of the Rydberg level to the adjacent n ± 1 principal quantum numbers is highly sensitive tostray microwaves from BBR. In quantum computing and simulation applications, three main types

44

b

a

Figure 2.12: (a) Room temperature microwave specteum (orange) and 45 K microwave spectrum(blue) in the range up to 100 GHz which is within the frequency range of Rydberg energy spacing
between adjacent principle quantum number. The ratio between the BBR from 300 K and 45 K,
roughly constant in this frquency range, is found to be around 6. (b) Bandpass filter transmission characteristics (plotted in black dashed line) and the filtered 300 K BBR spectrum (orange)
comparing to the 45 K BBR spectrum showing a lower power density outside of the filter passband.

of decoherence channels affect the Rydberg level: natural (vacuum-induced) decay to the groundstate, BBR-induced photoionization of Rydberg atoms, and BBR microwave-induced transitions
out of the Rydberg state. Natural decay represents the fundamental limit of the Rydberg state
lifetime, which ideally we aim to achieve.
The effect of BBR photoionization due to environmental photons can be estimated using
calculations from Ref. [119]. The plot in Fig. 2.13(a) shows that even at room temperature, thephotoionization rate is much smaller than the natural decay rate of Rydberg states (typically inthe range of hundreds of microseconds for for n > 40). However, for circular Rydberg states withsignificantly longer natural lifetimes (10-100 milliseconds for n > 40), the BBR photoionization
rate, even at 30 K (∼ 10 s−1 ), becomes a dominant decay source [47, 119].
Next, we consider the impact of BBR microwave-induced transitions out of Rydberg states.
Plotting the decay rates for both natural and BBR microwave-induced processes for n = 70S1/2 , weobserve in Fig. 2.13(b) that at room temperature (300 K), the decay induced by BBR microwaves
is several times larger than natural decay (red data vs. blue data). At 30 K (green data), the decay

45
is primarily dominated by the natural decay rate [blue in Fig. 2.13(b)]. As previously discussed,
for circular Rydberg states with much longer natural lifetimes, decay out of the Rydberg state is
predominantly caused by BBR-induced transitions to adjacent Rydberg states, even at a microwave
BBR temperature of 11 K [47].
In our current coldbox design operating at 45 K, the effective microwave BBR shielding for
decay out of normal Rydberg states is estimated to be approximately 1/3 (in terms of BBR-induced
decay rate) of an ideal 45 K BBR environment. This is due to the large optical access we put on
the coldbox. The estimation is done by finding the total solid angle that is not covered by metal in
the design. In reality, this fraction may be larger due to the size of the openings is similar to themicrowave transition wavelength, potentially reducing the fraction of microwaves able to penetrate.
In the future, in order to increase room temperature microwave BBR shielding, the windows on
the coldbox can be replaced with ITO coated windows to form an Faraday cage.

a

Decay rate (s-1)

bPrinciple quantum number (n)Figure 2.13: (a) BBR photoionization rate (s−1 ) of rubidium nS Rydberg state as a function ofprinciple quantum number n (plot from Ref. [119]). For n > 20 at least an order of magnitude
reduction in ionization rate of the Rydberg state in 77 K microwave environment vs. in 300 K
microwave environment is observed. (b) Decay rate out of n = 70S1/2 Rydberg state to nP1/2,3/2
Rydberg states. Three different configurations are simulated: (blue) is natural Rydberg state(vacuum induced) lifetime, (red) is decay induced by BBR microwave photon at 300 K, and (green)is the decay rate induced by BBR microwave photon at 30 K. At room temperature the BBR
induced decay is several times higher than the natural decay rate, and at 30 K this rate is reduced
to a level that is several times shorter comparing to the natural Rydberg lifetime.

46
2.2.9

Thermal design and analysis

Several important aspects need to be considered when designing the thermal isolation andthermal link of the apparatus. One consideration is how the tweezer is implemented in the cryogenic
environment. Some approaches have been tried with success; in particular, in Refs. [58], they placedhigh-NA aspheric lenses in a cryogenic environment. With this approach, they could successfully
generate an array of optical tweezers. However, it is hard to obtain a large uniform array due tothe limitation of the field of view (FoV) inherent to the aspheric lens design. With the single lensapproach, compromises need to be made on the optical performance between 780 nm and 850 nm,
FoV, and size of the lens.
From early on, we set the design criteria of the apparatus to be as versatile as possible. We
wanted to develop a set of technologies that could be easily adapted to other AMO experiments
requiring large optical access with precise alignment requirements. To maximize the optical access,large windows are needed; however, the thermal conductivity of glass is low and the emissivity isnear unity. This means that at the center of the large window, due to absorption of room temperature BBR, the temperature rises drastically compared to the edge. From COMSOL simulations,
we found that with our window size of > 40 mm, a single-layer window needs to be on the orderof 1 cm thick to achieve a temperature gradient of 10 K between the edges and the center. To
circumvent this, we designed a double-pane window structure [an example shown in Fig. 2.14(b)],where we integrate a thermal break with a spacer between two relatively thin glasses. This allows
the majority of the room temperature BBR to be shielded by the outer window, while the inner
window stays cold. This allows a low thermal gradient on the inner window while keeping the total
thickness of the structure less than that of a thick glass window. Simulating the thin double-pane
window with radiative coupling to room temperature, we found that the design can tolerate thewindow to be as thin as 2 mm and still provide a very low thermal gradient on the inner window[Fig. 2.14(a)].
Conventional cryostats use thin-wall (<0.5 mm) stainless steel tubes forming a truss structure

47

a

Outer

b

Inner

glass

K

55 K

31.6 KAl
indium foil

Figure 2.14: (a) COMSOL double pane window simulation with window thickness of 2 mm. Left istemperature of the outer window showing a 25 K temperature gradient. Right is the temperatureof the inner window, which is the surface that the atom sees, only 1.6 K difference from the metalheatsink. (b) Example design of the double pane window that is used in the Montana cryostat testin Sec. 2.4.4. The indium gaskets shown in green act as thermal interface material between theglass and metal heatsink.

to connect between stages of different temperatures. This approach, while very effective in tension(load hanging from the top), is not very stable in compression, which is how our coldbox will
be supported. Furthermore, due to space constraints in the room temperature vacuum chamber,we cannot use enough tubes to form a structure that provides sufficient rigidity. To overcome thischallenge, we decided to use a thin-wall cone with a platform attached to the top for supporting the
coldbox. The advantage of this method is twofold: first, the cone shape naturally nests well with theshape of the recessed viewport, and second, the cone provides a more symmetric support that createsless stress and torsional movement when cooled to cryogenic temperatures. The material is grade 5
Ti6Al4V alloy, which has one of the lowest thermal conductivities in metals [120] and has enoughrigidity for structural applications. The cone consists of three parts laser welded together [a colorcoded 3D render of the assembly is shown in Fig. 2.15(a)]. The bottom flange is 2 mm thick, themiddle section is a 0.45 mm thin-wall cone with circular perforations to reduce thermal conductivity
further, and the top flange is 1.5 mm thick. The top platform, also 1.5 mm thick, is attached to theflange with titanium screws and PEEK spacers to further reduce thermal conductivity. Finally, thecoldbox is attached to the platform with PEEK spacers and titanium screws. The PEEK spacers

48
serve two purposes: they act as thermal breaks and provide some compliance to accommodate
slight thermal expansion coefficient mismatches, ensuring that stresses are manageable and screws
do not loosen after the assembly is cooled to cryogenic temperatures.
To verify that the thermal conductivity of the entire structure is sufficiently low, the entire
structure is simulated with the thermal conductivities of the materials set to their room temperature
values, which are normally larger compared to at cryogenic temperatures. The temperature at the
coldbox attachment point and the chamber anchor point is set to 30 K and 300 K, respectively.
This yields a heat conduction of 0.67 W flowing out through the support, which is an overestimate.

ab
Ti-6Al4V
PEEK

Coldbox clamping point

30K

~100K

~185K

300K

Figure 2.15: (a) The titanium thermal standoff. Blue colored parts are grade 5 titanium 6AL4V,
which has thermal conductivity of 7 W/m · K lowest in structral metals. The cone portion of thetitanium is 0.45 mm thick. The coldox is clamped in between two PEEK standoff on 4 corners(pointed by red arrows). (b) Thermal analysis of the titanium standoff with COMSOL. The base is
set to be 300 K and the coldbox clappling point is set to 30 K. The temperature of various location
is marked with arrows. The total heat conduct through the titanium structure is around 0.67 W.
Not taking into account the temperature dependence of thermal conductivity as well as assuming
perfect thermal contact between parts.

To find the required cooling power and thermal conductivity for the copper braid that connects the cryostat’s 30 K stage to the coldbox, we need to first estimate the amount of thermalload on the coldbox. The major heat sources and thermal load budget on the coldbox are listed asfollows (see Zhenpu Zheng’s thesis for detailed calculations).

49
Conduction from titanium support: Setting the boundary condition to be 30 K on thecold side and 300 K on the hot side, and not considering temperature-dependent thermal
conductivity, the estimated upper bound for the heat load is 0.67 W.
Radiative coupling of metal on the coldbox: The total surface area of the gold-plated metal
part is 250 cm2 . The emissivity of matte gold is taken to be 0.47, and for polished stainlesssteel is 0.075. This gives a radiative heat load of 0.8 W.
Radiative load on BK7 glass: The total surface area of the glass window is 52 cm2 . The
emissivity of glass is near unity in infrared wavelength longer than 4 µm. The room
temperature BBR spectrum peaks at 10 µm. Since the glass on the coldbox mostly faces
the glass viewport on the room temperature chamber, we assume both emissivities to be
1. This gives a load of 2.5 W.
Conductive heatload from stainless coaxial cable: The length of coax is 10 cm. The outerconductor is 304SS, inner conductor is BeCu. Using 30 K and 300 K as boundary condition,
we find the load to be 0.2 W.
Conductive heatload from DC wires: The DC electrode, coldbox heater, and temperature
sensor are all connected with kapton-insulated phosphor bronze wire. There are a total of
15 wires, each with a diameter of 0.2 mm and a length of 20 cm. The total heat load is
0.045 W.
The total estimated heat load is 4.2 W. This is below the specified cooling capacity of the radiation
shield region of the cryostat which is 5 W @ 50 K. Now we have the number, we can estimate
the temperature gradient across the copper braid. The copper braid is a standard P6-506 style
from Technology Applications Inc. with braid length of 25 mm. This gives an estimated thermal
conductance of 1.1 W/K @ 40 K. This is the largest copper braid we can fit in the space available.
This gives a temperature difference of 4 K across the braid, which is a reasonable number.

50
2.2.10

Optical access

Since the window is now part of a high-NA optical system, the glass needs to maintain goodflatness and mechanical alignment with respect to the vacuum chamber. Therefore, we need tocarefully control the stresses on the glass when the coldbox is cooled to cryogenic temperatures. The
difference in thermal expansion coefficient between copper and glass is quite significant; copper has
a CTE of 16 ppm/K, while BK-7 glass has a CTE of 7 ppm/K. This amount of CTE mismatch cancreate stresses in the glass or, worse, cause mechanical failure. Typically, cryostats use compliant
mounting such as springs to press the glass against the cold shield. This results in poor thermal
contact between the glass and the metal surface, which is typically not an issue if there are multiple
layers of radiation shield and only a small portion of the opening is glass. However, in our coldbox,
about 30% of the surface area is glass, so a more efficient thermal link between the glass andmetal is needed. This requires a mounting method that provides better thermal contact and bettermechanical stability. Here, we choose to use indium foil as a thermal interface material due toit’s ductility at cryogenic temperature, however this limits the baking temperature of the coldbox
to 120 ◦ C. To process these foils into rings, we use specially made ’cookie cutters’ and IPA as alubricant to prevent the foil from sticking [as shown in Fig. 2.16(a)]. Indium remains soft even at
cryogenic temperatures, which can absorb the difference in CTE between glass and metal. The
glass essentially ’floats’ between two pieces of indium foil. Through testing, we found that withrelatively thin foils (0.2 - 0.3 mm) and a large number of screws to create even pressure, indium canmaintain good mechanical stability and thermal contact while absorbing the difference in CTE. A
completed double-pane window assembly is shown in [Fig. 2.16(b)].
With the efficient thermal contact between glass and metal part of the coldbox as well as
the double pane window design, we put a large amount if optical ports on the cold box [shown in
Fig. 2.17] is the available optical access on the coldbox. In total we have
Top/bottom: Two 0.57 NA (37 ◦ ) symmetric on z-axis
Side: Two 0.25 NA (15 ◦ ) symmetric on y-axis

a

b

51

Figure 2.16: (a) Top: The cookie cutter used to cut the indium seal rings from 0.2 mm thickindium foils. Bottom: the cutted indium foil rings, methanol or IPA are used as a lubricat toprevent the indium from sticking to the cutter. (b) Photo of one of the completed double pane
window structure, showing good thermal contact (flattening) between the indium seals and glass.
The two layers of glass are visible in the photo.

Side tweezer: Two 0.55 NA, <0.1 mrad relative parallelism alignment error and λ/5 flatness.
MOT ports: 7 dedicated ports at 20 ◦ angle for MOT beams and probe beams.

52

Window substrate used in cryogenic Rydberg tweezer apperatus.
ManufacturePart #

Material

Coating

Size

Flatness

Edmund Optics

49634

BK-7

VIS-NIR(400-1000
nm)

35x35x3 mm

λ/10

Edmund Optics

37018

BK-7

VIS-NIR(400-1000nm)

50ϕx3 mm

λ/10

Edmund Optics

83468

BK-7

NIR I (6001050 nm)

12.5ϕx3 mm

λ/4

Edmund Optics

83471

BK-7

NIR I (6001050 nm)

20ϕx3 mm

λ/10

Edmund Optics

37014

BK-7

VIS-NIR(400-1000nm)

50ϕx2 mm

λ/4

Edmund Optics

37018

BK-7

VIS-NIR(400-1000nm)

50ϕx3 mm

λ/4

Edmund Optics

15436

BK-7

VIS-NIR(400-1000nm)

25.4ϕx2 mm

λ/10

Window thickness tolerance are listed to be ±0.2 mm, but they are typicallywithin ±0.05 mm and very consistent between windows (< 10 µm difference)from batch production
Table 2.1: List of window substrate used in cryogenic Rydberg tweezer experiment

53

microwave
antenna
OP + Ry 420 nm

objective

MOT beam

4 K cold finger
electrodes
MOT beam<50 K
coldbox

Ry: 1013 nm
Ti pedestal

107 87Rb
15 µm

MOT beam

Figure 2.17: (a) Cross sectional cut of the CAD remder of the core of the cryogenic tweezer
experiment, showing the large optical access and high-NA tweezer access from the side of the
coldbox.

54

2.3

Microscope
The most important part of achieving single atom trapping and manipulation is the highnumerical aperture objective. These objectives, when performing optimally, can generate optical
dipole traps that provide sub-wavelength confinement of atoms. We refer to these microscopic
dipole traps as optical tweezers. The optical dipole trap uses far-detuned light from both the 87 Rb
D1 and D2 transitions (795 nm and 780 nm, respectively). When the light is red-detuned, a dipole
moment that is in-phase with the electric field of the light is induced, thus lowering the groundstate energy (AC-Stark shift). This creates a trapping potential that is proportional to the lightintensity. When the trap is sufficiently tight (typically when the trap Gaussian waist is smaller
than 2 µm), the trapped atoms become close enough to interact and form bound states. In thepresence of near-resonant cooling light (MOT/PGC light), the atoms acquire energy from the lightand can either both escape or remain in the dipole trap [121]. The result is either one or none of
the atoms left in the trap, depending on the even or odd parity of the initially loaded trap. We callthis process light-assisted collisions.
In our system, we use 830 nm light to create the optical tweezers and detection on the 87 Rb
D2 transition at 780 nm. Both the tweezers and atom fluorescence pass through the high-NA
objective. Ideally, the objective should be diffraction-limited at both wavelengths at the same focal
plane. This sets many of the requirements for our optical systems.

2.3.1

High NA objective

There are generally two categories of high-NA objectives: infinite and finite conjugate. In a
finite conjugate design, the objective takes the point source on the front focal plane and re-focuses
it on the back focal plane. This is common in many microscope designs because only one lens isrequired to project the image to the eye or camera. In an infinite conjugate system, the objective
takes the point source at the front focal plane and generates a collimated beam out the back, andvice versa. In our application, we use an infinite conjugate objective design because of our need55
to place many optical elements before the light enters the objective. It is generally bad practice
to focus light through plano-optical elements (waveplates, filters, PBS, etc.) since this creates
additional aberration.
Our objective is custom-designed and made by Special Optics. It has an effective focal length(EFL) of 30 mm and is achromatic from 700 to 1100 nm. It has a designed numerical aperture(NA) of 0.55 and a working distance of 3 mm for BK-7 + 11 mm in vacuum. The objective iscompensated to focus through the 3 mm window. The FoV of the objective is ±200 µm around its
optical axis. The objective is designed to be UHV compatible; therefore, the housing is made oftitanium with vent slots, and it is non-magnetic due to its proximity to the atoms. Additionally,
the objective is designed to withstand temperatures up to 150 ◦ C if we wish to bake the chamber.

2.3.2

Tweezer light source and optical rail

The light source we use to trap single atoms comes from a MOGLabs MOAL 830 nm tapered
amplifier. The TA is seeded by an IPS Photonics VBG external cavity diode laser. The light iscoupled to the tweezer optical rail through a Coastal Connections angled-flat end-capped fiber forhigh power handling and to reduce unwanted interference inside the fiber. The flat termination isrequired to launch light at the tweezer rail to avoid aberration. In this configuration, we have 1.2
W of optical power available at the tweezer rail [Fig. 2.18(a)].
The AOD tweezer array is generated by a 2-axis AOD with a 6 mm active aperture from AA
Optoelectronics (DTSXY-400-850.930). After the deflected array exits the AOD, it passes througha 1:1 relay lens and folding mirrors. Then it enters the final telescope with a magnification of 7. This
creates a beam with a 16 mm Gaussian waist to fill the input aperture of the objective [Fig. 2.18(b)].
The relay and telescope lenses in the rail are carefully chosen and simulated in ZEMAX to ensure
diffraction limited performance of the optical tweezer. The 1:1 relay is important since it creates
extra space between the AOD and the final telescope for PBS and waveplates, which are used forcleaning the polarization and sampling the tweezer power for active intensity noise cancellation.
The intensity noise is crucial for maintaining a long trap lifetime. We will discuss the heating effect56of the intensity noise in Sec. 3.3.1.
To create a static potential with SLM working along side the AOD movable tweezers forrearranging. A system of optical relays, similar to the AOD tweezers, is constructed [Fig. 2.18(c)].
The two trapping lights are combined with a PBS before entering the objective. The SLM system isconstructed but not yet implemented in the experiment, so all the atom experiments in this thesis
are done using AOD tweezers. In the longer term, a static potential generated with optical lattices
will be implemented to create large amount of traps in a laser power efficient way.

57
IPS Laser
830 nmMonitor PD

Moglabs
830 nm TA

l/2
Isolator 35 dB

f=100 mm f=50 mm

Isolator 35 dB

cyl. tele.
4:1

To tweezer~1.2 W
To TA seed

l/2 PBS

Intensity
servo

Intensity
servo AOM

From IPS lasera. Tweezer source

From rail PD

Path saver

ck
feedba acamer

Thorlabs
AC-250-075

c. SLM tweezeryIntensit D
servo P

Thorlabs
5
AC128-02

m TA
n
From 830

lectronic
AA Optoes AOD
2-axi

l/2

PBS

s
Thorlab00
AC254-2 m
f=200 mVacuum chamber

to monitor camera

Objective

Thorlabs
AC508-600
f=600 mm

PBS

SLM
l/2

l/2

l/2

f=75 mm

ED 49395
f=350 mm

Coldbox

ED 49366
f=250 mm

ED 49366
f=250 mm
PBS

l/2

Intensity
servo PD

ED 49396
f=500 mm

PBS
f=50 mmThorlabs
ACA254-050
Path saver

b. AOD tweezer

MaranacameraFigure 2.18: (a)The schematic of the tweezer light source including the 830 nm IPS seed laser and
MOGLabs tapered amplifier. (b) Tweezer optical rail setup of the AOD tweezer path including
relay lenses and part number. The optical perfomance of the whole rail system is verified withZEMAX to guarantee operation at diffraction limit. (c) The rail schematic for SLM tweezer, also
verified with ZEMAX to guarantee to be at diffraction limit. The two sets of tweezers are combinedwith a large PBS before entering the objective.

58
2.3.3

Pinhole testing of the objective

The objective is the most important optical element of the cryogenic tweezer apparatus. Oncethe objective is inside the vacuum chamber, any aberration would be hard to measure and correct.
Therefore, careful testing needs to be performed before mounting it inside the vacuum chamber.
Historically, there are two different types of measurements to assess the performance of the lens.
One is the same type of pinhole measurement described in Sec. 5.5.4; the other is an interferometricmeasurement using a reference sphere at the focus of the objective, as described in [122]. If the lenspasses the PSF measurement test, the interferometric test is not necessary, as it is only required
if the optical phase of the aberration needs to be measured. We measure the PSF created by theobjective with wavelengths of 780 nm, 826 nm, and 1013 nm to verify the objective’s performanceover the specified wavelength range. In the test setup, we align the objective precisely to within
10 µrad with respect to the test tweezer window, with the setup shown in [Fig. 2.19(a)], and obtain
the pinhole image. The PSF image [an example shown in Fig. 2.19(b)] is fitted with Eq. 5.6, andthe magnification is calibrated using the same method described in Sec. 5.5.4. The extracted onaxis NA is between 0.52 and 0.53, which is sufficiently close to the specified value of 0.55 fromthe manufacturer. We also measure the PSF over the FoV range of ±200 µm by translating thepinhole [as shown in Fig. 2.19(c)]. We observe a slight degradation of NA to 0.5 beyond 150 µm,
which might be due to the FoV limit of the imaging lens that collimates the light onto the camera.
Similar to Sec. 5.5.4, we also test the focal length dependence of the imaging lens, which is identicalto the one used in the experiment. This allows us to know how much the imaging lens needs to betranslated with respect to different wavelengths.

59af = 150 mm

objective

pinholeilluminationf = 500 mm
350 nmPinholeb50/50 BS

Camera3 mm BK-7
front
illuminationcFigure 2.19: (a)Schematic diagram of Special Optics objective pinhole test setup, the pinhole isilluminated with a laser on the back, the front illumination is for initial search of the pinhole substrate. (b) Example pinhole image taken with the Special Optics objective @ 826 nm illumination.
The imaging lens is the same as the one we used in the experiment for atom imaging. (c) The
extracted NA from images similar to (b) over the FoV up to ±175 µm and wavelengths of 780 nm,
826 nm and 1013 nm. Showing NA ranging from 0.51 to 0.53. This is close to the specified NA of
0.55 from the manufacturer.

60

2.4

Cryostat testing
In this section, we describe various tests on the mechanical aspect of the cryogenic tweezer

apparatus. This includes vibration testing, heat load on the cryostat, alignment of the coldbox
window versus temperature, stability of the anti-reflection (AR) coating under cryogenic temperatures, and characteristics of various DC electrodes and microwave filters when cooled to cryogenic
temperatures.

2.4.1

AR coating testing

The glass windows on the coldbox are all AR coated to minimize unwanted reflections andmaximize the power balance on the retro-reflected beams. Due to the extreme temperature range
to which we subject our AR coating, choosing the correct types of AR coating is important. Typical
AR coating types include electron-beam/thermal evaporation, ion-assisted deposition, magnetron
sputtering, and ion-beam sputtering. We choose windows with a coating that is ion-beam sputtered.
Ion-beam sputtering creates a coating that is fully dense, with no water absorption and goodadhesion to the substrate. To prevent the need to have custom coatings made, we choose touse commercially available windows from Edmund Optics, Thorlabs, and Newport for windows
on the coldbox. These large commercial optics vendors generally have their coating process wellcontrolled compared to small batch custom jobs, so their coating generally is more consistent with
better adhesion. We found that Edmund Optics provides a large selection of substrate sizes andcoating types that are robust against thermal cycling with very minimal shift in spectral properties.
The thickness tolerance and flatness from Edmund Optics are generally very good [Fig. 2.20(a)]as well despite having much looser tolerance specified on their website. The thickness tolerenc iswithin 50 µm, and the wedge tolerence is within 0.1 mrad from specification most of the time.
To test these windows before cutting them into the required shape, we thermally stress themin liquid nitrogen at 77 K. The window transmission is measured first. Then the windows are
dipped in liquid nitrogen for a few minutes to thermalize, after removing from liquid nitrogen let

61
it warm up to room temperature in air. This process is repeated three times, and the substrate iswiped clean with solvents, and the coating is visually inspected for cracks and using the Scotch tapemethod [123] to check for adhesion. Finally, the reflection spectrum is measured again and they
are compared [Fig. 2.20(b)]. We see very minimal change in the spectral bandwidth and reflectivity
after thermal cycling.

62

a

b

Figure 2.20: (a) The flatness and wedge measurement of the cold window substrate with the ZYGO
fizeau interferometer. Showing better than λ/10 flatness @ 633 nm and wedge less than 0.05 mrad.
(b) Reflection spectrum of the cold window substrates before and after thermal stressing in 77 K
liquid nitrogen environment. The reflection spectrum remains the same and there is no visible
damage of the coating. The dashed lines are wavelength of interest for our apparatus including
Rydberg excitation (420 and 1013 nm), trapping (532, 830, 1030 and 1064 nm) and imaging (780
nm).

63
2.4.2

Vibration measurement

The Stinger cryostat provides a way of transferring the cooling power of the cryocooler
without coupling vibration to the tweezer experiment. In AMO experiments, low vibration isparticularly important due to the precise alignment of atoms and light required in most experiments.
The main effect of vibration is that it changes the optical phase between the laser and the atoms.
In many AMO experiments, lasers are used to prepare and excite the quantum states. In ourparticular example, a pair of blue (420 nm) and NIR (1013 nm) lasers are used to excite the atoms
to the Rydberg state. If the relative position between the atom and the surrounding optics changes
even by a fraction of a wavelength, the optical phase seen by the atom will be drastically different.
For a detailed discussion of how vibration introduce excess phase noise and comparison to othercryogenic ion trap experiments [124, 125], see Appendix F.
Another effect, which is much less of a concern, is excess vibration can create excitation in
the motion of the atom inside the trap due to the jitter of the trap with respect to gravity, similar toshaking a ball sitting in a bowl. This heating effect is more prominent in optical tweezer and opticallattice experiments due to the trapping potenetial typically being very shallow, ∼ 1 mK or less.
This makes them sensitive to vibration at lower frequencies and smaller amplitudes. Although, in
our design we are second-order sensitive to these vibrations because the objective is mechanicallyreferenced to the same CF flange as the coldbox.
To measure displacement well below one optical wavelength between the coldbox and theoptical table, it is impossible to measure it through mechanical means. Therefore, we add asmall reflector on the coldbox [Fig. 2.21(a)] and construct an interferometer between the coldbox
and a reference mirror on the optical table. Figure 2.21(b) shows how the interferometer is set
up with respect to the vacuum chamber and the coldbox. The green dashed box represents theinterferometer breadboard fixed on the optical table.
The PZT mirror in [Fig. 2.21(b)] is used to calibrate and adjust the interferometer so that theoutput at the photodiode sits at zero crossing. This is done to maintain the maximum sensitivity

64

a

b

PZT mirror

reﬂector for interferometer

Coldbox

PD
50/50
BS
laser source
780 nmcoldboxInterferometer

Objective
Vacuum chamber

Figure 2.21: (a) Photo of the small reflector attached to the side of the coldbox as part of the interferometer for vibration measurement. (b) Schematic diagram of the interferometer setup between
the coldbox reflector and the optical table. The interferometer assembly (green dashed box) is on
an optical breadboard attached to the optical table and launches light onto the reflector picturedin (a).

of the interferometer, as the slope of the intensity (I) changes as a function of displacement (l) as
follows:
I(l) = I0 (1 + cos(2

2π
l))
λ

−→

dI  π 
4πV= I0, units :
dl 2
λ
nm

(2.7)

This gives us the calibration between the recorded photodiode voltage and displacement.
Since our vibration is well within λ/4, we do not need to consider the phase wrapping of the interferometer signal. Scanning the PZT voltage [orange trace in Fig. 2.22(a)], a sinusoidal oscillation inphotodiode voltage is observed [blue trace in Fig. 2.22(a)]. Fitting the photodiode amplitude (I0 ),we can obtain a conversion factor of 50.2 nm/V. With the calibration, a time trace of vibration
amplitude can be extracted [Fig. 2.22(b)]. Without damping of the stinger flex line with lead shots,
the GM vibration excites mechanical resonances on the optical table every time the GM cycles [bluetrace in Fig. 2.22(b)] with excursions roughly every second. With damping of the flex line, thoseexcursions disappear [orange trace in Fig. 2.22(b)]. Taking the displacement time trace, a power
spectral density of the vibration can be calculated [Fig. 2.22(c)]. The three regions in [Fig. 2.22(c)]correspond to different mechanical responses of the apparatus, including the optical table. Region(i), between 1 - 10 Hz, is the excitation due to GM cycle, mostly concentrated around a few Hz

65range. Region (ii), between 10 - 200 Hz, comprises the mechanical resonances of the optical breadboards and optics mounts. The blue trace shows peaks where the resonances are excited by theGM when there is no lead shot damping. Region (iii), larger than 200 Hz, is where there is no
more mechanical vibration, and we are measuring the noise floor where the blue (no damping) andorange (damped) traces become identical.

a

c

b

GM cycle

Figure 2.22: (a) Photodiode voltage (orange) vs. piezo voltage (blue). Linearly scanning the piezo
voltage, result in a sinusoidal oscillation in the interferometer signal. This curve is used to obtainthe calibration factor for converting volts on the PD to nm. (b) The time trace of the photodiodevoltage. Blue is with GM on, we can see the mechanical excitation around every second causedby 1 Hz GM operating frequency. The orange trance is with damping on the stinger line withlead shots. Showing the vibration being largely eliminated. (c) The PSD of the time trace in (b).
Region (i) is low frequency vibration from the operation of the cryostat. (ii) is the mechanicalvibration of the structures on the optical table excited by the GM. (iii) is the residual vibrationfrom the building and the optical table.

Without damping of the flexline, the peak-to-peak displacement of the coldbox relative to theoptical table is in the order of 30 nm, which is in good agreement with interferometric measurements
done in Ref. [126]. With damping, the peak-to-peak displacement decreases to around 10 nm,
similar to the residual vibration from the background environment when the GM is off. From
the time traces in [Fig. 2.22(b)] we can calculate the RMS displacement. For the blue trace, it
is 4.8 nm, and for the orange trace, it is 2.1 nm. The undamped RMS vibration agrees very wellwith Ref. [126] and shows that with rigid mounting Stinger offers pretty low vibration in general.

66
These vibration values are significantly smaller than the atom wavefunction distribution, typicallyin the order of ∼ 80 nm. We deem this level of vibration sufficient to maintain the coherence of theincident laser light and mostly limited by residual vibration from the building and optical table.

2.4.3

Heat load measurement

In Sec. 2.2.9, we discuss the expected heat load on the cryostat. However, precise measurements are still needed to confirm that we are not overloading the cryostat and that the copper
thermal braid is sufficiently sized and the cryopumping cold finger is at a sufficiently low temperature to provide efficient cryopumping. To provide accurate monitoring of the temperature, we placetemperature sensors in a few critical places, including the 4 K stage of the cryostat, the radiation
shield stage of the cryostat before the copper thermal braid, and the far side of the coldbox tomeasure the upper bound of the coldbox temperature.
There are two heat load measurements we conducted with the cryostat. In the first measurement, we attach a heater directly to the radiation shield of the cryostat with the cryostat
disconnected from the rest of the system. This allows us to measure the exact cooling power onboth the radiation shield and the 4 K stage. The heater and temperature sensor can be seen on
the radiation shield [Fig. 2.23(a)]. The end of the cryostat is capped with aluminum foil to eliminate any BBR load on the 4 K stage. With heat load applied on the 4 K stage, temperatures
are recorded [Fig. 2.23(b)]. Between applied loads of 0-0.5 W, the temperature of the 4 K stage
remains fairly constant, showing a cooling capacity of >600 mW is available. Beyond that, we see
a temperature rise of 2.5 K/W on the 4 K stage. The shield temperature rises with 1.75 K/W,
which is expected since the returned helium gas from the 4 K stage cools the radiation shield. The
same measurement is done on the radiation shield [Fig. 2.23(c)]. We observe an 8.5 K/W capacity
on the radiation shield with a base temperature of 12 K. The thermal link between the 4 K stage
and the shield is 2 mK/K, which is negligible for our application.
The second measurement is conducted with the coldbox attached. We measure the temperature dependence versus additional heat load applied with the heater on the coldbox. With

67atemperature sensorbheaterc

Figure 2.23: (a) Locations of the temperature sensors and heater for heatload testing of the barecryostat (b) Temperature of the 4 K stage and the radiation shield vs. load applied on the on the
4K stage. Beyond the rated 600 mW cooling power it shows a 2.5 K/W cooling capacity on the 4
K stage. (c) Temperature of the 4 K stage and the radiation shield vs. load applied on the on theradiation shield. The radiation shield shows a cooling capacity of 8.5 K/W.

temperature sensors attached to different parts of the system (Fig. 2.24), we can record the temperature difference between the radiation shield and the coldbox (gradient across the copper braid).
Then, we can extrapolate the total BBR load on the coldbox as well as the thermal conductivity of the copper thermal braid by plotting temperature as a function of applied load (Fig. 2.25).
The green data is from the radiation shield, and the orange data represent the cold box. The
dashed line is a linear fit to the data, and the crossing point is where the thermal gradient crosses
the copper braid is zero (equal to BBR load on the coldbox). We extract the thermal conductivityof the copper braid to be 2.2 W/K and the BBR load to be 2 W. However, if we use the load versus
temperature curve in Fig. 2.23(c), we get a load of 3 W. So, it is safe to assume the BBR load
is between 2-3 W, which is fairly close to our estimation in sec. 2.2.9. The discrepancy between
the two methods might arise from the nonlinearity of the copper braid thermal conductivity withtemperature or the unknown load on the 4 K cold tip raising the base temperature of the radiation

68
Thermal braid

ColdboxTshield

Tcoldbox

Radia�on shieldS�nger cryostat

4 K cold ﬁnger

TﬁngerCryo pumping element

Ti thermal stand-oﬀ

Figure 2.24: (a)Schematic showing the titanium thermal standoff and temperature sensor locations
on the coldbox. There are three temperature sensors, one on the coldbox, one on the radiation
shield and one on the 4 K stage of the cryostat. Two heaters, one located on the 4 K stage of thecryostat and another one located on the coldbox.

shield.

Figure 2.25: Temperature from three temperature sensors (described in Fig.2.24) vs. the applied
heat load on the coldbox. This allows us to extract the thermal load on the coldbox and the thermal
conductivity of the copper thermal braid. The crossing between the green and orange dash line
is where the temperature gradient across the copper braid vanishes. This allows us to extract thethermal load on the coldbox and the thermal conductivity of the copper thermal braid.

69
2.4.4

Tweezer cold window alignment measurement

Once the coldbox is installed, we set up a system to monitor the angular change of the two
coldbox windows and the objective with respect to each other. Since there is currently only one
objective installed, a reference beam is set up on the side adjacent to the objective for monitoring
the alignment. This beam defines the optical axis of the coldbox windows as well as the AOD
tweezer system. This beam is also used as the reference for aligning the objective and the tweezer
rail.

a

b

T=300 K

T=45 Kalignment monitor setup
coldbox50/50 BSwindow 1
window 2
objec�ve
f = 150 mm
vacuum chamber
0.1 mradCCD camera

0.0
Angluar deviation (mrad)

objec�ve

incoherent
light

1.4

Time (h)

2.6

6.7

0.3

0.2

0.1
300

250

200
150
100
Coldbox temperature (K)50

Figure 2.26: (a) Schematic of alignment monitoring setup for coldbox tweezer window alignment.
An incoherent light is launched and reflect off both cold windows and the objective front facecreating the total of three reflections (inset). With a lens focus onto the CCD, the reflections
are converted to angular change. (b) The relative alignment of tweezer windows vs temperaturerecorded during the cryostat cooldown. Inset shows the image on the CCD camera showing thereflection from all three surfaces. This gives an alignment error of ∼0.3 mrad between the coldbox
and the objetive.

The monitoring setup [Fig. 2.26(a)] uses a collimated incoherent light source with a 10 nm
bandwidth centering at 840 nm to eliminate interference from different sides of the windows. The
10 nm bandwith is sufficiently small to minimize chromatic focal shift of the focusing lens. The
reflected light from the windows and the objective is picked off by a 50/50 beam spliter and send toa focusing lens with an EFL=150 mm. This focusing lens focuses the colimmated beam into tight70
spots while converting the input angular change into position change on the CCD camera.
The focused spots have 25 µm Gaussian waist and the pixel size of the CCD camera is
3.45 µm. With typical fit uncertainty to the centroid of the Gaussian beam around 1 pixel, we canachieve < 20 µrad angular resolution on this monitoring system.
This system is also used to align the objective to the cold window. The mount of the objective
consists of three parts, the base plate, the vertical flange mount and the threaded flange where theobjective is mounted on [Fig. 2.17]. These three pieces separates translation degree of freedom
from the angular degrees of freedom of the objective. The bolted connection between the baseplate and the vertical flange mount is first shimmed with copper shims to the correct angle using
the monitoring system to within 0.2 mrad. Then the objective together with the threaded flange
plate can be slided to center the objective to the reference beam using the transmitted light. Finally,
the bolts that connects the threaded flange plate to the vertical mount is tightened.
During cooldown, we record the angular change of the cold tweezer window [Fig. 2.26(b)].
The results show that after the initial relaxation from the first few cooldowns, the angular alignment
stays relatively stable in subsequent cooldowns. We can see that the most alignment error between
the window and the objective is 0.3 mrad.
To verify the angular tolerance between the window and the objective, we simulate the opticalsystem in ZEMAX. We set up the simulation with collimated light input to the objective and plot
the Strehl ratio over FoV of the light focused through the tweezer window. Changing the tilt of
the tweezer window up to 2 mrad we can see how the Strehl ratio changes as the window tilt angle
increases [Fig. 2.27]. From this we can conservatively obtain within 1.5 mrad tweezer window tilt,
the objective can remain diffraction limited over the design FoV. Which is much larger than the
angular alignment error of 0.3 mrad between the tweezer window and the objective.

71

0.9

0.9

0.8

0.7

0

diﬀrac�on limit

0.045 0.09

0.135 0.18

0.225

Field of view (mm)715 nm

0.8

0.7

0

1 mrad2 mrad1.0

Strehl ra�o

1.0Strehl ra�oStrehl ra�oOn-axis1.0

diﬀrac�on limit

0.045 0.09

0.135 0.18

0.225

Field of view (mm)780 nm852 nm

0.9

0.8

0.7

0

diﬀrac�on limit

0.045 0.09

0.135 0.18

0.225

Field of view (mm)1064 nm

Figure 2.27: ZEMAX OpticStudio simulation of the strehl ratio of Special Optics objective vs.
angular alignment to the cold window. On-axis, showing near perfect Strehl ratio of > 0.95 acrossdifferent wavelength. 1 mrad alignment error, 3 times larger than what we measured from the
coldbox, shows no significant degredation of Strehl ratio (> 0.95) and still remain diffraction limited
over the full FoV (±225 µm). At 2 mrad misalignment, the strehl ratio is still > 0.9 with a slight
reduction in FoV to (±200 µm).

72
2.4.5

Cold filter testing

There are two types of filters we install in our apparatus. One is for DC electric field control,
and the other is a microwave bandpass filter. Since the DC electrodes are the closest non-grounded
metal objects to the atoms, it is essential to ensure that no unintended noise can couple to them.
To prevent RF and microwave noise from entering the coldbox, we designed an RC lowpass filteron an alumina PCB with a cutoff frequency of 80 kHz [Fig. 2.28(a)]. The cutoff frequency is chosen
so that the voltage at the DC electrode can settle to within 0.5% of the final value in around 10 µs.
The DC electrode filters are also thermalized with the coldbox to below 45 K to block any potential
BBR path. To make sure the filter characteristics remain unchanged at cryogenic temperatures,
we made a test filter and dunked it in liquid nitrogen at 77 K while measuring the filter transfer
function. The gain and phase of the filter are calculated by comparing the voltage Vin and Vout.
We found that with C0G type ceramic capacitors, the filter remains very stable even down tocryogenic temperatures [Fig. 2.28(b)].

ab

R = 2k

+
Vin
-

C
1 nF

+
Vout
-

Figure 2.28: (a) schematic of the test setup and photograph of the DC electrode filter installed on
the coldbox. (b) Filter transfer function at room temperature vs. submerged in liquid nitrogenat 77 K. The measurement shows essentially no change of the filter characteristics over the large
temperature range due to the excellent thermal stability of C0G dielectric material.

For the microwave bandpass filter, cryogenic characterization is also needed. Since the filter

73
has a steeper frequency response, it is expected to shift more when cooled to cryogenic temperatures. To simplify the filter construction and ensure UHV compatibility, we choose low-temperature
co-fired ceramic (LTCC) filters. These filters consist of fully sintered multi-layer metal encased ina ceramic monolith, which is nonporous and has very low outgassing. This ceramic dielectric material is very similar to the C0G ceramic capacitor used in the DC electrode filters described above,
providing excellent temperature stability. The actual filters are chosen based on insertion loss, passband, and stopband rejection. We tested two filters from Johanson Technology (6613BP44A1025
and 7000BP15A1600E), both providing a passband centered around 6.8 GHz and rejection over
30 dB beyond 20 GHz. Ultimately, we chose the 6613BP44A1025 due to its lower insertion loss.
To test these filters, we soldered them directly to a semi-rigid coaxial cable and measured their
S-parameters while immersing them in liquid nitrogen at 77 K [Fig. 2.29(a)].
The S-parameter measurements of the filter at room temperature versus 77 K show somechange in the center frequency of the passband [Fig. 2.29(b)], but it is not significant for ourapplication. The rejection and insertion loss remain unchanged between room temperature and 77
K. The sharp resonances that degrade the filter rejection between 15 and 20 GHz are measurement
artifacts caused by the coaxial cable being soldered close together on the filter.

74

a

b

Figure 2.29: (a) photo of the microwave bandpass filter both in test setup and soldered on the PCB
in the coldbox. For liquid nitrogen testing, the bandpass filter is soldered directly to semi-rigid
coaxial cable and fix on a PCB for structural support. (b) the measured S-parameter of the filter atroom temperature vs. 77 K. The result shows a small shift in passband frequency, but no significant
difference in rejection and insertion loss.

Chapter 3

Cryogenic tweezer atom characterization

3.1

Overview
Note: For the most up-to-date analysis of cooling, imaging, and vacuum lifetime measure-ments, please refer to the upcoming publication ”A High Optical Access Cryogenic Tweezer Array
with a 3000-Second Lifetime” (in preparation).
In the previous chapter, I described the design and construction of the cryogenic tweezer
apparatus. Additionally, I presented the initial test results of the mechanical stability, thermal
performance, and optical performance of the apparatus. The tests showed that the optical alignment is sufficiently stable between the cold box and the objective to achieve diffraction-limited
performance over the full FoV. Using a Michelson interferometer, we verified that the peak-to-peakdisplacement between the cold box and the optical table is around 10 nm, with an RMS of 2.1 nm.
This level of vibration of the cold box is sufficient for the required trap heating and optical phasenoise induced from vibration to achieve long trap lifetime and coherence. With these initial opticaland mechanical measurements, we are confident that the apparatus is performing well, and we canmove to the next step, which is to test the performance of the apparatus with trapped atoms.
In this chapter, I first present the loading and lifetime measurement of the MOT in a cryogenic
environment. The loading rate from the background flux of rubidium from the 2D MOT is alsomeasured, and its impact on the trapped atoms is estimated. Since the vacuum lifetime is expected
to be very long, all the loss mechanisms need to be carefully characterized to infer an accurate
vacuum-limited lifetime. We carefully measure the imaging and cooling losses as well as the trap

76
heating rate from Raman (inelastic) scattering of the tweezer light and other noise sources (RFand intensity noise). With the expected vacuum lifetime of τvac > 1000 seconds, we can measure
and characterize atom losses to the level of < 10−3 /s−1 , or with a typical tens of milliseconds
imaging time, to less than 10−5 per image/cooling pulse. This allows us to measure an imaging
loss as low as (2.7(3) × 10−4 /10 ms image time), with imaging infidelity of 10−6 , and cooling loss of(2(1.6) × 10−3 /s−1 ). After careful characterization, we demonstrate low loss cooling and imaging,enabling us to measure the vacuum-limited lifetime of τvac = 2800 seconds with the trapped single
atom array in a cryogenic environment.
In addition, the optical performance of the objective installed in the full system is measured
with trapped atoms. Analysis of trap frequency and depth shows a NA of 0.48 ± 0.01 without SLM
aberration correction. The FoV of the microscope is also tested, and we observe no significant
degradation of optical performance over distances greater than 100 µm, mostly limited by the scanangle of the AOD used in the tweezer rail. A second set of tweezers with SLM will allow us tocorrect for residual aberration and utilize the full FoV of the microscope.
Finally, we demonstrate that our apparatus is compatible with the LGM enhanced loading
technique previously developed in our group [84]. We show an average loading efficiency of 83%
across the full FoV of the microscope. We also observe good uniformity (±5%) of single atom
loading probability and trap depth over the FoV, compared to [127]. The enhancement of loading
efficiency provides a pathway to increase rearrangement efficiency and create larger tweezer arrayswith a higher probability of success. This can prove to be a useful tool in scaling up the size of thequantum system.

3.2

MOT lifetime
The first step in trapping a single atom is loading the science MOT. Before trapping atoms

with optical tweezers, we would like to check the vacuum condition using the MOT lifetime. The
reason is a much simpler loss mechanism in the MOT compared to a single atom trapped in
optical tweezers. Typically, in a room temperature vacuum chamber, MOT lifetimes of up to77
a few minutes are observed [96, 102], corresponding to pressures in the range of 10−10 torr. In
cryogenic experiments, a MOT lifetime as long as 1 hour was observed in Ref. [88]. This served as
an important benchmark, allowing us to compare our system against other cryogenic apparatus,
especially since the MOT lifetime reported in Ref. [58] was measured before the vacuum leak was
fixed on their apparatus so the MOT lifetime reported there might not necessarily correspond tothe vacuum-limited tweezer lifetime.
The loading/loss mechanism of the MOT can be described with the following equation [96,102, 128].
dN= R − αN − βN 2
dt

(3.1)

The term R corresponds to background rubidium that can cause loading of the MOT. This term
does not depend on the existing atom number. Due to very aggressive differential pumping between
the 2D MOT chamber and the coldbox, this term is expected to be small. The coefficient α = 1/τvac
represents how the loss due to residual background gas can be extracted. The term β, which is
proportional to N 2 , represents the two-body inelastic collision loss, e.g., light-assisted collision.
To measure the MOT lifetime, we start with a very dilute MOT containing fewer than 200
atoms. This is to suppress loss mechanisms other than the background loading and vacuum loss.
With this low density, we can consider β = 0. Then, we can fit the MOT number over time withthe following equation (which is a solution of Eq. 3.1 when β = 0).
t

N (t) = (N0 − Rτ ) e− τ + R τ(3.2)

After loading the MOT, the gate valve separating the source MOT and science chamber ispartially closed to obstruct the beam path and prevent residual Rb background loading. Closing
the gate valve is critical; without it, the final atom number actually increases over time, making
the measurement inaccurate. Once the MOT is loaded and the gate valve is closed, the MOT isheld in place, and an image is taken every 15 seconds with a 600 ms camera exposure time overthe course of 6000 seconds. Then, the counts are summed and fitted with Eq. 3.2.
Typically, during normal tweezer loading operations, we load around 107 87 Rb atoms within

78
a 500 ms loading time into the MOT, as shown in [Fig. 3.1(a)]. After acquiring a time trace andfitting the data (black circles) with Eq. 3.2, we extract a lifetime of τvac = 3800 seconds, as depicted
in [Fig. 3.1(b)]. This is the longest MOT lifetime we’ve ever observed in a cold atom experiment,
consistent with measurements reported in Ref. [88]. Using conversions from Ref. [96], we estimate
a pressure of 5 × 10−12 torr inside the cold box.

ba~107 87Rb

Figure 3.1: (a) Photo of the MOT of around 107 87 Rb atoms trapped between two tweezer cold
windows inside the coldbox. This is the typical MOT size during tweezer loading. The microwave
loop antenna is visible on the background on the top left. (b) The MOT atom number vs time.
Black solid circles is data, blue fit line on top is based on Eq. 3.2 where we extract a τvac = 3800 sec
The inset is the image of the dilute MOT used for the time trace where only about 100 atoms areloaded.

3.2.1

Residual flux of 2D MOT

In Ref. [59] and [58], a significant reduction in lifetime is observed due to background Rb flux
from the Zeeman slower atom source. They report a strong dependence of vacuum lifetime andthe source oven temperature. With this in mind, we designed three aggressive differential pumping
stages where the flux from the 2D MOT is well isolated from the cold box. To quantify the leakage
of Rb atoms from the 2D MOT, we use the background loading of the MOT to estimate the total
Rb flux that passes through all the differential pumping stages.
Typically, when a well-controlled atom source is used, e.g., a 2D MOT instead of a Zeeman
slower, MOT loading from background rubidium cannot be observed since α is generally much

79
larger than R. For example, in the vacuum system described in Sec. 5.2, this background loadingis never observed even with the ion pump pressure reading below 10−10 torr. However, due to theexcellent vacuum in our apparatus, we can observe background loading in the absence of 2D MOT
light. The loading is purely due to leakage from the Rb vapor that travels through the pinhole
and is captured by the 3D MOT. To measure the background loading rate, we turn on the 3D
MOT beam and open the gate valve. We record an image for 200 ms every 2 seconds for up to 600
seconds. We infer the atom number based on the scattering rate and solid angle of the imaging
system using the following equation [106].

dΩ
Ncount = Natom timg ηCCD
Rsc
4π

, where

Γ
Rsc =
2



6 I/Isat
1 + 4(∆/Γ)2 + 6 I/Isat


(3.3)

Then fitting the atom number over time with the loading curve Eq. 3.4 where both vacuum loss
and loading are considered we can extract the background loading rate R.
t

N (t) = R τ (1 − e− τ )(3.4)

Based on Eq. 3.4, we extract R = 0.26 ± 0.07 atoms/s and τvac = (1.8 ± 0.4) × 103 s [Fig. 3.2(a)].
To convert the captured atom flux to the total incoming flux, we need to determine the range
of velocities at which the leakage atoms can be trapped by the 3D MOT. This allows us to findthe ratio between all velocity classes and the trappable ones. The lower bound of the velocity isdetermined by the drop of the atom beam due to gravity. By plotting the atom beam trajectories
vs. distance with different initial velocities, we can observe that the cutoff is around 16 m/s [shownin Fig. 3.2(b)]. Any atom launched from the 2D MOT with an initial velocity lower than this will
miss the cold box completely. To calculate the maximum capture velocity of the 3D MOT, we use
the following equation [129, 130].
vmax =√

aRb wM OT ≈ 26 m/s

Where aRb =ℏkΓRb, wM OT = 5 mm
2 mRb

(3.5)

Now, by comparing the portion in the velocity range of 16-25 m/s [red shaded area in Fig. 3.2(c)]to the rest [grey shaded area in Fig. 3.2(c)], we find the ratio to be 1600, translating to 415 atoms/s

a

80

c

b
diﬀ. pumping tube

Trapped por�on~ 1:1600

Figure 3.2: (a) Mot atom number vs. time from loading of background Rb flux from 2D MOT
chamber. At t=0, the lasers are turned on and the gate valve between the 2D MOT and sciencechamber opens. (b) plotting the atom beam trajectories vs distance over different initial velocity.
The red boxes represent the differential pumping tubes. Due to gravity, atom with initial velocity
less than 10 m/s misses the middle differential pumping tube and initial velocity less than 16 m/s
misses the coldbox. (c) Probability density over all velocity (grey shaded) assuming 300 K Rb
vapor temperature. Only a portion of atom (red shaded) is slow enough to be captured by the 3D
MOt.

of background rubidium from the 2D MOT.

3.2.2

Dipole-push beam

With the typical exit velocity from the 2D MOT on the order of 10-20 m/s, most of the atoms
actually miss the entrance of the differential pumping on the coldbox [shown in Fig. 3.2(b)] The
effect of gravity during the free flight of the atom beam is not negligible, and a simple push beam
is no longer sufficient to guide the atom into the cold box due to the lack of confinement along thedirection of gravity.
In order to efficiently guide the atoms, we employ a dipole trapping push beam (dipolepush) that is close detuned from the 87 Rb D2 line. The close detuning simultaneously provides
the required trapping and scattering to push the atom along the axial direction of the dipole-pushbeam [131, 132].
Different from the typical push beam configuration where the atoms eventually move outof the path of the beam once they exit the 2D MOT cell, the atoms guided by the dipole-push81constantly scatter the dipole-push light, so the exit velocity is proportional to the scattering rate.
Since the trap depth scales as 1/∆ and scattering (thus velocity) scales as 1/∆2 , where ∆ is thedetuning, it is possible to choose a dipole-push detuning that satisfies the following condition: theminimum required trap depth is reached before the atom is accelerated beyond the capture velocityof the 3D MOT.
We choose the dipole-push beam to be 600 MHz red detuned from the F = 2 ↔ 3′′ coolingtransition with a Gaussian waist of 0.3 mm and power approximately 10 mW. The 600 MHz
detuning ensures that the dipole-push is red detuned from all the hyperfine states in the D2 excitedstate manifold, thus no anti-trapping occurs. We can estimate a lower bound on the trap depth
and scattering rate if we consider the atom mostly cycling between F = 2 ↔ 3′′ , which gives
approximately a 2 mK depth and a 0.16 MHz scattering rate. However, in reality, the atom also
cycles between F = 2 ↔ (1, 2)′′ with a smaller dipole matrix element but a closer detuning. Thus,
the trap depth and scattering rate can be up to 3 times larger than the estimation above.
The intensity of the dipole push beam, therefore the atom flux and velocity, is optimized byobserve the loading rate of the 3D MOT similar to the curve shown in [Fig. 2.10(c)].
The detailed calculation of the trap depth of this guiding push beam and the characterizationof the velocity of the atom beam pushed out from the 2D MOT using this method will be discussed
in Zhenpu Zheng’s thesis.

82

3.3

Single atom lifetime
In the previous section, we characterized the loss and loading of the MOT in a cryogenic

environment. Based on the measured MOT lifetime, we expect our single atom to have a lifetime
greater than 1000 seconds. However, measuring a long vacuum-limited lifetime is not trivial. During
the holding and probing of the trapped atom array, various losses are also introduced. If not treatedcarefully, these losses can introduce systematic errors in the inferred vacuum-limited lifetime of thesingle atom array.
For a single atom trapped in an optical tweezer, we consider the following loss mechanisms:
heating loss out of the trap, cooling loss, and imaging loss. The time-dependent loss of eachmechanism is specific to the method used to measure the single atom lifetime. In general, there are
two ways for lifetime measurement: one is a single-shot image with variable hold time [Fig. 3.3(a)],and the other one is continuous measurement [Fig. 3.3(b)].

abPvac(thold)
Pimg
Pcool
Pcool
Pheat(Δt)
Pheat(Δt)

Pimg

Pvac(thold)
Pcool
Pcool
Pcool
x
x
x
Pimg PPP
heat(Δt) img Pheat(Δt) img

pulse
cool

RPGC
img

cool
+
img

RPGC
img

Δt

Δt

pulse
cool

1 mK

Δt

cool
+
img

Δt

cool
+
img

1 mK

Uhold

Uhold
14 ms

tcool

tcool

14 ms

t

timg

timg

timg

t

thold = N x (Δt + timg)

thold = N x (Δt + tcool)Figure 3.3: Timing sequence for single atom lifetime measurement and the associated survival
probabilities of different stages during the sequence. (a) is the single shot method where thetweezer hold times varies and only two images are taken, one in the beginning and one at the end.
(b) is the continuous method where the atom in the array is repeatedly imaged over time.

We can write the time dependent loss function of these two different measurement method
as follows.
2
Psurvival (t) = Pimg
Pvac (t) [Pheat (∆t) Pcool (tcool )]N

for single shot
(3.6)
N

Psurvival (t) = Pvac (t) [Pimg (timg ) Pheat (∆t) Pcool (timg )]for continuous

83
Here, the first line in Eq. 3.6 is for the two-image method, where the finite imaging loss contributes
to a fixed loss factor that is time-independent. The second line is for the continuous imaging case,where an image is taken at a given interval. This makes the imaging loss time-dependent. However,
the exact form of the time dependence is sequence-specific, which we will discuss in Sec. 3.3.4.

3.3.1

Heating in optical tweezer

Before we can achieve a long-lived tweezer array, we need to verify whether the tweezers are
introducing any unnecessary heating to the trapped atom. More heating implies more frequentcooling, which might introduce loss during the holding of the atoms. Several mechanisms can cause
heating of the atom in tweezers, including intensity noise, beam pointing, and photon scatteringfrom the tweezer light. These mechanisms all have different dependencies on trap frequency, trap
depth, and power spectral density of the perturbation. Ultimately, we measured a recoil-limited
heating rate, but it was not an easy task, as disentangling the various sources of heating requires
careful characterization of the atom temperature under different conditions. In this section, I will
briefly describe the methods we used to characterize the atom temperature and different heating
mechanisms.
Recoil heating is the fundamental heating mechanism that atoms trapped in optical tweezers will experience. This is due to off-resonant photon scattering from the intense tweezer light.
The heating rate is linear in time and can be defined as follows [133].
αsc = 2 Trec Rsc = 2 Trec

ΓRb
Udip
ℏ∆

(3.7)

Rsc is the off-resonent scattering rate, Udip is the tweezer dipole potential, Trec is the recoil energyof trapping light, ∆ is detuning of the trapping light with respect to 87 Rb D2 transition (780 nm)
and ΓRb = 2 π × 6MHz is the natural linewidth of 87 Rb. For 1 mK trap depth at 830 nm thisheating rate is about 10.5 µK/s.
Assuming a Boltzmann distribution of the atom energy inside the trap, with a truncation

84
imposed by the trap depth, the probability of an atom remaining in the trap is given by [134].
Pthermal (t, Uesc ) = 1 − e

−η(t)



1
1 + η(t) + η(t)22

, whereη(t) =Uesc
kB T (t)(3.8)

Here E = Udip is the cutoff energy (trap depth) and T (t) is the atom temperature as a function oftime. In the absence of cooling, T (t) should equal to αsc t.
As a baseline, we measure the survival of the atom without cooling. Assuming no vacuum
loss, Eq. 3.8 fits the data nicely, and we extract a heating rate of 94 µK/s [Fig. 3.4] which iswell above the recoil limit. This issue needs to be addressed before we can obtain an accurate
measurement of the vacuum lifetime.

Figure 3.4: Dark tweezer lifetime without cooling. The atom survival is taken over different holdtime while no cooling is applied during the wait. The trap depth is 1 mK. Fitting the data (black)to Eq. 3.8 (blue dash) we extract a 94 µK/s heating rate, which is much larger than just the recoil
heating for this trap depth.

In order to measure the temperature of the atom more accurately over time, we use a release
and recapture (R-R) method. Here, the trap is turned off for a variable amount of time, typically a
few microseconds, then turned back on, and the atom survival is measured. The measured survivalis then fitted with a curve generated by a classical trajectory Monte Carlo simulation based on
the specific trapping parameters used in the experiment. This method allows us to investigate thedifferent dependencies of the heating rate and separate different heating mechanisms.
Release and recapture Monte-Carlo simulation: Here, we describe the steps taken to85
generate the R-R probability using classical trajectory Monte Carlo simulation, as described inRef. [134, 135].
Initialization: Considering the atom inside the trap follows a Maxwell-Boltzmann distributionpwith temperature T , the standard deviation of position and velocity are ∆X = kB T /mRb ω 2 ,p∆v = kB T /mRb . Then, a randomly sampled set of position and velocity (x, y, z, vx , vy , vz )ican be generated.
Evolution: The trap is turned off for time trr , which gives the updated position and velocity:(x + vx trr , y + vy trr − gt2rr /2, z + vz trr , vx , vy , vz )i
Determine probability: After the new position is calculated, the final energy, kinetic + potentialof the atom, is calculated. If the trap is perfectly harmonic, the total energy is 1/2mv 2 +1/2mω 2 x2 . In the case of Gaussian potential from the tweezer, assuming a harmonic trap
gives an underestimation of the atom temperature because the Gaussian potential provides
weaker confinement at the periphery of the trap. The difference between fitting with theGaussian and harmonic trap is discussed in Appendix G.
If the total energy is > 0 then the atom is considered escaped. The evolution is done overa range of trr , typically with trr up to 50 µs, and a survival vs trr curve can be obtained.
For most of the analysis done in this thesis, unless specified otherwise, the harmonic trap
shape is used in the Monte-Carlo simulation so it can be easily compared to other literature’s
such as in Ref. [134, 136].
Fitting: The data can be fitted with the numerically calculated curve to minimize the least squarevalue by varying the atom temperature T in the Monte-Carlo simulation.
The sequence for R-R consists of two atom images with the tweezer trap briefly off for trr in
between, as shown in [Fig. 3.5]. Varying both trr and thold allows us to measure the heating rateof the atom inside the trap either as a function of time or sequence of interest. As an example,
the temperature after our typical RPGC imaging is measured, and the best fit curve from the86Monte-Carlo simulation gives a temperature estimation of the atom around 13 µK, as shown in[Fig. 3.5(b)].

baImage 1

RPGC RPGC
cool img

Image 2
sequenceof interest

Release RPGC RPGC
Recapture cool img

Uhold

25 ms 14 ms

thold

tR-R

25 ms 14 mst

Figure 3.5: (a) Timing sequence for release and recapture measurement to measure the heatingrate. The atoms are first imaged and the release and trap is briefly turned off and the atom
recaptured. The second image is taken and the survival is recorded. (b) An example of the release
and recapture data (circles) after the typical RPGC imaging with the best fit probability curvefrom the Monte-Carlo simulation (solid lines) showing an atom temperature of 13 µK.

Anomalous heating due to RF Source: Initially, we observe a large heating rate that iswell beyond the tweezer recoil limit. We attribute this to intensity noise caused by the intermodulation of the RF tones used to generate the tweezer array. We employed an evenly spaced 9 × 9 gridwhere the RF intermodulation perfectly overlaps with the adjacent tweezers. In this configuration,
we measured a heating rate of 230 µK/s at a trap depth of 1.3 mK, whereas the recoil heating
should only be 15 µK/s. However, when the trap depth decreased to 0.38 mK, the heating rate
dropped to 7 µK/s, only twice the expected recoil heating rate of 4 µK/s, even though the RF
power driving the AOD did not change; only the total power from the fiber is decreased. This
indicates that the process is most likely parametric heating, and as the trap depth decreases, theresponse of the atoms to the intensity noise at a particular frequency also shifts, resulting in a lower
heating rate. To diagnose different sources of heating mechanisms, we measured the heating rate
in a series of different configurations, as shown in [Fig. 3.6]. The dashed line in Fig. 3.6 represents
the recoil limit, is calculated using Eq. 3.7. The tweezer recoil-limiting heating rate with a clean
synthesized RF source allows us to exclude the problem originating from the laser source and focus

87
our efforts on the AOD and RF generator.

Figure 3.6: Single atom heating rate of various tweezer configuration including aspheric lens andobjective, different RF source and different NA. The green dashed line is recoil limit calculated basedon the tweezer configuration. The data lines below this line is recoil limited heating, means theexternal source is not introducing additional noise significant enough to effect the atom temperature.

To drive the tweezer AODs, we use a custom-built RF synthesizer consisting of two evaluation
boards: the Xilinx KCU105 FPGA board and the Texas Instruments DAC39J84 4-channel 16-bit
DAC. This system is capable of synthesizing up to 56 RF tones in real-time with the ability tochange frequency, phase, and amplitude for rearrangement. The DAC is running at 614.4 MSPS,
and the output frequency is in the first Nyquist zone up to 307.2 MHz. This means that themodulation bandwidth equals the baseband sample rate of 614.4 MHz, which is sufficient for the
bandwidth requirement of rearrangement.
During our investigation, we found that even when the RF tones appear good on the spectrum
analyzer (> 60dB SFDR), there are still complicated dynamics that can arise due to nonlinearity in
the AOD and the RF source. Primarily, there are time-dependent power fluctuations where the total
RF power remains unchanged, but the individual tones fluctuate in a complementary manner. Thisis a subtle effect since our intensity servo only monitors the total power of the array and corrects forany intensity fluctuations, remaining oblivious to individual tweezer intensity fluctuations. Even
worse, due to the way the RF tones interact, the total power can remain the same, but only the

88ratio of power to the individual tweezers differs. To observe this effect in detail, one would need toconstruct a spatial filter to select the intensity of only one tweezer and measure it. This aspect is
not studied in detail in this thesis.
The specific configuration that gives a very low intermodulation depends strongly on thetype of RF source and the technique used to generate the signals. Eventually, an RF frequency
configuration specific to our custom-built RF generator (Gigamoog) was found that can provide alow heating rate, and our AOD tweezer can operate at the recoil heating limit [(5x5 array, 830 nm,
AOD Gigamoog new) in Fig. 3.6]. For a detailed discussion of the RF dynamics of the Gigamoog
RF synthesizer and construction, please refer to Zhenpu Zhang’s thesis.
Laser intensity noise: Another heating mechanism is parametric heating from the laser
intensity fluctuation. The atom in the tweezer with sufficiently cold temperature is trapped at
the bottom of the Gaussian trap which can be approximate as an a quantum harmonic oscillator.
Changes in laser intensity, ϵ(t) = (I(t) − I0 )/I0 , introduces small fluctuations in the spring constantof this harmonic trap. The perturbed part of this Hamiltonian can be written as follows [137].
1
H ′ (t) = ϵ(t)MRb 2πν 2 x2
2

(3.9)

Calculating the transition probability up the harmonic oscillator energy levels as a function on
single sided power spectrum of ϵ(t) the energy increase of the atom can be derived. Follow thederivation in Ref. [137], the frequency dependent atom heating rate as a function of intensity noise
power spectrum has the following relationship.
⟨Ė⟩ = π 2 ν 2 Sk (2 ν) ⟨E⟩(3.10)

ν represents the trap frequency, and Sk (2 ν) is the single-sided power spectral density of thelaser intensity. From the equation, we see that the rate of energy increase is proportional to theinitial temperature and the square of the trap frequency. The factor of 2 relationship between thetrap frequency and PSD indicates that with the parametric heating process, fluctuations of thetrap depth at frequency 2 ν will heat the atom at the trap frequency of ν.

89

a

b
axial x2

radial x2

axial

radialFigure 3.7: (a) Single sideband noise spectrum of tweezer RIN noise. Green is the noise floor of
the measuremetn setup. Orange (Blue) is with (without) the intensity servo engaged. With theintensity servo engaged, the low frequency noise is suppressed and small amount of noise is added
around 100 kHz. (b) Heating rate of the tweezer based on the RIN measured in (a). THis iscalculated using the parametric heating Eq. 3.10 and the initial atom temperature around 10 µK
which is typically what we have after cooling.

To rule out the possibility of heating due to laser intensity noise, we measure the tweezer
relative intensity noise spectrum (RIN) and calculate the heating rate. The tweezer RIN is measured in the final telescope just before entering the vacuum chamber, as shown in [Fig. 3.7(a)]. A
significant difference is observed between the locked [blue in Fig. 3.7(a)] and unlocked [orange in
Fig. 3.7(a)] intensities. We notice a slight servo bump around 200 kHz, which is near the 2 × νradialof the tweezer [black dashed lines in Fig. 3.7(a)]. The intensity servo suppresses the low-frequency
noise almost to the noise floor of the measurement setup without introducing significant noise in
the high-frequency part of the noise spectrum. To ensure that the heating rate is acceptable, we
calculate the heating rate with an initial atom temperature of 10 µK as measured in Fig. 3.5(b),using Eq. 3.10 and [Fig. 3.7(a)]. The resulting heating rate [Fig. 3.7(b)] is much lower than 1 µK/s
(and also much lower than the recoil limit), verifying that the laser intensity noise from the tweezer
source is not a limiting factor in our tweezer trap.
Heating from fluctuation of trap center: Mechanical vibration of the objective willcause the shaking of the tweezer trap with respect to gravity. Imaging shaking a bowl with a ball90
inside at the rate where the ball oscillate back and forth. Excessive perturbation of the trap center
will cause the atom to heat up. The frequency dependence of this heating is given by [137]⟨Ė⟩ = 4 π 4 mRb ν 4 Sk (ν)(3.11)

This heating rate ⟨Ė⟩ remains constant over atom energies and lacks the factor of 2 infrequency dependence compared to parametric heating. We can observe this (or rather the lack ofit) by holding the atom and measuring the temperature as a function of hold time. If the heating
is parametric, the energy would increase exponentially; if it’s due to shaking of the trap, it would
increase linearly. We see a linear increase in energy that closely follows the recoil limit [(the purple,
1x1 0.49 NA obj. dataset) in Fig. 3.6].
To further verify that the shaking of the trap center due to mechanical vibration is nota concern, we conducted heating rate measurements with different tweezer numerical aperturesby adjusting the input beam aperture [(1x1 0.28 NA obj. and 1x1 0.49 NA obj.) in Fig. 3.6].
This alters the trap frequency without changing the trap depth. The results show that in both
numerical aperture settings, the heating rates are recoil-limited. Combining this with measurementsof different trap depths in the given configuration, we can infer that over a broad range of trap
frequencies, we do not have intensity noise or vibration of the tweezer causing heating to a levelcomparable to the recoil heating from the tweezer light.

3.3.2

Cooling loss

Since the recoil heating is always present as long as the optical tweezer is turned on, coolingthe atom from time to time is necessary to achieve long trapping lifetime. In the absence of cooling,
the recoil heating from the trap light reduces the lifetime to less than 100 seconds [58, 59]. We canalso infer this with Eq. 3.8 and by scaling the heating rate in (Fig. 3.4) to 9 µK/s. This decay rate
is significantly faster than the expected vacuum-limited lifetime, indicating that cooling during theholding of the atom is necessary. To establish an optimal cooling duty cycle, we need to understandthe loss and final temperature associated with the cooling.

91
Firstly, we determine the saturation time for cooling to take effect by varying the coolingpulse length and measuring the survival at a fixed R-R time. It is generally preferable to choose an
R-R time such that the survival rate is 0.5, maximizing sensitivity to atom temperature changes.
From the measurements in [Fig. 3.5(b)], we can select an R-R time of 10 µs for a final temperatureof 15 µK.
Once the R-R time is established, a cooling pulse length needs to be chosen. Fixing at
the 50% survival R-R time and varying the cooling pulse length, we observe saturation of coolingtemperature in 10 ms, as shown in [Fig. 3.8(a)]. Therefore, choosing a standard cooling pulse lengthof 25 ms provides some margin, in case the saturation time varies during optimization.

a

c

b
XN
RPGC
img

RPGC
cool

RPGC
img

25 ms

20 s

1 mK
0.3 mKt

Figure 3.8: (a) Setting the R-R time to give 50% survival at nominal atom temperature we scan
the cooling pulse duration. The temperature reaches equalibrium after 10 ms so we set the coolingpulse length for 25 ms to be conservative. (b) Timing sequence for the cooling loss measurement.
The total sequence time is fixed, to eliminate the fluctuation in vacuum loss, and the number ofcooling pulsed are varied and the survival is recorded. (c) Measured atom survival vs number of
25 ms cooling pulses. Fitting a line to the survival extract a cooling loss of 5(4) × 10−5 per 25 ms
pulse or 2(1.6) × 10−3 s−1

After determining the cooling pulse length, we choose a cooling laser detuning of −60 MHz
red-detuned from free space resonance (−90 MHz in a 1 mK deep trap), and the repump laser istuned to in-trap resonance. We then optimize the cooling and repump power. The final coolingand repump intensities are 82 Isat and 0.04 Isat respectively, where Isat = 3.576 mW/cm2 .
To measure the cooling loss, we follow the procedure in [Fig. 3.8(b)], where the atom is held
for a constant time of 20 seconds between two PGC images, and a variable number of cooling pulses

92(shaded green box) are repeated during the 20 second hold time. The constant hold time ensuresthat the atom loss due to vacuum does not change from different number of cooling pulses. By
fitting a line to the survival versus the number of cooling pulses [Fig. 3.8(c)] we can extract theslope, which gives a cooling loss of 5(4) × 10−5 per 25 ms pulse or 2(1.6) × 10−3 s−1 .

baRPGC probe pulse
img heat cool
1 mK

R-R

RPGC
img

trr

14 ms

0.5 ms 25 ms

14 ms

t

Figure 3.9: (a) Timing diagram for measuring the atom temperature after pulse cooling. The atoms
are heated with a resonance beam and a cooling pulse is applied and R-R sequence is used to measure
the temperature. (b) The resulting release and recapture data with Monte-Carlo temperature fit
gives atom temperature of 7.9 ± 0.3 µK.

To verify that the cooling is at its optimal, we also measure the atom temperature after one
cooling pulse [Fig. 3.9(b)] using the R-R technique described in Sec. 3.3.1. To prepare the atomwith a higher temperature, we applied a short (0.5 ms) heating pulse after the first PGC image[Fig. 3.9(a)] to raise the atom’s temperature to about 100 µK. Then, a single 25 ms cooling pulse isapplied, followed by an R-R sequence and a second atom image. This gives an atom temperatureof 7.9 ± 0.3 µK, indicating that the pulse cooling is effective.

3.3.3

Imaging loss

In general, imaging loss is an important parameter since it directly impacts measurement
fidelity, especially if one wishes to use trapped atoms as qubits. Poor imaging fidelity will also
limit the success rate of forming a large atom array due to losses after rearrangement moves are
performed. This is non-deterministic because every time an observation is made, an atom is possibly
lost. Since the last step of rearrangement is always imaging, this loss is a fundamental limitation

93
for rearrangement fidelity. Furthermore, in our application where the lifetime is measured withrepeated imaging, a large imaging loss can affect the apparent vacuum-limited lifetime. Therefore,
careful characterization of imaging loss is necessary.
To find the imaging loss, one would need to determine an optimal imaging time so thatthe collected signal is large enough for good imaging fidelity. We typically use 14 ms in our red
PGC imaging. This imaging time is similar to what others have used [35, 118, 138] and perhaps
on the shorter side compared to the typical value (∼30-50 ms). In another case where a very low
imaging loss was reported [43], 80 ms imaging time is used which is almost twice as long as typical.
Now we need to determine the optimal imaging detuning and intensity where the photon counthistograms are well separated from the camera noise. Taking into account the light shift fromthe trap (0.8 mK × 28.8 MHz/mK = 23 MHz), we use the Imaging 2 parameter in table 3.1. Thisparameter is chosen based on the balance of three factors: collected photon number, heating (loss)
during imaging, and imaging fidelity.
Determining imaging fidelity: Before investigating the imaging loss, a method of determining the imaging fidelity needs to be established. It is important for the imaging infidelity to belower than the quantity of interest to be measured. The imaging fidelity we used follows the samedefinition as others in optical tweezer experiments: 1 − Pfalse , where Pfalse is the probability of false
detection. False detection probability is where an atom in the tweezer is mistakenly identified aspresent (1-atom) while in reality it is non-present (0-atom) and vice versa.
To calculate the imaging fidelity, a number of atom array images are taken, then the images
are averaged [Fig. 3.10(a)]. After averaging, each tweezer site is analyzed as follows.
• From the averaged atom image of each tweezer [Fig. 3.10(a)] three or four brightest pixels
are chosen and turned into a binary mask, example shown in [Fig. 3.10(c)]. The variable
number of pixels in the mask (3 or 4) is due to the possibility of the atom signal aligning
between pixels on the camera. To determine whether 3 or 4 pixels are needed in each mask,
the total photon count between the two cases is compared using the averaged atom image.

94
If the difference is larger than 5%, the configuration with more signal is used.
• The binary mask with chosen pixels is applied to each atom image (not the averaged image),and the counts on those pixels are summed. Then, these photon counts from each image
are binned into a histogram [Fig. 3.10(d)].
• The 0-atom distribution and 1-atom distribution in the histogram are fitted with Gaussianfunctions [orange line in Fig. 3.10(d)]. We choose Gaussian instead of Poissonian because,taking into account the camera noise, the distribution is a convolution of Poissonian andGaussian. In our case, the Gaussian standard deviation is larger than the Poissonian, thus
Gaussian fits better.
• The amplitude of the fitted Gaussians is normalized, and a threshold is determined [dashedline in Fig. 3.10(d)] by minimizing the region of the two Gaussians cut off by the threshold.
• The imaging fidelity is then determined by finding the area of the Gaussian outside of thethreshold and taking the ratio of this area to the total area of the Gaussian.

a

b

c

d

0
1
Camera count weight

Figure 3.10: (a) CCD image of an averaged 9x9 atom array. (b) Zoom in to a single tweezer image
to extract the three brightest pixels. (c) The mask applied to select the three brightest pixel inthe single tweezer image and the counts on those pixels are summed together. (d) The histogramof the count vs. instances after the mask is applied and the camera photo-electron count aresummed together. Gaussian functions are fitted to the histograms and a threshold is determinedby minimizing the overlap of the Gaussian tail. Dashed vertical line is the threshold to determine
whether or not single atom is detected. The fidelity of the imaging is determined by the area of
the Gaussian below the line.

95
There are a few different methods other than the binary mask can be used to calculate
imaging fidelity including, PSF weighted mask [95] and model-free approach [40, 43]. The PSF
weighted mask uses a pixel mask where the weight of the individual pixel is calculated according
to the integrated area of the tweezer PSF under each pixel, in other word, the PSF intensity iscoarsely sampled by the camera pixel.
For completeness, we compare the binary pixel mask with the PFS weighted pixel mask
extracted from PSF fitted to the averaged atom image. There are small differences in total photons collected, which is expected since the PFS weighted mask takes all photons into account.
However, in our configuration where photons are mostly in 3 or 4 pixels, we did not observe asignificant difference in infidelity between the two methods, and they are all below the imaging loss
we measured.
The model-free analysis of imaging fidelity, similar to the one used in Ref. [40, 43], takes
into account all the losses during imaging without any presumed distribution of photon counting
statistics. This method of estimating the imaging fidelity uses 3 atom images taken in sequence
without reloading of the tweezer. The measured probability of detecting an atom in each image
is determined by a combination of factors: L losses during imaging, F0 fidelity for successfully
detecting zero atoms, F1 fidelity for successfully detecting one atom, and iF0 and iF1, which
represent their respective infidelities. Due to the causality of the sequential atom images, each
image consists of a combination of these losses and fidelities. Taking many trials of the 3 image
sequence, a series of probability can be measured and associated with their respected combinationof losses and fidelities. By solving the set of these linear equations with the measured probabilities,
L, F0 and F1 can be extracted. One draw back to this method is the extracted fidelity is only
as good as the total number of trials of the 3 atom images. Therefore to measure 10−5 level ofinfidelity, one would need at least 105 trials or better to reach the error bar that is necessary for themeasurement. In this case we do not have enough statistics to resolve losses and fidelities below
10−4 with sufficient error bar using the model-free method. We can only bound the loss to be as
good as the value we measured.

96
In the end we choose to use the binary mask method for most of our analysis since it requires
the least amount of data processing overhead and less atom image statistics with only small difference in fidelity estimation that is sufficient for the level of loss we measured. For detail comparisonof PSF pixel mask and model-free method with the binary mask method please refer to Zhenpu
Zheng’s thesis.

baprobe lighttimg

Cameraexp

exp
196 msexp

N*196 ms 20 st

Figure 3.11: (a) Timing diagram for measuring the single atom imaging loss. The imaging light isturned on for 20 seconds and every 196 ms the camera is triggered to expose for timg = 14 ms andthe stom survival is recorded. (b) Atom loss vs imaging light duration. Red dash is exponential
decay of 0.027(3) s−1 or 3.8(4) × 10−4 per image loss rate. This time constant is not obtained fromthe fit shown in the plot. It is from the MLE estimation using the data.

After determining good imaging parameters, the imaging loss is assessed by exposing theatom to the imaging light for a long duration (20 seconds, equivalent to 1428 images) [Fig. 3.11(a)].
The camera is triggered every 196 ms (equivalent to every 14 images) for 14 ms, and a series of 102
images is captured. The survival of the subsequent images is determined by comparing them tothe very first image [Fig. 3.11(b)]. Using a maximum likelihood estimation (MLE), we extract theeffective imaging loss rate of 0.027(3) s−1 or 3.8(4) × 10−4 per 14 ms image. This rate is among thelowest observed in optical tweezer experiments [40, 43, 138]. The red dashed line in [Fig. 3.11(b)]represents an exponential decay with the loss rate extracted by MLE method, to guide the eye.

97
3.3.4

Pulsed cooling

Since we know that the vacuum-limited lifetime is much longer than the recoil heating-limited
lifetime, the atom must be cooled while held in the trap. In the previous sections, the cooling andimaging losses were determined to be very low. However, from Sec. 3.3.2, we know that with
continuous PGC cooling, the cooling-limited lifetime is around τcool = 500 s. This is significantly
lower than the expected vacuum-limited lifetime. Therefore, we need to determine the optimalcooling interval for the range of vacuum lifetimes that we expect to measure. We can write theprobability versus time that corresponds to our measurement sequence, [Fig. 3.12(a)], as follows.
Psurvival (N ∗ dt + t) = Pvac (N ∗ dt + t) × (Pthermal (dt) × Pcool&img )N(3.12)

N is integer of total cooling and imaging pulse applied. dt is the interval between pulses and t < dt
is the remainder that is shorter than the pulse interval. Pvac is the vacuum loss that does notdepend on the cooling and imaging pulses. Pthermal (dt) ,defined in Eq 3.8 is the atom survival due
to recoil heating of the tweezer while the cooling beam is off. Pcool&img is the cooling pulse where
the scattering is intentionally made stronger so the atom can also be imaged during cooling. The
imaging parameter is (Cooling 2 in table 3.1) and the loss is measured using method in Sec. 3.3.3.

a

b

c

Pvac
Pthermal Pcool & img
RPGC
...N
cool & img

RPGC
img
1 mK
0.55 mK
10 s

t

Figure 3.12: (a) Timing diagram of the pulse cooling and imaging used for atom lifetime measurement. (b) Probability vs cooling pulse interval showing an optimal range (where the survival isclose to unity) of interval for each vacuum lifetime based on Eq. 3.12. (c) Measurement of survival
vs pulse interval at a fix hold time of 60 seconds to determine the optimal cooling interval. Herewe choose one cooling pulse every 10 seconds.

98
Plotting Eq. 3.12 against the pulse interval, we can see there is a range of dt where thesurvival is optimized [Fig. 3.12(b)]. The behavior at short dt is dominated by the losses fromcooling and imaging applied too frequently, where Pcool&img becomes small. The loss at large dt
is dominated by high atom temperature due to recoil heating, causing Pthermal (N dt) to be small.
The optimal range for dt is where the (Pthermal (N dt) × Pcool&img ) << Pvac (t). To find the value ofdt experimentally, we take a set of data corresponds to the sequence shown in [Fig. 3.12(a)] while
keeping the total hold time constant and only vary dt and N . The result, [Fig. 3.12(c)], shows
similar behavior as predicted in Eq. 3.12, where at pulse cooling interval < 1 second, the survival
is low, and intervals larger than 20 seconds, the survival is low as well. With wait time between
5-20 seconds, the survival remains high, [inset of Fig. 3.12(c)]. Therefore, we decide that dt = 10 sseems to be a nice middle ground, and that is the parameter we use for the lifetime measurement.

3.3.5

Vacuum-limited single atom lifetime

Now that we have carefully characterized all the losses that can affect the long vacuum lifetime
measurement, we can measure the single atom lifetime and extract the vacuum-limited loss using
the sequence shown in [Fig. 3.12(a)]. To estimate the effect of cooling and imaging on the atom
lifetime, we use the loss measured in Sec. 3.3.2, where the following cooling/imaging parameter isused (Cooling 2 in table 3.1) with an interval between cooling set to 10 seconds. The effective decay
time constant solely from this cooling+imaging loss is τimg = (1/0.027) × 10 = 37000 s, which ismuch larger than the expected vacuum lifetime and shouldn’t be limiting our lifetime measurement.
To better understand the relationship between the measured lifetime (τeff ) with respect tothe cooling pulse interval (dt) and heating rate, we calculate τeff as a function of dt and heating
rate using Eq. 3.12. First, the atom survival vs time is calculated for a given heating rate and dt.
Then an exponential decay function a × Exp(−t/τef f ) is fitted to the calculated time trace using
different parameters, and an effective lifetime τeff is extracted [Fig 3.13(a)]. All other parameters
are fixed to the values we used in the lifetime measurement.
The result from [Fig 3.13(a)] shows that at dt = 10 s with small variation in heating rate, the99perceived τeff can change dramatically between 56% to 91% of τvac . Although the dt = 10 s seems
to be a reasonable choice, it is possible that this cooling interval is not frequent enough dependingon the heating rate. This dependence between τvac and τeff was not easily observed in the cooling
duty cycle measurement in [Fig. 3.12(b and c)], which we used to determine dt = 10 s.

ab

Figure 3.13: (a) Plotting the effective lifetime τef f as a function of cooling interval. under different
dark tweezer heating rate. We see a strong dependence of the τef f when the heating rate is onlychanged slightly. (b) Single atom vacuum limited lifetime using the sequence shown in [Fig. 3.12(a)].
We measure τef f = 2800 s using the MLE method to extract the time constant. THe red dashedline is, again, to guide the eye.

Finally, we carry out the single atom lifetime measurement in optical tweezer in a cryogenic
environment with pulse cooling+imaging every 10 seconds and a total hold time in the tweezer ofup to 500 seconds [Fig. 3.12(a)]. The 500-second hold time is mainly limited by the sequence length
limit of the experiment control system. Figure 3.13(b) shows the measured atom survival versus
time for up to 500 seconds. The errors are correlated between each image, so the τeff of the arrayis determined with the MLE method, and the data is plotted without error bars. We measured aτeff = 2825+502−432 s, and the red dashed line represents an exponential curve of τeff = 2850 s to guidethe eye. With the bound we get from [Fig. 3.13(a)] and dt = 10 seconds, we can set a lower boundon our vacuum-limited lifetime (τvac ) to be at least on the order of 3000 seconds.

100

Type

Cooling
Detuning

Cooling
Intensity

RepumpDetuning

RepumpIntensity

Loss

Cooling 1-90 MHzin-trap

82 Isat

0 MHz
in-trap

0.04 Isat2(1.6) × 10−3 s−1
5(4) × 10−5 /pulse

Cooling 2-90 MHzin-trap

71 Isat

0 MHz
in-trap

0.3 Isat1.2(4) × 10−2 s−1
3.0(7) × 10−4 /pulse

Imaging 1-88 MHzin-trap

35 Isat

0 MHz
in-trap

2 Isat2.7(3) × 10−2 s−1
3.8(4) × 10−4 /img

Imaging 2-68 MHzin-trap

17 Isat

0 MHz
in-trap

1 Isat2.0(2) × 10−2 s−1
2.9(4) × 10−4 /img

Table 3.1: Single atom cooling and detection parameters

101

3.4

Tweezer FoV atom charactrization
One of the most important aspect to achieve large atom array is the FoV of the objective.

During the design and construction of the cryogenic tweezer array, a lot of care is taken to ensure
the alignment and aberration from the coldbox is under control over the temperature range from100 K down to 45 K. However, the ultimate test of the optical performance is done using atom
trapped in the optical tweezer with the system at operating temperature. While SLM is a very
useful tool to eliminate aberration from the optical systems. It is often very difficult to compensate
if the aberration is not common mode across the array. For example, compensating severe coma
and field curvature that only exist on the edge of the FoV is proven to be a difficult task [59, 127].
Since the only reliable metric is the effective NA seen by the atom, the optimization of higher orderaberration correction is increasingly difficult. These field related aberration, if left uncorrected not
only impact the uniformity of the trap, but also effect the fidelity of moving and detecting of thesingle atoms [127]. Therefore, having a good starting point of the optical system alleviate a lot ofthese complicated optimization problem for the SLM.

3.4.1

NA measurement

To characterize the performance of the objective over the FOV, we generated an array of
5x5 grids spanning over the 125 µm × 125 µm square on the tweezer plane. By measuring the trap
depth and frequency of the tweezers, the effective tweezer focus waist seen by the atom, thus NA,
can be extracted. To measure the trap depth U0 , a resonance heating beam is used to drive the(87 Rb, D2, F = 2 → 3) resonance. This resonance drive heats the atoms out of the trap, and theloss versus heating beam detuning is recorded. Figure 3.14(a) shows the measurement sequence.
Due to the AC-Stark shift from the trapping light, the resonance shift U0 is 28.8 MHz/mK. By
probing the in-trap resonance and comparing it to the free-space resonances, the trap depth canbe measured [Fig. 3.14(b)].
The trap frequency νtrap is measured with parametric heating by applying a sinusoidal modu-

102
lation to the intensity of the trap that is 5 − 10% of the trap depth for about 50-100 ms [Fig. 3.14(a)].
We discussed the process of parametric heating in Sec. 3.3.1, and according to Eq. 3.10, we expect
to see heating at 2νtrap . By scanning over the modulation frequency and measuring the atom
survival, we find significant atom loss at 2νtrap [Fig. 3.14(c)].

aResonant hea�ng

Probe

probe
heat
5-10%

1 mK

1 mK
30-100 μst50-100 ms

Image 1

Probeb

Trap modula�on

RPGC RPGC
cool img

1 mK

t

c

Image 2hea�ngortrap mod.

RPGC RPGC
cool img
trap ramp

0.1 mK
25 ms 14 ms

15 ms

25 ms 14 mst

Figure 3.14: (a) Experiment timing sequence used for trap depth and trap frequency measurements.
(b) Typical trap depth data measuring atom survival over heating/kick out beam resonance. The
dip, shifting from zero detuning allows us to extract the trap depth with a conversion of 28.8
MHz/mK (C) Typical trap frequency measurement at 0.827 mK shows atom loss at the modulation
frequency twice the trap frequency.

After measuring the trap depth and trap frequency of each individual trap across the FoV,
we can determine the waist and NA using the following equation [115].
s
λtrap
1
4kB U0 1
w0 =and, NA = 0.427, U0 : in Kelvin
2πmRb νtrap
w0(3.13)

Plotting the NA of each tweezer at the corresponding location over the FoV of the objective
focal plane shows the distribution of the NA [Fig. 3.15(a)]. We observe that at one corner of the
FoV, the NA is slightly lower by 0.02, which is only less than 5% difference from the center, andtherefore does not pose significant impact on the trapping performance. Taking the average of thedata from the measurement [Fig. 3.15(b)] demonstrates, on average, diffraction-limited performanceof the objective up to NA=0.48 with a standard deviation of 0.01, which aligns well with the optical

103
characterization conducted in Sec. 2.3.3. It’s worth noting that the 125 µm × 125 µm FoV here islimited by the AOD scan angle, and in theory, the objective can achieve a 225 µm × 225 µm FoVwith the SLM setup, over four times as large.

a
b

0.49

0.47

Figure 3.15: (a) Plotting NA over the 125 µm × 125 µm FoV of the tweezer. This shows that our
tweezer is relatively diffraction limited over the FoV. (b) The distribution of the NA over the fieldof view showing an average NA of 0.48 and standard deviation of 0.01. The number in x axis
corresponds to the the tweezer location in (a).

3.4.2

LGM loading

In the typical optical tweezer array experiment, the probability of loading single atom into theoptical tweezer is typically around 60%. This is due to nearly equal probability of initially loadingeven and odd number of atoms into the trap and after light-assisted collisions, this probability is
projected into parity 0 and 1 atom remaining in the trap. The LGM loading described in Ref. [84]is a useful tool to overcome this barrier. In the application of creating a large array, higher initialloading probability can significantly reduce the rearrangement step required to assemble a defectfree array. It has been demonstrated that in single tweezer, loading rate as high as 90% can beachieved and for a larger 10x10 array loading rate can be as high as 80% [84] and 92.7% [95]. In
this section we demonstrate efficient LGM enhanced loading across the 100 µm sq FoV.
In the typical red PGC loading, when an atom is excited to a excited state of 5P3/2 from the104
87 Rb D2 transition it can form a molecule with another 5S
1/2 atom in the trap.

This molecularpotential is attractive and the atom moves closer to each other. When an atom emit an photon anddecays, this potential no longer exist and the extra energy gain kicks both atom out of the trap,
losing two atoms at a time [red solid and dash transitions in Fig. 3.16]. At the end, depends on theinitial atom number in the tweezer, even or odd, 0 or 1 atoms remains. Therefore, the single atom
loading probability equals to the initial probability of loading even or odd number of atoms in thetrap which is around 50-60%.
In the case of LGM, the cooling light cycles through the 87 Rb D1 transition and is blue
detuned. The resulting molecular potential between 5P1/2 and 5S1/2 is repulsive [blue solid anddashed transitions in Fig. 3.16]. Thus when the atom decays it doesn’t have enough energy tokick both atom out of the trap and only one atom is lost. This results in the final atom number
independent of the inital atom number in the trap. Thus achieving single atom loading probability
greater than 90%

Figure 3.16: Illistration of LGM molecular potential verses typical red PGC potential. The red
detuned coolign light creats bound state that ejects atoms in pairs during light assisted collisions.
Causing the initial even or odd atom number in the trap to be projected into 0 or 1 remaining in
the trap after cooling. The blue detuned cooling from the LGM only ejects one atom at a timethus only one atom remains after cooling independent of initial atom number. [84]Typically LGM light assisted collision process is slower comparing to the RPGC light assisted

105
collision process. Therefore, to really maximize the loading rate in the order of 90%, the bias
magnetic fields needs to be carefully compensated. The dynamic bias field adjustment is critical
for the LGM to work over the whole cooling time. Due to the strong eddy current being generated
by the copper parts at cryogenic temperature, the time constant for the eddy current to decay arein the order of 10’s to 100 ms depends on the direction of the applied field. To compensate thechange in bias field over time, we adjust the bias coil current every 40 ms over the 200 ms coolingtime. We perform a 2D scan of the loading rate as a function of bias field over differnt times andfind the optical field setting for each time step [Fig. 3.17(a)]. If we plot the optimized field verses
time, we can see the magnetic field decay that resemble an exponential decay due to eddy current[Fig. 3.17(b)]. A more detailed magnetic field switching behavior of the coldbox is measured inSec. 4.3 using the magnetic field dependent microwave transition.

ab

Figure 3.17: (a) LGM loading efficiency vs bias field scan at different times. We see the area of
maximum loading shift as the bias magnetic fields settles due to decay in eddy current. (b) Plotting
the x-bias and y-bias field as a function of time showing the decay of the eddy current. The blue
trace is x-bias field, the orange trace is y-bias field and the orange dash is y-bias field but re-scaled
to better compare it to the x-bias field. The more precise dynamics of the time constant for thecold box is measured in Sec. 4.3.

106
To verify that the tweezer is uniform enough to provide good loading across the full FoV. We
perform a loading rate scan as a function of LGM 2-photon detuning (from 87 Rb, D1, F = 2 → 2′ )and the trap depth [Fig. 3.18(a)]. The region of maximum loading are nearly identical across thearray, meaning good trap balancing and homogeneity across the FoV. Average the loading rate over
all tweezers, we can obtain a loading rate plot where the maximum array loading can be identified[Fig. 3.18(b)]. At the end we choose to operate at the detuning and depth identified by the redstar in [Fig. 3.18(b)]. Plotting the loading rate across the tweezer index [Fig. 3.18(c)] shows on
average of 83% single atom loading rate with a few traps achieving > 90% loading. More careful
optimization can probably push this number to > 90% but this level of enhancement is sufficientfor our application now.

ba

95 MHz, 1.2 mKc

95 MHz, 1.2 mK

Figure 3.18: (a) The 2D scan of loading rate as a function of single photon detuning of the coolingbeam and the trap depth. This allows us to see wheteher the traps are uniform enough to achieve
good loading over the entire FoV. (b) Average loading rate over all the tweezer into one plot tofind the optimal loading rate for the arrya over the full FoV. The star denote the final detuningand depth we choose to operate. (C) Plotting the loading probability over all the tweezers with theloading parameters (stat point in (b)). Showing an average of 83% loading rate with site-to-site
fluctuations of ∼ 5%. The maximum loading rate for single site can be as high as 93%.

107

3.5

Impact of vacuum, loading rate and imaging loss on achieving large
atom array
Recent progress in quantum simulation [44, 139] and quantum computation [32, 35] has

sparked a broad effort towards single-atom control of large atom arrays. There are significant
incentives to scale up fully programmable single-atom array platforms up to few thousand atoms.
Proposals to explore collective behavior in large ensembles [140] or to demonstrate verifiable quantum advantage [141] can greatly benefit from configurable arrays with thousands of atoms. In
quantum information, even with the most resource-efficient error correction protocols, with current
two-qubit gate errors in the order of 10−3 , encoding more than 100 error-corrected logical qubits
requires several thousand physical qubits [142, 143].
In this section we are going to discuss the impact of vacuum-limited lifetime, loading rate andimaging loss to the creation of large atom arrays. These losses effect the probability of achievinga large atom array in different ways. Some are deterministic and correctable, for example vacuum
loss and rearrangement loss. Some are non deterministic and not correctable, for example, coolingloss and imaging loss. First we will discuss the effect of cooling and imaging loss on achieving large
array.

3.5.1

Imaging loss vs detuning and intensity

In Sec. 3.3.3 we measure the imaging loss down to 3.8(4) × 10−4 /img using the (Imaging
1) parameters in table 3.1. However we would like to have a better understanding of what is thedependence of imaging loss with respect to different laser parameters. so expanding on the resultin Sec. 3.3.3, we explore larger range of detuning and intensity of the imaging beam.
We perform a series of 2-dimensional scans with the sequence used in imaging loss measurement [Fig. 3.11(a)], varying the intensity from 40 to 230 Isat and detuning from -25 to -65 MHz
from free space resonance of the imaging beam. The atom survival is recorded over the imaging
time under these different conditions and the loss rate per image is extracted [Fig. 3.19(a)]. The

108
red star in Fig. 3.19(a) marks the parameter (Imaging 1 in table 3.1). The data here is plottingwith normalized intensity (scattering rate). This takes into account the detuning dependence of
the imaging beam. With this, we found that with sufficiently large detuning (> 25 MHz from freespace) where the PGC imaging is cooling the atom sufficiently, the imaging loss follows a linear
relationship with the scattering rate in the range between 0.03 to 0.1 Γ. Outside of this range, theimaging loss rate becomes exponential to the scattering rate.

ab

Figure 3.19: (a) Imaging loss vs. detuning and intensity. Not normalized to 42 photo-electron counton camera. (b) Imaging loss vs. detuning and intensity after normalizing to 42 photo-electron counton camera by scaling the loss as a function of time. Assuming linear photon collection rate overthe single camera exposure time.

However, if we consider the goal of the imaging is to only collect enough photon for certain
fidelity. Once the amount of photon and imaging fidelity is reached, we stop the imaging process.
Then one can rescale the imaging loss in Fig. 3.19(a) based on the imaging time required to collectcertain camera photo-electron count. Here we normalize it to the camera photo-electron countwith the imaging parameter noted by the red star in Fig. 3.19(a), which is the parameter usedin Fig. 3.11(b). This shows a universal scaling of imaging loss, in the range of 0.03 to 0.1 Γ,
only depends on the collected photon and independent of detuning, intensity and scattering rate
[Fig. 3.19(b)]. One hypothesis of this fix loss rate is when atoms are in the excited state they are
anti-trapped by the tweezer, thus experience loss as a function of how long they stay at the excitedstate. A more systematic study would be an area of interest for future high fidelity imaging andrearranging of large atom arrays, but it is beyond the scope of this thesis. The result of further

109
study will be documented in Zhenpu Zhang’s thesis.

3.5.2

Effect of imaging loss on defect-free array

Imaging and cooling losses are important for creating large defect-free array because they are
essential steps during the atom rearrangement operation. During the rearrangement, the atoms
can heat up and potentially escape the trap, which requires cooling to reset the atom temperature
before subsequent operation whether it’s imaging or more atom move steps. In the case of imaging,
the last operation following rearranging is always to take an atom array image to verify the resultof the rearrangement. The cooling and imaging creates an nondeterministic loss that can not becorrected by more rearranging because these operations occurs after the atom rearrangement. To
visualize the impact of the cooling and imaging loss we perform a Monte-Carlo simulation of theprobability of obtaining a defect free array with rearrangement and the presence of imaging andcooling loss. The simulation is set up according to the sequence in Fig. 3.20 as the following:• The atoms are first loaded in an array with some loading fraction that is probabilistic.
Typically with loading fraction of 60-80%.
• An image sequence is applied and the atoms in the array are subjected to some loss (i.e.
remove from the array with some probability according to the imaging loss). This is to
identify the tweezer site that has atom loaded.
• Then a rearrange step is applied and the atoms are removed randomly according to rearrangement loss similar to the step before. The rearrangement time is set according to thealgorithm used and the defect-free atom number N.
• Then another image is taken and the image loss is applied to the array again. This is toverify the result of the rearrangement.
• During all the steps the vacuum loss is also applied according to the time it takes for each
operation.

110• Depends on the requirement of the simulation, either more rearrangement is performed, orthe final result of defect-free probability and average array filled fraction is recorded.
Unsorted arrayFirst
rearranging~ O(N)

load atoms
stochas�c
60%-80%Ini�al image

Defect-free probability
array ﬁll frac�on

Small defects
Subsequent
rearranging
10s of ms

cool
+
img

nrearrange

Veriﬁca�on image
Veriﬁca�on image
Cool + Img loss
Img loss
Img lossrearangemnt loss
rearangemnt loss
Vacuum loss

�me

Figure 3.20: The sequence diagram used to program the rearrangement Monte-Carlo simulation.
Atom are loaded stochastically into an array with 60-80% loading probability. An initial image istaken and the imaging loss is applied. Atoms are rearranged, during that time the rearrangement
loss and vacuum loss is applied. A verification image is taken and the imaging loss is applied
again. Depends on the simulation, extra rearrangement is performed or the final defect-free arrayprobability and array fill factor is extracted.

We simulate two imaging loss, one is the imaging loss (0.45%) commonly observed in othertweezer experiments [144, 145]. Another one is the imaging loss we have (0.03%) we measured inSec. 3.3.3. We see that even with rearrangement, the defect-free array probability drops dramatically from ∼ 60% to less than 1% (in the case of 1000 qubits) when the imaging loss changes from
0.03% [solid lines in Fig. 3.21(a)] to 0.45% [dashed lines in Fig. 3.21(a)]. We see that at 0.45%
imaging loss, the dominant source of defect is from imaging and not the vacuum lifetime that is
attainable at room temperature tweezer experiments (around 300 seconds) [dashed lines comparing
to green solid line Fig. 3.21(a)]. However, for imaging loss of 0.03%, a vacuum lifetime on the orderof 1000 seconds becomes the limiting factor. Improvements in vacuum lifetime therefore increase
the probability of a defect-free array [blue solid line and orange solid line in Fig. 3.21(a)].
If we perform additional round of rearrangement, we see that the probability of obtaining
a defect-free array remains almost identical [Fig. 3.21(b)] showing that this nondeterministic loss
is not correctable with more rearrangement rounds unlike vacuum or rearrangement losses. This

111
simulation result [Fig. 3.21(b)] matches quite close with the result from [144] where they observe
the defect-free probability saturates at ∼ 33% after multiple rounds of rearrangement with an arraysize of 225 atoms and imaging loss of 0.45%.

ab

Figure 3.21: (a) Monte-Carlo simulations of single round of rearrangement show results for 0.03%
imaging loss (solid lines) and 0.45% imaging loss (dashed lines) with different vacuum lifetimes(green: 300 s, blue: 1000 s, and orange: 3000 s). For 0.45% imaging loss, the vacuum lifetime is
not a limiting factor for creating a defect-free array (dashed lines). However, for 0.03% imaging
loss, increasing the vacuum lifetime from 300 seconds to 3000 seconds dramatically increases theprobability of a defect-free array (b) Monte-Carlo simulations of multiple rounds of rearrangementwith 0.45% imaging loss show that the imaging loss is not deterministic and cannot be corrected
through additional rounds of rearrangement.

3.5.3

Effect of vacuum loss on defect-free array

To see the impact of the vacuum lifetime on success rate of obtaining a defect free array. We
use the same code as before, but this time we turn off all the losses other than vacuum limited loss.
This allows us to isolate the effect of finite vacuum lifetime. We simulate the defect-free probability
with three different vacuum lifetime, 300 seconds (typical state-of-the-art room temperature tweezer
experiment), 1000 seconds (best record of room temperature tweezer experiment) and 3000 seconds(our cryogenic tweezer vacuum lifetime) [Fig. 3.22(a)]. In the simulation, the rearrangement time isdefined to be 50 + N ∗ 0.85 ms which is linearly increasing as a function of final array size (N) [27].
This time scaling was shown to be fairly efficient in the case of atom-by-atom rearrangement without
parallel move of multiple atoms for rearranging a compact array [145]. From the simulation we see

112
a substantial improvement (around a factor of 10) of obtaining a defect free array while vacuum
lifetime increases from 300 to 1000 second (only a factor of 3)[blue and orange line in Fig. 3.22(a)],meaning that a threshold for vacuum lifetime around 1000 seconds is needed to achieve usabledefect-free array with the size of ∼ 1000 atoms.
To further increase the defect-free probabilities, one can perform multiple round of rearranging. Since the subsequent round of rearranging requires less time, the vacuum limited lifetime hasless effect on the atom loss [Fig. 3.22(b)]. However, there is always a minimal wait time required
between subsequent rearrangement, therefore, the defect-free array probability will saturate by thevacuum lifetime limited loss during wait [dotted line in Fig. 3.22(b)].
Another way of increase the defect-free probability is with parallel move of the atoms. In thiscase, the rearrangement time can be further reduced and the threshold of ∼ 1000 s can be lowered,
but this is done in the expanse of lower rearrangement fidelity [144].

ab

Figure 3.22: (a) Monte-Carlo simulations of single round of rearrangement with different vacuum
lifetime (blue: 300 s, orange: 1000 s, and green: 3000 s) shows substantial improvement (around afactor of 10) of obtaining a defect free array at 1000 qubits while vacuum lifetime increases from
300 to 1000 second (only a factor of 3) (b) Monte-Carlo simulations of multi-round rearrangementwith different vacuum lifetimes (solid: 3000 s and dotted: 300 s) shows that, with a 300-second
vacuum lifetime, even with multiple rearrangement cycle, the defect-free probability saturates at
a value lower than that achieved with a single round of rearrangement and a 3000-second vacuum
lifetime.

Chapter 4

Cryogenic tweezer qubit manipulation

4.1

Overview
Note: For the most up-to-date analysis of ground state qubit rotation and Rydberg excitation,please refer to the upcoming publication ”A High Optical Access Cryogenic Tweezer Array with a
3000-Second Lifetime” (in preparation).
One of the important aspect of our apparatus is to be able to control the atoms in the relevant
state where the qubits are encoded and the entanglemets are realized. Typically the qubits are
encoded in the ground state of the atoms in alkali-earth atoms these are nuclear spins [95] and inalkali-metals these are hyperfine structures [29]. These states are less sensitive to environmental
perturbation, so they can have coherence time in the order of seconds to 10’s of seconds [32,35]. To
manipulate and prepare atom to these states, optical pumping with microwave or Raman transitions
are typically required. For the control interactions, these are typically done in Rydberg state where
the long range dipole-dipole interactions are available to provide entanglement [23–25]. These statesat sensitive to local electric field fluctuations and require a stable field either through electrodes ordissipation of accumulated charge.
In this section we show that our cryogenic apparatus is capable of such manipulations andpreparation of quantum states and provides good environmental stability for the ground state single
atom control. We will show the microwave rabi oscillations between ground state hyperfine levels.
This will provide insight about the magnetic field stability of the apparatus and also the switching
dynamics of the bias fields due to induced eddy current from metal parts in cryogenic temperatures.

114This is an important first step showing single particle quantum state control that is required forquantum simulations and quantum information.

115

4.2

Microwave
To drive microwave transition between |F, mf ⟩ = |1, 1⟩ ↔ |2, 2⟩ the atom needs to first be

prepared in |2, 2⟩ state with the optical pumping pulse. The optical pumping light is resonant
between F = 2 ↔ 2′′ in the 87 Rb D2 transition with a pure σ+ polarization. Combining with arepump resonant to F = 1 ↔ 2′′ . This creates a dark state at |2, 2⟩ where the atom eventually
accumulates. Figure 4.1 is the level diagram for optical pumping, repump and state selective pushout beam (same as MOT cooling beam). The push-out beam resonant to the cycling transition
F = 2 ↔ 3′′ allows us to push atoms out of F = 2 and detect population remaining in F = 1
manifold.

Rydberg
D2 780 nm
σ+ OP

σ+ OP

σ+ OP

σ+ OP

420 nm

Cool
RP

OP

σ− π σ+ 6.8 GHz
0.7 MHz/G

mF = -2

-1

0

1

2

Figure 4.1: Level diagram for the ground state qubit (green box) and Rydberg transition (bluedashed box). The ground state manifold consists of F=1 and F=2 Zeeman manifold with magnetic
quantum number mf = 0, ±1, ±2. The energy seperation between F=1 and 2 is around 6.8 GHz.
The repump and optical pumping for state praperation is within the 87 Rb D2 manifold. OP is onresonant with F = 2 ↔ 3′′ and purely σ+ polarization. Repump is on resonant with F = 1 ↔ 2′′ .
The Rydberg transition is a two photon transition with intermediate state detuning around 10 GHz
from 62 P3/2 . The intermediate state is driven with a blue 420 nm photon and the final state isdriven with a NIR 1013 nm photon.

Two components need to work in unison to achieve optimal optical pumping, the quantization

116
axis defined by a bias magnetic field, and polarization of the OP beam. To ensure polarization
purity, a Glan-Taylor polarizer is used to generate the initial linear polarization to a contrast of> 10000 : 1, following a l/2 and l/4 waveplates to compensate any birefringence from the viewports
and the coldbox windows. The σ+ polarization is first measured using a polraization analyzer, thisgives a good starting point where the magnetic field can be aligned to the OP beam following theprocedure described in Ref. [101]. Figure 4.2(a) is an example scan of atoms in F = 1 vs bias field.
Then the l/2 and l/4 waveplates are adjusted to maximize the depumping time of the atom out of
the |2, 2⟩ state.
After optimization, we measure the depumping rate out of the |2, 2⟩ using only the OP
beam without repump. If there are any unwanted polarization the population will leak out of|2, 2⟩. We measure a depumping rate of τdp = 0.95 ± 0.13 ms [Fig. 4.2(b)]. Comparing this ratewith our typical OP beam pumping rate of about τop 10 µs we get the steady state population
τop
P|2,2⟩ = 1 − exp − τdp
= 99%

a

b

c
τdp = 0.95 ± 0.13 ms

Figure 4.2: (a) Atoms in F = 1 vs bias field. When the bias field is optimized, the atom is mostly
in |2, 2⟩ therefore by minimizing the F = 1 population we can optimize the optical pumping bias
field. (b) Atom in F = 1 vs OP beam on time. Since atoms that fall to F = 1 manifold and in|2, 2⟩ will remain dark to the OP beam, we can use this to determine the impurity of polarizationin OP beam by looking at the dumping rate out of F = 2 manifold. We find a depumping rate ofτdp = 0.95 ± 0.13 ms (c) OP pumping time vs population in F=1 after a 0.5ms depumping time.
This is to used for optimizing the pulse duration of the OP beam to make sure it is sufficiently longto maximally pump the atoms into |2, 2⟩

117
4.2.1

B-field calibration and noise

We use the microwave antenna with a cold bandpass filter described in Sec. 2.2.8 to couple
6.8 GHz microwave into the coldbox. This is a split loop antenna with a loop circumference half the
microwave wavelength. Effectively acting like a half-wave dipole antenna folded into a circle. The
antenna is around 12 mm to the atoms. Figure 4.1(a) shows the ground state zeeman manifold andmicrowave drive coupling the states. Depending on the mf state the energy shift of the Zeeman
levels follows the ∆Zeeman = mf × B × 0.7 MHz/Gauss.
To perform microwave spectroscopy on the atoms, we first prepare atoms in the F = 2
manifold, not necessary in the stretch state (|2, 2⟩). To do this we simply turn on a repump pulse
for 50µs to pump the atom out of the F = 1 manifold. This allows us to simultaneously makesure the push-out beam is functioning as intended. Once the atoms are in F = 2, the microwave isturned on and the population is scrambled across F = 1 and F = 2 manifolds when the microwave
frequency is on resonance with one or more Zeeman sublevel. Finally the atoms in F = 2 arepushed out and population in F = 1 is detected, the sequence is shown in [Fig. 4.3(a)]. The long
750 ms wait is to ensure all magnetic field switching dynamics has settle to a steady state since wedidn’t know the decay time constant at that point. With different combination of Zeeman levels inboth F = 1 and F = 2 we expect to see peaks that follows ∆fµw = n × B × 0.7 MHz/Gauss, where
n=0,1,2,3.
With a non-zero B-field we see a series of microwave resonance peaks that are evenly spaced[Fig. 4.3(c) left panel]. As the field reduces we see the different resonances merge to a single peak.
The slope of frequency difference and bias coil control [dashed lines in Fig. 4.3(c middle)] can beused to calibrate the bias coil current to B-field ratio at the atom location. The calibrations for thecoil gives (x,y,z) = (5.52, 5.92, 12.52) G/A which is very close to the predicted value in Sec. 2.2.7.
Finally we can estimate the level of how well we can zero the field by looking at how well we canresolve the splitting in the Zeeman sublevel. The linewidth of microwave spectroscopy is roughly
20 kHz [Fig. 4.3(c) right] so 20 kHz/(3 × 0.7 MHz/Gauss) ≈ 10 mG is the field zero error.

aImage 1

Image 2probe RPGC set& OP img B-ﬁeld

750 ms

F=2
OP

118

b

F=2
push RPGC
out imgμw
RMS ~ 100 μA
25 ms

c

~ 1ms

40 μs 0.1 ms 40 μs

n=3
n=2

n=1

14 ms

t

n=3

n=2

n=1

n=0

~20 kHz

Figure 4.3: (a) Timing sequence for microwave spectroscopy. The 750 ms wait between setting thefield and optical pumping is to ensure the switching dynamics has already settled. We didn’t know
the decay time constant at that point hence the long wait time. (b) The measured current noise
spectrum in the bias field coil driver, integrate over the frequency gives 100 µA current noise. The
current to field strength conversion determined in (c) gives a RMS magnetic field noise of 1.2 mG.
(c) Resonance scan used to zero the field, showing the merges of the different Zeeman sub levels.
Microwave resonance vs applied current on bias field current controller.

Now with the bias coil current to B-field conversion obtained from the microwave spectroscopy. We can convert the current noise measured from the coil current driver to the B-field
noise. The current noise spectral density of the bias coil current driver is measured with a 4Ω
sense resistor connected on the output [Fig. 4.3(b)]. The integrated RMS current noise from thespectrum is ∼ 100 µA which translate to ∼ 1.2 mG RMS B-field noise.

4.2.2

Rabi

The polarization of the microwave is not controlled, but applying a bias magnetic field we canlift the degeneracy of different mf levels within F = 1 and 2 manifold. Then we can selectively drive
σ+,π and σ− transitions by tuning the frequency of the microwave to be resonant to a specific mflevel. We typically operate at a field of ∼ 3.5 Gauss in x-direction (perpendicular to the tweezer)

119which is sufficiently large to eliminate the fictitious field created by the tweezer. The fictitious fieldoriginates from tightly focusing of a linearly polarized light to a point that the Rayleigh range ofthe beam are in the order of a few wavelengths. Due to the sharp change in ⃗k, the polarization
creates a gradient when traveling through the focal point of the light and create a effective circularpolarization transverse to the polarization plane. This in term creates an effective magnetic field
pointed in the transverse direction w.r.t. tweezer propagation direction [146]. At 0.55NA and thetrap depth of 1 mK this field is around 0.1 G/µm near the tweezer focus.
To drive the rabi oscillation we first optically pumped to the stretch state |2, 2⟩. Depends
on the microwave transition we wish to drive, we either transfer atom to |1, 1⟩ with a π-pulse or
directly drive from |2, 2⟩ [Fig. 4.4(a)]. By changing the microwave pulse length and measure thepopulation remaining in F = 1 using the state selective push-out (push out atoms in F = 2) we
can record the Rabi oscillation of σ+, π and σ− transitions [Fig. 4.4(c)]. Fitting a decaying sinefunction to the data we extract the Rabi frequency of (σ+, π, σ−) = (5.98 ± 0.01, 22.6 ± 0.4,
2.05 ± 0.04) kHz.
From the COMSOL simulation given the drive power of 1 W into the antenna, we can plot
the microwave B-field strength vs distance to the loop antenna [Fig. 4.4(b)]. At a distance of 12 mm
the B-field is ∼ 20 mG. Scaling this to 6 W of microwave power we should have ∼ 2π × 42 kHz, notconsidering the polarization of the microwave field. We can calculate the equivalent Rabi frequency
from our measurement using the dipole matrix element of the microwave transition (σ+, π, σ−) =p√( 3/2, 3/2, 1/2) µB . Normalize the rabi frequency to the σ+ transition and take the quadrature
q
√ 2
√ 2
sum we have 5.982 + 22.6 2 + 2.05 6 = 33 kHz. This is in good agreement with the 42
kHz prediction considering error in distance from microwave antenna to the atom and losses in the
microwave cable and cold filter assembly.

Image 1b

Image 2

F=2
push RPGC
out imgprobe RPGC |2,2⟩& OP img OP
tRabi

μw
25 ms 40 μs

c

40 μs 14 mst

120
Magne�c ﬂux density norm (mG)ain experiment ≈ 20 mG @1W

Atom to antenna distance (mm)

𝝈−

π

𝝈+
5.98(1) kHz

22.6(4) kHz

2.05(4) kHz

Figure 4.4: (a)Sequence for Rabi oscillation measurement. Consists of optical pumping to |2, 2⟩,pulse on microwave for variable amount of time and push out atom in F = 2 manifold. (b) Simulation of the field strength from the half-wave loop antenna gives around 20 mG/W at the atom
location in our experiment. From this we can calculate 6 W microwave input to the antenna will
result in 42 kHz total Rabi frequency. (c) Rabi oscillation of σ+,π and σ− transitions from themicrowave input power of approximately 6 W into the antenna. Each transition are measured withthe same microwave power and on resonance to the particular transition that is being driven.

4.2.3

RamseyFinally we perform Ramsey spectroscopy at the atoms between |1, 1⟩ and |2, 2⟩ states. Again
the state is Initialized to |2, 2⟩ via optical pumping. Then a π/2 pulse is applied to rotate the state
to the equator of the Bloch sphere. After a variable wait time TR , a second π/2 pulse is applied andthe phase of this second pulse is scanned from 0 to 2π to obtain the Ramsey fringe. The sequenceis illustrated in Fig. 4.5(a). Once the second π/2 pulse is applied the population in the F = 2 isdetected with the push-out beam and the survival (Ramsey fringe) is recorded [Fig. 4.5(b)]. We also
perform a spin echo version of the same measurement by adding a π pulse in the middle of the wait
time TR /2 [Fig. 4.5(a)], to first order cancel out decoherence due to environmental perturbationsuch as magnetic field noise and we obtain a much longer coherence time. We plot the Ramsey

121
fringe contrast as a function of wait time, assuming the decoherence is caused by a random process
we can fit a Gaussian to the decay of the contrast [Fig. 4.5(c)]. The data is plotted in log scalein the wait time due to the large difference in decoherence time between echoed and non-echoedcases. For the measurement without spin echo we measure a T2∗ = 131(5) µs and with spin echo wemeasure a T2 = 0.95(5) ms almost an order of magnitude longer.
We can convert the T2∗ measurement to the magnetic field noise using the following conversion[147].
∆B = ∆f (2.1 MHz/G) , where∆f =1
2 π T2∗(4.1)

With the T2∗ = 131(5) µs and the time it took to take the data we have ∆B ≈ 0.5 mG in ∼ 10 min
time scale. This is comparable to the magnetic field noise measured in other cryogenic experiments,for example in Ref. [47] they measured a 1.2 mG with bandwidth up to 100 Hz.

c 1.0
Image 1

Image 2probe RPGC |2,2⟩& OP img OP
π/2
φ0

μw
25 ms 40 μs

tRamsey/2

π
echo

π/2
φ1
tRamsey/2

40 μs 14 ms

T2 = 0.95(5) ms

0.8

F=2
push RPGC
out img

Fitted contrast

a

t

0.6

b

TR=130 μs

T2* =131(5) μs

0.4
0.2
0.0

10−2

10−1
Ramsey time Tr (ms)100

Figure 4.5: (a) The experimental sequence for the Remsey measurment. (b) Typical Ramsey fringe,
atom survival in F=1 vs phase of second π/2 pulse. (c) The contrast vs. wait time between two
π/2 pulses. Fitting the decay to a Gaussian we can extract the coherence time with and without
echo. From this we infer a RMS magnetic field fluctuation of 0.5mG over few minutes time scale.

122

4.3

Eddy current measurement
In the apparatus, we use a large amount of copper for it’s excellent thermal conductivity.

For metal, if it has good thermal conductivity almost always means it also has good electrical conductivity. For standard OFHC copper used in typical cryogenic application, the residual resistanceratio is typically between 50-100. This means they are highly conductive with very low residual
resistance. This creates a problem when the external magnetic field is subject to change, for example, during the switching of the quantization axis. Due to it’s proximity to the atoms, the local
magnetic field seen by the atom is perturbed by the induced eddy current in the nearby copper. The
induced eddy current is almost unavoidable as long as the OFHC is used and the system is undercryogenic temperature. In Ref. [58], they did not observe significant adverse effect in loading andcooling of single atoms from the long decay time constant of eddy current in the OFHC used in thecryostat. As a precaution, we put slits on our coldbox and with a combination of higher resistancecopper alloys, the resistance of the current loop is increased hoping to reduce the effect of eddy
current. Initially we had the same observation as Ref. [58] where the loading of MOT and coolingof single atoms are okay, but as we start to setup optical pumping and microwave spectroscopy,
we observe instability in setting of the magnetic field. These operation requires a more dynamic
switching of the magnetic field up to a few Gauss to create a quantization axis. In addition, we also
see a substantially different magnetic field settings for cooling and tweezer loading between when
the apparatus is at 300 K vs at 45 K. The difference can be as large as several Gauss. This lead us
to believe that a substantial amount of eddy current is present in our system.
Therefore, we decided to to perform a careful study of the magnetic field switching characteristics with microwave spectroscopy we described in Sec. 4.2.1. After the first atom image, thefield is zeroed and wait for 750 ms for the field to settle. Then the atoms are prepared in F = 2 via
optical pumping following with a magnetic field jump. After the field jump a microwave pulse of 0.1
ms is turned on with variable delay (Twait ) and detuning, [Fig. 4.6(a)] is the experiment sequence
for the measurement. This gives a microwave spectrum [similar to the one in Fig. 4.2.1(c)] as a123function delay time.
We clearly observe a double exponential decay where a fast decay in the order of (1-2 ms)
corrospondes to coil current switching and a slower time constant (5-100 ms), depends on the axis,
corrosponds to the induced eddy current decay. Fitting to the time trace, we obtain the slow
switching time constant (τx , τy , τz ) = (100 ms, 60 ms, 5 ms) for three axis of the coldbox. Thesetime constant qualitatively match the coldbox geometry. The z axis which has the fastest decay iswhere we put the slit. The x axis that has the longest time constant has the thickest copper crosssection.

a Image 1

Image 2

RPGC zeroimg B-ﬁeld

jump
750 ms F=2 B-ﬁeld
OP

750 ms

Twait

25 ms

~ 1ms

40 μs

μw
0.1 msbτcx = 0.8 ms

τx = 100 ms

F=2
push RPGC
out img

40 μs

14 ms

t

τcy = 0.6 ms

τy = 60 ms

τz = 5 ms

Figure 4.6: (a) The experimental sequence for characterizing the eddy current of the coldbox in x,yand z axis. (b) The magnetic field at the atom location as a function of time after the bias field
is jumped. The time trace shows a double exponential decay where a fast decay (∼1 ms) is frommagnetic field coil switching and the slower (5-160 ms) decay is from the eddy current induced in
the coldbox. We measured the slow decay time constants (τx , τy , τz ) = (100 ms, 60 ms, 5 ms)

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4.4

OutlookThis thesis describe several efforts of creating a new type of experimental platform where

cryogenic environment is leveraged to provide better isolation of the atom arrays from the environment. We focus on two aspect of isolation, one is minimizing the collision between the atomwith residual background gas. Another one is isolating the Rydberg atom from the microwave
BBR background. The effort of reducing the effect of residual background gas collision throughtrapping atoms in a cryogenic environment, combining with atom rearrangement provide a way ofcreating long lived atom arrays for quantum simulation and computation. In the meantime, recent
advances in technique of driving Rydberg transition [148] in conjunction with the microwave BBR
shielding provided by the cryogenic environment further improves coherence comparing to at room
temperature. The benefit of cryogenic tweezer array will allow us to access experimental parameter
space previously limited by the vacuum lifetime and coherence.
Combining cryogenic environment with high NA optical access comes with it’s own set ofchallenges. Aside from the thermal considerations typically associated with cryogenic setup, maintaining stringent optical alignment over a large temperature range is required to maintain the
optical performance of the microscope objective. We have demonstrated a series of technique that
we developed to achieve stable optical alignment from 400 K to 40 K. These techniques are essential for future application of cryogenic environment to optical tweezer and quantum gas microscopeexperiments.
Beyond the experimental advances, we are able to study imaging and cooling losses to a very
low level (10−5 ) owing to the excellent vacuum lifetime the cryogenic tweezer is able to provide.
These losses are previously not accessible since the typical vacuum lifetime being in the order of
300 seconds and only until recently 1000 seconds [43]. Achieving low imaging and cooling losses
are essential for advances in high fidelity quantum gate operation, for example, in transport based
protocol where atoms are extensively manipulated by the tweezers. In the context of large array,
low imaging and cooling losses are important since these losses are nondeterministic and can not

125
be corrected with more rearrangement rounds as discussed in Sec. 3.5.2.
Furthermore, we carefully characterize the cryogenic environment with atoms including measuring the magnetic field stability and eddy current decay from the coldbox. The next step would
be setting up the Rydberg lasers and measuring the coherence and electric field stability inside the
coldbox with the Rydberg transition.

4.4.1

Atom Array, SLM and optical lattice

Currently there is only one set of trapping potential from AOD tweezers. AOD tweezer,
although very configurable dynamically, suffers from RF intermodulation and is not suitable forholding the atoms long term. In addition, the arrangement of the AOD tweezer trap is restricted
on a grid and can not be arbitrary. In the immediate future, we will add a secondary potential
generated by an SLM projected through the microscope objective (Fig. 2.18). We can estimate theachievable tweezer number with the current objective FoV of larger than ±200 µm and the atom
spacing of 3 µm, to be 10000 tweezers. Combining with rearrangement technique similar to theone used in [149], enhanced loading from Sec. 3.4.2 and Ref. [84, 95] and the long vacuum lifetimeof our cryogenic platform, defect free array of greater than 8000 atoms is possible. At this point
we will be limited by the laser power and heat dissipation of the absorbed trapping light from theobjective.
As part of the design consideration of the coldbox, ultimately we plan to include bi-chromatic
optical lattice to trap Rydberg atoms and ground state atoms simultaneously. This provide a power
efficient way of generating large number of traps. Currently, one of the limitation is anti-trappingof the Rydberg state during gate operation [57]. The trapping of Rydberg state eliminates atom
loss during Rydberg excitation, therefore increase the coherence of the Rydberg drive and reduce
the error during sequential gate operations. This can also reduce the amount of hardware overhead
required for error correction [31] and mid-circuit measurements.
Following the additional trapping potential, we will also install the second microscope objective. This allows us to send one of the Rydberg light tightly focused through the objective and126
achieve site selective Rydberg state manipulation [32].

4.4.2

Cryogenic enhancement: cryopmping

In the current coldbox design, we utilize a low outgassing environment created by the 45 K
cryogenic shield (discussed in Appendix D) that enclose the atom array from the room temperature
vacuum chamber almost entirely. While this shield is very low outgassing, it’s cryopumping ability
for common residual gas in UHV chamber, e.g., H2 , CO or CH4 , are relatively weak (Fig. D.2).
Therefore, we place a 4 K cold finger with slits cut out to increase the surface area inside the
coldbox with high conductance to the atom. However, this cryopumping surface did not havedirect line of sight to the atoms and the surface area is not enough to sustain efficient cryosorption
for a long time (see Appendix D for details about cryosorption). We typically see the lifetime
degrades after a few days from 3000 seconds to around 1500 seconds and remain at that value until
we regenerate the cold finger by heating it up to 20 K for a few hours. During normal operation,
we regenerate the 4 K cold finger daily.
Given that our design minimize the 4 K cryopumping surface protruding into the coldbox,
significant improvements can be made from improving the geometry of the 4 K cryopumping cold
finger. To further increase the saturation time of the cryopumping cold finger and to achieve more
efficient pumping. Two modifications can be done to the 4 K cold fingers. For longer saturation
time, we can increase the surface area of the 4 K cold finger with large volume to surface ratio
media, e.g., porous or mesh. Typically, activated charcoal or molecular sieves are used. However,thermalizing these materials to 4 K and attaching them to the cold finger requires the use of epoxy
which is not preferred in our chamber. Alternatively, metal or ceramic of various porosity andmeshes can be used. An extensive study was done in Ref. [150] where the cryosorption ability ofvarious materials are investigated. Here are some of the potential candidates for sintered metaland porous ceramic cryosorption materials (McMaster-Carr part number: 9224T83, 9446T32 and8978T793). These are example part numbers for sintered stainless porous discs, porous alumina
and copper mesh, specific mesh size or pore size depends on the application and needs to be chosen

127
accordingly.
Additionally, the 4 K cold finger can be extended further into the cold box, increasing line ofsight exposure to the atom trapping location (Fig. 2.2.10). This allows the residual gas that enters
the coldbox through the small openings having higher probability to collide with the cryopumping
surface before interacting with the trapped atoms.

4.4.3

ITO coating

The coldbox presented in this thesis only has normal AR coatings on the glass windows.
From other experimental observation [47] microwave BBR leaking through the window opening
can significantly effect the coherence of the Rydberg state. From our estimation, we can expect up
to a factor of 50% increase in Rydberg lifetime in our current coldbox design from the microwave
BBR leaking through the glass windows on the coldbox. In the future, if we wish to take the fulladvantage of the 45 K environment, ITO conductive coating can be added to the glass windows
on the coldbox. This forms a Faraday cage where the microwave up to 100 GHz can be effectively
shielded. The ITO coatings capable of shielding microwave signals typically has a very low sheet
resistance in the order of 10 Ω/sq. One drawback for these type of ITO coating is the optical loss
becomes fairly large (∼ 10%) because the coating needs to be thick (around 150 - 200 nm) [151],
so careful planning is needed in terms of increasing heatload and coating damage from high power
lasers. One potential solution to reduce the laser absorption on the ITO coating is to use the thinner
ITO coating with larger sheet resistance on the window where high power laser shines through,for example, the tweezer ports, and thicker coating with higher optical loss on the low power
windows (MOT and probe). We expect the tweezer windows might not need as much microwave
shielding because the opening is on the order of microwave wavelength in tens of GHz range, thus
the microwave might not couple through the tweezer window holes efficiently.

Chapter 5

Metasurface lens experiment

5.1

Overview
In neutral atom tweezer array high-NA optics are key for creating trapping potentials, theoptical addressing of individual atoms in quantum gate protocols, and imaging the fluorescenceof single atoms [3, 16]. Often multielement objective lenses are required to achieve the requisite
performance [3,16,152], although single aspheric lenses have also been instrumental in state-of-theart experiments studying interacting Rydberg atoms [153]. They suffer from relatively small FoV
and chromatic focal shift.
Optical tweezers require both low aberrations to achieve tight confinement and high focusing efficiency to achieve sufficient trap depth for a given trapping power and to efficiently image
single atoms. Achromatic properties are needed for simultaneously collecting atom fluorescence,
conservative trapping in a far off-resonance trap, and often also the delivery of excitation light that
controls the atomic state in individual tweezers [154]. Broadband operation is especially importantfor multispecies or molecular optical tweezer experiments [155–157]. Further, arbitrary and cleanpolarization control is increasingly desired to provide additional degrees of freedom in controllingthe internal state of the atoms.
A long working distance (WD) is required to allow access for laser cooling beams, to maintainsufficient distance between the lens substrate and atoms in high-energy Rydberg states that are
sensitive to surface electric dipoles, and to focus light into a complex vacuum chamber or cryogenic
environments [58]. In addition, stability of the optics is crucial, for example, in registration of

129
optical tweezers and lattices or for in-vacuum applications. Further, perturbations to the trap
focus due to multi-beam interference or scattered light need to be minimized, especially if they are
not static, as these fluctuations can drive atom motion.
In this section I describe a proof-of-principle experiment where, for the first time, a high-NA
focusing metasurface lens is used to create array of optical tweezer to trap and detect single atoms.
The optical tweezer trap is then tested with atoms for their diffraction limited performance. The
result is compared to the optical measurement using subwavelength pinhole with good agreement.
Finally, I will discuss future metasurface lens design with characteristics that are desirable forsingle atom trapping. Our successful demonstration of trapping single atoms with metasurface
lens shows indicats that potential deleterious effects, for example, scattered light, the presenceof undeflected zero-order light, or deformations due to absorption and heating of the lens makes
negligible contributions to the trapping performance.

5.2

UHV system
The goal of this apparatus is to create a platform that can accommodate wide range of differ-ent metasurface samples and rapid testing of these metasurface with atoms. Specific requirements
when designing the metalens UHV systems includes• Simple design, easily adapt to different science cells• Reusable science glass cell for multiple samples• Compact design for vacuum processing (assembly, baking and metalens sample swap)• Maximize optical access for tweezer experiment• Vacuum level in the 10−10 torr range.
This results in a simple UHV chamber consists of a source glass cell, a science glass cell and a pump,
CAD model of the UHV system shown in (Fig. 5.1). The source glass cell is purchased commerciallyfrom ColdQuanta and is connected to the rest of the system through a differential pumping pinhole.

130
The science glass cell is connected adjacent to the source cell and to an ion pump with pumping
speed of 25 l/s. The estimated differential pumping ratio is given by the conductance to the pumpand the source cell pinhole which is calculated to be around 10 l/s and 0.05 l/s respectively. This
gives a differential pumping ratio of 200, allow the science cell to maintain pressure in the 10−10
torr range while the source can operate at around 10−8 torr for efficient capture of 87 Rb background
vapor by the 2D+ MOT.

Figure 5.1: Main: CAD model of the metalens UHV chamber. Source cell (right) and science cell(left) are connected through a modified 3-way in-line all-metal valve to an 25 l/s ion pump. The
separation between source MOT and science MOT is about 30 cm and the 2D MOT magnet is on
a translation stage to align the center of the magnetic field gradient to the differential pumping
pinhole. (a) Image of science MOT (white patch at the center) in front of the metalens sample(square background). (b) Image of the source MOT (white dot at the center) view from the end of
the source cell.

The 3-way connection between the source cell, science cell and ion pump is done through a
modified all-metal inline valve (MDC vacuum, P/N 316000) where a third port is added on the
tube part after the valve body. This reduces the required vacuum hardware and keep the distance
between source MOT [Fig. 5.1(b)] and science MOT [Fig. 5.1(a)] short at around 30 cm. Keeping

131
the distance short allows a denser atomic beam from the 2D MOT to reach the science cell for more
efficient atom transfer to the 3D MOT. The all-metal inline valve serve to isolate the source cellfrom the rest of the system during sample swap. When swapping the metalens sample, the valve isclosed and the ion pump and the science cell are vented to air.

5.2.1

ColdQuanta source cell

The source cell is purchased from ColdQuanta. It consists of a special glass cell with a 0.75
mm diameter pinhole to achieve the desired differential pumping and four electrical connection for
two Rb dispenser from SAES Getters (model Rb/NF/3.4/12). The pinhole on the glass cell has aconductance of 0.05 l/s 1 to isolate the higher vapour pressure inside the source cell from the restof the UHV system during operation. The dispensers are activated during bake-out of the vacuum
chamber (see appendix B for additional detail). The source cell mounts on a stainless steel adapter
plate to allow the mounting of the the 2D+ MOT magnet translation stage, [Fig. 5.2(a)].
The required magnetic field gradient for the 2D+ MOT is created through a set of four
permanent bar magnets on a translation stage, [Fig. 5.2(b)]. This provides a 37 gauss/cm 2D
quadruple magnetic field gradient along the transverse direction (cooling laser direction) [Fig. 5.2(c)]and the atom are captured at the center of the gradient (zero B-field). The relative position of
the magnet assembly can be changed w.r.t. the glass cell to align the field gradient center to thepinhole.
To operate the 2D MOT, we use two pairs of retro-reflected beams in transverse direction withelliptical beam diameters of 12 x 25 mm in σ+ and σ- configuration. The transverse cooling beams
are red detuned 2.5Γ from 87 Rb D2 F = 2 → 3′′ transition with 27 mW in each beam. Typicaldispenser current ranges between 2-3A. Using a push beam with 1 mW power, same detuning as
the 2D+ MOT beam and a diameter of 2 mm, the atoms are pushed into the science glass cell. In
this configuration the source MOT delivers an atom flux in the order of 107 − 108 atoms/s in the1
The conductance is taken from the datasheet of ColdQuanta (Infleqtion) cold atom source cell (CASC) it’s a 0.7
mm diameter pinhole on 1 mm thick silicon substrate.
2
Image from ColdQuanta (Infleqtion) CASC datasheet

a

132

b

σ-

magnetsc
σ+magnetsσ+σ-

Figure 5.2: (a) The source glass cell on the stainless steel adapter plate 2 . The pinhole on thesilicon end plate is visible in the middle. The 0.7mm diameter pinhole creates a differential pumpingof 0.05 l/s to isolate the high vapour pressure of Rb when the dispensor is operating. (b) The 2D+
MOT magnet assembly 2 . The black aluminum base is the translation stage that attaches to thestainless steel adapter plate (in (a)) for aligning the 2D quadruple zero B-field location, which
determins the 2D+ MOT location, w.r.t. the pinhole. (c) The field line of the B-field gradientcreated by the permanent bar magnet viewing from the rear of the assembly (axial direction of the
2D MOT) 2 .

science cell. To measure the exit velocity of the atom beam as well as the transverse temperature,a retro-reflected probe beam is set up to measure the size and vertical drop of the atom beam near
the end of science cell. The probe beam, launched vertically along the gravity, is retro-reflected
to reduce momentum kick on the atoms. It is on resonance and has an intensity around 10 Isat
with 5% repump co-propagating. The measured vertical drop is 3 mm and transverse FWHM is
4.2 mm. Using the simple free fall equation ∆y = 21 gt2 we can calculate the time it takes for theatom beam to travel the distance between the 2D+ MOT and the end of the science cell, which isaround 30 cm. Thus we infer the atom beam exits the 2D MOT cell with a velocity of 12 m/s, andwith eq. 5.1 we calculate the transverse temperature of the pre-cooled atom beam to be ∼ 80 µK.
s
σf =

σi +

kB T 2
t
mRb

(5.1)

133

Science cell

Metalens sample holder

Push beam

Vx = 12 m/s

2D+ MOTy = -3 mm

4.2 mm

9 mm

y = 0 mmg

30 cm

Figure 5.3: A schematic showing the source MOT w.r.t. the science cell and probing of the Source
MOT atom beam that is cooled in transverse direction. The probe beam is launched vertically(along the gravity direction) and retro-reflected to minimize momentum kick of the probe photon.
Probing at the end of the science cell (30 cm from the 2D+ MOT) shows the atom beam had an
FWHM of 4.2 mm and a vertical drop of 3 mm. This allows us to calculate the atom velocity to
be 12 m/s using free fall equation and infer the transverse temperature of the atom beam to be∼ 80 µK from divergence of the beam. From the loading rate of our 3D MOT we estimate the atom
flux in the science cell is in the order of 107 ∼ 108 atoms/s.

5.2.2

Science glass cell AR coating

Uncoated glass surface on the science cell will create unwanted interference if input laser lightis collimated and nearly normal incident. This will create intensity fluctuation in the order of 10%
which will be detrimental for the optical tweezer. To circumvent this, we coat the outer surface ofthe science cell with AR coating. There are two types of AR coating, the coating on the top andbottom is designed for 35◦ AOI for the science MOT beams shining at that angle. For the front andback, the coating is designed for 0◦ AOI for the tweezer light incident almost perpendicular to thecell wall. The coating is designed using Thin Film Cloud software from LightMachinery 1 . Thena test coating is made on a glass sample and the reflection is measured using a spectrophotometer.
Due to the high temperature required for anodic bonding. The reflectivity of the coating on thesample is measured again after subject to various thermal cycling with temperature ranging from1
The Thin Film Cloud is a web based design software where the optimization is run on LightMachinery’s server.
It can be access through https://lightmachinery.com/optical-design-center/thin-film-cloud/.

134
200 ◦ C to 300 ◦ C. After thermal cycle, the coating shows minimal shift in reflectivity and bandwidth[Fig. 5.4(b)].
The AR coating consists of alternating layers of high and low index of refraction dielectricthin films. The thin films are deposited through ion beam sputtering in an argon atmosphere. The
AR coating layer stack up are described below, the high index material is TiO2 and low index
material is SiO2 and numbers are in nm.
• 0◦ AOI AR coating: Pyrex, H 39, L 50.6, H 79.2, L 163.6, Air• 35◦ AOI AR coating: Pyrex, H 9, L 154.7, H 2.6, L 208.6, H 39, L 24.5, H 121.4, L 151.6,
Air
To coat the cuvette that will become the science cell, a special holder is made to only expose
one side of the science cell to the sputtering target at a time [Fig. 5.4(a)]. The two different coating
display a different hue when viewing under room light [Fig. 5.4(a) bottom]. After coating thetransmission is verified using Agilent Cary 5000 spectrophotometer. The maximum transmissionof 96% is due to the inside of the glass cell not AR coated [Fig. 5.4(c)]. Taking that into account,the performance is similar to the typical AR coatings that are commercially available.

135

a

b

c

35° AR

0° AR

Figure 5.4: (a) Aluminum holder for holding the Pyrex cuvette in the ion beam sputtering chamber.
The holder only expose one side of the cell at a time by masking other three side of the cuvette.
Teflon gasket is used to press against the cell to prevent metal directly contacting the glass. (bottom
left) The side with AR coating optimized for 35◦ AOI showing a pinkish hue. (bottom right) The
side with AR coating optimized for 0◦ AOI showing a brown hue. (b) AR coating reflectivity afterthermal cycling to temperature uo to 300◦ C. The coating changes slightly after thermal cycling,
but not significant enough to effect the functionality. (c) Transmission curve measured with theAgilent Cary 5000 spectrophotometer. Measurement include losses from the uncoated side of the
cuvette. Blue trace is the transmission of AR coating optimized for 0◦ AOI. Brown trace is thetransmission of AR coating optimized for 35◦ AOI. The performance of the coating is very similar
to the coating that are commercially available.

5.2.3

Science glass cell

The science glass cell consists of three parts, the AR coated Pyrex cuvette, silicon adapter
and the glass to metal transition. The cuvette was purchased custom made from FireflySci Inc.
with wall thickness of 3.5 mm and outer dimension of 26 mm x 26 mm x 70 mm [Fig. ??(a)]. The
silicon adapter has an outer diameter of 38 mm with a square hole of 19 mm x 19 mm in the middle
and is machined from a 3 mm thick double side polished silicon wafer also shown in [Fig. 5.5(a)].

136

a

b

c

Polished

Pyrex cuve�e

Si adapterdeScience cell

+ HV
- HV

Figure 5.5: (a) The pyrex cuvette (left) is polished at the end for anodic bonding. The silicon
adapter (right) is double side polished, 4 mm thick, 38 mm in diameter with a 19 mm square
opening at the center. (b) Partially optically contacted (green shaded area) between pyrex cuvetteand silicon adapter before anodic bonding. (c) Fully anodically bonded pyrex cuvette and thesilicon adapter. The dark (red shaded part) area is the bonded region now fully extended aroundthe polished edge of the cuvette forming a UHV seal. (d) The setup for anodic bonding the sciencecell cuvette and the silicon adapter. The hotplate at the bottom is set to 350◦ C with temperature
ramp rate of roughly 3-4◦ C/min. The electrical connection to the pyrex (negative) and silicon(positive) is visible on the right. A brass weight is placed on the top to maintain pressure for goodthermal connection to the hotplate and prevent the stack from shifting. (e) The completed cellconnected to the helium leak checker. The leak rate is tested to be < 5 × 10−11 Atm · cc/s

The end of the glass cell and glass to metal transition are polished to an optical polish withflatness better then 1λ per cm. If the components are sufficiently polished and clean placing them
together will show regions that are optically contacted [green shaded in Fig. 5.5(b)]. To anodicallybond the parts, electrodes are attached to the Pyrex (negative) and the silicon (positive) parts andthe assembly is heated to 350◦ C at a rate of 3-4◦ C/min. A metal weight is placed on the assembly
to keep gentle pressure on the parts [Fig. 5.5(d)]. After the temperature stabilized at the bond

137
area, high voltage is applied and the bond is formed between Pyrex and silicon [red shaded regionin Fig. 5.5(c)] become a UHV seal. Two anodic bond are performed to bond the Pyrex cell tothe silicon adapter and then to the glass to ConFlat 2.75” adapter. For a more detailed procedureand description of anodic bonding see appendix C. After bonding the components together, thecompleted science glass cell is helium leak checked with a Pfeiffer Vacuum ASM 340 leak detector
to a level < 5 × 10−11 Atm · cc/sec [Fig. 5.5(e)].

5.2.4

Metalens sample mounting

In high NA application, any stress on the substrate will cause wavefront distortion on themetasurface. In order to maintain the flatness of the metasurface, the substrate is mounted to thesample holder through optical contact bonding. Optical contact bonding is an adhesive free bonding
method that rely purely on the Van der Waals force between the two surfaces. Due to this nature,
the flatness is preserved and the bond can withstand high temperature (typically up to 250 ◦ C)
during UHV baking from the lack of adhesive. One of the requirement to achieve optical contact
bonding is the surfaces needs to be optically flat. Typical surface flatness to achieve bonding isaround λ/4 per cm or better depends on the thickness of the substrate. In the metalens experiment
the sample holder is machined from a fused silica wedge window (Thorlabs P/N WW41050) witha 0.5◦ wedge, this eliminates any residual interference between the metalens substrate and theuncoated inner surface of science cell. The sample holder is glued to the inside of science glass cellwith Epo-Tek 353nd high temperature low outgassing UHV compatible epoxy. Then the metalens
substrate is optically contact bonded to the C-shaped sample holder [Fig. 5.6(a)]. Afterwards,
the cell is baked and UHV of 10−10 − 10−11 torr is reached [Fig. 5.6(b)] is metalens sample inexperiment after UHV baking.

138

a

b
BondedNot bonded

Figure 5.6: (a) Sample mounted with optical contact bonding on the C-shaped sample holder
epoxied onto the inside of the glass cell with Epo-Tek 353nd. The sample holder, machined froma Thorlabs fused silica window, has a 0.5◦ wedge to prevent interference. Most of the substratenear the center are bonded (dark region) and the outer part of the substrate are not bonded(interference fringes are visible). Even though the edges are not fully bonded, it provides enoughadhesion to secure the sample. (b) Sample mounted in the science cell connected to the rest of
the vacuum system and after the vacuum bake. The pressure after the bake is typically between
10−10 − 10−11 torr.

5.3

Metasurface Overview
Metamaterials are artificially engineered materials that exhibit properties that ordinarily do

not exist in the constitute materials. Electromagnetic metamaterials typically consists of subwavelength geometric structures that can significantly modify the resulting materials’ response toelectromagnetic waves depending on the geometry of these structures. Metamaterials was first
explored in the 1950’s in the form of artificial dielectric in the microwave spectrum [158–160] inapplications such as beam forming and antennas. During the study of artificial dielectric, it was
found that the response of these subwavelength structures to electromagnetic radiation, either as
lumped element resonators or particles, behave as dielectric medium with en effective permittivity
and permeability. The values of the permittivity and permeability can be designed to be arbitrarily
large or small even negative, which does not exist in nature. The electromagnetic properties of theseparticle ensembles solely depends on the shapes, sizes and arrangements and not the intrinsic properties of the material. For example, these artificial dielectric can be made into lenses [Fig. 5.7(a)] bychanging the geometry of the resonator structure and achieving dielectric constant as high as 225

139(index of refraction of 15) [159, 161] which allows the lens to be made very thin comparing to thefreespace wavelength of the operating microwave frequency. The dielectric constant of the aforementioned lens is not possible with ordinary materials. More recently, with advances in fabricationand computational numerical simulation of electromagnetic waves in late 1990’s, the first microwave
metamaterial exhibit negative refractive index is developed in 2001 [162]. This composite material
consists of coupled sub-wavelength split-ring resonators and is the first metamaterial that exhibit
physical properties that is not available in nature [Fig. 5.7(b)]. This is the first metamaterials
that uses the concept of meta-atoms similar to the one uses in modern metasurface which we will
introduce next.

Figure 5.7: (a) Artificial dielectric microwave lens formed with subwavelength metallic structure(dark square) arranged on support structure (large white disks) to create effective dielectric constantas large as 225. Image from ref [161] (b) Photo of composite microwave metamaterial that exhibit
negative refractive index. Image from ref [163] (c) Microwave Huygens metasurface designed todeflect 40 ◦ . Image from ref [164]5.3.1

Optical metasurface

optical metasurfaces are planar metamaterials that operates in optical spectrum. They areplanar photonic elements typically composed of periodic array of of subwavelength dielectric or
metallic nanostructures with their thickness normally ranging from 0.1λ to 1λ [165–167]. These
nanostructures couple to the incoming electromagnetic radiation either resonently or off-resonently,
and re-radiate with a transformed phase, polarization, and amplitude determined by the nanostruc-140
ture’s shape, size, and material composition. Due to fabrication challenges, optical metasurfaces
are not feasible until the last two decades where nanofabrication technologies are more accessiblein academic settings [165, 168, 169].

Figure 5.8: (a) Plot showing transmission and phase of light after transmitting through thenanopillar of diameter and pitch ratio. The optical transmission remains near unity while thephase varies from 0 to 2π. This is the basic requirement for metasurface that will be used asa phase shifting element [77]. (b) Scanning electron microscope image of the nanostructure thatcreates a spatially varying phase profile with sub-wavelength resolution. The pillars are siliconof 660 nm height with variable square cross section on fused silica substrate [115]. (c) Notional
illustration of metasurface lens where the incident plane wavefronts (pink dashed lines) interact
with the metasurface and creating secondary wavelets (black semicircles) with various phase profile
which interferes at the far field to create the desire wavefront [115].

Conventional optics relies on refraction and reflection where the wavefront shaping (opticalphase) arises from the propagation of light with varying optical path length, either through media
with different index of refraction or different path length. In the contrary, metasurface introduce
the optical phase shift via the interaction of the nanostructure (nanoscatterer) with the incoming
electromagnetic wave (this is also the distinction between metasurface and conventional diffractive
optical elements). These interactions, which is deterministic, can create abrupt changes in opticalproperties of the transmitted/reflected light in a distance that is a fraction of the operating wavelength of the metasurface. One criteria for the nanoscatterer is to achieve a full 2π phase shift
while maintaining high optical transmission, one method to achieve this is changing the pitch todiameter ratio of the pillars [Fig. 5.8(a)]. By changing the configuration of the nanoscatterer (for

141
example, size shape, spacing and orientation), the array of nanoscatterers create a spatially varying
optical response with subwavelength spatial resolution [77, 170], [Fig. 5.8(b)] shows an SEM imageof a metasurface with nanoscatterer arranged in subwavelength spacing for wavefront manipulation. The re-scattered light is then interfered at the far field to create the desire optical wavefront[Fig. 5.8(c)]. Thus metasurface are conceptually similar to reflectarrays and transmitarrays in
microwave and millimeter-wave community [171, 172].

5.3.2

Anisotropic plasmonic nano-antennas

When the optical metasurface was first demonstrated in the mid-IR [165] and NIR [168], it was
based on nanoscopic metallic metal antennas. In order to fully control the optical wavefront, thesesubwavelength antenna needs to be able to scatter with phase shift relative to the incident light up to
2π and the ability to change the polarization of the incident light. Ref. [173] shows that a plasmonicelement supporting two independently tunable resonance mode that are orthogonal can provide
enough degree of freedom to completely control the amplitude, phase and polarization of scattered
light, therefore, suitable for application in metasurface. The V-shaped design of the plasmonic
nano-antenna used in Ref. [165] is one example that support individually tunable resonance mode
[Fig. 5.9(a)]. The symmetric mode, excited by one linear polarization, creates current symmetric tothe bend. Th antisymmetric mode, excited by the orthogonal linear polarization, has the current
flowing across the bend over the whole antenna. By tuning the angle ∆ and the arm length h of theantenna and the relative angle of the antenna with respect to the incoming linear polarization (along
y-axis) a complete control of phase of scattered light in the opposite polarization can be achieved.
The grey-scale shading on the antenna in [Fig. 5.9(a)] is the current density of the resonatormode [173]. The simulation of an array of nano-antenna to find the angle needed to achieve a
specific phase delay. These antennas, when excited by a y-polarized incident light, each scatters in
x-polarization and creates λ/8 (λ= 8 µm) more phase delay than the previous element [165, 173]
[Fig. 5.9(b)].
An array of nanoantenna arranged to create a periodic 0-2π linear phase gradient over the142period of 11µm [highlighted yellow in Fig. 5.9(c)] allows efficient negative reflection and refraction(negative index of refraction) with up to 30% efficiency [165]. The same plasmonic nano-antenna
can also be arranged to create light with angular momentum [165]. A vortex waveplate of L=1
is created for λ=8µm. To create such waveplate, different zones consists of each types of theantenna from [Fig. 5.9(a)] are arranged to generate a phase shift that varies from 0-2π azimuthally[Fig. 5.9(d)]. The resulting vortex beam is analyzed by interfering with a Gaussian beam througha beamsplitter creating a interferogram with a distinct y-shaped fringe at the center [Fig. 5.9(e)].

Figure 5.9: (a) Scanning electron micrograph of gold nano-antenna of 50 nm thick. The highlighted
portion corresponds to nano-antenna that creates a phase shift spanning 0-2π [165]. (b) Scanningelectron micrograph of nano-antenna arranged in a vortex waveplate configuration. The eightsections creates a linear phase from 0-2π azimuthally [165]. (c) Interference between the vortex
beam and a tilted Gaussian beam through a beam splitter creating a distinctive interference patternwith a Y shape fringe at the center [165].

5.3.3

Dielectric metasurface

The plasmonic metasurface successfully demonstrate the fundamental principle of metasurfaces. However due to their plasmonic nature, they need to be fabricated with deep sub-wavelength

143
thickness. This causes large ohmnic losses and combining with the reflectivity of the metal limits
their use in practical applications especially in transmission mode [77, 174].
This leads to significant effort in developing dielectric resonators due to wide selection ofdielectric materials that are intrinsically low loss in optical wavelengths.

The first dielectriccounter part of plasmonic antenna are silicon dielectric antennas where they can be designed to
exhibit strong birefringence. The anisotropic response of the silicon nanobeam antenna and thePancharatnam-Berry phase optical elements enable highly efficient transmissive metasurface devices [175]. Similar to the plasmonic nano-antennas, Pancharatnam-Berry phase is a geometricphase added to the light, on top of the polarization conversion, when transmitted through a waveplate. The Si nanobeam antennas are designed to be half-waveplates that convert the helicity ofcircularly polarized light while adding the Pancharatnam-Berry phase according to the following
relation [175].
ϕpb (x, y) = ±2θ(x, y)
Here ϕpb (x, y) is the added phase between input and scattered light as a function of spacial coordinate (x,y) across the metasurface. θ(x, y) is the angle between the fast axis of the nanobeam
antenna and the incoming polarization.
Similar to the plasmonic nano-antenna, the two orthogonal mode (TE and TM) of the silicon
nanobeam antenna with a 2π phase delay, give rise to the birefringence [Fig. 5.10(a)]. Depends
on the angle of input polarization θ a linear combination of the two mode are excited, with the
2π phase shift between the TE and TM mode any arbitrary phase delay between 0 − 2π can berealized. To test the efficiency of the Si nanobeam antenna a grating is constructed by arrange
the Si nanobeam in close spacing with a periodic phase profile and the measured diffraction whenlight of different polarization incidents the metasurface [Fig. 5.10(b)]. The efficiency is high forthis grating [Fig. 5.10(b)] but the response of the metasurface is strongly dependent on the input
polarization and only work with circularly polarized light.
In order to create metasurfaces without polarization dependence, another type of dielectric

144
resonators were created. This type of resonators has isotropic response to the incoming light, thus
no birefringence. Instead, the phase control between the input and output are tuned throughthe pitch/diameter ratio of the silicon nanodisks [176]. However, due to the varying diameter ofthe nanodisk they inadvertently have resonances that will cause discontinuity in phase and lowtransmission. To eliminate the resonance reflection in the nanodisk dielectric resonator to maintainhigh transmission throughout the full 0-2π phase delay, both electric and magnetic dipole modes are
utilized. The TE and TM excitation [Fig. 5.10(c)] have similar resonance frequencies but opposite
phase destructively interfere, which creates near unity transmission around the resonances [176,177].
This type of metasurface is called Huygens’ metasurface due to lack of the use of PancharatnamBerry phase.
A Huygens metasurface grating is constructed with periodically varying diameter (in thiscase 50 nm spacing between the disk while the disk radius ranges from 115 to 155 nm) of densely
packed nanodisk resonators, visible within a unit cell in [Fig. 5.10(c)]. The measure deflection
showing high efficiency into the first order and it is polarization insensitive [Fig. 5.10(c) bottom].

a

b

c

Figure 5.10: (a) The electric field silumation of dielectric nanobeam antenna where the twopolarization excite the orthogonal TE and TM resonance mode that has 2π phase shift [175]. By
rotate the angle of the input polarization, different linear combination of the TE and TM mode canbe excited thus create arbitrary 0-2π phase delay. (b) A grating created with the nanobeam antennafrom [175]. Showing efficient diffraction, but the phase chirp direction is polarization dependent.
(c) A polarization independent Huygens metasurface utilizing silicon discs that are tightly spaced
together. This type of metasurface does not require polarization to achieve 0-2π phase shift, butthey suffer from inter resonator coupling due to close spacing [176].

145
Even though both Pancharatnam-Berry phase and dielectric Huygens’ metasurface overcome
the low transmission of metal metasurface there are still some challenges limit its practicality.
First, due to the constraint from the electric and magnetic dipole mode sometimes the height of
the nanodisk resonator needs to change to achieve full 0-2π phase coverage [77]. This make thestructure difficult to fabricate with standard nanofabrication technologies. Second, due to theresonance mode requirement of these resonators, they require subwavelength thickness which makethe resonance modes have a lot of evanescent components. Combinging with the dense spacing
required in these metasurfacesm coupling to the adjacent nanoscatterer in both PancharatnamBerry phase and Huygens’ metasurface are considerable [77, 175–177], which make the design ofmetasurface challenging without full simulation and the performance is also sensitive to fabrication
imperfections. The strong adjacent resonator coupling also makes achieving large deflection angle(thus phase gradient) while maintaining high transmission is difficult [77], which is required in highNA application.
As a result, more groups shift their attention to metasurfaces using high-contrast transmit/reflect array (HCA). These types of metasurfaces uses thicker high-index dielectric where theheight of the nanoscatterers are typically around 0.5λ to λ [77] surrounded by low-index medium
such as air. The first of high-contrast array structure are blazed binary optical elements describedin Ref. [178]. In the early HCAs the waveguides in the arrays are uncoupled so the spacing only
allow one transverse mode to propagate and the collective propagation is equal to effective index of refraction of the individual waveguide [77], this is also called effective field theory (EMT)
regime [178]. Later it is demonstrated that using higher-index refraction material can result in devices with higher efficiency for large deflection angles, despite having pillar spacing larger than the
structural cut-off but small enough that other transverse mode are suppressed. Structural cut-off
is the largest waveguide/resonator spacing where only one transverse mode is supported [178].
The frist high-NA demonstration of HCA metasurface uses Si-nanopillar where the TE andTM modes are well confined within the structure [167]. Even with different diameters the mode
tructure all mostly concentrated within the dielectric resulting in a very little cross coupling between

146
the adjacent resonators [Fig. 5.11(a)]. This high efficiency, high-NA HCA metasurface lens designedfor λ = 1550 nm [Fig. 5.11(b)] demonstrated focusing efficiency of 82% at the NA of 0.5 and 42%
at NA of 0.97 while maintaining a diffraction limited spot [Fig. 5.11(c)]. The parameter d in
[Fig. 5.11(c)] is the designed focal length of different metasurface lens all with lens diameter of 400µm, which translate to NA ranging from 0.5 to 0.97.
With the use of high-index materials, HCAs allows the nanoscatterers to be designed individually due to stronger mode confinement and they can be designed with constant thickness over thefull 0-2π phase while maintaining high transmission because multiple resonator mode contributeto a given phase delay [77, 179]. This makes them the primary design approach in modern state of
the art metasurfaces.

aD = 350 nm

bc

0.5 NA

1 µm
D = 500 nm

1 µm

0.97 NA

Figure 5.11: (a) Numerical mode simulation of HCA metasurface design. Shows the mode mostly
confined inside the dielectric results in small resonator cross coupling [167]. (b) A high NA metalensconstructed with HCA array showing a looser resonator spacing than shown in Fig. 5.10 [167]. (c)The efficiency and PSF measurement over samples with different NA shows high focusing efficiency
at NA >0.7 with diffraction limited performance [167].

147

5.4

Design principle of metasurface and metalens
Our nano-fabrication collaborators that is in charge of designing and fabricating the nano-pillar structures follows the general three step design flow [Fig. 5.12]. The design starts fromacquiring the desire optical phase profile ϕi (x, y) for a specific wavelength λi . Here (x, y) represent
the spatial coordinates on the metasurface device. The goal is to introduce a spatially varying
phase profile when the light propagate through the metasurface, so when the transmitted wavefront
interfere at the far-field it creates the desire effect. The phase profile can be obtained through several
different methods depends on the application. For holograms, Fourier optics methods can be used
to generate the phase profile. For gratings and beam deflectors, grating equations cam be used to
calculate the phase profile. For simple metalens, the following analytical form is used [78, 180].

ϕi (r) =−2π p 2
r + f2 − f
λi

(5.2)

This is the hyperbolic phase profile that give no aberration for on-axis incident ray to focus topa diffraction limited spot. r in Eq. 5.2 is the radial distance from center which equals to x2 + y 2 ,f is the desire focal length of the designed metalens and λi is the design wavelength.
Early metalens demonstrated in Ref. [167, 181] uses this type of phase profile. The metalensdemonstrated in this thesis also use the hyperbolic phase profile. Even though the on-axis performance of these type of lens is very good, the field-of-view is very limited and any off-axis incident
angle quickly result in coma and astigmatism optical aberrations similar to a singlet plano-convex
aspheric lens.
More advance simulation with ray-tracing software is required to generate metalens designthat contains multiple surfaces or non-rotational symmetric, for example, metalens doublet or
metalens achromatic lens. The tool we used in this thesis for ray-tracing simulation is OpticStudiofrom Zemax. For rotational symmetric phase profile, it is often approximate with the sum ofpolynomials described in the following form.

148

ϕ(r) =

NXann r 2nR(5.3)

For free-form optical elements such as cylindrical lens or phase plate, a generalized polynomial
expression is used.

ϕ(x, y) =M,NX

amn xm y n

(5.4)

m,n

M, N are maximum polynomial order user define for approximation, typically order less than 16 isused. an and anm are coefficients of different polynomial orders. R is lens diameter normalization
factor. With these expressions and parameters, a figure of merit can be defined and the coefficient
can be found through ray-tracing optimization by minimizing the optical path difference(OPD).
The second step is to construct a library of nanostructure with full-wave electromagnetic
simulation solver such as Lumerical or RSoft. This provide a connection between a given structure
and the transmission and phase it can provide. Fig. 5.12(step 2) shows a simulated transmission
and phase when light pass through a 660 nm tall silicon nanopillar of different diameter on fusedsilica substrate. This is the same library used in the metalens sample in Ref. [115].
The last step is the mapping of the nanopillar to the metasurface device. The phase of thepillar is digitized into few steps based on the steepest phase gradient in the phase profile ϕi (x, y)(mod 2π). The resulting phase vs. position along the metasurface needs to satisfy the Nyquist
criterion meaning the density of the pillar needs to be half the period created by the steepest phasegradient in the range of 0-2π [78]. Once the sampling coordinate (xi , yi ) of the phase profile isdetermined, the pillar type can be match to the phase value and a layout file can be generated for
lithography. Fig. 5.12(step 3) shows an example of a metalens layout file for lithography. Scale barat the bottom shows the typical sampling density for nanopillars in high-NA metalens application.
The quasi-periodic phase profile in the radial direction near the peripheral of the metalens is dueto the phase wrapping every 2π of the steeper phase gradient of hyperbolic phase Eq. 5.2 at large
radius.

149

Figure 5.12: Steps for designing a metasurface. Step 1: obtain the desire phase profile, either withanalytical form, computer simulation with Fourier optics method or comercial ray tracing software
such as ZEMAX. Step 2: generate the desired nanopillar design through FTDT software such as
Lumerical. Most common approach is parameter sweep of the pillar dimensions (diameter, height,
pitch etc...). Step 3: Take the phase profile from step 1 and create a list of coordinates and phase
where the nanopillars will be placed. The sampling depends on the conversion efficiency of themetasurface required and the steepest phase gradient. The sampling needs to satisfy the Nyquist
criterion to sufficiently sample the phase profile [78].

150

5.5

Metalens sample
In this section we present the metalens samples used in this thesis and the optical charac-terization of the samples before they are put into the vacuum chamber for atom trapping. The
samples are made in collaboration with Amit Agrawal and Wenqi Zhu from NIST Gaithersburg.
The nanopillar designs, numerical simulations and sample fabrications are done by the collaborators. The optical design of the future metalens samples, for example the doublet and hybrid lens
optical design are done in our group. There are several different combination of samples that wasused in the metalens experiment. Including NA of 0.5 and 0.8, polarization multiplexed and nonpolarization sensitive lenses and working distance (EFL) of 1 mm and 4 mm. We have attempted
atom trapping with several samples but only was able to successfully detect atom signal with the
0.5NA, 4mm WD, non-polarization sensitive lens due to some fabrication challenges which cause
the focusing efficiency to be low.

5.5.1

Non-polarization multiplexed lens

In this section we describe the metalens used in Ref. [115] for single atom trapping. Including
the phase profile, materials used, fabrication steps and the optical testing of the metalens. Before
designing the lens we need to identify the required NA, working distance, focal length and operating
wavelength of the metalens. Due to restriction in optical access that is required for creating a MOT
around the focus of the metalens we choose a less aggressive NA of 0.55 and a working distance of
3-5 mm. After taking into the consideration of electron-beam lithography time, which is directly
proportional to the area of the metalens and thus proportional to the focal length square of themetalens, a shorter focal length of 3 mm is preferred. once the parameter is chosen, the size of themetalens is set to be 4 mm in diameter to achieve the required NA.
For trapping and imaging of single Rb atom, a 852 nm optical tweezer trapping light is usedand detection of Rb D2 fluorescence at 780 nm is required. This set the two primary constraint on
wavelength for nanopillar material selection. Since the tight optical confinement is more important

151
in this application, we choose to design the metalens primary at the trapping wavelength λT = 850
nm and with reduced performance at the detection wavelength of λd = 780 nm.
The metalens described in this section uses amorphous silicon (a-Si) nanopillars with squarecross section. The square cross section allows for polarization independent phase delay. The
substrate is chosen to be fused-silica due to optical transmission, flatness and ease of processing forsample mounting in the vacuum chamber for atom testing. Each nanopillar unit has side width L
and height H = 660 nm, acts as a phase-delay waveguides that are weakly coupled to each other.
Fig. 5.13(a) shows the SEM image of the fabricated metalens where the quasi periodic phase wrap
over 2π can be seen as the width L changes. The zoomed in part of [Fig. 5.13(a)] shows the squarecross section of the pillar and spatially varying width to create the spatial varying optical phasedelay. The nanopillars are arranged in a square lattice with spacing of a = 280 nm, this ensure thepillars are sufficiently far apart to minimize cross coupling, but also dense enough to sample thefast varying phase gradient near the edge of the high NA phase profile.
To connect the nanopillar geometry to their corresponding phase delay, FDTD simulation
with Lumerical is used to determine the side width L of the pillars. By vary L between 85 nm to
185 nm we create a libary of unit-cell design that offers transmittance close to unity with relative
phase shift ϕ covering the full 0 to 2π range. Fig. 5.13(b) shows the simulated transmission (blueline, left y-axis) close to unity and phase delay between incident and transmitted light from 0 to
2π (yellow line, right y-axis).
This design library is then used to implement the hyperbolic phase profile given by Eq. 5.2[78, 180, 182]. where λ is the design wavelength (here, λ = 852 nm), f is the focal length (here,f = 3 mm), and r is the rotationally symmetric coordinates of sampled phase position relative tothe lens center.
However, field-of-view simulations that we will introduce later requires the use of Zemax
OpticStudio. Instead of using the analytical form of hyperbolic phase, OpticStudio uses the polynomial phase up to order N to approximate a given phase profile. Thus with good approximation
we represent the hyperbolic phase profile with 5.4 up to 8 polynomial terms (N =8) and normal-

152

Figure 5.13: (a)Scanning electron micrograph of the fabricated metasurface lens containing a periodic array (lattice constant = 280 nm) of amorphous-Si nanopillars (height 660 nm) of width
ranging from 85 nm to 185 nm (dark blue) on top of a 500 µm thick fused-silica substrate (lightblue). Red box shows the varying nanopillar width to achieve the desired phase shift. (b) Transmittance t and phase shift ϕ as a function of nanopillar side length L. (c) photo of the sample usedin single atom trapping experiment.

ization radius R of 2.3 mm. Then the coefficients an can be extracted for n up to 8. Fig. 5.14(a)shows the hyperbolic phase over the whole metalens before modulo-2π. Fig. 5.14(b) are the zoomin part with modulo-2π of different region in [Fig. 5.14(a)]. Near the center we see the phase wrap
over 2π with increasing spatial frequency [Fig. 5.14(b) top] meaning the phase getting steeper as afunction of radius. At the edge of the lens the phase are effectively a sawtooth profile [Fig. 5.14(b)bottom] of nearly the same spatial frequency over some large radius range (10-100 µm). For highNA metalens, the phase gradient difference can be as large as a factor of 20, ranging from 0.03(2π)/µm near the center of the lens to 0.6 (2π)/µm near the edge of the lens.

5.5.2

Metasurface fabrication

The metasurface optic is fabricated by depositing a layer of 660 nm thick amorphous silicon
film (a-Si) on a 500 µm thick fused silica (SiO2 ) wafer using plasma enhanced chemical vapor
deposition (PECVD) [Fig. 5.15(a)]. The film thickness and index of refraction is checked withellipsometry. Next, a 300 nm thick layer of electron beam resist (ZEP 520A) followed by a 20 nm
thick layer of anti-charging conductive polymer (ESpacer 300Z) are spin-coated onto the a-Si film
[Fig. 5.15(b)]. To write the nanopillar pattern a 100 keV electron beam lithography system is usedaGradient: 0.03 (2𝜋)/𝜇𝑚

100

0.5

0
-200

0
-400

-0.24

-0.12

1.0
-600

Gradient: 0.6 (2𝜋)/𝜇𝑚
-800

0

0.12

0.24

Lens diameter (mm)Phase (2𝜋)Phase (2𝜋)

153

1.0

Phase (2𝜋)

b

0.5

-1.84

-0.92

0

0.92

1.84

Lens diameter (mm)
0
2.290

2.292

2.294

2.296

2.298

2.300

Lens diameter (mm)Figure 5.14: (a) The phase profile of a 0.55 NA ideal lens with focal length of 3 mm and diameterof 4 mm. (b) The phase profile after modulus 2π. This is the actual phase used for fabricating
the metalens for single atom trapping. Top is the phase near the center, Bottom is the phase near
the edge of the lens. We can see after 0.5 mm from the center, the phase can be approximate as
sawtooth with gradually varying slope as a function of lens radius.

to expose the pattern onto the resist [Fig. 5.15(c)]. The ESpacer 300Z is removed with deionizedwater at room temperature, and resist development with hexyl acetate at 4 ◦ C. A 70 nm thick
Al2 O3 is electron-beam-evaporated on to the developed resist layer [Fig. 5.15(d)] and the patternis transferred to Al2 O3 after the ZEP resist is lift-off [Fig. 5.15(e)]. By using the patterned Al2 O3layer as an etch mask, the underlying a-Si film is eitch off with inductively-coupled-plasma reactive
ion etching (ICP-RIE) [Fig. 5.15(f)]. The following ICP-RIE parameters are used (gas mixture:
SF6 and C4 F8 ; ICP power: 1750 W; radio frequency (RF) power: 15 W) at etch temperature of 15◦ C to create the high-aspect-ratio a-Si nanopillars. The metasurface optics fabrication is finalizedby soaking the wafer in a mixture of hydroperoxide and ammonia hydroxide solutions at 80 ◦ C for
30 minutes to remove the Al2 O3 etch mask and any etch residue [Fig. 5.15(g)].

5.5.3

Optical testing

In the next few sections we present the methods and result of testing the fabricated metalensthrough optical means. Before the lens can be put into the vacuum chamber it needs to be verified

154

Figure 5.15: Metalens fabrication steps for the lens used in single atom trapping demonstrated inthis thesis, figure adapted from Ref. [183].

that the optical performance is as designed. Some of the most important metric to determine
whether the metalens is good enough to be put into the vacuum chamber are(1) Focusing efficiency = power at focus / total input power to substrate(2) Diffraction limited performance(3) Chromatic performance including chromatic focal shift and spot size vs wavelength
Testing of the focusing efficiency is rather tricky due to the high-NA design of the metasurface
lens. It is often difficult to separate light that is focused from forward scattered, non-diffracted, andabsorbed by the nanopillars. Initially a subtraction method is used where the power of transmitted
light at the far field (non-diffracted) is measured. However, it was later realized due to fabrication
variability, assuming negligible losses in the material is not valid. There are significant amount offorward scattering (not diffracted) component that is diverging, thus not being captured by the
far-field power measurement and counted as part of focusing efficiency by error.
To mitigate these errors and obtain a much more accurate focusing efficiency a spatial filtering

155
technique is used, where only the light around the focus is sampled with a pinhole. The size of thepinhole is chosen so that it is much larger than the focused spot, however still small enough that it
samples negligible portion of the scattered light and non-diffracted light.
A spatial filter is constructed with a collimated lens. This ensures the light that hits thepower meter has consistent input angle to eliminate any angle dependent power measurement
error. Fig. 5.16(a) shows the setup, the additional camera is used to check the spatial filter pinholealignment by imaging the collimated beam after the asphere. Fig. 5.16(b) is another spatial filter
constructed without the asphere. In this case the transmission efficiency of the asphere is eliminated,but the angular response of the power meter is added. We see the efficiency measurement differencein the order of a few percent between these two measurement methods [Fig. 5.16(a, b)]. Fig. 5.16(c)is the setup to measure the total transmitted power through the metalens. In this measurement
the power meter with large aperture is placed sufficiently close so all the light through the metalens
is captured.
The table below are the result of various metalens samples including some of the low NA
lenses (sample number 1 ans 2 in Table 5.1) for generating optical lattice and dipole trap. Thesesample exhibit higher transmission and focusing efficiency, but still not as high as what was reported
in literature such as Ref. [167]. Sample # 4 is the only sample that ended up being tested withsingle atom trapping. This sample has acceptable focusing efficiency of 54 % at the detection
wavelength of 780 nm. We will discuss some of the reason for low focusing efficiency in Sec. 5.6.6.

5.5.4

Metalens PSF with subwavelength pinhole

In this section we describe the method we use to test whether the metalens is diffraction
limited. The purpose of this test is two fold. One is to verify that the metalens is diffraction
limited at the design wavelength of 850 nm for tight single atom trapping. Second, is to test if theimage system is good enough to allow single atom detection.
Testing a high-NA lens is always difficult due to increase sensitivity in alignment and lackof reference target that has sufficiently high resolution. There are several different approach for

156

Figure 5.16: Two different types of spatial filter technique used to measure the focusing efficiencyof the metalens. (a) uses an conventional lens to form a telescope with the metalens and the spatial
filter pinhole is placed in between the telescope. This allows the power meter to be placed further
away from the metalens. (b) directly filter out the unwanted light at the focus with the spatial
filter and place a large power meter in a short distacne directly after. This might introduce angle
dependent error from the power meter and is not suitable for high NA lens. (c) Setup to capture
total power transmitted through the lens. A large area photodiode is placed as close to the metalens
as possible.

measuring the performance of an imaging system. Including, measuring the MTF of the system,
direct image of the PSF and using high resolution traget. Out of all the options, the most common
way is using an objective that has higher-NA to image the PSF of the device under test however
this method has several drawback including.
(1) Optical performance of the imaging objective is not guaranteed to be diffraction limited(2) Alignment can be difficult due to most commercial objective having no well defined opticalreference surface for alignment (i.e. flat front or back optical surface).
(3) Using aberration as a metric for alignment can be risky, due to possibility of some aberration being corrected by misalignment, for example, coma, astigmatism can be canceled byangular misalignment and spherical aberration can be partially compensated by defocus.
Due to high-NA for some of our metalens sample and potential drawback listed above, we

157
Sample EFL / NA /#Wavelength

Total
Transmission

Far-field
Efficiency

Spatial Filter
Efficiency

0-th order
Leakage

Other
Losses1

EFL=150 mm,
NA=0.001,
850 nm

72 %

48 %

43 %

24 %

5%

2

EFL=150 mm,
NA=0.001,
850 nm

70 %

44 %

38 %

26 %

6%

3

EFL=1.5 mm,
NA=0.8,780 nm

54 %

41 %

13 %

12 %

29 %

EFL=3 mm,
NA=0.55,
780 nm67 %notmeasured56 %notmeasured

11 %Same lens but
at 850 nm

63 %

not
measured58 %notmeasured

5%

4

Table 5.1: Metalens Focusing Efficiency

decide to measure the PSF of the metalens by imaging a flat high resolution target. The target that
we used is the same as in Ref. [101,122]. This target consists of 200 nm of silicon-nitride membrane
on top of a 300 µm thick silicon substrate and 600 nm of gold coated on top of the membrane. A
window throught the silicon substrate is etched to create a suspended area of silicon-nitride andgold membrane. Two sub-wavelength pinhole with diameter of 300 nm are created with electron
beam milling. The pinholes are 6.86 µm apart. Knowing the separation of the pinhole allow us
to calibrate the magnification of the test image independent of the expected magnification of the
optical system. Fig. 5.17(a) shows the test setup schematic for acquiring the image of the pinholewith the metalens. The two fiber launcher are for alignment of the metalens and illuminate thepinhole.
Once we have the pinhole image the following steps are taken to determine the performanceof the metalens(1) The image is converted to real size on camera sensor with the pixel size of the camera (i.e.

158pixel to length conversion). In this case each pixel is 4.4 µm × 4.4 µm(2) The magnification is calibrated by fitting the center of the two spots, calculate the distanceand divided by the known separation (6.86 µm) of the pinhole. Once the magnification is
known, the image is re-scaled to it’s real size based on this ratio.
(3) The gaussian waist ω0 is extracted by fitting a gaussian function to the imaged point-spreadfunction. The function is defined as−2r2
I(r) = I0 exp
ω0 2

(5.5)

(4) According to diffraction theory the image of a point source through a finite NA opticalsystem results in a point-spread function (PSF) that can be described by the Airy function
which is defined as below [101, 153]

I(r) = I0

2J1 (ζ)
ζ

2
where

ζ=

2π r N A
λ(5.6)

By fitting the imaged PSF to this function the effective NA of the optical system can beextracted.
Fig. 5.17(b) is the image of the pinholes through the metalens. Fitting the center of the two
PSF, we determine the magnification to be 44.5(1), which is within 10% of the designed value of
49. On the right is the corss-sectional cut of one of the pinhole after re-scaling with the measured
magnification. The red line is the fitted Airy function (Eq 5.6). A gaussian function (Eq 5.5) is also
fitted to the same data. We extract a 1/e2 Gaussian waist of 0.72±0.02 µm and an effective NA of
0.53±0.01. This is sufficiently close to the designed NA of 0.55. So we deem that the metalens isdiffraction limited at the tweezer wavelength of 850 nm.

5.5.5

Atom image system test

Another test we perform with the pinhole is the performance of imaging optics to compensate
for the chromatic focal shift of the metalens between the trapping (850 nm) and Rb fluorescence159

Figure 5.17: (a) Schematic illustration of the pinhole test to test the optical performance of themetalens. (b) The image of the pinhole from the metalens. The distance between the two pinholes
are well calibrated. This allows us to calibrate the magnification of the imaging system. Aftercalibration we can measure the Gaussian waist of 0.72 ± 0.02 µm

wavelength (780 nm). Opposite from conventional lenses, the focal length of the metalens increases
as the wavelength increase. Fig. 5.18(a) shows the measured focal length of the sample 4 inTable 5.1. The focal length changes from 3.25 mm at 780 nm to 3.04 mm at 850 nm. Thismeans the atom trapped at the 850 nm focus and emit 780 nm fluorescence will be at the location
shorter than the 780 nm focus of the metalens. Thus the light will exit the metalens diverging.
This diverging light once focused by a single imaging lens onto the camera will not be diffraction
limited, further more, the diverging beam will be too large before reaching the imaging lens. In
order to correct for this, a relay optical system is designed with two achromatic doublets (Thorlabs
AC254-150-B) [L1 and L2 in Fig. 5.18(c)]. The first lens L1 is placed 111 mm away from themetalens. This creates a collimated light between L1 and L2 making the space between L1 and L2
infinite conjugate. The light is then focused by L2 onto the camera. In order to eliminate the extra
degree of freedom introduced by adding the optical relay, the distance between L2 and the camera
is pre-aligned by focusing a collimated beam on the camera. The only remaining adjustment tofocus the whole image system is the distance between the metalens and L1.
[Fig. 5.18(c) right] are the Zemax OpticStudio simulation result of the imaging system showing
diffraction limited performance of the optical relay lenses both on axis and off axis.

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To test the imaging system, the green dashed portion in Fig. 5.17(a) is replaced by theimaging system from Fig. 5.18(c) without changing the distance between the metalens and thesubwavelength pinhole. Once everything is aligned, the illumination light is changed to 780 nm
and the image on the camera is focused by moving L1 w.r.t. the metalens.
Fig. 5.18(b) is the image of the pinholes at 780 nm. Following the procedure identical to
850 nm pinhole test, we measure the magnification to be 11.42. The measured PSF has 1/e2
Gaussian waist of 1.1±0.05 µm (corrospond to NA of 0.32±0.01) which is not diffraction limited(due to metalens has spherical aberration at 780 nm) but sufficient for single atom detection, andthe effective solid angle for light collection is equivalent to 0.55 NA. The precise determination of
the magnification is important because the magnification is not as easy to calculate in this more
complex imaging system.

Figure 5.18: Metalens pinhole test, NA = 0.323 mag=11.42 (d) Schematic illustration of opticaltweezer imaging path with lenses that compensate for out-of-focus imaging due to chromatic focal
shift introduced by the metalens. Insets are the ray-tracing simulation of the imaging system(object on-axis and 10 µm off axis) assuming metalens only has chromatic focal shift and no otheraberration. The result shows L1 and L2 does not introduce additional aberrations. Black circle isthe diffraction limited Airy-disk.

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5.6

Testing with atom
In this section we present the result of single-atom trapping in a metalens generated opticaltweezer array. After testing the 0.55NA metalens with the sub-wavelength pinhole and determine
the optical performance of the lens. The metalens is put into the science glass cell by opticalcontacting the substrate onto the sample holder. The chamber is then baked at 200 ◦ C for 5 days
resulting in a vacuum environment of < 10−10 torr measured at the ion pump.

5.6.1

Single atom loading and detection

Atoms are captured into the optical tweezers by overlapping the focus of the metalens witha magneto-optical trap (MOT) and applying polarization-gradient cooling (PGC) while the opticaltweezer traps are on. The single atom loading starts with the three-dimensional (3D) science MOT.
The atoms from the dispenser in the source cell are cooled in the transverse direction with MOT
laser red detuned from 87 Rb D2 F = 2 to F ′ = 3 transition (free-space atomic resonance) by 14
MHz and transported to the science cell via a push laser beam. The collimated atom beam has aflux up to 108 s−1 . Fig. 5.19(a) is the image of the 2D source MOT viewed from the end. The
science MOT loading lasts 500 ms with a typical MOT size of (3 × 107 ) atoms and a density of
109 cm−3 . Fig. 5.19(b,c) are image of the science MOT viewed from the front of the metalens andfrom the top perpendicular to the metalens substrate. This is also used to determine whether or
not the MOT is aligned to the metalens focus.
After loading the science MOT, the source MOT lasers are shut off and the magnetic field
gradient is turned off and the MOT lasers are changed to 144 MHz red detuned from free-spaceatomic resonance to perform PGC with σ+ - σ− configuration for 15 ms. During the PGC the opticaltweezer is turned on to load atoms into the tweezer. The typical free-space PGC temperature is
between 30 µK to 50 µK, and the tweezer trap depth is typically at 1.3 mK during loading. During
the PGC loading the laser is red detuned from the atomic resonance resulting in light assisted
collision that assures only a single atom is trapped [3].

162

Figure 5.19: (a) Image of the source MOT (bright spot in the middle) viewing from the push beam
direction. The diameter of the 2D MOT is ∼1 mm. (b) Image of the science MOT in from of
the metalens. Metalens (4 mm x 4 mm square) can be seen in the background. (c) Science MOT
looking from the top, verifying that the location overlaps well with the metalens focus.

To image a single atom in the tweezer, we utilize PGC imaging. The PGC configuration
with less detuning cools the atom while scattering photons. The trapped atom is illuminatedwith a 500 µm 1/e2 Gaussian waist and 150 µW PGC/probe beam propagating parallel to thesubstrate to avoid scattering off the metasurface (beam geometry shown in Fig. 5.20(c), ≈10Isat ,47 MHz red detuned from free-space atomic resonance) for 25 ms and the fluorescence is collectedby the metalens (Isat is the saturation intensity of 87 Rb D2 F = 2 to F ′ = 3 transition). After
passing through the metalens, the slightly diverging fluorescence is reflected by a dichroic mirror
and passed through the compensation and imaging lens (L1 and L2 in Fig. 5.18(c)) and focused
onto a Princeton Instruments PIXIS 1024B CCD camera. While a full analysis of fluorescencecollection efficiency requires calibration of the probe light intensity, trap depth and imaging pathefficiency we can roughly estimate the expected CCD count. The total fluorescence collected in
the experiment is around 2.5% which is consistent with the expected efficiency of 4.5% given someof the imaging elements not being calibrated. The data presented are in CCD counts and are notconverted to photon count.
We first analyze in detail one trap that is at the center of the metalens FoV. We plot ahistogram of the fluorescence counts collected through the metalens and registered on the camera(CCD counts) versus occurrence from a single pixel at the atom location (Fig. 5.20(a) and 5.20(b)).

163
The lower count histogram peak corresponds to background signal with no atom and the second
higher CCD count peak corresponds to the fluorescence of the single atom. Collisional blockade
prevents loading of more than one atom into the tweezer, as reflected in the absence of a twoatom peak [3]. We find a loading probability of (47±5)%. However, due to the probe beam being
perpendicular to the tweezer direction (Fig. 5.20(c)), there is no axial cooling during imaging.
Since the axial direction has weaker confinement from the trap we see (10±2)% atom loss during
imaging which is higher than typical. Taking this into account, a loading probability of (52±5)%
is comparable to typical loading efficiency from other optical tweezer experiments [184].

Figure 5.20: (a) Camera count histogram indicating presence of either 0 or 1 atoms in a tweezer trap.
A threshold is chosen to determine if an atom is trapped and to calculate the loading efficiency.
(b) Typical single shot fluorescence image of a single atom imaged through the metalens with
PGC imaging. (e) End view of the vacuum cell, showing the probe beam (also resonant heatingbeam) orientation in relation to metalens sample and tweezer beam. Probe beam has 500 µm 1/e2
Gaussian waist, shines vertically up and is 3 mm away from the metalens overlapping with theoptical tweezer focus.

We also determined the length of time a single atom remains trapped in the optical tweezer
focus (vacuum lifetime), with no cooling light present, by holding the atom with variable time between two consecutive images. The measurement gives a lower bound of exponential decay lifetimeof 10 sec; atom lifetime assessment in a metalens trap beyond this scale will require additional investigation of background gas collision rates due to finite vacuum level and potential atom heating
and loss due to inelastic light scattering from residual trapping light.

164
5.6.2

Trap depth

To measure the tweezer trap depth a resonant heating beam is used to measure the induced
AC-Stark shift from the tweezer trap. This beam is identical to the tweezer probe beam other thanthe detuning. The beam has 500 µm 1/e2 Gaussian waist and the geometry is shown in Fig. 5.20(c).
The experiment sequence for this measurement is in Fig. 5.21(a). The first PGC image is taken toverify the loading of single atom in a experiment run. Then the resonance heating beam is turned
on for 60 µs with intensity of 100 µW (≈6Isat ). This pulse time and intensity are carefully chosen
so that the atom is not being overly driven by the heating beam (i.e. survival not zero). Afterthe heating pulse, the trap intensity is lowered to 0.25 mK for 15 ms. This allows the hot atom toescape. Then a second PGC image is taken and the survival between the two images vs. heating
beam detuning is recorded. Fig. 5.21(b) shows the typical resonance scan. A gaussian fit (solid redline) is used to extract the center of the shifted resonance and the width is used to check whether
or not the atomic transition is over driven. The AC-Strak shift created by the 850 nm trapping
light on the 87 Rb D2 F = 2 to F ′ = 3 transition is 28.8 MHz/mK. Comparing to the free-spaceresonance (dotted gray line) allows the trap depth to be calculated.

Figure 5.21: (a)Experiment sequence of the light shift measurement. Heating beam is turned on
for for 60 µs with intensity of 100 µW between two PGC image separated by 300 ms. (b) Typicallight-shift measurement with a Gaussian fit (red line) to the shifted atomic resonance. Dashed line
corresponds to free-space 87 Rb D2 F = 2 to F ′ = 3 transition

165
5.6.3

Trap frequency

To characterize the effective tweezer waist seen by the atom we need to measure the trap
frequency (harmonic oscillator strength of atom moving in the optical tweezer light). There areseveral method that can be used to measure the trap frequency, all of them require measuring theatom’s frequency response to an excitation. Two methods are the most popular, the one used inRef. [153], excite the atom by temporarily switch off the tweezer trap and measure the oscillation
frequency of the atom afterwards. Another method, which is used in this thesis is the parametric
excitation method. Here the trap is modulated between 5% and 10% of its depth around thenominal value at variable frequency for 50 ms. After modulation, the trap is lowered to 250 µK for
15 ms which is much longer than the period of the atom oscillating in the trap. This allows the
hot atoms to escape the trap. Fig. 5.22(a) shows the experiment sequence of the trap frequency
measurement where the parametric excitation happens between two PGC images. On the rightis the illustration of how the modulation (blue dashed line) changes the trapping potential (blacksolid line) and the atom wave function (red and purple).
After the modulation and trap lowering, the atom survival vs. modulation frequency isrecorded. Fig. 5.22(b) shows the typical trap frequency measurement where the atom is heated attwice the trap frequency (νtrap ) and subsequently loss from trap lowering.
The trap waist can be deduced from the slope of a graph that plots the trap frequency versus
depth as per the equation below
1νtrap (U, w0 ) =
2π

r

4U
w0 mRb

(5.7)

U is the tweezer trap depth, mRb is the mass of Rb87 , w0 is the tweezer Gaussian waist and νtrap
is the trap frequency.
Taking trap depth and trap frequency measurements at different tweezer intensity and fitting
to the data we extract a 1/e2 Gaussian waist of w0 =(0.80 ± 0.04) µm at 852 nm (Fig. 5.22(C)),
which is consistent with the value determined from the sub-wavelength pinhole characterization(Fig. 5.17(b)). With the clipped Gaussian beam illumination used for the optical tweezer trapping

166(versus uniform illumination during characterization) we expect the tweezer to have a waist of 0.78µm, consistent with the measured value.

Figure 5.22: (a)Experiment sequence of the trap frequency measurement. Between the two PGC
image separated by 300 ms, the tweezer trap is modulated for 50 ms and the trap is lowered to 250µK to eject the hot atoms. (b) Typical parametric heating measurement with a Gaussian fit (redline) to extract the trap frequency (νtrap ) from the modulation resonance (2νtrap ). Each point is anaverage of 100 trap loading sequences. (c) Measured trap frequency versus trap depth (light shift)obtained from multiple measurements similar to (c) and (d). The solid red line is a model fit (seemain text) to extract the effective Gaussian tweezer waist seen by the trapped atom.

5.6.4

Tweezer array setup and FoV

An important metric for creating and imaging large atom arrays is the lens FoV. To study
the FoV of the metalens, we create an array of tweezers with an acousto-optic deflector (AOD).
AOD uses RF to diffract an input laser beam at an angle that is proportional to the RF frequency.
By injecting several different frequencies into the AOD input, one can create a series of diffracted
beams at different angles. After focused by the tweezer lens, it became an array of optical tweezers

167
for single atom trapping.
Fig. 5.23(a) is the schematic illustration of the tweezer setup. The tweezer light collimatedwith 1/e2 waist = 2 mm at 852 nm (pink shaded beams) is launched into the AOD (AA Optoelectronics Model: DTSXY-400-850.930-002). This produces a series of deflected beams with variable angle controlled by the AOD input RF frequencies. This array of angled collimated light isthen imaged with a 1:1 relay lens with focal length of 150 mm (Thorlabs AC254-150-B) onto theback aperture of the metalens substrate. The relay lens ensures all the deflected beams coincide on
the metalens to minimize asymmetric beam clipping. Fig. 5.23(b) is an average image of 3x3 array
after 100 loading cycles.

Figure 5.23: (a) Optical setup for trapping (pink) and fluorescence imaging (green) of single atoms
in an array created with multiple input beams generated using a 2-axis acousto-optic deflectors.
(b) Image of a trapped 87 Rb array created by averaging over multiple experiment iterations (100in this case) with ∼ 52% probability of a single atom in each trap per image. The variation in
the averaged intensity is caused by trap depth and shape variations that affect relative loading
probability and imaging signal in the array.

For this, we create four traps with the lower left tweezer at the center of the FOV (opticalaxis), and characterize the traps (with various spacing) the same way as in Sec. 5.6.2 and Sec. 5.6.3.
Fig. 5.24(a) is the tweezer image showing how the array is arranged in this measurement. In the168
presence of aberrations the traps become asymmetric, resulting in non-degenerate frequencies in theradial dimensions of the trap. This will manifest as a double-dip structure in the trap frequency
measurement (Fig. 5.24(b)). We characterize the FoV by plotting the waist determined fromthe trap frequency and depth measurements as a function of the distance from the optical axis
(Fig. 5.24(c)) and find the aberrations are consistent with FDTD calculations (see Sec. 5.6.5 fordetails of FDTD simulation) of tweezer intensity from our metalens optical field distribution (bluedash lines in Fig. 5.24(c)). Here FoV is defined as the distance to a point where the average waist
is 10% larger (Strehl ratio > 0.8) than at the center, and we find a FoV of ±11 µm (±0.2◦ ).

5.6.5

Focusing efficiency and FoV simulation

We use the grating averaging technique [185] to estimate the focusing efficiency, the fractionof incident light remaining as undeflected zeroth-order light, and the total reflectance of the mmscale diameter metalens. Following this technique, we approximate the metalens as a combinationof aperiodic beam deflectors. For an aperiodic beam deflector with a deflection angle θD (θD =sin−1 (λ/N a), where N was chosen to calculate θD between 0◦ to 50◦ ), the deflection efficiency η1 ,the fraction of light in the 0th-order η0 , and the reflectance ηrefl , for unpolarized input light, arecalculated (circles in Fig. 5.25(a)) using rigorous coupled wave analysis (RCWA), and fitted withparabolic functions (solid lines in Fig. 5.25(a)). Finally, the focusing efficiency of the metalens T1 ,the total fraction in the undeflected 0th-order T0 , and the total reflectance Trefl , are estimated as
the area average of η1 , η0 , and ηrefl , respectively, using:2ZZ
ηi ds = 2/R

Ti = 1/(πR )
S

2

Z R
ηi (r)r dr,

(5.8)

0

where i= 1, 0, or refl; r = f tanθD ; and R is the radius of the metalens.
To simulate the FoV the beam waist at the focal spot as a function of the distance fromthe metalens optical axis or, equivalently, the incident angle of the input beam, is calculated using
FDTD technique, with a minimum mesh size of 4 nm. Due to the millimeter scale size of themetalens, a cylindrical metalens is simulated instead, implemented by using one unit-cell along

169

Figure 5.24: (a) PGC fluorescence image of atom array trapped with metasurface optical tweezers.
Image is averaged over 100 experimental cycles. Bottom left tweezer is on optical axis of themetalens. The off-axis tweezer site typically have a lower loading probability and non-optimal
PGC imaging detuning resulting in a dimmer single atom signal. (b) Example of typical trap
frequency measurement data at ≈13.6 µm from FoV center where asymmetric aberrations in thetrap are present, along with a double Gaussian (green line) fit. (c) Extracted Gaussian waist as
determined from the atom trapping as a function of distance (r ) to the metalens optical axis (centerof FoV). The average waist extracted from a single Gaussian fit to the trap frequency data (red),and the major waist (green circle) and minor waist (green diamond) extracted from data similarto (b) when the two trap frequencies are distinguishable. We compare to theory of the major andminor Gaussian waist fitted from FDTD simulation (see Sec. 5.6.5).

the y-axis with periodic boundary condition. All the unit cells along the x-axis are included in
the simulation, and the oblique incidence angle is only applied along the x-direction. For a given
incident angle, a near-field monitor records the electric and magnetic fields of the output beam at
a distance of 50 nm from exit surface of the metasurface. A near-field to far-field projection is thenused to calculate the focal spot intensity profile at the focal plane (Fig. 5.25(b)). The major and170
minor waists of the focal spot are obtained as the distance from the intensity peak to the 1/e2 ofpeak intensity along the x-axis (blue dash lines in Fig. 5.24(c)).

Figure 5.25: (a) Calculated deflection efficiency η1 , the fraction of light in the undeflected 0th-orderη0 , and the reflectance ηrefl of aperiodic metasurface beam deflectors as a function of deflection angle
θD . Circles are data from RCWA simulations and solid lines are parabolic fit. (b) FDTD simulated
beam profiles of the focal spots as a function of the angle of incident light.

5.6.6

Discussion and challenges

From the focus efficiency measurement done in Ref. 5.5.3 (sample 4 in Table 5.1) the metalens
used for single atom trapping has focus efficiency of 58% at 852 nm and 56% for 780 nm. Thisis about 20% lower than the theoretical estimate done in Sec. 5.6.5 and Fig. 5.25(a). We foundthat the transmitted power through the substrate is significantly lower than the simulated value by
the amount close the loss in focus efficiency. Possible reasons for the reduction of the transmission
efficiency and increase of zeroth-order light in comparison to theory are likely due to fabrication
imperfections resulting in nonideal nanopillar cross sections and sidewall tapering which is evident
in some of the SEM images taken from low efficiency samples. The nonideal nanopilar cross section
due to lithography calibration or anisotropic etching was observed in sample 1&2 in Table 5.1 where
we see strong polarization dependent focus efficiency change of up to 30% from one polarization
to another. The material loss was ruled out by testing the a-Si film loss tangent before patterning
the nanopillars and using different silicon film deposition method including epitaxial, yields similar

171
focus efficiency.
As one comparison, we discuss the performance of a typical commercial asphere that has beenused in optical tweezer experiments. Reference [153] uses an aspheric lens with NA = 0.5, a working
distance of 5.7 mm, a focal shift of -40 µm from 852 nm to 780 nm, and a focal length of 8 mm.
This aspheric lens has a transverse field of view of ±25 µm (±0.18◦ ), an inferred beam waist of 1
µm for the trapping wavelength, and a 0.9 µm waist for the imaging wavelength. The metasurface
studied here has a worse focal shift than a standard asphere, but as discussed in 5.7 this was notof primary concern in our experiments given the prospects for future control, for example, using
wavelength polarization multiplexing. The singlet metasurface here achieves the Gaussian 1/e2
radius (waist) of the focused tweezer spot to be w0 =(0.80 ± 0.04) µms which is better than theasphere in Ref. [153] and FoV (average waist 10% larger or Strehl ratio > 0.8) of ±11 µm (±0.2◦ )which is consistent with a full theoretical model of the metalens (Fig. 5.24(c) and Fig. 5.25(b)) andcomparable to Ref. [153] in terms of focal length to FoV ratio, i. e. angle.

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5.7

Future sample
A complex high-NA objective lenses used for atom trapping and imaging can have FOV of afew 100 µm (≈ ±0.5◦ ) combined with achromatic operation over a wide range of wavelengths [16,186]. While the singlet metalens described in this thesis does not yet achieve these metrics, we
now discuss the horizon of prospects for design features of future metasurfaces. In the following
sections we will present some future metalens design ideas for a few different metrics desirable forsingle atom trapping such as FoV and Chromatic performance.
As discussed previously, with a metasurface it is possible to achieve a focusing response that iseither polarization selective [187] or one that transforms the polarization [188], which are functions
not offered by traditional optical lenses. For example, polarization multiplexing provides a method
to trap and collect fluoresence at the diffraction limit for two different wavelengths using a singlet
metasurface lens, and may find utility in combining additional multifunctional beams in complex
trapping experiments.
To illustrate this prospect, we have designed and optically tested a sample with rectangularshape cross-section nanopillars that achieves equal focal lengths for 780 nm and 852 nm light oforthogonal polarization with diffraction limited performance at NA of 0.8. This concept can beused to trap at 852 nm, and collect fluoresence at 780 nm, with a 50% efficiency due to the random
polarization of the scattered light from atoms. The design and optical testing of this metalens will
be described in the next section (Sec. 5.7.1).
More functionality can be achieved with expanding the number of surfaces offered in thedesign. To focus on FoV as one metric, an enhanced FoV up to ±25◦ has been achieved bypatterning both sides of the substrate to create a double-layer metasurface [81]. We estimate thatby using design components similar to the singlet lens presented here, expanding to a doublet can
improve the field angle to beyond ±5◦ at 0.7 NA. To demonstrate this capability, we present aphase map design of such doublet in Sec. 5.7.2 demonstrating mm scale FoV with working distancesimilar to objective used in Ref. [16, 186].

173
Further design improvements can be achieved through the use of an expanded unit-cell library
to include cross, donut and other exotic nanopillar shapes [189] or via inverse design [190]. Choosing
optimal materials and designs that are robust to nanofabrication variation is expected to offer higher
efficiencies that exceed that achieved in the experiments presented here [78]. Further, a hybrid lensdesign consisting of a curved refractive surface and a metasurface patterned on the same substrate
will offer additional prospects for enhanced design space [182, 191]. In Sec. 5.7.3 we present an
example design of this type of hybrid lens with less chromatic aberration.

5.7.1

Polarization multiplexed lens

The metalens sample used in this thesis and Ref. [115] utilize nanopillar with square crosssection (Fig. 5.13(a)), therefore the focusing response has no polarization dependence. One popularmethod for creating polarization dependent response is using asymmetrical cross-section nanopillars
such as in [79, 188].
When light incident to the nanopillars, a portion is confined in the nanostructure while therest leaks to the surrounding air. This is very similar to an optical waveguide, so we can consider
the nanopillars as truncated waveguides. By changing the size of the nanopillar, the ratio of themode inside the waveguide and in the air changes, thus the effective index of refraction seen by theincident light also changes. Due to the symmetry of the structure, two orthogonal linear polarization
excites two orthogonal TE mode in the waveguide. If the cross-section is anisotropic, the incident
linear polarized light will experience a different effective index, nl and ns , when passes throughthe long and short axis of the nanopillar [78]. Fig. 5.26(a) is the effective index calculated with
Lumerical for TiO2 nanopillar arranged in square lattice of 400 nm spacing and corss-section of
250 nm by 80 nm. When the incident light is polarized along the long axis, it experiences a larger
effective index of refraction (red curve). This is because most of the electric-field is localized inside
the nanopillar. Conversely, when the polarization is along the short axis, the mode is mostly in theair, resulting a low effective index (blue curve). Two subplot in Fig. 5.26(b) shows the structureof the electric-field of each mode, top panel is polarization along the long axis and bottom panel is174polarization along the short axis.
The complex transmission from the anisotropy can be written as [78]:
2π it = |t| exp(nef f − 1)h
λ

(5.9)

Where the nef f is the effective index calculated in Fig. 5.26(a) (i.e. nl and ns ) and h is the height of
the nanopillar. Fig. 5.26(c) is the calculated phase shift of h=600 nm TiO2 nanopillar on the longand short axis. The solid lines are from effective index in panel(a), square are FDTD simulation.
Deviation of phase at low wavelength might due to resonances and excitation of higher-order modes.

Figure 5.26: (a) Effective index of the nanopillar with different width along the long (250 nm) andshort (80 nm) axis. If the polarization of the input light lines up with the log axis will experiencea large effective refractive index comparing to the polarization aligned to the short axis. [78] (b)Simulated E-field distribution of the two different polarization alignment with respect to the long(i) and short (iii) axis of the nano-pillars. In case (i) the mode structure is mostly inside thedielectric so it experience larger effective index (red curve in (a)). For (iii) the mode is mostly in
air so it sees a lower effective index (blue curve in (a)). [78] (c) Phase shift of different linear
polarization alignment to the pillar. The solid line is calculated from the effective index in (a) thesquares are FDTD simulation. Deviation at shorter wavelength might due to higher order mode in
the pillars. [78]Following this general principle we designed a metalens to implement an ideal-lens phasefunction for x-polarized light at λ = 780 nm and for y-polarized light at λ = 850 nm, simultaneously,targeting a focal length of 0.5 mm and NA of 0.8, for both wavelengths (Fig. 5.27(c)). This isachieved by assembling the metalens with a library of a-Si birefringent nanopillar unit-cells, eachhaving an rectangular cross-section that can simultaneously impose the desired local phase shift
ϕx for x-polarized light at λ = 780 nm and ϕy for y-polarized light at λ = 850 nm based on

175
Eq. 5.2 (Fig. 5.27(b)). Fig. 5.27(a) shows the phase as a function of lens radius with respected towavelength of 780 nm in x-pol and 850 nm in y-pol. Fig. 5.27(b) shows input polarization w.r.t.
the nanopillar orientation. Fig. 5.27(c) shows the two wawelength with orthoronal polarization,
after passing through the polarization multiplexed metalens, creates identical focusing wavefront.

Figure 5.27: (a) Phase map of polarization multiplexed lens (b) amorphous-Si nanopillar (darkblue) with rectangular cross section on fused silica substrate (light blue) creates phase delay for two
orthogonal polarization independently (polarization multiplexing). (c) Illustration of polarization
multiplexed metasurface lens operation. Input wavefront (pink dash) with orthogonal polarizationfor 780 nm (green) and 852 nm (blue) propagates and interact with metasurface. Secondary wavelets(black semicircles) re-emitted by the nanopillars interfere and creates identical focusing wavefront
for both 780 nm and 852 nm.

A sample is fabricated, based on the phase profile in Fig. 5.27(a), we subject it to thesame focusing efficiency measurement and PSF imaging described in Sec. 5.5.3 and Sec. 5.5.4. We
found that the focusing efficiency is only less than 20% which is significantly lower than typical
expected value of 60-80%. This is most likely due to the same fabrication challenges we discussed
in Sec. 5.6.6. The polarization dependent PSF measurement is done by filtering out the unwanted
polarization with a PBS before the light entering the imaging lens and focused onto the CCD camera(by replacing the 50/50 BS with a PBS in Fig. 5.17(a)). The PSF are imaged without changing
the focal distance between the metalens and subwavelength pinhole to test if the polarization
multiplexing is working properly. The measured intensity distributions at the targeted focal plane
reveal diffraction-limited focal spots for both wavelengths (Fig. 5.28(a) and Fig. 5.28(b)). Measured

176
NA for 780 nm x-polarized light is 0.8 ±0.01 and 850 nm y-polarized light is 0.82 ± 0.01.

Figure 5.28: (a,b) Experimental PSF by imaging the subwavelength pinhole with the metalens for780 nm x-polarized and 852 nm y-polarized light. The image of two wavelenth configuration are
obtained without changing the focus distance between the polarization-multiplexed metalens andthe subwavelngth pinhole. 2D cuts of the PSF shows the fitted Airy function from which the NA
is extracted.

5.7.2

Doublet design

In this section we present a metalens-doublet design that takes a similar approach to Ref. [81].
The doublet in Ref. [81] has 0.44 NA, focal length ≈ 350 µm and a FoV of ±25◦ (± 150 µm).
This doublet consists of a aperture metalens that has very little focusing power and a focusing
metalens that has most of the focusing power. The aperture metalens, when placed at an optimal
distance, corrects for the aberration that can be introduced under large FoV angle. The functionof the aperture metalens behave exactly like a Schmmidt corrector plate which controls coma andastigmatism aberration. Normally this corrector plate is placed at the distance equals to the radiusof curvature of the lens on the opposite side of the focus which equals to f (n−1) where n is the index
of refraction of the lens. Fig. 5.29 shows the general shape of the Schmidt corrector plate and how
the rays are bend (left panel), combining with a spherical lens (middle) resulting in a diffraction
limited doublet (right) and when the aperture of the system is place at the correct position (typicallyat the Schmidt plate) the corrector plate also compensate for coma and astigmatism.
Here we choose a working distance that is similar to the one in Ref. [153] and NA of 0.7.
We setup the location of the Schmidt corrector plate and the focusing metalens. Once those177

Figure 5.29: Schematic illustration of the doublet design that consists of a Schmidt corrector plate
and a focusing metalens. The Schmidt corrector plate corrects for the field related aberration whenplaced at the back of the focusing lens f (nl − 1) away from the lens. figure from Ref. [81]parameters are determined, we set up the optimizer using Eq. 5.3 with maximum term (N = 9).
We let the optimizer in Zemax OpticStudio run for maximum FoV possible and we found that thisconfiguration gives a diffraction limited FoV of ±1.1 mm (±7◦ ). Fig. 5.30(a) shows the configurationof the surfaces for this doublet design, the substrate material is a 12.7 mm thick fused silica. The
aperture metalens is 12.7 mm in diameter and the focusing metalens is 15 mm in diameter. Inset
of Fig. 5.30(a) shows the ray converges to a diffraction limited point under different FoV settings.
Fig. 5.30(b) shows the simulated PSF at different FoV and their cross-sections. We can see theStrehl ratio remains high (>0.8) for up to FoV of 1.1 mm.
Table 5.2 is the coefficients used to generate the metalens doublet design. The resulting phaseprofile is shown in Fig. 5.31(a,b) for the aperture and focus metalens. This phase profile has thegeneral shape described in Fig. 5.29 and Ref. [81].
Normalized radius (R): 5 mm, Substrate thickness: 12.7 mm, Substrate: Fused silica
Surface r2

r4

r6

r8

r10

r12

r14

r16

r18

1

-2111.72

699.9

58.5

155.72-196.82188.4-101.7830.314-3.7182-1.056E4-3.69916.524-19.3

11.578-3.9740.793-0.083

3.4E-3

Table 5.2: Metalens Doublet Coefficients, NA 0.7, FoV= ± 1.1 mm, WD= 7 mm

178a
Aperture
metalens

Focus
metalens

7° 1.1mm
4.5°
3.5°

2 mm

7 mm

12.7 mmIntensity

2

-2
1.0

0

2

2

0

2

-2
1.0

0

0

2

7° 1.1 mm

4.5° 0.7 mm

3.5° 0.5 mm

2

2

0

-2

-2

On-axis00
Y – position (μm)
-2
1.0

-2

-2

b

-2
1.0

0.8

0.8

0.8

0.8

0.6

0.6

0.6

0.6

0.4

0.4

0.4

0.4

0.2

0.2

0.2

0.2

0
-2

0
X – position (μm)

2

0
-2

0
X – position (μm)

2

0
-2

0
X – position (μm)

2

0
-2

0

2

0
X – position (μm)

2Figure 5.30: (a) The lens setup to design the metalens doublet with FoV up to 2.2 mm and NA
of 0.7. The stop aperatue is set on the aperture metalens and the substrate is 12.7 mm thick. THezoom in part shows the ray converges to a diffraction limited point at the focus. (b) The simulated
PSF over the FoV of the lens up to 1.1 mm half FoV. The angular FoV is 7◦ . The position isdetermined by when the Strehl ratio drops to 0.8.

179

a

b

Aperture metalens

100

-2e3
Phase (2𝜋)Phase (2𝜋)

0

-100

-4e3

-6e3

-200

-300

Focus metalens

0

-5.08

-2.54

0
2.54
Lens diameter (mm)

5.08

-8e3

-8

-4

0
Lens diameter (mm)48Figure 5.31: Metalens doublet surface phase based on tabel 5.2. We see the shape of the phaseon the corrector metasurface looks like the curverture of the Schmidt corrector plate. The phaseof the focusing lens looks parabolic with very high gradient for strong focusing.

180
5.7.3

dual wavelength lensA conventional refractive focusing lens exhibit larger index of refraction for shorter wavelength
due to material dispersion. This results in chromatic aberration where shorter wavelength light
focus closer to the lens (Fig. 5.32(a)top). Contrary to refractive optical element, a diffractive lens
exhibit longer focal length for larger wavelength, due to the angle of diffraction is proportional to
the wavelength (Fig. 5.32(a)middle). This give rise to the possibility where chromatic aberrationof a refractive and diffractive optical elements can be balanced (Fig. 5.32(a)bottom).
We use the binary 2 surface from ZEMAX to design a hybrid lens with a conventional lensand a metasurface. The binary 2 surface models a diffractive surface that is infinitely thin with acontinuous phase change across the surface. It also take into account the wavelength dependent
diffraction angle. By balancing the optical (focusing) power of the two surfaces one can createa hybrid lens where the chromatic focal shift cancel out each other over a large spectral range.
The resulting chromatic behavior will be better than achromatic lens and closer to an superapochromatic lens. We present a preliminary design of an hybrid lens utilizing a commerciallyavailable asphere from Thorlabs (AL1210) and metalens on a N-BK7 substrate. The choice of
N-BK7 for metalens substrate is to match the CTE of the metalens to the CTE of the asphere, inmost cases are S-LAH64 from Ohara. CTE of N-BK7 is roughly 7.1 ppm/K and S-LAH64 is about
6.1 ppm/K so they are sufficiently close. The lens layout is illustrate in (Fig. 5.32(b)), the letalens
is on a 0.5 mm thick substrate in close contact to the asphere.
After optimizing the phase profile of the metalens using the same polynomial phase frombefore Eq. 5.3, we obtain a phase profile shown in (Fig. 5.32(d)) that resembles a weak focusinglens. The resulting optical performance is shown in (Fig. 5.32(c) where the Strehl ratio is plotted
over a wide wavelength range with several FoV configuration up to 20 mum. We can see the hybridlens is diffraction limited (Strehl ratio >0.8) from 760 nm to 870 nm continuously with a FoV of±18 µm (similar to the original asphere). This design has a much wider spectral range than the
typical achromatic lens where only two wavelength will be on focus.

181

a

b

AsphereThorlabs AL1210

Metasurface

10 mm

d

0

c 1.0
Phase (2𝜋)-20Strehl ratio

0.8
0.6
Field of View

0.4

on-axis
7 µm
10 µm
20 µm

0.2
0

750

775

800
825
Wavelength (nm)

850

-40

-60

-80
875

-4

-2

0
Lens diameter (mm)24Figure 5.32: (a)Hybrid metalens concept. The dispersion of a refractive and a diffractive element
have opposite sign. Therefore, by combining the two the dispersion can cancel and one can get aachromatic focusing lens. (b) The ZEMAX setup showing Combining a metasurface with a Thorlabs
AL1210 asphere to correct for chromatic abberation over a large range. (c)Simulated Strehl ratioof the hybrid design over wavelength and different FoV up to 20 µm. (d) Surface profile of themetalens, showing a relatively weak focusing lens which should give a high focusing efficiecny makethis lens suitable for atom trapping.

182

5.8

OutlookIn this chapter, we demonstrate that metasurface lenses hold promise for future atom trapping

experiments that requires a robust multifunctional surface. We successfully trapped single atoms
using a metasurface lens generated high NA optical tweezer up to 0.5 NA. Additionally, we showedthat the metasurface is capable of efficiently imaging single atoms with low background. Finally,
we studied the field of view (FoV) behavior of the metalens with an atom array generated by
combining the high NA metalens with an AOD. We found that the FoV of the singlet metasurface
lens is ±11 µm, similar to a singlet aspheric lens of the same focal length.
An immediate improvement could be achieved by increasing the efficiency of the current
sample through better fabrication techniques. In this work, our metalens achieved only 56% focusing efficiency, whereas efficiencies between 80-90% have been routinely demonstrated in theliterature [185, 190, 192]. This can be accomplished through improvements in the fabrication process, including better deposition of dielectric materials, improved etching homogeneity, and tighter
control over nanopillar dimensions.
Further improvements through design of the nano-pillars can be achieved through the use ofan expanded unit-cell library to include cross, donut and other exotic nanopillar shapes [189] or
via inverse design [190]. Choosing optimal materials and designs that are robust to nanofabrication variation is expected to offer higher efficiencies that exceed that achieved in the experiments
presented here [78]. Further, a hybrid lens design consisting of a curved refractive surface anda metasurface patterned on the same substrate (design shown in Sec.5.7.3) will offer additional
prospects for enhanced design space [182, 191].

5.8.1

Polarization multiplex

Looking forward, a few direction can immediately benefit the single atom trapping community. As discussed previously, with a metasurface it is possible to achieve a focusing response that iseither polarization selective [187] or one that transforms the polarization [188], which are functions

183
not offered by traditional optical lenses. For example, polarization multiplexing provides a method
to trap and collect fluoresence at the diffraction limit for two different wavelengths using a singlet
metasurface lens, and may find utility in combining additional multifunctional beams in complex
trapping experiments. To illustrate this prospect, we have designed and optically tested a sample
with in-plane rectangular shape pillars that achieves equal focal lengths for 780 nm and 852 nm
light of orthogonal polarization (see Sec. 5.7.1). This concept can be used to trap at 852 nm, andcollect fluoresence at 780 nm, with a 50% efficiency due to the random polarization of the scattered
light from atoms.
Another application of the multifunctional polarization-dependent response of the metasurface is to generate a fast-switchable trapping potential that depends on the input polarization to themetalens (Fig. 5.33). Unlike AOD or SLM devices, where switching occurs on the order of hundredsof kHz for AODs and approximately 100 Hz for SLMs, using EOMs can achieve nanosecond-level
polarization changes, enabling very rapid switching between different trap configurations. Some ofthese fast-switchable trapping potentials are being explored in Ref. [193], where metasurfaces areused as switchable holograms to project different trapping potentials with the use of a high NA
objective.

22°

trap B trap A

44°

184

68°

3 μm

Figure 5.33: A series of traps generated by light diffracted through a polarization dependent
metasurface hologram and focused through a high NA objective. The resulting trapping potential
is a function of input polarization (top three panels). A continuous switching between two configuration is done through the rotation of linear polarization of the input light. Monitoring theintensity of two trap location (trap A and B) one can obtain the extinction ratio between the two
configurations (bottom panel). This figure is reproduced from Ref. [193].

5.8.2

Large FoV doublet metalensMore functionality can be achieved with expanding the number of surfaces offered in thedesign. To focus on FoV as one metric, an enhanced FoV up to ±25◦ has been achieved by patterning
both sides of the substrate to create a double-layer metasurface [81]. This enhancement is of greatinterest in quantum gas microscope experiments [17, 194, 195], where high-resolution imaging overa large FoV is necessary to observe atoms trapped in optical lattices. In quantum simulation andcomputation applications, where atoms are trapped in large arrays of tweezers [44, 45] or shuttled
between different interaction zones separated by large distances [31, 35], increasing the FoV canalso provide significant benefits.
For comparison, state-of-the-art microscope objectives can achieve FoV around ±500 µm withan NA of 0.6 [Fig. 5.34(a)] [43]. In contrast, our metalens doublet design (see Sec. 5.7.2) is capableof ±1.2 mm, which is over four times larger [red dashed circle in Fig. 5.34(b)]. Additionally, themechanical robustness of the metalens is advantageous, while the laser power required to generate

185
thousands of tweezers can reach tens of watts [43], traditional large objectives may suffer fromoptical performance degradation due to thermal dissipation issues. Since the metalens is flat,
thermal deformation is not a concern, and heat dissipation is more efficient due to its simpler
mechanical structure and lower optical losses. Alternatively, using a sapphire substrate insteadof fused silica can further enhance thermal conductivity. These favorable qualities of metasurface
lenses can offer significant benefits for trapping and imaging large arrays of atoms, holding great
promise for future atom trapping experiments.

ab

Figure 5.34: (a) Image of over 6000 atoms loaded in optical tweezer projected from the state-fothe-art objective with the FoV around ±450 µm [43]. (b) The capability of the doublet design fromSec. 5.7.2 with FoV over ±1.2 mm (ref dashed circle) comparing to the state-of-the-art objectivewith multiple optical elements and 50 times the volume and weight. Figure reproduced from
Ref. [43].

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Appendix A

Montana cold window test

Overview:

In this appendix we discuss the test of cold windows done in Montana In-

strument CryoStation optical cryostat. Before designing the coldbox window structure, we need toverify the thermal performance of the double window and the dimensional stability that the indium
foil gasket can provide under cryogenic environments. The dimensional stability of the indium seal
is measured with the relative angular change between two windows in the window stack. The
angle between windows are measured optically with a resolution of better than 0.02 mrad. The
temperatures of the cold windows are measured with cryogenic sensors attached to the center ofthe window and the gradient is measured between the holder and the window.

A.1

Double window thermal test
To conduct this test, we created a simple structure resembling the thickness and separationof the coldbox windows [design shown in Fig. 2.14(b)]. The assembly was then attached to a test
cryostat from Montana Instruments and cooled to 60 K, which is the radiation shield temperatureof the test cryostat. The distance of the window to the room temperature shield was kept similar
to the coldbox configuration to ensure the thermal radiation coupling was as similar as possible.
Figure A.1(a) shows the schematic of the test setup, showing the room temperature enclosure
and windows, test window assemblies, and temperature sensor locations. Sensors T1 and T4 areattached to the metal parts of the test window assemblies, while sensors T2 and T3 are attached tothe center of the glass window with Stycast 2850 epoxy to ensure accurate temperature measure-

203

a

b
T sensor 1

T sensor 4

T sensor 2
double window
test

4.5 mmradia�on shield

tweezertest
17 mm
T sensor 3
room temperature enclosure

Figure A.1: (a) Schematic of double window and tweezer window test. Both the thermal and opticalalignment stability are tested. Various temperature sensors measures the temperature difference
between the glass and the aluminum heatsink. Showing good agreement between the simulated andmeasured value.(b) Relative alignment of windows on tweezer windows test assembly vs temperaturerecorded during the Montana Instrument cryostat test.

ments. After the system reaches thermal equilibrium at 60 K, the temperature differences between
T1/T2 and T3/T4 are measured. The results show a temperature difference of less than 3 K for the
double-pane window (T2-T1), which is similar to the simulated value in Sec. 2.2.9. However, thismeasurement represents an upper bound since the glass used here is much thinner (1.5 mm vs. 2
and 3 mm) compared to the ones used in the coldbox design. For T3-T4, we measure a temperature
difference of less than 6 K, which is also similar to the COMSOL simulation. This demonstrates
that the indium gasket provides sufficient thermal contact to thermalize the glass window to themetal structure.

A.2

Indium seal dimensional stability
To test the dimensional stability of the indium gasket for maintaining good angular alignment

between the two tweezer windows, we utilize light reflected from the surfaces of the windows. Since
the Montana Instrument cryostat provides optical access, this setup is relatively simple. The setup[similar to Fig. 2.26(a)] uses a collimated incoherent light source with a 10 nm bandwidth toeliminate interference from different sides of the windows. The 10 nm bandwith is sufficiently small

204
to minimize chromatic focal shift of the focusing lens. The reflected light from the windows arefocused with a lens of EFL=300 mm onto a CCD camera, converting angular changes of the beam
into positions on the CCD camera.
The focused spots have 60 µm Gaussian waist and the pixel size of the CCD camera is 4.4 µm.
The typical fit uncertainties to the centroid of the Gaussian is less than 15 µm (less than 4 pixels).
This translate to the angular uncertainties of < 50 µrad. Figure A.1(b) shows the angular changeof the test windows, indicating a deviation of less than 0.1 mrad between the two windows from300 K to 60 K. This demonstrates that the indium foil gasket can provide the required dimensional
stability sufficient for the objective to remain diffraction-limited.

Appendix B

SAES Rb Dispenser Activation

Overview:

In this appendix we discuss the operating principle of the Rb alkali metal

dispenser from SAES getter. We also document the activation steps of these type of dispenser andthe typical behavior during activation.

B.1

The Alkali Metal Dispenser
The main component of the SAES alkali metal dispensers (AMDs) are alkali metal salt andreducing agents. In this case the chromate is Rb2 CrO4 which is stable under atmosphere. The
reducing agent used in SAES AMDs is a type of non-evaporable getter material consists of Zr 84%
and Al 16% (st101) [196, 197]. To release the alkali metal the dispenser needs to be heated to asuitable temperature, typically between 550-850◦ C [197]. This starts the reducing reaction between
the chromate and the St101 alloy and cause the alkali metal to evaporate. Besides the reducing
reaction, the ST101 getter also absorbs most of the reactive gasses produced during the reaction
which allows a clean source of alkali metal vapor to be produced [197]. However, some amountof H2 and CO2 are released during normal operation of the dispenser [196, 197]. Even though therubidium chromate is stable at room temperature, the reducing agent oxidizes when exposed to air.
Therefore the dispenser needs to be activated under vacuum before first use. The st101 alloy needs
to be activate at a temperature beyond 600◦ C which requires a current of larger than 5A throughthe dispenser. The activation also serve to degas the dispenser.

206

B.2

Dispenser Activation
The activation of the dispenser depends on the operating current and vapor flux required

during normal operation. Typically for AMO experiment, the flux required are much lower than
the intended purpose of the AMDs which is coating of alkali metal film. So the AMDs are typicallyrun at <3.5A instead of 4.5-7 A stated in the SAES AMD manual. This results in a much lower
and shorter activation cycle (typically 5.5A @ 1 min repeated 3 times). The Rb dispenser in thesource cell was activated during vacuum system bakeout. The activation was done according to thefollowing steps.
(1) When the source cell reaches baking temperature (typically around 200 ◦ C) record thepressure. Turn the dispenser current to 3 A slowly and monitor the chamber pressure.
Leave current at 3 A until the pressure stabilized (typically in 15 min) and record theinitial, peak and end pressure as well as the voltage across the dispenser.
(2) Turn the dispenser current to 5.5 A for 1 min and record initial, peak and end pressure aswell as the voltage across the dispenser. Then set the current back to 3 A and wait for thepressure to stabilize.
(3) Repeat step 2 three times.
(4) Turn the dispenser current off, wait for the pressure to stablized then record the final
pressure.
(5) Typical dispenser resistance should be around 1 Ω or less when measured at the hightemperature lead.
The table below is the typical activation record for SAES AMDs for refernce. The operating
current for this dispenser is 3 A.

207

Before activation, Bake temperature: 170 ◦ C, Pinit : 1.8×10−7 torr
Current (A)Initial Pressure (torr)

Initial Voltage (V)Max
Pressure (torr)

Final Pressure (torr)

Final Voltage (V)Pre-activate@ 3A

1.8×10−7

0.9 V

2.8×10−6

4.3×10−7

0.9 V

4.5A @ 1 min

2.2×10−7

1.3 V

1.2×10−6

9.7×10−7

1.3 V

5.5A @ 1 min

2.2×10−7

1.6 V

1.3×10−6

8.0×10−7

1.6 V

4.5A @ 1 min

2.2×10−7

1.3 V

2.5×10−7

2.5×10−7

1.3 V

Dwell dispenser current at 3 A, Pend : 1.8×10−7 torr
Table B.1: Typical activation behavior of SAES alkali metal dispenser

Appendix C

Anodic Bonding

Overview:

In this appendix we discuss the basic principle of anodic bonding and the steps

we took to bond the metalens science glass cell.

C.1

Anodic Bonding
Anodic bonding is a technique to produce hermetic seals between glass and metal. The mostcommon types of seal is between silicon and alkali-glass, e.g. Pyrex, Schott Borofloat 33, Corning
7740, and is most widely used in MEMS industry for sensor packaging [198]. It is an electrochemical
process requiring heat and electric field to form a new oxide layer between silicon and glass. The
systematic investigation of the anodic bonding process was first published in 1969 with bonding
between glass to metals or silicon [199]. It has several benefits that makes it excellent for forming
vacuum tight seal for UHV application. These benefits including• both silicon and borosilicate glass are UHV competable materials• Borosilicate glass is commonly used as optical windows• the bonding process is relatively low temperature with no melting• no adhesive is needed to form the bond• relatively relaxed surface flatness (1-2λ per cm) comparing to other direct bonding process,e.g. optical contacting (λ/10 per cm or better)209
However there are several limitations for material selection. Due to the heating requred
during the bonding process, the substrate materials requires similar CTE. Also, the glass needs tohave high alkali content such as Na2 O or K2 O to achieve the required electrical resistivity. The mostcommon combination is silicon and borosilicate glass, both have CTE of 3 ppm/K. For successful
bonding, a RMS roughness of less than 10 nm on the substrate is preferred [198]. Although theanodic bonding process is fairly tolerant to surface contaminant, a mechanical scrubbing or chemicalcleaning can increase the bond quality [198, 199].

C.2

Bonding Principle
The principle of anodic bonding is using heat to disassociate and mobilize ionic compounds

added in the glass [Fig. C.1(a)]. In borosilicate glass these compounds are usually Na2 O or K2 O.
Once the ions are mobilized, the electric field is applied and draws the oxygen ion towards the bondinterface. During this process, a depletion layer is formed where there are no sodium ions in theglass [Fig. C.1(b)]. This cause an excess of oxygen near the bond interface and a new oxide layer
starts to grow and forming a permanent connection between the two bonded materials [Fig. C.1(c)].
The anodic bonding process roughly separates into three parts:• Heating: The bond interface is heated to temperature between 250-450 ◦ C. The rate of
temperature rise depends on the uniformity of the applied heat and geometry of the bonded
parts, typically around 2-10 ◦ C/min.
• Apply electric field: The electric field required to form the bond is applied throughelectrodes contacting the two bonded materials. The glass is negatively charged and silicon is positively charged. The electric field needs to be intense and uniform so typicallyseveral 100’s to 2 KV is applied through electrodes touching the substrates depends onthe geometry. During this process, the current flowing through the bond exponentiallydecreases and a bond front starts to propagate from the region nearest to the electrodes
outwards. Typically it takes a few minutes to form a complete bond, which depends on

210bond geometry.
• Cool down: After the completion of the bond, the electric field and the heat are removed
and the parts cool down. The cooling rate needs to be slow since the CTE of the two bonded
material are not identical and stress can accumulate if the part is cooled too rapidly. Typicalcooling rate is around 2-10 ◦ C/min depending on the geometry.

Figure C.1: (a) The borosilicate glass substrate (top light blue) and silicon substrate (bottom
magenta) are optically polished and put in physical contact with each other. Then the substrates
are heat up to 200-450 ◦ C. This cause the sodium oxide in the borosilicate glass to disassociate andbecome mobile. (b) A high voltage between 500 V - 2 kV is applied, with negative side connects
to the borosilicate glass and positive side connects to silicon. This cause the sodium ion to drift
away from the substrate interface creating a depletion layer. Excess oxygen then forms silicatebond with silicon across the substrate interface. (c) After the new layer of silicon dioxide is formed
across the interface, the high voltage and heat are removed. The bond between the two substrate
become permanent.

C.3

Metalens Cell Bonding Parameter
Here are the parameters and steps used for the anodic bonding of the metalens science glasscell.
• Heating: An oven is constructed to heat the parts to 350 ◦ C at the rate of 1 ◦ C/min forcoated glass cells and 5 ◦ C/min for uncoated glass cell.

211• Applied Voltage: Electrodes made of thin stainless steel wire is wrapped around theparts to ensure uniform electric field distribution. A voltage of 1000 V is applied in 100 V
increment to keep the bond current within managable level. The peak current is recorded
and when the current drops to 25% of the peak current the power supply is turned off.
• Cool down: The heater is turned off and the part is cooled at the rate of 5 ◦ C/min.

Appendix D

Basic Principle of cryopumping

Overview:

In this section we introduce the basic concept of cryopumping and how thetemperature effects the outgassing property of a surface. We describe the roles of the 45 K and4 K surface in our coldbox and the behavior of their ability to adsorb gas on the surface. We
first describe the basic principle of cryopumping and give a list of pressures of different gas species
under different temperature. Finally, we discuss the design of coldbox and cold fingers from thecryopumping point of view.

D.1

Cryopumping Principle
A cryopump is one of the most prominent types of entrapment pumps, similar to sputter

ion pumps and getter pumps, that captures and immobilizes gases without requiring an outlet.
Cryopumping leverages the most rudimentary aspect of reducing pressure by reducing the temper√
ature of the gas that is being pumped. The kinetic energy of the gas is reduced following T andthus the pressure in the system is reduced. When the temperature of the gases are low enough,they can be freeze onto the cold surface provided by the cryopump. Typically, the intermolecular forces involved in these low-temperature processes are relatively weak (Van der Waals type),
and no chemical reactions or chemical bonds are present, unlike in getter pumps. Consequently,
cryopumping can pump all gasses including noble gasses, for sufficiently low temperatures on thepumping surface. This property is the main distinction between a cryopump and other capture
pumps.

213
There are many factors that can affect the amount of gas molecules adsorbed and the equilibrium pressure attainable by a cryogenic surface including surface temperature, physicochemical
properties of gas, surface energy distribution and mechanical properties of the surface (roughness,porosity etc...). There are three main types of physisorption mechanism to achieve cryopumping(Fig. D.1), which we will describe below.

abphysisorp�onccondensa�oncryotrapping

Figure D.1: Three different cryopumping regime: (a) physisorption (b) condensation and (c) cryotrapping depends on the amount of gases, temperature and gas type. (Image from [110])

D.1.1

Cryocondensation

In the case of cryocondensation [Fig. D.1(b)], the temperature of the surface is kept coldenough that the gasses condenses. Typically the pressure where cryopump operates is below thetriple point pressure of all gasses, so the gas undergoes sublimation transition from gas to solidand vice versa. In order for the cryocondensation to occur, the cold surface must be saturated
by the gas being pumped. The pressure achievable in this case depends only on the gas species
and temperature of the surface, following the sublimation equilibrium pressure curve (Fig. D.2).
This pressure curve (Fig. D.2) is obtained when the rate of condensation and sublimation equals(zero net pumping speed) under a given temperature. In theory, all gases can be condensed to very
low pressure, but in practice the pressure is limited by the lowest attainable temperature by thecryogenic refrigerator typically around 4 K. From (Fig. D.2) we can see that temperature around
100 K is sufficient to condense water and all hydrocarbon to pressure below 10−12 torr. Below 20
K, all air components can be condensed out including most noble gasses other than neon. At 4 K
all gases other than hydrogen and helium can be condensed out.

Satura�on Pressure (Pa)214

Temperature (K)Figure D.2: Saturation curve (sublimation equilibrium pressure) of common gasses verses temperature. The saturation pressure is obtained when the rate of condensation equals to the rate ofsublimation at a given temperature. (Image from [111])

D.1.2

Cryosorption

When the amount of gas of the particular species being pumped is not sufficient to cover
the cryopumping surface up to a monolayer, the cryopumping follows another mechanism callcryosoprtion [Fig. D.1(a)]. Cryosorption is when gas particles come into contact with surfaces at
sufficiently low temperature, they can loose so much kinetic energy that they stay attached to thesurface by weak intermolecular forces [112, 200]. The equilibrium pressure of adsorbed gas is much
lower than that of condensed gas due to the higher binding energy between gas molecules and thesolid surface compared to the binding energy between gas molecules in the case of cryocondensation.
The difference in activation energy of gas molecules and solid surface is typically 2-3 times higher
comparing to activation energy between gas molecules. The difference is even more significant for

215
light gases, ∼ 10 for H2 and ∼ 30 for He. See Refs. [110, 113, 201] for list of activation energy
between common gases and between gases on cold surfaces. Therefore, once the gas molecules arecryoadsorbed onto the surface, the likely hood of them escaping is much lower, thus achieving lower
pressure. This mechanism is essential for pumping helium and hydrogen at 5 K, much higher thantheir condensation temperature.
However, the cryosorption process is quite complex and strongly dependent on surface condition (chemical, physical and geometry) and the gas coverage on the surface [110, 111, 202, 203].
Typically large surface area materials, such as activated charcoal or molecular sieves, are used toprovide large surface to volume ratio (∼ 1000 m2 /g) to prevent saturation of the surface. Sometimes in particle accelerator applications, technical surfaces such as metal, glass, or NEG surfaces
are also used for cryosorption (Fig. D.3). These surfaces has much lower porosity but provides
pumping along the whole beam line [204]. This is sometimes favorable since thermal regenerationof activated charcoal can be difficult and most of the cryopump outgasses significantly at room
temperature and can not withstand high temperature UHV bakeout.
Typically monolayer gas coverage is around 1015 H2 /cm2 [110]. However, even with low
monolayer coverage, the probability of gas molecules immediately binding to the surface is still
low. Therefore, the gas molecules typically takes a few bounces on the cold surfaces before being
captured. This means the pumping efficiency of cryosportion is lower comparing to cryocondensation, but with lower ultimate pressure [111]. Beyond a few monolayer surface coverage, the surface
effect becomes negligible and the pumping characteristics follows the cryocondensation behavior.
To leverage the increased pumping speed of cryosorption at higher temperatures (especially for H2
and He), one must provide a large pumping surface area to prevent saturation under the given gas
load.

Xenon pressure (Torr)

216

Xenon/cm2
Figure D.3: Xenon adsorption isotherm curve at 77 K of selected technical surfaces including
stainless, copper, aluminum (bare and anodized) and NEG. (Image from Ref. [204]) Note: It isappropriate to use xenon at 77 K because its saturated vapor pressure at this temperature issimilar to that of hydrogen at 4 K and follows a roughly equivalent curve. Thus, any enhancementin equilibrium pressure due to surface effects and cryosorption will be similar once the temperature
is scaled.

D.1.3

Cryotrapping

In the case of cryotrapping [Fig. D.1(c)], two (or more) species or gas are co-pumped. The
cryotrapping process involves using a dominant gas species that is condensable at a much higher
temperature to form a porous cryodeposit layer. Non-condensable gases, such as H2 and He, areentrapped during this process. A fresh layer of condensable gas continuously freezes onto the cold
surface, trapping the cryosorbed gas underneath and achieving a lower equilibrium pressure thancryosorption alone [111, 202]. Typically, Ar is used as the entrapment gas due to its higher freezing
point and being chemical inert [205, 206]. One main drawback of cryotrapping is a continuous
supply of condensable gas is needed and during regeneration of a cryopump, significant amount ofgas load is generated and needs to be pump away with mechanical pumps [111].

217

D.2

Outgassing at low temperature
From the discussion in the cryosorption section above, we know that as the temperature ofa surface becomes colder, the probabilities of gases trapped by the weak intermolecular potential
becomes greater. If we consider outgassing from the surface as the release of trapped gas after somesojourn time, it does not matter whether the gas source is diffusion through the wall or capture
from the environment. An infinite sojourn time on the surface implies that the gas is pumped out.
Then, at sufficiently low temperature, we can write the sojourn time τs as a function of temperatureaccording to the Frenkel equation as follows [113, 202, 207].

τs =1 E/kTe
ν0(D.1)

ν0 ∼ 1012 s−1 is the vibration frequency of the molecules in the intermolecular potential
well near the cold surface. E is the binding energy of the surface potential in kJ/mole. k and T
are ideal gas constant and temperature in Kelvin. To illustrate the effect of temperature on thesojourn time, for a binding energy of 0.4 kJ/mole which is typical for physisorbed molecules on thesurface [202] the average sojourn time is around 10−8 s at room temperature. However, at 77 K
this time would increase to around 1 year. Meaning the outgassing rate from a given surface anddiffusion rate of gas from the bulk are significantly reduced. This strong temperature dependenceof outgassing rate is what we rely on when we design the coldbox.

Appendix E

Stinger Cryostat

Overview:

In this appendix we discuss the basic principle of stinger cryostat and its

operation. We first discuss the construction and operation of the Stinger cryocooler. Then we will
discuss the clogging issues of the cryocooler and some of the additional filters we put in to prolong
the operation of the Stinger cold hold time. Finally, we will present solutions that can potentiallysolve the contamination problem that causes the cryocooler to be clogged.

E.1

Stinger cryocooler operating principle
The cryostat of our choice is the Stinger system from Coldedge Technologies. The stinger

cryocooler is a fully closed-cycle flow cryostat with a flexible interface to decouple the vibration
generated from the Gifford-McMahon (GM) cryocooler. The flexible coupling also allows a morerelaxed space constraint near the main experiment chamber. The Stinger cryocooler consists of threeparts, two separate helium loops and the UHV cryostat. The first loop is a standard Sumitomo
Gifford-McMahon cryocooler (Model RDK-408D2 with 1 W cooling power at 4.2 K and is driven
by a Sumitomo F-70 compressor). This is where the main cooling power is generated (orange
dashed box in Fig. E.1). The second helium loop (outline with blue dash in Fig. E.1) is driven
by a recirculating compressor (Sumitomo HC-4E1) and connects to a manifold before entering the
Stinger cryocooler.
The manifold contains valves to isolate the Stinger from the recirculating compressor andpressure regulator to control the flow and a vacuum port for pumping out the loop inside the219
Stinger cryocooler. The manifold can also be used to maintain the Stinger helium loop, including
refilling helium, and pumping and flushing out residual air in the system.
This secondary loop couples to the GM cooler via heat exchangers (HX1, HX2 and HX3)
located inside the Stinger cryocooler body. The cooled helium exit through a 1.5 m long flexible
helium transfer line (attached to the Stinger body) with temperature ∼ 6 K, pressure ∼ 110 psi
and flow rate around 20 liter/min. The Stinger cryocooler body and the transfer line are vacuum
insulated and this insulating vacuum is completely separated from the UHV part of the experiment.
At the end of the transfer line there is a capillary restriction that further reduces the heliumtemperature to below 4K through Joule-Thomson effect. This provides ∼700 mW of cooling power
at 4.2 K at the 4 K stage and 5 W of cooling power at 50 K at the radiation shield attachment point.
This transfer line is removable from the UHV cryostat without effecting the UHV portion of theexperiment. The transfer line tip is identical to the one used in most liquid helium flow cryostat.
This allows modular design of the UHV cryostat without changing the Stinger cryocooler. Dueto the complicated dynamics in the Joule-Thomson expansion orifice and the long helium flex line
used for vibration decoupling, the passive temperature stability of this system is worse than thatof the GM alone. Therefore, it might not be suitable for applications where precise temperature
control is required.

220

GM He loopaddi�onal Supelco ﬁlter

S�nger He loop

UHV cryostat

Figure E.1: Stinger block diagram adapted from the Stinger user manual. The stinger includes a
primary GM helium loop to drive the cryocooler that cools the helium through heat exchangers.
The secondary helium loop circulates the cooled helium to the UHV cryostat. And the UHV
cryostat separates the helium path and the UHV chamber while providing cooling attachment
points. (Image adapted from ColdEdge Stinger manual)

E.2

The UHV cryostat
The UHV cryostat attached to the main experiment functions as a heat exchanger, so its

construction is relatively simple, with no moving parts. It is designed to be UHV-compatible andcapable of withstanding UHV bakeout. The flexible helium transfer line eliminates the need fora copper thermal braid for vibration decoupling, which can be a potential source of outgassing in

221
the absence of UHV bakeout due to its large surface area. Since the flexible helium transfer line
provides most of the vibration decoupling, we put bags of lead shot on top of the transfer line
to dampen the residual vibration transmitted through the line [Fig. E.2(a)]. The copper braid
mentioned in Sec. 2.2.9 is for small alignment adjustments rather than vibration decoupling.
The UHV cryostat mounts on a standard 4.5” CF UHV flange. The cryogenic temperature
sensor wires are routed through the UHV feed through on the 4.5” CF flange. It contains a 4 Kstage and a radiation shield attachment point. The helium exit the Joule-Thomson capillary flows
through the 4 K stage heat exchanger in a spiral path. Then the returning helium flows throughthe radiation shield heat exchanger before exiting the UHV cryostat back to the transfer line
[Fig. E.2(b)]. The 4 K stage heat exchanger and radiation shield heat exchanger are connected toeach other and to the flange through thin stainless steel tube for thermal decoupling and structural
rigidity.

abhelical He pathstainless support tube

supply He

return He
rad shieldheat exchanger
lead shots
20 cm

4K heat exchanger

4.5" CF ﬂange
S�nger
ﬂex line

Figure E.2: (a) The photo showing how the lead shots are placed on top of the Stinger flex helium
line to dampen the residual vibration from the GM cryocooler transmitted throught the line. (b)The cut out of the CAD model of the UHV cryostat showing the 4 K stage attachment point andradiation shield attachment point as well as the helium flow path.

222

E.3

Compressor helium contamination issue
During the operation of the cryostat, we see gradual clogging of the helium transfer line. This

gradual clogging limits the cold hold time to around two weeks before the cryostat spontaneously
warms up. The sudden warm up of the cryostat is hard to predict even with a flow meter monitoring
the helium flow. The point where the clog will be triggered varies from run to run. The symptoms
typically consists of a slow drop of the helium flow (∼5 liter/min) over a period of time (a few
days) [Fig. E.3(a)]. Then a sudden clog will be triggered either with the perturbation of the line
or pressure change, and the flow completely stops within one hour [zoomed in part of Fig. E.3(a)].
Our hypothesis is that the contaminating gas in the helium loop freezes, causing ice to build up overtime inside the Stinger cryostat heat exchanger and the transfer line. Eventually, the ice breaks
loose due to helium flow or perturbations in the line, blocking the orifice of the Joule-Thomson
capillary.
Here are a few hypothesis we have from our observations when running the system.
• We see that every time we replace the helium in the loop with UHP (99.997%) pure helium
the Stinger will clog during the first few cooldown. This means that the UHP helium is
not sufficiently pure and the pump, flush and refill process might introduce contaminationfrom the outside.
• Although the system initially stays cold for a long time, we observe a gradual degradationin cold hold time. Warming up and pumping out the contaminant, then cooling down
without adding extra helium, appears to have no effect on resolving the clogging. This
indicates that the source of contamination is internal to the helium circulating loop andnot external.
• With freshly charged helium, after running the system for a while, we notice that the vented
helium has an odor similar to that of some organic solvents. This odor persists regardlessof how many times the helium is replaced, indicating that the contaminant may be coming

223
from the compressor oil breaking down or from gas dissolved in the oil.
With these observations and hypothesis we have come up with some solutions. One solution
we employ is to use a freshly serviced compressor and high purity helium filter (Supelco helium
purifier, part number: 27601-U) in the helium line between the manifold and the Stinger cryocooler(Fig. E.1). The Supelco filter can purify UHP grade (99.997%) helium to a cumulative level of 100
ppb (hydrocarbons + moisture + oxygen + carbon monoxide + carbon dioxide). By combining
this with regular warm-ups of the 4 K stage to 20 K, which helps to loosen the ice, we observe
cold hold times of up to several months. Eventually the flow drops significantly (∼5 liter/min)then we perform a full warn up to room temperature and use the manifold to pump out all thecontaminating gas in the Stinger cryocooler helium loop and the flow can be restored [Fig. E.3(b)].
However, the condition of the serviced compressor from Sumitomo is not consistent from one toanother. Although they are all within specification according to Sumitomo, not all of them areclean enough to sustain long cold hold time in our system.
A more permanent solution is to use a dry compressor, where a diaphragm compressor isemployed instead of an oil compressor. Due to the high pressure (7 bar or 100 psi) and high flowrate (30 liters/min) there are only a few compressor choices. At the end we settled on KNF N630.15
combine with the Supelco helium filter. This can potentially prolong the cold hold time to as longas the diaphragm service interval of the compressor (normally 3-5000 hrs of continuous operation).
Another advantage of the KNF compressor is that, unlike the HC-4E1, the diaphragm maintenancecan be done on our own without sending the pump back to the manufacturer, thus reduce the downtime of the cryostat. Initial tests with these pumps done by ColdEdge and NIST ion storage groupshows no degradation of cooling power and clogging over extended periods of time from the Stinger
cryostat. Therefore, we will soon replace our Sumitomo HC-4E1 with the dry compressor solution.

224

a

(hr)

b

Figure E.3: (a) The helium flow versus time during a typical clogging and warm up. We see
drops in helium flow over the course of days, and a sudden initiation of the clog and the flow
completely stops within 1 hour (zoomed in portion on the right). A partial clog that revived itself
was observed between 3/1 and 3/2. Illustrates the unpredictable nature of the Stinger cryostat
clogging behavior. (b) The helium flow during the long months scale operation of the cryostat
showing a slow degradation over a long time and partial revival of the flow after each regenerationof the 4 K cold tip at 20 K. A full warm up to room temperature was carried out between 12/9
and 12/10 and the helium flow was restored to the original value. No helium replacement is done
during full warm up.

Appendix F

Laser phase noise from cryostat vibration

Overview:

In this appendix we discuss the potential decoherence effect of mechanicalvibration between the coldbox and the optical breadboards where the Rydberg excitation light islaunched. The displacement caused by the cryostat is important as it can potentially introduce
optical phase error in the Rydberg excitation light or Raman light. Depending on the duration of
the optical pulse, the phase noise in different spectrum either introduce shot-to-shot fluctuation ornoise within the optical drive pulse.

F.1

Converting a vibration spectrum to phase noise
To calculate the phase noise introduced by the cryostat vibration, we can use the interfero-

metric spectrum [Fig. 2.22(c)] measured in Sec. 2.4.2. The conversion is the following.

P SDphase noise = (P SDvibration λ)2(F.1)

λ is the laser wavelength, for Rydberg it’s 420 nm and 1013 nm, and for Raman it’s 795 nm.
√ 
2The unit for P SDphase noise is (rad /Hz) and P SDvibration is nm/ Hz . With this relation wecan compare the phase noise from the cryostat vibration against the phase noise of the laser itself[Fig. F.1(a)]. We see that the cryostat vibration induced phase noise is much lower than the laser
phase noise by 2 orders of magnitude for both 420 nm and 1013 nm. Therefore, this residual
phase noise from the cryostat vibration will not be the limiting factor for the Rydberg and Raman
coherence up to 100s of milliseconds. As a comparison, we take the vibration time trace from one

226of the low vibration cryogenic ion trap from Jonathan Home’s group at ETH [124, 125, 208]. Thismeasurement was done by forming an interferometer between the 4 K stage and the breadboard
outside the cryostat. In their case, the RMS vibration is around 40 nm and pk-pk around 200 nm[Fig. F.1(b)], which results in a coherence time in the order of 10 ms [124] when addressing theindividual ions with free space beam launched from the breadboards. For us, the RMS vibrationis less than 3 nm [Fig. F.1(c)] and pk-pk is around 20 nm even with the interferometer drift. This
yields an expected coherence time (considering only vibration-induced phase noise) on the orderof 100 milliseconds, which is much longer than the time scales for Rydberg and Raman transitions(less than 1 millisecond) and other dephasing mechanisms such as magnetic field noise.

227

a

b

c

Figure F.1: (a) Vibration induced laser phase noise spectrum calculated from [Fig. 2.22(c)] in
Sec. 2.4.2. Solid lines are laser phase noise measured after locking to the stable cavity (blue datais 420 nm laser phase noise, orange data is 1013 nm laser phase noise). The dashed lines are phasenoise calculated from the cryostat vibration spectrum (Blue is with respect to 420 nm and orange
is with respect to 1013 nm). (b) Cryostat vibration time trace reproduced from Ref. [124, 125, 208]
corresponds to 40 nm RMS vibration amplitude this yields around 10 ms coherence time on thequbit drive. (c) Coldbox vibration time trace of our experimental setup. This corresponds to lessthan 3 nm RMS vibration amplitude, more than 10 times less comparing to Ref. [124].

Appendix G

Release-recapture Gaussian vs harmonic trap

Overview:

In this appendix we discuss the difference between fitting the release and recap-

ture data with Gaussian trap and Harmonic trap. We observe that fitting the same release-recapture
data with a harmonic potential results in a lower atom temperature comparing to fitting to a Gaussian trap. We discuss qualitatively the difference between two methods and discrepancies that wesee between fitting to the two different trap shapes.

G.1

Harmonic trap
Historically the release-recapture Monte-Carlo simulation is done assuming a harmonic trap

over the full tweezer trap depth [134, 136, 152]. If assuming a harmonic potential over the full trap
depth, the total energy of an atom in the trap is expressed as 1/2mvi2 + 1/2mω 2 x2f , where vi isthe initial atom velocity, xf is the final atom position after the release-recapture time and ω isthe trap frequency. If the initial atom temperature is low enough, typically less than 1/10 of thetrap depth, then the atom samples the bottom of the Gaussian function which can be approximate
as harmonic trap with good accuracy. In fact this is typically done when dealing with analysis of
atoms in tweezers. However, this is not accurate since in release and recapture the atom samples
significant portion of the periphery of the trap regardless of initial atom temperature. If we plot
the trap shape of the harmonic potential vs. Gaussian and Airy, we can see that the confinement
is much less near the brim of the trap for Gaussian and Airy shape (Fig. G.1).

229

G.2

Gaussian and Airy
To fit with the Airy or Gaussian trap shape, we first assume the atom initial temperature is

low so that the initial position and velocity can be sampled identically as in Sec. G.1. The difference
now is that after flashing off the trap, the final energy of the atom is the initial kinetic energy plus
the potential energy at the atom location after the trap is turned back on. The potential energy isnumerically determined based on the trapping parameters.

Figure G.1: The trap profile difference between Airy (blue), Gaussian (orange), and harmonic(green). The three trap shapes are normalized to the same trap frequency at the bottom of thetrap (intensity maximum). The grey shaded parts are where the atom is recaptured by the Gaussianand Airy trap profile, but will escape if assuming a harmonic trapping potential.

G.3

Comparison
For harmonic trap shape, we see good agreement of temperature measurements between

the R-R and Raman-sideband spectroscopy and free space time-of-flight imaging [152, 209]. If wecompare the harmonic trap shape to the Gaussian and Airy, we see that the harmonic shape hasa smaller recapture volume, which is shown in grey shaded regions in Fig. G.1. When fitted tothe same data, the temperature and heating rate with harmonic trap shape is consistently lower
by about a factor of 3 comparing to the Gaussian and Airy case. When we compare this tothe calculated recoil heating limit, we see harmonic trap goes below the calculated recoil limit(5 ± 2 µK/s vs. 10 µK/s), but the Gaussian (12 ± 4 µK/s) and Airy (12 ± 5 µK/s) agrees well230
with the calculated recoil limit, [Fig. G.2(a)].
Examining the actual fit of the data with different trap shapes, we see that the harmonic
trap curve lies above the data points to the left of the red dashed line and below the data points
to the right of the dashed line. In contrast, the Gaussian and Airy fits describe the data better[Fig. G.2(b)]. This systematic skew when fitting the data with the harmonic trap shape is also
observed in other studies (for example Ref. [59]) using release and recapture for atom temperature
measurement.
This comparison is to describe the discrepancies we observe while using release and recapture
as a method of measuring temperature and not an extensive study of the cause of this discrepancy.
For ease of comparison to other literature, we use harmonic fit in this thesis unless otherwise
specified. A more through investigation of the origin of this discrepancy will subject to future
study.

231

Airy

Gaussian

Harmonic

Time (s)Time (s)Hea�ng rate (μK/s)aTime (s)bFigure G.2: (a) Fitted recoil limited heating rate between between Airy, Gaussian, and harmonic.
The Gaussian and Airy gives similar heating rate of 12 ± 4 µK/s and 12 ± 5 µK/s respectively, while
the harmonic gives 5 ± 2 µK/s about twice as low as the prediction from recoil limit. (b) Release
and recapture data fitted with Monte-Carlo using Airy, Gaussian and harmonic trap shape. The
Airy and Gaussian gives similar temperature result of 38 ± 4 µK and 41 ± 5 µK while the harmonic
gives 13 ± 3 µK. The small difference between the Airy and Gaussian trap shape near the brim of
the trap have negligible effect on the resulting temperature. However, fitted with harmonic trap
consistently gives a lower temperature by a about a factor of 3. Visually we see that harmonic over
fit in the region to the left of the red dashed line and under fit in the region on the right, and theGaussian/Airy fits better to the data.

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